OF THE

AMERICAN MATHEMATICAL SOCIETY

Edited by Everett Pitcher and Gordon L. Walker

CONTENTS

MEETINGS Calendar of Meetings • . . • ...... Inside Front Cover Program for the April Meeting in Berkeley, California . 158 Abstracts for the Meeting: Pages A -469-A -482 PRELIMINARY ANNOUNCEMENTS OF MEETING . . . . 163 MATHEMATICAL SCIENCES EMPLOYMENT REGISTER . 166 A MATHEMATICAL VISIT TO CHINA. 167 LETTERS TO THE EDITOR ...... 170 MEMORANDA TO MEMBERS ...... 172 Combined Membership List, 1972-1973, Change of Address or Position SPECIAL MEETINGS INFORMATION CENTER. 173 SUMMER GRADUATE COURSES . . . 182 NEW AMS PUBLICATIONS ...... 183 NEWS ITEMS AND ANNOUNCEMENTS . 162, 165, 166, 172, 181, 184 PERSONAL ITEMS ...... 185 ABSTRACTS ...... A-289 SITUATIONS WANTED ADVERTISEMENTS A-482 INDEX TO ADVERTISERS ...... A-498 Six Hundred Ninety-Fourth Meeting University of California, Berkeley Berkeley, California April22, 1972

The six hundred ninety-fourth meet­ HOTEL DURANT ing of the American Mathematical Society 2600 Durant Avenue, Zip Code 94704 will be held at the University of Califor­ Phone: (415) 895-8981 nia, Berkeley, California, on Saturday, Single $12,00 up April 22, 1972. Double 15,50up By invitation of the Committee to Twin 16,50 up Select Hour speakers for Far Western HOTEL SHATTUCK Sectional Meetings, there will be two Shattuck Avenue and Allston Way, Zip one-hour addresses. Professor Roger Code 94704 Richardson of the University of Washing­ Phone: (415) 845-7300 ton will lecture at ll :00 a.m.; the title of his lecture is "On the variation of The Hotel Durant and the Hotel Shattuck isotropy subgroups." Professor Joseph L. are within easy walking distance of the Taylor of the University of Utah will give campus. Participants should write or the second address at 2:00 p.m.; he will telephone hotels or motels if reservations speak on "Functions of several noncom­ are desired. muting variables." Both lectures will be Several airlines serve the San Fran­ given in Room 10 of Evans Hall. There cisco and Oakland airports. Taxi fare will be sessions for contributed papers from Oakland Airport to downtown Berke­ in the morning and afternoon. ley is approximately $8.00, and lim::>usine The registration desk will be located service is about $3.00. The Hotel Durant in the ground level lobby of Evans Hall. is a terminal for Oakland Airport Limou­ Registration will begin at 8:30 a.m. on sine Service. Taxi service from San Saturday. Francisco Airport to Berkeley is approx­ There are numerous motels and ho­ imately $14.50. One can take a limousine tels in Berkeley, some of which are listed from the San Francisco Airport to the below: downtown San Francisco terminal, then a BERKELEY MOTEL taxi to the bus terminal, and a bus to 2001 Bancroft Way, Zip Code 94704 Berkeley. Helicopter service is available Phone: (415) 845-9644 from San Francisco Airport to the Berke­ Double $ 8.50 up ley Heliport. The fare, purchased locally, Twin 10.00 up is $9.00, WhentheSanFrancisco-Berkeley Family room 13,50 flight is purchased as a portion of a ticket from point of departure, the fare may be $4.50 BERKELEY TRAVELODGE reduced by an amount varying from 1820 University Avenue, Zip Code 94703 to $8.00, depending on the airline. Persons driving to the meeting on the Phone: ( 415) 843-4262 Single $11.00 up freeway from any direction should take Double 14.00 up the University Avenue turnoff. Drive east Twin 17.00up on University Avenue (toward the hills) to the cam pus; turn left off University onto FLAMINGO MOTEL Oxford Street. Turn right off Oxford onto 1761 University A venue, Zip Code 94703 Hearst Avenue and continue up Hearst to Phone: (415) 841-4242 LaLoma Avenue. Participants may park Single $12.00 up in the AB lots of the parking structure H Double 14.00 up which is located on the northwest corner Twin l6.00up of Hearst and LaLoma.

158 PROGRAM OF THE SESSIONS

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

SATURDAY, 9:00 A. M.

Session on Semigroups, Room 70, Evans Hall 9:00-9:10 (1) Exponent of semigroups and order-bounded groups Professor Takayuki Tamura, University of California, Davis (694-All) 9:15-9:25 (2) On exponential archimedean semigroups with idempotent. Preliminary report Professor Takayuki Tamura and Mr. Thomas Edward Nordahl*, University of California, Davis (694-A13) 9:30-9:40 (3) Quotient group of finitely generated N-semigroups. Preliminary report Professor Takayuki Tamura, University of California, Davis, and Professor John C. Higgins*, Brigham Young University (694-A14) 9:45-9:55 (4) Noncancellative congruences on N -semigroups. Preliminary report Dr. Robert P. Dickinson, Jr., Lawrence Radiation Laboratory, University of California, Livermore (694-A10) (Introduced by Professor Takayuki Tamura) 10:00-10:10 (5) The study of commutative semigroups with greatest group homomorphism Professor Takayuki Tamura and Mr. Howard B. Hamilton*, University of California, Davis (694-A12) 10:15-10:25 (6) Greatest regular images of tensor products of commutative semigroups Professor Thomas J. Head*, University of Alaska, and Professor Nobuaki Kuroki, Nikon University, Narashino, Japan (694-AS) 10:30-10:40 (7) Some languages derived from abelian groups Professor Edward J. Tully, University of California, Davis (694-A16)

SATURDAY, 9:00 A. M.

Session on Functional Analysis, Room 9, Evans Hall 9:00-9:10 (8) Convergence of positive sequences. Preliminary report Professor Philip M. Anselone, Oregon State University, and Professor Ralph L. James*, Stanislaus State College (694-B12) 9:15-9:25 (9) Continuity of derivations of Banach algebras into modules. Preliminary report Professor William G. Bade*, University of California, Berkeley, and Professor Philip C. Curtis, Jr. , University of California, Los Angeles (694-B14) 9:30-9:40 (10) Order-continuous states on Rickart algebras. Preliminary report Mr. Milton Philip Olson, Lafayette, California (694-B18) 9:45-9:55 (11) A note on locally c*-algebras. Preliminary report Professor Banshi D. Malviya, North Texas State University (694-B15) 10:00-10:10 (12) Extension of the Krein-Milman theorem. Preliminary report Mr. Stephan Michael Sperling, University of California, Los Angeles (694-B6) 10:15-10:25 (13) Generalized numerical range. II. Preliminary r13port Professor Charles F. Amelin, California State Polytechnic College, Kellogg-Voorhis (694-B2)

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

159 10:30-10:40 (14) Vanishing second commutators in operator theory and Lie group representations Preliminary report Professor Robert T. Moore, University of Washington (694-B13)

SATURDAY, 9:00 A. M.

General Session, Room 3, Evans Hall 9:00-9:10 (15) Equivalence of constant and time varying quadratic performance criteria Dr. Clyde F. Martin, NASA, Ames Research Center, Moffett Field, California (694-C1) 9:15-9:25 (16) On the spectral representation for symmetric stable random variables. Preliminary report Dr. Marek Kanter, Tulane University (694-F1) (Introduced by Professor Jacob Feldman) 9:30-9:40 (17) Pointwise bounds on the discrete Laplacian of a harmonic function Dr. Robert L. Meyer, California State College at Los Angeles (694-C3) 9:45-9:55 (18) Existence of solutions for singular nonlinear integral equations resulting from plane free surface flows. Preliminary report Mr. Gordon E. Ritchie, University of Santa Clara (694-C2) (Introduced by Professor Irving Sussman) 10:00-10:10 (19) Jackson type theorems for approximation with side conditions. Preliminary report Mr. Darell J. Johnson, University of California, Riverside (694-B3) 10:15-10:25 (20) A generalized hypergeometric integral transform. Preliminary report Professor Moses E. Cohen, Fresno State College (694-B8) 10:30-10:40 (21) Theorems on absolute Cesaro summability of integral Mr. Santiranjan Mukhoti, Council of Scientific and Industrial Research, Rafi Marg, New Delhi, India (694-Bll)

SATURDAY, ll:OOA. M.

Invited Address, Room 10, Evans Hall On the variation of isotropy subgroups Professor Roger Richardson, University of Washington

SATURDAY, 2:00P.M.

Invited Address, Room 10, Evans Hall Functions of several noncommuting variables Professor Joseph L. Taylor, University of Utah

SATURDAY, 3:15P.M.

Session on Number Theory and· Combinatorics, Room 81, Evans Hall 3:15-3:25 (22) Modules over orders and transformations of quadratic forms Dr. Bart Francis Rice, Naval Postgraduate School (694-A2) 3:30-3:40 (23) Consequences and generalizations of algebraic solution of ~ = yx (0 < x < y). Preliminary report Dr. Daihachiro Sato, University of Saskatchewan, Regina (694-A17) 3:45-3:55 (24) Some combinatorial formulas for partitions. Preliminary report Professor Hugo Sui-Hwan Sun, Fresno State College (694-A9) 4:00-4:10 (25) Self-complementary generalized orbits of a permutation group Professor Roberto Frucht, Universidad Tecnica Federico Santa Maria, Valparaiso, Chile, and Professor Frank Harary*, University of Michigan (694-A3)

160 SATURDAY, 3:15 P. M.

Session on Algebra, Room 70, Evans Hall 3:15-3:25 (26) On quasi-groups and quasi-loops satisfying the identity x • xy = yx and their isotopes. Preliminary report Dr. Volodymyr Bohun-chudyniv, Morgan State College (694-A7) 3:30-3:40 (27) Covering theorems for finite nonabelian simple groups. I. Preliminary report Professor J. L. Brenner* (University of Arizona), Palo Alto, California, Mr. M. Randall, Amoco, Calgary, Alberta, Canada, and Professor James Riddell, University of Victoria (694-A1) 3:45-3:55 (28) Free subgroups and Folner's conditions Professor Melven R. Krom, University of California, Davis, and Professor Myren Krom*, Sacramento State College (694-A15) 4:00-4:10 (29) Commutative idempotent generated rings as generalized Boolean rings. Preliminary report Dr. Martin K. McCrea, University of California, Davis (694-A6) (Introduced by Professor Takayuki Tamura) 4:15-4:25 (30) Henselian valuations on an ordered field. Preliminary report Dr. Michael J. Wright, Loyola University of Los Angeles (694-A4) 4:30-4:40 (31) Subalgebra maps induced by endomorphisms and automorphisms of algebras Dr. Matthew I. Gould, Vanderbilt University (694-A5)

SATURDAY, 3:15 P. M.

Session on Topology, Room 9, Evans Hall 3:15-3:25 (32) Homotopy chain functors Professor Donald W. Anderson, University of California, San Diego (694-G4) 3:30-3:40 (33) On the Spanier conjecture Dr. Remi P. Held and Dr. Denis Sjerve*, University of British Columbia (694-G2) 3:45-3:55 (34) On some fixed point theorems. Preliminary report Dr. Sankatha P. Singh, Memorial University of Newfoundland (694-G3) 4:00-4:10 (35) A countable, connected, Urysohn space with a dispersion point that is regular almost everywhere Professor Eldon J. Vought, Chico State College (694-G1) -SATURDAY, 3:15 P. M.

Session on Analysis, Room 3, Evans Hall 3:15-3:25 (36) Weyl's lemma for pointwise solutions of elliptic equations. Preliminary report Professor James R. Diederich, University of California, Davis (694-B7) 3:30-3:40 (37) Oscillation criteria for y" + P(t)f(y,y') = 0, with f homogeneous of degree one. Preliminary report Professor Grant B. Gustafson, University of Utah (694-B1) 3:45-3:55 (38) On ex-convex functions. II. Preliminary report Professor Petru Mocanu, Babes-Bolyai University, Cluj, Romania, and Professor Maxwell 0. Reade*, University of Michigan (694-B5) 4:00-4:10 (39) Bounded holomorphic mappings and generalized Bloch's theorem Professor Kyong T. Hahn, Pennsylvania State University and University of California, Berkeley (694-B10) 4:15-4:25 (40) Levi-flat two-manifolds in cc 2• Preliminary report Professor Michael B. Freeman, University of Kentucky (694-B16)

161 4:30-4:40 (41) On a class of invertible distributions Professor Carlos A. Berenstein*, Harvard University, and Professor Milos A. Dostal, Stevens Institute of Technology (694-B4) 4:45-4:55 (42) Rotation invariant measures on Euclidean space Professor David L. Ragozin, University of Washington (694-B9) 5:00-5:10 (43) A converse Steinhaus theorem for locally compact groups. Preliminary report Professor Steven M. Simmons, Eastern Washington State College (694-Bl'l

Kenneth A. Ross Associate Secretary Eugene, Oregon

NEWS ITEMS AND ANNOUNCEMENTS

TEACHING OPPORTUNITIES IN GERMANY

The National Carl Schurz Associa­ over, DM 100 ( $30) per month; Ph.D., tion, Inc. (NCSA), a nonprofitorganization DM 100 ( $30) per month; family allowance whose aim is to promote and sponsor the of DM 50 ( $15) per month for each depend­ German language and culture within the ent child under 18 with the maximum allow­ United States, has announced that it is ance for dependents DM 200 ( $60) per seeking fifty teachers of mathematics for month. Insurance and round-trip trans­ the secondary schools of the Land of portation for the participant only (not his Northrhine-Westfalia. Representatives of family) will also be paid by NCSA. the Ministry of Culture of Northrhine­ Those interested should write di­ Westfalia and the NCSA have worked out rectly to the National Carl Schurz Asso­ mm1mum criteria for the selection of ciation, Inc., 339 Walnut Street, Philadel­ these teachers. These criteria include phia, Pennsylvania 19106. In addition to (1) minimum age of 27; (2) minimum meeting the qualifications listed in the M.A. or M.S.; (3) certification to teach preceding paragraph, the completed ap­ mathematics, physics, or chemistry (biol­ plication must include the following: ( 1) ogy, if in connection with one of the pre­ signature of the applicant; (2) letter of ceding) on the senior high school level recommendation from mathematics, phys­ with practical teaching experience of at ics, or chemistry department; ( 3) letter least two years; or A.B.D. or Ph.D. with of recommendation from teaching super­ either two years of full-time undergraduate visor or persons who can testify to teach­ teaching or four years as a teaching as sis­ ing ability; (4) letter of recommendation tant; (4) knowledge of the German lan­ for character reference; ( 5) transcript of guage, enough to teach in German after an undergraduate work, and of graduate work; eight-week intermediate, intensive lan­ (6) medical statement from candidate's guage course. Applicants are advised to physician, in duplicate, stating that the keep in mind that the German schools are candidate is in good physical health; encountering serious disciplinary prob­ ( 7) additional proof of successful teaching lems different from those in the United experience in high school or undergraduate States, and teachers are only successful classes. Applicants will be notified of the if they know how to handle youngsters be­ Selection Committee's decision some time tween the ages of 12 and 18. Basic remu­ after a personal interview has been ar­ neration will be DM 170 0 ( $51 0) per month ranged. Deadline for applications to reach with the following added allowances: age NCSA is May l, 1972. 36-45, DM 50 ( $15) per month; 46 and

162 PRELIMINARY ANNOUNCEMENTS OF MEETINGS Six Hundred Ninety-Fifth Meeting University of Washington Seattle, Washington June 17, 1972

The six hundred ninety-fifth meeting of the meetings. of the American Mathematical Society Dormitory accommodations will be will be held at the University of Washing­ available on Thursday, Friday, andSatur­ ton in Seattle, Washington, on Saturday, day nights. These accommodations are June 17, 1972. The Mathematical Asso­ not recommended for families with chil­ ciation of America and the Society for dren under twelve. The rates are $3.50 Industrial and Applied Mathematics will per person per night on a double occupancy hold Northwest Sectional Meetings in con­ basis, and $5.50 per person per night in a junction with this meeting of the Society. single room. No maid service is provided. The Association will have sessions on Reservations should be sent, before June 2, Friday and Saturday, June 16 and 17; the 1972, to Professor Gomer Thomas, De­ Saturday sessions will be concerned pri­ partment of Mathematics, University of marily with community college problems. Washington, Seattle, Washington 98195, The SIAM sessions will be held on Friday, and should be accompanied by a check June 16. payable to the University of Washington By invitation of the Committee to and in the amount covering the length of Select Hour Speakers for Far Western time desired. Room requests should in­ Sectional Meetings, there will be two in­ clude the expected time of arrival and the vited addresses. Professor Donald E. type of accommodations desired. Sarason of the University of California, The following motels and hotels are Berkeley, will lecture at ll :00 a.m. on located within walking distance of the Saturday on "Algebras of functions on the campus: unit circle." Professor Peter Crawley of the California Institute of Technology and UNIVERSITY TOWER HOTEL Brigham Young University will address 4507 Brooklyn Avenue, N.E., Zip Code the Society at 2:00p.m. on Saturday. The 98105 title of his lecture is" Extensions of Ulm' s Phone (206) 634-2000 theorem for abelian p-groups." These Single $13,00up addresses will be given in room 134 of Double 15,50 up the General Engineering Building. There Twin l8.00up will be sessions for contributed papers UNIVERSITY INN on Saturday morning and afternoon. Ab­ 4140 Roosevelt Way, N.E., Zip Code stracts should be submitted to the Amer­ 98105 Phone (206) 632-5055 ican Mathematical Society, P .0. Box 6248, Single $10.50 up Providence, Rhode Island 02904, so as to Double 12.50 up arrive prior to the deadline ofMay4, 1972. Twin 15.00 up Late papers will be accepted for presen­ Double-double 19.00 up tation at the meeting, but late papers will not be listed in the printed program of UNIVERSITY MOTEL the meeting. Sessions for contributed 4731 12th Avenue, N.E., Zip Code 98105 papers will be held in the General Engi­ Phone (206) 522-4724 neering Building. Single $10.00 up The registration desk will be located Double l2.00up in room ill of the General Engineering Twin 12.00 up Building, and will be open for the duration Double-double l6.00up

163 UNIVERSITY TRAVELODGE tional Airport. Other air travellers can 47Z5 25th Avenue, N.E., Zip Code 98105 take the airport limousine to the Olympic Phone (206) 525-4612 Hotel in downtown Seattle and then take a Single $10.00 up taxi to the university district. Persons Double 13.00 up driving to the meeting on Interstate 5 Twin 14.00 up should take the 45th N.E. or 50th N.E. exit, drive east to 17th Avenue, N.E., Reservations should be made directly with turn right onto 17th Avenue, N.E., and the desired motel or hotel. proceed onto the campus. Campus officers Breakfast and lunch will be available on duty at the entrance will give further on Friday in the UnionBuildingoncampus. directions. Special parking permits may There are numerous restaurants in Seattle, be purchased at the registration desk, but and several of them are located in the will not be needed for the initial arrival area adjacent to the campus. on campus. Seattle is served by major airlines and bus companies. The University Tower Kenneth A. Ross Hotel provides transportation for its pa­ Associate Secretary trons from the Seattle-Tacoma Interna- Eugene, Oregon The Seventy-Seventh Summer Meeting Dartmouth College Hanover, New Hampshire August 29-September 1, 1972

The seventy-seventh summer meet­ junction with meetings of the Institute of ing of the American Mathematical Society Mathematical Statistics, the Mathematical will be held at Dartmouth College, Han­ Association of America, and Pi Mu Epsilon. over, New Hampshire, from Tuesday, The Mathematical Association of America August 29, through Friday, September 1, will meet from Monday, August 28, through 197Z. All sessions of the meeting will take Wednesday, August 30. The Earle Raymond place on the campus of the college. Hedrick Lectures, sponsored by the Asso­ Two sets of Colloquium Lectures will ciation, will be given by Professor Peter be scheduled: Professor Stephen Smale of Lax of the Courant Institute of Mathemati­ the University of California, Berkeley, cal Sciences, New York University. The will lecture on "Applications of global title of the lectures to be presented by analysis to biology, economics, electrical Professor Lax will be "Scattering theo­ circuits, and celestial mechanics"; Pro­ ries.~~ fessor john T. Tate of Harvard University COUNCIL AND BUSINESS MEETING will lecture on "The arithmetic of elliptic curves." The Council of the Society will meet It is expected that there will be ten at 5:15p.m. on Tuesday, August 29, in the invited one-hour addresses at the meeting. Drake Room of Hopkins Center. The Busi­ Sessions for contributed ten-minute papers ness Meeting of the Society will be held in will be held during the morning on Wednes­ the Spaulding Auditorium at 4:00 p.m. on day, Thursday, and Friday. Abstracts of Thursday, August 31. contributed papers should be sent to the REGISTRATION American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02904; Dartmouth College will operate a the deadline for receipt of abstracts is Housing Bureau as a special service to july 5, 197Z. There isnolimitonthe num­ those attending the Mathematics Meetings. ber of papers that will be accepted for Hanover and the surrounding country is a presentation at the 1972 summer meeting. popular tourist area, and hotels and motels Provisions will not be made for late papers. are generally filled the latter part of Au­ This meeting will be held in con- gust. The college has made arrangements

164 with several nearby motels to hold rooms There will be no extra charge for members for participants, in addition to the dormi­ of the families of registered participants. tory rooms. Persons desiring to make The unemployed status refers to any mem­ reservations for either dormitory or motel ber currently unemployed but actively accommodations through the Housing Bu­ seeking employment. It is not intended to reau will be required to preregister for the include members who have voluntarily re­ meeting. In addition to being ensured of signed or retired from their latest position. preferred living accommodations, those EMPLOYMENT REGISTER who preregister will pay a lower meeting registration fee than at the meeting, as The Mathematical Sciences Employ­ indicated in the schedule below. Preregi­ ment Register will be maintained from strants will be able to pick up their badges 9:00 a.m. to 4:00 p.m. on Tuesday, August and programs when they arrive at the 29, and from 9:00 a.m. to 5:40 p.m. on meeting. Full information, including rates Wednesday and Thursday, August 30-31, for dormitory and motel accommodations, in the Alumni Hall of the Hopkins Center. and a registration form will be included in Alumni Hall is located on the second floor the June issue of these cJiotiuiJ. of the center adjacent to the registration The registration fees for the meeting area. are as follows: EXHIBITS Preregistration At meeting Book exhibits and exhibits of edu­ Member $ 5 $ 7 cational media will be displayed on the Student or second level of the Hopkins Center. unemployed Walter H. Gottschalk member 1 Associate Secretary Nonmember 10 14 Middletown, Connecticut NEWS ITEMS AND ANNOUNCEMENTS

UNIVERSITY OF TENNESSEE ing, leaving the remainder free for study, POSTDOCTORAL attendance at professional meetings, re­ INTERN-FELLOWSHIPS search and publication. At the end of their one-year contract period, the University In response to the current shortage of Tennessee will consider them for its of jobs in teaching, the University ofTen­ permanent faculty, or assist in their job nessee is instituting a unique new plan for search. temporary employment. The Postdoctoral Eligibility is restricted to those Intern-Fellowship Program, developed by completing the doctorate in the 1971-1972 Kenneth L. Knickerbocker, Vice-president academic year. Letter of application, cur­ for Academic Affairs, is especially de­ riculum vita, and complete dossier should signed to aid new doctoral graduates who be sent no later than June 15, 1972, to have not found a permanent position for Dr. Kenneth L. Knickerbocker, Vice-pres­ September 1972, and, at the same time, ident for Academic Affairs, University of to strengthen the university's temporary Tennessee, Knoxville, Tennessee 37916. teaching staff. All five primary compuses of the University of Tennessee System (Chatta­ UNIVERSITY OF COLORADO nooga, Knoxville, Martin, Nashville, and the Memphis Medical Units) are partici­ The Board of Regents of the Univer­ pating in the program, offering positions sity of Colorado has approved a Ph.D. for superior candidates in thirty-eight program in computer science at the Uni­ disciplines, including mathematics. In ad­ versity of Colorado, Boulder. Applications dition to a good fellowship stipend, the for admission to this program and related Intern-Fellows will receive in-service information may be obtained by writing training and supervision by experienced to Professor Lloyd D. Fosdick, Chairman, faculty. Only three-fourths of the Intern­ Department of Computer Science, Univer­ Fellows' time will be required for teach- sity of Colorado, Boulder, Colorado 80302.

165 MATHEMATICAL SCIENCES EMPLOYMENT REGISTER

An open Register will be maintained Register is separate and apart from both at the joint Mathematics Meeting sched­ meeting registration and from the pub­ uled for Dartmouth College, Hanover, lished listings. It is imperative that both ·New Hampshire. The Register will be in applicants and employers who wish to operation from 9:00 a.m. to 4:00 p.m. on participate in the open Register sign in at Tuesday, August 29, and from 9:00 a.m. the Employment Register desk as early to 5:40 p.m. on Wednesday and Thursday, as they can on Tuesday morning. Appoint­ August 30-31, To assist the Register ments will be scheduled only for those personnel in planning for this open Regis­ people who have actually signed in at the ter, it is requested that those persons Register and obtained a code number. Re­ who expect to participate send a note to quests for appointments can be submitted the Providence office. Employers are re­ on Tuesday and Wednesday only, and these quested to indicate the number of positions interviews will be scheduled on Wednesday available and to specify which day or days and Thursday respectively. they will be available for interviewing. The May issue of the published Reg­ The changes instituted at the open Regis­ ister will be mailed from Providence in ter in Las Vegas, namely the establish­ late April. A subscription to the lists, ment of a registration fee of $10 for each which includes three issues (May, August, employer participating in the open Regis­ and January) of both the applicants list ter and the requirement that all partici­ and the positions list, is available for $30; pants (employers and applicants) MUST the single set may be purchased for $15. BE REGISTERED for the general Mathe­ A subscription to the applicants list alone, matics Meeting before registering for the or single copies of that list, are not avail­ open Register, will still be in effect. able. Copies of the positions list onlymay There is no fee for applicants participating be purchased for $5. A subscription to the in interview schedules. Registration fees, list of positions (May, August, January) is location of the registration area, and hours available for $12. of operation for the registration of par­ The lists are mailed book rate from ticipants for the Mathematics Meeting are Providence (normal delivery time to most listed in the preliminary announcement of locations in the U.S. or Canada is approx­ the meeting included in this issue of these imately fourteen days) unless the pur­ c}fo&:eD. chaser either indicates in advance a will­ Applicants and employers should ingness to pay the first class or airmail secure an instruction sheet explaining the charges, or includes the fee for this ser­ rules and operating regulations; these vice when prepayment is made. The ap­ will be available on request in the Alumni plicable postage charges, determined by Room of the Hopkins Center at 9:00 a.m. the location of the purchaser, will be on Tuesday. There will be no interviews furnished on request to those persons who scheduled for the first day. Please keep would like to take advantage of this service. in mind that registration for the open NEWS ITEMS AND ANNOUNCEMENTS

JOHN H. BARRETT morial Fund supporting this lecture series MEMORIAL LECTURES was established by friends and colleagues of Professor Barrett. For further infor­ The third series of John B. Barrett mation, please write to Professor D. B. Memorial Lectures in Ordinary Differen­ Hinton, Department of Mathematics, The. tial Equations will be delivered by Pro­ University of Tennessee, Knoxville, Ten­ fessor Garrett Birkhoff at The University nessee 37916. of Tennessee on May 2-4, 1972. The Me-

166 A MATHEMATICAL VISIT TO CHINA by Chandler Davis

Profoundly alike though China and the factory or other work unit where they North Viet Narn may be, in history, social worked before admission. structure, and official ideology, still con­ The occasion for this divergence is trasts leap to the eye. Here's one: the of course the Great Proletarian Cultural university mathematics curriculum at Revolution, which washed over China from Peking versus that at Hanoi. 1966 to 1970 utterly disrupting routines Mathematics concentrators at the and institutions -especially acadern ic. University of Hanoi take, in the first year, Mathematics was surely one of the calculus, analytic geometry, algebra, and fields most affected by the Cultural Revo­ non-mathematical sciences such as phys­ lution. The leading differential geometer ics. In the second year, more calculus and Su Pu-ching was one of the first main algebra, differential geometry, ordinary targets of Red Guard criticism at Futan differential equations, mechanics, numeri­ University; the topologist Wu Wen-tsiin cal analysis. Third-year courses cover was also individually attacked; and Hua real variables and integration, metric Lo-keng, who had been head of the Mathe­ spaces and Banach spaces, complex func­ matics Institute of the Academy, became tion theory, partial differential equations, the first leading intellectual to publish for more numerical analysis, probability, and foreign readers his self-criticism follow­ theoretical physics; the student also writes ing the Cultural Revolution. 1 But all sci­ a small thesis. In the fourth year he takes entific fields were involved. University more real analysis, general topology, and teaching stopped, scholarly journals other functional analysis, and optional subjects than medical suspended publication, and (such as linear programming and mathe­ most contacts with scientists of other matical economics); there is also another countries (even of North Viet Narn) were small thesis, in which some originality is broken off. expected. On completing undergraduate Now the near-isolation is corning to work, most students do some advanced an end. Within the five months May­ study and research, and a considerable September 1971, several North American number proceed to the equivalent of our scientists were guests of the Chinese: Ph.D. Ethan Signer and Arthur Galston in biolo­ Quite otherwise at Peking. First­ gy, Robert Coe in geophysics, and myself. year students take calculus and more My hosts arranged conversations for me elementary subjects, and work at produc­ with groups of mathematicians at the tive labour; this labour may take them off Mathematics Institute, at Peking Univer­ campus to (for instance) a radio factory, sity, and at Futan University in Shanghai; but it is considered an important part of had my time allowed, I was welcome to the curriculum and is to be integrated have more such conversations. It was with the intellectual work. Second-year possible-and rewarding-to set up in­ students take probability and statistics, formal small meetings on request. I spoke and learn to program in machine language, at length with (among others) Hua Lo-keng, then in Algol-and they work at productive Wu Wen-tsun, T'ien Fang-tseng, and Kwan labour. There are no third-year students Chao-chih in Peking, and Su Pu-ching and now, but next year when there are, they Hsia Tao-hsing in Shanghai. Many mathe­ will spend a still larger portion of their maticians spoke positively of the value of time at work off campus. After three years future contacts, or even sent greetings to of studenthood, almost all will return to individual mathematicians they knew from

Reprinted from Notes of the Canadian Mathematical Congress, vol. 4, no. 4, January 1972, with the permission of the Canadian Mathematical Congress. 1. China Reconstructs, November, 1969.

167 past stays in North America. in China, universities are run by Revolu­ The possibility has now opened to tionary Committees which are "3 -in-1 begin to see what the Cultural Revolution teams": cadres, Army, and revolutionary has meant for mathematics.z masses. Thus for instance Chow Pei-yuan How were the years of moratorium at Peking University, a senior mathemati­ spent? Professors and other intellectuals, cal physicist, though he performs adminis­ like Party functionaries, had been charged trative duties rather like a rector's, has with isolation from the masses; to rectify no such title. The power and responsibility this, they were all to work with the masses belong to the Revolutionary Committee, of for at least two years in physical labour. which he is simply a member, along with Whereas Party functionaries wentto "May representatives of the students and the 7 schools", often in the countryside, mathe­ Mao Tse-tung Propaganda Team. maticians went mostly to factories. Though How do the mathematicians spend they were there to shed intellectual snob­ their working day? Research in applied bism, they were not asked to shed their mathematics occupies a major part of intellects, but were encouraged to think everyone's time. Successful research ef­ mathematically about production problems forts in geometry (e.g., for design of gear­ in their factories. They are all back at grinding machine tools), fluid mechanics their institutions now, I was told, includ­ (e.g., intersecting jets of gas), and other ing junior teachers and assistants. Except fields were mentioned to me with pride. for their periods of labour, they have been It was emphasized that the object is al­ at their institutions the whole time, draw­ ways to solve the problem actually pre­ ing salary, devoting full time to political sented by the needs of production, not a discussions-which sometimes verged on modified problem more amenable to pre­ civil war. The students who went through conceived approaches. This sometimes this with them are now almost all gone, means returning to the factory which to factories and farms. Since late 1970, raised the question. It certainly means the new students have been arriving on that a mathematician trained in (say) to­ campus. pology or number theory will at present I was received, at both Peking and be doing most of his research in other Futan Universities, by committees repre­ domains. And it means that a mathemati­ senting all estates within the University. cian is supposed to be as satisfied with a Professors, yes -but the professors solution using no recondite reasoning at didn't do all the talking! I also heardfrom all as with an elegant theoretical one. I younger staff; from workers a~d soldiers wonder if they really go that far! of the Mao Tse-tung Propaganda Teams Secondly, the mathematicians teach. which entered the campuses in 1968-69 Teaching the new students is a new sort and are staying indefinitely to monitor of experience. They are not necessarily ideology; from the few Red Guards who are as mathematically apt, though quite likely staying around, apparently for like pur­ more enthusiastic, as those admitted under poses; and from some new students. Work­ a more conventional system. A graduate of ers without mathematical training feel lower middle school (roughly our 1Oth free to express opinions, and in particular grade, I gather) who has worked at least to criticize the former elitism of the three years in a factory or agricultural senior scientists; at the same time there commune may apply for university en­ is some deference by laymen to academics, trance. His application is judged by his and by junior to senior academics. fellow workers and the leader of his unit, This seems to represent the adminis­ primarily on criteria of "attitude"; though trative relations. Like other institutions the University subsequently must accept

2. A thorough study up to the present is Stephen Salaff' s biography of Hua Lo-keng, forthcoming in Isis. I have learned a great deal from conversations with Salaff both before and since my trip. For a reportage parallel to the present one, see Ethan Signer's "The biological sciences in China", Science for the People3 (1971), no. 4, or his joint article with Galston forthcoming in Science.

