CALENDAR

This Calendar lists all of the meetings which have been approved by the Council up to the date this issue of the c}I/;Jiiai) 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 meetings to which no numbers have yet been assigned. Abstracts should be submitted on special forms which are available in most departments of mathematics; forms can also be obtained by writing to the headquarters of the Society. Abstracts to be presented at the meeting in person must be received at the headquarters of the Society in Providence, Rhode Island, on or before the deadline for the meeting. Meeting Deadline for Abstracts* Number Date Place and News Items 737 August 24-28, 1976 Toronto, Canada June 15, 1976 (80th Summer Meeting) 738 October 30, 1976 Storrs, Connecticut September 7, 1976 739 November 6, 1976 Ann Arbor, Michigan September 7, 1976 740 November 19-20, 1976 Columbia, South Carolina September 28, 1976 741 November 19-20, 1976 All:uquerque, New Mexico September 28, 1976 742 January 27-31, 1977 St. Louis, Missouri November 3, 1976 (83rd Annual Meeting) March 31-Aprll 1, 1977 Huntsville, Alabama April15-16, 1977 Evanston, Wlnois April 22-23, 1977 Hayward, California August 14-18, 1977 Seattle, Washington (Slat Summer Meeting) November 11-12, 1977 Memphis, Tennessee Ja1Bl8ry 18-22, 1978 Atlanta, Georgia (84th Annual Meeting) January 11-15, 1979 Milwaukee, Wisconsin (85th Annual Meetlng January 8-12, 1981 Ban Francisco, Califomla (87th Annual Meeting)

*Deadline for abstracts not presented at a meeting (by title) August 1976 issue: June 8 October 1976 issue: August 31 November 1976 issue: September 21 OTHER EVENTS August 2-21, 1976 Summer Research Institute on Algebraic and Geometric Topology, Stanford University, Stanford, Califomla August 22-23, 1976 Short Course on Mathematical Economics, Toronto, Canada February 1977 Symposbun on Some Mathematical Questions ln Biology, Denver, Colorado

Please aflix the peel·afl label on th- cJ{oticti) to correspondence with the Society concerning fiscal maHers, changes of address, promations. or when placing orders for boob and jaumals. The cJ{oticti) of the American Mathematical Society it published by the American Mathematical Society, P. 0. Box 62<411, Pro¥idence, Rhode Island 029-40, in January, February, April, June, August, October, November, and December. Subscription prices lor the 1976 volume (Vplume 23) are list $19.00, member $9.50. The subscription price lor members is included in the annual dues. Back iuues of the cNotictiJ are available lor a twa year period only and cost $2.88 per issue list price, $2.16 per issue member price for Volume 21 (1974) and $3.65 per issue list price, $2.74 per issue member price lor Volume 22 (1975). Orders lor 111bscriptions or bad< issues must be accompanied by payment and should be sent to the Saciely at P. 0. Box 1571, An- Stotion, Providence, Rhode Island 02901. Other correspondence should be addreued to P. 0. Box 62<411, Providence, Rhode Island 029-40. Second cla11 postage paid at Providence, Rhodo Island, and addi~onal maHing aflicos.

Copyright@ 1976 by the American Mathematical Society Printed In the United States of America cNoticeiJ

OF THE AMERICAN MATHEMATICAL SOCIETY

Ed Dubinsky, Joseph B. Keller, ADVISORY COMMITTEE Robion C. Kirby, Yiannis N. Moschovakis, ON EDITORIAL POLICY Barbara L. Osofsky, Everett Pitcher (Chairman), Scott W. Williams

MANAGING EDITOR Gordon L. Walker ASSOCIATE EDITOR Hans Samelson

CONTENTS June, 1976

MEETINGS Calendar of Meetings . • . . • ...... • . Inside Front Cover Program for the June Meeting in Portland, Oregon ...... • . • . . . . 184 Abstracts for the Meeting: A-458-A-464 PRELIMINARY ANNOUNCEMENTS OF MEETINGS ...... • . • . . . 187 ORGANIZERS AND TOPICS OF SPECIAL SESSIONS...... • . • • ...... • • 199 INVITED SPEAKERS AT AMS MEETINGS. . . . . • ...... • . . • . . . . • • . • . 200 PERSONAL ITEMS . • • ...... • ...... • ...... • . • • 201 NOMINATIONS FOR VICE-PRESIDENT OR MEMBER-AT-LARGE...... 202 THE NOMINATING COMMITTEE FOR 1977 . . . • . . . . . • ...... • . • . • . . 204 THE CHANGING ROLE OF THE MASTER'S DEGREE . . . • . . . . . • . • . . • . . . 206 NEW AMS PUBLICATIONS . . . • . • ...... • . . . • . . . • . . . • • • • . • . 210 LETTERS TO THE EDITOR .•.••...•..... , . • • . . . . • ...... 214 NEWS ITEMS AND ANNOUNCEMENTS . . . . • . . . • • . . • • . • • ...... • • . 216 SPECIAL MEETINGS INFORMATION CENTER • . • . • . . . • . . . • . . • • • . . • . 219 QUERIES ...... •• , • ...... • . • • . . • • . • . • . . . • . • . . • • . . • . • . 223 ABSTRACTS ...... • • . . • . . . • . . • • . . . • ...... • . • . . . A-419 SITUATIONS WANTED .••..•...... •...... •.•.....•.•.... A-464 CLASSIFIED ADVERTISEMENTS . . . • . . • ...... • . . . • . • . . . . . • . . • . A-464 PREREGISTRATION FORM (Toronto Meeting) . • • . • . . . • • . . . • • . • . . . • . A-471 RESIDENCE HALL RESERVATION REQUEST FORM (Toronto Meeting) • • • . . A-472 736TH Portland State University Portland, Oregon MEETING june 18, 1976

The seven hundred thirty-sixth meeting of PORTLAND MOTOR HOTEL (503) 221-1611 the American Mathematical Society will be held at 1414 S. W. 6th Avenue (97205) Portland, Oregon, on Friday, June 18, 1976. The Rates $17. 50 up Mathematical Association of American and the So­ ciety for Industrial and Applied Mathematics will RAMADA INN-PORTLAND CENTER hold Northwest Sectional Meetings in conjunction (503) 221-0450 310 S. W. Lincoln with this meeting of the Society. All of their ses­ (97201) Single sions will be held on Saturday, June 19. Jn addi­ $19.50 up tion, Pi Mu Epsilon will sponsor a session of Double 24.50 up fifteen-minute papers on Friday afternoon. The following two hotels are in downtown The Mathematical Association of America Portland about one-half mile from the Portland will sponsor two hour talks. Richard M. Koch of State campus. A free Tri-Met bus runs from the University of Oregon will lecture at 9:00 a.m. downtown to the north end of the campus: on Saturday, June 19. The title of his lecture is "Invariant functions on matrices." Raymond A. HEATHMAN/PARK HAVILAND HOTELS (503) 228-5262 Mayer, Jr. of Reed College will lecture at 11:00 712 W. Salmon a.m. on "Some applications of nonstandard analy­ s. (97205) Single sis to number theory. " $10. 00- $19. 00 Double The Featured SIAM Lecturer will be Philip 12.00- 22. 00 M. Anselone of Oregon State University. His fifty­ CONGRESS HOTEL (503) 228-0181 minute address is entitled "Nonlinear operator S. W. 6th and Main street (97204) approximation theory and applications to integral Single $11. 00- $16. 00 equations," and will be given at 10:00 a.m. on Double 16.00- 20.00 Saturday, June 19. SIAM will also sponsor con­ tributed talks until3:30 p.m. on Saturday and The following is located at the Lloyd Center, three-and-one-half will schedule a no-host lunch at 12:30 p.m. miles from campus: By invitation of the Committee to Select Hour SHERATON MOTOR INN (503) 288-6111 Speakers for Far Western Section Meetings, there Lloyd Center (97232) will be two invited addresses. R. James Milgram Single $17.00- $23.00 of Stanford University will lecture at 11:00 a.m. Double 23.00- 29.00 on Friday on "Generalized cohomology theories Meals can and their applications." Thomas M. Liggett of be obtained at a variety of estab­ lishments near the University of California, Los Angeles, will the campus. More information will be available lecture at 2:00p.m. in the registration area. The on Friday. The title of his campus food talk is "The stochastic service will not be in operation dur­ evolution of infinite sys­ ing the meetings. tems of interacting particles." Both hour ad­ Portland International Airport dresses will be given in Room 71 of Cramer Hall. is served by There will be sessions for contributed pa­ several major airlines. Both limousine service pers scheduled Friday afternoon. Late papers will and taxis are available between the airport and downtown. Neuberger be accepted for presentation at the meeting, but Hall is located on Broadway will not appear in the printed program of the thirteen blocks south of Morrison Street. Grey­ meeting. hound and Trailways Bus Lines serve Portland. The registration desk will be located in Persons driving to the meetings may park Room 334 of Neuberger Hall, and will be open free in the parking structure across the street during the following periods: 10:00 a.m. to noon from Neuberger Hall. The freeway system in and 1:00 p.m. to 4:00p.m. on Friday; 8:30a.m. Portland can be very frustrating for people un­ to noon on Saturday. The registration fee will be familiar with it. Here are instructions on how to $2. get to the parking structure mentioned above. From the south There are numerous hotels and motels in on Interstate 5: At milepost 299 get in Portland. Reservations should be made directly the right lane. Take the exit with the sign 11 City Center-Beaverton. II with the hotel or motel; early reservations are ad­ On the ramp look for vised. The following are located near the Portland the sign "Portland State 2nd Right;" this ramp State campus: joins highway 26. You must move over two lanes to the right. Skip the 4th Avenue exit and take JAMAICA MOTOR HOTEL (503) 226-4751 the 6th Avenue exit. The entrance to the parking 415 S. W. Montgomery (97201) structure is two-and-one-half blocks north, on Single $12. 50 up the left side of the street. From the north on Double 13.50 up Interstate 5: Stay on the freeway all the way to Twin 14. 50 up the Marquam Bridge and keep in the center lane.

184 The left lane disappears as you approach the this lane, marked by "Salem-Beaverton" over­ Marquam Bridge. The two lanes on the ramp head signs, This will join Interstate 5. Get in change to four lanes on the bridge. Take the the middle lane. The left lane disappears as second lane from the left, marked on the signs you approach the Marquam Bridge, The two lanes above by highway 26. Skip the 4th Avenue exit and on the ramp change to four lanes on the bridge. turn right on the 6th Avenue exit. The entrance Take the second lane from the left, marked on to the parking structure is two-and-one-half the signs above by highway 26. Skip the 4th Ave­ blocks north, on the left side of the street. From nue exit and turn right on the 6th Avenue exit. the east on Interstate SON: After you pass th_e __ The entrance to the parking structure is two­ 21st Avenue viaduct, get in the left lane. (The and-one-half blocks north on the left side of the left lane is restricted up to that point. ) Stay in street.

PROGRAM OF SESSIONS The time limit for each contributed paper in the general sessions is ten minutes. To maintain the schedule, the time limits will be strictly enforced, FRIDAY, 11:00 A. M. Invited Address, Room 71, Cramer Hall (1) Generalized cohomology theories and their applications. Professor R. JAMES MIL­ GRAM, Stanford University (736-G3) FRIDAY, 2:00 P. M. Invited Address, Room 71, Cramer Hall (2) The stochastic evolution of infinite systems of interacting particles. Professor THOMAS M. LIGGETT, University of California, Los Angeles (736-F2) FRIDAY, 3:15 P. M. Session on Geometry and Topology, Room 341, Neuberger Hall 3:15-3:25 (3) An irreducible Hahn-Mazurkiewicz theorem. Professor L. E. WARD, Jr., University of Oregon (736-G2) 3:30-3:40 (4) A topological approach to the four-color problem. Mr. ALLEN D. ALLEN, Algorithms, Incorporated, Northridge, California (736-Gl) 3:45-3:55 (5) Semi-regular maps. Preliminary report. JOSEPH ZAKS, University of Haifa, Israel (736-D4) 4:00-4:10 (6) Metric circles and bisectors on surfaces without conjugate points. Preliminary report. Dr. PAUL E. EHRLICH, Bonn University, West Germany, and Dr. HANS-CHRISTOPH IMHOF*, University of California, Berkeley (736-Dl) 4:15-4:25 (7) Perfect colorings of transitive tilings. Professor BRANKO GRUNBAUM*, University of Washington, and Professor G. C. SHEPHARD, University of British Columbia (736-D2) 4:30-4:40 (8) The eighty-one types of transitive tilings of the plane. Professor BRANKO GRUNBAUM, University of Washington, and Professor G. C. SHEPHARD*, University of British Columbia (736-D3) FRIDAY, 3:15 P. M. Session on Analysis, Room 385, Neuberger Hall 3:15-3:25 (9) Integral equations with semidegenerate kernels. Preliminary report. Professor MICHAEL A. GOLBERG, University of Nevada, Las Vegas (736-B4) 3:30-3:40 (10) The sum of a multiple hypergeometric series. Preliminary report. Professor H. M. SRIVASTAVA, University of Glasgow, Scotland and University of Victoria (736-B5) 3:45-3:55 (11) On the nonuniqueness of the "optimal order" of approximation of polynomial operators. Preliminary report. Professor A. S. B. HOLLAND*, Dr. BADRI SAHNEY, University of Calgary, and Dr. P. D. KA THAL, Government College, Manilla, India (736-B3) 4:00-4:10 (12) On the stability of minimum points. Preliminary report. Professor D. H. HYERS, University of Southern California (736-B7) 4:15-4:25 (13) An example pertaining to local vs global essential cluster sets. Dr. J. H. MATHEWS*, California State University, Fullerton, and Dr. C. L. BELNA, Western lllinois Uni­ versity (746-B2) 4:30-4:40 (14) Comparison of order topologies with the topology of an ordered topological vector space. Preliminary report. Professor ROGER W. MAY*, Walla Walla College, and Professor CHARLES W. McARTHUR, Florida State University (736-Bl) (15) WITHDRAWN.

*For papers with more than one author, an asterisk follows the name of the author who plans to present the paper at the meeting. 185 FRIDAY, 3:15 P. M. General Session, Room 387, Neuberger Hall 3:15-3:25 (16) Covering theorems for finasigs. VIII. Almost all conjugacy classes in An have exponent:;:;; 4. J. L. BRENNER, Palo Alto, California (736-A1) 3:30-3:40 (17) Laws of Cp Wr Cp2. Dr. MARSHALL L. CATES, California State University, Los Angeles (7~6-A2) 3:45-3:55 (18) Rings with a nil ideal yielding polynomial constraints. Professors D. L. OUTCALT and ADIL YAQUB*, University of California, Santa Barbara (736-A3) 4:00-4:10 (19) Is every submartingale a convex function of a martingale? Professor DAVID GILA T, University of Minnesota (736-F1) (Introduced by Professor Bert E. Fristedt) 4:15-4:25 (20) An intriguing numerical relationship between employment and the consumer price index. Preliminary report. Professor NATHANIEL MACON*, American University, and Mr. PAUL BRYANT, Federal Preparedness Agency, Washington, D. C. (736-C2) 4:30-4:40 (21) A:g. iteration scheme for determining equations of dynamical systems. Dr. DAVID A. SANCHEZ* and Dr. DANIEL SWEET, University of California, Los Angeles (736-C1) Kenneth A. Ross Eugene, Oregon Associate Secretary

186 PRELIMINARY ANNOUNCEMENTS OF MEETINGS 80TH University of Toronto SUMMER Toronto, Canada MEETING August 24-27, 7976

Please read section on RESIDENCE HALL HOUSING on page 190 very carefully, since this information has been considerably re­ vised from that contained in the April issue of these c}/oticti). The eightieth summer meeting of the Ameri- (tentative); W. T. TUTTE, University of Water- can Mathematical Society will be held at the Uni- loo, Chromatic polynomials and related topics; versity of Toronto, Toronto, Ontario, Canada, and G. T. YANG, University of Pennsylvania, from Tuesday, August 24, through Friday, August Transformation groups. 27, 1976. All sessions of the meeting will take There will be sessions for contributed ten- place on the campus of the university. minute papers on Tuesday afternoon, Wednesday A set of Colloquium Lectures, consisting of morning, early Wednesday afternoon, Thursday four one-hour talks, will be presented by JUrgen afternoon, and Friday afternoon. Abstracts of K. Moser of the Courant Institute of Mathematical contributed papers should be sent to the American Sciences, New York University. The title of the Mathematical Society, P. 0. Box 6248, Provi­ series is "Recent progress in dynamical systems." dence, Rhode Island 02940; the deadline for re­ The first lecture in the series entitled "A survey," ceipt of abstracts is June 15, 1976. No provision will be given at 10:00 a.m. on Tuesday, August 24. will be made for late papers. The second lecture entitled "Closed orbits," will If a sufficient number of requests is re- be given at 9:00 a.m. on Wednesday, The third cewed, a poster session will be organized. For lecture entitled "Integrable systems," and the a description of poster sessions, please see the fourth entitled "Unstable orbits in particular for news item on page 217 of this issue of these then-body problem," will be given at 1:30 p.m. on c}/oticti). Individuals who wished to have their Thursday and Friday. In accordance with a recent papers considered for a poster session should Council decision, no other sessions are planned to have submitted their abstracts to the Society by conflict with the Colloquium Lectures. June 1, 1976. By invitation of the Society's Program Com- The AMS Committee on Employment and mittee, there will be six invited one-hour ad- Educational Policy (CEEP) will sponsor an open dresses. The names of the speakers, the titles of meeting on the job market on Tuesday, August 24, their addresses, and the times of presentation are at 8:30p.m. The meeting will consist of a brief as follows: Eugenio Calabi, University of Pennsyl- report on the state of the job market, followed by vania, "Nearly flat triangulations of Riemannian an open discussion with comments and suggestions manifolds," 1:30 p.m. Tuesday; Per Enflo, Uni- welcomed from the audience. Speakers will in- varsity of Stockholm, title not yet available, 2:45 elude Charles W. Curtis, Wendell H. Fleming p.m. Tuesday; Lawrence A. Shepp, Bell Telephone (moderator), John W. Jewett, and John A. Nohel. Laboratories, "Optimal reconstruction of a func- Professor Jewett will report on the current sur- tion from its projections," 4:00 p.m. Tuesday; vey of the mathematical profession by the Con- Michael Aschbacher, California Institute of Tech- ference Board of the Mathematical Sciences. nology, "Determining the finite simple groups," This meeting of the Society will be held in 2:45 p.m. Thursday; Edward Nelson, Princeton conjunction with the annual summer meetings of University, "Principles and practice of nonstan- the Mathematical Association of America (August dard analysis," 4:00p.m. Thursday; and Marian 26-28), and Pi Mu Epsilon, Participants should B. Pour-El, University of Minnesota, Minneapo- note that the Society and Association are meeting lis, "New directions in computability theory" in order opposite from the usual schedule for a (title tentative), 5:15p.m. Thursday. summer meeting. The twenty-fourth series of The following special sessions are being or- Earle Raymond Hedrick Lectures, sponsored by ganized: HENRY W. GOULD, West Virginia Uni- the Association, will be given by Martin D. Davis versity, Combinatorial identities; JOHN W. GRAY, of the Courant Institute of Mathematical Sciences, University of illinois at Urbana-Champaign, Cate- New York University. The title of his lecture gory theory; EDWARD J. MAYLAND, Jr., York series is "Some mathematical applications of University, Knots and three-manifolds; Z, H. NI- logic." At the Business Meeting of the Associa- TECKI, Tufts University, Differentiable dynami- tion at 10:00 a.m. on Friday, August 27, the cal systems (tentative); ANATOL RAPOPORT, Lester R. Ford Awards will be presented, there University of Toronto, Mathematical psycholor.v; will be a tribute to a distinguished Association RAY SHIFLETT, California State University, Ful- member, and gift to the Association will an- lerton, Doubly stochastic measures and their nounced. operators; JOSEPH R. SHOENFIELD, Duke Uni- There will be a dinner at 6:30 p, m. on Fri- versity, Recursively enumerable sets and degrees day, August 27, for those who have been members

187 Short Course on Mathematical Economics August 22-23, 1976 The American Mathematical Society will theory of general economic equilibrium"; Wer­ present a one-and-one-half day short course on ner Hildenbrand, Department of Economics, Mathematical Economics on Sunday and Monday, University of Bonn, will speak on "Measure August 22 and 23, in Room 2135 of Sidney Smith spaces of economic agents"; David Gale, Depart­ Hall on the campus of the University of Toronto. ments of Economics and Mathematics, Univer­ The course will present an introduction to sev­ sity of California, Berkeley, and Center for Ad­ eral problems (existence and computation of vanced Study in the Behavioral Sciences, Stan­ equilibria, structure and dependence on param­ ford, California, will speak on "The role of pri­ eters of the set of equilibria, game models and ces and interest rates in dynamic economics"; the theory of the core, dynamic economics) il­ Andreu Mas-Colell, Departments of Mathemat­ lustrating the application to economics of var­ ics and Economics, University of California, ious fields of mathematics (in particular, alge­ Berkeley, will speak on "The theory of econom­ braic topology, measure theory, and differential ic equilibrium from the differentiable point of topology). It is intended to present both mathe­ view"; Robert J. Aumann, Departments of Eco­ matically challenging aspects and their applica­ nomics, Hebrew University and Stanford Univer­ tions in current economic theory. sity, will speak on "Some game models in eco­ The program is under the direction of nomics"; and Stephen Smale, Department of Gerard Debreu, Departments of Economics and Mathematics, University of California, Berke­ Mathematics, University of California, Berke­ ley, will speak on "Computation and existence ley, in cooperation with Hugo F. Sonnenschein, of Walras equilibria. " Department of Economics, Northwestern Univer­ Summaries of these talks and accompanying sity, as codirectors. This short course was reading lists appeared on pages A-405 through recommended by the Society's Committee on A-407 of the April issue of these cJ{oticei). Employment and Educational Policy (CEEP), This short course is open to all who wish whose members are David Blackwell, Charles to participate upon payment of the registration W. Curtis, Wendell H. Fleming (chairman), fee. This fee has been reduced for students and Martha K. Smith, and Daniel H. Wagner. unemployed individuals, with a modest increase The program will consist of six seventy­ for other registrants. Please refer to the sec­ five minute lectures, as follows: Hugo F. Son­ tion entitled MEETING REGISTRATION AND nenschein will speak on "Price formation: The PREREGISTRATION for details. of the Association for thirty years or more. The 7:00p.m., and a panel discussion on "Other dinner will be followed by a short program with mathematical models in economics" at 8:00p.m. Carroll V. Newsom as toastmaster, and H. L. on Thursday, August 26. Alder, Burton W. Jones, Henry 0. Pollak, and Rooms 2050 and 2054 in Sidney Smith Hall G. Bailey Price as speakers. The dinner has been (@on the map on page 192) have been set aside planned as a pleasant occasion which recalls the as informal discussion rooms, and will be open past services of the Association's senior mem­ daily from 8:00 a.m. to 10:00 p.m. for small bers and their spouses, informs them about cur­ g-roups desiring a quiet room with blackboard rent activities and future plans, and will provide space to discuss mathematics. Room 2050 is an opportunity for renewing friendships. Individ­ available on a first-come, first-served basis; uals who have been MAA members for thirty years room 2054 is available for one-hour periods only, or more and would like to attend this dinner will and must be reserved in advance. A reservation find tickets on sale at the Joint Meetings registra­ form will be posted on the door to room 2054 for tion desk until 5:00p.m. on Thursday. Tickets individuals to sign up for use of this room. It is for the dinner are $7. 50 each, and spouses are in­ requested that discussion groups not be planned vited. The ticket price includes sales tax and ser­ which conflict with the business meetings or vice charge. major lectures. Sutherland Frame Lecture will be de­ The J. COUNCIL AND BUSINESS MEETING livered to Pi Mu Epsilon on Wednesday, August 25, at 7:00p.m. The name of the speaker and The Council of the Society will meet at title of the lecture are not yet available. 2:00p.m. on Monday, August 23, in the Council The Association for Women in Mathematics Chamber in the Galbraith Building (@ on the will hold a panel discussion on the "History of map on page 192). The Business Meeting of the women in mathematics" on Thursday, August 26, Society will be held in Convocation Hall (@ on at noon. Lenore Blum will serve as moderator. the map) at 4:00p.m. on Wednesday, August Speakers will include Lida K. Barrett, Mary W. 25. The secretary notes the following resolution Gray, Linda Keen, Emiliana Noether, and Martha of the Council: Each person who attends a Busi­ K. Smith. The panel discussion will be immediate­ ness Meeting of the Society shall be willing and ly followed by AWM•s Business Meeting. There able to identify himself as a member of the So­ will be an open meeting of the AWM Executive ciety. In further explanation, it is noted that Committee at 5:30p.m. on Wednesday, August 25. "each person who is to vote at a meeting is there­ The Council of the Conference Board of the by identifying himself as and claiming to be a Mathematical Sciences will meet on Friday, Au­ member of the American Mathematical Society. " gust 27, at 2:30 p.m. The Council recommends to the Business The Mathematicians Action Group will hold Meeting changes in the bylaws establishing a. cate­ its Business Meeting on Tuesday, August 24, at gory of "foreign member." Certain individuals,

188 the class to be defined precisely by the Coun- DENCE HALL HOUSING and HOTELS. cil, may elect to be foreign members, with all Meeting registration and preregistration fees privileges accorded to ordinary members except partially cover expenses of holding the meetings. the right to vote. The dues, as set by the Council The preregistration fee does not represent an ad­ with the approval of the Trustees, would not ex­ vance deposit for lodgings. - ceed two-thirds of the dues of an ordinary member. Please note that separate registration fees The Council recommends to the Business are required for the short course and for the Joint Meeting that Article IV, Section 8, of the bylaws Meetings. These fees are as follows: be deleted. Mathematical Economics Short Course These two paragraphs constitute the "notice of proposed action and of its general nature" as Preregistration At Meeting required by Article XIII of the bylaws. (by mail prior to 8/6/76) MEETING PREREGISTRATION AND REGISTRATION Members/Nonmembers $18 $20 Student or unemployed 3 5 Registration for the short course~ will One day fee for second begin on Saturday, August 21. Lecture notes and day 10 other short course material will be distributed Joint Mathematics Meetings before the first session at the short course registration desk. Those individuals who do not Member $12 $15 preregister for the short course are strongly Student or unemployed 1 2 urged to register and pick up their material on Nonmember 20 24 Saturday evening so as not to miss the start of There will be no extra charge for members the lecture on Sunday morning. General meeting of the families of registered participants except registration will commence on Monday, August that all professional mathematicians who wish to 23, at 2:00p.m. Participants who are not at­ attend sessions must register independently. tending the short course are advised that no The unemployed status refers to any parti­ general meeting information or registration ma­ cipant currently unemployed and actively seeking terial will be available prior to the time listed employment. It is not intended to include partici­ below for the Joint Mathematics Meetings reg­ pants who have voluntarily resigned or retired istration. Upon arrival at the University of from their latest position. Students are considered Toronto campus, participants should proceed to be only those currently working toward a de­ directly to the dormitory to which they have been gree who do not receive an annual compensation assigned in order to check in to their accommo­ totaling more than $7, 000 from employment, fel­ dations before registering for the meetings. lowships, and scholarships. Mathematical Economics Short Course Registration Checks for the preregistration fee(s) should be mailed to arrive in Providence not later than Date and Time Location August 6, 1976. Participants should make their Saturday, August 21 Hallway outside of room own reservations directly with hotels in the 4:30 p.m.-7:30p.m. 2135, Sidney Smith area (cf. section titled HOTELS). It is essential, Hall (Saturday through however, to complete the Meeting Preregistra­ Sunday, August 22 Monday) tion Form on the last page of these c){oticei) to 8:00 a.m.-5:00p.m. take advantage of the lower preregistration fee(s) Monday, August 23 and to obtain dormitory accommodations. 8:00a.m. -noon A fifty percent refund of preregistration fees Joint Mathematics Meetings Registration will be made for all cancellations received in Providence prior to August 20. There will be no Date and Time Location refunds granted for cancellations received after Monday, August 23 Entrance lobby, Sidney that date or to persons who do not attend the meet­ 2:00 p.m.-8:00p.m. Smith Hall (Monday ings. through Saturday) MATHEMATICAL SCIENCES Tuesday, August 24 EMPLOYMENT REGISTER 8:00 a.m. -5:00p.m. At last summer's meeting at Western Michi­ Wednesday, August 25 to Friday, August 27 gan University, an experimental variant of the 8:30 a.m.-4:30p.m. Employment Register was operated (successfully) on Saturday, August 28 a limited basis. No interviews were scheduled by the staff. Instead facilities were provided for 8:30 a.m.-1:30 p.m. applicants and employers to display resumes and Participants who wish to preregister for the job listings. Message boxes were set up for indi­ meetings should complete the Meeting Preregis­ viduals to leave messages for one another re­ tration Form on the last page of these c){oticei). questing interviews. Tables and chairs were pro­ Those who preregister will pay lower registration vided in the room for interviews. fees than those who register at the meeting, as It is planned to repeat this form of the Em­ indicated in the schedule below. Preregistrants ployment Register at the University of Toronto. will be able to pick up their badges and programs Employers are encouraged to attend the meetings when they arrive at the meeting after 2:00p.m. and participate, if possible. Applicants should on Monday, August 23, at the Joint Mathematics recognize that the MSER cannot guarantee that any Meetings registration desk. Complete instructions employers will, in fact, attend the meeting or be on procedures for making hotel or dormitory res­ able to participate in the Employment Register. ervations are given in the sections entitled RESI- The AMS-MAA-SIAM Committee on Employment 189 Opportunities will, however, request employers Student, Unemployed, and Children Age Ten or Over listing in the July and August 1976 issues of Em­ Single or ployment Information for Mathematicians to sig­ Double nify in their listing their intention to participate $9/Dight per person* $8/night per person** in the Employment Register at the summer meet­ ing. All Other Participants not Accompanied by Any Children under Ten Years of Age EXHIBITS Double The book and educational media exhibits will ~ $11/night* $9/night per person* be displayed in Room 2096 of Sidney Smith Hall $10/night** $8/night per person** at the following times: August 24 (Tuesday), 1:00 p.m. to 5:00 p, m.; August 25 and 26 (Wednesday *with air-conditioning and Thursday), 8:30a.m. to 4:30p.m.; August 27 **without air-conditioning (Friday), 8:30 a.m. to noon. All participants are Participants accompanied by any children encouraged to visit the exhibits sometime during under ten ears of a e will be assigned rooms in the meeting. Victoria College ( K on the map). These rooms RESIDENCE HALL HOUSING are not air-conditioned. Each room contains two beds. Children ten years of age or over must oc­ Several residence hall facilities have been cupy a bed and will be charged the appropriate set aside for the use of participants in the Joint rate (single or double). For children between the Mathematics Meetings and the Mathematical Eco­ infant stage and nine years of age, cots without nomics Short Course. According to regulations sides are available at $5 per cot per set by the university's housing office, these fa­ night. Chil­ dren in this age bracket may also occupy a bed, cilities are divided into sections for male parti­ but will be charged the same rate cipants only, female participants only, couples, as an adult. Participants wishing to rent these cots should so families accompanied by children all of whom are indicate in the space provided on the ten years of age or over, and families with any Residence Hall Reservation Request Form. For small children under ten years of age. In order for your in­ fants, cribs with sides can be rented for dormitory assignments to be made correctly, you a flat charge of $12, regardless of the must be explicit when completing the preregistra­ number of nights required. Participants wishing to rent tion and reservation forms on the last two pages cribs with of these c}/oticei) • sides should write to Thomas H. Callahan, De­ partment of Mathematics, University The rates quoted below are in Canadian of Toronto, Toronto, Ontario, Canada M5S 1A1, funds, and are subject to a seven percent Provin­ giving spe­ cific details as to the number of cribs required, cial Sales Tax. These.are now firm rates. arrival and departure dates. Please mark Participants accompanied by children, all the out­ side of the envelope "CRIBS." No deposit of whom are ten years of age or over will be is re­ quired. At most, one cot or crib will be permitted assigned rooms in Wetmore Hall in New Colleg~ in a room (maximum room occupancy is three and Whitney Hall in University College <@and\:!) persons). Participants with small children will be respectively on the map on page 192). New Col­ requested to sign a waiver on property damage, lege is air-conditioned; University College is not. and are advised to bring plastic sheets. Light Only one person may occupy a single room, and kitchen facilities are available in the basement only two people may occupy a double room. Sleep­ at Victoria College so that families accompanied ing bags, cots, and cribs are not allowed. Fami­ by small children may prepare light lies with more than two members will be assigned meals, warm bottles, etc. Participants should bring to adjacent rooms . Children will be charged the their own cooking Student/Unemployed rate. Single rooms for male utensils. Rates at Victoria College are as follows: participants are in Wetmore Hall, New College; Sir Daniel Wilson Hall, and Whitney Hall, both in Rates at Victoria College University College; and in Devonshire House <® Cot Crib on the map). New College is air-conditioned; Uni­ night $12/flat charge versity College and Devonshire House are not. $57 Participants requesting air-conditioned rooms in ~__E3 Double this category should be aware of the fact that the $10Tnight* $8/night per person* number of these rooms available is extremely *All participants (regardless of whether limited. Air-conditioned rooms in this category student or unemployed) including children ten years of will be assigned on a first-come, first-served age or over basis, and your confirmation will tell you whether you have been successful in obtaining an air-con­ Please note that payment in full for your ditioned room. Double rooms for male oartici­ dormitory accommodations must be made at the pants are in Wetmore Hall, New College; and time of check-in.in Canadian funds. Participants Devonshire House. New College is air-conditioned; coming to the meeting from countries other than Devonshire House is not. Single and double rooms Canada are advised to exchange their currency for female participants are in Wilson Hall, New for Canadian money either before they leave for College, and are air-conditioned. Double rooms the meeting or at the airport immediately upon for couples are in Wetmore Hall, New College; arrival in Toronto, since clerks in the dormi­ and Whitney Hall, University College. New Col­ tories will not be authorized to accept U.S. funds lege is air-conditioned; University College is not. in payment, nor to allow occupancy of the room Rates for all of the facilities mentioned above for even the first night until a participant can have are: his or her money exchanged. There are several

1~)() banks nearby which will exchange U.S. for Cana­ HOLIDAY INN (Downtown)-@ dian funds. The Canadian Imperial Bank of·Com­ 89 Chestnut Street merce on the southwest side of Bloor and St. Single $33. 50 Double $40. 00 Twin $43. 00 George Streets is open from 10:00 a.m. to 4:30 Extra person (12 years) $6.00 p.m. Monday through Thursday, and from 10:00 Code: RT CL AC TV SP FP a.m. to 6:00p.m. on Friday. There is also a Telephone: 416-367-0707 branch of the Bank of Montreal on the northwest corner of moor and St. George. No banks are open KING EDWARD SHERATON-@ on Saturday or SUnday. 37 King Street East Clerks will be at the check-in desks in the Single $25.00 Double $28.00 Twin $30.00 dormitories from 8:00a.m. until midnight only. Extra person (17 years) $6.00 Participants arriving after midnight will not be Code: RT CL AC TV FP able to occupy their rooms until 8:00 a.m. the Telephone: 416-368-7474 next morning. Residence hall rooms may be oc­ 25% discount on rates for students and faculty of cupied from 8:00a.m. on Saturday, August 21, any university, upon proof of same. until10:00 a.m. on Saturday, August 28. Under no circumstances will participants be allowed to oc­ ROYAL YORK HOTEL-@ cupy rooms past 10:00 a.m. on Saturday, since Front and York Streets at Union station the university staff must begin at that time to Single $32.00 Double $41.00 prepare these rooms for students arriving the Extra person (14 years) $7.00 next week. Facilities for checking baggage will be Code: RT CL AC TV; Parking $3.50 (24 hours) available at the Joint Meetings Registration Desk. Telephone: 416-368-2511 Pay telephones are located on each floor. Each dormitory has a fully equipped laundry room WINDSOR ARMS HOTEL -@ with coin-operated washers and dryers. Ironing 22 St. Thomas Street at Bay and moor facilities are also available. There are no private Single $24. 00 Twin $30. 00 baths; generally speaking, there are two large Extra person (12 years) $6.00 bathroom facilities on each floor. Toilet paper Code: RT CL AC TV FP and soap will be provided. Light kitchen facilities Telephone: 416-921-5141 are available in some dormitories. Beds in all dormitories except Victoria College will be made Y. M. C. A. (men only) -(j) daily, Monday through Friday; however, only one 40 College street at Yonge set of sheets and towels will be furnished for the Single $10.00 Double $6.36 per person per day. meeting period. Pets are now allowed in the These prices include 7% sales tax. residence halls. To be assured of a room, participants should FOUR SEASONS MOTOR HOTEL -@ register in advance. Please use the Residence 415 Jarvis Street at Carlton Hall Reservation Form provided on the last page Single $29. 00 Double $37. 00 of these c}/oti.cei). Residence hall reservation re­ Extra person (14 years) $6.00 quests will be acknowledged by the Mathematics Code: RT CL AC TV SP FP Housing Bureau. Do not include payment for dor­ Telephone: 416-924-6631 mitory accommodations with your preregistration form, since this will only cause a delay in the LORD SIMCOE HOTEL -@ processing of your preregistration and housing re­ 150 King Street West at University quest. Single $19. 00 Double $25. 00 Extra person (12 years) $6.00 HOTELS Code: RT CL AC TV SP FP (overnight) Blocks of rooms have been set aside for use Telephone: 416-362-1848 by participants at the Park Plaza Hotel, the Hotel Plaza II, and the Chelsea Inn. Participants should WESTBURY HOTEL - UOl make their own reservations with these hotels di­ 475 Yonge Street at College rectly, and should identify themselves as partici­ Single $29. 50 Double $36. 50 pants in either the Mathematical Economics Short Extra person (14 years) $7. 00 Course or the Joint Mathematics Meetings. Also Code: RT CL AC TV Parking $3. 50 (24 hours) listed below are several other hotels in the area Telephone: 416-924-0611 of the campus. All prices are subject to change without notice and a seven percent Provincial SUTTON PLACE HOTEL -@ Sales Tax. The extra person charge is for a roll­ Bay at Wellesley Street away cot. The age limit for children under which Single $32.50 Double $40.50 there is no charge, provided an extra cot is not Extra person (12 years) $8.00 required, is shown in parentheses. The following Code: RT CL AC TV SP Parking $3. 00 (24 hours) codes apply: FP- Free Parking; SP- Swimming Telephone: 416-924-9221 Pool; AC - Air-Conditioned; TV - Television; CL- Cocktail Lounge; RT- Restaurant. The num­ PARK PLAZA HOTEL -@ bers at the end of the first line correspond to the Bloor at Avenue Road numbers on the maps on pages 192 and 193. Single $35.00 Double $43. 00 Hotels numbered 3, 6, 7, 10, and 11 are within Extra person (14 years) $8. 00 a 10-15 minute walk of the campus. Hotels Code: RT CL AC TV Parking $3. 00 (24 hours) numbered 4, 5, 8, 9, 12, 13, and 14 are within Telephone: 416-924-5471 a 20-30 minute walk, or short subway ride.

191 I I I I I I I Wellesley Sta. SPADINA

1 Park 1 Sta. ~ CARLTON ST. I i5 +++++++++++++++ I ~ : METROPOLITAN TORONTO i r---~N~~----~------~---GERRARD ST. + and I @@4 + CAMPUS : +++++++++++++++

DUNDAS ST. St. Patrick 1 Sta. lCD ~ I;>< ~ ~ ~N ~ "'.a < "' Is;: "';>< z "' I~ < "0 ~ ;>< ~ !tt:J- I;;; ...,< QUEEN ST.

KING ST.

FRONT ST.

-.LAKESHORE BLVD.

-LAKESHORE BLVD (Rt 2)

0 -@ HOTELS (see list) @ Sidney Smith Hall @ - ® identijled in text ----SUBWAY

19:2 Toronto

®- ® HOTELS (see list)

N~gara Falls 7?·~>.6! 8-mr. ./ ~ Buffalo I' 96 mi. Erie, PA 201 mi.

