OF THE

AMERICAN

MATHEMATICAL

SOCIETY

VOLUME 12, NUMBER 4 ISSUE NO. 82 JUNE 1965

OF THE

AMERICAN MATHEMATICAL SOCIETY

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

CONTENTS

MEETINGS Calendar of Meetings • • • • . • • . • • • • • • • • . . • • • • • • • • • • • • • • • • • • • 400 Program of the Meeting in Eugene, Oregon ••••••••.••.•.•••.•.•.• 401 Abstracts for the Meeting - Pages 449-456 PRELIMINARY ANNOUNCEMENT OF MEETING. • • • • . • • . • • • • • . • • • • . • • 404 ACTIVITIES OF OTHER ASSOCIATIONS •• , . • • • • • • • • • • • • • • • • • • • • • • • • 409 MEMORANDA TO MEMBERS Request for Names of Visiting Foreign Mathematicians ••• , • • • • • • . • • • 408. Mathematical Sciences Employment Register • • • • • • • • • • • • • • . • . • . • 411 Notice of Change of Address • • • . • . • • . • • • . • • • • • • • • • • . • • • • • • • • 411 NEWS ITEMS AND ANNOUNCEMENTS ...... 410, 412, 418 PERSONAL ITEMS • • • . • . • • . . • • • • • • • • • • • • • • • . • • • • • • • • • . • • • • • • 414 NEW AMS PUBLICATIONS. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 417 DOCTORATES CONFERRED IN 1964. • • . • • • • • • • • • • • • • • . • • • • • • . • • • • 419 SUPPLEMENTARY PROGRAM- Number 32.... • . • • • • • • • • • . • • . . • • • • • 443 ABSTRACTS OF CONTRIBUTED PAPERS ••••••••••••.•••••••••.•••• 447 INDEX TO ADVERTISERS • . • . • • • • • • • • • • • • • • • • . • • • . • • • • • • . • • • • • 494 RESERVATION FORM ••••••••••••••..••••••••••••••••••..•••• 495 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 cNoticeiJ 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*

625 August 30 - September 3, 1965 Ithaca, New York July 9 (70th Summer Meeting) 626 October 30, 1965 Cambridge, Massachusetts Sept. 13 627 November 12-13, 1965 Lexington, Kentucky Sept. 28 628 November 26-27, 1965 Iowa City, Iowa Sept. 28 NOVEMBER 1965 Southern California Meeting. None scheduled 629 December 29, 1965 Berkeley, California Sept. 28 630 january 24-28, 1966 Chicago, Illinois (72nd Annual Meeting) April 9, 1966 Honolulu, Hawaii August 29 - September 2, 1966 (7lst Summer Meeting) New Brunswick, New jersey january 24-28, 1967 (73rd Annual Meeting) Houston, Texas August 28 - September 1, 1967 (72nd Summer Meeting) Toronto, Ontario, Canada January, 1968 San Francisco, California August 26-30, 1968 (73rd Summer Meeting) Madison, Wisconsin *The abstracts of papers to be presented in person at the meetings must be received in the Head­ quarters Offices of the Society in Providence, Rhode Island, on or before these deadlines. The dead­ lines also apply to news items. The next two deadline dates for the by title abstracts are .July 2 and September 6. 1965.

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ThecNoticeiJ of the American Mathematical Society is published by the Society in January, February, April, June, August, October and November. Price per annual volume is $7.00. Price per copy $2.00. Special price for copies sold at registration desks of meetings of the Society, $1.00 per copy. Subscriptions, orders for back numbers (back issues of the last two years only are available) and inquiries should be addressed to the American Mathematical Society, 190 Hope Street, Providence, Rhode Island 02906. Second-class postage paid at Providence, Rhode Island, and additional mailing offices. Authorization is granted under the authority of the act of August 24, 1912, as amended by the act of August4, 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©, 1965 by the American Mathematical Society Printed in the UnitedStatesofAmerica

400 Six Hundred Twenty-Fourth Meeting University of Oregon Eugene, Oregon June 19, 1965

PROGRAM

The six hundred twenty-fourth meet­ Dormitory space will be available on ing of the American Mathematical Society campus for the nights of June 17, 18, and 19. will be held at the University of Oregon in The rates are $3.50 per person for adults Eugene, Oregon, in conjunction with a and $1.00 for each child under 10. Reserva­ meeting of the Pacific Northwest Section of tions for dormitory accommodations also the Mathematical Association of America should be sent to Professor Beelman at the and the Society for Industrial and Applied address above. Requests should include the Mathematics. The American Mathematical number and names of the adults and child­ Society will meet on Saturday, June 19, 1965, ren, and the dates and times of arrival and and the Association and SIAM will hold departure. their sessions on Friday, June 18. The following is a list of Eugene By invitation of the Committee to motels which are within easy walking dis­ Select Hour Speakers for Far Western tance of the meetings. Sectional Meetings, the Society will be ad­ Flagstone Motel dressed at 11:00 A.M. on Saturday by Pro­ 1601 Franklin Boulevard fessor Murray Protter of the University of Single Double Twin California at Berkeley. The title of Pro­ $7.50 $9.00 $10.00 fessor Protter's talk is "The maximum Hyatt Chalet Motel principle". This address will be given in 1857 Franklin Boulevard Room 123 of the Science Building. There will be sessions for contributed $7 .oo $10.00 $12.00 papers at 9:30 A.M. and 2:00 P.M. in New Oregon Motel Rooms 16 and 30 of the Science Building. 1655 Franklin Boulevard Abstracts of the papers to be presented at these sessions appear on pages 449-456 $8.00 $10.00 $13.00 of these c}/ofi.ai) . There are cross refer­ Travel-Inn Motel ences to the abstracts in the program. 2121 Franklin Boulevard Registration for the meeting wilJ" take place in the lobby of the Science Building, $8.00 $10.00 $12.00 beginning at 9:00 A.M. on Friday and Anyone who wishes to stay in a motel Saturday. should write directly and as soon as pos­ On Friday night, June 18, there will sible to the motel for reservations. be a no-host banquet. Persons who plan to Meals will be available in Erb Mem­ attend are asked to make reservations in orial Union. advance. The desired number of tickets Eugene is served by the Southern should be requested from Glenn T. Beelman, Pacific Railway, United Airlines, West Department of Mathematics, University of Coast Airlines, and the Greyhound Bus Oregon, Eugene, Oregon. On Saturday after­ Company. Persons who drive to the meet­ no.on, a tea will be held. ings will find ample free parking on campus.

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

SATURDAY, 9:30A.M. General Session, Room 16, Science Building 9:30 - 9:40 (l) How to distinguish 5-dimensional space from 4-dimensional space Professor J. R. Reay, Western Washington State College (624-15) 9:45 - 9:55 (2) Trees in polyhedral graphs Mr. D. W. Barnette, University of Washington (624-11) (Introduced by Professor V. L. Klee) 10:00 - 10:10 {3) A generalization of the Cartan-Kahler theorem. Preliminary report Mr. Larry Mansfield, University of Washington ( 624-14) 10:15 - 10:25 {4) The visual cortex as a fiber bundle Dr. W. C. Hoffman, Boeing Scientific Research Laboratories, Seattle, Washington (624-2)

SATURDAY, 9:30A.M. Session on Analysis, Room 30, Science Building 9:30 - 9:40 {5) Spatially homogeneous Markov operators Professor J. R. Brown, Oregon State University (624-23) 9:45 - 9:55 (6) On the transformation of integrals in measure space Dr. R. W. Chaney, Western Washington State College (624-12) 10:00 - 10:10 (7) On a class of non-linear differential-difference equations Dr. M. L. Slater, Sandia Corporation, Albuquerque, New Mexico (624-17) 10:15 - 10:25 {8) A generalization of a result of A. Tihonov Professor R. K. Juberg, University of Minnesota (624-22)

SATURDAY, 11:00 A.M. Invited Address, Room 123, Science Building The maximum principle Professor Murray Protter, University of California, Berkeley

SATURDAY, 2:00 P.M. Session on Logic, Algebra and Theory of Numbers, Room 16, Science Building 2:00 - 2:10 (9) Hyperarithmetic incomparability Mr. S. K. Thomason, Cornell University (624-3) 2:15 - 2:25 ( 10) A combinatorial problem connected with number theory Professor Ivan Niven, University of California, Berkeley, and University of Oregon (624-18)

402 2:30 - 2:40 ( 11) A limited arithmetic on simple continued fractions Professor j. H. Jordan* and Professor C. T. Long, Washington State University ( 624-13) 2:45 - 2:55 ( 12) A construction in category theory. Preliminary report Professor D. F. Sanderson, Western Washington State College (624-16) 3:00 - 3:10 (13) A note on inseparability Professor j. D. Reid, Syracuse University (624-20) 3:15 - 3:25 ( 14) The existence of inertial subalgebras Mr. E. C. Ingraham, University of Oregon (624-6) 3:30 - 3:40 ( 15) On finite p-groups with maximal automorphism groups Mr. R. A. Morris, Reed College (624-8) (Introduced by Mr. T. C. Brown)

SATURDAY, 2:00P.M. Session on Analysis and Topology, Room 30, Science Building 2:00 - 2:10 ( 16) Invariant integrals on locally compact semigroups Dr. L. N. Argabright, University of California, Berkeley (624-10) 2:15- 2:25 ( 17) Invariant metrics Professor Hidegoro Nakano, Wayne State University (624-7) 2:30 - 2:40 ( 18) Function spaces of quasi-uniform spaces ProfessorS. A. Naimpally, Iowa State University (624-5) 2:45 - 2:55 (19) Dense, nowhere dense, dense in a set, and dense-in-itself Professor z. z. Yeh, University of Hawaii (624-4) 3:00- 3:10 (20) A general fixed point theorem Professor L. E. Ward, University of Oregon (624-9) 3:15 - 3:25 (21) On a question concerning fixed-points Professor A. L. Yandl, Seattle University (624-19) 3:30 - 3:40 (22) Even-integer connectivity and half-integer genus Mr. K. Demys, 307 S. 7th, Artesia, New Mexico (624-21) R. S. Pierce Seattle, Washington Associate Secretary

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

403 PRELIMINARY ANNOUNCEMENT OF MEETING

Seventieth Summer Meeting and Forty-Third Colloquium Cornell University Ithaca, New York August 31-September 3, 1965

The American Mathematical Society 02904 so as to arrive prior to the deadline will hold its seventieth summer meeting of July 9. According to authorization of at Ithaca, New York from Tuesday through the Council, the number of contributed Friday, August 31-September 3, 1965. papers will be limited to 170. Contributed All sessions will be held in lecture papers which meet the standard set in rooms and classrooms on the campus of Article X, Section 5 of the By-Laws will Cornell University. be accepted in order of their receipt un­ Professor A. P. Calder6n of the til 170 have been accepted or July 9 has University of Chicago will present the arrived. There will be no provision for Forty-Third Colloquium in a set of four late papers. lectures with the title "Singular integrals." Several organizations will cooper­ The first lecture will be given in the Alice ate in holding meetings or council meet­ Statler Auditorium of Statler Hall on Tues­ ings on the same campus as the Society day, August 31 at 2:00 P.M. The second and at approximately the same time. These will be in the same auditorium on Wednes­ include Pi Mu Epsilon, Mu Alpha Theta, day, September 1 at 9:00 A.M. The third and the Society for Industrial and Applied and fourth will be on Thursday, September Mathematics. In particular SIAM will 2 at 2:00 P.M. and on Friday, September present the John von Neumann Lecture in 3 at 9:00 A.M., respectively, but in Room the Alice Statler Auditorium on Thursday, B 17 of Upson Hall. September 2 at 8:00 P.M. The invited By invitation of the Committee to speaker is Professor Freeman J. Dyson Select Hour Speakers for Summer and of the University of California at San Annual Meetings, Professor George Lor­ Diego, whose topic is "Applications of entz of Syracuse University will address group theory in particle physics." the Society on Friday, September 3 at The Mathematical Association of 2:00 P.M. in the Alice Statler Auditorium. America will hold their forty-sixth sum­ He will speak on "Applications of entropy mer meeting from Monday, August 30 to approximation." through Thursday, September 2. This There will be sessions for contrib­ meeting marks the fiftieth anniversary of uted papers on Tuesday, August 31 in the the Association, an event which will be afternoon; Wednesday, September 1 in the heralded in the special character of their morning; Thursday, September 2 in the program. In deference to the Association, afternoon; and Friday, September 3 both the Society has yielded a half day of time morning and afternoon. normally devoted to Society sessions in Abstracts of contributed papers order to permit the Association to present should be sent to the American Mathemati­ an expanded anniversary program. cal Society, Providence, Rhode Island In many of the meeting rooms there

404 is a NO SMOKING rule which must be The registration fees will be as strictly enforced as a safety measure. follows: Member $2.00 COUNCIL AND BUSINESS MEETINGS Member's family .50 for the first such registration and no The Council of the Society will charge for additional registration, meet at 5:00 P.M. on Tuesday, August 31 in Room 320 of the Industrial and Labor Student No fee Relations (ILR) Conference Center. There Others $5.00 will be an intermission for dinner at The preferred procedure is to 6:30 P.M. (This paragraph has been register in advance, as described in the changed materially from the earlier ver­ section titled ADVANCE REGISTRATION, sion.) and to complete the process by picking The Business Meeting of the Society up the badge and information packet at will be held in Room B1 7 of Upson Hall on the registration desk. It is desirable to Thursday, September 2 at 4:45 P.M. have one's local address already estab­ On recommendation of the Council, lished when completing registration as there will be proposed changes in the this information will be recorded at the by-Laws to introduce Mathematics of Com­ registration desk for the visual index. putation as a Society publication, and to In particular, persons with dormitory make the members of the Editorial Com­ reservations should go first to University mittee members of the Council Hall I before completing registration. See the section on DORMITORY HOUSING. ADVANCE REGISTRATION It is possible to register at the desk The advance registration procedure without advance registration. However, will be used. On the inside back cover of one may find that the facilities for which this issue of the Notices is a registra­ attendance is estimated and guaranteed tion form. The same form appeared in the will be sold out. April Notices but it will not appear in the August issue. The form provides for EMPLOYMENT REGISTER registration together with badge, regis­ The Mathematical Sciences Em­ tration fee, and information packet for all ployment Register will be in Rooms 200 persons attending the meeting. It also and 205 of the ILR Conference Center. provides for a parking permit and for dor­ (This is a change from the previously mitory room reservations, barbecue announced place.) It will be open Tuesday tickets, and the non-mathematicians ex­ through Thursday, August 31 throughSep­ cursion to Corning Glass, together with tember 2, from 9:00 A.M. to 5:00 P.M. relevant advance payments. on each of the three days. Attention is A copy of the registration form may invited to the announcement of the Em­ be obtained by request from the American ployment Register on page 411, in parti­ Mathematical Society, Providence, Rhode cular to the deadline dates for application Island 02904. to the register and to the necessity for prompt registration at the Employment Register Desk by both applicants and REGISTRATION employers. The Registration Desk will be in EXHIBITS the Memorial Lobby of Willard Straight Hall. This is at the north end of the main Book exhibits and exhibits of edu­ floor. It will be open on Sunday, August29 cational media will be displayed in the from 2:00 to 8:00 P.M.; on Monday, Memorial Room of Willard Straight Hall August 30 from 8:00 A.M. to 5:00 P.M.; on Tuesday, Wednesday, and Thursday, on Tuesday through Thursday, August 31 August 31, September 1 and 2. The Mem­ through September 2, from 9:00 A.M. till orial Room is in the northwest corner of 5:00 P.M.; and on Friday, September 3 the main floor, to the west of the lobby from 9:00 A.M. till 3:30 P.M. which contains the registration desk.

405 BOOK SALE week and portacribs at $2.50 per week. Their local supply is limited, so that it Books published by the Society will is advisable to begin negotiations with them be sold for cash prices somewhat below early. the usual prices when these same books There will be a short list of baby are sold by mail on invoice. sitters at the registration desk. DORMITORY HOUSING MOTELS AND HOTELS Dormitory rooms will be available There are a number of motels in in University Halls and Baker Dormi­ the Ithaca area, including the following: tories. The rates are $5.50 per day for a Collegetown Motor Lodge - 312 College single room and $3.50 per day per person Avenue- 41 rooms- singles $9, doubles for a double room. Soap, towels, and maid $12 - 10 minute walk from campus - service are provided. air-conditioned. Rooms will be available from Satur­ day, August 28 to Saturday, September 4. Howard .Johnson Motor Lodge - North Rooms must be vacated by 9:00 A.M. on Triphammer Road and Route 13 - 72 September 4. rooms- swimming pool- singles $10.50 Dormitory room reservations are to $16.50, doubles $14.50 to $18.50 and to be made on the advance registration family units - 15-20 minute drive from form already described. The reservation campus - air-conditioned. is to be accompanied with payment for Meadow Court - 529 South Meadow Street- one night, as noted on the registration 50 rooms - singles $8 and up, doubles form. $12 and $13, family units $20 - 15 Reservations will be confirmed. A minute drive from campus - air-con­ local map will accompany the confirma­ ditioned. tion. When people with dormitory re­ Plaza Motel - corner Meadow Street and servations arrive, they should go directly Elmira Road - 84 rooms - swimming to University Hall I, without going first to pool - singles $8, doubles $12 to $13, the meeting registration desk. The office for three $15 to $17, for four $17- in University Hall I will be open 24 hours 15 minute drive from campus - air­ a day to give room assignments and keys conditioned. and to collect the remainder of the room Wonderland Motel - 654 Elmira Road - rent. 27 rooms - swimming pool - singles Persons arriving without dormitory $8 to $13, doubles $10 to $14, family reservations may be able to obtain dor­ units $19 to $21 - 2 5 minute drive mitory accommodations through a dor­ from campus - air-conditioned. mitory representative at the registration desk during the hours that desk is open Hillside Inn - 518 Stewart Avenue - 41 and at University Hall I during other hours. rooms - singles $6, doubles $8, family This is not the recommended procedure. units - 35 rooms have private baths During the day, bellhops will be and are air-conditioned - 10 minute available. They will accept tips. It is not walk from campus. necessary to use their services. Ithaca Hotel - 219 East State Street - Dormitory rooms are not air-con­ 73 rooms, of which about three-quar­ ditioned. ters are air-conditioned - singles There are no special provisions $7.50 to $8.50, doubles $12 to $14 - on for small children, For an older child bus line which goes to campus. (but not for an adult) a cot 30 inches by 6 feet can be put in a double room at a Persons wishing to reserve motel cost of $2.00 per night. or hotel accommodations should write United Rent-Ails, 363 Elm ira Road, before July 1 directly to Mr. B. Anderson, Ithaca, New York (Phone Area 607, AR3- Meadow Court, 529 South Meadow Street, 1807) has standard cribs at $3.50 per Ithaca, New York 14850, listing three

406 choices and mentioning the meeting of For non-mathematicians, there will mathematicians. Beyond the date of July 1 be an excursion by bus on Thursday, Sep­ it is less reasonable to suppose that tember 2 to the Corning Glass factory. The accommodations will be available. round trip by bus will cost $1.7 5 per person, payable in advance through the CAMPING advance registration procedure. Departure time will be immediately after lunch at There are three state parks in the 12:45 P.M. The trip takes about four hours area with tent and trailer sites for camp­ but the time of return is not guaranteed. ing. The location and the mail address for Persons registered at the meeting information and reservations follows: will be able to use the Cornell Golf Buttermilk Fails State Park - 2 miles Course by showing their badges and pay­ south of Ithaca on route 13. R. D. 5, ing a greens fee of $2.00 on weekdays and Ithaca, New York 14850. $3.00 on weekends. Golf clubs can be rented. Robert H" Treman State Park - 5 miles There is swimming and picnicking south of Ithaca on route 13" R. D. 5, at the state parks and at other places. Ithaca, New York 14850. There are tennis courts adjacerit to the Taughannock Falls State Park - 8 miles dormitories and bowling alleys near the north of Ithaca on route 89. R. D. 3, campus. Information about these diver­ Trumansburg, New York 14850. sions may be obtained at the registration desk. MEALS TRAVEL The Willard Straight cafeteria will be open all day for meals, beginning at 7:00 Ithaca is centrally located in the A. M., and through the evening for light Finger Lakes region of New York State. refreshment. The Sage Hall cafeteria will Mohawk Airlines serves the region with be open all day for meals. Both of these connecting flights from the principal sur­ operate on a cash basis. rounding large cities--New York, Newark, The Statler Hall dining room will Boston, Pittsburgh, Cleveland, Buffalo, be open for breakfast, lunch, and dinner. Detroit. The Tompkins County Airport is One should realize that it is more expen­ two miles from the University with taxi, sive than the cafeterias and that reserva­ limousine, and car rental service avail­ tions may be necessary. able" American Airlines serves this area A list of local restaurants will be at through Syracuse. Empire State Airlines the registration desk. connects Binghamton, Ithaca, Syracuse, Elmira, and New York, thus connecting ENTERTAINMENT Ithaca with major air carriers. Also, Commuter airlines provide service be­ The Society for Industrial and Ap­ tween Binghamton and Washington, D" C. plied Mathematics will sponsor their tra­ The Airline Guide should be consulted ditional Beer Party at Noyes Lodge on the for flight times. evening of Monday, August 30 at 8:00P.M. There is no direct railroad service Tickets will be sold through the advance to Ithaca, but the New York Central goes registration form at a price of $1.25 each. to Syracuse and the Erie Lackawanna to It is possible that some additional tickets Binghamton and Owego" Buses connect will be sold at the registration desk. Syracuse and Binghamton with Ithaca, There will be a chicken barbecue but there is no public transportation be­ on Wednesday evening at 6:00 P.M. in tween Owego and Ithaca. Owego is approxi­ Alumni Field, or in Barton Hall in case mately 30 miles from Ithaca. of rain. Tickets will be sold through the The Greyhound Bus Company runs advance registration form at $2.75 per several buses daily that connect with New person and for children up to the age of York, Buffalo, Rochester, Syracuse, Bing­ 12 at $1.50 for ahalfportion.Itis possible hamton, and Scranton. that some additional tickets will be sold Ithaca can be reached by private at the registration desk.

407 car by using an excellent system of con­ be charged, at a reasonable fee, to the necting highways. Coming from the west individual seeking medical care. one uses the New York Thruway to Water­ In general, those requiring hos­ loo and then connects with New York Route pitalization and night-time emergency 89 to Ithaca; coming from theN ew England care will rely on private medical facilities area one uses the New York Thruway to of the community. The Gannett Clinic Syracuse where U. S. Route 11 and New physician who is on emergency call will York Route 13 lead to Ithaca; coming from assist in arranging such care if his help New York City one uses New York Route is needed, and can be contacted by calling 17 to Owego where New York Route 96 Sage Hospital,AR2-6962,or Gannett Clinic, connects directly to Ithaca. From a south­ Ext. 3493. erly direction, one can take the northerly extension of the Pennsylvania Turnpike ADDRESS FOR MAIL from Philadelphia to Scranton and then AND TELEGRAMS Interstate Highway 81 to Binghamton, and The address for mail and telegrams then proceed as if from New York City. is in care of Mathematics Meetings, The city of Ithaca operates a bus Willard Straight Hall, Cornell University, line which provides infrequent local ser­ Ithaca, New York 14850. Individuals should vice within the city. Persons employed by check for mail from time to time in the certain universities and government agen­ vicinity of the registration desk. cies are entitled to car rental discounts. Participants should determine if they COMMITTEE qualify for this discount. Car rental agen­ cies exist both in Ithaca and in the sur­ The committee on arrangements consist of rounding cities of Syracuse, Binghamton, H. L. Alder A. Rosenberg, Chairman and Elmira. S. U. Chase G. Sacks R. Greenblatt G. L. Walker PARKING May Kinsolving R. J. Walker E. Pitcher S. Wainger The Safety Division (campus police) G. S. Rinehart H. Widom will issue cards which will allow parking on the campus. Permits will be made Everett Pitcher available to those who indicate on the ad­ Associate Secretary vance registration form that they will Bethlehem, Pennsylvania drive to the meeting.

WEATHER The mean temperature during the week of August is 67°. During this period the average maximum is 79°, and the MEMO TO MEMBERS average mm1mum is 55°. However, maxima as high as 90° are not impossible. Minima as low as 40° have also been REQUEST FOR NAMES OF known to occur. The average rainfall is VISITING FOREIGN MATHEMATICIANS 69/100 inches for the week. The humidity is between 50 and 60 percent in the after­ The editors of the c}/oticei) would noon and rises to an average of 90 per­ appreciate receiving the names and ad­ cent at night. dresses of foreign mathematicians who will be visiting the United States in 1965-1966. The information then will be published in MEDICAL SERVICES the August is sue of the cNoticeiJ, in time for mathematics departments to extend The staff of the Gannett Clinic will speaking invitations for the coming aca­ provide people attending the meeting with demic year. The regular annual list of daytime out-patient medical care for acute foreign mathematicians will appear in the illness and injury. Such medical care will November issue.

408 ACTIVITIES OF OTHER ASSOCIATIONS

MAA AND SIAM june 19, 1965, Eugene, Oregon

There will be a joint meeting of the Pacific Northwest section of the Mathe­ matical Association of America and the Society for Industrial and Applied Mathematics on Friday, june 19, 1965, in conjunction with the Society meeting at Eugene, Oregon.

PROGRAM

9:00 - 10:00 Some aspects of celestial mechanics which have application to space travel Dr. Andre Deprit, Boeing Scientific Research Laboratories 10:00 - 10:30 Coffee break 10:30 - 1Z:OO Panel discussion: Mathematics in the two year community college Mr. j. A. Carlson, Everett Junior College Mr. john Knutson, Portland Community College Mr. W. S. Sims, British Columbia Institute of Technology Z:OO - 3:00 Rings and radicals Dr. Nathan Divinsky, University of British Columbia 3:00 - 3:30 Coffee Break 3:30 - 4:30 Control theory and mathematical programming Dr. Alan Goldstein, University of Washington

THE ASSOCIATION FOR SYMBOLIC LOGIC Leicester, England- September 9-1Z, 1965

A meeting of the Association for to Dr. j. N. Crossley, St. Catherine's Symbolic Logic will be held in the Uni­ College, Oxford and should not exceed versity of Leicester, England from Sep­ 300 words. Persons wishing to attend the tember 9-1Z, 1965 in conjunction with the meeting or the Summer School may re­ last few days of a Summer School in ceive details from Mr. R. F. Wheeler, Mathematical Logic, August Z6-September Mathematics Department, The University, 1Z. Abstracts of papers to be presented Leicester, England. should be submitted before july 15, 1965

409 1965 SIAM NATIONAL MEETING

The Society for Industrial and Ap­ at the Symposium were Professors plied Mathematics held its 1965 National W. Feller, Princeton University; N. Grad, Meeting in New York City on june 7, 8., Courant Institute; C. Lanczos, Dublin Insti­ and 9 at the Courant Institute, New York tute for Advanced Studies; C. C. Lin, University. The meeting which included M. I. T.; M. M. Schiffer, Stanford Univer­ invited lectures and contributed papers, sity; and G. B. Whitham, California In­ featured a special symposium in honor of stitute of Technology. The Symposium was Professor Richard Courant. Dr. A. S. organized by a committee consisting of Householder of the Oak Ridge National Professors K. 0. Friedrichs, F. john, Laboratory presented his retiring presi­ H. B. Keller, and P. D. Lax. It was sup­ dential address on the afternoon ·of ported by a grant from the Air Force june 7, at the opening session of the Office of Scientific Research. meeting. On june 8, Dr. V. L. Klee, Jr. Those at the meeting were invited of the University of Washington and to a reception on the afternoon of june 7 Boeing Mathematical Research Laboratory and to a banquet, held in honor of Profes­ gave an invited one-hour lecture entitled sor Courant, on the evening of june 8. "Some Unsolved Geometric Problems The speaker at the banquet was Dr. Warren from Science and Technology." Weaver of the Alfred P. Sloan Foundation. The Courant Symposium began on The general chairman of the 1965 National the afternoon of june 8 and continued Meeting was Dr. D. L. Thomsen, Jr. through june 9. Invited to present papers

NEWS ITEMS AND ANNOUNCEMENTS

INTERNATIONAL SYMPOSIUM ON DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS

An International Symposium on Dif­ differential equations and some special ferential Equations and Dynamical Systems topics in the theory of partial differential will be held at the University of Puerto· equations. Also there will be a limited Rico, College of Agriculture and Mechanic number of contributed papers on the Arts, Mayaguez, Puerto Rico on Decem­ above topics. ber 27-30, 1965. The purpose of the sym­ This symposium is to be sponsored posium is to discuss the developments in jointly by the Air Force Office of Scien­ the theory of ordinary differential equa­ tific Research, the University of Puerto tions and related topics that have trans­ Rico and Brown University. pired since the 1961 International Sympos­ Those wishing further information ium on Nonlinear Differential Equations are requested to write to the Center for and Nonlinear Mechanics held at the Dynamical Systems, Brown University, United States Air Force Academy, Colo­ Providence, Rhode Island. rado Springs, Colorado. Abstracts should be from two to A number of invited addresses will three typewritten pages and should be survey the developments in oscillations, submitted to the above address no later stability and control, hereditary pheno­ than September 1, 1965. mena, the qualitative theory of ordinary

410 MEMORANDA TO MEMBERS

MATHEMATICAL SCIENCES of appointments. EMPLOYMENT REGISTER The Mathematical Sciences Em­ ployment Register is sponsored jointly by The latest compilations of available the American Mathematical Society, the positions and of applicants for positions Mathematical Association of Ame·rica, and in the mathematical sciences, published the Society for Industrial and Applied May 15, 1965, may be purchased from the Mathematics. Mathematical Sciences Employment Re­ gister. The List of Applicants is available for $7 .50; the List of Positions costs NOTICE OF CHANGE OF ADDRESS $3.00. At the 1965 Summer Meeting in Effective June 1, 1965allcommuni­ Ithaca, New York, the Employment Regis­ cations directed to the American Mathe­ ter will again schedule interviews and matical Society should be addressed to: distribute a listing of applicants and po­ sitions. The Register will be open from American Mathematical Society 9:00 A.M. to 5:00 P.M. on Tuesday, P .0. Box 6248 August 31 through Thursday, September 2 Providence, Rhode Island 02904, in Rooms 200 and 205 of the Industrial U.S.A. and Labor Relations (ILR) Conference Center. Although the main headquarters There is no charge for registration, offices of the Society will remain at either to job applicants or to employers, 333 Grotto Avenue, correspondence and except when the late registration fee of orders should be directed to the above $5.00 for employers is applicable. Pro­ address. vision will be made for anonymity of ap­ The editorial offices of Mathemati­ plicants upon request and upon payment cal Reviews will move from Providence, of $5,00 to defray the cost involved in Rhode Island to Ann Arbor, Michigan on handling anonymous listings. June 1, 1965. The new address will be: job applicants and employers who Mathematical Reviews wish to be listed will please write to the University of Michigan Employment Register, P. 0. Box 6248, Ann Arbor, Michigan 48103, U.S.A, Providence, Rhode Island 02904 for appli­ cation forms or for position description Effective June 1, 1965 all corre­ forms. These forms must be completed spondence concerning MATHEMATICAL and returned to Providence not later than REVIEWS except for subscriptions, sales, July 1, 1965, in order to be included in and missing issues should be addressed the listings at the Summer Meeting in to the Ann Arbor office. Correspondence Ithaca. Position Description Forms which pertaining to actual distribution of current arrive after this closing date, but before and back issues should be addressed to: August 1, will be included in the register Sales and Subscription Department at the meeting for a late registration fee American Mathematical Society of $5.00. The printed listings will be P. 0. Box 6248 available for distribution, both during and Providence, Rhode Island 02904 after the meeting. U.S.A. It is essential that applicants and employers register at the Employment Professor W. J. LeVeque of the Register Desk promptly upon arrival at University of Michigan is now Executive the meeting to facilitate the arrangements Editor of Mathematical Reviews.

411 NEWS ITEMS AND ANNOUNCEMENTS

FIFTH U.S. NATIONAL CONGRESS awards were made upon the recommenda­ OF APPLIED MECHANICS tion of the NAS-NRC Committee on Math­ ematics Advisory to the ONR. The re­ cipients, their fields of research, and the The U. S. National Congresses of institutions making the appointments are Applied Mechanics are held every fourth listed below: year and are planned to supplement the International Congresses of Applied Mech­ William F. Hammond-Modular func­ anics. The Fifth U.S. National Congress tions, Brandeis University will be held at the University of Minne­ Micha A. Perles-Convexity, University sota from June 14-17, 1966. of Washington The Congress will be concerned Gerald J. Porter-Algebraic topology, with experimental and theoretical applied Brandeis University mechanics, including mechanics of rigid Michael I. Rosen ~Algebraic number bodies and deformable solids, mechanics theory, Brandeis University of fluids and gases, thermodynamics, Chung-Tuo Shih~Probability, Cornell and heat transfer. There will be invited University survey lectures and sessions for contrib­ Cleve B. Maier-Numerical Analysis, uted papers. Eidgenossischen Technischen Hoch­ Research workers are encouraged schule, Zurich to submit abstracts covering their current work. The abstracts should be no longer GUGGENHEIM than 200 words and should be submitted FELLOWSHIP AWARDS, 1965 by January 15, 1966 on special forms available from the Secretary of the Con­ The John Simon Guggenheim Mem­ gress. Each accepted paper will be allot­ orial Foundation has announced 1965 ted time for presentation and discussion fellowship awards totaling $2,115,700. at an appropriate session of the Congress. This sum, the largest ever granted by the To encourage new contributors in the Foundation in its annual competitions, field, the Editorial Committee will select exceeds the 1964 total by more than a a group of research papers from among quarter of a million dollars. The awards the abstracts for presentation and dis­ were made to 313 scholars, scientists, cussion in fulL and artists, who were chosen from among Inquiries regarding the Congress 1,869 applicants by the Foundation's Com­ should be addressed to Professor Robert mittee of Selection. Plunkett, Secretary of the Applied Mech­ Nine of the awards were made to anics Congress, 107 Aero Building, Uni­ mathematicians. The recipients, their in­ versity of Minnesota, Minneapolis, Min­ stitutions, and their proposed studies are nesota 55455. listed below. James Ax, Cornell University Studies in algebraic number theory ONR RESEARCH ASSOCIA TESHIPS IN Edgar H. Brown, Jr., Brandeis University MATHEMATICS Studies in algebraic and differential topology The Office of Naval Research has announced the award of six postdoctoral Robert Finn, Stanford University research associateships in mathematics Studies in the field of partial differen­ for the academic year 1965-1966. These tial equations

412 James Glimm, Massachusetts Institute of Dr. Hyman Bass, Barnard College; Technology Dr. Gilbert Baumslag, City University of Studies in the differential equations of New York; Dr. Glen E. Bredon, University the motion of gases of California, Berkeley; Dr. Edwin H. Rice University; Dr. Louis de Serge Lang, Columbia University Connell, University; Dr. Robert Studies in algebraic number theory Branges, Purdue W. Gilmer, Jr., Florida State University; Edward L. Reiss, New York University Dr. Mark Mahowald, Northwestern Uni­ Mathematical studies of the propagation versity; Dr. Daniel R. McMillan, Jr., of waves through solid media University of Virginia; Dr. George J. of Michigan; Dr. Calvin Murray Rosenblatt, University of Cali­ Minty, University California, Berke­ fornia, San Diego C. Moore, University of Massachusetts Nonlinear analysis of random processes ley; Dr. James R. Munkres, Institute of Technology; Dr. Richard S. Frank Spitzer, Cornell University P alais, Brandeis University; Dr. Hugo The theory of stochastic processes Rossi, Brandeis University; Dr. Robert jacob W olfowitz, Cornell University T. Seeley, Brandeis University; Dr. Studies in mathematical statistics and William G. Strang, Massachusetts Insti­ information theory tute of Technology; Dr. Joseph A. Wolf, University of California, Berkeley.