168 him, that does not seem to be a serious were smiles and vigorous nods around the hurdle, and no conventional academic table. criteria or tests were mentioned to me. Thus they are reading our papers. These students have trouble understanding Futan's math library, which I visited, is classroom explanations, and blame the well stocked and up-to-date, and it is difficulty on the teachers, not on them­ clearly well used. "We can learn from selves; their teachers thank them for you," I was told. "We can learn from you revealing the inadequacies of the explana­ too," I replied. "When will you publish tions. 3 The curriculum includes problems your results?" arising from current industrial practise, The international journals Acta Math­ which teacher and students are to attack ematica Sinica and Progress in Mathe­ jointly. (At least in principle it does; no matics are to resume publication in an case was reported to me of an undergrad­ unspecified future. Meanwhile, Chinese uate class solving a new practical prob­ mathematicians' applied results are com­ lem.) Organized expositions of standard municated but not published. Any other topics are also included, but they are theorems they may be finding are appar­ being completely re-thought to bring them ently not communicated at all, for the closer to practise. Epsilon and delta are proud presentation of possibly-useless banished from first year. theorems is one of their former bourgeois Beside university teaching, mathe­ habits which they most emphatically re­ maticians are engaged in efforts to bring nounce. Their new-style textbooks will go certain methods into extremely wide use through much testing and criticism before by non-specialists. It is intended, for publications. Even articles ~their new instance, that workers with secondary educational methods are still meager. education will be able routinely to use Especially frustrating to me is their linear programming or the critical-path reluctance to publish anything of their method in planning their own units' work. "summaries" until they are sure their For the present, although some re­ analyses are correct. Though these criti­ search institutes exist, there is nothing cisms have been going on all over China corresponding to our graduate education for over a year now, theyhavenotreached in mathematics. the stage of drawing conclusions or of Does all this mean that Chinese holding nation-wide conferences ("maybe mathematicians expect to do only narrow­ next year", guessed one professor). They ly practical work, leaving generalization haven't even reached the stage where one and theory to the capitalists and revision­ university's study group will issue theses ists? Not at all; they say, "Ourtheorywill for debate by the others. I ventured, and be better because it will be based on prac­ after reflection I repeat, my opinion that tis e." What are they doing to prepare the in such an elusive and wide field of mathe­ resurgence of theory? matical philosophy and policy as they are This is their third main activity: the covering, any quest for perfection is stul­ writing of "summaries" of mathematics. tifying. They should publish some of their Each field is being surveyed by special­ past polemics, even though outdated. They ists, in collaboration with non-specialists. should publish their draft "summaries", The aim is perspective on the whole ter­ even though tentative and imperfect, and rain, but also evaluation. "It is hard to benefit from others' reactions. anticipate what parts of mathematics will They will at any rate converse, as I be applicable," I was told, "but we are was most pleased to find, and I trust they certain that not all parts are equally use­ will also correspond. If the Cultural Rev­ ful." I ventured that a motto for this am­ olution in Chinese society is being con­ bitious project might be, "Be bold to solidated, the revolution in Chinese mathe­ analyse, be cautious to discard"; the in­ matics has hardly begun, and it will be terpreter must have rendered this into a fascinating and important to follow it. satisfactory Chinese proverb, for there

3. Peking Review, Sept. 3. 1971, pp. 17-18, gives one teacher's testimony.

169 LETTERS TO THE EDITOR

Editor, the c}/oticei) work out a scheme and actually try it. I imagine the best approach would be to I propose the following partial solu­ first get study funds from an interested tion to the problem of unemployment of foundation for a projected scheme, to young Ph.D. mathematicians. The sugges­ determine the right makeup for the group tion proceeds from two observations: (size, field of interest) and to scout actual 1. A young Ph.D., at a critical point properties, determining costs and condi­ in his career in terms of momentum for tions of lease or purchase and remodeling. research, would be willing to make sac­ The AMS Committee on Employment rifices for an opportunity to continue his and Educational Policy could perhaps help research in a congenial atmosphere. determine the amount of interest on the 2. Many young people these days are part of prospective participants. I am willing to live a communal life focused on sending the members copies of this letter mutual interests, certainly in lieu of wal­ for their consideration. lowing in unemployment or underemploy­ ment. George Pedrick Suggestion: "Mathematical research com­ California State College, munes" should be organizedtobringyoung Hayward research mathematicians together for a Editor, the c}/oticfi) year or so. This would require a suitable build­ In his absorbing piece on Chinese ing for living quarters and minimum in­ mathematicians and mathematics, Chand­ comes for the participants, not to mention ler Davis is "frustrated" because very access to a good library, some secretar­ little is being published. Specifically, he ial help, office space, seminar space and asks that they publish draft versions of the like. Many colleges hire part-time their "summaries, even though these are teachers and others would be willing to tentative and imperfect, in order that their split some positions, say three ways, to views may have the reaction of others. provide the minimum incomes (three to It is apparent that Professor Davis four thousand for teaching one course is entirely unfamiliar with the harsh ethos seems a fair guess). of the Communist credo, with the hostility The lease or purchase and remodel­ to the individual, and with the vicious sup­ ing of the building would require founda­ pression of personal liberty to think, be­ tion money. Foundation money would also lieve and speak, as these freedoms prevail be desirable for secretarial support and in countries of the running-dog imperial­ other such amenities, some of which the ists. host institution might provide. The thousand flowers no longer With housing provided and living bloom, and the hundred views no longer costs reasonably low (I conjecture at most contend. Only the dried Mao-flower re­ one thousand for the year per member) mains, pressed between the pages of the the living conditions should compare fa­ Little Red Book. vorably with those of graduate assistants. It might be thought by the naive that The host institution gains a signifi­ a commission of algebraic topologists cant injection of new young blood each would have the liberty of suggesting an year-a minor research institute attached outline appropriate for a course in this to the department. In fact, some of the subject. This, however, would be an ex­ part-time teaching might be generated by pression of the most infantile naivete, of regular faculty members voluntarily taking simplistic innocence of the actualities of reduced loads and salaries to work on re­ the Communist autarchy. To venture an search with the group. opinion on a subject unknown to the work­ Given the basic idea, interested peo­ ers, unfamiliar to the Army, and myste­ ple will soon see many possible variations rious to the revolutionary masses would in detail. I hope to see some institutions almost certainly convert the commission

170 of algebraic topologists to a commission there is a question as to whether persons of migrant farm workers. . who have not established themselves in It is that simple-and that unbellev- the first rank ought to vote-or to be ex­ able. pected to pay dues: Such an approach seems Alexander Doniphan Wallace doomed; in this connection let me recom­ University of Florida mend the remarks (really quite conserva­ tive) of Warren Weaver in the last AAAS Editor, the c!loriaiJ Bulletin to your attention. Another possi­ bility is to break the Society into two bodies; The letters of Anderson and Halmos but isn't the Society-Association split (not, in the FebruarycJiotice.Dbring out an issue to my knowledge, paralleled in other which must be faced, without delay, bythe fields) already enough of a weakening of Society. As Anderson puts it: "Is the So­ our forces? ciety an organization primarily for mathe­ I have too much respect for the moral matics or primarily for its members?". and intellectual capacity of my fellow I think that this phrasing is somewhat un­ mathematicians to believe that we can't sound; I prefer to think of the choice as in fact perform both roles within the same between mathematics in abstracto and the Society. I suggest that, in the interest of community of people who do mathematics. the survival of mathematics in our Society In any case, both writers optfortheform­ (does anyone who was at Las Vegas doubt er, and hold that therefore the Council and that the issue is that serious?) the Coun­ officers ought to be chosen purely or sole­ cil ought to be chosen to represent the ly according to their research accomplish­ members as persons who do mathematics, ments. and deal with their role and obligations in Surprisingly, our By-laws make no the real world. The mathematics which clear statement of purpose. The choice they do ought, quite independently, to be "for mathematics" is clearly implicit in judged by officers (e.g. editors) chosen Art. IV, Section 2: "The Council shall for and operating on the most solid scien­ formulate and administer the scientific tific standards. Then Halmos' objections policies of the Society ... "; but the other to (still hypothetical:) counter bias in favor outlook is reflected in Section 8 of the of blacks, women, and other groups fully same Article: "The Council shall also stand as regards the scientific officers; have power to speak in the name of the they need not stand with regard to the Society with respect to matters affecting Council. Likewise it is to be hoped that mathematics and mathematicians such as new procedures will obviate the admittedly proposed ... legislation; conditions of em- futile and non-representative resolutions ployment ... ; regulations, policies or acts procedure (Anderson's second issue); his­ of governmental agencies or instrumental­ torically this procedure arises from the ities; and other items which tend to affect firm determination of many in the Society the dignity and effective position of mathe­ to deal with the real world (again as dis­ matics" (All emphasis supplied. I suggest tinguished from the Real Line) before that that, for example, recently proposed reso­ world "deals with" the Society. lutions on the Southeast Asian war fall within this rubric.) W. M. Lambert, Jr. Having in mind that in the real world (as distinct from the Real Line:) a failure EDITORIAL COMMENT: Section III of the to directly act is per se a form of action Certificate of Incorporation of the Society I fear that the recent history of the Society begins with the following statement: (and of its effects on the profession) clear­ "III. The particular business and ly shows that a Council elected purely on objects of the Society are the furtherance the basis of research accomplishments of the interests of mathematical scholar­ cannot be expected to deal adequately with ship and research." the broader duties assigned to it. The remainder of Section III concerns It may be that the situation ought to election of members and empowers the be resolved by limiting the Society to a corporation to make rules to carry on its strictly "scientific" role; in this case business.

171 MEMORANDA TO MEMBERS

COMBINED MEMBERSHIP LIST, 1972-1973 CHANGE OF ADDRESS OR POSITION In the past, it has been customary for a return postcard to be sent to each mem­ her of the Society with a request for changes in the listing in CML or a confirmation of the current listing. Since we must reduce the cost of the CML operation, this procedure is being discontinued. If there have been any changes in address or position, and the Society has not been notified, members are requested to fill in the form below andre­ turn it to the American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02904, no later than May 15, 1972.

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NEWS ITEMS AND ANNOUNCEMENTS

LEHIGH UNIVERSITY Professor and director of the Center for the Application of Mathematics. The pro­ aca­ Beginning with the 1972-1973 gram leading to these degrees will be will offer demic year, Lehigh University interdepartmental and will stress the ap­ and doctor of master of science (M.S.) plication of mathematics to the particular in applied philosophy (Ph.D.) degrees areas in the physical, social, biological, leading to the mathematics. Programs or engineering sciences; the program is degrees in applied mathe­ M.S. and Ph.D. designed for candidates who have basic be administered by the Com­ matics will training, either at B.S. or M.S. levels, in Mathematics, headed by mittee on Applied mathematics or in other fields. Ronald s. Rivlin, Centennial University

172 SPECIAL MEETINGS INFORMATION CENTER

The purpose of this center is to maintain a file on prospective symposia, colloquia, institutes, seminars, special years, meetings of other associations, and to notify the organizers if conflicts in subject matter, dates or geographical area become apparent. A first announcement will be published in the cJ{oticei) if it contains a call for papers, place, date, and subject, where applicable; a second announcement must contain reasonably complete details of the meeting in order for it to be published. 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.

Third Saturday of each month P. J. Hilton, G. Papy, P. Ven Praag INFORMAL SEMINAR IN FUNCTIONAL (partial list) ANALYSIS Information: Centre beige de Pedagogie Kent State University, Kent, Ohio de la Mathematique, 224 avenue Program: Two one-hour lectures with Albert, B-1180 Brussels, Belgium breaks for discussion. Aimed at pro­ viding functional analysts from the May 8-12, 1972 Ohio-Pennsylvania-Michigan region CONFERENCE ON COMMUTATIVE with a place to congregate and dis­ ALGEBRA cuss their work. University of Kansas, Lawrence, Kansas Information: C. Cleaver, J. Diestel, or Program: Invited speakers, sessions R. Lohman, Department of Mathemat­ for twenty-minute contributed papers ics, Kent State University, Kent, Ohio Speakers: S. Abhyankar, D. Buchsbaum, 44242 R. Gilmer, I. Kaplansky, P. Samuel Support: NSF (pending) May 1-3, 1972 Proceedings: To be published ACM SYMPOSIUM ON THEORY OF Information: Commutative Algebra Con- COMPUTING ference, Department of Mathematics, Heart 0' Denver Motel, ll50 East Colfax University of Kansas, Lawrence, Kan­ Avenue, Denver, Colorado sas 66044 Program: Twenty-nine talks in the gen­ eral areas of program schemata, May 10, 1972 computational complexity, theory of S YMP.OSH:fM DN 'FRAF F IC algorithms, automata, formal lan­ ENGINEERING AND PLANNING guages, and theory of compiling. The Polytechnic, Queensgate, Sponsor: SIGACT Huddersfield, England proceedings: To be published and dis­ Program: Lectures: W. D. Ashton, Uni­ tributed to registrants at the confer­ versity of Surrey, "Gap-acceptance ence. After the conference, proceed­ problems at a traffic intersection"; ings will be available from the Asso­ C. B. Winsten, University of Essex, ciation of Computing Machinery. "A geographical incidence of traffic Information: Professor Jeffrey D. Ull­ accidents"; D. A. Quarmby, London man, Department of Electrical Engi­ Transport, "The use of mathematical neering, Princeton University, models and operational research in Princeton, New Jersey 08540 the planning and operation of urban bus services." Discussion periods are May 8-ll, 1972 planned. FIRST INTERNATIONAL CONGRESS Information and application forms: Sec­ OF ZWIN retary and Registrar, The Institute of Knokke, Belgium Mathematics and Its Applications, Speakers: Frederique, R. Diesbourg, Maitland House, Warrior Square,

173 Southend-on-Sea, Essex SS1 2JY, Information: Professor H. Renggli, De­ England partment of Mathematics, Kent State University, Kent, Ohio 44242 May 12-15, 1972 CONFERENCE ON ORDERS AND May 25 - August 13, 1972 GROUP RINGS SEMINAR IN PROBABILITY AND The Ohio State University, Columbus, MATHEMATICAL STATISTICS (in con­ Ohio junction with the Carleton branch of the Program: Half of each day will be de­ Summer Research Institute of the Canadian voted to formal talks, and half to in­ Mathematical Congress) formal discussions. Carleton University, Ottawa, Ontario, Speakers: Twe1 ve invited speakers in­ Canada cluding Maurice Auslander and Irving Subjects: Stochastic processes; weak Reiner. convergence and applications to non­ Information: Professor Thomas Ralley, parametric statistics Department of Mathematics, The Ohio Application deadline: April 15, 1972 State University, Columbus, Ohio Information: Professor D. Dawson, De­ 43210 partment of Mathematics, Carleton University, Ottawa, Ontario, Canada May 22-26, 1972 K1S 5B6 REGIONAL CONFERENCE ON PARTIAL DIFFERENTIAL EQUATIONS May 31 -June 2,1972 Texas Tech University, Lubbock, Texas FOURTEENTH MEETING OF Program: Principal speaker, Louis THEORETICAL PHYSICISTS AND Nirenberg. Sessions for contributed MATHEMATICIANS papers. Institute of Advanced Research Mathe­ Support: NSF (pending) matics, Strasbourg, France Participants: Travel and subsistence Information: Secretariat de la R.C.P. allowances for twenty-five partici­ n° 25, I.R.M.A. 7, rue Rene Descartes, pants. 67 Strasbourg, France Information: Professor Dalton Tar­ water, Department. of Mathematics, June 4-12, 1972 Texas Tech University, Lubbock, FIRST CIME SESSION 1972 Texas 79409 Bressanone, Italy Speakers: L. Cesari, Ju. A.M. Mitropol­ May 25-27, 1972 ski, T. Vogel (partial list) SYMPOSIUM ON THE SONIC BANG Information: Professor R. Conti, lstituto Aachen, Federal Republic of Germany Matematico "U. Dini," 67/A viale Information: Professor K. Oswatitsch, Morgagni, 1-50134, Firenze, Italy DVLR, Theatercasse 13, 51 Aachen, Federal Republic of Germany June 5-9, 1972 CONFERENCE IN COMBINATORICS May 25-27, 1972 The George Washington University, Wash­ REGIONAL CONFERENCE ON ington, D. C. CONFORMAL AND QUASICONFORMAL Program: Ten lectures by Gian-Carlo MAPPINGS Rota entitled "The combinatorics of Kent State University, Kent, Ohio the symmetric group." The lectures Program: Professor 0. Lehto, Case will range from the combinatorial as­ Western Reserve University, will de­ pect of the theory of representations liver three main lectures. Other lec­ of the symmetric group and its appli­ tures, as well as shorter reports, will cations to a variety of subjects in­ be given by selected speakers from cluding combinatorics, geometry, and Ohio and neighboring states. various aspects of physics and chem­ Participation: Open. Limited funds for istry. The central part of the lectures support are available for mathema­ will consist of the theory of Young ticians in nearby states. tableaux. In addition, seminars will

174 be organized on the current research fields of applied mathematics over of the participants. the past twenty years, and current Support: NSF and CBMS (pending) problems which appear to have sig­ Participants: Applicants for support nificance in future years. Emphasis should include a brief vita and indica­ will be given to the encouragement of tions of recent research interests. more mathematical effort in the so­ Advanced graduate students should cial, economic, and biological sci­ have major professor send a letter ences, and in other vital areas in of recommendation. which mathematics can make a con­ Information and applications: Confer­ tribution, Sessions for contributed ence Directors, Ruth A. Sari and papers. Irving J. Katz, Department of Mathe­ S-peakers: D. F. L. Bauer, Technisch matics, The George Washington Uni­ Hochschule, Munich; G. E. Forsythe, versity, Washington, D. C. 20006 Stanford University; J. K. Hale, Brown University; P. Henrici, Eidgen June 5-10, 1972 Technisch Hochschule, Ziirich; M. CONGRESS FOR THEORETICAL AND Kac, Rockefeller University; C. C. APPLIED MECHANICS Lin, Massachusetts Institute of Tech­ Ba¥ko Polje (between Split and Maka-rska), nology; L. W. Neustadt, Brown Uni­ Yugoslavia versity; E. J. Polak, Buchnell Uni­ Application deadline: Was March 1, but versity; A. Rapoport, University of those interested should write to Yugo­ Toronto; S. M. Ulam, University of slavian Society for Mechanics, Kneza Colorado Milosa 9, P.O. Box 218, 11000 Bel­ Information: Society for Industrial and grad, Yugoslavia Applied Mathematics, 33 South 17th Street, Philadelphia, Pennsylvania June 10-22, 1972 19103 ADVANCED STUDY INSTITUTE ON DISCRIMINANT ANALYSIS AND June 12-16, 1972 APPLICATIONS GORDON RESEARCH CONFERENCE ON Athens, Greece THEORETICAL BIOLOGY AND Program: Advanced instruction in an BIOMATHEMATICS atmosphere which will promote inter­ Proctor Academy, Andover, national scientific fellowship and co­ New Hampshire operation. Program: Invited speakers and discus­ Support: NATO (operating expenses); sion periods. NSF (travel) Participation: Applications must be sub­ Participant support: Advanced graduate mitted in duplicate on standard ap­ and postdoctoral students and junior plication form obtainable from the faculty who are U.S. citizens are eli­ office of the Director (address below). gible for round-trip air travel support. Accommodations: Participants expected All awards made upon the specific to live at conference location. nomination of the Institute Director; Information and application forms: Dr. final commitment by NSF. Alexander M. Cruickshank, Director, Information and application: Professor Gordon Research Conferences, Pas­ T. Cacoullos, Statistical Unit, Univer­ tore Chemical Laboratory, University sity of Athens, Panepistimiopolis of Rhode Island, Kingston, Rhode Couponia, Athens (621), Greece Island 02 881. After June 12, Colby Junior College, New London, New June 12-14, 1972 Hampshire b3257. SIAM 1972 TWENTIETH ANNIVERSARY MEETING June 12-16, 1972 Benjamin Franklin Hotel, Philadelphia, SYMPOSIUM ON GENERALIZED Pennsylvania RECURSION THEORY Program: Symposia which will survey University of Oslo, Blindern, Oslo, Norway important developments in various Program: Invited lectures on axiomatic,

175 recursion theory, hierarchy theory, sity of Minnesota; Marcos Moshinsky, theory of inductive definitions, higher Universidad Nacional Autonoma de order functionals, and constructibility Mexico; Eugene P. Wigner, Princeton theory. University Speakers: P. Aczel, j. Barwise, j. E. Information: Society for Industrial and Fenstad, R. Gandy, T. Grilliot, P. Applied Mathematics, 33 South 17th Hinman, Y. Moschovakis, W. Richter, Street, Philadelphia, Pennsylvania G. Sachs, S. Simpson, F. Ville 19103 Participation: Open Information: Professor j. E. Fenstad, june 18-27, 1972 Department of Mathematics, Univer­ ADVANCED STUDY INSTITUTE ON sity of Oslo, Blindern, Oslo 3, Norway FINITE GEOMETRIC STRUCTURES Bressanone, Italy june 14-25, 1972 Program: Advanced instruction in an ADVANCED STUDY INSTITUTE ON atmosphere which will promote inter­ MATHEMATICAL PROGRAMMING IN national scientific fellowship and co­ THEORY AND PRACTICE operation. Coimbra, Portugal Support: NATO (operating expenses); Program: Advanced instruction in an NSF (travel) atmosphere which will promote inter­ Participant support: Advanced graduate national scientific fellowship and co­ and postdoctoral students and junior operation. faculty who are U.S. citizens are eli­ Support: NATO (operating expenses); gible for round-trip air travel support. NSF (travel) All awards made upon the specific Participant support: Advanced graduate nomination of the Institute Director; and postdoctoral students and junior final commitment by NSF. faculty who are U.S. citizens are eli­ Information and application: Professor gible for round -trip air travel support. E. Bompiani, Director, C.I.M.E., In­ All awards made upon the specific stituto Matematico, Universita di nomination of the Institute Director; Roma, 00185 Rome, Italy final commitment by NSF. Information and application: Professor june 19-23, 1972 A. dos Santos Goncalves, Faculty of CONFERENCE IN HARMONIC ANALYSIS Science, University of Coimbra, University of Montana, Missoula, Montana Coimbra, Portugal Progr_~Jil_: Two lectures each day by joseph L. Taylor, University of Utah, june 15-16, 1972 on aspects of his work on measure DREXEL UNIVERSITY-SIAM JOINT algebra;. Two hours each day for MEETING thirty-minute talks by conference Drexel University, Philadelphia, participants. Pennsylvania Support: NSF, CBMS, Rocky Mountain Program: Symposia on practical com­ Mathematics Consortium putational methods and various ap­ Proceedings: To be published by the plications of Lie algebras. Sessions AMS for CBMS for contributed papers. Meeting com­ Participants: Travel grants and a modest memorates founding of SIAM. subsistence allowance will be paid by Speakers: A. 0. Barut, University of the NSF. Preference will be given to Colorado; Robert E. Beck, Villanova prospective participants from the University; johan G. F. Belinfante, Rocky Mountain and Pacific North­ Carnegie-Mellon University; Bruno west regions. It is hoped that thesis Gruber, Southern Illinois University students, junior mathematicians, and at Carbondale; Brian R. Judd, Johns senior mathematicians will partici­ Hopkins University; Bernard Kolman, pate. Drexel University; james R. McCon­ Abstracts: Abstracts of proposed thirty­ nell, Dublin Institute for Advanced minute talks are invited. Studies; Willard Miller, Jr., Univer- Information: Professor Michael j.

176 Fisher, Conference Coordinator, june 26-30, 1972 Rocky Mountain Mathematics Consor­ SYMPOSIUM ON THE MATHEMATICAL tium, Room 203, Mathematics Build­ FOUNDATIONS OF THE FINITE ELE­ ing, University of Montana, Missoula, MENT METHOD WITH APPLICATIONS Montana 59801 TO PARTIAL DIFFERENTIAL EQUA­ TIONS june 19-23, 1972 University of Maryland, Baltimore County CONFERENCE ON TOPOLOGICAL Campus, Ba:ltimore, Maryland DYNAMICS Program: A comprehensive review of Yale University, New Haven, Connecticut the mathematical foundations of the Program: Approximately ten one-hour finite element method, designed to ~tures and a limited number of acquaint workers in the area of nu­ contributed papers. merical analysis, and in particular Speakers: R. L. Adler, joseph Ausland­ the Navy mathematicians, with recent --;;-:r~- ·.L.ouis Auslander, Anotole Beck, advances in the field of the finite Harry Furstenberg, W. H. Gottschalk, element method. The symposium is L. W. Green, J. C. Oxtoby, and W. A. planned to be international in scope, Veech bringing together researchers from Support: NSF (pending) various parts of the world. Guest Pirtlcipation: All interested mathema­ speakers, small number of contrib­ ticians are invited to attend. uted papers, and discussions. Information: Conference on Topological Sponsors: Division of Mathematics of Dynamics, Department of Mathemat­ the University of Maryland and the ics, Yale University, New Haven, Office of Naval Research Connecticut 06520 Proceedings: To be published Information: Professor A. K. Aziz, June 19-28, 1972 Division of Mathematics, University INTERNATIONAL SYMPOSIUM ON of Maryland, Baltimore County Cam­ PARTIAL DIFFERENTIAL EQUATIONS pus, Baltimore, Maryland 21228 AND GEOMETRY OF NORMED SPACES The Hebrew University, jerusalem, Israel july 2-15, 1972 Speakers: E. Alfsen, E. Asplund, H. ADVANCED STUDY INSTITUTE ON Brezis, F. Browder, D. Dacunha­ FACIAL STRUCTURE OF COMPACT Castelle, L. Ehrenpreis, R. C. james, CONVEX SETS AND APPLICATIONS W. B. Johnson, V. Klee, J. L. Lions, Swansea, England A. Martineau, S. Mizohata, L. Nachbin, Program: Advanced instruction in an L. Nirenberg, R. Phelps, H. Rosen­ atmosphere which will promote inter­ thal, H. Schaefer, J. Serrin, and national scientific fellowship and co­ F. Tr-eves operation. Sponsors: International Mathematical Support: NATO (operating expenses); Union, Israel Academy of Sciences NSF (travel) and Humanities, and several scientific Participant support: Advanced graduate institutions in Israel. and postdoctoral students and junior Accommodations: Limited number of faculty who are U.S. citizens are eli­ hotel rooms have been reserved as gible for round-trip air travel support. well as space in the university resi­ All awards made upon the specific dence halls. nomination of the Institute Director; Information: Mrs. R. Dicker, Authority final commitment by NSF. for Research and Development, The Information and application: Dr. A. J. Hebrew University (Mathematical Ellis, Department of Pure Mathemat­ Symposium), jerusalem, Israel ics, University College of Swansea,

177 Singleton Park, Swansea SA2 8P P, July 9-29, 1972 Wales, United Kingdom SYMPOSIUM ON ANALYSIS Federal University of Pernambuco, Recife, July 2-15, 1972 PE., Brazil ADVANCED STUDY INSTITUTE ON Program: Invited speakers; sessions PADE APPROXIMANTS AND THEIR for contributed papers APPLICATIONS Speakers: F. Treves and additional Kent, England speakers to be announced at a later Program: Advanced instruction in an date atmosphere which will promote inter­ Organizing Committee: J. Barros-N eto, national scientific fellowship and co­ Fernando Cardoso, Ruy L. Gomes, operation, Chaim S. Honig, Leopolda Nachbin, Support: NATO (operating expenses); Roberto Ramalho NSF (travel) Information: Professor Roberto Ramal­ Participant support: Advanced graduate ho, Instituto de Matematica, Univer­ and postdoctoral students and junior sidade Federal de Pernambuco, Caixa faculty who are U.S. citizens are eli­ Postal 2672, Recife, PE., Brazil gible for round -trip air travel support. All awards made upon the specific July 10-12, 1972 nomination of the Institute Director; INTERNATIONAL COLLOQUIA ON final commitment by NSF. THEORY OF PERMUTATIONS AND Information and application: Professor PERMUTATIONS AND SOCIAL SCIENCES J. S. R. Chisholm, Cornwallis Build­ Rene Descartes University, Paris, France ing, University of Kent, Canterbury, Program: Papers on finite permutations Kent, England under the "combinatory" aspect; in­ finite permutation groups; permuta­ July 3-7, 1972 tions in the fine arts (architecture, COLLOQUIUM ON INFORMATION poetry, music, ornaments); psychol­ TECHNIQUE ogy (the perception of permutations); Rocquencourt, France sociology (permutations and elemen­ Information: !RIA, Domaine de Voluceau, tary kinship relations structures); 78, Rocquencourt, France history of the theory ofpermutations. Proceedings: To be published July 3-22, 1972 Abstracts: In English, French, German, SUMMER SCHOOL ON CALCULUS OF or Russian; thirty lines maximum. PROBABILITY Information: Secretariat des Colloques Foyer des Planchettes, Saint Flour "Permutations"; U.E.R. de Mathema­ (Cantal), France tiques, Logique Formelle etlnforma­ Subject: Ergodic processes and trans­ tique, 48, rue Saint-Jacques, 75 Paris formations 5e, France Speakers: S. Dubuc, University of Mon­ treal; G. Kallianpur, University of July 10-14, 1972 Minnesota; M. Keane, University of REGIONAL CONFERENCE ON THE Rennes NUMERICAL SOLUTION OF NONLINEAR Application deadline: April 15, 1972 ALGEBRAIC SYSTEMS WITH APPLICA­ Information: P. L. Hennequin, Bilite TIONS TO PROBLEMS IN PHYSICS, Postale n° 45, 63 Aubiere, France ENGINEERING, AND ECONOMICS University of Pittsburgh, Pittsburgh, July 3-22, 1972 Pennsylvania SUMMER SCHOOL ON NUMERICAL Program: Werner Rheinboldt, Univer­ ANALYSIS sity of Maryland, will give ten lec­ Breau-sans-Nappe (near Rambouillet), tures. Talks by other participants France will be solicited. Speakers: W. Eckhaus, C. Franc;ois, Support: NSF and CBMS (pending) P. Van Dyke Participation: Open, with travel and Information: M. Steinberg, 17 avenue de subsistence allowances for a limited la Liberation, 92, Clamart, France number of participants.

178 Application deadline: May 12, 1972 August 21-27, 1972 Information and applications: Professor SYMPOSIUM ON MATHEMATICAL C. A. Hall, Conference Director, De­ FOUNDATIONS OF COMPUTER SCIENCE partment of Mathematics, University Jablonna (near Warsaw), Poland of Pittsburgh, Pittsburgh, Pennsyl­ Program: Scope will be limited to prob­ vania 15213 lems that have been developed in Poland, i.e. semantics of program­ August 7-18, 1972 ming languages and foundations of ADVANCED STUDY INSTITUTE ON computing systems. In particular the MATHEMATICAL PHYSICS: GROUP following topics will be dealt with: THEORY IN NONLINEAR PROBLEMS proving properties of programs, pro­ Istanbul, Turkey gram transformations, primitives of Program: Advanced instruction in an programming languages, and comput­ atmosphere which will promote inter­ ing systems. The program will include national scientific fellowship and co­ addresses, contributed papers, and operation. discussion sessions. Support: NATO (operating expenses); Sponsors: Computation Center of the NSF (travel) Polish Academy of Sciences and the Participant support: Advanced graduate Institute of Computing Machines of and postdoctoral students and junior the Warsaw University. faculty who are U.S. citizens are eli­ Participation: Open. Assistance with gible for round-trip air travel support. visas and accommodations will be All awards made upon the specific given by Organizing Committee. nomination of the Institute Directors; Organizing Committee: A. Blikle, P. final commitment by NSF. Dembinski, A. Mazurkiewicz, E. Information and application: Professor OrYowska, Z. Pawlak, H. Rasiowa, A. 0. Barut, Department of Physics A. Salwicki, S. Turski, A. Walig6rski, and Astrophysics, University of Colo­ J. Winkowski rado, Boulder, Colorado 80302 Information: Dr. Andrzej Blikle, SUM­ MER SCHOOL, Computation Center, August 20 - September 10, 1972 Polish Academy ofSciences, Warsaw, CONFERENCE ON APPROXIMATION PKIN, P .0. Box 22, Poland THEORY August 24 - September 2, 19 72 A. Mickiewicz University, Poznan, Poland THIRD CIME SESSION 1972 Program: Forty-five minute lectures Varenne, Italy and twenty minute communications Theme: Geometric theory of measure Sponsors: Mathematical Institute of the and minimal surfaces Polish Academy of Sciences and In­ Speakers: W. K. Allard, F. J. Almgren, stitute of Mathematics of the A. Jr., E. Bombieri, E. Giusti, M. Mickiewicz University Miranda Information: Professor Dr. Julian Information: Professor R. Conti, Istituto Musielak, Institute of Mathematics, Matematico "U. Dini," 67/A viale A. Mickiewicz University, Poznan, Morgagni, I-50134, Firenze, Italy Matejki 48/49, Poland August 28 - September 2, 1972 August 21-25, 1972 THIRD CZECHOSLOVAKIAN SIXTH INTERNATIONAL CONFERENCE CONFERENCE ON DIFFERENTIAL ON OPERATIONS RESEARCH EQUATIONS AND THEIR APPLICATIONS Dublin, Republic of Ireland Brno, Czechoslovakia Information: A.F .C.E. T. (Section R.O.), Information: Equadiff III, Purkyne Uni­ Immeuble Centre Dauphine, Place versity, Department of Mathematics, Marechal de Latre de Tassigny, 75, Janackovo nam. 2a, Brno, Czecho­ Paris 16, France slovakia

179 August 28- September 8,1972 case) centered on page, name of author CONFERENCE ON ALGEBRAIC and his city of residence in lower case K-THEORY on separate lines. Battelle Seattle Research Center, Seattle, Inform!J:.!!_c>_g: Professor J. Gani, Depart­ Washington ment of Probability and Statistics, Program: Focus on computations of The University, Sheffield S3 7RH, topological obstruction groups and England, or Dr. Vincze (address related algebraic problems; construc­ above). tion of higher algebraic K-functors; and connections of K2 (and Kn) with number theory. September 5-7, 1972 Support: NSF CONFERENCE ON RECENT Participants: Partial support for a lim­ MATHEMATICAL DEVELOPMENTS IN ited number of participants, including CONTROL some Europeans. Descriptions of any University of Bath, England papers applicant wishes to present ?rogram: Main topics will be algebraic should be included with application. systems theory, stability of nonlinear Proceedings: To be published systems, optimal control, filtering Information and applications: Professor theory, control of systems governed Hyman Bass, Department of Mathe­ by partial differential equations. Ses­ matics, Columbia University, New sions for contributed papers. York, New York 10027 Inf~E_l!l~ti~n and applicati~!l__ forms: Sec­ retary and Registrar, The Institute of August 31 - September 5, 1972 Mathematics and Its Applications, EUROPEAN MEETING OF Maitland House, Warrior Square, STATISTICIANS Southend-on-Sea, Essex, SS1 2JY, Budapest, Hungary England Sponsors: The Hungarian Academy of Sciences, the International Associa­ tion for Statistics in the Physical September 6-15, 1972 Sciences Section of the International THIRD CONGRESS OF BULGARIAN Statistical Institute, European Region MATHEMATICIANS of the Institute of Mathematical Sta­ International House of Scientists, Druzba tistics (near Varna), Bulgaria .~I"ganizer: Bolyai Janos Mathematical Program: One-hour and half-hour in­ Society vited papers, as well as scientific Hour speakers: 0. Bunke, T. Cacoullos, communications in the following ar­ K. Dietz, J. Hajek, J.F .C. Kingman, eas: mathematical structures (foun­ A. N. Kolmogorov, D. V. Lindley, dations, logic, topology, algebra, A. N. Shiryayev, K. Urbanik, I. Vincze structures of analysis, structures of Half-hour speakers: W. Buhler, D. R. geometry); mathematical foundations Cox, M. Girault, M. Iosifescu, D. G. of computer science and cybernetics Kendall, J. Neveu, F. Papangelou, (automata theory, control theory, soft­ F. Spitzer ware, numerical methods); mathemat­ Participation: Deadline for receipt of ical modelling (probability and sta­ registration form is April 10. Those tistics, operational research, mathe­ wishing to try to register now should matical models in mechanics, physics, write to Dr. I. Vincze, Bolyai Janos economics, biology and medicine, so­ Mathematical Society, Budapest, V. cial sciences, linguistics). Symposia Szabadsag ter 1 7, Hungary. and panel discussions. Contributed papers: Deadline is April 30, Participation: Send note of intention to 1972. Send to Dr. Vincze at the above attend to Bulgarian National Commit­ address a summary not longer than tee for Mathematics, P .0. Box 373, one or two pages: typed, symbols and Sofia, Bulgaria. Include family name, formulas inscribed in black ink or first name, institution, address, type, numbered pages, title (in upper country.

180 Academic year 1972-1973 (beginning Papers: Copy of each accepted paper, September 10, 1972) typed on special forms, will be due Mittag-Leffler Institute, Djursholm, at the aboye address by August 14. Sweden Information: Professor James H. Pugs­ Program: Emphasis will be on proba­ ley, Electrical Engineering Depart­ bility, in particular with regard to ment, University of Maryland, College applications in analysis. Park, Maryland 20742. Participation: Open, including postgrad­ uate students Information: Professor Lennart Carle­ November 8-10, 1972 son, lnstitut Mittag-Leffler, Aura­ JOINT MEETING OF OPERATIONS vagen 17, S-182 62 Djursholm, Sweden RESEARCH SOCIETY OF AMERICA, October 4-6, 1972 THE INSTITUTE OF MANAGEMENT SYMPOSIUM ON CONTROL AND FEED­ SCIENCES, SYSTEMS ENGINEERING BACK MECHANISMS IN FLOW NOISE GROUP OF THE AMERICAN INSTITUTE Gottingen, Federal Republic of Germany OF INDUSTRIAL ENGINEERS Information: Professor E. A. Muller, Chalfonte-Haddon Hall Hotel, Atlantic MPI Stromungsforschung, Bbttinger­ City, New Jersey strasse 6/7, 34 Gottingen, Federal Program: Sessions entitled Business Republic of Germany and Industry, Government, Method­ ology, Operating Control Systems, October 25-27, 1972 and Special Sessions SYMPOSIUM ON SWITCHING AND Abstracts: Abstracts for contributed AUTOMATA THEORY papers received by April 15 will be University of Maryland, College Park, considered for the subject area ses­ Maryland sions (unless the author indicates Abstracts: Deadline May 17, 1972. Areas otherwise). Those received by May 15 include switching theory, automata will be considered for a "Work in theory, and the theoretical aspects of Progress" session (Special Session). computers, computation, and pro­ Abstracts received after May 15 will gramming. Abstracts must provide be declined. Send abstracts, which sufficient detail to allow the commit­ must be typewritten, double spaced, tee to apply uniform criteria for ac­ and a single paragraph, to James C. ceptance, and should include appro­ Grace, Manager, Operations Re­ priate references and comparisons search, Room 505A East Building, with extant work. Include brief inter­ Bethlehem Steel Corporation, Bethle­ pretation of major results and ex­ hem, Pennsylvania 18016. planation of their significance. A total of five to eight typewritten pages is suggested, six copies, no word limit. ERRATA: SIAM has requested that their Send to Professor Richard M. Karp, 1972 fall meeting, an announcement of Department of Computer Science, 573 which appeared in the FebruarycNo!keU, Evans Hall, University of California, be indicated as the SIAM-SIGNUM 1972 Berkeley, California 94 720. FALL MEETING.

NEWS ITEMS AND ANNOUNCEMENTS

OPPORTUNITIES FOR preparation of proposals for research NAVAL RESEARCH support from ONR, and should prove useful in determining the kind of research pro­ The Office of Naval Research has jects that would be of interest to this agen­ recently published a booklet entitled "Op­ cy. Copies may be obtained from the Chief portunities for Naval Research" ( ONR -l 7). of Naval Research, Code 432, BOO North This booklet contains information on the Quincy, Arlington, Virginia 22217.

181 SUMMER GRADUATE COURSES

The following is a list of graduate courses being offered in the mathematical sciences during the summer of 1972. It is a continuation of the list given in the February issue of these c/lfolicei).