HOTEL PLAZA II - @ HOLIDAY INN EAST -@ Bloor at Yonge street Highway 401 at Warden Avenue Single $37. 00 Double $44. 00 Single $28. 50 Double $33. 50 Twin $38. 00 Extra person (10 years) $5.00 Extra person (12 years) $4.00 Code: RT CL AC TV Parking $3. 75 (24 hours) Code: RT CL AC TV SP FP Telephone: 416-961-8000 Telephone:416-293-8170 CHELSEA INN - ~ SKYLINE HOTEL-@ Gerrard at Bay street 655 Dixon Road (near International Airport) Single $22. 00 Double $27. 00 Single $28.00 Double $34.00 Extra person (13 years) $5.00 Extra person (12 years) $6. 00 One Bedroom Suite with Kitchen: Code: RT CL AC TV SP FP $47/day based on 2 adults and 2 children Telephone: 416-244-1711 $54/day based on 4 adults Code: RT CL AC TV Parking $1.75 (24 hours) THE CONSTELLATION HOTEL-@ 900 Dixon Road (near International Airport) Telephone: 416-595-1975 Single $32. 00 Double $40. 00 The following hotels are in the suburbs, and Extra person (16 years) $6. 00 are within an hour's drive of the campus: Code: RT CL AC TV SP FP Telephone:416-677-1500 CANADIANA MOTOR HOTEL -@ Kennedy Road and Highway 401 INN ON THE PARK -@ Single $23. 00 Double $31. 00 Leslie and Eglinton East Extra person (12 years) $7. 00 Single $32. 00 Double $37. 00 Code: RT CL AC TV SP FP Extra person (18 years) $6.00 Telephone: 416-291-1171 Code: RT CL AC TV SP FP HOLIDAY INN WEST -@ Telephone: 416-444-2561 Highway 427 at Burnhamthorpe SEAWAY TOWERS MOTOR HOTEL-@ Single $27. 50 Double $32. 50 Twin $37. 00 2000 Lakeshore Boulevard West Extra person (12 years) $4.00 Single $26.00 Double $32.00 Code: RT CL AC TV SP FP Extra person (12 years) $4.00 Telephone: 416-621-2121 Code: RT CL AC TV SP FP (overnight) Telephone: 416-763-4521

193 HOLIDAY INN YORKDALE -@ George and College Streets, and has summer Dufferein Street at Highway 401 hours of 9:00a.m. to 8:00p.m., Monday through Single $29.00 Double $35.00 Twin $38.00 Friday, and 9:00a.m. to 5:00p.m. Saturday. Extra person (12 years) $4. 00 Code: RT CL AC TV SP FP MEDICAL SERVICES Telephone: 416-789-5161 The University Health Service (®on the map) FOOD SERVICES is open from 9:00a.m. to 4:30p.m. daily for Two cafeterias will be in operation on cam­ medical attention. Emergencies occurring during the evening or weekends can be handled at the pus durin~the meetings. Wilson cafeteria in New College (@ on the map on page 192) will be open Emergency Department of any of the local hospi­ for breakf.ist from 7:00a.m. to 8:30a.m., lunch tals: Toronto General Hospital, College at Uni­ from 11:30 a.m. to 1:30 p.m. , and dinner from versity; Women's College Hospital, 76 Grenville 5:30p.m. to 7:30p.m. throughout the meetings, Street (College at Bay); The Hospital for Sick Child­ beginning on Sunday, August 22, through Friday, ren, 555 University Avenue. In addition, the Aca­ August 27. New College cafeteria will be open for demy of Medicine can advise of local doctors who breakfast and lunch only on Saturday, August 28. are on emergency call. Their telephone number is The cafeteria in the Medical Sciences Building 922-1134. Dental service can be arranged through <@on the map) will be open from 7:30a.m. to4:00 the University Health Centre. p.m. for breakfast and lunch, Monday through Friday. The average costs of meals in both these DAY CARE CENTRES facilities are $1. 70 breakfast; $2. 60 lunch; $3. 45 The Campus Cooperative wUI accept up to dinner. Individual meal tickets or daily tickets fourteen children. The cost there is $11 per day, ($7. 75)willbe on sale at the Joint Meetings Regis­ plus two hours time donated by a parent. Inter­ tration Desk. Participants may also pay for meals ested parties should write directly to Ms. Marilyn on the spot; however, a slight saving may be rea­ Wilcoxen, Campus Cooperative, 12 SussexAvenue, lized if tickets are purchased in advance. Toronto, Canada, giving the dates they will uti­ A snack bar, selling soup, sandwiches, lize the Centre, and include a deposit equal to beverages, light desserts, etc. will be operated one day's fee. Margaret Fletcher Daycare Centre in Room 5025 of Sidney Smith Hall, and will be will accept up to fifteen children. The cost there open from 8:30a.m. to 3:30p.m. Monday through is $11 per day, but no donated time is required Friday. In addition, there are a number of catering of parents. Participants should write directly to trucks which park outside Sidney Smith Hall; pic­ Mrs. N. Lupton, Margaret Fletcher Daycare nic tables are provided. Centre, University of Toronto, to make reserva­ CAMPING tions, again giving dates. An $11 deposit is re­ quired. Participants interested in utlizing day There are no suitable camping sites located care facilities are asked to check the appropriate near the University of Toronto. Those persons box on the Residence Hall Reservation Request wishing to camp should contact their local KOA Form. office for the current issue of "Handbook and Di­ rectory for Campers." ENTERTAINMENT BOOKSTORES The University of Toronto is planning en-: There are three bookstores located on cam­ tertainment for mathematicians and their families pus. The University of Toronto Bookroom, located during the meetings. At 8:00p.m. on Wednesday, on King's College Circle <@on the map) and the August 25, there will be an evening beer party University of Toronto Textbook store, located on in Wetmore Hall Cafeteria, New College. Tickets Huron Street <@on the map), are both open from to this event will be sold in advance at the Joint 8:45a.m. to 4:30p.m., Monday through Friday. Meetings Registration Desk. The price per ticket The Student Christian Movement Bookstore, lo­ is $5. Light sandwiches and snacks will be served. cated on the edge of campus at the southwest cor­ During the week of the meeting, there will be ner of moor and St. George, is open 9:00a.m. to many entertainment events in Toronto and vicinity. 6:00p.m. Monday through Friday, and 10:00 a.m. A number of theatres will be giving regular per­ to 6:00p.m. on Saturday. formances at this time. At Niagara-on-the-Lake LffiRARIES there is a Shaw festival, and in Stratford, a Shakespeare festival. Both of these places are The Mathematics Department Library, lo­ relatively close to Toronto and return transporta­ cated on the second floor of Sidney Smith Hall, tion for any evening is easy to arrange. In addi­ will be open from 9:00a.m. to 9:00p.m. for the tion to theatrical events, Toronto has a wide duration of the meetings. Information concerning variety of musical activities in the summer. The books located in other libraries is available from Canadian National Exhibition will be in progress the Department Library. The main collection of during the week of the meetings, and it is easily books is in the Science and Medicine Library, accessible from the university by public trans­ located on King's Colle~Circle <®on the map). portation. Tours can be arranged during the day­ The Robarts Library ( G on the map) houses the time to local places of interest such as the On­ Humanities Collection. mmer hours for univer­ tario Science Centre, the McMichael Collection, sity libraries are 8:00 a.m. to 11:00 p.m., Mon­ and the large, new Metropolitan Zoo. Participants day through Friday, and 8:00a.m. to 6:00p.m. interested in these events should check with the on Saturday. The Metro Toronto Central Public Local Information section of the Joint Meetings Library is located at the southwest corner of St. Registration Desk.

194 TRAVEL AND LOCAL INFORMATION however, are urged to drive as little as possible Toronto is served by Air Canada, Allegheny, between dormitories and the meeting area. American, C. P. Air, Eastern, Great Lakes, WEATHER Nordair, North Central, Quebecair, Transair, and United airlines to Toronto International Air­ The normal daytime high temperature during port. There are several ways of getting from the this period is 79°F. Normal night-time low is airport to campus: (1) There is bus service avail­ 61°F. Rainfall in August averages 2. 65 inches, able to the Islington subway station. From there, with a 30 percent probability of precipitation each subway and surface transportation is available in­ day. Humidity ranges from a daytime high of 67 to the campus area. (2) The airport bus goes to percent to a night-time low of 55 percent. The the Royal York Hotel and the Sutton Place Hotel. record high and low temperatures for August are The cost is about $2. 50. (3) Cab service from the 102oF and 39°F, respectively. Light sweaters airport costs up to $10 per cab. (4) There is also and jackets are advised for evening wear. Tem­ a limousine service which costs up to $12 per air­ peratures in Canada are now given in the Celsius conditioned luxury car. scale, so the preceding temperatures would read: Rail service to Toronto is by Canadian Na­ normal high 26°; normal low 16°; record high tional and Canadian Pacific Railways, with good 39°; record low 4°. connections from Detroit, Buffalo, and Montreal. MAIL AND TELEPHONE MESSAGES Limited access highways (#401, #427, and Queen Elizabeth Way) connect Toronto with Detroit, Buf­ All mail and telegrams for persons attending falo, Kingston, or Montreal. the meetings should be addressed in care of Entering Canada is usually no problem for Mathematics Meetings, Department of Mathemat­ American citizens, and involves nothing more than ics, University of Toronto, Toronto, Ontario, answering questions about where they were born, Canada M5S 1A1. Mail and telegrams so ad­ where they are going, and how long they will stay. dressed may be picked up at the Joint Meetings To be assured of entry, however, it is advised Registration Desk located in the entrance lobby of that participants bring with them some proof of Sidney Smith Hall. citizenship, such as a voter's, baptismal, or birth A telephone message center will be located certificate. Permanent U.S. residents who are in the same area to receive incoming calls for not citizens are required to bring their alien regis­ registrants during the hours the desk is open, tration receipt card (U.S. form 1-151). Entry re­ cf. the section entitled MEETING PREREGIS­ quirements vary for people coming to Canada from TRATION AND REGISTRATION, on a previous countries other than the United States. As a gen­ page. Messages will be written down, and the eral rule, the visitor should have a valid national name of any participant for whom a message has passport. been received will be posted until the message is picked up at the message center. The telephone PARKING number of the center is (416) 978-4856. Parking throughout the campus is extremely LOCAL ARRANGEMENTS COMMITTEE limited as the campus was not designed for motor traffic. Parking stickers for pay lots will be on sale David F. Andrews, Edward J. Barbeau, Jr., at the Joint Meetings Registration Desk for $1. 50 H. Botta, Thomas H. Callahan, Walter H. Gott­ perday. There are also several areas where on­ schalk (ex officio), Lee Lorch, Stephen J. Pierce, street parking is free from 9:00 a.m. to 4:00p.m. Paul G. Rooney (chairman), David P. Roselle Maps indicating these parking areas will also be (ex officio), Roderick A. Ross, James R. Van­ available at the Registration Desk. Participants, stone, and Gordon L. Walker (ex officio).

195 SUMMARY OF ACTIVITIES The purpose of this summary is to provide assistance to registrants in the selection of arrival and departure dates. The program, as outlined below, is based on information available at press time.

AMERICAN MATHEMATICAL SOCIETY

SATURDAY, August 21 SHORT COURSE ON MATHEMATICAL ECONOMICS 4:30p.m. - 7:30p.m. REGISTRATION (Short Course Only)

SUNDAY August 22 8:00a.m. - 5:00p.m. REGISTRATION (Short Course Only) 9:00 a.m. - 10:15 a.m. Price formation: The theory of general economic equilibrium Hugo F. Sonnenschein 10:45 a.m.- noon Measure spaces of economic agents Werner Hildenbrand 2:00p.m. - 3:15p.m. The role of prices and interest rates in dynamic economics David Gale 3:45p.m. - 5:00p.m. The theory of economic equilibrium from the differentiable point of view Andreu Mas-Colell

MONDAY, August 23

8:00a.m. - noon REGISTRATION (Short Course Only) 9:00a.m. - 10:15 a.m. Some game models in economics Robert J. Aumann 10:45 a.m. - noon Computation and existence of Walras equilibria Stephen Smale

AMS - MAA SUMMER MEETINGS MONDAY, August 23 American Mathematical Society Other Organizations 2:00p.m. - 8:00p.m. REGISTRATION 2:00p.m. Council Meeting

TUESDAY, August 24 AMS Other Organizations 8:00a.m. - 5:00p.m. REGISTRATION 10:00 a.m. - 11:00 a.m. COLLOQillUM LECTURE I Recent progress in dynamical systems: A survey Jiirgen K. Moser noon - 6:00p.m. Sessions for Contributed Papers Special Sessions 1:00 p.m. - 5:00p.m. EXHffiiTS 1:30 p.m. - 2:30p.m. INVITED ADDRESS I Nearly flat triangulations- of Riemannian manifolds Eugenio Calabi 2:45p.m. - 3:45 p.m. INVITED ADDRESS Title to be announced Per Enflo 4:00p.m. - 5:00p.m. INVITED ADDRESS Optimal reconstruction of a function from its projections Lawrence A. Shepp 7:00p.m. - 8:00p.m. Mathematicians Action Group Business Meeting 8:30p.m. - 10:00 p.m. Committee on Employment and Educational Policy Open Meeting: The state of the job market

WEDNESDAY, August 25 AMS Other Organizations 8:00a.m. - 9:00a.m. Sessions for Contributed Papers Special Sessions 8:30a.m. - 4:30p.m. REGISTRATION 8:30a.m. - 4:30p.m. EXHffiiTS 8:30a.m. - 4:30p.m. EMPLOYMENT REGISTER

196 SUMMARY OF ACTIVITIES

WEDNESDAY, August 25 American Mathematical Society other Organizations

9:00 a.m. - 4:00 p.m. Mathematical Association of America Board of Governors Meeting 9:00 a.m. - 10:00 a.m. COLLOQUWM LECTURE II Recent progress in dynamical systems: Closed orbits Ji:irgen K. Moser 10:00 a.m. - 3:45 p.m. Sessions for Contributed Papers Special Sessions noon - 1:00 p.m. Pi Mu Epsilon - Council Luncheon 2:00p.m. - 4:00p.m. liME Contributed Paper Sessions 4:00p.m. Business Meeting 5:30p.m. - 6:30p.m. Association for Women in Mathematics Open Executive Committee Meeting 5:30p.m. - 6:30p.m. liME Banquet 7:00p.m. - 8:00p.m. liME J. Sutherland Frame Lecture Speaker and title to be announced 7:00 p.m. - 8:00p.m. MAA Committee on Two-Year Colleges Informal Meeting: Mathematical education at the two-year college level 8:00p.m. BEER PARTY

THURSDAY, August 26 AMS Other Organizations 8:00 a.m. - 9:00 a.m. InME Dutch Treat Breakfast 8:30 a.m. - 4:30 p.m. REGISTRATION 8:30 a.m. - 4:30 p.m. EXHffiiTS 8:30 a.m. - 4:30 p.m. EMPLOYMENT REGISTER 9:00 a.m. - 9:10 a.m. WELCOME ADDRESS G. DeB. Robinson 9:10 a.m. - 10:00 a.m. MAA- THE EARLE RAYMOND HEDRICK LECTURES I: Some mathematical applica­ tions of logic: Unsolvable problems Martin D. Davis 10:10 a.m. - 11:00 a.m. MAA INVITED ADDRESS Stamp out math boredom N. G. Gunderson 11:10 a.m. - noon MA:A INVITED ADDRESS Geometrical optics and the singing of whales Cathleen S. Morawetz noon - 1:30 p.m. Sessions for Contributed Papers Special Sessions noon - 1:15 p. m. AWM Panel discussion: History of women in mathematics Lenore Blum (moderator) AWM Business Meeting 1:30 p.m. - 2:30 p.m. COLLOQUIUM LECTURE ill Recent progress in dynamical systems: Integrable systems Jiirgen K. Moser 2:30 p.m. - 6:00 p.m. Sessions for Contributed Papers Special Sessions 2:45 p.m. - 3:45 p.m. INVITED ADDRESS Determining the finite simple groups Michael Aschbacher 3:00 p.m. - 5:00 p.m. liME Contributed Paper Sessions

4:00 p.m. - 5:00 p.m. INVITED ADDRESS Principles and practice of nonstandard analysis Edward Nelson I 5:15 p.m. - 6:15 p.m. INVITED ADDRESS New directions in computability theory (title tentative) Marian B. Pour-El I

197 SUMMARY OF ACTIVITIES

THURSDAY, August 26 American Mathematical Society Other Organizations

7:00 p.m. - 9:48 p.m. MAA - Film Program All films are from the MAA Mathematics Today Series; unless noted otherwise, all films are in color 7:00p.m. - 8:01 p.m. Let us teach guessing 8:05 p.m. - 9:00 p.m. Challenge in the classroom 9:05 p.m. - 9:48 p.m. Gottingen and New York 7:00 p.m. - 10:00 p.m. MAA Section Officers 8:00 p.m. - 10:00 p.m. MAG Panel discussion: Other mathematical models in economics

FRIDAY, August 27 AMS Other Organizations

8:30 a.m. - noon EXHIDITS 8:30a.m. - 4:30 p.m. REGISTRATION 8:30 a.m. - 4:30 p.m. EMPLOYMENT REGISTER 9:00 a, m. - 9:50 a.m. MAA- THE EARL RAYMOND HEDRICK LECTURES II: Some mathematical applica­ tions of logic: Diophantine sets Martin D. Davis 10:00 a.m. - 10:50 a.m. MAA Business Meeting Presentation of Lester R. Ford Awards and tribute to a distinguished Association member 11:00 a.m. - 11:50 a.m. MAA Retiring Presidential Address Convergence, divergence, and the computer Ralph P. Boas noon - 1:30 p.m. Sessions for Contributed Papers Special Sessions 1:30 p.m. - 2:30 p.m. COLLOQUIUM LECTURE IV Recent progress in dynamical systems: Unstable orbits in particular for the n-body problem Jiirgen K. Moser 2:30p.m. - 6:00 p.m. Sessions for Contributed Papers Special Sessions 2:30 p.m. - 6:00 p.m. Conference Board of the Mathematical Sciences Council Meeting 6:30 p.m. - 9:00 p.m. MAA Banquet for 30 year members 7:00 p.m. - 10:03 p.m. MAA - Film Program All films are from the MAA Mathematics Today Series; unless noted otherwise, all films are in color 7:00 p.m. - 7:48 p.m. Pits, peaks, and passes 7:55 p.m. - 8:55 p.m. Fixed points 9:00 p.m. - 10:03 p.m. John Von Neumann, A documentary (b & w) 8:00 p.m. - 10:30 p.m. CBMS - Council Meeting

SATURDAY, August 28 Mathematical Association of America

8:30 a.m. - 1:30 p.m. REGISTRATION 9:00 a.m. - 9:50 a.m. THE EARLE RAYMOND HEDRICK LECTURES III: Some mathematical applica­ tions of logic: Nonstandard analysis Martin D. Davis 10:00 a.m. - 10:50 a.m. INVITED ADDRESS Mathematics of genetics and evolution Samuel Karlin 11:00 a.m. - 11:50 a.m. INVITED ADDRESS Geometrical models H. S. MacDonald Coxeter 1:30 p.m. - 2:20 p.m. INVITED ADDRESS Tbe role of mathematics in the design of an x-ray Tomographic Human Body Scanner Lawrence A. Shepp

198 SUMMARY OF ACTIVITIES

SATURDAY, August 28 Mathematical Association of America

2:30 p.m. - 3:20 p.m. INVITED ADDRESS J. C. Fields, the Fields Medal, and mathe­ matical research Henry S. Tropp 3:30 p.m. - 4:20 p.m. INVITED ADDRESS Modern algebraic. approach to combinatorial designs N. S, Mendelsohn

Walter H. Gottschalk Associate Secretary Middletown, Connecticut

ORGANIZERS AND TOPICS OF SPECIAL SESSIONS Abstracts of contributed papers to be considered for possible inclusion in special sessions should be submitted to the Providence office by the deadlines given below. The latest abstract form has a section for indicating special sessions. Lacking this, be sure your abstract form is clearly marked "For consideration for special session title of s ecial session)." Those papers not selected for special sessions will automatically be considered for re ar sessions unless the author gives specific instructions to the contrary. Deadline Toronto, Canada, August 1976 Expired Henry W. Gould, Combinatorial identities John W. Gray, Category theory Edward J. Mayland, Jr., Knots and three-manifolds Z. H. Nitecki, Differentiable dynamical systems (tentative) Anatol Rapoport, Mathematical psychology Ray Shiflett, Doubly stochastic measures and their operators Joseph R. Shoenfield, Recursively enumerable sets and degrees (tentative) W. T. Tutte, Chromatic polynomials and related topics C. T. Yang, Transformation groups Albuquerque, New Mexico, November 1976 September 7, 1976 David Fox, The estimation of eigenvalues Pramad Pathak, Probability and statistics Stanly Lee Steinberg, Partial differential equations

199 739TH University of Michigan Ann Arbor, Michigan MEETING November 6, 1976 The seven lmndred thirty-ninth meeting of afternoon, Abstracts should be submitted to the the American Mathematical Society will be held at American Mathematical Society, P. 0. Box 6248, the University of Michigan, Ann Arbor, Michigan, Providence, Rhode Island 02940, so as to arrive on Saturday, November 6, 1976. The sessions of prior to the deadline of September 7, 1976. the meeting will be held in the Auditorium Unit of There will probably be several sessions of James B. Angell Hall. selected twenty-minute papers. These will be an­ By invitation of the Committee to Select Hour nounced in the August issue of these c}/otiui). Speakers for Western Sectional Meetings, there Information on travel and accommodations will be two one-hour addresses. Philippe M. Ton­ will be given in the August issue of these c}/otictJJ. deur of the University of Ulinois at Urbana-Cham­ Blocks of sleeping rooms have been reserved in paign will address the Society at 11:00 a.m.; his the Michigan Union and in the Campus Inn. subject is "G-foliations and their characteristic classes." M. Pavaman Murthy of the University of Chicago will speak at 1:45 p.m. on the topic, "Serre's problem and complete intersections." Paul T. Bateman There will be sessions for the presentation Associate Secretary of contributed ten-minute papers both morning and Urbana, Ulinois

INVITED SPEAKERS AT AMS MEETINGS This section of these c}/otiui) lists regularly the individuals who have agreed to address the So­ ciety at the times and places listed below. For some future meetings, the lists of speakers are incom­ plete. Toronto, Canada, Augu.st 1976 Michael Aschbacher Edward Nelson Eugenio Calabi Marian B. Pour-El Per Enfl.o Lawrence A. Shepp Jtfrgen K. Moser (Colloquium Lecturer) Ann Arbor, Michigan, November 1976 M. Pavaman Murthy Philippe M. Tondeur Columbia, South Carolina, November 1976 Frank T. Birtel Thomas G. Hallan Thomas A. Chapma:J)

200 PERSONAL ITEMS

REINHOLD BAER of the EidgenHssische To Chairman, Department of Mathematics. Technische Hochschule, ZUrich, Switzerland, has Cornell University: CLIFFORD J. EARLE, Jr.; been awarded the honorary degree of Dr. phil. nat. Indiana University Southeast: CHURL S. KIM. h. c. by both the University of Giessen (1975), and the University of Ki.el (1976). SIEGFRIED K. GROSSER of the University of To Professor. Purdue University: RICHARD Vienna has been elected President of the Austrian B. HOLMES; University of North Carolina at Mathematical Society. Greensboro: JERRY E. VAUGHAN; University of CHARLES R. HADLOCK of Amherst College Tulsa: J. BEE BEDNAR; Virginia Polytechnic will be visiting the Universidad Nacional de Co­ Institute and State University: CHARLES D. lombia, Bogota, from June 1 to September 1. FEUSTEL. DONALD HARTIG of Ohio University has been appointed to an assistant professorship at the To Associate Professor. Case Western Re­ U.S. Naval Academy. serve University: WALTER J. HENDRICKS; ROGER HAYNE of the University of Califor­ Ohio University: ROBERT E. ATALLA, DAVID nia, Riverside, has been appointed to a visiting N. KECK, M. S. K. SASTRY, and SHIH-LIANG assistant professorship at Indiana University South­ WEN; University of Santa Clara: VLADIMIR east, New Albany, Indiana. DROBOT; University of Texas at Arlington: JAMES F. HURLEY of the University of Con­ FRANK N. HUGGINS; Virginia Polytechnic Insti­ necticut has been appointed a Visiting Fellow in tute and State University: WILLIAM GREENBERG, the Institute of Advanced Studies, Research School BRUCE E. REED, ROBERT L. SNIDER, and of Physical Sciences, Australian National Univer­ ROBERT E. SPENCER; Worcester Polytechnic sity. He spent the 1975-1976 academic year at Institute: MAYER HUMI. the University of Philippines under the National Science Foundation Scientists and Engineers in DEATHS Economic Development Program. Professor Emeritus WILLIAM L. AYRES JOHN G. KEMENY of Dartmouth College will of Southern Methodist University died on January become the 25th recipient of Dickinson College's 24, 1976, at the age of 70. He was a member of Priestley Memorial Award, for his contribution to the Society for 51 years. the welfare of mankind through science. Professor DELBERT R. FULKERSON of JOHANNES H. B. KEMPERMAN of the Uni­ Cornell University died on January 10, 1976, at versity of Rochester was among thirteen Ameri­ the age of 51. He was a member of the Society cans elected this year to the International Statis­ for 15 years. tical Institute. Professor LEO LAPIDUS of Lewis & Clark LAWRENCE MAND of the University of Ken­ College died on December 21, 1975, at the age tucky has been appointed to a visiting assistant of 65. He was a member of the Society for 27 professorship at Indiana University Southeast, years. New Albany, Indiana. Professor WILSON LEE MISER of Adrian KENNETH W. NEVES of Babcock & Wilcox College died on February 28, 1974, at the age of Company has been appointed a senior specialist 87. He was a member of the Society for 59 mathematician with the Boeing Computer Services, years. Inc. , Seattle. Professor PAUL McCARTNEY SWINGLE RICHARD O'NEIL of SUNY at Albany has of the University of Miami died on March 29, been appointed to a visiting professorship at the 1976, at the age of 76. He was a member of the University of Firenze, Italy. Society for 48 years. JONATHAN D. SONDOW of the Pennsylvania Dr. WILLIAM L. G. WILLIAMS of McGill State University has been appointed a research University died on January 31, 1976, at the age staff member at the Institute for Defense Analyses, of 88. He was a member of the Society for 38 Communications Research Division Princeton years. New Jersey. ' ' Professor Emeritus LEVIT. WILSON of PROMOTIONS the U.S.NavalAcademy, JacksonvUle State Uni­ versity, Alabama, and Jacksonville University, To Associate Director of Research; Bat- Florida, died on December 20, 1975, at the age telle Columbus Laboratories: FREDERICK J. MIL­ of 89. He was a member of the Society for 60 FORD. years.

201 NOMINATIONS FOR VICE-PRESIDENT OR MEMBER-AT-LARGE

One position of vice-president of the Society 2. The name of the candidate must be given and member of the Council ex officio for a term of as it appears in the Combined Membership List. two years is to be filled in the election of October If the name does not appear in the list, as in the 1976. The Council has nominated two candidates case of a new member or by error, it must be for the position, namely as it appears in the mailing lists , for example on the mailing label of these c}/oticei). John T. Tate 3. The petition for a single candidate may [Second candidate withdrew. E. P. ] consist of several sheets each bearing the state­ Additional nominations by petition in the manner ment of the petition, including the name of the described below are acceptable. position, and signatures. The name of the candi­ Five positions of member-at-large of the date must be exactly the same on all sheets. Council for a term of three years are to be filled 4. On the facing page is a sample form for in the same election. The Council has nominated petitions. Copies may be obtained from the Sec­ eight candidates for these positions, namely retary; however, petitioners may make and use photocopies or reasonable facsimiles. Theodore W. Gamelin 5. Richard J. Griego A signature is valid when it is clearly Karl P. Hofmann that of the member whose name and address is Raymond L. Johnson given in the left-hand column. At least fifty valid Robert P. Langlands signatures are required for a petition to be con­ Henry P. McKean, Jr. sidered further. Martha Kathleen Smith 6. The signature may be in the style chosen W. T. TUtte by the signer. However, the printed name and address will be checked against the Combined Additional nominations by petition in the manner Membership List and the mailing lists. No at­ described below are acceptable. The Council in­ tempt will be made to match variants of names tends that there shall be at least ten candidates with the form of name in the CML. A name not for the five positions and will bring the number up in the CML or on the mailing lists is not that of to ten if the number of nominations by petition does a member. (Example: The name Everett Pitcher not do so. is that of a member. The name E. Pitcher ap­ Names of these candidates are published to pears not to be. Note that the current mailing assist those who may wish to make nominations by label of these c}/oficei) can be peeled off and af­ petition. fixed to the petition as a convenient way of pre­ The name of a candidate for the position of senting the printed name correctly. ) vice-president or of member-at-large of the Coun­ 7. When a petition meeting these various cil may be placed on the ballot by a petition that requirements appears, the Secretary will ask conforms to several rules and operational consi­ the candidate whether he is willing to have his derations, as follows: name on the ballot. His assent is the only other 1. To be consi

202 NOMINATION PETITION FOR 1976 ELECTION The undersigned members of the American Mathematical Society propose the name of.___ -.- ...... ,.....--,,..----:---...,-...,.,...--...,.-:--.....-:~as a candidate for the position of * of the American Mathematical Society for a term beginning January 1, 1977. Name and Address (Printed or typed, or c/'loticeiJ mailing label) Signature

*Specify "vice-president'' or "member-at-large of the Council".

203 THE NOMINATING COMMITTEE FOR 1977 The Council has approved the continuation consist of several sheets each bearing the state­ for another year of the procedure of filling places ment of the petition, including the name of the on the Nominating Committee by election. There position, and signatures. The name of the candi­ will be four continuing members of the Nominating date must be exactly the same on all sheets. Committee, namely 4. On the facing page is a sample form for petitions. Copies may be obtained from the Sec­ Phyllis J. Cassidy retary; however, petitioners may make and use Edward B. Curtis photocopies or reasonable facsimiles. Davis Chandler 5. A signature is valid when it is clearly R. 0. Wells, Jr. that of the member whose name and address is There will be four places filled by election in a given in the left-hand column. At least 100 valid preferential ballot. The president will name at signatures are required for a petition to be con­ least six candidates. When this issue of the sidered further. cJ/otiai) went to press, the secretary had not se­ 6. The signature may be in the style cured the assent of all of the tentative candidates, chosen by the signer. However, the printed name so none of the names is released now. Nominations and address will be checked against the Combined by petition, in the manner described below will be Membership List and the mailing lists. No at­ accepted. Should the final number of candidates tempt will be made to match variants of names be less than eight, the President will bring it up with the form of name in the CML. A name not to eight. in the CML or on the mailing lists is not that of The name of a candidate for member of the a member. (Example: The name Everett Pitcher the Nominating Committee may be placed on the is that of a member. The name E. Pitcher ap­ ballot by a petition that conforms to several rules pears not to be. Note that the mailing label of and operational considerations, as follows: these cJ/otiai) can be peeled off and affixed to the petition as a convenient way of presenting the 1. petitions must be ad­ To be considered, printed name correctly. ) dressed to Everett Pitcher, Secretary, Box 6248, 7. When a petition meeting these various 02940, must arrive Providence, Rhode Island and appears, the Secretary will ask by August 13, 1976. requirements the candidate whether he is willing to have his 2. The name of the candidate must be given on the ballot. His assent is the only other as it appears in the Combined Membership List. name condition of placing it there. Petitioners can If the name does not appear in the list, as in the the procedure by accompanying the case of a new member or by error, it must be facilitate the candi­ as it appears in the mailing lists, for example on petitions with a signed statement from the mailing label of these CJ{otiai). date giving his consent. 3. The petition for a single candidate may

204 NOMINATION PETITION FOR 1976 ELECTION (Nominating Committee of 1977) The undersigned members of the American Mathematical Society propose the name of-..,...,,..-­ as a candidate for the position of Member of the '='N=-o-m"'"'i"'"na-t'"'i-ng--,C""o-mm-...,.itt,...,....ee-o-.f...,.t,...he-,A"'"m-er-:i,-c-an--:M"'a""'t,....h-e-m-a:....,.;tical Society for the year 1977. Name and Address (Printed or typed, or c/{oticti) mailing label) Signature

205 THE CHANGING ROLE OF THE MASTER'S DEGREE

At the San Antonio Meeting in January 1976the employment prospects that bleak? Exactly what do AMS Committee on Employment and EducationalPol­ we have to offer? A variety of answers have been icy sponsored a panel discussion on the Master's De­ given to this question in terms of doctoral train­ gree in mathematics and its role in departments ing. An answer to this question in terms of gradu­ across the country. Members of the panel were: Jo­ ate education that terminates at the master's level seph E. Adney ,Jr., Chairman of the Department of is pet"haps provided by our considerations. Mathematics at Michigan State University ,John W. We look at the variety of programs offered Kenelly, Chairman of the Department of Mathematical at the master's level. Some of the schools with Sciences at Clemson University, Lester H. Lang, top rated Ph. D. programs now offer a master's Dean of the School of Science, San Jose State Univer­ degree only as a consolation prize. other Ph. D. sity and Erik Hemmingsen, Chairman of the Depart­ programs attract students from smaller schools ment of Mathematics at Syracuse University. The whose initial objective is the master's degree. moderator was Lida K. Barrett, Head of the Depart­ What type of employment do these individuals ment of Mathematics at the University of Tennessee. find? What sort of future do they have? Do they, The texts of the panel members' talks follow: in spite of their initial objective, go on to a Ph. D. ? The question will be addressed by Erik Hemming­ LIDA K. BARRETT, Introduction sen based on his experiences at Syracuse Univer­ First, some background information: Each sity. Erik's presentation will be followed by the year the December issue of the AMS c}/oficei) con­ one by Lester Lang, who will discuss the situa­ tains a listing of teaching assistantships and fel­ tion at the California universities offering lowships. This year there are listings from 426 the master's degree only. Following these, we departments. Inside the front cover of this year's will hear from two schools which have specialized December issue, somewhat like the warning from master's programs. One, the program at Clem­ the Attorney General that appears on each package son University, involves a master's in mathe­ of cigarettes, is AMS President Lipman Ber•s let­ matics aimed primarily at industrial employment. ter pointing out current predictions for academic The other, the program at Michigan State Univer­ employment for a mathematics Ph. D. sity, involves permitting graduate students to Is the mathematics community guilty of at­ spend an extra year of study for the Ph. D. in tempting to entice with funds-fellowships and mathematics in order to receive a master's de­ assistantships-bright young people into graduate gree in an outside field. Here are a few statistics study in our field on a self-serving basis? Does from the December 1974 and 1975 cNoticeiJ show­ our need for Teaching Assistants cause us to ig­ ing how many master's degrees are awarded in nore their bleak employment prospects? Are the relation to other degrees.

Number of Departments Number of Bachelor• s Number of Master's in Mathematics and Degrees awarded by Degrees awarded by Mathematical Sciences these departments these departments 1973-1974 1974-1975 1973-1974 1974-1975 1973-1974 1974-1975 Ph. D. granting deEart- ments 231 251 8,190 8,571 3,269 3,092 De~rtments grantigg Masters only 164 194 5,808 5,616 1,539 1,478 TOTAL 395 445 13,998 14,187 4,808 4,570 = = = = Total Ph. D.'s awarded in the last three Iears 3,990 3,788

Note that in either of these years the number of Master's Degrees awarded is greater than the num­ ber of Ph. D. •s granted in a three-year period. Data for 1973-1974 and 1974-1975 is taken from the December 1974 and 1975 issues of these cNoticeiJ.

ERIK HEMMINGSEN of four of our graduate students used to attempt the Ph. D. degree, and by no means all of these Our master's degree program has never been achieved it. Of course, more than one in four, a consolation prize. Most students working here on at one time, probably hoped for further study, a Ph. D. have gotten the master's degree as an in­ and some had to be discouraged. dication of their completion of one stage of the Whatever their motive, the students we had Ph. D. program. For many others, however, it has five and more years ago took course wo.>rk that been the degree they came to get on their road to was principally preparatory to a Ph. D. They industrial employment. Indeed only about one out studied real and complex variables, linear and

206 abstract algebra, point set and algebraic topology, by studying for a master's degree!) There has etc. A thirty-hour master's program for the been a decrease in the numbers of those headed average graduate student had only six hours of purely for the Ph.D. degree. In several regions, electives although, because of the tuition arrange­ there has been a cancellation of support by in­ ments for graduate assistants, most of these stu­ dustry of employees• enrollments in our courses. dents had thirty-six hours by the time of the mas­ (Such students used to get pay increments for ter's degree and hence twelve hours of electives. such study on their part. ) There has been a great A properly chosen set of twelve hours of electives increase in service course efforts by our depart­ in computing, numerical analysis, statistics, or ments. There are some students who are going to education together with a firm groundwork in school because there is a lack of work out in the mathematics used to give people with our master's world. degrees a good edge in the employment market, It Heads of Mathematics Departments of sev­ still does. eral universities in California, public and private, Things have changed in the last four years. were contacted by telephone. The questions asked We no longer encourage (or are able to encourage) and summaries of the responses from eight de­ so large a proportion of our graduate students to partment heads follow: stay on for a Ph. D. A number of good students I. who might have made it to a Ph. D. now leave with Are there changes in the last year or so at a master's as described above. The most impor­ your school in the REQUffiEMENTS for the tant change, however, has been that we are now master• s degree ? What changes ? getting students whose interest in topology, Le­ Most departments responded "no". One de­ besgue measure, or Galois theory is minimal. partment tightened up on the qualifying exam, and They came to graduate school to learn statistics, introduced a seminar course for which the final numerical analysis, and optimization. We require exam serves as the qualifying exam for the mas­ them to take a graduate course in analysis and one ter's degree work to follow, Another recently in linear algebra in addition to two semesters of strengthened the undergraduate requirements numerical analysis and two semesters of proba­ and changed the master's qualifier to reflect bility and statistics. The rest of their programs these four areas: algebra, real analysis, complex can be made up of various courses in statistics, analysis, and topology. It also dropped the thesis. linear programming, logic, combinatorics, com­ Yet another department added a thesis require­ puter science, and operations research. Almost ment and dropped the language requirement. all of these students leave at the end of their second year. II. Are there some CURRICULUM changes of If they eventually want a Ph. D. , they will note? need to stay a little longer than usual to get it. Few dramatic changes were reported, but This is also true if they want to switch to a pro­ there has been a steady evolution toward more gram in computer science. At the same time, offerings in applied mathematics. One depart­ some of our present Ph. D. candidates are taking ment introduced a master's degree in computer an extra semester or two to study and in some science. Another university introduced a mas­ cases even to get a degree in computer science ter's degree in computer and information sciences, or statistics. which heavily involves mathematics as well as Now that we have fewer Ph. D. candidates, electrical engineering courses. A third depart­ the average level of teaching experience of our ment has introduced more options at the under­ graduate assistants has been dropping. We have graduate level: solid mathematics courses plus had to sharpen our supervision and provide more offerings in statistics and computer science. A assistance with the problems of instruction. We fourth added a graduate course in optimization, have also tended to encourage good students to and considerable changes in the undergraduate remain an extra year beyond the master's to take curriculum (including new courses in computer courses of a professional nature. The long term science, statistics, service courses in business, trends here are hard to predict, but in the short sampling theory, more numerical analysis). term we have had to place a greater burden on the faculty. One difficulty in the way of long term III. Are there ENROLLMENT changes? Over- prediction is that we do not know how all? More pressure for certain kinds of many good courses? students will stay in graduate school as an alter­ -- native to unemployment. Several departments reported pressure for computer science related courses. This is part of LESTER H. LANGE a trend toward more applied mathematics, which has led to some new programs and hiring new A telephonic survey of some typical Califor­ staff. Several indicated declining demand for pure nia schools giving master's degrees but not the mathematics and master of arts in teaching pro­ Ph. D. in Mathematics, clearly shows that the re­ grams, and two reported fewer students headed cent economic difficulties have affected these for the Ph. D. programs. There has been a clear increase in the offering of courses in applied mathematics and in IV. Are you getting fewer applications for teach­ computer related courses, both in the graduate ing assistants? Is the quality holding up? curricula and the undergraduate areas. There is Almost all eight departments reported no de­ a decrease in enrollments connected with high cline in numbers of applicants, with some ·in­ school teaching careers. (In fact, some students creases. Quality was believed to be holding up. are being advised to get the teaching job first, get tenure and then make themselves expensive V. Any other comments that come to mind?

207 Responses included the following: At Clemson University the institution's malium size "Our Master of Arts in Teaching (M.A. prevented the typical growth of T.) several competing is down. A beginning teacher with a M.A. T. departments and the institu­ tion's engineeringcharacterfostereda prices himself out of the market," and so the strong in­ cur­ terest in applied rent advice is: "Start teaching, get tenure, THEN mathematics. As a result, in the expansion period of the sixties the department make yourself expensive by getting a master's developed as a closely integrated degree!" Other responses were the following: group of indi­ viduals with research interests in "Our master's degree is a pre-Ph. D. de­ the full range gree, and the demand for it of the mathematical sciences. This atmosphere is way down. " "Our attracted and overall enrollments are down, held faculty members with an ap­ perhaps because preciation for local industries are no longer supporting such a comprehensive program, so a lot of the current study by their employees, who used to get pay negative thinking is out of in­ character with crements for such study. No more true ..•... our impression of the broad role of graduate education. Mathematics There is still interest in becoming junior college has tended to limit its graduate teachers; not so many interested in high school programs to the needs of the academic community and its teaching." "We hope to build our graduate pro­ full potential has gram. There has been a change not been realized. In particular, the master's in the power struc­ degree is an example ture in the department. It was a controversial of neglect and we appreciate the Committee's interest, thing, but we have now pointed more toward ap­ and sponsorship of this panel. plied mathematics. " Even though our assumptions might be in­ JOHN KENELLY dividually questioned, we still advance them as the points on which our program is based. They The National Science Foundation has awarded are: a three-year grant to Clemson University for the 1. A development and implementation major source of employment for mathemati­ of an alternative cal scientists approach to graduate education in the mathematical in the future will be industrial firms and governmental sciences. This grant is a part of a national effort agencies. and there are currently three universities involved: 2. These employers will require more than a Clemson University, Washington State University, bachelor's degree in mathematics and less and Rensselaer Polytechnic Institute. All three than a Ph.D. are developing alternative approaches to the cur­ 3. Employers will prefer diversity rent masters and Ph. D. programs in the mathe­ of experience in the matical sciences. The objective is to produce mathematical sciences and will expect graduates better equipped than the current gradu­ more than superficial knowledge in at least ates to satisfy the need for mathematical scien­ one area. tists. 4. Graduates should be experienced in communi­ A new, expanded master's degree at Clem­ cating with persons in fields of activity other son has been designed both for the industrially than the mathematical sciences. oriented master's degree student and as a founda­ 5. It is tion for the broadly trained Ph. D. student. desirable to obtain a broad education in The the mathematical master's program is diversified in that eve:cy stu­ sciences prior to special­ izing dent is required to have a minimum knowledge in for the Ph. D. degree. each of the areas of the mathematical sciences (al­ Clemson's program is attracting undergradu­ gebra, analysis, computer science, operations ate mathematics majors with an interest in com­ research, statistics). In addition, each student puting, operations research, and statistics. As wUl take a concentration of courses in one of the mathematics majors, they are anxious to diver­ areas of mathematical sciences and at least one sify but in an atmosphere that is essentially interdisciplina:cy mathematical models course. mathematical. The graduates are generalists- Models courses in biomathematics, management e. g. , the typical programs include courses in sciences, and environmental systems a,re being exploratory data analysis , deterministic and sto­ developed under the grant and other models chastic methods in operations research, the courses will be developed in the future. A mini­ architecture of computer systems and software, mum of thirty-six semester hours is required for numerical processes, simulation, analysis and the master's program. The minimum would be algebra. The same graduates are specialists- sufficient for only the well-prepared students. i. e. , six graduate level courses are required in Many students will have to take forty-two hours. a concentration area. With this broad background All students accepted into the graduate program and concentrated expertise, they are attractive to should be able to complete the degree in two indust:cy. academic years and one summer. Graduate as­ The reorientation of graduate education sistants in the master's program are supported from academic careers to industrial careers has for this time period (twenty-one months). major implications in the area of graduate as­ An integral part of the grant provides for sistant support mechanisms, i.e. , is the teach­ the collaboration with a similar program at Wash­ ing assistantship appropriate for a nonteaching ington State University. Portions of the programs career? However, that is another important developed at each university will be tested at the question for another time. other institution. Don Bushaw is director of the Interestingly enough the faculty's concen­ Washington State University grant and c. V.Aucoin trated efforts at the master's degree level has is the director of the Clemson University grant. had a positive effect on our Ph. D. program. The

208 graduate program is alive and it has a very posi­ In an effort to address the employment tive atmosphere. Since we honestly feel that a problems of Ph. D. students, the department has broad background is important to Ph. D. •s as well, broadened the areas of study that are available the very best and dedicated students are enthusi­ to all graduate students. Within the department astic about staying on for the Ph. D. The economic we have, or are in the process of developing, and intellectual value of graduate study is appar­ master's level work in: biomathematics, mathe­ ent, and the faculty is enthusiastic about encour­ matical economics and operations research. aging bright students to study mathematics. These are in addition to the more classical areas In order to increase the career opportuni­ of applied mathematics such as numerical analy­ ties of our graduates we should refine and pro­ sis, etc., that are already in existence. Exter­ mote the master's degree. Its wide acceptance nally we have arrangements with departments would lead to a deeper industrial interest in our such as statistics and computer science for a Ph. D. graduates. This is apparently the case in master's degree in those areas. These agree­ chemistry, and Clemson's experience suggests ments allow some mathematics courses to be that this could be the case in mathematics. After supplemented by courses in statistics or compu­ we had placed several master's graduates with ter science. the Deering-Milliken Corporation, they upgraded The main consequences of these small ad­ the position and employed two of our Ph. D. •s. A justments to date are: year later, they changed the job description to 1. An increase in breadth has improved the in­ require a Ph. D. and, to date, five Ph. D. •s have dustrial employment possibilities of the mas­ this posi­ started their industrial careers through ter's degree students as well as the Ph.D. tion. students. Indeed, we now have some diffi­ J. E. ADNEY culty keeping Ph. D. students in the program, since there are excellent opportunities for in­ the stan­ In the past Michigan State offered dustrial jobs. dard master's degree. Ph. D. students took a master's degree "en passant". Students who were 2. Those Ph.D. students who have already taken only interested in a master's degree had individ­ advantage of these opportunities have had very ual programs of study that fitted their needs. good success in obtaining teaching jobs.