CONFERENCE ON CATEGORICAL ALGEBRA ALFRED P. SLOAN RESEARCH FELLOWSHIPS The University of California at San Diego held a conference on Categori­ Ninety-one young scientists in cal Algebra from June 7-13, 1965, with American and Canadian universities re­ the sponsorship of the Air Force Office cently were named to receive unrestric­ of Scientific Research. Participants in­ ted grants for basic research, totaling cluded M. Auslander, M. Barr, H. Bass, nearly $1.4 million, from the Alfred P. J. M. Beck, D. Buchsbaum, S. U. Chase, Sloan Foundation. S. E. Dickson, V. H. Dyson, S. Eilenberg, The grants, which are effective in E. Engeler, P. Freyd, J. W. Gray, D. K. September, 1965, are for fundamental Harrison, A. Heller, P. J, Hilton, J. R. research in chemistry, mathematics, Isbell, D. M. Kan, G. M. Kelley, ]. B. physics, and interdisciplinary fields, such Leicht, F. E. J. Linton, S. MacLane, as geochemistry and astrophysics. The G. S. Rinehard, H. Rohrl, J. Sonner, J. L. awards are, in general, for a two-year Verdier, C. E. Watts, and 0. Wyler. period. The Organizing Committee for the Sixteen of the awards were given conference consisted of S. Ellenberg, for research in mathematics. The recip­ D. K. Harrison, S. Mac Lane, R. G. ients are listed below. Pohrer, and H. Rohrl.

413 PERSONAL ITEMS

Professor WINIFRED ASPREY of professorship at the Graduate Mathematics Vassar College will be on leave for the aca­ Center of The City University of New York. demic year 1964-1965 at the University of Dr. J. M. GANI, a visiting Professor of California, Los Angeles. She has received a Statistics at Michigan State University, has National Science Foundation Faculty Fellow­ been appointed to the Chair of Statistics ship. at Sheffield University, Sheffield, England. Professor LOUIS AUSLANDER of Professor F. W. GEHRING of the Uni­ Yeshiva University has been appointed to a versity of Michigan has been appointed to professorship at the Graduate Mathematics a visiting professorship at Harvard Univer­ Center of The City University of New York. sity from July 1964 to July 1965. Professor M. L. BALINSKI of the Uni­ Mr. M. C. GEMIGNANI of the Univer­ versity of Pennsylvania has been appointed sity of Notre Dame has been appointed to to an associate professorship at the Gradu­ an assistant professorship at the State ate Mathematics Center of The City Uni­ University of New York at Buffalo. versity of New York. Dr. H. H. GOLDSTINE of the Inter­ Mr. 0. G. BECKER of the International national Business Machines Corporation, Telephone and Telegraph Company has Yorktown Heights, New York has been accepted a position as Project Manager, named Director of Scientific Development Operations Research with the Philco Cor­ for the Data Processing Division of the poration, Fort Washington, Pennsylvania. International Business Machines Corpora­ Professor R. H. BING of the Univer­ tion, White Plains, New York. He has also sity of Wisconsin has been elected to accepted an adjunct professorship at the membership in The National Academy of Graduate Mathematics Center of The City Sciences. University of New York. Professor D. H. BLACKWELL of the Mr. G. S. GOODMAN of Stanford Uni­ University of California, Berkeley has been versity has been appointed to a visiting elected to membership in The National assistant professorship at the Aarhus Uni­ Academy of Sciences. versity, Aarhus, Denmark for the current Dr. L. E. BLUMENSON of the Univer­ academic year. sity of Chicago has been appointed Senior Professor ALEX HELLER of the Uni­ Cancer Research Scientist, Mathematical versity of Illinois has been appointed to a Biology, at Roswell Park Memorial Institute. professorship at the Graduate Mathematics Professor G. F. CLANTON has been Center of The City University of New York. appointed Associate Dean at the College Dr. EINAR HILLE, Professor Emeritus of Arts and Sciences at Vanderbilt Univer­ of Yale University. has been appointed a sity. Senior Lecturer at the University of Cali­ Dr. R. G. DOUGLAS of the University fornia, Irvine. of Michigan has been awarded an NAS-NRC Dr. A. J. HOFFMAN of the International Postdoctoral Research Fellowship. He will Business Machines Corporation, Thomas J. study at the Institute for Advanced Study. Watson Research Center, Yorktown Heights, Dr. W. B. EVANS, Department of En­ New York has accepted an adjunct pro­ gineering, University of California at Los fessorship at the Graduate Mathematics Angeles has been appointed Director of the Center of The City University of New York. Biomedical Data Processing and Analysis Professor MARK KAC of the Rocke­ Center and Associate Professor in the De­ feller Institute has been elected to member­ partments of Mathematics and Biometry, ship in The National Academy of Sciences. Emory University Professor G. K. KALISCH of the Uni­ Dr. E. A. FELDMAN of Princeton Uni­ versity of Minnesota has been appointed to versity has been appointed to an assistant a professorship at the University of Cali-

414 fornia, Irvine. accepted a visiting professorship at the Dr. STANLEY KAPLAN on a Fulbright Graduate Mathematics Center of The City Fellowship at the University of Pisa, Pisa, University of New York. Italy has been appointed to an assistant Professor J. C. C. NITSCHE of the professorship at the Graduate Mathematics Institute of Technology, University of Min­ Center of The City University of New York. nesota has been appointed to a visiting Dr. S. S. KOH of the University of professorship at the University of Hamburg, California at Berkeley has been appointed Hamburg, Germany for the period May 1, to an assistant professorship at Washington 1965 to July 31, 1965. University. Professor RUFUS OLDENBURGER, Dr. A. G. KONHEIM of the International Director of the Automatic Control Center Business Machines Corporation, Thomas J. at Purdue University was elected an honor­ Watson Research Center, YorktownHeights, ary member of the Japan Society of Mech­ New York has accepted an adjunct pro­ anical Engineers. fessorship at the Graduate Mathematics Professor M. R. P ARAMESWARAN Center of The City University of New York. of Michigan State University has been ap­ Mr. J. E. LAGNESE of the University pointed Professor and Head of the Depart­ of Maryland has been appointed to an assist­ ment at the University of Kerala, Trivan­ ant professorship at Georgetown University. drum, India. Dr. H. J. LANDAU of Bell Telephone Dr. HARRY POLACHEK,formerTech­ Laboratories, Murray Hill, New Jersey nical Director of the Navy's Applied Mathe­ has accepted an adjunct professorship at matics Laboratory at David Taylor Model the Graduate Mathematics Center of The Basin, has accepted a position as Specialist City University of New York. in Mathematics and Computer Technology Professor ARNOLD LEBOW of New with the Division of Research of the United York University has been appointed to an States Atomic Energy Commission, Wash­ assistant professorship at the University ington, D. C. of California, Irvine. Dr. H. 0. POLLAK of Bell Telephone Dr. K. M. LEE of the University of Laboratories, Murray Hill, New Jersey Michigan has been appointed to an assist­ has accepted an adjunct professorship at ant professorship at the University of the Graduate Mathematics Center of The Rhode Island. City University of New York. Professor JEAN LERA Y of the College Mr. R. D. PURINGTON III of Case of France, Paris, France has been elected Institute of Technology has accepted a a foreign associate to the National Aca­ position as Mathematician with the United demy of Sciences. States Naval Weapons Laboratory, Dahlgren, Dr. H. E. LACEY of Abilene Christian Virginia. College has been appointed to an assistant Professor GIAN-CARLO ROTA of the professorship at the University of Texas. Massachusetts Institute of Technology has Professor GEORGE MCCARTYofHar­ been appointed to a professorship at the vey Mudd College has been appointed to an Rockefeller Institute, effective in July. assistant professorship at the University of Professor J. P. RUSSELL of the Poly­ California, Irvine. technic Institute of Brooklyn and Adminis­ Dr. ROBERT MALTZ of San Diego trative Officer of the Department of Math­ State College has been appointed to an niatics has been named Associate Dean of assistant professorship at the University the Institute's Long Island Graduate Center of California, Irvine. at Farmingdale. He will continue also as Dr. G. E. MARTIN of the University of Professor of Mathematics. Michigan has been appointed to an assistant Dr. BERNARD RUSSO of the University professorship at the University of Rhode of California, Los Angeles has been ap­ Island. pointed to an assistant professorship at Professor G. J. MINTY of the Univer­ the University of California, Irvine. sity of Michigan has been appointed to a Professor RICHARD SACKSTEDER of professorship at Indiana University. Columbia University has been appointed to Professor Emeritus MARSTON MORSE an associate professorship at the Graduate of the Institute for Advanced Study has Mathematics Center of The City University

415 of New York. University has been appointed to an asso­ Professor W, R, SCOTT of the Univer­ ciate professorship at the University of sity of Kansas has been appointed to a pro­ California, Irvine. fessorship at the University of Utah. Dr. NEAL ZIERLER of the Mitre Dr. ARNOLD SEIKEN of Michigan State Corporation has been appointed a Member University has been appointed to an asso­ of the Staff of the Communications Research ciate professorship at the University of Division at the Institute for Defense Rhode Island. Analyses. Mr. SAKAI SHOICHIRO of Yale Uni­ Dr. ISRAEL ZUCKERMAN of Queens versity has been appointed to an associate College has been appointed to an assistant professorship at the UniversityofPennsyl­ professorship at Vassar College. vania. Dr. D, C. SODA of Yale University has The following promotions are announced: been appointed to an assistant professorship at Washington University. A. K. AZIZ, Georgetown University, Dr. H. A. SIMMONS of Northwestern to a professorship. University has been appointed to a visiting DAVID GORMAN, Washington Univer­ professorship at Wartburg College for the sity, to Associate Professor of Applied academic year 1964-1965. Mathematics. Dr. J. F. SLIFKER of the University G. B. HENSEL, The Catholic Univer­ of Notre Dame has been appointed Lecturer sity of America, to an assistant professor­ at the University of Manchester, Manches­ ship. ter, England. R. H. McDOWELL, Washington Uni­ Dr. WILLIAM SMOKE of the University versity, to an associate professorship. of California, Berkeley has been appointed M. W. OLIPHANT, Georgetown Uni­ to an assistant professorship at the Uni­ versity, to a professorship. versity of California, Irvine. PAUL SCHAEFER, State University of Professor ALFRED TARSKI of the New York at Albany, to a professorship. University of California, Berkeley has been G. F. SIMMONS, Colorado College, to elected to membership in The National a professorship. Academy of Sciences. JOHN THERRIEN, State University of Professor E. 0. THORP of New Mexico New York at Albany, to an associate pro­ State University has been appointed to an fessorship. acting associate professorship at the Uni­ versity of California, Irvine. The following appointments to Instructor­ Mr. A. G. VASSALOTTI of Hofstra ships are announced: University has been awarded a Science Faculty Fellowship from the National Sci­ College of St. Thomas: J. W. FEND­ ence Foundation. RICH, G. P. SCHWARTZ; State University Dr, G. R. VERMA of Birla College, of New York at Albany: EDMUND MEM­ India has been appointed to an assistant MOTT; Swarthmore College:ERIKA MARES. professorship at the University of Rhode Island. Dr. W. L. W ALTMANN of Iowa State Deaths: University has been appointed to an assist­ ant professorship at Wartburg College. Mr. V. J. DOBERL Y (DOBROLIUBOFF) Mr. R. J. WERNICK of Mechanical of New Rochelle, New York died on Febru­ Technical Incorporated has been appointed ary 21, 1965 at the age of 66. He was a to an assistant professorship at the State member of the Society for 13 years. University of New York at Albany. Professor HSIEN-CHIANG HSIEH of Dr. N. M. WIGLEY of Los Alamos Northwestern University died in August, Scientific Laboratory, Los Alamos, New 1964 at the age of 34. Mexico has been appointed to an assistant Professor JOHN RALEIGH of Temple professorship at the University of Arizona, University died on December 26, 1964 at starting next fall. the age of 63. He was a member of the So­ Professor JAMES YEH of Rochester ciety for 20 years.

416 NEW AMS PUBLICATIONS

MEMOIRS Number 54 GENERALIZATIONS OF A THEOREM OF CARATHEODORY By John R. Reay 52 pages; List Price $1.50, Member Price conv X; the set X may have certain con­ $1.13. nectedness or symmetry conditions; the point p may be replaced by a finite set of Caratheodory has shown that if a distinct points. In each case the general point p is in the convex hull of a set X in an minimal upper bound is found for the car­ n-dimensional linear space, then plies in dinality of a subset of X whose convex hull the convex hull of some n + 1 or fewer similarly contains the given point (or points). points of X. The present Memoir considers The theory of positive bases for linear various extensions and generalizations of spaces proves to be a useful tool in estab­ this theorem. The hypothesis may be modi­ lishing these theorems. The theory is ex­ fied in one or more of the following ways: tended by several useful results found by the point p may be interior to conv X or the author. "well-embedded" in the boundary of

Number 55 FOUNDATIONS OF RELATIVE HOMOLOGICAL ALGEBRA By S. Eilenberg and J. C. Moore 40 pages; List Price $1.50, Member Price as established in Chapter I, is that of a $1.13. projective class of sequences in an arbi­ trary pointed category. Chapter II presents This Memoir deals with the exist­ the "adjoint theorem", which is used in ence and properties of projective resolu­ Chapter III to construct various projective tions with a view to defining the concept of classes. The fourth and final chapter a derived functor, which is basic to homo­ studies the category, together with its logical algebra and will be discussed in subcategories, of complexes over an subsequent articles. The main notion here, abelian category.

SELECTED TRANSLATIONS, Series II

Volume 47 Volume 48 304 pages; List Price $8.00, Member Price Approximately 300 pages. Prepublication $6.00. Prices, before June 1, 1965: List $9.90, Member $7 .43; after June 1: List at least Thirteen papers on Functional Analysis $11.00. and Partial Differential Equations, by M.S. Brodskit, Ju. L. Dalecki!, D. M. ETdus, Fourteen papers on Logic, Algebra, I. S. Iohvidov, S. G. Krein, 0. A. Lady:len­ Complex Variables and Topology, by L. M. skaja, V. B. LidskiT, Ju. I. Ljubic, V. I. Abramov, V. S. Cernjavskii, E. S. Go1od, Macaev, A. Ja. Povzner, L. A. Sahnovic. T. D. Ma!strova, A. I. Mal'cev, S. P. Novikov, Ju. L. Smul'jan, I. V. Suharevskir, N. N. B. I. Penkov, V. A. Roblin, I. R. Safarevic, Ural'ceva. V. V. Sazonov, B. H. Sendov, V.I.Sestakov, A. S. Svarc, V. S. Varadaraln, V. S. Vladimirov.

417 SELECTED TRANSLATIONS IN MATHEMATICAL STATISTICS AND PROBABILITY

Volume 5 Approximately 400 pages; List Price $11.60, B. M. Levitan, Ju, V. Linnik, S. V. Nagaev, Member Price $8,70, V. V. Petrov, I. V. Romanovskii, N. A. Sapogov, 0. V. Sarmanov, A. V. Skorohod, Nineteen papers on Statistics and N. V. Smirnov, I. Vincze, V. A. Volkonskif, Probability, by D. L. Berman, A. A. Borov­ V. M. Zolotarev. kov, A. Ja, Dorogovcev, D. V. Gusak, I. A. Ibragimov, M. M. Judin, V. S, Koroljuk,

Recent Reprints

COLLOQUIUM PUBLICATIONS

Volume 38 THEORY OF GRAPHS By Oystein Ore

280 pages; List Price $9 ,20, Member Price systematic interest; the second volume will $6.90. deal with such topics as the four-color conjecture, the New fields of application, such as game theory of flow, electrical networks, theory and programming, communications and games. The fiften theory, electrical networks and switching chapters of the first volume present, in a well-organized circuits, and problems from biology and setting, the results of psychology, have given an intense stimulus Cayley, Ramsey, Frucht, Hall, Mann, Ryser, Dirac to the development of graph theory in the and many other out­ standing researchers past twenty years. The present volume, the in the subject. This volume first of a projected two-volume work, is reprinted without changes from the gives an almost complete treatment of the first edition of 1962. basic concepts and the results of particular

NEWS ITEMS AND ANNOUNCEMENTS

INSTITUTE ON INTEGER PROGRAMMING AND NETWORK FLOWS

NATO and the Office of Naval Re­ R, E. Gomory, T. C. Hu, E. L Johnson, search are jointly sponsoring an advanced C. Witzgall and R, D. Young. study institute on Integer Programming Single rooms with 3 meals will be and Network Flows to be held at the Uni­ $20.00 per day; double rooms with 3 versity of California's Alumni Center at meals will be $15,00 per day, Lake Tahoe, California from June 21st For further information write to: through July 1st. Professor T. C. Hu, Operations Research The speakers will be M. L. Balin ski, Center, 4193 Etcheverry Hall, University E. M. L. Beale, P. Carnian, G, B. Dantzig, of California, Berkeley, California 94720. J, Edmonds, D. R. Fulkerson, F. Glover, Attention, Barbara Orlando,

418 DOCTORATESCONFERREDIN1~4

The following are among those who have received doctorates in the mathematical sciences and related subjects from universities in the United States and Canada during 1964. After the name of each university is given the number of doctorates granted by that institution. Each entry includes the dissertation title, non-mathematical minor if applicable, and, when available, the name of the thesis advisor. 97 universities are listed with a total of 643 names.

ADELPHI UNIVERSITY {1) Sandberg, Rollin Theodore Cannonito, Frank Benjamin On the compatibility of integrals Hierarchies of computable groups and the (B. Schweizer) word problem (D. Solitar) AUBURN UNIVERSITY (5) Clanton, Donald Henry UNIVERSITY OF ALABAMA (3) Minor: Physics Allsbrook, Janet Scott Some algorithms for the calculation of the Equicontinuity and recursive transformation characteristic roots and vectors of a groups normalizable matrix (R. W. Bagley) (N. Macon) Fitzpatrick, Marjorie Higgins Horner, James Melvin Kronecker and other special products of The Pochhammer differential equation: A matrices unified approach to a class of special (E. V. Haynsworth) functions (M. 0. Gonzalez) Ford, Jo Wharton Imbedding closed collections in continuous Stegenga, Martin Eugene collections of arc On the topologies of N-ordered sets (B. Fitzpatrick, Jr.) (L. G. Novoa) Ingram, William Thomas UNIVERSITY OF ALBERTA, CALGARY (1) Concerning nonplanar circle-like continua (H. Cook) Varma, Arun Kumar Lacunary interpolation through trigonomet­ Whipple, Kenneth Edward ric and power polynomials Cauchy sequences in a Moore space (A. Sharma) (B. Fitzpatrick, Jr.) BRANDEIS UNIVERSITY (3) UNIVERSITY OF ALBERTA, EDMONTON (1) Freedman, Marvin Irving Murdeshwar, Mangesh Ganesh Extremal overlap with respect to translation Non-coercive estimates for forms over (L. Moser) vector-functions (J. J. Cohn) THE AMERICAN UNIVERSITY (1) Kim, Ki Myong Marshall, Louis Clinton Deformations, related deformations and a Approximately continuous transformations universal subfamily on compact metric spaces (T. Matsusaka) (E. S. Johnson) Riley, John Astwood Ideal theory in noetherian rings UNIVERSITY OF ARIZONA (2) (M. Auslander) DeVore, Robert Henry Best possible inequalities for the lowest UNIVERSITY OF BRITISH COLUMBIA (1) points of the fundamental domains for the Hilbert modular groups for RfV5) and Lee, Richard Way Mah R(V'Z) The motion of a self-excited rigid body (H. Cohn) (E. Leimanis)

419 BROWN UNIVERSITY (12) Foster, Lorraine Minor: Philosophy Al-Khozaie, Said Mehsin On the characteristic roots of the product On Green's functions and Saint-V enant' s of certain rational integral matrices of principle in linear theory of viscoelasti­ order two city (0. T. Todd) (E. Sternberg) Gross, Fletcher Brothers, John Edwin Minor: Physics Integral geometry in homogeneous spaces The 2-length of a finite solvable group (H. Federer) (M.Hall, Jr.) Edelstein, Warren Stanley Kruse, Robert Some topics in the linear theory of dynamic Minor: Economics viscoelasticity Rings with periodic additive groups in which (M. E. Gurtin) all subrings are ideals Friedman, Nathaniel Abraham (R. A. Dean) A class of point transformations of the unit Lopes, Louis interval and their associated isometries Operator differential equations in Hilbert (R, V. Chacon) space Hughes, Thomas Haley (C. R. DePrima) Problems in hydrodynamic stability Mack!, Jack (W. H. Reid) Minor: Physics Krieger, Henry Alan Singular perturbations of a boundary-value Toeplitz operators on locally compact problem for a system of nonlinear differ­ Abelian groups ential equations (M. Rosenblatt) (A. Erdlilyi) Kumpel, Paul Gremminger, Jr. Sawyer, Stanley On the homotopy groups of the exceptional On inequalities of weak type Lie groups (C. R. DePrima) (B. Harris) Shampine, Lawrence Niiler, Pearn Peter Asymptotic L2 inequalities of Markoff type Asymptotic analysis of thermal instability (J. Todd) in a rotating sphere (F. E. Bisshopp) UNIVERSITY OF CALIFORNIA, BERKELEY (35) Ruh, Ernst Alfred Amaral, Leo Huet On the automorphism group of a G-structure Hypersurfaces in non-Euclidean spaces (K. Nomizu) (S. S. Chern) Van Ness, John Winslow Anderson, Donald Werner Estimates of the bispectrum of stationary A new cohomology theory random processes (P. E. Thomas) (M. Rosenblatt) Banchoff, Thomas Francis Wineman, Alan Stuart Tightly embedded two-dimensional polyhe­ Material symmetry restrictions on con­ dral manifolds stitutive equations (S. S. Chern) (A. C, Pipkin) Cooper, William Secord Witterholt, Edward John Minor: Logic and Methodology of Science The interaction of thermal and ultrasonic A formalization of the logic of ordinary waves in solids conditionals (R. Truell) (W. W. Adams) Cottle, Richard Warren Nonlinear programs with positively bounded Jacobians CALIFORNIA INSTITUTE OF TECHNOLOGY (B) (R. S. Lehman and G. B. Dantzig) Andrea, Stephen Creese, Thomas Morton Minor: Physics Maximal subgeometric semi-norm inter­ On the embedding of homeomorphisms of polation in Lebesgue spaces and function the plane in flows algebras (F. B. Fuller) (E. A. Bishop)

420 Dachslager, Howard Lee Olson, Milton Philip On sums of two-valued functions of special A characterization of conditional probab­ type ility (D. Blackwell) (E. W. Barankin) F arooqui, Asghar Saeed Osgood, Charles Freeman Water waves in triangular canal Some theorems on Diophantine approxima­ (R. S. Lehman) tion (R. S. Lehman) Feder, Samuel Immersions and embeddings in complex Pickett, Harry Eldon projective spaces Subdirect representation of relational sys­ (E. H. Spanier) tems (A. L. Foster) Fickas, Ernest Theodore, Jr. Studies on convergence and summability Raghavachari, Madabhushi of Dirichlet series The two-sample scale problem when lo­ (H. Helson) cations are unknown (E. L. Lehmann) Filippenko, Vladimar On the reflection of harmonic functions and Ramachandramurty, Ponnapalli Venkata of solutions of the wave equation On some nonparametric estimates and tests (H. Lewy) in the Behrens-Fisher situation (E. L. Lehmann) Fisher, Charles Samuel Accretive operators and an evolution equa­ Ross, Leonard Erwin tion On graded Lie algebras (T. Kato) (G. P. Hochschild) Garland, Howard Roussas, George Gregory On the cohomology of lattices in Lie groups Some extensions of Dubins' and Savage's (S. S. Chern) work on red and black (D. Blackwell) Greenberg, Vida Lazarus Robust inference in some experimental Sherman, Malcolm Jay designs Vector value analytic functions (E. L. Lehmann) (H. Helson) Gupta, Milan Kumar Stark, Harold Mead An asympototically nonparametric test of On the tenth complex quadratic field with symmetry class number one (E. L. Lehmann) (D. H. Lehmer) Herman, Eugene Alexander Suryanarayana, Ponnaluri Generalized Fredholm conditions in anal­ Higher order differentiability of solutions gebra of singular integral operators of abstract evolution equations (H. 0. Cordes) (T. Kato) Jaffa, Robert Edward Tracewell, Theodore Nelson Extremal doubly stochastic operators and A general theory of mediality in groupoids related matrices (G. P. Hochschild) (J. Feldman) Ucci, John James Jayanthi, Chidambaraswamy Skew homotopy theory of bundles Divisibility properties of certain factorials (M. W. Hirsch and E. H. Spanier) (D. H. Lehmer) Venti, Richard James Koh, Sebastian Su Some changes of variables theorems On affine symmetric spaces (S. P. Diliberto) (S. Kobayashi) Yasuhara, Mitsuru Moskowitz, Martin Allen Studies on generalized quantifiers Locally compact Abelian groups and homo­ (R. L. Vaught) logical algebras (G. P. Hochschild) UNIVERSITY OF CALIFORNIA, DAVIS (1) Moyer, Robert Dale Merkel, Rudolph Burton Comments on the differentiability of solu­ Semigroups which are the union of their tions of boundary value problems [0-) minimal left ideals (H. 0. Cordes) (T. Tamura)

421 UNIVERSITY. OF CALIFORNIA, LOS ANGELES Mookini, Edwin ( 18) Necessary conditions and sufficient for an Akasaki, Takeo optimal control problem Maps into an absolute neighborhood retract (M. R. Hestenes) (S. T. Hu) Silvers, Abraham Arveson, William B. Differential geometric methods in the study Prediction theory and group representations of mappings into abstract Riemann sur­ (H. A. Dye) faces (L. Sarlo) Balluce, Lawrence P. The theory of infinite valued predicate logic Strauss, Frederick B. (C. C. Chang) Additive commutators of matrices (B. Gordon) Boehm, Barry Existence characterization and convergence Westman, Joel J. of best rational Tchebycheff approxima­ Induced mappings of differential operators tion (R. Arens) (E. W. Cheney) CARNEGIE INSTITUTE OF TECHNOLOGY (10) Eng, Fook H. Minimal surfaces Borchers, Dietmar Rudolf (E. F. Beckenbach) Second order stochastic differential equa­ tions of the Ito type Feldstein, Morley A. (M. M. Rao) Discretization method for retarded ordinary differential equations Bragg, Lincoln Ellsworth (H. J, Stetter) On relativistic worldlines and motions, and on non-sentient response Figa-Talamanca, Alessandro (W. Noll) Multipliers of P-integrable functions (P. C. Curtis, Jr.) Feldman, Jerome Arthur A formal semantic for computer-oriented Gragg, William B. languages Repeated extrapolation to the limit in the (A. J, Perlis) numerical solution of ordinary differen­ tial equations Karlovitz, Les Andrew (P. K. Henrici) Theory and application of duality in infinite programming Gray, Alfred (R. J. Duffin) Minimal varieties and Kahler submanifolds Lazer, Alan Cecil (B. O'Neill) The behaviour of solutions of the differen­ tial equation y"' + p(x)y' + q(x)y ~ 0 Guinn, Theodore (R. A. Moore) On first order necessary conditions for variational and optimal control problems London, Ralph Leslie (M. R. Hestenes) A computer program for discovering and proving sequential recognition rules for Johnson, William L. well-formed formulas defined by a re­ Harmonic functions of bounded character­ cursive grammar istics in locally euclidean spaces (A. Newell) (L. Sarlo) Mangad, Moshe Asymptotic development of Fourier inte­ Larsen, Kenneth grals Extremal harmonic functions in locally (R. J. Duffin) euclidean n-spaces (L. Sarlo) Patrick, Merrell Lee Inequalities concerning Jacobi polynomials Levine, Arnold (R. J, Duffin) Optimal weighting and choice of observation Peterson, Elmor Lee points in prediction problems Some properties of complex valued lattice (P. G. Hoel) functions (R. J, Duffin) Meehan, Robert Capacity problems in locally Euclidean Porsching, Thomas August spaces Network models for conductance (L. Sarlo) (R, J. Duffin)

422 CASE INSTITUTE OF TECHNOLOGY (5) Kuo, Tzee-Char Universal objects for spectral sequences Abrego, Jose Trevino (S. Mac Lane) Purchasing raw material on a fluctuating market Mountjoy, Robert Harbison (S. Sengupta) Abelian varieties attached to representa­ tions of discontinuous groups James Thomas Ross Jackson, (S. Mac Lane and W. L. Baily) Some speculative strategies in the stock market Oklander, Evelio (S. Sengupta) On interpolation of Banach spaces (A. P. Calder6n) King, William Richard Personnel evaluation and optimal assign­ Park, Heebok ment Improving the robustness of inferences (R. Ackoff) (D. L. Wallace) Nathan, Gad Solovay, Robert Martin An optimal matching process A functorial form of the differentiable (E. Marks) Riemann-Roch theorem (S. Mac Lane) Pruzan, Peter Mark An application of control theory to national Towber, Jacob economic planning Reducibility in the exterior algebra over a (S. Sengupta) free module (I. Kaplansky) THE CATHOLIC UNIVERSITY OF AMERICA(3) UNIVERSITY OF CINCINNATI (6) Conover, William Jay Minor: Anthropology Alonso, Father James A k-sample model in order statistics On differential equations of fractional order (E. Batschelet) (A. J. Macintyre) Jordan, Sister Mary Ignatius Casey, Joseph Kenneth Minor: Physics Functions harmonic in cones and similar On the convergence of trigonometric series infinite regions and integrals (A. J. Macintyre) (T. Kawata) Chiang, Pou-Shun Maisel, Herbert Estimates for constants in annular forms Minor: Psychology of Bloch's theorem Restricted sequential test for comparisons (A. J. Macintyre) (E. Batschelet) Kuo, Yueh-er Wang Relationships between tensor products and (12) UNIVERSITY OF CHICAGO inequalities Barsky, Marvin (A. Jaeger) convergence of series On repeated Meyer, Kenneth Ray (A. Zygmund) Functional equations and control theory Brown, Robert Bruce (A. J. Macintyre) of types E and E Lie algebras 6 7 Warlick, Charles H. (A. A. Albert) On fundamental solutions for a class of Carlborg, Frank William polyharmonic finite difference operators A procedure for selecting independent (C. Saltzer) variables in multiple regression (W. H. Kruskal) COLORADO STATE UNIVERSITY (5) Chang, Chao Ping Barr, Donald Roy On certain exponential sums arising in Sequential decision rules for a multiple conjugate multiple Fourier series choice theory (A. Zygmund) (K. Sea) DeSapio, Rodolfo Connell, Terrence Lee Embedding Tr-manifolds Sample sizes for specified precision (R. K. Lashof) (F. A. Graybill) Hamelink, Ronald Clare Owen, William Burrage Lie algebras of characteristic 2 General confidence intervals (I. Kaplansky) (P. W. Mielke, Jr.)

