OF THE

AMERICAN MATHEMATICAL SOCIETY

Edited by Everett Pitcher and Gordon L. Walker

CONTENTS

MEETINGS Calendar of Meetings .•••••••••...•••..••...... ••••... 582 Program for the June Meeting in Portland, Oregon ..•.•••..••.. 583 Abstracts for the Meeting- Pages 613-622 PRELIMINARY ANNOUNCEMENT OF MEETING •••••••••••••••••••••• 588 AMS MATHEMATICAL OFFPRINT SERVICE. • • • • • • • • • • • • • • • • • • • • • • • 595 THE CONFINEMENT OF A RUSSIAN MATHEMATICIAN •.••••••••••••••. 597 MEMORANDA TO MEMBERS Resolution on the Draft •••••••..•...••••••••.••..•..... 587 Different Categories for By-Title Abstracts ••...•....•...... 587 Request for Names of Visiting Foreign Mathematicians .•...•.... 587 NEWS ITEMS AND ANNOUNCEMENTS ••••••••••••••• 594, 596, 601, 607, 610 PERSONAL ITEMS •••••••••••••••••••••••••••••••••••••••••• 606 LETTERS TO THE EDITOR •••••••••••••••••••••••••••••••••••• 608 ABSTRACTS OF CONTRIBUTED PAPERS •••••••••••••••••••••••••• 611 INDEX TO ADVERTISERS 667 DORMITORY RESERVATION FORM • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 668 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 cffoliuv 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*

658 August 26-30, 1968 Madison, Wisconsin July 1, 1968 (73rd Summer Meeting) 659 October 26, 1968 Baltimore, Maryland Sept. 6, 1968 660 November 8-9, 1968 Clemson, South Carolina Sept. 25, 1968 661 November 16, 1968 Riverside, California Sept. 25, 196E 662 November 29-30, 1968 Evanston, Illinois Sept. 25, 1968 663 January 23-27, 1969 New Orleans, Louisiana Nov. 8, 1968 (75th Annual Meeting) August 25-29, 1969 Eugene, Oregon (74th Summer Meeting) January 22-26, 1970 Miami, Florida (76th Annual Meeting) August 1970 Laramie, Wyoming (75th Summer Meeting) January 21-25, 1971 Atlantic City, New Jersey (77th Annual Meeting) *The abstracts of papers to be presented in person at the meetings must be received in the Head­ quarters Offices of the Society in Providence, Rhode Island, on or before these deadlines. The dead­ lines also apply to news items. The next two deadlines for by-title abstracts will be June 24, 1968, and August 30, 1968,

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The cflolictAJ of the American Mathematical Society is publisiied by the Society in January, February, April, June, August, October, November and December. Price per annual volum!\ is $12.00. Price per copy $3.00. Special price for copies sold at registration desks of meetings of the Society, $1.00 per copy. SUbscriptions, orders for back numbers (back issues of the last two years only are available) and inquiries should be addressed to the American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02904. Second-class postage paid at Providence, Rhode Island, and additional mailing offices. Authorization is granted under the authority of the act of August 24, 1912, as amended by the act of August-4, 1947 (Sec. 34,21, P. L. and R.). Accepted for mailing at the special rate of Postage provided for in section 34,40, paragraph (d). ·

Copyright ©, 1968 by the American Mathematical Society Printed in the United States of America

582 Six Hundred Fifty-Seventh Meeting Reed College Portland, Oregon June 15,1968

The six hundred fifty- seventh meet­ ACCOMMODATIONS ing of the American Mathematical Society will be held on Saturday, June 15, 1968, Dormitory space will be available at Reed College in Portland, Oregon, in at the rate of $2.75 per single room per conjunction with meetings of the Pacific night. Reservations for dormitory rooms Northwest Section of the Mathematical should be sent to ProfessorBurrowes Hunt, Association of America and the Society Department of Mathematics, Reed College, for Industrial and Applied Mathematics. Portland, Oregon 97202. The Association and SIAM will meet on The first two of the following hotels Friday and Saturday, June 14 and 15. and motels are on the east side of Port­ By invitation of the Committee to land (as is Reed College), while the last Select Hour Speakers for Far Western three are in downtown Portland. Sectional Meetings, there will be two hour Rose Manor Motel addresses at this meeting. Professor Single $ 8.00 Double $10.50 William G. Bade of the University of Cali­ Twin 11.50 fornia, Berkeley, will address the Society (These are commercial rates.) at 11:00 a.m. on Saturday in Room 314, Eliot Hall. The title of his lecture is Sheraton Hotel "Sources of equicontinuity in functional Single $ 9.00 - $15.00 analysis." Professor W. A. J. Luxemburg Double 15.50 - 19.00 of the California Institute of Technology (Slightly lower rates are available to fac­ will present a lecture entitled "On some ulty members with a Sheraton ID card.) recent developments in the theory of Riesz Benson Hotel spaces" at 2:00 p.m. on Saturday in Single $12.00 Double $16.00 Room 314, Eliot Hall. Twin 17.00 There will be sessions for con­ (These are faculty rates, requiring only tributed papers at 9:15a.m. and 3:15p.m. personal identification.) in Rooms 200, 314, and 416, Eliot Hall. Late papers may be added to the program. Heathman Hotel Information on late papers and program Single $ 7.50 Double $11.00 changes will be available at the registra­ Twin 12.50 tion desk. (These are faculty rates, requiring only personal identification.) REGISTRATION Hilton Hotel Single $ 9.00 Double $16.00 The registration desk for the meet­ (These are. faculty rates, requiring only ing will be located at the main entrance to personal identification.) Eliot Hall. It will be open from 3:00 to 6:00 p.m. on Thursday, June 13; from MEALS 8:30 a.m. to 5:00 p.m. on Friday, June 14; and from 9:00 a.m. to 12:00 noon on Luncheon will be available in the Saturday, June 15. College Commons on the days of the meet-

583 ings. If enough dormitory reservations are Persons arnvmg by automobile received, breakfast and dinner may also be from the north on Interstate 5 should fol­ served in the Commons. Information con­ low signs toward Oregon City, then break cerning local restaurants will be available off right at the sign to East Moreland and at the registration desk. Reed College. Automobiles from the south on Interstate 5 should exit across the TRAVEL Ross Island Bridge and follow Powell to S. E. 39th, then right on 39th to Wood­ Portland is served by the major stock and right to Reed. Persons driving west coast airlines and railroad lines. from the east on the Columbia River Taxi service from the railroad station to Freeway should exit at about 40th, take a Reed College costs approximately $3,00, sharp right on 46th, right to Gleason, and from the airport the cost is about right to 39th, then left to Woodstock, and $6.50, There is an airport limousine to right to Reed College. all hotels.

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

SATURDAY, 9:15A.M. Session on Topology, Room 314, Eliot Hall 9:15-9:25 ( 1) Some generalizations of the contraction mapping theorem. Preliminary re­ port Professor B. T. Sims, Eastern Washington State College (657-25) 9:30-9:40 (2) On a problem of Borsuk, Preliminary report Professor N. R. Gray, Western Washington State College (657-14) 9:45-9:55 (3) On three problems of Franklin and Wallace Professor L. E. Ward, Jr.•, and Mr. E. D. Tymchatyn, University of Oregon (657-6) 10;00-10:10 ( 4) Regularity for convergence structures Mr. B. V. Hearsey, Washington State University (657-21) (Introduced by Dr. D. C. Kent) 10;15-10:25 ( 5) A representation theorem for uniform convergence structures Professor D. C. Kent, Washington State University (657-16)

SATURDAY, 9:15A.M. Session on Number Theory, Room 200, Eliot Hall 9:15-9:25 (6) The parity of the integer (2n- 2):/n:(n- 1): Mr. D. M. Silberger, Western Washington State College (657-7)

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

584 9:30-9:40 ( 7) Prime divisors of the binomial coefficient Mr. E. F. Ecklund, Jr., Western Washington State College (657-11) (Introduced by Professor M. L. Faulkner) 9:45-9:55 (8) "Decimal" expansions to negative bases Professor C. T. Long, Washington State University (657-5) 10:00-10:10 (9) The distribution on quadratic residues in fields of order p2 Professor J. H. jordan*, Washington State University, and Mr. N. R. Hardman, Harvard University ( 657-4) 10:15-10:25 ( 10) Pisot sequences and Pisot-Vijayaraghavan numbers. Preliminary report Mr. M. J. DeLeon, Pennsylvania State University (657-15)

SATURDAY, 9:15A.M. Session on Analysis, Room 416, Eliot Hall 9:15-9:25 ( 11) A note on mini-gaps Dr. B. G. Eke, University of California, San Diego (657-18) (Introduced by Professor S. E. Warschawski) 9:30-9:40 (12) Bounds for the number of deficient values of entire functions whose zeros have angular densities Professor Ki Ghoul Oum, State University College at Buffalo (657-19) 9:45-9:55 ( 13) Reduction of order for a system of differential equations at a turning po-int. Preliminary report Dr. R. Y. Lee, University of Minnesota (657-22) 10:00-10:10 ( 14) Lower bounds for the essential spectrum of fourth order differential opera­ tors Mr. Kurt Kreith, University of California, Davis ( 657-3) 10:15-10:25 (15) A theorem relating generalized Hankel and Whittaker transforms. Prelimin­ ary report Professor H. M. Srivastava*, West Virginia University, and Mr. Om Dutta Vyas, Jodphur University, India (657-9) 10:30-10:40 ( 16) Uniform convergence of spherical harmonic expansions Mr. D. L. Ragozin, Massachusetts Institute of Technology (657-29)

SATURDAY, 11:00 A.M.

Invited Address, Room 314, Eliot Hall Sources of equicontinuity in functional analysis Professor William G. Bade, University of California, Berkeley

SATURDAY, 2:00P.M.

Invited Address, Room 314, Eliot Hall OJ:!. some recent developments in the theory of Riesz spaces Professor W. A. J. Luxemburg, California Institute of Technology

585 SATURDAY, 3:15P.M.

General Session, Room 314, Eliot Hall 3:15-3:25 ( 17) WITHDRAWN.

3:30-3:40 ( 18) Complementing pairs of subsets of the plane Professor R. T. Hansen, Montana State University (657-20) 3:45-3:55 (19) Projections off-vectors of 4-polytopes Professor J. R. Reay, Western Washington State College (657-24) 4:00-4:10 (20) On finite decompositions of (2n - !)-spaces Mr. Joseph Zaks, University of Washington (657-23) 4:15-4:25 (21) Concerning the ideal and order topologies on a lattice. Preliminary report Mr. Charles Atherton, Washington State University (657-27)

SATURDAY, 3:15P.M.

Session on Algebra, Room 200, Eliot Hall 3:15-3:25 (22) On finite groups with a cyclic Sylow subgroup Professor Marcel Herzog, University of California, Santa Barbara(657-2) 3:30-3:40 (23) A class of modules over a left-noetherian ring Professor G. R. Krause, Washington State University (657-28) (Introduced by Professor J. H. Jordan) 3:45-3:55 (24) On commutative splitting rings Mr. J.D. Fuelberth, University of Southern California (657-13) (Introduced by Professor S. E. Dickson) 4:Q0-4: 10 (25) Decomposition of atomic and orthogonally complete rings Professor Alexander Abian, Iowa State University (657-31)

SATURDAY, 3:15P.M.

Session on Analysis, Room 416, Eliot Hall 3:15-3:25 (26) Singular integrals in several variables over a local field Professor Keith Phillips*, California Institute of Technology, and Profes­ sor Mitchell Taibelson, Washington University, St. Louis (657-32) 3:30-3:40 (27) On uniformly approximable Sidon sets. Preliminary report Professor R. W. Chaney, University of California, Santa Barbara (657-1) 3:45-3:55 (28) On the essential set Professor D. R. Chalice, Western Washington State College (657-26) 4:00-4:10 (29) Linear operators on spaces of continuous functions Professor R. E. Atalla, Ohio University (657-30)

586 4:15-4:25 (30) p-strictly singular operator;:; Professor D. R. Horner, Eastern Washington State College (657-8) 4:30-4:40 ( 31) The strict singularity of T and S does not imply the strict singularity of T + S Professor D. R. Horner and Mr. W. L. Buck*, Eastern Washington State College (657-lZ) R. S. Pierce Seattle, Washington Associate Secretary

MEMORANDA TO MEMBERS

RESOLUTION ON THE DRAFT

The following resolution was passed graduating seniors and first year gradu­ at the Council meeting of April 19, 1968: ate students. The possible partial or total "The Council of the American elimination of two successive graduate Mathematical Society regrets the recent classes can cause a dangerous hiatus in government decision about the drafting of the future production of scholars."

DIFFERENT CATEGORIES FOR BY-TITLE ABSTRACTS

By-title abstracts will be grouped quest. The subject fields correspond to according to the subject fields, starting those already in use, and the old abstract with the August issue of cNotiaiJ; abstracts form may be used. Authors who are not classified in "other fields," or in more aware of this new procedure will be given than one field, will appear in the miscel­ an opportunity to provide the subject field laneous category. This change is being when receipt of the abstract is acknow­ made in response to requests by several ledged. Papers for which subject classifi­ members. cations are not provided by the author by A new abstract form is available the deadline for by-title abstracts will be and will be sent to individuals upon re- placed in the miscellaneous group.

REQUEST FOR NAMES OF VISITING FOREIGN MATHEMATICIANS

The editors of these cNoticeiJ would The information will be published in the appreciate receiving the names and ad­ August issue of these cJ{otiui), in time for dresses of foreign mathematicians who mathematics departments to extend speak­ will be visiting the United States in 1968- ing invitations for the coming academic 1969. Information should include the name year. The regular annual list of foreign and country of each mathematician, his mathematicians will appear in the Novem­ U. S. address and the period of his visit. ber issue.

587 PRELIMINARY ANNOUNCEMENT OF MEETING

Seventy- Third Summer Meeting University of Wisconsin Madison, Wisconsin August 27-30, 1968

The American Mathematical Society Friday, August 30, at 1:45 p.m. His topic will hold its seventy-third summer meeting will be "Recent developments in the theory in Madison, Wisconsin, from Tuesday, of pseudogroups." All three talks will be August 27, 1968, through Friday, August given in the Union Theater. 30, 1968. All sessions will be held in lec­ There will be numerous sessions ture rooms and classrooms of the Univer­ for the presentation of contributed ten­ sity of Wisconsin. Times are CENTRAL minute papers on Wednesday at 10:45 a.m., DAYLIGHT SAVING TIME throughout. on Thursday at 10:15 a.m., and on Friday There will be two sets of Collo­ at 10:15 a.m. and 3:00p.m. All of these quium Lectures each consisting of four sessions will be held in Van Vleck Hall. lectures. Professor Donald C. Spencer Abstracts of contributed papers should be of Stanford University will speak on the sent to the American Mathematical Society, subject "Overdetermined systems of par­ P. 0. Box 6248, Providence, Rhode Island tial differential equations." His lectures 02904, so as to arrive prior to the dead­ will be given on Tuesday, August 27, at line of July 1. Abstract blanks can be ob­ 1:30 p.m., on Wednesday at 8:30 a.m., tained on request from the same address. and on Thursday and Friday at 9:00 a.m. There will be no limit on the number of The other Colloquium Lecturer will be contributed papers. No provisions will be Professor John W. Milnor of Princeton made for late papers. University and the University of California, This meeting will be held in con­ Los Angeles. His topic will be announced junction with meetings of several other later. He will speak on Tuesday at 2:45 organizations. These include the Mathe­ p.m., on Wednesday at 9:30 a.m., and on matical Association of America, the So­ Thursday and Friday at 9:00 a.m. The ciety for Industrial and Applied Mathe­ initial lecture in each series will be de­ matics, the Institute of Mathematical Sta­ livered in the Union Theater, but the sub­ tistics, Pi Mu Epsilon, and Mu Alpha Theta. sequent lectures will be given in Rooms The Mathematical Association of America B102 and B130 in Van Vleck Hall. will meet from Monday through Wednes­ By invitation of the Committee to day. The Association will present Profes­ Select Hour Speakers for Annual and Sum­ sor Hyman Bass of Columbia University mer Meetings, there will be three one­ as the Earle Raymond Hedrick Lecturer hour addresses. Professor W. C. Hsiang on the general topic of "Algebraic K­ of Yale University will speak on Thursday, theory." His three lectures will be given August 29, at 1 :45 p.m. His talk will be on Monday, August 26, at 9:15a.m. and at entitled "Non-simply connected differen­ 1:30 p.m. and on Tuesday at 9:00a.m. in tial topology." Professor P. A. Griffith the Union Theater. The individual titles of the University of Houston and Prince­ will be "Stability theorems in topology and ton University will address the Society on in algebra," "Grothendieck groups and Thursday, August 29, at 3:00 p.m. His Whitehead groups," and "Congruence sub­ subject will be" Some transcendental prob­ groups and reciprocity laws." The Society lems in algebraic geometry." Professor for Industrial and Applied Mathematics V. W. Guillemin of the Massachusetts will present Professor Peter D. Lax of the Institute of Technology will speak on Courant Institute of Mathematical Sci-

588 ences, New York University, as the von EMPLOYMENT REGISTER Neumann Lecturer. He will speak on The Mathematical Sciences Em­ Wednesday, August 28, at 8:00 p.m. in ployment Register will be held in Room Room 6210 of the Social Science Building 226 and the second floor Lake Lounge of on the subject "Nonlinear partial differen­ the Wisconsin Center. It will be open tial equations." The Institute of Mathemati­ Tuesday through Thursday, August 27 cal Statistics will meet from Tuesday through August 29, from 9:00 a.m. to 5:00 through Friday. The Institute will present p.m. on each of the three days. Attention Professor Herman A. Chernoff of Stan­ is invited to the announcement of the Em­ ford University as the Wald Lecturer on ployment Register on Page 451 of the the general subject "Continuous time stop­ April cJioticei), in particular to the dead­ ping problems and optimal stochastic con­ line for applications and to the necessity trol." His three lectures will be given on for prompt registration at the Employ­ Tuesday, August 27, at 4:00 p.m. and on ment Register desk by both applicants Wednesday and Thursday at 3:00 p.m. in and employers. Room 1351 of the New Chemistry Build­ ing and will have the individual titles EXHIBITS "Stopping problems and the heat equation," Book exhibits and exhibits of educa­ "Sequential analysis and the one-armed tional media will be displayed in the bandit problem," and" A stochastic control Wisconsin Center on Tuesday, Wednesday, problem and the two-armed bandit prob­ and Thursday. lem." The Institute of Mathematical Sta­ tistics will also present a special invited BOOK SALE address by Professor Gian-Carlo Rota of Books published by the Society will the Massachusetts Institute of Technology. be sold for cash prices somewhat below He will speak on Tuesday, August 2 7, at the usual prices when these same books 11:00 a.m. on "Recent advances in com­ are sold by mail on invoice. binatorial theory." Pi Mu Epsilon and Mu Alpha Theta will meet concurrently with DORMITORY HOUSING the Society. Dormitory rooms will be available COUNCIL AND BUSINESS in both the University Residence Halls and MEETINGS in private dormitories. Reservations for dormitory rooms should be made in ad­ The Council of the Society will meet vance, using the form provided in these on Tuesday, August 27, at 5:00p.m. in the cNotimJ. Although it is possible that rooms Lake Shore Room of the Wisconsin Center. will still be a vail able to persons who have The Business Meeting of the Society will not registered in advance, this is not be held on Thursday, August 29, at4:30 p.m. guaranteed. Please note that the university in the Union Theater. dormitories are not air-conditioned, al­ REGISTRATION though some of the rooms in the private dormitories are. If for reasons of health, The Registration Desk will be in air-conditioning is necessary, please indi­ the lobby of the Wisconsin Center. It will cate on the housing form so special pro­ be open on Sunday from 2:00p.m. to 8:00 visions can be made for an air~condi­ p.m.; on Monday from 8:00 a.m. to 5:00 tioned room. p.m.; on Tuesday, Wednesday, and Thurs­ The University Residence Halls to 5:00 p.m.; and on day from 9:00 a.m. (Sellery, Ogg, and Witte) are modern, at­ to 1:00 p.m. Friday from 9:00a.m. tractive, quite new, and within walking dis­ The registration fees will be as tance of the various meeting rooms. Rates follows: (including meals) are: $8.00/dayjperson Member $3.00 for a double room; $9.75/day/per­ family 0.50 Member's son for a single room; children 2-11 Students No charge $3.00/day; children 2 and under no charge. Others 7 .so

589 L AI<. E. MENDOTA

1. Wisconsin Center 2. Lowell Hall 3. Carroll Hall 4. Wisconsin Memorial Union 5. Sellery Hall 6. Ogg Hall 7. Gordon Commons 8. Witte Hall 9. Lot 60 Cots and cribs cannot be supplied by the Lowell Hall, 610 Langdon Street, is university, but participants may make ar­ a private dormitory, also within easy walk­ rangements for rental of this equipment ing distance of the Wisconsin Center. Rates by writing to: A to Z Rental, 2620 East (not including meals): $5.00/day/person Washington Avenue, Madison, Wisconsin, in a double room; $7.00/day/person in a or AAA Rental Center, 4233 West Beltline single room; children 2-11 are half price; Highway, Madison, Wisconsin. An infant children under 2 are free. Cribs can be may occupy the parents' room without ad­ rented with advance notice. All rooms are ditional charge if the parents provide cribs air-conditioned. and bedding. Sleeping bags and pets are not Upon arrival on campus all guests permitted in the Residence Halls. Linen, who have made advance dormitory re­ blankets, towels, and soap are furnished, servations should go directly to the resi­ and daily maid service is provided. Please dence hall indicated on the confirmation note that there is no air-conditioning. slip. Guests who arrive without reserva­ Dormitories may be occupied from Satur­ tions should go directly to the Meetings day, August 24, at 10:00 a.m. to Saturday, Registration Desk in the Wisconsin Cen­ August 31, at 10:00 a.m. As meal service ter if they arrive during the regular does not begin until dinner (evening) on registration hours as indicated in the Sunday and terminates with lunch on Fri­ section of this announcement entitled day, the charge for rooms only on Saturday, 'Registration.' Those arriving when the August 24, and on Friday, August 30, will Registration Desk is closed, should go to be $5.75/day/person for asingleroomand the main desk on the ground floor of Ogg $4.00/person in a double room. Charges Hall. Guests are urged, however, to arrive will be adjusted for late arrivals and early during the normal registration hours. departures on a pro rata basis. For those Payment should be made in the occupying the University Residence Halls Southeast Halls Office for those staying in for just a few days, one of those days being the University Residence Halls. Those August 26, the day of the picnic, a small staying in any of the halls in the Carroll additional charge will be levied. (For Hall complex will make payment in the example, a person arriving after lunch on respective hall office. Monday and leaving after lunch on Thurs­ FOOD SERVICE day will pay $24.85 for double occupancy. The extra $0.85 will go toward the picnic.) For those staying in the University The University will not make refunds for Residence Halls, meals will be served meals that may be missed or accommoda­ cafeteria style in the air-conditioned din­ tions not used during the period arranged ing rooms of Gordon Commons (adjacent for. There is limited parking in one of the to the housing buildings) beginning with university lots one to five blocks from the breakfast on Monday, August 26, and con­ Southeast Residence Halls; permits will be tinuing through lunch on Friday, August 30. issued for a charge of $1.00 for the meet­ Hours for food service are as follows: ing period to those residing in the Univer­ Breakfast 7:30 a.m. 8:30 a.m. sity Residence Halls. If space in one of Lunch 11:30 a.m. 1:00 p.m. these lots is desired, please so indicate Dinner 5:30 p.m. 6:30 p.m. on the housing form on Page 668 of these c}/otit:ci). The Wisconsin Center Coffee Shop Carroll Hall, 620 North Carroll will serve coffee, juice, and doughnuts Street, is a complex of private dormi­ from 9:30a.m. to 11:00 a.m. andluncheon tories all within easy walking distance of from 11:30 a.m. to 12:30 p.m. the Wisconsin Center. Rates (not including The Wisconsin Memorial Union will meals): $4.50/day/person in a double serve meals at the following times: room; $6.50/day/person in a single room; Breakfast 7:30 a.m. 9:00 a.m. children 11 and under are half price. Cots Lunch 11:30 a.m. 1:15 p.m. and cribs may be obtained for a small Dinner 5:00p.m. -· 6:45 p.m. charge with advance notice. Many of these rooms are air-conditioned. Occupancy is A list of nearby restaurants will be the same as for the University Residence available at the Registration Desk in the Halls. Wisconsin Center.

591 MOTELS AND HOTELS *EDGEWATER HOTEL {608) 256-9071 666 Wisconsin A venue. 103 rooms There are a number of motels and Single $11.00 - $22.00 hotels in the area, some of which are Double 16.00 - 28.00 listed below with coded information which is explained at the end of the list. Those Twin 16.00 - 28.00 Extra person $1.00 motels preceded by an asterisk {*) are Code: RT-CL-FP-TV-AC within walking distance of the Wisconsin Center. Participants should make their own NATIONAL MOTOR INN {608) 257-5341 reservations with hotels and motels. 350 W. Washington Avenue. 100 rooms Single $10.00 - $12.50 *TOWN-CAMPUS MOTEL {608) 257-4881 Double 14.50 - 16.50 441 North Frances. 46 rooms Twin 16.50 - 18.50 Single $9.00 - $12.00 Extra person $3.00 Double 12.00 - 15.00 Code: RT-CL-FP-SP-TV-AC Twin 15.00 - 16.00 5 minutes drive from campus Extra person $3.00. Code: RT-FP-TV-AC Located on campus {State and Frances) *LORAINE HOTEL {608) 256-0231 123 W. Washington Avenue. 400 rooms *IVY INN MOTEL {608) 233-9717 Single $ 6.50 - $10.50 2355 University Avenue. 60 rooms Double 10.00 - 14.00 Single $1 0.00 Twin 13.50 18.00 Double 12.00 Code: RT-CL-TV-AC Twin 14.00 Double-double $18.00 MIDWAY MOTOR LODGE {608) 244-2424 Extra person $4.00, rollaway. 3710 E. Washington Avenue. 100 rooms Code: RT-CL-FP-TV-AC Single $10.50 - $13.00 Double 14.00 - 16.00 RAMADA INN {608) 244-2481 Twin 16.00 - 20.00 3841 E. Washington Avenue. 198 rooms Extra person $2.00 Single $10.00 Code: RT-CL-FP-SP-TV-AC Double 13.00 3 miles from campus Twin 15.00 Extra person $2.00. HOLIDAY INN {608) 244-4703 C9de: RT-CL-FP-SP-TV-AC 4 l/2 m NEon US 151. 208 rooms 4 miles from campus Single $ 7.00 - $9.75 Double 9.00 - 11.75 STERLING MOTEL {608) 256-0691 Twin 13.75 915 W. Beltline. 40 rooms Extra person $2.00 Single $8.00 Code: RT-CL-FP-SP-TV-AC Double 10.00 - 14.00 6 1/2 miles from campus Twin 12.00 Extra person $2.00 R T - Restaurant SP - Swimming Pool Code: RT-CL-FP-TV-AC CL- Cocktail Lounge TV - Television 2 miles from campus FP - Free Parking AC - Ai:~:-Conditioned HOWARD JOHNSON {608) 244-6265 ENTERTAINMENT 4822 E. Washington Avenue. 80 rooms There will be a bus excursion to Single $10.00- $15.00 Spring Green on Wednesday, August 28. Double 13.00 The tour will be from 9:30 a.m. to 4:00 Twin 17.00 - Extra person $3.00 p.m. and will include a guided visit to the Code: RT-CL-FP-SP-TV-AC Frank Lloyd Wright School of Architecture 6 miles from campus and buildings designed by Wright of local TRA VELODGE {608) 256-8365 sandstone, stucco, and natural wood com­ 910 Ann Street. 100 rooms binations. Lunch will be served at Apple Single $ 8.00 - $10.00 Hill Farm and will be followed by a visit Double 10.00 - 12.00 to the "House on the Rock." The charge Twin 12.00 - 14.00 will be $6.00 for adults with a slight re­ Extra person $2.00 duction for young people between the ages Code: FP-SP-TV-AC of 10 and 16. The trip is not recommended 3 miles from campus for children under 1 0. Tickets will be on 592 sale in the registration area. Milwaukee Steamship Company, 500 N. The traditional SIAM Beer Party Harbor Drive, Milwaukee, Wisconsin; and will be held in the Wisconsin Memorial Chesapeake and Ohio Autoferries, jones Union Cafeteria on Wednesday, August 28, Island Outer Harbor, Milwaukee, Wiscon­ at approximately 9:00 p.m. following the sin. von Neumann Lecture, If weather per­ mits, guests are welcome to use the Union CAMPING Terrace. Tickets will be $2.00 and will be on sale at the registration area. There is a large number of both A picnic will be held on Wednesday, private and public campgrounds within August 28, at 5:15 p.m. at the Athletic driving distance of Madison. There are no Field, In case of rain, the picnic will be campgrounds within the city limits. For a held in the Gordon Commons Dining complete list of public and private camp­ Rooms. Tickets will be $2.60 for adults grounds, one should write to the Vacation and $1.30 for children 11 years of age and and Travel Service, Wisconsin Conserva­ under and will be on sale at the registration tion Department, Box 450, Madison, Wis­ area. consin 53701. Those interested in summer Conducted nature walks of the Wis­ cottages or resorts should write to the consin Arboretum will be available. Trail Madison Chamber of Commerce, Inc., maps will be provided in the registration 615 E. Washington Avenue, P. 0. 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594 AMS Mathematical Offprint Service

The AMS has received excellent an offprint. He will make this specification responses to its Mathematical Offprint by writing a Boolean equation. The com­ Service. It was first introduced in the puter will first comply with all of the sub­ cffotim.J in January 1968 on pages 44-48. scriber's requests in the new free part of In answer to requests from many mathe­ the form before it answers the general maticians, we have incorporated several specifications. The diagram below explains improvements into our system. the procedure used to produce, for the The most important improvement is subscriber, an offprint or a title listing. an addition to the profile form. In the ori­ The subscriber may elect to have all ginal form the subscriber made the speci­ the articles which meet his offprint speci­ fications that he desired as to authors, fications generated only as title listings languages, journals, and fields of interest. for a period not to exceed six months; his Now he may take the specifications that statement will indicate which title listing he has made in the four categories and would have been offprints. This option combine them in any way that he chooses allows the subscriber to determine how to produce a desired result. He may make closely his profile specifications reflect exceptions to the rules he has already his actual interests and to see if he has made or he may add to them. To achieve been accurate or specific enough in filling his desired result, the subscriber will out his profile form. formulate a Boolean expression which will The subscriber's statements will state a condition for which the subscriber be generated on the last Friday of every will indicate the result that he wishes-­ month and mailed to the subscriber on the offprint, title listing, or nothing. A simple following Wednesday. Each statement will example for an exception would be the include a listing of all offprints enclosed following: A subscriber has indicated that with the statement, and the title listings he does not wish to receive any items in of all articles in which the subscriber has Spanish, but if an article appears in Span­ a peripheral interest or which match with ish written by Brown then he would like his profile as a title listing. The sub-

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NEWS ITEM

CONFERENCE ON COMBINATORIAL THEORY

A Conference on Combinatorial Gian-Carlo Rota, The many lives of Theory was held at Yale University on lattice theory. May 1 0-ll, 1968, in honor of Professor Melvin Dresher, Distribution of pure Oystein Ore who is retiring this year. The equilibrium points in n-person. conference was organized by Professors Claude Berge, Some problems of the G. C. Rota and B. L. Rothschild of the composition of a permutation with the Massachusetts Institute of Technology and use of graph theory Nathan Jacobson of Yale University, and Garrett Birkhoff, Heterogeneous alge­ included the following speakers: bras. J. W. Moon, The distance between points Marshall Hall, Groups and combinator­ in random trees. ial designs. R. P. Dilworth, The partition lattice. W. T. Tutte, Towards a theory of cross­ G. B. Dantzig, Complementary pivot ing numbers. theory, extensions and graph inter­ H. H. Crapo, On Galois connections. pretation. Frank Harary, Enumeration techniques D. R. Fulkerson, Flows and infinite in graph theory. graphs (joint work with J. Folkman). M. D. Plummer, Covering concepts in Gerhard Ringel, The Heawood map graphs. coloring theore~ A. J. Hoffman, Some integer polyhedra A. M. Gleason and J. G. Stemple also B. L. Rothschild, A matching theorem gave talks. for graphs. M. E. Watkins, Connectivity of vertex transitive graphs.