209 NEW AMS PUBLICATIONS

CBMS REGIONAL CONFERENCE The second half of the volume deals with SERIES IN MATHEMATICS problems in neurobiology. Hirsh Cohen and John Rinzel survey a large body CLASS GROUPS AND PICARD GROUPS OF of the literature con­ cerning mathematical contributions GROUP RINGS AND ORDERS by Irving Reiner to neuro­ physiology, beginning from the cellular level Number 26 and classical Hodgkin-Huxley theory for the 44 pages; list price $6. 80; member price $5. 10 transmission of a voltage pulse and proceeding to ISBN 0-8218-1676-4 more modern theory and work on dendritic inte­ Publication date: July 31, 1976 gration. H. B. Barlow, in the final paper, pro­ To order, please specify CBMS/26 vides a counterpoint with a somewhat more nega­ tive view of the role The aim of the lectures is to provide an in­ of mathematics and a caveat tblit to make troduction to recent developments in the an impact the mathematician must theory of become class groups and Picard groups. The techniques a neurobiologist. employed come from three main areas: algebraic number theory, representation theory of algebras MEMOIRS OF THE AMERICAN and orders, and algebraic K-theory. MATHEMATICAL SOCIETY The book is divided into the following sec­ tions: explicit formulas, change of orders, class MULTIPLIERS OF PEDERSEN'S IDEAL by A. J. groups of p-groups, Mayer-Vietoris sequences, Lazar and D. C. Taylor calculations, survey of specific results, induction Number 169 techniques, Picard groups, references, and an 75 pages; list price $7. 60; member price $5. 70 index. ISBN 0-8218-1869-4 Publication date: April 30, 1976 LECTURES ON MATHEMATICS To order, please specify MEM0/169 IN THE LIFE SCIENCES Let A be a C*-algebra and M(A) its two­ SOME MATHEMATICAL QUESTIONS IN BIOLOGY. sided multiplier (double centralizer) algebra. VII, edited by Simon A. Levin When A is commutative, that is, A= Co(X), the Volume 8 algebra of all complex valued continuous functions 182 pages; list price $20. 00; member price $15.00 which vanish at infinity on some locally compact ISBN 0-8218-1158-4 Hausdorff space X, then M(A) = Cb(X), the al­ Publication date: May 31, 1976 gebra of all bounded complex valued continuous To order, please specify LLSCI/8 functions on X. The C*-algebra A together with its multiplier algebra M(A) is the noncommuta­ This volume contains lectures given at the tive analogue of the above relationship between Ninth Symposium on Some Mathematical Questions Co(X) and Cb(X), and is generally con~Jdered as in Biology, held in New York on January 29-30, the algebraic counterpart of the Stone-Cech com­ 1975, in conjunction with the annual meeting of the pactification of X. This noncommutative ana­ American Association for the Advancement of logue suggests that it is possible to obtain a non­ Science. The Symposium was cosponsored by the commutative generalization of the relationship American Mathematical Society and by the Society between c00 (X), the functions in Co(X) with com­ for Industrial and Applied Mathematics under the pact support, and its multiplier algebra C(X), auspices of Section A, Mathematics, of the AAAS. the algebra of all complex valued continuous func­ The first three papers in this volume, by tions on X. This can be done by studying the two­ Simon Levin, George Oster, and Brian Charles­ sided multipliers of an ideal contained in the C *­ worth, deal with problems in ecology and evolution­ algebra A which plays a role similar to that of ary biology. The classical mathematical theory of the ideal COO(X) in Co(X). Such an ideal was ecological dynamics has for the most part treated shown to exist by Pedersen and is denoted here the population as a collection of identical individ­ by K. uals, each with the same requirements and poten­ In this paper the authors make an extensive tial for growth. However, real populations of this study of the multipliers of Pedersen's ideal, kind are hard to find; individuals differ, with re­ which they denote by r(K). In Chapters 1 and 2 gard to age, size, spatial location, social rank, they include some basic concepts and definitions and other traits. The first three papers relax in on multiplier algebras and Pedersen's ideal. In various ways the constrictive assumption of ho­ Chapter 3 they study r(K) under the K-topology, mpgeneity, and place emphasis on the critical role a noncommutative analogue of the compact open of population structure. The paper by Levin intro­ topology. In Chapter 4 they give some examples duces· spatial structure, and investigates the of r(K), and in Chapter 5 they develop a com­ changes which occur in ecological theory. In the prehensive spectral theory and functional calcu­ second paper, Oster surveys the implications of lus for r(K). They characterize the dual of the introduction of age structure; and in the third, r(K) under the K-topology in Chapter 6. In Chap­ using a somewhat different framework, Charles­ ter 7 they study homomorphisms of r(K) and worth explores the interface between demographic they prove extension theorems in the sense of and evolutionary parameters. Tietze. They study order and ideals in r(K) in

210 Chapter 8 and they prove a decomposition theorem Algorithms are constructed for solving equations in the sense of Riesz-Pedersen. In Chapter 9 they in three unknowns and systems of equations in two study derivations of r(K), and in Chapter 10 they unknowns. The structure of the solutions of equa­ study PCS-algebras, an algebraic counterpart to tions is investigated. pseudocompact topological spaces. Word equations are connected with Dio­ phantine equations. Namely, to each word equa­ PARAMETRIZED KNOT THEORY by Stanley tion it is possible to assign a Diophantine equation Ocken in such a way that there exists a one-to-one cor­ Number 170 respondence between the solutions of both equa­ 114 pages; list price $7. 60; member price $5. 70 tions. ISBN 0-8218-1870-8 The investigation of word equations may be Publication date: June 30, 1976 useful in the study of equations in free groups, To order, please specify MEM0/170 a problem having great significance for the ele­ mentary theory of free groups. Let %(M) be the set of cobordism classes This monograph is intended for advanced of embeddings f: sU X M--.sn+2 X M which are undergraduates, graduate students and scientists homotopic, relative to sn X aM, to the standard interested in algorithmic problems. inclusion. This work computes Gt(M) for M an arbitrary comract manifold, and shows that in TRANSACTIONS OF THE MOSCOW most cases Gn(M) admits a geometrically defined abelian group structure induced by a natural bi­ MATHEMATICAL SOCIETY jection j: Gt-1(MXI)--.Gt(M). The cobordism Volume 31 (1974) equivalence relation is quite close to the classical 306 pages; list price $38. 80; member price $29. 10 notion of concordance of embeddings. The group ISBN 0-8218-1631-4 Gt(M) is the middle term of a short exact se­ Publication date: July 31, 1976 quence involving a classical Wall surgery group To order, please specify MOSCOW/31 and a new Cappell-Shaneson homology surgery group. This volume is a cover-to-cover transla­ The group Gt(M) embeds in a larger group tion of Transactions of the Moscow Mathematical of embeddings Gn (M), which satisfies fourfold Society for the year 1974, Volume 31. This trans­ periodicity, and which is defined by studying em­ lation was prepared jointly by the London Mathe­ beddings in sn+2 X M of manifolds homotopy matical Society and the American Mathematical equivalent to sn X M. When M is a point, Gt(M) Society. The book is dedicated to Ivan Georgievic and Gn(M) correspond respectively to Kervaire's Petrovskil, and contains seven papers on partial and Levine's claasical knot cobordism groups. differential geometry. Results are obtained on unknotting up to cobord­ The articles are "In memoriam Ivan ism and other geometric problems. This work Georgievic Petrovski'l" by P. s. Aleksandrov, used the Capell-Shaneson approach to the theory A. N. Kolmogorov, and 0, A. Olelnik, "On of codimension two embeddings. conditions for the existence of nonanalytic solu­ tions of linear partial differential equations of PROCEEDINGS OF THE arbitrary order" by 0. A, Oleinik and E. V. Rad­ STEKLOV INSTITUTE kevic, "On the behavior of solutions of higher­ order elliptic equations in unbounded domains" EQUATIONS IN FREE SEMIGROUPS by Ju. I. by E. M. Landis, "On sufficient conditions for Hmelevskii the local solvability of pseudodifferential equa­ Number 107 tions of principal type" by Ju. V. Egorov, 272 pages; list price $50. 40; member price $37. 80 "Positive solutions of linear partial differential ISBN 0-8218-3007-4 equations" by V. A. Kondrat•ev and S.D. Eidel'man, Publication date: May 31, 1976 "The energy method in the Cauchy problem for To order, please specify: STEKL0/107 Petrovskif-correct differential operators" by L. R. Volevic, "Energy estimates connected with the This monograph is devoted to an investiga­ Newton polyhedron" by S. G. Gindikin, and "De­ tion of equations and systems of equations in a generate elliptic pseudodifferential operators and free semigroup with a finite number of generators. the oblique derivative problem" by V. G. Maz 1ja Such equations are also called word equations. and B. P. Panejah.

211 "Au II

REVIEWS IN NUMBER THEORY edited by William J. LeVeque

Excerpts from a review by D. J. Lewis, University of Michigan " ... We have found this collection to be an indispensable tool, and after having it for a year none can imagine the old days when we had to have recourse to many sources to get much less informa· tion. The time saved is worth its cost a hundred fold. In terms of literature search it is equivalent to replacing a hand calculator by an HP65."

From Paul T. Bateman, University of Illinois, Urbana " ... Its value to anyone interested in recent research in number theory is hard to overestimate ... These volumes contain reviews of practically every article in number theory since the demise of the jahrbruch iiber die Fortschritte der Mathematik, which is no mean accomplishment. The arithmetical community owes Professor LeVeque· a tremendous debt of gratitude for his dedication in fashioning this important research tool."

From Berry G. Richards, MART Science and Engineering Library, Lehigh University " ... These well-assembled review volumes represent major time-saving reference tools for students, researchers, librarians. Strongly recommended for every good mathematics collection."

212 REVIEWS IN NUMBER THEORY edited by William J. LeVeque

A total of 14,426 reviews in all; 3,376 pages. All reviews of number theoretic interest that appeared in Volumes 1-44 (1940-1962) of MATHEMATICAL REVIEWS, as well as reviews of papers of an arithmetical nature.

VOLUME I -CHAPTER A. Congruences; arithmetic functions; primes, factorization, continued fractions and other expansions. B. Sequences and sets. C. Polynomials and matrices VOLUME 2 -CHAPTER D. Diophantine equations E. Forms and linear algebraic groups F. Discontinuous groups and automorphic forms G. Diophantine geometry VOLUME 3-CHAPTER H. Geometry of numbers j. Diophantine approximation K. Distribution modulo 1; metric theory of algorithms VOLUME 4- CHAPTER L. Exponential and character sums M. Zeta functions and L-functions; analysis related to multiplicative and additive number theory N. Multiplicative number theory P. Additive number theory; lattice point problems Q. Miscellaneous arithmetic-analytic questions VOLUME S- CHAPTER R. Algebraic number theory: global fields S. Algebraic number theory: local and p-adic fields T. Finite fields and finite commutative rings U. Connections with logic VOLUME 6-CHAPTER Z. General TO THE READER Subject Index Author Index Institutional Individual List Members Members Student Code Volume 1 $50 $35 $20 $10 REVNUM/1 Volume 2 so 35 20 10 REVNUM/2 Volume 3 40 28 16 8 REVNUM/3 Volume 4 so 35 20 10 REVNUM/4 Volume 5 40 28 16 8 REVNUM/5 Volume 6 40 28 16 8 REVNUM/6 Complete set Volumes 1-6 $190 $133 $76 $38 REVNUM Orders must be prepaid. Please use codes when ordering.

AMERICAN MATHEMATICAL SOCIETY P. 0. BOX 1571, ANNEX STATION PROVIDENCE, RHODE ISLAND, 02901

213 LETTERS TO THE EDITOR

Editor, the c}/oti.cri) Editor, the c}/oticti) Soviet mathematician Andrei L. Rukhin [Ru­ There seems to be a recent fashion of jocu­ hin] was born in 1946; he is married and has one larly referring to Zorn's Lemma and saying "Of child. He was employed at the Mathematics Insti­ course, it wasn't invented by Zorn... " with no tute of the Academy of Sciences of the USSR until further reference. The principle appears (in November 1974, when at that time, he applied for rather quaint and ancient language) in the article permission to emigrate from the Soviet Union to of M. Zorn, "A Remark on Method in Transfinite Israel. His application was refused. Since then, he Algebra", Bull. Amer. Math. Soc. 41 (1935), has been without a job. Mr. Rukhin has worked 667-670, where the reference is not to a "generally with ProfessorYu. Linnik; many of his works have partially-ordered system" but to a system of sub­ been published, mostly on Probability Theory and sets of a given set "closed under the union-opera­ related subjects. Because of his knowledge of tion". English, he has translated many Russian works in­ If there are earlier papers in which the to the Western languages. Any American mathe­ principle is so clearly enunciated, I would appre­ matician planning to visit Leningrad is invited to ciate hearing about them. If we are going to contact him at Opotchinina 15 a. 72, Leningrad "make jokes in our lectures" about this, then 199106. His telephone number is 17-23-27. there had better be some scholarly backup. Ignacy I. Kotlarski George J. Minty Editor, the c}/otirti) Editor, the c}/otirti) I would like to inform all colleagues that J. Denes, of the Institution of Coordination Leonid Plyushch, a Kiev computer mathematician and Computing Techniques, Budapest, Hungary, and scientist who had been confined to the Dnepro­ and I are writing a research monograph entitled petrovsk Psychiatric Prison, has been released "Combinatorial Properties of Binary Relations". and permitted to leave the USSR together with his We would appreciate being informed of any recent wife and children. He is now in Paris. work ·on binary semigroups, digraphs, distribu­ This was due to all the colleagues who sup­ tive lattices,and Boolean matrices. My ported address the campaign of the International Defense is Box 69, Alabama State University, Montgomery, Committee of Mathematicians. I believe that the Alabama 36101. actions of our French colleagues, in particular, of our Paris Secretary, Professor Michel Broue Kim Ki-Hang Butler were particularly effective. The large meeting Editor, the c}/otirti) and mutualte had important political repercussions. In particular, after that meeting the French com­ The ever increasing proliferation of pub­ munists joined the campaign for the liberation of lished literature suggests a serious responsibility Plyushch. for referees. They should try to prevent the pub­ lication Lipman Bers of false, old, or trivial results which would waste the time of lmndreds of future readers. Editor, the c}/otiu.i) And they should try to improve the quality of those I was distressed to read of the difficulties of papers which they do recommend to occupy space our colleague, Professor Laurent Schwartz, in in thousands of libraries around the world for pos­ coming to and through the United States, and terity. pleased to learn that the Society had succeeded in That is a lot to ask of people who get no re­ obtaining a measure of redress in his case. No ward or recognition for such work, and who would doubt there are other colleagues, possibly some often prefer to spend the time on their own re­ of them not so eminent as Professor Schwartz, search. who also suffer the same exclusion, and for whom Here is a suggestion: relief might also be sought. I think the Society When a paper is finally accepted and pub­ should express a willingness to seek relief for all lished, let the name(s) of the referee(s) appear of these, and should solicit information about any somewhere on the paper---perhaps at the end in such cases as are still being barred from the small print. If a paper is rejected, let the ref­ United States for holding inconvenient opinions. eree(s) remain anonymous as is the present cus­ It is all well and good to reproach the Soviets tom. and the Chilean junta about a respect for scientific Such a policy should improve the quality of and intellectual freedom, but we should not forget refereeing. For on the one hand, the referee will that in our own country, in this bicentennial year, know, as he or she reads a paper, that if it is pub­ the cause of liberty is in less than perfect repair. lished and turns out to be junk, then the referee Let us take on, as our project in honor of our will share the blame. On the other hand, a referee, Nation's bir.thday, the ending of political restric­ finding a paper worthy of publication, often sug­ tions on freedom of movement of scientists in our gests improvements. If the author then chooses to own land. acknowledge "suggestions of the referee", credit will go where it belongs. Anatole Beck Another potential benefit from identifying

214 the referee on a published paper is that it would would even condescend to think about computational then be easy, if desired, to call upon the referee methods. to also review the same paper for Math. Reviews. Later, as supervisor of a large number of Note that this proposal can be considered excellent engineers and scientist-turned-engineers quite independently of the question of blind ref­ who were engaged in the development and analysis ereeing. of guidance and navigation techniques on the Apollo and post-Apollo programs, I had the opportunity Rodney D. Driver to participate in this great research and develop­ ment adventure. We even called on denizens of Editor, the ciloticeiJ that most isolated of academic ivory towers, the I should like to register an objection to the university astronomy department. In their youth Society's total neglect of the bicentennial. It they gladly forsook the pragmatic professions for seems to me that an event of major impor­ a life filled with research adventures on the most tance is being neglected. Not one special lec­ lofty plane. None of them dreamed, while studying ture, nor one special seminar celebrates this F. R. Moulton's 1902 text, that their knowledge important anniversary. In 1776, on 17th of Octo­ of orbital mechanics would some day become a ber, L. Euler presented a paper to the St. Peters­ critical engineering resource for the creation and burg Academy in which the Euler conjecture on management of artificial celestial bodies. Nor did Latin Squares was first presented. The final they imagine, unless they had secret science-fic­ settling of this conjecture was not achieved until tion daydreams, that this celestial mechanics 1960. Surely a problem which remained unsolved tome contained many hints of how one could ex­ for 184 years deserves recognition on its bicenten­ pand old, or develop new, orbital and trajectory nial year. computational methods for use with radar systems, optical systems and their associated spaceborne E. Mendelsohn and ground-based computer processing systems. The excitement of learning from COMMENT astronomers from J. J. Kohn, Chairman of the past and present; solving new problems; develop­ AMS Bicentennial Program Committee: ing new methods; arguing endlessly and vigorously "The Bicentennial Program Committee of the over accuracy, reliability, and probability of AMS was asked to find an appropriate way to com­ success relative to astronaut survival; critiquing memorate the Bicentennial of America's indepen­ and revising design documents and finally seeing dence. We are planning an exchange of speakers our ideas "launched" with a thundering roar and with the London Mathematical Society. Perhaps under the watchful eyes of hundreds of millions another committee ought to consider the matter of around the world will never be repeated. celebrating the bicentennial of Euler's problem." Hopefully before memories fade and signif­ icant documentation is lost, this significant phase of engineering-mathematical history will be proper­ Editor, the ciloticeiJ ly chronicled by someone possessing the appropriate depth of understanding, requisite wit, and literary The articles and letters regarding the em­ descriptive talents. There is nothing comparable ployment situation and lack of training in applied on the horizon for the open-minded mathematician. mathematics seem somewhat strange to me be­ Applications of mathematics and computers to the cause they should have been written years ago and major problems of clinical medicine could become appear to arise solely due to the economic situa­ the next great adventure. Curiously, although the tion. They remind me that perhaps the greatest needs are more immediate and vital(!), this field adventure and challenge for mathematicians of all has not excited the engineering-Scientific com­ time has already passed unnoticed by the mathe­ munity, the executive-legislative branches of matical community. While you ponder what that government, nor the general public sufficiently to was, let me provide a hint in the following incident. bring about appropriate long-term efforts. From In 1962 I was advised by my superior, a rather my experience on small-scale research efforts in good applied mathematician, to turn down an invi­ one such area, that of accurate analysis of elec­ tation to work in the guidance and navigation field trocardiograms, I know that such problems could for the prime Apollo Program contractor. He sug­ be quickly solved by engineering-scientific teams gested it would be far better for me to return to one-tenth the size of the Apollo community. school and complete the Ph. D. requirements. "Af­ Unless mathematicians interact with those ter all," he stated, academically, "what can you in computer science, electrical engineering, and do on this program, since there can be nothing new physics as R. Hermann suggests in the August in the guidance and navigation field because Gauss issue of these cNoticeiJ there will be no realistic has done it all. " (!) Due to my high regard for him, approach to participation in applied mathematics although this statement was patently false, I re­ in times of economic recession or recovery. Per­ plied with silence, respectfully. The fact, for haps even now it is too late, because the engineer­ example, that Gauss employed a slate for his nu­ ing community has taken the "high ground" of merical experiments instead of that fantastic genie applied mathematics and has established sover­ of our day, the large-scale computer, and could eignty over it. Mathematicians may find them re­ not consider methods that employed range and luctant to issue entrance visas. range-rate measurements had somehow escaped this man and certainly other mathematicians who H. Stephen E. Schloss

215 Editor, the cf/oticeU Editor, the cf/oticeU At the Council Meeting in San Antonio, the Igor Shafarevich (Safarevic), one of the following motion was introduced and passed: world's most respected mathematicians, has been barred from lecturing at Moscow University. No The Society should urge research oriented de­ official reason was given, but it is known that partments to offer unemployed mathematicians, Shafarevich belongs to Sakharov• s Human Rights and mathematicians in jobs with no research Committee and is the author of several essays facilities, opportunities to remain associated deviating from official Soviet ideology. We re­ with the research community. This can be gret this violation of academic freedom done by offering such mathematicians which nominal, deprives Moscow University students of a great unpaid research appointments which would teacher and makes their school a less illustrious provide office space (when feasible) and access center of learning. to the library as well as an opportunity to participate in seminars and, in general inter­ Lars V. Ahlfors, Harvard act with members of the department. Michael Artin, MIT Michael Atiyah, I appeal to all colleagues responsible for re­ Oxford Lipman Bers, Columbia search oriented departments to try to implement Armand Borel, lAS this recommendation. I am aware that in many Raoul Bott, Harvard cases this is already happening and that many re­ Richard Brauer, Harvard search departments are "second home" for mathe­ Heisuke Hironaka, Harvard maticians who have no "first home" or whose Nathan Jacobson, Yale "first home" has no research facilities. Still, it Deane Montgomery, may be desirable to make such informal arrange­ lAS Marston Morse, lAS ments more formal, and to indicate in some offi­ cial way how welcome such colleagues are, es­ G. D. Mostow, Yale I. M. Singer, MIT pecially since in many cases they contribute signi­ ficantly to the mathematical life of a department. Shlomo Sternberg, Harvard John Tate, Harvard Please give serious thought to this matter and if Andre Weil, IAS you feel like it, share your experiences with the Hassler Whitney, officers of the Society. lAS Oscar Zariski, Harvard Lipman Bers

NEWS ITEMS AND ANNOUNCEMENTS

AMS RESEARCH FELLOWSHIPS AWARDED sity; Joan Kathryn Plastiras, Acting Assistant Professor, University of California, San Diego; Recipients of the AMS Research Postdoctor­ and Paul C. Yang, Evans Instructor, Rice Uni­ al Fellowships for 1976-1977 have been announced versity. by the Selection Committee on Postdoctoral Fel­ The AMS Research Fellowship Fund was lowships. There are two award winners this year: established three years ago because of the scar­ Fredric D. Ancel, a Graduate Student at the Uni­ city of funds for postdoctoral fellowships. The versity of Wisconsin, and Joseph A. Sgro, Gibbs fellowships are awarded to individuals who have Research Instructor at Yale University. In the recently received the Ph.D. degree, who show course of making the awards, one was offered to unusual promise in mathematical research, and Eloise H. Carlton, Instructor of the University of who are citizens or permanent residents of a California, Berkeley, but the award was declined. country in North America. Serving on the Com­ Each of the fellowships carries a stipend of mittee this year were: Leonard Gillman, Daniel $10,000 with an additional $500 allowance for tra­ Gorenstein, Peter J. Hilton, Mark Kac, Alice T. vel expenses. Schafer, and William P. Thurston. An Honorable Mention has been awarded by The continuation of the Research Fellowship the Committee to the following applicants: William Program depends on the contributions the Society I. Bertiger, Student, University of California, receives. It is hoped that every member of the Berkeley; Sy D. Friedman, Student, M. I. T.; Society will contribute to the Fund. Contributions Marjorie G. Hahn, Lecturer, University of Cali­ are, of course, tax deductible. Checks should fornia, Berkeley; David L. Johnson, Teaching be made payable to the American Mathematical Associate, University of Minnesota; Kent W. John­ Society, clearly marked "AMS Research Fellow­ son, Teaching Assistant, Brown University; Orin ship Fund" and sent to the American Mathematical M. Linden, Student, Yeshiva University; Larry Society, P. 0. Box 1571, Annex Station, Provi­ M. Manevitz, Postdoctoral Fellow, Yale Univer- dence, Rhode Island 02901.

216 MATHEMATICIANS NAMED TO POLISH ACADEMY OF SCIENCES BULLETIN NATIONAL ACADEMY OF SCIENCES The Bulletin of the Section of Logic of the The National Academy of Sciences (NAS) has Institute of Philosophy and Sociology of the Polish announced the election of the following mathemati­ Academy of Sciences is now accepting short pa­ cians to its membership: Theodore W. Anderson pers for publication in addition to abstracts. The (Stanford University), Raj Chandra Bose (Colorado purpose of the Bulletin, published quarterly, is State University), Wassily Hoeffding (University the publication of short papers and abstracts of North Carolina, Chapel Hill), George Polya dealing with sentential calculi, their methodology, (Stanford University), Julia Robinson (University and algebraic interpretation. The inclusion of of California, Berkeley), Jacob T. Schwartz short papers began with Volume 5, Number 1 (Courant Institute of Mathematics), and Richard G. (March, 1976). Swan (University of Chicago). In addition the NAS has elected several dis­ POSTER SESSIONS of the tinguished scientists as foreign associates Poster sessions have become a popular and Academy, among them Lars V. Htlrmander (Uni­ effective means of relating information at mathe­ versity of Lund) and Sir James Lighthill (Univer­ matical meetings. As a result, they will be of­ sity of Cambridge). fered at annual and summer meetings held by the NATIONAL ACADEMY OF ENGINEERING American 'Mathematical Society. ELECTS NEW MEMBERS The primary purpose of a poster session is to disseminate information and introduce new Among the new members recently elected to ideas to as many people as possible. Poster ses­ the National Academy of Engineering are Julian D. sions offer an unlimited amount of discussion Cole (University of California, Los Angeles), time to authors, i. e. , they are not restricted to Ruth M.Davis (U.S. Department of Commerce), a ten- or twenty-minute talk, but are, instead, Solomon W. Golomb (University of Southern Cali­ limited only by the length of the poster session fornia, Los Angeles), Eric Reissner (University (usually of two-hour duration); consequently the of California, San Diego), and J. Ernest Wilkins, sessions induce more informal and in-depth dis­ Jr. (Howard University). cussion. In addition, ideas exchanged by author and viewer may stimulate further mathematical VON NEUMANN THEORY PRIZE investigations and discoveries. The Institute of Management Sciences and An important feature of displays at poster the Operations Research Society of America have sessions is the ease with which viewers choose jointly awarded the second annual John von Neu­ subject matter of interest to them. The basic for­ mann Theory Prize to Richard E. Bellman, of mat of a poster display is a 28" X 44" poster set the University of Southern California. The prize on a tripod stand. Displays, however, are open includes $1,000 from each of the sponsors, and to a great deal of flexibility-authors may use be was awarded in recognition of Dr. Bellman's more than one poster if necessary and will pioneering development of dynamic programming assisted in obtaining any special equipment and his contributions to control theory. needed for their displays. A sign with the author's name and title of paper will be supplied at each display; posters will be arranged in the same or­ SCIENTISTS AWARDED NSF ENERGY-RELATED der as they appear on the program. Authors POSTDOCTORAL FELLOWSHIPS participating in a poster session should be pres­ Four mathematicians were among 118 scien­ ent at their displays for explanations and discus­ tists awarded National Science Foundation Post­ sion. doctoral Energy-Related Fellowships. The awards, Deadlines and requirements for poster ses­ made to United States citizens or nationals who sions are the same as for ten-minute papers. A have received the doctorate degree, were given standard abstract form should be completed. The strictly on the basis of merit. The awards are de­ form below should be attached to an abstract signed to aid talented scientists capable of con­ form unless the form includes a poster session ducting studies of critical scientific and techno­ information request. logical problems related to the uses of energy. Fellowship recipients are provided a stipend of American Mathematical Society $12, 000 per year for full-time study or research POSTER SESSION ' with a tenure period of six months to one year. (attach to a standard abstract form) The four mathematicians, their current af­ filiations and their fellowship institutions are: Name: Richard E. Ewing, Oakland University (University of Chicago); Thomas J. Mahar, Michael Williams, Meeting: January___ _ August:,__ __ and Barry E. Willner, New York University (New York University). Number of poster tripods required: NEW RESEARCH CENTER FOR Special equipment needed: APPLIED MATHEMATICS The University of Georgia has formed a Check one: Center for Applied Mathematics. The Center is [ 1 Poster Session only engaged in research in the modeling of dynamical systems involving nonlinear and stochastic be­ [ 1 Standard presentation as a ten-minute havior in applications to physics, biology, eco­ paper is acceptable if poster session nomics, engineering, ecology, etc. not available

217 JOHN SIMON GUGGENHEIM RECENT AMS APPOINTMENTS FELLOWSHIP AWARDS John W. Jewett has been appointed to the Thirteen individuals in various mathematical Committee on Academic Freedom, Tenure, and fields have been awarded John Simon Guggenheim Employment Security. Continuing members are fellowships. Three hundred scholars, scientists, Murray Gerstenhaber, Paul J. Sally, Jr. , and were selected from 2, 953 applicants. and artists Paul Mostert. The thirteen recipients and their proposed studies are: William B. Arveson, professor of Burton Rodin has been appointed to the Com­ mathematics, University of California, Berkeley: mittee to Select Hour Speakers for Far Western Studies in operator theory; Raoul Bott, professor Sectional Meetings. Continuing members are of mathematics, Harvard University: Studies in Kenneth A. Ross, chairman, and Shoshichi Kobo­ geometry and topology; Wendell H. Fleming, pro­ yashi. fessor of mathematics and applied mathematics, NEW AMS COMMITTEES Brown University: Studies in applied probability; Robert W. Floyd, professor of computer science, Committee on the Future of the Steele Prize. Stanford University: Architectural and design is­ President Lipman Bers has appointed the follow­ sues in computer programming; H. Jerome Keis­ ing members: Walter Feit, Phillip A. Griffiths, ler, professor of mathematics, University of Paul R. Halmos, John W. Milnor, and James B. Wisconsin, Madison: Studies in mathematical Serrin (chairman). logic; Joseph J. Kohn, professor of mathematics, Committee to Write Rules for the Operation Princeton University: Theoretical studies in of CAFTES. CAFTES is the Ad Hoc Committee mathematical analysis; Alan C. Newell, profes­ on Academic Freedom, Tenure, and Employment Tech­ sor of mathematics, Clarkson College of Security. Members appointed by President Lip­ mathematics; Allen Newell, pro­ nology: Applied man Bers to the new Committee to Write Rules fessor of computer science, Carnegie-Mellon are: Charles W. Curtis, Ronald G. Douglas, A systematic treatment of the field of University: Murray Gerstenhaber, Edwin E. Mo'ise, M. Su­ profes­ artificial intelligence; Robert Osserman, san Montgomery, Karl K. Norton, and Charles Studies sor of mathematics, Stanford University: E. Rickart (chairman). in differential geometry and complex analysis; Hugo Sonnenschein, professor of economics, NATIONAL SCIENCE FOUNDATION Northwestern University: Studies in the theory of GRADUATE FELLOWSHIPS monopolistic competition; Elias M. Stein, pro­ fessor of mathematics, Princeton University: The National Science Foundation has awarded Studies in mathematical analysis; Frank W. War­ 550 Graduate Fellowships to students of outstand­ ner III, professor of mathematics, University of ing ability in the sciences, mathematics, and Pennsylvania: Studies in differential geometry; engineering. Fifty-six awards were made to stu­ and Geoffrey S. Watson, professor of statistics, dents in various mathematical fields. Princeton University: Statistical studies in geo­ More than 5, 330 students competed for the physics. fellowships, which were awarded on the basis of merit. Panels of scientists, appointed by the National Research Council, evaluated applica­ MEETING HONORING W. V. D. HODGE tions; final selections were made by NSF. In ad­ The London Mathematical Society will hold dition to the fellowship awards, NSF accorded a meeting in Cambridge, England, in honored honorable mention to 1, 980 applicants. memory of Sir W. V.D. Hodge. President Lipman Each fellowship, awarded for three years Bers has appointed Phillip A. Griffiths of the of graduate study, carries a stipend of $3, 900 University of California, Berkeley, to be the per year for full-time study. Graduate Fellows delegate of the AMS to this meeting on June 18, may attend any appropriate nonprofit United 1976. States or foreign institution of higher education.

218 SPECIAL MEETINGS INFORMATION CENTER

THIS CENTER maintains a file on prospective symposia, colloquia, institutes, seminars, special years, and meetings of other associations, helping the organizers become aware of possible conflicts in subject matter, dates, or geographical area. AN ANNOUNCEMENT will be published in these c}/olictiJ if it contains a call for papers, place, date, subject (when ap­ plicable), and speakers; a second full announcement will be published only if there are changes or necessary additional information. Once an announcement has appeared, the event will be briefly noted in each issue until it has been held and a reference will be given in parentheses to the volume and page of the issue in which the complete information appeared. IN GENERAL, SMIC announcements of meetings held in the United States and Canada carry only date, title of meeting, place of meeting, speakers (or sometimes general statement on the program), deadline dates for abstracts or contributed papers, and name of person to write for further information. Meetings held outside the North American area may carry slightly more detailed information. Information on the pre-preliminary planning will be stored in the files, and will be available to anyone desiring information on prospective conferences. All communications on special meetings should be sent to the Special Meetings Information Center of the American Mathematical Society. DEADLINES are the same as the deadlines for abstracts. They appear on the inside front cover of each issue. January !-December 16, 1976. Mathematisches For­ Information: Seminaire de mathematiques superieures, schungs Jnstitut Oberwolfach, Federal Republic of Ger­ Departement de Mathematiques, Universite de Montreal, many (Weekly Conferences) (22, p. 295) Case Postale 6128, Montreal, Quebec, Canada H3C 3J7. 1976-1977. Arne Beurling Year, Jnstiint Mittag-Leffler, 7-July 9. GROUP THEORETICAL METHODS IN PHYSICS, Sweden (23, p. 82) Montreal, Quebec, Canada. Euromech Kolloquia 1976. Fifteen colloquia in Europe on Program: Analyse harmonique sur les groupes, theorie Applied Mathematics and Mechanics (23, p. 119) des representations, groupes de Lie, sous-groupes conti­ nua et discrete' applications a la physique des particules May 17-August 13. WORKSHOPS OF THE CANADIAN elementaires' nucleaire' atomique, de 11 etat solide et a la MATHEMATICAL CONGRESS-SUMMER RESEARCH IN­ relativite generale; symetries et super-symetries. STITUTE, Dalhousie University, Halifax and University of Information: R. T. Sharpe, C.R.M., University, P.O. Victoria, Victoria, Canada. Box 6128, Montreal, H3C 3J7, Quebec, Canada. Program: There will be workshops on mathematics in 18. SIR W. V.D. HODGE MEMORIAL MEETING, Cam­ medicai sciences (Halifax, June 21-25) with speakers E. J. bridge, England. Davidson, A. R. Feinstein, W. Forbes, G. Hill, L. Jn­ Speakers: M. F. Atiyah (Oxford), P. A. Griffiths (Har­ drenyi, G. Karreman, V. LiCko, and R. Rosen; automata vard), and W. Schmid (Columbia). theory and applications (Halifax, June 14-July 9) with Information: J. L. Britton, Queen Elizabeth College, speakers J. E. Hopcroft, J. Lipson, Z. Manna, and J. D. London, W8 7AH, England. Ullman; associative rings and algebras (Halifax, June 28- July 9) with speakers M. Artin, V. Dlab, A. Jategaonkar, 24-July 2. HYPERBOLICITY, Cortona, Italy. J. Lambek, s. Montgomery, and L. Small; mathematical Sponsor: CIME, Ministero della Pubblica Istruzione, CNR ecology (Victoria, June 7-11 and July 12-16) with speakers (Italy). K. Cooke, V. Gallucci, D. Jones, N. Kazarinoff, D. Lud­ Program: This course will be dedicated to the study of hy­ wig, G. Oster, R. Peterman, and C. Walters; combina­ perbolic partial differential equations, both classical and torics and applications (Victoria, May 25-June 4 and abstract using the modern techniques of functional analysis August 9-13) with speakers A. Hoffman and V. Kl.ee. and also numerical methods, in particular, for the hyper­ Information: s. Swaminathan, Dalhousie University, Hali­ bolic equations of mathematical physics. fax, Nova Scotia, Canada, or 0. P. Noble, University of Information: Antonio Mora, CIME, Istituto Matematico Victoria, Victoria, British Columbia, Canada, "Ulisse Din!", Viale Morgagni 67/A, 50134 Firenze, Italy. JUNE 26-30. SUMMER CONFERENCE ON OPERATIONS RE­ SEARCH, Northern Michigan University, Marquette, 7-July 2. S~MINAIRE DE MATH:tMATIQUES SUP:tRIEU­ Michigan RES, Universite de Montreal, Montreal, Quebec, Canada. Program: The purpose of this conference is to introduce Program: This international seminar is meant particularly mathematicians and other interested parties to certain as­ for mathematicians, theoretical physicists and students pects of the problems and methodology of operations re­ engaged in study and research at the predoctoral or post­ search. A series of ten expository lectures will be given. doctoral level. This year, the seminaire will be held jointly Except for the speakers, no support money is available. with the Summer School of the Theoretical Physics Division Speakers: Erhan Cinlar, Industrial Engineering Department, of the Canadian Association of Physicists. Northwestern University "Optimal stopping and decisions ~: Henri-Fran9ois Gautrin, Universite de Montreal: over Marcov chains"; and Allan Hoffman, IBM Research Etats coherents et representations principales; Sigurdur Hel­ Center "Applications of linear programming to combina­ gason, M. I. T. : Topics in the theory of invariant differential torics". equations; Louis Michel, Jnstiint des Hautes :tindes Scienti­ Contributed Papers: Papers in any area of mathematics fiques: Les symetries brisees; Marcos Moshinsky, Univer­ and its applications may be submitted for inclusion in the sity of Mexico: Groups of canonical transformations and program. their representations in quantum mechanics; Jin Patera, Sponsor: Northern Michigan University and the Michigan Universite de Montreal: Semisimple subgroups of semi­ Section of the MAA. simple Lie groups: general theorems, practical methods, Information: Robert B. McNeill, Department of Mathe­ and results; Abraham Pais, Rockefeller University: (Title matics, Northern Michigan University, Marquette, Michi­ to be announced); Robert T. Sharp, McGill University: gan 49855. Methods of labeling internal states; Pavel Winternitz, Uni­ versite de Montreal: Conformal group of space-time, its 28-July 2. REGIONAL CONFERENCE ON MATHEMATICS subgroups, their invariants, and applications; Hans Zas­ OF OPTII\IIAL FACILITY LOCATION, St. Olaf College, senhaus, Ohio State University: Lie algebras and their Northfield, Minnesota. representations. Program: Alan J. Goldman (National Bureau of Standards) Sponsor: Ministry of Education of Quebec, the National Re­ will deliver a series of ten lectures on the mathematical search Council of Canada, the North Atlantic Treaty Or­ aspect and applications of location theory (in operations ganization, the Institute of Particle Physics of Canada, the research). There will also be informal discussions and Atomic Energy of Canada Limited and the Universite de sessions for a limited number of contributed papers. Montreal. SUpport: (Anticipated) National Science Foundation and