SAMFORD UNIVERSITY UNIVERSITY OF SOUTH CAROLINA Birmingham, Alabama 35209 Columbia, South Carolina 29208 Application deadline: June 5 Application deadline: Late Fee after May 1 - Information: W. D. Peeples, Jr., Head, De­ 1st session; June 1Oth - 2nd session. partment of Mathematics Information: W. H. Caldwell, Director, Grad­ uate Studies in Mathematics Tune 5 - Tuly 12 Math 513 Higher Geometry Tune 2 - Tuly 7 504 Number Systems July 13 - August 18 505 Foundations of Geometry Math 405 Special Functions 507 Advanced Calculus for Engineers 511 Theory of Probability 515 Statistical Methods HOFSTRA UNIVERSITY I 543 Algebraic Structures I Hempstead, New York 11550 551I Introduction to Analysis I Application deadline: June 9 700I Set Concept in Mathematics Information: Harold M. Hastings, Graduate Curriculum Coordinator, Department of Mathematics 74 7 Combinatorics Tune 14 - Tuly 19 753 Complex Analysis Math 298 Functional Analysis Tuly 13 - August 17 Seminar 508 Advanced Calculus for Engineers 512 Introduction to Mathematical Statistics I UNIVERSITY OF TOLEDO 516 Statistical Methods II Toledo, Ohio 43606 544 Linear Algebra Information: H. L. Bentley, Department of 552I Introduction to Analysis II Mathematics 572 Modern Geometry Tune 20 - Tuly 21 743 Abstract Algebra Math 701 Rings of Continuous Functions 742I Introduction to Modern Algebra 758 Distribution Theory

DREXEL UNIVERSITY UNIVERSITY OF WISCONSIN-MADISON Philadelphia, Pennsylvania 19104 Madison, Wisconsin 53706 Information: Dale W. Lick, Head, Department Application deadline: June 12* of Mathematics Information: Summer Sessions Office, 602 State Street, Tune 19 - September 1 Madison, Wisconsin 53706 Vector Analysis Tune 12 - August 5 Partial Differential Equations Math(725) Hilbert Space** Mathematical Ecology Math(845) Rings and Fields** Numerical Solutions of Differential Equations (521) Advanced Calculus Algebraic Theory of Automata (541)-(542) Modern Algebra Approximation Theory ( 551) Elementary Topology Integral Equations (567) Elementary Number Theory (623) Complex Analysis

* "Permit to Register" must be obtained earlier, preferably by May 1. **Strictly graduate courses.

182 NEW AMS PUBLICATIONS

TRANSLATIONS-SERIES II emergence of a new facet of biology: the theoretical biology of development. Simple Volume 99 mathematical ideas are very much involved in this aspect of biology, as they are in FIVE PAPERS ON LOGIC AND many other aspects of the life sciences. FOUNDATIONS 275 + iv pages; List Price $14.00; Member Volume 4 Price $10.50 SOME MATHEMATICAL QUESTIONS This volume of the American Mathe­ IN BIOLOGY. III matical Society Translations-Series 2 Edited by Jack D. Cowan contains the following papers: "A method 151 +vi pages; List Price $6.80; Member of presenting the theory of algorithms and Price $5.10 enumerable sets" by G. S. Ceitin; "A con­ structive calculus of statements with This volume contains lectures given strong negation" by N. N. Vorob'ev; "Cal­ at the Fifth Symposium on Some Mathe­ culi of constructive logic with subordinate matical Questions in Biology, held in variables" by A. V. ldel'son; "Concerning Chicago on December 27, 1970. The lec­ a generalized principle of constructive tures in this volume are concerned main­ selection" by G. S. Ceitin and I. D. Zaslav­ ly with the topic introduced in Volume 3 skii; and "Concerning the constructive of this series: the theoretical biology of interpretation of auxiliary formulas.!" by development. This series of volumes N. A. Sanin. serves to show how vigorous an inter­ action there has been between biology and LECTURES ON MATHEMATICS mathematics, and to indicate how this IN THE LIFE SCIENCES interaction will continue.

Volume 3 PROCEEDINGS OF THE SOME MATHEMATICAL QUESTIONS STEKLOV INSTITUTE BIOLOGY. II IN Number 113 (1970) Edited by Jack D. Cowan PROBLEMS IN THE CONSTRUCTIVE 121 +vii pages; List Price $6.00; Member TREND IN MATHEMATICS. V Price $4.50 Edited by V. P. Orevkov and N. A. San in This volume contains lectures given 287 + iv pages; List Price $22.1 0; Member at the Fourth Symposium on Some Mathe­ Price $16.58 matical Questions in Biology, held in Boston on December 27, 1969. The purpose This book is the fifth collection of of this series of symposia is to stimulate articles discussed in the seminar on con­ a closer relationship between mathemati­ structive mathematics in the Leningrad cians and biologists with some mathe­ branch of the Steklov Mathematics Institute matical background. The lectures in this of the Academy of Sciences of the USSR volume are concerned with some of the (which is at the same time the Research most interesting and basic problems of Seminar of the Mathematics -Mechanics contemporary biology: the molecular and Department of the University of Lenin­ cellular control of differentiation and grad). The collection consists of articles development in multicellular organisms. on the theory of algorithms and on con­ These lectures clearly demonstrate the structive mathematical logic and analysis.

183 MEMOIRS OF THE AMERICAN group, for an almost-periodic continuous MATHEMATICAL SOCETY positive-definite function provided the Bohr topology on the subgroup is that Number 116 induced by the Bohr topology of the whole group, for an arbitrary subgroup of any CONSTRUCTIVE MEASURE THEORY group with a closed locally compact abelian By Errett Bishop and Henry Cheng subgroup of finite index, and for positive­ 85 + v pages; List Price $1. 70; Member definite functions in the Fourier algebra. Price $1.28 It is also shown to be true for the center of a group with small invariant neighbor­ The authors hope that the present hoods, and evidence is collected to sup­ paper is close to being a definitive treat­ port a conjecture that the same is true ment of constructive measure theory. The for the center of an arbitrary locally com­ subject is approached by means of the pact group. Truth of this conjecture is Daniell integral, which permits a general equivalent to truth of the converse of an and flexible theory and constitutes a sig­ often used result of Douady. nificant advance over previous treatments. First a measure is constructed from an integral, than an integral is constructed TRANSACTIONS OF THE from a measure, and the two constructions MOSCOW MATHEMATICAL SOCIETY are compared. Convergence theorems, Volume 22 (1970) Fubini's theorem, and integrals in locally compact spaces are among the topics TRANSACTIONS OF THE MOSCOW treated. MATHEMATICAL SOCIETY 280 + iv pages; List Price $21.80; Member Number 117 Price $16.35 EXTENSIONS OF POSITIVE-DEFINITE This volume of the TRANSACTIONS FUNCTIONS OF THE MOSCOW MATHEMATICAL SO­ By John R. McMullen CIETY contains the following papers: "Factor representations of the antic om­ 71 + iv pages; List Price $1.90; Member mutation relations" by V. Ja. Golodec; Price $1.43 "Facial characterization of convex sets; In this volume conditions are sought extremal elements" by D. P. Mil 'man; under which a continuous positive-definite "On the formulation of problems in sta­ function on a closed subgroup is a restric­ tistical physics" by A. S. Svarc; "Cate­ tion of such a function defined on the whole gories with involution and correspondences group. This is found to be true for an in 'Y-categories" by M. S. Burgin; and arbitrary compact subgroup of a locally "Many-valued mappings and Borel sets.I" compact group, for an arbitrary open sub- by M. M. Goban.

NEWS ITEMS AND ANNOUNCEMENTS

MATHEMATICS IN ISRAEL AND ITALY ics in Italy" in ONR London Report R -41- 71 (15 October 1971). A limited number Articles on mathematics in Israel of copies are available. Copies of the and Italy by Professor Robert H. Owens former may be obtained by writing to of the Department of Applied Mathematics Miss Sandra D' Allesandro, American and Computer Science of the University Mathematical Society, P .0. Box 6248, of Virginia have been published by the Providence, Rhode Island 02904. Copies Office of Naval Research. "Mathematical of the latter may be obtained by writing Science at Selected Israeli Institutions" to Commanding Officer, Office of Naval appeared in ONR London Report R-31-71 Research, Branch Office, Box 39, FPO (20 August 1971), and "AppliedMathemat- New York 09510.

184 PERSONAL ITEMS

KENNETH F. ANDERSEN of Royal in a seminar on global analysis at the Uni­ Roads Military College, Victoria, British versity of Warwick, England, and a con­ Columbia, Canada, has been appointed to ference on differential geometry at Ober­ an assistant professorship at the Univer­ wolfach, Federal Republic of Germany. sity of Alberta, Edmonton, Alberta, Can­ He will also visit the University of Paris ada. and the University of Bonn and attend a GREGORY F. BACHE LIS of Kansas conference in Montreal. State University has been appointed to an PHILIP H. JOHNSON of Franklin, associate professorship at Wayne State New Hampshire, has been appointed a University. teacher at Lebanon High School, Lebanon, ADRIANO BARLOTTI of the Univer­ New Hampshire. sity of Perugia, Florence, Italy, is visit­ SEYMOUR KASS of the Illinois Insti­ ing Colorado State University for the tute of Technology has been appointed to spring semester 1972. an associate professorship and to the RONALD F. BARNES of Syracuse chairmanship of the Department of Mathe­ University has been appointed to an assis­ matics at Boston State College. tant professorship at SUNY, College at STANISLAS L. KLASA of Northern Brockport. Illinois University has been appointed to RAYMOND E. BEASLEY of Rose­ an associate professorship at McGill Uni­ Hulman Institute of Technology has been versity. appointed to the chairmanship of the De­ BERNARD H. KORTE of the For­ partment of Mathematics, Computer Sci­ schungsgruppe fiir empirische National­ ence and Statistics at Central State Uni­ okonomie has been appointed to a profes­ versity, Edmond, Oklahoma. sorship at the Universitat Bonn, Federal DENIS L. BLACKMORE of the Poly­ Republic of Germany. technic Institute of Brooklyn has been TAP ANI J. KUUSALO of the Univer­ appointed to an assistant professorship sity of Helsinki has been appointed to an at Newark College of Engineering. associate professorship at the University LOKENATH DEBNATH of East Car­ of Jyvaskyla, Finland. olina University has been appointed to a PAOLO LANZANO of the North visiting professorship at the Center of American Rockwell Corporation, Space Advanced Study in Applied Mathematics, Division, Downey, California, has been University of Calcutta, India. appointed head of the Mathematics Re­ HOWARD GARLAND of Cornell Uni­ search Center at the U.S. Naval Research versity has been appointed to a professor­ Laboratory, Washington, D.C. ship at Columbia University. BOO SANG LEE of the University of MICHAEL A. GAUGER of the Uni­ Kentucky has been appointed to an assis­ versity of Notre Dame has been appointed tant professorship at Trenton State Col­ to an assistant professorship at the Uni­ lege. versity of Massachusetts. ERNEST W. LEGGETT, Jr., of the JAMES L. GEMMA of the Battelle University of Illinois has been appointed Memorial Institute, Columbus Labora­ a programmer with the IRCC at Ohio tories, has been appointed to an assistant State University. professorship at Wichita State University. JAMES N. McNAMARA of the Uni­ ROE GOODMAN of Massachusetts versity of Rochester has been appointed to Institute of Technology has been appointed an assistant professorship at SUNY, Col­ to an associate professorship at Rutgers lege at Brockport. University. SANFORD S. MILLER of the Univer­ CHUAN C. HSIUNG is on academic sity of Kentucky has been appointed to an leave from Lehigh University for the assistant professorship at SUNY, College spring semester 1972. He will participate at Brockport.

185 DONALD R. MORRISON of Sandia To Professor. University of Massa­ Corporation has been appointed to a pro­ chusetts: HELEN F. CULLEN, SAMUELS. fessorship at the University of New Mexico. HOLLAND, Jr.; San Diego State College: ALEXANDER PEYERIMHOFF ofthe HUNG-TA HO; Ibaraki University, japan: Universitat Marburg has been appointed SIGERU OKAMOTO. to· a professorship at the Universitiit To Associate Professor. DePaul Uni­ Ulm, Federal Republic of Germany. versity: j. MARSHALL ASH, CONSTAN­ JOHN F. PRICE of the Universita di TINE E. GEORGAKIS, MICHAEL H. WICH­ Genova, Italy, has taken up a research MAN; Indiana State University: ANDRE fellowship at the Institute of Advanced deKOR VIN; University of Minnesota, Min­ Studies of the Australian National Univer­ neapolis: SIEGFRIED K. GROSSER; SUNY, sity, Canberra, Australia. College at Brockport: RICHARD T. MA­ BAYARD RANKIN of Case Western HONEY; Sir George Williams University, Reserve University has been appointed Montreal: KAILASH K. ANAND. mathematics editor with Encyclopaedia Britannica, Chicago, Illinois. To Assistant Professor. City College SEYMOUR SHERMAN on sabbatical (CUNY): DAVID K. HSIEH; University of leave from Indiana University will be Massachusetts: CHAN-NAN CHANG, RON­ visiting the Weizmann Institute of Science, ALD P. MORASH; Villanova University: Rehovot, Israel, until june 1972. ROBERT M. DeVOS, DON GOELMAN. MEIR SMORODINSKY of Warwick INSTRUCTORSHIPS University has been appointed a senior lecturer at Tel-Aviv University. Fayetteville State University: JER­ NIKOLA SVILOKOS of the Allied YIH LIN; Solano Community College: BEN Chemical Company has been appointed an HECHT. analyst with the New York Stock Exchange and Evening Math, Incorporated, New DEATHS York, New York. THOMAS R. TERRELL of Ohio State Professor Emeritus THOMAS L. University has been appointed to an assis­ DOWNS of Washington University died on tant professorship at the University of january 12, 1972, at the age of 64. He was Cincinnati. a member of the Society for 35 years. FRANK A. WATTENBERG of Har­ vard University has been appointed to an Professor Emeritus JOSEPH M. assistant professorship at the University FELD of Queens College (CUNY) died on of Massachusetts. August 4, 1971, at the age of 70. He was KJELL-OVE H. WIDMAN of Uppsala a member of the Society for 43 years. University has been appointed to a pro­ fessorship at Linkopings Hogskola, Lin­ Dr. STANLEY KATZ of the City koping, Sweden. College (CUNY) died on February 19, 1972, at the age of 51. He was a member of the PROMOTIONS Society for 24 years.

To Acting Head. Department of Math­ ematics and Statistics, University of Mas­ Professor GERALD R. MA.CLANE sachusetts: HASKELL COHEN. of Purdue University died on March 10, To Associate Head. Department of 1972, at the age of 52. He was a member Mathematics and Statistics, University of of the Society for 30 years. Massachusetts: LORRAINE D. LAVAL­ LEE. Dr. EDWARD RUSSELL STABLER To Distinguished Professor. Naval of Dover, New Hampshire, died on Sep­ Postgraduate School: FRANK D. FAULK­ tember 26, 1971, at the age of 65. He was NER. a member of the Society for 29 years.

186 ABSTRACTS PRESENTED TO THE SOCIETY

Preprints are available from the author in cases where the abstract number is starred.

The papers printed below were accepted by the American Mathematical Society for presentation by title. The ab­ stracts are grouped according to subjects chosen by the author from categories listed on the abstract form. The mis­ cellaneous group includes all abstracts for which the authors did not indicate a category. An individual may present only one abstract by title in any one issue of the cflotiai] 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 an issue.

Algebra & Theory of Numbers

72T-A74. JOEL A. WINTHROP, University of Missouri, Columbia, Missouri 65201. The homological characterization of modules.

A classifying scheme is introduced for .any category of modules whereby almost all well-known classes of modules are realized as r; -flat with respect to a sequence r; of bifunctors. In addition almost all well-known classes of extensions are realized as the !: -pure extensions with !: as above. Finally the notions of relative projectivity and injectivity are treated in a similar way, and the general notion of an "iterated relative homological algebra" is studied. (Received November 26, 1971.)

72T-A75. KIM Kl-HANG BUTLER, Pembroke State University, Pembroke, North Carolina 28372.

Enumeration of generalized inverses of Boolean relation matrices. Preliminary report.

Let ,Bn denote the set of all n X n matrices over the Boolean algebra of order 2. Let A E {3n and consider the matrix equation (*) A = AXA. We follow R. Plemmons ["Generalized inverses of Boolean relation matrices," SIAM J. Appl. Math. 20(1971), 426-433] in calling any solution to (*) in ,Bn a generalized inverse of

A. The common value of the row and column rank is called the rank of A in ,Bn. Incidentally, the row and column rank of A in ,Bn is not necessarily equal. Plemmons has shown in the above paper that the matrix A in f3n has a generalized inverse only if the row rank of A is equal to column rank of A. In this paper we count the number of generalized inverses of members of f3n which have rank r (r< 7). (Received December 6, 1971.)

*72T-A76. WALTER D. WALLIS, University of Newcastle, New South Wales 2308, Australia. Room

squares.

A room square of side n is a way of arranging the numbers 0, 1, .•. , n in a square array of side n such

that: (i) every cell of the array contains an unordered pair of numbers or is empty; (ii) every possible unordered

pair occurs precisely once; (iii) every number appears once per row and once per column. The square is called

skew if it further satisfies: (iv) the ith diagonal cell contains {o, i}; (v) if i cJ j then the cell in row i and

column j is empty or the cell in row j and column i is empty but not both. Theorem. If there is a room square

of side n (n cJ 1) then there is a room square of side 3n. Theorem. If there is a skew room square of side

n (ncJ 1) then there are room squares of sides 2n -1 and 2n + 1. (Received December 10, 1971.)

A-425 *72T-A 77. JOHN DA UNS, Tulane University, New Orleans, Louisiana 70118. Simple modules and centralizers.

A class of modules generalizing the simple ones is constructed. Submodule structure and centralizers of quotient modules are completely determined. The above class of modules is used to study the primitive ideal structure of the tensor products of algebras. (Received December 27, 1971.)

72T-A78. THOMAS L. JAGER, Indiana University Northwest, Gary, Indiana 46408. Steinberg characters and Gallagher characters.

Theorem. Let G be one of the following groups: GL(n,q), n ~ 1; Sp(2n,q), n ~ 2; SU(n,q2), n ~ 3; OE(2n,q), n ~ 4; 0(2n+1,q), n ~ 3 and 2 Aq. Then the Steinberg character of G is a Gallagher character except in the following cases: GL(2,q), q + 1 is a prime power; Sp(4,2); Sp(4,3); SU(3,2\ SU(4,22). (Received

November 5, 1971.)

72T-A79. GEOFFREY T. JONES, Lorna Linda University, Riverside, California 92505.

Pseudocomplemented semilattices. Preliminary report.

The following results have been obtained for the category of pseudocomplemented semilattices (PCSL).

(1) The injectives are the complete Boolean algebras. (2) Let A be a finite PCSL, P be the set of meet irreducible elements of A - (1} , and M the set of maximal elements of P. Then A is projective if and only if the following hold: (a) If Q r;; P, then fi(Q) = 0 iff for each m E M, there is a q E Q such that m ~ q.

(b) If Q r;; P, p E P and 0 < fi(Q) ;§ p, then q ;§ p for some q E Q. (c) There exists an m E M such that m ~ p for all p E P - M. (3) The embedding, amalgamation and congruence extension properties hold. (4) The subdirectly irreducible elements are B U [1}, where 1 > b for all b E B, and B is a Boolean algebra. (5) The only nontrivial proper equational class is the class of Boolean algebras. (6) The HSP diagram is determined.

(7) The free PCSL's are described and certain properties are established. For example, every free PCSL is complemented, and its chains and disjointed sets are countable. (Received January 21, 1972.) (Author introduced by Professor Alfred Horn.)

*72T-A80. HENRY w. GOULD, West Virginia University, Morgantown, West Virginia 26506. The case of the strange binomial coefficient identities of Professor Moriarty.

The identities :Bk(~~~1 )(~)= 2n-2j and~\(;)(~)= 2n-2j-1

University, San Antonio, Texas, 1962, p. 71. The history of these identities is traced back to 1826 and applications of them are given together with a definitive bibliography. (Received January 21, 1972.)

A-426 72T-A81. R. M. RAPHAEL, Sir George Williams University, Montreal107, Quebec, Canada. Some homological results on certain finite ring extensions.

All rings are commutative with identity and all modules are unitary. A ring R is connected if 0 and 1 are the only idempotent elements of R. R is semiconnected if the number of idempotents in R is finite.

Proposition. Suppose that R is connected, that I is a principal ideal of R[x], and that R[x]/1 is a finitely generated R-module. Then Ri:X]/1 is a freeR-module. PropoSition. Suppose that R is semiconnected, that

I is a principal ideal of Ri:XJ, and that Rli!/I is a finitely generated R-module. Then R[x]/1 is a projective

R-module. These results are applied to simple algebraic extensions in the sense of Enochs and the author.

(Received January 24, 1972.)

72T-A82. WITHDRAWN.

*72T-A83. PHILIP G. BUCKHIESTER, ·Clemson University, Clemson, South Carolina 29631. Gauss sums and the number of solutions to the matrix equation XAXT = o.

Let A be an n X n symmetric matrix over F q, a finite field of characteristic two. If A has 0 diagonal,

A is said to be an alternate matrix. Otherwise, A is nonalternate. A symmetrical bilinear form f on V n

V (F ) is said to be an alternating bilinear form if f(x,x) 0 for every x in V (F ). By developing Gauss sums n q = n q for alternating bilinear forms and by using Albert's canonical forms for symmetric matrices over fields of characteristic two, the number Ns(A,O) of s X n matrices X over Fq such that XAXT = 0 is found. In finding this number, the cases of A alternate and A nonalternate are treated separately. (Received January 27, 1972.)

72T-A84. GEORG J. RIEGER, University of Munich, Munich, Federal Republic of Germany, An estimate for the number of amicable numbers below x.

Denote by a(n) the sum of the positive divisors of n and by logr the r-fold logarithm. Two natural numbers a, b are called amicable if and only if a(a) = a(b) = a + b. Denote by B(x) the number of pairs of amicable numbers a ;!;; b with a ;!;; x. Erdos (Publ. Math. Debrecen 4(1955)) proved B(x) = o(x). Refining his -1/2 method we show B(x) = O(x(log4 x) log5 x). (Received January 31, 1972,)

*72T-A85. FRANCIS CHI-YU TANG, University of Waterloo, Waterloo, Ontario, Canada, On the

Frattini subgroups of certain generalized free products of groups. Preliminary report.

Let G = simple group then the Frattini subgroup 4>(G) is trivial.

Theorem 2. If G is residually finite and the groups Ai have compatible H-filters then 4>(G) is contained in the

maximal G-normal subgroup contained in H. Applying known results on residually finite generalized free products

we immediately have Corollary, W(G) is contained in the maximal G-normal subgroup contained in H if (1)

the groups Ai are finitely generated torsion-free nilpotent groups and H is closed in Ai; or, (2) Ai E :J • a

(:J being the class of finite groups and a the class of finitely generated abelian groups). (Received January 31,

1972.)

A-427 72T-AB6. ULRICH SCHOENWAELDER, Rhein. -Westf. Technische Hochschule Aachen, Templergraben

55, Federal Republic of Germany. Finite groups of Sylow 2-type M24• Preliminary report.

A finite group is called of Sylow 2-type M24 if it has a Sylow 2-subgroup T isomorphic to that of the 7 Mathieu simple group M24 . Thus, T is an extension of an extra-special group (dihedral type) of order 2 by a

dihedral group DB of order 23. Theorem 1. Suppose H is of Sylow 2-type M24 with Z(T) ~ Z(H). Then O(H)E

of an involution one obtains Theorem 2. If G is a simple group of Sylow 2-type M24 with O(CG(Z(T))) = 1,

then G =< M24 , L 5(2), or Held's group H2 . (Received February 1, 1972.)

*72T-AB7. CHRISTIAN HERRMANN, Technische Hochschule Darmstadt, 61 Darmstadt, Federal

Republic of Germany. Decomposition of modular lattices of finite length.

Theorem 1. Let M be a modular lattice of finite length. The skeleton L = (x IM there is x* E M such

that fX,x*] is a maximal atomistic interval in M} is a V-subsemilattice of M with 0. M is reconstructed by

Theorem 2. Let L and Lx (x E L) be lattices of finite length. Assume the lattice structures of Lx and LY

agree on L L = L , L V for all x, y E L; for all x § y in L, L n L is a principal filter (ideal) of X yn Xny n X y X y L (L ) ; if y covers x then L n L = fl. Then U (L Ix E L} is a lattice whose cover relation is the union of X y X y X those of the Lx; it is modular (n-distributive) iff the Lx are so. Applications. (1) In a subdirectly irreducible

modular lattice any two prime quotients are in 2 •l(M) - 1 steps projective. The class of modular lattices of

primitive length ~ n (n ~ 6) has infinitely many covers in the lattice of lattice varieties. (2) There is a

subdirectly irreducible modular lattice with five generators which has length ~ 6 and breadth ~ 3. (3) The

skeleton of a modular lattice of length ;,;; 6 is modular up to one exception. (4) Modular lattices of breadth ;,;; 2

and finite length are embeddable in projective geometries. Finitely generated ones are finite. (Received

February 2, 1972.) (Author introduced by Dr. Peter Burmeister.)

*72T-AB8. RAYMOND F. COUGHLIN and MICHAEL RICH, Temple University, Philadelphia,

Pennsylvania 19122. Scalar dependent algebras.

Let A be a nonassociative algebra over a field F and let g: A X A X A _, F such that (xy)z = g(x,y, z)

•X(yz) for all x, y, z in A. Assume also that (x,x,x) = 0 for all x in A. Theorem. If A is simple with an

idempotent e and if the characteristic of F -1 2, 3 then A is associative. This generalizes a previous result on

scalar dependent algebras (Abstract 71T-A90, these c!fotiai] 18(1971), 621). (Received February 4, 1972.)

*72T-AB9. N. S. MENDELSOHN and STEPHEN H. Y. HUNG, University of Manitoba, Winnipeg,

Manitoba R3T 2N2, Canada. Further results on handcuffed designs.

For a definition of a handcuffed design H(v, k, X, u) see "On Steiner handcuffed designs," Abstract

72T-A50, these cJfoticfi) 19(1972),A-295, by the present authors. Necessary conditions on the parameters for

the existence of a handcuffed design are (1) Xv(v -1) = 2(k -1)b; (2) (X(v -1) -u)v = kb, These necessary

conditions are sufficient in the following cases: (a) X= 1 or 2; (b) k = 3; (c) k is even and (X, 2h -1) = 1;

(d) k is odd and (X,4h) = 1; (e) k is odd and (X,4h) = 2. (Received February 7, 1972.)

A-428 *72T-A90. HARRY LAKSER, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada. Principal congruences in pseudocomplemented distributive lattices.

Let L be a pseudocomplemented distributive lattice. If a, bE L let 19Lat(a,b) denote the principal lattice congruence generated by a,b, and let 19(a,b) denote the principal pseudocomplementation-preserving lattice congruence. Theorem 1. If a,;; b then x =y (19(a, b)) iff x A a= y A a, and (x V b) A (a* A b)*= (yV b) A (a* A b)*. By a result of A. Day, Algebra Universalis 1(1971), 234-235, we get Corollary 1. The class of pseudocomplemented distributive lattices has the congruence extension property. Corollary 2. If L is a

Stone algebra and a ,;; b then 19(a, b) = 19Lat (a V b*, b V a*). Corollary 3. In aey Stone algebra the intersection of two principal congruences is principal, An example is given to show Theorem 2. The largest equational class of pseudocomplemented distributive lattices in which the intersection of two principal congruences is principal is the class of Stone algebras. Corollary 1 was first proved, in a different manner, in G. Gratzer and

H. Lakser, Trans. Amer. Math. Soc. 156(1971), 343-358. Corollary 3 is due to Kirby Baker (oral communication). (Received February 7, 1972.) (Author introduced by Professor George A. Gratzer.)

*72T-A91. ROBERT GILMER, Florida State University, Tallahassee, Florida 32306. On commutative rings of finite rank.

Let R be a commutative ring with identity and let k be a positive integer. Then R has rank k if each ideal of R has a basis of k elements; this is the terminology of I. S. Cohen [Duke Math. J. 17 (1950), 27 -42] •

In particular, Cohen has characterized integral domains with identity of finite rank. Theorem, The following conditions are equivalent. (1) R has finite rank; (2) R is Noetherian and R/P has finite rank for each prime ideal p of R of height O; (3) R is Noetherian and there is a positive integer t such that RM has rank t for

each maximal ideal M of R. More specifically, in (3), if each RM has rank t, then R has rank t + 1; this

followsfromatheoremofO. Forster [Math. Z. 84(1964), 80-87]. (ReceivedFebruary7, 1972,)

*72T-A92. PAUL E. BLAND, Easteru Kentucky University, Richmond, Kentucky 40475. Quasi­

corational extensions by a module.

In this paper quasi-corational extensions of modules are introduced and via this concept a new description

of finite direct sums of complete matrix rings over perfect completely primary rings is given. It is shown that

the class of rings over which every module is quasi-corationally complete is precisely that class of rings over

which every module is corationally complete. A module is called quasi-corationally complete if it is free of

proper quasi-corational extensions. It is shown that the quasi-corational completeness of a module over a

perfect ring yields a weakened form of quasi-projectivity and that over a commutative perfect ring no proper

corational extensions by a simple module exist. (Received February 7, 1972.)

A-429 *72T-A93. DOUGLAS D. SMITH, Central Michigan University, Mt. Pleasant, Michigan 48858. The spectra of groupoid equations. Preliminary report.

For each n ~ 2, the sets [1]n, [o]n U [1]n and (m: m > n} U (1} (where [a\ denotes the set of positive integers congruent to a modulo b) are shown to be spectra of finite sets of equations in a language of groupoids (i.e., with one binary operation symbol), as is [o]n for n prime or n = 4 or 6. Trevor Evans

("The spectrum of a variety, 11 Z. Math. Logik Grundlagen Math. 13(1967), 213-218) has shown that sets of the form Lo]n for all positive integers n and (m: m > n} U [1} for n !"!: 2 are spectra of sets of equations in other languages. For a given class K of algebras of some fixed type, we define the K-spectrum of a set :!: of equations, SpK(!:), to be the set of cardinalities of finite members of K which are models of :!:. If :!: is any set of groupoid equations and K the class of all semigroups, commutative semigroups, groups or abelian groups

(treated as groupoids), then SpK(!:) is either (1}, w- (oJ, or the multiplicative closure of a finite set of primes.

If K is the class of all bands or all semilattices, then SpK(!:) is [1} or w - [oJ. (Received February 8, 1972.)

(Author introduced by Professor Richard St. Andre.)

72T-A94. A. M. W. GLASS, Bowling Green State University, Bowling Green, Ohio 43403. The lattice of convex directed subgroups of a directed interpolation group.

The partially ordered group G is an interpolation group if whenever a, b § c, d in G, there exists g E G such that a, b § g § c, d. Let C be a convex subgroup of G and /-I(C) the group generated by C+. 1-!(C) is the largest convex directed subgroup of G contained in C. Theorem • The convex directed subgroups of a directed interpolation group G form a complete lattice, C.(G), under inclusion which is distributive. It is generated by its meet irreducible elements, e-1 (G), which are the set of values of positive elements of G (ME C.(G) is a value of g E G if g f: M and g E N whenever M c N E C.(G)). Theorem. If G is a directed interpolation group, the following are equivalent: (1) C.(G) is completely distributive. (2) e-1 (G) freely generates

C. (G). (3) Each member of e-1 (G) is special (is the only value of some positive g E G). They all imply (but are not implied by) (4) Each positive member of G has a finite number of values. Theorem. For any partially

ordered set X, there is a directed interpolation group G such that e-1 (G) is order-isomorphic to X. Consequently, any finite distributive lattice is lattice-isomorphic to C.(G) for some directed interpolation group G. (Received

February 8, 1972.)

*72T-A95. AURELIO SARTORELLI, Federal University of Parana, Curitiba, Brazil. A characterization of Boolean algebras by means of one operation. Preliminary report.

Let B be a set and U a binary operation defined in B, such that the axioms below hold for any elements

a, b, c in B. Axiom 1. (aU b) U c = a U (b U c). Axiom 2. a U b = b U a. Axiom 3, a U a= a, Axiom 4.

There exists 1 E B such that a U 1 = 1. Definition 1. For each a in B, Ca = [x E B: xU a= 1}. Axiom 5. Ca

= Cb implies a= b. Axiom 6. The family (Ci\Eca ordered under set-inclusion has a least element.

Definition 2. Let b be an arbitrary element in B. An element b' E B is called the complement of b iff Cb, is the least element of the family [C.}. EC ordered under set-inclusion. Definition 3. For any elements a, b in 11 b

A-430 B, a n b = (a' u b')'. It is shown that B is a Boolean algebra with respect to the operations u' n and I. (Received February 14, 1972,) (Author introduced by Professor Jayme Machado Cardoso.)

72T-A96, GEORGE MARKOWSKY, St. Mary's College of Maryland, St. Mary's City, Maryland 20686.

The number of idempotents in the semigroup of binary relations. Preliminary report.

Let A be a regular element of the semigroup of binary relations on some finite set, Then we have the following formulas: (1) IE(DA>I = (1/ IH,AI> E~=O (~1)i ~) (z -i)\ (2) IE(LA)I = ~= 1 «~I:t>ni - ('lt -1)ni) _a ni I 1 k rllj_ r 11_ a ,11_ •ui=k+1 (ll:t) ; (3) E(RA) = ni=l ((ll:i) - (~ -1) ) ni=k+1 (ll:t) ; where E(DA), E(LA), E(RA) are the sets of idempotents in the set as JJ-, .t-, R-classes of A, HA is the W-class of A, k is the rllllk of A, and the number!

a, a., n., q~, n~ and z depend on A, The number z actually only depends on the JJ-class of A, as does the "1. 1 1 1 number a, The numbers 'lt and ni depend only on the £-class of A, while the numbers '1f and ni depend only on the R-class of A. There is a straightforward way for calculating the various quantities above, once A is

given. (Received February 10, 1972.)

*72T-A97. JOHN T. BALDWIN, Michigan State University, East Lansing, Michigan 48823. Equational

classes categorical in uncountable power.

An equational class of algebras K is categorical in power x if all members of K with power x are

isomorphic. Theorem. If the similarity type of K is countable and K is categorical in power ~ 1 • then K is ~ 0 -categorical if and only if K has no finite members. The proof relies ou theorems by Jonsson (Math. Scand. 8(1960), 137-142), Lindstrom (Theoria 30(1964), 183-196), and Ryll-Nardzewski (Bull. Acad. Polon. Sci. Ser.

Sci. Math. Astronom. Phys. 7(1959), 545-548). (Received February 11, 1972.)

*72T-A98. WALTER F. TAYLOR, University of Colorado, Boulder, Colorado 80302. Characterizing

Mal 'cev conditions.

A class K of varieties of algebras is defined by a strong Mal'cev condition iff there exists a conjunction ex of equations in function variables fi and individual variables xj such that a variety V is in K if and only if there exist terms fi in the language of v such that v I= y~ ... xna' where Cit is formed from ex by replacing each occurrence of each fi by fi. K is defined by a Mal'cev condition iff K is expressed as a countable nested

union of classes defined by strong Mal'cev conditions. Theorem. A nonempty K is definable by a !i;ltrong]

Mal'cev condition if and only if K satisfies all of the following five conditions: (a) K is closed under the

formation of equivalent varieties; (b) K is closed under the formation of subvarieties; (c) K is closed under

the formation of finite products [arbitrary products) of varieties; (d) if V E K and V is generated by all

reducts of members of V1 to a fixed type, then V1 E K; and (e) if the equations !: define a variety in K of

type 0 , then there exist finite subsets !:0 and 0 0 of !: and 0 with !:0 defining a variety in K of type 0 0• This theorem answers a question of G. Gratzer [J. Combinatorial Theory 8(1970), 334-342], who remarked

that (a)- (d) are necessary conditions. Various allied results are given. (Received February 14, 1972.)

A-431 72T-A99. MARJORY JANE JOHNSON, University of South Carolina, Columbia, South Carolina 29208.

Maximal right ideals of transformation near-rings. Preliminary report.

Let T 0(G) denote the (left) near-ring of transformations from a group (G, +) into itself which map 0 into o. In his paper "One-sided ideals in near-rings of transformations" (J. Austral. Math. Soc., to appear),

H. E. Heatherly has shown that if x E G, x,; o, then A((x}) = {a E T0(G) I (x)a = o} is a maximal right ideal of T0(G). He raises the question: Is C = (a E T0(G) I Isupport al < loll a maximal right ideal of T0(G), where support a = {x I (x)a ,; o} ? The following results can be used to provide a negative answer. For w E

T0(G), let Bw = {x E G I (x)W = o}. Let R = {R IR is a right ideal of T0(G) and Ba n Bf3 is infinite, for all a, BE R}. Theorem 1. S is a maximal right ideal of T0(G) if and only if either S = A( {x}) for some nonzero x E G or S is a maximal element in the collection R. Theorem 2. Right ideals of T0(G) satisfy the ascending chain condition if and only if G is finite. (Received February 14, 1972.)