423 Rounding, Robert Campbell Green, Paul Stephen A two-step sample size problem KSC*(X), An extraordinary cohomology (F. A. Graybill) theory (I. 8 erstein) Urquhart, Norman Scott Minor: Genetics Shih Kuo, Chia-Hui On the permutation distribution of a multi­ On secondary operations in homotopy and variate test homology (J. S. Williams) (:P~ J. Hilton) Siegel, jerrold Norman UNIVERSITY OF COLORADO (1) Generalized homology with continuous co­ Guenther, Ronald Bernard efficients On positvive solutions of parabolic equations (I. Berstein) (W. B. Fulks) Whitman, Walter William Some strong laws for random walks and COLUMBIA UNIVERSITY (8) Brownian motion Adams, William Wells (F. Spitzer) Transcendental numbers in the p-adic do­ Wolf, Carol Euwema main Partial recursive operators on recursively (S. Lang) closed sets Bank, Steven Barry (A. Nerode) On the instability theory of differential Wong, Yuen-Fat polynomials '11"-manifolds and homotopy type of manifolds (W. C. Strodt) (I. Berstein) Beck, Jonathan Mock Triples, algebras and cohomology DUKE UNIVERSITY (2) (S. Eilenberg) Purcell, Wilbur Hallan, Jr. Gardner, Laurence Terrell, Jr. Minor: Physics On isomorphisms of C*-algebras A calculus-of-variations approach to the (R. V. Kadison) eigenvalues of a finite difference equation (F. G. Dressel) Glickfeld, Barnett Weil Contributions to the theory of holomorphic Wenner, Bruce Richard functions in commutative Banach alge­ Minor: Philosophy bras with identity A Nagata-type metric on infinite-dimen­ (E. R. Lorch) sional spaces (J. H. Roberts) Linton, Fred Ernest J. The functorial foundations of measure theory UNIVERSITY OF FLORIDA (1) (E. R. Lorch) LeVan, Marijo O'Connor Mayer, Raymond Allen Minor: Philosophy Localization and summability for Fourier Generalized totients over certain classes of series on compact groups n-square matrices (R. V. Kadison) (J. E. Maxfield) Meyer, Paul Richard Topologies on spaces of real-valued func­ FLORIDA STATE UNIVERSITY (1) tions Schaufeie, Christopher Baldwin (E. R. Lorch) On spanning surfaces for links (M. L. Curtis) CORNELL UNIVERSITY (9) Barnes, Bruce Alan GEORGE PEABODY COLLEGE FOR TEACHERS Modular annihilator algebras (2) (R. A. Sonic) Kay, james Frazier Minor: Science Education Brown, Lawrence David An investigation in the teaching of quadratic On the admissibility of invariant estima­ equations using programmed instruction tors of location parameters (J. H. Banks) (J. Kiefer) Speed, George Maurice Cornell, Ross Hunter Minor: Comparative Education Commutator product in function spaces History of projective geometry (I. 8 erstein) (F. L. Ceiauro)

424 GEORGE WASHINGTON UNIVERSITY (1) Isaacs, Irving Martin Finite p-solvable linear groups Lilliefors, Hubert Whitman (R. Brauer) An information theoretic application of lin­ ear discriminant functionals to stochastic Johansen, Donald Edward processes Optimal control of linear stochastic sys­ (S. Kullback) tems with complexity constraints (A. E. Bryson) UNIVERSITY OF GEORGIA (1) Kripke, Bernard Britain, Joel Williams Modules of holomorphic vectorvalued func­ The effect of truncation on tests of hypothe­ tions ses for normal populations (L. H. Loomis) (A. C. Cohen, Jr.) Kronstein, Karl M. Characters and systems of subgroups (R. Brauer) HARVARD UNIVERSITY (29) Lichtenbaum, Stephen Curves over discrete valuation Baum, Howard Richard (J. T. Tate) On hypersonic small disturbance theory for steady flows Lindzen, Richard (G. F. Carrier) Radiative and photochemical processes in strato- and mesopheric dynamics Black, Fischer Sheffey (R. M. Goody) A deductive question answering system (P. Fischer) Mann, James Edward The baffled resonator Bloom, Joseph Morris (G. F. Carrier) The effect of a riveted stringer on the stress in a sheet with cracks or cutouts McBride, Lyle Erwin (J. L. Sanders) Determination of optimal parameters for multivariable systems Dauns, John (K. S. Narendra) Convolution transforms whose inversion functions have complex roots Passman, Donald (D. V. Widder) Group ring properties of groups (R. Brauer) Drew, Philip Garfield A rapid-acting parameter-perturbation Plath, Warren Joseph adaptive control Multiple-path syntactic analysis of Russian (R. E • Kronauer) (A. G. Oettinger) Dreyfus, Stuart Ernest Quillen, Daniel Grey Variational problems with state variable Formal properties of over-determined sys­ inequality constraints tems of linear partial differential equa­ (A. E. Bryson) tions (R. H. Bott) Fields, Jerry Rational approximation-s to generalized Ramamoorthy, Chittoor hypergeometric functions Generating functions of abstract graphs with (J. L. Walsh) systems applications (D. Tufts) Fine, Terrence Leon Properties of an optimum digital system Rossettos, John Nicholas (D. Tufts) Toroidal shells under internal pressure in the transition range Goldwyn, Roger Martin L. Sanders) Generation of quadratic type Liapunov func­ (J. tions Schlessinger, John Michael (K. S. Narendra) Infinitesimal deformations of singularities T. Tate) Haimo, Deborah Tepper (J. Integral equations associated with Hankel Streeter, Donald Nelson convolutions Adaptive control using correlation methods (D. V. Widder) (K. S. Narendra) Holzsager, Richard Sussenguth, Edward Henry On spaces with multiplication and comulti­ Structure matching in information proces­ plication sing (R. H. Bott) (G. Salton)

425 Turyn, Richard Joseph Diekhans, Herbert Henry Character sums and difference sets Abstract density theorems under hypotheses (A. M. Gleason) of mean regularity (W. Trjitzinsky) Van Dyke, Peter Stresses about a circular hole in a cylin­ Eby, Edward Stuart drical shell Abstract Vitali-Carath~odory theorem with (B. Budiansky) infinite measures (W. Trjitzinsky) ILLINOIS INSTITUTE OF TECHNOLOGY (10) Gudder, Stanley Phillip Allgower, Eugene Leo A generalized probability model for quan­ Generalized derivatives tum mechanics (H. Reingold) (R. Bartle) Czarnecki, Adam Zbigniew Haggstrom, Gus Wendell Physical systems in a Finsler space Optimal stopping and experimental design (J. De Cicco) (D. Burkholder) Farkas, Herbert Moe Ihrig, Ann Harris Quadratic forms in quadratic fields Applications of recursive function theory to (G. Pall) algebra (W. Boone) Lemos, Anthony Huang-Rhys factors for absorption and emis­ Lanier, Lyle Hicks, Jr. sion of light by F-centers Semi-groups of scalar operators (J. Markham) (R. Bartle) Ling, Joyce Cho-Hsin Pak, Jingyal Representation of associative functions Action of transformation groups on lens (A. Sklar) spaces (D. Bourgin) Mitchell, Theodore Invariant means on semigroups and the Phelan, James Frederick constant functions Metabelian groups of order p1 ° (R. A. Silverman) (H. R. Brahana) Penner, Sidney Pu, Lena Chang Bi- and tri-operational algebras of func­ Group representations tions (I. Reiner) (A. Sklar) Rehm, Allan Stanley Pranger, Walter Anthony Schwartz distributions Certain functional equations and their series (P. W. Ketchum) solutions Rowe, Keith Aylwin (M.A. McKiernan) On Borel-Moore homology theory Synowiec, John Adelbert (A. Heller) Some theorems in pseuto-conformal geom­ Russell, Peter Bradley etry Primes generated by polynomials in one (J. De Cicco) variable Tsai, Chester Eng-Shi (P. T. Bateman) Flexible partially stable algebras Wilde, Carroll Orville (L. A. Kokoris) On amenable semigroups and applications UNIVERSITY OF ILLINOIS (19) of the Stone-Vech compactification (M. M. Day) Al-Hussaini, Ata Nari Some almost everywhere convergencetheo­ Shult, Ernest Edward rems for positive operators Finite groups admitting abelian fixed point (D. Burkholder) free operator groups (M. Suzuki) Berton, John Andrew Asymptotes in straight line planes Singletary, Wilson Eugene (H. Levy) Decision problems in logic and algebra (W. Boone) Bollman, Dorothy Ann On the periodicity of states in a linear Thomas, Robert Jay sequential machine having a composite Classificatinon and properties of three valued modulus logics (F. E. Hohn) (F. E. Hohn)

426 INDIANA UNIVERSITY {10) Lomen, David Orlando Finite deformation of elastic membranes Buckley, Joseph Thaddeus with application to the stability of an Polynomial mappings on groups inflated sphere {E. Snapper) {H. J. Weiss) Conway, Edward Dairo, III Peterson, Leonard Dean Hamilton-Jacobi theory and generalized Stability of solutions of non-linear diffusion solutions of partial differential equations problems {E. Hopf) {C. G. Maple) Graham, Phyllis Ann Ramsey, Fred Lawrence Cohomology of dihedral groups of order 2 Effect of trend elimination on tests for {G. Whaples) time series Graver, Jack Edward {R. Buehler) An analytic triangulation of an arbitrary Rogge, Thomas R. real analytic variety Nonlinear sloshing {A. H. Wallace) {H. J. Weiss) Parr, James Theodore Secres~ Bruce G. Cohomology of cyclic groups of prime square The propagation and growth of discontin­ order uities in magnetogasdynamics in the pre­ {G. Whaples) scence of finite conductivity Pendleton, Robert Leon {G. A. Nariboli) Structure theorems for torsion free coher­ Waltmann, William Lee ent sheaves over the Riemann manifold Tridiagonalization of an arbitrary square of a field matrix {E. Snapper) {R. J. Lambert) Prasad, Balarm Fracture of thick flat plates of elastic­ UNIVERSITY OF IOWA {9) plastic material under uniform stress Barr, David Ross {T. Thomas) On the power function of the likelihood ratio Robertson, James Bryon test for cases in which the range depends Multivariate continuous parameter we kly upon a parameter stationary stochastic processes {R. V. Hogg) {P. Masani) Bradley, John Spurgeon Rosenberg, Milton Adjoint quasi-differential operators of The spectral analysis of multivariate weakly Euler type stationary stochastic processes {W. T. Reid) {P. Masani) Fugate, Joseph Braucher Yang, Kung Wei On decomposable chainable continua Realization of cohomology of certain finite {S. Armentrout) groups Holt, John Melvin {G. Whaples) Integral inequalities related to non-oscil­ lation theorems for differential equations {W. T. Reid)

IOWA STATE UNIVERSITY {9) Johnson, Roy Andrew On product measures and Fubini's theorem Bueker, Robert Clark in locally compact spaces Existence of almost-periodic solution of {S. K. Berberian) functional- differential equations {G. Seifert) Kegley, Junius Colby Convexity with respect to Euler-Lagrange Dayhoff, Eldred Eugene differential operators Generalized polykays and application to {W. T. Reid) obtaining variances and covariances of components of variation Kerr, Gerald Lee {0. Kempthorne) Almost-Gaussian domains {H. T. Muhly) Graber, Leland Dale Some stability properties of linear operator Lininger, Lloyd Lesley equations Some results on crumpled cubes {R. H. Homer) {S. Armentrout)

427 Schori, Richard Miles Cranford, Richard Henry Some results on inverse limit spaces Relations between classes of idea in an (S. Armentrout) integral domain (H. S. Butts) JOHNS HOPKINS UNIVERSITY (1) Crownover, Richard McCranie Ireland, Kenneth Frederick Concerning function algebras On the zeta function of an algebraic variety (P. Porcelli) (B. M. Dwork) Kellogg, Charles Nathaniel On certain algebras of centralizers UNIVERSITY OF KANSAS (6) (H. S. Collins) Durbin, John Riley Nunnally, Ellard Velpeau Minor: Physics Stable homeomorphisms of SN Decompositions and automorphisms of (R. D. Anderson) wreath products (W. R. Scott) Poynor, George Vaughn Valuations on rings Gerlach, Eberhard G. P. (H. S. Butts) Minor: Physics On spectral representation for selfadjoint Taylor, Joseph Lawrence operators; expansion in generalized The structure of convolution measure alge­ eigenelements bras (N. Aronsajn) (P. Porcelli)

Gray, Mary Lee Wheat UNIVERSITY OF MARYLAND (7) Minor: Philosophy Katz, Irving Jack Radical subcategories On EPr and normal EPr matrices (W. R. Scott and L. M. Sonneborn) (M. H. Pearl) Johns, Buddy Ava, Jr. Kiwan, Abdul Razzak Minor: Physics The Cauchy problem for systems of partial Norlund summability of orthogonal series differential equations (S.M. Shah) (J. B. Diaz) Liebnitz, Paul William, Jr. Levin, Simon Asher Minor: Physics Uniqueness and non-linearity Neighborhood extensions of continuous maps ( M. H. Martin) (C.J. Himmelberg and A. H. Kruse) Nielsen, Robert Maurice Nymann, DeWayne Stanley Completions Minor: Physics (J. W. Brace) Dedekind groups (W. R. Scott) Poole, John Terry Coefficient extremal problems for schlicht UNIVERSITY OF KENTUCKY (l) functions (J. A. Hummel) Distler, Raymond Jewel Minor: Electrical engineering Schaefer, Philip William The domain of univalence of certain cl!lsses On the Cauchy problem for an elliptic sys­ of meromorphic functions tem (A. W. Goodman) (L. E. Payne) Wilson, Frank Wesley, Jr. LEHIGH UNIVERSITY (l) Homotopy properties of vector fields on Sloyer, Clifford Wilson, Jr. the three sphere On the continuation of meromorphic func­ (B. L. Reinhart) tions and fractional ideals of meromor­ MASSACHUSETTS INSTITUTE OF TECHNOLOGY phic functions on complex analytic mani­ (Zl) folds Anderson, Peter Gordon (S. Gulden) Minor: Linguistics and Logic Some results on oriented Bordism groups LOUISIANA STATE UNIVERSITY (7) (F. P. Peterson) Brechner, Beverly Lorraine Blum, Manuel On the dimensions of certain groups of A machine-independent theory of the com­ homeomorphisms plexity of recursive functions (R. D. Anderson) (M. L. Minsky)

428 Bobrow, Daniel Gureasko Schlesinger, James William Minor: Physics Minor: Political Science Natural language input for a computer prob­ The semi-simplicial free Lie ring lem solving system (D. M. Kan) (M. L. Minsky) Schneider, David Ira Crapo, Henry Howland Minor: Economics Minor: Literature The unitary group over generalized valua­ On the theory of combinatorial independence tion rings (G. C. Rota) (K. Iwasawa) Hastings, Stuart Pendleton Sherman, Thomas Oakley Minor: Philosophy Skew symmetric representations of real Boundary value problems for one differen­ split semi-simple Lie algebras tial equation with a discontinuity (B. Kostant) (N. Levinson) Soland, Richard Martin Holmes, Richard Bruce On some renewal processes in one and two Minor: Modern Languages dimensions Approximation theory in Banach spaces (G. P. Wadsworth) (W. G. Strang) Starr, Norton Isles, David Frederick Minor: Economics and Social Science Minor: Modern Languages Limit theorems for positive operators Consistency proofs of extensions of ENT (G. C. Rota) via ordinal mappings Ton, Bui An (H. Putnam) On the asymptotic distribution of eigen­ Johnson, Robert Wells values and eigenfunctions of a general Some arithmetic properties of certain class of elliptic boundary-value problems r X f extensions of algebraic number (F. E. Browder) fields Trojan, Allan Peter (K. Iwasawa) Minor: Functional Analysis Kahn, Daniel Stephen On the integral extensions of isometries of Minor: Music quadratic forms over local fields Some results on generalizedhomologytheo­ (N. C. Ankeny) ries (G. W. Whitehead) McGILL UNIVERSITY (2) Newberger, Stuart Marshall Berthiaume, Pierre Minor: Electrical Engineering Rational completion of monoids The CJ- symbol of the singular integral op­ (J. Lambek) erators of Calder6n and Zygmund Kotze, Wessel J. {J. G. Glimm) Iterative solution of equations in linear Park, David Michael Ritchie topological spaces Minor: Philosophy (H. Schwerdtfeger) Set-theoretic constructions in model theory (H. Rogers) McMASTER UNIVERSITY (1) Raphael, Bertram Tewari, Krishna Minor: Physics Complexes over an algebra SIR: A computer program for semantic (Bernhard Banaschewski) information retrieval (M. L. Minsky) MICHIGAN STATE UNIVERSITY (7) Rosenbaum, Peter David Chartrand, Gary Theodore Minor: Philosophy Graphs and their associated line-graphs Eigenfunction expansions for a class of (E. A. Nordhaus) singular linear, integral equations Fleck, Arthur Charles (N. Levinson) Algebraic structure of automata (G. P. Weeg) . Rosencrans, Steven Ira Minor: Analysis Kronk, Hudson Van Etten Energy propagation in a stratified gas Continuous near-homogeneity (C. C. Lin) {J. G. Hocking)

429 Mandrekar, Vidyadhar S, Morrill, John Elliot Multiplicity and representation theory of Discrete economic survival game model purely non-deterministic stochastic pro­ for insurance surplus distribution cesses and its applications (C. J. Nesbitt) (G. Kallianpur) Nordgren, Eric Albert Naimpally, Somashekhar Amrith Some contributions of the theory of Toeplitz Essential fixed points andalmostcontinuous operators functions (P. R. Halmos) (J. G. Hocking) Rogak, Earl Daniel Osborne, Richard Paul A mixed problem for the wave equation in Embedding Cantor sets in manifolds a time dependent domain (J. G. Hocking) (N. D. Kazarinoff) Van Ryzin, John R. Sanchez, Daavid Alan Asymptotic solutions to compound decision Calculus of variations for integrals depend­ problems ing on a convolution product (J. F. Hannan) (L. Cesari) Stoddart, Arthur William John UNIVERSITY OF MICHIGAN ( 17) Integrals of the calculus of variations Becker, James Cyril (L. Cesari) Homotopy theory of cross-sections and Votruba, George Frank equivariant maps in the stable range On generalized inverses and singular equa­ (C. N. Lee and E. Halpern) tions in functional analysis Brooks, James Oliver (L. Cesari) Classification of representation modules West, Robert William over quadratic orders Representations of multiplications on con­ (D. G. Higman) travariant functions and some exact se­ Coburn, Lewis Alan quences of spaces Function algebras and Hilbert space. (E. Halpern) (P. R. Halmos) Eiseman, Richard Leo UNIVERSITY OF MINNESOTA (14) On solutions of alliance games Fillmore, Jay Paul (R. M. Thrall) Minor: Physics Frisinger, Hubert Howard, II On Maccaulay modules and abstract Hil­ The role of mathematics and mathemati­ bert polynomials cians in the development of meteorology (H. Rohrl) to 1800--With special emphasis on the Finlayson, Charles Henry pressure-height problem AgeneralizationofCameron's "A Simpson's (P. S. Jones) Rule" for the numerical evaluation of Grobe, Charles Alfred, Jr. Wiener's integrals in function space Some results on Sheffer's A-classification (R. H. Cameron) for simple sets of polynomials Grossman, Nathaniel (E. D. Rainville) Geodesics on Hilbert manifolds Grobe, Elizabeth Mendel (L. W. Green) Some z-adic and g-adic versions of Roth's Grove, Larry Charles theorem Minor: Philosophy (W. J. LeVeque) Tensor products of H*-algebras Lee, King Mei (B. R. Gelbaum) A mixed problem for hyperbolic equations Hora, Rajinder Bir with time-dependent domain Fiducial probability theory for distributions (N. D. Kazarinoff) with a group structure Leysieffer, Frederick Walter (R. J. Buehler) Functions of finite Markov chains. Hsieh, Po-Fang (A. B. Clarke) A turning point problem for a system of Martin, George Edward linear ordinary differential equations of On arcs in the finite projective planes the third order (Donald Livingstone) (Y. Sibuya and H. L. Turrittin)

430 Lardy, Lawrence James MONTANA STATE COLLEGE {2) Minor: Philosophy Alberda, Willis John Tensor products involving semigroup alge­ Two central limit theorems and their ap­ bras plication to the estimation of both para­ {B. R. Gelbaum) meters in the binomial distribution MacEachern, Donald Gregory {C. J. Mode) Minor: Educational Psychology Loftsgaarden, Don Owen Confidence intervals from the Bayesian Nonparametric classification and nonpara­ viewpoint metric density function estimation {I. R, Savage and R. J, Buehler) {C. P. Quesenberry) Milgram, Richard James Minor: Physics UNIVERSITY OF MONTREAL {1) The homology ring of symmetric products Gagnon, Francais-Pierre of Moore spaces Demi-groups, congruences et semi-anneaux {E. Calabi) {G. Thierrin) Natzitz, Boaz Minor: Physics UNIVERSITY OF NEBRASKA {2) Tensor products of group algebras Cobb, Ernest Benton {B. R, Gelbaum) The characterization of the solution sets Russell, David Lewis for generalized reduced moment prob­ Minor: Physics lems and applications Analytic simplification of second order sys­ {B. Harris) tems with transition points Frederickson, Paul Oliver {W. A. Harris, Jr. and H. L. Turrittin) Series solution of an almost periodic differ­ Thornby, John Ingram ential equation Minor: Economics {A. M. Fink) A study of two ranking criteria for selecting a subset containing the best of k bivariate NEW MEXICO STATE UNIVERSITY {12) {M. Sobel) Arterburn, David Roe Wilson, Howell Kenneth Weak perfect compactness and generalized Minor: Physics conjugates A boundary value problem in the structure {E. Thorp) of solutions to the Karman swirling flow Ball, Ralph Wayne problem and a canonical form for linear Maximal subgroups of the infinite symmet­ transformations of En under nonlinear ric groups substitutions {E. D. Gaughan) {J. B. Serrin) Carver, Alfred Curtis Yackel, James William Topologies for function spaces Semi-Markov processes {J. B. Giever) {S. Orey) Christenson, Charles Orlyn A study of the knotting number and a class of knots {J. B. Giever) OF MISSOURI {3) UNIVERSITY Cutler, Doyle Otis England, James Walton Quasi-isomorphism for infinite abelian Continuous flows on 2-cells and 2-spheres p-groups {W. R. Utz) {J. Irwin) Gioia, Anthony Alfred Howard, Edgar Joseph Generalized Dirichlet products of number­ First and second category abelian groups theoretic functions with the n-adic topology {M, V. Subbarao) {J. Irwin) Heckenbach, Alan John Mader, Adolf Gustav N-systems: a generalization of dynamical Automorphism group and endomorphism systems ring of abelian groups {W. R. Utz) {J. Irwin)

431 Mitchell, Alfred Richard Fischer, Emanuel Some properties of basic and semi-basic Some properties of the Watson transforma­ subgroups tion (J. Irwin) (W. Magnus) Mitchell, Roger Wightman Grossmann, Kenneth Gerald An extension of Ulm's theorem A finite difference solution for singular (J. Irwin) integral equations of the first kind, with a Cauchy type kernel Walden, William Earl (E. Isaacson) Solution of games by computation (E. Thorp) Keen, Linda Jo Goldway Canonical polygons for finitely generated Whitley, Robert James Fuchsian groups Compact and strictly singular operators (L. Bers) (E. Thorp) Lewin, Jacques Williams, Francis Merrill Residual properties of loops and rings Imbedding in summands of infinite abelian (G. Baumslag and W. Magnus) groups (J. Irwin) Loveland, Donald William Recursively random sequences UNIVERSITY OF NEW MEXICO (2) (P. Ungar) Franck, Wallace Maskit, Bernard On the optimal search problem On Klein's combination theorem (J. R. Blum) (L. Bers) Swarup, Chaitanya Matthews, Jane Some information-theoretical and empirical The solution of the conjugacy problem for techniques in statistical inference the finitely generated free metabelian (J. R. Blum) groups (W. Magnus) Morton, Keith William NEW YORK UNIVERSITY (24) Finite amplitude compression waves in a Anastasio, Salvatore collision-free plasma Maximal abelian subalgebras in hyper­ (H. Grad) finite factors Quarles, Donald Aubrey, Jr. (J. Schwartz) A moving coordinate method for shock wave Bailyn, Paul Merrill calculations- -stability theory including Doubly-connected minimal surfaces effect of shock boundary conditions (R. Courant) (R. D. Richtmyer and P. Lax) Bein, Donald Robertello, Raymond Anthony Schauder estimates for parabolic equations On a knot cobordism invariant (L. Nirenberg) (J. Schwartz) Bloom, Clifford 0. Sacker, Robert John Diffraction by a hyperbola On invariant surfaces and bifurcation of (J. B. Keller) periodic solutions of ordinary differential equations Conn, Joan Ann (J. Moser) Some buckling and boundary layer problems of circular plates Schatz, Alfred Harry (H. Keller) Boundary value problems in nonlinear plane elasticity Dowd, Robert Edward (F. John) Asymptotic description of the cusps of a hydromagnetic figure of equilibrium Schechter, Murray (P. Garabedian) Buckling of an elastic membrane on a sur­ face of revolution Ercolano, Joseph Louis (E. Isaacson) Spectral theory for operators generated by elliptic boundary problems with eigen­ Sibner, Lesley Millman value parameter in the boundary condi­ Some boundary problems for equations of tions abruptly changing type (M. Schechter) (L. Bers)

432 Sirovich, Carole Hochman Shimi, Ismail Nabih On the distribution of elements of groups Inventory problems concerning compound generated by conjugates of a number products in a real cyclic algebraic number field (W. L.Smith) (L. Ehrenpreis) Somerville, William Henry Snow, Wolfe An integral equation in a certain semi­ Existence, uniqueness, and stability for lattice nonlinear differential-difference equa­ (J. S, MacNerney) tions in the neutral case Van Zyl, Gideon Johanes Jacobus (J. Moser) Inventory control for perishable commodi­ Van Goethem, Pierre Robert ties Bounds for some infinite products of trans­ (W. J. Hall and W. L. Smith) formations Whyburn, Clifton Thomas (L. Ehrenpreis) On the second smallest quadratic non­ residue UNIVERSITY OF NORTH CAROLINA, CHAPEL (A. T. Brauer) HILL (15) Burton, Robert Corey NORTH CAROLINA STATE OF THE UNIVERSITY An application of convex sets to the con­ OF NORTH CAROLINA at RALEIGH (13) struction of error correcting codes and Brawley, Joel Vincent, Jr. factorial designs forms under similarity for in­ (R. C, Bose) Canonical volutory matrices over residue class Crabtree, Douglas Everett ring of integers Some new results on the characteristic (J. Levine) roots of matrices Dalton, Robert Edgar (A. T. Brauer) Minor: Industrial Engineering Embry, Mary Rodriquez Discrete programming with Gomory cuts Subspaces associated with contractions in (R. Llewelyn and J, Levine) Hilbert space John Henry (J. S. MacNerney) Heinbockel, Minor: Engineering Mechanics Etgen, Garrett Jay The construction of periodic solutions of On systems of non-linear differential equa­ nonlinear oscillators tions (R. A. Struble) (W. M. Whyburn) Kuehl, Robert Otto Fridy, John Albert Major: Statistics and Genetics Divisor summability methods Estimators for genetic parameters ofpopu­ (J. S. MacNerney) lations derived from parents of a diallel mating Herod, James Victor 0. Rawlings) Solving integral equations by iteration (J. (J. S. MacNerney) Magistad, John Gilbert Some steady-state and transient solutions Koh, Kwangil On prime rings for sampled queues L. Anderson) (A. C. Mewborn) (R. Reneke, James Allen Orense, Marcelo Macabiog of the exponentials in a double Integral equations in linear spaces Estimation regression curve (J. S. MacN erney) exponential (A. H. E. Grandage) Roescher, Frank Alexander Some properties of Euler's phi functions Overton, Walter Scott (A. T, Brauer) Minor: Zoology On estimating the mean ordinate of a con­ Ross, Clay Campbell tinuous function over a specified interval Singular differential systems (A. L. Finkner) (W. M. Whyburn) Proctor, Thomas Gilmer, Ill Scott, Mackinley Minor: Physics A study of some single-counter queueing Almost periodic solutions for two weakly processes coupled nonlinear oscillators (P. P. Naor) (R. A. Struble)

433 Rhyne, Alfred Leonard, Jr. Wilde, Shirley Marie Some multiple comparison sign tests A quaternary relation as the primitive (R. G. D. Steel) notion in several geometries (B. Sobocinski) Roberts, Jerry Allen Approximate methods for the solution of certain dual Fourier-Bessel series re­ lations OHIO STATE UNIVERSITY (7) (J. W. Cell) Anderson, Charles Timon Rohde, Charles August On an identity common to Lie, Jordan, and Contributions to the theory, computation quasiassociative algebras and application of generalized inverses (I. Kleinfeld) and H. R. van der Vaart) (H. L. Lucas Brabenec, Robert Lee White, Kathleen Elizabeth A set-valued measure for a certain product Minor: Public Health space An investigation of small sample properties (R. G. Helsel) estimate of a general Markov of Zahl's Brooks, James Keith model for follow up studies process A transformation theory for Banach spaces (B. G. Greenberg) (P. V. Reichelderfer) Welch, John Tunstall Chaney, Robin Ward Physics Minor: Decomposition theorems for weight func­ of an integral equation Numerical treatment tions in transformation theory elasticity arising in a mixed-boundary (P. V. Reichelderfer) problem (J. W. Cell and I. N. Sneddon) Houghton, Charles Joseph Finite-coherent Peano spaces NORTHWESTERN UNIVERSITY (3) (E. J. Mickle) Helzer, Garry Maxwell, John Glenn On divisibility and injectivity Derivative in measure space: abstract and (D. Zelinsky) applied (P. V. Reichelderfer) Lewis, John Invariants of higher order G-structures Pu, Hwang-Wen (H. C. Wang) A set-valued integral (R. G. Helsel) Yuan, Shuen On Amitsur cohomology for inseparable fields (D. Zelinsky) OKLAHOMA STATE UNIVERSITY (5) Gillett, Billy Earl DAME (6) UNIVERSITY OF NOTRE The determination of some optional experi­ Bond, James W. mental designs The structure of Lie algebras with large {J. L. Folks) generating sets minimal Hagan, Melvin Roy (H. J. Zassenhaus) Properties of peripherally continuous func­ Hughes, Sr. Mary Rosanna tions and connectivity maps An application of the domain of action to (0. H. Hamilton) norm-distance the Mordell cubic Tennison, Robert Lee (H. J. Zassenhaus) On almost-linear spaces, an almost-meas­ Markham, Sr. M. Justin ure of symmetry and curves of constant A group-theoretic characterization of the width ordinary and isotropic Euclidean planes (R. B. Deal) (H. J. Zassenhaus) Thomas, Harold Lee, Slifker, James F. Partitioned factorials Exotic multiplications on S (D. L. Weeks) (J. D. Stasheff) White, David Weidman, Donald R. Construction of confounding plans for mixed The character ring of a finite group factorial designs (H. J. Zassenhaus) (R. A. Hultquist)

434 UNIVERSITY OF OKLAHOMA (l) Cover, Alan Seymour Properties of harmonic functions in Hodges, Billy Gene n (?; 3) variables Recurrent tensor field and conformal Rie­ (J. Mitchell) mannian manifolds (T, K. Pan) Levitz, Hilbert On the ordinal notations of Schutte and the OREGON STATE UNIVERSITY (5) ordinal diagrams of Takeuti Henderson, Mason Edward (W, A. Howard and H. B. Curry) Finite geometries without the axiom of Mares, Erika Adrienne parallels Semi -perfect modules (H. E. Goheen) (R. G. D. Ayoub) Johnson, Ben Clarence Sandomierski, Francis Louis Integral equations involving special func­ Relative injectivity and projectivity tions (L. W. Anderson) (R, G. Buschman) Shore, Samuel David Krogh, Frederick Thomas On homomorphisms of structures of con­ Stability and accuracy of predict-correct tinuous functions methods of differential equations (S. Mrowka) (W. E. Milne) Smith, Sigmund Arnold Kapila Soni, Kusum Lata Prediction of achievement in analytic geo­ On some relations which are equivalent metry and calculus to functional equations of the type of (R. G. D. Ayoub) Riemann's zeta functions, and tables of divisor transforms Willard, Earl Ray (F. Oberhettinger) Functors between categories of modules (R, G. D. Ayoub) Wyse, Frank Oliver Nets in nets in uniform spaces: monotoneity, limit theorems (B. H. Arnold) UNIVERSITY OF PENNSYLVANIA (4) Andrews, George W. UNIVERSITY OF OREGON (4) On the theorem of Watson dragonette for Baartz, Arne Peter Ramanujan's mock theta functions The measure algebra of a locally compact (H. Rademacher) idempotent commutative semi-group Rockoff, Maxine Lieberman (K. Stromberg) Comparison of some iterative methods for Crittenden, Richard Byrd solving large systems of linear equations A theorem on the uniqueness of the (C,l) (M. Gerstenhaber) summability of Walsh series Goldsmith, Donald (V. L. Shapiro) An elementary proof of the prime number Lindahl, Robert John theorem for arithmetic progressions with On the second conjugate algebra of the group an error term algebra for locally compact abelian (E. Grosswald) groups Smith, Harvey Alvin (P. Civin) Tesnor products of topological algebras Stafney, James Dean (C. Ionescu Tulcea) Arens multiplication and convolution (P. Civin) OF PITTSBURGH (13) PENNSYLVANIA STATE UNIVERSITY (10) UNIVERSITY Allen, Stephen Ives Beidleman, James Clark of algebraic functions of a On near-rings and near-ring modules Representation sur­ (L. W. Anderson) complex variable by two dimensional faces in four space and application to Bernstein, Neil Riemann surface theory On the cohomology of semigroups (J. S. Taylor) (R, P, Hunter) Angotti, Rodney Bolger, Edward Martin A pseudo-Euclidean geometry in the three­ Exponential distributions space (W, L. Harkness and J, B. Bartoo) (M. Benedicty)