219 Conference Board of the Mathematical Sciences; travel and Additional Lecturers: In addition tbe following speakers subsistence allowance for twenty-five invited participants. will give one-hour lectures: T. Fiegiel, Polish Academy For possible financial support, please provide curriculum of Sciences; J. Garnett, University of California, Los vitae. Preference will be given to applicants interested in Angeles; W. B. Johnson, Ohio state; J. Lindenstrauss, presenting contributed papers. Hebrew University, Jerusalem; R. Rochberg, Washington Information: Stuart Th-ching Hsu, Department of Mathe­ University; W. Rudin, University of Wisconsin; A. Shields, matics, St. Olaf College, Northfield, Minnesota 55057. University of Michigan; E. L. Stout, University of Wash­ ington; and L. Tzafriri, Hebrew University, Jerusalem. JULY SUpport: (Tentative). National Science Foundation and 2-12. LMS Durham: Symposium on Potential Theory and Conference Board of the Mathematical Sciences. Conformal Mapping, England (22, p. 368) Information: Joe Diestel, Jolmnie Baker or Charles Clea­ ver, Department of Matbematics, Kent State University, COLLOQUIDM ON MODULAR 2-14. INTERNATIONAL Kent, Ohio 44242. FORMS, Bonn, Germany. Information: D. Zagier, Mathematisches Institute der 12-23. LMS Durham Symposium on Partial Differential Universit!tt Bonn, Wegelerstr. 10, 5300 Bonn, Germany. Equations, England (22, p. 368) OF ALGEBRA, Instituto de Matematica 4-17. HIGHER COMBINATORICS, Berlin, Germany. 12-31. IV SCHOOL de Sao Paulo, Sao Paulo, Program: A survey of some recent progress in combinator­ e Estatfstica da Universidade ial tbeory and applications. Sections: counting tbeory, com­ Brazil. is to stimulate re­ binatorial group theory, combinatorial geometries and Program: The goal of tbe conference There will be survey lectures order tbeory, designs and configurations, coding theory. search in algebra in Brazil. theory, cohomology of finite Information: M. Aigner, II. Math. Inst. Freie Universit!tt, on integral representation in­ Ktlnigin-Luise-Str. 25, D-1 Berlin 33, Germany. groups and trends in representation tbeory. In addition structional courses will be held on group rings, topics on 5-9. REGIONAL CONFERENCE ON COMPLEX MANI­ finite groups and quadratic forms. There will be con­ FOLD TECHNIQUES IN RELATIVITY, The University of ferences and sessions for short announcements. Pittsburgh, Pittsburgh, Pennsylvania. Principal Lecturers: Vladistmil Dlab (Carleton University), Program: Two lectures by the main speaker each day. Irving Reiner (University of lllinois), and Klaus Roggen­ SUpplementary one-hour lectures each day by invited kamp (Universitut Stuttgart). speakers. Contributed Papers: Accepted papers and research an­ Lecturer: Roger Penrose, Oxford University. nouncements will be published in tbe school's proceedings. SUpport: Anticipated from the National Science Foundation Contributed papers should be sent before June 13, 1976. and tbe Conference Board of tbe Matbematical Sciences, Information: Cesar Polcino Milies, Instituto de Mate­ Travel and subsistence allowed for twenty-five partici­ matica e Estatfstica, Universidade de Sao Paul, Caixa pants. Postal 20.570 (Agencia Iguatemi), Sao Paulo, Brazil. Information: Alan Thompson or John Porter, Department of Mathematics and Statistics, University of Pittsburgh, 12-31. NEW DEVELOPMENTS IN QUANTUM FIELD Pittsburgh, Pennsylvania 15260. THEORY AND STATISTICAL MECHANICS, Institut d1etudes de Cargese (Corsica), France. Functional Differential Scientifiques 5-15. SUmmer School on Nonlinear Program: Interdisciplinary study of recent results and ex­ (23, p. 180) and Volterra Integral Equations, Belgium ploration of future possibilities in phase transitions and 9-13. PROBLBMES COMBINATOIRES ET TH£0RIE DES infra-red behaviour, short-distance behaviour and non­ GRAPHES, Universite Paris-SUd, Orsay, France. linear solitary waves in field tbeory and statistical mechan­ Sponsor: Centre National de la Recherche Scientifique, ics .. Societe Mathematique de France. Information: M. Levy, Lab. de Physique Theorique, Univ. Information: Jean-Claude Bermond, C. M.S., 54 boule­ Pierre et Marie Curie, Tour 16, 1er Etage, 4, Place vard Raspail, 75006 Paris, France. Jussieu, 75230 Paris Cedex 05, France. 11-17. THIRD LATIN AMERICAN SYMPOSIDM ON MATHE­ 14-18. THIRD MEE'l'ING OF SEAMS, "THE ROLE OF MATICAL LOGIC, University of Campinas, Brazil (23, MATHEMATICS IN PLANNING AND DEVELOPMENT", p. 123) Institut Teknologi Bandung, Indonesia. Program: Sessions and invited speakers are as follows: Sponsor: Southeast Asian Mathematical Society, Institut Model tbeory: J. Shoenfield, c. Pinter, R. Fra"isse, Teknologi Bandnng, Indonesia. M. Guillaume, R. Chuaqui; Nonclassical logics: R. Rout­ Information: Drs. Rawnh, Departemen Matematika, In­ ley, J. Kotas, F. M. Quesada, N.C.A. daCosta, andA.I. stitut Teknologi Bandung, Jalan Gauesha 10, Bandung, Arruda; Applied logic: R. M. Solovay and F. G, Asenjo. Indonesia. The symposium is a meeting of tbe Association for Sym­ Colloquium, Scotland (22, p. 297) bolic Logic. 14-24. St. Andrews Participants: Autbors are invited to send contributed papers 19-23. Nonlinear Systems and Applications to Life Sci­ of five to thirty pages in length (typewritten double-spaced) ences, Texas (23, p, 84) with abstracts of no more than 300 words. OF Information: Ayda I. Arruda, Instituto de Matematica, 19-30. THE 1976 EUROPEAN SUMMER MEETING Universidade Estadual de Campinas, C.P. 1170, 13100 THE A. S. L. (Logic Colloquium '76), University of Oxford, Campinas , Brazil. England. Program: The following is a schedule of sessions and pro­ 11-31. Course on Banach Algebras and Operator Theory, visional list of invited speakers. There will be no extended England (23, p. 180) lecture courses. History of modern logic (July 19-20): functionals 12-13. OPERATIONAL RESEARCH IN DECISION MAKING, R. 0. Gandy and J. van Heijenoort; Continuous Y. L. Ershov, Singapore. (July 21-22): J. Y. Girard, M. Hyland, (July 23 and 26): Sponsor: Operational Research Society of Singapore, Lee and S. Feferman; Applied model theory North Hol­ Kong Chian Institute of Mathematics and Computer Science, A. Macintyre, G. Sabbagh, and H. Simn1ons; in Logic Singapore National Academy of Science. land Symposium celebrating 25 years of Studies (July 27): J. Barwise, H. J. Keisler, G. Kreisel, and 12-16. FIRST NATIONAL CONGRESS OF MATHEMATI­ P. SUppes; Complexity of computation (July 28-29): M. 0. CIANS IN INDONESIA, Bandung, Indonesia. Rabin and R. Statman; General session (July 30): L. Har­ Information: M. Ansjar, Departamen Matematika, Insti­ rington, T. J. Jech, P. Krauss, A. R. D. Mathias, and tut Teknologi Bandnng, Jalan Ganesha 10, Bandung, In­ J. Truss. donesia. Information: Logic Colloquium '76, Matbematical Insti­ St. Giles, Oxford OX1 3LB, England. 12-16. REGIONAL CONFERENCE ON BANACH SPACES tute, 24-29 OF ANALYTIC FUNCTIONS, Kent State University, Kent, 19-30, NATO Advanced Study Institute on Computer-Based Ohio. Science Instruction, Belgium (23, p. 85) Principal Lecturer: Aleksander Pel'czynski will give on Harmonic Analysis of Functions, ten lectures. 25-31. Conference

220 Measures and Convolution Operators on groups, Poland Support: (Anticipated) National Science Foundation and the (23, p. 85) Conference Board of the Mathematical Sciences. Travel and subsistence allowances for twenty-five participants. 26-August 20, M.A. A. Workshop on Modules in Applied Information: Joseph Gall ian, Department of Mathematics, Mathematics, New York (22, p. 297) University of Minnesota, Duluth, Minnesota 55812. 26-31. FIRST PAN-AFRICAN SYMPOSIUM IN MATHE­ 16-21. Third International Congress on Mathematical Edu­ MATICS, Rabat, Morocco, cation (ICME), Germany (22, 197; 23, p, 180) ~: International Mathematics Union, UNESCO. .:;~~::=':=iti:;;on=:: I. Khalil, Faculty of Sciences of Rabat, 17-20. Third Australian Statistical Conference, Australia Avenue Moulay Cheriff, Rabat, Morocco, (23, p. 180) AUGUST 23-27. FOURTH PRAGUE SYMPOSIUM ON GENERAL TOPOLOGY AND ITS RELATION TO MODERN ANALYSIS 1-21. CONFERENCE ON QUADRATIC FORMS, Depart­ AND ALGEBRA, Prague, Czechoslovakia. ment of Mathematics, Queen's University, Kingston, On­ Sponsor: Czechoslovak Academy of Science, IMU, tario, Canada, Information: Josef Novak, Mathematical Institute, Zi1:na ~: Part I (August 1-14) will be an instructional 25, 115 67 Praba 1, Czechoslovakia. conference, There will be several short courses including "Quadratic forms over fields" taught by T. Y. Lam (Uni­ 23-27. NINTH INTERNATIONAL SYMPOSIUM ON MATHE­ versity of California, Berkeley) and "Symmetric bilinear MATICAL PROGRAMMING, Budapest, Hungary, forms over some commutative rings and algebraic varie­ Sponsor: Bolyai Janos Matematiksi Tarsulat. ties" taught by M. Knebusch (University of Regensburg). Information: c. Szabados, Secretary, Bolyai Janos Mate­ This program will be of interest to advanced graduate stu­ matikai Tarsulat, Pf 240, 1368 Budapest, Hungary. dents and any mathematicians wishing to learn about 23-28. Eighth AICA International Congress on Simulation of quadratic forms. Part II (August 15-21) will be concerned Systems, The Netherlands (22, p. 297) with recent development in quadratic forms. There will be at least ten invited lectures by experts in this field, 24-27. Fifth Australian Conference on Combinatorial ~: National Research Council of Canada. Mathematics, Australia (22, p, 369) contrl.liU.ted Papers: There will be some fifteen-minute 24-27. SITUATIONS PE':DAGOGIQUES, Jambes, Belgium. presentations of appropriate papers. Sponsor: Centre Belge de Pedagogie de la Mathematique. Information and to Contribute a Paper: Grace Orzech, Information: CBPM, Avenue Albert 224, 1180 Bruxelles, Department of Mathematics, Queen's University, Kings­ Belgium. ton, Ontario, Canada. 26-September 1. Second Los Alamos Workshop on Mathe­ 4-11. CONFERENCE ON TRANSFORMATION GROUPS, matics in the Natural Sciences, New Mexico (23, p, 180) University of Newcastle upon Tyne, England (23, p. 123) Information: Czes Kosniowski, School of Mathematics, 27-September 4. Centro Internazionale Matematico University of Newcastle upon Tyne, Newcastle upon Tyne, Estivo 1976. Third Session: Differential Topology, Italy NE1 7RU, England. (23, p, 180) 9-12. First Conference on the European Co-operation in 30-September 4. Fourteenth International Congress of informatics, The Netherlands (23, p. 123) Theoretical and Applied Mechanics, The Netherlands (21, p. 279; 23, 1. 123) 10-12. 1976 ACM Symposium on Symbolic and Algebraic Computation, New York (23, p. 85) SEPTEMBER 11-13. Second Cape Town Symposium on Categorical 1-10. Advanced Study liB titute on Combinatorics, Ger­ Topology, Republic of South Africa (22, p. 368) many (23, p. 181) 11-20, GALTON-WATSON PROCESSES AND RELATED 5-10, Conference on Finite Groups and Geometries, TOPICS, Centre de recherches mathematiques, Univer­ Belgium (23, p. 123) site de Montreal, Quebec, Canada. 6-11. European Meeting of Statisticians, France (23, p. 85) Information: Anatole Joffe, Conference on Galton-Watson Processes and Related Topics, Centre de recherches 6-11. Eighth International Congress on Cybernetics, Bel­ math6matiques, Universite de Montreal, Case Postale gium (22, p, 298; 23, p. 124) 6128, Montreal, Quebec, Canada. 6-17, NATO Advanced Study Institute on Boundary Value 14-20. VACATION SCHOOL IN LOGIC, Victoria Univer­ Problems for Evolution Partial Differential Equations, sity of Wellington, Wellington, New Zealand. Belgium (22, p. 369) Program: Introductory series of five talks each on de­ 20-24. The Second COMPSTAT Symposium on Computa­ grees of unsolvability; modal logic; formal semantics for tional Statistics, Germany (22, p. 298) natural languages; category theory and toposes; compu­ ters and logic; model theory. Reports on current re­ 24-25. Fourth Annual Mathematics and Statistics Con­ search in relevance logic; formal semantics for algorith­ ference, Ohio (23, p. 181) mic languages; constructive versions of set theory. 25, Pi Mu Epsilon Student Conference, Ohio (23, p, 181) Speakers: M. Cresswell, D, Lewis, W, Malcolm, R. Ep­ 26-0ctober 1. CONFERENCE ON FINITE GEOMETRIES stein, R. Goldblatt, and M. Jorgenson. AND GROUPS, Domaine des Masures, Han-sur-Lesse, Contributed Talks: Hour long talks on any area of mathe­ Belgium. matical or philosophical logic are welcome. Deadline is Program: Invited lectures on topics of current research July 1, 1976. of the relation between finite (simple) groups, finite Sponsors: Mathematics and Philosophy Departments of geometries and combinatorics. Victoria University of Wellington, New Zealand Mathe­ Participation: Partjcipation is limited to about 60 attenders. matics Society, Australasian Association of Logic. Some limited financial support will be available. Information: Dick Epstein, Department of Mathematics, Information: J, Van Bugganhaut, CSOO, Vrije Universiteit Victoria University of Wellington, Private Bag, Welling­ Brussel, Pleinlaan 2, B-1050-Brussels, Belgium. ton, New Zealand. OCTOBER 16-20. Conference on Numerical Analysis, Ireland (23, 4-6. SECOND INTERNATIONAL WORKSHOP ON MODEL­ p. 85) LING AND PERFORMANCE EVALUATION OF COMPUTER 16-20. REGIONAL CONFERENCE ON FACTORIZATIONS SYSTEMS, Stresa, Lago Maggiore, Italy. IN LOCAL SUBGROUPS OF FINITE GROUPS, University Program: The emphasis is on recent research results and of Minnesota, Duluth, Minnesota. applications in the area of performance methodology, of Program: George Glauberman (University of Chicago) will mathematical and simulation models and measurements of give ten one-hour lectures on the conference theme. There computer systems including operating systems, architec­ will also be a number of talks by participants. ture, networks and data-bases, The following is a partial

221 list of topics: Modelling methodology; modelling and mea­ of the following areas: geometry and relativity; algebra surement of existing systems, operating systems, com­ and functional analysis; inference and econometrics; ap­ puter networks, architecture and data-bases; measurement plications to space dynamics; applications to environmen­ methods and tools; scheduling; and probabilistic models of tal sciences; and applications to demography. Short com­ computer systems reliability. munications of about fifteen minutes will also be delivered, Contributed Papers: Thirty-minute presentations of each Proceedings of the seminar will be published. accepted paper will be scheduled. Proceedings containing Information: R. s. Mishra, Head of the Department of all papers presented at the workshop will be published. Mathematics and Director of the Seminar, Banaras Hindu Information: H. Fangmeyer, J. Larisse-JRC Euratom­ University, Varanasi 221005, India. CETIS, I-21020 Ispra (Va) Italy, 27-January 1, 1977. ADVANCED MATHEMATICAL 18-20. SIAM 1976 FALL MEETING, Georgia Institute of TECHNIQUES IN PHYSICAL SCIENCES, Centre of Ad­ Technology, Atlanta, Georgia. vanced Study of Physical Sciences, University of Calcutta, Program: There will be four symposia on numerical solu­ Calcutta, India. tion of initial value problems for partial differential equa­ Information: M. Dutta, Centre of Advanced Study in Ap­ tions; deterministic and stochastic operator equations; plied Mathematics, 92, Acharya Prafulla Chandra Road, mathematics and environmental health (SIMS); and Anni­ Calcutta-700009, India. versary Symposium on Applied Mathematics honoring the 30th anniversary of founding of the Office of Naval Re­ search. A special feature will be a panel discussion on computing practice in industry. Participants will be com­ puter management oriented scientists from industry and JANUARY government. Deadline for Contributed Abstracts: July 22, 1976. 4-7. SECOND CARIBBEAN CONFERENCE IN COMBINA­ Information: H. B. Hair, SIAM Headquarters, 33 South TORICS AND COMPUTING, University of the West Indies, 17th Street, Philadelphia, Pennsylvania 19103. Cave Hill, Barbados, West Indies, Sponsor: Departments of Mathematics of the University. 20-22. ACM 1976 Annual Conference, Texas (23, p, 85) Program: The conference is designed to bring together 25-27. Seventeenth Annual IEEE Symposium on Founda­ mathematicians interested in combinatorics, graph theory, tions of Computer Science, Texas (23, p, 181) and computing and will consist of invited lectures and a series of twenty-minute contributed papers. 25-27. SECOND ERDA STATISTICAL SYMPOSIUM, Oak Abstracts: Persons wishing to contribute a paper should Ridge National Laboratory, Oak Ridge, Tennessee, send a title and abstract (10-20 lines) to the address below Program: There will be presentations of problems en­ so as to arrive before August 31, 1976, countered by statisticians in attempting to aid scientists, Information: Charles C. Cadogan, Department of Mathe­ engineers or other investigators who are working on the matics, University of the West Indies, P. 0. Box 64, nation's energy problems. The purpose is to present the Bridgetown, Barbados, West Indies. problems for discussion and input by the symposium attendees for possible solutions or improvement in solu­ APRIL tions. There will also be presentations of statistical re­ 18-23. Uppsala 1977 International Conference on Differen­ search results particularly relevant to solutions to the tial Equations, Sweden (23, p, 181) nation's energy problems. Reports of research motivated by energy problems or clearly applicable to energy prob­ JUNE lems are especially desired. 3-4. GAUSS BICENTENNIAL SYMPOSIUM, Ontario Science Abstracts: Please send four copies of an abstract of the Centre, Toronto, Ontario, Canada. problem br of the technical paper to the address below. Speakers: H. s. M. Coxeter (Toronto)-Geometry; J, A, Abstracts should .not be more than one page in length and Dieudonne (Nice)-Algebra and analysis; E. Forbes (Edin­ should be identified with the author's name, address, and burgh)-Astronomy; G, Garland (Toronto)-Geo-magnetism; telephone number. K. 0. May (Toronto)-Historical introduction; A, Selberg Information: Donald A. Gardiner, Union Carbide Cor­ (Princeton)-Number theory; and D. A, Sprott (Waterloo)­ poration, Nuclear Division, P. 0, Box Y, Oak Ridge, Statistics. Tennessee 37830, Information: G, de B. Robinson, Ontario Science Centre, DECEMBER Toronto, Ontario, Canada, 27-January 1, 1977. INTERNATIONAL DEDICATION AUGUST SEMINAR ON RECENT ADVANCES IN MATHEMATICS International Conference on General Rela­ Banaras Hindu University, 7-13. Eighth AND ITS APPLICATIONS, Canada (23, p, 85) India. tivity and Gravitation, ~: There will be invited one-hour and half-hour 16-27. International Conference on Combinatorial Theory, surveyTectures by distinguished mathematicians in each Australia (23, p. 85)

222 QUERIES Edited by Hans Samelson

QUESTIONS WELCOMED from AMS members regarding mathematical matters such as details of, or references to, vaguely remembered theorems, sources of exposition of folk theorems, or the state of current knowledge concerning published conjectures. REPLIES from readers will be edited, when appropriate, into a composite answer and published in a subsequent col· umn. All answers received will ultimately be forwarded to the questioner.

QUERIES AND RESPONSES should be typewritten if at all possible and sent to Professor Hans Samelson, Amer· ican Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940.

@ QUERIES "addition theorems," whereby f(a +b) is related to f(a), !(b), 94. Jay I. Miller (Department of Mathematics, University of !'(a), and /'(b); see Harris Hancock's Lectures on the theory Wisconsin, Milwaukee, Wisconsin 53201). In a freshman calcu· of elliptic functions (W"dey, 1910; reprinted by Dover, New lus course, one can easily list several functions, e. g., ex 2 , which York, 1958, p. 34, MR 20 #6540). As limiting cases the expo­ do not have elementary antiderivatives. Are there any criteria nential, trigonometric and rational functions are mentioned here. one might apply to an arbitrary function to determine whether Can anyone refer me to some place in the literature where the or not it has an elementary antiderivative? I have seen a book addition theorems for the rational functions are developed in by J. F. Ritt (Integration in finite terms. Liouville's theory of detail? In particular, are /'(a) and /'(b) needed? elementary methods, Columbia Univ. Press, New York, 1948, 100. John L. Leonard (Department of Mathematics, Building MR 9, 573), but it did not help me. 89, University of Arizona, Tuscon, Arizona 85721). Can anyone 95. George M. Bergman (Department of Mathematics, Univer· tell me of a popular, or even semipopular, exposition of the sity of California, Berkeley, California 94720). Let G be a Banach-Tarski paradox, which would be suitable for junior­ group, written multiplicatively, with identity e, and let R be a senior level students? G-graded ring: R = (9 Rg, R~h f. Rgh. In addition to the 0 RESPONSES Jacobson radical J(R), one can defme the "graded Jacobson @ radical of R," a homogeneous ideal J 0 (R) = the intersection The replies below have been received to queries published in of the annihilators of all simple G-graded R-modules = the recent issues of these c/{otictD. The editor would like to largest homogeneous ideal I= (9 Ig such that I. f. J(R.) thank all who have replied. (among other characterizations). I can prove that if G is finite 79. (vol. 23, p. 80. Jan. 1976, Anshel). The latest work I am solvable, J (R) J(R), while if G is torsion-free abelian, 0 f. aware of on the decision problems for finitely generated com­ J (R) ;;:? J(R). Either inequality can be strict. I would be in­ 0 mutative semigroups is: M. A. Tai'clin, On the isomorphism terested to know of any overlapping work, or even any referen­ problem for commutative semigroups (Russian), Mat. Sb. (N. S.) ces where J (R) is considered. 0 93 (135) (1974), 103-128, 152 (MR 48 #8663; English 96. Alexander Abian (Department of Mathematics, Iowa State translation, Math. USSR Sbornik 22 (1974), 104-128). University, Ames, Iowa 50011 ). 1. Let R be a ring in which There are numerous other papers by the same author published every element has a unique cube root. Would R be necessarily in 1962-1973. The isomorphism problem is unsolved as yet. commutative? The word problem is decidable (see: V. A. Emelirev, On algo­ 2. Let n ~ 2 be a fixed natural number. If every ele­ rithmic decidability of certain mass problems in the theory of ment of a ring R has a unique nth root, would R be necessarily commutative semigroups (Russian), Sibirsk. Mat. Z. 4 (1963), commutative? 788-798 (MR 28 #2146); A. I. Mal'cev, On homomorphisms 3. If for every natural number n ~ 2, every element of onto finite groups (Russian), tJrenye Zapiski Ivanovsk. Gos. a ring R has a unique nth root, would R be necessarily com­ Ped. Inst. 18 (1958), 49-60. There are also other papers by mutative? Would R be necessarily a Boolean ring? Emeli~ev published in 1958-1966. (B. M. Schein) 97. T. W. Ma (Department of Mathematics, University of 84. (vol. 23, p. 80, Jan. 1976, Rosinger). Taking f(x) = Western Australia, Nedlands, Western Australia, 6009, Austra­ exp(-21xl-1) • sin(x-1) andg(x) = exp(-lxi-1) (all func­ lia). Letf(x) = :E~=Oakexpiu,.x be a trigonometric polyno­ tions defined as 0 in x = 0), then f, the difference be­ mial of a real variable x where ak are complex and uk are dis­ tween g and h = g - f, is obviously not monotonic near tinct reals. What is the best estimation of the sup-norm 11!11~ x = 0. But all g{P>(x) and h

(x) are of constant sign deter­ in terms of ak and uk? mined by a dominating term of form c • ix 1-m • exp(-lx 1-1) 98. T. W. Ma (Department of Mathematics, University of (coming from g{P>(x)); outside x = 0 they are analytic and Western Australia, Nedlands, Western Australia, 6009, Austra­ hence monotonic near every point. Therefore, the set A in lia). I would like to have either a proof or a counterexample question is not a vector space. (Th. Bang) of each of the following statements. Let H be an equicontinu­ 85. (vol. 23, p. 125, Feb. 1976, Cater). Part (a). Many such ous set of functions from a separated locally convex space E ringS are given in a preprint Artinian non-Noetherian rings by into itself. (a) There exists li > 0 such that U ;=1 (liH)" is L. Levy (available from the author, Van Vleck Hall, Univer­ also equicontinuous. (b) There exists X> 0 such that every sity of Wisconsin, Madison, WI 53706); an example (G. M. element in I lli is a topological isomorphism where I is the + Bergman) is R 0 x R 1 /(e13 -x), where R0 = subring of the identity map on E. 3 x 3 matrix ring over Z/pZ, p prime, spanned by e12, e13, 99. Dennis Weeks (1630 McGee Ave., Berkeley, California e22, e23, R 1 = ZP~ with 0-multiplication,x an element of 94703). Elliptic functions can be studied via their algebraic ZP~ of order p.

223 Part (b). The ideal Fe 11 + Fe 12 in the ring of 2 x 2 at the 82nd Annual Meeting of the Society at San Antonio, matrices over infmite field F. January, 1976. (This list was presented by the organizer of the Part (c). The ideal Rx in the polynomial ring R = F[x] , special session, H. Elton Lacey, Department of Mathematics, infmite F (ifF= Q, e.g., there is the chain (x) c (2- 1x) C University of Texas, Austin, Texas 78712.) (2- 2x) c ·· · of ideals of 1). (G. M. Bergman, L. Levy, J. C. 1. W. J. Davis. Let x = (x , . .. , x n) be in R2n and put Robson) 1 2 Ox= {(±xp(l)' ... , ±x (2 n)) :pis a permutation on 86. (vol. 23, p. 125, Feb. 1976, Wichmann). The answer to {1, ... , 2n}. For which i•s is there an orthonormal basis in Ox? part (a) is "No," as shown in a paper, Function algebras with· [It is known to be true for all x when n .;;;; 3, and not all x for out nontrivial positive linear forms by A. Lenard (Dept. of n = 4.] Math., Indiana University, Bloomington, IN 47401). 2. W. J. Davis. Let a be a tensor norm. What conditions on a Part (b). Let Banach space E be the conjugate Banach pair of Banach spaces E and F and on a imply that E ®"' F is space E = F* of Banach space F. For~ E F and x = (x1, x2, reflexive when E and Fare? Is there an Ci such that when E ... ) E r(E) let ~ • X E zoo denote the scalar sequence ~ • X = and F are uniformly convex Banach spaces (with the m. a. p.) ((~, x 1), (~, x ), .•• ), with properties 2 then E ®"' F is any one of the following: (a) uniformly convex; (i) II~. XII.;;;; ll~llllxll, ~ E F, X E r(E). (b) superreflexive; (c) B·convex? (ii) If x E zoo(E) is such that xn -+ (limx) E E then ~ • x E c C ZOO with lim(~ • x) = (t limx). 3. H. P. Rosenthal. Does every normalized weakly null sequence Let

p·absolutely summing operator from 1 to X p·integral? It is dore H. Sweetser). A similar result holds for V smooth reflex­ 1 known to be true for 1 p 2. ive Banach and W closed, by way of uniqueness of norm-pre­ < < serving extensions in this case. (F. Deutsch) 7. John Whitfield. Let X be a smooth Banach space and S, S* be the unit spheres of X and X* respectively. When is the J 87. (vol. 23, p. 125, Feb. 1976, Raphael). The main counter­ map of S to S* Borel measurable? (Jx = unique support func· example appears to be the bicyclic sernigroup (generated by p, q tiona! at x; Borel sets are w. r. t. the norm topologies.) with relation pq = 1, or also N x N, with (m, n) · (p, q) = (m + p- r, n + q - r), r = min(n, p)) (see A. H. Clifford and 8. R. C. James. Is there a quasi-reflexive space X of order one G. B. Preston, The algebraic theory of semigroups. !, AMS Math. such that X is isomorphic (isometrically isomorphic) to X*? Surveys, No.7. 1961). (H. d'Alarcao, G. M. Bergman, J. W. 9. W. B. Johnson. Does there exist a polyhedral space Eof Crawley, Jr., D. W. Miller, W. W. Williams) Also the finite ex­ dimension n such that P(E) = max{P(F): dim F = n}, when ample ({e, f, x, y, 0} with ee = xy = e, ff = yx = f, ex = xf P(F) is the projection constant ofF? When E is the 2-dirnen· = x, fy = ye = y, all other products equal to 0) (Crawley) and sional space whose unit sphere is the regular hexagon it is a general example (S =semi-direct product G • P with G = per· known that P(E) = 4/3. Is this best possible for 2-dirnensional mutation group and P = power set of a set X of cardinality > 1; spaces? (g, p)(h, q) = (gh, ph n q)) plus modifications thereof. (Bergman) 10. W. B. Johnson. Find a Banach space condition A such that if X is a quotient of LpCp > 2) and X satisfies A, then X is a 88. (vol. 23, p. 125, Feb. 1976, Heatherly). In a paper in prep· quotient of lP Ell 1 . aration, R. M. Guranick (Dept. of Math., Univ. of California, 2 Los Angeles, CA 90024) shows the following: If (i) the order of 11. W. B. Johnson. Are XP and lP the only LP subspaces of G is < 96, or (ii) that of G' < 16, then every element of G' is lP Elll2 for p > 2? a commutator. These bounds are minimal; a counterexample to 12. W. B. Johnson. Is lP the only LP subspace of lP Elll2 for (i) is a suitable semi-direct product of Q x K and Z ; for 8 4 3 1 2? (ii) there is an old example of W. B. Fite (Trans. Arner. Math.

224 ABSTRACTS

The abstracts are grouped according to subjects chosen by the author from categories listed on the abstract form and are based on the AMS (MOS) Subject Classification Scheme (1970). Abstracts for which the author did not indicate a category are listed under miscellaneous. * Indicates that preprints are available from author. •Indicates invited addresses.

Abstracts for papers presented at Appear on page 735 meeting in Reno, Apri/23-24, 1976 A-458 736 meeting in Portland, june 18, 1976 A-458 Abstracts Presented to the Society

The abstracts printed below were accepted by the American Mathematical Society for presentation by title. An individual may present only one abstract by title in any one issue of the c}foticei], but joint authors are treated as a separate category. Thus, in addition to abstracts from two individual authors, one joint abstract by them may also be accepted for the same issue.

Algebra and Theory of Numbers (05, 06, 08, 70, 72-7 8, 20) 76T-A112 VERNER E. HOGGATT, JR., San Jose State University, San Jose, California 95192. A second variation of a problem of Jones. Preliminary report.

Let Fn be the nth Fibonacci numbers; then the set F2n+ 1, F 2n+3 , F 2n+5 ' X is such that F 2n+1F 2n+3 - 2 2 2 2 2 2 = S1; F2n+1F2n+5- 1 = S2; F2n+3F2n+5- 1 = S3; F2n+1 X+ 1 = S4; F2n+3X + 1 = S5; F2n+5X + 1 = s6, where X-

4(F2n+2 F 2n+3 F 2n+4). Uniqueness of the X for each n is conjectured. (Received January 26, 1976.)

*76T-All3 Dwight Duffus and Ivan Rival, University of Calgarv, Calgary, Canacl.;,, Crn1.vns------in dismantlable partially o'rdered sets.

Let P be a finite partially ordered set. Let I(P) clenote the SHbset of Rll il•.rvclud/'1..-

elements of P, that is, all elements of P with precisely one u~1ner cnver or prec:iselv on12 lower cover. A nonempty subset Q of P is obtained from P by dismantling by irredud.h [,-,,

provided that P-Q ~ {a 1 ,a2 , ... ,am} where ai E I(P-{a1 ,a2 , ... ,ai_1 )) fori= 1,2 ... .,r-1. Theorem. Let S and T be subsets of a finite partially ordered set P :md let boLh ,; and 'i' h~ obtained from P by dismantling by irre<1ucibles. If I(S) = '/J I(T), then S

a centr!'ll rol<> in the study of finite partially ordered sets P satisfying I(F) = 0. \·le

introduce the related concept of a tower which, roughly speakinR, is a sub~eL n[ :· •'(':i:-~ ; !·· of a linear sum of crowns of order four. We say that P is disma:ntlcible by il'Y'Cduet.bl.eB if there is a one-element subset of P which is obtained from P bv dis"lantling bv irreducibles.

Theorem. Every finite partially ordered set which contain3 neither a spanninp. crown nor ,q spanning tower is disrnantlable by irreducibles. (Received February 9, 1976.)

*76'r-All4 N. Smythe, Australian National University, Canberra, Australia. A geneTaiization of Grushko's theorem to mapping cylinder groups.

Let G be the group of the, mapping cylinder of a diagram of groups (D,A) , as defined in R.H. Crowel and N. Smythe The J.>u.bg!Lou.p theO!Lem fio!L ama.igama:ted fi!Lee p!Lodu.w, HNN-c.oM.:tltu.c::.tioM and c.o-V.mw, PJWc.. Sec.ond Int. Con6. Theo!Ly o6 G!Lou.pl.l, Ca.nbe.Ma., 797 3, pp. 241-280; such gro'lps include free products with an amalgamation, HNN-groups, tree products and graph products, al--hough we do not ;require here that the homomorphisms of the diagram be monol'1vrphisms. Let f : :' _, G • e a 'homomorphism of a finitely generated free group F into G Then F can be w~itten as a free

product of groups Fv and a group F1 , in such a way that each Fv is generated by elements

mapping into conjugates of the image in G of the vertex group Av , and F1 is mapped

monomorphically by f ; moreover f(F 1 ) does not meet any conjugate of the images of the vertex groups. The proof is based on an algebraic version of Stallings' method of "binding ties", and a

A-419 generalization of Britton's Spelling Lemma to mapping cylinders. C.F. Miller has informed me he has proved a similar theorem for HNN-groups by topological methods. (Received February 13, 1976.)

76T-A115 G.J. RIEGER,Technische Universitat,D-3 Hannover, On the Harris modification of the euclidean algorithm. V.C. Harris (Fibonacci Quaterly §(1970), 102-103) modified the euclidean algo­ rithm (= algorithm by greatest integers) for finding the gcd of two odd inte­ gers a> b > 1. The conditions a= bq + r, lrl < b, 2lr define the integers q,r uniquely. In case r = 0, stop. In case r ~ O, divider by its highest po­ wer of 2 and obtain c (say); proceed with b, lei instead of a,b. This algorithm stops after H(a,b) (say) steps. Example: 83 = 47·1 + 4·9, 47 = 9•5 + 2·1,9=1•9; H(83,47) = 3. Denote by E(a,b) resp. N(a,b) the number of steps in the algo­ rithm by greatest resp. nearest integers for a> b > 0. According to Kronecker, N(a,b) ~ E(a,b) always. Let c 0 := 1, cn+1=2cn+5(n ~ 0); obviously E(cn+ 1 'cn)~5 (n ~ 0); since cn+2 = 3cn+1 - 2cn, 2 fen (n ~ 0), the choice ak=ck,bk=ck_1 (k > 0) gives Theorem 1. For every integer k > 0 there exist odd numbers ak > bk > 0 with E(ak,bk) ~ 5, H(ak,bk) = k. Using a Fibonacci sequence in the construction, we also prove Theorem 2. For every integer k > 0 there exist odd numbers gk > hk > 0 with N(gk' hk) > k , H(gk' hk) = 2. (Received February 16, 1976.)

*76T-A116 Ivan Rival, University of Calgary, Calgary, Canada. Varieties of nonmodular lattices.

There ~re 0~ecisely sixteen vRrieties of lattices, each generated by its

snhrl.i.rectlv irreducible members r.vhich contain no infinite chains, and each covering

the 1-e.o,st nonnodular va)Cietv in the lattice of varieties of lattices. (Received February 16, 1976.)

*76T-A117 STEPHANE FOLDES and PETER L. HAMMER, Department of Combinatorics and Optimization, Univ. of Waterloo, Waterloo, Ontario, Canada. A Class of Matroid - Producing Graphs.

A finloe graph G is called threshold if a hyperplane separates the characteristic vectors of its stable (edgeless) sets from those of the others. V. Chvatal and P.L. Hammer proved that a graph is threshold if and only if there are no 4 distinct vertices v1 , v 2 , v 3 , v 4 , with v1 adjacent to v 2 but not to v 3, and v4 adjacent to v3 but not to v2 . A natural generalization of threshold graphs are the matl'ogenic graphs, i.e. those graphs for which the induced threshold subgraphs form the independence system of a matroid. Proposition 1. A graph is matrogenic if and only if it has no 5 distinct vertices v1 , v 2, v 3 , v 4 , v 5 such that v1 is adjacent to v2 and to v3, but not to v4 , while v5 is adjacent to v4, but not to v2 and not to v3 . Let N(x) be the set of vertices adjacent to a vertex x. Proposition 2. A graph is matrogenic if and only if its vertex-set can be partitioned into•3 disjoint sets, A, B, C, such that: (1) A induces a complete graph, (2) B is stable, (3) C induces either a pentagon, or a

perfect matching, or the complement of a perfect matching, (4) for any three elements x,y,z E B, such

that INCx) I = IN(y) I = IN(z) I, if N(x) = N(y) then N(y) = N(z), (5) for all x,y E B, if neither

N(x) 5_ N(y) nor N(y) 5_ N(x), then Card [ (N(x) u N(y))\(N(x) n N(y))] = ·2, (6) every x E C is adjacent to every vertex in A and to no vertex in B. (Received March 3, 1976,.) *76T-A118 HARTt1UT HDFT and PAUL E. HOWARD, Eastern Michigan University, Ypsilanti, Mich. Representing multi-algebras~ algebras, the axiom of choice and the axiom of dependent choice. Multi-algebras are sets witn set-valued operations. We give a short proof of the Representation Theorem (Gr\Hzer): For every multi-algebra A of type ~ , there is an algebra B of type <.:> and an equivalence relationE on B such that A and 8/E are isomorphic multi­ algebras. We further prove Theorem 1: The axiom of choice is equivalent to the Representa- tion Theorem for unary multi-algebras. For any well-ordered cardinal number k, let RMl,k be the Representation Theorem for A-420 multi-algebras whose multi-operations are unary and indexed by k. Theorem 2: DCk implies RMl,k. Theorem 3: RMl,l implies DC~ 0 • This last theorem corrects the false Theorem 2 in [Notices AMS, 21, No.7, *719-Al4, p. A635 ]. (Received February 23, 1976.)

*76T-All9 MiaHA.EL SLATER, The University, Bristo 1 BS 8 lTW, England. Strongly semiprime alternative rings, ~J! Goldie theorems. l:!u}lpose throughout that R is an alternative rine;, and aRa = (0) im"91ies that a=O, for aER. (1) If R is prime but not associative, then it is a CD ring; i.e. a central order in a Cayley algebra. (2) If R has e.• c.c. on 2-sided annihilator ill.eals and is purely alternative (no non-zero nuclear ideals) then it is a central order in a finite direct sum of Cayley algebras. (3) If R has a.c.c. on 2-sided annihila.tor ideals, and satisfies a p.i., then it is a central order in a finite direct sum of simple rings, each finite-dimensional over its center. ( 4) If R satisfies the right Goldie conditions, it is a r1.ght order in a finite d.irect sum of rings, each of which 1S simple with d.c.c. (5) If R is purely alternative, it is isomorphic to a subdirect product of CD rings. (6) If R is arbit­ rary, it is isomorphic to a subdirect product of CD rings and prime asso­ ciative rings. The converse of each of the above results also holds. (Received February 23, 1976.)

76T-AJ.20 MAX MLYNARSKI, Kingsborough Community College, Oriental Blvd., Brooklyn, N.Y. 11235. Matricial Norms.

A vector norm ~ on en and a direct-sum decomposition ~ of en induce a vectorial norm of order k:p(x)=(~(Eix))i=l, ••. ,k where the Ei's are the projections associated with~. We write p=(~;~) and call p equilibrated if lu)il~(Ei)=sup~(Eix)/~(x)=l TJ Hx;o!O). We denote lub~ by I 1/cj>' An equilibrated vectorial norm p=(~;~) induces a matricial norm of order k:lubp(A)={iEiAEjl~i, j=l, ••• ,k, for all nxn comple~ matrices A. Denoting r(A)=max{ jAj :AeA(A)} we will say that lubp is minimal on A if r(A)=r(lubp(A)). Theorem. Let A be a complex nxn matrix, let A1 , ... ,As be its distinct eigenvalues lying on the boundary of the convex hull of the spectrum of A. Denoting by kj the multiplicity of A· in the minimal polynomial of A (j=l, ... , s), there exists an equilibrated vecforial norm p on en of order k=max{k1 , ... , ks} such that lubp is minimal on {A-I; In : l; e C}. (Received February 23, 1976.)

*76T-Al21 D. Suryanarayana and V. Siva Rama Prasad, Univ. of Georgia, Athens, Georgia 30602, Andhra Univ., Waltair, India and A.M. Jain College, Madras, India. The number of semi-k-free divisors of an integer.

Let k be a fixed integer ~ 2. A divisor d of n is called semi-k-free if d is not s* divisible unitarily by the k-th power of any prime. Let T (k) (n) denote the number of semi-k-free divisors d of n. In this paper (to appear in Ann. Univ. Sci. Budapest, s* Eotvos Sect. Math.) we prove the following: Theorem 1: ~ r(k)(n) = n"'X s* 1/k -8/5 ~*x /iog x + 2y- 1 + :r (kp;!:;:l)log p\+ A~=) (x), where A(k)(x)=O(x expf-A(l-ka)k ~ p (p -p+l) ) log315x (log1og x)-115 }) or O(xa), according as k=2,3 or k~4. In the above y is s* -k -k-1 Euler's constant, A is an absolute positive constant, ~ = n (1-p +P ) and a p is the number which appears in the Dirichlet divisor problem (~

A-421 76T-A122 G.V.CBOODNOVSKI, !l!arasovllka;ya .. 10&1, ~q>.1?, Kiev, U.S.S.B..,252033• On Sebanuel•s blpolihesis.!l!bree alsebraicallY independent D.UIIbe~s II. Ptvlim1na":T "epo"t• Tbeo~m 1. I:f ot..=/=0)1 is algebraic, fo is algebraic of 5th degree, thell among ~fo, ... .)ot..fo4 there are 3. algebraic~ly_independent (a.i.) nu•N be"s. Tbis result is wide:cy ge11er~~~d. Let ~ J fo..,..., be the s~t~fro• C , C M satisfying tbe estimates l(:x:. J o(,) I ~ ex,p(- Lio II X II).) I(~' ..fo) I ~ :::::exp(-7:ojfyll):ror xezN\{O},YEZM\{O},where (., .) and II II- a~ scala~:' product and norm. Theorem 2. I:fMN>Z(M+N) the11amo11g the numbera

We prove Theorem 6. If the equation xp+yp+zp = 0, p an odd prime > 3 and (xyz,p) = 1, has an integral solution, then each of the following conditions must be satisfied. (1) (2:) = 2 (mod p4) or (2p;l) = 1 (mod p4). (2) [(p:~~/ 2 ) = 22(p-l)(-l)(p-l)/2 (mod p4).

(5) H - H 0 (mod pJr+J) r r-1 = and n > 1.

The above theorem follows from the combination of well known results. For several other results of the above kind, reference may be made to the author's book Fermat's Last Theorem, Vol. I (elementary methods). The work is supported by the James Vaughn Jr. Foundation. (Received February 26, 1976.)

76T-A124 Jarmila Chvatalova, University of Alberta, Edmonton, Alta. and University of Waterloo, Waterloo, Ontario. The bandwidth problem for infinite graphs. Preliminary report.

If G is a graph with denumerable number of vertices and at least one edge, we set ~(G) = min maxlf(u)-f(v) I, where the maximum is taken over all edges uv and the minimum over all one-to­ one mappings f: V(G) + Z+; V(G) denotes the set of vertices of G and ~(G) is often called a bandwidth of G. Some necessary conditions and sufficient conditions are given for infinite graphs to have finite bandwidth. Theorem 1. If cp(G') .::_ k for every finite subgraph G' of G then

~(G) _::_ 2k. Theorem 2. If G has infinitely many disjoint infinite paths then ~(G) is unbounded. The following has been conjectured by Erd!ls in 1973: "Let G be an infinite graph. If there exists a constant c such that for every v E V(G) and for every k E Z+ we have l{x: x E V(G) and d(x,v) _::_ kll _::_ ck then HG) is bounded". We give a construction of an infinite graph GE satis­ fying the conditions of the above conjecture which bandwidth is proved to be infinite. The following theorem follows from the construction of GE. Theorem 3. If a graph G contains as its subgraph a subdivision of a binary tree of depth k then ~(G) ~ k. Further, we can prove the following Theorem 4. Let G be a tree of maximum degree 3. If G contains (as a subgraph) no subdivision of

a binary tree of depth k then there exists a suhdivision G' of G such that ~(G') < k. (Received March~. l976.) (Author introduced by A. Meir.) 76T-Al25 M.E. ADAMS and J. SICHLER, University of Manitoba, Winnipeg, Manitoba R3T 2N2 Free products of hopfian lattices A lattice is hopfian iff every onto endomorphism is an automorphism. Theorem: Let !

be any nontrivial variety of lattices. There exist hopfian lattices 10 ,11 whose ,Y-free product is no.t hopfian. Theorem (G.C.H.): Let ! be any nontrivial variety

of lattices. For any set I there exists a family of lattices Li (i E I) such

that the V-free product of (til i E I 1 s; I) is hopfian iff I' f. I. (Received March 1, 1976.)