*72T-A100. ELVIRA STRASSER RAPAPORT, state University of New York, Stony Brook, New York

11790. Finitely presented groups. The deficiency.

Let G e. P = (x1, ••• ,xn;R1 , ••• ,~) = F/M, k minimal for n. Then dn = n- k is the deficiency of

G for n generators, d =max dn its deficiency. Let m = min n, Pm = (x1, ••• ,xm;R1, ••• ,Rm-dm) and such -1 -1 that the set R contsins the largest possible number, N, of elements fgf g , f c F, g c M. Then (1) d :'!! dm

+ N; (2) G has 2. presentation of deficiency d = n - k such that ~, ••• ,xn-m c M and the set R contains them modulo [)' ;¥]; (3) if G is non-Hopf then there is a presentation F/M on k relators and G"" F/M* with M* ::> M ,; M* but M* is also closure of k elements of F; (4) if F ,; F c in the lower central series of F, then qua factor group of F/Fc' G, has deficiency d = dm; (5) there exist groups whose every presentation contains elements of CF,MJ. (Received February 14, 1972.)

72T-A101. MICHAEL DOOB, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada.

A spectral characterization of the line graph of a BIBD with X = 1.

In recent years various authors have characterized the line graphs of finite affine planes, finite

projective planes, and Steiner triple systems by their spectra. Theorem. If r + k > 18, then the line graph of

a BIBD with parameters (v, b, r, k, X = 1) is characterized by its spectrum among all regular connected graphs.

(Received February 17, 1972.)

72T-A102. L. N. GUPTA, State University of New York at Buffalo, Amherst, New York 14226.

A Galois theory for quasi-coherent algebras over a quasi-compact, quasi-separated prescheme. Preliminary report.

Let Y be a q compact, q separated prescheme. Then (C,Oy lc> is a connected prescheme for any connected component C of Y. It can be shown that for any finitely presented q coherent Oy-algebra R the canonical map: AutOy-algebra (R)-+ Aut0 Yic-algebra (R lc> is onto. A faithful, finite etale Qy-algebra R is said to be weakly Galois if for any affine open set U of Y, G(U)R(U) HomOy(U)-mod (R(U), R(U)), where G =

A-432

A-433 A-433

1972.) 1972.) 23, 23, February February (Received (Received socle. socle. large large has has R R or or compact compact not not is is topology topology this this Then Then ideals. ideals. right right maximal maximal

of of number number countable countable a a only only has has R R that that such such and and zero zero radical radical Jacobson Jacobson with with ring ring a a be be R R Let Let 4. 4. Proposition Proposition

socle. socle. the the contain contain not not does does M M iff iff point point isolated isolated an an is is M M Then Then ring. ring. regular regular a a be be R R Let Let 3. 3. Proposition Proposition

obtained. obtained. is is 3 3 Proposition Proposition ideal, ideal, right right maximal maximal the the to to equal equal is is it it iff iff ideal ideal right right maximal maximal one one only only in in contained contained is is

ideal ideal right right cyclic cyclic R a a R ring ring regular regular a a in in that that fact fact the the of of use use By By topology. topology. Zariski Zariski the the is is topology topology this this then then ring ring

regular regular Neumann) Neumann) (von (von commutative commutative a a is is R R If If ideals. ideals. right right of of lattice lattice the the in in complements complements of of number number finite finite a a

only only has has ideal ideal right right simple simple every every that that is is compact compact be be to to topology topology this this for for condition condition necessary necessary 2. 2. Proposition Proposition A A

point. point. isolated isolated an an is is M M then then R R of of socle socle the the contain contain not not does does E E 1. 1. M M ?71 ?71 If If Proposition Proposition ideal. ideal. right right generated generated

= = I I ~::: ~::: finitely finitely a a is is I I where where I} I} ?71, ?71, M M M E E M (M (M 0 0 where where 0 0 all all of of JJ JJ set set the the is is by by given given topology topology the the for for basis basis ?71. ?71. A A

on on class class convergence convergence a a is is }} }} M. M. ( ( nn~kS(k)) nn~kS(k)) UnES UnES that that such such ideal ideal = = right right maximal maximal a a is is M M ?71 ?71 and and in in net net a a is is S S

where where (S,M), (S,M), psirs psirs ordered ordered all all of of set set .17 .17 the the be be Let Let classes. classes. convergence convergence of of concept concept the the ?71 ?71 using using by by on on put put is is

topology topology Hausdorff Hausdorff ideals. ideals. right right A A maximal maximal all all of of collection collection ?71 ?71 the the be be Let Let unity. unity. with with ring ring a a be be R R Let Let

report. report. Preliminary Preliminary ideals. ideals. right right maximal maximal of of set set the the on on topology topology A A

14454. 14454. York York New New Geneseo, Geneseo, York, York, New New of of College College University University State State LEVINE, LEVINE, M. M. JEFFREY JEFFREY 72T-A104. 72T-A104.

1972.) 1972.) 21, 21, February February (Received (Received possible. possible. best best sense sense some some in in is is (1954)) (1954)) Wiss. Wiss.

Akad. Akad. Bayer Bayer (Sitzber (Sitzber Lenz Lenz to to due due geometries geometries affine affine of of axiomatization axiomatization the the 3, 3, order order have have geometry geometry incidence incidence

finite finite a a of of lines lines all all where where that that show show examples examples These These unique. unique. is is points points 81 81 on on design design Hall Hall the the (3) (3) and and geometry, geometry,

affine affine an an of of lines lines and and points points the the to to isomorphic isomorphic not not is is which which points points 3n 3n with with system system triple triple affine affine an an is is there there 4 4 ;;; ;;; n n

every every for for (2) (2) n, n, integer integer some some for for points points 3n 3n has has system system triple triple affine affine every every (1) (1) that that here here shown shown is is it it 3, 3, exponent exponent

of of loops loops Moufang Moufang commutative commutative of of existence existence the the to to equivalent equivalent is is systems systems triple triple affine affine of of existence existence the the that that fact fact

the the Using Using AG(4,3). AG(4,3). geometry geometry affine affine an an of of lines lines and and points points the the to to isomorphic isomorphic not not is is which which points points 81 81 on on system system

triple triple affine affine an an constructed constructed has has 1962) 1962) 6, 6, vol. vol. Math., Math., Pure Pure Sympos. Sympos. (Proc. (Proc. Hall Hall Marshall Marshall plane. plane. affine affine 3 3 order order

an an generate generate triple triple a a in in not not points points three three every every if if system system triple triple affine affine an an system system triple triple Steiner Steiner a a call call We We

report. report. Preliminary Preliminary systems. systems. triple triple Affine Affine 10036. 10036. York York

New New York, York, New New Center, Center, Graduate Graduate York, York, New New of of University University City City YOUNG, YOUNG, PEYTON PEYTON HOBART HOBART 72T-A103. 72T-A103.

1972.) 1972.) 18, 18, February February (Received (Received 35(1969)). 35(1969)). J. J. Math. Math. Nagoya Nagoya idempotents," idempotents," many many infinitely infinitely with with theory theory

. .

ra ra

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Oy-

("Galois ("Galois Villamayor-Zelinsky Villamayor-Zelinsky of of theory theory affine affine corresponding corresponding the the generalizes generalizes theory theory This This (R). (R). b b al al Aut Aut

G G of of = = subgroups subgroups distinguished distinguished and and R R of of Oy-subalgebras Oy-subalgebras separable separable quasi-coherent, quasi-coherent, between between correspondence correspondence

bijective bijective inverse inverse an an is is there there then then 0y-algebra; 0y-algebra; Galois Galois weakly weakly a a and and R R prescheme prescheme separated separated q q compact, compact, a q q a be be Y Y

cJVotiai] cJVotiai] Let Let Theorem. Theorem. following following the the things things other other amongst amongst prove prove we we 636, 636, 18(1971), 18(1971), these these 71T-A139, 71T-A139, Abstract Abstract

prescheme,' prescheme,' connected connected a a over over sheaves sheaves quasi-coherent quasi-coherent for for theory theory Galois Galois "A "A our our and and others others among among above above mentioned mentioned

= = results results the the Using Using RF. RF. RH RH t. t. s. s. H H of of F F subgroup subgroup finite finite a a exists exists there there and and closed closed is is H H if if distinguished distinguished

be be to to said said = = is is H H lc· lc· lc lc t. t. s. s. h E H H E h exists exists there there fh fh fg fg lc(Y), lc(Y), of of f f idempotent idempotent minimal minimal any any and and R R Y Y of of

ra ra ge ge Oy-

C C component component connected connected any any and and g E G G E g any any for for if if closed closed be be to to said said is is G G of of H H A A subgroup subgroup (R). (R). b b al al Aut Aut 72T-A105. ALBERT J. SADE, 364 Cours de la Republique, (84) Pertuis (Vaucluse), France. Une nouvelle construction des quasigroupes orthogonaux a leur conjoint.

Deuxgroupoides Q= E() et R= E(•) sontconjointssixy= y•x. Unquasigroupeestanti.-abelien s'il est orthogonal a son conjoint. La seule methode actuellement connue pour construire un quasigroupe anti-abelien (A. Sade, Ann. Soc. Sci. Bruxelles Ser. I 74(1960), 91.-99; MR 25 #4017) consiste a prendre R = ·E(•), x•y = ax+ by+ c sur Z/h, a, b, a+b, a-b premiers avec h, ou sur GF(n), a, b, a+b, a-b fc o. Voici une nouvelle construction: Soit G = E(+) un groupe abe'iien, quelconque, d'unite u, T un automorphisme de G ne contenant aucun cycle binome et aucun cycle monome en dehors de u, enfin S un automorphisme de

G, permutable avec T; alors le quasigroupe isotope a G, Q = E(•) qui a pour loi de composition x•y = xS + yTS + c, c = Cte, est anti-abelien. (Received February 23, 1972.)

*72T-A106. ROGER C. ENTRINGER and DOUGLAS E. JACKSON, University of New Mexico, Albuquerque

New Mexico 87106 and JOSEPH A. SCHATZ, Sandia Corporation, Albuquerque, New Mexico 87115. On nonrepetitive sequences.

The following results are proved. Theorem 1. There exists an infinite binary sequence having no identical adjacent blocks of length 3 or greater. Theorem 2. Every binary sequence of length greater than

18 has identical adjacent blocks of length 2 or greater. Theorem 3. Every infinite binary sequence has arbitrarily long adjacent blocks that are permutations of each other. (Received February 23, 1972.)

72T-A107. OLGA TAUSSKY-TODD, California Institute of Technology, Pasadena, California 91109.

The factorization of an integral matrix into two integral symmetric matrices. I.

It is known that every n x n matrix A with elements in a field F can be expressed as the product of two symmetric matrices s1 , s2 over F. If the elements of A lie in a ring, e.g. Z, the elements of the Si may have to lie in Q . Let A have an irreducible characteristic polynomial f(x) with 0! as a zero. There is a 1-1 correspondence between the classes of matrices with the same f(x) under unimodular similarity and the ideal classes in Z ~]. Assume that Z ~] is the maximal order. Then previous work of Faddeev, Taussky, Bender implies that A will have a factorization of the type studied if the ideal class is of order 1 or 2. However, this condition is not necessary. For f(x) = x2 - m, m E Z, square free, > 0, "' 2, 3 (4) ideal classes of order 4 also lead to an integral factorization. (Received February 25, 1972.)

72T-A108. GEORGE PIEGARI, Vanderbilt University, Nashville, Tennessee 37235. On subalgebra lattices of unary algebras.

An algebra!:_= (A;F) is unary if F is any set of unary and nullary operations on A. Let S@) denote the subal\l'ebra lattice of !!c· Theorem. For i = 1, 2, 3, let Li be lattices and ~i unary algebras such that IL.I > 1 and L. '=" S(A.). Then there exist unary algebras B and B such that L. '=" S(B. X B.) for i = 1, 1 1 -1 - 1 - 2 1 -1 -1 2 and L3 '=" 8(~1 x ~ 2 ). The proof is based on work of W. A. Lampe presented in Abstract 653-295, these c}/oticei) 15(1968), 165, and Awad A. Iskander presented in Abstract 691-08-5, these c}/oticei) 19(1972), A-45.

(Received February 28, 1972.)

A-434 *72T-Al09. SIEMION FAJTLOWICZ, University of Colorado, Boulder, Colorado 80302. Categoricity in varieties.

Theorem. Let V be a variety such that for every cardinal m > 0, V has exactly one model of cardinality m. Then, up to definitional equivalence, V is either the variety of sets or the variety of pointed sets. (Received February 28, 1972.)

*72T-Al10. PHILIP KELENSON, City University of New York, John Jay College of Criminal Justice,

New York, New York 10010. Regu1ar Schrier varieties of universal algebras. Preliminary report.

A variety K of algebras is called a Schrier variety if each subalgebra of a K-free algebra is K-free.

An element of an algebra is prime if it does not belong to the range of any of its primitive operations. A variety

K is regu1ar if each algebra which is K-free has at least one prime element, equivalently, if K has a basis of equations in which no si.de is an indeterminate. In thi.s case each K-free algebra B has a unique set of free generators--the set pr(B) of its prime elements. Regular Schrier varieties are characterized. Some examples are provided. The result of Meakin, "On some Schrier varieties of universal algebras", J. Austral. Math.

Soc. 10(1969), is easily obtained as a special case. (Received February 28, 1972.)

*72T-Alll. STEVE GALOVICH, IRVING REINER and STEPHEN ULLOM, University of lllinois,

Urbana, lllinois 61801. Class groups for integral representations of metacyclic groups.

Let ZG be the integral group ring of a metacyclic group G of order pq, where p is an odd prime, and

q is any divisor of p - 1. The locally free class group C(ZG) is determined explicitly, in terms of the ideal

class group of some cyclotomic field, and the class group C(ZH) of a cyclic group H of order q. Let K =

Q(ll/p) :::::> L :::::> Q, R = alg. int. (K), S = alg. int. (L}, where (K: L) = q. Let R o H be the twisted group

ring of H over R, with H acting on R via the isomorphism Gal(K/L) ==' H. By viewing the group ring ZG as

a fibre product, and using a Mayer-Vietoris sequence, it is shown that there is an epimorphism C(ZG) _, C(S)

.f. C(ZH), whose kernel is a cyclic group of order q/(q, 2). The proof depends on the decomposition of the

hereditary ring R o H into minimal left ideals; this decomposition permits one to describe the units of R o H

explicitly. (Received February 28, 1972.)

*72T-A112. ROBERT WILLIS QUACKENBUSH, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada. Generalizations of idempotent functions and their compositions.

For k ~ 1, f: An _, A is k-idempotent if f(a1, ••• , an) E (a1 , ••• , an) whenever I (a1 , ... , an} I ~ k;

f is a quasi -projection if f(a1 , ..• , an) E (a1 , .•• , an} for all a1 , ••• , an E A. Theorem 1. If k + 2 ~ lA I then

every k-idempotent function on A is a composition of k + 1-ary k-idempotent functions; if 2 ~ IAI < N0 then every quasi-projection on A is a composition of ternary (but not binary) quasi-projections. Theorem 2. Let lA I = n ~ 3 be prime (respectively odd, even). Then there are 2 (respectively 3, 5) ternary idempotent (i.e. !-idempotent) functions on A such that every idempotent function on A is a composition of them. Theorem 3. Let k~ 2, IAI = n~ k+2, q= 2<2k-l), and p= [logq(n-l)J. Thenthereare p+2 k+l-aryk-idempotent functions on A such that every k-idempotent function on A is a composition of them; no set of p k + 1-ary

A-435 k-idempotent functions on A has this property. Theorem 4, Let IAI = n;;; 2 and p = [log8(n -1)]. Then there are p + 2 ternary quasi-projections on A such that every quasi-projection on A is a composition of them; no set of p ternary quasi -projections on A has this property. (Received February 29, 197 2. )

72T-Al13. JACOB T. B. BEARD, JR., University of Texas, Arlington, Texas 76010 and ROBERT M.

McCONNEL, University of Tennessee, Knoxville, Tennessee 37916. Matrix fields of (Z/(m))n. Preliminary report.

Let Z denote the integers and, for m E Z, m > 0, let Z/(m) denote the residue class ring. For an integer n > 0, let (Z/(m))n denote the complete matrix ring of all n x n matrices over Z/(m) under normal matrix addition and multiplication. If a subring M of (Z/(m))n is a field, then M is called a matrix field.

Using a result of R. W. Davis [Duke Math. J, 35(1968), 49), we determine the number of matrix fields contained in (Z/(m))n· The techniques used are similar to those used to obtain the results announced in Abstract 691-12-1, these cJYofice,i) 19(1972), A-54. (Received March 2, 1972.)

Analysis

*72T-B94. H. M. SRIVASTAVA, University of Victoria, Victoria, British Columbia, Canada and

J. P. SINGHAL, University of Jodhpur, Jodhpur, India. Certain generating functions for Jacobi, Laguerre and

Rice polynomials.

Let F(3) [x, y, zJ denote a general triple hypergeometric series, introduced earlier by H. M. Srivastava

[Proc. Cambridge Philos. Soc. 63(1967), 425-429], which provides a unification of Lauricella's fourteen 7(1893), hypergeometric functions F 1, ... , F 14 of three variables (G. Lauricella, Rend. Circ. Mat. Palermo 111-158J and the additional functions HA, HB, He [H. M. Srivastava, Rend. Circ. Mat. Palermo (2) 16(1967), 3 99-115]. The object of the present note is to derive an expansion for the function F( ) Cx, t, yt] in a series of powers of t involving the products of two generalized hypergeometric functions. As particular cases of this expansion, the authors deduce a large number of bilinear and bilateral generating functions for Jacobi, Laguerre and Rice polynomials. (Received December 13, 1971.)

72T-B95. JOHN C. MORGAN II, Department of Statistics, University of California, Berkeley,

California 94720. Baire category from an abstract viewpoint. Preliminary report.

The "duality" between Baire category and Lebesgue measure is explained by the existence of a general theory under which the analogous results can be subsumed. This abstract theory is developed by the author, and various examples and methods of classifying perfect sets are investigated. (Received December 27, 1971.)

(Author introduced by Professor David Blackwell.)

*72T-B96. CHARLES F, DUNKL and DONALD E. RAMIREZ, University of Virginia, Charlottesville,

Virginia 22903. Characterizations of Sidon sets and Ap sets on compact groups.

For a compact group G with dual G, a set E c: G is a Sidon set if and only if any unitary operator valued function on E can be interpolated almost surely by a Fourier-Stieltjes transform. Further, E is a AP

A-436 set for p > 2 if and only if any unitary operator valued function on E can be interpolated by an element of the multiplier algebra of Lp(G). (Received January 10, 1972.)

*72T-B97. H. M. SRIVASTAVA, University of Victoria, Victoria, British Columbia, Canada. On the

Konhauser sets of biorthogonal polynomials suggested by the Laguerre polynomials.

In recent papers, Joseph D. E. Konhauser [J. Math. Anal. Appl. 11(1965), 242-260; see also Pacific J.

polynomial sets !Ya(x; k) J and !Za(x;k)\. which are biorthogonal with Math. 21(1967), 303-314] discussed two n n respect to the weight function xae -x over the interval (0, oo), where a > -1 and k is a positive integer. The

object of the present note is to derive a class of bilateral generating functions for the polynomials Y~(x;k), and

to show that the polynomials Za(x;k) can be expressed as a finite sum of Za(y;k). As one of the applications of n n the results obtained in this paper, the following bilateral generating function is deduced: 1 1 1 (*) :0:0 !n!/r(,B + .tn+ 1)\ Y~(x; k) Z~(Y ;,l,)tn = (1- t) -(a+ )/kexp !x[1- (1- t) - /k] J H[x(1- tf /k, -y,Q,t/(1- t)],

where, for convenience, H[x, t] = :0:0 Y~(x ;k) !tn/r(/l+ ,l,n+ 1)\. For k = ,Q, = 1 and a= B, the generating relation (*) would evidently reduce to the well-known Rille-Hardy formula for the Laguerre polynomials.

(Received January 11, 1972.)

*72T-B98. HOWELL K. WILSON, Southern Illinois University, Edwardsville, Illinois 62025. Limit

sets for solutions of a second order linear differential equation. Preliminary report.

Consider the differential equation x" + q(t)x' + p(t)x = 0, where q is nonnegative and continuous and p

is absolutely continuous, non increasing, and bounded below with p(t) ~ a as t ~ + oo. If a> 0, then every

solution x = cp(t), y = cp' (t) has for its w-limit set precisely one (possibly degenerate) ellipse y2 + ax2 = c ?;; 0 in

the xy-plane. If a::f 0, then every w-limit set must be a subset of either a pair of (possibly coalescing) parallel lines y =:!: c or a (possibly degenerate) hyperbola i - \a\ x2 = c ::f 0. (Received January 14, 1972.)

*72T-B99. MARTIN M.LIPSCHUTZ, William Paterson College, Wayne, New Jersey 07470. On a

multi-point initial valued problem. Preliminary report.

An existence and uniqueness theorem is given for the differential equation y(n) = f(t, y, ... , y(n-1)) with

general linear multi-point initial conditions :0k m~y(j- 1 ) (~) = hi, i, j, k = 1, ... , n, ~ E [a, b], for a sufficiently

small interval [a, b] and initial conditions satisfying det:0km~.P .. (t.) i 0, where P .. (t) = Di-1(ti-1). Proof is lj lj "k l) via the Banach fixed point theorem. (Received January 21, 1972.)

72T-B100. PHILIP J. BOLAND, University College of Dublin, Belfield, Dublin 4, Ireland. Weighted

spaces of nuclearly entire functions on a Banach space. Preliminary report.

Let E be a Banach space and San a-sequence space. HNb(E,S) is the space of complex valued

nuclearly entire functions on the space E weighted by S. With its natural topology, HNb(E, S) is barrelled iff

HNb(E,S) is bornological iff S is a perfect a-sequence space. If S is perfect, then Hkh(E,S) is in the strong

topology isomorphic to Hb(E', S*), the latter being the space of entire functions on the dual E' of E weighted by

S*. Using this isomorphism, it can be shown that for a large class of these spaces HNb(E, S), any nonzero

A-437 convolution operator is onto, and that the kernel of any convolution operator is the closure of the subspace generated by the exponential polynomial functions in the kernel. (Received January 24, 1972.)

72T-B101. T. WALSH, University of Florida, Gainesville, Florida 32601. The dual of Hp(Rn_:-1) for E....::..l.·

The dual of Hp of the unit disk for 0 < p < 1 has been characterized by Duren, Romberg and Shields. Let HP (R~+ 1) denote the Hp space of systems of n + 1 conjugate harmonic functions in the half space R~+ 1. For a > 0, N ~La] and k = N + 1, the Lipschitz space Aa is defined to consist of all residue classes of functions modulo polynomials of degree N at most such that \\f\1 (Aa]~ sup 1\h\-a \1 Ak(h)f\\00 : hE Rnl < oo. Suppose (n-1)/n

n(1/p-1) and for F EHP(R~+l),

*72T-B102. MYRON GOLDSTEIN and JOSEPH L. WALSH, University of Maryland, College Park,

Maryland 20742. Approximation by rational functions on Riemann surfaces.

The authors consider the class Lp(k, 0!) of functions possessing a kth derivative on the boundary of a

compact bordered Riemann surface R which satisfy a pth mean integrated Lipschitz condition of order 0! and

show that every function in this class can be written uniquely as the sum of a function in Hp (k, a) and a function

in Gp(k,a). Furthermore, it is shown that every function in the class Lp(k,a) can be approximated in the mean

of order p by rational functions Rn(z) on R union its double to within A/nk+a • These results generalize

earlier ones obtained for a plane region bounded by a single analytic Jordan curve. (Received January 27, 1972.)

72T-B103. JEFFREY A. KUESTER, University of Wisconsin, Milwaukee, Wisconsin 53201. An

unusual region. Preliminary report.

A simply connected bounded region G is constructed such that the impression of every prime end of the

region is exactly the boundary of the region. This region is also shown to have prime ends whose set of principle

points is exactly the entire boundary of G. Hence, it is shown that there exist bounded schlicht holomorphic

functions defined in the unit disk with exactly the same cluster set at every boundary point of the disk. The

class K of all such functions is described. The region G is also generalized as follows: there exist countably

many disjoint bounded simply connected regions all with the same boundary, with the property that the impression

of any prime end of any of these regions is equal to some fixed set K, namely the boundary of any region.

(Received January 28, 1972.)

72T-B104. ANTHONY W. LEUNG, University of Cincinnati, Cincinnati, Ohio 45221. Studies on

doubly asymptotic series solutions for differential equations in unbounded domains. Preliminary report.

n (n) n-1 (n-1) The paper is concerned with the equation (1) E u (z) - E pn_1 (z) u (z) ... - Ep1 (z)u' (z) -

A-438 p0(z)u(z) = 0, where pi(z), i = 0, .•• ,n -1, are polynomials. The pi(z)'s satisfy hypotheses which essentially ensure that the n solutions for the characteristic e~ua__tion asso~iated ~th the equivalent first order system of

- qi/Pi (qi -1)/Pi < . < "' A /A "' (1) have series at z = oo of the form \(Z) - ai0z + a11 z + ••• , 1 ~ 1 = n, where 0 - q 1 P1 - ••• ;!'; ~/pn, and each pair <\, pi are relatively prime integers. Formal power series solutions to (1) are constructed. Such series are "doubly" asymptotic (in the sense that successive terms become smaller in order of magnitude both as t: ... o+ or z ... oo) to solutions of (1) in certain unbounded sectors in the z-plane, 0 < f:

< t: 0. There is an analysis of the conditions which guarantee that a fundamental system of n such asymptotically known solutions can be found in a given sector. This paper improves the result of the author's thesis (at the

University of Wisconsin, Madison) which studies equation (1) with the restriction that pi(z) has degree

(m/n)(n-i), i= 1,2, .•. ,n-1, where m isthedegreeofp0(z). (ReceivedJanuary31, 1972.)

72T-B105. DONALD R. WILKEN, State University of New York, Albany, New York 12222. The integral means of close to convex functions.

Let C denote the class of close to convex functions analytic in the unit disc A= [z: lzl < 1}, normalized so that iff E C, f(z) = z + L:;~= 2 anzn. Let k(z) = z/(1-z)2 be the Koebe function. Let 0 < p < 1. Theorem. If f E C, then s;IT if(peiS) jP d9/21T ;!'; r!T ik(peiS )jP d9 /21T for any real number p ~ 1. Recently (Abstract 72T -B39, these cAfoliCt.O 19(1972), A-306), MacGregor established this result for p a positive integer. Independently, the author had previously established the case p = 1. (Received February 1, 1972.)

72T-B106. CARLOS AUGUSTO SHOLL ISNARD, Instituto de Matema'tica Pura e Aplicada, Rio de Janeiro, Brazil, Transversality and topological degree.

1 1 If X andY are real or complex c -Banach-manifolds and f:X ... Y is a proper c -4>0-map transversal 1 1 to some submanifold M of Y, then the restriction f- (M) ... M of f is a c -.P0-map that has as regular values exactly the y E M that are regular values for f:X ... Y, Furthermore, the degree at those y is the same for both maps. This transversality property is used as a dimension-reducing tool to define a degree for proper 1 c -4>0-maps (and for proper c1 -maps, when dim X= dim Y < oo). Using the property when Y is open in a 1 Banach space, a degree is defined for proper maps f + c :X ... Y, f c -.P0-map, c continuous locally compact, generalizing the Leray-Schauder degree. Taking f constant, we have a degree for continuous proper maps between n-dimensional c1-manifolds (= Brouer's degree if manifolds are oriented). (Received February 3, 1972.)

72T-B107. A. R. REDDY, University of Missouri, St. Louis, Missouri 63121. A contribution to best approximation with some square norms. Preliminary report.

The set of all functions f(z) which are singlevalued and analytic in the unit circle B and for which

JBJ!f(z)l 2 dxdy < oo will be denoted by L 2(B). Set for any f E L 2(B), [An(f)J2 = 2 min~JJB1f(z)-a0 -a1 z- ... anznj dxdy wherethenormisthe L2 normontheunitcircle B. Let f(x) be a real valued continuous function on [-1, 1] and w (x) be a nonnegative Riemann integrable weight function on 2 2 1 2 [-1,1]; then set E~ ) (f)= minPEITn CJ:1 jf(x) -p(x)i W(x)dx] / , n = 0, 1, 2, ••. , where the norm is the L2-

A-439 norm on [ -1,1] and 'II' denotes the class of all polynomials of degree at most n. Further, let E (f) "' n n minpE1Tn lf(x) -p(x) I, n = 0,1, ... , where the norm is the uniform on [-1,1]. We are interested primarily in knowing how An(f) is related to the order and type of an entire function. We also investigate how An(f) is

of f. Secondly we discover how E(2) (f) is related to the related to the (n + l)th coefficient of the Taylor series n order and type of an entire function. Finally, we obtain an asymptotic relation between En(f) and E~2 ) (f) for a sequence of values of n. (Received February 4, 1972.)

72T-Bl08. S. OWEN BURKINSHAW, Purdue University, West Lafayette, Indiana 47907. Weak

compactness in the order dual of an Archimedean Riesz space. Preliminary report.

Let E be an Archimedean Riesz space. E is called a-complete if for each order bounded sequence

[xn) the sup (xn} exist in E. Let E0 and Ecrc be the spaces of linear functionals on E that are continuous

with respect to order convergence of nets and sequences respectively. Let Eb be the space of linear functionals

are order bounded. Then E0 c E ere c Eb. A sequence [x in E will be called an t 1-sequence if on E that n J thereexistsan x in E such that :0~ lxnl;:;; x, m = 1, 2, .... For a set A c Eb, let II·IIA. denote the Minkowski functional on E defined by A • . A set A c Eb will be called t 1 -equicontinuous on E if lim llxniiA. = 0 foreach t 1-sequence [xn} in E. Wemakeastudyoftherelationshipof t1-equicontinuityto compactness, The following is proven. Theorem. Let E be a-complete and A c Eac. A is relatively

com pact in the vague topology on E ere defined by E iff A is t 1-equicontinuous on E. In general, the vague

compactness of a set A in Eb is not equivalent to A being J, 1 -equicontinuous on E. An ideal I of a vector

lattice is a linear subspace with the property that a E I, lbl ;:;; Ia! implies b E I. Let I be the ideal in (Eb)c

generated by E, so E c I c (Eb) c. Theorem. Let E be an Archimedean Riesz space and A c Eb. A is

relatively compact in the weak topology on Eb defined by I iff A is J, 1 -equicontinuous on E. We apply the

above to obtain some results on the vague convergence of sequences in Eb. (Received February 7, 1972.)

*72T-Bl09. PETER HESS, University of California, Berkeley, California 94720. A strongly

nonlinear elliptic boundary value problem.

On the open, bounded, smoothly bounded subset 0 of Rm (m ~ 1) the generalized Dirichlet problem

foranequation (.7"u)(x) = (au)(x)+p(u(x))= f(x) isconsidered. Here au"' :01a1;;;1 (-l)ladDaAa(x,u,Du),

and p: R - R is a continuous function on which we impose only the sign condition p(t) t ~ 0 (t E R). It is

assumed that the (nonlinear) Dirichlet form a(u, v) "' I: Ia 1§ 1 Jo Aa(x, u,Du) Davdx is defined for all u, v in the Sobolev space V"' ~,p (0) for some 1 < p < co, and that the mapping A: V- V* induced by (Au, v) =

a(u, v) (u, v in V) is of the monotone type (M) : un -" u in V, Aun-' X in V*, and lim sup(Aun, un -u) § 0

imply that X= Au. Note that this is in particular true if a satisfies the conditions of Leray-Lions (Bull. Soc.

Math. France 93(1965), 97-107). Let further V1 "' [uE V:p(u) E L\0), p(u) • uE L\0)}. Theorem, Under if llvll -1 [a(v,v)+ Jop(v)vdx} - +co as llvll -co, v E then the Dirichlet the above assumptions, 1,p 1 ,p C~(O), problem for the equation .ru = f has a variational solution for each f E V*, i.e. there exists u E V1 with a(u, v) + Jop(u)vdx = (f, v) for all v E C~(O). This theorem generalizes recent results by F. E. Browder

and by the writer; its proof involves an elliptic regularization "of singular perturbation type". (Received

February 7, 1972.)

A-440 72T-Bl10, RAYMOND M, REDHEFFER, University of California, Los Angeles, California 90024.

On a nonlinear functional of Berkowitz and Pollard.

Let 2~ = 3 + (33)1/2• Define 2A = ~ + (3~ +2l/2, a= 5A3 - 20A2 - A+ 4, b = 67A3 + 75A2 + 43A-

17, c= A3 -4A2 +4A-1, d= A2 -4A+1. Thentheminimumvalueof llul\i+ llu"ll; on (O,oo) subjectto u(O) 0, u'(O) = 1 is (2a/45b//5 (c/d), (Received February 9, 1972.)

*72T-Bl11. MARTIN G. BUNTINAS, Loyola University, Chicago, lllinois 60626. Toeplitz sections in sequence spaces.

n Let T be a row finite matrix whose columns tend to 1. Let t = (tn1, tn2, tn3, ••• ) be the nth row of

T. The nth T -section of a sequence x is tn • x = (tn1x 1 , tn2x2 , tn3x3, ..• ) • Let E be a locally convex sequence space. E has the property TB (FTK) if, for every x E E, (tn • x} is a bounded subset of E (for every x E E, 6 f E E 1 , limn f(tn • x) exists). Let e k be the sequence with 1 in the kth place and zeros elsewhere, E =

{x: for some fE E', x. = f(ek)}, JtT = {x: for each y E E, sup I'B t .x.y.l < oo), EBT = {x: for each y E E, K n nJ J J limn'B tnjxjyj exists). Theorem 1. (E6?T = 'ETB"' {x: {tn • x} is a bounded subset of E) and (Eti){3T =

'EFTK"' {x: for each fEE', limnf(tn • x) exists}. Theorem 2. If Eisa barreled K-space, E'YT c Eti.

Theorem 3. The following statements are equivalent for a barreled K-space: (a) E has the property TB;

(b) E c (E6?'r; (c) E 'YT = Eti; (d) (E'YT?T = 1ETB' The same holds if yT and TB are replaced by {JT and FTK respectively. Theorem 4. A barreled K-space has the property FTK if and only if E6 = E'YT = Ef3T,

Theorem 5. If E is an FK-space, then 'ETB and 'EFTK are FK-spaces, (Received February 9, 1972.)

72T-Bl12. SUDHANSHU KUMAR GHOSHAL and A. MASOOD, Jadavpur University, Calcutta-32, India.

Investigation of the oscillatory solution of strictly nonlinear differential equations. Preliminary report,

The alternate plane concept was applied by W. R. Utz and G. DiAntonio to investigate oscillatory solutions, if any, of quasi -linear differential equations. The procedure has been extended to examine the oscillatory behaviour of the solutions of strictly nonlinear ordinary differential equations of second order. As examples, we have applied the theory successfully to the two differential equations viz: (i) x•2 x" = r- x + sx2 and (ii) x•2x" + K2 sin x = o. (Received February 10, 1972.)

72T-Bl13. SUDHANSHU KUMAR GHOSHAL and MADAN CHATTERJEE, Jadavpur University,

Calcutta-32, India. Stability of a system of nonlinear ordinary differential equations in a critical case.

Preliminary report.

The stability of the following system of nonlinear ordinary differential equations about the origin is

discussed here: z' = Az + Z, where zT = (z1, z 2, ... ,zn+Zp)' A is an (n+2p) X (n+2p) matrix, zT = (Z1, z 2,

••• , Zn+2p) each ZK is a suitable holomorphic function of the variables z1 , z2, , , , , zn+2p and begins with the terms, at least, of the second order. The characteristic determinant, of the corresponding linearized equation,

viz lA -A Tl has 2p number of roots equal to zero and the nonzero roots have negative real parts. The procedure

of investigation is based on Lyapunov's second method. A suitable Lyapunov function is formed and then

A-441 applying Chetaev's instability theorem (Ustoichovost dvizheniia, Gostekhizdat, vol. 50, 1953) we have shown that

the system is unstable at the origin. (Received February 10, 1972.)

72T-B114. RAYMOND L. JOHNSON, University of Maryland, College Park, Maryland 20742. Extension of some theorems of Stein and Zygmund. Preliminary report.

Extension of the results of Stein-Zygmund (Ann. of Math. (2) 85(1967), 331-349) from Lipschitz spaces of

Besov-Taibleson to those considered by Herz (J. Math. Mech. 18(1968), 283-324). Let k E A~,+CD (i.e., suppose J!f(x+h)- f(x) I dx~A !hi'\ and let T be the translation invariant operator defined by Tf = k*f, for smooth functions. Theorem 1. T is of strong type (p,q), 1/q = 1/p - 01./n. Theorem 2. If f E L , Tf is p,+CD of bounded mean oscillation. Theorem 3. If f is of bounded mean oscillation, f is in A0 , and thus, if f is CD,CD ofboundedmeanoscillation, Tf E AOI. • Remarks. (1) The Riesz kernel lxl-n+OI. E AOI. , but is not in CD, CD 1, +CD 11{01.; 1,+CD). (2) In fact, in Theorem 1, T: LP- L q,p, the Lorentz space. (3) Analogues of both theorems concerning translation invariant operators on LP in the appendix of Stein-Zygmund are given. (Received February 11, 1972.)