435 Becker, Stanley Jason PRINCETON UNIVERSITY {13) Axisymmetric cylindrical stress waves in Andrews, Peter Bruce a hollow elastic cylinder without axial A transfinite type theory with type variables effects {A. Church) {A. M. Bryson) D' Atri, Joseph Eugene Dillon, Thaddeus Michael The kernel of a homomorphism of continu­ Statistical hypothesis modification ous pseudogroups {E. Fels) (D.C. Spencer) Hardman, Hunter Easton, William Bigelow Projective coordinates on a line and non­ Powers of regular cardinals affine transformations in complex two­ {A. Church) space interpreted in Argand four-space Giffen, Charles Herbert {J. S. Taylor) Fibered knots and periodic transformations Miller, Douglas Leo {R.H. Fox) Linear groups over semi-local rings Gilmartin, Michael Cooper {M. Benedicty) Every analytic variety is locally contract­ Ortner, Gene Merle ible Equiconvergence of Riesz and Cesaro trans­ {R. Gunning) forms of infinite series Hsiang, Wu-Yi {G. Laush) The classification of differentiable actions Park, Richard Miller of the classical groups on ?r-manifolds Representation of complex three-space in {J. C. Moore) Argand six-space Koch, Richard Moncrief {J. S. Taylor) On pseudogroups associated with the one Schaffer, Paul Theodore dimensional foliation group On summability methods proposed by Fekete {R. C. Gunning) {G. Laush) Lomonoco, Samuel James, Jr. Stringer, Loren Frank An algebraic theory of local knottedness On the mathematical theory of the rolling of {R. H. Fox) metals at elevated temperatures May, J. Peter {A. M. Bryson) The cohomology of restricted Lie algebras Terrell, James and of Hopf algebras; application to the Minor: Biostatistics Steenrod algebra A criterion for choosing between alternative {J. C. Moore) sample designs when the primary interest Moore, Robert Thorpe is in relationships between variables Lie algebras of operators and group repre­ Tezel, Akin sentation on Banach spaces Minor: Mechanical Engineering {E. Nelson) The application of finite integral transform Petrie, Ted Edgar techniques to thin plates constrained and/ Cohomology of loop spaces of Thorn com­ or supported at several points plex Waidman, Louis {W. Browder) Minor: Chemical Engineering Spivak, Michael David Digital computer solution of the coupled On spaces satisfying Poincare duality non-linear equations governing spherical {J. W. Milnor) phase growth Wells, Robert Immersion cobordism groups POLYTECHNIC INSTITUTE OF BROOKLYN {2) {J. W. Milnor) Capobianco, Michael F. Minor: Chemical Engineering PURDUE UNIVERSITY {11) On the binomial distribution and related Bruening, Leonard Francis problems Generalization of nilpotency {A.J. Terzuoli) {E. Schenkman) Smuckler, Abraham M. Buchanan, John Edward On the algebraic elements of a commutative On the stability of solutions of a differential­ Banach algebra difference equation with variable lag {G. Bachman) {J. Lillo)

436 Cooley, Robert Lee Tsao, Sherman Free topological groups Stability of flow between arbitrarily spaced (M. Shanks) concentric cylindrical surfaces and re­ lated mathematical problems Eidswick, John Allan (R. C. DiPrima) A hereditary class of Hilbert spaces of entire functions RICE UNIVERSITY (3) (L. deBranges) Barth, Karl Frederick Faudree, Ralph Jasper Jr. Asymptotic values of meromorphic functions Subgroups of the multiplicative group of a (G. R. MacLane) division ring (E. Schenkman) Ford, Wayne Timothy Mathematical programming and integra­ Foster, Louis Ashley differential equations Tests for homogeneity of variances {J. Douglas, Jr.) (I. W. Burr) McMillan, John Emerson Hunter, Ulysses Asymptotic values of functions holomorphic Essential cluster sets in the unit disc (C. Goffman) (G. R. MacLane) Prullage, Donald Leon Summability in topological groups UNIVERSITY OF ROCHESTER (1) (C. Goffman) Bick, Theodore Avery Rodine, Robert Henry Determining sequences in Euclidean n-space On mixtures of perfect probability measures (D. M. Stone) (L. J. Cote) RUTGERS - THE STATE UNIVERSITY (11) Snyder, Larry Eugene Continuous Stolz extensions and Stolz angle Barback, Joseph limit functions Contributions to the theory of isols (C. J. Neugebauer) (J. C. E. Dekker) Trutt, David Ferguson, Donald Carvel Some mean squares of entire functions Infinite products of non-negative integers (J. Rovnyak) (J. C. E. Dekker) Hogben, David QUEENS UNIVERSITY (2) Some properties of Tukey's test for non­ Atkinson, Harold R. additivity Semi-normed linear lattices with duals of (M. B. Wilk) integral type Jones, William Richard (H. E. Ellis) Differential systems with integral boundary McDonald, R. Ian conditions Banach spaces of operators (H. J. Zimmerberg) (I. Halperin) Kirwan, William English II Extremal problems for certain classes of analytic functions RENSSELAER POLYTECHNIC INSTITUTE (4) (M. S. Robertson) Livingston, Albert Edward Davis, Stephen Howard p-valent close to convex functions Effects of free boundaries and property (M. S. Robertson) variations in thermal convection (L. A. Segel) Osofsky, Barbara Langer Homological properties of rings and mod­ Goodwin, Bruce Edward ules On the theory and application of Fredholm (C. Faith) integral equations and on integral equa­ tions whose kernels depend on the eigen­ Pelios, Angelo value parameter Completion of contiguity spaces (W. E. Boyce) (S. Leader) Griswold, Robert Gray Sansone, Fred Jay Connectivity and pseudo-trees in directed The summation of certain infinite series of graphs isols (P. Slepian) (J. C. E. Dekker)

437 Shapiro, Samuel Sanford Epling, Mary Lena An analysis of variance test for normality A multivariate stochastic approximation (complete samples) procedure (M. B. Wilk) (L. E. Moses) Wang, Chung-Lie Gieser, Leon Jay On the degenerate Cauchy problem The comparison of multivariate tests of (R. Carroll) hypothesis by means of Bahadur efficiency (I. Olkin) SAINT LOUIS UNIVERSITY (l) Grace, Donald Wayne Padberg, Harriet, R.S.C.J. Computer search for non-isomorphic con­ Minor: Philosophy vex polyhedra Computer composed canon and fugue (G. Polya) (W. A. Vezeau) Hahn, Kyong Taik Minimum problems of Plateau type in the UNIVERSITY OF SASKATCHEWAN (l) Bergman metric space Artiaga, Lucio (S. Bergman) Theory of functions and integral transfor­ Hinich, Melvin Jay mations Large sample estimation of an unknown (A. P. Guinand) discrete waveform which is randomly repeating in Gaussian noise UNIVERSITY OF SOUTH CAROLINA (1) (H. Chernoff) Lampton, George Benjamin Klauber, Melville Roberts Category theory Some estimates of location (J. Sonner) (L. E. Moses) UNIVERSITY OF SOUTHERN CALIFORNIA (3) Larson, Ronald John Almost invariant measures Eckert, Ernest John (K. deLeeuw) Positive polynomials on Banach spaces (D. H. Hyers) Motteler, Zane Clinton Existence theorems for quasi-linear ellip­ Resprepo, Guillermo tic partial differential equations inn vari­ Differentiability of the norm in Banach ables spaces (D. Gilbarg) (J. Dugundji) Press, Sheldon James Storer, Thomas Frederick Some hypothesis testing problems involving A family of generalized difference sets multivariate normal distributions with (A. L. Whiteman) unequal and intraclass structural covari­ ance matrices STANFORD UNIVERSITY (23) (I. Olkin) Baranchik, Alvin John Samuels, Stephen Mitchell Multiple regression and estimation of the On a Chebyshev-type inequality ·for sums of mean of a multivariate normal distribution independent random variables (C. M. Stein) (S. Karlin) Bhattacharyya, Ramendra Kumar On some affine connections Schaerf, Mirella Casini (Charles Loewner) Estimation of the covariance and autore­ gressive structure of a stationary time Causey, Robert Lewis series On closest normal matrices (E. Parzen) (G. E. Forsythe) Donio, Jean Isaac Selliah, Jegadevan Balendran Some stochastic learning models with time Estimation and testing problems in a as a continuous parameter Wishart distribution (P. C. Suppes) (I. Olkin)

Efron, Bradley Srivastava, Muni Shanker Problems in probability of a geometric Optimum procedures· for classification and nature related problems (R. G. Miller, Jr.) (C. M. Stein)

438 Thiebaux, Helen Jean Terzaghi LaTorre, Donald A randomness problem in macromolecular On the radical of a bemiring sequences (D. D. Miller) (L. E. Moses) Ray, David Wallenius, Kenneth Ted Extensions of homeomorphisms in three Biased sampling: A noncentral hypergeo­ dimensions metric probability distribution (0. G. Harrold, Jr.) (G. J. Lieberman) Robinson, Richard Weiner, Howard Jacob An indefinite product factorization for a On age dependent branching processes class of generating functions defined by (R. G. Miller, Jr.) power series (E. Cohen) Wetzel, John Edwin A compactification theory with potential­ Wiginton, Carroll L. theoretic applications Simplicial mappings of the three-sphere (H. Royden) (L. K. Barrett) Zador, Paul Laszlo Zettl, Anton Development and evaluation of procedures Related two-point boundary value problems for quantizing multivariate distributions and associated harmonic matrices (H. Solomon) (J. W. Neuberger)

STEVENS INSTITUTE OF TECHNOLOGY (1) UNIVERSITY OF TEXAS (11) Strumpf, Albert On a class of transformations leading to Murray, Christopher Brock 'similar' solutions of the steady, two­ On the mean integral dimensional, Navier-Stokes equations (H. S. Wall) (P. D. Ritger) Robinson, Charles Dee Orthogonality in normal linear spaces SYRACUSE UNIVERSITY (3) (H. S. Wall) Brualdi, Richard Anthony Rodin, Ervin Yechiel-Laszlo Combinatorial aspects of the direct product A generalization of the Laplace transform of matrices (W. T. Guy, Jr.) (H. Ryser) Ryan, Donald Edwin Crawford, Douglas Houston A Banach algebra of analytic functions Minor: Education (J. E. Scroggs) An investigation of age-grade trends in understanding the field axioms Schawe, David Paul (V. J. Glennon) Cauchy sequences in abstract spaces (H. S. Wall) Reddy, William Lawrence Expansive homeomorphisms and their re­ Stewart, A. D. lationship to transitive homeomorphisms On the Abel equation in n-dimensions, n ii; 2 (E. Hemmingsen) (H. J. Ettlinger) Stocks, Douglas Roscoe, Jr. UNIVERSITY OF TENNESSEE (9) Concerning certain lattice paths and lattice Bacon, Philip point theorems Coincidences of real-valued maps from the (R. E. Greenwood) n-torus Vick, George Robert (W. S. Mahavier) Pseudo-metric axioms ofaEuclideangeom­ Bennett, Ralph etry Rectangularly chainable continua (D. E. Edmondson) (W. S. Mahavier) Wang, Hsuan-Heng Hampton, George On the tridiagonalization of a non-Hermitian Green's relations and dimension in abstract matrix using similarity transformations semigroups (R. T. Gregory) (D. D. Miller) Whitmore, Ralph Martin johnson, Gordon On the mixed partial derivative operator Concerning the moment problem and its inverse (J. W. Neuberger) (H. J. Ettlinger)

439 Zund, Joe David Marx, Morris Leon Degenerate fields with twisting rays On extensions of normal curves (A. Schild) (G. S. Young) Mathews, Harry Thomas UNIVERSITY OF TORONTO (10) Cluster sets of analytic functions Banerjee, Sourendra Kumar (G. S. Young) Contributions to order statistics Scarborough, Charles Thomas (D. A. S. Fraser) Minimal topology on Hausdorff spaces Benedetto, John Joseph (M.P. Berri) The Laplace transform of generalizedfunc­ Teller, John Roger tions On ordered algebraic structures (C. Davis) (P. F. Conrad) Bouwer, Izak Zurk Standard representations of simple Lie UNIVERSITY OF UTAH (3) algebras Kurtz, Lynn Clifford (A. J. Coleman and B. Abrahamson) Vector-valued summability methods on a Breach, Derrick Rodney linear normed space Axisymmetric viscous flow past finite Robertson, Jackie Max bodies Some topological properties of certain (A. F. Pillow) spaces of differentiable homeomorphisms Deakin, Alan Stanton Uherka, David Jerome Reflection and transmission problems for A Riesz representation theorem hyperbolic systems (G. F. D. Duff) VANDERBILT UNIVERSITY (1) Garner, Cyril Wilbur Luther Christy, John Harlan Regular polyhedra and honeycombs in hyper­ Minor: Physics bolic space Expansive transformation groups (H. S. M. Coxeter) (B. F. Bryant) Mathai, Arakaparampil Mathai Missing values in statistical analysis VIRGINIA POLYTECHNIC INSTITUTE(4) (R. Wormleighton) Barnett, Frederic Charles Miller, Donald Richard The problem of classifying members of a Viscous flow past a paraboloid of revolution population on a continuous scale (A. F. Pillow) (J. G. Saw) Plante, FranTois-Andre Bowman, Kimiko Osada Random orderings Moments to higher orders for maximum (D. A. S. Fraser) likelihood estimators with an application to the negative binomial distribution Tanyi, Gregory Eno (L. R. Shenton) Explosions in elastic spheres (R. A. Ross) Myers, Raymond Harold Minor: Chemical Engineering Orthogonal statistics and some sampling of properties of moment estimators for the TULANE UNIVERSITY (8) negative binomial distribution Babcock, William Warren (L. R. Shenton) On linearly ordered topological spaces Wolock, Fred Walter (L. B. Treybig) Cyclic designs Bell, Harold (H. A. David) On fixed-point properties of plane continua (G. S. Young) UNIVERSITY OF VIRGINIA (4) Clark, William Edwin Hoffman, Frederick Affine semigroups over an arbitrary field Finite groups admitting fixed-point-free (A. H. Clifford) automorphisms (E. C. Paige) Lloyd, Justin Thomas Lattice-ordered groups and a-permutation Lawrence, Sidney Herbert groups Stochastic differential equations (P. F. Conrad) (E. J. McShane)

440 Weaver, David William UNIVERSITY OF WATERLOO (1) Cohomotopy theory for spaces with operators Mullin, Ronald Cleveland (E. E. Floyd) An enumerative survey of triangular maps Williams, George Kenneth (W. T. Tutte) On a class of functions of two complex vari­ UNIVERSITY OF WESTERN ONTARIO (1) ables (G. T. Whyburn) Nanjundappa, Rudraiah Hydrodynamic and magnetohydrodynamic WASHINGTON UNIVERSITY (5) channel flows (J. H. Blackwell) Brown, Thomas Craig On semigroups which are unions ofperiodic WAYNE STATE UNIVERSITY (4) groups Kabe, Dattatraya Govind (E. E. Lazerson) Some contributions to multivariate statisti­ Cholewinski, Frank Michael cal analysis A Hankel convolution complex inversion (C. K. Tsao) theory Nemeth, Abraham (I. I. Hirschman, Jr.) Digital enciphering of English into braille Hattemer, Jimmie Ray (C. Briggs) Boundary behavior of temperatures in sev­ Pernavs, Nora eral variables A uniqueness theorem for then-dimensional (G. Weiss) Helmholtz equation Montgomery, Philip Reid (0. G. Owens) Extension of endomorphisms Schreiner, Erik (F. Haimo) Modular pairs in orthomodular lattices Obrock, Arthur Edgar (D. Foulis) Certain regions of values for bounded uni­ UNIVERSITY OF WISCONSIN (17) valent functions (J. A. Jenkins) Barnhill, Robert Numerical contour integration UNIVERSITY OF WASHINGTON (7) (P. Hammer) Borges, Carlos Jorge Do Rego Glaser, Leslie Stratifiable spaces Contractible complexes in sn (I. Namioka) (R. H. Bing) Kannappan, Palaniappan Giesy, Daniel The functional equation f(xy) On convexity in Banach spaces 2f(x)f(y) for Abelian groups (A. Beck) (E. Hewitt) Green, Charles Peck, Newton Tenney On some properties of finitely additive Topics in nonlocally convex spaces measures (V. L. Klee) (S. Coleman) Phillips, Keith Lowell Hanson, Richard Joseph Hilbert transforms for the p-adic and p­ Reduction and classification of certain turn­ s erie s fields ing point problems for systems (E. Hewitt) (W. Wasow) Pozsgay, Lawrence James Henderson, David Wilson Go del's second theorem for elementary Properties of locally Euclidean spaces arithmetic (R. H. Bing) (R. W. Ritchie) Hessler, Paul A topic in algebraic analysis Sundaresan, Kondagunta (R. C. Buck) Some geometrical properties in normed Hosay, Norman linear spaces Characterizations of tame continua in E3 (V. L. Klee) (R. H. Bing) Wu, Ling-Er! Eileen Ting Miller, Richard A topological generalization of quasi-Fro­ Asymptotic behavior of solutions of differ­ benius rings ential equations (J. P. Jans) (J. Nohel)

441 Murtha, James Brown, James Russell A generalization of Goldies' theorems Approximation theorems and integral re­ (L. Levy) presentation for Markov operators (S. Kakutani) Price, Thomas Upper semi-continuous decompositions of E3 Brunell, Sister Mary Zachary (R. H. Bing) Some problems in transfinite diameters (E. Hille) Rider, Daniel Gap series, and measures on spheres Constantine, .Alan Graham (W. Rudin) Hypergeometric functions of matrix argu­ ment Robinson, Daniel Alfred (A. James) Bol loops (M. Osborn) Greenleaf, Frederick Paul Characterization of group algebras in terms Rowbottom, Frederick of their translation operators Large cardinals and small constructible (C. E. Rickart) sets (J. Keisler) Greiner, Peter Charles Eigenfunction expansions and scattering Stout, Edgar theory for perturbed elliptic partial dif­ Algebra of analytic functions ferential equations (W. Rudin) (F. E. Browder) Strauss, Aaron Soda, Dominic Carmen Asymptotic behavior of differential equa­ Groups of type D4 defined by Jordan alge­ tions bras (J. Nobel) (N. Jacobson) Todd, Christopher Watkins, Mark E. The strict topology on C*(X,E) A characterization of the planar geodetic (R. C. Buck) graph and some geodetic properties of non-planar graphs YALE UNIVERSITY (12) (0. Ore) Alvarez, Laurence Richards Basic graphs of finite modular lattices and their associated undirected graphs (0. Ore) Anderson, George Albert Asymptotic expansions for the distribution YESHIVA UNIVERSITY (4) of the latent roots of the covariance matrix and for a hypergeometric function Clark, Arthur Albert of matrix argument Algebraic solutions of Waring's conjecture (A. James) (W.L. Zlot) Beals, Richard William DiPaola, Robert Arnold Non-local boundary value problems for On pseudo-complements of recursively elliptic partial differential equations enumerable sets (F. E. Browder) (M.Davis) Berger, Melvyn Stuart Mann, Jonah An eigenvalue problem for non-linear ellip­ Hausdorff means and the Gibbs phenomenon tic partial differential equations (D. J. Newman) (F. E. Browder) Sherry, Edwin John Bingham, Christopher Methods for obtaining upper and lower Distributions on the sphere and on the pro­ estimates of the capacity of the square jective plane plate (A. James) (B. Epstein)

442 SUPPLEMENTARY PROGRAM-Number 32

During the interval from February 18, 1965throughApril 28, 1965 the papers listed below were accepted by the American Mathematical Society for presentation by title. After each title on this program there is an identifying number. The abstracts of the papers will be found following the same number in the section on Abstracts of Con­ tributed Papers in this issue of these cNotiaiJ. One abstract presented by title may be accepted per person per issue of the cNotiaiJ. Joint authors are treated as a separate category; thus in addition to abstracts from two authors individually one joint abstract by them may be accepted for a particular issue. ( 1) On the existence of an empty set in {10) A k-complex in En (n ~ 2k + 2) is set-theoretical models tame if each simplex is tame Professor Alexander Abian and Mr. Mr. J. L. Bryant, University of S. E. LaMacchia, The Ohio State Georgia (65T-233) University (65T-223) ( 11) Restricted second order theory of (2) On automorphisms of compact Rie­ ordinals mann surfaces. Preliminary report Professor J. R. Biichi, The Ohio Professor R.D.M. Accola, Brown State University (65T-206) University (65T-234) ( 12) Extremal problems for sums of ( 3) Sequences in topological spaces. II powers of complex numbers. II Professor C. E. Aull, Kent State Professor J.D. Buckholtz, Univer­ University (65T-227) sity of Kentucky (65T-235) ( 4) Asymptotic values of meromor_phic (13) Extreme positive definite functions functions and Choquet' s Representation Theo­ Lieutenant K. F. Barth, Ballistic rem Research Laboratories, Aberdeen, Professor R. S. Bucy and Pro­ Maryland (65T-236) fessor G. J. Maltese, University of ( 5) An extension of a theorem of Bern­ Maryland (65T-237) stein to meromorphic functions (14) Oriented manifolds fibered over the Mr. R. C. Basinger, Sandia Corpo­ circle ration (65T-258) Mr. R. 0. Burdick, University of (6) The subgroups of PSL(3,q) for odd q. Virginia (65T-247) Preliminary report (15) n-ality theory of rings Professor D. M. Bloom, Brooklyn Professor J. M. Cardoso, Univer­ College (65T-277) sidade do Parana, Brazil (65T-264) (7) On a nonlinear implicit operator in (16) The square root problem for trans- the space of continuous functione formations having contirtuous spec­ Professor Witold Bogdanowicz and trum Professor Michael Mullen, Catholic Professor R. V. Chacon, Ohio State University of America (65T-241) University (65T-245) (8) The Kakutani problem ( 17) On k-complementary graphs Professor D. G. Bourgin, Univer­ Professor C. Y. Chao, University sity of Illinois (65T-289) of Pittsburgh (65T-230) (9) Solutions of restricted cases of the ( 18) Local homeomorphisms of Euclidean halting problem used to determine space onto arbitrary manifolds particular values of a non-computable Mr. M. M. Cohen, University of function Michigan (65T-211) Professor A. H. Brady, University (19) Some properties of two systems of of Notre Dame (65T-270) set theory

443 Professor N.C.A, da Costa, Uni­ Professor S, P. Hastings, Case versity of Parana, Brazil(65T-256) Institute of Technology (65T-208) (20) Parallel families in Riemannian (31) Ideals generated by products space V n Professor A. P. Hillman and Pro­ Professor John De Cicco and Mr. fessor D. G. Mead, University of R. V. Anderson, illinois Institute of Santa Clara, Professor K. B. Technology (65T-244) O'Keefe, University of Washington (21) On the solution of a system of equa­ and Dr. E. S. O'Keefe, Boeing Cor­ tions of the modified Guernsey­ poration, Seattle, Washington(65T- Balescu type 240) Mr. J. E. Dutt, Cornell Aeronautical ( 32) Generalizations of some standard Laboratory, Bailey's Cross Roads, theorems on recursive functions Virginia (65T-218) Mr. P. G. Hinman, University of (22) Embeddings of topological products of California, Berkeley and University circularly chainable continua of Warsaw, Poland (65T-238) Professor Laurence Fearnley, Uni­ (33) Uniform distribution of sequences versity of Wisconsin (65T-281) in GF [q,x]. Preliminary report (23) Invariant subspaces for certain quasi- Professor J, H. Hodges, University nilpotent operators of Colorado and University ofCali­ Professor Jacob Feldman, Univer­ fornia, Berkeley (65T-212) sity of California, Berkeley (65T- (34) The undecidability of the immortality 250) problem for semi- Thue systems (24) Sets with 0-dimensional kernels Mr. P. K. Hooper, Harvard Uni­ Professor N. E. Foland and Pro­ versity (65T-263) fessor J. M. Marr, Kansas State (Introduced by Professor Hao Wang) University (65T-285) (35) Expansion theorems for solutions of (25) Existence of Abelian subgroups in two generalizations of the heat equa­ Sylow 2- subgroups of finite groups tion Dr. George Glauberman, University Mr. A. T. Hopper, Case Institute of Wisconsin (65T-276) of Technology (65T-265) (26) Realizations of Hasse diagrams (36) Further applications of Kalmar's Mr. R. M. Grassl and Miss R. F. techniques in proof theory Torretto, University of Santa Clara, Professor D. F. Isles, Tufts Uni­ and Mr. M. T. Stroot, University versity (65T-228) of Maryland (65T-215) (37) Connected G0 graphs (Introduced by Professor A. P. Hillman) Professor F. B. Jones and Mr. (2 7) An Lp estimate for rearrangements E. S. Thomas, Jr., University of of trigonometric sums California, Riverside (65T-229) Mr. C. A. Greenhall, California (38) On the stable homotopy of a Moore Institute of Technology (65T-213) space (28) Approximate solution of the exterior Professor D. W. Kahn, University Dirichlet problem with application to of Minnesota (65T-283) the estimation of capacity (39) First order properties of pairs of Professor Donald Greenspan, Math­ cardinals. Preliminary report ematics Research Center, Univer­ Professor H. J, Keisler, Univer­ sity of Wisconsin (65T-275) sity of Wisconsin (65T-291) (29) Integral representation of general­ (40) Locally compact Hausdorff spaces ized temperature functions in nega­ with open cone neighborhoods tive time Mr. V. M. Klassen, Virginia Poly­ Professor D. T. Haimo, Southern technic Institute (65T-278) Illinois University and Harvard Uni­ {41) On the nonregularity of certain gen­ versity and Professor F. M. Chole~ eralized Lototsky transforms. Pre­ winski, University of North C'aro­ liminary report lina (65T-255) Professor C, F. Koch, Kansas State (30) Growth of solutions to a nonlinear University (65T-251) equation (42) Fields with infinite cyclic automor-

444 phism group (55} Almost retracts Professor Willem Kuyk, McGill Professor S. A. Naimpally, Iowa University (65T-284) State University (65T-207} (43) A lattice-theoretic correspondence (56) The differential ideal [y 1 ••• yJ Mr. W. A. LaBach, University of Professor K. B. O'Keefe, Univer­ Illinois (65T-274) sity of Washington and Dr. E. S. (44) An extension of the intuitionistic O'Keefe, The Boeing Company, propositional calculus Seattle, Washington (65T-290) Mr. E. G. K. Lopez-Escobar, (57} Embedding Cantor sets in a manifold Massachusetts Institute of Tech­ Mr. R. P. Osborne, University of nology (65T-220) Idaho (65T-210} (45) A singular homology characterization (58} A characterization of tame 2-spheres of the 2-sphere among Hausdorff Lieutenant C. A. Persinger, Air spaces Force Institute of Technology, Dr. M. C. McCord, University of Wright-Patterson Air Force Base, Georgia (65T-279) Ohio (65T-257) (46) Regressive sets with r .e. complement (59} On tensions in a revolving disk Dr. T. G. McLaughlin, University Professor Vaclav Pfeffer, George of Illinois (65T-269) Washington University (65T-219) (47) Local properties of the embedding of (Introduced by Professor David Nelson) a graph in a three-manifold ( 60) A note on topologies of lattices Professor D. R. McMillan, Jr., Miss Mary Powderly, Universityof University of Virginia (65T-216) Connecticut (65T-219) (48) Countable compactifications (61) A number-theoretic estimate Professor K. D. Magill, Jr., SUNY Professor B. S. Randol, Yale Uni­ at Buffalo (65T-224) versity (65T-268} (49) The failure of Craig's theorem in (62) The differential of a primary matrix languages with uncountable conjunc­ function tions Professor R. F. Rinehart, Case Mr. J. I. Malitz, University of Institute of Technology (65T-254) California, Berkeley (65T-239) (63} On extending the Grzegorczyk Hier­ (50) Continuous images of ordered com­ archy pacta and a new dimension which Mr. J. W. Robbin, Princeton Uni­ neglects metric subcontinua versity (65T-217) Professor Sibe Mardesic, Univer- (64) On the associative admissibility of sity of Zegreb, Yugoslavia (65T- antiflexible algebras 272) Professor D. J. Rodabaugh, Van­ (51} Primary abelian groups which are derbilt University (65T-261) neither transitive nor fully transitive ( 65} A result on Saint-Venant' s principle Professor C. K. Megibben, Univer­ Mr. J. J. Roseman, New York Uni­ sity of Washington (65T-266) versity (65T-282) (52) A converse of Banach's contraction (66) Multi-head finite automata theorem Mr. A. L. Rosenberg, Harvard Mr. P.R. Meyers, NationalBureau University (65T-232) of Standards, Washington, D. C. (Introduced by Professor P. C. Fischer) (65T-267) (67) Multivariate prediction with continu­ (53} Transposed complementarity and ous time. Preliminary report regular polytopes Mr. Habib Salehi, Indiana Univer­ Professor T. S. Motzkin, Univer­ sity (65T-246) sity of California, Los Angeles (68) Special functions on locally compact (65T-288} fields (54) On algebraic structure in rings of Mr. P. J. Sally andProfessorM.H. arithmetic functions Taibleson, Washington University Mr. A. A. Mullin, University of (65T-253) California, Livermore ( 65T-225} ( 69) Function systems. Preliminary re-

445 port Mr. M. C. Thornton, University of Dr. Berthold Schweizer, The Uni­ Illinois (65T-248) versity of Arizona and Dr. Abe (78) Solution to a research problem Sklar, Illinois Institute of Tech­ Mr. P. C.-C. Wang, Wayne State nology (65T-209) University (65T-252) (70) Supersolvable wreath products (Introduced by Professor Everett Pit­ Professor W. R. Scott and Pro­ cher) fessor L. M. Sonneborn, Univer­ (79) Extensions of completely 0-simple sity of Kansas (65T-214) semigroups by completely 0-simple (71) A fixed point theorem for local con­ semigroups traction mappings Professor R. J. Warne, West Vir­ Mr. V. M. Sehgal, WayneStateUni­ ginia University (65T-286) versity (65T-222) (80) On the generalized Lienard equation (72) On the existence of a class of par­ Professor D. W. Willett and Pro­ tially balanced incomplete block de­ fessor J. S. W. Wong, University signs of Alberta (65T-231) Professor S. S. Shrikhande, Uni­ (81) Flat homogeneous manifolds versity of Bombay, India (65T-259) Professor J. A. Wolf, University (73) Dualities of class 2 groups of ex­ of California, Berkeley (65T-249) ponent p corresponding to 2::- spaces (82) Almost-periodic solutions of some Professor R. F. Spring, Ohio Uni­ second-order differential equations versity (65T-280) in Hilbert spaces (74) The topological complementation Professor Samuel Zaidman, Uni­ problem versity of Montreal (65T-221) Mrs. A. K. Steiner, University of (83) The endomorphism ring of a torsion- New Mexico (65T-273) free module (Introduced by Professor D. W. Dubois) Mr. J. M. Zelmanowitz, University (75) Some properties of the group ring of Chicago and University of Wis­ over rational integers of a finite consin (65T-287) group (84) Some simple sufficient conditions for Professor Shuichi Takahashi, Uni­ a transformation to sum a divergent versity of Montreal (65T-226) series (76) On functions of Baire class 1 Mr. Shimshon Zimering, Battelle Mr, E. S. Thomas, Jr., University Memorial Institute, Switzerland of California, Riverside (65T-271) 65T-262) (77) Singularly fibered manifolds (Introduced by Dr. S. H. Gould)

446 ABSTRACTS OF CONTRIBUTED PAPERS

The April Meeting in New York Aprill2-15, 1965

622-77. W. P. REID, U. S. Naval Ordnance Laboratory, White Oak, Silver Spring, Maryland. Temperature in a triangle.

An analytical solution is given for the temperature as a function of x, y and t in a 30° right triangle. The initial temperature of the region is prescribed, and the sides are maintained at zero degrees. (Received February 25, 1965.)

622-78. KWANGIL KOH, North Carolina State of the University of North Carolina at Raleigh, Raleigh, North Carolina and A. C. MEWBORN, University of North Carolina , Chapel Hill, North Carolina. A class of prime rings.

A ring R of linear transformations of a (left) vector space V over a division ring D is weakly transitive provided there is a right order K in D and a (K,R)- submodule M of V such that M is uniform as R-module and such that if {mpm2 , ••• ,mn} is a finite D-linearly independent subset of M and {w 1,w2, ••• ,wn} is a sequence from M, then there exists r E Rand k E K, k =I 0, such that mir; kwi' 1 ~ i ~ n. Theorem. A ring R is a weakly transitive ring of linear transformations if and only if R has a faithful, irreducible right ideal I such that (1) if a E R and [I: a] ¥I, then a E I, and (2) if J is a right ideal of R, J ¥I, then there exists r E J, r tj. I such that ri <;;;I, and if a E R, [J: a] 2 I, then [J: a J ¥I, The rings satisfying the conditions of the theorem form a class of prime rings containing the primitive rings and the prime rings with maximal annihilator and maximal complement right ideals. (Received February 25, 1965.)

622-79. JOSE BARROS NETO, University of Montreal, Montreal, Quebec, Canada. On differentiability of general kernels.

Let R be an open set in R n X R m; let K be a compact in Rn and n; cK its complement,

Denote by rl' the open set of all y ERn such that K X {y} C R. Suppose that Kx,y is a distribution kernel in the sense of L. Schwartz and denote by L: gy---> g~ and tL: gx----> 9'~ the natural mappings associated to Kx,y· Theorem. Under the above assumptions, suppose that K 1 and K2 are two arbitrarily given compact subsets of R n and R m, respecdvely. If for each T 1 E ~· (R n) and for each T 2 ES:f'(Rm) differentiable in n 1 and n 2, respectively, we have tLK(T1) differentiable inn~ and LK(T 2) differentiable in rlz, then the kernel Kx,y is a differentiable function in R. Converse of the theorem. Suppose K is differentiable in R. If T E ~· (R n) is differentiable in n, then tLK (T) x,y is differentiable in n•. These theorems extend well-known results of L. Schwartz about very regular kernels. (Received February 25. 1965.)

622-80. G. L. ITZKOWITZ, 213 Michael Hall, State University of New York at Buffalo, BufJ!alo, New York 14214. Homomorphisms that are essentially Fourier transforms.