596 The Confinement of a Russian Mathematician

On February 15, 1968, Aleksandr S. posedly helped in the typing of it. The pro­ Esenin-Vol'pin, a famous logician and secution also charged the four defendants mathematician, was taken from his home with associating with agents of theN.T.S., in Moscow and forcibly committed to a Narodno-trudovo'f sojuz--People's Labor mental institution 45 miles from Moscow. Alliance, a Russian anti-Soviet emigre Some 95 Soviet mathematicians and Z4 organization in West Germany founded to American mathematicians have separately carry out subversive activity against the protested this action against a colleague, Soviet government. The four were sen­ in written petitions. In order to present tenced for from one to seven years. some of the facts leading up to Esenin­ According to an article in the Vol'pin's commitment tothementalhospi­ January 11 issue of The Times, 31 writers tal at Stolbovaja, the following report has and scientists signed an appeal urging the been compiled from articles in The New Moscow City Court to permit a" full public York Times of December 10 and ZZ, 1967, airing" of the trial in the press and urging and January 11, 17, February 18, ZZ, that an impartial selection of witnesses for April Z4, and May 5, 14, 1968. the defense be made. (At this time, the Four young members of Moscow's Soviet press had not mentioned the trial.) literary underground went on trial in The appeal was sent to the chairman of the January 1968 on charges of "agitation or Moscow City Court on the eve of the trial. propaganda carried out with the purpose Among the 31 reported to have signed the of subverting or weakening the Soviet appeal was Igor R. Safarevic, a noted regime." They were tried under Article 70 mathematician .who won a Lenin Prize in of the Russian penal code dealing with state 1959. Copies were sent to Premier Kosy­ crimes. The trial followed a year of im­ gin, Party Leader Brezhnev, and President prisonment during which time the three Podgorny. men and young girl were allowed no com­ The appeal related to charges munication with anyone except their law­ against Ginzburg, expressing concern over yers. his year-long detention before the trial. One of the young men was Aleksandr (According to Article 97 of the Code of Ginzburg, a 30 year old poet, who had Criminal Procedure, detention before trial compiled a "white book" containing records may not exceed nine months.) The appeal and papers relevant to the case and trial contended that the compilation of the of Andre! D. Sinjavski~ and Julil' M. material on the trial of Sinjavski'r and Daniel', writers who were tried and im­ Daniel' was not adequate ground for pros­ prisoned in a labor camp in 1966 under the ecution. same article of the Russian penal code as Shortly before the Ginzburg trial were Ginzburg and his companions. This was concluded, Pavel M. Litvinov, grand­ 400 page book was printed in Frankfurt, son of the late Soviet Foreign Minister West Germany, in German, French, and Maxim M. Litvinov, and the wife of JuliY Italian, and also in a small pocket edition M. Daniel' stood outside the courthouse in Russian for smuggling into the Soviet and denounced the trial and treatment of Union. There is no evidence, however, that the four dissidents before foreign jour­ Ginzburg took part in getting the book out nalists. of the Soviet Union to be printed. The On January 16 after the trial had charges against the other defendants arose ended, a long account appeared in Izvestia. from their reported role in producing a This report dealt mainly with the charges political literary review entitled Phoenix that the four intellectuals were agents of 1966 in which the trial of Sinjavskfi' and theN. T. S. Daniel' was condemned. The two men con­ The February 18 issue of The Times tributed to the magazine and the girl sup- reported that internal security authorities

597 of the Soviet Union had warned relatives the gifted Soviet mathematician Aleksandr and friends of Ginzburg and his associates Sergeevi~ Esenin-Vol'pin, a well-known to "stop their protests or face the conse­ expert in the field of mathematical logic, quences." This issue also reported that has been forcibly committed, withoutpre­ Aleksandr S. Esenin-Vol'pin, a mathe­ liminary medical investigation and without matician, author of poetry, and friend of the knowledge or consent of his relatives, Ginzburg, was taken from his home on to Psychiatric Hospital No.5 at Stolbovaja, February 15 and was not returned. (!twas 45 miles from Moscow. learned a week later that he had been "The forcible commitment of a confined to a mental institution at Stol­ talented mathematician, in full possession bovaja. He had been incarcerated in the of his powers, to a psychiatric hospital for past as "mentally unstable.") Esenin­ the seriously ill, and the conditions to which Vol 'pin was reported to be the originator he has been subjected by the very nature of one of the most current slogans of the of such a hospital, are extremely harmful protesters, "Respect Your Own Constitu­ to his mental and physical health and to his tion." Pavel M. Litvinovwas summoned by dignity as a human being. the K. G. 8 ., the Soviet internal security "In view of the humane purposes of police, to report to them the same day our laws, and still more of our Public that Esenin-Vol'pin was taken, but did not Health services, we regard this action as go. The Times reported that he was still a gross violation of medical and legal at liberty. Ginzburg's mother and his standards. fiancee were called to the K. G. 8. on "We urge you to intervene at once February 10 and warned about spreading and to take such measures as will ensure slanderous rumors about his trial. The that in the future our colleague can con­ same Times article stated that the action tinue his work under normal conditions." against the protesters was viewed as The following are the 95 Russian mathe­ "a result of nervousness by the security maticians who signed the petition: agent over a number of petitions and Member of the Academy of Sciences letters of protest" circulating in Moscow. of the USSR: Lenin Prize Laureate P. S. On February 22, The Times stated Novikov. that 200 Soviet political prisoners in two Corresponding Members of the Aca­ labor camps and one prison had gone on a demy of Sciences of the USSR: Lenin and hunger strike in support of their demands State Prize Laureate I. M. Gel'fand; State for better conditions. They demanded bet­ Prize Laureates Lazar' Ljusternik, Dmi­ ter living conditions, better food, abolish­ trii' Men'~ov; Lenin Prize Laureates S. P. ment of forced labor, and the right to send Novikov, I. R. Safarevi~; Andrei' Markov. and receive letters. Ninety-five Soviet Professors, Doctors of the Mathe­ mathematicians, reported elsewhere as 99, matical and Physical Sciences: Lenin some of whom are members of the Academy Prize Laureates Vadim Arnol'd, Anatolii of Sciences, protested to the Government Vituskin, Juri! Manin; State Prize Laur­ against the confinement of Esenin-Vol'pin eates Aleksandr Kronrod, N. M. Merman; in a mental institution. Two of the signers F. F. 8okste!n, D. A. 8o~var, V. A. of the Russian petition were P. S. Novikov Efremovi~, Ljudmila Keldys, A. A. Kiril­ and his wife Ljudmila Keldys, sister of lov, V. A. Kondrat'ev, A. G. Kuro~. E. M. the President of the Academy of Sciences; Landis, A. M. Lod~ic, A. Ja. Povzner, they were supposed to visit the University N. V. Zbolinskii', I. I. P jateckii-Sapiro, of Illinois, Urbana, for about two months, V. P. Palamodov, Ju. M. Smirnov, S. V. but their visits have been canceled. Their Fomin, G. E. Silov, A. M. Jaglom, I. M. petition follows: Jaglom. To the Minister of Public Health of the Doctors of Mathematical and Physi­ USSR cal Sciences: M. S. Agranovi~, Docent To the General Procurator of the USSR A. V. Arhangel'ski'l. Copy to the Chief Psychiatrist of the City Senior Research Associate: V. of Moscow Ponomarev, Ja. G. Sinai. Candidates in Mathematical and "It has come to our attention that

598 Physical Sciences: Senior Research Asso­ courage further protests of this kind. ciates N. Vvedenskaja, L. Volevi~, N.H. "The undersigned American mathe­ Konstantinov, 0. S. Kulagina, R. A. Minlos, maticians and logicians know of the ex­ B. Poljak, D. B. Fuks; Research Associate cellent mathematical work of Esenin­ S. Gindikin; Docent B. G. Averbuh, Docent Vol'pin, and of his helpful contributions to B. M. Alekseev, L. M. Balakina, Docent the mathematical community by way of T. M. Baranovic, L. Basoalygo, N. M. translation and editorial work. We also Be skin, Blincevski!, N. M. Bruslinskaja, know of his great integrity and are greatly Docent A. D. Ventcel', Docent T. D. concerned over the reports that he may Ventcel', F. Ja. Vetuhovski'l, Docent E. have been subjected to pressure simply for Vinberg, E. Vul', V. L. Golov, Docent expressing his views on public matters. M. I. Grabar', S. A. Kabakov, K. V. Kim, "We would greatly appreciate it, if Docent L. A. Kronrod, S. N. Kry~kov, you would advise us concerning the cor­ A. L. Krylov, V. Lev~enko, A. L. Lund, rectness of the published accounts about K. A. Miha'llova, Docent A. L. Onif!iclk, E senin- Vol 'pin and concerning his present V. P. Orevkov, A. Skobeev, M.A. Smol­ status. janskir, Docent V. M. Tihomirov, Docent "Kindly address your reply to Pro­ L. Tutubalin, L. Flitman, Docent Ju. I. fessor Martin Davis, Courant Institute, Hmelevski'l, L. L. Cinman, A. Cernavskit, New York University, 251 Mercer Street, Z. Ja. Sapiro, I. Sarygin, V. A. Jankov. New York, New York." Candidates in Engineering Sciences: The letter was signed: Paul Axt, L. Ja. Kelperks, S. A. Panov. Paul J. Cohen, Chandler Davis, Martin Cand~date in Science: Senior Re­ Davis, Calvin C. Elgot, S. Feferman, search Associate G. Tjurina. Jacob Feldman, Thomas J. Grilliot, Mel­ Candidate in Pedagogical Sciences: vin Hausner, Nicholas Kazarinoff, Saul A. Ju. A. Sihanovic. Kripke, M. R. Krom, Bruce Lercher, Junior Research Associates: K. Hilbert Levitz, Louis Nirenberg, W. V. Babicki1, E.E.Glusberg,A.Gvozdev, V. V. Quine, Hugo Ribeiro, Dana Scott, Harold Kun, G. S. Pod'japol'skil, V.N.Osaulenko, S. Shapiro, Norman Z. Shapiro, Allen G. E. Cekin. Shields, Stephen Smale, Sherman Stein, Research Assistant: F. L. Varpa­ Ann Yasuhara. hovski!. On April 24 The Times reported Senior Lecturer: G. A. Sestopal. that "the Soviet Communist party expelled N. Vil'jams, Ju. A. Lecturers: some of the country's leading scientists, Gastev. including several working on military and Computer Engineer: I. G. Kristi. space programs, for having participated in Senior Engineer: V. K. Fink. a protest against the detention of the mathe­ for the cosigners was given The address matician Aleksandr Esenin-Vol'pin."Some Department of Mathematics and Mech­ as: of the most prominent of the scientists Moscow State University (Lomono­ anics, mentioned were: I. M. Gel'fand, famous Hills, Moscow 234, USSR. sov), Lenin for the use of mathematical methods in On March 19, following reports in biology and winner of a Lenin Prize; The New York Times, 24 American mathe­ Sergei Fomin, head of the cybernetics de­ maticians wrote a letter to the Embassy of partment at Moscow University; Juri'! the Union of Sovi~t Socialist Republics in Manin, winner of a Lenin Prize; and Igor Washington, D. C., asking for a clarifica­ R. Safarevil!, who had signed the appeal tion of Esenin-Vol'pin's status. The letter concerned with the Ginzburg trial and also read: the petition concerning E senin- Vol 'pin. As stated in The Times report, this "On February 18 and February 22 was the first time that the Communist Times printed stories about The New York party had taken steps against" intellectual made to A. S. Esenin-Vol'pin, and threats leaders of the scientific establishment his confinement. These stories sug­ about whose military importance has usually actions were based upon his gest that such given them privileged status. The authori­ the sentencing of Soviet protests against ties ignored their participation in protest intellectuals, and were designed to dis-

599 meetings and their demands for intellectual of the most outstanding mathematicians in and artistic freedom." the Soviet Union: N. N. Bogoljubov, L. S. The May 5 issue of The Times quoted Pontrjagin, A. A. Markov, I. G. Petrovski'f, part of a speach given by Party Leader P. S. Novikov, P. S. Aleksandrov, M. V. Brezhnev a few weeks earlier in which Keldy~, S. N. Mergeljan, A. N, Tihonov, he stated that foreign anti-Soviet organi­ Two of these were among the 9 5 protesters. zations use, to further their propaganda, The text of the 1 etter follows: Soviet citizens who are "morally un­ "We acknowledge receipt of your stable, weak and politically immature letter concerning A. S. Esenin- Vol 'pin people. Sometimes they catch in their net and answer it herewith. The information persons avid for self-publicity, those you quote from the newspaper "The New willing to make themselves known as loud­ York Times" about his present situation ly as possible,,,by whatever dubious politi­ does not correspond to the facts. cal means they find at their disposal, and "In reality, Esenin-Vol 'pin has been the praise of our ideo­ without scorning under medical observation for mental ill­ logical opponents. The Soviet public sharp­ ness during the course of several years shameful acts of these ly condemns the and, as a result of his illness, he has been hypocrites. The renegades can not expect confined repeatedly to psychiatric hospi­ to go unpunished," tals. At the present time he is again hos­ Times article stated that The same pitalized in the clinic of the Psychiatric Mstislav Keldy~, President of the Academy Institute of the USSR Academy of Medical of Sciences of the USSR, "warned protest­ Sciences, where he is under the care of ing scientists that Soviet science could go first-class specialists. ahead without them, and that therefore "We regret that this case has been their scientific abilities would not protect brought to your attention. You will agree followed them. His words were reportedly that if a person is ill and in need of medi­ several distinguished by the expulsion of cal assistance, it should be given to him," scientists ...from the Communist party." On May 8, 1968, a letter was written The May 14 New York Times re­ to Professor Martin Davis in answer to the ported that Esenin-Vol'pin had been re­ March 19 letter of the 24 American mathe­ leased from the mental hospital on the maticians who questioned Esenin-Vol 'pin's previous day. confinement. The letter was signed by nine

600 NEWS ITEMS AND ANNOUNCEMENTS

FIFTH PRAGUE CONFERENCE ON L. Le Lam (Berkeley): On the INFORMATION THEORY, asymtotic behaviour of posterior distribu­ STATISTICAL DECISION FUNCTIONS, tions. AND RANDOM PROCESSES P. Mandl (Prague): The elimination of Killing measures by transformations of The Czechoslovak Academy of Sci­ diffusion processes (presented in absen­ ences Institute of Information Theory and tia). Automation is preparing the Fifth Prague P. A. Meyer (Strasbourg): Some Conference on Information Theory, Statis­ recent advances in martingale theory. tical Decision Functions, and Random Pro­ J. Neveu (Paris): New results on cesses which will be held in Prague from Gaussian processes. September 9 to 15, 1968. A. Perez (Prague): Reduction pro­ The purpose of the conference is cedures in statistical decision. the presentation of new results dealing M. S. Pinsker (Moscow): New meth­ with the above mentioned disciplines and ods of information transmission through discussions of important problems con­ channels with feed-back. nected with them. From the invited scien­ L. Telksnis (Vilnius): The deter­ tists, the following have tentatively agreed mination of the change in the structure of to give papers: random processes under uncertainty about statistical characteristics. A. A. Borovkov (Novosibirsk): Theorems on convergence to diffusion Those wishing to participate in the processes. Conference are invited to present 15 min­ H. Dinges (Frankfurt): Wiener-Hop£ ute communications in any of the above factorization of substochastic transition mentioned areas. Ample time will be al­ functions in ordered state spaces. lowed for discussions. The papers pre­ R. L. Dobru~in (Moscow): Problems sented at the Conference will be published. of the theory of random fields arising in The summaries of lectures and statistical physics. communications should be submitted to the J. Gani (Sheffield): Recent develop­ Organizing Committee, in English, not ments in the theory of storage. later than June 30, 1968. The address of R. Z. Hasminski1 (Moscow): Some the Committee is Prague 2, Vyi!eh3'adska topics in the theory of differential equa­ 49. For more information write to Cedok­ tions containing random functions. Intravel, Congress Department, Prikopy J. Havel (Prague): Special devices 18, Praha 1. for stochastic problems solution. V. Imedadze ( Tbilisi): A set of specialised computing devices for the SYMPOSIUM ON COMPLEX ANALYTIC treatment of random processes. METHODS IN MATHEMATICAL PHYSICS M. Iosifescu (Bucharest): Sequen­ tial analysis of random systems with com­ A symposium on complex analytic plete connections. methods in mathematical physics, sup­ D. G. Kendall (Cambridge): On ran­ ported by the Air Force Office of Scienti­ dom sets. fic Research, is being held at Indiana F. Kozin (Brooklyn): Approximation University, Bloomington, Indiana, on June 3 techniques for nonlinear stochastic sys­ through June 6, 1968. The symposium terns. stresses the use of complex analytic meth­ K. Krinkeberg (Heidelberg): Recent ods in problems of modern and classical results on mixing in topological measure physics, and the theory of partial differen­ spaces. tial equations. The program features one-

601 hour invited addresses by leading mathe­ ships to holders of doctoral degrees in the maticians and physicists working in this mathematical, physical, engineering, general area, and also a session of fif­ social, and life sciences. Each of the teen-minute contributed papers. The in­ awards includes an annual stipend of vited papers and selected contributed $6,500, an allowance for dependents, and papers will be published as a volume. an allowance to help defray travel costs. The list of invited speakers and Fellowship recipients in this program will their tentative address titles is as follows: study or carry on research at institutions in the United States and abroad. V. Bargman, Princeton University, The following are the recipients of Group representations on Hilbert spaces of the Postdoctoral Fellowships in mathe­ analytic functions. matics: R. H. Berk, University of Michi­ L. Bers, Columbia University, gan; L. J. Billera, City University of New Teichmiiller spaces. York; P. R. Chernoff, Harvard University; J. Bros, Institute for Advanced L. J. Corwin, Harvard University; F. T. Study, Analyticity properties implied by Farrell, Yale Cniversity; J. D. Lubin, two-particle structure in general quantum Brown University; D. B. Meredith, Bran­ field theory. deis University; D. P. Niebur, University R. J. Eden, Cambridge University, of Wisconsin at Madison; B. A. Taylor, Consistency problems in S-matrix theory. University of Michigan; H. M. Taylor, A. Erdelyi, University of Edinburgh, Cornell University; and S. V. Ullom, Uni­ Uniform asymtotic expansions of integrals. versity of Maryland. L. D. Faddeev, Leningrad Univer­ sity, Many-dimensional generalization of the inverse scattering problem. DEPARTMENT OF STATISTICS P. Garabedian, New York Univer­ UNIVERSITY OF MANITOBA sity, Analytic methods for the numerical computation of fluid flows. The University of Manitoba has J. M. Jauch, University of Geneva, created a separate Department of Statis­ Scattering theory in general quantum tics within the Faculty of Arts and Sci­ mechanics. ence. The Department presently offers M. Z. Krzywoblocki, Michigan State courses leading to a B. S. Major in Statis­ University, Integral operator methods in tics, a B. S. (Honours) in Statistics, and elasticity, fluid dynamics, and electro­ an M. S. degree. The planning for a Ph. D. magnetics. G. Mahoux, State University ofNew program has been started. The emphasis is on both the theoretical and applied as­ York at Stony Brook, Extensions of analy­ pects of statistics. Enquiries concerning ticity domains of functions with positivity the program, financial assistance for properties. study, and teaching or research positions F. Rohrlich, Syracuse University, may be directed to Dean A. L. Dulmage, The coherent state representation and Acting Head, Department of Statistics, quantum field theory. University of Manitoba, Winnipeg 19, A. Weinstein, American University, Manitoba, Canada. Some theoretical ramifications of the intermediate problems for eigenvalues. J. A. Wheeler, Princeton Univer­ TRAINING AIDS FOR sity, Quantum geometrodynamics. EQUATION TYPING D. V. Widder, Harvard University, Homogeneous solutions of the heat equa­ Sandia Corporation has announced tion. the reprinting of Sandia's Training Bulle­ tin No. 1, Development of Training Aids for Equation Typing, SC-M-67-617. A free NSF AWARDS FOR POSTDOCTORAL copy of the Bulletin may be had by sending WORK IN THE SCIENCES requests, including full title and number of the Bulletin to Sandia Office of Industrial The National Science Foundation Cooperation, Sandia Corporation, P .0. Box has awarded 120 Postdoctoral Fellow- 5800, Albuquerque, New Mexico 87115.

602 RECIPROCITY AGREEMENT WITH first year level of graduate study, $2,600 SVENSKA MATEMATIKERSAMFUNDET for study at the intermediate level, and $2,800 for the terminal year. They include Svenska Matematikersamfundet has additional allowances for dependents and announced that the fee for membership has for limited travel. In lieu of tuition and been raised and is now 15 Swedish Kronor fees, United States institutions receive a for the fiscal year 1967-1968. Under standardized $2,500 cost-of-education reciprocity agreements, most foreign allowance for each Fell ow. At its discre­ members pay half this amount or approxi­ tion each fellowship institution may aug­ mately $1.50. (The fee for lifetime mem­ ment the stipend by not more than $1,000 bership is now 200 Swedish Kronor and is per year. not subject to reductions under reciprocity The Foundation plans to reopen the agreements.) A member of the AMS wishing Graduate Fellowship program for 1968- to become a member may send a check, 1969 in October 1968. Application forms payable to Svenska Matematikersamfundet, will be available when the program is to Folke Eriksson, Svenska Matematiker­ announced. samfundet, Nackrosgatan 6, Molndal, Sweden. Your attention is also called to the following fact, taken from the inside cover of Mathematica Scandinavica: A sufficient condition to obtain the reduced subscrip­ LA TIN AMERICAN tion rate to Mathematica Scandinavica is SCHOOL OF MATHEMATICS to be a member of the AMS or any other society having a reciprocity agreement The first Latin American School of with Svenska Matematikersamfundet. Mathematics will be held at the Instituto de Matematica Pura e Aplicada, Rua Luiz de Camoes 68, Rio de Janeiro 58, GB, NSF AWARDS FOR ADVANCED Brazil, from July 1 to 26, 1968.Itis being SCIENCE GRADUATE STUDY organized and financed by the National Research Council of Brazil, with financial TheN ational Science Foundation has support of other institutions. The scientific announced awards of 1,925 Graduate Fel­ program will consist of the following lowships in the sciences, mathematics, and series of lectures at research and post­ engineering for the academic year 1968- doctoral level: 1969. The awards are made to accelerate Spectral theory of differential oper­ the output of highly qualified scientists ators, Jean Dieudonne of the Universite by encouraging outstanding students to ob­ de Nice, France. tain full-time advanced graduate training Desintegration of measures and in the sciences. All fifty states, the applications to stochastic processes, Lau­ and Puerto Rico are District of Columbia, rent Schwartz of the Universite de Paris, represented. NSF Graduate Fellows may France. attend any appropriate nonprofit U.S. or Singular integrals and elliptic oper­ of higher learning. foreign institutioiF ators, R. T. Seeley of Brandeis University. of the fellowships Forty-two percent Recent and classical results on the awarded are for tenure periods of two Cauchy problem, Fran~ois Tr~ves of Pur­ years. Continuation of the second year is due University. dependent upon satisfactory academic pro­ gress of the Fellow during the first year There will also be one-hour lec­ and the availability of appropriated funds. tures and sessions devoted to short re­ The new awards are in addition to search communications. continuation awards offered to 575 Gradu­ The organizing committee is J. ate Fellows who received fellowships in Adem (Mexico), J. Barros-Neto (Brazil) March 1967. and L.A. Santal6 (Argentina). The chair­ Graduate Fellowships provide basic man of the local committee is C. S. Honig stipends (for 12 months) of $2,400 for the (Brazil).

603 HISTORY OF MATHEMATICS AT NEW OFFICERS AND MEMBERS OF THE UNIVERSITY OF TORONTO THE NATIONAL ACADEMY OF SCIENCES

With support from the Canada Coun­ The 1 OSth annual meeting of the cil, a mathematical information bank is National Academy of Sciences was held being developed at the University of Tor­ April 22-24, 1968, in Washington, D. C. onto under the direction of Kenneth 0. Emanuel R. Piore, Vice President and May. Filed under key words are defini­ Chief Scientist of the International Bus­ tions, results, historical facts, citations to iness Machines Corporation, was elected bibliographies, and titles of biographies, to a four-year term as Treasurer of the historical studies, expositions, and ori­ Academy. John W. Tukey, Chairmanofthe ginal sources. The bank will serve as a Department of Statistics at Princeton Uni­ source for historical research, for an­ versity and Associate Executive Director swering inquiries on a cost basis, and for of Research in Communication Sciences at the preparation of reference publications. Bell Telephone Laboratories, was elected The Ph.D. in mathematics may be as one of the new members of the Council. earned at the University of Toronto with Fifty new members were elected to a thesis of an historical character. In ad­ the Academy in recognition of their dis­ dition to meeting the usual requirements tinguished and continuing achievements in for the doctor's degree, candidates will original research. The following were find facility in two or more foreign lan­ elected for their contributions in mathe­ guages essential. The recently established matics: Garrett Birkhoff, Harvard Univer­ Institute for the History and Philosophy of sity; A. P. Calder6n, University of Chicago; Science and Technology offers related William Prager, University of California, graduate work. Inquiries about the program San Diego; I. M. Singer, Massachusetts In­ and fellowships should be addressed to stitute of Technology. Professor May, Department of Mathemat­ Carl Ludwig Seigel, Professor ics, Toronto 5, Canada. Emeritus at the University of Gottingen, was one of ten distinguished scientists who are not citizens of the United States to be elected as a foreign associate, one INTERNATIONAL CONFERENCE ON of the highest honors that can be bestowed COMPUTING METHODS IN by the Academy on a foreign scientist. OPTIMIZATION PROBLEMS

The second International Confer­ ence on Computing Methods in Optimiza­ tion Problems will be held in San Remo, MAA INFORMATION BOOKLET Italy, September 9-13, 1968, for the pur­ pose of disseminating new computing tech­ The Mathematical Association of niques in the broad area of optimization America has released its new Information problems. The conference will be spon­ Booklet for the Spring 1968. It describes sored by The Society for Industrial and the services and materials available Applied Mathematics with the cooperation through the MAA for the improvement in of The University of California and The mathematics and the teaching of mathe­ University of Southern California. There matics in colleges and universities. The will be both invited and contributed papers. Booklet lists information on publications, Special sessions will be devoted to specific films, employment, meetings of the Asso­ application areas such as nuclear reactors ciation, lecturers and consultants, and and transportation systems. guidance of students. An address for fur­ Inquiries on travel arrangements ther information on each category appears should be addressed to L. W. Neustadt, in the booklet. For a copy of the booklet Department of Electrical Engineering, write to Harry M. Gehman, Mathematical University of Southern California, Los Association of America, State University Angeles, California 90007. of New York, Buffalo, New York 14214.

604 BACKLOG OF MANUSCRIPTS must worry about the backlog of papers WILL BE ELIMINATED piling up. But also, as a collection of pub­ lishing mathematicians, the Society has On April 19, 1968, the Council ap­ been concerned with the long lag between proved the recommendation of the Com­ completion of a piece of research and its mittee to Monitor Problems in Communi­ appearance in print, and indeed, one of cation that the backlog of manuscripts the charges of this Committee was to accepted for publication in AMS journals investigate ways of speeding up publica­ be eliminated. The recommendation con­ tion." tained the statement: Beginning in 1969, Proceedings and "The backlog does not reflect, as Transactions will be increased in size in some would suggest, a flooding of the order to eliminate the backlog. Once the journals with low quality work. Rather, backlog is depleted, the journals will pub­ it is a simple consequence of the fact that lish manuscripts as soon as possible after the mathematical profession has grown acceptance by editors. The following is a more rapidly than its publishing mechan­ comparison of the 1968 and 1969 sizes and ism. With the help of Federal funds, and prices of the two journals. in the conviction that there is a serious Proceedings is now a bi-monthly, need for more mathematicians in our so­ one volume per year journal which will ciety, mathematicians have in recent contain 1536 pages for 1968, In 1969 it years set to work to reproduce their own will be published monthly, being divided kind, with notable results; the annual pro­ into four volumes with 3960 pages budgeted duction of Ph.D.'s in mathematics has for the year. The list price for 19 69 will grown from about 250 to about 850 in the be $80.00 as compared with $12,00 in 1968: past five years. These young people have Transactions is now a monthly jour­ all been imbued with the importance of nal divided into 5 volumes with 2800 pages publishing research, and they are pro­ for 1968, In 1969 it will consist of 12 vol­ ceeding to do so. In the view of the Com­ umes with 6720 pages budgeted for the year. mittee, it is inconsistent deliberately to The list price for 19 68 is $60,00, and for enlarge the profession while attempt­ 1969 will be $175.20. ing to hold down the volume of research The price to members for both published. There are unquestionably more journals is one-half the list price. papers published now that are of no interest Mathematics of Computation and to any given reader than there were ten Bulletin are operated at zero backlog now years ago, but it cannot be deduced that and will continue to do so in the future. most of these papers are low quality. On Mathematics of Computation will be in­ the contrary, standards for acceptance creased in size to 1000 pages if enough have certainly risen during this period; manuscripts are received. the lower density of "interesting" papers is due instead to the larger corpus of knowledge and the proliferation of fields of specialization, "With the help of page charges, Erratum larger sales to libraries and higher sub­ scription prices to the latter, it is now ST. ANDREWS COLLOQUIUM, 1968 possible to publish mathematics at no cost to the publisher. It is therefore recom­ At the Edinburgh Colloquium, St. mended that the Society plan to publish as Andrews, Scotland, July 10-20, 1968, four many pages in each of its journals as are courses will be given, each consisting of necessary to keep the backlog essentially eight lectures. The four courses are at zero, while keeping acceptance stand­ jordan Algebras, NormedAlgebras, Four­ ards where they are now. This will of ier Analysis and Generalized Functions, course have to be accompanied by a pro­ and Weak Convergence Methods in Proba­ portional rise in annual subscription bility, and will be given by Professors prices. Nathan Jacobson, F. F. Bonsall, N. G. de "As a publishing house, the Society Bruijn, and P. Billingsley, respectively.