A-422 Herbert S. Wilf, University of Pennsylvania, Philadelphia, Pa .• 19114. A Unified Setting for Selection Algorithms. Recursive algorithms for the construction of combinatorial objects are often special cases of the following: a directed acyclic graph G is given, with multiple, numbered edges, without infinite paths, and with a single terminal vertex T If V E G, .an object of order V is a walk from V to T . The edge numbering induces lex order on the objects. Hence we can produce them in sequence or at random, rank them or construct them if the rank is given. Included are n-permutationswithk cycles, k-subsets of n-sets, k-subspaces of n-space over GF(q) , partitions of n into k parts, partitions of an n-set into k classes, Young tableaux, etc. (Received April 15, 1976.) .~,...UoW ALBERT A. MULLIN, 6840 Todd, Patton Park, Ft. Hood, Texas 76544, ~ value results for generalized multiplicative arithmetic functions. This note extends several basic mean-value results for multiplicative arithmetic functions to the broader class of generalized multiplicative arithmetic functions. Lemma 1. Let f(•) be any complex-valued generalized multiplicative funcrtion such that lf(n)l~1 for all n (e.g., the author's modified MClbius func-tion).J.* as defined in Amero Math. Monthly 74 (1967), 1100 satisfies this condition). Then lim (~ f(m))/n exists and is non-zero iff ~P(f(p) - 1)/p over·all primes p n+- m"" n 2 ( ) is converge;t and for some expori·entiated sequence of 2' s (e.g., 22, 22 , "•), e 2 , f(e( 2)) ::/= -1. Lemma 2. Let f(•) be any complex-valued generalized multiplicative arithmetic function. Then f(•) has a mean-value M( 1 )+ 0 and a mean-squared-absolute· value M( 2 )

Let X be trre ·set o:'' inter~>rs [1,2, ••• ,~ ,n~ 2 ana lP.t '.f bP. a family

of subsets of X.In t~i~ ~an0r we ~rove ~~~ Pollowi~~= THEOR"Bl'-~.SupposP. -tr.at fo~::-- 8.n~J. F 1 ,F 2 ,F-;;;~-j: \'):J.~ 2n-2 ann equ.alif:7r ,·oJ.as i!' ~-1'"'.(1- rY·J~_:v

1 ~ i < j .~ n "''' c\-, 7.1- "-t. 1 = { "S: :: I c € F , .' E !<'}. (Received March 8, 1976. ) (Author introduced by M. Deza.)

Ralph McKenzie, University of California, Berkeley, California 94720. On elementary proper- ties of the variety and quasi-variety generated by a finite algebra. (1) The variety generated by a para primal algebra is finitely axiomatizable. This is a corollary of Theorem. ll CL is a finite algebra, ·VUJ. has only finitely many subdirectly irreducible (S.I.) members, and '1/(JL. has first-order definable principal congruences, ~ ·v:r~.,- is finitely axiomatizable. (2) The only varieties of lattices having definable principal congruences are the two varieties of distributive lattices. (3) We have an example of a finite algebra whose generated variety is finitely axiomatizable, and generated quasi-variety is not. This gives, by Boolean power constructions, a countable ~0 -categori­ cal theory whose universal part is not finitely axiomatizable. (Received March 10, 1976.) 76T-Al30 DAVID ZEITLIH,1650 Vincent; AYe, lonh,Mtrmeapolla,MI.,55411e Parpat;ric aoluticmt !s!£ 1!2 !.Q!W. S1 ~ 1'7]&quadz!Ua, II, Prel.DiJiarT repon.

Let; '\+2 =~ + ~ ,k=0,1, ••• , wh81'8 P,W0 , and w1 are integera.Then,tor k.=0,1, ••• ,w haYe (1) (2~+5)4+(49l'l\+,>4+(1~,>4+(1~)4+(20Wk+2)4+(3~)4+(1~)4+(21l'Wk+2)4 + (S6PWk+2)4+(~)4+(28P'wk-12)4+(~)4+(1 ~)4+(~)4+(~)4+(~)4 + A-423 <~>4. c~4>4+C1~4>4+c~4>4+c~4>4+c~4>4+c~>4+4 + (2011k+3)4+(32\:+3)4+(UPVk+3)4+(21'\...3)4+(~)4+(~)4+(~3)4+(4~)4 +

(1~)4+(~)4 eREMARKBol'ol' P=2,(1) 1Uplli'iea -.o ($) (~)4+(~)4+(t~4 )4 + (1~4)4+(~2)4+(1~)4+(21~)4+(~)4+(~)4+(21~)4+(~)4. (21~)4 + (281lkf4)4+('7ifk+3)4+(1~,>4+(21\...,>4+(42Wc+,>4+(~)4+(~)4+(10\.+,)4.(1~)4+("'\)-\ See I in tbeae ~ 23(1976),.A..353. Add1Uonal. npOI'ta are twth4101W1c, :tDeludiug a genwal.iHUon of I and II.Additional reaul.118 OD nbu haTe been obtdaect(to be &DilOIDUied). (Received April 9, 1976.)

*76T-A131 J. H. YANG, Ohio State Univers:i,ty,.. Columbus, Ohio 43210. Positive Definite ~dratic Forms Under Same Field Extnesions, Preliminary report.

N. C. Ankeny raised the question on the behavior of the genus of a positive definite integral quadratic form under a totally real algebraic extension over Q • For related results on spinor genus, see A. G. Earnest and J. s. Hsia's paper, [B.A.M.S., Sept., 1975]. Let L, K be p::>sitive definite quadratic Z/:-lattices, E a totally real algebraic number field, '\: its ring of integers. Put L = L ® OE and K= K ® ~. The conjecture is: Arry isometry from L onto 'j{ induces an isometry of L onto K via restriction. Here we give some evidences that the conjecture seems to be true:

THEOREM: The conjecture holds when E is real quadratic, totally real Dirichlet's biquadratic, the maximal real subfield of the cyclotomic subfield of pa-th root of unity, where p is a prime, or produ<;:t of these maximal real subfields over distinct primes.

COROLLARY: Let E/il. be as in Theorem. Then any positive definite indecampo!able zz: -lattice L lifts (via tensor product) to an indecomposable OE -lattice L • (Received March 15, 1976.) *76T-A132 Timothy P. Donovan, University of Colorado, Boulder, Colorado, 80309. Enumeration of solutions to certain matric equations mod pn. This paper develops a canonical form under equivalence for matrices with integer entries mod pn.

This canonical form is then employed to simplify the congruences AX = B (mod pn} and UA + BV = C

(mod pn}, for given matrices A. B, and C. Appropriate partitioning and basic combinatorial techniques

are then used to count the solutiop.s. (Received March 15, 1976.) (Author introduced by John H. Hodges.)

76T-A133 MANOHAR MADAN, Ohio State University Columbus, Ohio 43210 and SAT PAL, Ohio State University, Columbus, Ohio 432io. Abelian Varieties and a Conjecture of R. M. Robinson. A conjecture of R. M. Robinson states that the polynomial 2 Gm(Y) = n[y2 - (4 + 2 cos ~k)Y + 1] 1 0 S k :S ~ 1 (k,m) = 1 is irreducible over the field of rational numbers for all natural numbers m except 2, 7 and 30. This conjecture is proved for the case when m is a prime and it is shown that Gm(Y) is, almost always, irreducible. Assuming Robinson's conjecture, the isogeny classes of simple abelian varieties defined over finite fields with q elements and with one rational point are determined in terms of the minimal polynomials of the corresponding Weil numbers. It turns out that for q = 2, there are infinitely many such isogeny classes. For q = 3 and q = 4, in each case, there is one isogeny class. For q > 4, there are no such varieties. Finally, the above results are applied to improve the results of Leitzel and Madan, (Acta Arithmatica, to appear), concerning the following question: Let F/K, E/L be algebraic function fields in one variable over finite fields of constants such that E/F is finite and separable. Let F, E be the fields obtained by extending the fields of constants to the algebraic

closure. Let J(F) 1 J(E) be the Jacobian varieties of the non-singular projective curves associated to F, E. When do J(F), J(E) have the same number of rational points? (Received March 15, 1976.) A-424 Dr. Brian J. Day, Department of Pure Mathematics, University of Sydney, Sydney, N.S.W. 2006, Australia. Note on monoidal monads.

The representation theory of categories is used to embed each promonoidal monad in a monoidal biclosed monad, The existence of a promonoidal structure on the ordinary Eilenberg-Moore category generated by a promonoidal monad is examined. Several results by previous authors (notably A. Kock and F.E.J.

Linton) are reproved and slightly extended. (March 15, 1976. )

*?af~Al~~ A. Holleman, Institute for Secondary Teacher Training VL-VU, Amsterdam, The Netherlands. Inseparable Algebras.

Recently Sweedler defined the notion of a purely inseparable algebra. Unfortunately Sweedler's concept is far from being functorial. Call an algebra B over a commutati- ve rjng A ~dicial (or purely inseparable) if the kernel of the multiplication map m : B @A B ~ B is contained in the prime radical of B @A B. Using this definition,

inseparability is unaffected by taking tensor products or direct limits. The pro-

perty of being radicial is a local one (i.e. B radicial over A iff Bx radicial over

Ax for all x E Spec A), even a pointwise one (i.e. B radicial over A iff B@ k(x)

radicial over k(x) for all x E Spec A). Remarkably, the prime radical Prm B of a

radicial algebra B contains the commutator ideal so that the quotient ring B/Prm B

is commutative. (Received March 16, 1976.) (Author introduced by Professor Willem Kuyk.)

*76T-A136 Thomas 0. Hand, Indiana State University, Terre Haute, Indiana 47809 Boolean Rings lb. Let R be a ring and n be a positive integer. Theorem: If x2n( 2n-l) = x, for all x in R, then R is a Boolean ring. 22 Theorem: If n E 0 mod 2 and x (2n-l) = x, for all x in R, then R is a Boolean ring. 22(2n-l) However, if n 10 1 mod 2 and x = x, for a 11 x in R, then R need not be a

Boolean ring. (Received March 17, 1976.) HOWARD KLEIMAN, Queensborough Community College, Bayside, N.Y. 11364, Hamil ton lines,

ln'whiiit •t6:U'o'iifl;· IG· is a simple planar ~ph of order n , It V(G) = fv 1,v2,, .. ,v J is the set of vertices of G and F(G) = t ~(1),f(2),, .. 1f(s)} is its se\ of faceR, let Q(f(j)) be the length of the minimal circuit of f~j) , S(vi) , the set of faces ~f G which border v and T(f(j)), the set of faces of G which bijrder f(j) , THI!:OREJI, The followin~aj:!!- necesl!la:ey: and sufficient condi tiona for G to have a hamilton ~ ' There exists a subset F' of F , say F' = {r(1),f(2), ••• ,f(r)!, such that r+1 " (1)f- 8(r(j)) • n + 2r (2) S(vj)(j:_ F• for j=1,2, ... ,n 1 (3) If T'(f(j)) • T(f(j)) = F' and T'(f(rt-1)). { f(a1),f(a2), ... ,f(ao<.)} , T'(f(ai)) ={f(a11 ),f(a12), ... ,f(aip)} • (i=1, ... ,o() ' [ T'(f(aij)) = f(a1j 1),f(a1j 2), ••• ,f(a 1 j~)} (i:1,2, ... ,

For some years it has been known that every singular square matrix over an arbitrary field F is a product of idempotent matrices over F. This paper quantifies that A-425 result to some extent. Main result: for every field F and every pair (n,k) of positive integers, an n x n matrix S over F is a product of k idempotent matrices over F iff rank(I-S) ~ k•nullity S. The proof of the "if" part involves only elementary matrix operations and may thus be regarded as constructive. Corollary: (for every field F and every positive integer n) each singular n x n matrix over F is a product of n idempotent matrices over F and there is a singular n x n matrix over F which is not a product of n - 1 idempotent mat rices. (Received April 12, 1976.)

76T-Al39 JAMES W. CARTER, San Gabriel, California, 91776. Some observations of number theory. I. Consider .:!: Jtil: the exact number of lattice points contained in the sphere of radius p is "" K~l L l:fJ eoo • [ ., 'l. J N(f) = 4 IC-1} [;z.l<.-d+il L ~ H)l+l ti-~1 +:l[fl +1 a' • J'"l u. Tt ,--... e·h ( ;. ~~... +.) (for n large).

('VI !>:1. Ill. 1l '"'-' ( ;1. \'), .... , ) n (for n large) (Received March 22, 1976,)

*76T-Al40 HEIKO HARBORTH, Technische Universitat Braunschweig, D 3300 Braunschweig, West Germany. Number of odd binomial coefficients.

Denote by F(n) the number of odd binomial coefficients in the first n rows of Pascal's triangle. Let 8 = (log 3)/(log 2), a= lim sup F(n)/n8 , and a= lim inf F(n)/n8 • Then K.B. Stolarsky (see these NOTICES 22 (1975), A-669- A-670, Abstract 728-A 7) showed 0.72 ~a~ 38/7 < 0.815, and 1 ~ :-~ 1.052. We improve this to a = 1 and a = 0,812 556... (Received March 22, 1976.) 76T-Al41 Alan H. Mekler, Stanford University, Stanford, California 94305. The number of x-free abelian groups and the cardinality.~£ Ext. Preliminary report

By "group" we shall mean "abelian group" and will be uncountable. A group is 7r -free if all sub­ groups of cardinality less than X are free. A group is strongly ')(-free if every subset of cardin­ ality < :>c 1 s contained in a >r -pure free subgroup. Given a X -free non-free group of cardinality Jt:, we construct non-free strongly X'-free groups of cardinality :><1. Theorem 1: If there exists a X-free non-free group of cardinality K, there exist 2,.. strongly X-free groups of cardinality :JC. Recall that a group is a Whitehead group (W-group) if Ext(A,'Ze.) = 0. Theorem 2: If all W-groups~of cardinalityXare free and A is a >(-free non-free group of cardinality :X, then f Ext(A,~) / = 2) • This improves a result due independently to Eklof and Mekler ( Notices AMS 23 A-273 Abstract fF76T-A66) and Shelah. (Received March 22, 1976.) 76T-Al42 Carl Bumiller, St. John's University, Jamaica, New York 11~39 ! ~ for partial geometries. Theorem: For any (R,S,T) partial geometry [Bose, Pac. J, Math 13(1963), 389-~19], (R-1)(S-2T) < (S-2)(S-T) 2 • 'l'he proof uses the Krein condition for a rank three graph [D.G. Higman, Coherent configurations I, Geom. Dedicata ~(1975), no. 1, p. 29]. Suppose the 0-1 adjacency matrix of an arbitrary rank three graph has eigenvalues k, r, and s where k is the valence. 'rhen the Krein-condition inequalities given by Higman reduce to: (r+1)(k+r+2rs) < (k+r)(s+1) 2 • For a partial geometry, k = R(S-1), s = S-T-1, and r = -R; and thus the theorem. If r = S-T-1 and s = -R, then nothing results, Note that R and S can be inter­ changed since a partial geometry has a dual {S,R,T) partial geometry. If T = 1, then the partial geometry is a generalized ~-gon and the theorem reduces to the well-Jcnown result: R-1 _::: (S-1) 2 • (Received March 23, 1976.)

*76T-Al43 C. S. Johnson, Jr, and F. R. McMorris, Bowling Green State University, Bowling Green, Ohio 43403. Commutative nonsingular semigroups. Let S be a commutatjve semigroup with o. S is said to be nonsingular if every intersection large ideal is dense. Recalling that a commutative ring R is nonsingular iff R is semiprime and that A-426 R is semiprime iff its multiplicative semigroup is separative, we ask if commutative nonsingular semigroups are precisely commutative separative semigroups. We show that separative is necessary but not sufficient and offer a characterization of commutative nonsingular semigroups in which every ideal is finitely generated. We also improve a result of Lopez and Luedeman (Semigroup Forum, to appear) to read that if S is nonsingular then the maximal quotient semigroup of S is regular. (Received March 24, 1976.)

*76T-A144 David J. Saltman, Yale University, New Haven, Connecticut 06520. Splittings of cyclic p-algebras.

We say a group G appears over a field K if there is a Galois extension L/K with Galois group

G. If A is a finite dimensional central simple algebra over K then we say G appears over K in

A if such an L can be found which is also a maximal subfield of A. Theorem: If K is a field of characteristic p and A is a cyclic p-algebra over K of degree pn, then a group G of order pn appears over K if and only if it appears in A. With K as above, define P(K)

{ xp-x1 x £ K} and call NK the dimension of K/P(K) over the field of p elements. Witt showed that a finite p-group appears over K if and only if it is generated by fewer than NK elements.

Theorem: 1) If NK is finite, the p-primary part of the Brauer group Br(K) is zero; 2) If NK is infinite and A is as above, then all groups of order pn appear in A. (Received March 26, 1976.)

7&rl~45 Louis Halle Rowen, Bar Ilan University, Ramat Gan, Israel. Finite dimensional representations o£ nonassociative rings. A nonassociative polynomial f(X1 , ••• ,Xm) is t-~ (tSm) i£ f is linear on ~, ••• ,Xt and alternating on ~, ••• ,Xt. R denotes a (not necessarily associative) ring. Definition: R is NPI of degree St i£ every t+l- normal polynomial is an iden­ tity of R. (Note: every subring of at-dimensional algebra is NPI of degree t.) Theorem 1. The "central closure" (Erickson-Martindale-Osborn Pac. J. Math. 60 (1975)) o£ a prime NPI-ring of degree t is central, of dimension t. Theorem 2. Suppose R is semiprime Jordan NPI. (1) Every nil subring is nilpotent. (2) Every nonzero ideal intersects the center nontrivially. (3) R has a central polynomial. (4) R[\] is a subdirect product o£ central simple algebras. (5) I£ R is also prime o£ degree t, then its ring o£ central quotients is central simple, o£ dimension t. (Received March 29, 1976.) *76T-A146 GEORGE HUTCHINSON, National Institutes of Health, Bethesda, Maryland 20014, A test for identities satisfied in lattices of submodules.

Suppose R is a ring with 1 of characteristic k. A lattice is representable by R-modules if it is embeddable in the lattice of submodules of some unitary left R-module; the class lt(R) of all lattices representable by R-modules is known to be a quasivariety. The divisibility condition D(m,n) for integers m,n ~ 0 is the formula ( x)(m•x = n·l), where i•y is given by O·y = 0 and (i+l)·y = (i·y)+y for i ~ 0. For example, D(2,1) for R means 1+1 is invertible in R, and D(O,l2) for R means that k divides 12. For any lattice polynomials d and e, a divisibility condition D(m,n) iR rer.ursively

constructed such that the lattice identity d = e is satisfied in every !attic~ in al(R) if

and only if D(m,n) is satisfied in R. So, the word problem for the free at(~)-lattice on denumerably many generators is recursively solvable if there is a decision ])ro•·edure for divisibility conditions in R. This is true for rings with nonzero characteri~: ic, matrix rings over fields, the ring of integers, and many other rings. A set of repres:mtacive rinj:!;s

is constructed so that, for each ring R with 1, there is a unique representa•ive rin~ S such that the varieties of homomorphic images of lattices representable by R-modules and by S-modules are the same. There is exactly one representative ring of characteristic k for each k ~ 1, and there are continuously many representative rings with zero characteristic. (Received March 30, 1976.) A-427 *76T-Al47 K. CHANG and D. K. BAY-CHAUDHURI, The Ohio State University, Columbus, Ohio 43210, On the existence of lattice designs.

Let n,m and A be given positive integers and K a set of positive integers. An (n,m,K,A)-lattice design is a quadruple (X,?r,w,a) where (l) X is a finite set of nm elements (called points), ( 2) 'lr is a class of n-subsets of X which partition X (called vertical groups), (3) ~ is a class of m-subsets of X which partition X (called horizontal groups), ( 4) IV n HI = l for every V E 'lr and H E ~ , ( 5) a is a family of subsets of X whose coordina.lities are in K (called blocks), ( 6) no block meets a vertical or horizontal group in more than one point, ( 7) every pair (x,y} of elements of X, not contained in a vertical or horizontal group, is contained in exactly A blocks. For any subset K of positive integers, define ~(K) = gcd (k(k-l)lk E K} and a(K) = gcd (k-llk E K}. The necessary conditions for the existence of an (n,m,K,A)-lattice design are (a) An(n-l)m(m-l) ~ 0 (mod ~(K)) and (b) A(n-l)(m-l) = 0 (mod a(K)). It is proved that, given a subset K of positive integers, there is a constant Mo = Mo(K) with the property that if m > Mo and A. are given positive integers, then there is a constant C = C(m,K,A) such that an (n,m,K,A) - lattice design exists for all n > C satisfying (a) and (b) above. (Received March 31, 1976.)

76T-Al48 WITHDRAWN

*76T-Al49 EDWARD M. CORWIN AND CLIFFORD S. QUEEN, LEHIGH UNIVERSITY, BETHLEHEM, PA. 18015 "RINGS OF EUCLIDEAN TYPE" Let A be an integral domain. We say that A is of Euclidean Type if there is a function ~ from A - {o} into a well ordered

set where if a, b E A, b f 0 and (a, b) = aA + bA = A, then .there

exists q, r ., A such that a = bq + r, r = 0 or ~(r) < ~(b). We prove some general results for subrings of global fields and establish the importance of the above property when investigating

SL2 (A), where A is the ring of integers of a real quadratic number field. (Received April 8, 1976.)

*76T-Al50 Yehiel Ilamed, Soreq Nuclear Research Centre, Yavne, Israel. On a characterization of the standard polynomial of even degree.

Let R be the real field, Mn(R) be the algebra of nxn matrices with real entries, and let (n! (n) X1 ,X2 •···E Mn(R) be generic matrices. Let P be the algebra over R generated by the noncommuting

indeterminates x1,x2, ..• , and let sk(x1, ••• ,xk) be the standard polynomial of degree k. We say that

P = p(x1, ••• ,xhJeP is orthogonal to q = q(x1, ••• ,xk)f.P and indicate this by writing pJ.q if ( (n) (n) (n) (n} tr p(X1 , ••• ,Xh )q(X1 , •.• ,Xk )) = o, n=1,2, •••

THRORE.~ 1. Let p(x1, ... ,xk) ~P be k-Unear. Then xiJ.p(x1, ... ,xk), i=l, ... ,k, implies:

(i} foro even k=2n , p(x1, ... ,x2n; = a.s 2nrx1, ... ,x2n; , aE R; (ii) for odd k=2n+1

i=1, ... ,2n (Received April 9, 1976.)

*76T-Al5l B.N. DATTA-IMECC-UNICAMP-Campinas-Brasil MATRICES SATISFYING SILJAK'S CONJECTURE

The following conjecture was made by D.O. Siljah [stability of Large - -Scale Systems, Proc. 5th Int. Fed. Automatic Control Congress, Paris,France, June 12-17,1972, paper C-32] and recently reported Matemayer and Womack [IEEE Trans.Automat. Contr. Vol.AC-20, pp. 572-573] : Let A be a stability matrix. Then there exists a symmetric positive defi

A-428 nite matrix H = (hl..) that satisfies the Lyapunov matrix Equation: H A+ ATH - W J und has the following scructure: h. > 0 if i ;, j 1. J hii > 0 In this paper, it is shown that this conjecture is true in three important cases when the given matrix A is (i) a companion matrix of a polynomial (ii) a Schwarz matrix (iii) a nonderogatory

~trix in its Routh-Canonical form. The conjecture is shown to be true in all these cases by actually constructing the matrix H in each case. Also, given a companion matrix A, a matrix B, similar to A ,is constructed such that B satisfies Siljak 1 s conjecture. (Received April 13, 1976.) *76T-Al52 IRA J. PAPICK, Adelphi University, Garden City, NY 11530 Coherent Overrings.

In a recent paper, announced in these Notices, April 1976,

*76T-Al06, we proved: (a) if R is a Noetherian domain and each overring

of R is coherent, then the Krull dimension of R is at most 1. We also

raised the following more general question: (b) if each overring of R

is coherent, is R treed? (Spec(R) under ~ is a tree.) Using a con­

struction modelled after Nagata's counterexample to the saturated chain

condition, we show that question (b) above has a negative answer in

general, and we determine wnen this construction yields coherent domains.

An application of these techniques shows that if each overring of a

domain R is coherent, then the integral closure of R is Prufer,

hence generalizing (a) above. (Received April 14, 1976.)

76T-Al53 Khalid Benabdallah and Robert Bradley, Universite de Montreal, Montreal, Canada. Torsion and Torsion-free quasi-pure-projective groups.

A reduced p-group G has the property P if for any basic subgroup B of G and for any idempotent endomorphism cp of G/B there exists an endomorphism 6 of G such that v8. e ~

if for any pure subgroup H of G and for any homomorphism f: G ~ G/B there exists an endomorphism

Khalid Benabdallah and Ad~le Laroche, Universite de Montreal, Montreal, Canada. Quasi­ p-pure injective groups. Preliminary report.

An abelian group G is said to be quasi p-pure injective. (q.p.p.i.) for a prime rtumber p, if Hom (G,G)

~Hom(K,G) + 0 for every p-pure subgroup K of G. We have the following: Theorem-~· Let G be q.p.p.i. and p-reduced, then qG = G ( (q,p) =1 ) and G1 = 0. Further G = II m K where II is t!J<' largest fully in-

variant pure subgroup of G containing T(G), and K is torsion~free and both are q.p.p.i •• Theorem 2. A p-reduced group H with torsion p-hasic subgroup B is q.p.p.i. if and only if it is isomorphic to a pure fully invariant subgroup of the p-adic completion of B. Theorem 3. If a torsion-free reduced group K is q.p.p.i. then K is an R-module where R is a p-pure subring of the ring Jp of p-adic inte-

A-429 gers such that, g E Rand g- 1 E Jp => g- 1 E R. Theorem 4. For such R's as in Th. 3, a freeR-module is q.p.p.i. if and only if it is of finite rank. Theorem 5. If T(G) = G then G is q.p.p.i. if and p only if its reduced part is q.p.p.i .• Theorem 6. If G = H ffi K where His a reduced adjusted cotorsion group and K is torsion free q.p.p.i. then G is q.p.p.i .• Theorem 7. If H is q.p.p.i. p-reduced with torsion basic subgroup and K is a finite rank free R-module where R is as in Th. 3 then H ffi K is q.p.p.i.. (Received April 16, 1976.) (Authors introduced by Dr. Paul Gauth~er.)

76T-Al55 JOEL BERMAN and G. GRATZER, Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2. Uniform representations of congruence schemes.

Let T be a type of algebras. A congruence scheme S of type T is a sequence of polynomials p0 , • · ·, pn _ 1 of type T , where pi is a polynomial in the variables x, y3 , 0 ,; j < ~; (these variables are assumed to occur in p. , variables with different indices are distinct, ni = 0 is permitted) and a function t: (o, ~1, ... ' n -1} + ro' 1} . A nontrivial equational class ~ of type ~ is a uniform representation of S iff T ~ ~ (that is, every operation in T is an operation in ~ and therefore every polynomial of type T can be regarded as a polynomial of type ~) and for every algebra m E K the following equivalence holds: for a0 , a1 , b0 , b1 E A, b0 = b1 (B(a0 , a1)) iff there ~xist c~ E A (0,; i < n, 0,; j < n.} satisfying 0 . . J ~ bo =po(at(O)' co, c~, ... ), ... , P/al-t(i)' c~, c~, ... ) = i+l i+l n-1 n-1 Pi+l(at(i+l)' c0 , c1 , '"), ... , Pn_ 1Cal-t(n-l)' c0 , c1 , "') = b1 (in other words, principal congruences can be described by S). Proposition. If S has a uniform representation, then ni > 0 for all 0 ,;;; i < n . Our main result is a partial converse: Theorem. Let S be a congruence scheme whose polynomials p. contain no constants. If S satisfies n. > 0 for all 0 < i < n , then S has a uniform repfesentation. Theorem. The scheme Po= 0 + CtO +x) +yg) t(O) = 0 has no uniform representation. (Received April 19, 1976.) Brian A. Davey and Moshe S. Goldberg, La Trobe University, Bundoora, Victoria, *76T-JU56 Australia, 3083. Free distributive double Stone algebras. Prel~minary report. Denote the category of double Stone algebras by a; the category of bounded distributive lattices by D,and the two, three and four element chains by 2, 3 and 4, respectively. It is known that a= Equ(4). THEOREM 1. Let (L. I i E I) be a-fa~ily of double Stone algebras. Then the D-free ~ product L of the Li is a double Stone algebra. In fact, L is the a-free product of the family (Li I i E I). 0 Fa (1) 2 x 2 x 3 = (F (0)) 2 x F (1). Let Q be the disjoint union of a two- - - - D D element unordered set and a two-element chain; Q is the poset of join-irreducible elements of Fa(l). THEOREM 2. For every cardinal K, Fa(K) is isomorphic to the lattice of clopen order-ideals of QK, where QK is endowed with the product topology. D In the finite case, Fa(n) can be described ~xplicitly. THEOREM 3. n (RJ 2n-m Fa (n) n [F0 (m)] 0 m=O (Received April 19, 1976.)

*76T-Al57 Alexander ABIAN, Dept. of Math. Iowa State univ. Ames, Iowa 50011 Conditionally complete and conditionally orthogonally complete rings.

Let R be a ring in which every element has a unique cube root. R is called a cuberoot ring. A cuberoot ring R necessarily has no nonzero nilpotent ele­ ment and as such (R,~) is a partially ordered set (see Proc. Amer. Math. Soc. 52(1975)45-49) where x ~ y is defined as xy.= x 2 . A subset S or R is called orthogonal iff the product of every two unequal elements of S is zero. Theorem 1. Let R be a cuberoot ring. Then (R,~) is conditionally complete (i,e., every nonempty bounded above subset of R has a lub) iff is conditionally orthogonally complete (i.e., every nonempty bounded above orthog­ onal subset of R has a lub). Theorem 2. Let R be a cuberoot ring. Then every countable sublattice of (R,~) has a lub iff every countable orthogonal subset of (R,~) has a lub. The results can be generalized to the case of n-root rings (if n L 2 is a fixed natural number then a ring R is called an n-root ring iff in R every element has a_ unique n-th root). (Received April 19, 1976.) A-430 ROGER MADDUX, University of California, Berkeley, California 94720

~ nonrepresentable relation algebras, Preliminary report. For notations see McKenzie, Michigan Math. J. 17 (1970), 279-287. Let At be the set of atoms of an atomic Boolean algebra b· r are dual, in a certain sense, to those of Lwndon, Michigan KatJJ.,.J. 8 (1961), 21-28. (Received April 20, 1976.)

Analysis (26, 28, 30-35, 39-47, 49) ELEMER E. ROSINGER, Technion, Haifa, Israel - Dept. of Computer Science *76T-B78 A class of nonunique solutions of the Schroedinger equation with the potential a positiv& power of the Dirac delta distribution

The autnor gave (Notices, 22,5,1975, A-514, Abstract 75T - Bl67) function solutions u(x)

00 = u_{x) + (u+(x)-u_(x)) • H(x), X E R1 , with u_,u+ E C (R 1), H the Heaviside function, for the Schroedinger equation (D2+k2+a(o(x))m)u(x) 0, x E R1 , in the case of mE (0,1] u [2,"').

The solutions were not unique for mE (2 ,"'), ct E (0,00 ). The present paper gives a wider class

of nonunique solutions of the above type, for the same values mE (2,00), ct E (0,"'). Form in­ teger, the equation and the solutions are considered within the associative and commutative algebras containing D'(R1 ), introduced by the author (Notices, 22,2,1975,A-310, Abstract 75T­ B39). For m arbitrary, the construction is understood in the usual "weak" sense. (Received January 12, 1976.) (Author introduced by Professor F. Treves.) C. Diminnie and A. 11hite, Gt. Ilon'lvrmture Univ., :)t. 5orcT'li:ntur··', ·:.'., lle778. Gener'ltine; 2-inner product'' in 2 normor1 GP'' CP3. Pr~,; i :-:i rn ~·.·: report. A 2-functional F on a line!1r 2-normed flp:1ce (L, I I·, •I I) is non-'.len·en"'~-. tr:~ -: f F(a+b,c)=F(a-b,c) for all b implies th'tt a :mel c r! on L. See \illite (Math.N?.chr. pLtLr-GO, 190/J)

An operator between two locally convex spaces is a ~+-operator if it is open, has a closed graph, a closed range, and a finite dimensional kernel. By using duality, a tlleorem of Gokhberg and Krein [Amer. Math. Soc. Transl., (2), 13, 185-264 (1960)] and an argument of L. Schwartz [C.R. Acad. Sc. Paris, 236, 2472-2473 (1953)], the following is proved. Proposition. Let E, F be two locally convex Hausdorff topological vector spaces, and T, P

two operators from E into F. Assume that Tis a ~+-operator, P a continuous operator such

that PU c K + N, for some neighbourhood U, a precompact set K and a finite dimensional sub­

space N. Then T + P is almost open and dim N(T+P) < oo

If either E or the range R(T) is complete then T + P is a ~+-operator and K(T+P) K (T), where K(T) =dim N(T) - codim R(T) is the index ofT. (Received February 13, 1976.) A-431 *76T-B81 J.S. BRADLEY, M::Master university, Hamilton, ontario, canada. Interpolation and Lipschitz Classes,

we present some sufficient conditions for the absolute convergence of the Fourier series of functions belonging to certain Lipschitz classes on totally disconnected groups. The tech­ nique used is one of interpolating between certain endpoint results which are proven directly. The results given are shown to be best possible. (Received February 16, 1976.)

H. GUGGENH.iiiiNER, Polytechnic Institute of New York, Brook~ NY 11201. Distribution of Zeros and Limits of Solutions of Differential Equations.

The results are valid for n-th order homogeneous linear ordinary differential equations :iefined on an interval (a,oo) for which all initial value problems have unique solutions. Proposition 1: If r = oo for a third order equation then either limt y1 (t)/y.(t) 21 ·400 J exists for any two solutions or y(t) = (y1 : y 2 : yJ) is asymptotic to a closed, projectively convex curve in the projective plana. Proposition 2: If r 21 = co and the solutions defined by y(a) = y'(a) = 0 do not tend to 0 for t ->CO then for all t > a there exist solutions disconjugate after t and with a simple zero at t

~ The adjoint of a third-order equation with r 21 = oo is one with r 12 = oo • Preposition 4-: :;;:: 2 then for any t* > a there

exists u. bas.is of solutions where y1 is disconjugata after t* , yn is nonoscillatory and tends to 0 for t-w, and for avery £ > 0, \Yi (t)\ <' E an infinity of til'les, 1 < i < n • In the case r = co lim y (t) = 0 • n-2 11 ' i Froposition 6: If n is even, r 1 n-1 1 = w implies rl n-1 = oo • (Received February 18,1976)

*76T-B83 Ed Dubinsky and William B. Robinson, Clarkson College, Potsdam, New York 13676. Quotient Spaces of (~) with Bases.

Let E be a Kothe space. E is (d4) if there exists a basis (xn) for E and a fundamental system of norms

76T-B84 MARTIN H. ELLIS, SUNY, Albany, NY 12222. A necessary condition for attaining d by a Markov joining. Preliminary report. A joint process is an n-step Markov joining if it is an n-step Markov process on the joint atoms. A condition is examined which any n-step Markov joining of two m-step Markov processes must satisfy to attain the d-distance between the two processes. Using this condition it is shown that if a one-step Markov joining of two one-step two-state Markov processes attains the d.-distance between them, then the d.-distance between them is the partition distance (the converse is false). An example shows that a one-step Markov joining of two one-step three-state Markov processes can sometimes attain d when d. is not the partition distance. (Received February 20, 1976.) A-432 Micha Sharir, Tel-Aviv University, Tel-Aviv, ISRAEL. A non-nice extreme operator.

Let the scalars be real. Let X be the unit ball of ~l(r)' for an uncountable r, with the w* -topology induced by and let s denote the unit ball of C(X) * , co(r)' X with the w* -topology. Then the operator T:C(X) + C(Sx)' given by Tf(]l) = ]l(f), for each 11 e:··sx' f e: C(X), is a non-nice extreme operator. (Received February 23, 1976.) (Author introduced by A. Lazar.) *76T-B86 LAWRENCE A. FIALKOW, Western Michigan University, Kalamazoo, Mich. ~ note on guasisimilarity of operators ~ The following results extend those given in [Abstract *-76T-B3, these Notices, 23(1976), A-10.]. Let H be a complex, infinite dimensional, separable Hilbert space and let L(H) denote the algebra of all bounded linear operators on H. Theorem 1. If T and S are quasisimilar operators in L(H), then the Fredholm essent1al spectra of T and S have nonempty intersection. Acknowledgement: In a preliminary version of this paper, the author was unable to prove Theorem 1, and instead posed it as a question. L.R. Williams, meanwhile, independently found a somewhat different proof of Theorem 1, which will appear in his note Quasisimilar operators have overlapping essential spectra. Theorem 2. If T and S are quasisimilar injective bilateral weighted shifts in L(H), then T and Shave equal spectra. Theorem 3. Let Q s denote the set of operators that are quasisimilar to some quasinilpotentq operator. Then Q is closed under countable direct sums. qs (Received March 5, 1976.) A. Y. W. LAU, North Texas State University, Denton, Texas 76203 76T-B87 ~-Measures on Compact Metric Spaces An H-measure m on a space X is a nonnegative normalized regular Borel measure on X such that m(V)>o for each nonempty open set v. A short proof is given for the theorem: if X is a compact metric space, then there is an H-measure on X. The theorem fails for compact Hausdorff spaces, e.g., the long line, but it holds for compact groups (Haar measure). The author invites comments or other proofs of this theorem. (Received February 26, 1976.) *76T-B88 Catherine L. Olsen, State University of New York at Buffalo, Amherst, N. Y. 14226. Approximation by unitary operators. Preliminary report. Let T be a bounded linear operator on a complex separable Hilbert space H • Define the index of T to be zero if the kernels of T and T* are of the same dimension; otherwise, the index of T is nonzero. Let u(T) denote the distance from T to the group of unitary operators on H • Donald Rogers ["Approximation by unitary and essentially unitary operators," preprint] has shown that a necessary condition for the existence of a unitary operator of minimum distance from T is: either T has zero index or u(T) > 1 • We show that this condition is sufficient; for T of nonzero index with u(T) > 1, a unitary operator closest to T is constructed. Related results. are obtained for approximation by unitary elements in a von Neumann algebra. (Recei~ed March 1, 1976.) Athanassios G. Kartsatos, University of South Florida, Tampa, Florida 33620. Perturbed evolution eguations and Galerkin's method.

Let H be a real separable Hilbert space with basis u1,u2, •••• Let Hn' n = 1,2, ••• be the subspaces of H generated by the elements u1,u2, ••• ,un' and let Pn be the projection from H onto Hn. 1 Consider the problems (*) x (t) + A(t)x(t) = G(t,x(t)), x(O) = x0 and (**)n x~(t) + PnA(t)xn(t) ~

PnG(t,xn(t)), xn(O) = Pnx0 • Conditions are given for the nonlinear operators A(t) : D(A(O))~H,

G: [O,T]XH-H (D(A(O)) ~ H, x0 e. D(A(O))) so that the Galerkin approximants xn(t), tE. [O,T], exist as solutions of (**)nand converge strongly and uniformly to the unique solution x(t), t(; [O,T] of (*). This solution is strongly continuous, weakly continuously differentiable and satisfies (*) on [O,T]. Roughly speaking, A is supposed to be m-accretive, and G Lipschitzian-like in t and pseudo­ contactive in u. (Received March 5, 1976.) (Author introduced by Professor M. N. Manougian.) A-433 Wayne C. Bell, North Texas State University, Denton, TX 76203. A Note On A Class of Linear Transformations.

The setting is as in two papers by W.D.L. Appling (J. London Math. Soc. 44(1969), 385-396; Trans. A.M.S. 199(1974), 131-140). Theorem: If Tis in~ and T=T2 then T is the nearest point map for a c-set. (Received March 3, 1976.)

Richard C. Gilbert, Gahfornia State University, Fullerton, California 92634. .!!:._ synup.etric ordinary differential operator whose deficiency nuznbers differ by two. 3 3 Theorem. Su;ppose 0 < r2 < rl, rl f. 2r2. Suppose Po = X • pl = X (-3rl - r2), p2 = 3 2 -2 3 3 2 -2 3 3 2 = x [(3r1 + 3r{2 ) + 27x ], p 3 x [-(r1 + 3r1 r 2 )- 18(3r1 + r 2 )x ], P 4 = x [r1 r 2 +12(3r1 + 2 2 1 + 3r1r 2 )x- ], p 5 = x [-6(r: + 3r:r2 )x- ], P 6 = x[3r:r2 ], P 7 = O. Let Ly = ~=Oik~y, (r) (r) I ) r (r))(r+l) + (Pn-"r-ly(r+l))(rl}. Then L where LzrY = fP7 _2ry } , LZr+ly = (1 2 t

for linear systems y' = A(x)y, A(x) analytic in a sector, to obtain asymptotic formulas for

a fundamental set of solutions for Ly = A y • (Received March 5, 1976.)

N.K. GOVIL and 'l.K. JADJ, Indian Insti<.:;ute of •rechnolo.:;;'~ !-Iauz Khas, New Delhi-11 0029, On Enestrorr.-Kakeya ':.'he ore~ II.

Govil ar.d Rahman [Tohoku t'ie-th. Jour. 20(1'Jc;.'J) i:C•· -1 )!·] have ottained a generali­ sation of Enestrom-i{akeya ':"he ore;,J for polynomials :.-; :i. ~<·1 c omplEJX coefficients. As an C• .•. improvement of this result we prove: :..et p(7.) ~ ~.. ::.,. /(!: o) lle a polynomial of deg- ree n with cocJplex coef'::'icients suci1 that (i)la;;t.. u..c~rl :e;; a :e;; ~. k = 0,1, ••• ,n for some real nur(jbers a and~.~. and (:a) Ia- Ia. ·I;;,: ••• ~ then p(z) has its 11 I;;;, Ti-l 1a 0 I• all zeros in R3 :e;; JzJ ~ 11 2 , wi1ere R. = --1:;- f-R~I''i (.:..-j~' 1;.(41a In~ ;,_;;.,. R~lbl 2 (r.J...-Ia 1 )2 ) ;:>' 1- ' 0 0 c "' c ·c. I Q 1 c ' 1 1 c 2 . 1 - ,. -- 1 ·, . flo I ]2], R2 ~ 2 \ ) + [~; i.lanJ '.:. ~ ~2 ,_ P.] ]·, :·:::: ~ la:~.IR~l[R +R2- ra;J( 1\J '"1 t:I-1 +Sino.-;:: s.i.na. J c Ja a Ia d · jal· ;./ .. 1-'•·,, • - ~~ '·n- n-1 n

76T-B93 A. P. BLOZINSKI, Ball State University, Muncie, Indiana 47306, The modulus of continuity of a

function, Lipschitz spaces, L(p, r, g) spaces and class Exp L(p1 r, g). The function f(x) is Lebesgue measurable on Rn. f*(t) is the nonincreasing rearrangement of f(x) on · (O,oo). f**(t) = (1/t) jt0 f*(u) du. L'if(x;t) = lf(x + t) - f(x)J. Q(x;t) = [x1, x1 + t] X ••• X [x ,x + t]. We put F(t) = . n n k1/ suphE Q(O·t:J;D.)(1/t) J0 lif(• ;t)*(u) du. Our primary Theorem. For t > 0 and ~n > t (we may choose A = 2 t n, k = 1, 2,.:.) lf(xll ~ 2n JAn F(s) ds/s + f**~ n) + (1/t) JQ(O;tl/n) lf(x + y) - f(x)l dy, Consequences. Norms are obtained in terms of the fuodulus of a function, and equivalent to the L(p,q) norms. Simplified proofs of some known embeddings for the Lipschitz A(a,r,q), a> 0, spaces are given. It refines and extends a somewhat similar inequality (A.M. Garsia, Indiana Univ. Math. J. 25(1976), 85-102) from [0,1] to En space. The method there is a [0, 1] proof by combinatorial techniques. The method here is by classical argumenta. We point out strongly: though similar the two results do not compare in all respects on [0, 1]. (Received March 10, 1976.) (Author introduced by Professor Frank W. OWens.)

w. R. Utz, University of Missouri, Columbia, Missouri 65201. Limiting evaporation velocities for vertical soil columns.