It is shown that there exists a large class of group algebra homomorphisms that are

447 essentially Fourier transforms. More specifically, a homomorphism it: LI(G1) ->M(G2) is essen­ tially a Fourier transform ifitf = i(a)o0 , where a E GI and o0 is the point mass at the origin. Theorem I. Let GI and G2 be LCA groups. Let GI be topologically isomorphic to Rn X Go where 00 is 0-dimensional, and let G2 be a discrete torsion free group. Then every homomorphism it: L I(GI)----? M(G2) is essentially a Fourier transform. Lemma. Let GI, G2 be nontrivial compact connected Abelian groups. If GI is torsion-free there exists a nontrivial open continuous homomor­ phism of GI into G2. Theorem 2. Let G I' G 2 be LCA groups and let G2 be a discrete torsion-free divisible group. Then every homomorphism it: LI(GI) ----.M(G2 ) is essentially a Fourier transform iff G1 is topologically isomorphic to Rn X G0 where Go is 0-dimensional. These theorems are consequences of P. J. Cohen's group algebra homomorphism theorem and the structure theory for LCA groups. (Received February 25, 1965,)

The April Meeting in Stanford, California April24, 1965

623-46. L. S. BOSANQUET, University of Utah, Salt Lake City, Utah 84ll2. Abel's integral equation for an infinite range; fractional integrals and derivatives,

For 0 < a. < I, write I~f(x) = I/r(a.} J~(t - x)a.-If(t}dt (x ;'i; w}, n;:;f(x} = - (d/dx)I~- '4:(x) (x ;'i; w},

Ia.f(x) = limw_,00r;f(x), Da.f(x) = limw~ooD~f(x). It is well known that f(x) is representable in the form (I) f(x) = I~g(x) p.p. for x ;'i; w if and only if I~- a.f(x) is absolutely continuous in every interval [X, w], a. a.-I - oo < X < w, and that when (I) holds then (2) g(x) = Dwf(x) p.p. for x < w. When f(x) = (w- x) , (2) holds but (I} does not. I prove that F (x) is representable in the form (3) F (x) = I" G(x) p.p. if and only if (i) I~-a.F(x) is absolutely continuous in every interval [X,w], - oo

623-47. T. E. FRAYNE, 32 Rivoli Street, San Francisco, California 941I7. The generalized Souslin problem.

A partially ordered set is a.- separable if it has a dense subset of power < a.; a.- saturated if every disjointed family of intervals {x: a < x ;'i; b} has power < a.; a.- CD if every chain (simply ordered subset) and every disjointed subset has power < a.. For additional terminology see Keisler-Tarski, (Fund. Math. 53, pp. 234, 297). Consider the conditions: (i) Every a.-saturated chain is a.-separable. (ii) Every a.-saturated chain includes some a.-separable interval. (iii) The family of a.-separable subsets of any chain is different from the family of nowhere dense subsets. (iv) Every a.- saturated, a.-distributive Boolean algebra is atomic. (v) Every a.-CD set in which every two elements are either comparable or disjoint has power < a.. (vi) Every a.-CD ramification system has power < a.. For a.= w1 (i) is Souslin' s hypothesis; the generalized Souslin hypothesis is the statement: (i) holds for every a. ~ w. Previously known theorems: (i) - (vi) are equivalent for a.= wi; (i) holds if a.= w or a. is singular; (v), (vi) fail if a. is singular (see Gillman, Ann. of Math. 56, p. 446; Gaifman, Pacific J.

Math. I4, p. 70}. By generalizing we obtain: I. For every regular a. ~ w (i) - (vi) are equivalent. II. (i) - (iv) hold if a.= w or a. is singular. (Received February 26, I965,)

448 The June Meeting in Eugene, Oregon June 19, 1965

6Z4-1. WITHDRAWN.

6Z4-Z. W. C. HOFFMAN, Boeing Scientific Research Laboratories, P. 0. Box 3981, Seattle Z4, Washington. The visual cortex as a fiber bundle.

The visual manifold is a replica, in the visual cortex, of the image projected upon the surface of the retina that is augmented by various proprioceptor inputs that determine the metric tensor of the manifold. The psychological constancies of visual perception: "shape constancy", "size constancy", and the invariance of recognition of an object even when in motion, may be expressed as Lie groups of transformations over the visual manifold (the affine group, dilatation group, and Z-dimensional Lorentz group, respectively). Lie's fundamental theorems then require two additional perceptual "constancies": invariance under efferent binocular distortion, and what appears to be some sort of circulating memory. At the microscopic level, the cell body of the neuron constitutes, apart from higher order effects and random perturbations, a Lie group germ, and there must exist a cortical analogue of the Lie product as well. The processing of an afferent valley of nerve impulses by the visual cortex thus constitutes a fiber bundle (or at least a fibering), with the visual manifold as base manifold. (Received April 19, 1965.)

6Z4-3. S. K. THOMASON, Cornell University, Ithaca, New York 14850. Hyperarithmetic incomparability.

A ~T B (A ~h B) means that A is Turing-reducible to (hyperarithmetic in) B. (OA) is the set of notations for the constructible ordinals (ordinals constructible relative to A). Theorem 1. There are sets A, B hyperarithmetic in 0 such that A ~T oB and B ~T oA. (It follows that A is not expressible in either one-function-quantifier form relative to B, and vice versa; note also that

449 A < h 0 and· yet A ¥T 0.) In addition, the only sets recursive (hyperarithmetic) in each of A, B are the recursive (hyperarithmetic) sets. Theorem 2. There are sets A, B strictly hyperarithmetic in 0 with 0 ;;;T A,B. Hence the hyperdegree of 0 is the 1. u.b. of its strict predecessors. Theorem 3. If C is not hyper arithmetic, there is a set A with C ~h A and A ;fiT oC; C ~T oA only if C ;;;T 0. Each of these theorems can be relativized. The use of the "forcing" method in the proofs follows Feferman (Some applications of the notions of forcing and generic sets, Stanford University, Mimeo). However, we work entirely with a first-order language, obtained from the language of first-order A B arithmetic by adjoining symbols for the "generic" sets A, B and for the relations < 0 , < 0 • (Received March 3 1, 1965.)

624-4. z. z. YEH, University of Hawaii, Honolulu, Hawaii. Dense, nowhere dense, dense in a set, and dense-in-itself.

The usual definition of "dense" and its logical extension "dense in a set" lead to some awkward situations: a set may be dense somewhere (i.e., in some nonempty set) and yet be nowhere dense. The definition of a density point is given as the basis for redefining dense, nowhere dense, dense in a set, and dense-in-itself, thus bringing coherence to these definitions. Fortunately the new definitions of dense and nowhere dense are equivalent to the usual ones for all spaces that are at least T 1 and free from isolated points, but not so for other spaces. Counterexamples are considered. The new definition of "dense in a set" differs radically from the old one. The new definition of dense-in-itself is derived from the new definition of "dense in a set" via subspace topology, but otherwise is equivalent to the standard one. Paradox. The Cantor set is dense-in-itself, but not dense in itself. (Received April 21, 1965.)

624-5. S. A. NAIMPALLY, Iowa State University, Ames, Iowa 50010. Function spaces of quasi-uniform spaces.

Let (X,%') and (Y,o/) be quasi-uniform spaces (Pervin, Math. Ann. 147 (1962), 316-317). For X X each V E 'r'let W(V) = { (f,g) E Y X Y J(f(x), g(x)) E V for all x EX}. The family {W(V) IV Eo/} is a basis for a quasi-uniformity :ll'for Yx. (1) Lemma. :Iris larger than the quasi-uniformity of pointwise convergence. (2) Lemma. If (Y,o/) is a Tcspace i = 1,2 then so is (Yx,~. (3) Lemma. If C C YX consists of continuous functions then (C, :II) is jointly continuous and so contains the k-topology. (4) Theorem. If (Y,o/) is a complete (Sieber and Pervin, Math, Ann, 158 (1965), 79-81) Hausdorff space then (Yx,:ll} is complete, (5) Theorem. (X,%') is a regular space iff for each U E%' there exists a symmetric V E "'r such that V "V[x]C U[x]. (6) Theorem, If (Y,o/) is X regular then C is closed in (Y ,:1/'). (7) Theorem, If (Y ,o/) is a regular Hausdorff space which is complete then (C,:I/') is complete. (Received April 26, 1965.)

624-6. E. C, INGRAHAM, University of Oregon, Eugene, Oregon. On the existence of inertial subalgebras.

Let K be a commutative ring with 1 and R a finitely generated K-algebra. A subalgebra S of R is called an inertial subalgebra if S is K-separable and S + rad(R) = R. K is called an inertial coefficient ring if every finitely generated K-algebra R such that R/rad(R) is K-separable contains

450 an inertial subalgebra. Necessary and sufficient conditions are obtained for the existence of an inertial subalgebra in a finitely generated, commutative, faithful K-algebra R having a finite group G of automorphisms such that R G = \x E R !u(x) = x, V a E G} = K, where K contains no idempotents but 0 and 1. Azumaya has proved (Nagoya Math.]., 2 (1951), 119-150) that every Hensel ring is an inertial coefficient ring. The following partial converse is proved: Let K be a noetherian, semi-local ring such that either (i) K contains no idempotents but 0 and 1, and all residue class fields of K are perfect, or, (ii) K is an integrally closed integral domain. Then, if K is an inertial coefficient ring, K is a Hensel ring. Finally, it is proved that a Dedekind domain K is an inertial coefficient ring if and only if either K has zero radical or K is a Hensel ring. (Received April 26, 1965.)

624-7. HIDEGORO NAKANO, Wayne State University, Detroit, Michigan 48202. Invariant metrics.

Invariant metrics on topological groups were constructed by G. Birkhoff, A note on topological ~· Compositio Math. 3 (1936) and S. Kakutani, Uber die Metrization der topologischen Gruppen, Proc. Imp. Acad. Japan, 12 (1936) independently. More generally we consider invariant metrics on uniform space with transformation groups. As an immediate consequence of Fundamental Theorem 3 in the book by H. Nakano, Uniform spaces and transformation groups, which will be published in the U.S.A., I would like to prove a Theorem: for a uniformity U on a space S with a transformation group G there is an invariant metric m on S such that U is the induced uniformity of m if and only if U is sequential and G is equicontinuous for U. (Received April 27, 1965.)

624-8. R. A. MORRIS, Reed College, Portland, Oregon 97202. On finite p-groups with maximal automorphism groups.

The Burnside Basis Theorem provides an upper bound for the automorphism group of a finite p-group with ad element basis. If this bound is attained, a p-group is said to have the property --ft. P. Hall (Proc. London Math. Soc. (2), 36 (1933), p. 37) mentions a class of groups with_,/{. This paper generalizes this example and provides a simple proof as follows: G has --It if the kernel of the natural homomorphism of Fd, the free group on d symbols, onto G has fully invariant kernel. Now choose K to be the subgroup generated by all p 1th powers and c-fold commutators, 1 and c fixed integers. K is fully invariant in Fd and Fd/K is finite and has 1. Because the examples of P. Hall are regular, they are exactly those groups given here with c ;;; p. (Received April 29, 1965.)

624-9. L. E. WARD, University of Oregon, Eugene, Oregon. A general fixed point theorem.

A fixed point theorem is proved for a large class of partially ordered spaces, from which it is possible to deduce a number of classical results and/or generalizations thereof, including the Borsuk fixed point extensibility theorem and the Eilenberg-Montgomery fixed point theorem. If X is a partially ordered set we define L (x) = {y: y ;;; x}, M(x) = {y: x ;;; y }. C (x) = {y: M(x) n L(y) = jx, y}}. An element p of X is a zero of X if M(p) = X. In what follows X is assumed to be a compact continu­ ously partially ordered space having a zero, such that each set L(x) is simply ordered and each set M(x) - x is open. Let Q be a property of uppersemicontinuous mappings such that point images are

451 connected, and which is preserved under composition with isotone retractions, Then X has the fixed point property for Q(X) if and only if each set C(x) has the fixed point property for Q(C(x)). (Received May 3, 1965,)

624-10, L. N. ARGABRIGHT, University of California, Berkeley, California. Invariant integrals on locally compact semigroups.

Let S be a locally compact (Hausdorff) semigroup satisfying the condition: (#) for each compact set K C S and each element a E S the set Ka- 1 = {x E S: xa E K} is compact, A right invariant integral on S is a positive right invariant linear functional on the space of continuous (real) functions on S having compact support. Theorem. The following are equivalent: (i) S admits a right invariant integral; (ii) S admits positive regular Borel measure f.L satisfying f.L(Ba -l) = f.L(B) for all Borel sets

B C S and all a E S; (iii) S contains a unique minimal left ideal. This generalizes a theorem of W. G. Rosen (Proc. Amer, Math. Soc, 7 (1956), 1076-1082), Conjecture. Let S be a locally compact semigroup not necessarily satisfying (#). Then (ii) above is equivalent to: (iii) S contains a right ideal which is a left group, The conjecture is correct in the case that S is discrete, (Received May 3, 1965,)

624-11, D. W. BARNETTE, 4005 15th N.E., Apartment 302, Seattle, Washington 98105. Trees in polyhedral graphs.

A d-polyhedral graph is one which is isomorphic with the graph formed by the vertices and edges of a bounded d-dimensional convex polyhedron, A k-tree is a connected acyclic graph in which each vertex is of valence ~ k. Theorem 1. Every 3-polyhedral graph G can be covered by a 3-tree; that is, G admits a subgraph T such that Tis a 3-tree and each vertex of G is a vertex ofT. Theorem 2, For every k there is a 5-polyhedral graph which cannot be covered by any k-tree. Theorem 1 settles a problem raised by Grunbaum and Motzkin [Longest simple paths in polyhedral graphs, j. London Math. Soc., 37 (1962), 152-160] and also yields partial solutions to several unsolved problems on paths in polyhedral graphs. (Received May 3, 1965.)

624-12. R. W. CHANEY, Western Washington State College, Bellingham, Washington 98225, On the transformation of integrals in measure space,

Let T be a function whose domain is a measure space { S,M, f.L} and whose range is a measure space {x.~. 11}. Assume that ~ and ~ are sub-u-fields of ~ and~ respectively. A nonnegative extended real valued function 'W defined on X X~ is termed a weighing function for T if for each

B in!! it is true that 'W(x,B) = 0, a, e., 11 off TB, 'W(.,B) is ~-measurable, and 'W(x,B) = ~'W(x,Bi) when B is the union of a countable number of pairwise disjoint sets Bi• Given 'W, T is said to be ACW if fx 'W(x,B) d 11 < oo and if there is a no1;1negative real valued f.!- integrable function f defined on S such that JBf(s)df.L = fx 'W(x,B) d11 for each B in~· Under certain standard conditions it is shown that if T is ACW then a transformation formula holds as follows: If H is a real valued N-measurable function then the integrals fBHT(s)f(s)df.L and fxH(x)'W(x,B)dll both exist and are equal if either exists. Conditions are given which are necessary and sufficient to guarantee that an arbitrary f.!- integrable function f defined on S admits a weighing function 'W such that JBf(s) df.L = fx 'W(x,B) dll

452 for each B. A "Lebesgue-type decomposition" theorem for weighing functions is proved. Finally there are applications to earlier work. (Received May 3, 1965.)

62.4-13. j. H. JORDAN and C. T. LONG, Washington State University, Pullman, Washington 99163. A limited arithmetic on simple continued fractions.

Let r be the golden ration, Fj the jth Fibonacci number, and Lj the jth Lucas number. The following results are established for integers m. (i) mr = (a,b) if and only if m = F Z.k- 1, a = F 2k and b = L 2k_ 1 for some positive integer k; (ii) mr = (a,i,~) if and only if m = Fz.k• a= F2k+ 1 - 1, and c = L 2k- 2.. It is further established that (iii) L 2kr = (L2k+ 1' F 2k' 5F2k); (iv) L 2k_ 1r = (L2k- 1, 1, FZ.k- 1 - 1, 1, 5F2k_ 1 - 2). More general results are obtained replacing r by certain quadratic irrationals. (Received May 3, 1965,}

624-14. LARRY MANSFIELD, 416 West Lee, Seattle, Washington. A generalization of the Cartan-Kiihler theorem. Preliminary report.

An involutive differential system S, generated by one forms and two forms, is defined to be hyperbolic if it satisfies certain conditions. These conditions are chosen so that an existence theorem for the solution of a hyperbolic system of partial differential equations can be used in the proof of the following Theorem: LetS be a ck-hyperbolic differential system (k ~ 1) with genus g,

and Ir_ 1 be an (r - I)-dimensional integral Ck-variety of S, then at each regular point p E Ir-l of S there exists an r-dimensional integral ck-variety containing Ir-l' for r such that r - 1 < g. (Received May 3, 1965.}

624-15. j. R. REAY, Western Washington State College, Bellingham, Washington. How to distinguish 5-dimensional space from 4-dimensional space.

Suppose B is a positive basis for the linear space L, i.e., B is a minimal subset of L which positively spans L. M is a spanned linear subspace if M = pos(M n B). If card(M n B) = 1 + dim M as well, then (MnB) is called a minimal positive basis forM, and M = pos(MnB) is called a minimal subspace of L. L is always the linear sum of its minimal subspaces. The following conditions on B may hold. (a) L is the direct linear sum of its minimal subs paces, equivalently, distinct minimal positive bases for minimal subspaces are pair-wise disjoint. (b) B admits a partitioning into pair­ wise disjoint subsets, each of which is a minimal positive basis for a minimal subspace of L. (c) B

admits a partitioning as in (b) which is unique. Clearly (a) ~(c)~ (b). Theorem: B ~(a) if

dim L ~ 4, and (c) ~(a) if dim L ~ 5. Examples in 5- and 6-dimensional space are given which show that these results are best. Thus 4- and 5- dimensional linear spaces may be characterized by considering their positive bases and conditions (b) and (c). (Received May 3, 1965.)

624-16. D. F. SANDERSON, Western Washington State College, Bellingham, Washington 982.25. A construction in category theory. Preliminary report.

Let A and Ba. be objects in a category, with a. in some index set ~.and let fa. be a map from

A to Ba.' for each a.. We slightly extend a definition of Hilton and Eckmann and define (C,ga..~) to be

453 the equalizer of the system (A,Ba,fa,2l) if ga E Map(Ba,C), if gia = gfJffJ for all a,/3 E 2{, and if whenever (D,h , 21) is such that h E Map(B ,D) and h f = hf.!f f.!, for all a,f:J E 21, then there is an a a a. a.a /JIJ hE Map(C,D) such that ha = gga. We define coequalizers dually. Applications are given to the theory of direct and inverse limits and extensions of the Gelfand-Mazur representation theorem. (Received May 3, 1965.)

624- 17. M. L. SLATER, Division 5251, Sandia Corporation, Sandia Base, Albuquerque, New Mexico. On a class of non -linear differential-difference equations.

Suppose K(x) measurable and 0 < K(x) & 1 for x €: (- oo, oo). Suppose f(u) convex for u E [0,1], f(O) = 0, f(u) > 0 for u E (0,1), and f(u) = 1 - f'(1)(1 - u) + 0(1 - u) 1+ 0 as u ---->1 for some o > 0. (Example: f(u) = up, p ~ 1.) Theorem: The equation (*)(x) = J:+ 1 K(~)f jdl(~) jd~ has a solution {jl(x) satisfying 0 < (x) & 1 for x E (- oo, oo) if and only if J00 eax [1 - K(x)]dx < oo where a is the largest real root of a= f' (1)(1 - e- a). Furthermore, if is any such solution of (*), then the limits (±oo) exist and satisfy ({jl(+ oo) - (- oo))/2 = f_<~, ({jl(x)- K(x)f[{jl(x}]dx. (These results are an extension of the 1960 Pacif. J. Math. paper of M. L. Slater and H. S. Wilf in which the linear case (f(u) = u) was studied.) (Received May 3, 1965.)

624-18. IVAN NIVEN, University of California, Berkeley, California. A combinatorial problem connected with number theory.

Let u be a permutation carrying 1, 2, 3, ••• , n into i 1, i 2 , i3'"'' in. Corresponding to u is an increase-decrease pattern {3 = A 1A2 ... An_ 1 where Aj = I (for increase) in case ij+1 - ij > 0 and Aj = D (for decrease) otherwise. Thus {3= {3(u) is a mapping of the n! permutations onto the 2n- 1 increase-decrease patterns. Typical results are as follows. Given any specific {3, a closed formula is given for N(/3) in terms of the structure of {3, where N({3) is the number of permutations u that map onto {3. For fixed n, N(/3) is a maximum in case A1 A2 ... An_ 1 is made up of alternate I's and D's; in such a case write bn for N(/3). The integers bn are closely related to the Euler numbers if n is even, to the Bernoulli numbers in case n is odd, (Received May 4, 1965.)

624-19. A. L. Y ANDL, Seattle University, Broadway and Madison, Seattle, Washington. On a question concerning fixed-points.

Suppose that X and Y are continua which have the fixed-point property (f.p.p.) and that the intersection X nY is also a continuum. Several authors have sought additional conditions to guaran­ tee that the union XUY also has the f.p.p. The following question, apparently due to L. Ward, is answered in the negative. Must X UY have the f.p.p. if X nY is a retract of X UY 7 Specifically, it is shown that there exist two 2-dimensional continua X andY in E 3 such that X, Y and X ny all have the f.p.p. and such that X n Y is a retract of X UY, although X UY lacks the f.p.p. Two other examples related to the same problem are given. Namely, it is shown that there exist two (non­ compact) plane sets X andY such that X, Y and X ny all have the f.p.p., in fact X ny is an arc, and yet XUY fails to have the f.p.p. The third example, based on one of Klee, exhibits two arcwise con­ nected one-dimensional plane continua X and Y such that X, Y and X nY all have the f.p.p. and xny is in fact a chainable continuum, although XUY lacks the f.p.p. (Received May 4, 1965.)

454 624-20. J. D. REID, Syracuse University, Syracuse, New York 13210. A note on inseparability.

For certain fields E (of characteristic p) there exist finite extensions P /E that are not separable, yet contain no purely inseparable elements. It seems natural to ask for further information on such extensions, which we call here exceptional extensions.. Concerning the separable part of such an extension, one has: Theorem 1. A separable extension l:/E is the maximal separable subfield of an exceptional extension if and only if Ux EE.l:P(x) f. :2:. Theorem 2. K(x}, x an indeterminant, is the maximal separable subfield of an exceptional extension P /E if and only if K is not perfect. Other results obtained imply Theorem 3. The field E admits no exceptional extension if and only if every extension P /E of exponent l splits: P = l: ®E L, where l: is the maximal separable subfield of P /E and exponent 1 for P means pP ~ l:. (Received May 5, 1965.)

624-21. K. DEMYS, 307 South 7th, Artesia, New Mexico. Even-integer connectivity and half­ integer genus.

For any simply or multiply connected map or complex, K, in parabolic n-space, Nk being the number of k-dimensional elements inK, and~ the genus, S = L~=-l (- l)n+lNk= 2h + s, p = h + (l/2}s, where hand s are new parameters defining p and hence the connectivity, 2p+ 1, more pre­ cisely,§. being the number of strictures inK (points at which 2 sheets of K's hyper-surface are pinched together}, K being transformable into a hyper sphere with.!! hyperhandles and.§ omphali. U s = 1, h = 0, we have p = 1/2, S = s = 1, which for n = 3 can be represented by (x2 + y2 + z 2)2 j(x2 + y 2) = 2 (2a) , the umbilicoid or omphaloid we may term it, thus disproving by Gegenspiel the assertion that K(p = 1/2} could not exist. The lacuna of hitherto not recognizing the existence of half-integer genus is hereby rectified. There are immediate connections with m-sheeted Riemann surfaces, since m + h - 1 = 0 and s = l:(w), where the sum is over the orders of the winding or branch-points. These relations also make the construction of non-self-intersecting Riemann surfaces in 3-space now possible. Omphaloids in general are distinguished further by the fact that suitably defined hypo­

omphaloids exist for~ n < 0 only, while hypospheres exist only for odd n < 0. (Received May 5, 1965.)

624-22. R. K. JUBERG, University of Minnesota, Minneapolis, Minnesota 55455. A generaliza­ tion of a result of A. Tihonov.

Let u = u(x,t) solve the equation D;mu + (- l)mDtu = 0 in IR+ X JR.+, IR+ the positive half-line. Suppose that u is bounded in finite time intervals; that it is zero initially and satisfies B(Dx )u(o+,t) = f(t) where f is an m-vector and B(Dx) has components Bj (Dx) (j = l, ••• ,m): Bj (Dx) are linear differen­ tial operators. Let p be the order of z = 0 as a zero of det(Bj(wkz)), wk (k = l,••• ,m) the 2mth roots of(- l)m+l with negative real parts, and let q be the least order of z = 0 as a zero of the (m - I)­ minors of (Bj(wkz)). Let Z be the set of zeros, different from 0, of det(Bj(wkz)) and set a.= p - q. Under the assumption det(Bj(wkz)) 'f 0, we prove the Theorem. Assume f E C00 (i[+) with support in IR+ and limt f(t) = C. Then for each x > 0, Dxa.u(x,t) has a finite limit at t = oo, if and only if -- ~oo Z C {z: jarg zl > r/4m}. Moreover, a.~ m- 1. The uniqueness of solutions for such general mixed problems, needed for this complete characterization, is provided in a paper by the author. Moreover, the results of that paper allow for the assumption of much less smoothness of the data here, However

455 such refinements would seemingly add little to the essential nature of this result" (Received May 5, l965o}

624-23" ]o R" Brown, Oregon State University, Corvallis, Oregon" Spatially homogeneous Markov operators"

Let G be a metrizable, compact abelian group with normalized Haar measure m, and let T be a Markov operator on L 2 (m), ioeo, a positive linear operator such that Tl = T*l = L Tis said to be spatially homogeneous if JA+x(TXB+x)(y)m(dy) = JA (TXB)(y)m(dy) for all x E G and all Borel sets A and B" It is shown that T is spatially homogeneous iff there is a (necessarily unique) probability measure J.L on the Borel sets of G such that Tf(x) = fxf(x - y)J.l(dy) a"e" for each f E L2 (m)o It is noted that the extreme points of the convex set 1: of spatially homogeneous operators are the trans­ lations and that this gives a topological and algebraic embedding of G in 1:" It is further shown that a

Markov operator T on L 2 of a finite, separable measure space (X, {3, v) is spatially isomorphic to a spatially homogeneous operator Ton a compact abelian metric group G (that is, T = UTU-l, where Uf(x) = f(cp(x)) is given by an invertible measure-preserving transformation cp) iff there exists a complete orthonormal system C for L2 (v) consisting of bounded proper functions of T and forming a group under pointwise multiplication" This extends a familiar result for measure-preserving trans­ formations with pure point spectrum" (Received May 5, 1965")

456 ABSTRACTS PRESENTED BY TITLE

65T-206, J, R. BUCHI, The Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210. Restricted second-order theory of ordinals.

For an ordinal a., let RS(a.) be the interpreted formalism containing a constant < for the order­ ing relation on a., individual variables with range a., unary predicate variables ranging over all subsets of a., and quantifiers for both types of variables, If 'Y is a cardinal, let RS(a,-y) be like RS(a.), except that predicate variables are restricted to range over sets of cardinality less than 'Y. Theorem 1,

RS(a., w~ is decidable, for any a.. In particular, Theorem 2, RS(a.) is decidable for any countable a.. Actually we obtain complete information on definability in these formalisms, and on the relation

a.E {3: [a.,<] is RS( w1)-extension of [{J, < ). The crucial part of the proof is the theorem on transfinite automata recursions, stated in these c}(oticei), Abstract 65T-l 7-7., We do not know whether RS(w1) is decidable, (Received January 21, 1965.)

65T-207, S. A. NAIMPALLY, Iowa State University, Ames, Iowa 50010, Almost retracts.

Stallings introduced the concept of an almost continuous function (Fund, Math, 47 (1959), 249-263), A function f: X --->Y is almost continuous iff corresponding to each open set U in X X Y containing the graph G(f) off, there exists a continuous function g: X --->Y such that G(g) CU. An example is constructed to show that the composition of two almost continuous functions is not necessarily almost continuous, Y C X is an almost retract of X iff there exists an almost continuous function f on X onto Y such that for all y in Y, f(y) = y. Theorem, If X is a Hausdorff space with fixed point property and Y is an almost retract of X then Y has the fixed point property. An example of a nonlocally connected set with the fixed point property is constructed. (Part of Doctoral Dis­ sertation submitted to Michigan State University, Supervisor Professor John G. Hocking,) (Received February 15, 1965,)

65T-208. S. P. HASTINGS, Case Institute of Technology, University Circle, Cleveland 6, Ohio. Growth of solutions to a nonlinear equation,

Let g(x) be an odd, nondecreasing function which is continuous for all x and positive for x > 0.

Let a(t) be positive for t ~ 0 and absolutely continuous on finite sets, and let h(t) be continuous, but

not necessarily differentiable, for t ?; 0. Let x(t) be a solution to (1) i(t) - x(O) + JJa(s)g(x(s))ds = h(t) defined at t = 0, (If h(t) is differentiable, then ii. + a(t)g(x) = h' (t).) Theorem I. If J:la' (s) l/a(s)ds < oo, then x(t) can be extended to [O,oo) and x(t) = 0(1 + fo jh(s) jds) as t goes to infinity, Theorem II. If a(t) is nondecreasing, then x(t) can be extended to [O,oo) and there is a c such that jx(t) I ;;; c + 3 JJih(s)jds, all t ~ 0. (Received February 19, 1965,)

457 65T-209. BERTHOLD SCHWEIZER, The University of Arizona, Tucson, Arizona and ABE SKLAR, Illinois Institute of Technology, Chicago, Illinois 60616. Function systems. Preliminary report.

A function system is a quadruple (S,o,L,R) consisting of a nonempty set S, a mapping 'o' from S X S into S and 2 mappings L, R from S into S, subject to the following axioms: (1) (S,o) is a semi­ group; (2) For any a in S, LRa = Ra, RLa = La, and La o a = a= a o Ra; (3) For any a,b in S, L(a o b) = L(a o LB), R(a o b)= R(Ra o b), La o Rb = Rb o La, and Ra o b = b o R(a o b). Function systems are natural generalizations of function semigroups [B. Schweizer and A. Sklar, Math. Ann. 143 (1961), 440-447) and in addition include inverse semigroups, 'small' categories and Brandt semi­ groups as special cases. The definition: a ~ b iff a = b o Ra induces a partial order on S. Under this partial order, a function system is order-isomorphic to a system of concrete functions closed under composition and ordered by restriction; this representation theorem is a simultaneous gen­ eralization of those of Schweizer and Sklar for function semigroups, Vagner and Preston for inverse semigroups, and Ellenberg and MacLane for categories. (Received February 19, 1965.)

65T-210. R. P. OSBORNE, University of Idaho, Moscow, Idaho. Embedding Cantor sets in a manifold.

It is shown that every Cantor set C in a polyhedraln-manifoldlies on the boundary of ann-cell which is locally polyhedral except at points of C. Using a construction similar to that used by Blankinship a wild Cantor set is constructed in E 4 which, when embedded in s2 X s2, lies in no open 4-cell. Using this we get that s2 X s2 contains a simple closed curve which lies in no open 4-cell but which bounds a topological disk. (Received November 25, 1964.)

65T-211. M. M. COHEN, University of Michigan, Ann Arbor, Michigan. Local homeomor­ phisms of Euclidean space onto arbitrary manifolds.

Theorem l. If Mn is a connected topological n-manifold then there exists a local homeomor­ phism, f, of Rn (Euclidean n-space) onto Mn. Theorem 2. If Mn is a closed connected topological n-manifold other than then-sphere, and if Mn can be covered by q open cells, then there exists a local homeomorphism, f, of Rn onto Mn such that, for every x in Mn, f- 1(x) consists of no more than q points. If Mn is a combinatorial or differentiable manifold then the map f in Theorems 1 and 2 may be chosen to be a piecewise linear or differentiable immersion, respectively. (Received February 22, 1965.)

65T-212. J. H. HODGES, University of California, Berkeley, California 94720. Uniform distribution of sequences in GF [q,x]. Preliminary report.

Let (} = {Ai} be an infinite sequence of elements of 4> = GF (q,x], the ring of polynomials in an indeterminate x over an arbitrary finite field of q elements and let M E 4> be monic of degree m > 0.

For any B E4> and any integer n G; 1 define fJ(n,B,M) as the number of integers 1 ;;;; i;;;; n such that Ai = B (mod M). Then 8 is called uniformly distributed modulo M if for all B, 8(n,B,M)/n ----+q-m as n----+ oo, and uniformly distributed if it is uniformly distributed modulo all such M. All of the theorems

458 proved concerning uniform distribution of sequences in are the analogs of results proved by Ivan Niven (Trans, Amer. Math, Soc. 98 (1961), 52-61) for sequences of integers. Investigated are the relationships between uniform distribution of a sequence modulo M and (1) density properties possessed by certain ()and their "complements", (2) uniform distribution of () modulo K and MK,

(3) an exponential sum corresponding to 0, (4) generation of ()by certain "irrationals" and "rationals" in an extension field of <1', A subsequent paper will contain the analogs of results proved for sequences of integers by Burke Zane (Amer. Math. Monthly 71 (1964), 162-164) and Stephan R. Cavior (to appear). (Received February 22, 1965,)

65T-213, C. A. GREENHALL, California Institute of Technology, Pasadena, California. An Lp estimate for rearrangements of trigonometric sums,

Let {k:k = 1,2, ... ,n} be a system of complex-valued functions on an int_erval I such that the integral of any finite~product of the k is nonnegative, In particular, k(x) = e mkx or cos nkx on

[0,21r]. Let T(x) = L~= f kk(x) (ck complex numbers) be any finite sum, and let T(x) = L~= 1 1ck lk(x), lf u= (u , ... ,u) is any permutation of (l, ... ,n), set S*(x,u) = max < IL:k~lc if> (x)l. The result 1 n 1 :::;;m=n - uk uk is that (1/n!) L ufr [S*(x,u)jP dx ~ Ap JI IT(x) lp dx for all positive even p, where Ap depends only on p. This gives what may be the best possible extension of the corresponding inequality for p = 2, proved in a paper of A. M. Garsia (Existence of almost everywhere convergent rearrangements for Fourier series of L2 functions, Ann. of Math. (2) 79 (1964), 623-529). There, {d was any orthogonal system, and of course Jlfl2dx = fiTI2dx. (Received February 22, 1965,)

65T-214, W. R. SCOTT and L. M. SONNEBORN, The University of Kansas, Lawrence, Kansas 66045. Supersolvable wreath products.