605 PERSONAL ITEMS

Professor STEFAN BERGMAN of Dr. J, R. MCLAUGHLIN of Wayne Stanford University has been appointed to State University has been appointed to an a visiting professorship at the College of assistant professorship at Pennsylvania France, Paris, for 2 months, State University. Professor WARD BOUWSMAofPenn­ Professor ROBERT MOYER of Penn­ sylvania State University has been ap­ sylvania State University has been ap­ pointed to an assistant professorship at pointed to an associate professorship at Southern Illinois University. the University of Kansas. Professor J. W. BREWER of Florida Dr. ARNOLD OBERSCHELP of the State University has been appointed to an Technical University of Hannover, Ger­ assistant professorship at Virginia Poly­ many, has been appointed to a professor­ technic Institute. ship at the University of Kiel, Germany. Dr. W. J, ECKERT, former Director Mr. ALBERT SOGLIN of Soglin and of the IBM Watson Laboratory at Columbia Associates, Chicago, Illinois, has been ap­ University, will receive the honorary Doc­ pointed to an associate professorship at tor of Science degree from Oberlin College Chicago City College, Loop Campus. on June 10, Dr. G. J. TEE of the University of Professor H. B. ENDERTON of the Lancaster, England, has been appointed University of California, Berkeley, has Senior Lecturer at the University of Auck­ been appointed Assistant Editor of Re­ land, New Zealand, views, Journal of Symbolic Logic, and Dr. C. L. VANDEN EYNDEN of Miami Lecturer at the University of California, University, Oxford, Ohio, has been ap­ Los Angeles. pointed to a visiting assistant professor­ Professor R. S. FREEMAN of the ship at Pennsylvania State University. University of Maryland has been appointed to an associate professorship at the Uni­ PROMOTIONS versity of Oregon, Dr. T. J, GRILLIOT of Duke Univer­ To Professor. University of Oregon: sity has been appointed to an assistant R. F. TATE; Pennsylvania State Univer­ professorship at Pennsylvania State Uni­ sity: PAUL AXT; University of Santa versity. Clara: K. L. DE BOUVERE. Dr. R. F. JOLLY of the University of California, Riverside, has been appointed To Associate Professor. Pennsyl­ to an assistant professorship at Pennsyl­ vania State University: GEORGE AN­ vania State University. DREWS, A. M. KRALL, MARIO PET­ Dr. JEROME KARLE of the Naval RICH; University of Santa Clara: G. L. Research Laboratory, Washington, D. C., ALEXANDERSON; Sogan Jesuit College, has been named to the newly-created Chair Seoul, Korea: H. C. LEE. of Science for the Structure of Matter at the Laboratory. To Assistant Professor. Pennsyl­ Professor D. E. KNUTH of the Califor­ vania State University: BERTHA MATHER. nia Institute of Technology has been ap­ pointed Professor of Computer Science at INSTRUCTORSHIPS Stanford University. He will be on leave during the 1968-1969 academic year as a Mary Washington College: C. A. staff mathematician at the Institute for KEMMLER; Purdue University, Calumet Defense Analyses, Princeton, New Jersey. Campus: SIGRID WAGNER. Professor GUNTER LUMER of the University of Washington will be on leave DEATHS as a visiting professor at the University of Strasbourg, France. Professor B. A. AMIRA of the He-

606 brew University, Jerusalem, Israel, died age of 62, He was a member of the So­ on January 20, 1968, at the age of 72, He ciety for 41 years. was a member of the Society for 21 years. Professor C. J. TREMBLAY of Bard Dr. ELI GOURIN of New York, New College died on January 1, 1968, at the York, died on February 4, 1968, atthe age age of 53. He was a member of the Society of 85, He was a member of the Society for for 21 years. 41 years. Professor F. J. HAHN of Yale Univer­ sity died on January 14, 1968,attheage of 38. He was a member of the Society for 13 ERRATA years. Professor R. E. LANGER of the Uni­ Professor L. L. LOWENSTEIN of versity of Wisconsin died on March 12, Arizona State University died on August 23, 1968, at the age of 73, He was a member 1967, at the age of 64, He was a member of the Society for 47 years. of the Society for 39 years. Dr. MILTON ROSENTHAL of Culver Dr. C. E. STENARD of the Institute City, California, died on December 11, for Defense Analyses, Princeton, New 1967, at the age of 50, He was a member Jersey, has accepted a position as a of the Society for 8 years. Member of the Technical Staff with the Dr. R. G. SANGER of Kansas State Bell Telephone Laboratories, Whippany, University died on March 13, 1968, at the New Jersey.

NEWS ITEMS AND ANNOUNCEMENTS

INTERNATIONAL SUMMER COURSE 188TH ANNUAL MEETING OF ON ALGEBRAIC MODELS IN THE AMERICAN ACADEMY OF PSYCHOLOGY ARTS AND SCIENCES

The Nether lands Universities The 188th annual meeting of the Foundation for International Co-operation American Academy of Arts and Sciences (NUF FIC) announces its 1968 International was held May 8, 1968, in Boston, Massa­ Summer Course on Algebraic Models in chusetts. A new Vice-President, Roger Psychology to be held at 'Het Oude Hof', Revelle, Director of the Center for Popu­ The Hague, the Netherlands, on August lation Studies at Harvard University, was 5-17, 1968. The course is the third in a elected to replace Garrett Birkhoff as succession of summer courses in psy­ Vice-President representing the Mathe­ chology organized by NUFFIC and spon­ matical and Physical Sciences. Also elec­ sored by NATO. The Course, directed by ted were 103 new Fellows and 20 Foreign Professor C. Flament, will deal with alge­ Honorary Members. Elected as Fellows braic systems, mathematics and alge­ for their endeavors in the field of mathe­ braic models as used in psycholinguistics, matics were A. A. Albert, University of social psychology (graph theory), cognition Chicago; H. P. Greenspan, Massachusetts and genetic epistemology. Applications Institute of Technology; J. G. Kemeny, are invited from young graduates in psy­ Dartmouth College; Howard Raiffa, Har­ chology and related disciplines who are vard Business School; Stephen Smale, Uni­ interested in furthering their knowledge versity of California at Berkeley, in this specific field. For an application Two other members of the AMS write to Professor R. Doop, Course Re­ were elected: Valentine Bargmann of gistrar, Netherlands Universities Founda­ Princeton in physics, and P. C. Suppes of tion for International Co-operation, 27 Stanford in philosophy and theology. Molenstraat, The Hague.

607 LETTERS TO THE EDITOR

Editor, the cJ/otiaiJ take public issue with the vote of the Council. There appeared in these cJ/oticeiJ not The first point is that Grothen­ long ago an expository article on the cur­ dieck' s article is not a political document, rent state of mathematics in mainland unless one wants to argue that an article China. It contained information of legiti­ about, say, mathematics in China, is a mate concern to large numbers of mathe­ priori political. It is concerned with maticians, information which was not exactly what its title suggests. Indeed, one otherwise easily available. Among the of its striking conclusions is that there items discussed were: The major centers does exist serious mathematical activity of scientific activity; the work and in­ in North Vietnam. Moreover it is as per­ fluence of some of the principal mathe­ ceptive and as independent in judgment as maticians; the areas of specialization re­ its authorship would lead one to expect. It ceiving greatest attention; the nature of covers facts of precisely the type dis­ state scientific policy, particularly as it cussed in the China article, on the basis of affects conditions of research and inter­ direct observation by a distinguished national scientific communication. scientist. There is no strong reason to com­ It cannot be denied that political mission a series of such reports, no more significance might, by some, be attached so than there is to reject competently re­ to the publication of Grothendieck' s article ported information of this type when, in just as I now attach political significanc; the course of events, it becomes available. to the Council's refusal to do so. That is After a recent trip to North Vietnam, as a problem with which we all must, and do, a guest of the Mathematical Society of live. Members of the Council do not escape Vietnam, A. Grothendieck gave a lecture, it by pretending that negative action re­ "La vie mathematique en Republique De­ presents political innocence or neutrality. mocratique du Vietnam", at the invitation The essential fact is that the article of the University of Paris. When a copy which I contend to be of legitimate an~ of it came to the attention of Serge Lang timely scientific interest, was rejected, and myself, we agreed that it would be of not by intrinsic or scientific criteria, but serious interest to a large number of on the basis of speculation about its re­ American mathematicians. With Grothen­ ception in the present political climate. dieck' s consent, we therefore proposed its Thus, a portion of the Council has trans­ publication in one of the journals of the formed itself into a self appointed board AMS. Those who have read the article of censors whose action constitutes pre­ and with whom I have spoken, have en~ cisely the improper intrusion of political dorsed this action, and the editors with criteria into areas of scientific communi­ whom we first corresponded concurred. cation which it is pretending to eschew. There was concern, nevertheless Of course, certain members of the that an undesirable political constructio~ Council may contest my evaluation of the might be put on such an act of publication, article as a non political document. Un­ and the proposal was therefore put to the fortunately, they have taken steps to guar­ Council of the AMS for final approval. antee that very few other mathematicians That approval was denied at the April will have an opportunity to judge for them­ meeting of the Council in Chicago. selves. I feel that this matter involves is­ sues of general importance to the Society, Hyman Bass so I would like to present my assessment of the essential facts in the matter, and to

608 Editor, the cNO!iceiJ Editor, the cJ{o!icei) We are members of the Council of The Grothendieck report mentioned the AMS who attended the Council meeting, in the foregoing letter appears as an at­ April 19. At this meeting there was much tachment to the Council agenda for April discussion about publishing in the c)loriceAJ 19, 1968, and will also be included in the an article by A. Grothendieck, on his trip minutes of the meeting. The Secretary of to North Vietnam and Hanoi, where he the Society has pointed out that these min­ spent three weeks lecturing on mathemat­ utes are distributed to over 100 Society ics last fall. Grothendieck wrote mainly members. I urge that the recipients make about the state of affairs of North Vietna­ the report available to their colleagues in mese mathematics, and the effect of Amer­ order that as many mathematicians as ican bombing on the life of North Vietna­ possible may have an opportunity to inspect mese mathematicians. By a very close it. vote, the Council decided not to publish Grothendieck' s report has unfor­ Grothendieck's article in the cJ{o!icei). (A tunately been treated like a political hot majority of the members voting voted to potato rather than as an item of informa­ publish, but the motion failed when, after tion of interest to mathematicians. Publi­ an appeal to Council rules, fractional cation is not endorsement. The Society votes were counted,) should put the report before every mem­ We would like clearly to disasso­ er, letting each, in his own wisdom, ciate ourselves personally from this act of evaluate it for himself: the Council of the AMS, and we hope that Murray Gerstenhaber the Council will reconsider the matter at a subsequent meeting. P. T. Bateman R. Palais M. Gerstenhaber M. Protter Editor, the cNO!iceiJ J. J, Kohn S. Smale Bass was not at the Council Meet­ S. Lang B. Yood ing, and his second-hand account of our reasons for rejecting the article is not The Secretary supplies background correct. The essence of the arguments at for the parenthetical remark. the meeting was well-summarized by the By-laws, Article IV, Section 4: Secretary, Everett Pitcher, who wrote in a "The method for settling matters before letter to the Council members: the Council ... shall be by a majority vote "There were two lines of argument of the members present. If the result of a presented or implicit at the meeting ... vote is challenged, it shall be the duty of "I, Men of good will who are politi­ the presiding officer to determine the true cally aware should seize any available vote by a roll call .... Fractional votes platform to promote the views which they shall be counted," (See Bulletin 73 (1967), know are right. 989.) "II. The declared purpose of the Council Minutes, April 19, 19 68: Society is 'the furtherance of the interests "It was moved and seconded that the Editors of mathematical scholarship and re­ be instructed to puBlish the article. Initially search.' Each proposed action of the So­ the Presiding Officer ruled that a majority ciety should be judged by that standard vote of the members present favored the and no other. motion. The result was challenged and the "I think they are the lines over true vote was determined, as specified in which the division in the vote took place." the By-laws, by a roll call in which frac­ tional votes were counted. ---the motion Of course, each side insists that lost," it is acting on the scientific merits, where-

609 as the other is politically inspired. Argu­ reports in the cNo!iceiJ. If this means re­ ments over whether the article is "politi­ signing ourselves to the inevitable crisis cal" or not end up in semantics: in parti­ of "censorship," that's something with cular, whether a document can be non­ which editorial boards everywhere must, political in content and yet be political in and do live. its effect. In this regard, we feel the over­ Arthur Matuck Gian-Carlo Rota all impression the report makes is approximately that of Harrison Salisbury's well-known New York Times articles on life in Hanoi, except that it focuses on a Editor, the cJ{oticei) narrower aspect of North Vietnamese life. I am writing about an experience We voted against printing the report which must be common to many editors of because as Pitcher suggests, we construe mathematical periodicals. After sending the Society's function narrowly and felt out an author's paper with the referee's that the article was of minimal interest to suggestions, one receives a revised ver­ mathematicians ~ mathematicians. In­ sion from the author, with a letter expres­ deed, it seems ludicrous to us to seriously sing thanks to the referee, but no mention compare it in mathematical interest to the in the paper itself. report on Chinese mathematics. Such ac­ It seems to me that if the referee's counts of mathematical life in various work on the paper was such as to merit an small and underdeveloped countries have acknowledgment, this should be a public been appearing from time to time in the one, even though he, of course, remains Monthly, and that seems to be a reasonable anonymous. outlet for them, but we would vote in the same way against printing any of these P.M. Cohn

NEWS ITEM

ASSOCIATION FOR MATHEMATICAL GEOLOGY

A thirteen member committee with Elections of officers for the new Associa­ international representation is working on tion will be held in Prague this summer the formation of the Association for Mathe­ during the lUGS annual meeting. For more matical Geology. The Association will be information write to Professor Richard affiliated with the International Union of Reyment, Paleontologiska lnstitutionen, Geological Sciences (lUGS) and will work Uppsala Universitet, Fack, Uppsala 1, closely with the International Association Sweden. for Statistics in the Physical Sciences.

610 ABSTRACTS OF CONTRIBUTED PAPERS The March Meeting in Los Angeles March 21-23,1968

654-34. BASIL GORDON, University of California, Los Angeles, California, and LORNE HOUTEN, Washington State University, Pullman, Washington 99163. Asymptotic results in plane partitions.

In Abstract 642-38, these cJVoticei) 14 (1967), 71-72, the notion of a plane partition, monotonic on rows, is defined. The authors have shown that the generating function, for b(n) the number of such partitions, is r::o b(n)xn = n~ 1 (1 - X p) -[( v + l)/2]. The authors obtain in this case that b(n) ~2- 3 / 4 (311'" 5(3))- l/2 (n/ I (3)) - 49172 exp [3 /2 1(3)(n/ 1{3 )2/ 3 u 2 (n/ w)) l/3 /24 - 'II" 4/3456 1(3) + J~

654-35. H. B. RIBEIRO, Pennsylvania State University, University Park, Pennsylvania 16802. On atomisticity in dependence algebras.

Basic results immediately related to the notion of dependence for vector spaces had been established within the framework of Boolean algebras with an idempotent and monotone unary operator in a purely elementary way- -arithmetically, non set theoretically, without the help of the finite character of dependence or the completeness of the Boolean algebra. However, the atomisti­ city of the Boolean algebra was used to obtain, within such framework, that each maximal free element generates. It is now shown that this ad hoc assumption may also be removed. (Received January 29, 1968.)

654-36. A. F. ROBERT, Stanford University, Stanford, California 94305. A numerical procedure for the determination of compressible fluid flows.

In J. Rational Mech. Anal. 4 (1955), 883-905, Bergman indicated a procedure for the deter­ mination of the flow pattern of compressible fluids. This method involves the evaluation of double integrals and solutions of a boundary value problem. The author discusses the computational program for solving the above problems using high speed computing machines. In this work he considers the case of subsonic flows near the sonic line. In many cases the flow can be continued into the super­ sonic region. In this way one obtains flows of mixed type (transonic flows). (Received February 1, 1968.) (Author introduced by Professor Stefan Bergman.)

654-37. MACIEJ SKWARCZYNSKI, Stanford University, Stanford, California 94305. The invariant distance in the theory of pseudo-conformal transformations and the Lu-Qi-Keng conjecture.

For a bounded domain D C en the correspondence t-> K(z,t•) (where tED and K(z,t•) is the Bergman kernel function of D) defines an analytic imbedding of the domain D into a Hilbert space L 2 H(D). As was pointed out by Kobayashi [Geometry of bounded domains, Trans. Amer. Math. Soc. 92 (1959)] that the norm in L 2H(D) induces in a natural way the pseudo-conformally invariant distance

611 p(p,t) in D. The author shows that p(p,t) can be expressed by the kernel function, namely, p(p,t) = (Z - Z(K(t,p*)K(p,t*)/K(p,p*)K(t,t*)) 1/Z)!/Z. This fact yields a new and simple proof of the theorem that the Bergman metric is positive definite. Also it yields a new formulation of the Lu-Qi-Keng conjecture on zeros of the kernel function. !l;ee On Kaehler manifolds with constant curvature, Acta Math. Sinica (Z) 16 (1966).] (Received February 1, 1968.)

654-38. H. M. LIEBERSTEIN, Wichita State University, Benton, Kansas 67017. Viscous flow properties in the operation of a nephron.

The salient feature of a nephron is its tiny size, six to ten microns radius, since the forth power law requires a tiny flow rate even from a large pressure gradient. Along closely folded hair­ pin loops, present in the nephrons of birds and mammals and called loops of Henle, a salt concentra­ tion gradient forms in the ambient medullary tissue. Urine collects in this tissue in ducts, equili­ brates with it osmotically, and produces a final product hypertonic to blood. Other authors explain the mechanism of this loop in terms of an hypothesis of active extrusion of a small amount of sodium from one branch of the loop and operation of a countercurrent multiplication principle. By close attention to realistic physical principles, we construct a model that does not use this hypothesis but produces in numerical studies the observed concentration gradient and an amplification of this effect with length. A much weaker assumption than similar ones made in other models, that two salt concentrations take on stationary values, causes a linear initial value problem for a (Z X Z) first order ordinary differential equation system to replace a (4 X 4) first order partial differential equation system. Basic mechanisms used such as back diffusion were of no importance in previous square law models. Work supported by NASA Contract NSR 39080001. (Received February ZO, 1968.)

654-39. R. R. DOUGLAS and F. SIGRIST, University of British Columbia, Vancouver, Canada. Sphere bundles over spheres which are H -spaces.

It follows from a theorem of J. F. Adams (Topology 1 (!96Z), 67-72) and our theorem (below) that if a q-sphere bundle over the n-sphere is an H-space (q, n positive), then either both q and n belong to the set {1,3,7}, or (q,n) is one of the pairs (!,Z), (3,5). (All of these possibilities are realized by either products of spheres or S0(3), SU(3), respectively.) Theorem. If the integral' cohomology ring H*(X), of the space X, is a torsion-free, exterior algebra on two generators whose dimensions are either (7,11) or (7,15), then X is not an H-space. An interesting consequence of this theorem concerns Stiefel manifolds. The manifold W 3,Z of (3 X Z) orthogonal quaternionic matrices, and the manifold Y Z,Z of (Z X Z) orthogonal matrices over the Cayley numbers are 7-sphere bundles o.ver the !!-sphere and the IS-sphere, respectively. Therefore, these Stiefel manifolds are not H-spaces. (Received March 18, !968.)

612 The June Meeting in Portland, Oregon June 15, 1968

657-1. R. W. CHANEY, University of California, Santa Barbara, California 93106. On uniformly approximable Sidon sets. Preliminary report.

Let G be a compact abelian group and let I' be its character group. Suppose E C I'. E is a uniformly approximable (u.a.) Sidon set if it is a Sidon set and if there is a sequence of measures in the measure algebra M(G) whose Fourier-Stieltjes transforms converge uniformly to the characteris­ tic function of E. Then E is a u.a. Sidon set if and only if for every E > 0 and every bounded function f onE there exists 11. in M(G) such that P. = f onE and 1;11 < E off E, or, equivalently, if and only if for

every E > 0 there exists B > 0 such that if g is any function on E which vanishes at oo then there exists f in L 1 (G) so that f = g on E, llf 111 ;:;! B llg I~ , and lfl < E llg 11 00 off E. A third equivalent condition is that whenever f is a bounded function on r which vanishes off E and {An j is a norm bounded sequence of measures in M(I') so thl!t lim An = 0 pointwise on G, then lim Jr fdAn = 0. All known (to the author) Sidon sets are u.a. Sidon sets. (Received November 16, 1967.)

657-2. MARCEL HERZOG, University of California, Santa Barbara, California 93106. On finite groups with a cyclic Sylow subgroup.

A detailed information concerning the nonexceptional irreducible characters of a finite group G with certain cyclic Sylow subgroup is contained in: Theorem. Let G be a finite group containing a. subgroup M of order m and satisfying the following conditions: (i) for every element h of M#, CG(h) is contained in M; (ii) a Sylow p-subgroup of M is cyclic and nontrivial; (iii) NG(M) f. M,G. Then the nonexceptional irreducible characters of G, non vanishing on M #, are of one of the following

two types: (I) Pi(i = l, •.• ,y) where Pi(l) = r 1m + 1, Pi (b.)= 1 for all h in M#; (II) Qj(j = l, ••. ,q- y) where Q.(l) = s .m - 1, Q.(h) = - 1 for all h in M#; where q = I_NG(M): M] and r ., s. are nonnegative J J J 1 J integers. As a consequence, the following characterization of the simple groups PSL(2,p) and PSL(2,2b) is .obtained: Corollary. Let G and M satisfy the assumptions of the theorem. In addition, let G' = G and [G: NG(M)] < (m 2 + 3m + 2)/2. Then G is isomorphic either to PSL(2,p), p > 3, or to PSL(2,2b), where m = 2b + 1 > 3. (Received February 5, 1968.)

657-3. KURT KREITH, University of California, Davis, California 95616. Lower bounds for the essential spectrum of fourth order differential operators.

For a second order Sturm-Liouville operator k defined by ku = - (pu')' + qu on Y 2(0,oo), there is a :well-known relation between the infimum of ue(k) (the essential spectrum of k) and the oscillation properties of ku = 0. By establishing a similar relation for the fourth order differential operator l defined by lu = (pu")" + qu on Y 2(0,oo), one is able to establish estimates for inf ue(l) similar to those established by K. 0. Friedrichs for inf u e (k). For example: If there exists a real n such that lim x-2 -np(x) = p > 0 and if Z(x) = q(x) + p ((1- n2)2/16)xn-2 , then lim inf Z(x) X-tOO 0 0 X-+00 ;:;! inf u (l) ;:;! lim sup Z(x). (Received February 22, 1968.) e. x-+oo

613 657-4. J. H. JORDAN, Washington State University, Pullman, Washington 99163, and N. R. HARDMAN, Harvard University, Cambridge, Massachusetts. The distribution of quadratic residues in fields of order p 2.

Let p be a prime of the form 4k + 3. The lattice points in the square with vertices at (±. p/2, ± p/2) form a model for the field of order p2. The squares of this field are called quadratic residues of the field. If we define (a/p) = l if a is a square and - 1 otherwise, then the following are established by elementary methods. (i) (a/p) = a

657-5. C. T. LONG, Washington State University, Pullman, Washington 99163. "Decimal" expansions to negative bases.

Consider the following infinite sequences of integers lmd i~l' jMJi~l' lsdi;;;l' jtdi;;;l'

luili ;;; 1, lvili;;;l' and jvJi;;;l where, for each i;;; 1, mi;;; 2, Mi = Ilj=l mi' si = 0 or 1 with infinitely S· i for which si = 0 and infinitely many i for which si = 1, ti = (- 1) I and ui = 0 for si = 0, ti = 0 and S· si-1 oo sk ui. = (- 1) I for s. = 1, v. =min(- 1) M. l"k .(- 1) y /M , 0 ;;; yk < mk, and vi.= I I I-£...., =I k k max(- l)si-1Mi_ 1 :L~ 1 (- l)sky~Mk, 0;;; yk < mk. The purpose of this paper is to show that every real number a with v 1 ;;; a < V 1 is representable in the form a= L~ 1 (- l)siy./M., 0 ;;; y. < m. 'Vi S· S· - I I I I and that if a= ~ 1 (- 1) Iy/Mi = L~ 1 (- 1) Iz/Mi are two such representations for a where y r > zr are the first unequal coefficients, then Yr = Zr + 1 and Yi = 0, zi = mi - 1 if si = sr (mod 2), yi = mi - 1, zi = 0 if si ¢ sr (mod 2) for all i > r. (Received March 8, 1968.)

657-6. L. E. WARD, JR., and E. D. TYMCHATYN, University of Oregon, Eugene, Oregon 97403. On three problems of Franklin and Wallace.

In their paper The least element map (Colloq. Math. 15 (1966), 217-221) Franklin and Wallace

ask these questions. Let X be a compact Hausdorff space, R a reflexive relation on X and ~the family of compact subsets of X having an R-least element. (1) If R is a closed quasi-order, is the set of

maximal numbers of ~closed (in the Vietoris topology)? (2) If R is a closed partial order, is the set of maximal R-chains closed? (3) If R is a closed quasi-order, under what conditions will R contain a closed partial order which is chain-equivalent to R? It is shown that (1) and (2) have negative answers. In (1) it can be shown that if R is a partial order, then the answer is yes if and only if the

set N of R- minimal elements is closed and the map x - [ y: x ~ y} on N is continuous. In (2) the answer is yes if (X,R) is a topological lattice. A variety of information is available for (3), but none of these answers is sufficiently succinct to state here. (Received March 18, 1968.)

614 657-7. D. M. SILBERGER, Western Washington State College, Bellingham, Washington 98225.

The parity of the integer (2n - 2)~/n~(n - 1)~.

Analytic proofs have been given by P. Quarra and by j .H.M. Wedderburn that the number of formally distinct nth powers of an element x in a groupoid is (2n - 2)~/n~(n - 1)~. There is an elementary proof of this fact by Ivan Niven, and one of an equivalent fact by C. F. Gum mer. We present yet another elementary proof thereof which relates this powers question to the number of cyclic permutations of n black objects and n - 1 white objects by noting that each such permutation corresponds to exactly one formal nth power of x, and that for each such permutation there is exactly one object from which to start counting clockwise so that the count of black objects always exceeds the count of white objects. We then prove the Theorem. The integer (2n - 2)~/n~(n- 1)~ is odd if and

only if n is a power of 2. (Received March 18, 1968.)

657-8. D. R. HORNER, Eastern Washington State College, Cheney. Washington 99004. p-strictly singular operators.

Let E [t 1] and F [t2] be (Hausdorff) locally convex topological vector spaces with Q(t1) and Q(t2) collections of seminorms generating t 1 and t 2 respectively. Let p E Q(t1). A continuous linear operator T: E [t 1] - F[t2) is said to be p-strictly singular if for each (closed) infinite-dimensional vector subspace M of E [tJ] and q E Q(t2), q(Tm) < p(m) for some m E M. A continuous linear

operator T: E [t 1] - F [t2] is p-strictly singular if and only if for each (closed) infinite-dimensional vector subspace M C E [t 1] and q E Q(t2), there is an infinite-dimensional vector subspace N C M with q(Tn) < p(n) for all nonzero n EN. The collection pSS(E,F) of p-strictly singular operators

T: E [t 1] - F [t2] is a linear subspace of C(E,F), the class of appropriate continuous linear operators. The following is also such a linear subspace: PSS(E,F) = Up E Q(tl)pSS(E,F). In fact, PSS(E,E) is a two sided ideal in C(E,E). (Received April 1, 1968.)

657-9. H. M. SRIVASTAVA, West Virginia University, Morgantown, West Virginia 26506, and OM DUTT A VY AS, jodhpur University, jodhpur, India. A theorem relating generalized Hankel and Whittaker transforms. Preliminary report.

Recently, an elegant unification of the various generalizations of the classical Laplace trans­ form due to c. S. Meijer [?roc. Kon. Nederl. Akad. Wetensch. 43 (1940), 599-608; ibid. 44 (1941), 727-737] and R. S. Varma [Current Sci. 16 (1947), 17-18; Proc. Nat. Acad. Sci. India Sect. A 20 (1951), 209-216] was introduced by H. M. Srivastava by means of the integral equation (1) S(pk,u) [f(t): p) (pt)u-l/2exp (- (l/2)qpt)Wk m(ppt)f(t)dt, where, if f(t) = O(t0eEt ), then q, ,rn = Fno , R [(q + p)p - 2 f] > 0, and R(u + o :1:. m + 1) > 0. In the present note the authors establish a theorem that relates the Whittaker transform (1) with the generalization of Hankel transform also given by H. M. Srivastava jMatematiche 21 (1966), 199-202]. Various specialized cases of the theorem in­ clude the results proved earlier by H. M. Srivastava [Math. japon. 11 (1966), 11-13], R. K. Saxena [Math. z. 81 (1963), 414-415], K. C. Sharma [Proc. Glasgow Math. Assoc. 6 (1963), 107-112], and B. R. Bhonsle [!'roc. Glasgow Math. Assoc. 5 (1962}, 114-115]. (Received March 29, 1968.)

615 657-10. WITHDRAWN.

657- 11. E. F. ECKL UNO, JR., Western Washington State College, Bellingham, Washington 98225. Prime divisors of the binomial coefficient.

n The Sylvester-Schur Theorem implies that the binomial coefficient (k), with n ~ 2k, has a prime divisor p > k. The present paper considers "complementary" results: If n ~ 2k, then the bi­ nomial coefficient (~) has a prime divisor p :"' f r. Theorem II. If n ~ 2k, then(~) has a prime divisor p ,; max[n/k, n/2 }, with the exception of . (Received April 1, 1968.) (Author introduced by Professor M. L. Faulkner.)

657-12. W. L. BUCK and D. R. HORNER, Eastern Washington State College, Cheney, Washington 99004. The strict singularity ofT and S does not imply the strict singularity ofT+ S.

H. E. Lacey posed the question: Is the set SS(E,F) of all strictly singular operators from

E [t 1] to F [t 2] a (closed) vector subspace of C(E,F), the set of continuous linear operators from E ~1J to F ~ 2 ]? The answer is known to be affirmative for E ~ 1 ] a normed space. SS(E,F) need not be closed in C(E,F). Furthermore, SS(E,F) need not be a linear space. DefinedT,S: 11 -> !1 ~k(c 0 , 11 )] X 11 by Tx = (x,x) and Sx = (- x,O). The operators T and S are strictly singular while T + S is not. (Received April 4, 1968 .)