72T-Bl15. WILLIAM D. L. APPLING, North Texas State University, Denton, Texas 76203. A Fubini- type theorem for finitely additive measure spaces.

Suppose (u1 X u 2, F 3, m 3} is the product space of the finitely additive measure spaces (u1 , F 1 , m 1} and (U 2, F 2, m 2). Integrals in this abstract are limits of appropriate sums with respect to refinements of subdivisions. L and G, with appropriate subscripts, denote, respectively, sum supremum and sum infimum functionals. Very simple inequalities involving L sums and G sums with respect to F1 , F 2 and F3 imply the following Theorem. Suppose that for each (I,J) in F1 x F2 and (x,y) in I x J, a(x,I,y,J) (;; :m. Suppose his a function from F3 into exp(:Bl) with bounded range union such that if (I,J) is in F1 X F2, then h(IX J) ::<

UIXJa(x,I,y,J). If JSu1xu2 hm3 exists and B is Lor G, then each of Ju1 CJu2 B2(a(x,I, •,•)m2)Jm1(I) and

Ju2Cfu1 B1(a(•, •,y,J)m1)Jm2(J) exists and is Jfu1xu2hm3• Note, for example, that one consequence of this theorem is the equation 0 = fu1 CJu 2 (L2(a(x,I, •, •) m 2) - G2(a(x, I, •, •) m2)}] m1 (I) which immediately implies that if w(x, •, •) = a(x,I, •,•) = a(x,u1 , •, •) for x in I and I in F1 , then the set of all x in u 1 for which w(x, •, •) is not m 2 integrable has m1 outer measure 0. (Received February 14, 1972.)

*72T-Bl16. ATHANASSIOS G. KARTSATOS, University of South Florida, Tampa, Florida 33620.

Convergence in perturbed nonlinear systems.

Consider the nonlinear system (*) x' = A(t,x) and the perturbed system (**) y' = A(t,y) + B(t,y). A method developed by the author in Abstract 691-34-9, these cNoticeiJ 19(1972), A-128, is employed here in order to study the existence of convergent solutions, and their properties, for systems of the form (**), in connection with their associated unperturbed systems (*), For example, sufficient conditions are given for the convergence, equi-convergence, equi-uniform convergence, or coalescence of the system (**), provided that (*) has the same property. The results are intimately related to those of Avramescu [Ann. Mat. Pura Appl. 4(1969), 147 -168] and Hallam [Ann. Mat. Pura Appl., to appear]. (Received February 16, 1972.)

A-442 72T-Bl17. R. P. SINGAL, Guru Nanak College, Ferozepore Cantt, India. A type of transformation of the double hypergeometric function.

In previous papers (Abstract 70T-B228, these cNoticei) 17(1970), 959, and Abstract 71 T-B201, ibid.

18(1971), 950) we obtained transformation formulae for Kampe' de Feriet hypergeometric function at (1, 1). The object ofthis paper is to obtain transformation formulae for the double hypergeometric function of a type which transforms functions with argument ([-X/(1-X-Y)Jk, [-Y/(1-X-Y)]k) into functions With arguments (X, Y), For k = 1, a known transformation of F 2 into F 2 and new transformations for Fi: ~· F 1, F 2, and F 4 in terms of Fi: ~, F 1 ,

F 2, and F 4 respectively are deduced; and for k = 2, a transformation due to Bailey for F 4 in F 2 is deduced (Math. Student, to appear). (Received February 16, 1972.) (Author introduced by Dr. Brij M, Nayar.)

*72T-B118. V. R. GOPALA RAO and TSUAN W, TING, University of illinois, Urbana, illinois 61801,

Solutions of pseudo-heat equations in whole space.

Cauchy problems for the pseudo-heat equations, (A -I)ut + Bu = 0, for the solutions u(x, t) in the whole space Rm x R, m ;;;:; 1, have been considered, where I is the identity operator and where A and B are second- order elliptic operators with constant coefficients. The existence and uniqueness of the solutions have been established for initial data in Sobolev spaces wk•P(Rm), k;;;:; 2, 1 < p < ro, andinSchauderspaces ck+OI(Rm), k;;;:; 2, 0 < 01 < 1. It has been found that if the restriction of the solution to any instant t belongs to a Sobolev space or a Schauder space, then it stays there for all time t. Moreover, the solutions are just as regular in the space variables as the initial data, but no more. Also, the solutions, together with their time derivatives of all

orders, decay (grow) exponentially as t .... ro (as t""' -ro). (Received Feburary 17, 1972,)

*72T-Bll9. WOLFGANG J. STROBELE, Lehrstiihle fiir Numerische und Angewandte Mathematik,

Universitat Gottingen, 34 Gottingen, West Germany. On the representation of the extremal functionals on

Let T be a locally compact Hausdorff space, X a normed linear space and let C0 (T, X) be the linear space of all continuous functions f: T .... X "vanishing at infinity" with the sup-norm. Let M be a linear

subspace of X and let f1 f T0 ~ T and !DlTo ~ C0(T ,X) the subspace of all f with f(T0) ~ M. !DlTo is a Cc(T)­

module (Cc(T):= span (C 0(T), const. functions)). Let L E ext(SX*nM.L), where SX*:= [LEX*: \\L\\ ;§ 1).

Let x0 E T0 and define a functional on C0(T ,X) by .P(f) = L(f(x0)). By an application of the characterization theorem for extremal functionals of Buck and Phelps an elementary proof is given for the following Theorem. q, E ext (Sc (T X)* n !Dl-!j. ) . As a corollary one obtains that in the case M: = {o}, T0 : = T, !DlT = (o} the 0 ' 0 "generalized evaluation functional" q, is an extremal point of the unit-ball SCo(T ,X)*. (Received February 1,

1972.) (Author introduced by Professor R. C. Buck.)

*72T-B120. JOHN DAVID LOGAN, University of Arizona, Tucson, Arizona 85721. Higher dimensional

problems in the calculus of variations.

Let !rmnl be a finite sequence of real numbers, m ~ M1 - 1, ... , M2 and n ~ N1 - 1, ... , N2, and let Lm

and Ln be difference operators defined by Lmrmn= rm-1,n and Lnrmn= rm,n-1. The fundamental problem is

A-443 M2 N2 to select among all sequences lrmnl the one which extremizes J = L:M1 L:N1 F(m, n, rmn' Lmrmn' Lnrmn> where F(x, y, u, v, w) is a given continuously differentiable function. A necessary condition that lrmnl afford an extreme

value to J is that \rmnl satisfy a second order partial difference equation, which is in general nonlinear. From

this condition, it is possible to state a variational principle for certain discrete systems, i.e. systems governed

by second order partial difference equations. The necessary condition is derived by applying a discrete version

of Green's theorem for partial difference operators to calculate the first variation of J. Such a procedure

generates boundary terms, i.e. terms defined on the boundary of the given mesh, which lead to natural boundary

conditions. Conservation laws can also be obtained from these boundary terms by investigating the invariance

properties of F. In particular, it is shown that if F is invariant up to a difference under the one parameter

family of transformations rmn = rmn + t A(n,m, rmn>• then conservation laws can be obtained explicitly. (Received February 22, 1972.)

72T-B121. GERALD M. LEIBOWITZ, University of Connecticut, Storrs, Connecticut 06268. Hausdorff

transformations and shifts. Preliminary report.

Theorem. Let H be a regular Hausdorff transformation with absolutely continuous mass distribution.

Then the following are equivalent: (1) H = r~ for some a > 1/2; (2) there is some c > 0 such that (I- cH~*

acting on L 2(0, ro) is a simple bilateral shift; (3) there is some c > 0 such that (I- cH)* acting on L2(0, 1) is a

simple unilateral shift. The fact that (1) implies (2) and (3) has been proved by N. Sharma (to appear in Acta

Sci. Math. (Szeged)). My argument uses a Wiener-Hopf approach. (Received February 24, 1972.)

*72T-B122. MORRIS SNOW, City University of New York, Queens College, Flushing, New York 11367.

A joint (Browder) essential spectrum. Preliminary report.

Let A E B(x) the set of bounded linear operators on a Banach space X. X E a es(A) if X- A is Fredholm

and :S: a deleted open neighborhood of A contained in the resolvent set of A. X E ae5(A) (the Browder essential

spectrum of A) if A (/.a 65(A). Let A1 and A2 be commuting operators in B(X), Z the double commutant of A1

and A2, and Z the almost double commutant of A1 and A2• We will say (X1, X2) E pe5(A1,A2) if CX 1, X2) E

Pw(A1,A2) in Zw and :ii a deleted open neighborhood of (X 1, X2) contained in the joint resolvent p(A1,A2) in Z.

Theorem. For f analytic in a neighborhood of a(A1) Xa(A2) we have f[ae5(Al'A2)J = ae 5[f(Al'A2)J. As a consequence of this theorem we can prove that if X and Y are Bauach spaces, A E B(X), BE B(Y), X® Y the

completion of the tensor product of X and Y with respect to some crossnorm, A1 =A® I and B1 =I® B, then

ae5(A1 ,A2) = [ae5(A).X a(B)J U(a(A)X ae5(B)J and if f is a function analytic in a(A) x a(B) then ae5[f(A1,A2)J =

Ef(a e5(A), a(B))] U(f(a(A), a es(B))]. (Received February 25, 1972.)

*72T-B123. EVELYN MARIE SILVIA, Clark University, Worcester, Massachusetts 01603. Some

properties of functions alpha-starlike of order {3. Preliminary report.

Let a !!!! 0, 0 lii (3 < 1 and suppose that f(z) = z + L::2 anzn is holomorphic in U = \z/lzl < 1} with f(z) •f'(z) i 0 in 0 < lzl < 1. If Re[(1- a) zf'(z)/f(z) + a(zf"(z)/f'(z) + 1)] > {3 for z E U, then f(z) is said to be alpha­

starlike of order {3 and we write f(z) E Sa(/3). The class Sa(O), 0 lii alii 1, was introduced by P. Mocanu

A-444 (Mathematica (Cluj) 11(1969), 127-133] and was extended by S. Miller to 0 ~IX< oo [Abstract 691-30-19, these cJfoticei) 19(1972), A-115). The author shows that many of the results of S. Miller can be generalized to SIX(P).

We show the following Representation Theorem. The function f(z) E SIX(P), IX !!; 0, 0 ~ P < 1, if and only if there exists a function F(l;:) starlike of order P such that f(z) = [(1/1X)J~LF(C}J 1 /IX C 1dl;: ]IX. A distortion theorem for

lf(z)l and a coefficient bound for la2l are also proved for the functions in this class. (Received February 28, 1972.)

72T-B124. LAWRENCE A. ZALCMAN, Stanford University, Stanford, California 94305. Functions whose Fourier coefficients vanish.

2 9 Let f E C(JR ) and write f(z+ei ) = r;:,_ 00 ~(z)eikB (Fourier series). If 0 ~ k < n and a -k(z) = a -n(z) = -k ..1 -k o for all z E C, then (o I oz) f = o. If -k < 0 < n and k,.. n, then ('21/oz) f = 0 and ('21/oz)nf = 0; in particular, f is a polynomial in z and z. It is actually enough to assume f E r.ioc (1! 2) and that the Fourier coefficients in question vanish for almost all z. A crucial ingredient in the proof is Bourget's hypothesis. (Received February

28, 1972.)

72T-B125. JAMES S. BYRNES, University of Massachusetts, Boston, Massachusetts 02116. Functions which multiply bases. Preliminary report.

For a fixed p, 1 < p < oo, consider the space LP(-1T,1T). If cp(x) is a function on (-1T,1T) satisfying cp E

L 00 and 1/cp E L 00, then it is clear that !cp(x) exp (inx)!., -oo < n < oo, is a basis for Lp. We show that this result

is best possible in the following sense. Theorem. Let q and r be any 2 positive numbers, at least one of which

is less than oo. Then there is a function cp(x) on (-1T,1T) satisfying (1) cp E Lq; (2) 1/cp E Lr; and (3) !cp(x)exp(inx)!

is not a basis for LP. (Received February 28, 1972.)

*72T-B126. RICHARD A. ALO, Carnegie-Mellon University, Pittsburgh, Pennsylvania15213 andANDRE

de KORVIN and LAURENCE E. KUNES, Indiana State University, Terre Haute, Indiana 47809. Linear operators

on £~. I.

For the notations see "A topology associated with .ePspaces. I," Abstract72T-B85, these c:J.kJticei) 19

(1972), A-319. Let c.cr,f be the tribe of !J.O'-finite subsets ofT. Let 1~p~ oo. Let U be a continuous linear

operator from £~ into F (ol~ denotes the space of E-valued !J.p-integrable functions). Assume U < < !J., i.e. if

!J.(A) = 0 then U is the zero operator when restricted to functions of £~ vanishing outside A. Theorem. If pI

oo and T E C. a, f then U is compact if and only if (0'*, lim, ql is compact. If p = oo and U is a continuous linear

operator from .e: into F then U is compact if and only if (0'*, lim, 1) is compact (R is the scalar field). (Received February 28, 1972.)

*72T-B127. ANDRE de KORVIN and LAURENCE E. KUNES, Indiana State University, Terre Haute,

Indiana 47809. Linear operators on .e~. IT.

For the notations see "A topology associated with .eP spaces. I," Abstract72T-B85, these cJfoticei) 19(1972), A-319. Let C. be the a-ring generated by compact G !i subsets of the locally compact space T.

Theorem. Let pI oo and fiiq be finite on C.cr,f' The right continuity of fiiq is equivalent to (1) there exists a

A-445 sequence of open Baire sets Un such that IJfndml"" 0 uniformly for every sequence {fn} with Np(fn) l!5 1 and fn = 0 off Un. Theorem. Let U be a continuous and compact operator from .t~(~) into F with U << ~· Then the representative measure of U is c. a. (Received February 28, 1972.)

*72T-B128. ANDRE de KORVIN, Indiana State University, Terre Haute, Indiana 47809 and CHARLES

CHENEY and RICHARD A. ALO, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213. A Vitali-Hahn­

Saks theorem. II.

For the notations see "A Vitali-Hahn-Saks theorem. I", Abstract 72T-B84, these cJfoticti) 19(1972),

A-319. Let F be a collection of linear maps from M into N, then F is uniformly weakly completely additive

(u. w. c. a.) if for every commutative collection Pa of projections in M with inf PIX = 0, if x E H, then for every f > 0 there is an a 0 such that l(f(Pa)x,x)l < t for all fE F and a~ a 0 • Theorem. Let cpn be a sequence of continuous linear maps from M into N. Suppose that p0cpn is in M* for every cpn and every p in N*. Assume that lim cpn(P) = cp(P) exists for every projection P in M. Then {cpn} is u. w. c. a. If each cpn is positive then cp is positive, normal and completely additive. (Received February 28, 1972.)

*72T-B129. ROIDT J. PARIKH and MILTON N. PARNES, State University of New York at Buffalo,

Amherst, New York 14226. Relative probability and uniform sets.

In the following we identify the unit circle with the interval [ -i, i;;) with addition mod 1. ~(A) is the Lebesgue measure of A. p(A, B) is, intuitively, the probability of A relative to B. Theorem 1. There exists afunction p(A,B) definedforallsubsetsA,B of(-i;;,t) withvaluesin[0,1J suchthat (a) p(A,A)=1,

(b) p(A U B, C)= p(A, C) + p(B, C) if An B = t1, (c) p(A,B)• p(B, C)= p(A, C) if As;; B s;; C, (d) p(A+x, B+x) = p(A, B), (e) p(A, B) • ~(B) = ~(A) if A, B are measurable and A s;; B, (f) p(A, B)= 0 if A is finite and B is not.

There is no function p satisfying(a) -(f)andalso(g) p(A•x, B•x) = p(A,B), Definition. As;; L-t,iJ) is almost invariant if :i! (i ... 0 such that A + ri =A for all i. A is p-uniform if for all a, b, if -i < a < b < t then p((a, b),A) = b- a. A is uniform if A is p-uniform for all p satisfying (a)- (f) above. Theorem 2. Every almost invariant set is uniform. The methods of proof involve use of nonstandard analysis. Tbe results hold also for the real line. (Received February 28, 1972.)

*72T-B130. ROGER D. NUSSBAUM, Rutgers University, New Brunswick, New Jersey 08903. A fixed point theorem.

Theorem. Let G be a closed, convex subset of a Banach space X and assume the interior of G, G, is nonempty. Let f: G "" G be a continuous map. Suppose that there exists a compact set M c G with f(M) c M such that for any compact set A c G and any open neighborhood U of M there exists an integer n = n(U,A) such that fiD(A) c U for m i!l n. Finally, assume that there exists an open neighborhood V of M and a fixed integer

N i!t 1 such that rNI V is a k-set-contraction, k < 1 (e. g., a compact map) and flv is continuously Frechet differentiable. Then f has a fixed point. Corollary. Let G be a closed, bounded convex subset of a Banach space and assume G'I /J. Let f: G "" G be a continuous map and assume that rN is a compact map for some integer N i!ll. Suppose that M = nni!l1 cl(f(G)) c Gand that f is continuously Frechet differentiable on an open

A-446 neighborhood of M. Then f has a fixed point. The interest of this corollary is its relation to the following long- standing Conjecture. Let G be a closed, bounded convex subset of a Banach space and f: G .. G a continuous map. Assume that :rN is a compact map for some integer N l!! 1. Then f has a fixed point? It should be remarked that A. and v. Istratescu (Rev. Roumaine Math. Puree Appl. 16(1971), 1073-1076) have claimed this conjecture, but their proof appears incomplete. (Received February 28, 1972.)

72T-B131. PAUL CARLTON WOODS, Florida State University, Tallahassee, Florida 32306. Banach­

Steinhaus spaces. Preliminary report.

Let (E, T) be a locally convex Hausdorff space with topological dual E'. E is a Banach-Steinhaus space iff each O'(E', E) bounded subset of E' is {J(E', E) bounded. It is apparent that E is barrelled iff E is both infrabarrelled and a BS-space. Also, if E is sequentially complete, E is a BS-space. Proposition 1. The following are equivalent: (a) (E, T) is a BS-space; (b) (E, O'(E, E')) is a BS-space; (c) The barrel topology on E and T have the same bounded sets; (d) (E', O'(E', E)) is a BS-space; (e) Each pointwise bounded family (or sequence) of continuous linear mappings of E into an arbitrary locally convex Hausdorff space is uniformly bounded on bounded sets. Proposition 2. Let M be a subspace of a BS-space. If M is complemented or if M is of finite codimension, then M is a BS-space. Proposition 3. The locally convex direct sum of a family of

BS-spaces is a BS-space. Proposition 4. Let E be the strict inductive limit of a sequence E1 c E2 c ... of its subspaces with En closed in En+1' n = 1,2, •..• If each En is a BS-space, then E is a BS-space. Proposition 5. Being a BS-space is a strictly weaker property than property (S) [Levin and Saxon, Proc. Amer.

Math. Soc. 29(1971), 97J. Examples are given of a BS-space which is neither barrelled nor sequentially complete and of a dense subspace of a barrelled space which fails to be a BS-space. (Received February 28, 1972.)

*72T-B132. ROBERT W, CARROLL, University of Illinois, Urbana, Illinois 61801. Some growth theorems for differential equations with operator coefficients. Preliminary report

Let A be the generator of a locally equicontinuous group T(t) in a complete locally convex space E and 2 write cosh(At) = ~(T(t) + T(-t)). Solutions of w:+ (2m+ 1) coth(t)wf+ m(m+ 1)wm = A wm with wm(O) = w0 E D(A2) and wf(O) = 0 are constructed using an extension of a technique of Hersh (J. Differential Equations 8(1970),

570-579); the case when m ;!!; 0 is an integer and A 2 = t:. in SL(2, R)/80(2) is treated by Carroll and Silver (cf. C.R. Acad. Sci. Paris 273(1971), 979-981). A Bonine type formula connects the solutions with m > -t real to the m = -t case (telegraph equation), and for suitable w0 and E, growth properties result. For example, if 2 2 2 cosh(At) (Am w0) l!! 0 with m l!! -t real then wm is nondecreasing in t (Am = A - m(m+1)). Concrete examples are given to show when cosh (At)v0 ~ 0 follows from (realistic) hypotheses of the form A 2nv0 ~ 0 for 0 ;!5 n ;!5 N. Similar growth results hold for EPD equations (cf. A. Weinstein, Ann. Mat. Pura Appl. 43(1957), 325-340, and

R. Carroll, Ann. Mat. Pura Appl. 56(1961), 1-31); the relevant Bonine formula obtained in this case was also established for Banach spaces by Donaldson (Proc. Nat. Acad. Sci. U.S.A. 66(1970), 269-274) using a different method. Related growth theorems can also be obtained for a wide class of differential equations with operator

coefficients. (Received February 29, 1972.)

A-447 72T-B133. MORRIS MARDEN, University of Wisconsin, Milwaukee, Wisconsin 53201. Kakeya's

problem for polynomials.

The problem is that of finding ljl(n,p) such that, if f is any nth degree polynomial having p zeros

(2 ~ p ~ n) in lzl~ R, its derivative f' has at least p - 1 zeros in lzl~ Rljl(n, p). Exact values of ljl are known

for p = 2, n- 1 and n. Upper bounds, but not the smallest, have been found for IJl(n,p) by Marden and Biernacki.

Conjecturing that optimally the p zeros of f must be concentrated at just two points on lzl = R, this paper deduces that ljl(n,p) = [2- (p/n)i/~ with the smaller ljl(n,p) = (2np) -1 11+ (2n-p)p + (p2 -1)1/2L(2n -p)2 -1i/2l valid if

p is an odd integer 3 ~ p ~ n. (Received March 2, 1972.)

72T-B134. T. K. PUTTASWAMY, Ball State University, Muncie, Indiana 47306. Stokes' multipliers

for a certain ordinary differential equation of w. B. Ford's type. Preliminary report.

In this paper, the author has solved in the large the differential equation (1) ~J=O zj (aj+bjz+ cjz2)djy/dzj

=0. The variable z isregardedascomplexaslikewisetheconstants aj,bj,cj(j=0,1,2, •.. ,n) with ano/0 and

en of o. It is assumed that the equation (1) has no finite irregular singular point. The indicial equation at z = 0

isfoundtobe (2) ao+L:~~an-in].:~-1 (h-j)= o. Itisalsoassumedthattheroots hi' i=1,2,3, ••• ,n, of equation (2) are such that no two of them differ by an integer. This paper extends the results of the author's

earlier paper "Solution in the large of a certain third order ordinary differential equation of W. B. Ford's type," Abstract 691-34-8, these cNotie.eD 19(1972), A-127. (Received March 2, 1972.)

Applied Mathematics

*72T-C16. MARTIN SCHECHTER, Belfer Graduate School, Yeshiva University, New York, New York

10033. An improvement of the Schwinger bound.

For real valued functions on R3 set (u, v) = (47T) - 2 SJ u(x) v(x) lx -yj-2 d3xd3y and S(u) = (u, u) - supv~o {(v,v)-1 max[(u,v),o]}. If u(+)(x) = max[u(x),O], then S(u) ~ (u,u)- (u(+)' u(+{1 .max [ (u(+)' u),o]. Theorem. Let V(x), W(x) be functions such that V(x) ~ W(x) and ( lvl.lvl> < ro. Then

the Hamiltonian with potential W(x) has at most S(V) negative eigenvalues (counting multiplicities). This

improves estimates due to Schwinger CProc, Nat. Acad. Sci, U.s.A. 47(1961), 122-129] and Ghiradi-Rimini

[J. Math. Phys. 6(1965), 40-44]. (Received January 7, 1972.)

*72T-C17. C. J. EVERETT and E. D. CASHWELL, Los Alamos Scientific Laboratory, University of

California, Los Alamos, New Mexico 87544. Relativistic charged particle ballistics in electrostatic and

magnetic fields.

The relativistic trajectory of a charged particle with arbitrary initial position and velocity is completely

determined in the presence of (1) a uniform constant electrostatic field, (2) a uniform constant magnetic field,

and (3) two such fields superimposed, in parallel and in perpendicular orientation, the latter in the three cases

of relative magnitude of the fields. (Received February 3, 1972.)

A-448 '~<72T-C18. C. E, SIEWERT and ERNEST E. BURNISTON, North Carolina State University, Raleigh,

North Carolina 27607. An exact closed-form solution of Kepler's equation.

The methods of linear discontinuous boundary problems for sectionally analytic functions are used to develop an exact closed-form solution to Kepler's equation in both the elliptic and hyperbolic forms. (Received

February 14, 1972.)

*72T-C19. KAILASH CHANDRA, Division of Physical Sciences, Institute of Advanced Studies, Meerut

University, Meerut, u. P., India. A note on the paper of Luis N. Howard and A, S, Gupta.

The results of Luis N. Howard and A. S, Gupta (J. Fluid Mech. 14(Hi62), 463-476) have been generalized as listed below: (i) The region for unstable wave velocity vectors is given by [Cr -!(a+b)J2 + C~ l!'i t(a -b)2 + maximum of (0, - wr2) in the C -C. plane, as a generaliZation of the result on page 466, (ii) The region for r 1 unstable wave velocity vectors is given by [cr -!(a+ b)J2 + C~ ;§ tea -b)2 +maximum of (0, - >l'i) in the Cr - ci plane, as a generalization of the result on page 471. (iii) It is proved that the strong uniform axial magnetic field can stabilize the fluid flow, provided the condition v!- 4(V2 + (!J.H~/4'1Tp)) + >l'r2 ;:; 0 is satisfied everywhere in the domain of the flow. If it is not satisfied even then the system is stable for the wave number satisfying k2 ;;; max [(4/VXr 2)(V2 + (~J.H~/4'11 p)) - (>1' /V!) - 1/r2l as a generalization of the result on page 474.

(iv) The region for unstable wave velocity vectors is given by c; + C~ ;§ max (4(V2 + (!J.H~/4'1Tp)) - >¥r2 - v!l in the Cr - Ci plane, with reference to the case discussed on page 474. (Received February 14, 1972.)

(Author introduced by Vice-Chancellor J. N. Kapur.)

*72T-C20. KAILASH CHANDRA and LAKSHMINARAYAN, Division of Physical Sciences, Institute of

Advanced Studies, Meerut University, Meerut, U. P. , India. Thermal instability of a density stratified fluid layer.

The stability of a density stratified, incompressible and inviscid fluid layer, statically confined between two horizontal parallel planes (at Z = 0 and Z = d, where Z is the vertical coordinate) and heated or cooled uniformly from below, is investigated under linear theory by the normal mode technique. The resultant density

stratification assuming the principle of superposition is taken to be p = p0(e -ez -01{3Z), where p0 e -oz is the basic density of the fluid and f3 is the temperature gradient which is maintained. It is proved that the neutral

modes may exist, in contrast to Banerjee's statement (Indian J. Pure Appl. Math. 2(1971)). In case f3 is positive, the necessary and sufficient condition for the stability is that 01{3 + Oe -Od;;; o. When f3 is negative,

o is positive, and the inertial effects of the density variation are neglected, a sufficient condition for the stability of nonoscillatory modes is found to be Oe -Od + 01{3 ;;; maximum of (- rxf3, o3/2A. 2J where A. is the wave number

of the perturbation, Some bounds on the growth rate of unstable modes are obtained in different situations.

(Received February 14, 1972.) (Authors introduced by Vice-Chancellor J. N. Kapur.)

A-449 *72T-C21. R. K. RATHY and KAILASH CHANDRA, Division of Physical Sciences, Institute of Advanced

Studies, Meerut University, Meerut, u. P., India. Hydrodynamic stability of density stratified spiral flows with a radial gravitational force.

The stability of an incompressible, inviscid fluid with density P(r), velocity :!!. = (0, V(r), W(r)), between two fixed coaxial cylinders, in the presence of a radial gravitational force g towards the axis of the cylinders, is discussed under linear theory by the normal mode technique against nonaxisymmetric perturbations.

A sufficient condition for the stability is found to be X > 0 and Y2 - 4XZ < 0 everywhere in the domain of the flow, where X= (V2/r) dp/dr - g dp/dr - !~d/dr) (V /r)i r 2p, Z = (1/r) (d/dr) (PV2r 2) - gr2 dp/dr -

~(dW/dr) 2 r 2 p, and y = -(p/2r2) (dW/dr) (d/dr) (r3V). (Received February 14, 1972.) (Authors introduced by

Vice-Chancellor J. N. Kapur.)

*72T-C22, LOKENATH DEBNATH, East Carolina University, Greenville, North Carolina 27834, On an unsteady hydromagnetic channel flow.

This is a sequel to the author's recent work (Abstract 691-76-1, these cJioticei) 19(1972), A-241, and Bull. Calcutta Math, Soc. 62(1970), 173-182) where an investigation is made of an unsteady hydromagnetic boundary layer flow induced in an incompressible electrically conducting fluid between two infinite parallel disks in the presence of a uniform magnetic field by the elliptic harmonic oscillations of the disks. Taking into account the finite electrical conductivity of the fluid, the initial boundary value problem is solved by the Heaviside operational calculus combined with the theory of residues. The steady and the transient components of the velocity as well as the induced magnetic field are calculated explicitly. The transient effects are shown to die out exponentially in the limit t- ro, and the ultimate steady flow is established. Attention is given to the analysis of the structure of the associated hydromagnetic boundary layers adjacent to the disks. The skin friction, the rate of energy dissipation and the induced electric field have been calculated and their significance has been discussed. Several limiting cases of interest are recovered. (Received February 16, 1972. )

72T-C23, ELIAHU SHAMIR, IBM Research Center, Yorktown Heights, New York 10598. Relations between tape and time complexity via pushdown machines, Preliminary report.

Theorem 1. Every set accepted by a nondeterministic auxiliary pushdown Turing machine M (S. Cook,

J. Assoc. Comput. Mach. 18(1971), 4-18), with storage L(n) ~ Log n can be accepted by a deterministic

Turing machine with storage L(n)2 • This generalizes a result of Savitch. Proof. As Cook has shown, one can simulate M by performing T = 2cL(n) "convolution" operations on realizable pairs of configurations of M. In our simulation at stage i, one guesses a realizable pair, verifies it in T/2i convolutions using Log T -i blocks of memory of length cL(n) each, and records the verified pair in block i. Then to test if the recorded pairs in the first i blocks yield an accepting pair in T/2i convolutions, using the (now released) Log T -i blocks of memory, one proceeds to stage i + 1. The details of backtracking are standard, The total space required is cL(n)2. Corollary 1, If A is accepted by a deterministic Turing machine in time 2cL(n), then A is accepted by a deterministic Turing machine in storage L(n)2• Corollary 2, If A is accepted by a two way pushdown automata then A is in the deterministic complexity class (Log n)2• (Received February 18, 1972.)

A-450 72T-c24. J. L. BRENNER (University of Arizona), 10 Phillips Rd., Palo Alto, California 94303 and

JOHN de PILIJS, University of California, Riverside, California 92502. Partitioned matrices and Seidel convergence.

Without using spectral resolution, we give an elementary proof of convergence of Seidel iteration. The proof is based on the lemma (generalizing a lemma of P. Stein): If (A+A *)- B*(A +A*)B > 0, where B= -(P+ L) - 1R,

A = P + L (lower)+ R (upper), then Seidel iteration of AX= Y0 converges if and only if A+ A*> 0. This lemma has as corollaries not only the well-known results of E. Reich (Ann. Math. Statist.) and Stein (Math.

Comp.) but also applications to a matrix that can be far from symmetric, e.g. M = [Aij]~, where A21 = -Ai2• 1 All, A22 are invertible; All + Ah = A22 + A~ 2 ; and the proper values of Ai~ A11, Ai; A22 are in the interior of the unit disk. This article will be published in Numer. Math. (Received February 25, 1972,)

*72T-C25. ADOLPH SELZE.R, 51 Linden Street, New London, Connecticut 06320. A unified theory of gravitational potential and nuclear potential. Preliminary report.

In a previous paper, Yukawa's suggestion of a nuclear field was used to show the possibility of photon capture by applying suitable boundary conditions. This was achieved by the reduction of the formula for the pion field to a scalar potential. In the present paper, the idea is extended to the gravitational field theory of general relativity. It is shown that the difficulties connected with the Einsteinian field equations can be removed if one abandons the restrictions caused by Newton's constant of gravitation. The boundary conditions, which are taken from the classical theory of partial differential equations, are sufficient to connect the field of the nucleon with that of gravitation. (Received February 28, 1972.)

*72T-C26, ALDO DeLUCA and FAUSTO ADRIANOPOLI, Laboratorio di Cibernetica del C. N. R.,

80072 Arco Felice, Napoli, Italy. Some algebraic properties of measures of computational complexity,

Let cp"' Icpi !~ 0 be a Godel numbering of all partial computable functions and 'i>"' Iil>i !~ 0 a computational complexity measure (Blum, J. Assoc. Comput. Mach. 14(1967), 322) of cp. 'i> is a complexity measure of cp if

and only if, for any integers i and n, domain il>i =domain cpi and 'i>i(n) = ~yP(i, n,y) where P is a recursive

predicate. If 0 is the set of all complexity measures with respect to a given Godel numbering, it is possible to

prove that 0 is a distributive, but noncomplemented lattice with respect to the lattice operations V and A

associating to any pair of measures 'i> and ~ the elements 'i> V ~ and 'i> A ~ of 0 defined as ('i> V ~)i(n) = Maxl~i>i(n), :i.i(n)}, ('i> A ~)i(n) = Minl~i>i(n), ii(n)}. Furthermore 0 is an L-semigroup with respect to the binary operations (+) and (•) defined as ('i> + ~)i(n) = il>i(n) + ~i(n), ('i> • i)i(n) = 'i>i(n) • ~i(n). (Received February 28,

1972.) (Authors introduced by Dr. Giorgio M. Germano.)

A-451 Geometry

*72T-D9. BANG-YEN CHEN, Michigan State University, East Lansing, Michigan 48823, On the mean curvature of surfaces in euclidean 4-space. Preliminary report.

Theorem 1. Let M be a compact surface in E4 such that the Gauss curvature does not change its sign.

If the mean curvature H satisfies JM H2 dV ~ (2 + !T)!T, then M is homeomorphic to a sphere, where dV is the area element of M. Theorem 2. Let M be a compact surface in E4 with nonpositive Gauss curvature. Then

we have JMH2 dV;;;; 21r 2 • The equality sign holds when and only when M is a Clifford torus. These results

give a partial answer to a problem proposed by the author ~'On the total curvature of immersed manifolds. II:

Mean curvature and length of second fundamental form," Amer. J, Math. (to appear)] • (Received January 24,

1972.)

72T-D10. FLORINDA KATSUME MIYAOKA, Federal University of Parana, Curitiba, Brazil. Common tangents of two central conics. Preliminary report.

If two ellipses or two hyperbolas have parallel focal axis and have equal eccentricities, then they are

said to be of the same form (seeN. A, Childress, Math •. Mag. 38(1965), 147-149), The principal circle of a

central conic is the circle whose center is the center of the conic and contains the points in which the conic

intersects the focal axis. It is shown that the common tangents of two central conics of the same type and the same form meet in the centers of similitude of its principal circles (see Abstract 70T-D17, these c:J.kJtir.tiJ 17(1970), 670). (Received January 28, 1972,) (Author introduced by Professor Jayroe M. Cardoso.)

Logic and Foundations

72T-E33. BARUCH GERSHUNI, Ibn Gvirol Street 43, Tel Aviv, Israel. Some achievements of the simplified Elementary Theory of Totalities (ETT), II.

The set so obtained has the same elements as the original class, but is nevertheless another totality.

(I) There are totally 10 kinds of primitive totalities (while in the nonsimplified ETT there are 20): (1) abc ••• ;

(2)abc ... ;(3) alblcl ••• ; (4) a-b-c-... ; (5) a,b,c, ... ; (6) (abc .. .}; (7) [abc .. .}; (8) {alblcl ... };

(9) {a-b-c-... }; (10) {a, b, c, ... }. (II) All these kinds of totalities are different from one another. (III) The

above condition of von Neumann is modified and louds: only singulars are allowed to be elements of sets, while elements of classes may be also plurals. This is stated in order that the mathematical analysis not be unnecessarily hampered. Singulars may be not only sets, but also improper plurals, i.e. plurals not having

more than one element (classes fulfilling the same condition) and individuals, Individuals are defined as entities

which are considered as having no elements and no parts. They are the converse of totalities and are singulars. Totalities are entities which have elements (building-stones) or at least parts, and whose elements are separated

or connected with each other. The author considers however in all his publications hitherto only such totalities

which have elements (and therefore also parts). (Received November 22, 1971.)

A-452 *72T-E34. G. P, MONRO, "Laputa", Seaview Avenue, Femy Creek, Victoria 3786, Australia. The

strong amalgamation property for complete Boolean algebras.

If B, C are complete Boolean algebras (cBA's), we say B ~;; C if B is a regular subalgebra of C, i.e. B is a subalgebra of C and, if X is a subset of B, (VX)B = (VX)C. The strong amalgamation property is

the following. Given cBA 1s A, B, C such that A ~;; B, C and B n C = A there is a cBA D such that B, C ~;; D. We give a proof of this by the method of forcing. The proof adapts a result of Jensen's given on p, 76 of

T. J. Jech, "Lectures in set theory", Lecture Notes in Math., vol. 217, Springer-Verlag, Berlin and New York,

1971. (Received December 21, 1971.) (Author introduced by Professor John E. Shepherdson.)