Durbin (Abstract 65T- 58, these c){oticei) 12 (1965), 143) has found all finite standard wreath products which are supersolvable. Theorem. If A is a nontrivial group and B is a nontrivial (faithful) permutation group on a set S, then A wr B (nonstandard) is supersolvable if and only if: (1) A is nilpotent, (2) S is finite and (3) A is finitely generated and infinite and B is an elementary Abelian 2-group, or (3') A is finite and for every prime p I o(A) and every Sylow p- subgroup P of B, we have P :::JB' and p = 1 (modexp B/P). (Received February 22, 1965,)

65T-215, R. M. GRASSL and R. F. TORRETTO, University of Santa Clara, Santa Clara, California and M. T. STROOT, University of Maryland, College Park, Maryland, Realizations of Hasse diagrams.

Using extensions of methods in Hillman (Pro c. Amer. Math. Soc. 6 (1955), 542- 548) and the theorems of Stroot and Grassl (Abstract 64T-426, these c){oticei) ll (1964), 672) and of Hillman and Mead, Related pairs of Hasse diagrams. II (see Abstract 622-44, these c){oticei) 12 (1965), 338)~ the formulas f(n) are obtained for the numbers of realizations of all Hasse diagrams with 5 vertices by families of 5 subsets of a set of n elements, (Received February 22, 1965.)

459 65T-216. D. R. McMILLAN, JR., University of Virginia, Charlottesville, Virginia. Local properties of the embedding of a graph in a three-manifold.

Let G be a finite graph topologically embedded in the interior of a 3-manifold M, and let p be a point of order n of G. We say that "M - G has 1-FLG" (free local fundamental groups) at p if: For each sufficiently small open set U containing p, there is an open set V such that p E V C U and if W is any connected open set such that p E W C V then the image under the inclusion homomorphism i*: 1r1(W- G) ---->7r1(U- G) is a free group on (n- 1) generators. It is known that if G is tame then M- G has 1-FLG at each point of G. Theorem. If M - G has 1-FLG at each point of G and if each arc ill G pierces a 2-cell, then G is tame. Corollary. If G is locally tame except possibly at a zero-dimensional subset and if M - G has 1-FLG at each point of G, then G is tame. (Received February 22, 1965.)

65T-217. J. W. ROBBIN, Princeton University, Princeton, New Jersey 08540. On extending the Grzegorczyk hierarchy.

For each a. < ww define a function wa. as follows: wo (x) = 2x; w a.+1 (x) = w~ (1) (i.e., wa.+ 1 (0)

= 1, W a.+ 1 (x + 1) = W a.(W a.+l (x))); W a.(x) = W a.[x](x) for a. a limit ordinal. Here the function a.[x] is defined so that a.[O], a.[1], a.(2], ••• is a strictly increasing sequence of ordinals with limit a.; namely, write a.= w k+ 1( fJ + 1) and set a.[x] = wk+ 1 fJ + wkx. Let ifa.= if(W a.>• the set of functions elementary recursive in W a.· Theorem. (1) a. ;f fJ implies .l:fd' <:;; _G>fJ· (2) a. ~ fJ implies ~f. Si{3· (3) _A/n = U a.~ (a. < wn). Here .A/n denotes the class of nested n-fold recursive functions. (Received February 22, 1965.)

65T-218. J. E. DUTT, Washington Projects Department, Cornell Aeronautical Laboratory, Bailey's Cross Roads, Virginia. On the solution of a system of equations of the modified Guernsey­ Balescu type.

A class of simple solutions are obtained for a system of N kinetic equations of the modified Guernsey-Balescu type for spatially homogeneous plasmas. A mathematical technique which is capable of generalization to the non-homogeneous case has been found that allows the original set of kinetic equations to be replaced by a much simplified set of differential equations in certain parameters of the solution (multivariate distribution). Certain properties of the plasma including stability are discussed for the case of N = 2 species, viz. electrons and ions. (Received February 22, 1965.)

65T-219. MARY POWDERLY, 369 Main Street, Portland, Connecticut 06480. A note on topologies of lattices.

The author studies a class of lattices which are compact and Hausdorff in their order topologies. These lattices have nets which do not have order convergent subnets. Recent methods developed by Tong and his collaborators, Lanford and Kozlowski, are used. This note also answers a question raised in a paper by A. J. Insel [Proc, Amer. Math. Soc. 15 (1964), 847-850]. (Received February 23, 1965.)

460 65T-220. E. G. K. LOPEZ-ESCOBAR, Massachusetts Institute of Technology, Cambridge, Massachusetts. An extension of the intuitionistic propositional calculus,

Let ~ be the extension of the (finitary) propositional language (with the connectives

7, A, v, -->) obtained by adding the two (infinitary) connectives A, V and modifying the definition of a formula so that if (An) n< w is an w-sequence of formulas then A nAn and VnAn are also formulas • .%• is the set of formulas obtained by adding to the usual set of axioms for the intuitionistic propositional calculus certain infinitary formulas which express natural conditions about the infinite conjunctions and disjunctions. t- *A means that there exists a countable well-ordered sequence of formulas (B17 ). 71;;;,p. such that B 11 = A and for each 11 ;;;, !J., either B11 E .%• or is obtained from previous formulas in the sequence either by modus ponens or by conjunction (i.e. from (en) n· Theorem 1, 1f_ A is a finitary formula, then t-* A iff A is an intuitionistic tautology. Theorem 2, _..!!. 1-*VnAn, then for some n, t-* An. Let [A] be the formula obtained from the formula A by replacing the subformulas of the forms BvC and VnBn by-(- BA- C) and- An- Bn resp. and then replacing every propositional variable p by - p. Theorem 3. A is a classical tautology iff t-* [A]. (Received February 24, 1965.)

65T-221. SAMUEL ZAIDMAN, University of Montreal, Montreal, Quebec, Canada. Almost- periodic solutions of some second-order differential equations in Hilbert spaces.

Let H be a Hilbert space, A a self-adjoint, non-negative (unbounded} operator in H, with domain D A• and f(t) an almost-periodic function (in Bochner's sense}, from - oo < t < + oo to H. Theorem 2 2 2 2 Suppose that u(t}, du/dt E L10c(- oo,oo; D(A+I}1/2}, d u/dt E L10c(- oo,oo;H), u(t) E DA almost- 2 2 everywhere, and d u/dt - Au = f almost-everywhere in - oo < t < oo. Then, if ju(t) IH < L, - oo < t < + oo, then (A + I)112u(t) and du/dt are almost-periodic in H. We have two proofs of this result; the first one uses spectral decomposition for self-adjoint operators, the second one, indicated to us by L. Nirenberg, is based on some estimates; both of them reduce the problem to a theorem by L. Amerio on almost-periodicity of bounded functions in Hilbert spaces which have almost-periodic derivative. (Received February 24, 1965.)

65T-222. V. M. SEHGAL, Wayne State University, Detroit, Michigan 48202, A fixed point theorem for local contraction mappings.

Let {X,d} be a complete metric space, A mapping f: X -+X is said to be a local contraction mapping at the point ~EX if d(fn(~)(y).~(~)(l;)) ;;;, kd(y,O for ally EX, where k is a real number (0 ;;;, k < 1) and n(~) is a positive integer depending on t In the above fn denotes the nth iterate of f. In this paper we prove the following Theorem. Iff is any continuous, local contraction mapping of a complete metric space {X,d l into itself, then (i) there exists a unique point u E X such that f(u) = u, and (ii) if x0 is an arbitrary point of X and xn is defined inductively by xn+1 = f(xn>• n = 0,1,2,.,., then lim x u. We remark that if 1 for each EX, then f is a contraction mapping and n---ioo n = n(~) = ~ the above result yields the well-known Banach fixed point theorem. (Received February 25, 1965.)

461 65T-Z23. ALEXANDER ABIAN and S. E. LaMACCHIA, The Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210. On the existence of an empty set in set-theoretical models.

Assuming in the models considered that the axiom of extensionality (and no other Zermelo­

Fraenkel set-theoretical axioms) is satisfied, the following are proved, where Pn(s) with n > 0 means the power-set of Pn-l(s) and P 0 (s) = s. Lemma. In a model let there be a singleton-sets such that Pn (s) exists and is not a singleton for some n > 0. Then there exists an empty set. Theorem. In a model let a set s exist such that (i) for every set x, x = Pn(s) for some n ;;:: 0, (ii) Pn(s) exists and s ~ Pn (s) for some n > 0. Then there exists an empty set. (Received February 26, 1965.)

65T-224. K. D. MAGILL, JR., State University of New York at Buffalo, Buffalo, New York. Countable compactifications.

All topological spaces discussed here are locally compact and Hausdorff. By ann-point compactification of a space X we mean a compactification a.J<: where nuX - X consists of n points (n denotes a positive integer). A compactification a.X is said to be a countable compactification if a.X - X is countable. Theorem. A space X has a countable compactification if and only if there exists

a sequence { a.nXf: 1 of n-point compactifications of X with the property that for each n there is a continuous function fn+ 1 mapping a.nt 1 X onto a.nX such that fn+ 1 (x) = x for each x E X. Corollary. Suppose there exists an integer N such that every compact subset of X is contained in a compact subset whose complement consists of at most N components. Then X has no M -point compactification forM >Nand hence has no countable compactification. Corollary. Let (X,d 1} and (Y,d2} be two unbounded, connected metric spaces such that for all points x 0 E X and Yo E Y and every positive

number r, the sets {x EX: d 1 (x,x0) ~ r} and { y E Y: d2 (y,y0 ) ~ r I are compact. Then X X Y has no M-point compactification for M > 1 and hence has no countable compactification. The latter two cor­ ollaries imply that no Euclidean N -space has a countable compactification. (Received February 17,

1965.)

65T-225. A. A. MULLIN, University of California, Box 808, Livermore, California 94551. On algebraic structure in rings of arithmetic functions.

By analogy to the notions of a divisor and its complementary divisor, one can define the complementary submosaic to a prescribed submosaic within a given mosaic. Consider the com­ mutative ring of all arithmetic functions defined on a Gaussian monoid (with a single unit, viz., the multiplicative identity 1} and taking values in an integral domain. Addition of functions is component­ wise addition in the domain. Multiplication of functions is by analogy to "Dirichlet multiplication" of arithmetic functions, but with the summation over submosaics and their complements instead of over divisors and their complementary divisors. The identity function is the identity of the domain, if its argument is 1 and the zero of the domain otherwise. Theorem. The arithmetic functions modulo the generalized multiplicative functions form a (compatible) Abelian quotient monoid. The structures of this monoid and of its modular lattice of submonoids remain open questions, as do the structures of the multiplicative functions and completely multiplicative functions. (Received

February 10, 1965.)

462 65T-226. SHUICHI TAKAHASHI, University of Montreal, Montreal, Province of Quebec, Canada. Some properties of the group ring over rational integers of a finite group.

G. Higman in his paper on the units of group rings (Proc. London Math. Soc. (2) 46 (1940), 231-248) studied some properties of the group ring of a commutative group by using characters. His idea can be generalized to general noncommutative groups by using representations directly. For instance, one can prove: Theorem, Let r denote the group ring over rational integers of a finite group G, and let a = LuEGa(u) ·u be an element of r. Suppose for some pEG, a(p) f. 0 and a· p • 1 is a unit of finite order in r, then a(p) = 1 or - 1 and a(u) = 0 for all u f. p. This explains the unit of order 2: - u + u2 - T + UT + TU and another unit of infinite order: 1 + u - u 2 + uT - TU of the symmetric group of 3 letters {1,u, i,uT,u2-r} discovered by Olga Taussky (Bull. Amer. Math. Soc, 66 (1960), 327-345).Another instance is a direct proof of Theorem (of Nakayama). r has no idempotent except the identity. (Received February 17, 1965.)

65T-227, C. E. AULL, Kent State University, Kent, Ohio 44240. Sequences in topological spaces. II.

A side point, y, of a sequence, {xn} is a cluster point such that no subsequence converges toy.

A sequence is highly divergent if no subsequence converges. The following are proved, If a topo­ logical space is locally sequentially compact, no highly divergent sequence has a side point. If no highly divergent or convergent sequence has a side point, then every countably compact set is sequentially compact. If every sequentially compact set is closed in a locally sequentially compact space, then the space is sequential (see author's Abstract 65T-79, these cNOtiuiJ 12 (1965), 222). If every sequentially compact set is closed, sequences converge to at most one point and convergent sequences have no side points. In sequential spaces, in addition, no highly divergent sequence has side points. (Received March 1, 1965.)

65T-228. D. F. ISLES, Tufts University, Medford 55, Massachusetts. Further applications of Kalm!!r's techniques in proof theory.

This notice supplements that appearing in_ Abstract 65T-18, these cNoticeiJ 12 (1965), 130. These results are not new; what is of interest is that so many proof-theoretic facts should be derivable in such a uniform and direct manner, By ramified logic we mean the logical system ap­ pearing in Schutte (Beweistheorie, pp. 244-249, Springer, Berlin, 1960) having the usual quantification rules and axioms plus the axioms of concretion. Theorem. Given a proof of a formula F, in ramified logic, a proof of it may be primitively recursively constructed in which every formula is a sub­ formula of F. The notion of a first-order formula having property C of order pis described in Dreben, Andrews and Aanderaa (Bull. Amer. Math. Soc. 69 (1963), 699-706). Theorem. Given a (first-order) proof of a formula F, the order of F may be computed in primitive recursive manner. (Received March 1, 1965.)

463 65T-229. F. B. JONES and E. S, THOMAS, JR., University of California, Riverside, California. Connected G0 graphs.

All functions under consideration have domain I (the unit interval) and range contained in I; the word "graph" means the graph of such a function. Consider the following two questions. U the graph of the function f is a connected G 0 set then is f the pointwise limit of continuous functions? With the same hypotheses on f, if U is an open subset of the plane containing the graph off then does U contain the graph of a continuous function? Although bearing closely on the same problem, the two questions are independent. An example is given which answers both questions in the negative. In addition, the authors characterize connected G 0 graphs as dense subsets of certain "nice" continua. It is then proved that in the first question one can get pointwise convergence on a dense G0 set; the example mentioned above shows that this is "best possible". (Received March 1, 1965.)-

65T-230. C. Y. CHAO, University of Pittsburgh, Pittsburgh, Pennsylvania 15213. On k-complementary graphs.

G. Ringel in Selbstkomplementlire Graphen (Arch, Math. 14 (1963), 353-358) gives an algorithm for constructing self-complementary undirected graphs when the number of vertices is either

"' 0 mod 4., or = 1 mod 4, The purpose of this note is to generalize Ringel 1 s result to directed graphs and to k-complementary graphs which are defined as follows: Let x 1,x2,. .. ,Xk be k ( ~ 2) directed graphs of n vertices, and E(Xi) be the set of directed edges of Xi. Then x 1,x2, ... ,Xk are said to be k-complementary if U}= 1E (Xi) is the set of edges of the complete directed graph of n vertices, E(Xi)nE(Xj) = !,J, i = j, and i,j = 1,2, ... , k, and Xi are pairwise isomorphic. The self­ complementary graphs are 2-complementary. The main results are: (1) U k ( ~ 2) divides either nor n - 1, then there exist k-complementary directed graphs of n vertices. (2) Let x 1 and x 2 be 2-complementary directed graphs of n vertices, and u be an isomorphism of X 1 onto x 2 then no transitive set determined by the cyclic group generated by u has an odd number of letters > 1, From (1) and (2) we can determine all of the isomorphisms of 2-complementary graphs (directed and undirected). The proofs are constructive, and the method is similar to Ringel 1 s. (Received March 4, 1965.)

65T-231. D. W. WILLETT and J. S. W. WONG, University of Alberta, Edmonton, Alberta,

Canada. On the generalized Li~nard equation.

Different sets of conditions are determined which imply boundedness, unboundedness, or global asymptotic stability of the solutions of the system i = y, y = - f(x,y) - g(x). The main assump­ tions are the following: (i) For (x,y) in some neighborhood of oo, the solution starting at (x,y) is unique, yf(x,y) ~ 0, and (x,y) is not a singular point; (ii) The function equal to g(x)VO when x < 0 and

- g(x)vO when x ~ 0 has finite integral over the whole real line. For the most interesting case, which is when y- 2f(x,y) is a nondecreasing function of y and Jg(x)J has finite integral from- oo to+ oo, all solutions will be bounded, if there exists an f > 0 and for any c ~ 1, there exists xc such that f(x,~) ~ [(f + l}c - l)g(x) for all x ~ xc' and the functions - f(- x,- y) and- g(- x) satisfy a similar condition. For this case, there will exist an unbounded solution if there exists c 0 ~ 1 and x 0 such that f(x,../zc0 J: g) ;;; (c0 - 1)g(x) for all x ~ x 0 . This result and others solve the boundedness

464 problem completely for equations, when lf(x, y) I ~f 0 lx l11 ly Ia. as lx I-> oo, y--> 0 and lg(x) I ~ g0 lx 1tl as ·lxl,_.oo, (Received March 3, 1965.)

65T-232. A. L. ROSENBERG, 298 Pierce Hall, Harvard University, Cambridge 38, Massachusetts. Multi-head finite automata •

.Ln is the class of sets of tapes defined by finite automata with n reading heads. Theorem. For n > 1, there are sets in .Ln which are not context"free (CF). There exists a CFL which is not in .Ln for any n. Theorem. Vn, ~is closed under complementation. For n > 1, ...i'n is not closed under union or intersection. Ur.Lr is a Boolean algebra. Theorem. Vn, .Ln+1 properly includes _,/{ • Let g be the class of sets of n-tuples of tapes defined by n-tape finite automata (cf. Abstract n n 64T-343, these cN"ot:iuiJ 11 (1964), 469, and Abstracts 65T-47, 65T-48, ibid. 12 (1965), 139); and let En be the set of n-tuples of tapes defining the n-ary diagonal relation. Theorem. A set J is in .Ln iff there exists a set A E g n such that J is the projection on any coordinate of An En. Theorem. The following problems are unsolvable for sets A,B in .Ln: (a) Is A= r}? (b) Is A= 2:*? (c) Is A finite? (d) Is A~ B? (e) Is A = B? (f) Is A regular? (g) Is A CF? Theorem. The following problems are unsolvable for A in gn: (a) Is A nEn = CJ? (b) Is En~ A? (c) Is A ~En? (Received March 3, 1965.)

65T-233. J. L. BRYANT, University of Georgia, Athens, Georgia. A k-complex in En (n ~ 2k + 2) is tame if each simplex is tame.

Theorem. Suppose (X, Y) is a polyhedral pair, with dim X = k, and f is an embedding of X into En, where n ~ 2k + 2, such that flY is piecewise linear and fi(X - Y) is locally tame, Then,

given e > 0, there is an e-isotopy ht (0 ~ t ~ 1) ~En onto itself such that h0 = 1, h 1f: X--> En~ piecewise linear and htlf(Y) = 1 for all t, 0 ~ t ~ 1. Denote by Hom(X,X - Y;En) the set of all em­ beddings of X into En which agree with f on Y. It is known that the union of any two dense, solvable subsets of Hom(X,X - Y;En) is dense and solvable (cf. H. Gluck, E.mbeddings in the trivial range, Bull. Amer. Math. Soc. 69 (1963), 824-831). The result follows by showing that f is contained in a dense, solvable subset of Hom(X,X - Y;En), since the set of piecewise linear embeddings in Hom(X,X - Y;En) is known to be dense and solvable. Corollary. J!..K is a k- complex and f: K--> En (n ~ 2k + 2) is an embedding which is tame on each simplex of K, ~ f is tame. john Cobb has previously obtained these results by different methods, except with n ;;; 3k + 1 (Abstract 618-11, these cN"ot:iuiJ 11 (1964), 762). (Received March 4, 1965.)

65T-234. R. D. M. ACCOLA, Brown University, Providence, Rhode Island. On automorphisms of compact Riemann surfaces. Preliminary report,

For g ~ 2, let N(g) be the largest order of a group of conformal self-maps on a surface of

genus g. Then N(g) ~ 8(g + 1). If 3lg, then N(g) ~ 8(g + 3). These lower bounds are sharp for

infinitely many g's. The largest abelian group on a surface of genus g ~ 2 is of order 4(g + 1), a bound attained for each g by a hyperelliptic surface. The only nonhyperelliptic surface attaining this bound is a surface of genus 3. (Received March 5, 1965,)

465 65T-235. ]. D. BUCKHOLTZ, University of Kentucky, Lexington, Kentucky 40506. Extremal problems for sums of powers of complex numbers. II.

Given n complex numbers {zj}J= 1, let sk denote the sum of their kth powers, k = 1,2, .•. ,n. For each r > 1 let Vn(r) = inf L~=dskl' where the infimum is taken over all n-term complex sequences {zj}J=l such that maxlzjl ~ r. For fixed r, the asymptotic behavior of Vn(r) as n->oo is determined, and a lower bound for Vn (r) is obtained. Theorem. 0 < Vn(r)- nrC(r) = O(logn), where C(r) = max {(log t)/(1 + t) }. 1 < t ~ r. Furthermore, the estimate O(logn) cannot be replaced by o(log n) for those values of r which are less than the number t0 = 3. 5911 .•• which maximizes (logt)/(1 + t). (Received March 8, 1965.)

65T-236. K. F. BARTH, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland 21005. Asymptotic values of meromorphic functions.

G. R. Mac Lane has considered the class Slt' of functions which are nonconstant and holomor­ phic in {lz I< 1} and have asymptotic values at a dense set on { lz I= 1 f. We consider the correspond­ ing class Slt' of meromorphic functions. The following theorems extend results of MacLane and m F. B agemihl. Let f be nonconstant and merom orphic in { lz I < 1}. Theorem 1. If there exists a complex number a (possibly oo) such that N(r,a) = 0(1) and J~(l- r)T(r)dr < oo, then f E Slt'm. Theorem 2. Iff is normal (Lehto and Virtanen) and there exists a complex number a (possibly oo) f EN • Theorem 3. Iff EN and there exists a complex number a such that N(r,a) = 0(1), then m m (possibly oo) such that N(r,a) = 0(1), then on any subarc 'Y of {lz I= 1} on which f does not have the asymptotic value· a, f has asymptotic values on a set of positive measure. Theorem 4. Let f E Nm and there exist complex numbers a,b (a i b) such that N(r,a) = 0(1) and N(r,b) = 0(1), then f has .!!.Q arc tracts. (Received March 5, 1965.)

65T-237. R. S. BUCY and G. J. MALTESE, University of Maryland, College Park, Maryland. Extreme positive definite functions and Choquet' s Representation Theorem.

Let G be a locally compact abelian group. The famous theorem of Herglotz-Bochner-Riesz­ Weil states that the set of positive definite functions P in the unit ball of L 00 (G) is isometrically isomorphic to the set of positive Radon measures defined on the character group and having norm at most one. A new and elementary proof of this theorem is given based on a direct identification of the extreme points of P (without the use of Banach algebra or group representation theory) and on the Choquet Representation Theorem for which Bonsall (J. London Math. Soc. 38 (1963), 332-334) has recently given a concise and elementary proof. The methods employed give a clear picture of the natural order structure of the class P. (Received March 8, 1965.)

65T-238. P. G. HINMAN, University of California, Berkeley, California 94720. Generalizations of some standard theorems on recursive functions.

Feferman (Berkeley Symposium, 1963) and Addison.(Abstract 622-71, these cNoticeiJ 12 (1965), 347) have applied Cohen's notions of forcing to problems of first-order definability in number theory. The methods allow in general, as Addison notes, extension of results about recursive

466 sets to apply to all levels (2:11 : v < w1) of the effective Borel (hyperarithmetic) hierarchy. A strength­ ened and extended form of Addison's theorem (I) can be put as a basis result: Theorem l, For any

2: 11 -t-2 subset A of NN, if A is non-meager (i.e, of second category) then A contains a 2:11 function. Corollary 1. For any (hyper)arithmetic subset A of NN, if A contains all (hyper)arithmetic functions, then A is co-meager. For v < w1 the notion of .6.11-comparability is defined by relativization by analogy with relative recursiveness, Theorem 2, For any a E NN, if a is not .6.11, then the set of

{J E NN which are .6. 11-comparable with a is meager. Corollary 2. For any given non- (hyper)arith­ metic function a there exist functions not (hyper)arithmetically comparable with a. Theorem 3.

(Generalized Friedberg-Muchnik Theorem). For any v > 0 there exist two 2:11 subsets of N which are not .6.11-comparable. (Received March 8, 1965,)

65T-239, ]. I. MALITZ, University of California, Berkeley, California 94720. The failure of Craig's theorem in languages with uncountable conjunctions.

Let a,{J be arbitrary infinite cardinals, a ~ {J. By L(a,{J) we mean the first-order language with equality that allows conjunctions involving less than a conjuncts and quantification over less than {J variables. Theorem. Suppose u is a sentence of L(a,{J) having no relation symbols other than equality and suppose A ~B. X ~ {J. Then u is true in A if and only if it is true in B. Using

this theorem one can show that there are sentences u and p of L(w2, w) such that u-->p is valid but no formula f/J exists in any language L({J,w) with the property that u-->f/Jand f/J-->p are both valid.

Hence Craig's interpolation theorem fails for L( {J, w) whenever {J > w1• This solves a problem posed by E.G.K. Lopez-Escobar in Infinitely long formulas with countable quantifier degrees, Doctoral dissertation, Berkeley, California, 1964, p. 7, (Received January 29, 1965,)

65T-240. A. P. HILLMAN and D. G. MEAD, University of Santa Clara, Santa Clara, California, K. B. O'KEEFE, University of Washington, Seattle, Washington and E. S. O'KEEFE, Boeing Company, Seattle, Washington. Ideals generated by products,

The following generalizes Levi's results on the differential ideal [uv] in (Trans, Amer. Math. Soc. 51 (1942), 532-568). Let Yij(i = l, .. ,n and j = 0,1, ... ) be independent indeterminates over a field F. Let xk be a linear combination, with nonzero coefficietnts in F, of all the products

y 1hy2j2'"Ynjn with weight h + ... +in= k. Let It be the ideal (x0 ,x1, ... ,xt) in the ring over F of polynomials in they ..• A monomial Min the y.j has signature b = (d , ... ,d ) if its degree in the ~ 1 1 n y ij with i = h is dh. Functions f(D) and g(D) are obtained such that M E It if its weight is less than

f(D) and t ~ g(D) and such that for every D and w ~ f(D) there is an M of signature D and weight w

that is not in (x0 ,xl' ... ). The g(D) is minJ!j[di + dj- 2) and the f(D) is as follows: Without loss of generality let d 1 ;:;; d2 ;:;; ... ;:;; dn· For 2 ;:;; i ;:;; n let qi = (d1 + ... + di)/(i - 1) and let k be the small­ est i for which qi assumes its minimum. Let q and r be integers defined by d 1 + ... + dk = (k - 1)q + r and 0 ;:;; r < k - l, Let ci = q - di for 1 ;:;; i ;:;; k. Let s 1 = c 1 + ... + ck and s 2 =

Li

467 65T-241. WITOLD BOGDANOWICZ and MICHAEL MULLEN, The Catholic University of America, Washington, D. C. 20017. On a nonlinear implicit operator in the space of continuous functions.

Let C(Q) be the set of all continuous real-valued functions defined on Q = I1 X I2 , Ij = IJJ,aj]. The norm in C(Q) is defined as usual. A continuous, increasing, subadditive function E defined on [O,oo) such that E(O) = 0 is called a modul. A modul E is a modul of continuity for a function v E C (Q) if lv(x,y)- v(i,Y)I ;;; E(lx- il + IY- 'YI) for (x,y), (x,y) in Q. Assume f(x,y,p,q) is continuous from

I 1 X I2 X R X R into the space R of real numbers. For (x,y) in Q and p,p,q,q in R assume that lf(x,y,p,q)- f(x,y,p,cj)l;;; w(IP- iii+ lq- where w is a modul such that J~(w(~))- 1 = oo. Define the operator F by (F(s,v))(x,y) = f(x,y,J~s(t,y)dt,v(x,y)) for s,v in C(Q). Theorem. There exists a unique operator H from the space C(Q) into itself such that H(v) = F(Hv,v) for all v in C(Q). This operator H is continuous and I!Hv II ;;; A(l + llv II> for all v in C (Q) and some constant A > 0. For each constant M > 0 and a modul E there exists a modul 11 such that if llv II ;;; M and E is a modul of continuity for v, then 11 is a modul of continuity for the image s = Hv. An explicit formula for the modul 11 has been found. This type of an operator one meets in some problems of the theory of the partial differential equations. (Received March 11, 1965.)

65T-242. WITHDRAWN.

65T-243. WITHDRAWN.

65T-244. JOHN De CICCO and R. V. ANDERSON, Illinois Institute of Technology, 3300 South Federal Street, Chicago, Illinois 60616. Parallel families in Riemannian space Vn.

For a conservative field of force 0, in a Riemannian space V n' if a set of oo' Faraday lines of force L is tangent to an admissible arc AB in V n' and if this set is orthogonal to an admissible arc DE, then the measure of the magnitude I1 = 1, this reduces to the jacobi string condition for V n· Also there is derived an extension of the jacobi envelope theorem. A family of equipotential surfaces V(x) = const, with A1(V) 'f 0, is parallel if and only if A1(V) = H(V), where H depends on the nonconstant potential function V(x) alone. For a., (3 fixed and for all such a.,{J = 1,2, ••• ,n, the family V(x) = const is parallel if and only if the covariant vector V,f3V•ja.- V,a. V,j/3 is orthogonal to V(x) = const. The preceding results are extensions of the Hamilton-jacobi partial differential

468 equation to V n• A cartogram T between two Riemannian spaces V n and Vn' is homothetic if and only if every parallel family in V n corresponds to a parallel family in Vn· Finally the above results can be extended to wave fronts in a Riemannian space Vn. (Received February 22, 1965.)

65T-245. R. V. CHACON, Ohio State University, Columbus, Ohio. The square root problem for transformations having continuous spectrum.

An example is constructed of an invertible measure-preserving transformation of the unit interval having continuous spectrum and which admits no square roots in the class of nonsingular transformations, thus negatively answering a question raised by Halmos. The transformation can be made to also satisfy certain side conditions, such as not being strongly mixing (the nth root problem may be considered using extensions of the methods given here). (Received March 15, 1965.)

65T-246. HABIB SALEHI. Indiana University, Bloomington, Indiana. Multivariate prediction with continuous time. Preliminary report.

It is shown that a q X q non-negative hermitian matrix-valued function F 1 on (- oo,oo), such that

F 1 (A) and log det F 1 ( A)/(1 + AZ, E L 1 (- oo,oo) can be factored in the form F 1 = <1><1>* a. e., (A) = J~C(t)eiAtdt E L2 (- oo,oo) where is an outer function in the upper half-plane. In case F 1 (A)= 2 {I + M(A) ~/(1 + A2 ) where J.L = ess. lub. -oo< A can be determined by an iterative procedure analogous to that obtained by Wiener and Masani [Acta Math. 99 (1958), 93-137]. In case

F 1 is the spectral density of a q-variate stationary stochastic process this algorithm yields the linear predictor in the spectral domain. A similar factorization algorithm is given for the case in

which F 1 is not necessarily hermitian. Let M 1' M2 be subspaces of a Hilbert space H spanned by countably additive, orthogonally scattered measures ~ 1 • ~ 2 • An expression involving ~ 1 • ~ 2 is obtained for the projection of a vector x in H on the subspace @5(M 1 + M2) under a suitable restriction. When ~ 1 • ~ 2 are the canonical measures occuring in the Wold decomposition of a bivariate stationary SP with continuous time, these expressions yield algorithms for the linear predictor. (Received March 8, 1965.)

65T-247. R. 0. BURDICK, University of Virginia, Charlottesville, Virginia 22903. Oriented manifolds fibered over the circle.

Let M be a closed oriented C00 -differentiable n-manifold. Let Diff+ (M) be the group of orientation-perserving C00-diffeomorphisms of M onto itself, R the reals, and Z the integers. If f E Diff+(M), let Z act on M X R by (x,t)•n = (fn(x), t + n). Then M X R/Z -->S1 is a bundle over the

circle with fiber M. Define the homomorphism .B: Diff+(M)--> n n+ 1, the (n + I)-dimensional oriented cobordism group, by f/(f) = 1M X R/Z]. This can be used to show the following Theorem. If E is a closed oriented C00 -differentiable manifold fibered differentiably over the circle with compact Lie group as group of the fibration, then E represents a torsion element of n •. the oriented cobordism ring. In showing .B is a homomorphism a construction is made which can be used to show the following Theorem. If an oriented cobordism class in n. can be represented by a manifold fibered differentiably over then-sphere for n positive, then it can be represented by a manifold fibered differentiably over the circle. (Received March 15, 1965.)

469 65T-248, M. C. THORNTON, 1909 Meadow Drive, Champaign, Illinois. Singularly fibered manifolds,

Let D be a closed cell, M, F, B be topological manifolds. Given a homeomorphism f: Sn-q- 1 X Fq --->s X {o} X F C S X I X F, a core space Cf, homeomorphic to Dn -q X Fq, is defined by identifying the sets f(S,y), Cf has an induced fibering by Fq which may have a singularity depend­ ing on the isotopy class of f. Let Mn be fibered by Fq over Bn-q and suppose aM= sn-q- 1 x Fq. Let h: acf ---->aM be a fiber-preserving adjoining homeomorphism. Then Mn UCf is a manifold with one singular fiber. By removing the interior of a tubular neighborhood of a fiber this procedure can be iterated, defining a singularly fibered manifold. This procedure is seen to be equivalent to the identification of certain Sn-q- 1 fibers on Sn-q- 1 X Fq boundary components of Mn. For Fq an

(n - 2)-torus (the product of n - 2 circles) the types of singularities of the cores are determined, In special cases with orientable M and B, trivial characteristic and one or no singular fibers, this leads to a classification of the Mn up to homeomorphism. Thus some results of H. Seifert (Acta Math. 60 (1933), 147-238) and E. J. Brody (Quart. J. Math, Oxford Ser. (2) 13 (1962), 161-171) are extended to higher dimensions. (Received March 17, 1965.)