657-13. j. D. FUELBER TH, University of Southern California, Los Angeles, California 90007.

On commutative splitting rings.

The concepts. of torsion and T-rings are as inS. E. Dickson's paper Decomposition of modules. II: Rings without chain conditions, Math. Z. (to appear). All rings are commutative with unit and all R-modules are unitary. Definition. A ring R is a splitting ring iff for all modules the torsion sub­ module is a summand. In his paper, Noetherian splitting rings are Artinian, j. London Math. Soc. 42 (1967), 732-736, Dickson conjectures that all splitting rings are torsion and shows that the conjecture holds for commutative Noetherian rings. The following theorems show that the conjecture is valid for a larger class of rings. Theorem. Let R be von Ne4_mann regular. R is a splitting ring iff R is tor­ sion. Theorem. Let R beaT-ring. Then the following are equivalent: (i) R is a splitting ring,

(ii) R has a finite number of maximal ideals and is a splitting ring, (iii) R is torsion. (Received

April 10, 1968.)

616 657-14. N. R. GRAY, Western Washington State College, Bellingham, Washington 982Z5. On a problem of Borsuk. Preliminary report.

A metric space (X,d) is said to be SC (strongly convex) provided that, for every pair of points x,y E X, there is a unique point z EX such that d(x,z) = d(z,y) = (l/2)d(x,y): A subset of a metric space is SC if it is SC in the relative metric. It is well known that a compact SC metric space is contractible and locally contractible, and thus, if finite-dimensional, is an absolute retract for the class of m·~trizable spaces. (Abbreviation: X E AR(1) .) Also, a finite- dimensional compact metric space in which each point has at least one SC neighborhood is an absolute neighborhood retract for the class of metrizable spaces. (Abbreviation: X E ANR(.£).) Borsuk asks (K. Borsuk, Theory of retracts, Ars Polona, 1967, p. 219) whether in either of these cases the condition of finite-dimen- sionality may be dropped. Introducing the concept loc Sc (locally SC, meaning that every point of the space has a neighborhood basis consisting of SC neighborhoods), we get Theorem 1. If X is a locally compact metric space which is loc SC, then X E ANR(..L). Theorem 2. If X is a compact metric space which is both SC and loc SC, then X EAR(.£). (Received April 10, 1968.)

657-15. M. j. DeLEON, Pennsylvania State University, University Park, Pennsylvania 16802 Pisot sequences and Pisot-Vijayaraghavan numbers. Preliminary report.

Definition. A sequence of natural numbers a0 , a 1, ... ,an, ... with 2;;; a 0 < a 1 is called a Pisot Sequence, denoted by PS(a ,a ), iff- 1/2

values of d (which depends on a0 ) such that PS(a0 ,a 1) satisfies a linear recurrence relation if a 1 = d (mod a~. Definition. A Pisot-Vijayaraghavan (PV-) number is an algebraic integer greater than one such that all of its conjugates, with the exception of itself, have moduli strictly less than one. Definition. The polynomial P(z) is called a PV-polynomial iff it is the minimal polynomial of a PV-number. Vijayaraghavan [!'roc. Cambridge Philos. Soc. 37 (1941), 350] showed that there exist PV-polynomials of all degrees. The roots of the polynomials he presented were not given explicitly. For every natural number n, a PV-polynomial P n (z) of degree n is given. Also the roots of P n (z), two real if n is even but only one real if n is odd, are given explicitly. (Received April 12, 1968.)

657- 16. D. C. KENT, Washington State University, Pullman, Washington 99163. A representation theorem for uniform convergence structures.

Theorem. The following statements about a convergence space (S,q) are equivalent: (1) The space is uniformizable (in the sense of being compatible with the generalized uniformity defined by C. H. Cook and H. R. Fischer). (2) q belongs to the closure with respect to the order topology of the set of all completely regular topologies in the lattice C(S) of all convergence structures on the setS. (3) q is a limit space with the property that Y n !§ q-converges to x andy whenever Y is a filter

which q-converges to x, ~is a filter which q- converges to y, and Y and ~ fail to contain disjoint sets. (Received April 12, 1968.)

617 657-17. WITHDRAWN.

657-18. B. G. EKE, University of California, San Diego, La Jolla, California 92037, A note on mini- gaps.

Suppose f(z) = L:oanzn is a really mean univalent in lz I < 1, with an = 0, if n = nk (k ~ 0), and suppose n k+l - nk :? C < + oo if k " k0 . A discussion of the behaviour of the sequence {nk }~ will be given for the subclass of functions whose maximum moduli grow as swiftly as possible. (Received April 22, 1968 .)

657-19. KI CHOUL OUM, State University College, Buffalo, New York 14214. Bounds for the number of deficient values of entire functions whose zeros have angular densities.

Let f(z) be an entire function of finite order X. Arakeljan has shown that, for every X > 1/2, f(z) may have infinitely many deficient values in the sense of R. Nevanlinna. The author shows that this cannot happen if (i) the zeros of f(z) have an angular density (in the sense of Pfluger and Levin) and (ii) X is not an integer. Under these two assumptions the number of deficient values cannot exceed 2X + 1. If f(z) is of completely regular growth (in the sense of Levin), the result also holds for integral values of the order. The proofs rely heavily on a lemma of Edrei and Fuchs [?roc. London Math. Soc. 12 (1962), 321, Lemma I]. (Received April 22, 1968.)

657-20. R. T. HANSEN, Montana State University, Bozeman, Montana. Complementing pairs of subsets of the plane.

Let A and B be subsets of zn, where Z denotes the set of integers and n is a natural number. Pair (A,B) is called a complementing pair for A+ B provided each element of the sum is uniquely re­ presented and A n B contains the "zero" element of Zn, Such a pair is denoted by (A,B) ~A + B, Complementing pairs for A+ B equal to N = [0,1,2, ... } and Nm = {0,1,2, ... , m - 1} have been charac­ terized by N.G.de Bruijn [?ubl.Math,Debrecen l (1950),232-242; Nieuw Arch. Wisk.4 (1956), 15-17] and C. T, Long [to appear in Pacific]. Math.]. Their addition theorems generalize to the so-called proper pairs in the plane which have the property that (x,y) is in a component iff both (x,O) and (O,y) are in the same component, A complementing pair (A,B) is proper iff the respective projections on each axis form a complementing pair for the axis interested with A + B, All complementing pairs

for Nn X Nm are proper, Those complementing pairs of a quadrant for which the components are

finite on distinct axes are also proper. Pair (A,B) ~ C is called unique provided that for any pair

(A' ,B) ~ C we have A= A'. Complementing pairs for subsets of N X N are unique. Also, if (A,B) ~

C C Z X Z and Cis finite, then (A,B) is unique. (Received April 18, 1968.)

657-21. B. V. HEARSEY, Washington State University, Pullman, Washington 99163. Regularity for convergence structures,

Let C(S) be the set of convergence structures on S, ordered in the usual manner. Theorem l. The closure with respect to the order topology on C(S) of the set of regular topologies on S is equal

618 to the closure of the completely regular topologies (which is known to be equal to the set of all Cook­ Fischer uniformizable convergence structures). Theorem 2. The closures with respect to the order topology on C(S) of the T 2, T 3, and T 4 topologies are equal, and equal to the set of all T 2 convergence structures. Theorem 1 is used to define order regularity for a convergence structure. Four other possible definitions for regularity of a convergence structure are given, including the Cook- Fischer definition. These five definitions are compared for convergence structures and for pretopologies. In particular, it is shown that the Cook- Fischer concepts of regularity and uniformizability are inde­ pendent. On productivity and heredity the following is proved: Theorem 3, Strong, Cook- Fischer­ Biesterfeldt, and order regularity are hereditary; t and weak regularity are not hereditary. Cook­ Fischer- Biesterfeldt and order regularity are productive; t, weak, and strong regularity are not productive. (Received April 18, 1968.)

657-22. R. Y. LEE, University of Minnesota, Minneapolis, Minnesota 55455. Reduction of

_£Eder _for a syste~-~~iffe,E<:!!tial equatio!!~':...':!~rni~~· Preliminary report.

Consider equation (l) fdy/dx = A(x,f)y, where xis a complex variable, £a small complex parameter, y ann-dim. vector, A(x, f) an n-by-n matrix function holomorphic in x, fin the domain lx I ;§ xo· 0 < I (I ~ to· larg (I ;§ fQ • Let A(x, f) = Ao (x) + fA(x, f). Assumptions. (i) Ao(x) =

;§ (ii) (x) xPA:0 (x), where pis a positive integer and 'A0 (x) is holomorphic in lxl x0 with A0 (0) t- 0. A0 is a 2-by-2 block diagonal matrix in which none of the eigenvalues of the upper left diagonal block of real a. and for every A0 (0) is equal to that of the lower right diagonal block. (iii) For a fixed /3 (0 < 8 <1r/(p + 1)), A(x, £) admits an asymptotic expansion, as £- 0, valid uniformly in the region of A (x) S(x0 ,a., {3) = [xl lxl ;§ x0, larg x- a.l ;<> {3}. Assumption (i) asserts that all eigenvalues 0 coalesce at x = 0, making x = 0 a turning point of(!), Theorem, If assumptions (i), (ii), (iii) are satisfied, then there exists a function P (x, f) such that the transformation y = P (x, f)z takes ( 1) into

(2) £dz/dx = B(x,(jz, where B(x,£) is a 2-by-2 block diagonal matrix having the same partition as as A0 (x) and B(x,d = A0 (x) + n3(x, £). The matrices B(x,£) and P(x,£) have asymptotic expansions £ - 0, which are valid uniformly for x in any proper subsectors of S(x1,a.,{3), 0 < lx 1 1 ;§ x0 . An appli­ cation of the theorem will be given. (Received April 25, 1968.)

657-23, JOSEPH ZAKS, University of Washington, Seattle, Washington 98105. On finite decompositions of (2n- I)-spaces.

It is known that if G is an u,s,c. decomposition of Em, with only finitely many nondegenerate 2 elements, then the decomposition space Em /G of G is embeddable in Em+ (L. V. Keldy!!, Proc. Sympos. Prague, 1961, p. 230-234). The codimension 2 is the best possible one, at least for all odd dimensions m, in the following sense: Theorem, For each integer n, n > l, there exists a monotone decomposition G of E 2n-l, with only 2n + 3 nondegenerate elements, such that E 2n- 1 /G is not em­ beddable in E 2n; moreover, the nondegenerate elements of G can be so chosen as to be p.w.l. homeo- morphic to the (n- l) skeleton of the (2n + !)-simplex (and hence of dimension n- 1). For n = 2, there are better results, for example D. Gillman's A five circle decomposition of 3- space (Abstract 636-65, these cNoticeiJ 13 (1966), 594). (Received April 8, 1968.)

619 657-24, J, R. REAY, Western Washington State College, Bellingham, Washington 98225, Projections off-vectors of 4-polytopes,

If Pis a polytope in Rn, its f-vector is defined as f(P) = (f0 ,f1, .. .,fd- 1) where fi is the number of i-dimensional faces of P. It is a well-known unsolved problem to determine the set of all fi i-facesJ. (f0 ,f1,f2 ,f3 ) which represent a 4-polytope. Let 11"(0,2) = [(\) ,f2) 1:[ 4-polytope P with Lemma. If (f0 ,f2 ) ~ 11"(0,2), then 10 "" f 2 :'§ f6 - 3f0 and fo :'§ f 2 - [(1 + Jl+Sfz)/2], The converse is false, and the complete set of (~ ,f2 ) which satisfy the equations but do not represent any 4-polytope are given for various cases. (Received April 29, 1968,)

657-25, B. T, SIM3, Eastern Washington State College, Cheney, Washington 99004. Some generalizations of the Contraction Mapping Theorem, Preliminary report.

The Contraction Mapping Theorem, which asserts that every contraction mapping on a complete metric space has a unique fixed point, has been generalized in a number of ways in recent years. J. Mathews showed in his Thesis (Iowa State University) that every contraction mapping on a develop­ able space with a strong development has a unique fixed point, Two elementary generalizations of the Contraction Mapping Theorem, which require only minor alterations in the standard proof (cf, Ko1mogorov and Fomin, Functional analysis, I, p. 43), are the following results: Theorem 1,

If f is a contraction on a bounded and complete semimetric space (S,d), then f has a unique fixed point, The or~~ 2. If f is a contraction on an a- metric space ( S,d), where a > 0, then f has a unique fixed point, Moreover, the nonnegative iterates off determine for each point of S a finite semiorbit terminating at this fixed point, (Received April 29, 1968,)

657-26, R. CHALICE, Western Washington State College, Bellingham, Washington 98225. On the essential set,

Define the essential set E of a function algebra A on a compact Hausdorff space X as the minimal closed set in X supporting every measure in A .L. Using the fact that any representing measure has as closed support a set of antisymmetry, one finds that the result of Hoffman and Singer that every representing measure on X which is not a point mass lives on the essential set is immediate, From this the results of H. S, Bear follow immediately. If A is approximately normal on X, then the P where P [x E XI :[a closed nbhd V of x3: Al'if = C(V )}. Using this, it essential set E =X- = X X X follows that if X= U~ F., where the F. are closed and AIF. = C(F.), then A= C(X). This result has 1= 1 1 I I 1 also been proved by Mullins and by Gamlin and Wilken in a different fashion. (Received April 29, 1968,)

657-27, CHARLES ATHERTON, Washington State University, Pullman, Washington, Concerning the ideal and order topologies on a lattice, Preliminai'y report,

If A c L, let c(A) denote the closure of A in Frink's ideal topology. Theorem. Let B be a

Boolean algebra and A c B. (i) If A is closed under finite joins (meets), then x E c(A) iff x E c(A nx*) (x E c(A n xl). (ii) If A is closed under finite joins (meets), then x E c(A l·lx*) (x E c(A nx+)) only if xis a finite meet (join) of elements from A n x• (An x+). For a discussion of autd-topologies, see

Rema (J, Indian Math, Soc. 30 (1966)), Theorem, Let B be a Boolean algebra, I an ideal of B and D = [a' E B: a E I}. The topology TID having a subbase of all sets of the form a• n b+ (a E I, b ED) is

620 auto-uniformizable and coincides with T D" Corollary. The order topology on an atomic Boolean algebra is auto-uniformizable. Kent (Illinois J. Math. 10 (1966), 90- 96) defined three conditions genera ted; the (c 1, c2 ,c3) on a lattice. Theorem. A lattice L satisfies c 1 and c 3 iff L is bicompactly compact elements of L form an ideal and the cocompact elements of L form a dual ideal. Theorem.

On any complete lattice satisfying c 1 and c 3 the ideal topology is finer than the order topology. (Received April 30, 1968.)

657-28. G. R. KRAUSE, Washington State University, Pullman, Washington 99163. A class of modules over a left-noetherian ring.

Let R be a left-noetherian ring with identity and M a unitary left R-module. A prime ideal P of R is called associated with M if it is the left annihilator of all submodules t- 0 of a nonzero sub­ m::>dule N of M. If P is furthermore left annihilator of all elements t- 0 of N, P is called strictly associated with M. M is called admissible if all its associated prime ideals are strictly associated and it is shown that M is admissible if and only if the injective hull E(M) is the discrete direct sum of injective indecomposable modules of the form E(R/P), where P runs through the set of all primes associated with M. This result is a generalization of a result obtained by E. Matlis l!'acific j. Math. 8 (1958), 518]. Theorem. The following properties of Rare equivalent: (l) R is a finite direct sum of local, left-artinian rings. (2) R has the Artin-Rees-property for left ideals and all prime ideals are maximal left ideals. (3) All unitary left-R-modules Mare admissible and each prime ideal containing the left annihilator of M is associated with M. (Received April 26, 1968.)

657-29. D. L. RAGOZIN, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. Uniform convergence of spherical harmonic expansions.

Let M = Sq (q "'2) and decompose L 2 (M) as $'fHn where Hn is the set of spherical harmonics .1. of degree n. (Alternatively, Hn = ~ n @ n _ 1 where @n = (p tM: p is a polynomial, deg p ~ n }.) Each f in L 2 (M) has a unique L 2-expansion f = I; f., f. E H., with partial sums L f = 2) nlfJ.• For f E ck(M), J J J n let w(f(k); 6) be the sum of the moduli of continuity of some ample set of kth derivatives of f.

E @ such that Jlf- p Jl = Previous work of the author yields Theorem 1. Iff E Ck(M), then 8pn n n oo O(n-kw(f(k); 1/n)). The use of known estimates involving jacobi polynomials gives Theorem 2. The map Ln: C(M)- C(M) has 1\Lnll = O(n(q-l)/2). Following Lebesgue, these two results are combined

to prove the main result of the paper which is the following analog of the Dini- Lipschitz theorem on uniform convergence of Fourier series. Theorem 3. If k = [(q- l)/2] and f E Ck(M), then Lnf- f uniformly, provided either (i) q is odd or (ii) q is even and w(f.kl; 6) = o( 6112). (Received May l,

1968.)

657-30. R. E. ATALLA, Ohio University, Athens, Ohio 45701. Linear operators on spaces of continuous functions.

Let X,Y,Z be completely regular T 2-spaces, C(X), C(Y), C(Z), the corresponding spaces of bounded complex-valued continuous functions. Let A be a continuous linear operator from C(X) to C(Z), and D a continuous linear operator from C(X) to C(Y) such that range (D) is dense in C(Y).

621 If S c V, where Vis a topological vector space, we define the k-hull of S, [SJk' to be the closure of elements of the form I:tiSi, Si E S, ti real, I:ti =I, I; /ti/ ~ k, I~ i ~n. Theorem. Let A and D be as above. Then the following are equivalent: (a) There is a linear operator B from C(Y) to C(Z) such that 1/B /1 = k, B I = I, Bf is real valued iff is, and A= BD. (b) For each f E C(X), Af(Z) c [pf(Y)l, and this fails fork' < k. Letting X= Y = Z, A= B, D be the identity, and k = I (!-hull= closed convex hull), we obtain a special case of R. R. Phelps [Proc. Amer. Math. Soc. 16 (1965), 381-382]. Using the representation of matrix operators on sequence spaces given in Sequential cores and a theorem appear in these we can prove Theorem. If (a ) is a real of Phelps, with J. Bustoz fro cJ{oticei) ], nm valued regular matrix, then lim supm _00 I;: 1/anm / is the least number k such that core (Bx) c [core xJk for each bounded sequence x of complex numbers. (Received May I, 1968.)

657-31. ALEXANDER ABIAN, Iowa State University, Ames, Iowa 50010. Decomposition of atomic and orthogonally complete rings.

Let R be a ring in which for every x E R there exists a natural number n(x) > I such that xn(x) = x. Lemma. The ring R is partially ordered by ~ where x ~ y if and only if xy = x 2 for every x, y E R. Definition. A nonzero element a of R is called an 2.!.£.!!1 of R provided x ~ a implies x = a or x = 0, for every x E R. Moreover, R is called atomic if for every nonzero element r of R there exists an atom a of R such that a ~ r. Furthermore, R is called orthogonally complete if for every setS of pairwise orthogonal elements of R the supS exists. Theorem. The ring R is isomorphic to a direct sum of fields if and only if R is atomic and orthogonally complete. (Received May !, 1968.)

657-32. KEITH PHILLIPS, California Institute of Technology, Pasadena, California 91109 and MITCHELL TAIBLESON, Washington University, St. Louis, Missouri. Singular integrals in several variables over a local field.

Singular integral transforms of the Calderon-Zygmund type are constructed for Kd, K a non­ discrete zero dimensional locally compact field. The transforms have the form Lf = limk - 00 1/Jk *f, where the kernel ¢k vanishes in a neighborhood of 0, J/x /= 1 'IJc (x)dx = 0, 1/!k is homogeneous of degree 0, and 1/J.: satisfies certain smoothness conditions. In general, 1/Jk ~ L 1• Convolution is with respect to additive Haar measure. Pointwise and Lr (I < r

(Received May I, 1968.)

622 ABSTRACTS PRESENTED BY TITLE

68T-4ll. H. LAKSER, University of Manitoba, Winnipeg, Manitoba, Canada. Free lattices generated by a partially ordered set. II.

For the notation see Part I, Abstract 68T-356, these c}foticei) 15 (1968), 544. Theorem 2. A lattice L generated by a poset Pis FL(P;M,N) if and only if (i) for all x E L, (x] n P is an (M,N) ideal and dually; (ii) for all x, y E L, (xvy] n P = ((x] n P)v((y]nP) (join of ideals in P) and dually; (iii) givenx0,xl'y0,y 1 EP suchthatx0Ax1 ~ y0 vy 1, x 0 Ax 1 ~yi' xi ;ty0 vy1 , i= 0,1, then there is apE P such that x 0 Ax1 ~ p ~ y 0 vy 1. The characterization of CF(P), R. A. Dean, Trans. Amer. Math. Soc. 83 (1956), 246, Theorem 6, is a corollary of this result. Theorem 3. Let L be a lattice and L* a completion of L. Any isotone f: P -> L has an isotone extension F: FL(P;M,N)

---> L*; if all pseudo-principal (M,N)-ideals and -dual ideals are finite set unions of principal ones, then Im(FL(P;M,N)) <; L. A special case is in Yu. Sorkin, Mat. Sb. N. S. 30 (1952), 677-694. Let L be the free product of lattices Li' i E I, amalgamated by a finite sublattice M. Theorem 4. There exists an algorithm to determine whether A ~ B holds in L in terms of the lattice structure of the Li and M. (Received January 15, 1968.) (Author introduced by Professor G. A. Gratzer.)

68T-412. C. J. MOZZOCHI, University of Connecticut, Storrs, Connecticut. On the completeness of a symmetric generalized uniform space. I.

Let (X,~) be a symmetric generalized uniform space (Abstract 67T-612, these c}foticei) 14

(1967), 839). Definition l. (X,~) is totally bounded iff for every V in 'U there exists x 1, ... ,xn in X such that X= V(x 1) U ... U V(xn). Definition 2. A filter ff in X is weakly Cauchy iff for every U in G2£ there exists x in X such that U[x] E ff. Theorem l. If (X, UZl) is a symmetric uniform space, then a filter ff in X is Cauchy iff it is weakly Cauchy. Theorem 2. If (X,3'-) is a symmetric, connec­ ted topological space, then there exists a totally bounded, symmetric generalized uniformity iJ!l on X such that .5'-(IU) = 5 and every filter in X is weakly Cauchy. Definition 3. (X,i&) is complete iff every weakly Cauchy filter in X has a cluster point. Theorem 3. (X, 02() is compact iff it is complete and totally bounded. (Received February 2, 1968.)

68T-413. LUDVIK JANOS, University of Florida, Gainesville, Florida. On equivalent families of pseudometrics.

Let X be a compact Hausdorff space and I:>= IP(x,y;a)la E 21l a family of pseudometrics on X inducing the given topology of X. Let f: X ->X be a continuous map of X into itself and c > 0. For each a E 'n, we define p*(x,y; a.,f,c) = sup p(f(~), f(~); a.) where the supremum is taken over the set of all nonnegative integers. For each f: X ---> X and c > 0, the family Il(f,c) = IP*(x,y; a,f,c) Ia E 21l is a family of pseudometrics on X and conditions have been investigated under which Il*(f,c) induces the same topology as Il. Theorem l. Iff: X --->X is continuous and c E (0,1), then Il*(f,c) induces the same topology as I:>. Theorem 2. Let F: X---> X be continuous and such that the inter­ section n fn(X) of all iterated images fn(X) of X is a singleton. Then the preceding theorem is valid also for c = l. (Received February 12, 1968.)

623 68T-414, J, N. McDONALD, Rutgers, The State University, New Bruns wick, New Jersey 08903. Maximal measures and the Dirichlet problem of the Choquet boundary,

Let X be a compact T 2 space, let H be a point-separating linear subspace of CR (X) 3 1 E H, and let B denote the Choquet boundary of X w.r.t. H. For x EX, f E CR(X), g ECR(B) with suplg(B)I < co, let f(x) = inf Ih(x) lh E H, h f:: f), let g(x) = inf I h(x) lh E H, h IB f:: g), let J:(x) = ( -=-fXx), and let ~(x) = - (- g)(x). If J.1. is a regular Borel measure on X, J.1. is said to represent x E X, written

J.l. "-'Ex' if fx h dJ.I = h(x) '

J.1. rv fx and Ixf dJ.I. = f(x)<=> there exists a measure t with t rv fx and Ix J. d t = f(x). Theorem. If g is a bounded and continuous function on B 3 giB = ~f8, then the following are equivalent: (1) g can be extended to an H-harmonic function on X; (2) If JJ. 1 and JJ.2 are maximal measures representing x, then I B g dJ.1 1 = I B g dJ.I.?j (3) g = !!:· Using this result, one can derive Alfsen's characterization of functions g E Cc(B) 3 g = hiB for some hE H, where His a uniform closed linear subspace of Cc(X). See Erik M. Alfsen, On the Dirichlet problem of the Choquet boundary, Math. Scand. (to appear). (Received February 21, 1968.)

68T-415. Y. MATRAS, University of Florida, Gainesville, Florida 32669. On a particular functional equation over groups.

Let G= G U lo) be a group with zero, e being the unit of G. By a I-F-system we mean a subgroup F of G and a system I of left representatives modules F verifying ZF !;::; Z where Z = G\IF. Theorem, A function f of G into itself, not identically 0 or e, is a solution of the functional equation f(xf(y)) = f(x)f(y) if and only if there exists an I-F -system such that (1) f(x) = 0 if x E Z; (2) f(x) = i -lx if x E iF (i E I). A similar characterization is given for groups. Each solution generates a semlgroup of solutions, If this semigroup is of finite order, it is a group. (Received February 21, 1968,) (Author introduced by Professor Ludvik Janos.)

68T-416, E. G. CALYS, University of Kentucky, Lexington, Kentucky 40506. On the radius of univalence and starlikeness of certain regular functions.

In this paper, the following theorems are proved, Theorem 1. Suppose that f(z) = 2 z + a 2z 2 + ... and g(z) = z + b2z + ... are regular in lz I < 1 and g(z) is univalent and convex for lz I < 1. If the function F, defined by F(z) = (2/z) I~ f(t)g(t)dt/t, is univalent and starlike in lz I < 1, then f(z) is univalent and starlike for lz I < 2 - '\13. This result is sharp. Theorem 2, Let f(z) and F(z) be as in Theorem 1 with g(z) = z + b2z 2 + ... regular and univalent in lz I < 1. If F is univalent and starlike in lz I< 1, then f(z) is univalent and starlike for lz I < 1/5. This result is sharp. 2 Theorem 3. Suppose that f(z) = z + a 2z + ... and g(z) = 1 + b 1z + ... are regular in lzl < 1 and Re lg(z)) > 0 for lz I< 1. If the function F, defined by F(z) = (2/z)I~ f(t)g(t)dt, is univalent and starlike in lz I < 1, then f(z) is univalent and starlike for lz I< (5 - .yT'f)/4. This result is sharp. (Received February 22, 1968.)

624 68T-417. W. A. LAMPE, University of Manitoba, Winnipeg 19, Manitoba. Representations of some transformation semigroups as endomorphism semigroups of universal algebras. I. Preliminary report.

Let E be a set of transformations (i.e. mappings) on the set A. E is algebraic iff E = E( ~)

(the set of endomorphisms of ~)for some ~.where ~ = (A;F) is a universal algebra. For a,b E A, (a,b) is an E -pair iff a a= aT implies bu = br for all a, TEE. For a E A, define ¢a by x¢a=aforallxEA. SetCE= /¢alforeverybEA,af-b, (a,b) isnotanE-pair}. Call¢aan

E -constant iff E ~ E(~) implies ¢a E E( ~)for any ~- E is locally closed (B. J6nsson) means that a E E iff for every finite X ~ A there is {3 E E such that al X= {31 x· Let G be a group of permutations on A. Theorem. Let K be a set of constant maps. G UK is algebraic iff G is locally closed KG ~ K, and c 0 ~ K. Corollary 1. G is algebraic iff G is locally closed and c 0 = (if. Corollary Z. For any G, c 0 is the set of~ G-constants. Remark. In proving the theorem one defines an operation fab for every G-pair (a,b) with ¢a EE K by fab(a¢) = b¢ for ¢ E G and fab(z) = z otherwise. The other opera­ tions used are from B. j6nsson, Algebraic structures with prescribed automorphism groups (to appear). (Received February Z3, 1968.) (Author introduced by Professor G. A. Gratzer.)

68T-418. G. A. GRATZER and W. A. LAMPE, University of Manitoba, Winnipeg 19, Manitoba, Canada. Representations of some transformation semigroups as endomorphism semigroups of universal algebras. II.

Theorem Z. There exists a transformation semigroup E and an E -constant ¢a such that Ilia EE CE. Theorem 3. Let a transformation group G be transitive and locally closed. Then G is algebraic iff there exists a G-pair. Let us call G symmetric if there exists a G-pair ( a,b) such that (b,a) isalsoaG-pair. ForX~A,letGx= laiXIaEG,Xa~X}. Theorem4. LetGbe transitive. Then the following conditions are equivalent: (i) G is symmetric; (ii) if lA I > 1, then there exists X~ A, IX I > 1, with GX abelian; (iii) same as (ii) with GX cyclic. If A is finite, then G is always symmetric, hence (ii) gives a result of P. Goral~ik, Z. Hedrl(n and j. Sichler (to appear). (Hint to Theorem 4: assume (i), and take a ¢ E G with a¢= b where (a,b) is a symmetric G-pair, and set X= I a¢n In= 0, ±.1, ±l, ... } .) Theorem 5. There exist nonsymmetric transitive transformation groups. (Received February Z6, 1968.)

68T-419. H. V. KRONK, State University of New York, Binghamton, New York, and D. R. LICK, Western Michigan University, Kalamazoo, Michigan 49001. On graphs with prescribed connectivities.

Let K(G), A(G), and JL(G) respectively denote the connectivity, line-connectivity, and minimum degree of a graph G. Chartrand and Harary [Graphs with prescribed connectivities, 1966 Sympos. on Graph Theory, Tihany, Acad. Sci. Hung., 1967, 61-63] showed that for all integers b,c, and d with

0 < b ;i; c ;;; d, there is a graph G having K(G) = b, A(G) = c, and JL(G) = d, and also constructed a smallest such graph in terms of the number of points. Let G 1 and Gz be two graphs, where Gi has pi points and qi lines, i = l,Z. Then G1 is smaller than G 2 if either (i) p 1 < Pz• or (ii) p 1 = Pz and q 1 < q2• A smallest graph H with K(H) = b, A(H) = c, and JL(H) = dis constructed. (Received February 29, 1968.)

625 68T-420. SHIMSHON ZIMERING, Battelle Institute, Advanced Studies Center, Geneva, Switzerland. Some mercerian theorems for slowly varying sequences.