*72T-E35. STEVEN K. THOMASON, Simon Fraser University, Burnaby 2, British Columbia, Canada.

A new representation of S5.

We consider first a modal propositional language having propositional constants (and no propositional

variables). We show that there is exactly one set H of formulas of that language satisfying certain syntactic

consistency and completeness conditions, and present a rather simple formal system in the language, whose

theses comprise the set H. We then show that the theses of S5 are characterized by the fact that all their

substitution instances (in the language with constants) are in H. (Received January 7, 1972.)

*72T-E36. WILLIAM BOOS, State University of New York at Buffalo, Amherst, New York 14226.

Infinitary compactness without strong inaccesibility.

Notation. c(X, A) , for w ~ A ~ X, means that for each !: ~;; LX A of power X, if each !: ' ~;; !: of power < x has a model, so does !: . c(X, x) + strong inaccessibility is usually called weak compactness •

.BWIJ is the algebra for adding jj Cohen reals. By extending unpublished ideas of Solovay and Kunen one can

prove the following Theorem. (i) Assume the G.C.H. holds and 1.1 is subtle; then for every 11,., JJ there are

!L cardinals x < !L such that [c(X, X)] Bwv = 1. (ii) If c(X, w1) and 2~ 0 = jj > ~~+a' then (X is a-indescribable)L. (iii) By contrast with (ii), c(X, W) is equivalent without use of strong inaccessibility to ni -indescribability, and therefore is preserved under adjunction of > X Cohen reals. (iv) One cannot in + general prove c(X, A) => c(X, A ) for any regular A < x . (i) and (ii) are special cases of a somewhat more

general result, best stated in terms of strong indescribability conditions on X in natural models R(T) and L('T)

for 'T ,., !L. These conditions evolved in an attempt to refine (i) and (ii) toward a common necessary and B sufficient condition for ([c(X,X)] W!L = 1)L. (Received January 10, 1972.)

72T-E37. EUGENE M. KLEINBERG, Massachusetts Institute of Technology, Cambridge, Massachnsetts

02139. An absoluteness theorem. Preliminary report.

ZF denotes Zermelo-Fraenkel set theory. A formula fi'(X) is said to be absolute if given any two

standard transitive models M and N of Z F having the same ordinals, and given any member !!:. of both M and

N, M I= fl'(!!:) iff N I= fl'(!!:). A sentence 1/J is said to be generic over ZF if given any standard transitive model

M of ZF there exists a partial ordering fJ in M such that any generic extension of M with respect to {} satisfies

A-453 1/J. Finally, a sentence a is said to be a standard consequence of a theory T (written T 1-' a ) if a is true in every standard model of T. Theorem. Suppose that cp (x) is absolute and that 1/.J is generic over ZF. Then

ZF + 1/.> ..... :R~ xtp(x) iff ZF .... , :R~ xcp(x). All of the above remains valid With ZF replaced by any sufficiently rich theory. (Received February 8, 1972.)

*72T-E38. ROIITT J. PARIKH, State University of New York at Buffalo, Amherst, New York 14226 and JOHN MAYBERRY, Bristol University, Bristol BS8 1 TW, England, D-structures and *-structures.

Preliminary report.

D-structures have been defined in a previous (Abstract 72T-E10, these c}/oticei) 19(1972), A-329) abstract of R. Parikh. *-structures have been defined by Ehrenfeucht et al. in an unpublished report. The folloWing results are stated forD-structures but a very similar argument gives analogous results for *-structures.

Theorem 1 (Skolem -Lowenheim Theorem). Let M = F, H be a D-structure. There exist countable F' !;;; F,

H' !;;; H such that for D E F' and A a closed formula of LJ.l* lnl, if M' = F', H' then M, D I= A iff M', D I= A. Theorem 2 (Existence of a proof theory). The set of all closed LJ.l formulae holding in all M, D is recursively enumerable. Theorem 3 (Compactness). Let F be any set of LJ.l sentences such that for any finite subset [A1 , ... ,An} ofF, there are M, D such that M, D I= A1 "· • .1' An. Then there is M, D such that ('!A E F) (M, D I= A). It should be mentioned that the results for *-structures have been obtained independently, and simultaneously, by J. Geiser and R. Maddux at Berkeley. (Received January 26, 1972.)

72T-E39. JUDITH M. HARPER, Purdue University, West Lafayette, Indiana 47907. Principles of cofinality and variations of Zorn's lemma. Preliminary report.

Let Z(Q,S) denote: if x is a nonempty Q-ordered set in which every S-ordered subset has an upper bound, then x has a maximal element; and let C(Q,S) denote: every nonempty Q-ordered set has a quasi-cofinal

S-ordered subset. A subset z of a partially ordered set x is quasi-cofinal if z has no strict upper bound in x.

Theorem. Z(Q,S) = C(Q,S) if Q is at least a partial order and S is a property on sets which holds for any singleton set. Proof. If x is Q-ordered and has a maximal element, m, then {m} is a quasi-cofinal S-subset.

If x has no maximal element, then by Z(Q,S), x has an S-subset with no upper bound, so it is clearly quasi­ cofinal. Conversely, if x is Q-ordered and has a quasi-cofinal S-ordered subset, an upper bound for this set is a maximal element. Q.E.D. The theorem shows that the concept of quasi-cofinality is an important one in relation to maximal principles and the axiom of choice (AC). For many Q and S one of the forms, but not the other, has been shown to be equivalent to AC. The theorem also shows that principles of cofinality are not a truly new category of equivalents of AC, as was suggested by U. Feigner (Compositio Math. 18(1967), 170-180).

(Received January 27, 1972.)

72T-E40. JUDITH M. HARPER and JEANE, RUBIN, Purdue University, West Lafayette, Indiana

47907. Variations of Zorn's lemma, Preliminary report.

Let Z(Q,S) denote: if x is a nonempty Q-set in which every S subset has an upper bound, then x has a maximal element, For Q substitute P: partially ordered; L: linearly ordered; D: directed; R: ramified;

A-454 F: forest; and T: tree. (For definitions see u. Felgner (Math. z. 111(1969), 221-232),) For S substitute W: well ordered; L, D, R, F, and T. Let (wx, I) denote the tree of well ordered subsets of a set under the initial segment relation. The statement M: (x) ((wx, I) has a maximal element) is equivalent to the well ordering theorem, Theorem 1. Each of the following implies M and hence is equivalent to the axiom of choice (AC): Z(Q ,S) for Q = P, R, F and T, S = W, L and D. Now let OE denote: every partial order can be extended to a linear order. Theorem 2. If OE is assumed, then the following are equivalent to AC: Z(Q,S) for Q = P, L,

D and R,S= F and T, Z(L,W) and Z(D,W). TheassumptionofOE isnecessaryintheproofsthat Z(L,S) for S = W, F and T, and Z(R,F) imply AC. Similar results have been obtained concerning class forms of variations of Zorn's lemma and the axiom of choice. However, in some cases the axiom of regularity is used, as well as a class form of OE. (Received January 31, 1972.)

*72T-E41. ROBERT TOWNSEND, Pennsylvania State University, University Park, Pennsylvania 16802.

The decidability of extensions of Th( Zn: n in N). Preliminary report.

If cp is a sentence of ring theory, the following is a procedure for deciding if cp is true in all Zn for n

in N. Let A = (Zq: q in Q•) where Q' is a finite set of prime powers no two of which are powers of the same

prime (so that rings of the form Z $ Z m f. Z +mare not encountered), and by Theorem 3.1 of Feferman and pn p pn Vaught, ("The first order properties of products of algebraic systems," Fund. Math. 47(1959), 57-103) the

validity of cp the product is equivalent to an intuitive statement S in the Boolean algebra P(Q'). By a method of

elimination of quantifiers for S, a canonical form is obtained in which Ax's procedure for Th(Z q : q any power of any prime) found in (Ann. of Math. (2) 88(1968), 239-271) can be used to decide cp. This method can be used to

prove the decidability of certain extensions of Th(Zn: n in N) of the form Th(Zr : r in P ~:: N). It can also be

shown that there are uncountably many undecidable extensions of Th(Zn: n inN) of the above type, (Received

January 27, 1972.)

72T-E42. LESLIE H. THARP, Rockefeller University, New York, New York 10021. The

characterization of monadic logic,

See Lindstrom's "On extension of elementary logic," Theoria 35(1969) for definitions. Let EM be the

usual elementary monadic logic. Let L be a monadic logic. Let L -Sw be the condition: If a countable set of

formulas has a model, then it has a countable model. (1) If EM ~:: L and L is compact and satisfies L -Sw,

then EM "' L. (2) D. A. Martin has shown that "compact" can be weakened to "compact for recursive sets."

(3) There is a monadic logic L properly extending EM which is decidable and satisfies L-S w. (4) There is

a compact monadic logic L properly extending EM such that: If M satisfies cp in L, then there is a finite

submodel J ~:: M (whose size is calculable from cp) such that J satisfies cp in L. (Hence L is decidable for

validity,) (5) Suppose EM~:: L and there is a function f into w such that if M satisfies cp in L, then M has

a submodel J of cardinality f(cp) such that if J ~:: J' ~:: M, then J' satisfies cp. Then EM"' L. (Received

February 7, 1972,) (Author introduced by Mr. John Riordan.)

A-455 72T-E43. HERMAN RUBIN and JEANE. RUBIN, Purdue University, West Lafayette, Indiana 47907.

A weakened form of Zorn's lemma and some of its consequences.

Let P and Q be the following propositions. P: For every set x of cp , if R is a transitive relation on x such that every R-antisymmetrically ordered subset has an R-upper bound, then x has an R-maximal element.

(u is an R-maximal element if ('ifv) (uRv- vRu).) Q: There is a choice function on every well-ordered nonempty set of nonempty sets. All theorems are in ZF without AC. Clearly Zorn's lemma implies P. We have shown

P- Q - (for each infinite cardinal m there is an ordinal 0! such that ~(m) == ~Ol+l)- (~ 0 ~ m for all infinite cardinals m). Also Q- (the union of a well-ordered set of well-orderable sets is well-orderable)- (the union of a countable set of countable sets is countable). It is unknown to us whether P or Q implies AC. (Received

February 11, 1972.)

72T-E44. MATATJAHU RUBIN, Hebrew University, Jerusalem, Israel. Theories of linear order.

Preliminary report.

T denotes a fixed complete theory of linear order. !U == (A,<) is a linearly ordered set. n < w.

S (T) is the set of complete types with n variables over T. Theorem 1. Either T is ~ ~ 0 categorical or T n ~ has 2 ° nonisomorphic countable models. Theorem 2. \\S1(T)\\;;; ~ 0 => 'In \\Sn(T)\\ ~ 1:{0• Theorem 3, If T has an infinite model and x is regular, then T has a x-saturated model of power 11. iff 11. ;;;; x2i, !U k-saturated and x singular=> !U is x+-saturated, Theorem4. (!U~\.<11. are x-satur.atedelementarilyequivalentmodels of power ~ 11. . Then there is a x -saturated model !U of power ~ 11. such that 'I; !U; < !U • Q , Z, ll. denote the rationals, the integers and the ordered set with a single element respectively. Let M be the smallest set

containing 1 and closed under the following operations: (I) (!U, 1!1) - !U + Ill; (TI) !U - !U • Z; (Ill) (!U1, ••• , !Un) - ~ EQ!Ur where [r\!Ur = !U.} = Q. is dense in Q and U?- Q. = Q, Theorem 5. lls (T)II < ~O iff there is r 1 1 1=1 1 1 !U E M such that !ll I= T. Theorem 6. lls1 (T) II < ~ 0 => T is finitely axiomatizable. Theorem 7.

Vn II (T\IIs1(T)II ~ n}\\ < 1:{ 0, (Received February 14, 1972.) (Author introduced by Professor Michael 0. Rabin.)

*72T-E45. CHARLES c. PINTER, University of California, Berkeley, California 94720. Cylindric and

substitution algebras.

A substitution algebra is an algebra !ll =(A,+,•,-,0,1,S~)x,>.

designate the dimension set of x. Consider the following statements: ('IT1) for each x E A and 1\

!yErgX:y;;:xl hasaleastelement;('IT2) forall K,).

if ll of X,>., where c is defined by (B), below; ( 'IT4) if x E rg 1t, y E rg A and x §; y, then there is some z E rg X ll n rg A such that x §; z §; y; ('IT5) for all x, A< a, AdxA. i a. The following are definitions: (A) S~x = x if x = >.,

and S~x= cx(x•dx>..> if 1!.of A; (B) cltx=min!yErg 1\:y;;: xl; (C) d11). =min!x:S~x= 1\. Theorem. If m is a cylindric algebra and S~ is defined by (A), then !U is a substitution algebra with ('IT1)--('IT5). Conversely, if m

is a substitution algebra, and cK and dltA are defined by (B)--(C), then ~I is a cylindric algebra provided that

A-456 either ('11'1)--('11'3) hold, or ('11'1), ('11'2), ('11'4) and ('11'5) hold; furthermore, if Ax I o: for all x E A, then ti is a cylindric algebra provided only that ('11'1)--('11'2) hold. (Received February 28, 1972.)

72T-E46. JOHN MYHILL, School of Mathematics, University of Leeds, Leeds LS2 9JT, England.

Embedding of classical theories in intuitionistic ones.

The Glidel translation A ... A* of classical formulas into intuitionistic ones has the property that if A is provable in the classical propositional calculus (reap. classical arithmetic), then A* is provable in the corresponding intuitionistic system and conversely. This property fails for the predicate calculus. If however we modify his translation by setting ((1fx)A(x))* = (1fx)-,-, A(x))* this defect is overcome and we have the Theorem.

Let T be an intuitionistic theory such that (1) Tl- A* for each axiom A of T; (2) if A is atomic and Tl--,-, A, then T 1- A. Then A is provable in T + excluded middle iff A* is provable in T. (For the Glidel translation (2) is replaced by the stronger condition: if A is atomic, then T 1-,-, A ... A.) The theorem gives embedding results for theories to which Godel' s original translation is not applicable--e. g. type-theory based on arithmetic, but without extensionality; also the predicate milculus itself. (Received February 28, 1972.)

*72T-E47. A. EHRENFEUCHT, SIEMION FAJTLOWICZ and JAN MYCIELSKI, University of Colorado,

Boulder, Colorado 80302. Slender algebras.

For every cardinal n we denote by mn the first measurable cardinal larger than n. Let ti be an algebra. We say that a subset x 0 s;; X is a support of a homomorphism H of tiX into ti if for every f, g E tiX whenever flx0 = glx0 then H(f) = H(g). ti is called slender!!! the power m if every homomorphism of tim into ~ ij has a finite support. Theorem. If ti is slender in the power Iijl 0 then for every k < m1 til, ij is slender in the power k. (Received February 28, 1972.)

72T-E48. WITHDRAWN.

72T-E49. ARTHUR L. RUBIN and JEAN E. RUBIN, Purdue University, West Lafayette, Indiana 47907.

Dedekind finite cardinals. Preliminary report.

Let A be the set of infinite Dedekind finite cardinals. Several years ago A. Tarski conjectured that = ~0 = there was a model of ZF- AC such that A= c = 2 . Tarski has shown that A I cp ... c;:;; A (Abstract 65T-432, these c/{oticei) 12(1965), 719). While we have not proved the Tarski conjecture, we do have the following

results: mE A & m can be linearly ordered ... c • e(m);:;; A, where e(m) is the cardinal number of the set of all

finite subsets of m. Consequently, it follows from a result of E. Ellentuck (Z. Math. Logik Grundlagen Math.

14(1968), 143-158, Theorem 4) that there is a Fraenkel-Mostowski model of ZF- AC in which A= c • e(m),

where m is the cardinal number of the set of urelements. In the standard Cohen model of ZF - AC, if m is the

cardinal number of the set of generic subsets of w, c • e(m) = c and A is the number of countably infinite partitions of e(m). It follows that in this model c;:;; ~;:;; 2e(m). We conjecture that 2e(m);:;; * c, but have not yet

been able to prove it. (Received February 29, 1972.)

A-457 72T-E50. ROBERT I. SOARE, University of Illinois at Chicago Circle, Chicago, Illinois 60680.

Automorphisms of the lattice of recursively enumerable sets. I: Maximal sets.

Let tJ be the lattice of recursively enumerable (r. e.) sets under inclusion (as explained in Rogers,

"Theory of recursive functions and effective computability," Chapter 12). Let A~;;;* B denote that A- B is finite. For any r.e. set A let C.(A) = \W:W r.e. and wr;;,* A or W *;

D. A. Martin, and negatively answers the question raised by A. H. Lachlan of whether there are any nontrivial elementary classes of maximal sets ("On the lattice of recursively enumerable sets, "Trans. Amer. Math. Soc.

130(1968), 36). Corollary 2. Turing degree is not invariant under automorphisms of 8. (Corollary 2 also has a more direct proof.) (Received February 29, 1972.)

Statistics and Probability

72T-F5. DANIELRICHARDSON, PolytechnicoftheSouthBank, LondonS, E,1, England. Random

growth in a tessellation. Preliminary report.

Let n dimensional space S be divided into cells by a regular tessellation. Assume that at time 0 the

cell at the origin is black and all other cells are white, Blackness spreads by the following rule: If a white cell

has a black neighbor at time t, then it becomes black at time t + 1 with probability p. Let S(t) be the black

shape at time t. Theorem. There is a norm N on S so that for any E: > 0, the N ball of radius (1- E:)t is p p contained in S(t) and S(t) is contained in the Np ball of radius (1 + E:)t with probability tending to 1 as t tends to infinity. A similiar result holds for a family of related growth processes, including the k = oo case of the Williams Bjerknes model for the spread of an abnormal clone· in the basal layer of the epithelium. This theorem

confirms a conjecture of Murray Eden. (Received February 24, 1972.) (Author introduced by Dr. David

B. Singmaster.)

A-458 Topology

*72T-G51. GEORGE MICHAEL REED, Ohio University, Athens, Ohio 45701. On metrizability of complete Moore spaces.

R. W. Heath (Colloq. Math. 12(1964)) showed that each strongly complete, separable Moore space is metrizable. In a recent paper (Abstract 689-G3, theseGM;aa.D 18(1971), 1064), the author defined strong star-screenability, a property shared by separable and metrizable spaces, and conjectured that each strongly star-screenable, strongly complete Moore space is metrizable. D. R. Traylor has defined a sequentially complete Moore space to be one having a development (G1, G2, •.• ) such that if M1 , M2, •.• is a monotonic decreasing sequence of closed sets where for each i, Mi c st(pi' Gi) for some point pi' then n Mi ~ ~. Lemma 1. Each sequentially complete Moore space is strongly complete. Lemma 2. Each Moore space which

admits a continuous, strongly complete semimetric is sequentially complete. Theorem 1. Each normal,

strongly star-screenable, sequentially complete Moore space is metrizable. Theorem 2. Each strongly star­

screenable, sequentially complete Moore space with the three link property is metrizable. Theorem 3. Each

strongly star-screenable Moore space which admits a continuous, strongly complete semimetric is metrizable.

It is also noted that Theorem 3 does not hold if either strong star-screenability or strong completeness is

removed from the hypothesis. (Received December 22, 1971.)

*72T-G52. LOUIS M. FRIEDLER, University of Alberta, Edmonton, Alberta, Canada. Properties of

the elementary proximity. Preliminary report.

Let (X, 6) be a separated proximity space, ~(6) normal and 6 the elementary proximity 60, where

A60B iff A n B ~ ~. (1) o is metrizable iff there is a sequence of locally finite closed covers \Fi\~ 1 such that

whenever A = B, there is some N satisfying A ~ St(A, F N) ~ B. This extends the well-known metrization theorem of Morita. The theorem is used to prove a result for proximity spaces similar to the Hanai-Morita-

Stone Theorem. (2) Given a collection ((Xa, 6 a>l of proximity spaces, the elementary proximity on the product

is defined in terms of this collection. (3) Given a collection ((X a, 6a)J of Wallace separation spaces, their

product is defined. (Received December 27, 1971.)

*72T-G53. JOHN WILLIAM GREEN, University of Oklahoma, Norman, Oklahoma 73069. Filters and

C- and C*-embeddings.

In an earlier paper of the author (Abstract 691-54-9, these GM;ticei) 19(1972), A-198) a characterization

of the Stone-Cech compactification was used to characterize the C- and C*-embedded subsets of Tychonoff

spaces. Here the notion of generalized boundary of A. D. Myskis is used to show that all the results in the

earlier paper, except Theorem 6, hold in arbitrary spaces. A number of related results are obtained and the

well-known tangent disk space is investigated in regard to these results. (Received January 24, 1972.)

A-459 *72T-G54. HAROLD W. MARTIN, 5137 Portage Drive, Vermilion, Ohio 44089. Perfect coherent maps of symmetrizable spaces. Preliminary report.

A surjection f: X _, Y is coherent if X is symmetrizable via a symmetric d such that whenever d(an,bn) _, 0 and f(an) _, y, then f(bn) _, y. Every regular map of A. v. Arhangel'skil is coherent. It is known that the perfect image of a metrizable space is metrizable. As for perfect maps of symmetrizable spaces, the following can be shown: Theorem 1. If f: X _, Y is a perfect map from a symmetrizable space X onto a space Y, then Y is metrizable iff f is coherent. Using the technique of symmetries, a new proof may be given of the following theorem of A, V. Arhangel'ski'i: Theorem 2. If f:X _, Y is a perfect map of a symmetrizable

Hausdorff space X onto a metrizable space Y, then X is metrizable. (Received January 28, 1972.) (Author introduced by Professor Peter W. Harley ill,)

*72T-G55. CHARLES DANA FEUSTEL, Virginia Polytechnic Institute and State University, Blacksburg,

Virginia 24061. A generalization of Kneser's conjecture. Preliminary report.

Let F be a closed connected surface. Let M be a closed, connected 3-manifold such that 1T2(M) = o.

It is proved that if 1T1 (M) is the free product of groups A1 and A2 with amalgamation across w1 (F), there is an embedding of F in M realizing the free product with amalgamation. A somewhat more complicated theorem is also proved in case M has boundary. (Received February 2, 1972.)

*72T-G56. JOSEPH R. MORRIS, Virginia Commonwealth University, Richmond, Virginia 23220,

Essential maps between n-spheres.

An application of the homotopy extension property gives a generalization of the intermediate value theorem. The generalization has the same hypothesis and is a corollary to the following theorem which is proven by induction. Theorem. Let Sn and :En ben-spheres, These two n-spheres are triangulated as follows: n+l ) . n+l n n+l n Let cr = (x0, ••• ,xn+l be a standard (n+l)-slmplex and h1 : ocr _, S and k1 : ocr _, I: be the homeomorphisms which determine the triangulations so that h1 (xi), for i = 0, ••• , n + 1, is a vertex on Sn and k1 (xi), for i = 0, .•• , n, is a vertex on r;n. Let f: Sn _, :En be a vertex map which satisfies the following condition: for each j-simplex (h1 (x10), .••• ~ (xij)) on Sn, f((h1 (x10), ••• ,h1 (xij))) c (k1 (x10), ••• ,k1 (xij)) where j = 0, .•. ,n. Then f is an essential map. (Received February 3, 1972.)

72T-G57. CHANDRA MOHAN PAREEK, University of Saskatchewan, Regina, Saskatchewan, Canada.

Product spaces and paracompactness. Preliminary report.

In this note we prove the following results: (1) Let f be a pseudo-open mapping of a topological space

X onto a paracompact space Y satisfying the following conditions: (i) for each y in Y, e1y is normal; (ii) for each y in Y and C a closed subset of e1y, if C c U where U is open in X, then there is a V open in X such that C c V c cl(V) c U. Then X is a normal space. (2) Let f be a pseudo-open mapping of a topological

space X onto a paracompact space Y satisfying the following conditions: (i) for each y in Y, f-ly is

paracompact; (ii) for any locally finite family of subsets (F 0! Ia E A} of e1y for y in y there exists a locally

A-460 finite open collection of subsets {Gala E A} of X such that Fa c Ga for a in A. Then X is paracompact, Using above results various product theorems have been proved, (Received February 3, 1972.)

*72T-G58. RAYMOND YEN-TIN WONG, University of California, Santa Barbara, California 93106.

Involutions on the Hilbert space,

Two homeomorphisms f, g :X .... X are said to be conjugate if there is a third homeomorphism h :X ....

X such that h • f = g • h. Let S denote the unit sphere of an (infinite-dimensional) Hilbert space, Theorem.

Every fixed point free involution on S is conjugate to the antipodal map, If S is separable, the above theorem also holds in the Cw -smooth category. An interesting consequence is the following corollary. Corollary, There is a homeomorphism h: L2 \(o) .... :Rw\(o} satisfying h(x) + h(-x) = o, Theorem. Let a, fj be fixed point free involutions on S. Then the orbit spaces S/a and S/fj are homeomorphic. We actually prove these theorems in a much more general setting than Hilbert spheres. Furthermore, it is possible to generalize all these results to periodic homeomorphisms by classifying all infinite-dimensional (Hilbert) lens spaces. (Received

February 7, 197 2.)

72T-G59. LUDVIK JANOS, University of Florence, Firenze, Italy. A characterization of topological completeness,

Theorem. If (X, P) is a metric space and ~J a nonconverging Cauchy sequence on X, then there exists a metric P* on X topologically eqnivalent to p for which the sequence (xn) is no longer Cauchy, and moreover there is no sequence on X which is p*-Canchy without being p-Cauchy, Let X be a metrizable topological space and M(X) the set of all metrics on X generating the topology of X. Introducing the quasi-order relation;§ on M(X) defined by: p1 ;§ p2 iff aey p 2-Caucby sequence is also p1-caucby, then the above result implies that a metrizable topological space is topologically complete if and only if M(X) has a maximal element,

(Received February 7, 1972,)

72T-G60, PETER FLETCHER, Virginia Polytechnic Institute and State University, Blacksburg, Virginia

24061 and WILLIAM F, LINDGREN, Slippery Rock State College, Slippery Rock, Pennsylvania 16057.

Orthocompact and quasi-metric spaces. Preliminary report.

A Q-collection of a space (X, '1') is a collection C- c 'I' such that for each x E X, n{c E C-: x E C) E T [Trans. Amer. Math, Soc. 123(1966), 275-309], A space is (orthocompact) a-orthocompact provided that every open cover has a refinement which (is a Q-cover) is the union of countably many Q-collections. Theorem. A countably metacompact a-orthocompact space is orthocompact. Corollary. Let (X, T) be a Moore space, The following conditions aTe equivalent: (a) (X, T) admits a nonarcbimedian quasi-metric, (b) (X, T) admits a strong

nonarchimedian quasi-metric 0 (i.e. To c 'To -1), (c) (X, '1') is orthocompact. A space (X, T) is~ Cech complete proVided there is a countable collection of open covers such that every open Cauchy filter converges

[Czechoslovak Math, J. 13(1963), 127-138]. Theorem. Every Tychonoff space which is either strongly Cech

complete or which admits a compatible complete quasi-metric is Cech complete. Theorem. An orthocompact

(metacompact, paracompact) space (X, '1') is strongly Cech complete if and only if (X, T) admits a complete

A-461 nonarchimedian quasi-metric (and has a 0'-point finite base, 0'-locally finite base). Corollary. A space admits a complete metric if and only if it is paracompact and strongly Cech complete. Theorem. In a completely regular

Moore space strongly Cech completeness, completeness and Cech completeness are equivalent. Every completely regular metacompact strongly Cech complete space is a complete Moore space. (Received February 7, 1972.)

*72T-G61. BRIAN J. SMITH, Florida State University, Tallahassee, Florida 32306. More on non-Euclidean factors of En. Preliminary report.

Using certain shrinking isotopes constructed by Andrews and Curtis, Kwun proved the Theorem. If a c Em and fJ c En are arcs, then Em/a X En/fJ"" Em+n. This theorem can be generalized to prove the following: Theorem 1. Suppose that a c Em and fJ c En are compact, UV00 continua such that a X E1 and fJ X E 1 are shrinkable. Then Em/a X En/fJ =:. Em+n. AnideaintheseproofsisabstractedinTheorem2. Let A = (a J and B = [{J) be upper-semicontinuous collections of continua such that (1) A* c Em and B* c En are compact, and (2) A X E 1, B x E1 , and A x B are shrinkable collections. Then the collection {A x B}

U (Ax (En -B)} U ((Em -A) x B} is a shrinkable collection. Theorems 1 and 2 may be used to expand the class of factorizations of Euclidean space En =< X x Y where neither X nor Y is Euclidean. Such applications apply to nearly all known examples in which En/G x E1 "" En+1• (Received February 9, 1972.)

*72T-G62. KENNETH R. VAN DOREN, Auburn University, Auburn, Alabama 36830. Closed, continuous

images of complete metric spaces. Preliminary report.

The author has previously shown that a closed, continuous image of a metric space need not contain a

dense metrizable subspace. It is shown herein that every closed, continuous image of a complete metric space

contains a dense subspace which is metrizable in a complete manner. It follows as a corollary to this result that

the Baire category theorem is valid for closed, continuous images of complete metric spaces. It is also shown

that a closed, continuous image of a complete metric space need not be the union of countably many closed,

metrizable subspaces. (Received February 14, 1972.)

*72T-G63. ANTONIO 0. FARIAS, Eastern Michigan University, Ypsilanti, Michigan 48197. Immersions

of the circle and extensions to orientation -preserving mappings.

The author extends his results about immersions of s1 and extendible mappings into R2 [Bull. Amer.

Math. Soc. 77(1971), 422-424] to the case when the target is an oriented 2-manifold, M, with orientation w.

All manifolds and mappings are C00 • For definitions see the paper above. It is proved that the set of positively

extendible mappings is open and the set of normal mappings is C00 -structurally stable in the c1 topology of

Q = C00(S1, M). As a consequence, a normal mapping is extendible iff it is positively extendible. Let A be a vector field on M divergence-free relative to w, ~(A, z, t} its flow and 01 a nonnegative function on s1; define

a function, T, from -Q into itself by (Tf) (B)= ~(A,f(B},a(B)w[A"f(B), f'(B)J; aT-operator is any finite

composition of such functions on Q. A degenerate mapping is one that can be factored through R1 (the complement

of its image is connected) and a T-mapping is the image of a degenerate mapping by a T-operator (it can be

A-462 obtained from a degenerate mapping by a finite number of displacements along integral curves of vector fields on M). Theorem. Every T-mapping is extendible and every positively extendible mapping is aT-mapping; as a consequence, a normal mapping is extendible iff it is aT-mapping. (Received February 11, 1972.)

*72T-G64. H. E. WINKELNKEMPER, Institute for Advanced Study, Princeton, New Jersey 08540. Manifolds as open books. Preliminary report.

Let V be a manifold with OV ~ ~ and h: V .... V a diffeomorphism such that hI o V = Id. Form the mapping torus Vh which has oVh = S1 X oV; for each x E oV, identify (t,x), t E s\ to a point obtaining a closed manifold which we call an "open book." The fibers of the mapping torus are the "pages" and the image of s1 x oV under the identification is a codimension 2 closed manifold diffeomorphic to oV, called the "binding."

Theorem, Let n > 6 and ~ be simply connected. (1) If n i 0 (mod 4), M is an open book. (2) (a) If n = 0 (mod 4) and M is of type I, M is an open book if and only if the index 'T(M) = 0; (b) If n = 0 (mod 4) and if M is of type II, M is an open book if T = 0 and if the Wu class Vn/2 = o. Corollary 1. In every dimension n there exists a closed simply connected manifold Mn such that any self-diffeomorphism which is the identity outside of an n-ball is isotopic to the identity. Corollary 2. A manifold ~ (n> 6) is cobordant in 0 to a manifold fibered over s2 if, and only if, T(M) = o. our construction of the cobordism shows that if M is a spin manifold the cobordism and the fibered manifold are also spin. All the above hold in the PL case.

(Received February 15, 1972.)

*72T-G65. GERHARD X. RITTER, University of Florida, Gainesville, Florida 32601. Free actions 3 of Z 2k on S • Preliminary report.

3 It has been shown that for k = 1, 2, 3 or 4, every free action of Z2 k on s is topologically equivalent to an orthogonal transformation. In this paper we prove the following generalization: Theorem. Every free action of z2k, k a positive integer, is equivalent to an orthogonal transformation, (Received February 15, 1972.)

*72T-G66. ELDON J. VOUGHT, Chico State College, Chico, California 95926 and University of

Oklahoma, Norman, Oklahoma 73069, Monotone decompositions of continua into arcs and simple closed curves,

In [Rozprawy Mat. 50(1966), 1-74], Thomas defined a compact, metric continuum M that is irreducible between two of its points to be~ if M has a monotone, u.s. c. decomposition whose quotient space is an arc.

Extend the definition so that a continuum M that Is separated by no subcontlnuum is~ If M has a monotone, u.s. c. decomposition whose quotient space is a simple closed curve (B. c. c.). A collection {G} of subsets of

M is hi-saturated if for each G E (G) and p ¢ G there exist G1, G2 E {G) such that G1 U G2 separates G from p in M, Theorem. A compact, metric continuum that is separated by no subcontinuum is type A if and only if there exists a hi-saturated collection (G) each of whose elements is a continuum with void interior. It is also shown that the elements Q in the decomposition with void interiors are of the form T(T(x)), x E Q, where

T is the aposyndetic set function. The structure of the elements with nonvoid interiors is described and the methods are used to characterize continua of type A that are irreducible between two points. (Received February

16, 1972.) A-463 *72T-G67. RONALD C. FREIWALD, Washington University, St. Louis, Missouri 63130. Cardinalities of metric completions.

lA! denotes the cardinality of A. Suppose X is a metrizable space of infinite weight k and that X is a 1:{0 1:{0 completion of X in some compatible metric. lx I can only be k or k . For k > k, we discuss the possible values of I'X I for different compatible metrics on X. Since I'XI must be kl:{ 0 if lx I > k, we also assume that 1:{ lxl = k. Theorem. X has a completion of cardinality k 0 iff X contains a closed discrete subset of cardinality p, where p is the least infinite cardinal such that p ~ 0 k~ 0 Theorem. X has a completion of cardinality k iff every nonempty set in X contains a nonempty (relatively) open set of cardinality < p, where ~ ~ p is the least infinite cardinal such that p 0 k 0• These theorems also obviously give characterizations of ~ those spaces X which have only completions of cardinality k or only completions of cardinality k 0• In the ~0 latter case, the theorem can be reformulated roughly as follows: X has every completion of cardinality k iff

K (X) )il, where K (X) is a certain transfinitely defined "kernel" of X, reducing, in the case k ~ • to the p -1 p = 0 ordinary perfect kernel of X. (Received February 17, 1972.)

*72T-G68. EDWARD C. TURNER, State University of New York, Albany, New York 12222. Rotational

symmetry: basic properties and application to knot manifolds. Preliminary report.

A simply connected manifold is rotationally symmetric if 'IT0(Diff(M))""' 'IT0(Diff(M,Dn)). This property was studied by Levine for homotopy spheres where it was shown to be an invariant of framed cobordism. An

obstruction to carrying rotational symmetry through a surgery is described and used to decide which manifolds of the form Sk x D.t Uf Sk x D.t (knot manifolds) are symmetric: most are, nonsymmetry being a very unstable phenomenon. A counterexample has yet to be found to the framed cobordism invariance of rotational symmetry.

In another vein: all nonspin manifolds are rotationally symmetric. (Received February 18, 1972.)

72T-G69. WITHDRAWN.

*72T-G70. JOHN SAWKA, Harvey Mudd College, Claremont, California 91711. A note on almost-

countably paracompact spaces.

M. K. Singal and Shashi Prabha Arya [Math. Ann. 181(1969), 119-133] define a topological space X

to be almost-countably paracompact if and only if for each countable open cover a of X there exists a locally

finite family B of open subsets of X such that each element of B is contained in some element a and

LB: BE B} is a cover of X. Subsequently, M. K. Singal and Asha Rani [Froc. Japan Acad. 43(1967), 856-857]

gave several characterizations of almost-countably paracompact spaces. In this note it is shown that, in fact,

every topological space is almost-countably paracompact. (Received February 21, 1972.) (Author introduced

by Professor John Greever.)

A-464 72T-G71. T. G. RAGHAVAN, Thiagarajar College of Engineering, Madurai-15, India. Bitopological

P-spaces.

If (X, 7'1 , T2) is a b. t. s. then we call the b, t. s. a bitopological P-space if and only if 7'1 , 7'2 are

P-spaces. If (X, 7'1 , 7'2) is a pairwise regular (P)-space, then (X, 7'1 , 7'2) is pairwise completely regular. (Received February 22, 1972.) (Author introduced by Professor M. Rajagopalan.)

72T-G72. CHRJSTOPHER F. COX, University of illinois, Urbana, illinois 61801. Two questions of

Borsuk concerning movability and fundamental retractions. Preliminary report.