65T-249. J, A. WOLF, University of California at Berkeley, Berkeley 4, California. Flat homogeneous manifolds.

2 2 R~ denotes euclidean space with the indefinite metric - L~= 1 (dxi) + ,Lf=s+ 1 (dxi) . A complete structure theorem is given for discontinuous groups of isometries acting freely on R~ with homo­ geneous quotient manifold. This completes earlier results of the author (Trans. Amer. Math. Soc. 104 (1962), 462-469J, giving a classification of complete connected homogeneous pseudo-Riemannian manifolds of zero curvature. Because of some other results [Comment. Math. Helv. 36 (1961), 112-147; ibid, 39 (1964), 21-64] it also completes the classification of the complete connected homo­ geneous locally isotropic pseudo-Riemannian manifolds. (Received March 11, 1965.)

65T-250, JACOB FELDMAN, University of California at Berkeley, Berkeley, California. Invariant subspaces for certain quasi-nilpotent operators.

If A is a bounded linear operator with spectral radius zero on a Hilbert space, and if the uniformly closed algebra generated by A contains some nonzero compact operator, then A has a nontrivial invariant subspace. The proof is a variant of the Halmos simplification of the Bernstein­ Robinson proof that every polynomially compact operator has a nontrivial invariant subspace. (Received March 18, 1965.)

65T-251. C. F. KOCH, Kansas State University, Manhattan, Kansas 66504. On the nonregularity of certain generalized Lototsky transforms. Preliminary report.

In a recent paper by V. F. Cowling and C. L. Miracle (Canad. j; Math. 14 (1962), 418-435) . ill n . "oo - 1 "oo 11 it is conJectured that for dn = Pne ; rf "-'n= 1Pn = + oo,"-'n= 1 Pn = + oo, and limn->oo n = 0 then the generalized Lototsky (or [F,dn]> summability matrix, with entries Pnk defined by Poo = 1, Pok = 0 fork i 0, and II~= 1 x + dj/1 + dj = ,L_:,0 P nkxk for n <;: 1, is regular. The following theorems give conditions under which this fails to be the case. Theorem 1. If 2::: 1 p~ 1 = + oo, 2::: 1pn = + oo,

470 7r/2 > (J > 0, lim fJ = 0, and "'00 02 p - 1 diverges in such a way that there exists a sequence "n n n n ~n= 1 n n satisfying Ok ~ a.n fork ~ n and limn~ooa.~L~= 1p_k 1 = + oo then the [F ,dn] transformation is not regular. Theorem 2. If in addition to the hypotheses of Theorem 1, M ~ pn ~ N > 0, and (Jn = O(n - 112logf(n + 1)), where f > 0, then the [F ,dn] transformation is not regular. (Received March 22, 1965.)

65T-252. P. C.-C. WANG, Wayne State University, Detroit, Michigan 48202. Solution to a research problem.

W. R. Utz has given a research problem (Bull. Amer. Math. Soc. 71 (1965), 138) to determining conditions for the existence of a real function f(x), not identically zero, satisfying f' (x) = af(g(x)), where a is a given constant and g(x) is a given real function. The following theorem is proved.

Theorem: If there exists a real number x 0 such that g(x) is continuous and x0 ~ g(x) ~ x 0 - d for x E [x0 ,x0 - d) where d, 0 < d < 1/ Ia 1. Then there exists a unique continuous real function f(x) defined in [x0 ,x0 - d] which satisfies f'(x) = af(g(x)). (Received March 22, 1965.)

65T-253. P. ]. SALLY and M. H. TAIBLESON, Washington University, St. Louis, Missouri 63130. Special functions on locally compact fields.

In a recent paper (Uspehi Mat. Nauk 18 (1963), 29-100), Gelfand and Graev have discussed gamma and Bessel functions on a locally compact, totally disconnected field K. Let 0 be the ring of integers inK, 9 the maximal ideal in 0 and q the order of 0/9. If 'Iris a unitary character on K*, then r(1r) may be defined as a distribution on K*, the Mellin transform of a certain additive unitary character X(x). If 7r is unramified (7r(x) = lx Ia.), r(7r) = p. V. [(I - qa.-I)/(1 - q- a.)] + 7r/(q'ln q) •ll(a.).

Setting 1r(x) = lxla.7r1(x) (1r1 a character of ramification degree h on the group of units of 0) we have r(7r) = c7r1 qh(a.- 1/ 2). IC7r~ = 1. In either case r(7r) may be computed as a principal value integral. The computation here is similar to Tate's local computation for the zeta function. The r-function

may be defined for other 1r by means of the Fourier transform or analytic continuation. The Bessel function J7r(u,v) is defined for u,v inK* and 1ra unitary character on K* as the Mellin transform of X(ux + v/x). For all u,v inK*, J7r(u,v) may be computed explicitly as a principal value integral.

It can also be defined as the Fourier transform of X(v/x)7r(x)lxrl. For 1r unitary, J7r(u,v) is bounded.

With J7r(u, v) as a kernel, a Hankel transform can be defined which is unitary on L 2 (K) and bounded

from Lp to Lp' for 1r"" 1, I ~ p ~ 2. (Received March 24, 1965.)

65T-254. R. F. RINEHART, Case Institute of Technology, 10900 Euclid Avenue, Cleveland, Ohio 44106. The differential of a primary matrix function.

Let F(X) be a function with domain and range in the algebra of n X n complex matrices. If the elements of F (X) are differentiable functions of the elements of X on some open set, then it is well known that F(X) is Hausdorff-differentiable, i.e., dF(X) is expressible in the form

"'~ H dXK. where dX = (dx ) and H.,Ki are matrices which are independent of dX. Explicit for- "'-'1= 1 i 1 rs 1 muias for the ~, Ki have not been developed in general. For the special type of function f(X) arising from the extension of a scalar function f(z) of a complex variable to complex matrices and for do-

471 mains restricted to matrices X with distinct eigenvalues, the following formula is developed: df(X) = f' (X)dX t (dP)P -lf(X) - f(X)(dP)P -l - f' (X}[(dP)P -IX - X(dP)P -I), where P is a matrix such that P- 1XP is diagonal. The matrix (dP)P-l depends only on X, not on the function f(X). It is further shown that for a large class of open subdomains of the set of matrices with distinct eigenvalues, the matrix P can be so chosen that the coefficients of the dxrs in the elements of (dP)P-l are rational functions of the elements of X. Analogous results are obtained for functions with domain in the set of real n X n matrices and range in the set of complex n X n matrices. (Received March 24, 1965.)

65T-255. D. T. HAIMO, 77 Snake Hill Road, Belmont, Massachusetts and F. M. CHOLEWINSKI, 411 Westlley Drive, Chapel Hill, North Carolina. Integral representations of generalized temperature functions in negative time.

In a paper of F. M. Cholewinski and D. T. Haimo, The Weierstrass-Hankel convolution trans­ form (to appear in Journal d'Analyse Mathematique) and in another of D. T. Haimo, Generalized temperature functions (to appear in Duke Mathematical Journal), characterizations were derived for generalized temperature functions, defined for positive time, which may be represented by Poisson­ Hankel-Stieltjes integral transforms. These results can be extended, by an appeal to Appell trans­ forms, to yield integral representations for generalized temperature functions defined in negative time. The generalized heat equation is (*) <1xu(x,t) = (a;ax)u(x,t), where ~f(x) = f"(x) t 2(q/x)f' (x), q a fixed positive number. H denotes the class of generalized temperature functions, that is, of c 2 2 2 solutions of(*). The fundamental solution of(*) is the function G(x,y;t) = (l/2t)qtl/2e -(x +Y )/4t .zr. '"" q-1/2 1/2-q • __.r(xy /2t), where __.r(z) = 2 r(q t l/2)z Iq-l/2 (z), Ir(z) being the Bessel function of imaginary argument of order r. Theorem. A necessary and sufficient condition that u(x,t) = 2 Jg:>ety Y(xy)du(y), - oo < t < c, with u(y) i, is that, for - oo < t < c, u(x,t) E H, u(x,t) ~ 0. 2 Theorem. A necessary and sufficient condition that u(x,t) = J~ /Y Y(xy)da(y), t < c < 0, with jg:>eCY2 ida(y) I < oo, is that u(x,t) E H for t < c < 0, and jg:> iu(x,t) IG(x;c - t)d!L(X) < M, where d!L(x) = (2q-l/2 r(q t l/2)f 1x2qdx. (Received March 24, 1965.)

65T-256. N. C. A. da COSTA, University of Parana:, Caiza Postal 1963, Curitiba, Parana:, Brazil. Some properties of two systems of set theory.

As it is well known, the usual systems of set theory are not appropriate to serve as foundations for category theory. Sonner (On the formal definition of categories, Math. Z. 80 (1962), 163-176) and Grothendieck (cf. Gabriel, Des categories abeliennes, Bull. Soc. Math. France 90 (1962), 323-448) have proposed new systems to this purpose. To meet the difficulties of category theory, the author has formulated (Newton C. A. da Costa, On two systems of set theory, to appear in Nederl. Akad. Wetensch. Proc.) two other systems, T and T*, intermediate between the Kelley-Morse system (Kelley, General topology, appendix, Van Nostrand, New York, 1955) and the theory of types. Employ­ ing ideas of Grothendieck, the author has proved the existence of strongly inaccessible sets in T and T*, and also other interesting properties of these systems. (Received March 26, 1965.)

472 65T-257. C. A. PERSINGER, Air Force Institute of Technology, Air University USAF, Wright­ Patterson Air Force Base, Ohio 45433. A characterization of tame 2-spheres.

Let K be a finite topological 2-complex in E 3. If vis a vertex of K, Stv is the star of v, ~tv 0 is the open star of v, and Lk v ~ St v - St v is the link of v. The triviall-dimensional homology group

of K will be denoted by H 1(K) ~ 0. Theorem 1. If K is a finite, connected topological 2-complex in E 3 such that (i) Lk v is connected for each vertex v in K and (ii) K contains no wild arcs, then K is a compact 2-manifold with boundary. Theorem 2, If (i) K satisfies the conditions of Theorem 1,

(i) H 1 (K) ~ 0, and (iii) no arc inK separates K, then K is a 2-sphere. (Received March 29, 1965.)

65T-258. R. C. BASINGER, Division 5253, Sandia Corporation, Box 5800, Albuquerque, New Mexico 87115. An extension of a theorem of Bernstein to meromorphic functions.

A. J. Macintyre and S.M. Shah (J. Math. Anal. Appl. 3 (1961), 351-354) give sufficient conditions whereby a meromorphic function which is uniformly bounded on the axis of reals has a derivative which is also bounded there. The following theorem (cf. Abstract 65T-114, these cJVotiuiJ 12 (1965)., 233) extends their result. Theorem. Let f(z) be a meromorphic function which satisfies: -1 p (a) lim supr~oor T(r ,f) ~ t < oo, (b) lf(x) I ~ A exp(B lx I ) for - oo < x < oo (A,B > 0; 0 ~ p < 1), (c) f(z) has no poles in IIm(z) I < h, and (d) Ln;;; ~Im(bn) llx - bn 1- 2 ~ C < oo for - oo < x < oo, where

b 1 ,b2 , ... are the poles of f(z). Then for every ~ > 0, If' (x) I ~ K exp [(1 + E)B lx IP] for - oo < x < oo,

where K ~ K(A,B,C,h,t,f). (Received March 29, 1965.)

65T-259. S. S. SHRIKHANDE, New University Building, 164 Backbay Reclamation, Road No. 3, Bombay - 1. (India) On the existence of a class of partially balanced incomplete block designs.

It is shoWn that when n is even (n t 8), the existence of any one of the four partially balanced incomplete block designs [Bose and Nair, Sankhya 4 (1939), 337-372) implies the existence of the A 1 2 rest. (A 1): v ~ cn,2 ' b ~ (n- 1)(n- 3), r ~ n- 3, k ~ n/2, \ = 0, 2 ~ 1, n 1 ~ 2(n- 2), p 11 ~ n- 2,p11 2 4; (Ai): v ~ (n- 1}(n- 3), b = cn, 2 ' r = n/2, k = n- 3, A1 = 0, A2 ~ 1, n 1 = (n- 2) /2, p~ 1 = p~ 1 = 2 (n- 2) /4; (A2):v = Cn- 1,2 ,b = (n- 1)(n- 3), r = n- 3, k = (n- 2)/2, \ = 0, A2 = 1, n 1 ~ 2(n- 3),

pl 1 = n- 3, py 1 = 4; (Ai): v = (n- l)(n- 3), b ~ Cn- 1, 2 , r = (n- 2)/2, k = n- 3, A1 = 1, Az ~ A3 = 0, 1 3 2 2 1 2 n 1 = (n - 2)(n- 4)/2, n 2 ~ n- 4, p 11 ~ p 11 ~ (n- 4) /4, p 11 ~ (n- 2)(n- 6)/4; p 12 = (n- 6)/2, p 12 = 0, Piz = (n - 4)/2, p~ 2 = p~ 2 ~ 0, p~ 2 ~ n - 5. Utilising a result of Ray-Chaudhari, it is shown that these designs exist for n ~ 2m + 2. For n ~ 8, it can be shown that (~) with triangular association

scheme Tn' (Ai) with association scheme Tn for its dual, (A2) and (Az) are nonexistent. (Received March 29, 1965.)

65T-260. VACLAV PFEFFER, The George Washington University, Washington, D. C. On tensions in a revolving disk.

The radial and tangential tensions un and ut of a revolving circular disk are described by the following boundary problem in the interval [a,b), 0 ~a< b: (d/dx)[(xl+ 11/E)(ut- IIUn)] = 2 II l+V 2 . (1 - v )(x /E)un- x (dTjdx), (d/dx)(xhun) ~hut- Ghx , un(b) = Sn' ut(a)- un(a) 1s 0 or A according

as a= 0 or a > 0, respectively. Here G, Sn' A, II are given real constants, 0 < 11 < 1, and E, T,h are

473 given real-valued functions on [a,b), E > 0, h > 0. Let us assume that E, T ,h have bounded derivatives in (a,b) and that E • h is analytic in some neighborhood of a if a = 0. Under these assumptions is proved the convergence of a very simple numerical analogy. The system of linear equations in this analogy has essentially the triagonal matrix with units in the main diagonal. (Received March 3, 1965.)

65T-261. D. j. RODABAUGH, Box 1631, Vanderbilt University, Nashville, Tennessee 37203. On the associative admissibility of antiflexible algebras.

A conjecture of Frank Kosier is that a simple not associative power-associative antiflexible algebra is associative admissible. The following three theorems are proved. Theorem. An anti­ flexible algebra A is associative admissible if and only if it is quasiequivalent to an antiflexible algebra A(h) for h 1 0,1. Theorem. A simple antiflexible algebra which satisfies (x,x,x) = 0 and one of the identities (x2,y,x) = 2x' (x,y,z), (z,y,x2) = 2x'(z,y,x), or (y,x2,z) = 2x"(y,x,z) (note: see Frank Kosier, A generalization of alternative rings, Trans. Amer. Math. Soc, 112 (1964), 32-42) is associative admissible, Theorem. If Z (= { x:xy = yx for all y in A f) contains no nilpotent elements in a semisimple power-associative antiflexib1e algebra, then A= C + D where C is an associative ideal and D is associative admissible. All known examples satisfy the hypothesis of this third theorem. Several conditions equivalent to this hypothesis are given. (Received April 1, 1965,)

65T-262. SHIMSHON ZIMERING, Battelle Memorial Institute, Department of Mathematical Physics, 7, route de Drize, Carouge-Geneva (Switzerland). Some simple sufficient conditions for a transformation to sum a divergent series.

Let tn = L~=Oan,ksk, n = 0,1,2, ... , (1) be a triangular, reversible and convergence-preserving transformation, i.e., an 1 0, n = 0,1,2, ... , and s ---> =>t --> n-->co (2), Theorem, If {a kl .n n f n f', n, f satisfies (2) and one of the three following conditions: (i) There exists a diverging sequence { s; f such that t; = o(sri) (n --->co); (ii) The element of the matrix {an,kf is positive; the Schmidt function f(t) =lim '\" ::!> ta k exists and is continuous fort= 1, i.e., a k ~ 0, n = 0,1,2, ... , n=oo~-n n, n, 0 ~ k ~ nand f(l - 0) = f(I); (iii) The matrix {a kf contains a sub-matrix {a k f such that n, n, i limn=coMax0 ~ki~n1an,kil = 0, then the transformation (1) is stronger than convergence, i.e., tn---> f' {';> sn---> f, n--> co. The part (iii) is an extension of a theorem of R. P, Agnew (A simple suffi­ cient condition that a method of summability be stronger than convergence. Bull. Amer. Soc. 52 (1946), 128-132). (Received April I, 1965.)

65T-263. P. K. HOOPER, Harvard University, 48 Conant Hall, Cambridge 38, Massachusetts. The undecidability of the immortality problem for Semi-Thue systems.

It is a well-known result that a Semi-Thue system (ST) may be constructed to simulate any Turing Machine, showing the halting problem forST's undecidable. This, however, sheds no imme­ diate light on the immortality problem for these systems, the problem of deciding, for each ST, whether or not there is a word to which an arbitrarily large number of rules may be successively applied, an "immortal" word. For this result, it is sufficient to augment the alphabet of the ST by replacing each "tape" symbol Si by two new symbols, sf' and sf, modifying the rules of the system as follows. The superscript on a "tape" symbol is to indicate in which direction a "state" symbol

474 can occur, and no "state" symbol can shift left to scan an S~ or right to scan an S~. This prevents any interaction between occurrences of "state" symbols in a word of the ST, and ensures that an immortal word of the ST must represent at least one immortal instantaneous description of the simulated TM. Thus the result becomes a consequence of the undecidability of the immortality problem for TM's. (Abstract 620-3, these cJioticeiJ 12 (1965), 209-210.) (Received April 2, 1965.)

65T-264. J. M. CARDOSO, Rua Cel. Mena Barreto Monclaro, 461, Curitiba, Paran.!, Brasil. n-ality theory of rings.

The author develops the n-ality theory of rings initiated with n-rings by A. L. Foster (Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 31-38). Ann-ring R (n =prime) is a commutative ring with identity, such that for all z E R, zn = z and nz = 0. A permutation of rank k is defined over a semi­ group as a permutation the decomposition of which in disjoint cycles has only cycles of length one and k. As the main result it is proved that any permutation h of rank 3 is a triality in a semigroup (G, £) relative to the operations £, $and &, where x $ y = h2(hx £ hy) and x &y = h(h2x £ h2y), for all x,y in G. It is also shown that the example given by Foster is a permutation of rank 3 that may be generalized by replacing the ring identity by any fixed element in the 3-ring. (Received April 2, 1965.)

65T-265. A. T. HOPPER, Case Institute of Technology, 10900 Euclid Avenue, Cleveland, Ohio 44106. Expansion theorems for solutions of two generalizations of the heat equation.

Theorems are developed which give conditions under which classical solutions of the equation (*) au(x,t)/at = (- l)j+1a 2ju(x,t)/ax2j, j = 1,2, ... , can be represented in terms of either a set {v~j(x,t)}: 0 of polynomial solutions of (*) (v~j(x,O) = xk /k!) or an associated set {w~j(x,t)}:o of solutions of (*). The w~j(x,t) are related to the fundamental solution of (*). Some typical results are: Theorem 1. If u(x,t) = L:f= 0 akv~j(x,t) converges absolutely for lxl < oo and It I < cr then u(x,t) is a solution of·(*) and satisfies a Huygens' principle in the integral It I < min(o;cr /m) where m is a positive constant related to equation (*). Theorem 2. Let ~(x) be an entire function with crlx 12j -1/2Joo t 2j · l~(x) I ;;; Be , cr ~ 0. Then u(x,t) = (211") _00 ~(s)e- s -lXSds is a solution of (*) for lx I < oo, 0 ;;; cr < t and this u(x,t) can be represented in a convergent series of the {w~j(x,t)\ for 0 ;;; cr < t. Extensions of these results to equations other than (*) are noted. The work is related to that of Rosenbloom and Widder [Trans. Amer. Math. Soc. 92 (1959), 220-266; Math. Reviews 21 (1960), no. 5845) and uses certain of their terminology. (Received April 5, 1965.)

65T-266. C. K. MEGIBBEN, University of Washington, Seattle, Washington 98105. Primary abelian groups which are neither transitive nor fully transitive.

Countable p-groups and p-groups without elements of infinite height are known to be both transitive and fully transitive (see Kaplansky's Infinite abelian groups, Ann Arbor, Michigan, 1954), and Kaplansky has suggested that "it seems plausible to conjecture" that all p-groups are such. However, we have the following examples: (1) If G = H

475 p-group such that G1 is an elementary p-group and E(G/G1) = Rp EB Es(G/G1) (see Pierce, Homomor­ phisms of primary abelian groups, Topics in Abelian Groups, Chicago, 1963), then every subgrou-p of G1 is fully invariant-in particular, if G1 is not cyclic, then G is neither transitive nor fully transitive, (Received April 5, 1965.)

65T-267. P. R. MEYERS, U. S. Department of Commerce, National Bureau of Standards, Washington, D. C. 20234, Room 140 North Building. A converse of Banach's contraction theorem.

Let X be a topological space admitting a metric (complete metric), and f: X -+X a continuous map with a fixed point x0 obeying the purely topological conditions (a) fn(x) -+x0 for all x EX, (b) f(U) C U, fn(U) ----> {xo} for some neighborhood U of x0• Theorem. X admits a metric (complete metric) for which f is a contraction. This generalizes L. Janos' result (Abstract 64T-469, these

c}(oficei) 11 (1964), p. 686), for the case of compact X under the hypothesis n fn(X) = {x0}. (Received April 7, 1965.)

65T-268. B. S, RANDOL, Yale University, New Haven, Connecticut, A number-theoretic estimate.

Suppose k is a positive integer greater than 1. Denote by N(x) the number of integral lattice­ 2 2 points (nl'n2), such that n~k + n~k ;> x, and denote by A the area of the region defined by x k + y k ;> I, Theorem. N(x)- Axl/k = O(xR), where R = (2k- I)/4k2, and this is the best possible exponent, I.e., N(x)- Axl/k f. O(xR-E), for any E > 0, (Received April 9, 1965.)

65T-269. T, G. McLAUGHLIN, 244 Illini Hall, University of Illinois, Urbana, Illinois, Regressive sets with r. e, complement,

In this note we prove, by means of an exceptionally straightforward priority argument, the following Theorem. There exists a recursively enumerable set whose complement is regressive but not retraceable. Corollary I. There exists a hyperimmune regressive set which fs not retrace­ able. Corollary 2, Retraceability is not necessarily preserved by recursive permutations, even in No the case of sets with r.e. complement. (It can be shown that there are 2 retraceable sets with non-r.e. complement whose retraceability is not preserved by all recursive permutations,) The sets of the above theorem can be required to have degree 0; we do not at present know exactly what degrees are possible for them (it seems likely thatJ!:!!.r.e. degrees > 0 are possible). (Received April 9, 1965,)

65T-270, A. H. BRADY, University of Notre Dame,.Computing Center, Notre Dame, Indiana, Solutions of restricted cases of the halting problem used to determine particular values of a non­ computable function.

The integer function l:(k), defined as the maximum number of marks appearing on the output tapes of all k-state binary Turing machines which eventually halt after starting on a blank input tape, has been shown by Rado to be non-computable, A heuristic program was written for a digital com­ puter to "solve" the blank input tape halting problem and determine the value of l:(k) and the asso-

476 elated "shift number", S(k), for k = 2,3, and 4. In the program an efficient method for the generation of all non-redundant k-state machines is coupled with a heuristic method for applying some simple theorems and procedures which can establish that certain machines will never halt because of con­ ditions arising in their history of operation or because of their peculiar construction. It was deter­ mined that S(2) = 6 and that ~(2) = 4, ~(3) = 6 and S(3) = 21. The latter three values corroborate results of Lin and Rado and also hold when the center move is permitted. It is conjectured that

~(4) = 12 (or 13 using another stopping convention) and S(4) = 106. These are known to be lower bounds, and the 4-state case has been reduced to deciding the individual cases of a set of about 6,000 distinct machines. (Received April 12, 1965.)

65T-271. E. S. THOMAS, JR., University of California, Riverside, California, On functions of Baire class 1.

A function f on a topological space X into -I (the unit interval) is of Baire class 1 provided f- 1(F) is a G0 whenever F is closed. The object here is to characterize such functions by properties of their graphs. Definition. A subset G of X X I has property B iff G = n~ 1 Ui where each Ui is open in X X I and, for each i and each x in X, {y E Ij(x,y) E Utl is an open interval in I. Sample result: J!..f:X->I is of Baire class 1 then the graph off has property B; if X is metric the converse

~ A purely topological characterization is obtained in the following special case: If the graph G .!!!_ f: I--> I is connected then f is a Baire class 1 iff G = n~ pi where each U i is open and simply connected in I2• (An example shows that connectedness of G cannot be removed.) (Received April 12, 1965.)

65T-272. SIBE MARDESIC, University of Zagreb, Zagreb, P. P., 314, Yugoslavia. Continuous images of ordered compacta and a new dimension which neglects metric subcontinua.

Definition. Let Y be a Hausdorff compact space and let IDl denote the class of metric compacta. Ind(Y, ID/) = 0 means that all components of Y are metric. Ind(Y ,IDI) ;;!;; n means that, for every closed set F C.Y and every open set U CY, F CU, there exist an open set V, F CV CU, such that Ind(Fr V,IDl) ;;!;; n - 1. Let X denote the image of a totally ordered compact space K under a mapping f: K--> X onto X. Theorem 1. Ind(X,IDI) ~ 1. Corollary. If X contains no metric subcontinua, then

Ind X ~ 1. Theorem 1 depends on: (i) The density s (least number of points in a dense set) is mono­ tone on closed subsets of X, (ii) The frontier of an open Fu - set U C X is separable, i.e., s(Fr U);;!;; )(O, (iii) (L. B. Treybig). If X is connected, then the weight w(X) = s(X). Let u(X) denote the least number of closed subsets of X which separate X between any two disjoint closed subsets

Fl'F2 CX. (iii) Is a consequence of: Theorem 2. u(X);;!;; s(X), and Theorem 3. If Y is connected, then w(Y) ;;!;; u(Y). (Received April 12, 1965.)

65T-273. A. K. STEINER, University of New Mexico, Albuquerque, New Mexico. The topological complementation problem.

The family of topologies on a given set is a lattice. Gaifman (Abstract 61T-l6l, these cNoticeiJ 8(1961), 356) has shown that this lattice is complemented if the set is countable. Also, in

an unpublished paper, he reduces the general problem to finding complements for T1 -topologies. The

477 T 1-topologies constitute a sublattice. Let G be a relation on a set S and T a topology defined as: 0 C S is open if and only if x E 0, (x,y) E G implies y E 0. Topologies constructed in this way will be called principaL The principal topologies also constitute a sublattice. It can be shown that each

T 1-topology has a lattice complement which is principal. Thus, the lattice of topologies on any ground set is complemented. (Received April 12, 1965,)

65T-274. W. A. LaBACH, 909 South First Street, Champaign, Illinois. A lattice-theoretic correspondence.

Theorem. Let H and K be lattices and let u:H-->K and T:K-->H be isotone mappings. Define for x E H, lf= x!TT and for y E K define y* = yTu. Suppose that x;;; x andy & y* for each x E Hand y E K, respectively. Then x-->x is a dual closure operation, y-->y* is a closure operation, and the lattices of closed sets of H and K are lattice isomorphic. This theorem may be applied to algebraic systems and their automorphism groups and endomorphism semigroups. (Received April 12, 1965,)

65T-275. DONALD GREENSPAN, Mathematics Research Center, University of Wisconsin, Madison 6, Wisconsin. ApJ?roximate solution of the exterior Dirichlet problem with application to the estimation of capacity.

Let (~,17, v) be the coordinates of (x,y ,z) under inversion. Define (t 1), v) = cf>(x,y ,z) .(~ 2 +1)2 +v 2 f1 1 2 , where cf>(x,y,z) is given. Then since the Laplace equation maps into the Laplace equa­ tion under inversion, the exterior Dirichlet problem with boundary function cf> is equivalent to an in­ terior Dirichlet problem with boundary function <1>. Standard finite difference methods then apply to yield an approximate solution to the interior problem, and hence in a natural way to the exterior problem. If U is the analytical solution of the constructed interior problem, then the capacity of the boundary S of the given exterior problem is given by c = l(47r) -l ffs,(~ 2+ 1J 2+ vl- 2 '(U(~,1J,V) a;an(~ 2 +17 2 +i)l/Z + (~ 2 +1)2 +i) 1 / 2 au;an)dA'I, where S -->S' under inversion, The quan­ tity c then can be readily estimated by means of the approximate solutio!). described above. (Received April 12, 1965.)

65T-276, GEORGE GLAUB ERMAN, University of Wisconsin, Madison 6, Wisconsin. Existence of Abelian subgroups in Sylow 2-subgroups of finite groups.

LetS be a Sylow 2-subgroup in a finite group G. Let K(G) be the maximal normal subgroup of odd order in G. Theorem. Let x E Z(S). Assume K(G) = 1 and x t/- Z(G). Then there exists an element g of odd order in G and an Abelian subgroup U of S such that g normalizes both U and Cs(U) and g does not centralize x. Corollary. Let xES. Assume K(G) = 1 and x ¢ Z(G). Then S contains some conjugate of x that is different from x and centralizes x. These results yield non­ simplicity criteria for finite groups. For example, J, Alperin and J. Thompson have pointed out that if K(G) = 1 and S is a direct product of generalized quaternion groups, then Z(S) n Z(G) >"' 1, so G is not simple. (Received April 12, 1965.)

478 65T-277. D. M. BLOOM, Brooklyn College, Brooklyn, New York 11210. The subgroups of PSL(3,q) for odd q. Preliminary report.

Let q = pa be a power of an odd prime p and let G be a subgroup of PSL(3,q) of order > 1. Assume G has no normal elementary-abelian subgroup of order > 1. Then G is isomorphic to one of the following types: (1) PSL(3,l), bla; (2) PU(3,pb), 2bla; (3) a group containing the subgroup of type (1) with index 3, if 3(pb- 1)l(q- 1); (4) a group containing the subgroup of type (2) with index 3, if 3(pb + 1)l(q- 1); (5) PSL(2,pb) or PGL(2,pb), with bla and pb > 3; (6) PSL(2,5) with q = ± 1 (mod 10); (7) PSL(2, 7) with q3 = 1 (mod 7); (8) A6, A7, or a group containing A6 with index 2, with p = 5 and a = 0 (mod 2); (9) A6, with q = 1 or 19 (mod 30). All of these cases do occur. All groups isomorphic to a given group of types (1)- (8) are conjugate to each other in GL(3,q), but for the type (9) there are two classes of conjugate subgroups. Explicit representatives of all conjugacy classes can be given. The above strengthens results in preceding abstracts (these cJVoticeiJ, Abstracts 63T-18, 64T-314, 64T-317, 64T-453). (Received April 12, 1965.)

65T-278, V. M. KLASSEN, Virginia Polytechnic Institute, College of Arts and Sciences, Blacksburg, Virginia. Locally compact Hausdorff spaces with open cone neighborhoods.

Let X be a locally compact Hausdorff space, with non-null invert set I(X). Suppose that x E I(X) has an open cone neighborhood U. Then if the inverting homeomorphism for x and U moves x, X is a suspension. If I(X) = {x}, and C 1(U) - U is not connected, then X is a suspension, with sus­ pension points identified. Also, if the inverting homeomorphisms fix x, X is continuously invertible at x. (Received April 12, 1965.)

65T-279. M. C. McCORD, University of Georgia, Athens, Georgia. A singular homology characterization of the 2-sphere among Hausdorff spaces,

For any space X, let Hq(X) denote the q-diJllensional, reduced, singular homology group of X, with integer coefficients. A space X is acyclic if Hq(X) = 0 for all q. Call a Hausdorff space M an S-space if the complement of each point in M is acyclic. Theorem 1. Suppose M is an S-space. Then (a) the complement of every cell in M is acyclic; (b) if Sm is an m-sphere (m ;;; 0) contained

as a proper subset of M, then for every integer q, Hq(M - Sm) ""Hq+m+1 (M). Theorem 2. Let

0 ~ n ~ 2. If M is an S-space and Hn(M) "I 0, then M is ann-sphere. Theorem 3, If M is an n-dimensional polyhedron (n ;;; 0), then M is an S-space if and only if M has the same local and global

homology groups as then-sphere. To give an idea of what can happen in Theorem 2 if the condition that M be Hausdorff is omitted, the following theorem is given. Theorem 4. For every n ;;; 0 there is a space, with only finitely many points, which has the same homology as the n-sphere, and in which the complement of every point is acyclic. (Received April 16, 1965.)

65T-280. R. F. SPRING, Ohio Universfty, Athens, Ohio. Dualities of class 2 groups of exponent p corresponding to 2:-spaces.

Let Md be the class 2 group of maximal order generated by d elements of order p, (p = an odd prime). Any other class 2 group of exponent p with d generators is isomorphic to Md/S, where S is some subgroup of Kd, the commutator subgroup of Md. A subgroup S in the subgroup lattice L(Kd)

479 is called a2:-space if Sis also in L(Ki)' i

65T-281. LAWRENCE FEARNLEY, University of Wisconsin, Madison, Wisconsin. Embeddings of topological products of circularly chainable continua.

The principal results of this paper are the following solutions of the embedding problems for topological products of finite collections of circularly chainable continua and finite collections of planar circularly chainable continua. Theorem 1. The topological product of any collection of n circularly chainable continua can be embedded in Euclidean space of dimension n + 2. Theorem 2. The topological product of any collection of n planar circularly chainable continua can be embedded in Euclidean space of dimension n + I. These results were developed in response to questions raised by R. H. Bing in a research seminar at the University of Wisconsin, 1964. The methods used in establishing Theorem 1 involve a series of preliminary results on the approximation of mappings of nerves which are generalized tori by homeomorphisms which satisfy certain extendability condi­ tions and certain Lipschitz conditions. The establishment of Theorem 2 involves the development of a particular inverse limit characterization of planar circularly chainable continua. It is shown that Theorem 1 is the best possible general result and that, for each positive integer n, Theorem 2 is the best possible result for planar circularly chainable continua. (Received April 19, 1965.)