Definition. A sequence \snl (n = 1,2, •.. ) is said to be slowly varying if limn ~ 00(L [tn]/Ln) = for every oo > t > 0. Theorem l. Let (an,k) be a triangular matrix (i.e. an,k = 0, k > n). If lim supn~ 00 (n11 I;~= 1 1an,klk- 11) < oo and lim infn~oo 0 for some 11 > 0, I:kn a k = l, then s L ~ I:kn a ksk ~ L (n ---->co) for every slowly varying and if lim n---+oo = 1 n, n n~ = 1 n, n sequence \L nl· Theorem 1, which is a generalization of a result of R. P. Agnew [Eigenvalence of methods for evaluation of sequences, Proc. Amer. Math. Soc. 3 (1952), 550-556], is used in the proof of Theorem 2. Let a. be a real number and \ Lnl a slowly varying sequence. Then sn ~ Ln

¢> a.sn + ((! - a.)/n)L~= 1 sk ~ L n (n----> oo) if and only if a. > 0. Theorem 2 is an extension of Mercer's theorem. Theorems 1 and 2 were suggested by Professor R. Bojanic. (Received March 6, 1968.)

68T-42l. R. E. ATALLA and JOAQUIM BUSTOZ, Ohio University, Athens, Ohio. On sequential cores and a theorem of R. R. Phelps.

The core of a sequence of functions is defined in these cNoficei) 15 (1967), 938. The bounded core theorem characterizes those regular matrices for which the core of a transform sequence at

a point z0 is contained in the core at z 0 of the original uniformly bounded sequence. This bounded core theorem is shown to be a special case of a theorem of R. R. Phelps [Proc. Amer. Math. Soc. 16 (1965), 381-382] as follows: The space of suitably defined equivalence classes of uniformly bounded sequences Is n l· s n E C(D), D completely regular, is a Banach algebra with norm defined by lim sup(n ----> oo) llsn 11. where llsn II is the sup norm of C(D). This Banach algebra is isometric to C(X) for suitably defined compact X in such a way that limiting values of sequences are the same as values of corresponding functions in C(X). A matrix operator on sequences whose rows are a bounded subset of 11 and whose columns are null sequences then defines an operator on C(X) with

the same operator norm. If the matrix operator is (aij)' the operator norm is lim sup (i ----> oo) Ljlaij 1. Phelp's theorem is then applied. (Received February 26, 1968.)

68T-422. ETHAN COVEN, Wesleyan University, Middletown, Connecticut 06457. Compactness of limit sets and semiorbit closures.

Let T be a generative topological group and let K be a compact, symmetric neighborhood of the identity which generates T. Let P be a replete semigroup in T and define an equivalence relation

on P as follows: a ~ b provided that there exists kl' ... 'kn E K such that (I) b = ak 1 ... kn and

(2) ak 1 ... ki E P fori= 1,2, ... , n. Definition. A subset Q of P is said to be K-connected provided that every element of Q belongs to the same equivalence class. Theorem. pP is K-connected for some p E P. Theorem. Let (X, T,1r) be a transformation group where X is a locally compact Haus­ dorff space, and Tis generative. Let P be a replete semigroup in T and let x EX. Then Px is compact and nonempty if and only if xP is compact. (Received February 26, 1968.)

626 68T-423. WITOLD BOGDANOWICZ, The Catholic University of America, Washington, D.C. 20017. On analytic extensions of holomorphic functions with values in the space of continuous functions.

Let Q be a compact set in the Euclidean spa•:e R n. Assume that the interior of the set Q is dense in the set Q. Let C(Q) be the space of all continuous complex-valued functions on the set Q. Let L(Q) be the space of all Lebesgue summable complex-valued functions on the set Q. Denote by H(Q,D) the set of all continuous complex-valued functions f on the product Q X D such that for every fixed q E Q the function f(q, •) is holomorphic on the set D being an open domain in the complex plane. This space of functions can be identified with the space of all holomorphic functions from the domain

D into the Banach space C(Q). Let D 1 be another domain in the complex plane such that D C D 1. The set D 1 is also assumed to be open. Theorem. Let f be a function from the space H(Q,D). If for every Lebesgue summable function g E L(Q) the function Jg(q)f(q, • )dq has a holomorphic extension on the domain Dl' then there exists a unique function f 1 in the space H(Q,D 1) being the extension of the function f. The proof of the theorem makes use of the category argument. Some related results will soon follow. (Received February 26, 1968.)

68T-424. ALBERT SADE, 364 Cours de la Republique, Pertuis Vaucluse, France. Quasigroupes parastrophiques: Groupe des automorphismes gauches.

Pour qu'un quasigroupe soit isomorphe 11 son transpose il faut et il suffit qu'il soit isotope d'un quasigroupe possedant une autotopie gauche (a,b,c) telle que !'equation x 3 = cba ait une solution en x. Si un quasigroupe, Q = E( ), est isomorphe 11 son transpose, cet isomorphisme appartient au normalisateur du groupe d'automorphisme, Sf!, de Q dans ~E' et les isomorphismes qui appliquent

Q, soit sur lui-m~me, soit sur ses transposes forment un groupe dans lequel Sf! est normal, d'indice 3 ou l. L'ensemble de toutes les applications J de Sf! dans Sf!, J = X --> (XT)3 , T = constante, X decrivant E, est Ie produit par l'une d'elles du groupe jD.j des translations de .s¥'. Pour que cet ensemble soit un groupe--.alors necessairement isomorphe 11 .s¥'--il faut et il suffit qu'il existe une J identique. Dans ces conditions tous les automorphismes de Q sont involutifs et il existe un automor- phisme gauche de Q dont le cube est l'identique. (Received February 26, 1968.)

68T-425. R. C. COURTER, Wayne State University, Detroit, Michigan 48202. Corational completeness; a shadow of a neo-classic theorem. Preliminary report.

All modules are right modules for a ring R with identity. Definition I. A module M is cor a­ tionally complete if and only if the functor HomR (M, ... ) induces only onto maps when applied to epi­ morphisms N -N/V in the restricted situation that zero is the only R-homomorphism of M to epimorphic images of V. Definition 2. A module W is a corational extension by its factor module W /T if and only if T is a small submodule of W and only the identity map among elements of HomR (W, W) induces the identity map of W /T, Definition 3. A ring R is right perfect if and only if every right R-module has a projective cover. Theorem 1. If M is a corationally complete module, then there are no proper corational extensions by M. A partial converse is: Theorem 2. If R is a right perfect ring and if there are no proper corational extensions by right R- modules, then every right R- module is corationally complete. Theorems and 2 are shadows of the theorem of Findlay and Lambek (Canadian Bulletin of Mathematics, Vol. l (1958), 156): A module is rationally complete if and only if it has no proper rational extension. (Received February 26, 1968 .)

627 68T-426. HILBERT LEVITZ, New York University, Bronx, New York 10453. On the Finsler and Doner-Tarski arithmetical hierarchies. Preliminary report.

Finsler (Comm. Math. Helv. 25 (1951), 75-90) and Doner-Tarski (Fund. Math. (to appear)) have set up hierarchies jocl and J¢c I resp. of binary operations on the ordinals. The operations of these hierarchies satisfy a generalization of the recursion relations a(b + l) = ab +a and ab+l = aba.

Theorem I. If c ~ 4 and a~ w, then 0-l+c(a, I+ b)=¢ .(b,a) where c* = c +I if c = d + n, dis a c limit ordinal, and 0 ~ n < w; c* = c otherwise. Let r be the Schiitte-Feferman ordinal, i.e. the least solution of the equation gx (0) = x where g s is defined inductively as follows: g0 (x) = wx, and gs enum­ erates in order those numbers which are fixed points of gv for all v < s. Theorem 2. a,b,c < rim­ plies Oc (a,b) < f. Furthermore, w < d < f implies d = 0 c (a,b) where a,b,c < d. The author had previously shown (Comm. Math. Helv. 41 (1967), 273-286) that f has the same behavior with respect to Finsler's hierarchy. (Received February 20, 1968.)

68T-427. R. DE SAPIO, University of California, Los Angeles, California 90024. Differentiable structures on a product of spheres.

Let IJn be the group of homotopy n-spheres, let+ denote the connected sum operation, and let sn denote the standard n-sphere. Theorem I. If M is a differentiable manifold that is homeomorphic to Sk X Sn (2 ~ k ~ n, n + k ~ 6), then there exist An E IJn, vn+k E IJn+k such that M is diffeomorphic k n n+k n n+k IJ . . . k n n+k to (S X A ) + V . (a) If B E IJn, U E n+k such that M ts dtffeomorphtc to (S X B ) + U , then Sk X An all'd Sk X B0 are diffeomorphic. (b) If k ~ n - 3, then M is diffeomorphic to (Sk X Sn) + Vn+k and Vn+k is unique. Now let Tn,k : IJn ®1rk(SO(n - l)) ---> IJn+k denote the pairing of Milnor- . n IJ n+k c IJ k n n+k k n Munkres- N ovtkov. Theorem 2. If A E n, V c: n+k , then (S X A ) + V and S X A are diffeomorphic if and only if there exists a. E 1rk(SO(n - l)) such that Vn+k = T n,k (An 0 a.). In particular, if k ~ n - 3, then Tn,k = 0. Now let ~+ 1 denote the subgroup of IJn consisting of those homotopy n-spheres that embed in (n + k + !)-space with trivial normal bundle. Theorem 3. If An, Bn E IJn (k > l, n > 4), then sk X An and sk X Bn are diffeomorphic if and only if An=± Bn mod ~+l. Thus the differentiable structures on Sk X Sn can be classified in terms of IJn/<1>~+ 1 and the pairing

Tn,k. (Received February 26, 1968.}

68T-428. ]. N. PANDEY, University of Alberta, Edmonton, Alberta, Canada. An extension of Haimo's form of Hankel convolutions.

Define PN(D.x) = TI! 1 (I - D.x/a~) where x > 0 and D.x is the differentiation operator (D2 + (2-y /x)D). It is assumed that 0 < "(, 0 < a 1 ~ a 2 ~ a 3 ••• and :Lf= 1 a i/ < oo. Let the functions J.L(x) and G(x,y) be the same as defined by D. T. Haimo in her work on Handel convolutions. We then define a generalized function space H' (I) where I is the open interval (O,oo) and establish the following result. Theorem. Let F(x) = (p'(y)f(y), G(x,y)) where f(y) E H'(I). Then, (J.L'(x)PN(D.x)F(x),¢(x)) ---> ( p.'f, ¢ ), as N---> oo for all¢ E D(I). D(I) is the space of smooth functions defined on I having compact support. (Received February 21, 1968.) (Author introduced by Professor S. A. Naimpally.)

628 68T-429, N. A. FRIEDMAN, Westfield College, University of London, London, N. W., England. On entropy and mixing transformations.

Let (X,_q(,JL) be the unit interval with Lebesgue measure. A class of ergodic measure pre­ serving transformations on X were constructed by the author and D. S. Ornstein in (1): On induced transformations, Tech. Rep. , Univ. of New Mexico, 1967, As in (l), let T(T) be the transformation corresponding to the tower T = I Ij,k : l ;;; j ;;; n, l ;;; k ;;; m(j) j. Theorem l. The entropy of T(T) is given by h(T(T)) = (n log n)/L,~ 1 m(j). Theorem 2. For each T = T(T}, the sets A such that the induced transformation TA is mixing are dense in _w'with respect to the metric d(A,B) = JL(A 6 B). (Received February 28, 1968,)

68T-430. W. J. GORDON, General Motors Research Laboratories, 12 Mile and Mound Roads, Warren, Michigan 48090. Cardinal expansions of bivariate functions.

Let the K -dimensional linear space

68T-43l, BARUCH GERSHUNI, Frans van Mierisstraat 37h, Amsterdam, Nederland. Linear axiom -systems.

There are axioms and axiom -systems ~ which have the property that if two sets S 1 and s 2 are solutions (models) of ~. then also the set S = S 1 U s 2 is a solution of ~. P 2, the second axiom of Peano, (3f)(Vx)(x E S => fx E S), is such an axiom. A setS is here defined as a class C together with

a function f defined inC and associated to C: S = ( C,f ). A fundamental solutionS of ~ is a solution

representable in the form of a sequence of pairs (a 0' fa0 = a 1 ) , (a 1,fa 1 = a 2 ) , ... or abbreviated in the form of the sequence: a0 , a 1, ... , where ai+l = fai. We suppose that if ~has a solutionS, then each set isomorphic to S (with respect to f) is also a solution of ~. We write the union T = Uusu' u EM, where each Su is isomorphic to one and the same setS, in the form {3S, where {3 is the cardinal of M, {3 ;;; 'Y ('Y the cardinal of the continuum). We call an axiom or axiom-system

~ linear if together with the solutions S 1 and s 2 of ~. also f31S 1 U {3 2S 2 is a solution of ~. Then P2 is a linear axiom. (Received February 28,1968.)

629 68T-432. SHMUEL KANTOROVITZ, University of Illinois, Chicago, Illinois. The Ck-classifica­ tion of certain operators in Lp.

For z complex, let Tz = M + zJ, where M: f(x) ----> xf(x) and J: f(x) ----> J~ f(t)dt act in Lp (0, 1), 1 < p < oo. Using the terminology of our previous papers (Trans. Amer. Math. Soc. 115 (1965), 194- 224; J, Math. Mech. 17 (1967), 181-188), we prove Theorem l. Tz is of class en if IRezl;;;; nand only if IRez I< n + 1 (the Cn -operational calculus for T z• IRez I ;;;; n, is explicitly given). Theorem 2. The W k-manifo1ds of T z are dense for Rez < 0 and trivial for Rez <:: 1 and k < [kez }. Corollary 3, Tz and Tw are not similar if [Rez) f. [Rew] (they are similar if Rez = Rew; the stronger result

"Tz is similar to Tw if and only if Rez = Rew" is obtained in another paper by a different method). Corollary 4. Tz is not spectral for i}tez) <:: 1 (it is spectralfor Rez = 0). (Received January 29, 1968.)

68T-433, P. C. ROSENBLOOM, Teachers College, Columbia University, New York, New York. Recursion algebras.

The functions on N lo, 1,2, ... 1toN form an algebra with respect to the operations of+ g, f o g, and Nf defined by (Nf)(x) = fx(O). By the results of Robinson, the set of primitive recursive functions is the subalgebra generated by S and E, where (Sx) = x + 1 and E(x) = x - [X 1/ 2]2. The theory of degrees of recursive unsolvability can be formulated in terms of the inclusion relations among algebras of this type. The study of general recursive functions is the theory of a certain type of algebraic extension of these recursion algebras. This paper also initiates the study of the structure and representation of recursion algebras. (Received March 4, 1968.)

68T-434. J, P. HUNEKE, University of Minnesota, Minneapolis, Minnesota 55455. Mountain climbing.

Let function mean continuous function mapping the closed unit interval onto itself. For a function f and a pointy in [0, 1], let fc(y) denote the set of all x in [0, 1] for which there exists o <:: 0 such that: x - o and x + o are in [9, 1]; f( [x - o, x + o]) is a singleton; and for every f > 0, y is in the interior of f(x - o - f, x + o + €), either x - o = 0 or f( [X - o - f, x]) is not a singleton, and neither x + (J = 1 or f([x,x + o+ €]) is not a singleton, Let f and g be functions. Technical necessary and sufficient conditions for the existence of functions h and j such that fj = gh (i.e. f(j(x)) = g(h.(x)) for each x in [0, 1]) offer the following results, Proposition. Iff and g have fixed points 0 and 1, if g locally constant at x implies that fc(g(x)) is a finite set, and iff locally constant at y implies that gc(f(y)) is a'finite set, then there exist functions h and j with fixed points 0 and 1 such that fj = gh. Proposition. If each off and g is either a nowhere constant function or has the property that the inverse of each point in [9,1] is a set with a finite number of components, then there exist functions h and j such that fj = gh. Proposition. For every real number f > 0, there exist functions h and j such that Jlfj - gh Jl < f, (Received March 1, 1968.)

68T-435, C. E. FALBO, Sonoma State College, Rohnert Park, California 94928. Reducing the order of some nonlinear second order differential equations.

Use properties of the relative derivative in the monograph of M. Petrovic (Edan Differentialniye Algoritam i Negobe Primenye, Srpska Kral.evska Akademija. Posebna Izdanja, Book CXI Belgrad,

630 1936) and assume throughout this discussion that u, y, andY are reversible functions on R 1, with u E class c 1, andy Ec 2 , and YE c2 such that y(l)(x) = u(y), and for f E c 0 , y(l)(x) = u(Y) • c • exp[- Jx f(r)dr], c const • .!f_ {3is nonlinear, then Y satisfies y(2)(x) + c·y(l)(x) = {3(y) iff c • u(t) + u(t) • u(t) = {3(t). Also, Y satisfies Y (2) (x) + f(x) • Y(l )(x) = F (x, Y) .ill_ c 2 • u(t) • u(t) = exp[2Jxf(r)dr]F(x,t). See I. Bandic (Sur !'invariant d'une t!'quation differentielle non-!int!'aire du second ordre, Glasnik Mat.-Fiz. Astron. Dru!1tvo Mat. Fiz. Hrvatske Ser. II. 16 (1961), 161-165). 0 In the second theorem, F is of class c in R 2• (Received March 1, 1968.)

68T-436. P. F. DUVALL, JR., University of Georgia, Athens, Georgia. Weakly flat spheres.

If 2:k is a topologically embedded k-sphere in Sn, 2:k is said to be weakly flat if Sn - 2:k is homeomorphic to Sn-k-l xRk+l. If X is a closed subset of a manifold M, M- X~ l-Ie !!.!_X if for each open set U containing X the:ce is an open set V, X C V C U, such that each loop in V - X is null homotopic in U - X. Theorem 1. If ::!:k C Sn is an embedded k-sphere, 2 ;;; k ;;; n - 3, ::!:k is weakly flat if and only if Sn - 2:k is 1-lc at ::!:k. Theorem 2. LetS, Y, Z, and A be compact, !-connected ANR's, where X= Y U Z, Y n Z =A, and (Z,A) is homeomorphic to (CA,A) where CA is the cone over A. If h: X --> sn is an embedding, n ~ 5, dim X ;;; n - 2, and sn - h(X) is 1-lc at h(X), then Sn - h(Y) is 1-lc at h(Y). Several facts about weak flatness and cellularity follow from Theorems and 2, together with McMillan's cellularity criterion [Ann. of Math. 79 (1964), 327-337]. For example Theorem 3. If 2:k is the boundary of a cellular (k + I)-cell in sn, 2 ;;; k ;;; n - 3, then 2:k is weakly flat. (Received February 29, 1968.)

68T-437. HARRY POLLARD, Purdue University, Lafayette, Indiana 47906, and D. G. SAARI, Yale University, New Haven, Connecticut 06520. Singularities of then-body problem. l.

In the problem of n point masses governed by the Newtonian law of attraction, necessary and sufficient conditions for singularities to be due to collisions are found. The main result is the following Theorem. A singularity as t-+ 0+ is due to collisions if and only if U ~At - 2/ 3, t----+ 0+ for some positive constant A. U is the self-potential of the system. (Received March 1, 1968.)

68T-438. W. L. REDDY, State University of New York, Albany, New York 12203. Pointwise expansive homeomorphisms.

A homeomorphism f of a compact metric (d) space X is said to be expansive at x in X if there is a positive constant C(x) such that if y t x there is an integer n with d(xfn, yfn) ~ C(x). If a positive number C exists with the above property for each x in X, f is called expansive. In answer to a question ofT. S. Wu (Topological dynamics, Benjamin, to appear) examples of homeomorphisms, which are expansive at every point but not expansive, are constructed on spaces including the Cantor set and a continuum. Theorem 1. Iff is expansive at each point of X and p,q are points of X, only countably many points are positively asymptotic to p and negatively asymptotic to q. Theorem 2. Iff is expansive at each point of X and X is self-dense, there exist a positively proximal pair and a negatively proximal pair of points. Iff is transitive, proximal can be replaced by asymptotic. (Received February 29, 1968.)

631 68T-439. SOLOMON GARFUNKEL, Cornell University, Ithaca, New York. On the undecidability of certain classes of abelian groups with an extra unary predicate.

Theorem I, The theory of finite elementary abelian groups with an additional unary predicate is hereditarily undecidable. Corollary, The theory of finite abelian p-groups with an additional unary predicate is hereditarily undecidable. Theorem 2, The theory of periodic abelian groups with an additional unary predicate is hereditarily undecidable. Theorem 3, The theory of finite cyclic groups with an additional predicate is hereditarily undecidable. (Received March 4, !968,)

68 T -440. G. I. GAUDR Y, Mathematics Institute, University of Warwick, Coventry, England. Changes of signs of restrictions of Fourier -Stieltjes transforms.

Let G be a compact Abelian group with ordered dual X, r n the nth Rademacher function. Theorem I. Suppose that X is countable and that¢ is a function defined on X+ and vanishing off

S = lx 1,.x2 .... ). Suppose that¢ has the following property: for each t E (6' C(O,l) where m(1f) > 0, the function which is 0 off S and rn(t)¢(Xn) when X= Xn is the restriction to X of a Fourier-Stieltjes 2 I + transform. Then ¢ E l (X+). Theorem 2, If X is ordered but not necessarily countable and ¢ has the property that w¢ is the restriction to X+ of a Fourier-Stieltjes transform for every ±1-valued function won X+' then¢ E L2 (X+). (Received February 29, !968,)

68T-44l. R. C. GILBERT, California State College, Fullerton, California 92631. Symmetric operators with singular spectral functions.

Let A be a simple closed symmetric operator with deficiency indices (1, I) in the Hilbert space H. Let Ao be a selfadjoint extension of A with a pure point spectrum having no finite limit points, Let A+ be a minimal selfadjoint extension of A, Let R(;\) be the generalized resolvent of A corresponding to A+. Then, (R(A)f,h) = I(JL+ iT)dJL, 4>(,\) = @(,\)Q1 (,\) - l] 1 • [0(,\) t Q 1 (,\)r • From this representation certain facts about the spectrum and spectral multipli­ city of A+ follow. In the case that A is a singular Sturm-Liouville operator one can assums that Ao is a selfadjoint extension with a singular spectral function. (Received March 4, 1968.)

68T-442. D. W. COHEN, University of New Hampshire, Durham, New Hampshire. Extension of locally compact local groups to global groups.

In a locally compact local group there are arbitrarily small neighborhoods of the identity which split into the direct product of a compact group and a local Lie group. This extension of a result for global groups by V. M. Gluskov [Amer, Math, Soc, Trans. (2) 15 (1960), 55-93] proves that every locally compact local group is locally isomorphic to a global group. (Received March 5, 1968,) (Author introduced by Professor A. R. Jacoby.)

632 68T-443, ]. W. LAYMAN and G. G. PITTS, Virginia Polytechnic Institute, Blacksburg, Virginia 24061, Infinite exponential interpolation. Preliminary report.

Exponential approximation in the form f(x) ~ L~= 1 a~)mX, by interpolation at x = 0,1,2, ... ,n - 1, can be shown to result in Ia~) 1----> oo as n ----> oo for the case f(x) = x. A permanent biorthonormal system is provided by U[(x) = [~i-lnx /(i - l)!]n = 1, where ~is taken with respect to n, and L['(f) = (ES)i-lf(O), where Sis the diagonal difference operation (Math, of Comp. 18 (1964), 113-118). Inter­ polation in this system yields f ~ L ~= 1 Lk(f)Uk(x) where now the coefficients remain finite as n --> oo, giving a formal series converging to fat x = 0,1,2, .... For f(x) = x, this gives x ~ L~ (- l)k(k - Z)!Uk(x). A numerical check of the approximating ability of this expansion be­ tween interpolating points was made for selected values of x E [0,1]. Double precision (l6D) calcula­ tion suffices to calculate 40 terms giving as a typical result x = l/2 ~ .4993. (Received March 4, 1968.)

68T-444, ISIDORE FLEISCHER, Queen's University, Kingston, Ontario, Canada. Correction to "Completeness in valued spaces and algebras".

The assertion, in Quart. J, Math. Oxford Ser. 15 (1964), 346, to the effect that an incomplete valued module over a ring with projective ideals always admits an immediate extension is corrected as follows: The result holds as stated when the value set is subjected to an order theoretic hypothesis satisfied, for example, by linearly ordered sets (in conformity with Abstract 60T-2l, these cJ{otiai) 7 (1960), 999) and by partially ordered sets with the ascending chain condition. In the general case, projectivity of the ideals should be replaced by injectivity of the kernels of the projective system maps defining the valuation, and of their intersections. This is deduced from the possibility of mapping any element in a valued extension of such a module value homomorphically over it on an element immediate over it. Details as well as some further consequences will be furnished in a forthcoming Queen's Paper. (Received March 6, 1968.) (Author introduced by Mr. S. M. Fakhruddin.)

68T-445. R. C. BRIGGS, University of Houston, Houston, Texas 77004. More about weak and strong cover compactness.

The statement that a space S is compact means that every infinite subset of S has a limit point. For definitions of weak cover compactness (w.c.c.) and strong cover compactness (s.c.c.), see

Abstract 650-24, these cJ{oticei) 14 (1967), 922, If a first countable (or locally compact) regular T1 space is s.c.c. (w.c.c.), it is collectionwise normal with respect to any discrete collection of degenerate point sets. (A locally compact, s.c.c. (w.c.c.) regular T 1 space need not be normal; Abstract 68T-326, these cJ{oticei) 15 (1968), 396.) If a first countable (or locally compact) regular T 1 spaceS is s.c.c, (w.c,c.), each open cover G of S has an open refinement H such that H' =

l C l (h) lh E H l is a s. c. c. (w. c. c.) closed refinement of G. If a locally compact, regular T 1 space S is s.c.c. (w,c.c.), if B is any basis for S, and if G is an open cover of S, then some subcollection of B is a s.c.c. (w.c.c.) refinement of G. (There exists a metric spaceS and a basis B for S such that no subcollection of B coveringS is s.c.c.) If rruEA Sa is s.c.c. (w.c.c.), then each Sa is s.c.c, (w.c.c.). However, the Cartesian product of two s.c.c. (w.c.c.) spaces need not be s,c.c. (w.c.c,). (Received

March 6, 1968.)

633 68T-446, WALTER ROSENKRANTZ, Courant Institute of Mathematical Sciences, New York University, New York, New York 10012. Some theorems on large deviations.

Let x0 , Xl' x2 , ... ,Xn denote the random variables of a Markov chain with state space the nonnegative integers and transition probabilities given by P(Xn+l = j + IIXn = j} = pj' P(Xntl =

j - 11Xn = j}= q. where Pj = (1/2}(1 + ,\/(j + ,\))and q j = (1/2)(1 - ,\j(j + ;.\)), Po= 1, x0 = 0 and J fx 2.\ -t2/2 .\+1/2 2n - l/2 < ,\ ~ l. Let 4>(.\,x) = 0 !6(,\,t)dt where !6(,\,t) = 2t e /2 f(,\ + 1/2). Let P 2j = P (X 2n = 2j IX 0 = 0). Using the Karlin- McGregor integral representation for these random walks we have been able to obtain the following results. Theorem l. y"ii72P~~ ~!6(,\,t.) where tj = 2j/Vln J J 2 and limn_,oo tf /Vn = 0. Theorem 2, P(X2n;y'2ri ;;:; tjn) = 0(1 - <11(,\,tjn}}, tjn-> oo, tjn/Vn-> 0,

Theorem 3, P(lim supn_,00 (Xn/v'2n log log n) ~ 1) = l. The case A= l/2 has been discussed by the author (cf. Ann. Math, Statist. (4) 37 (1966)}. (Received March 6, 1968.)

68T-447. R. D. M. ACCOLA, Brown University, Providence, Rhode Island 02912. On Riemann surfaces of genus two admitting an automorphism of period three.

Let W be a Riemann surface of genus two. Let O(u) be the usual first order theta function as so­ ciated with the Jacobian J(W) of W. Theorem. W admits a conformal self-map of order three if and 2 only if there are two vectors in c , e 1 and e 2, so that (1) e 1 "/; ± e 2 (mod periods of J(W}}, (2) 3e 1 = 3e2, (3) 6e 1 = 6e2 = 0, and (4) O(e 1) = 0(e2) = 0. The proof depends on Riemann's solution to the Jacobi inversion problem- -the fact that if W admits such an automorphism then it is 3 2 2 the Riemann surface of the algebraic equation y = (x- a 1) (x- a 2) (x - b 1}(x- b2}, and that on such a Riemann surface two linearly independent abelian differentials of the first kind are dx/y and ((x - a 1)(x - a 2}dx)/l. Remark. The codimension of such surfaces in Teichmueller space for genus two is two. (Received March 6, 1968.)

68T-448, NIEL SHILKRE T, Polytechnic Institute of Brooklyn, Brooklyn, New York 1120 I. Non-Archimedean Banach algebras, Preliminary report,

Let X be a commutative non-Archimedean Banach algebra with identity over a nontrivially valued field F. The algebraic structure of the rings V = I x EX I llx II ~ I} and W = I x E X I infm EM llx + m II ~ I, VM, M a maximal ideal of X} has been investigated, Results include the following: Theorem. If V or W is a Noetherian ring, X is a Noetherian ring and F has discrete valuation. Theorem. If M is a maximal ideal of X and X/M is isomorphic to F, then M n V + P, where P = I x E X I llx II < lj, is a maximal ideal of V and V /M n V + P is isomorphic to the residue class field of F. Theorem. Every maximal ideal of V contains P. (Received March 6, 1968.)

68T-449. H. M. SRIV AST AVA, West Virginia University, Morgantown, West Virginia 26506. An extension of Hille-Hardy formula. Preliminary report.

In the course of an attempt to give extensions of the well-known Hille-Hardy formula [cf., for instance, equation (20) on p. 189 of A. Erdelyi et al., Higher transcendental functions, Vol. II, McGraw­ Hill, New York, 1953) for the generalized Laguerre polynomials IL~a)(z}} defined by (I - t)- l- aexp [- xt/(i - t)] = L ~O L~a)(x)tn, the author invokes the Laplace and inverse Laplace transform techniques and the method of finite mathematical induction to obtain several bilinear

634 generating functions for certain classes of generalized hypergeometric polynomials. Some of the formulas derived here extend the results presented by the author at the annual meeting of the Bharata Ganita Parisad at Lucknow (India) on April 16, 1967, while their specialized and limiting cases appear in the earlier works of N, Abdul-Halim and W. A. Al-Salam [Duke Math. J, 30 (1963), 51-62], F. Brafman [Pacific J, Math. 7 (1957), 1319-1323], T. W. Chaundy [Quart. J, Math, (Oxford), 14 (1943), 55-78], ]. Meixner [Deutsch Math. 6 (1942), 341-349], L. Weisner [pacific]. Math. 5 (1955), 1033- 1039], and others. (Received March 6, 1968,)

68T-450, RAYMOND BALBES, University of Missouri, St. Louis, Missouri 63121,

~(J, M, m)-extensions of order sums of distributive lattices.