For definitions of the concepts involved see the references below. Theorem 1. A movable component of a compactum X is a fundamental retract of X. This answers affirmatively a question of Borsuk ("On movable compacta", Fund. Math. 66(1969/70), 137-146, Problem 6.1). Theorem 2. There exists a decreasing sequence of compacta (xi li;;; 1} such that Xi+1 is a retract of Xi for every index i but X= nxi is not a fundamental retract of (in fact not fundamentally dominated by) x 1 • This provides a negative answer to another Borsuk problem ("Fundamental retracts and extensions of fundamental sequences", Fund. Math. 64(1969), 55-85,

Problem 9.7). (Received February 22, 1972.)

*72T-G73. R. CHRlSTOPHER LACHER, Florida State University, Tallahassee, Florida 32306.

Mappings between odd-dimensional manifolds. Preliminary report.

Let f: M- N be a surjective mapping between closed (2k+1)-dimensional manifolds. Theorem 1. If each f-1(Y) is strongly acyclic in dimensions ~ k- 1 and f is nonalternating in dimension k then f is strongly acyclic in all dimensions. Theorem 2. If each f-1(Y) is strongly acyclic in dimensions ~ k - 1 and the (Cech)

Euler characteristic of f-\y) is one for each y then {yEN lf-1 (Y) is not strongly acyclic in all dimensions} is -1 k-1 -1 a finite set. Theorem 3. If each f (Y) has property UV and the Euler characteristic of f (Y) is one for each y then {y E N IC1(y) is not cellular in M} is a finite set. Note that there are no restrictions on k in any of the above. Theorem 1 is a generalization of an obvious fact about functions of a real variable. Theorem 3 is an analogue of the result that a uvk-1 map between (2k)-dimensional manifolds is cellular except at a finite set (cf. Michigan Math. J. 17(1970), §7). (Received February 24, 1972.)

*72T-G74. VINCENT W. GIAMBALVO, University of Connecticut, Storrs, Connecticut 06268.

Tangential pin cobordism.

Pin is the double cover of 0 whose identity component is Spin. Anderson, Brown, and Peterson

[Comm. Math. Helv. 44(1969), 462-468] computed the cobordism group O~in of manifolds with a Pin structure on the stable normal bundle. In this note the group O~in' of manifolds with a Pin structure on the stable normal bundle is computed, and several corollaries are given. Corollary 1. ~in' ® Z 0 for p odd, p = pin' pin' pin pin' 0* ® Q = 0 and 20k = 0 if k ¢ 0 mod 4. (Note. 20k = o if k F- 2 mod 4.) Corollary 2. Image 0*

... !ll* is the set of those classes all of whose Stiefel Whitney numbers divisible by w2 + wi vanish. pin' S i Corollary 3. The subgroup of 0* spanned by elements of order greater than 2 is generated over 0/ n by [:DI.P4k]}. The same is true for O~in with [Jkp4k] replaced by [:$P4k+~. (Received February 24, 1972.)

A-465 *72T-G75. NEIL B. ffiNDMAN, California State College, Los Angeles, California 90032. On the natural map from {3(N X N) to f3N X f3N. Preliminary report.

This paper deals with the size of the preimages of points in {3N X {3N under the continuous extension,

T, of the identity map on N X N. It is easily seen that if (p,q) E {3N x {3N and either p or q is in N then IT- 1(p,q)l = 1. The following summarizes the author's .knowledge of I r-\p,q)l where p and q are in {3N\N. (1) There exists p in {3N\N such that I r-\p,q)l = 3. This p is necessarily aP-point. (2) There exist p and q in {3N\N such that lr-1(p,q}l = 2. These are necessarily distinct P-points. (3) There exists aP-point p of {3N\N such that 1-r-\p,p)l = 2c. (4) ThereexistdistinctP-points p and q of {3N\N suchthat

1-r-\p,q)l = 2c. (5) There exists a point p of {3N\N such that lr-1(p,q)l = 2c for every q in {3N\N. The proofs of (1), (2), (3), and (4) use the continuum hypothesis. The author has recently learned that results (1) and (2) follow from more general results of A. Blass. (Received February 28, 1972.)

*72T-G76. B. J. BALL and RICHARD B. SHER, University of Georgia, Athens, Georgia 30601.

Extending cell-like maps on manifolds.

Let X be a closed subset of a manifold M and G0 be a cell-like upper semicontinuous decomposition of X. We consider the problem of extending G0 to a cell-like upper semicontinuous decomposition G of M such that M/G "' M. Under fairly weak restrictions (which vanish if M = En or Sn and n fc 4) we show that such a G exists if and only if the trivial extension of G0, obtained by adjoining to G0 the singletons of M -X, has the desired property. In particular, the nondegenerate elements of Bing's dogbone decomposition of E3 are not elements of any cell-like upper semicontinuous decomposition G of E3 such that E3 /G"' E3; this answers negatively a question raised by Bing [Trans. Amer. Math. Soc. 149(1970), 351-3691 Call a cell-like upper semicontinuous decomposition G of a metric space X simple if X/G "" X and say that the closed set Y is simply embedded in X if each simple decomposition of Y extends trivially to a simple decomposition of X. We show that tame manifolds in E3 are simply embedded and, with some additional restrictions, obtain a similar result for a locally flat k-manifold in an m-manifold (k, m fc 4). Examples are given of an everywhere wild simply embedded simple closed curve in E3 and of a compact absolute retract which embeds in E3 yet has no simple embedding in E3 • (Received February 28, 1972.)

72T-G77. ANTONIO CONDE, Institute for Advanced Study, Princeton, New Jersey 08540. Best immersions for some real flag manifolds. Preliminary report.

Let F(il, .•• , ir) = 0( !:ij)/O(i1) X ••• X O(ir), and

Stiefel-Whitney class of F(i1 , ••• , ir) is zero if and only if the numbers i1 , .•• , ir are all of the same parity. The normal bundle of F(2, .•• ,2), where 2 is repeated n times, classifies in BO(n). So this proposition gives us the geometric dimension of the (stable) normal bundle of this manifold, which is n. The known result of

Hirsch gives us then Corollary 2. The manifold F(2, .•• , 2), 2 repeated n times, immerses in codimension n, and this is best possible. A consequence of the second part of 1 is that F(i1 , ••• , ir) is orientable if and

A-466 only if the numbers i1, ••• , ir have all the same parity. Also that tCF (i1, ••• , ir) is a spin manifold if and only if the numbers ~, ••• , ir have all the same parity. (Received February 28, 197 2.)

*72T-G78. MELVIN C. THORNTON, University of Nebraska, Lincoln, Nebraska 68508. Equalizers and closure operations in topalogical categories.

Equalizers, epi and onto maps are characterized in various topological categories using the notion of the double of a space along a subset. The characterizations are in terms of certain closure operations, Closure

conditions on the diagonal determine the category in some cases, (Received February 29, 1972.)

72T-G79. WITHDRAWN.

*72T-G80. DOUGLAS R, ANDERSON, Syracuse University, Syracuse, New York 13210, The

Whitehead torsion of a fiber homotopy equivalence.

Let ~ = (E,p,B, F) be a PL fiber bundle and f: B' ... B be a PL homotopy equivalence. Then the

induced bundle f*~ is a PL bundle and the natural map g: E(f*~) ... E is also a homotopy equivalence, Let

'T(f) E Wh 1r 1 (B) and T(g) E Wh 1r 1 (E) denote the Whitehead torsions of f and g respectively. Theorem. The

correspondence T(f) ... T(g) defines a homomorphism p*:Wh1r1(B)-+Wh1r1(E). Theorem. Let ~1 = (Ei, pi, Bi, F i) i = 1, 2, be PL bundles with connected base and fiber, and let g: E1 ... E2 be a fiber homotopy

equivalence covering f: B1 ... B2 and inducing h: F1 ... F2. Then T(g) = p*T(f) + X(B2)j2*T(h) where j 2*:

Wh'rr1(F2)-+ Wh 1r1(E2) is induced by the inclusion j 2 : F2 ... E2. (Received March 1, 1972.)

*72T-G81. ROBERT D. EDWARDS and RICHARDT, MILLER, University of California, Los Angeles,

California 90024. Local connectivity of spaces of embeddings.

Let EmPL~' Qq) be the semisimplicial complex of PL embeddings of a compact polyhedron X into

a PL manifold-without-boundary Q. Generalizing previous work of the authors and of Lusk (Abstract 674-23,

theseeAfo/iui) 17(1970), 527), we have Theorem. Suppose x ~ q- 3 and suppose f:X X if.-+ Q X if. is a

k-simplexin EmPL(X,Q) such that rlxxo~k=f0 x identity(o~k) where f0 :X-+ Q. Given ( > o, there exists 0 > o such that if each fs, s E ~k, is within 0 of f0, then there exists a ~k -indexed ambient isotopy

Ft : Q x I x -+ Q x I x within ( of the identity such that Ft = identity(Q) on 0 x U I x and ,s ~k ~k ,s ~k o~k F1,sfo = fs. Corollary 1. If x;:;; q- 3, then EmPL(X,Q) is locally n-connected for all n, by ambient isotopy

of Q, Corollary 2. If in addition q ~ 5, then EmPL(X,Q) is locally n-connected at points of the space of

topological embeddings EmTOP(X,Q) and is dense. Corollaries 3 and 4 (Cernavskii for X a manifold), If X

is a compact polyhedron [resp. compactum], Q is a topological manifold, x ~ q - 3 l:resp. 2x+2 ~ q] and

q ~ 5 then the space EmLT(X,Q) of locally tame embeddings [embeddings with the 1-ULC property] is locally

n-connected at points of EmTOP(X,Q), by ambient isotopy of Q. The proofs are accomplished by sharpening

and extending known PL techniques, and by using sliced engulfing for Corollaries 2, 3, and 4, The usual

relative, noncompact, with-boundary versions hold. (Received March 1, 1972.)

A-467 *72T-G82. MARLON C. RAYBURN, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada.

A characterization of realcompact extensions.

Let X be a Tychonov space and g E C(X) have Stone extension g*: {3X- R*. As in Exercise SB of

and Jerison, "Rings of continuous functions", let 11 X= (p E {3X: g*(P) f. ooJ. Suppose T is any Tychonov Gillman g space containing X densely and f: f3X- f3T the continuous map fixing X pointwise, CT ~ C(X) is the family

For any such T, f-[T] ~ £11 X: g E CT}. of functions that can be continuous extended to T. ---Lemma, n g Theorem. T is realcompact if and only if{"[T] = n £vgX: g E CT). (Received February 7, 1972.)

*72T-G83. JOSEF K. H. DORTMANN, Federal University of Parana, Curitiba, Brazil. The semi- interior operator.

Levine in his paper "Semi-open sets and semi-continuity in topological spaces", published in Amer.

Math. Monthly 70(1963), 36-41, has defined semi-open subsets of a topological space in the following way: a set S in a topological space X will be termed semi-open if and only if there exists an open set 0 such that 0 c

S c cS, where c denotes the closure operator on X. Semi-interior operator is defined as an application which associates to a set S c X the union of all semi-open subsets contained in S. It is shown that it is possible to find different topologies on X giving the same collection of semi-open sets. Therefore, if given the collection of semi-interior sets, it is impossible to define the semi-interior operator. (Received March 2, 1972.) (Author introduced by Professor Jayroe M. Cardoso.)

Miscellaneous Fields

72T-Hl. ROBERT VAIL HARRIS, Computer Science Department, Cornell University, Ithaca, New

York 14850. A polynomial bound on Davis-Putnam analysis of symmetrizable propositions.

Let A be a (quantifier-free) DNF proposition in the variables a1 , ••. , an. Call A "symmetrical" if it is invariant under every permutation of indices of the variables. Call A "symmetrizable" if it is symmetric or becomes so when certain variables are replaced by their negations throughout. Let "DP" mean the dual of the

Davis-Putnam procedure (thus DP recognizes DNF tautologies), augmented by the rule: If the set of literals in any clause c1 contains the set of literals of c2, delete c1 • Suppose A is a symmetrizable DNF proposition of c clauses in n variables; then the space required for DP to halt on A is at most 2n2c, and the time is at most

36n6 c 2• (Received February 15, 1972.) (Author introduced by Professor Robert L. Constable,)

A-468 The April Meeting in Berkeley, California April22, 1972 Algebra & Theory of Numbers

694-Al. J. L. BRENNER (University of Arizona), 10 Phillips Road, Palo Alto, California 94303,

M. RANDALL, Amoco, Calgary, Alberta, Canada and JAMES RIDDELL, University of Victoria, Victoria,

British Columbia, Canada. Covering theorems for finite nonabelian simple groups. I. Preliminary report.

(I) For n > 6, there is no class C of period 2 in An (alternating group) such that CC covers An.

(II) For n = 12k + 10, there is no such class of period 3 in An. (III) For every class C in An (n> 4), CC covers C. This is an extremely delicate assertion; it includes the corollary that for every type T, TT covers

T. (IV) Theorems are given relating covering by classes and covering by commutators of a fixed element, the latter (used by L. E. Dickson in proofs of simplicity) being only slightly more rapid. (V) Let C be a nontrivial class; let 11 = II{C) be the smallest exponent such that c 11 covers the nonabelian simple group G. II(C) is

always finite if G is finite; it is tabulated for all C in A5 , A 6, A 7 , A 8, PSL(3,4). The (doubtful) conjecture mine II(C) = 2 is verified in PSL(3,4), Also maxcii(C) (in An) is very large (n-+co), viz. [(n-1)/2],

(VI) Finally some covering criteria involving lei are given. Results supplement and extend work of Xu, Bertram, Brenner; extend and furnish alternative proofs of theorems of Ore, N. Ito that every element of An

is a commutator (n> 4), In particular, every element of PSL(3, 4) of order 20, 160 is a commutator; but

CC :::> PSL(3, 4) is a stronger result. (Received September 27, 1971.)

*694-A2. BART FRANCIS RICE, Naval Postgraduate School, Monterey, California 93940, Modules

over orders and transformations of quadratic forms.

A natural relationship exists between automorphic transformations of binary and quaternary quadratic

forms f (which take f into a multiple of itself) and ideals (viewed as Z-modules) in quadratic and quaternion

rings R. Factorization of such transformati

Q-module is an associative composition algebra. It is shown that if f is a definite n-ary quadratic form and

S is a set of automorphic transformations of f with the above property (whence dim S: Q = 1, 2 or 4), then

dimS :Q = 4 implies n = 0 (mod 4) and dimS :Q = 2 implies n is even. (Received February 7, 1972.)

*694-A3. ROBERTO FRUCHT, Universidad Tecnica Federico Santa Maria, Casilla 110-V, Valparaiso,

Chile and FRANK HARARY, University of Michigan, Ann Arbor, Michigan 48104. Self-complementary generalized

orbits of a permutation group.

A permutation group A of degree n acting on a set X has a certain number N1 of orbits, each a subset of X. More generally, A also induces an equivalence relation on x(k), the set of all k-subsets of X, and the

resulting equivalence classes are called k-orbits of A. Thus orbits are 1-orbits and we may refer to k-orbits

as generalized orbits. A subset S of X is self-complementary if X and X - S are members of the same

A-469 generalized orbit. A self-complementary k-orbit is one for which every k-subset in it is self-complementary.

Our main result is an inclusion-exclusion formula for the number s(A) of self-complementary generalized orbits of an arbitrary permutation group A in terms of its cycle index. We show that self-complementary graphs, digraphs, and relations provide special classes of self-complementary generalized orbits. (Received February 11, 1972.)

*694-A4. MICHAEL J. WRIGHT, Loyola University, Los Angeles, California 90045. Henselian valuations on an ordered field. Preliminary report.

In this note we examine those (nonarchimedean) valuations on an ordered field K which have henselian prolongations to its real closure Krc' the unique maximal algebraic extension of K prolonging the order of K.

Let cp be the set of all noninfinitely large elements of K. Theorem 1. Let K be a real closed field, v a valuation on K. Then v is henselian if and only if the nonunits of v are infinitely small. Corollary. The ordered field K has a nontrivial valuation with a henselian prolongation to Krc if and only if K is nonarchimedean ordered. Theorem 2. The valuation v on the ordered field K has a henselian prolongation to Krc if and only if its valuation ring A is of the form cpp, for some prime ideal P of cp. (Equivalently, if and only if whenever x, y E K are such that x t' A and 0 ~ y ~ x, then y EA.) (Received February 24, 1972.)

*694-A5. MATTHEW I. GOULD, Vanderbilt University, Nashville, Tennessee 37203. Subalgebra maps induced by endomorphisms and automorphisms of algebras.

Given a universal algebra !II, its endomorphism monoid End(!l!) and automorphism group Aut(!l!) induce, respectively, a monoid End*(!l!) and a group Aut*(!ll) of mappings of the subalgebra lattice S(!l!) into itself.

Specifically, for a E End(~) we define a*: S(!l!) ... S(!l!) by setting xa* = !xalx Ex} for all X E 8(21). We then define the monoid of closure endomorphisms by End*(!l!) = Ia* Ia E End(!l!)} and the group of closure automorphisms by Aut*(!l!) = !a*la EAut(!l!)}. It is clear that End*(!l!) and Aut*(!l!) are homomorphic images of End(!l!) and Aut(!l!) respectively. Given a monoid M and a homomorphic image M*, say that M induces M* if there is an algebra !II such that Me. End(!l!) and M* e. End*(!l!). In previous work of the author and c. R. Platt LAlgebra Universalis

1(1971), 90-92] it was shown that every monoid induces itself. Theorem 1. A monoid M induces all its homomorphic images if and only if M is leftcancellative. Theorem2. Every left cancellative monoid induces all its homomorphic images in which each element is either right cancellative or a right zero. Corollary. Given a group G and a homomorphic image G*, there is an algebra !II such that G ""Aut(!l!) and G* ""Aut*(!l!). (Received

February 28, 1972.)

*694-A6. MARTIN K. McCREA, University of California, Davis, California 95616. Commutative idempotent generated rings as generalized Boolean rings. Preliminary report.

In this communication all rings are commutative. Let R be a ring generated by its Boolean ring of idempotents B. Let 1'/: B ... N take each x E B to its additive order in R (where Tj(x) = 0 if there exists no n EN, n f. 0, for which nx = 0). It is shown that the pair (B, Tj) determines R to isomorphism in the sense that if R' is another idempotent generated ring with corresponding pair (B', Tj') and cp: B ... B' is a Boolean ring isomorphism

A-470 satisfying 11'. cp = 11· then R and R' are isomorphic. A subcategory of the category of pairs (B, 11>. where B is a Boolean ring and 11 : B .. N is an upper semilattice homomorphism into the natural numbers ordered by divisibility, is shown to be equivalent to the category of idempotent generated rings. Finally, Stringall's result, that for primes p and q the categories of p-rings and q-rings are equivalent, is generalized to hold in the context of idempotent generated rings. Corollary. The category of torsion-free idempotent generated rings is equivalent to the category of Boolean rings. (Received February 22, 1972.)

694-A7. VOLODYMYR BOHUN-CHUDYNlV, Morgan State College, Baltimore, Maryland 21212. On quasi-groups and quasi-loops satisfying the identity x• xy = yx and their isotopes. Preliminary report.

The aims of this paper are: (1) to determine algorithms for constructing quasi-groups and quasi-loops satisfying the identity (1) x • xy = yx; (2) to find algorithms for constructing loops and groups which are isotopes to quasi-groups and groups respectively satisfying identity (1); (3) to find sets of isotopic or phenotopic quasi- loops or quasi-groups from given quasi-loops or quasi-groups satisfying identity (1), Phenotopic operators were introduced in a previous paper by the author (Abstract 691-17-6, these cMitict.019(1972), A-82). (Received

January 28, 1972.)

*694-A8. THOMAS J. HEAD, University of Alaska, Fairbanks, Alaska 99701 and NOBUAKI KUROKI,

Nikon University, Narashino, Japan. Greatest regular images of tensor products of commutative semigroups.

Let A be a commutative semigroup. It is known that there is a regular commutative semigroup Reg(A) and a map A .. Reg(A) with the property that A .. Reg(A) is a factor of every map of A into a regular semigroup.

Likewise it is known that there is a commutative group Gro(A) and a map A .. Gro(A) with the property that A ..

Gro(A) is a factor of every map of A into a group. The commutative semigroup A is said to have a greatest

regular (group) image if there is a regular semigroup (group) G and a map of A onto G which is a factor of

every map of A onto a regular semigroup (group). Theorem 1. If A has a greatest regular image th~n so does

A ® B and Reg(A) ® Reg(B) is such an image. Theorem 2. If A has a greatest group image then so does A ® B

and Gro(A) ® Gro(B) is such an image. Theorem 3. If A has a greatest group-with-zero image then A ® B has

such an image if and only if B is archimedean. If A has a" as a greatest group-with-zero image and if B is

archimedean, then (G ® Gro(B))c is a greatest group-with-zero image of A ®B. (Received March 6, 1972.)

*694-A9. HUGO SUI-HWAN SUN, Fresno State College, Fresno, California 93710. Some combinatorial formulas for partitions. Preliminary report.

Graphical methods and the sieve formnla are used in deriving the following Theorem. Let n be a 2 positive integer. If n is even, the number of self-dual partitions of n is !:~= 1 P2i (n-(2i) /2), where Ps(t) is the number of partitions of t with summands ~ s, and k is the largest integer such that 4k2 a!! n. If n is odd, 2 the number of self-dual partitions of n is !:~ 1 P2i+1 (n-(2i+1) /2), here k is the largest integer such that 2 (2k+1) ~ n. (Received March 8, 1972.)

A-471 *694-A10. ROBERT P. DICKINSON, JR. , Computation Department, Lawrence Radiation

Laboratory, University of California, Livermore, California 94550. Noncancellative congruences on N- semigroups. Preliminary report.

This paper describes a constructive method of finding all congruences p (T4-congruences) on commutative cancellative archimedean semigroups S without idempotents (N-semigroups) such that S/p is noncancellative and without idempotents (T4-semigroups). Tamura has given a representation of an N-semigroup by a pair (G, I), where G is an abelian group and I is a function from G X G into the natural numbers. In terms of this representation, we study N-congruences (congruences 17 on S such that S/17 is an N-semigroup) and refinements of N-congruences. It is shown that an N-semigroup S can have a T4-congruence if and only if it has a proper N-congruence. It is also shown that all T4-congruences are refinements of proper N-congruences, and every proper N-congruence has such a refinement. T4-congruences have representations in terms of

Tamura's representations of N-semigroups and N-congruences. (Received March 8, 1972.)

*694-All. TAKA YUKI TAMURA, University of California, Davis, California 95616. Exponent of semigroups and order- bounded groups.

Let S be a semigroup and E(S) the set of all positive integers n satisfying ~)n = xnyn for all x, y

E S. E(S) is a multiplicative semigroup and is called the exponent of S. Let (S~: ~ E :E:} be a class of semigroups. The exponent of a subdirect product of (S~ : ; E :E:} is the intersection of E (S;), ; E :E: • If G is a group, then E(G) is defined to be the multiplicative semigroup of integers n satisfying the above identities. Let

Z be the set of all integers and let m -f 0, mE Z, M(m) = (mx,mx+1 :xE Z}. Theorem. Let G be a nontrivial order-bounded group. Then there is a finite set of integers (m1, ••• , mk} such that E(G) = n~= 1 M(mi). Conversely if (m1, ••• ,mk} is given, then there is an order-bounded group G such that E(G) = n~ 1 M(mi). A multiplicative semigroup C of integers is called a complementary semigroup if (1) 0,1 E C; (2) x E C implies 1 - x E C. The exponent of a group is a complementary semigroup. (Received March 8,

1972.)

*694-A12. TAKA YUKI TAMURA and HOWARD B. HAMILTON, University of California, Davis,

California 95616. The study of commutative semigroups with greatest group homomorphism.

A commutative semigroup S has a greatest group homomorphism if and only if for every a E S there

are b, c E S such that abc = c. The following question is raised: Is a minimal group congruence on a semigroup

S a smallest? If S is commutative or medial then it is true; but still open in general case. (Received March 8,

1972.)

*694-A13. TAKA YUKI TAMURA and THOMAS EDWARD NORDAHL, University of California, Davis,

California 95616. On exponential archimedean semigroups with idempotent. Preliminary report.

A semigroup S is called exponential if (xy)n = xnyn for all x, y E S and all positive integers n. If

S is an exponential archimedean semigroup with idempotent, then S is an ideal extension of I by N where I is

A-472 the direct product of an abelian group G and a rectangular band B, and N is an exponential nil-semigroup. This paper discusses how to construct S when N and I are given. (Received March 8, 1972.)

*694-A14. TAKA YUKI TAMURA, University of California, Davis, California 95616 and JOHN

C. ffiGGINS, Brigham Young University, Provo, utah 84601. Quotient group of finitely generated N-semigroups.

Preliminary report.

By an N-semigroup we mean a commutative cancellative archimedean semigroup without idempotent.

Theorem. An N-semigroup S is finitely generated if and only if the quotient group Q(S) is isomorphic to the direct sum of the infinite cyclic group and a finite abelian group. Let S = (G ;IP) be Tamura second representation of an N-semigroup S where G is a finite abelian group and cp is a nonnegative rational valued function on G satisfying certain conditions. Let s1 = (G1 ;IP1), s2 = (G2 ;cp2) be two finitely generated N-semigroups. As a consequence of the above theorem we can describe when (G1 ;IP1) and (G2 ; IP2) are isomorphic, (Received March 8, 1972.)

*694-A15. MELVEN R. KROM, University of California, Davis, California 95616 and MYRENKROM,

Sacramento State College, Sacramento, California 95819. Free subgroups and Folner's conditions.

It has been conjectured (F. P. Greenleaf, "Invariant means on topological groups", Van Nostrand,

New York, 1969, p. 9) that a group has a free nonabelian subgroup if and only if it is not amenable. Possibilities

are considered for further limiting the conditions given by Folner ("On groups with full Banach mean value",

Math. Scand. 3(1955), 243-254) to characterize the class of nonamenable groups. It is shown that extreme

instances of Folner's conditions identify finite sets of free generators for free subgroups. Theorem. For any

group G and any finite F c G the subgroup generated by F is free on F if and only if for every finite E c G, ~aEF IE nEal < lEI where lsi denotes the cardinality of s. (Received March 9, 1972.)

*694-A16. EDWARD J. TULLY, University of California, Davis, California 95616. Some languages

derived from abelian groups.

Every uniquely decipherable language closed under juxtaposition in which all rearrangements of an

occurring string always occur (or, in other words , every free subsemigroup of a free semigroup which, along

with ab always contains ba) can be constructed by taking the alphabet to be a family (with repetitions allowed) of

generators of some abelian group, and letting the language consist of every string of symbols whose product is

the group identity. Corollary. If in addition the alphabet is finite, the language can be constructed by taking an

m X n matrix (a.. ) of integers, and letting the language consist of all strings formed from the symbols 1, 2, ••• , lJ m - 1 for which the sum of the corresponding rows is element-wise congruent to zero modulo the elements in

the last row (or, in other words, all those strings x1 .•• xr such that, for all j, amj divides ~~= 1 axi· The methods involve semigroup theory. A preliminary result states that a commutative semigroup containing a

unitary subsemigroup T such that a cf b implies either ax E T, bx ~ T or ax ~ T, bx E T, for some x, must

be either a group or a group with zero. (Received March 9, 1972.)

A-473 694-A17. DAIHACHffiO SA TO, University of Saskatchewan, Regina, Saskatchewan S4S OA2, Canada.

Consequences and generalizations of algebraic solution of ~ = yx (0 < x < y). Preliminary report.

In the Shorter Notes of the Proc. Amer. Math. Soc. (31(1972)), characterizations of solutions of an exponential equation ~ = yx (0 < x < y) in the field of algebraic numbers and in the ring of algebraic integers were given. In this report, several consequences and generalizations of the above mentioned solutions will be presented and applications to the solutions of certain diophantine equations will be discussed. (Received March

9, 1972.) Analysis

*694-B1. GRANT B. GUSTAFSON, University of utah, Salt Lake City, Utah 84112. Oscillation criteria for y" + p(t)f(y,y') = 0, with f homogeneous of degree one. Preliminary report.

Let 3' be the set of functions f E C(R2 - R) which satisfy (1) xf(x,y) > 0 for x f 0, (2) f is locally

Lipschitz, (3) f(tx,ty) = tf(x,y). For p E C[O,oo] and f t; 3', consider the equation y" + p(t)f(y,y') = o. The equation is said to be strongly oscillatory if y" + Ap(t) f(y ,y') = 0 has only oscillatory solutions for all A > o.

It is established that strong oscillation of the nonlinear equation is equivalent to strong oscillation of the linear equation y" + p(t)y"' 0 in the case p(t) ~ 0. The case when p(t) changes sign and J00p = oo is studied, and a number of criteria are developed. An example is given which shows that J00p = oo and f t; ;; is compatible with the existence of an eventually positive solution; hence Bihari's oscillation theorem [I. Bihari, "An oscillation theorem concerning the half-linear differential equation of the second order," Magyar Tud. Akad.

Mat. Fiz. Oszt. Kozl. 8(1963), 275-280] cannot be extended to include the case when p(t) changes sign.

(Received November 26, 1971.)

*694-B2. CHARLES F. AMELIN, California State Polytechnic College, Pomona, California 91768.

Generalized numerical range. n. Preliminary report.

Let T, A E L(X, Y) where X and Y are both Hilbert spaces, Banach spaces, or Frechet spaces. We define a generalized numerical range, W(T ,A), using an inner product if X and Y are Hilbert spaces and using

Lumer structures to define a semi-inner product if X and Y are B-sp or lcs. W(T ,A)= w1 (T ,A) U W2(T ,A).

Cf. Abstract691-47-18, these C!Vottee.D19(1972), A-184. Lemma. If P(T,A) f ¢,then pt O"(T,A)c w1(T,A) and res O"(T ,A) c W2(T ,A). In what follows assume X and Y are both Hilbert or both Banach spaces.

Theorem. If A is (semi-) Fredholm and ~ (/_ W(T ,A), then T - ~A is (semi-) Fredholm so that cont a(T ,A) c

W(T ,A). It is possible for a(T ,A) = n1 and def(T- ~A) > n2• It is possible to define both parts of the GNR for closed operators T and A. Provided the domains of T and A satisfy certain regularity conditions, e.g. re density and inclusion, some of the above results will hold for the closed T and A. (Received February 7, 1972.)

A-474 694-B3. DARELL J. JOHNSON, University of California, Riverside, California 92502. Jackson type theorems for approximation with side conditions. Preliminary report.

Let [Mk} be an increasing sequence of linear subspaces of a normed linear space X, whose union is dense. Suppose f E X and £.ti}~= 1 ~ X*. We obtain some convergence rates for approximating f by elements mk E Mk satisfying .timk = .lif (i = 1, .•• , n) and llmk \1 ;;; II f\1 in terms of li(f;Mk), the distance of f to the subspace Mk. In particular, for SAIN approximation by polynomials on C [a, b], with point evaluations, we get the estimates Ek(f) ;;; Cwt

694-B4. CARLOS A. BERENSTEIN, Harvard University, Cambridge, Massachusetts 02138 and MILOS

A. DOSTAL, Stevens Institute of Technology, Hoboken, New Jersey 07030. On a class of invertible distributions.

We studyaclass R of entire functions for which good lower estimates can be found everywhere in ern. If

R is the class of distributions of compact support T, with Fourier transform TE R, then one can show that the convolution mapping of cp.., T * cp is surjective for cp E .IT'(l!n) and other spaces of distributions. Furthermore, the

"Titchmarsh-Lions formula" for singular supports holds for T E R and 1fS E a'. Concrete examples of such distributions are, for instance, distributions with finite support; however, the class R also contains distributions with nondiscrete support. Different modifications of the class R can be introduced and studied in the same context. It is interesting to point out that using such classes one can prove the most general form (known to date) of the "fundamental principle" of L. Ehrenpreis (cf., e.g., Lecture Notes in Math, vol. 256, Springer-

Verlag). (Received February 28, 1972.)

*694-B5. PETRU MOCANU, Babes-Bolyai University, Cluj, Romania and MAXWELL 0. READE,

University of Michigan, Ann Arbor, Michigan 48104. On Cl!-convex functions. II. Preliminary report.

The authors prove the following Theorem. If f(z) is analytic in the unit disc A, if f(z)f'(z)/z -1 0 there, and if for some real 0' the inequality Re[(1-0!)z(f'(z)/f(z)) + C1!(1+zf''(z)/f'(z))J ;;,: 0 holds in A, then f(z) is univalent and starlike in A. If 0! ;;; -1, then 1/f(1/z) is convex for z E A. (Received March 1, 1972.)

694-B6. STEPHAN MICHAEL SPERLING, University of California, Los Angeles, California 90049.

Extension of the Krein-Milman theorem. Preliminary report.

By the methods of nonstandard analysis the Krein-Milman extreme point theorem, which is known for

locally convex spaces, is extended to Hausdorff topological vector spaces. The proof uses the following separation

theorem, which generalizes the strong separation theorem for locally convex spaces. Theorem. Let A and B

be two nonempty disjoint convex subsets of a Hausdorff topological vector space X. Suppose A is closed and B

is compact. Let *R be a nonstandard extension of the reals R and let T be the subtopology of the Q-topology

Q on *R defined by T = [u E Q: U n R is open in RJ. Then there exists a continuous linear function f mapping

X into (*R, T) strongly separating A and B. Proof. A subspace E of *X is constructed so that X is contained

A-475 in E and the dimension of E is an infinite natural number n. E is shown to be isomorphic with (*R)n. Then the proof is done in E and pulled down to X. (Received March 3, 1972.)

*694-B7. JAMES R. DIEDERICH, University of California, Davis, California 95616. Weyl's lemma for pointwise solutions of elliptic equations. Preliminary report.

iJ. is said to be a pointwise solution of (1) :Ea.. IJ. •• + :Eb. iJ.. + ciJ. = 0 at x if iJ. has a total l) l) l l 0 differential of order 2 in the L1 , Lebesgue class, sense at x0 whose coefficients satisfy (1). Theorem. Every

L1, pointwise solution of (1) is a classical solution. (Received March 3, 1972.)

694-BS. MOSES E. COHEN, Fresno State College, Fresno, California 93710. A generalized hypergeometric integral transform. Preliminary report.

We introduced the square of the Legendre polynomial integral transform in [Math. Z. 108(1969), 121) and studied it in subsequent papers. Here, the polynomial is generalized and a Rodriguez formula is presented which permits its use as an integral transform. Jacobi, Legendre, the square of the Legendre and other known and new polynomials are particular examples of the transform. (Received March 6, 1972.)

*694-B9. DAVID L. RAGOZIN, University of Washington, Seattle, Washington 98195. Rotation invariant measures on Euclidean space.

Let G = SO(n) , n !!; 2, and let MG (lln) = !1-1 E M(Rn) : iJ. (gS) = iJ. (S) for all Borel sets S (; lRn and all g E G l denote the rotation invariant finite Borel measures on JRn. MG(Rn) is a subalgebra of the convolution algebra

M(Jin). Its convolution structure is much simpler than that of M(:Rn) in light of Theorem 1. If iJ., 11 E MG(lln) are continuous measures, then 1-1 * 11 is absolutely continuous (with respect to Lebesgue measure). This theorem is proved using a simple geometric consequence of the irreducibility of the identity representation of G. Since the only discrete measures in MG(lln) are multiples of 60 and since the maximal ideal space of Lr(ll~ (=the of rotation invariant L1-functions) is lln/G ""[0, w), we easily derive Theorem 2. The maximal ideal space MG(lln) is [O,w], the one point compactification of the maximal ideal space of Lr(lln). These theorems are special cases of more general results on MK(:Rn), the algebra of K-invariant measures on :Rn, where K (; G is any closed subgroup with no nontrivial invariant subspaces in ]ln. (Received March 6, 1972.)

694-B10. KYONG T. HAHN, Pennsylvania State University, University Park, Pennsylvania 16802 and

University of California, Berkeley, California 94720. Bounded holomorphic mappings and generalized Bloch's theorem.

Let w = f(z) be a holomorphic mapping of the unit ball B c en onto f(B) c en. For each z0 E B, let ) cf o, df(z0) be the radius of the largest univalent ball in f(B) centered at w 0 = f(z0) if Jf(z0) = det(of/oz) (z0 where (of/oz) (z0) denotes the Jacobian matrix of f at z0. Define df(z0) = 0 if Jf(z0) = o. The author obtains the following Theorem. Let w = f(z) be a univalent mapping of B onto f(B) and let >.f be the positive square root of the smallest characteristic value of (of/oz) *(of/oz). Then, df(z) ~ (1 -JzJ2) >.f(z) ~ df(z)/r0(B) for z E B, where r 0(B) denotes the generalized Koebe constant of B [Trans. Amer. Math. Soc. 162(1971)] and Jzl the

A-476 euclidean norm of z in en. Lemma. Let w = f(z) be a holomorphic mapping of B into a bounded domain c BM' ball of radius M with center o in en, such that f(O) = 0 and (of/oz) (0) =In. Then w = f(z) is univalent in BR, where R = 2./3/9M, and f(B) contains a univalent ball of radf.us ,f3/9M. Uslng thls lemma the author obtains

Theorem. Let w = f(z) be a univalent mapping of B into a bounded domain c BM. Then, dr(z) ~(1-lzi 2 >Xr(z) ~ (6./3 M df(z))1/ 2 for z E B. The following theorem gives a generalization of Bloch's theorem and a lower bound for the constant of Bloch. Theorem. Let w = f(z) be a holomorphic mapping of B into a bounded domain c BM with Af(O) !; a> 0. Then there exists a point w0 = f(z0) and an absolute constant {3 > 0, independent of f, such 2 that df(z0) !; f3 ~ ./3 a /72 M. (Received March 8, 1972.)