The differential equation

dT m dz = 1: + a: a, a > 0: m > l. occurs in the study o£ evaporation velocities for vertical soil columns. For a sample of applications see the references given by Dale- Swartzendruber, The flow of water in unsatu­

rated soils in Flow Through ~Media, Academic Press, 1969, pp 215-292. In this paper A-434 limiting cases, as T + 00 , are treated for all real m > 1 and then it is observed that traditional simplifying assumptions may be ignored in determining an upper bound for evaporation velocities. (Received March ll, 1976.)

*76T-B95 RUSSELL RIMMER, La Trobe University, Bundoora, Victoria, Australia, 3083 Bifurcation of Periodic Solutions in Symmetric Hamiltonian Systems, Breliminary Report A Hamiltonian H: U CR4 -+R satisfies a symmetry property if there is a linear map M, with matrix± diag(l,-1,-1,1) or ± diag(l,l,-1,-1), such that H o M = H. A periodic

solution ~ : lR -+ U of the Hamiltonian differential equations y! = -D.H(u) i = 1, 2 (1) xf = Di+2H(u) 1+2 l is symmetric if M~(-t) = ~(t). Suppose that the multipliers of~ are land exp{±).}. Suppose

that ~ has le~st period T > 0. When )./v = Ao/v0, where v = 21Ti/T and 1. 0 and vo are mutually prime with AQ < vo,the paper describ'Os the g">neric behavicur of periodic solutionB of (l)

which bifurcate from~- Generically, all bifurcating periodic solutions of (1) are

symmetric when v0 ~ 2. However, when Ao = 0 two generic situations arise : either, (i) there is one family of symmetric periodic solutions of (1), or (ii) there are two families of unsymmetric periodic solutions of (l) and one

family of symmetric periodic solutions bifurcating from ~. The author can also describe generic bifurcations when the Hamiltonian H has two symmetry properties using the techniques developed for the above work. (Received March 15, 1976.) (Author introduced by Dr. Brian Davey.)

*76T-B96 Steve Wright, Indiana University, Bloomington, Indiana 47401. Uniqueness of the Adjoint in B(H). Preliminary report.

Let A be a Banach algebra with involution *· We say that * is a C* - involution if 1/ "<"'"' 1/ :::: /I at// L J va, EA. Let H be a Hilbert space, and let B (H) denote the algebra of all bounded linear operators on H. Suppose A is a Banach subalgebra of B(H). Then the following theorem holds: Theorem. The only C* - involution that can be admitted by A is the operator adjoint. The theorem is first proved under the assumption that A is commutative, using "··

direct-integral type decomposition of representations of C0 (X) due to G. W. Mackey. The general case follows by localizing to commutative subalgebras and then apply- ing the Hermitian decomposition to • elements in A. (Received '-larch 15, 1976.)

76T-B97 MARZUQ,M.MAHER,Iuwait University,Kuwait.Integrability of trigonometric cosine series with quasi-monotone coefficients.

Concerning integrability of trigonometric cosine series,•ve prove a general theorem which generalizes a result of Yong [Monatsh. ~~th.,69 (1965),4~1- 430].By lf(x)~[a, b] , O'a ~b(oo or -"'-·(a(_ b~o,v:e denote a non-negative function !f(x), not identically zero, such that x-a lf(x) t and x-b lf(x) i as x increases in {O,oo). By lf(x)-(a, b) , we denote a functioa \J(x) such that for some positive E., lf(x)~[aH, b-io] Chen [studia Math., 24 (1964), 61-88 ].Theorem. Let \f(x)~(-1,0), and let ia \be a q·~a<;i-monotone ~ L n sequence. Then ~~~~ lf(t) ak is convergent if and only if

1 00 2 a 0 + 2:~}n cos nx converges everywhere to .f(x) except pc-c.:ibly at x=O, and !f(x) f(x) ( L(0,7i) • (Received March 26, 1976.)

76T-B98 Jeffrey F. Jones, Brown Unlversity, Irovic:.epcf', >:IJouc r:· 'l ') '.1] ~:. Generators of A(D) (II) .. Ireliwinary ~eport.

Let D denote the closed unit

functions continuous on ll, holomorphic on iJ 0 • For l', G < ;, (:J) , let [F, G 1 denote A-435 the smallest closed subalgebra of A(D) containing F, G, and the constants. Suppose and C separate points on iJ and are smooth on D. It is known that if for each :.: ( .J, !:'' (z) 'I 0 or G' (z) 'I 0, then [F,G] = A(D). It was noted previously that there is 1;" i1(u) separatinc; points with (z-1) 3 and smooth on D such that [H,(z-1) 3 ] t A(u) [aLstract 76'.i.'-B30, AHS t;.£_tice:!!, Feb. '76]. II was constructed so that H(z) - 1, (:.:") nus an inner sing..1lar factor on a subrec;;ion of D, where z and z* are points iuei,tifiec. under (z-1) 3 • 'l'lleoren.: If this situation does not occur, i.e. if g (z) - ,,.(z*) is outer, .:1nu 'i is a sir;ooth function in l'.(D), then [g,(z-1) 3] = A(D). (Received March 29, 1976.) 76T-B99 J. s. Byrnes, University of Massachusetts at Boston and ETH-Zurich. Polynomials with coefficients of Modulus one. Preliminary report. We consider the class ~n of all polynomials of the form gn(a) = I~=oexp(aki)zk , where the ak are real constants and z = exp(2xia) • We show that the function p E ~ given by P(a) = INk- 1 I~- 1 exp(2xijkN-1)zj+kN satisfies: (1) For any N2-i 1 =o J=O -1 e > o (e <2), IP(a)l = N + E for e ~a~ 1-e , where IEf < 1 + 5(xe) ; and

(2) IP(a) 1 ~ 2N for all a • Using these P 1 s , we then show that for any positive

integer n there is a function fn which is "almost" in ~n and which satisfies If (a) I = (n+:l) l/2 + 0 (n1/ 4 ) for all a • The former result improves upon a result n of J. E. Littlewood [J. Lon. Math. Soc. 41(1966), 367-376]. (Received March 29, 1976.) *76T-Bl00 ARTHUR D. GRAINGER, Louisiana State University, Baton Rouge, Louisiana 70803. On the nonstandard duality theory of locally convex spaces. Preliminary report. Let lK be either the reals or the complex numbers, let E and F be vector spaces over IK and let <· · •, • · ·> be a bilinear functional on E XF such that (E,F,(···,···>l is a dual system. For A c *E , a nonstandard model of E , C(A) c A is defined as follows: z E C(A) if and only if z A and Sz <. A for some infinite S E *lK , a nonstandard model of lK A The set fJ.(A) is called the pseudo monad of A • Relationships among C(A) , Fin(A) and Ai are developed, where are those z * F for which (a,z) is an infinitesimal for each a E A • These relationships are then used to obtain a necessary and sufficient condition for a locally convex, linear space to have invariant nonstandard hulls. Also pseudo monads are used to obtain the following result: Theorem. Let E be a vector space over IK and let ~:IK XE ~ E denote the scalar multiplication map. Let e be a topology on E for which vector addi­ tion is continuous. If ~ is continuous at (0,0) then the map X~ AX is

I)-continuous on E for each A E IK such that lA I ~ 1 • (Received March 29,1976.) *76T-Bl01 Professor Stephen Grossberg, Boston University, Boston, Massachusetts 02215. Pattern Formation E,r_ the Global Limits of ~ l'lonlinear Competitive Interaction in .!!. Dim~. Preliminary report. Let xi= A(x)f(xi)- B(x)g(xi), x = (x1 ,x2, •.• ,xn); i = l,2, •.. ,n; n;;, 2. This system generalizes ~- = -~x. + (S-x. )f(x.)- x. E f(xk) which describes a neural network undergoing mass action inter- 1 l l l lk>'i actions via the signals f(xi) in an on-center off-surround anatomy. Theorem. Suppose f(w) and g(w) are continuous, satisfy f(w) ;;, f(O) = 0 = g(O) < g(w), w > 0, and have bounded piecewise-derivatives; A(w1 , ... ,w) and B(w1 , ... ,w) are positive, continuous, and have bounded piecewise-derivatives such aA n aB n that -- ,; Q :5. -- , k = l ,2, ... ,n; f o;; gh where h( w) describes "a sequence o:t; hillE; and yalle;y-E; awk awk with successive hills no steeper and no higher than previous ones "as w increases; and lim sup h(w)

A-436 76't-B102 Gloria Potter Gagola, Joseph B. Harkin, and James N. McNamara, State University College of New York at Brockport, Brockport, New York 14420. Pick's Measure in the Plane. Preliminary Report. Let r be a positive integer. We define a grid, Gr, of length 1/r in the plane by Gr = {(x,y)

x = n/r, y = m/r for some integers m,n}. Then a polygon, Pr, is said to be an r-grid polygon if each

vertex of Pr is at a grid point in Gr. The area of Pr is calculated using Pick's Formula, A= l/r2 ·

[B/2 + I - 1] where B is the number of grid points in the boundary of Pr and I is the number of grid

points in the interior of Pr. The collection of all such r-grid polygons forms a sequential covering

class of the plane and we define an outer measure~*, which we call Pick's outer measure by

~*(A) = inf {n~l (Area of Pr) [ nQl Pnrj A} where A is any subset of the plane. Let M be the collection

of measurable sets with respect to~*· Then M is equivalent to the collection of Lebesque measurable

sets and the restriction ~*1M is equivalent to Lebesque measure. (Received March 31, 1976.)

Oklahoma, Norman, Oklahoma 73069. Existence of *76T-Bl03 JAMES R. WARD, University of strong solutions to quasilinear differential equations in a Banach space.

Let X be a real, separable, reflexive Banach space with norm II· II and dual x'' We use + (_J) to denote strong (weak) convergence in X , and R+ = [O,oo) Consider the quasilinear Cauchy problem (1<) x' + A(t,x)x = f(t,x),t::: O,x(O) =a EX • Assume A(t,u) is , for each (t,u) E R+ x X , a positive bounded linear operator on X such that the map (t,u,v) + A(t,u)v is weakly continuous on R+ x X x X into X i.e., if Let f : R+ x X + X t + t 0 , u _J u0 , v--' v0 then A(t,u)v--' A(t0 ,u0 )v0 • Theorem. be weakly continuous. If there is a real function g(t) E L1(R+) with c ·~ ~ such

that [[f(t,ulll S g(t)[[u[[ for (t,u) E R+ x X , then for each a EX there is a strong solution x(t) to (1<) with [[x(tlll S (1 - 2c)-1 1[a[[ for all t E R+ . The proof ls based on the Schauder-Tychonov theorem. (Received March 31, 1976.)

*76T-Bl04 Kenneth J. Preskenis, Boston College, Chestnut Hill, Hass. 02167. Functions with Negative Jacobian determinant. Preliminary report.

Let f be a complex valued function on D = (I z I ~1), Pf = uniform closure on D of

polynomials in z and f. Theorem. Let g be a complex valued function of a real

variable which is differentiable in a nbd. of [0,1). Assume f = zkg(!z[ 2k) for z in

a nbd. of D where k is a positive integer. If f has negative Jacol.Jia'1 determinant,

then Pf = C(D). Theorem. If f is an ACL 2 function in a nbd. of D wh.1ch satisfies

Refz~lfzl a.e. in D, then Re[(t-s) (f(t)-f(s)))~O for all s,t in D with sit. This

last conclusion together with the condition f-l(f(a)) is countable for each acD implies Pf = C(D). It is an open question whether an arbitrary smooth function f

with negative Jacobian determinant satisfies Pf = C(D). (Received April 1, 1976.)

Texas 76T-Bl05 WILLIAM D.L. APPLING, North Texas State Univ"rsity, Dc-nt,lil, 7620?, :g~and~~er Dj,stribu!_ion Fu!!ctj,£~·

Notation and notions as in previous abstracts. Suppose k is a real-val­

ued continuous function on IR, A in pB' min p~, and for all (x,r· in IRXF,

B(x)(I)~{O,l}, l in B(x)(I) iff there is y in A(I) such that y < x, C in

B(x)(I) iff there is yin A(I) such that x S y, g(x) = )UG(B(x)rr:, L), u(x)

ruL(B(x)m)(I). Theorem l. If r < t, then (u(r)-g\t))(t-r) S )uLU Am)-G(

Am)](I). Theorem 2. If k is nondecreasin~ and (p,h,q) is (u,k,~) or \g,-k,

A-437 u), then)~: hdp ,:s )uG(h(A)m)(I) ,:s \uL(h(.A.)m)(I) ,:s ):: hdq. Theorem 3· If k(x) ~ 0 on ll., then ~~~min{dg,du} ,:S fuG(k(A)m)(I) ,:S fuL(k(A)m)(I) ,:S r:: k max{dg,du}. Corollary 1. The following are equivalent: 1) luA(I)m(I) ex­

ists, 2) g(X+) = U(X+) in It, 3) ~:: xdg(x) = \:: xdu~x). Corollary 2. If )UA(I)m(I) "xists, then ruk(A(I))m(I) = 5~:kdg = r::kdu. (Received April 2, 1976.)

76T-Bl06 ROBERT M. HARDT, University of Minnesota, Minneapolis, Minnesota 55455· On boundary regularity for integral currents or flat chains modulo two minimizing the integral of an elliptic integrand. Preliminary report.

Suppose ~ is a positive elliptic integrand of degree m and class q;:: 3 on Rn , T e Im(Rn) is absolutely ~ minimizing [H. Federer: Geometric measure theory, Springer, NY, 1969, §5.1], and spt aT is a submanifold of class q of Rn Theorem. Near each x in spt aT with IElm(IIT\\,x) =~ , spt T is a submanifold with boundary of class q -1 Theorem. If n = m + 1 and spt aT is contained in a class 2 hypersurface bounding a uniformly convex set, then, near spt aT , spt T is a submanifold with boundary of' class q- 1 Similar facts hold for flat chains modulo 2 minimizing the integral of a positive even in­ tegrand which is elliptic modulo 2 [Ibid., 5.3.21]. All these results generalize some work of W.K. Allard on the area integrand, [On boundary regularity for the Plateau problem. Brown University dissertation, 19')8] or [On the first variation of' a varifold: boundary behavior, Ann. of' Math. 101 (1975), 418-446]. (Received April 2, 1976.)

*76T-Bl07 A. R. Reddy, Institute for Advanced Study, Princeton, N.J. 08540. A Contribution to Rational Approximation.

Recently, Erdos, Newman and Reddy, Advances in Math. (to appear); Erdos and Reddy, Advances in Math., 20(1976) (in press); Freud, Newman and Reddy, Quart. J. Math. (to appear); Meinardus, Reddy, Taylor and Varga, Trans. Amer. Math. Soc., 170 (1972), 171-185; Newman and Reddy, Pacific J. Math. (to appear) studied the problem of approximating certain classes of functions on the positive real axis by rational functions of degree n. In this note we announce the following:

THEOREM: Let f(z) = ~ akzk be an entire function of order p (l:::_p -=-ce=------,-----,-- (Received April 5, 1976.) f(x) Q(x) Loo[O,oo)- 8n 6 m(7T)n/Pw-n/P

*76T-Bl08 r:.vr:~~KATl'.HA~f.l\_:··-i, .'.adura Col.lc<0, ·radnr.3i f1~5n11 r!N1 ,~,_.• ~-r~S.PA'-f,~SKAl·fY, A .. V .. C.College, ~·fayurar:! !'0930'1(Tndi.:q). t~Y!0Z...~..!:~ 1 .1rr. ~,r,-.l_i_-..·!·,~rv J-,:nnrt~

Lt?t: Y ~1c thf) c1asc; nl. .'"'J 11 fuz"':v sPts :~n_ 2 -.,-i_vpn .sP.t X ov~r the set J of all rational

nu:nhers in I, the unit interval. Ur first ;~f'fine th{': f"tt7.7.V -.~-nut€'r ....,0ACJun" on a set A in F vdth

respect to a fixed point x in

fuzzy set outer measure on A. ;.;·ith rcSfH''C~ to ;~11. th0 r-n-!!~tf; ir! f .• :~ere ~·re n~:'.~ark that (1) the fuzzy

(Received April 5, 1976.) (Author introduced by Dr. T.V. Lakshminarasimhan.)

*76T-Bl09 ERIC P. KRONSTADT, University of Michigan, Ann Arbor, Michigan 48109 , and CHARLES W. NEVILLE, Central Connecticut State College, New Britain, Connecticut 06050. Interpolat_~.ng Sequences for Hardy Classes in Polydisks. Preliminary report. Let Un be the uni.t. polydisk in ltn, let Hp (Un) (1 ~ p ~,_,a) be the usual Hardy spaces of .., functions on Un (as de_f"-.;"l.~d, e.g. l·n R d' ' b k) d 1 t A l - .,.~ u ln s oo , an e = z.~a i ;i=l ,./' - }'(a1.. il' · · · ' a in ) >"' i=l ,""

A-438 be a sequence in un. We will say A is uniformly separated (US) if there exists a uniformly bounded sequence f 1 ,f2, ... of lf"'C~l functions such that f.(a.) = 0 if i;olj and f.(a.) = 1. (o I 2 1J l. ., l. l. For fe HP(~) define TPA(f) = l (1 - I a· ) ... (1 - fa. I ) 'l~'f(a.) J . 1 . ' 2 •n 2 l. l.= ,oc . . Fix ti"D and let D =~·cz , ••• ,z lean: (1- (z.l l//1- z. />t, for j=l;2, ... ,n-l] z. 1 n J J Theorem: If AC D then ~ (Hp (un) ) =j p iff A is uniformly separated.

The above theorem generalizes results of Carleson, Shapiro ind Shields, and Kronstadt. The following counter-example indicates the differences between u and un.

~: There is a sequence Ac u5 such that T! (H2 cu5) ) = } 2, but A is not uniformly separateli. (Received April 5, 1976.) ry€fl! .. B110 Tienchien Chu, Columbia University, New York, N.Y. 10027. 'l'he Weil­ Petersson metric in the moduli space. Preliminary report. 'l'he Weil-Petersson metric in Teichmuller space is known to be incomplete (see Abstract 75T-Bl98), so is the Weil-Petersson metric in the Riemann space. We show that in the Weil-Petersson metric the completion of the Riemann space is the augmented Riemann space, that is, the standard compactification of the Riemann space, and that the Weil-Petersson volume of the Riemann space is finite. 'l'he calculations proceed in the mixed Schottky Teichmuller space and use a basis of quadratic differentials constructed by Bers (Ann. of Math. Studies 79). 'l'he proofs are to be included in the author's Ph.D. dissertation. It has been communicated to the author that Howard Masur has found a similar result about the completion of the Riemann space. (Received April 12, 1976.) S.M. Shah, University of Kentucky, Lexington, Kentucky 40506. Approxi­ mations to meromorphic functions on the positive real axis. Preliminary report. Let f(z) be an entire function such that f(n) (0) ~ 0. Let ~n denote the col­ lection of all polynomials p(x) of degree not exceeding n, and 1 1 Let L(x) be any non-decreasing positive and II f(x) - p(x) slowly changing function on [N ,co). Theor.em 1. If E; 1/nL (n) < oo then

lim inf(A ) exp(n/L(n))=O. Theorem 2. Suppose (i) log L(x)/loglog x 4 1/c, n + co o,n

o (x) is any positive function and L ( (x))/L (lo0 x) 4 t, t (x) c . wh ere o<:t ( x l -_ ( logx) where t (x) +t as x+oo and conversely. 'J:'nen

l!!\!1i!.lol.P loglog (l/A0 ,nl /log n>l+A-c where A=lh!l!ii1.UP L(n)/L(~ log(n!/f(n)(OJ)). ·rnese theorems extend some recent results of P. Erdos and A.R. Reddy (Received Aprill2, 1976.)

*76T-Bll2 ROBERT KNO'tiLES, The University of Connecticut, \ve.terb::.ry, CT (/710. Dound-finite Locally Convex Spaces.

If' a locally convex sp~:,ce E has tl:.e property U;e.t ever~· :cu';·.>.

subset is finite-dimensional, we say that E is a~-~ e;•c.c... In

this paper we discuss permanence properties, exr..mples, ~:nd equivd•?nt cl:ar- acterizations of such spaces. There is a connection with the theory cf'

bornologies which we use to obtain a necessary and eufficient concl!.t.io!·l ror bound-finiteness in terms of topologies on the dual space. (Received J\.pril 13, 1976.)

76T-Bll3 ALUA.NDER G• .RWI, Institute of Fine Mechanics & Optics, Leningrad, 197101. Nonlinear egU§tions of the oscillation theory. !fhia ia the improvement of the JLrevious results { thfseN otices 22, 1975, 75f-:S172). Consider eq. Tu.::; Au.+ FU=f (!},where 1) P: E-+[""is a hemicontinuous map, A-439 being a re:nexive BanacJl sP,ace wijh a dual map J /l]u/1 =1/U/1 1 VIIGF; A is a closed linear operator, cet~(A H:£ ,2(r =2J(A)j Re(Ttt-Tv;u-JJ=O/:!Y u = zr, YU, II G 2(T) ,· there exists a sequence of linear bounded operators A" such that A.. u -+ A l<. V'I.<~~{A), A: u. ...., A*u., VIA'f!l!J(It'fJ•·{#-'f"'f!."t(..,fi;?I;'U.:;,. ?f) :i-0' "'1/tl, zrE E; 7.:. _:. A,,:t- F ;.' ' «e.(T,.U,Lt.h ~(/IU.II)IIu.l(·l·''@'i!, "r(t~ Whm''ht':t'P) IJ(t:l-Hoo q..j t -Hoo; 2.)/le(Tu-Tv; «-zrk lfe~t;''\ lllO, lf4E..,~)/ ~~~~1/~c/1141/+C{O, vc,o, t.tt:E, F:E...., E ) != being monotonic;/IFu-F?r/#CfP)I/ll-11/1,/11.{/l~f,llllll.~.f. Denote I<~::.(>.J +At!·E~E1 ..\'0. Theorem 2. If 3) holds then the solution of- · eq. {1) exists, J.s uni9.u.e,,.the. map;,~i:t;IIAis, ,c:l(?ntinll,ous~ )!ere~ is the space with the normJu.Jt= Re'flfli,iu.1l ','1th~&1f 'l\j: ltlht('!Com...,Pl'etioi'l ot~VA)relative to thi~ norm. Moreover 'the IIJtftatlilffApl!'OdiiifSJU~;,<:r,.\l<'~U.~,.-RxFui<.,.R,d, li, EE bemg arbitrary, A being sufficiently large, converges not slower than the geometry progression with the denominator o< q. " 1 to the solution of ••• (,). Theorem 2 assumptions hold for the oscillation in the loop consisting of a passive· stable linear twoport L and a nonlinear twoport.;V, .f. being the source of the current, A being the operator admittance of L, F being the characte­ ristic i = f: U. of ./{. (Received April 12, 1976.)

*76T-B114 K.P.R.Sastry, DeDt. c;f Maths, Andhra University,Waltair 530003, INDIA and S.V.R.Naidu, Dept. c;f A!)Dlieil Maths, An<'lhra University, 'lialtair, INDIA. Cc;nvexity cDnditic;ns in narmed linear soaces. Preliminary report. A new cc;nvexity cc;nditic;n, 0-canvexity, on a '3amtch soace X is nefineil as fc;llows: X is 0-cc;nvex if there exist a oositive integer nann 0<£<2 such that whenever x 1 , •• ,xn are of unit norm, '3 1~i

LOHELL J. HANSEN, University of Maryland, College Park, Md. 20742 76T-Bl15 On the growth of entire functions which are large on small sets. Prellmlnary report.

Let f be an entire function and M(R) = MaxiZi=R lf(z)l. For c > O, put E(c) {z: lfCz)l > cL P. Erodes [Hayman, New Problems, f12.40] has raised the question of what growth is imposed on M(R) by the assumption that E(c) has finite area for some C• W.K. Hayman has conjectured that oo R f log log MCR) dR < ""· Our theorem affirms this conjecture, and considers as well the consequences of weaker hypotheses on the area of E(c).

Theorem: Let Q be a component of E(c) and put A(R) = Area of [Q n {lzl < R}]. Suppose that A(R)R-2 --+ 0 as R--+ +oo. Then (i) There exists a constant K = K(f) so that

log log M(R) > KR 2/A(R) for a11 large R; and,

(ii) If A(R) is bounded, then J"" log JogR M(R) dR < "" · We also obtain analogous results for subharmonic functions. (Received April 14, 1976.) 76T-Bll6 S.K.Khasbardar and N.K.Thalmre,Department of Hathematics,Shivaji Uni­ versity,Kolhapur;India-416oo4. Reducing subspaces of operators vrith !lQ_ijl ometric p;otr'ts. In this note ,.,e prove the result : Sup:;Jose that an operator T e (B (H) has no isometric part (i.e. there is no nonzero vector x e H such that A-440 " ~~~ =II x 11 for every n = 1 , 2 , - - -) • Let K be the closure of the range of I -T*T A subspace M of H reC:,_·ce:o T iff H = closed span { T*nr : f e s , n ~ o for some unique subspace S of K \vhich is invD.ri:?Jlt un<'.cr ( I-T*T) TmT*n ~ *n for every m 1 n = 0 , 1 , 2 , - - - • In this C8.Se 11 = closed spfill { T' f : f is in the orthogonal complement of S in K , n ~ 0 }.

As an application vTe have the corollary If T is an oper<•.tor \-ri th no isometric part and ( I - T*T ) has 1-dimensional range , then T is irreducible • [Ref-. Gtiyker : Proc .A.ll. s. 45,1974,411-413 .j (Received April 15, 1976.) (Authors introduced by Professor P. A. Fillmore.) University of New Orleans, New Orleans, La. 70122. Entire *76T-Bll7 GARY G. GUNDERSEN, Functions with the ~Zero-One set, Preliminary report.

Define the zero-one set of f(z) as the pair of sequences Ean 3, Ebn 3 such that f(an)=O

and f(bn)=l, where multiple occurrences in the sequence correspond to points of corresponding

multiplicity, and such that f assumes 0 or 1 at no other points.

Now suppose two entire functions f,g have the same zero-one set and, say, f has finite

order. Then I have found the necessary and sufficient form of both f and g when they are

unequal. (Received April 19, 1976.)

*7~'1'~oB118 A.T. Dash, University of Guelph, Guelph, Ontario, Canada NlG 2Wl. As an operator grows, does the boundary of its spectrum grow?

Let T be an operator on a complex Hilbert space H and Let M be an invariant subspace

of T. We denote the restriction of T toM by TIM. The spectrum of T and its

boundary are respectively denoted by cr(T) and o(cr(T)). Theorem. Let M be an

invariant subspace of an operation Bon H. If cr(B)ccr(BIMl, then o(cr(BIM))c3(cr(B)).

Corollary. If A is a subnormal operator and B is its minimal normal extension, then

3(cr(A))c3(cr(B)). Various examples are discussed which make the result of our theorem

sharper. (Received February 20, 1976.) 76T-B119 D. S. JERISON, Universite de Paris-Sud, 91 Orsay, France. A remark on a theorem of R. Nevanlinna. Preliminary report. Let f be a function in H00 of the disc, and let b be a Blaschke product. R. Nevanlinna (Ann, Acad, Sci, Fenn. Ser, A 32,7 (1929), 48) has proved that if the coset f + bH00 contains at least two functions of norm :§ 1, then it contains a unimodular function, It is shown that the conclusion can fail if the hypothesis is weakened by replacing "two functions" by "one function". More precisely, Theorem. If f belongs to B, the closed unit ball

in H00, and if f is continuous and of modulus one on an arc of the unit circle, then there exists a Blaschke product b such that (f + bH00) nB = [f}. In particular, there are functions f in B that are not unimodular but for which (f + bH00) n B = (f}, The proof makes use of an implicit condition on f (due to Denjoy, ibid, p, 42) that guarantees that (f + bH00) n B = (f}, (Received March 8, 1976,) (Author introduced by Professor Stylianus Pichorides. )

Applied Mathematics (65, 68, 70, 73, 76, 78, 80-83, 85, 86, 90, 92-94)

1&r-'C~8 PADAM C JAIN and B.S.GOEL, Indian Institute of Technology,Pow;o:·j, Bombay-4000?6 • .A NUMERICAL STUDY OF UNSTEADY LAMINAR FORCED CONVEC~ION FROH A CIRCULAR ..(:YLINDER A numerical investigation of an unsteady laminar forced convectic-n from a circular cylinder is presented. The Navier-stokes equations and the enerf!.Y equ-

A-441 ation for an unsteady incompressible fluid flow are solved by the finite diff­ erence method. The results are obtained at Reynold number 100 and !00. The temp­ erature field around the cylinder is obtained throughout the region of computation and is sho-wn by isotherms at different times. The variations of the local Nusselt number around the cylinder at different times are computed. and shown lly graplls. The mean Nussel t number am the strouhal number are also calculated.. The computed results are compared with the other available experimental. and theoretical results and are found to be in good agreement with them. {Received January 21, 1976.)

76T-C29 EDUARDO D. SONTAG, Center for !48-th.System Theory, U.of Florida, Gainesville,Fla. On Realizations of Discrete-Time Nonlinear Systems. Preliminary report. Fix an infinite field k. A (scalar, polynomial) input output map is a sequence f = f1, f2, ••• where ft: kt ->k is a polynomial map for all t, such that ft 0, ••• ,0)=0 and ft+l(O,.)=ft· {Interpret ft{ul, •.. , ut) as output at time t+l if input is Uj at time j.) The free monoid on the nonnegative integers is A=N*. The formal power series s(f) on the variables A has c_oefficient s (f) a of a={ a1, ••. , at) equal to the coefficient of utl, ••. , uit in the polyno­ mial ft(uJ_, ••• ,ut). The (Hankel) matrix with rows and columns indexed by A and {a,b)-th entry s(fJa.b is H(f). f is bounded if s{f) is a series in L* for some finite subset L of N. A state-affine system is given by an n>l and equations x(t+l)=F(u{t)~+G{u{t)), x{O)~, y{t)=Hx(t), where x(t),t=O,l, .•• is in- kn, F(.) is a matrix and G(.) a. vector of polynomial functions, G(O)=O, and H is a {constant) 1 by n matrix; it is span-canonical if it is observable and the states reachable from 0 span Jtll. RESULTS: {I) The following are equi­ valent for an I/Q map f: {1) f is realizable by a state-affine system; {2) H(f) has finite rank; (3) s(f) is a rational power series; {4) f is bounded and realizable by a system of {arbitrary) polynomial difference equations; (5) the space of observables of f is finite di­ mensional. {II) Arry realizable f has a span-canonical realization, unique up to a linear change of coordinates in kn. Corresponding definitions and results hold for maps with arry number of input and output channels. An algorithm provides a span-canonical realization direc­ tly from H(f). The above results 'and algorithm generalize and unif'y those known to date, including linear, internally-bilinear, and I/O-bilinear systems. Furthermore, condition (5) above expresses an approximation to arbitrary I/O maps by realizable bounded maps, in the context of a realization theory for arbitrary polynomial maps, since state spaces of canonical realizations are characterized by the algebra of observables ·of f. {Received February 18,1976.)

T.Karunakaran, Deptt.Elect.Engg., Indian Institute of Technology,New Delhi- 11 0029, India. Tolerance realisation and conditions for finite termination Preliminary repor .

For an automaton whose output space is endowed ~1ith a tolerance structure (a reflexive symmetric relation) the realisation theory has been developed (T. Karunakaran: 'Toler­ ance Realisation', National Systems Conference, NS0-71f, New Delhi; also in 'Advances in Systems Engineering 'Tata-Mc Graw Hill, in press, (Eds) s.c.Dutta Roy et.al.) which results in (non-unique) reduced models and yieldS the well-known Nerode realisation as a special case. The theory is adapted for line2.:• s:\'stems by using the ~orrespon­ ding notion ::.f additive tolerance (S.MacLane : ~ouologc·, Springer,1965). Further, by imposing one more conditinn in addition to the finite termination conditions needed for the corresponding Nerode realisatinn (M.A. Arbio and H.P. zeiger :'On the releva­ nce of abstract algebra to control theory.,' Auto:uatica .5_, pp 58q-6o6, 1969) we prove that finite termination is possible in the tolerance case also. This involves the consideration of a particular tolerance relati'ln on the input string space induced by the output space tolerance and further the assur~ption that there exists a right inva­ riant equivalence relation cc:>arser than the Nerode equivalence relation. {Rec~ived March 8, 1976.) (Author introduced by Dr. S. N. Patnaik.) 76:r-C31 B. D. Sharma, University of Delhi, Delhi, India and Siri Krishan Wasan, Ramjas College, University of Delhi, Delhi, India. Weak Cyclic and Quasi Cyclic Codes. Preliminary report.

We introduce a class of weak cyclic codes and a class of weak quasi cyclic codes which have cyclic and quasi cyclic structure not over the whole word length but over its different parts. It is shown that an abelian code can be regarded as a weak cyclic code. It is shown that the product of two quasi­ cyclic codes is a quasi cyclic code and the shift length preserved by the product code is the product of the shift lengths preserved by the subcodes. It is shown that the product of two weak cyclic codes is also_a weak cyclic code. A product different from the Elias product is considered for quasi cyclic codes of constant rate. (Received March 15, 1976.) (Authors introduced by Frank 'Anderson.)

Andrew Wohlgemuth, University of Maine at Orono, Orono, Maine 04473. A Histocompatibilicy Model. Preliminary report. Consider a 0-1 matrix Bas a binary relation between the sets of rows A1(B) and columns A2(B) of B. Let Lk(B) be the lattice of subsets of Ak(B) closed under the closure operation X~ X defined by B for k = 1, 2. B is called reduced if for {a}, X~ Ak(B), ~=X implies a EX. For 0-1 matrices A, B de- A-442 fine A~ B if Lk(A) ~ Lk(B). Define A~ B if Lk(A) is isomorphically a subposet of Lk(B). A labeled reaction matrix is a triple (A, ~.B) where A= (aij) and B = (bij) are 0-1 matrices and~ is a rela­ tion between A (A) and A (B) such that a •. = 1 iff b = 1 for all s E i~, t E j¢. (A,¢, B) is monic k k ~J st _ _ if B is reduced and for each s E Ak(B) there exists i E Ak(A) such that i¢ = s. Theorem. For a given A there exists a unique (up to a permutation of rows and columns) reduced B giving a monic (A, ¢, B). A ~ B. Theorem. For given A, B there exists a relation ¢ such that (A, ¢, B) is a labeled reaction matrix iff A < B. In one application A represents histocompatibility testing data and B defines specificities of antigens and antibodies. (Received March 22, 1976.) GAIL A. CARPENTER, MIT, Cambri.dge, Ma. 02139. Nerve impulse equations: periodic bursting phenomena. Preliminary report. Traveling wave ·solutions of the generalized Hodgkin-Huxley model: V= R(cav + g(V,m,n,h)), a~= (1-m)a (V) - ma (V), an= (1-n)a (V) - n~ (V), ah = (1-h)a (V) - hah(V) -- (HH) , m m n l""n n have previously been studied using a geometric approach to singular perturbation problems. rn·that analy- sis we showed that (HH) exhibits single pulse solutions; elongated plateau pulse solutions; and fam­ ilies of periodic solutions which converge to the corresponding pulse solutions as the periods become infinite. We have now proved that a large class of systems of the form (HH) exhibits finite wave train solutions with any number of pulses and a corresponding class of periodic solutions. The wave trains and their periodic solutions may be either normal or plateau. The periodic solutions appear as bursts of activi.ty with arbitrarily many pulses separated by long intervals of rest. More generally, given any sequence of positive integers {Ni} solutions of (HH) with Ni pulses in the ith bursting interval have been constructed. One major criticism of the Hodgkin-Huxley system has been that it could exhibit only single pulse and infinite wave train solutions, missing the often-observed inter­ mediate phenomena which our analysis now includes. (Recei~ed March 26, 1976.) M.R.GAREY, R.L.GRAHAM, D.S.JOHNSON, Bell Laboratories, Murray Hill, N.J. 07974 and D.E. KNUTH, Stanford University, Stanford, California, 94305, The Complexity of Bandwidth Minimization For a graph G and any ordering a of the vertices of G, the bandwidth B(G) of G is defined by B(G) = ma.x la(i)-a(j) 1. We present a linear- (i, j)EE(G) time algorithm for arbitrary graphs which constructs a bandwidth 2 ordering whenever one exists. Given a graph G and integer b, we .show that the problem of determining whether there exists a bandwidth b ordering is NP-complete,

even when specialized to the case of trees. (Received March 31, 1976.)

76T-C35 E.A. GALPERIN, NP Research, P.O.Box 24, Station 1 CDN 1 , Montreal, Que. Canada. Duality between Stabilization and Asymptotical Observation, Preliminary report. The two problems mentioned are duals of each other in the following sense: (i) If the system dw/dt=A'vr+Hu, (')-transpose, with a regulator u = - Q'w becomes asymptotically stable, then the model dz/dt=(A-QH')z + Qy delivers an asymptotic observer for the system dx/dt=,\x •sith unknown x(O) and measu­ rements y(t)=li'x which yields the state: z(t)-x(t) as t-oe, whatever z(O), x(O), -and vice verse.•

(ii) The system dx/,lt,J\x is asymptotically observ

*76T-C37 Charles G. Nelcon, Harvard University, Cambrid~e, Mass. 02138. Multi-headed One-way Automata Let Lk denote the language (1) {f-anbm-1 such that ¢(n, m)} with ¢(n, m) =n =morn= 2m or ••• n km. Theorem: for each N 2 0, there exists a k such that no N-headed finite state automaton (one-way, deterministic) can recognize Lk, but some (N + 1)-headed FSA can recognize Lk.

The first step is to show that a language of the for·m ( 1) is accepted by S

(Soil Sci.Soc.Amer.Proc.l973 0Vol.37 pp.l97-199) solution obtained for the first type condition at z = 0 using the same met.'lod. (Received March 2, 1976.)

76T-C39 MICHAEL NEUMANN, University of Nottingham, Nottingham NG7 2RD, U.K. Partial orderings and iterative methods for least-squares solutions of minimal norm to linear systems. Let BE RmXn and let K and L be solid cones in Rn and Rm, respectively, Three concepts: (i) a K­ Ieft subgeneralized inverse of B, (ii) (K, L)-semipositivity of B and (iii) (K, L)-monotonicity of B, are defined, and applied to obtain convergent iterative schemes for least-squares solution of minimal norm to the system (*) Ax= b, A E Rmxn. OUr results, which extend, to the singular and rectangular cases, earlier results of Vandergraft for a nonsingular system (*), offer a different set of conditions for the convergence of the schemes from those of Berman and Plemmons. (Received April13, 1976.)

*76T-C40 S,K. SIM, Universidad Central Venezuela, Caracas, Venezuela. The uniqueness of global-response function of a linear system, Following M. Mesarovic andY, Takahara ["General Systems Theory", Academic Press, New York] a sub­ space S of the product space of two vector spaces X, Y over a field k is called a linear system. A 4-tuple (C, h, f, g) where C is a k-vector space, f: C .... Y, g :X .... Y are linear transformations and h : C X X .... Y is a function with the properties that h(c,x) = f(c) + g(x) and (x,y) E S iff there exists c E C such that h(c,x) = y is called a linear global-response function of s. A simpler proof of a theorem of Mesarovic and Takahara asserting

A-444 that every linear system has a linear global-response function is presented. Furthermore, (C, h, f, g) is said .- • reduced If h(c,x) = h(d,x) for all x EX implies that c = d and it is shown that every linear system has a reduced linear global-response function which is unique in the following sense: if (C,h,f,g) and (D,.t,m,n) are two reduced linear global-response functions of S then there exists an isomorphism u: D .... C such that m = f • u. (Received Aprill6, 1976.)

*76T-C41 JOHN HAYS, National Wildlife Federation, Vienna, Va. 22180. A gene­ ralization of. the Hardy-Weinberg equilibrium law of population-gen6- t1CS. Prel1m1narv reoort. Def. \Classical) t-genetics. considers only type ("kind") 1 o-g'enetics, also or­ der (' ~gree"). Metatheorem. o- is conservative extension of t-genetics. Lem­ ma 1. An observation set differing in type and order structures a distributive latticeD, derivable from basis J of join-irreducibles, with subset A of maxi­ mal rank. Lemma 2. Only regulars in Dare complemented; but each dED has unique supplement (generalization of complement, cf. "Matrices for free lattice logics~ NOTICES 22(1975),A647). Lemma 3, Dis equivalent to factor lattice Fn1 IIUEJ?lement.f a,. ot ae En i.s n;a. (.l;>J;obabil;i,ties can be assigned to system equi­ valent to complemented lattice; this extends to D by following algorithm, am­ plified elsewhere.) Lemma 4. (1) To elements of A, assign probabilities srua­ ming to 1; (2) if afA, of rank r in Fn, is assigned probability p, assign p/r to atom of a (in J)# distribute probabilities along their chain in a~ithmetic prog~ession; (3) measure~ for universal lower bound; (4) if g =1TfT1 and P(f. 1 ) =e., then P(g) =~ei; this satisfies (latticized) Kolmogo~ov axioms. (By1Fund. Th. of Arithmetic.) Cor. For any ft Fn, P (f) = 1 - P (f). Theorem. Given alleli9 pair, A, B, with probabilities p,q (p + q = 1), ranging in max­ tensity to AJ, Bk, with orobabilities jp, kq (jp + kq = 1); under usual H-W assumptions, equilibrium prevails in one generation. ("Usual" proof.) (Since evolutionary theory, with selection coefficients, etc., is a-theory, does its matching by a-genetics engender new insiqhts?) (Received A:pril 19, 1976.)

Geometry (50, 52, 53}

G. P. Graham, Jr., Indiana State University, Terre Haute, Indiana 47809 Homogeneous coordinates in.!!_ plane of prime power order. Let P be a projective plane of order pm. Let F and F3 be the Galois fields of order pm and p3m respectively, and let G and G3 be their respective multiplicative groups. Since

[G3:G] = p2n + pn + 1 (just the number of points (and lines) in P), define 1-1 correspond­ ences, e and$, between the points of P and the cosets of Gin G3, and the lines of P and these cosets, respectively. Definition: xGiyG iff (xG)e-l I(yG)$-1. Note that the cosets of G are sets of triples of numbers in F. Theorem: For the field plane P over F the corre- spondences e and $ can be chosen so that the cosets are just the usual homogeneous coordi­ nates for points and lines in P. In any case, for all P ot order pm, the incidence relation can be described in terms of the field operations in F and the cosets of G can be viewed as homogeneous coordinates of points and lines in P. (Received February 11, 1976.)