The order sum of a family I La luES of distributive lattices, indexed by a poset, was defined in Abstract 66T-488, these c}/oticeiJ 13 (1966), 738-730, In the first section of this paper a characteri- zation of order sum is given which is analogous to the characterization of a free distributive lattice as one generated by an independent set, We then consider the collection Q of order sums obtained by taking different partial orderings on S. A natural partial ordering is defined on Q and its maximal and minimal elements are characterized, Let J and M be collections of nonempty subsets of a distributive lattice L, and m a cardinal. We define a (J, M, m)-extension ('lt,E) of L, where E is a m -complete distributive lattice and it: L---> E is a (J, M)-monomorphism. In the last section we define am-order sum of a family of distributive lattices IL ul uES· The main result here is that the m -order sum exists if the order sum L of I L ul uES has a (J, M, m) -extension, where J and M are certain collections of subsets of L. These results are analogous toR. Sikorski's work in Boolean algebras. (Received March 7, 1968.)

68T-45l. L. R. BRAGG and]. W. DETTMAN, Oakland University, Rochester, Michigan 48063. A class of related Dirichlet and initial value problems.

Relationships are exhibited between solutions of the initial value problem (P 1) ut(x,t) = P(x,D)u(x,t), t > 0, u(x,O) = {ll(x), and the Dirichlet type problem (P2) v yy 0. v(x,O) = {li(x), where P(x,D) is a finite order linear partial differential operator. We obtain the following results: Theorem l. Let u(x,t) be a solution of (P 1) with lu(x,t) I ;;;; M or lu(x,t) - {li(x) I ~At a, A > 0,0 < a < 1/2. Then there exists a solution of (P 2) given by v(x,y) = y/ ~ Jgo e-i~ ~-l/ 2 u(x,1/40d~. Theorem 2. Let v(x,y) be a solution of (P 2) such that s- 112v(x,s 112 ) and s -l/~s(x,s l/2 ) are invertible Laplace transforms in the classical sense. Then 1 1 2 1 2 u(x,t) = ( yi';M)Sf~ Is - / v(x,s / >1 s-> 1/ 4t satisfies the equation in (P 1). If (P 2) is well posed limt_,0u(x,t) = {li(x). These results are applied to the heat equation and half space Dirichlet problems. See Bull. Amer. Math. Soc. 74 (1968), 375-378, for earlier results of the authors' that related initial­ boundary value problems. (Received March 11, 1968.)

68T-452. MOSES GLASNER, California Institute of Technology, Pasadena, California 91109, and R. KATZ, California State College, Los Angeles, California. On the Royden compactification for Riemannian manifolds.

Let r be the Royden boundary of a Riemannian manifold R and Ll. its harmonic part. Denote

635 by G a subregion of R, by G its closure in R U f, and by aG its relative boundary in R. Theorem. Suppose that u E HD(G). _!! limxEG,x->a u(x) ~ m, for every a E (G n~) U ac, then u ~ m. This seems to give the unique orthogonal decomposition of any Dirichlet finite function on R into an HD­ function and one which vanishes on~. which is the counterpart of M. Nakai's result for Riemann surfaces (Nagoya Math.]. 17 (1960), 181-218). (Received March 8, 1968.)

68T-453. S. A. GREIBACH, Harvard University, Cambridge 38, Massachusetts. Checking automata and one -way stack languages. Preliminary report.

A checking automaton is a one -way nonerasing stack automaton which, once it has entered its stack, never writes on it again. Theorem. If L <;;a • is an infinitez --cal (checking automaton lan­ guagc), then L contains an infinite regular set. Corollary. L = jan Jn ~ 1} is a one-way nonerasing stack language that is not a cal. Theorem. The cal form a full AF L closed under substitution.

Theorem. Let Y' be the one-way stack languages and let ..'1"1 and 5£2 be sub AFL of Sf'. Then !:/'is closed under substitutions by members of -'z1: into members of Y 2 if and only if either ..'1"1 is a family of cal or Y 2 is a family of context-free languages. Corollary. The one -way stack languages are not closed under substitution. Theorem. The one-way stack languages properly contain both the one-way nondeterministic quasirealtime stack languages and the one -way nonerasing stack languages. (Received March 11, 1968.)

68T-454. W. 0. ]. MOSER, McGill University, Montreal, Canada, and MORTON ABRAMSON, York University, Toronto, Canada. Enumeration of certain classes of restricted combinations.

By elementary methods, the number of p-combinations 1 < x 1 < x 2 < ... < x ~ n satisfying the conditions l ~ n + x 1 - xp ~I', k ~xi - xi-l ~ k', i = Z, ... ,p, is found to be cL·.t.(n,p) = Li=O(- l)i(p;l) l[(n- (p- l)(k -1)- i(k'- k + 1))/p] (n-p(k-l)+k~~-/(k'-k+l)-1)-

~(n-~ (p- l)(k- l• - 1)- t"(k' - k + 1))/p] (n-p(k-l}tkp--1!1-i(k'-k+l)-Z )j·, recurrence re1 atwns . . are gtven. Special cases are noted, e.g., Fibonacci and Lucas numbers and their generalizations. Another class of p-combinations, satisfying the conditions x 1 = 1 + k 1 (mod m), xi'=" xi-l + 1 + ki (mod m), . [(nt(m-l)p-(k1 + ... +k ))/m] j = Z, ... ,p, are enumerated and the1r number found to be ( · p P ). When the condition x 1 '=" 1 t k 1 (mod m) is deleted, the number is [(mu t p t vp)/(u t p)] (u;p), where u = ((n - p - (kz + ... t kp))/m] and v = n - p - (kz + ... t kp) - mu. These are generalizations of Terquem's problem. (Received March 8, 1968.)

68T-455. N. A. TSERPES, Wayne State University, Detroit, Michigan 48ZOZ. A characterization of idempotent probabilities on left groups.

We consider a measure space (S,m) where S is a locally compact Hausdorff semigroup and m a probability on S satisfying fsf(x)m(dx) = fsfsf(xy)m(dx)m(dy) for every real valued function f on S with compact support. Let F be the support of m. Theorem 1. If S is a left group (S has an idempotent element and Sx = S for all xES), then F is a closed left subgroup of S and m on F is r*-invariant and right invariant. If in addition A-lB is compact for every pair of compact A, B C S, then F is also compact. Corollary 1. Let S be bicancelative and £loA -l be compact for all compact A,B C S; then F is a compact subgroup of S and m=normed Haar measure on F. (One shows that for some q E F, qF

636 is compact, and since it is bicancelative it is a group, so that qFqx = qF or Fqx = F for all x E F, and so F is a bicancelative left group and hence also a group.) The proof of Theorem 1 uses the result

(J. G. Wendel) that the conclusion of Corollary 1 is valid for S= a compact group, and thus also provides a new proof of the well-known Corollary 2, Every idempotent probability on a locally com­ pact group is the Haar measure on a compact subgroup and conversely. (This follows from Theorem by observing that a left group with a two-sided identity is a group.) (Received March 7, 1968.)

68T-456, WITHDRAWN.

68T-457. N, D. GUPTA, University of Manitoba, Winnipeg, Canada, On two theorems of ]. M. Gandhi,

In Abstract 68T-169, these cJiotiaiJ 15 (1968), 349, Gandhi has announced: Theorem 2, The law [x,y]2 = [!c,y,x] in a group implies the law [x,y] = 1. Theorem 3, The law [x,y]z = [!< 1•x2,,.,,xn] in a finite group or in a soluble group implies the law ~.y] = 1. Theorem 2 is false since the dihedral group of order 8 satisfies the law [x,y]2 = [x,y,x]. The conditions in Theorem 3 [x,y] = 1 (replacing are superfluous as the law [x,y]z = [x 1, ... ,x0 ] trivially implies the law z, x1, .. .,xn by 1). (Received March 11, 1968.)

68T-458, R. G. MONTGOMERY, Clark University, Worcester, Massachusetts 01610, Spec A as a set with algebraic structure. Preliminary report,

Let A be a commutative ring with unity and S be a set, Then each point b E {JS (S having its discrete topology) defines an S -ary operation b: (spec A)S ->spec A. For T E (spec A)S, the function 'T: {3S-> spec A defined byr(b) = b(r) is continuous and restricts tor; it follows that b-> b is continu­ ous, Let K: Y• -> IK be the equational theory of compact, Hausdorff spaces (see F. E. j. Linton, Some aspects of equational categories, Proc, Conference on Categorical Algebra - La jolla, Springer­ Verlag, New York, 1966, pp. 84-94), Theorem 1. Spec A is a K-algebra. Specifically, A(S) = (spec A)s, A(f) ='To f defines the necessary functor AE yK, and the K-operations A(S-> 1) on spec A are precisely the b (bE {3S). Theorem 2. Let A= C(X), where X is a completely regular, Hausdorff space, The prime z- ideals are invariant under K -operations; the minimal primes under countable K-operations, (Received March 13, 1968.)

68T-459, K. F. BARTH and W. j. SCHNEIDER, Syracuse University, Syracuse, New York 13210. On a problem of Collingwood concerning meromorphic functions with no asymptotic values,

In Hayman's book Research problems in function theory, p. 34, Problem 5,19, the following problem (attributed to Collingwood) is posed: Must every function f(z), meromorphic in D = liz I < 1}, with no asymptotic values take on every value (finite and infinite) in every neighborhood of every point of C = liz I = 1} ? (It is known (Collingwood and Cartwright) that globally f(z) must take on infinitely often every value (finite and infinite) and that locally f(z) can omit at most two values.) This question Is answered in the negative by the following: Theorem. There exists a func­ tion f(z), meromorphic in D, whose zeros and poles cluster only to the point z = 1; and, in addition,

637 on a sequence of closed jordan curves Jn (Jn C [nt lntl CD with Jn tending to C), it has the properties (i) limk_,00 maxz E 12k lf{z) I = 0; (ii) limk_,00 min 12k+ 11f(z) I = oo (i.e. except at one point f{z) can omit the maximum number of values). The proof consists· of writing f as the quotient of two functions g and h where g and h are constructed by a method quite similar to one used in a recent paper of the authors [On a problem of Bagemihl and Erdos concerning the zeros of an annular func­ tion, ]. Reine Angew. Math. (to appear)]. (Received March 12, 1968.)

68T-460. A. H. SIDDIQI, Aligarh Muslim University, Aligarh. (UP), India. Fourier series with positive coefficients. I.

A continuous function f{x) will be said to belong to the class A *(X(x)) if f(x t h) t f{x - h) - 2f(x) O(IX{h) 1), h--+ 0 uniformly in x, where X(x) is a function satisfying the condition J3 (IX(x) l/x)dx < oo. For X(x) = x the class A*(X(x)) becomes A* class. The following theorem has been proved which generalizes a theorem of Boas (J. Math. Anal. Appl. 17 {1967), 463-483); Theorem. Let ,\(x) be a positive monotonic increasing function such that for o > 0, ,\(x)jxo lo, L~n ,\(k)/kl+o = 0(,\(n)/no), and L~=lAk/k = 0(,\(n)), n_,oo. If an;;:; 0 and f ~ :Lfan cos nx, then f E A*{X(x)) if and only if L~nak = O(IX{l/n)i), n--+oo, where X(n) and n2 1X{l/n)l satisfy the same condition, as ,\(x) above for o = l and 2 respectively. (Received March 14, 1968.) (Author introduced by Dr. S. M. Mazhar.)

68T-461. D. R. CHILDS, General Dynamics, 97 East Howard Street, Quincy, Massachusetts 02169. Rigorous solution of the monoenergetic transport equation. I: Slab geometry.

A rigorous solution of the monoenergetic transport equation with constant source has been obtained. The macroscopic absorption, scattering and total cross sections, are assumed to be constant in the region of interest. Solutions of the form eaxf(!l) and e -axf{~) are obtained, thus trans­ forming the integra-differential equation into a pure integral equation. The eigenvalue equation for the parameter a is exhibited explicitly for all orders of the scattering cross section in the form of a determinantal equation and the f(!l) is shown to involve Legendre polynomials of the first and second kind. Previous work yields an eigenvalue equation for the isotropic scattering cross section. This paper extends the work for all orders of the scattering cross section and gives a simple scheme for determining the element in the determinantal equation. The determinant which becomes larger for each order of the scattering coefficient considered can be reduced and factored so that the eigenvalue equation involves the ratio of two series, the coefficients of the series being determined by a recur- sion relationship. (Received March 13, 1968.)

68T-462. A. A. MULLIN, Aberdeen Proving Ground, 222 Bynum Ridge Road, Forest Hill, Maryland 21050. Intermediate topics in number theory. Preliminary report.

These results, the third chapter of a draft-monograph entitled Elementary Unique Factoriza­ tion: Novae Disquisitiones Arithmeticae, extend the author's primitive -recursive -function -theory version of elementary unique prime factorization (see, e.g., Amer. Math. Monthly 74 (1967), 1100- 1102) to intermediate algebra, viz. to {1) the elementary theory of ideals in the senses of Dedekind, Hilbert, and Artin and {2) to the theory of rings with special emphasis on regular local rings (e.g., the ring of p-adic integers) and free ideal rings (e.g., group algebras of free groups over a field),

638 so as to supplement the Euler-Kronecker infinite products for various zeta -functions, all analytic versions of unique prime factorization. Developed are new analogues of classical functions (e.g., a "mosaic analogue" of Mobius' function for Gaussian integers and its relation to the distribution of Gaussian primes). Further, use is made of the author's (1) ordinary modified Mobius' function and modified totient (these cJ./oticeiJ II (1964), 680) for reestimations with sieve techniques and (2) modified Liouville and modified Mobius functions in reexaminations of Siegel's theorem on the class numbers of binary quadratic forms. (Received March 19, 1968.)

68T-463. C. L. BELNA, Michigan State University, East Lansing, Michigan 48823. A result on the n -segment property.

Let f be a continuous complex-valued function defined on the open unit disk D, and let C denote the unit circle. LetS 1, ... ,Sn be n distinct rectilinear segments in D terminating at z = 1, and denote the image of~ under the rotation W(z) = eiaz by~ (a). We say that f has then-segment pro­ perty at eia E C relative to then segments s 1, ... ,Sn if n~=l C~ (a) (f,eia) = {.J, where CSk(a)(f,eia) denotes the cluster setoff at e 1a on Sk(a). Theorem. Let f be an arbitrary complex-valued function defined on D. Then the set of points eia E C at which f possesses the n -segment property relative to a fixed set of n segments is of measure zero. (Received March 18, 1968.) (Author introduced by Professor P. A. Lappan,)

68T-464. MIHA' L Y MAKKAI, Mathematical Institute, Hungarian Academy of Sciences, Budapest, V., Realtanoda u. 13-15, Hungary. Model theoretic results on infinitary sentences.

"Sentence" means a sente.nce of the language L(w 1,w) (i.e. a sentence formed by countable conjunction and disjunction and finite quantification), Let Ll. be the set of sentences defined in Abstract 68T-24, these cJ./oticeiJ 15 (1968), 196, and let F be any sentence, The following is an analogue of Theorem 2 of Ioc. cit, Theorem 1. The class of direct factors of countable models of F is identical with the class of countable models of a subset of Ll.. Similar results hold in case of other algebraic situations, e.g. those treated by H. I. Keisler [J. Symbolic Logic 30 (1965), 339-349]. Combinations of preservation statements, such as the following, are proved, Theorem 2. The

following conditions on sentences F 1, F 2 , H are equivalent, (i) There is a model A of H such that A is an homomorphic image of a (countable) model of F 1 and A is a substructure of a (countable)

model of F 2. (ii) If F 1 ""G1, F 2 \o G2, G 1 is positive and G2 is universal, then IG1,G2 , H} is cons is tent. (Received March 18, 1968.) (Author introduced by Professor Robert Vaught.)

68T-465. L. R. ANDERSON, Case Western Reserve University, Cleveland, Ohio 44106. Integral manifolds of a class of 3rd order autonomous differential equations.

Consider the differential equation y"'= ay" +by'+ g(y) where a< 0, b > 0, g E C', yg(y) ;:; 0,

g' (y) > 0 for y i 0 and lg(y) I --> oo as IY I --> co, e.g. g(y) = cy3 , c > 0. There exist subsets M2 and 2 M 1 of R3 homeomorphic to R and R respectively such that: (1) If (y(O), y'(O), y"(O)) E M 2 ( E M 1), therr y(t) is continuable on !9, + oo) (on (- co,O]) and y(k) (t) --> 0 as t--> + oo (t --> - oo), k = 0,1,2,3, 3 (2) If (y(O), y'(O), y"(O)) E R 3 - M 2 (E R - M 1) either y(t) is not continuable on [0, + oo) (on (-oo,O]) or y(t) is unbounded as t--> +co (as t-->- co). (3) M1 n M2 = I

639 68T-466. R. C. BUCK, University of Wisconsin, Madison, Wisconsin 53706. Decidable semigrouQS.

An extension of the Bernside problem is to determine those presentations of semigroups which, by virtue of the nature of their defining relations, permit a decision algorithm, and in particular, those that are finite. One such result is that of Green and Rees [On semigroups in which xr = x, Proc. Cambridge Philos. Soc. 48 (1952), 35-40). The present report deals with another special class of semigroups, extending the results announced earlier [Abstract 68 T-237, these cJ.foticeiJ 15 (1968), 370]. Let S(n,r) be the semigroup generated by two elements.!!. and E. subject only to the relations (i) aba = bn, (ii) bab = ar. By symmetry, assume n ~ r. In a paper to appear in the Monthly, it was shown that S (l,r) is finite with order 5r + 3. Theorem. S (2,r) is finite for r = 2,3,4, but not for r = 5. Specifically, ord(S(2,2)) = 31, ord(S(2,3)) = 74 and ord(S(2,4)) ~ 2130. All the semigroups

S(n,r), with n ~ 3, are infinite. If the commuting relation ab = ba is adjoined, then the semigroup S(n,r) becomes finite in every case except n = 2, r = 5. (Received March 20, 1968.)

68T-467. V. C. CATEFORIS, University of Wisconsin, Madison, Wisconsin 53706. Flat regular quotient rings.

Let R be an associative ring with l and singular right.ideal zero. LetS be a (von Neumann) regular ring, contai-ning R and such that RR is large in SR. Let M be a right R-module: (1) M is essentially finitely generated (EFG) if M contains a finitely generated large submodule, (2) M is essentially finitelyrelated (EFR) if there exists an exact sequence of right R-modules 0--> K --> F --> M--> 0 with F finitely generated free and K EFG. Theorem l. For arbitrary finitely generated right ideals I and J of R, the following are equivalent: (a) S is flat as a left R -module, (b) I is EFR, (c) (I: a)= {r E R I ar E I} is EFG for every a E R, (d) (0: a)= {r E R I ar = 0} is EFG for every a E Rand In J is EFG. Theorem 2. For any finitely generated free (right) R-module F and finitely generated submodules M and N of F, the following are equivalent: (a) S is flat as a leftR-module, (b)MisEFR, (c)(M:x)= {rERixrEM} isEFGforeachxEF,(d) (O:x)= {r E R I xr = 0} is EFG for each x E F and M () N is EFG. (Received March 20, 1968.)

68T-468. P. H. DOYLE, Michigan State University, East Lansing, Michigan. Alexander's cell theorem without direct limits.

• W. Alexander's proof (that if Pis a k-cell in euclidean n-space En, then H (En - P) Q< J - -- -~ q (Sn- 1)) is the usual proof given of this fact. It is inductive and explicitly uses a limiting argument. Hq This is another proof. From the Mayer-Vietor is sequence if En+ 1 :J En, we have Hq(En+ 1 - P) Q< ,-...; n """ n+r ,.... n - P). A lemma of Klee ensures that for r ;?; k, P is H q- 1 (E - .P). Whence H q (E - P) Q< H q-r (E. . . n+r · ~ n+r ~ n+r-1 ~ n equivalent to a k-simplex in E . So for r ~ k, H (E - P) Q< H (S ) Q< H (E - P). Since q q q-r ~ n+r-1 ~ n ~ n-1 Hq(S . ) is 0 except for q = n + r- 1, we have Hq(E - P)Q

640 68T-469. WITHDRAWN.

68T-470. D. D. BOOTH, University of Wisconsin, Madison, Wisconsin 53706. Some consistency questions in topology.

Let ZFC be set theory ZF along with the axiom of choice. Theorem. If ZF is consistent, so is ZFC & (no separable metric continuum is the union of X 1 mutually disjoint closed sets). Theorem. If ZF is consistent, so is ZFC & (no separable metric space of positive dimension is v' the unhn of X1 closed sets of dimension zero). Let N* be the Stone-Cech compactification of the integc without its isolated points; one calls a point of N* ap-point if every countable set of open sets <:bout it has an open set in its intersection. Theorem. If ZF is consistent, so is

ZFC · 2Xo =X 2 & (there are p-polnts inN*). These propositions can be given the form of iterated Coh·;n extensions. (Received March 21, 1968.}

68T-471. D. E. LEACH, Case Western Reserve University, 34800 Vine Street, Eastlake, Ohio 44094. On a class of second-order differential systems with bounded coupling terms.

Consider the second order system(*} x"k + gk(xJJ = hk(t,x 1, ... ,xn' xl, ... ,x~}, k = 1, ... ,n. Assume the functions hk are continuous and bounded and hk(t + 21r, x 1, ... ,xn' y 1, ... ,yn) = hk(t, x 1, ... ,xn' y 1, ... ,yn)' k = l, ... ,n. Assume the functions gk have continuous derivatives and there exist integers mk, k = 1, ... ,n, and o > 0 such that (mk + o) 2 :;:;; gk(x):;:;; (mk + 1 - o}2, k = 1, ... ,n. The gk are not necessarily assumed to be odd, Theorem. There exists at least one 27r-periodic solution of the system (*). If in the one-dimensional case x" + g(x) = h(t,x,x'}, h depends only on t, the 27r-periodic solution is unique. (see Abstract 68T-329, these cJ{oticei) 15 (1968}, 397). This partially generalizes a result of W. S. Loud (Abstract 636-119, these cJ{oficei) 13 (1967), 611). (Received March 21, 1968.)

641 68T-472. L. C. HIGBIE, National Center for Atmospheric Research, P. 0. Box 1470, Boulder, Colorado 80302. When best rational Chebyshev approximations are polynomials.

Let Tn be the nth degree Chebyshev polynomial normalized to the interval [a,b] and let Rn,m be the set of rational functions with no poles in [a,b] expressible as the ratio of a polynomial of degree n to one of degree m. Theorem. Let PN, P N" P N" denote polynomials of best approxima­ tion to f E C [a,b] of degrees N, N', and N", respectively, from RN ,O• RN -l,O , and RN'- l,O (provided N' > 0), respectively. Suppose P N' t f and that P N' Chebyshev alternates with f N' + D' + 2 times on [a,b ]. Then D' = Max I d: P N' is the best approximation from RN',dl and the test can be iterated

replacing N' by Nand finding a new value of D'. If PN' = f, then PN' = f = CTN' + PN.,. Remark. By reversing the orders of the subscripts on R, one obtains a similar theorem for reciprocals of polynomials. Several theorems in the paper of B. Boehm (Numer. Math. 6 (1966), 235-242) follow from this theorem as special cases. (Received March 21, 1968.)

68T-473. A. P. HALLSTROM, Brown University, Providence, Rhode Island 02912. Bounded point derivations and analytic capacity. Preliminary report.

Let X be a compact plane set, 0 E boundary X, R(X) the uniform closure on X of rational functions with poles off X, An= lz: l/2n+l ;;: Jz J ~ 1/2 nl• and -y(U) the analytic capacity of any bounded plane set U. A bounded point derivation for R(X) at 0 is a continuous linear functional D: R(X)---> C such that Dfg = f(O)Dg + g(O)Df for all f, g E R(X). Theorem. There is a nonzero bounded point derivation for R(X) at 0 iff I::o 4n'Y(An \X) < oo. The proof uses a result of Melnikov [Mat. Sb. 71 (113) (1966), 503-515] and a modified argument due to Curtis [Peak points for algebras of analytic functions (to appear)]. The result generalizes to higher order bounded point de­ rivations. An example is constructed for which int X t ~. R(X) t A(X), such that there is no bounded point derivation for R(X) at any point on the boundary of X. (Received March 21, 1968.)

68T-474. PETER COLWELL, Iowa State University, Ames, Iowa 50010. Subordination and a class of meromorphic functions.

Let D = liz J < 11 and C = liz J = 11 and let f(z), F(z) be functions defined in D. We say that f(z) is subordinate to F(z)--and write f-< F--if there exists ¢(z), analytic in D with J¢(z)J ~ jzJ, such that f(z) = F[¢(z)] everywhere in D. J. V. Ryff [Duke Math. J. 33 (1966), 347-354] studied Hp functions in D related by subordination. Let G be a domain whose boundary f has positive logarithmic capacity. If f(z) is meromorphic in D, takes all its values in G, and at almost every point of C has a radial limit on f, we call f(z) a function of class (L) (with respect toG and f). This class of functions was studied first by 0. Lehto [Ann. Acad. Sci. Fenn. Ser AI 160 (1953), 1-14]. Results similar to Ryff's are obtained for functions of class (L). Theorem l. If F(z) is of class (L) and f-< F, then f(z) is of class (L) if and only if ¢(z) is of Seidel's class (U), ¢(0) = 0, where f(z) = F [¢(z)]. In terms of Lehto's definitions of normal and exceptional values for functions of class (L), we prove Theorem 2. If f(z) and F(z) are of class (L) and f-< F, then every value in G exceptional for F(z) is exceptional for f(z). A value exceptional for f(z) need not be exceptional for F(z). (Received March 25, 1968.)

642 68T-475. JOSEPH ZAKS, University of Washington, Seattle, Washington 98105. On countably infinite decompositions of n-spaces.

It is known that if G is an u.s.c. decomposition of En, with only finitely many nondegenerate elements, then the decomposition space En/G of G is embeddable in En+2 (L. V. Keldysh, Proc. Sympos., Prague, 1961, pp. 230-234). The following is an extension to this theorem, and a partial affirmative answer to question #!6 of S. Armentrout's Monotone decompositions of E 3 (Topology Seminar Wisconsin 1965): Theorem. If G is an u.s.c. decomposition of En, H = jGuluEA the set of all the nondegenerate elements of G, H* = U uEA G d then En/G is embeddable in En+2 provided H • is closed, and for each u E A, there exists a neighborhood V u of G u such that u i {3 implies V u () V{3 = jil. (Received March 25, 1968.)

68T-476. E. D. TYMCHATYN, University of Oregon, Eugene, Oregon 97403. Partially ordered spaces with the product topology.

Let X be a compact metric space with a closed partial order. For each x EX, let f(x) =

jy EX lx ~ y or y ~ x \. An antichain is a totally unordered subset of X. An order arc is a closed, connected, totally ordered subset of X. Let L (resp. M) denote the set of minimal (resp. maximal) elements of X. Theorem l. If L and M are closed and, for each x E X, f(x) is a nondegenerate order arc, then X is homeomorphic to M X [0, 1]. Theorem 2. If L and M are closed and, for each x E X, f(x) is connected and nondegenerate, then Ythe family of maximal order arcs of X is a com­ pact subspace of the space of compact subsets of X (with the finite topology). Let Y X [0, l] have the

partial order (m,x) ~ (n,y) iff m = nand x ~ y. There exists a map f: YX [0, 1]----> X such that for

each T EY fljT\x[O,l] is an order preserving homeomorphism of {T l X [9,1] onto the maximal arc T C X and, for each y E [0, 1], f( !f X { y p is a compact maximal antichain in X. (Received March 25, 1968.)

68T-477. R. R. COLBY and E. A. RUTTER, JR., University of Kansas, Lawrence, Kansas 66044. QF -rings with zero singular ideal.

A ring R (with unit) is left QF -3 if it has a minimal faithful left module, i.e. a faithful left module which is (isomorphic to) a summand of every faithful left module. Theorem. The following are equivalent. (l) R has zero left singular ideal and is left and right QF-3. (2) There exist idempo­

tents e, f E R, such that Re and fR are faithful projective-injective left and right ideals and eRe is semisimple. (3) R has zero left singular ideal, contains no infinite set of orthogonal idempotents, and has a faithful projective- injective left ideal and a faithful projective injective right ideal. (4) R is a subring of a semisimple ring Q and R contains a left ideal I and a right ideal J such that I and J are, respectively, faithful left and faithful right ideals of Q. (5) R has a two-sided semisimple general­ ized ring of quotients and both the left and the right socles of R are essential. (Received March 25, 1968.)

643 68T-478. R.I. SOARE, University of Illinois, Chicago, Illinois 60680. Sets with no subset of higher arithmetic degree.

If X £; w, let [X)w denote I Y £;X I IY I= w}. A partial function cf> on 2w (with standard topology) is Borel measurable if cf>- 1(U) is Borel for every open set U £; 2w. Theorem. Let I cf>e} be any countable class of Borel measurable partial functions, and let A E [w)w. Then there exists B E [A]w such that: (e)(T)(S)[T E [B]w & S E [TJw & cf>e(S) = T => T- Sis finite]. Corollary. If A E [w]w, then there exists B E [A]w such that if S £; T £; B and T is arithmetic in S, then S is arithmetic in T. The proof of the theorem depends upon the recent result of F. Galvin and K. Prikry that every Borel set is Ramsey. This generalized a theorem of Ehrenfeucht that every open set is Ramsey which suffices to prove our theorem for continuous partial functions, and which we earlier used (Abstract 656-37, these c){oticei) 15 (1968), 513) to obtain the special case of the Corollary for Turing degrees instead of arithmetic degrees. (Received April 1, 1968.)

68T-479 •.. N. A. TSERPES, Wayne State University, Detroit, Michigan, and ARUNAVA MUKHERJEA, Eastern Michigan University, Ypsilanti, Michigan. On a theorem of Hewitt.

The following result is proved: Theorem 1. Let (S,2:,m) be a probability space such that S is topological, and every f E C(S), the space of real valued continuous functions on S, is a random variable with finite expectation. Then every f E C(S) is bounded almost surely; moreover, if S is realcompact or normal, admitting a complete uniform structure, then the support of m, F = I x; every 2:-measurable open neighborhood of x has positive measure}, is a compact subset of S. Hewitt (Fund. Math. 37 (1950)) proved the following: Theorem 2. LetS be completely regular and let I be a bounded linear functional on C(S) with I(l) = 1. Then: (a) There exists a Baire probability measure m* on S such that I(f) = Jsf(x)m*(dx) for every f E C(S). (b) Every f E C(S) is bounded al­ most surely (m *) and if in addition, S is realcompact, then the support of m *, F • = (x; every Baire open neighborhood of x has positive m •- measure } is compact. If part (a) of Theorem 2 is granted, then Theorem 1 implies clearly Theorem 2. Conversely, if m is outer regular relative to the open sets of S, or if F* is a G0 (or equivalently a zero set), then also Theorem 2 implies Theorem 1. (Received March 26, 1968.)

68T-480. C. R. SMITH, Northeast Louisiana State College, Monroe, Louisiana 71201. A measure-theoretic generalization of convexity. Preliminary report.

Consider the triple, (E,2:,!-1), where E is a real linear space, 2: is au-algebra of subsets of E, and 1-1 is a nonnegative, extended real valued, countably additive set function on 2:, with 1-1(~) = 0.