65T-282. ]. ]. ROSEMAN, New York University, Courant Institute, 251 Mercer Street, New York, New York 10012. A result on Saint-Venant's principle.

Consider a cylinder which has a self-equiliberated stress distribution on one end but is otherwise stress free. The cross section occupies a simply connected two-dimensional domain g with boundary ~. ~E0 and has a minimum radius of curvature R0• At every point P of 9, a circle of radius a ~ l/2R0 can be drawn tangent to ~at P but otherwise lying wholly within Q' Then, if Q is any point in the cylinder at a distance s Q from the stressed end, maxi,j lti/Q) I ~ (K/a312)[U(sQ- a) - U(sQ +a)] where tij is the stress tensor, K is a constant which depends on the geometry of Q and U(s) is the elastic energy in that part of the cylinder consisting of all points whose distance from the stressed end is greater than s. Combined with a result of Toupin (Arch. Rational Mech. Anal., May, 1965), this estimate, which is good up to the boundary, provides a verification of Saint-Venant's Principle. (Received April 19, 1965,)

65T-283. D. W. KAHN, University of Minnesota, Minneapolis, Minnesota. On the stable homotopy of a Moore space.

A study is made of bigraded exact couples over the Steenrod algebra Ap, p an odd prime. Theorem 1: There is exactly one graded Zp -vector space M such that there exists an exact couple (A, B, f, g, h) with (1) AO,O = A0•1 = ZP' AO,i = O,i > 1, A1•0 = 0, his an isomorphism for bidegrees

480 (0,0) and (0, l); (2) B = M 0 z A ; (3) in the associated spectral sequence, E 00 is trivial except for p p bidegrees (0,0) and (0, l). The example of such a couple is furnished by the stable spectral sequence of a Postnikov system for an appropriate Moore space. The interest in M is the Theorem 2: The dual of M, Hom(M,Zp), is the stable homotopy of a space with a single nonvanishing homology group ZP' with p odd. (Received April 20, 1965.)

65T-284. WILLEM KUYK, McGill University, Montreal, Quebec, Canada. Fields with infinite cyclic automorphism group.

The author shows the existence of (non-algebraic) field extensions L/K, the automorphism groups of which are isomorphic to the infinite cyclic group C 00• An example of such an extension is obtained by taking L = K(t0 ,t1,t_ 1,t2 ,t_2 , •.• ), where t 0 is an indeterminate and where ti are chosen in an algebraic closure of K(t0), satisfying the relations tf = ti-l + l. If K is any field of character­ istic p f 2, then it can be shown that any automorphism of L/K is of the form n (n E Z), where is the automorphism defined by the substitution: ti_ 1->ti. (Received April 26, 1965.)

65T-285. N. E. FOLAND and J. M. MARR, Kansas State University, Manhattan, Kansas 66504. Sets with 0-dimensional kernels.

The following theorem suggested by problems of F. A. Valentine (Convex Sets, McGraw-Hill, 1964) is proved. Theorem. Let S be the subset of a topological linear space L with the property that if x, y, z are noncollinear points of S, then there is a unique point p of S such that the three segments xp, yp, zp all lie in S. If S contains a noncollinear triple of points, then the convex kernel K of S consists of a single point and hence is zero dimensional. The proof is obtained by showing that such a set S can not contain a closed polygon of four line segments. (Received April 26, 1965.)

65T-286. R. ]. WARNE, 428 Cedar Street, Morgantown, West Virginia. Extensions of com­ pletely 0-simple semigroups by completely 0-simple semigroups.

For terminology and notation see these c}/oticei) Abstract 622-7, 12 (1965), 326-327. Theorem. An extension V of a completely 0-simple semigroup S by a completely 0-simple semigroup Tis given by a partial homomorphism if and only if under each non-zero idempotent of T there exists at most one non-zero idempotent of S. (Received April 26, 1965.)

65T-287. ]. M. ZELMANOWITZ, University of Chicago, Chicago, Illinois. The endomorphism ring of a torsion-free module.

Let R be a ring which possesses a semisimple (simple) two- sided quotient ring Q. Let M be an arbitrary finitely generated torsion-free left R -module, and set E = HomR (M,M), E* =

HomR (Q ® M, Q ® M). Theorem. E * is a right and left quotient ring of E. It follows from a theorem of A. W. Goldie that E is a semiprime (prime) ring with the ascending chain condition on annihilator ideals and direct sums of ideals. This generalizes the result of E. H. Feller and E. W. Swokowski IJ. London Math. Soc., 1964], who proved the theorem for the case where R is an integral domain. (Received April 27, 1965.)

481 65T-288. T. S. MOTZKIN, Los Angeles, California 90024. Transposed complementarity and regular polytopes,

If the rows of an r by r + s matrix A and of an s by r + s matrix B form bases of complement­ ary subspaces then their columns are said to form transposedly complementary point systems, a concept important also in the theory of systems of linear inequalities. In particular, if the columns of A form the vertices of a regular polytope, and if one row of B consists of ones, then the columns of the rest of B are said to form a transposedly regular point system in d-space, d = s - 1. Such a system consists of d + 1 double points, or else all its points are distinct. The number of affinely . k-1 different systems is 3 ford= 8, 16, 19, 115, 595 and ford= 2 - k, k ~ 4, and 2 for all other d .6: 2. (Received April 2 7, 1965.)

65T-289. D. G. BOURGIN, University of Illinois, Urbana, Illinois. The Kakutani problem.

For prime n the circumscribing cubes, D, for a convex body in R n constitute a set with n - 2 non trivial homology groups at least. In particular dim D ~ (n - 1)(n - 2)/2. There is always a

subset of circumscribing cubes of the same edge length with dim D' ~ n(n - 3)/2 and the lower bound can be attained, (Received April 27, 1965.)

65T-290. K. B. O'KEEFE and E. S. O'KEEFE, 2428 Boyer Avenue East, Seattle, Washington

98102. The differential ideal [y 1, ... ,ynJ•

In these c){oticei), A. P. Hillman, D. G. Mead, K. B. O'Keefe and E. S. O'Keefe have announced a sufficient condition for a monomial P to be in a differential ideal generated by a product.

The condition is stated in terms of the weight w of P and the signature of P, D = (d1, ... ,dn). In the special case of a Ritt algebra Rfyl''"'Yn]' there is a function f, whose arguments are d 1(P), ... ,dn(P), n;?; 2, such that if w(P) < f(d 1(P), ... ,dn(P)), then P E [y 1, ... , yn). This condition is found to be necessary as follows: If P E fy 1, ... , Yn], then for some factor Q of P, w(Q) < f(d1(Q), .. .,dn(Q)). The proof qses the recursive property f(d1, ... ,dn)= mino~p~mindi(f(d1- p, d2- p, ... ,dn-1- p) + pdn). This property is known to hold for all combinations of (d1, ... ,dn). (Received April 28, 1965,)

65T-291. H. J. KEISLER, University of Wisconsin, Madison, Wisconsin 53706. First order properties of pairs of cardinals. Preliminary report.

Let L be a first order logic with an identity symbol, predicates P, Q ,. .. , and individual variables vi'wi' xi' yi' zi' i < w. Let f(v0), g(v0) be formulas of L in which only v0 is free. Let f(A) = {!!:.!!satisfies f(v0) in the model A}. lXI =power of X. LetS be the set of all sencences of L which hold in every model A such that w ~ lg(A) I < lf(A) 1. Vaught has shown (Fund. Math. 54, 303-304) that S is recursively enumerable, and has proposed the problem of finding a specific set of axioms for S. We use the abbreviations xn for x0, ... ,~ and A~i for h0& ... &hm. For each n < w, let hi(xn,yn), i < w, be a list of all formulas of L in which only x0 ,. .. ,xn' y0, ... ,yn occur. Theorem. A set of axioms for S consists of the following sentences of L, where m, n < w.

(1) vxn3v0(g(v0)&Anv0 t- xi). (2) 3v0 Vw0y0 3x0z 0 ... Vwnyn3xnznf(v0)&Anv0 t- xi &[Ang(yi) ___, Am(h.(xn,yn) t--th.(wn,zn))]. (Received April 28, 1965.) 1 1

482 When the job is too big for this ... and this is too big for the job

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483 New Books in the INTERNATIONAL SERIES OF MONOGRAPHS ON PURE AND APPLIED MATHEMATICS General Editors: I. N. Sneddon, M. Stark, and s. Ulam Available now ... Volume 74: RADIATIVE TRANSFER ON Volume 65: A COLLECTION OF PROBLEMS DISCRETE SPACES OF MECHANICS By Rudolph W. Preisendorfer, By 1. V. Mcshcherskii University of California, San Diego One thousand three hundred and sixty-three This stimulating, original work shows how, by problems arc discussed in this newly translated using discrete spaces instead of a continuum in volume. They cover a wide range of topics, radiative transfer problems, practical numerical including problems on analytical mechanics, solutions may be obtained without distorting the dynamics of systems with variable mass, and radiance function or the equation of transfer. the theory of stability of motion. The continuous and the discrete theory of radia­ 520 pages, $12 ..50 tive transfer are developed from the notion of the "natural mode" of solution, a geometric con­ cept, from the local interaction principle, a vec­ Volume 64: COLLECTION OF PROBLEMS ON A tor space concept, and from an axiomatic point COURSE OF MATHEMATICAL ANALYSIS of view. The connections between radiative transfer theory and the main body of electro­ By the late G. N. Berman magnetic theory are indicated and the principle Written as a companion to A Course of Mathe­ of interaction which is basic to all branches of matical Analysis by A. F. Bermant, this inter­ the theory is discussed. 462 pages, $20.00 esting work has many valuable applications for students of mathematical analysis. It supplies Volume 71: SOME PROPERTIES OF POLYHEDRA instructive examples of problems in areas such IN EUCLIDEAN SPACE as functions, limits, functions and curves,- defi­ nite and indefinite integrals, improper integrals, By V. J.D. Baston., University of Southampton series, functions of several variables, multiple This pioneering work considers a special prob­ integrals and iterated integration, differential lem in three-dimensional space derived from equations, and elements of the theory of fields. Crum's Problem. A set of n bodies in three­ 275 pages, $12.50 dimensional space is said to possess Crum's Property if each pair of them have a common boundary of positive area but no pair of them Volume 50: VARIATIONAL METHODS IN overlap. The inquiry seeks the maximum number of tetrahedra possessing such a J:>roperty and MATHEMATICAL PHYSICS shows, by an ingenious argument, that the num­ By S. G. Mikhlin, Leningrad ber is either eight or nine, and in all probability This book describes applications of the so-called is eight. 212 pages, $10.00 energy method to problems involving the use of variation methods, and to the calculation of Volume 70: LECTURES IN GENERAL ALGEBRA eigenvalues. The nature of convergence and the By A. G. Kurosh estimation of error are dealt with thoroughly. A broad introduction to modern algebra which The problems are mainly in the field of elasticity, surveys a number of essential topics illustrating but the methods can be generalized. different aspects of modern algebra as well as 616 pages, $14.50 other branches of mathematics. Topics covered include relations, groups and rings, universal algebras, lattices, o_perator groups and rings, Volume 46: FUNCTIONAL ANALYSIS IN modules, linear algebras, ordered and topologi­ NORMED SPACES cal groups and rings, and normed rings. By L. V. Kantorovich and 364 pages, $7.50 G. P. Akilov, Moscow Volume 69: MATHEMATICAL ANALYSIS: Covers the main facts of the theory of normed spaces as developed by Riez and Banach, and Functions, Limits, Series, Continued Fractions also takes later work into account. It covers the Edited by L. A. Lyusternik and theory of normed spaces, the theory of opera­ A. R. Yanpol'skii, et al. tions, and the theory of functional equations. A basic work on the fundamentals of analysis Considerable attention is also given to non-linear which discusses general problems of the theory operations and equations, and, in addition to the of continuous functions of one and several vari­ general theory, concrete functional spaces and ables and its geometrical basis, the theory of o_perations are discussed at length. In ~articular, limiting values for sequences of numbers and there is a detailed treatment of differentiable vectors, and also the theory of numerical series functions of several variables introduced by S. L. and series of functions and other analogous infi­ Sobolev. Journal and monobrraph references are nite processes, particularly infinite continued supplied at the end of the book fractions. 404 pages, $10.00 456 pages, $20.00 Order from your bookseller, or directly from: THE MACMILLAN COMPANY, 60 Fifth Avenue, New York, N. Y. 10011 Distributors of Pergamon Press Books

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Published by Springer-Verlag Berlin · Heidelberg · New York: 3. Heft: Stable Homotopy Theory Lecture By J. Frank Adams, Department of Mathematics, University of Manchester (Lectures delivered at the University of California at Notes Berkeley 1961. Notes by A. T. Vasquez.) • In English. IV, 74 pages DIN A 4. 1964. Stiff paper tn bound US $1.95 4. Heft: Groups of Homotopy Classes (Rank formulas and homotopy-commutativity) Mathematics By M. Arkowitz and C. R. Curjel, Dartmouth College, Hanover, N. H., University of Washington, Seattle, Wash., Forschungsinstitut fur Mathematik, Eidgenos­ An informal series of special lectures, seminars and sische Technische Hochschule, Zurich reports on mathematical topics In English. IV, 36 pages DIN A 4. 1964. Stiff paper Edited by A. Dold, Heidelberg and B. Eckmann, Zurich bound US $1.20 This series aims at speedy and informal high level 5. Heft: Cohmologie Galoisienne information on new developments in mathematics Par Jean-Pierre Serre, College de France, Paris (Cours au College de France, 1962-1963, Seconde Published earlier: edition, 1964) En fran~ais. VIII, 214 pages DIN A 4. 1964. Broche 1. Heft: Seminar iiber Funktionen-Aigebren us $4.50 Von John Wermer, Professor an der Brown University, Providence, R. I. In preparation: (Gerhalten an der Eidg. Technischen Hochschule, Forschungsinstitut fUr Mathematik, Zurich, Winter­ Seminar on control theory. By Felix Albrecht semester 1963/64) Cohomologie und kommutative Gruppenmannigfaltig­ IV, 30 Seiten DIN A 4. 1964. Stiff paperbound US $1.00 keiten. Von P. Cartier Halbexakte Funktoren. Von A. Dold 2. Heft: Cohomologie des espaces localement com• pacts d'apres J. leroy lectures on injective modules and quotient rings. By Par Armand Borel, Institute for Advanced Study, C. Faith Princeton, N. J. Eine Termlogik mit Auswahloperator. Von H. Hermes de Topologie algebrique (Exposes faits au Seminaire Differentialgeometrie im Grossen. Von H. Klingenberg de L'Ecole Polytechnique Federale au printemps 1951, Troisieme Edition, 1964) lectures on spherical harmonic. By C. Muller

En fran~ais. IV, 93 pages DIN A 4. 1964. Broche US lectures on statistical mechanics and the kinetic theory $2.30 of gases. By C. Truesdell

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485 A new series of reprints of classics in mathematics Bibliotheca Mathematica Teubneriana

Uber die Theorie des Kreisels Available Spring 1965 By Felix Klein and Arnold Sommerfeld (1st edition} 4 parts in 1 vol., xx, 964 pp., 143 ill us., cloth (Reprint 1965) .... $20.00 Transzendenz von e und 1r Available Spring 1965 By Gerhard Hessenberg (1st edition, 1912} x, 106 pp., cloth (Reprint 1965} ... $4.50 Vorlesungen uber die Theorie der automorphen Funktionen By Robert Fricke Available Spring 1965 and Felix Klein (1st edition 1897, 1912), 2 vols., cloth (Reprint 1965} ... $27.50 Elementare Mechanik Available July 1965 By Georg Hamel (2nd edition, 1922}. xvii, 634 pp., 265 illus., cloth (Reprint 1965) ...... $15.00 Vorlesungen uber Geschichte der Mathematik Available July 1965 By Moritz Benedikt Cantor (1900-1908} 4 vols., cloth (Reprint 1965} ...... $100.00 Vorlesungen uber die Theorie der elliptischen Modulfunktionen By Felix Klein Available Fall1965 and Robert Fricke (1890-92)2 vols., cloth (Reprint 1965} ...... $35.00 Lehrbuch der Kristallphysik Available Fall1965 By Woldemar Voigt (1st edition, 1910} xxiv, 964 pp., 213 illus., cloth Reprint 1965) ...... $25.00 Geschichte der Mathematik im 16. und 17. Jahrhundert Available Fall1965 By H. G. Zeuthen (1st edition, 1903), viii, 434 pp., cloth (Reprint 1965} ...... $12.00 Die Mathematik im Altertum und im Mittelalter Available Fall1965 By H. G. Zeuthen (1st edition, 1912), iv, 95 pp., cloth (Reprint 1965} .. $4.00

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ALGEBRA: A MODERN INTRODUCTION J. L. Kelley, University of California at Berkeley

Based on the paperback, Introduction to Modern Algebra, this text - with a larger format and greatly revised and expanded text- provides a modern pre­ calculus course in algebra for college students. May, 1965, 326 pp., about $6. 75.

CALCULUS WITH ANALYTIC GEOMETRY Albert G. Fadel/, State University of New York College at Buffalo This text for the first-year calculus sequence is designed for students with a solid background in high school mathematics. The student learns to under­ stand mathematical ideas through analogies close to his experience, and to understand the theory of limits by way of the mapping diagram and the neighbor­ hood concept. These devices, together with excellent illustrations, precise yet informal exposition, and numerous examples, facilitate the transition from elementary to advanced mathematics. 1964, 733 pp., $9.75.

and now available-pre-prints of Volume II VECTOR CALCULUS and DIFFERENTIAL EQUATIONS Part 1, 308 pp., $3.50 Part 2, 300 pp., $3.50 Vector Calculus presents the usual content of third-semester calculus, while the second pre-print volume offers a differential equations course with strong connections to the previous calculus sequence.

VAN NOSTRAND MATHEMATICAL STUDIES In paperback, edited by Paul R. Halmos and Frederick W. Gehring

#1 LECTURES ON BOOLEAN ALGEBRA- Paul R. Halmos #2 LECTURES ON ELLIPTIC BOUNDARY VALUE PROBLEMS- Shmuel Agmon #3 NOTES ON DIFFERENTIAL GEOMETRY- Noel J. Hicks

Summer, 1965 ...

TOPOLOGY AND ORDER- Leopolda Nachbin NOTES ON LOGIC- Roger c. Lyndon EXERCISES IN SET THEORY- L. E. Sigler NOTES ON SPECTRAL THEORY- Sterling K. Berberian

487 IMPORT ANT NEW TEXTS LIE ALGEBRAS I IN MATHEMATICS AND FROM McGRAW-HILL LIE GROUPS University HOW TO GAMBLE IF YOU MUST: Lectures given at Harvard Serre. •Inequalities for Stochastic Processes by Jean-Pierre 253 PAGES PAPERBOUND $3.95* By LESTER E. DUBINS, University of Calif­ ornia, Berkeley, and LEONARD J. SAVAGE, Yale University. McGraw-Hill Series in ALGEBRAIC Probability and Statistics. Off Press. FUNCTIONS A unique semi-monograph on some ad­ Lectures given at Columbia University vanced aspects of probability theory and by Serge Lang. stochastic processes, using the art of 48 PAGES PAPERBOUND $1.95* gambling-and risk taking in general­ as a point of departure. *Prepaid orders will be gi•e11 a 20% dis

We specialize in reproducing out-of­ Send for examination copies fodoy print books, as well as typewriter com­ McGRAW-HILL BOOK COMPANY position. Let us quote on your next 330West42ndStreetjNewYork, N.Y. 10036 printing.

488 from ADDISON- WESLEY ...

FUNCTIONS OF SEVERAL VARIABLES BY WENDELL H. FLEMING, Brown University Intended for courses in advanced calculus, this text presents a systematic treatment of differential and integral calculus for functions of several variables. The standard topics are presented, but the treatment differs from the traditional one in several important respects. For example, the text presents the elements of the Lebesgue theory of integrals in some detail. The introduction of convex and concave functions makes possible a more thorough treatment of extrema. Instead of the traditional vector analysis, the text offers a treatment of differential forms and manifolds in n dimensional euclidean space. 337 pp., 91 illus., $9.75 LINEAR GEOMETRY BY RAFAEL ARTZY, Rutgers, The State University Written at the advanced undergraduate level, this book is intended for courses in higher geometry. The emphasis is generally theoretical and will be of prime interest to students majoring in mathematics. However, prospective teachers and students of the physical sciences will also find the book valuable. In Press INTRODUCTION TO NUMERICAL ANALYSIS BY CARL-ERIK FROBERG, University of Lund, Sweden Requiring a background of elementary calculus and differential equations, this text is intended for introductory courses in numerical analysis. In the selection of topics, stress is placed on modern and efficient methods, and, to a large extent, the book's content reflects the important role played by electronic computers in recent years. A brief account of the theory of matrices is presented, while applications of matrix methods have been treated in considerable detail. 340 pp., 22 illus., $8.95 DISTRIBUTIONS, COMPLEX VARIABLES, AND FOURIER TRANSFORMS BY HANS J. BREMERMANN, University of California, Berkeley The basis of this graduate level text is Schwartz's theory of distributions. It is developed from its very beginning to the point where many of the important theorems are proved. The book contains several new and unpublished results pertaining to the representation of distributions as boundry values of analytic function. In Press ALGEBRAIC NUMBERS BY SERGE LANG, Columbia University The purpose of this book is to give an exposition of the classical and basic algebraic and analytic number theory. In a certain sense, the plan of the book is still that used more or less by Hilbert in his Bericht, although of course both the algebraic and the analytic aspects of number theory have been updated (and the class field theory is omitted). 176 pp., 2 illus., $7.00 COLLECTED PAPERS OF EMIL ARTIN EDITED BY SERGE LANG, Columbia University AND JoHN TATE, Harvard University Intended as a memorial to the late mathematician, this volume contains forty-eight papers, the complete collected works, reprinted in the original language of publication. Included is the here­ tofore unpublished address given for Hilbert's one-hundredth birthday in Gottingen. In Press WRITE FOR APPROVAL COPIES ...... ADDISON-WESLEY PUBLISHING COMPANY, INC. Reading, Massachusetts

489 Applied Mathematicians Operations Operations Analysts Applied Physicists Research Analysis CAREER APPOINTMENTS Opportunities at Boeing

The Boeing Company, world leader in jet trans­ portation, has immediate openings for opera­ tions research analysts. These are long-range career opportunities involving the study of a wide variety of business, industrial and military problems. Responsibilities include consulting with all levels of management in matters involving problem formulation through to implementation. Engineering and Military Studies Assignments involve developing synthesized models to evalu­ ate relative importance of military systems in their operational environment at various levels Response of effectiveness and economic value. Business and Industrial Studies Responsibilities include financial and economic analysis, market research, inventory control and production planning, new product development and long­ Booz•AIIen Applied Research today serves many of the nation's top gov­ range planning. ernmental, military and private or­ Requirements for these positions include an ganizations. Responding swiftly to in operations research, or in their most complex technical prob­ advanced degree lems, we have worked to solve them mathematics, statistics or physical sciences with a blend of interdisciplinary tal­ with a knowledge of operations research. Addi­ ents, modern analytical techniques, tional experience in computer programming and and wide-ranging research experi­ ence. economic analysis is desirable. This has proven a most success­ Salaries are competitively commensurate with ful combination, both for our clients your experience and educational background. and for BAARINC. A good indicator of our success is our current need Travel and moving allowances are paid to newly for new talent to respond to still hired personnel. Boeing is an equal opportunity greater challenges ... in astronau­ employer. tics, communications, transporta­ tion, computer technology and naval These assignments are with Boeing's Airplane warfare, to cite just a few. Creative, Division facilities at Renton, (near Seattle), perceptive scientists and engineers Washington, in the uncongested, evergreen are invited to investigate these new mild avenues of career opportunity. Pacific Northwest, noted for year-round Please send your resume to Mr. climate, nationally famous recreational facili­ Robert ._H. Flint, Director of Profes­ ties, excellent schools and housing, and health­ sional Appointments. ful outdoor Western living for the entire family. BOOZ•ALLEN Send your resume, today, to Mr. Thomas Shep­ APPLIED RESEARCH Inc. pard, Airplane Division, The Boeing Company, 135 South LaSalle Street P. 0. Box 707- BCA, Renton, Washington 98055. Chicago, Illinois 60603 New York • Washington • Cleveland Chicago • Los Angeles An equal opportunity employer BOEING AIRPLANE DIVISION

490 IDA

Where two worlds meet The nature of the problems assigned to the Institute for Defense Analyses provides our staff members with the best of two worlds-one of intellectual challenge, the other of national purpose. In one world, the analysis of military systems and processes plunges us deep into development of new and penetrating methodological techniques, an exciting and fulfilling activity. At the same time, our staff is squarely in the world of national defense. Offices from which many of the nation's basic decisions must come-the Joint Chiefs of Staff, the Director of Defense Research and Engineering, and others-look to IDA for advice and research on which to base these decisions. We presently need scientifically trained people with grounding in operations research and systems analysis. If you require both challenge and purposefulness, write toT. J. Shirhall, Institute for Defense Analyses, 400 Army-Navy Drive, Arlington, Virginia 22202 (near the Pentagon). An equal opportunity employer sponsored by twelve of the nation's leading universities.

491 APPLIED MATHEMATICIAN American Optical Company has a career position 2 I 4 3 I 2 3 4 for an Applied Mathematician to conduct research 6 5 8 7 5 6 7 8 in methods of optical design. Candidates should 10 9 12 II 9 10 II 12 have an advanced degree and an interest and ex­ 13 14 15 I 13 14 15 rl perience in numerical analysis. Some knowledge of optical design is desirable but not essential. The position is at the Corporate Research Labo­ ratory in Southbridge; Massachusetts. This man If you thrive will have as collaborators established specialists in optical design, diffraction theory, thin films, lasers, on challenge ... and other areas of modern optics. The company activities supply contacts with real optical problems The sea has always challenged extending from microscopes to the largest and man. most sophisticated space optics systems. Throughout history, bold and creative imagination has met For further information, please contact the challenge by finding new Donald C. Jones, Research Laboratory Manager. ways to control the use of the sea for maritime advantage. (An Equal Opportunity Employer) Today, the age-old challenge remains. New concepts of ships and weapons design, of tactics • AMERICAN OPTICAL COMPANY and strategy are still needed to SOUTHBRIDGE, MASSACHUSETTS 01551 assure national security now and in the future. Discovery and evaluation of these new concepts are among the tasks of the Center for Naval Analyses of The Frank­ lin Institute. CNA is a private scientific organization engaged in opera­ tions research and broad-based studies for the United States Navy. STANFORD RESEARCH INSTITUTE A few CNA staff appointments Mathematical Services Department are available to operations ana­ located on the beautiful lysts, mathematicians, physical Monterey Peninsula in California scientists, and research engi­ has IMMEDIATE OPENINGS for neers of superior competence. For more information, write: • MATHEMATICIANS To program scientific calculations tor computing systems Director utilizing Burroughs 5500 and the IBM 7090 computers and data link transmission. Some problems formulation and CENTER FOR NAVAL ANAlYSES analysis involving mathematics and/or mathematical Dept. AM statistics. Participation in selection of new, large scale 1401 Wilson Blvd., Arlington 9, Va. computing systems. Requires at least a BS, preferably in mathematics or mathematical statistics ... plus 1 or more years experience in computor programming. • RESEARCH MATHEMATICIANS Will involve the use of mathematics and/or mathematical statistics in both formulation and solution of a variety CENTER FOR NAVAL ANALYSES of problems arising in physics, operations research, and OF THE FRANKLIN INSTITUTE design and evaluation of large-scale field experiments. Requires PhD, preferably in mathematics or mathematical OEG • OPERATIONS EVALUATION GROUP statistics plus several years experience in the formula· INS • INSTITUTE OF NAVAL STUDIES tion and solution of problems. NAVWAG • NAVAL WARFARE ANALYSIS GROUP Interested applicants are An equal opportunity employer invited to submit resumes to: R. P. Cantu STANFORD RESEARCH INSTITUTE 333 Ravenswood Menlo Park, California -an equal opportunity employer-

492 Computer Science Openings for

Numerical Analysts

Research in the area of automatic problem solving. Positions open at Ph. D. and M.S. level. Computer oriented experience desirable.

• University Research Laboratory atmosphere • Salaries competitive with industry • Westchester location convenient to New York City

Send resume to:

Mr. M.G. Fritz Hudson Laboratories Columbia University 145 Palisade Street Dobbs Ferry, N. Y.

493 MONASH UNIVERSITY Asymptotic Analysis and Approximations Clayton, Melbourne, Australia to the Functions of Mathematical Physics PROFESSOR OF APPLIED MATHEMATICS A Two-Week Short Course (July 19-29, 1965) -Applications for a full professorship in applied at Midwest Research Institute. Write: mathematics are invited from mathematicians Mr. Y. L. Luke, Senior Advisor for Mathematics whose interests lie preferably in one or more of 425 Volker Blvd., Kansas City, Mo. 64110 the fields of partial differential equations, fluid mechanics, plasma physics, and astronomy. Monash University accepted its first students in 1961 and has at present established six full professorships in the Department of Mathe­ matics. Interests of members of the department THE UNDECIDABLE-&.ic Papers include semigroups, groups, functional analysis, on Undecidable Propositions, Unsolooble Prob­ probability, and stochastic processes. The lems and Computable Functions; by Godel, university has installed a CDC 3200 and a Church, Turing, Rosser, Kleene, Post, incl. 4000-word Ferranti Sirius computer. An ener­ first publication of Gtidel's 1934 lectures. getic postgraduate programme is under way and Martin Davis, ed. 440pp. $7.95. Rauen Press there is opportunity to put into practice new thinking about undergraduate teaching. Enquiries about the department may be addressed to the Chairman, Department of Mathematics. For opportunities in Salary: £A5200. Superannuation on F.S.S.U. basis. 0. R. ANALYSIS Closing date: 15th July 1965. Information on PROGRAMMING conditions of appointment and application pro­ Contact: Mr. William O'Connell cedure is available from the Registrar of the Scientific, Engineering & Executive Consultants University. 24 Crescent St. Waltham, Mass.

INDEX TO ADVERTISERS Addison-Wesley Publishing Company . 489 American Optical Company. . 492 W. A. Benjamin, Inc...... 488 The Boeing Company ...... 490 Booz ·Allen Applied Research Inc. . 490 Center for Naval Analyses ...... 492 Cushing-Malloy, Inc...... 488 Hudson Laboratories of Columbia University . 493 Institute for Defense Analyses . '491 Johnson Reprint Corporation . 486 McGraw-Hill Book Company . 488 The Macmillan Company .. . 484 Midwest Research Institute . 494 Monash University . . 494 Raven Press ...... 494 S.E.E.C ...... 494 Springer-Verlag New York Inc. . 485 Stanford Research Institute .. . 492 D. Van Nostrand Company, Inc. . 487 Wyle Laboratories ...... 483

494 ADVANCE REGISTRATION, DORMITORY HoUSING RESERVATION AND ENTERTAINMENT TICKET FORM Ithaca, New York Meeting August, -September, 1965 Please fill out the form below for Advance Registration, Dormitory Housing Reservation and tickets for the three recreational events listed in the Preliminary Announcement. Return the form with your payment no later than August 1, 1965 to: Mathematical Meeting Arrangements Section American Mathematical Society Providence, Rhode Island 02904

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REGISTRATION: The registration fees will be as follows: Member $2.00 Member's family 0.50 for the first such registration and no charge for additional registrations. Student No fee Others $5.00 HousiNG: (Persons wishing to reserve motel or hotel accommodations should write before July 1 directly to Mr. B. Ander­ son, Meadow Court, 529 South Meadow Street, Ithaca, New York 14850. Rates are given in the Preliminary Announcement.) All reservations for dormitory accommodations are to be accompanied with payment for one night. Reservations will be confirmed. Rooms will be available from Saturday, August 28 to Saturday, September 4. Rooms must be vacated by 9:00 A.M. on September 4. There are no double beds or private baths in the dormitories. Cots can be put in a double room for an older child (but not an adult) at a cost of $2.00 per night. Dormitory housing rates are as follows: Single $5.50 per day Double 3.50 per day per person ENTERTAINMENT: SIAM Beer Party-Monday, August 30, 8:00 P.M.-Tickets $1.25 each. Chicken Barbecue--Wednesday, September 1, 6:00 P.M.-Tickets $2.75 each; children under 12, $1.50 each. Excursion-Coming Glass Works-Thursday, September 2-Round trip bus tickets at $1.75 per person.

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TRANSLATIONS OF MATHEMATICAL MONOGRAPHS Additive Theory of Prime Numbers byL. K. Hua

A detailed exposition, from the beginning, of the W aring-Goldbach problem of representing positive integers as sums of a given number of kth powers of primes. The work ofVinogradov on trigonometric sums is supplemented by many original results of the author, including his improved form of the Vinogradov mean-value theorem and his extension of the Waring problem to the representa­ tion of integers as sums of polynomials with integral coefficients. Volume 13 192pages List Price $11.00 Member Price $8.25 Special Chapters in the Theory of Analytic Functions of Several Complex Variables by B. A. Fuks

The five "special chapters" deal respectively with: approximation of functions and domains, the fundamental problems of Cousin and Poincare, domains which are convex in the sense of Hartogs, holomorphic extension of domains, and biho­ lomorphic mappings. Emphasis is placed on the discoveries of the last two decades, and in particular on the methods of holomorphic extension of domains which have recently become important in quantum field theory. Volume 14 Approximately 400 pages List Price $15.90 Member Price $11.93 Order from AMERICAN MATHEMATICAL SOCIETY P.O. Box 6248, Providence, Rhode Island 02904