*76T-D10 VJURK A. PINSKY, Northwestern University, Bvanston, illinois b0201. An Individual. Ergodic Theorem for the Diffusion on a Jiianifold of .legatiVe Curvature.

Let l< be a 2-dimensionaJ., complete, simply connected fd<•:r,f1.111lian manifold of non-positive curvature, In a system of geodesic polar 'cor.rdi"ates G is the density of the ldemannian volume element and the Brownian motion is ,;ritten X(t) "(r(t), 9(t)), Theorem 1: lim r(t)/t =lim {f~Gdr/ S Gdrl'/~ alJr.ost smuJ.y • .(;-"JDO l'"-'JOQ D 0 J Theorem 2: If the latter nuantity is strictly positive, then the SI.ectrun, of the La; lace- Beltrami operator is contained in a haJ.f-line~oo1 ~) with ~ < 0, (Received March 4, 1976.)

A-445 *76T-Dll Jli:R

~: let 0 be a point in Euclidean 3 space a3 • let X and Y be smopth :surfaces in a3-o.

Suppose that central projection from 0 is a 1-1 conformal map a£ X onto !, Then localll

either (i) Y is similar to X with center of similitude O, or (11) there is a 2 sphere

centered at 0 such that Y is obtained fran X by inversion with respect to th:ls sphere,

Corollary: let II: be a smooth closed positively curved surface in a3 , let p and q be

dismetrically opposite points on X (i.e, the tangent planes ~ and Tq to X at p and q are

plll'allel), Define 11 stereographic projection" s: (K-p) ~Tq to be the map vhich assigns

to each point x£ M-p the point s (x) € Tq in vhich the straight line joining p to x

intersects Tq , Then 11 is conformal if and onzy if K is a 2 sphere, (Received March 12,1976)

*76T-D12 JAYME M CARDOSO, Universidade Federal do Parana, Caixa Postal 1963, Curitiba, PR, Brazil. Non metrical ~y~ of projection.

We are concerned with a projective space S. Let IT be a plane of S and r and s two straight lines of S but not on IT. If P is a point of S then the straight line p through P and meeting both r and s is said the projecting line of P. If p is not on IT then p interse~ts IT in one point P' that is called the projection of P in this system of projection; if p is on IT then the line p is the pro­ jection of P. We prove the following results: (i) This system is the most general system in which the determination of the projecting lines does not involve metrical problems (the central and the parallel systems of projection are particular cases of the system we are presenting); (ii) The projection of a straight line is a conic (a necessary and sufficient condition is given for this conic to be degenerate). (Received March 17, 1976,)

*76T-D13 R, E, GREENE, Institute for Advanced Study, Princeton, New Jersey 08540 and University of California, Los Angeles, California 90024, and K, SHIOHAMA, Mathematics Institute, University of Copenhagen, Copenhagen, Denmark, Volume-preserving diffeomorphisms ~ embeddings of noncompact manifolds, Preliminary report, We announce generalizations to the case of noncompact orientable manifolds of well-known results of J, Moser concerning volume forms on compact manifolds, (We have obtained similar results for odd forms on noncompact nonorientable manifolds and for noncompact manifolds with boundary, which results are for brevity omitted here.) Definition: An end E of a noncompact orientable manifold M with a volume form w has infinite w-volume if for every compact set K the component of M - K containing

E has infinite w volume; E has finite volume if it does not have infinite volume, ~ 1: If w and T are C~ volume forms on a (noncompact) oriented manifold M, if JM w = JM T ~~ and if each end E of M has either both its T volume and its w volume infinite or both finite, then there is a C00 diffeomorphism ~: M~ M such that ~*w = T. Examples show that the end condition is essential, Theorem 2: If M is a (noncompact) oriented manifold which has a proper embedding (immersion) in ak, k fixed, then for each C00 volume form w on M there is a proper embedding (immersion) of M in ak such that the volume form determined by the induced Riemannian metric is w. (Received April 12, 1976.) *76T-D14 Ralph Alexander, University of Illinois, Urbana, Illinois 61801. Linearly additive plane metrics. Preliminary report. If d is a plane metric which satisfies d(p,r) = d(p,q) + d(q,r) f'or all collinear p,q,r with q between p and r, we say that d is linearly additive, Theorem. Let d be a linearly additive plane metric which is continuous with respect to the usual topology, Then there is a unique Borel measure on the lines of' the plane such that d(s,t) is the measure of the lines which cut the segment st • (Received April 16, 1976.)

A-446 Logic and Foundations (02, 04)

76T-E29 Leo Harrington, University of California, Berkeley, california 94720 and John Steel, State University of New York at Buffalo, Amherst, N. Y. 14226. Analytic ~and Borel isomorphisms. Preliminary report.

Theorem 1 Vx (x4fo exists) ., VA,B ~ WW (A,B are analytic non-borel => :tlf: WW -t WW (f a borel

bijection 1\ f (A) = B)). Let W 2 have the Cantor topology. For A ~ w2 , say A is epa iff 4fo w A n I is analytic non borel, for each interval I • Theorem 2 Vx (x exists) ., VA,B ~ w

(A,B epa and meager => :tlh: W2-+ w2 (h a homeomorphism A h (A) = B)) The r.h.s. of

Theorem 2 can be proved for A,B meager and everywhere .!}~ - lf~ (01. > 2) without assuming

Vx (x* exists). (Received February 23, 1976.)

*76T-E30 STEPHEN H. HECHLER, Queens College of the City University of New York, Flushing, New York 11367. On refining with almost-disjoint families.

Let N be the set of natural numbers, let P *S = {T!! S : IT I = IS I } , and define a family F!! P N* to be

sepa;able iff there exists an almost-disjoint family { GF E P *F : F E F}. Theorem 1. Every family F!! PN of cardinality less than !; is separable. Theorem 2. Martin's Axiom implies that a union

of fewer than !; separable families is separable. Theorem 3. It is consistent with ZFC + ~H + (£ regular) that PN* be decomposable into 1\v separable subfamilies. Define a family F!! PN* to

be complete iff F E F & F!! G + G E F, to be saturated iff F U G £ F + F E F V G E F, and to be ~ongly inseparable iff 3FE F(PF!:F).* Theorem 4. Every separable family can be extended to a complete saturated separable family. Let Mbe the hypothesis that every infinite maximal almost-disjoint family has cardinality !:· J. Reitman recently announced (No:Ueu, 22 (1975), A328) that M im­ plies the separability of every ultrafilter over N. Theorem 5. Mimplies that a family is sep­ arable iff it has a complete saturated extension which is not strongly inseparable. Corollary !• Mimplies that the union of a family of ultrafilters is separable iff it is not strongly inseparable. Define a subset S of I!N- N to be a £::.!!.!U: iff there exists a family U of ~ dis­ joint open sets such that U e U + (Un S = 0 & s,eu). Corollary 2. M implies that a subset of I!N-N is a c-set iff it is nowhere dense. Theorem 6. The conclusions of Theorem 5 and its corollaries are consistent with the negation of M. (Received February 25, 1976.)

76T-E31 Allen T. Retzlaff, Cornell University, Ithaca, N.Y., 14853. Hyper-hyper simple subspaces of V00 • Preliminary report. Let';[ be the lattice of all re. (recursively enumerable) subspaces of an

infinite dimensional fully effective vector space V00 over a recursive field (see Metakides-Nerode, R. e. vector spaces, Ann. Math. Logic (to appear)). An A E ,i_

is called a recursive (or decidable) subspace of V00 if A is complemented in ;f_. Def. A E~ is hyper-hyper simple if there is no recursive list (Fe}eEw of finite ~ensional Fe E't. such that for all e, Fer1 (_v Fi) = (o) and dim(Femod(Av.v Fi)))o. ~~e ~~e Theorem. Suppose A E .t, and the super spaces of A in ;;f_. form a complemented modular lattice. Then A is hyper-hyper simple. Theorem. Suppose A,B E t., A~ B, and A is hyper-hyper simple. Then there exists a recursive subspace C with A + C = B. The priority method proofs require analogues for vector spaces of devices of Yates and Lachlan for r. e. sets, but do not reduce to their theorr:ms. (Received March 1, 1976.) (Author introduced by A. Nerode.)

Iraj Kalantari, Cornell University, Ithaca, New York 142.'-)3. Major subspaces of recursively enumerable spaces. Preliminary ~eport. ----- Following Metakides-Nerode (R.E. Vector spaces, Ann. Math. Logic, to appear) let 1, (V.,) be the lattice of r.e. subspaces of an infinite dimensional fully effective vector space V~ over a recursive field. An R in :t(v.J is a reeursive subspace

(or decidable subspace) if it is complemented in '£.. (V00 ). If A, B E J..(V00 ), B ~ A, B A-447 is a major subspace of A if dim(A mod B) = oo and for every W E ~(VM) with = Span(WvA) V00 we have dim(V00 mod Span(WvB)) < oo. Theorem. Every r.e. non recursive subspace of V~ has a major subspace. The Theorem is analogue of Lachlan's for r.e. sets, but does not reduce to the latter. It requires a different finite injury priority argument suited to vector spaces. Cor. There exists an r-maximal subspace which is not maximal. (Received March 9, 1976.)

(Author introduced by A. Nerode.)

*76T-E33 Murray A. Jorgensen, Biometrics Section, Ministry of Agriculture and Fisheries,Wellington, New Zealand. ZFC minus Powerset as the 'classical hull' of constructive mathematics. Interpreting "set" to mean "hereditarily constructibly enumerable set" we examine the validity of the ZFC axioms. Pairing, Union, Foundation, Infinity and Choice are valid. Extensionality and Powerset fail (diagonal argument). The Separation and Replacement schemas fail if the underlying logic is classical but are valid in weaker logics. This. suggests a "middle ground" for matheinaticiims who are concerned by the highly non-constructive flavour of some modern mathematics, but who resist weaning from the mother's milk of extensional set theory and classical logic. ZFC- Powerset shares many of the restrictions of constructive mathematics, but not the necessity for radical reformulation called for by a rigidly constructivist position. The continuum problem is seen as a penalty for straying too far from the constructive straight and narrow. (Received February 27, 1976.)

76T-E34 Joseph Sgro, Yale University, New Haven, Connecticut 06520. Decision problems for topological abelian groups.

We settle (positively or negatively) the decision problem for the L(Q), L(Qn)new' L(I), L(In) and Ltop theories of topological abelian groups. This new work uses results announced in Abstract 76T-E2, these Notices 23 (1976), Abstract 75T-E79 in these Notices 22 (1975) and results of W. Bauer. Among the results are the undecidability of the Ltop theory and the decidability of the L(Q) theory. We also give some generalizations of the decision procedures to other topological

algebraic structures. (Received March 25, 1976.)

76T-E35 Barry E. Jacobs, Queens College, City University of New York, Flushing, New York 11367. The ~-Union Theorem and Generalized Primitive Recursive Functions.

A generalization to ~-recursion theory of the McCreight-Meyer Union Theorem is proven. Theorem. Let~ be an ~-computational complexity measure and {fclc <~}an ~-r.e. strictly increasing sequence of ~-recursive functions. Then there exists an a-recur­ sive function k such that C~ = c~~ c: . The proof entails a no-injury cancellation atop a finite-injury priority constru~tion and necessitates a blocking strategy to insure proper convergence. Two infinite analogues to (w-) primitive recursive functions are studied. Although these generalizations coincide at w, they diverge on all admissible ~ > w. Several well known complexity properties of primitive recursive functions hold for one class but fail for the other. It is seen that the Jenson Karp ordinally primitive recur­ sive functions restricted to admissible ~ > w cannot possess natural analogues to Grzegorczyk's hierarchy. (Received March 26, 1976.)

*76T-E36 Stanley Wagon, Smith College, Northampton, Massachusetts 01060. The universe constructible from a K-saturated ideal on K.

A well-known theorem of Kunen states that if I is a normal K+-saturated ideal on K then K is measurable in L[I], while an unpublished example, also due to Kunen, shows that there can be a K+-saturated ideal I on K (all ideals are assumed to be K-complete) such that K is non-measurable in L[I]. We study the question: if I is a K-saturated ideal on K, must K be measurable in L[I]? Using the methods of

A-448 Abstract #727-El (these Notices, ~ (1975) p. A-665) we prove the following. Theorem. If I is a K-saturated ideal on K and for some bounded subset of K, Y, lying in L[I], K++ is the same in L[I] and L[J,Y] (where J is a normal K-saturated ideal on K) then K is measurable in L[I]. Corollary. If M is the K-model and V is a K+(M)_chain condition generic extension of M and I is a K-saturated ideal on K then K is measurable in L[I]. Question. Can K bear a K-saturated ideal that does not satisfy the hypothesis of the theorem? (Received Aprill, 1976.)

76T-E37 SAHARON SHELAH, The Hebrew University of Jerusalem, Jerusalem, Israel. On powers of singular cardinals, compactness of second order logic. t'\ Th. 1: E.g. (A) If t'\a is a strong limit cardinal, cfl;\ = a ~· then 2 a< ~(K+) , K a lal + t{~K : cf ~ = ~· ~ a cardinal}. (B) If t'\(K) is the first cardinal t'\ • a of cofinality and Chang conjecture holds (i.e. is a ~ + [~]~ t'\ ), A= t'\(w1) -l· 0 A strong limit then 2 < l:\(w 2). Th. 2: (G. C. H.) E.g. if we add to 1-order logic qf. over a•.tt. of ordered fields, Boolean algebras, (and even .for we get a compact logic; which is stronger than first order even for finite models. E.g.' if T "says" the model is rigid over P, (P, R) have the strong independence property (~x 0 ,+ ... , xn' y0 , ... , yn ( P)[/' xi fo yj + (~)(A xiRz A-, yiRz)]) A strongly inaccessible, 2 • A , then T has a model of cA!dinality A with no undefinable automorphism. Th. 4: If e.g. ITI is regular P (i < t < 2A) complete types in J,(T), then T has a i 0 model omitting each Pi, Th. 5: If T has A independent formulas, T complete, T ~ T1 , lr11 = A, ~~A and there is a A - Kucepa tree with K branches, then I(~, T1, T) ~ min{z~, 2K}, Th. 6: E.g. if A(l) A++, A(n + 1) = (2A(n))++, n 22 a X space then X has a discrete subspace of card. .>.. ~ a a hausdorf (Received April 5, 1976.)

76T-E38 BARUCH GERSHUNI, Bloch Street 39, Tel Aviv, Israel, A justification of the way of writing a class in the form "a, b, c, .•. " (without brackets). The author wishes to justify his way of writing classes (with a first member) in the form "a, b, ... " without any brackets, This is a modification of the way John von Neumann, the discoverer of the concept "class" (1925), wrote it. A set is, nearly according to G. Cantor (1895) and H. Bachmann (1967? ), the result of collecting together ("zusammenfassen") a (suitable) totality into a collective, i, e. a unit. This is achieved by means of the collectivistic operator { } consisting of a pair of braces acting upon the said totality and symbolizing the wrapping of its members by an envelope or a cover. Now the members of the so-called shoreless totalities such as the Cantor totality of all sets are not capable to be collected together and contained in a cover, i.e. to become sets according to the just given definition of set. We give therefore to the shoreless totalities another name than "set" and call them "classes" and write them without braces for they do not allow a cover. We call now~ totality which is written without any collectivizing brackets and is consequently not supplied with a cover a "class". According to classical logic ("tertium not datur") any totality with a first member is either a class or a set. The case of totalities infinite and shoreless in both directions must be considered extra. (Received March 15, 1976.)

76T-E39 P. GIBONE, 62 Avenue Jean Perrin, Sceaux 92330, France. On the stability of free groups, It is easy to see that any nontrivial free monoid is unstable (in fact, this is one of several analogues for monoids of results known for rings, for instance, and more generally one has that a stable monoid with can­ cellation is a group). We prove Theorem, Any free group on at least two generators is not superstable, Of course all free groups can be embedded into w-stable groups (consider groups of matrices; cf. C. R. Acad, Sci, Paris Ser, A-B 280(1975), 603). (Received April 8, 1976.) (Author introduced by Professor G. Sabbagh.)

76T-E40 Heinrich Herre and Helmut Wolter. Decidability !?.!_ the theory T< (Qo:) !?.!_ linear ordering with ~ generalized quantifier _Jk. for regular wo:, Let T< be the elementary theory of linear order and T<(Qa) the extension of T< by the quantifier "there exist at least wa many". A-449 Theorem 1. T<(Qa) is decidable for any regular Wa

Theorem 2. a) If a,a successor ordinals, then T<(~) T<(Qa)·

b) (GCH) If a,a regular limit ordinals >0, then T~(Qg) = T<(Qa)·

c) (GCH) For any regular limit ordinal a>O is T<(Ql) ~ T<(Qa).

(Received April 12, 1976.) (Authors introduced by Professor Kenneth A. Bowen.)

76T-E41 John R. Cowles, Institute for Advanced Study, Princeton, New Jersey 08540. Ordered structures and logics with Ramsey quantifiers. Preliminary report.

For n ~ 1, let Q~ be the logic obtained from first order logic by adding the formation rule: H qJ

is a formula and x1, ••• , xn are distinct variables, then Qx1 •.. xn({J is also a formula.

IJ{ F Qx1... xn({J is given the meaning, "There is an infinite subset I of A such that whenever the

variables x1, ..• , xn are interpreted by distinct elements a1, ••• , an of I, then qJ holds in or·"

Theorem 1. For n ~ 1, the Q~ -theory of infinite linear orderings is decidable.

Theorem 2. The Q~-theory of Archimedean real closed fields is complete, decidable, and model­

complete. (Received April15, 1976.)

76T-E42 DAN MAULDIN, University of Florida, Gainesville, Florida 32611. The boundedness of certain analytic sets. Let X and Y be Polish spaces and A a subset of X X Y. Theorem 1. If A is an analytic set and for each x in X, there is an ordinal O!(x) such that the O!(x)-derived set of the x-section of A is empty, then there is an ordinal /3 < w1 such that O!(x) ~ /3, for all x. A key theorem for the proof of this theorem is the following generalization of a theorem of Lusin: Theorem 2. If A is analytic and each x-section of A is countable, then A is a subset of a Borel set each x-section of which is countable. (Received March 8, 1976.)

Statistics and Probability {60, 62)

T. E. Harris University of Southern California, Los Angeles, California 76T-F8 90007. A Correlation Inequality for Markov Processes in Partially Ordered State Spaces {Xt] is a Markov process whose state space E is partially ordered and finite.

{Ttl is the semigroup of {Xt}. F is the set of increasing functions f: E -+ R1• Assume {X I is monotone; that is , for each t, T maps F into itself. {Xt} is said t t to be positively correlated if Tt(fg) ~ Tl • Tt g whenever f, g e F, for all t ~ 0. THEOREM. In order that the monotone process {Xt I should be positively correlated, it is necessary and sufficient that all trans.itions should be up or down. The last expression means that if {Xt I can jump directly from x to y, then either x < y or x > y. (Received March 23, 1976.)

Topology (22, 54, 55, 57, 58)

*76T-G49 ROBERT A. HERRMANN, Math. Dept. U.S. Naval Academy, Annapolis, MD. 21402 The nonstandard theory of semi-uniform spaces.

In this paper, we develop the nonstandard theory of semi-uniform spaces in

the sense of Steiner and Steiner [Fund. Math. 83(1973), no. 1, 47-58]. As a by­

-product, we establish some new standard results. These include a theory of

minimal and maximal Cauchy families, a simple necessary and sufficient condition

for a set to be a cluster, as well as "pre-compact" semi-uniform sets with their A-450 associated structures. Other analogies to uniform spaces are also investigated. We study the nonstandard theory of the above concepts, among others, as well as a nonstandard completion of such spaces. We apply this theory to semi-uniform continuous mappings, to topological and semitopological semi- groups and the lattice of semi-uniform topologies on a set. (Received December 22,1975)

*76T-G50 RONALD c. FREIWALD, Washington University, St. Louis, Missouri 63130. Images of B(k)

There are a number of classical results concerning the existence of Borel isomorphisms of

specified class between the space of irrational numbers and other separable (metric) absolute

Borel sets. In this paper we consider analagous results for nonseparable (metric) absolute

Borel sets. In particular, after proving a technical decomposition lemma for certain complete

spaces, we characterize those complete spaces which are (0,1) homeomorphs of B(k), and prove

that, roughly, every (metric) absolute Borel set of multiplicative class a+ 1 which is of

weight k and not cr-local weight < k is, after the deletion of a "small" subset, a (O,a + 1) image of B(k). (Received February 13, 1976.)

E. K. van DOU!JEN, University cf Pittsburgh, Pittsburgh, Pa. 15260 and H. H. WICKE, Ohio University, Athens, Ohio 115701. !l real, weird topology on the reals.

In this article the following real (= using no axioms beyond ZFC) example is constructed.

EXAMPLE. There is a spacer, with the reals as underlying set, which is each of the followin~:

first countable, locally compact, w1-compact, submetrizable, separable, hereditarily weaklv 8-"PP.fin­ able, locally countable, scattered, quasi-developable, isocompact, a space having A-bases hered­ itarily, but which is none of the following: developable, meta-Lindelcf, countably metacomoact, weakl: &9-refinable, irreducible(of order w1), a w~-space, a a-space, countably orthocompact, perfect, and,unfortunately, normal. (The redundancy is intentional.) The usual topology on the reals is a subset of the topology of r. The example r greatly improves the example of Wicke [Abstract 729-621, These NOriCES 22 (1975) A-734]. r has a simple"geometric"structure making a quick check of a:u the

properties stated possible. The construction combines the basic idea of Wicke,~oc. ci~.,with a technique developed by van Douwen [fl. technique for constructing honest locally compact submetriz­ able examples (to appear)]"" which is related to an important example of A. Ostaszewski. Several applications of the example are given and two variations of it are discussed. (Received February 16, 1976.)

*76T-G52 3DSEF DORTMANN and FLORINDA K. MIYAOKA, Univsrsidads Federal Parana,Curitiba, c. p. 1963, Brasil. On the intersection of open sets in topological spaces. We are concerned with the problem of identifying topological spaces with the property that the intersection of an arbitrary family of open sets is also an open set. The main result of this paper can be stated as follows: there exist~ and can be constructed non trivial topological spaces in which the intersection of an in­ finite CXJ family of open sets is always (or never) an open sst. These Results are based on the concept of maximality of sets and on the cardinal number of o ps n sets · (Received February 18, 1976. )

*76T-G53 Eric K. van Douwen, University of Pittsburgh, Pittsburgh, Pa. 15260, USA and Tsodor c. Przymusinski, University of Pittsburgh and Institute of Mathematic> rAN, Warsaw, Poland. Countable and first countable spaces all compactifications of whicl, contain SN. Preliminary report.

Example 1. A regular first countable cosmic space ~ all compactifications 0f which contain SN.

Example 2. A regular countable space L with one non-isolated point all compactifications of which

contain SN. In particular: (1) ~ is a first countable Lindelof space having no first countable

A-451 compactification; (2) E is a countable space all compactifications of which have cardinality 2c

and uncountable tightness; (3) E is a scattered space having no scattered compactification.

Complicated examples of spacessatisfying one of the conditions (1)-(3) have been recently con-

structed by v. Ul 1 janov, B. Efimov, v. Malyhin and P. Nyikos. The spaces a and E are much simpler

and have better properties. The space a is the union of the unit interval I and of a countable

discrete space Nand E = a/I. (Received February 20, 1976.)

David J. Lutzer, University of Pittsburgh,·Pittsburgh, Pa. 15260 and Teodor C. Przymusinski, University of Pittsburgh and Instytut Matematyczny PAN, Warsaw. Continuous extenders in normal and collectionwise normal spaces. Preliminary report.

For a Banach space B and a space Z denote by c*(z,B) the space of continuous, bounded mappings

$ : Z ~ B with the sup-norm topology and by p*(z) (resp. p* (Z)) the space of continuous, bounded wo (resp. bounded and separable) pseudometrics p : Z x Z ~ R on Z with the topology of the subspace

of c*czxz).

Theorem. Let F be a closed subset of a collectionwise normal space X. There exist continuous extenders e : c*(F,B) ~ c*(X,B) and E: p*(F) ~ p*(x). Corollary. Let F be a closed subset of a

normal space X. There exist continuous extenders e : c*(F) ~ c*(x) and E : p* (F) + p* (X). wo wo The above extenders are homeomorphic (but, in general, neither linear nor isometric)

embeddings. (Received February 20, 1976.)

76T-G55 J. A. GUTHRIE, University of Texas, El Paso, Texas 79968, There exists a maximally connected Hausdorff space, Preliminary report. It is shown that there exists a connected topology jJ. for the reals which is finer than the usual topology, and every topology finer than jJ. is disconnected, (Received February 20, 1976,)

*76T-G56 Teodor C. Przytausit{ski, University of Pittsburgh, Pittsburgh, PA 15260 and Instytut ..Iatematyczny rlL:I, lJarsaw, Poland. An interesting and useful in topology set-theoretic condition. Preliminary report.

For ~

carJinality k of subsets of a set X of cardinality 1 there exists a separable metric li on X such

t1>at each A c,,..A'is an F subset of (X,o), Clearly T(k,r) implies T(k',T'), if k > k' and 1: ~ •'· ...... 0

~- J. T(k,T) T(T,k). 2. T(w 1,w1) is true. 3. Consistent (Zwo = 2wl and ,T(w1 ,w2)).

l,, Consistent (w 1 < 2wo .,z"'J and, T(w1 ,w2)). 5. T(2wo,2wo) is independent of the axioms of set

thl"ory. 5. T (w1 , z"'l) iff there exists a normal nonmetrizable separable lloore space. 7. Martin 1 s

Ax!on implies T(k, ~k), for each k < z"'o. 8. The negation of T(k,k) implies the existence of a

:1oore sr,ace of weight k which cannot be embedded into any regular separable first countable space.

9. T(k,T) is equivalent to eight other set-theoretic conditions. (Received March 25, 1976.) (Author introduced by Professor David J. Lutzer.)

*76T-G57 C. G.. Mendez, Metropolitan State College, Denver, Colorado 80204. On the Oxtoby-Ulam Theorem.

Let ~(~) denote the family of subsets of the unit square defined to be of first category (Lebesgue

measure zero) in almost every vertical line in the sense of measure (category). Theorem 1. There is

a homeomorphism of the unit square onto itself mapping a given set in ~(~) onto a set of Lebesgue·

A-452 measure zero. Theorem 2. There is no homeomorphism of the unit square onto itself mapping a given

set in w(Y) onto a set of first category. (Received March 1, 1976.)

J. van Mill, Free University, Amsterdam, The Netherlands. The superextension of the closed unit interval is homeomorphic to the Hilbert cube. If (X,d) is a compact metric space, then the superextension AX of X denotes the space of all

maximal linked systems consisting of closed subsets of X (a system is called linked if every two of

its members meet; a maximal linked system is a linked system not properly contained in another

linked system) topologized by the metric d(M,N) =sup min ~1 (S,T). We show the following SEM TEN THEOREM: The superextension of the closed unit interval is homeomorphic to the Hilbert cube,

thus answering a question of J. DE GROOT. (Received March 15, 1976.) (Author introduced by P. c. Baayen,)

*76T-G59 CHAN~, S. VOAA,,, $5. Kucheh Farshid, Intersection of Kakh Shomali and Elihabeth dil~Juleva·rd, 'l'ehrar, 14. Uniqueness of the Homology Theory on the Categ~;~;ry gf.. Simp-1-icial--Compl~es and Weighted Maps. Preliminary report. In one of tne recent papers, the author constructed a simplicial homology theory on the category of simplicial complexes and weighted maps. In this paper, uniqueness of the homology theory on the category of simplicial complexes and weighted maps is proved. (Received March, 15., 1976.) CHARLES L. HAGOPIAN, Ca_lifornia State University, Sacramento, California 95819. 76T-G60 ~characterization of solenoids.

We prove that every homogeneous continuum having only arcs for proper subcontinua is circle-like. It follows that a continuum M is a solenoid if and only if M is homogeneous and every proper subcontinuum of M is an arc. This result answers in the affirmative a question raised by R. H. Bing [~simple closed~~ the.'?~ homo2:~ bounde$ pla~ continuum_!:~ contains~-!!..!:£_, Canad. J. Math. 12 (1960), 219].

(Received March 15, 1976.)

*76T-G61 J. W. MAXWELL, Oklahoma State University, Stillwater, OK 74074. Embeddings of Non-compact Manifolds. Let X be a closed PL subspace of the PL space Y. The pair (Y, X) is said to be

(m, n) connected provided (Y, X) is m connected and n locally connected at A

proper map f is (m, n) connected provided the pair (Mf, X) is (m, n) connected, where

Mf denotes the mapping cylinder of f. The following embedding theorem is proven. Theorem:

Let Wn and Qq be PL manifolds with q - n ~ 3 and n ~ 6. Let f: ~ ~ Qq be a

(2n- q + 1, 2n- q + 1) connected proper map such that flaw: aw ~ aQ is a proper PL em-

bedding. Then if ~r(W) 0 for r < 3n - 2q + 3, f is properly homotopic, -r:elative to

aw, to a proper embedding of W into Q. The theorem is a generalization of a well known

theorem of Hudson. Its proof uses a generalizat.ion of engulfing to non-compact PL spaces

and a generalization of Zeeman's notion of inessentiality. (Received March 23, 1976.)

76T-G62 Eric K. van Douwen, University of Pittsburgh, Pittsburgh, PA 1326•:'. An invariance space.

A space X is an invariance space if for each continuous f :X ~ X there is a dosed I c X with 0 I I I X and f[I]ci. Let Q and P be the rationals and the irrationals, respectively. LetS P + Q

(+ denotes topological sum). We show that S is an invariance space, thus answerinr; 3 question raised in [A. M. Bruckner, J. Ceder and M. Rosenfeld, On invariant sets for functions, Bull. Inst. Hath. Acad. Sinica 3(1975) 333-347].

A-453 Let f:S..,. S be continuous. If f[P]c P we are done. If not, there is a nonempty open Uc:S with UcP and f[U]cQ. Since U is a Baire space and f[U] is countable, there is an open Vc:S with Vc:U and lf[VJI = 1. Define (an)n as follows: a0 £ V arbitrarily, an+l = f(an). Let A= {an:n ~ 1}. Then f[A]cA, hence f[A-]cA- by continuity. If Anv = \J then A-nv = 0 hence A-,. S. If Anv.,. Ill then A is finite, and again A- ,. S. QUESTION. Let X be neither Baire nor meagre (=first category). {Then there are disjoint nonempty open B,Mc.X with B llaire, H meagre and BUM dense in X.} Is X an invariance space? (Received March 25, 1976.) Gary Gruenhage, Auburn University, Auburn, Alabama 36830. Stratifiable spaces are M2. For the definition of Mi -spaces, see [J. G. Ceder, Pacific J. Math. 11(1961 ), 105-126]. Theorem 1. Stratifiable spaces are M2. Theorem 2. Let X be stratifiable. Suppose X=Kl/M, where

K is closed in X, and M is metrizable. If every closed subset of K has a ~-closure-preserving

base inK, then X is M1· (Received March 31, 1976.)

*76T-G64 Czes Kosniowski, University of Newcastle upon Tyne, Newcastle upon Tyne. NE1 7RU BiP Manifolds with Fixed Point Set having Trivial Normal Bundle.

Let p be an odd prime. Let M be a 4n dimensional oriented ~p manifold satisfying: (i) no component of M has trivial ~p action, and (ii) each component of the fixed point set has trivial normal bundle. Theorem. Suppose sn (M) I 0 mod p. If k(O s; k < n) is an integer such that k= 0 or 2k= 2n mod(p-1) ther: there exists a 4k dimensional component Kin the fixed point set such that sk (K) I 0 mod p. 1"heorom. Suppose 2n = pt-1 and sn(M) I 0 mod p2• If k(O s; k ~ n) is an integer of the form (pm-1 )/2 then there is a 4k dimensional component Kin the fixed point set such that sk(K) I 0 mod p2• (Received March 29, 1976.)

*76T-G65 THOMAS M. PHILLIPS, Auburn University, Auburn, Alabama 36830. Primitive extensions of Aronszajn spaces.

The concept of a primitive representative of a primitive sequence is used to describe an internal property of a space which permits the construction of a model for a completion of an arbitrary com­ pletable Aronszajn space. Consequences of this construction include the following results. Theorem 1. If Sis a completable Aronszajn space and x1, X2 ,··· are completions of S, then there is a com­ pletion X of S such that for each n, X is homeomorphic to a subspace of Xn. Corollary. If P is a hereditary topological property in the class of Aronszajn spaces and S is a completable Aronszajn space some completion of which has property P, then every completion of S contains a completion of S having property P. Corollary. Every Aronszajn completion of a metric space S contains a metric com­ pletion of S. Every Aronszajn completion of a semicompletable Moore space S contains a Moore space semicompletion of S. Theorem 2. If S is a completable Aronszajn space having either a point­ countable base or a a-disjoint base, then S has a completion having such a base. Remark. Analogous results are obtained for semicompletable Moore spaces. Thus from Theorem 2 it follows that a screen­ able semicompletable Moore space has a screenable semicompletion. (Received March 29, 1976.)

*76T-G66 LUDVIK JANOS, Washington State University, Pullman, Washington, gg163, Self-maps which shrink compact sets. A continuous self-mapping f:X ..,. X of a topological space X is said to shrink compact sets if n~ f"(Y) is a singleton for.every nonempty compact £-variant subset Y of X. Theorem 1. Assume X metrizable, f:X..,. X continuous and such that the sequence of iterates

{fn (x)} converges for every x E X . Then the following three statements are equivalent: (A) f shrinks compact sets. (B) f is contractive in the sense of Edelstein relative to a suitable metrization of X . (C) For every c E (0,1) there exists a generating family

A-454 {di} of pseudometrics di an X such that f is a c-contraction relative to each di .

Denoting by !l the minimal cardinality of the family satisfying (C) we have Theorem 2. If

X in Theorem 1 is locally compact then ~X = 1 and (C) reads: f is a Banach contraction relative to a suitable metrization of X. (Received April 1, 1976.)

RONNIE LEVY, Washington University, St. Louis, Missouri 63130. Orderability of compact Hausdorff spaces, Preliminary report.

For X a topological space, C(X) denotes the ring of real-valued continuous functions with domain X, (C(X),+) denotes the additive group of C(X). If A~ C(X), A-A denotes the set

{a-b:a,b E A}. Theorem. Suppose X is a compact Hausdorff space, Then the following are equivalent: (i) X is linearly orderable. (ii) There is a subsemigroup A of (C(X),+) such that (a) for each f E C (X) and for each unit c of C (X), there is an a E A-A such that for some g,h E C(X), f·a+c2 • g2 and e2-f+a = h2 and (b) for each ideal I of C(X), there is a maximal ideal M2I such that for every fE A and every maximal ideal N2 I, either there. are units £,5 of C(X) such that f·c EN, f-c-5 2 EM, or there is a unite such that f-e <: M n N, or fEMnN.

We observe that condition (ii) is stated in purely algebraic terms. (ii) can be restated as

(11'): There is a subsemigroup A of C(X) such that A-A is dense in C(X) (with the uni.form norm topology) and such that for every closed s~bset [or zero set] B of X, there is a pR ~ n such that for each f E A, f (p8) = min (fl B). (Received April 5, 1976. )

F. THOMAS FARRELL, The Pennsylvania State University, University Park, Pennsylvania 16802 and WU-CHUNG HSIANG, Princeton University, Princeton, New Jersey 08540. The rational calculation of certain surgery obstruction groups. Preliminary report.

Let 1 _,. S + ·r + G + 1 be an exact sequence of groups with G finite, S free abelian of finite rank m, and r torsion-free; let ro : r + {±1} be a group homomorphism with s c ker ro.

Define an automorphism a of zr by a(y} = ro(y}y-l for y E r; define a right r-module Q00 by xy = ro(y)x for y E r, X E Q where Q denotes the additive group of rational numbers.

Theorem. For all n

Ls(zr,a) ® Q"' Ell H. (r ,Q ) • n i=n mod 4 ~ 00

Corollary. If ro is determined by the first Stiefel-Whitney class of Bf, then

Ls(zr,a) ® Q ~ Ell Hi(r,Q). n ian-n mod 4

This result when G = {1} has been known for some time; due to work of Shaneson and Wall. (Received April 7, 1976.)

Harold Widom, University of California, Santa Cruz, California, 95064. Families of pseudodifferential operators.

A symbolic calculus is developed for certain families of pseudodifferential operators on manifolds. Roughly speaking the families are locally pseudodifferential operators on Euclidean space with symbol of the form a(x, ~/a) where a is a

parameter running through a subset of [1,~). (If the parameter set is [1), the case of a single operator, what is obtained is a new treatment of the complete symbol.) By-products are a manifold version of the Szego limit theorem for

A-455 Toeplitz determinants and a procedure for computing (in princip1e) as many terms

as desired of the asymptotic expansion for the heat operator on compact

Riemannian manifolds. (Received April 7, 1976.)

*76T-G70 Allan Calder, Birkbeck College, London_, England and Jerrold Siegel, University of Missouri, St. Louis, Missouri 63121. ~ech extensions are Kan extensions. Preliminary report.

Let (T,P) be a pair of topological categories. That is, T and P are full subcategories of TOP and P cT. For a functor F: P ~ .6e.t&~P (dual category of .6e.t&), let /-,F : T + .6e.t& denote the right Kan and the Cech extensions, respectively, of F to T. THEOREM 1 : FoP (T,P) any of the following paiPB (a) (C Reg r2,i6 Poi), aompleteZy Pegular hausdoPff spaaes. loaaUy finite polyhedzoa. (b) (Comp r 2,6 Poi), aompaat hausdoPff, finite polyhedzoa. (c) (C Reg T2,6 Poi), and any homotopy funatoP F : P + .6e.t& op, FK and ~ are natuPaUy equivalent. (A homotopy functor is one that is constant on homotopy classes of maps.) Note that in contrast to Hilton et a1 , /- is the Kan extension with respect to categories of continuous functions, not homotopy classes. In fact for (c) y<- :¥ F is not in general a homotopy functor. THEOREM 2 : 1.'he q-th Ceah aohomotopy fun

*76T-G71 STEVE J, KAPLAN, University of California, Berkeley, California 94720, Constructing framed 4-manifolds with given almost framed boundaries. The main purpose of this paper is to provide an algorithm for finding a framed, simply connected 4- manifold whose boundary is a given almost framed 3-manifold, (The framings are on the tangent bundles.) This enables one to compute the j.l.-invariant of an almost framed 3-manifold, We regard an oriented 3-manifold as the boundary of a 4-dimensional handlebody obtained by attaching 2-handles to B4• The main tools are those of Kirby's calculus of framed links, Two constructions are provided. The first gives many distinct 4-manifolds with a given boundary. The second yields a manifold with comparatively low second betti number. Closed, spin, 1-connected 4-manifolds of index 16 and betti numbe:r; 22 arise in many ways, but no smaller form of index 16 is realized. A new definition is given for the z2 knot invariant defined by Robertello by means of Bohlin's theorem. It is shown that every knot in aS bounds a certain type of immersed disk and that from any immersed disk of this type one can easily read off the z2 invariant of its boundary knot, These considerations are later applied to links in which any pair of components link each other zero times. This is tied in with the questions discussed above. (Received April 8, 1976,)

*76T-G72 Dix H. Pettey, University of Missouri, Columbia, Missouri 65201. Some new examples of minimal regular spaces.

A T3 space is said to be R-closed if it is a closed subspace of each T3 space in which it is em­ bedded and~ regular (MR) if no T3 topology on the underlying set is properly contained in the given topology. A T3 space X is said to be strongly minimal regular (SMR) if there is a base V for X such that for each V in V, X - V is an R-closed subspace of X. It is known that

(for a T3 space) compactness is strictly stronger than the SMR property, the SMR property is strictly stronger than the MR property, and the MR property is strictly stronger than the R-closed

property. THEOREM 1. There exists a noncompact T3 space whose every T3 continuous image is SMR. THEOREM 2. There exists a non-SMR T3s.Pace whose every T3 continuous image is MR. THEOREM 3. There exists a decreasing chain {xaJ!X < Q} of nonempty SMR spaces such that n X is empty. !X

*76T-G73 BRIAN R. ill'JMEL, University of Wisconsin-l"'ilwaukee, l"'ilwaukee, Wisconsin 53201. Products of ~-planar Complexes Do ~ Imbedd in ±-Space. In this paper a proof is given of the following Theorem: If K1 and K2 are finite simplicial complexes neither of which is homeomorphic to a subset of euclidean 2-space, then their cartesian product K1 xK2 is not homeomorphic to any subset of euclidean 4-space. (Received April 12, 1976.)

A-456 W. W. C

We cOMider only canpletely regular 1 Hausdorff' spaces, Responding to a question of

R. Levy and R. H. McDowell [Proc. Ar/ler. Math. Soc. 49 (1975) 1 426-430] we show that for CD ~ y ~ z3D there ia a separable apace equal to the (appropriately topologized) disjoint union of y copies of tbe"stone-Cech rema~" •N\N. More g~erally, denoting density character by d and weight by w, we prove this Theorelil: The following statements about infinite cardinal numbers v and 01 are equivalel\t: (a) t' .S zV and y ~ zt'; (b) For every flllllily {XI:~ < y) ~f spaces, with w(x1) ~ fl for all 1 < y 1 the set-theoretic disjoint union X = U~

Paul Bankston, McMaste~ University, Hamilton, Ontario, Canada Topological Ultraproducts and the GCH.

For any cardinal K ~ w let UPK be the statement that it is any K~sequence

of regular T1 spaces of weight $ exp(K) and if D is any regular ultrafilter on K then the topological ultraproduct ITDXa is paracompact. Then we have the Theorem: UPK iffexp(K) = K+. (Received April 19, 1976.)

Miscellaneous Fields (00, 01, 96-99)

A. iCHIII!i1IIMI!a1,.,ciJIDDUI.1.ity Colle&!!. of ~ilade1phia, ~iladelphia, Pennsylvania, 19110'1v·"·

*76T-H5 A. ~OWN and A. SCHREMMER, Community College of ~iladelphia, Philadelphia, Pennsylvania 19107. ! geometric, multivariate precalculus. Preliminary report. The geometric study of transformations from EP to Eq and variables on EP (i. e. partial maps from EP to R)(p, q = 1, 2) has been found to constitute an effective preparation to the Calculus, far superior to the usual study of functions from R to R (as. in, for instance, the CUPM's Math, 0). The main advantages of this approach are that it does not require any a priori computafional ability, it helps create and develop early a geometric intuition, it is an excellent introduction to the modern linear treatment of the multivariate calculus, it intro­ duces early the notion of approximation and it is psychologically re~li~tic in the case of very unprepared students. The course starts with a minimum of vector .geometfy in the Euclidean plane E2• The pasic ideas then appear in the study of images and preimages under parti~l maps in a series of examples introduced in the following optimal order: transformations frmn E2 to Ez, transformations from E2 to E, variables on E2, transformations framE to E2, paths in Ez, transformations from E to E, variables on E, functions from R to R. We then consider trans­ formations and variables· in relation to each other. After the elementary study of affine transformations and polynomial functions, we introduce the notion of approximation by limited polynomial expansions, the order of which, o[xn] at 0 and u.{xn] at co, are defined by dominaticn. This.is applied to: the definition of continuity and differentiability, the study of rational functions by their singular points, the approximate.solution of polynomial equations hy perturbation techniques, the study of the element~ry transcendental functions an•l their Taylor expansions, Help in specific computational techniques is provided, as needed, th-cough precise references to a separate student guide, (Received March 8, 1976.)