Definition. A subset, S, of E is said to be w convex iff for x,y E S, N(x,y) = (a.x + {jy: a, {j ~ 0, a+ {3 = 1} nsc E 2;(where sc is the set-theoretic complement of S) and ~t(N(x,y)) = 0. This definition gives the usual convexity if 2: is the power set of E and 1-1 is the measure which assigns to a set A in E the cardinality of A if A is finite and oo if A is infinite, hence a generalization. The~. The intersection of a countable collection of 1-1-convex sets in E is a wconvex set. (Received March 28, 1968.) (Author introduced by Professor George Berzsenyi.)

644 68T- 481, CAROLYN FARRAR, Northeast Louisiana State College, Monroe, Louisiana 7120 l, A sufficiency condition for uniform continuity,

Concerning the last sentence in a paper by E. Elyash, G. Laush, and N, Levine [On the product of two uniformly continuous functions on the line, Amer. Math, Monthly 67 (1960), 265-267], the following more general sufficiency condition for uniform continuity was found, Theorem. Let X be a metric space with metric d x• Iff: X ~ [O,oo) and f2 is uniformly continuous on X, then f is uniformly continuous on X. The proof is based upon the Lemma, If a "' 0 and b ~ 0, then I~- vbl ~ ~1. (Received March 28, 1968.) (Author introduced by Professor George Berzsenyi.)

68T-482, J, W. BREWER, Virginia Polytechnic Institute, Blacksburg, Virginia 24061. A note on the ideal transform and integral closure in Noetherian domains.

Let R be a commutative ring with identity having total quotient ring F. If A is an ideal of R, the transform of A, T(A), is defined to be U~ 1 Bn where Bn = R: An= [x E FlxAn ,;;; R}. We say that T(A) is a proper transform of R provided R C T(A). Theorem l, Suppose that D is an integral domain with A a finitely generated ideal of D. Let [P ul be the collection of all prime ideals P of D such that P ;!! A. Then T(A) = n Dp . Corollary l, If D is a Noetherian domain and A is an ideal (l (l of D, then the transform of A is determined as the intersection of those Dp 's where P is prime and P ::j? A. The converse to Corollary l is false, Theorem 2. If Dis a Noetherian domain which is not a local ring, then D is integrally closed if and only if each proper transform of D is integrally closed, The hypothesis that D is not local cannot be dropped, (Received March 2 7, 1968 .)

68T- 483. R. N. CAIN, Courant Institute of Mathematical Sci~nces, New York University, New York, New York 10012, The Leray spectral sequence for gene_Ealized cohomology,

The Leray generalization of the Mayer- Vietoris homology or cohomology exact sequence to a spectral sequence for the case of several subspaces is found to depend only on the first six Eilenberg- Steenrod axioms. Applying this to coverings of a space X by preimages of open sets in the range Y

of a mapping f: X ~ Y, one obtains a new derivation of both the Leray-Serre homology or cohomology spectral sequence of a fibration and the original Leray sheaf-cohomology spectral sequence for an arbitrary mapping, all without using the dimension axiom, Convergence in the general case is insured by keeping the covering dimension of the space Y finite, (Received March 27, 1968,)

68T-484. SEYMOUR KASS, Illinois Institute of Technology, Chicago, Illinois 60637, and W. G. WITTHOFT, University of New Hampshire, Durham, New Hampshire 03824, Some new classes of anticommutative algebras. Preliminary report.

The methods introduced by j. M. Osborn (Canad. j. Math, 17 (1965), 78-92) have been adapted to prove the following Theorem, Let A be an anticommutative algebra over a field of characteristic

not two, and let A satisfy an identity of degree ;§ 4 not implied by the anticommutative identity. Then A satisfies at least one of the following four identities: (I) ((yx)x)x = 0; (2) j(x,y,xy) = 0; (3) u(xy)(xz) t /3[((xz)x)y- ((xy)x)z] t 'Y[((xz)y)x- ((xy)z)x] + (/3 + -y)((zy)x)x = 0, where u, /3 ,'Y E F are not all zero; (4) j(x,y,z)w- j(y,z,w)x t j(z,w,x)y- j(w,x,y)z = 0, where j(x,y,z) = (xy)z + (yz)x + (zx)y, There exist non- Lie, non- Malcev antic om mutative algebras satisfying these identities. (Received March 28, 1968.) 645 68T-485. A. E. ROSENBERG, Brown University, ProvidencE, Rhode Island 02912. On the semisimplicity of the group algebra.

A ring is semisimple if its jacobson radical is zero. The following theorems give sufficient conditions on a group Gin order that Q[G] be semisimple. Theorem 1. Let H be a group and K an Abelian group. Let p: H - Aut (K) be a homomorphism. Via p, we can form the semidirect product

G; K XS H. Suppose the image of pis finite Abelian, and that Q[HJ is sernisimple. Then Q[G) is semisimple. Definition. A group G is countably supersolvable if it has an invariant series with cyclic factors whose intersection is the identity. Theorem 2. If G is countably supersolvable, then Q[GJ is semisimple. Theorem 3. If G is countably nilpotent, then Q[GJ is semisimple. Theorem 4. If G is any group and Z: is the group of integers, then Q[G X z:J is semisirnple. (Received March 29, 1968.)

68T-486. jOSEPH AUSLANDER, University of Maryland, College Park, Maryland 20742. Homomorphisms of minimal sets.

Let T be a topological group, let (M, T) be a universal minimal set for T, .and let G be the automorphism group of (M, T). If (X, T) is minimal, and 'Y: M - X is a homomorphism, let G(X, 'Y) ;

[a E G I ')'a; Y]. (i) If ')' 1 : M - X is a homomorphism, then G(X,'Y) and G(X 1 , ')' 1 ) are conjugate sub­ groups of G. If (Y, T) is minimal, and 1r: X - Y is a homomorphism, 1r is called~ if

G(X,')') 'f G(Y,7r'Y) (by (i) this is independent of 'Y). (ii) 1r is proper if and only if there exist x and x' not proximal such that 1r(x) ; 1r(x') if and only if there exist x 'f x 1 in X such that (x,x 1 ) is an almost periodic point of (X X X, T), and 1r(x) ; 7r(x'). 1r is said to be of distal type if whenever x 'f x 1 and 1 1r(x); 1r(x 1), x and x 1 are not proximal. (iii) 1r is of distal type if and only if 7r- (y) is an almost periodic set (y E Y). (iv) G(X, ')') is a normal subgroup of G if and only if (X, T) is a nonproper homomorphic image of a regular minimal set. (v) G(X,y) C G(X 1,')' 1) if and only if there is a minimal set (Y, T) and homomorphisms 1r: Y - X, 1r 1 : Y - X 1, where 1r is nonproper. (vi) (Notation as in Ellis, Group-like extensions of minimal sets, Trans. Amer. Math. Soc. 127 (1967), 125-135.)

Let .w'C!il C _w'(u). Then~ is a group-like extension of J.lfif and only if the homomorphism of l.'fi I to l.s¥1 induced by inclusion is of distal type. (Received March 29,1968.)

68T-487. A. H. LACHLAN, Simon Fraser University, Burnaby 2, British Columbia, and E. W. MADISON, University of Iowa, Iowa City, Iowa 52240. Computable fields and arithmetically definable ordered fields.

Computable (arithmetically definable, denoted AD) structures are algebraic structures whose relations can be viewed as recursive (arithmetical) number theoretic relations. Using the decidability of the elementary theory of the real numbers, we show Theorem 1. The fields of real algebraic, constructible and solvable numbers are computable fields. Also, concerning the fields of computable reals, denoted ~C, and arithmetical reals, denoted ~D, we prove the following. Theorem 2. If K is a computable (AD) subfield of the field of real numbers, then K is a proper subfield of ~C (~D). Corollary 3. ~C (~D) is not a computable (AD) field. However, we have Theorem 4. ~C is an AD-field. Finally, we prove Theorem 5. Let K be a (countable) subfield of real numbers which is not a proper subfield of ~D. No ordered field containing K (whose ordering is compatible with natural

646 ordering of K) is AD. Corollary 6. No ordered field containing Yi'D (whose ordering is compatible with the natural ordering of Yi'D) is an AD-ordered field. (Received April l, 1968.)

68T-488. j. W. HARDY and H. E. LACEY, University of Texas, Austin, Texas 78712. On the existence of Silov boundaries.

In this paper, necessary and sufficient conditions are given to insure that a linear· subspace of C(X) (X compact and Hausdorff) containing l has a Silov boundary. Also, necessary and sufficient conditions are given in order that the Silov boundary exist and be the closure of the Choquet boundary. These are given in terms of the smallest topology on X that makes each element of M continuous.

In particular, if M separates the points of the Choquet boundary of M, then the Silov boundary exists and is the closure of the Choquet boundary. An example is given where the Silov boundary exists, but is not the closure of the Choquet boundary. (Received April l, 1968.)

68T-489. P. ROSENTHAL, University of California, Berkeley, California 94720. On complemented and quasi-complemented subspaces of quotients of C(S) for Stonian S.

This work improves some of the results announced by the author in Abstract 68T-74, these cJVoticeiJ 15 (1968), 212, and in Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 361-364. m denotes the Banach of all bounded sequences of scalars (sometimes denoted by l~. LetS be an F-space (S is a compact Hausdorff space such that disjoint open Fu's have disjoint closures), let X be a Banach space which is a continuous linear image of C(S), and let Y be a closed linear subspace of X. Theorem 1. If X/Y is not reflexive, there exists a bounded linear operator mapping X/Y onto m. Theorem 2. If Y* is separable in the weak* topology, Y is quasi-complemented. In particular, if x• is weak* separable, every subspace of x• is quasi-complemented. Corollary 3. Every subspace of m is quasi-comple­ mented. Theorem 4. If S is u-Stonian (the closure of every open Fu is open) and X is not reflexive, then m is isomorphic to a subspace of X. Corollary 5. If X is an infinite-dimensional injective Banach space, m is isomorphic to a subspace of X, and every subspace of X with weak* separable dual is quasi-complemented in X. (Received April 1, 1968.)

68T-490. RUDOLF WILLE, Universitat Bonn, 53 Bonn, Wegelerstr. 10 Germany. On a problem of B. j6nsson.

Let Y 1 = ( { a,b,c,O }, +) be the semigroup where a + a = b + b = c are the only sums unequal 0; let Y 2 = ( (a,b,c,O}, +) be the semigroup where a + a= a + b = b +a= c are the only sums unequal 0. Then Y 1 and Y2 are two nonisomorphic, subdirectly irreducible, commutative semigroups which generate the same primitive class of semigroups. This gives a solution for Problem 67 in G. Birk­ hoff, Lattice theory, 3rd edition (1967). (Received April 4, 1968.) (Author introduced by Professor

K. W. jlinich.)

68T- 491. T. Y. LAM, University of Chicago, Chicago, Illinois, and IRVING REINER, University of Illinois, Urbana, Illinois 61801. Relative Grothendieck groups.

For n a field of characteristic p and H a subgroup of G, let a(G,H) denote the Grothendieck group of nG-modules, modulo H-split, G-exact sequences. This generalizes the notions of ordinary

647 Grothendieck groups and Green's ring of modular representations. Results concerning Z-bases for a(G,H) are therefore generalizations of the Krull-Schmidt and Jordan-Holder theorems. Theorem.

Let H Ll G be a cyclic p-group of order h, and Ul'···•Us be the nonisomorphic principal indecompos­ able, riG-modules. In each Uj, there exists a chain of G-submodules Vkj' I ~ k ~ h, such that ![Vkj]: I ~k ~h, I~ j :?sj form a free Z-basis for a(G,H). As a ring, a(G,H) has no nilpotent elements and can be described explicitly as an extension ring of a(G,l). If B is a p-complement of G, then the restriction functor induces a (ring) isomorphism a(G,H) "" a(H • B, H • B) if and only if each Brauer character of B can be lifted to a generalized Brauer character of G. Among the corollaries, we note: For Has above and Gap-group, a(G,H) ~ a(H) (and is thus independent of G). Other "excision" theorems can also be obtained. (Received April 3, 1968.)

68T-492. D. K. RAY-CHAUDHURI and R. M. WILSON, Ohio State University, Columbus, Ohio 43210. Solution of the Kirkman's school girl problem.

Let b,v,r,k and A be positive integers such that r = A(v- 1)/(k- !) and b = Av(v- 1)/k(k- 1). A (v,k, A)- Balance Incomplete Block Design (BIBD) consists of a finite set X of v elements called treatments and b subsets (called blocks) x 1,x2 •••. ,Xb such that (l) every treatment occurs in exactly r blocks, (2) every block contains exactly k treatments, and (3) every pair of treatments occurs together in exactly A blocks. A parallel class of blocks consists of a set of disjoint blocks such that every treatment occurs in one block of the class. A (v,k,A)-BIBD is said to be resolvable if the b blocks can be partitioned into r parallel classes. A (v,3,l)-resolvable BlED is said to be a Kirkman design of order v. In this paper it is proved that a necessary and sufficient condition for the

existence of a Kirkman design of order vis v == 3 mod 6. This problem is known as Kirkman's school girl problem and was introduced by Rev. Thomas J. Kirkman in an article, On a problem in combinations, Cambridge and Dublin Math. J. 2 (1847), 191-204, and attracted many mathematicians since 1847. (Received April 4, 1968.)

68T-493. P. L. ROSENTHAL, Stanford University, Stanford, California 94305. On the z~~ of the Bergman function in doubly-connected domains.

M. Skwarczynski in a recent paper entitled The invariant distance in the theory of pseudo­ conformal transformations and the Lu- Qi- Keng conjecture defines a Lu- Qi- Keng domain as a domain D C en for which the Bergman function KD(z,t) does not vanish in D X D. Further, he gave an example of a domain in c 1 which is not a Lu-Qi-Keng domain. Using some known results from the theory of elliptic functions, the author is able to extend M. Skwarczynski's results to conclude every doubly­ connected Lu- Qi- Keng domain in C 1 is pseudo- conform ally equivalent to a disc with the center de- leted. (Received April 4, 1968 .)

68T-494. SURJEET SINGH, Kirori Mal College, Delhi University, Delhi-?, India. Pseudo- injective modules. Preliminary report.

Let R be any ring. Then an R- module M is said to be a pseudo- injective R- module if for each submodule N of M every R- monomorphism of N into M can be extended to an R -endomorphism of M. The concept of pseudo-injective module was introduced by the present author and S. K. Jain in (On

648 pseudo-injective modules and self pseudo-injective rings, J, Math. Sci. 2 (1967), 23-31). In the above referred paper the problem that under what conditions a pseudo-injective module is quasi- injective was studied. The study of this problem is continued in this paper. In this paper all the rings are supposed to be with unity and the modules are taken to be unital right modules. Following are the main results proved. (1) Any pseudo-injective module over a commutative hereditary ring is quasi- injective. (2) Let K be any right Ore-domain, D the quotient ring of K, n any positive integer and R any ring such that Kn c;; R c;; Dn. Then any torsion free pseudo-injective R-module is injective.

(Received April 8, 1968.) (Author introduced by Professor R. S. Varm~

68T-495. S, K. BASU, Southern University in New Orleans, New Orleans, Louisiana. A supplementary note on comparison of the total strength of some Hausdorff methods. II.

ca and Ha being respectively the Ces1lro and Holder methods of order a> -1, the following result is proved as a supplement to a recent paper of the author (Math. Z. 103 (1968), 358-362). Theorem, For 0 < {3< a

68T-496. S.M. MAZHAR, Aligarh Muslim University, Aligarh, India. On the local property

~ IR, log n, llk summability of Fourier series.

A sequence [sn} is said to be summable IR, log n,llk' k ~1. if Lnk-lltn- tn-llk 1, it follows that summability IR, log n,llk of a Fourier series is a local property. On the other

hand it is well known that summability IR, log n,111 is not so. (Received April 8, 1968.)

68T-497. WILLIAM VOXMAN, University of Iowa, Iowa City, Iowa. On the shrinkability of decompositions of 3- manifolds.

An upper semicontinuous decomposition G of a metric space M is said to be shrinkable if

for each open covering U of H(;, each f > 0, and each homeomorphism h from M onto M, there exists

a homeomorphism f from M onto M such that (1) f = h on M - U* and (2) for each g E G, diam f(g) < f and there exists DE U such that h(D) :) h(g) U f(g). G is said to be weakly shrinkable if for each

open set W containing H~ and f > 0, there is a homeomorphism from M onto M such that h = id on M- Wand for g E G, diam h(g)

of ann-manifold M, then G is cellular and M/G = G. In fact, any shrinkable used of a manifold is cellular. A 0- dim. used G of a metric space M is locally shrinkable in case for each g E G, there

exists an arbitrarily small open set Ug containing g such that (1) Bd H(; n Ug = 13 and (2) if K is

compact, g C K C U , and f > 0, then there exists a homeomorphism h from M onto M such that g (a) if g' E HG' g' n K "f 13, then diam h(g') < f and (b) h(x) = x on M - Ug. Theorem 3. Let G be a

649 0-dim. monotone used of ann-manifold M such that H(; is a G0. Then (1) if G is locally shrinkable, 'Y > 0, .and G' is the used of M where HG' = (g E HG: diam g !" 'Y}, then M/G = M/G' = M, and (2) if n = 3, then G is locally shrinkable iff G is weakly shrinkable. (Received April 10, 1968.)

68T-498. NORMAN LEVITT, Courant Institute of Mathematical Sciences, New York, New York 10012. Transversality for spherical fibrations. Preliminary report.

Let Nn be ann-dimensional Poincar!!-duality space, A splitting of Nn is a triad (X;Y,Z), X= Y U Z, such that (Y, Y n Z) and (Z, Y n Z) are n- dimensional Poincar!!- duality pairs, together with a

homotopy equivalence !D: Nn ~X. A k-dimensional Poincar!! duality-subspace of Nn is a splitting of Nn such that the inclusion Yn Z <:;; Y is, up to homotopy, a fibration ~with fibre ~ sn-k-l.

If Mm is an m-dimensional Poincar!! duality space, f: Mm- Nn, then for a given splitting (X;Y,Z) of Nn, we say that f splits if there is a splitting (A: B,C) of Mm and a map of triads (A: B,C)- (X: Y,Z) homotopy consistent with f. If (X: Y,Z) is a k-dimensional Poincar!! subspace of Nn, we say that f: Mn- Nn is homotopic to a transverse-regular map if f splits so that the map (B, B n C) - (Y, Y n Z) is up to homotopy a map of sn-k-l_fibrations (thus making (A: B,C) an (m - n + k)-dimensional sub- Poincare-duality space of Mm). Theorem. Let f: Mm- if with (X: Y, Z) a k-dimensional Poincar!!-duality subspace of Nn and v(M) the Spivak normal fibration for Mm. If there exists a

sphericai fibration 11 over Nn with (1) f* 11- v(M) fibre-homotopically P L, (2) ~ + 11IY is fibre­ homotopically PL, (3) 2m 1" n + 2k + 2, then f is homotopic to a transverse regular map. (Received April 10, 1968 .)

68T-499. P. L. ANTONELLI, University of Tennessee, Knoxville, Tennessee 37916. Some results on stable diffeomorphism of spheres.

It is known that for any homotopy n-sphere ::!:n, the product ::!:n X sP is diffeomorphic to sn xsP provided p 1"n. But, for the exotic 16-sphere, ::!:16 xs 12 I' s 16 xs 12 . Calculations with

the h-enclosability group ~.plead to Theorem. Let n ;;i 2 (mod 4) and i = 1,2 or 3, Then for any homotopy n-sphere,~n, ::!:n X Sn-i ""Sn X Sn-i. Corollary. For n ;;i 2 (mod 4), ~n imbeds in RZn-Z

with trivial unique normal bundle. Clearly, these results are best possible in general. Theorem. There are exactly eight exotic 17-spheres which imbed with nontrivial normal bundle in the stable range (we say these are strange) and eight which do not. Only two of those which are not strange bound 11"-manifolds, If we restrict ton= 1,:3,5 (mod 8), then we can prove2:n xsn-i ""Sn xsn-i,

0 ;'i ;'6, This would indicate that the strangest spheres occur for n == 0 (mod 4). (Received April 8, 1968 .)

68T-500. ERIK ELLENTUCK, Rutgers, The State University, New Brunswick, New jersey 08901. Almost combinatorial functions.

Let ZFU 0 =set theory admitting urelemente and with AC replaced by AC 0 (= AC for sets of finite sets); t!. = Dedekind cardinals, and w = integers. Let 'H be an arbitrarily quantified positive first order sentence in functors for + and • . Let f 1, ... ,fk be function variables, and 'Hf the universal sentence obtained from 'H by replacing existential quantifiers by the fi as Skolem functions. Theorem, (i) (ZFU 0 ~ 'H holds in t!.) if and only if (ii) (ZFU0 ~there exist almost combinatorial fi

650 such that 'Hf holds in Ll), Using the methods of [Ann, of Math, 82 (1965), 225-248], (ii) can be reduced to prove-ability of a~: sentence about w. Several examples show that the theorem fails either (a) for Horn sentences, or (b) if we drop AC 0• We believe our result implies the 'naturalness' of almost combinatorial functions in axiomatic set theory. (Received April 12, 1968.)

68T- 501. J. D. MASON, California State College, San Bernardino, California. A necessary and sufficient condition that the limit distribution of a sequence of normed, centered sums of indepen­ dent random variables involving r distributions be a convolution of r stable distributions.

Theorem. Let (Xn 1 be a sequence of independent random variables such that the distribution function of each Xn is one of r distribution functions F p ... ,Fr· Assume that there exist normalizing constants (A } such that B- 1(X + ... + X ) - A converges in coefficients 0 < Bn-oo and centering n n 1 n n law to a distribution function G. In order that G be a convolution of r stable distributions with characteristic exponents 0 < '\ < ... < -\ :"' 2, it is necessary and sufficient that (i) F i belongs to the domain of attraction of a stable law with characteristic exponent \• 1 :"' i :"' r (r>ossibly requiring a permutation of subscripts of F 1, ... ,F rl; (ii) nr(n)/n - 1 as n - oo, where ni (n) denotes the number of

random variables among X 1, ... ,Xn whose distribution function is F i; and (iii) B (i,ni (n))/B (r ,nr(n)) - (some) pi' 0

1968.)

68T-502. G. P. WELLER, University of Chicago, Chicago, Illinois 60637. Locally flat

imbeddings of topological manifolds.

Let Mn be a closed, orientable topological manifold, n ;, 2, and let Qq be a simply connected

p .L. manifold. If M is 2n - q connected, Q is 2n - q + 1 connected and 2q > 3(n + 1), then any map f: M - Q is homotopic to a locally flat imbedding. If 2q > 3 (n + 1) + 1, M is 2n - q + 1 connected, and

Q is 2n - q -+' 2 connected, then any two locally flat imbeddings f ,g: M - Q which are homotopic are concordant, that is, there is a locally flat proper imbedding H: M X I- Q X I with H(x,O) = (fx,O) and H(x,l) = (gx,l). The proof of these facts makes use of a polyhedral imbedding theorem of A. V. Cernavski1 and engulfing theorems for topological manifolds due to M. H. A. Newman and]. A. Lees, and proceeds by induction on the number of coordinate neighborhoods of M. (Received April 12, 1968.)

68T- 503. Y AU-CHUEN WONG, University College of Swansea, Singleton Park, Swansea,

Great Britain. Order-infrabarrelled Riesz spaces.

A locally convex Riesz space (X,c,3') is said to be order-infrabarrelled if each barrel in X which absorbs all order-bounded sets in X is a Y-neighbourhood of 0. There are bornological

Riesz spaces which are not order-infrabarrelled. Theorem 1. An infrabarrelled Riesz space (X,C,Y) is order-infrabarrelled if and only if the topological dual X' of X is a band in the order­ bound dual Xb of X. Theorem 2. An order-infrabarrelled Riesz space (X,C,Y) is barrelled if and

only if the dual cone C', i,e. the set of all positive Y-continuous linear functionals on X, is a

651 ~-cone in (X', C', u(X', X)) (for definition see H. H, Schaefer: Topological vector spaces, Macmillan, New York, 1966). Corollary. Let the order-bound dual Xb of a Riesz space (X,C) be total over X, and let 3~ be the order topology on X (for definition see Schaefer's book), Then (X,C, _9j;) is barrelled if and only if C* is a ~-cone in (Xb,C*,u(Xb,X)), where C* is the set of all positive linear functionals on X. Theorem 3, An infrabarrelled Riesz space (X,C,Y) is barrelled if and only if C' is a ~-cone in (X', C', u(X',X)) and X' is a band in Xb. (Received April 12, 1968,) (Author introduced by Professor

Kung-Fu Ng.)

68T-504, P, F, DUVALL, University of Georgia, Athens, Georgia 30601, Topological embeddings in codimension one.

If X is a closed subset of a manifold Q, we say that Q- X~ 1 - lc at X if for each open set U containing X there is an open set V, XCV C U, such that each loop in V - X is null homotopic in U- X, Let Mn be a closed, !-connected topological manifold of dimension n, n ~ 5, and suppose M is embedded in the interior of a P L (n + !)-manifold Qn+l. The main theorem of this paper is the following analog of the regular neighborhood theorem. Theorem, If Q- M is 1 - lc at M, then every neighborhood of Min Q contains a compact PL submanifold wn+l of Q such that (1) M C int W, (2) W - M is P L homeomorphic to bd W X [9,1), (3) W is P L homeomorphic to the product of some closedPL manifold with [0,1], and (4) M is a strong deformation retract of W, Corollary. If Mn,n? 5, is a closed, !-connected PL submanifold in the interior of a PL (n + !)-manifold, then any regular neighborhood of M is P L homeomorphic to the product of a closed P L manifold with [0,1]. The proof of the theorem uses the Browder, Levine, and Livesay boundary theorem [Amer, j. Math. 87 (1965), 1017-1028]. (Received April 15, 1968.)

68T-505. WITHDRAWN.

68T- 506. B. TRE YBIG, Tulane University, New Orleans, Louisiana 70118. An approach to the polygonal knot problem using projections and isotopies,

D. E. Penney has defined for an oriented polygonal knot Kin regular position a "word" W(K), See these cNotiaiJ 12 (1965), 793, We extend Penney's idea to define also a boundary collection C(K). For trefoil T with projection double points a,b,c, W(T) = ab- 1ca- 1bc-l and C(T) = {(c- 1a, ca- 1}, {ab -l, a- 1b}, {bc- 1, b- 1c}, {c- 1a, a- 1b, b- 1c}, {ca- 1, ab- 1, bc- 1 }}. C(T) describes the complementary domains of projection 1r(T) with ordered segments which indicate over and under crossings. Given K, K' there is natural definition of C(K) ~ C(K'). Theorem A. C(K) ~C(K') ==>

K ~ K'. Now define simple transformations S: K - K' of type (a) I(I') add (delete) a small loop ... xx- 1 ... , (b) II(II') add (delete) a two crossing loop ... ab ... b- 1a- 1 ... , (c) III changing tri- angular domain ... xy ... y- 1z _... x-lz-l ... to ... rs ... zr-l ... z-ls- 1 .... Theorem B. If S: K - K'

652 is simple, then K ~ K'. We say T: C(K) -->C(K') is of type • if there exist L, L', S so that

C(L) ~ C(K), C(L') ~ C(K') and S: K' - L' is of type •. Theorem C. If his an orientation preserving homeomorphism of E3 onto E 3 taking K onto K', there exists C(K) = C(Ko) '"!J.c(K 1) ~ ... ~C(Kn) = C(K') or C(- K') where each Tp is simple. Conjecture. If K, K' are as in Theorem C and 1rK, n-K' have ,;; N crossings, there is a Theorem C sequence where 1rKP has ,;; 2N crossings. (Received April 19, 1968 .)

68T-507. R. J, DAVERMAN and W. T. EATON, University of Tennessee, Knoxville, Tennessee 37916. An equivalence for the embedding of cells in E 3•

Let B denote the set of points in E3 whose norm is less than or equal to 1, and 1r the map of B onto [- 1,1] given by projection onto the first coordinate. Theorem, Given a topological 3-cell K 3 in E3 and an open set U containing K, there is a map f of E3 onto E satisfying (1) f(K) is an arc, 3 onto E 3 - f(K), (3) f I is the identity, and (4) there are (2) f is a homeomorphism of E - K E 3 -u homeomorphisms g of B onto K and h of [- 1,1] onto f(K) so that h1r = fg. A similar result holds for topological 2- cells in E 3 . Corollary. Let C be a topological cell in E 3. Then there exists an arc A in E 3 so that E 3 - C is homeomorphic to E 3 - A. (Received April 19, 1968.)

68T-508. R. L. SEIFERT, JR., University of California, Berkeley, California, On prime binary relational structures,

Let ~ be the class of structures- (A,R) with one binary relation R <; A X A. If .5e <; ~. we say a structure '!I is St'-prime, or '!I E Pr(Ye}, iff: (i) '!I E .5e; (ii) '!I has more than one element; and (iii) if .8.~ EX and ~l(\8 X ~).theneither '!11\Bor 'HI~. [ll.8means l X 2J ~.8forsome 2J.] Let 1/" = [ (A,R) E ~: Dom(R) U Rng(R) = A}, ~ = [( A,R) E ~: Dom(R) = A}. Y = [ (A,R)E ~: Dom(R)nRng(R)=A}, f_prime: (i) the identity relations (A,= ), where A has a prime number of elements; (ii) reflexive closures (A, f U = ) of finite connected unary algebras (A,f) with more than one element; (iii) finite linearly ordered structures with more than one element. (Received April 19, 1968.)

68T- 509. B. W. HUFF, University of California, Riverside, California 92502, The loose subordination of differential processes to Brownian motion,

A differential process [X(T)/T E [0, oo]} is said to be loosely subordinate to the standard Brownian motion [W(t)} if it is independent of [W(t)} and if there exists a loose subordinator [S(T) }, i.e. a differential process whose sample paths are a,s. nondecreasing, such that W(S(T)) = X(T) a.s. Any differential process that has sample paths a.s. of bounded variation over [0,1) is loosely subordinate to the standard Brownian motion, The LE!'vy parameters of the loose subordinator

653 have been obtained in terms of those of the loosely subordinate process when [X(T)} is symmetric stable or is itself a subordinator. The common form of the Levy spectral function of the loose sub­ ordinator in these cases is given by Ms{.\) =- ~JJxl {2/Vz1rX")exp [- t 2/2.\} dtdMx(x) for A> 0. (Received April 2, 1968 .)

68T-510. H. j. KEISLER, University of Wisconsin and University of California, Los Angeles, California 90024. On the quantifier "There exist uncountably many".

Let L be a countable, first-order logic with identity, and form L(Q) by adding the quantifier (Qx) which means "There exist uncountably many x such that". Give L(Q) the usual first- order axioms and rules of inference, plus the Axioms: I. 1 (Qx)(x = y v x = z). 2. ('tx)(¢ - if;) - ((Qx)¢- (Qx)lf). 3. (Qx)¢(x) <--+(Qy)¢(y), where y does not occur in ¢(x). 4. (Qy)(3x)¢ -(3x)(Qy)¢ v (Qx)(3 y)¢. Completeness Theorem. A set T of sentences of L(Q) is consistent if and only if it has a model. The proof uses the completeness theorem for w-logic. A different completeness theorem for L(Q) is given by Ebbinghaus (these c){otiai) 15 (1968), 547) using methods of Vaught and Fuhrken. Now form the "w-logic" Lw(Q) by adding constants 0,1,2, •.• and a predicate N(x). Give L w(Q), the infinite rule ¢(0), ¢(1), ••. ~ N(x) - ¢(x) and the new Axioms: 5. N(O), N(l), .•.• 6. i(Qx)N(x). w-completeness Theorem. A set T of sentences of Lw(Q) is consistent if and only if

T has an w- model. Let T be a consistent theory in L w (Q). By a class in T we mean a set r of formulas ¢(x) with at most x free. Theorem. Let C be any set of pairwise inconsistent classes in T. Then the set of all r E C, such that r is realized in all w-models ofT, is countable. (Received April 22, 1968.)

68T-5ll. PAUL SCHAEFER, State University College, Geneseo, New York. Transformations of almost convergent sequences.

Regular matrices which transform almost convergent sequences into almost convergent sequences having the same F-limit were characterized by G. M. Petersen in Proc. Amer. Math. Soc. ll (1960), 469-477. We extend his results to the class of almost regular matrices, defined by j. P. King in Proc. Amer. Math. Soc. 17 (1966), 1219-1225. Theorem. Let A= [ank] be an almost regular matrix and let A(x) = [Li<:ankxk}. Then F-lim A(x) = F-lim x for every almost convergent sequence x if and only if limqL~o[I:E(: 0 (an+i,k - an+i,k+l)l/(q + l)] = 0 uniformly inn. We base our proof on the following analogue of a theorem of Schur. Lemma. Let A(i) = [ank(i)], i = 0,1, .•• , be a sequence of infinite matrices such that Lk lank(i) I < H < + oo for all n and i, and such that for each k, limn ank(i) = 0 uniformly in i. Then, limn Lk ank(i)xk = 0 uniformly in i for each bounded sequence x if and only if limn Lk lank(i) I = 0 uniformly in i. (Received April 2, 1968.)

68T-512. C. P. BRUTER, 33 Bld Dubreuil, 91-0rsay, France. Le Crit~re de Minimalit

Nous d

654 .les objets de ~. Designons par B la famille d'objets minimaux des¥ par rapport aux caract~res i E I(~), et qui ne sont pas objets de .d. Une classe :ff d'objets de~ n'est composee qui d'objets des¥ si et seulementsi :ff ne contient pas d'objets de B. Le crit~re que nous venons d'enoncer

manque de precision; il n'en est pas moins un guide tr~s utile pout la recherche de caracterisation de classes d'objets. (Received April 19, 1968.)

68T- 513. JOHN DE CICCO and R, CITRON, Illinois Institute of Technology, Chicago, Illinois. Concerning some relations between contact transformations and solutions of partial differential equations.

In a real space V n+l of (n + 1)?;; 2 dimensions the possible solution of a first order partial differential equation F(z;x;p) = 0, where F ¢ 0, is of at least class two in a region of the opulence

n 2n+l and may be obtained by means of the group of contact transformations T of the opulence. If such a T, for which F(z;x;p) is a component, can be found, then the complete solution is given by its Lie directrix equation. If z = xn+ 1 is considered to be an i,ndependent variable, then F = 0 may be written in a homogeneous form G(x;p) = 0, for which any solution is u = u(x) with Pi = au/ axi,

where i = 1, .. ,, n. In Vn+l a contact transformation Tis considered in which the Lie directrix equation is of the form u = u + H(x;x), where H is of at least class two in a (2n + 2):;; 4 dimensional region of points (X;x), such that the matrix (a2H;axi axj) is of rank (n + 1). Then Tis given by the set of equations ii = u + H(x;p), xi= xi(x;p), where i = 1, ... , n, for which ox/opt 0, and all of the

Poisson parentheses are zero. If xn+l = F, then solving the systems xi= xi(x;p), xn+l = 0, for the

Pi in the form Pi= Pi(x), the related Pfaffian equation Pidxi = 0 is integrable, of which an integral is the complete solution of G(x;p) = 0. (Received April 19, 1968,)

68T- 514, PIERRE LEROUX, Universite de Montreal, Montreal 3, Quebec, Canada. Une car;:tcterisation de la categorie des groupes.

11 est bien connu que si B est une variete de groupes satisfaisant le theor~me de Shreier, alors B est, ou bien la variete de tousles groupes, ou bien la variete de tousles g.roupes abeliens, ou bien la vatiete de tous les groupes abeliens d'exposant p pour un entier p premier (?. M. Neumann and M. F. Newman, On Shreier varieties of groups, Math, z. 98 (1967), 196- 199]. D'autre part, on sait que tout groupe est sous-groupe d'un groupe simple [w. R. Scott, Contributions to the theory of

~· University of Kansas, 1956]. La categorie de tousles groupes est alors caracterisee comme etant une categorie algebrique d'un type particulier, (en fait une variete de groupes) satisfaisant 1l l'axiome de Shreier et dans laquelle tout objet peut etre plonge dans un objet simple, ces notions ayant ete adequatement generalisees. D'autres categories de groupes sont aussi caracterisees, (Received April 19, 1968,)

68 T- 515. J. A. LEES, Rice University, Houston, Texas 7700 1, A sufficient condition for the triangulation of manifolds. Preliminary report.

Let M be a topological manifold without boundary. A neighborhood of a subset X of M is called a p.l. neighborhood if it can be triangulated as an open p.l. manifold containing X as a subcomplex, Theorem, Let Mq be a closed simply connected, topological manifold, q ;;. 6. Let Kn be ann-complex,

655 q- n;:; 3, and f: K ~ M- pt. an embedding. Iff is a homotopy equivalence of K and M-pt, and fK has a p.l. neighborhood in M-pt, then M is triangulable as a p.l. manifold. The proof proceeds along the following lines. First, Newman's Engulfing Theorem for Topological Manifolds is used to show that M- pt. is triangulable as a p.l. manifold. A splitting theorem of Browder and Stallings' proof of the Generalized Poincare' conjecture in dimensions E: 5 are then used to show that M is triangulable as a p.l. manifold. (Received April 18, 1968.)

68T- 516. B. L. OSOFSKY, Institute for Advanced Study and Rutgers, The State University, New Brunswick, New jersey 08903. Homological dimension in rings of continuous functions. Preliminary report.

Let R be the ring of all continuous functions on [0, 1]. Theorem. There exists a subring

S ~ R such that gl.d (S) = 3 and S has a localization with zero divisors. Theorem. Let f E R. Then

hd (fR) E: 3. Theorem. Let f, g E R. Then the continuum hypothesis implies hd (fR + gR) = 0 or 3. (Received April 18, 1968.)

68T- 517. J. T. WOOD, Swarthmore College, Swarthmore, Pennsylvania 19081. Algebraic reduction theory of W*- algebras.

If M is a von Neumann algebra with commutant M' and center Z, then R(M,M') is type I. Let e be a maximal abelian projection in R, T(ze) = z the isomorphism between Ze and Z, and P(x) = T(exe) for x E M. Identifying Z and C(K), and choosing t E K, one defines a state on M by f!St(x) = P(x)[t}. Then f6t induces a C*-representation ?rt(M) of M on Ht• Using the decomposition of a type In algebra into a sum of type I algebras, one obtains Theorem I. The direct sum representation (2::?rt)(M) of M on 2:: E9Ht is faithful. The definition of equivalent projections gives Theorem Z. If e and f are maximal abelian projections in R with u•eu = f, then ¢t,e and f!St,f induce unitarily equivalent representations of u *Mu and M, respectively. Theorem 3. If M is type In' then the weak closure of 1rt(M) is a type I factor, no matter what maximal abelian projection e is chosen. Theorem 4. If M is November ZO, 1967 .) type n 1, then e may be chosen so that 1rt(M) is a type n 1 factor. (Received

68T-518. J. L. CHRISLOCK, University of California and Adlai E. Stevenson College, Santa Cruz, California 95060, A certain class of identities on semigroups.

If f(xl''"'xm) = g(xl' ... 'xm) is an identity (f and g are words in the variables xl' ... 'xm)' then V(f) and V(g) will denote the set of variables used in f and g, respectively. A product of variables

h(xl'""xm) will be called~ if each variable in V(h) occurs only once in h(x1, ... ,xm). LetS be a semigroup. Theorem. The following arlO! equivalent: (1) S satisfies an identity f(x 1, ... , xm) simple semigroup whose = g(x1, ... ,xm) with V(f) "f V(g). (Z) S is an ideal extension of a completely structure group satisfies xr = e by a semigroup that satisfies yr = 0. (3) S satisfies an identity (ztwtzt)t = zt. Theorem. The following are equivalent: (1) S satisfies an identity f(x 1, ...,xm) = simple semigroup g(x1, ...,xm) with V(f)- V(g) "f IJ and g free. (Z) Some power of Sis a completely whose structure group satisfies xs =e. (3) S satisfies an identity (z 1 ••• zt w 1 ... wt z 1 ... zt)t • f(x , ... ,xm) = z 1 ••• zt = z 1 ... zt' Theorem. The following are equivalent: (1) S satisfies an identity 1

656 g(xl'''''xm) with V(f) - V(g) 'I II and f free, (Z) Some power of S is a rectangular band. (3) S satisfies an identity w 1 ... wt v 1 ... vt z 1 ... zt = w 1 ... wt. z 1 ... zt• (Received April 19, 1968.)

68T-519. G. J, RIEGER, State University of New York at Buffalo, Amherst, New York 14ZZ6. A number theoretic application of a Tauberian theorem with remainder term.

A classical real Tauberian theorem of Hardy and Littlewood (19Z9) is proved here with remainder term (for a special case of this, see G. Freud, Acta. Math. Acad. Sci. Hungar. Z (1951), 299-308). As an application, the following generalization of a theorem of E. Wirsing (Arch. Math. 7 (1956-1957), Z63-Z7Z) is proved: Let T be an infinite set of pri~es p with Lp"' ET p- 1 log p = -XrP (a. + O((log x) -t))log x and a.= a.(T) > 0, t = t (T) > 0; then Lm :;;;x,p lm~ E T m- 1 = (c + O((log log :x)- 1)) ET(1- p- 1)- 1 with c = c(a.) > 0. (Received April 17, 1968.) •IT p-x,p,.

68T-5ZO. D.P. ROLFSEN, Institute for Advanced Study, Princeton, New jersey 08540. A generalization of an isotopy invariant of Chen and Milnor.

A link in a connected n-manifold M is an embedding f: Q - M, Q being a finite disjoint union of spheres of various dimensions. Define G(f) = 11'1 (M - f(Q)) and let K(f) be the kernel of the inclusion homomorphism G(f) - 11'1(M). Then in the piecewise linear category, assuming dim Q:;;; n - Z: Theorem. If f,g: Q- M .are isotopic links in M, then G(f)/K(f)9 ~ G(g)/K(g)9 for each q = 0,1,Z, ••• , oo.

Here Ko ::) K 1 ::) Kz::) ••• ::) K00 denotes the lower central series of the group K, defined by K0 =' K, 3 Kq+l = [K, K9), K00 = nKq. This extends results of K. T. Chen (dim Q = 1, M = R , PL links) and j. Milnor (dim Q = 1, dim M = 3, topological category). (Received April 17, 1968.)

68T- 5Zl. A. C. LAZER and L. R. ANDERSON, Case Western Reserve University, Cleveland, Ohio 44106, On the asymptotic behavior of a class of linear differential equations.

Consider the differential equation y(n) + p(t)y = 0, where p E C(- oo,oo), M l!: p(t) l!: m > 0, and where n is either odd or of the form 4k, Let s1 (Sz) denote the subspace of solutions which tend to zero along with their first n derivatives as t ... + oo (t ... - oo). If n = 4k, then S 1 and S2 each have dimension Zk, and s 1 n s2 consists of the null solution. If n = 4k + 1, s 1 has dimension Zk.+ 1, s2 has dimension Zk, and S 1,n Sz consists of the null solution. If n = 4k + 3, S 1 has dimension Zk + 1 and the dimension of Sz is Zk + Z, with s 1 n s2 = {OJ. In each of the above cases, if y EE s 1 (y EE S2), then y is unbounded as t ... + oo (t ... - oo). For n odd and 0 > m l!: p(t) l!: m, the foregoing results are easily

applied to characterizing S 1 and s2 in this case, (Received April ZZ, 1968 .)

68T- 5ZZ, D. J. CAVENY, Florida State University, Tallahassee, Florida 32306. Bounded variation in LP ..BQ!.!!!.

Kaczmarz defined yP to be the subspace of Lp = Lp(O,Zr) consisting of those functions f(t)

for which the sums L(f(bk- t)- f(ak- t)) remain bounded in Lp norm as (a1,b1), ••• , (an,bn) ranges over all finite collections of nonoverlapping subintervals of (0,211'). Theorem. yP = V1 for l:;;; p;!; z. (Received April 17, 1968.)

657 68T-523. T. R. BERGER, University of Minnesota Institute of Technology, Minneapolis, Minnesota 55455. Odd p groups as fixed point free automorphism groups.

The theorem stated here extends results announced earlier (Abstract 68T-14, these c}l/otiaiJ 15 (1968), 192; also in Illinois]. Math. (to appear)). Tl)eorem. Suppose AG is a finite solvable group with normal subgroup G. Assume A is an odd p group. Suppose pc t- rd + l for any pc :;§ exp A and any primer where r 2d+ 1 divides IGI (and if A is irregular we also require that 2pc t- rd + 1 and pc 'I rd - 1 where rd divides IGI). If CG(A) = 1, then the Fitting length of G is bounded by the power of p dividing lA 1. The result was obtained by extending the methods of the earlier theorem. (Received April 22, 1968.)

68T-524. L. C. SIEBENMANN, University of Paris, ~athe'matiques, Faculte' des Sciences, 91-0rsay, France. Two consequences of topological engulfing.

Let (Wn; V, V') be an h- cobordism between closed topological (n - 1)- manifolds V and V'. 2k-l 2k-1 Let = denote homeomorphism. Theorem A. W X S = V X [0, 1] X S except perhaps if

n = 3, k :;§ 5 or n = 4, k :;§ 3. Theorem A follows from the case of the 1- sphere S 1 (i.e. k = ;1) since s2k-l is the k-fold join of S 1s. The case of S 1 follows formally from Connell's engulfing result W - V' = V X [0, l) (Illinois J. Math. ll ( 196 7), 300- 309). Hence Corollary. If An is the n 2k-1 n-1 2k-1 closed region in Rn between two locally flat (n - 1)- spheres, then A X S = S X JP, l] X S • Details will appear in Arch. Math. Some fresh engulfing proves Theorem B (cf. E. Luft, Invent. Math. 4 (1967)). If Mn, n ;; 5, is a contractible open separable topological n-manifold, and Mn is 1- LC at oo, tfien Mn = Rn. The property l - LC at oo is this proper homotopy invariant: for compact K C M, there exists compact L :J K so that every composed map S 1 - M - L '-"M - K is contractible. A generalization of Theorem B tries to characterize open collars. (Received April 22, 1968.)

68T- 525. P. A. FUHRMANN, Tel- Aviv University, Tel- Aviv, Israel. Maximal ideals in

OH 00 (N).

Let N be ann-dimensional Hilbert space. We denote by OH 00(N) the Banach algebra of all norm bounded N-operator valued analytic functions in the unit disc, B(N) the algebra of all linear

operators on N. Let ¢ be a complex homomorphism of H00 • We define a map cf> ¢,R of OH 00 (N) into

B (N) by the relation ( ¢,R (A)~ ,1]) = ¢(R- 1 A(z)R t11) where ~. 1] E N, A E OH00 (N) and R an invertible element of B(N). We write A(¢) for <1>¢,! (A). It is easy .to see that A({ll) = A(jil)* where A(z) = A(z)* and i the complex homomorphism of H 00 defined by i(f) = {ll(f). Theorem. is a homomorphism of OH00 (N) onto B (N) if and only if = cf> {II,R for some !II and R as above. Theorem. M is a maximal left, right or two sided ideal of OH 00 (N) if and only if it is of the form [A IA({II) ~ = 0}, [AlA(¢) • = 0 1 or

[ AIA(!Ii) = 0 }, respectively, ~being any nonzero vector in N. A similar pair of theorems holds true for the algebra OA(N), the subalgebra of OH 00 (N) of all functions having continuous extension to the unit circle. (Received April 22, 1968.)

658 68T- 526. F. E. SIWIEC and JUN-ITI NAGATA, University of Pittsburgh, Pittsburgh, Pennsyl­ vania 15213. A note on nets and metrization,

A collection ~of subsets of a topological space X is a net for X if for each ,oint x in X and open neighborhood U of x, there exists a B E ~ such that x E B C U. A space with a u-loeally finite net is called a u- space, and a T 3- space with a countable net is called cosmic, For other background see Abstract 68T-lll, these c}/oticeiJ15 (1968), 224, For a T 3-space X the following are equivalent: (1) X has a u-closure preserving net; (2) X is a u- space; (3) X has a u-discrete net. Improving A. Okuyama's theorem (Sci, Rep, Tokyo Kyoiku Daigaku Sect. A 9 (1967), 60-78), we can prove that for a T 1-space X the following are equivalent: (1) X is metrizable; (2) X is T3, an M-space and a u-space; (3) X is collection wise normal, a wA- space and a u-s pace; (4) X is collection wise normal, au-space and has a point countable base, A space is a cosmic space iff it is T 1, collectionwise ormal, separable, and au-space. (Received April 23, 1968.)

68T-527. H. H. WICKE and J. M. WORRELL, JR, Sandia Corporation, Albuquerque, New

Mexico 87115, On monotonically complete subs paces of T 2- spaces having X- bases locally.

The authors have shown that the class C of T 2-spaces having ;\-bases locally is equivalent to the class of T 2 open continuous images of metrically topologically complete spaces and also to the class of T 1- spaces which are locally monotonically complete and which, have bases closurewise of countable order. (The phrase 'having property P locally' means that there exists a base for the space each of whose elements has property P .) The purpose of this abstract is to state an analogue of the following theorem first proved by P. S. Alexandroff for the separable case [C. R. Acad, Sci. Paris 178 (1924), Ui5-187] and thenby Vedenisov (who gives credit for an independent discovery by Hausdorff) for the general case [J. de Math. Ser. 9, 9(1930), 377-381]. Theorem (Vedenisov-Hausdorff). A subspace of a metrizable space is metrically topologically complete if and only if it has a countably monotonically complete base. Theorem, A subspace of a space in class C is in class C if and only if it has a monotonically complete base, or, equivalently, if and only if it has a countably monotonically complete base, (For terminology see Duke Math. J, 34 (1967), 255-272, 813- 814; and Abstracts 66T-159, 66T-331, these c}/oticeiJ 13 (1966), 255, 510,) (Receiv~d April 23, 1968,)

68T- 528, C. J. McCANN, Washington University, St. Louis, Missouri 63130, Wiener-Hopf operators on locally compact groups.

Let G be a locally compact not necessarily Abelian group which has a Borel measurable linear order relation compatible with its group structure. Let M(G) be the Banach algebra of all complex bounded Radon measures on G with convolution as multiplication. For 71 E G let x+(.,, ~) and x-(.,, ~)be the characteristic functions of the sets {~ E G: ~ ~ 71} and {~ E G: ~ ~ 71}, respectively. For f E M(G), define (E+ (71)f)(B) "' JB x+ (.,, 0f(d~) for all Borel sets B. Define E- (1/)f simila:dy. If E+('T/)M(G) "' {f E M(G): E +(1/)f"' f} and E -(1/)M(G) "' {f E,M(G): E- (1/)f =' f1 then E+ (1/)M(G) and E- (71)M(G) are Banach spaces. For C E M(G) we consider the half-infinite Wiener-Hop! operators + +X + +p + -X - Wc'P, We' acting onE (e)M(G) where We'!£= E (e){!£• c), We'!£= E (e)(c *!£),etc. Here e is the -1 -1 -1 -1 -1 -1 + identity of G. We show that if c =Up • Vx = Ux • Vp where Up, Up , Ux, Ux E E (e)M(G) and VP' v; 1, VX' VAl EE-(e)M(G), then the theory developed by I. I. Hirschman, Jr. (Trans, Amer, Math,

659 Soc. 121 (1966), 133-159), in the case where G is Abelian, for finite section Wiener-Hopf operators, such as w~•P(11)P. = E+(e)E- (11)(P. *c), w~· >.(11)/l = E+(- 11)E- (e)(c * !l), etc., continues, with certain necessary modifications, to be valid. (Received April23, 1968.)

68T- 529. H. H. WICKE, Sandia Corporation, Albuquerque, New Mexico 87115. A sum theorem

~ T 2-spaces having >.-bases locally.

Worrell and Wicke characterized the class of T 2 - spaces having A- bases locally as a class of T 2 open continuous images of metrically topologically complete spaces and as the class of T 1- spaces which are locally monotonically complete and which have bases closurewise of countable order. (For references and terminology see Abstracts 66T-159, 66T- 331, 67T- 249, these c}/oticriJ 13 (1966), 255, 510; 14 (1967), 290; and Duke Math. J. 34 (1967), 255-272, 813-814. The term 'locally' here refers to the existence of a base whose elements have the property modified by the word

'locally'.) Theorem. If a T 2-space Sis covered by a locally finite collection of closed subsp;aces having >.-bases locally, then S has A-bases locally. (Received April 23, 1968.)

68T- 530. FRED GALVIN, University of California, Berkeley, California. Partition theorems for the real line.

I use the notation of Erdos and Rado, A partition calculus in set theory, Bull. Amer. Math. 2 Soc. 62 (1956), 427-489. Theorem. If 111- (11)! and a < w1, then Ill- (a, a) • Corollary. If 2 2 r6 ~ ''\• 111- (w, w + 1) , and a< w1, then 111- (a,a) • Corollary. If Ifill> M0 , r6 t; w1, Ill f; w;, and 2 a < w1, then r6 - (a, a) • This generalizes Theorem 6 of Hajnal, Some results and problems on set theory, Acta. Math. Acad. Sci. Hungar. 11 (1960), 277-298. Theorem. The relation >.-<>->!holds with respect to Borel partitions, for all finite r and n. More precisely, if E is the real line,

[E]r = K 1 U ••• U Kn and if ((x1, ••• ,xr): (x1, ••• ,xr} E Ki} is a Borel set in Er for each i, then A E lKi] for some i. (Received April 24, 1968 .)

68T- 531. P. M. GAUTHIER, Universitt! de Montrt!al, Montreal 3, Canada. Unbounded holomorphic functions bounded on a spiral.

Let w = f(z) be an unbounded holomorphic function in the unit disc which is bounded on a spiral a, jia denotes the tightness of a in the sense of W. Seidel. A sequence of points (zn} of the unit disc is a sequence of p-points for f(z) if (zn} are the centers of a sequence of non-Euclidean cercles de remplissage. Theorem. If iia = 0, then each sequence of points (zn}, lzn I- 1, is a sequence of p-points. An analogous theorem holds in the plane. Bagemihl and Seidel have shown the existence of functions satisfying these hypotheses. Thus we have the existence of extremely wild holomorphic functions (in the disc or the plane). Our theorem generalizes a theorem of V. I. Gavrilov. (Received April 24, 1968.)

68T-532. H. W.DAVIS, University of New Mexico, Albuquerque, New Mexico 87106 and F. j. MURRJ,\.Y and J. K. WEBER, JR., Duke University, Duke Station, Durham, North Carolina. Inductive

and projective limits of Lp~·

Since the spaces Lp = Lp I!J,l], 1 i!§ p < oo, regarded simply as linear vector spaces, have the

660 property Ls J Lt if s < t, it is natural to consider the spaces Lp- = n {Lt: l ;§ t < p} and Lp+ = U { Lt: p < t }, which are distinct from Lp. The spaces LP± are then topologized in a natural way as projective and inductive limits of the spaces Lt• where Lt may be given either its strong or weak topology. L p- and Lq+ are mutually dual (l < p ;§ oo, 1/p + 1/q = l and q = l if p = oo) and the pro­ jective and inductive limit topologies on LP± can be identified as the appropriate strong and weak topologies. With their strong topologies, the spaces LP± are separable, barrelled and reflexive; Lp- is complete but not normable; Lp+ is complete but not metrizable. With their weak topologies, the spaces Lp±. are separable, but neither complete nor metrizable. A set B C Lp+ is strongly bounded if and only if B is a strongly Lounded subset of some Lt, where p < t < oo. A similar result holds for Lp-. (Received April 2·5, 1968 .)

68T-533. J. M. WORRELL, JR., Sandia Corporation, Albuquerque, New Mexico 87115. Paracompactness as a relaxation of full normalcy.

Paracompactness as taken here does not require Axiom T2 • A spaceS is said to have Property Q if and only if for any collection H of open sets covering S there exists a collection K, of open sets covering S such that if P is a point, there exist a finite subcollection F of H and an open set D containing P such that every element of K intersecting D is a subset of some member of F. The fully normal spaces are those spaces S satisfying the strengthened form of Property Q in which it is required that F = l. Theorem. A topological space S is paracompact if and only if it has Property Q. Corollary. If fz' is a closed continuous mapping of a topological space S having Prop­ erty- Q and either (l) ¢is peripherally bicompact or (2) S is essentially T 1 and¢ is first countable or (3) ¢is open, then ¢(S) has Property Q. (Received April 24, 1968.)

68T-534. S. N. PATNAIK, University of Montr-eal, Montreal 3, Quebec, Canada. The Lefschetz number associated with set-valued maps.

Let F: X - Y be a upper semicontinuous set-valued map from a compact polyhedron X into another such space Y. Let C (Y) denote the space consisting of a class of nonempty closed subsets of Y with a suitable topology. Let the continuous single-valued map f: X- C(Y) be defined by f(x) = F(x) for all x EX. Then f induces the homology homomorphisms f*i: Hi(X,L)- Hi(C(Y),L) (L a field of coefficients). With respect to suitable chosen basis, f,.i has a rectangular matrix repre­ sentation iMkl' (If Hi (C(Y),L) is not finite dimensional, this could still be achieved by choosing proper minimal finite-dimensional subspaces.) Let f\. = mini(k,l) =rank iMkl' Considering the square matrices p Xi p formed from the rectangular matrices i Mkl' we define the Lefschetz numbers of the set-valued map F: X - X by L(F) = Li (- l)i trace p xi p. Theorem. If for a choice of the matrices p Xi p formed from the matrices i Mkl• L(F) f- 0, there exists a fixed point under F, i.e. ayE F(y), y EX. Fixed-point theorems of Eilenberg-Montgomery and B. O'Neill follow from this.

Corollary. If the space X is homologically acyclic and if the induced homomorphism f•o (as defined in above) is nontrivial, then there exists a fixed point under the set-valued map F: X - X. (Received April 24, 1968 .)

661 ANALYSIS I ELEMENTS OF THE THEORY OF

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668 0 Topological Dynamics Algebraic Geometry: An International Symposium Introduction to Schemes Edited by I. G. Macdonald, Oxford University Joseph Auslander, University ot Maryland About 128 Pages Walter H. Gottschalk, Wesleyan University Approx: $3.95 Paper/$8.50 Cloth About 384 Pages Approx: $11.50 Cloth Based on a series of lectures delivered at proceedings of an This volume contains the the University of Sussex in 1964-65, this international symposium m topological dy­ namics held at Colorado State University, book presents a brief introduction to the Fort Collins, in August 1967, and sponsored language of schemes. Of primary interest to by the Air Force Office of Scientific Re­ postgraduates in classical geometry, it is search. The papers presented deal with cur­ useful as well to pure mathematicians. An rent research in topological dynamics, re­ elementary knowledge of algebra and topol­ lated fields such as ergodic theory, and the ogy is assumed. qualitative theory of ordinary differential equations. This volume is an excellent refer­ ence for research workers, as well as gradu­ ate students. Ergodic Problems of Induced Representations Classical Mechanics V. I. Arnold, University of Moscow of Groups and A. Avez, University of Paris About 288 Pages Approx: $15.00 Cloth Quantum Mechanics Translated from the French, this mono­ George W. Mackey, Harvard University graph is designed to supplement a first- or About 136 Pages second-year graduate course in classical Approx: $3.95 Paper/$8.50 Cloth mechanics. It constitutes a report of re­ This text contains a set of lectures that cent applications of ergodic theory to dif­ were given at the Scuola Normale, Pisa in ferential geometry and classical mechanics, April 1967. Addressed to mathematicians and is written for physicists, astronomers, and physicists, these lectures deal with the mathematicians, graduate students, and nature of the theory of induced representa­ tions and its application to quantum research workers in these fields. mechanics. Science: Battelle Rencontres Men, Methods, Goals Edited by Edited by Baruch A. Brody Cecile DeWitt, University of North Carolina Massachusetts Institute of Technology John A. Wheeler, Princeton University Nicholas Capaldi 640 Pages $14.75 Cloth of New York The City University These lectures are based on a meeting held 288 Pages in Seattle in the summer of 1967, spon­ Approx: $2.95 Paper/$8.00 Cloth sored by the Battelle Memorial Institute. By making readily available some of the The purpose of this meeting was to enlarge most important philosophical statements the lines of communication between mathe­ help about science, this new anthology will maticians and physicists. The papers pre­ science student reflect the scientist and variety of topics, upon the nature, methods, and goals of the sented deal with a wide scientific enterprise. Among the readings ranging from Lie groups to the structure of included in the volume are selections from space-time, from homotopy theory to the the writings of Copernicus, Herschel, Kant, quantization of the gravitational field, and Mach, Maxwell, Mill, Newton, and Poincare. from differentia I geometry to S-matrix theory. W.A. BENJAMIN, INC. ONE PARK AVE. • NEW YORK, N.Y. 10016 AMERICAN MATHEMATICAL SOCIETY SECOND-CLASS POSTAGE PAID AT P.O. Box 6248 PROVIDENCE, RHODE ISLAND Providence, Rhode Island 02904 AND ADDITIONAL MAILING OFFICES

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JUST PUBLISHED GROUP THEORY AND ITS APPLICATIONS edited by ERNEST M. LOEBL, PolytechniC lnsCitute, Brooklyn, New York CONTRIBUTORS: R. E .. Behrends, L. c. Bieden· physics, including quantum mechanics, harn, A. J. Coleman, Stlg Flodmar_k, J. Holman 111 · solid state atomic spectroscopy nuclear J. M. Jauch, B. R. Judd, D1rk Kle1ma, P. Kramer, • . • F. A. Matsen, M. Moshinsky, L. O'Raifeartaigh, theory, elementary part1cle theory and the T. o. Phillips, o. R. Plummer, Eugene P. Wigner. theory of relativity. Development and de- Applies the principles of symmetry through scription of recent advances in group thee- the use of group theory to many areas of retical techniques are also included. 1968, about 690 pp., $19.50

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