A-457 University of Nevada 735TH Reno, Nevada MEETING April 23-24, 7976

*735-Dl Anthony E. Labarre, Jr., California State University, Fresno, Fresno, California 93740. Volume of an oriented polyhedron;extension. Let F be a face of polyhedron n, Assume F is oriented so that when the consecutive vertices P1 ,P2, .•• ,Pk ofF are traversed, the area enclosed by the boundary of F is on the left as viewed from the exterior of TI, With Pi=(xi'yi'zi) define an "area" vector A(F) by \ xl yl zl i yl zl lxl zll xl yl \ al bl \ where A(F) = ,i:k Yk zk ltk zk ' -,Jck zki' xk yk I ak bk I ixl Y1 zl IY1 zl xl zl! xl yl I al bl (alb2+a2b3+···+akbl) - (a2bl+a3bz+···+albk). Then if F., j 1,2, ..• ,n are the faces of n, the volume V of TI is .1 1 n V(n) = 0 ~ A(F.)·P. j=l J J where P. is any vertex of face F., and A( F.)· P. means the dot product of A(F.) and P.• J J J J J J The formula is extended to a closed regular surface. (Received April l6, l976.)

*735-Gll Larry Baggett, University of Colorado, Boulder, Colorado 80309. A Characterization of Heisenberg groups; when is a particle free?

Let H be a connected Lie subgroup of a connected Lie group G . Sufficient conditions are given so that the smallest closed normal subgroup of G containing H should be a "Heisenberg Group."

The condi.tions are that there should exist an irreducible unitary representation T of G and a non-zero vector v in the space of T such that a) g--> (Tgv,v) vanishes at infinity on G and b) the family h --> I (T11T g v T g v) I of functions on H should be equicontinuous at the identity.

Applications are made to quantum mechanics to conclude that a particle is free of external forces providing some nonzero state is transformed appropriately by the group of symmetries. (Received March 2, 1976.)

736TH Portland State University Portland, Oregon MEETING june 78, 7976 Algebra and Theory of Numbers (05, 06, 08, 70, 72-7 8, 20) 736-Al J.L. Brenner, 10 Phillips Rd, Palo Alto, California 94303. Covering Theorems for Finasigs. VIII. Almost all conjugacey classes in An have exponent ~ 4 •

The product of two subsets C, D of a group is defined as CD = { a {3 I a ~ C, {3 f n} The power Ce is defined inductively by c1 = C, Ce = C Ce-l = Ce-l C .It is known that in the alternating group An n > 4, there is a con­ jugacy class C such that C C ~ An . On the other hand, there is

a conjugacy class D such that not only D D ~ An , but even for e < [n/2~ . It may be conjectured that as n -> CD , almost all classes C

A-458 satisfy In this article, it is shown that as n->co,~

all classes C satisfy (Received Feb. 12' 1976

*736-A2 Marshall L. Cates, California State University at Los Angeles, Los Angeles, California 90032. Laws of C Wr C 2 p p The wreath product plays a strong role in the study of split extensions, hence in the study of product varieties. If C!GI , !HI) = 1 then the variety generated by GWr H is well known. The variety

generated by C Wr C is also well known. One of the next minimal cases would be C Wr C 2 A p p p p basis for the laws of Cp Wr Cp2 will be presented. This also sheds some light on the lattice of sub- varieties of ¥P2 • (Received March 8, 1976.)

*736-A3 D. L. OUTCALT, University of California, Santa Barbara, Santa Barbara, California 93106 and ADIL YAQUB, University of California, Santa Barbara, Santa Barbara, California 931o6. Rings with a nil ideal yielding po1ynornial constraints.

The following theorem is proved: Suppose R is an associative commutative ring, and I is a nil ideal ~n R. Suppose, further, that for every element x in R/I, there exists a positive integer n = n( x) and a polynomial f(),) = f x (}..) with integer coefficients such that xn = xn+l f( x). Then, R is isomorphic to a subdirect sum of subdirectly irreducible rings Ri (i E r), where each Ri is either a nil ring, or Ri has the property that x in Ri implies x is nilpotent or x-l is in Ri • (Received April 19, 1976.)

Analysis {26, 28, 30-35, 39-47, 49)

ROGER W. MAY, Walla Walla College, College Place, Washington 99324, & CHARLES W. McARTHUR, Florida State University, Tallahassee, Florida 32306. Comparison of Order Topologies with the Topology of an Ordered Topological Vector Space. Preliminary Report.

Let (E,~) be a complete, metrizable topological vector space ordered by a cone K. (E,T) is said to have Property (A) if each sequence which decreases in order to 8 converges to 8 in T. The following are shown. Theorem 1, If the cone K is T-closed, the following statements are equivalent: (1) K is generating

and ~-normal. (2) For each sequence (xn) in E and x € E, (xn) relatively uniformly star converges to x iff (xn) converges to x in ~. (3) For each sequence (xn) in E and x E E, (xn) converges to x in the

relatively uniform topology ~ru iff (xn) converges to x in T. (4) ~ru T. (5) Tru2 T and K is gen­ erating. (6) ~ru s=. ~ and K is ~-normal. Theorem 2. The following statements are equivalent: (1) K is ~-closed, generating, and T-normal, and

(E,~) satisfies Property (A). (2) For each sequence (xn) in E and x E E, (xn) order star converges to x iff(xJ converges to x in T. (3) For each sequence (xn) in E and x E E, (xn) converges to x in the

order topology ~ 0 iff (xn) converges to x in ~. (4) ~0 T, (5) T0 i2 ~and K is closed and genera­

ting. (6) T0 E ~, K is ~-normal, and (E,~) has Property (A). (Received November 24, 1975.)

*736-B2 J. H. MATHEWS, California State University, Fullerton, California 92634 C. L. BELNA , Western illinois University, Macomb, Illinois 61455 An Example Pertaining to Local ~ Global Essential Cluster Sets. A subset of the unit disk D is constructed which has global density zero and local up)iler density 1 at all points on the unit circle C. An outcome of this resnl• is the following: Let M and N be compact subsets of the Riemann sphere 0 with M ·- N , then utilizing Runge's approximation theorem a holomorphic function f(z) can 'w constructed in D which has the properties (i) Ce(f) = M where Ce(f) is the glohal essential

cluster set, and (ii) C e (f, !;) = N at all points ~ on C where C e (f, !;) '"' •·he local

essential cluster set of f at the point s . {Received February 24, 1976.)

A-459 *736-B3 TONY HOLLAND, University of Calgary, P.D. KATHAL, Govt. College, Mandla, India BADRI SAHNEY~ University of Calgary, Alberta, Canada, TZN 1N4. On the Non- Uni ueness or the "o timal order" of A roximation of ol omial o erators. Pre- nary report.

The Feje'r kernel of a Fourier Series is a positive operator and M. Zamanski [Ann. Sc. Ecole Normale Sup, 66 (1949) pp. 19-93] has given a set of necessary and sufficient conditions to determine the order of approximation of the functions belonging to the class Lip 1. It has

also been shown that the optimal order 0 ~} cannot be replaced by o (*). Thus the order of approximation is best possible. However, the uniqueness problem has not yet been studied.

In this direction, it is shown here that the optimal order is not unique. (Received March 30, 1976.)

*736-B4 MICHAEL A. GOLBERG, University of Nevada, Las Vegas, NV 89154 Integral Equations with Sei:nidegenerate kernels, Preliminary report We consider the class of integral equations (1) u(t) g(t)+AfbK(t,s)u(s)ds, = a where K(t,s) =.f1 a. (t)b. (s), a

*736-B5 H.M. SRIVASTAVA, University of Glasgow, Glasgow Gl2 8Qri, U,K, and University of Victoria, Victoria, British Columbia, Canada VSW 2Y2, The sum of a multiple hypergeometric series. Preliminary report,

A transformation formula is given for the generalized hypergeometric function pFq in series of similar functions. It is also shown how easily this formula can be applied to deduce various classes of summation theorems for multiple hypergeometric series, Some of the main results thus derived would provide elegant multiple-series analogues, for instance, of a well-known summation theorem [}>roc. London Math. Soc. (2) 29 (1929), p.512, Eq.(Cl], and a recent result [}>roc. Amer, Math, Soc, 52 (1975), p.137, Eq.(3)]. (Received April 19, 1976.)

736-B6 WITHDRAWN

A-460 736-B7 D.H. HYERS, University of Southern California, !~s ~ngel.es, California, ~0007 On the stability of minimum points. Preliminary report.

Let p be a real valued continuous and strictly increasing function on [ O, oo) '1ith

p(O) = 0. Let B be an open ball in Rn centered at X0 • P. function :f' c.efined on !J clearly has

an absolute minimum at Xa when f (X) - f (X0 ) ~ p ( \\X - X0 1\ ) for all X E B. Theorem. For a

E > given 0 there corresponds 6 > 0 such that \g(x) - f(x) J < o (X E B) implies that q(v.1 ) is

a relative minimum for g where l\x1 - x0 1\ < E, providing that g is lower semi-continuous on B. Some analogous results are obtained for certain variational prohlems. (Received April 23, 1976.)

Applied Mathematics (65, 68, 70, 73, 76, 78, 80-83, 85, 86, 90, 92-94)

Davia A. "San';hez and Daniel Sweet, University of' California, Los Angeles, Los Ange.Les, California, 90024. An Iteration Scheme f'or Determining Eguations of Dynamical Systems.

This paper considers the problem of determining solutions for the equation (1) F(u)=O where n-1 F:~ ... ~ and has the form: F(u)=(f'1 (u), ... ,fn_1 (u), L:i=l ai(u)f'i(u))T. This situation arises naturally when one attempts to value the determining equations when searching f'or periodic solutions of' ordinary differential equations possessing a first integral. Under appropriate differentiability conditions, (1) will generate a family of' solutions where dF(u*) will be singular at any solution u·*. Thus, in order to establish a convergent iteration scheme to determine solutions of (1), a modified form of Newton's method is employed. Provided certain natural and appropriate conditions are met, this procedure establishes a contraction map on an invariant (n-1)-dimensional byperplane of Rn proVided the initial estimate is close enough to the solution curve. (Received March 1, 1976.)

NATHANIEL MACON, The American University, washington, DC 20016 and PAUL BRYANT, Federal Preparedness Agency, Washington, DC 20504. An Intriguing Numerical Relationship Between Employment and the consumer Price Index. Preliminary Report

After manipulation of a large econometric date base on a computer, the results of one algorithm suggested that, with certain exceptions, the product of total u.s. employ­ ment and consumer price index yields a simple exponential function of time. The relationship was tested at national, state and local levels in both the U.S. and a number of foreign economies and was found to hold within a percent or two. The exceptions were primarily as a result of redefinition or revision of qata. (In Seattle a discontinuity was noted in conjunction with the economic disruptions which occurred there in 1970-72.) Noting that the product of CPI and labor force in the U.S. has an exponential trend led to a phenomenological description of the relationship between prices and unemploy­ ment. (The "long-run Phillips' curve can be derived as a by-product.) The results suggest a new class of curve fitting techniques for time series. (Received April 26,1976)

Geometry (50, 52, 53)

736-Dl PAUL E. EHRLICH, Bonn University, D5300 Bonn, West Germany and HANS-CHRISTOPH IMHOF, University of California, Berkeley, California 94720. Metric circles and bisectors on surfaces without conjugate points, Preliminary report. Let M be a complete, simply connected Riemannian two-manifold without conjugnte points. For distinct points p,q, EM, we determine the family of all metric circles through p and q, generalizing classical results in Euclidean and hyperbolic geometry. Define as usual the bisector M(p, q) of p and q by M(p, q) : = { m ~ M \d(p, m) = d(q, m)}. If M(p, q) n M(q, r) +II, p, q, r distinct points in M, then p, q, and r lie on a metric circle. Thus our results on metric circles imply a necessary and sufficient condition for M(p,q) and M(q, r) to have exactly one point of intersection or no intersection. In particular, this gives a simple geometric proof of

A-461 P. Eberlein's Proposition 2. 8 ("The cutlocus of noncompact finitely connected surfaces without conjugate points" Comment Math. Helv. (to appear)), which shows that the two bisectors M(p,q) and M(q,r) intersect at most once. (Received March 15, 1976.)

*736-D2 Branko Grllnbaum, University of Washington, Seattle, Washington 98195, and G.C. Shephard, University of British Columbia, Vancouver, B.C., Canada V6T 1W5. Perfect colorings of transitive tilings. A k-coloring of a tiling is a partition of the set of tiles into k subsets called colors. Such a coloring is called perfect if each symmetry of the tiling induces a permutation of the colors. Familiar examples of perfect colorings are the checker­ board 2-coloring of the square tiling, and the 3-coloring of the regular hexagonal tiling. Theorem 1. Every transitive tiling admits perfect k-colorings for infinite­ ly many values of k. Theorem 2. The square tiling admits a perfect k-coloring if and only if k is of the form n 2 or 2n2 . The regular hexagonal tiling admits a perfect k-coloring if and only if k is of the form n2 or 3n2 • For non-regular tilings there are many other possibilities. For example, a perfect 5-coloring of a tiling with symmetry group p4 will be shown. (Received April 19, 1976.)

*736-D~ Branko Grllnbaum, University of Washington, Seattle, Washington 98195, and G.C. Shephard, University of British Columbia, Vancouver, B.C., Canada V6T 1W5. The eighty-one types of transitive tilings of the plane. A transitive tiling is a covering of the plane by congruent closed topological discs called tiles, without gaps or overlaps, and such that the symmetry group of the tiling acts transitively on the tiles. Two tilings are of the same type provided that in each, every tile bears a similar relation to its set of neighbors by symmet­ ries of the tiling. This relation is codified by a "neighborhood symbol" which was introduced in a special case by B. N. Delone (Izv. Akad. Nauk SSSR, Ser. Mat. 23(19- 59),365-386). Examining all possible neighborhood symbols for each of the eleven "regular nets" one arrives at 93 combinatorial possibilities (U. Sinogowitz, Z. fllr Kristallographie 100(1938),461-508; H. Heesch, RegulMres Parkettierungsproblem, Westdeutscher Verlag, K8ln 1968), but not all of these are geometrically realizable. Theoren:. There exist 81 types of transitive tilings of the plane. (Received April19, 1976.)

736-D4 JOSEPH ZAKS, University of Haifa, Haifa, IaJ>ael, Semi-regular maps. Preliminary report. Aslight strengthening of a theorem of T.R.S. Walsh (Geometria Dedicata 1(1976),U7-l23) will be presented: If S=(p1 , ... ,pk) is the .... _yclic sequence of a semi­ regular map, then for every j, l~j,;k, S contains pj triples of the form Cx1pjx2), Cx2pjx3), ... (xp.pjxl) , where x1=pj-l and x2=pj+l (indices are modulo k). J Here a map is on an orientable 2-manifold and it is called semi-regular if the cyclic order of the faces around every vertex iE thr same, to within rotation and reflactions. An attempt to help a research on infinite regular polyhedra by the two architects Wachman and Kleinemann will be described. (Received April 19, 1976.)

Statistics and Probability (60, 62)

*736-Fl David Gilat, University of Minnesota, Minneapolis, Minnesota 55455· Is every submartingale a convex function of a martingale?

It is shown that every non-negative superfair process (in particular a non-negative submartingale) is the absolute value of a symmetric fair process (martingale). For the more general question

posed in the title, the evidence is inconclusive. If however the adjective ~ is omitted from

the title, an affirmative answer is provided, Furthermore, transforming functions ~ , such that

A-462 every superfair process (submartingale) is that ~ of a fair process (martingale), are shown to

exist. Results are first proved for discrete-parameter processes and then extended to continuous-

parameter submartingales with right-continuous sample functions. (Received March 10, 1976.) (Author introduced by Professor Bert Fristedt.)

736-F2 THOMAS M. LIGGETT, University of California, Los Angeles, California 90024. The stochastic evolution of infinite systems of interacting particles. • Classical statistical mechanics is concerned with the equilibrium theory of certain physical systems. During the last few years, several types of Markov processes have been proposed as models for the temporal evolution of these systems, Some of these processes have in turn been given economic or sociological interpre­ tations. We will describe these processes briefly and will discuss some of the basic problems associated with them. There are two techniques which have been of primary importance in solving these problems: (a) exploiting properties of processes which are in a sense dual to the processes of interest, and (b) coupling two or more processes together. We have chosen two particular contexts in which to illustrate the use of these techniques, as well as to indicate some of the results which can be obtained from them. In each of these, the state space is {0, 1}zd, where Zd is the d-dimensional integer lattice. For the first process, which is called the voter model, we imagine that there is an individual at each lattice point who favors one of two possible positions (denoted by 0 and 1) on some political issue. At certain random times, which are independent from site to site, each individual reevaluates his position and will change it with a probability which is proportional to the number of his neighbors who support the opposing position. The second process is the simple exclusion process, in which particles are distributed on Zd with at most one per site. These particles move on Zd according to independent continuous time random walks, except that transitions to occupied sites are suppressed. In both cases, the main problem is to determine the asymptotic behavior of the system for large times. (Received April19, 1976.)

Topology (22, 54, 55, 57, 58) *736-G1 Allen D. Allen, Algorithms, Inc., Northridge, Calif. 91324. A topological approach to the four-color problem.

A ~-diagram of graph G is constructed as follows: (1) The vertices v are partitioned

into X(G) independent sets Ai. (2) The vertices are embedded in the plane such that

for each Ai, one can draw a circle ci around all v E Ai, no v E Aj # Ai being inside

Ci. l3) The edges are constructed such that for each two sets A 1 and A 2 , not every

edge joining a v AJ and a v E A 2 passes between two vertices in a distinct set A 3 •

Conjecture: If G is a planar graph, then one can construct a planar ~-diagram of G.

Theorem: Assume the conjecture is true; then Y natural numbers n, 3 planar G with

X(G) = ~ iff Kn is planar. Note that the only open question on the conjecture is whether step (3) in the construction of a TI-diagram can preclude planarity in all

~-diagrams of some planar graph. (Received April 5, 1976.)

*736-G2 L. E. Ward, Jr., University of Oregon, Eugene, Oregon 97403. An irre­ ducible Hahn-Mazurkiewicz theorem.

A space Y is an irreducible image of a space X if there exists a con­ tinuous surjection f: X -+ Y such that f maps no closed proper subset of X onto Y. The Hahn-Mazurkiewicz theorem asserts that a Peano continuum is the continuous image of [0,1], but it is easy to find examples of Peano continua which cannot be the irreducible image of [0,1]. Theorem. A Hausdorff space is a Peano continuum if and only if it is the irreducible image of some dendrite. Some related open questions are discussed. (Received April 12, 1976.)

A-463 736-G3 R. J. MILGRAM, Stanford University, Stanford, California 94305. Generalized cohomology theories and their applications. • We summarize recent developments in this area. These include the structure and Adams spectral sequences for the theories associated to the various connective K theories as well as Mahowald•s examples associated to various Thorn spectra. We discuss applications in such areas as non-immersion theorems and other nonexistence theorems, for example, the recent result of the speaker and P. Zvengrowski. Theorem. The Whitehead product [J2n•J2nl is projective if and only if n = 1,2,4. (Received May 7, 1976.)

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A-464 ~ sents the basic phenomena and the pertinent mathematics of wave propagation in elastic Books from solids, within the framework of the classical theory of linear elasticity. Each chapter is supplemented by a set of problems, which pro­ North-Holland vides a useful test of the reader's understand­ ing as well as further examples of the funda­ mental principles. Optimization over Leontief "· . .- There is a wealth of clearly presented and import­ ant material . . . this book is strongly recommended .to Substitution Systems all serious workers in e/astodynamics." by GARY J. KOEHLER, ANDREW B. WHINSTON Journal of Applied Mechanics and GORDON P. WRIGHT, U.S.A. 1975 x+221 pages Discrete-Parameter Martingales Price: US $19.00/Dfl. 45.00 by J. NEVEU, France The material presented here is organized as 1975 viii+236 pages over 300 lit. refs. follows. After the introduction in Chapter 1, Price: US $23.95/Dfl. 62.50 Chapter 2 describes the iterative method for solving large scale linear programs over Leon­ The aim of this book is to present the funda­ lief Substitution Systems; several applications mental results of discrete martingale theory and example calculations are included. and a synthesis of the important theoretical Chapter 3 studies several extensions of pro­ results which have been recently obtained. gramming problems with Leontief constraint Applications to measure theory, Markov chains, sets. Several of the more detailed proofs of sequential decision and game theory are also the results in Chapters 2 and 3 are given in considered at length. Chapters 4 and 5 respectively. Further results of optimization of large scale linear programs The first chapter defines conditional expecta­ over Leontief Substitution Systems and matrix tions and their various operator properties. iterative methods are also presented in The basic results of convergence and stopping Chapters 4 and 5. of martingales are then proved. Chapter 3 is devoted to applications while several useful extensions of the concept of martingales are Convex Analysis and taken up in Chapter 5. Next, an optimisation Variational Problems problem which has been of great interest in recent years is treated. The final chapters by I. EKELAND and R. TEMAN, France cover recent developments of martingale 1976 ix+402 pages theory. Some Banach spaces of martingales Price: US $29.50/Dfl. 85.00 connected with their quadratic variations and maximal functions are studied. The interplay In recent years there has been a considerable between martingales and increasing proces­ expansion in the field of convex analysis, in ses provides a number of interesting theorems. conjunction with the development of various Throughout the book exercises are provided. mathematical tools of which some have be­ come standard. The book describes recently developed tools of convex analysis and applies Intensional and Higher-Order them to optimisation problems in function Modal Logic spaces. It is largely selfcontained, requiring only some knowledge of Banach and Hilbert With Applications to Montague Semantics spaces, and familiarity with distribution theory. by D. GALLIN, U.S.A. CONTENTS: Part. 1. Fundamentals of Convex Analysis. Chapters I. Convex functions. II. Minimisation of con­ 1975 xi+148 pages vex functions and variational inequalities. Ill. Duality in convex optimisation. Part 2. Duality and Convex Price: US $10.00/Dfl. 26.00 Paperback Variational Problems. IV. Applications of duality to the calculus of variations (I). V. Applications of duality to In part I the author provides an introduction to the calculus of variations (II): problems of the type Montague semantics, sets out in detail the minimal hypersurfaces. VI. Duality by the minimax theorem. VII. Other applications of duality. Part 3. syntax and semantics of Intensional Logic, and Relaxation and Non-Convex Variational Problems. treats completeness, persistence and normal VIII. Existence of solutions for variational problems. forms, with applications to the Montague pro­ IX. Relaxation of non-convex variational problems (1). gram. In part II an alternative system of higher­ X. Relaxstion of non-convex variational problems (II). Commentaries. Bibliography. order modal logic is developed.

Wave Propagation in Elastic Solids NORTH-HOLLAND ( c by J. D. ACHENBACH, U.S.A. . 1973 1st. repr. 1976 xiv+425 pages PUBLISHING CO. ~ 88 illus. I Hardbound price: US $46.25/Dfl.120.00 P.O. Box 211, 1/n the U.S.A.!Canada Paperback price: US $20.00/Dfl. 50.00 Amsterdam, 52 Vanderbilt Ave., Intended as a reference source for engineers The Netherlands New York, N.Y. 10017 N and sc,entists in the broad sense, and as a The Dutch gui'der prico is detiniti·1e. us $ priws are textbook fer graduate courses, this book pre- sub;ect to exchange rate fluctuations. 8 ~------..) A-465 THE CENTER FOR APPLIED MATHEMATICS of the University of Georgia has immediate openings for experienced re­ searchers in applied mathematics. The Cen­ ter will conduct research in mathematics applicable to problems of physics, biology, ecology, economics, engineering and in the mathematical analysis of nonlinear or sto­ REGIONAL CONFERENCE SERIES chastic differential, partial differential, or IN MATHEMATICS integral equations arising in the modeling of dynamical systems in applications. Out­ Number 25 standing researchers with an international RELATION MODULES OF FINITE GROUPS reputation are invited to apply. Junior po­ by KARL W.GRUENBERG sitions are also available. Joint positions with interested depart­ ments are possible. Qualified applicants are The aim of the ten lectures in this little invited to write to the Center for Applied book is twofold: On the one hand, to Mathematics, 612 Graduate Studies Research show group theorists how the presentation Center, University of Georgia, Athens, Geor­ theory of finite groups can be successfully gia 30602. approached with the help of integral repre­ The University of Georgia is an equal oppor­ sentation theory; on the other hand, to tunity/affirmative action employer. persuade ring theorists that here is an area of group theory well suited to applications THE AUSTRALIAN of integral representation theory. NATIONAL UNIVERSITY 1. Introduction invites applications for appointment as Senior 2. The Gaschtitz theory Research Fellow or Research Fellow, Depart­ 3. Module theoretic preliminaries ment of Mathematics, Research School of 4. Projective modules Physical Sciences. The Department has respon­ 5. Relation cores sibility for postgraduate research and study in 6. The presentation rank 7. Swan modules mathematics. Principal interests are functional 8. Heller modules analysis, foundations, group theory, ordinary 9. Decomposition of relation cores differential equations and control theory. Ap­ 10. Epilogue pointees expected to participate in supervision of research students. Appointment for three years in first instance; possible extension to five years. Salaries: SRF $A 18,282-$A21 ,335 86 pages, 1976; List $8.40; Individual $6.30 To order, specify CBMS/25 p.a.; RF $A12,834-$A17,113 p.a.; exchange Orders must be prepaid rate $A1: $US1.26. Reasonable travel expenses. Superannuation benefits. Further information AMERICAN MATHEMATICAL SOCIETY from Academic Registrar, ANU, PO Box 4, P.O. BOX 1571, ANNEX STATION Canberra, ACT, 2600, Australia, with whom PROVIDENCE, RHODE ISLAND 02901 applications close 28 June 1976.

A-466 DIRECTORY OF WOMEN MATHEMATICIANS

Under the direction of the AMS Committee on Women in Mathematics the Society has compiled a directory, first published in 1973, with supplements compiled in 1974 and 1975, of all women in mathema­ tics located through a search of the Combined Membership List, lists of Ph. D.'s granted, and word of mouth. All were sent a questionnaire. Those who returned it were included in the first Directory. The supplements were compiled by sending questionnaires to all new women recipients of the Ph. D. whose names could be obtained. Each supplement also contains changes of address from the previous years. Each entry contains the following information: 1. Name, previous names, and address. 2. Title, employer's name and address. 3. Highest degree, year earned, thesis title, thesis supervisor. 4. Fields of mathematical and other professional interest and bibliographic information on publica­ tions. The Directory is intended to aid department heads in complying with equal opportunity guidelines, and to help in implementing resolutions of the Council of the American Mathematical Society with respect to women. The supplements were published under the direction of the AMS-MAA Committee on Women in Mathematics whose present members are, Dorothy L. Bernstein, jane K. Cullum, Mary W. Gray, I. N. Herstein, Shirley A. Hill, Cathleen S. Morawetz, Charles B. Morrey, Jr., jacqueline C. Moss, Alice T. Schafer (Chair­ person), and Gail S. Young. Orders must be prepaid. The prices include both supplements. List Price $6.00; Member Price $4.50. American Mathematical Society P. 0. Box 1571, Annex Station Providence, Rhode Island 02901

Pitman Advanced Publishing Programme 1[ in Mathematics disseminates important new material, ranging over the whole spectrum of mathematics -for specialists and postgraduate students. NEW TITLES INCLUDE:

Trends in Applications of Pure Quasilinear Hyperbolic Systems. Mathematics to Mechanics and Waves Edited by G Fichera, A jeffrey, University of Newcastle­ Universita di Roma upon-Tyne Twenty-three timely papers-- an Paperback ISBN 0 273 00102 7 international collection -presented $15.90 Publication: June at the University of Leece in May 1975, and treating aspects of Bifurcation Problems in Nonlinear continuum mechanics, differential Elasticity equations, functional analysis and R W Dickey, materials and structures. University of Madison, Wisconsin Cased ISBN 0 273 00129 9 Paperback ISBN 0 273 001 03 5 $45.00 Publication: July $11.30 Publication: June

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A-461 ssettes Tapes Audio Recordings of Mathematical Lectures.

57. Missed opportunities, FREEMAN J. DYSON 72. The horseshoe crab eye: A little nervous system whose 58. Frobenius and the Hodge filtration, BARRY MAZUR dynamics are solvable, BRUCE KNIGHT 59. Set theory and general topology, MARY ELLEN RUDIN 73. Irregularities of distribution in Euclidean n·space, 60. Some applications of an axiomatic theory of sheaves, WOLFGANG M. SCHMIDT MYLES TIERNEY 7 4. The closing lemma revisited, CHARLES C. PUGH 61. Equivoriant real algebraic differential topology, 75. A general theory of identities of the Rogers·Ramanujan RICHARD S. PALAIS type, GEORGE E. ANDREWS 62. Foliations of compact manifolds, H. BLAINE LAWSON 76. Submanifolds of Riemannian manifolds, CHUAN C. HSIUNG 63. lie dynamical systems, LAWRENCE MARKUS 77. Bo~.:nds for the chromatic number, HERBERT S. Wllf 64. The deficiency index problem for ordinary selfadjoint 78. Some biological and mathematical aspects of cellular operators, ALLEN DEVINATZ dynamics, JOSEPH J. HIGGINS 65. On automorphism groups of induced CR structure 79. Dynamical systems, filtrations, and entropy, MICHAEL SHUB HUGO ROSSI 80. Noncummutative localization, JOACHIM .. tAMBEK 66. The structure of ottractors, ROB;:RT F. WILLIAMS 81. Applications of global analysis to economics, electrical 67. Complexity of statement, computation and proof, circuits, and celestial mechanics, STEPHEN SMALE JACOB T. SCHWARZ (Colloquium Series, 3 lectures) 68. Powers of singular cardinals, ROBERT M. SOLOVAY 82. Prebiatic polymer synthesis and evolution as a stability 69. Determination of codon frequencies and sequence structure problem, AGNESSA BABLOY ANTZ of palynucleotides, HANS J. BREMERMANN 83. Representations of t10n·Desarguesian projective planes 70. A global dynamical scheme for vertebrate embryology, in projective hyperspace, RAJ C. BOSE RENE THOM 84. Monotonicity and magnetism, SEYMOUR SHERMAN 71. Groups with a single defining relation, GILBERT BAUMSLAG 85. The index of elliptic operators, MICHAEL F. ATIYAH (Colloquium Series, 4 lectures) IIIIIIIIIIIIIIIIIIIIIIIIIIIIHIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIHIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIflllllllllllllfllllllllllllllllllltlllllllllllllllllllllfllllllllllt Please circle numbers when ordering.

The following lectures are on one cassette and are available at $12.00 each: 2 4 7 8 9 10 11 12 13 14 16 20 22 23 24 25 26 27 30 31 32 34 35 36 37 39 40 41 42 43 44 45 46 47 49 50 51 52 54 55 58 59 60 61 63 64 65 66 67 68 70 71 72 73 74 75 76 77 78 79 80 82 83 84. The following are on two cassettes and are available at $15.00 each: 1 3 15 17 18 19 21 28 29 38 48 57 62 69. The following are Colloquium Lectures on four cassettes and are $27.00 each: 5 6 33 53 81 85. Each lecture is accompanied by a manual. Extra manuals are $.30 each. NAME ______CASSETTES ( ) TAPES ( ) 1 7/8"/second {4.75 em/second) TAPES ( ) 3 3/4"jsecond (9.5 em/second) ADDRESS------

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A-468 Audio Recordings of Mathematical Lectures. Number l: The role of vector and operator valued measures in functional analysis and probability, PES! R. MASANI 2: On local solvability of linear partial differential equations, FRAN~OIS TREVES 3: The theory of decision problems in group theory: a survey, WILLIAM W. BOONE 4: Recent developments in fixed·point theory, EDWARD R. FADELL 5: The periodicity theorem for the classical groups and some of its applications, RAOUL H. BOTT (4 lectures) 6: Harmonic analysis on semisimple Lie groups, HARISH-CHANDRA (4 lecutres) 7: On the existence and uniqueness of triangulations of manifolds, ROBION C. KIRBY 8: Vector fields and Gauss-Bonnet, PAUL F. BAUM 9: K2 of global fields, JOHN T. TATE 10: K2 of global fields, HYMAN BASS II: Systems of ordinal functions and functionals, SOLOMON FEFERMAN 12: Free boundary problems for parabolic equations, AVNER FRIEDMAN 13: Univalent functions with univalent derivatives, SWARUPCHAND M. SHAH 14: Shifts and Hilbert space factorization problems, MARVIN ROSENBLUM IS: Higher derivations and automorphisms of complete local rings, NICKOLAS HEEREMA 16: Recent results on doubly transitive groups, ERNEST E. SHULT 17: Foliations and noncompact transformation groups, MORRIS W. HIRSCH 18: Acyclicity in 3-manifolds, DANIEL R. MCMILLAN, JR. 19: Rigorous quantum field theory models, JAMES G. GLIMM 20: The concept of torsion and Gorenstein rings, MAURICE AUSLANDER 21: Vector fields on manifolds, MICHAEL F. ATIYAH 22: Integration of complex vector fields, JOSEPH J. KOHN 23: Finite groups generated by transvections, JACK E. MCLAUGHLIN 24: Geometry of numbers of convex bodies, KURT MAHLER 25: Bounded convergence of analytic functions, LEE A. RUBEL 26: Finite-dimensional H-spaces, MORTON L. CURTIS

27: Ordinal solution of G6 games, DAVID BLACKWELL 28: Aspects of the theory of spherical functions on symmetric spaces, RAMESH GANGOLLI 29: Some probability results connected with Diophantine approximation, PATRICK P. BILLINGSLEY 30: Differential algebraic geometry and transcendence, JAMES B. AX 31: How functional analysis and approximation theory mix today with numerical analysis, RICHARD S. VARGA 32: Representations of algebras by continuous sections, KARL H. HOFMANN 33: Topology of 3-manifolds, R. H. BING (4 lectures) 34: Light open maps on n-manifolds, ERIK HEMMINGSEN 35: The isomorphism problem in ergodic theory, DONALD S. ORNSTEIN 36: Simple groups of low 2-rank, DANIEL GORENSTEIN 37: Some nonlinear stochastic growth models, HARRY KESTEN 38: Some open questions in the theory of singularities, OSCAR ZARISKI 39: Ergodic theory and the geodesics on surfaces of negative curvature, EBERHARD HOPF 40: Recent developments in infinite dimensional holomorphy, LEOPOLDO NACHBIN 41: Recent results in the algebraic theory of minimal sets, ROBERT ELLIS 42: Symmetry in manifold theory, DENNIS P. SULLIVAN 43: Automorphisms of linear groups, 0. TIMOTHY O'MEARA 44: Linearly compact Lie modules, ROBERT J. BLATTNER 45: Eichler cohomology and the Fourier coefficients of automorphic forms, MARVIN I. KNOPP 46: Fixed point theorems in functional analysis, KY FAN 47: Infmite loop spaces, JOHN M. BOARDMAN 48: The structure of the centralizer of involutions in finite simple groups, JOHN H. WALTER 49: Local Diophantine equations, SIMON KOCHEN 50: Developments in the theory of Schauder bases, CHARLES W. MCARTHUR 51: Conjugacy problems in discrete dynamical systems, JOEL W. ROBBIN 52: Recent progress on the isomorphism problem in ergodic theory, BENJAMIN WEISS 53: Uniformization, moduli and Kleinian groups, LIPMAN BERS ( 4 lectures) 54: Quasi-analyticity and semigroups, JOHN NEUBERGER 55: Locally flat embeddings of manifolds, JAMES C. CANTRELL

AMERICAN MATHEMATICAL SOCIETY, P. 0. Box 1571, Annex Station, Providence, R.I. 02901

A-469 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS

Volume 18, Part 1, NONLINEAR FUNCTIONAL ANALYSIS Part 2, NONLINEAR OPERA TORS AND NONLINEAR EQUATIONS OF EVOLUTION IN BANACH SPACES

A symposium on Nonlinear Functional Analysis held in Chicago in April 1968 has resulted in the publication of these two volumes edited by Felix E. Browder. The first part, published in 1970, contains papers contributed by the following speakers:

Edgar Asplund Tosio Kato Melvyn S. Berger William A. Kirk Haim Brezio J. L. Lions Lamberto Cesari Umberto Mosco Jane Cronin Richard S. Palais James Eels, Jr. W. V. Petryshyn K. D. Elworthy and A. J. Tromba P. H. Rabinowitz D. E. De Figueiredo and L. A. Karlovitz R. T. Rockafellar Hiroshi Fujita Erich H. Rothe Benjamin R. Halpern Guido Stampacchia G. Stephen Jones Walter A. Strauss

The second part, just published in 1976, contains the book-length text of a paper by Felix E. Browder composed in its entirety during 1968. It is a detailed treat­ ment of most of the major branches of nonlinear functional analysis as they had developed up to that time. The manuscript of this work has had wide circulation in mimeographed form and has been referred to in a considerable number of re­ search papers. It is now at last available.

PSPUM/18.1, 296 pages, LIST PRICE $19.60; MEMBER PRICE $14.70. PSPUMI18.2, 308 pages, LIST PRICE $30.00; MEMBER PRICE $22.50 ORDERS MUST BE PREPAID

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A-470 PREREGISTRATION FORM

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A-472 TOPICS IN TOPICS IN DIFFERENTIAL NUMERICAL ANALYSIS II Proceedings of the Royal Irish Academy GEOMETRY Conference on Numerical Analysis, 1974 edited by HANNO RUND and edited by JOHN J. H. WILLIAM F. FORBES MILLER During the summer CONTENTS: R. S. Clark, of 1974, the Natio~al Evan Tom Davies. Committee for Mathematics of the Royal lnsh E. Bompiani, Reminiscences of E. T. Davies. Academy held their se~ond. conference on ~u­ /. M. Anderson, The Uniqueness of the N~u­ merical analysis at Umvers1ty College, Dublin. trino Energy-Momentum Tensor and the Em­ This stein-Weyl Equations. volume contains, in complete form, all F. Brickel et a/., (G,E) the papers delivered by the invited Structures. G. W. Horndeski, speakers Tensorial Con­ -save one, which was a summary of work comitants of an Almost Complex Structure published A. Llchnerowicz, elsewhere-and covers a wide Vari~t~s Symplectiques, Va­ range of topics in the area of numerical ana­ ri~t~s Canoniques, et Systemes Dynamiques. lysis. The conference was truly international, D. Lovelock, Divergence-Free Third Order bringing together nt;~arly 200 participants Concomitants of the Metric Tensor in Three r~p­ Dimensions. resenting 26 countnes under the organization M. A. McKiernan, A Functional and directorship Equation in the Characterization of Dr. Miller. Professor Cor­ of _Null nelius Lanczos, who delivered the opening Cone Preserving Maps. H. Rund, Generalized address at the 1972 Clebsch Representations Conference and died in of Manifolds. A. G. Budapest in June 1974, is remembered Walker Note on Locally Symmetric Vector in the Fields 'in final session, a memorial tribute to his life a Riemannian Space. T. J. Willmore, and work. In Mean Curvature of Immersed addition to appealing to every Manifolds. Y. C. numerical analyst, this volume should prove Wong and K-P. Mok, Connections and M­ of value to many Tensors on the Tangent Bundle scientists (notably other TM. K. Yano, mathematicians and engineers) who use nu­ Differential Geometry of Totally Real Submani­ merical analysis. folds. 1976, 280 pp., $22.75/£9.20 1976, 206 pp., $19.50/£10.70 ISBN: 0-12-496952-7 ISBN: 0-12-602850-8 ANALYTIC COMPUTATIONAL THEORY OF COMPLEXITY APPROXIMATIONS edited by J. F. TRAUB With Applications This book presents the Proceedings of the edited by ALAN G. LAW and Symposium on Analytic Computational Com­ BADRIN.SAHNEY plexity held by the Computer Science Depart­ ment, Carnegie-Mellon University, Pittsburgh, This book contains full proceedings of the Pennsylvania, on April 7-8, 1975. Conference on the Theory of Approximation held at the University of Calgary, August CONTENTS: J. F. Traub, Introduction. 11- S. Wino­ 13, 1975. It features research and survey ar­ grad, Some Remarks on Proof Techniques ticles discussing the in Analytic Complexity. ongoing development of J. F. Traub and H. the general theory, a number of shorter papers Woznlakowski, Strict Lower and Upper Bounds on current research in approximation on Iterative theory Computational Complexity. H. T. and practice, theoretical numerical analysis, Kung, The Complexity of Obtaining .starting functional analysis, expansions, and applica­ Points for Solving Operator Equat1ons by tions. Topics Newton's covered include: recent develop­ Method. R. P. Brent, A Class of Op­ ments in spline theory and applications; timal-Order Zero-Finding Methods Using Deri­ generalized polynomial vative Evaluations. approximation; conver­ H. Woznlakowski, Maximal gence and algorithms for rational approxima­ Order of Multipoint Iterations Using n Evalua­ tions; estimation tions. R. D. Meersman, of functionals; approximations Optimal Use of Infor­ in function spaces; interpolation theory; engi­ mation in Certain Iterative Processes. B. neering applications. Kacewicz, The Use of Integrals in the Solution of Nonlinear Equations in N Dimensions. M. H. 1976, 432 pp., $17.50/£9.60 Schultz, Complexity and Differential Equations. ISBN: 0-12-438950-3 R. P. Brent, Multiple-Precision Zero-Finding Methods and the Complexity of Elementary Function Evaluation. H. Wozniakowski, Numer­ N.B.: Postage plus 50¢ handling charge on all ical Stability of Iterations for Solution of Non­ orders not accompanied by payment. linear Equations and Large Linear Systems. Prices are subject to change without notice. J. R. Rice, On the Computational Complexity of Approximation Operators II. D. Y. Y. Yun, Hensel Meets Newton-Algebraic Construc­ ACADEMIC tions in an Analytic Setting. R. P. Brent and H. T. Kung, O(n log n)"'") Algorithms _for PRESS Composition and Reversion of Power Senes. A Subsidiary of Contributed Paper Abstracts. Harcourt Brace Jovanovich, Publishers 1976,247 pp., $11.001£6.05 111 FIFTH AVENUE, NEW YORK, N.Y. 10003 ISBN: 0-12-697560-4 24-28 OVAL ROAD, LONDON NW1 7DX

Notices of the Ams June 76 Ill 3 ~ Ill -....~

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