694-Bll. SA NTIRANJAN MUKHOTI, Council of Scientific and Industrial Research, Rafi Marg, New Delhi,

India. Theorems on absolute Cesaro summability of integral.

We suppose that A(u) is of bounded variation in every interval 1 ~ u~ U and that A(1) = 0, and we consider Cesaro summability for integral orders of the Stieltjes integral Jfc,o(u) dA(u), and establish theorems.

Theorem 1 (the case k=O). Suppose that cp(t) is positive, continuous fort~ 1, and is of bounded variation in

[1,oo), and (i) cp(x) ... 0 as x ... oo, (ii) 9(t) J~ldc,o(x)l = 0(1) for all t!; 1 (B; 1, x!; 1, t

and (i) 4>k(x) = o(x~ as x ... oo, (ii) ft00lo/ox[¢(x, t)/xk] I dx ~A, for all t ~ 1 (t!!; 1, x ~ 1, t < x) where 4>k(x) = x k-1 Jt k I k I Jx k-1 (1/I'(k)) J1 (x-u) cp(u)du; 1/>(x,t) = 1 u (o au) Wk(x,u)du, Wk(x,t) = (9'(t) I'(k)) t (x-u) cp(u)du (9(x) = 1/cp(x)). Then necessary and sufficient conditions for Jfc,o(u) dA(u) to be summable lc,kl to B, are that

-J~c,o'(u)A(u)du should be summable lc,kl to B, and A(x)c,o(x) = o(1) lc,kl as x ... oo. (Received March 8, 1972.)

694-B12. PHILIP M. ANSELONE, Oregon State University, Corvallis, Oregon 97331 and RALPH L.

JAMES, Stanislaus State College, Turlock, California 95380. Convergence of positive sequences. Preliminary

report.

Let E be an ordered Banach space and x E E, x ~ 0. A set H c E* is x-regular iff for each f > 0 the

following holds: For each hE H, there are h,, h fEE* such that h '

{h f: hE Hl are totally bounded. If x ... x weakly and x , x ~ 0, then h(x ) .. h(x) uniformly for h in any n n n x-regular set H c E* [of. P. M. Anselone, "Abstract Riemann integrals, monotone approximations and

generalizations of Korovkin's theorem," International Series in Numerical Mathematics 15(1970), 107-114;

R. L. James, "Uniform convergence of positive operators," Math. z. 120(1971), 124-142]. Now let E = L1 [0,1]

and H = {sgn(xn-x): n E Z +l L00[0,1]. Theorem. Let xn ""' x weakly, xn' x !; 0. If H is x-regular then xn ... x strongly. In particular H is x-regular if there exists M < + oo such that, for each n E z+, sgn(xn-x) is a. e. a

step function with at most M discontinuities. (Received March 8, 1972.)

A-477 *694-B13. ROBERT T. MOORE, University of Washington, Seattle, Washington 98195. Vanishing second

commutators in operator theory and Lie group representations. Preliminary report.

A classical theorem asserts that if P and Q are bounded selfadjoint Hilbert space operators such that

p commutes with the commutator [p, Q) = PQ - QP, then P must commute with Q as well. Singer (Ann. of Math.

56(1952), 242-247) generalized this to show that a finite-dimensional Lie algebra of skew-adjoint operators necessarily Levi-decomposes into a compact semisimple factor and a central abelian radical. We give a very

short exponential proof of the first fact which still applies when P is a hermitian Banach space operator (not

necessarily bounded) in the sense of Lumer-Vidav and Q is merely bounded. (See our announcement Bull. Amer.

Math. Soc. 77(1971), 224-229.) Singer's result similarly generalizes. The notions introduced in our announce­

ment (op. cit.) supply generalizations of the first to P hermitian (not necessarily continuous) and Q continuous

on a general locally convex space, while extending Singer's result to Lie algebras of "skew-hermitian" continuous locally convex space operators. These last results have important applications in the theory of c00 locally convex representations of noncompact (semi) simple groups, especially on Frechet spaces of test functions or of

C00 vectors for classical unitary representations. (Received March 9, 1972.)

694-B14. WILLIAM G. BADE, University of California, Berkeley, California 94720 and PHILIP c. CURTIS, JR., University of California, Los Angeles, California 90024. Continuity of derivations of Banach

algebras into modules. Preliminary report.

Let A be a Banach algebra (with unit 1). A Banach space M is a Banach A-module if M is an A-module

and the bilinear maps (x, m) "'xm and (x,m) .. mx are continuous (and 1•m = m• 1=m, mE M). A derivation of

A into M is a linear map D: A ... M such that D(xy) = xD(y) + D(x)y for all x, yEA. Theorem 1. Let A be a

Banach algebra having dense socle. Let M be a Banach A-module satisfying the condition: if mE M and xm =

mx = 0 for all x E A, then m = 0. Then every derivation D: A .. M is continuous. Theorem 2. Let A be a

commutative Banach algebra with unit which is regular in Silov's sense and has the property that each closed

primary ideal in A has finite codimension. Let M be a Banach A-module which contains no one dimensional

subspaces invariant under action of A. Then every derivation D:A"' M is continuous. (Received March 9, 1972.)

*694-B15. BANSHI D. MALVIYA, North Texas State University, Denton, Texas 76203. A note on locally

c*-algebras. Preliminary report.

A *-algebra A is called a locally m-convex *-algebra if there exists a family IPiliEA of seminorms on

A such that (i) !p.j.EA defines a Hausdorff locally convex topology on A, (ii) p.(xy) < p.(x)p.(y) for every x, 1 1 1 1 1 yEA and every i E A and (iii) pi(x*) = pi(x) for every x E A and every i EA. A complete locally m-convex

*-algebra is called a locally c*-algebra if pi(x*x) = pi(x)2 for every x E A and every i E A. (See [E. A. Michael,

"Locally multiplicatively convex topological algebras," Mem. Amer. Math. Soc. No. 11(1952)], and [A. Inoue,

"Locally c*-algebra," Mem. Fac. Sci. Kyushu Univ. 25(1971), 197-235].) In this note we obtain certain

results on *-representations of locally c*-algebras by using positive linear functionals. We also obtain an

inequality connecting the positive functional and the spectral radius. This inequality is then used to show that the

direct sum of *-representations is defined. (Received March 9, 1972.)

A-478 694-B16. MICHAEL B. FREEMAN, University of Kentucky, Lexington, Kentucky 40506. Levi-flat two-manifolds in c::2• Preliminary report.

A real smooth 2-dimensional submanifold M of c2 is defined locally as the zero set of a smooth function 2 p: C ... C::. The real tangent space to M at p has a largest complex-linear subspace HPM, and M is Levi-flat at pif the Levi forms "'L.Jo 2 p I oz, 2. (Received March 9, 1972.)

*694-B17. STEVEN M. SIMMONS, Eastern Washington State College, Cheney, Washington 99004. A converse Steinhaus theorem for locally compact groups. Preliminary report.

Let G be a locally compact topological group, and let 1.1 be a complex regular Borel measure on G with total variation l~~ol. The following statements are shown to be equivalent:(!) For each Borel set E in G with l~~oi 0, the set EE-l= \xy-l: x E E and y EEl has nonvoid interior. (2) For each Borel set E in G with

11.11 (E) > 0, the set EE-l contains a neighborhood of the identity in G. (3) For each pair A, B of Borel sets with II.II(A)>O and II.II(B)>O, thesetAB-1 =\xy-11xEA andyEBI hasnonvoidinterior. (4) Themeasure 11o is absolutely continuous with respect to left Haar measure on G. Thus, a strong converse is obtained to the well-known theorem of Steinhaus which asserts that statement (4) above implies the previous three statements.

(Received March 9, 1972.)

*694-BlB. MILTON PHILIP OLSON, 1037 Los Arabia Lane, Lafayette, California 94549. Order- continuous states on Rickart algebras. Preliminary report.

Rickart algebras (also called B;-algebras) have been shown to have a-conditionally complete lattices of selfadjoint elements in a new ordering called "spectral order". It is shown that if a set of states on a C*-algebra with identity separates points, then convex combinations of these states are weak-star dense in the set of all states. A Rickart algebra has a separating family of states which are a-continuous in spectral order if and only if its selfadjoint elements form a sub-a-lattice of the s. a. elements of the von Neumann algebra which is its double dual. A similar result holds for Kaplansky algebras (also called AW*-algebras) if "a" is replaced by

"complete" or ''completely". In fact, such a Kaplansky algebra is already a von Neumann algebra. (Received

March 9, 1972.)

A-479 Applied Mathematics

*694-Cl. CLYDE F. MARTIN, Systems Analysis Branch, Ames Research Center, NASA, Moffett Field,

California 94035. Equivalence of constant and time varying quadratic performance criteria. t Let, i = 1, 2, J1(u, t1) = Jt:[x'(t)Qi(t)x(t) + u'(t)Ri(t)u(t)]dt; where x(t) is an n-vector, u(t) is an m-vector and Q., R. and R::-1 are nonnegative definite analytic matrix valued functions of appropriate dimension. Let ui* 1 1 1 be the extreme point of Ji(u, t1) at t1 subject to the linear dynamic constraint (L): x(t) = F(t)x(t) + G(t)u(t), x(t0)

= x0; F, and G analytic matrix valued functions. J 1 is equivalent to J 2 at t1 iff uf = u!· Let cp(t, t0) be the fundamental solution of (L) and then (dk/dtk)[cp(s, t)G(t)Ri-\t)J = cp(s, t)Pk .(t) where Pk . is defined recursively '1 '1 by pk .(t) = -F(t)Pk . (t) + (d/dt)Pk .(t), P i(t) = G(t)Ri-\t). Theorem 1. J is equivalent to J for all finite +1,1 ,1 ,1 0 ' 1 2 t1 iff for all k, Q1 (t)Pk, 1 (t) = Q2(t)Pk, 2(t). Corollary 1. If R1, R2, F and G are constant functions then J1 is k -1 k -1 ,;: equivalent to J 2 for all finite t 1 iff Q1 (t)F GR1 = Q2(t)F GR2 for 0;;;; k- n - 1. Theorem 2. If F, G, Qi and

Ri are constant matrices and J 1 is equivalent to J2 for some t1 > t0 then J1 is equivalent to J2 for all t1 > t0.

Corollary 2. If G is n )( 1 and (L) is controllable then J1 is equivalent to J2 at t1 iff Q1 = aQ2 and R1 = aR 2, a> 0. Proofs and related results will appear elsewhere. (Received February 28, 1972.)

694-C2. GORDON E. RITCHIE, University of Santa Clara, Santa Clara, California 95053. Existence of solutions for singular nonlinear integral equations resulting from plane free surface flows. Preliminary repor

The steady two dimensional flow of an inviscid, irrotational, incompressible fluid in a constant gravitational field is considered. The flow is bounded below by an infinite plane and above by a semi-infinite plane with a sluice gate inclined downward at an angle k1T/2, where 0;;;; k;;!; 1. Existence of the flow is shown to depend upon the solution of a singular nonlinear integral equation. The corresponding integral operator is shown to be uniformly approximated by operators corresponding to finite jet approximations. Then the existence of a solution of the integral equation for 0 < k;;!; 1 is proved using the Schauder fixed point theorem. Under mild restrictions the solutions can be constructed by successive approximations. Solutions for which the free surface falls are shown to exist with prescribed speeds at the sluice gate tip and at infinity; and, provided that the ratio of the given speeds is not too large, the solutions are unique and can be constructed. The limiting case k = 0 is shown to have no practical solutions for which the free surface is rising or falling, but mathematical solutions wit prescribed speeds do exist provided the ratio of the speeds is not too close to unity. (Received March 6, 1972.)

*694-CS. ROBERT L. MEYER, California State College, Los Angeles, California 90032. Pointwise bounds on the discrete Laplacian of a harmonic function.

Let u denote the solution to a Dirichlet problem for Laplace's equation defined on a polygon in the xy-plane with prescribed polynomial boundary data on the sides of the polygon. Let A~u denote the usual 5-point discrete approximation to the Laplacian operator used in formulating a finite difference analogue of the continuous problem. Fourier series and Poisson integral representations of u are used to obtain pointwise bounds on I~ul at all mesh points P whose distance from a vertex exceeds the mesh constant h. These bounds are in terms of

A-480 known data and do not involve derivatives of the unknown function u. The bounds for l6.tul may be used together with known bounds on data interpolation error for grid points near vertices to compute pointwise bounds on the discretization error of a finite difference analogue of the continuous problem which is defined so that all grid points are at a distance greater than h from a vertex. (Received March 6, 1972.) Statistics and Probability

*694-F1. MAREK KANTER, Tulane University, New Orleans, Louisiana 70118. On the spectral representation for symmetric stable random variables. Preliminary report.

Let 2 ;;.: p > 1. Suppose that (Y 0, Y1 , ••• ) is a sequence of jointly symmetric p-stable random variables,

and suppose that Y0 is measurable with respect to the completion of the a-field generated by the random variables (Y 1, Y2 , .•• ) . We prove that Y 0 is a limit in probability of linear combinations of the random variables

(Y1 , Y2, ••. ). (Received November 18, 1971.)

Topology

*694-Gl. ELDON J. VOUGHT, Chico State College, Chico, California 95926. A countable, connected, Urysohn space with a dispersion point that is regu1ar almost everywhere.

A countable, connected, Urysohn space with a dispersion point was constructed by P. Roy (Duke Math.

J. 33(1966), 331-333). It is well known that such a space cannot be regular but Roy asked in his paper whether it can be regular at every point of a dense subset. In this paper an example of such a space is constructed. Let

X be the set in Roy's space and let Z be the integers. A topology 0! is defined for the set Y = X U Z for which the subspace X is Roy's space. Then a topology 0! 1 coarser than 0!, i.e., 0! 1 c 0!, is constructed for Y so that the resulting space is the desired space of the title. In this topology Z is dense in Y and the space is regular at every point of Z. (Received pecember 6, 1971.)

, *694-G2. RENE P. HELD and DENIS SJERVE, University of British Columbia, Vancouver 8, British Columbia, Canada. On the Spanier conjecture.

Let L Zn+ 1 (P) be the standard lens space associated to an integer p and let f: L2n+ 1 (P) _, C Pn be the map obtained by factoring the Hopf map s2n+1 _, CPn through L Zn+l(p). The Spanier conjecture says that the

mapping cone Cf is stably homotopically equivalent (S-equivalent) to some stunted complex projective space

CPn+l+d/CPd. We say that an integer d satisfies condition S(n,p)<-> the coefficients of z0 , •• , ,zn in the power

series [log(1 + z)/z]d • ((z + 1)p -1)/pz are integers and we say that p satisfies condition (H ) <--> :B: an odd prime n q ~ n such that p ;t 0 (mod q). If w denotes the canonical complex line bundle over CPn and r is the realification functor, then our main theorem is: Theorem. (i) If Jr(WP) = (d+1)Jr(w) as elements of J(CPn) then d satisfies condition S(n,p) and Cf isS-equivalent to CPn+l+d/CPd. (ii) If n ;t 1 (mod 4) and d satisfies condition S(n,p) then Jr(Wp) = (d+1)Jr(w) and therefore Cf is S-equivalentto CPn+l+d/CPd. (iii) If Cf is

S-equivalent to CPn+l+d/CPd and n;;.: 6 then d satisfies condition S(n -1,p), Moreover, if either n is odd or p satisfies condition (Hn)' then d also satisfies condition S(n,p). (Received February 21, 1972.)

A-481 *694-G3, SANKATHA P. SINGH, Memorial University of Newfoundland, St. John's, Newfoundland,

Canada. On some fixed point theorems. Preliminary report.

Let X be a reflexive Banach space and C a closed, bounded convex subset of X. Let T: C .... c be

1-set contraction and (1-T) be convex. Then T has a fixed point. Let X be a complete metric space and

T:X""' X beadensifyingmapping. Letthereexistarealvaluedfunction g:X+X .... [O,w) suchthat g(x,y)=

0 (=) x = y. If f:X--+ R, a function defined by f(x) = g(x, Tx), is lower semicontinuous on X and {xn}' the

sequence of iterates, is bounded, then T has a fixed point (=) inf. {f(x)/x E A} = 0, where A= ~=Oxn. This

result generalizes a result due to Furi and Vignoli, (Received February 21, 1972.)

694-G4. DONALD W. ANDERSON, University of California at San Diego, La Jolla, California 92037.

Homotopy chain functors.

A covariant homotopy functor .P :_Q ... §. from a category _Q to a category of spaces.§ is a function from

the objects and morphisms of Q to those of §. which preserves sources, targets, and identity morphisms,

together with "higher homotopies" H(a0, ... , an): «P(A 0) X Dn .... .P(An+l) whenever ai: Ai ... Ai+l for 0 if i if n in _Q, which satisfy the obvious relations of compatibility on the faces of the n-cube Dn. A homotopy chain functor w is a covariant homotopy functor from the category of finite basepointed sets to _Q such that W(n+) ... (W(l+))n

is a weak equivalence for all n ~ 0. Theorem I. If X is a space with Dyer-Lashof maps which are strictly

compatible under wreath product, then there is an induced homotopy chain functor W with W(n +) = Xn.

Theorem II. If W is a homotopy chain functor for which 1T 0(W(1+)) is a group, then W(l+) is an infinite loop

space, and H(X) = ?TdwpqiJ is a homology theory. (Received March 6, 1972.)

SITUATIONS WANTED

Correspondence to applicants listed anonymously should be directed to the Editorial Department, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02904. The code which appears at the end of each anonymous listing should appear on an inside envelope in order that corre­ spondence can be forwarded by the Editorial Department.

MATHEMATICS PROFESSOR, RESEARCH, MATHEMATICS PROFESSOR, TEACHING AND TEACHING OR ADMINISTRATION. Ph.D. 1967. RESEARCH. Ph.D. 1970. Age 32. Specialty: Age 30. Specialty: analysis. Eight published algebra (ring theory). One article submitted papers. Five years of experience in research, for publication. Eight years experience in teach­ teaching and administration. References and ing and research. References and vita available vita on request. Available June 1972. V. Drobot, upon request. Available immediately. Pushpa 4246 Ridge Lea Road, Amherst, New York 14226. Gupta, Black Lantern Apt. 16, Oswego, New York 13126.

MATHEMATICS PROFESSOR, TEACHING AND MATHEMATICS PROFESSOR, TEACHING AND RESEARCH. Ph.D. 1970. Age 42. Specialty: RESEARCH. Ph.D. 1970, Age 34, Specialty: statistics. One published article and two sub­ statistics and probability. Two articles published mitted. Eleven years experience in teaching in abstract form and seven others submitted and research in U.S. East, midwest and south for publication. Thirteen years experience in preferred. References and vita available upon teaching and research. References and vita request. Available September 1972. Devendra available upon request. Available June 1972. Gupta, 118 West Seneca Street, Oswego, New Ramesh Gupta, Black Lantern Apt. 16, Oswego, York 13126. New York 13126.

A-482 STATISTICS AND MATHEMATICS PROFESSOR, cialities: algebraic topology, topology, algebra. TEACHING AND RESEARCH. Ph.D. 1960. Age Eight years teaching and research experience 43. Specialty: statistics, distributions. Seven in universities. Can teach all graduate core published articles. Eleven years experience in courses in pure mathematics. Extensive com­ teaching and academic research in U.S., Canada, puter experience. Anywhere in U.S. Available India. Eastern U.S. preferred. References, vita July 1972. References and vita upon request. upon request. Available immediately. Marakatha SW3 Krishnan, 1763 Kimball Avenue, Willow Grove, Pennsylvania 19090.

MATHEMATICS PROFESSOR, TEACHING OR MATHEMATICS PROFESSOR. Ph.D. 1970. Age TEACHING AND RESEARCH. Ph.D. 1971, Uni­ 31. Nine years experience in college teaching versity of Minnesota. Age 28. Specialty: ergodic and research. Specialty: topology. Postdoctoral theory. One published paper, one submitted. One study in computer science and applied mathe­ year full-time teaching experience in temporary matics. One published article. Available June position at Bowling Green State University. 1972. Vita and references upon request. SW4 Qualified to teach statistics and computer sci­ ence. References and vita available upon re­ quest. Available June 1972. Stephen A. McGrath, MATHEMATICS PROFESSOR, TEACHING, RE­ I 041 Varsity West, Bowling Green, Ohio 43402. SEARCH AND/OR ADMINISTRATION. Ph.D. Age 34. Ten years teaching experience. Numer­ MATHEMATICS PROFESSOR, TEACHING OR ous publications including several books. Spe­ TEACHING AND RESEARCH. Ph.D. 1971. Age cial interest: point-set topology, foundations of 27. Specialty: projective geometry and related geometry. Excellent references. Experience in­ subjects. One paper submitted, one joint paper cludes some in administration. Special qualifica­ accepted. Six years experience in teaching. One tions in curriculum development. SW5 year temporary position at the University of Iowa. References and vita available upon request. Available July 1972. Mark D. Pankin, 9 57-D MATHEMATICS PROFESSOR AND/OR ADMIN­ Westhampton, Coralville, Iowa 52240. ISTRATOR. Age 56. Thirteen years teaching experience, six as department head. Fourteen MATHEMATICIAN. Ph.D. Specialty: operations years as industrial mathematician and scientist. research. Nineteen years industrial experience, Interested in quality teaching and general educa­ accompanied by part-time graduate teaching to tion. Qualified for department head, dean, or full professorship level. Over a dozen research academic vice president in small liberal arts papers published; associate editor of two math. college or junior college. SW6 journals. Proved researcher; outstanding teach­ er; qualified administrator. B. L. Schwartz, COMPUTER SCIENCE PROFESSOR, TEACHING 1626 Macon Street, McLean, Virginia 22101. AND RESEARCH. Ph.D. 1968. Age 40. Interests: automata, theory of computation, formal lan­ MATHEMATICS PROFESSOR, TEACHING AND guages. Thirteen years experience in teaching RESEARCH. Ph.D. Summer 1972. Age 30. Spe­ and research. References and vita available cialty: general topology. Three published arti­ upon request. Available June 1972. SW7 cles, others submitted. Six years experience in teaching and research. New York area preferred. References upon request. Available June 1972. Frank Siwiec, 164-20 Highland Avenue, Apt. 6L, MATHEMATICS PROFESSOR, TEACHING AND .Jamaica, New York 11432. RESEARCH. Ph.D. 1967. Age 31. Speciality: analysis and topological algebra. Six published ANONYMOUS articles. Five years experience in teaching graduates and undergraduates and in academic MATHEMATICS PROFESSOR, TEACHING AND research, including one year in a British uni­ RESEARCH OR TEACHING. Ph.D. 1966. Age 32. versity. References and vita upon request. Avail­ Four published articles, one submitted. Spe- able immediately. SW8

A-483 K~eyszig is c:vmiRg!

Here's what's new in this edition: -The problems have been changed. -There's a new chapter on numerical methods for engineers. -The material on linear algebra and analysis has been revised and extended. -And there's also an elementary introduction to vector spaces, inner product spaces, and other concepts important in functional analysis and its applications.

Contents: Ordinary Differential Equations of the First Order. Ordinary Linear Differential Equations. Power Series Solutions of Differential Equations. Laplace Transformation. Linear Algebra Part 1: Vectors. Linear Algebra Part II: Matrices and Determinants. Vector Differential Calculus-Vector Fields. Line and Surface Integrals- Integral Theorems. Fourier Series and Integrals. Partial Differential Equations. Complex Analytic Functions. Conformal Mapping. Complex Integrals. Sequences and Series. Taylor and Laurent Series. Integration by the Method of Residues. Complex Analytic Functions and Potential Theory. Numerical Analysis. Probability and Statistics. Ap­ pendixes. Index.

ADVANCED ENGINEERING MATHEMATICS 3rd Edition By Erwin Kreyszig, Ohio State University 1972 In Press

For more information, contact your local Wiley representative, or write Ben Bean, Dept. 909-B, New York office. Please include your course title, enrollment, and present text. All prices subject to change without notice.

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A-485 FUNDAMENTALS OF ELEMENTARY MATHEMATICS

NUMBER SYSTEMS AND ALGEBRA MERLYN J. BEHR and DALE G. JUNGST, both at Northern Illinois University

REVIEW: " ... The authors have succeeded admirably in writing a unique book to meet the needs of elementary and junior high school teachers .... A com­ bination of deep treatment and intuitive motivation, a direct effort to develop the student's skill in writing proofs, and very extensive problems are the distinguishing features of this important book." -Science Books, December 1971

The authors believe that the prospective elementary or junior high school teacher should have, in addition to an appreciation of and skill with deduc­ tive reasoning, and understanding of mathematics at a level of sophistication much higher than the specific content he will teach. Therefore, this textbook emphasizes the structural and developmental nature of mathematics. It covers logic, sets, relations and functions, number and numeration systems, solution sets and graphs of mathematical sentences, abstract systems, and other appropriate topics. Terminology and symbolism are carefully intro­ duced, defined, illustrated, and used meaningfully. The exceptionally large number of examples, figures, and exercises will facilitate mastery of concepts.

The book is flexible in at least two ways: First, the format enables the instruc­ tor either to proceed in a rigorous and careful manner or to teach a more informal course, omitting proofs and concentrating on examples. Second, it will be useful in courses of varying length, from one quarter to two semesters. The exercises are of many types and varying degrees of difficulty. Not only do they serve to clarify and extend the textual material, they lead the student to discover basic concepts and to develop the ability to construct proofs. They also demonstrate the relevance of the mathematics to the teaching situation.

CONTENTS: An Introduction to Logic and Mathematical Reasoning. Sets Relations, Functions, and Operations. The System of Whole Numbers. Numeration Systems. Algorithms for Computation with Whole Numbers. The System of Fractional Numbers. The System of Integers. The System of Rational Numbers. Decimal Numerals for Fractional and Rational Numbers. Introduction to the Real Numbers. Abstract Sys­ tems. Answers to Odd Numbered Exercises. Answer booklet available to instructors. 1971, 419 pp., $9.00

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A-486 FUNDAMENTALS OF ELEMENTARY MATHEMATICS

GEOMETRY MERLYN J. BEHR and DALE G. JUNGST, both at Northern Illinois University The topics, especially chosen for prospec­ tive, and Equivalence Relations. Functions. tive junior high school and elementary FINITE GEOMETRY AND AN INTRODUCTION TO school teachers, include finite geometry; a EUCLIDEAN GEOMETRY: Introduction. A Finite short introduction to a deductive deve!op­ Geometry. A Deductive Introduction to Euclidean Geometry. General Comments and ment of Euclidean geometry; separation Exercises. SUBSETS OF LINES: Introduction. Lines and Line and subsets of lines; distance and its mea­ Segments. Convex Sets, Separation of Lines, sure; measure and congruence of line More Subsets of Lines. Separation of Planes, segments with careful attention given to the Angles. DISTANCE, MEASURE, CONGRUENCE application of the theory of measure; defini­ OF LINE SEGMENTS AND ANGLES: Number tion and classification of curves; mea­ Systems. Distance and Its Measure. Measure of sure of curves; congruence and similarity; Line Segments, Congruent Line Segments. Appli­ straightedge and compass constructions; cation of the Theory of Measure. Measure of definitions, classification and measure of Angles, Congruent Angles. Classification of Angles, Perpendicularity. Transversals and Par­ regions; definition, classification and mea­ allel Lines. CURVES: Definition and Classification sure of solids; and transformation geometry. of Curves. Some Special Closed Curves, Interior The material is organized sequentially, and Exterior. Measure of Curves. The Practice of and the authors stress the structural and Approximating the Measure of Union-of-Line­ developmental nature of mathematics. Ter­ Segments Curves. CONGRUENCE AND SIM­ ILARITY: Classification of Polygons. Congruent minology and symbolism are carefully intro­ Triangles. Minimum Conditions for Triangle Con­ duced, defined, illustrated and meaning­ gruence. Congruence of Polygons, Circles, and fully used. The exposition is characterized Congruence in General. Similarity. Straightedge by attention to detail. The exceptionally and Compass Constructions. REGIONS, MEA­ large number of examples, figures, and SURE OF REGIONS: Definition of a Region. Mea­ exercises will facilitate the learner's mas­ sure of Regions. The Practice of Approximating tery of mathematical concepts. Like its the Measure of Regions. Measure of Triangular predecessor, which was devoted to "Num­ Regions. Measure of Other Regions. SUBSETS OF SPACE-SURFACES ber Systems and Algebra," this textbook AND SOLIDS: Definition of Surface and Sphere. Definition and Classifica­ exemplifies the authors' philosophy that tion of Polyhedrons. Measure of Polyhedrons. prospective teachers should have an ap­ Circular Cylinders, Circular Cones, and Their preciation of and skill with deductive rea­ Measures. Solids, Measure of Solids. In Retro­ soning and must have an understanding of spect. OPERATIONS, GROUPS, TRANSFORMA­ mathematics at a level of sophistication TIONS, TRANSFORMATION GROUPS: Opera­ higher than the specific content they will tions. Definition of Group. Functions and Groups. teach. TRANSFORMATIONS OF A PLANE-MOTION GEOMETRY: Introduction and General Remarks. CONTENTS: Preface. Acknowledgments. Glos­ Rigid Motion of a Plane. Translations of a Plane. sary of Symbols and Notations. SETS, RELA­ Relations of a Plane. Reflections of a Plane. Sym­ TIONS, FUNCTIONS, AND LOGIC: The Develop­ metry. Overview. ANSWERS AND SUGGES­ ment of Mathematics. The Set Concept and Set TIONS FOR SELECTED EXERCISES. Subject Notation. Relations on Sets. Finite and Infinite Index. Sets. A Brief Introduction to Logic. Operations Answer booklet available to instructors. on Sets. Relations. Reflexive, Symmetric, Transi- February 1972, 326 pp., $9.50

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A-487 Major Mathe~natics Texts froiD Mac~nillan

PROBABILITY AND STATISTICS FOR ENGINEERS AND SCIENTISTS By Ronald E. Walpole, Roanoke College, and Raymond Meyers, Virginia Polytechnic Institute This text is designed to introduce probability theory, statistical inference, and analysis-of­ variance techniques to students majoring in engineering, math, and science. Numerous illustra­ tive examples and exercises are included. 1972 approx. 512 pages $13.95

MATHEMATICAL STATISTICAL MECHANICS By Colin J. Thompson, Northwestern University Concerned with the mathematical rather than the physical aspects of statistical mechanics, this text requires little knowledge of physics and features the mathematical aspects of kinetic theory, statistical mechanics, and biology. 1972 278 pages $12.75

ELEMENTS OF DECISION THEORY By B. W. Lindgren, University of Minnesota Probability decision theory, game theory, and the simplex method of solving linear program­ ming problems are covered. Previous knowledge of calculus is not required, and important concepts in probability are discussed. 1971 292 pages $8.95

STATISTICAL DESIGN AND ANALYSIS OF EXPERIMENTS By Peter W. M. John, The University of Texas A text for statistical design or design of experiments courses, it requires a solid background in matrix algebra and introductory mathematical statistics for complete understanding. The development of statistical logic for the design and analysis of research is covered, including analysis of variance and the complete fractional experiment. 1971 356 pages $14.95

For further information write to: THE MACMILLAN COMPANY Department C, Riverside, New Jersey 08075 In Canada, write to Collier-Macmillan Canada, Ltd., 1125B Leslie Street, Don Mills, Ontario

A-488 McGraw-Hill Book ComPany

ELEMENTARY LINEAR AND MATRIX ALGEBRA John T. Moore, University of Western Ontario 1972, 288 pages, {04291 0-3), $9.95 This text considers matrices both as rectangular arrays of numbers and as representations of linear transformations of vector spaces. Linear algebra, therefore, is approached basically through linear transforma­ tions (or operations) rather than through matrices. Much of the mate­ rial discussed in the book is motivated by geometry, but geometric analysis is not limited entirely to the second and third dimensions. Complex numbers are excluded as scalars so that the "reality" of the vector spaces will be more apparent.

UNDERSTANDING STATISTICS Arnold R. Naiman, Gene Zirkel, and Robert Rosenfeld, all of Nassau Community College, Garden City, New York 1972, 256 pages, (045790-5), $8.50 Promoting an understanding of the proper use of statistics and statis­ tical terms, this text is at once elementary and mathematically sound. Students with little mathematics background can comprehend the ideas presented within the scope of a one-semester course. An intuitive approach to basic concepts is utilized, and concrete examples are drawn from many fields, such as biology, education, and political science. The humorous treatment of many of these exercises renders them more than just uninspiring lists of numbers, and contemporary issues are also highlighted to arouse the student's interest.

COLLEGE ALGEBRA, Sixth Edition Paul K. Rees, Professor Emeritus, Louisiana State University, and Fred W. Sparks, Professor Emeritus, Texas Tech University 1972, 576 pages, (051696-0), $9.95 Containing a thorough treatment of the number system, fundamental operations, fractions, exponents and radicals, and equations and in­ equalities, this sixth edition uses the theory and terminology of sets throughout and employs field axioms in all proofs and procedures. Each new concept is accompanied by at least one example in order to clarify the idea involved. The exercises are a lesson apart, and each exercise contains similar problems in groups of four for ease in pre­ paring class assignments. The treatment of inequalities has been expanded and modernized, particularly systems of linear inequalities.

AND THERE'S MORE, ••

A-489 CALCULUS AND ANALYTIC GEOMETRY

William H. Durfee, Mount Holyoke College 1971, 736 pages, (018378-3), $12.95

Guiding the student toward an intuitive understanding of the calculus without sacrificing accuracy, this introductory text develops the sub­ ject logically, assuming only the properties of the real numbers. Every effort has been made to simplify the calculus and its proofs. The easy readability and detailed explanations, as well as the many figures, relieve the instructor of the necessity of completely covering each topic in class. Intended primarily for a one-year course at the college level in liberal arts or engineering curriculum, the book is also suitable for advanced placement courses in secondary schools or for a one semester introductory course.

FINITE MATHEMATICS: A Liberal Arts Approach

Irving Allen Dodes, Kingsborough Community College, City University of New York 1970, 352 pages, (017250-1), $8.95

This book provides a liberal arts approach to finite mathematics, the mathematics employed in industry, commerce, business, and data processing. The author has minimized the discussion of those topics which are mainly of professional interest and has emphasized "living topics" such as linear programming and the theory of games. All the material is relevant to our twentieth-century technological society. The book is exciting and stimulating from the start; direct involvement with the subject is maintained throughout.

PRECALCULUS MATHEMATICS

Kenneth Henderson, University of Illinois; Zalman Usiskin, The University of Chicago; and Wilson Zaring, University of Illinois 1971,720 pages, (028152-1), $11.95

Prepares the student to take a course in calculus or in calculus and analytic geometry. The first part of the text reviews material the stu­ dent has previously covered from approaches designed for best calculus preparation. Matrices, vectors, and transformations are used where appropriate to develop some analytic geometry. Limits and sequences are covered in detail. Mathematical induction and other types of proofs are introduced early and are used extensively throughout.

A-490 PLANE TRIGONOMETRY WITH TABLES, Fourth Edition Gordon Fuller, Professor Emeritus, Texas Tech University 1972, 271 pages, (022608-3), $8.95 This fourth edition is well suited for students who need sufficient mastery of trigonometry for use in analytic geometry, calculus, and more advanced mathematics. All topics and especially appropriate features contained in the third edition have been retained. The organ­ ization of the text has been changed, and the exposition improved in many places. In particular, some of the worked examples have been revised and others added to achieve a closer correlation with the exercise problems.

FUNDAMENTALS OF FRESHMAN MATHEMATICS, Third Edition Carl B. Allendoerfer, University of Washington, and Cletus 0. Oakley, Haverford College 1972, 648 pages, (001366-7), $10.95 This edition continues to serve the pre-calculus course or the termi­ nating course for business and liberal arts students in general. It assumes a knowledge of basic intermediate algebra (and includes adequate review for those students whose background is weak), but does not presuppose any trigonometric experience. Presenting a judi­ cious balance between theory, manipulation, and application, the text incorporates recent advances in mathematical education without dis­ regarding the virtues of many traditional courses.

OPERATIONAL MATHEMATICS, Third Edition Ruel V. Churchill, Professor Emeritus, University of Michigan 1972,496 pages, (010870-6), $10.95 This revised and expanded third edition is a textbook and reference on the theory and applications of those integral transformations that are most useful in solving problems in linear differential equations. The first half of the book presents a careful revision of the theory and applications of Laplace transforms. Additional material includes chap­ ters on finite and exponential Fourier transforms, Fourier transforms on the half line, and Hankel, Legendre, and other integral transforms. Previously unpublished material on general integral transforms, some of which was developed by the author, is also contained.

Address all orders to: McGraw-Hill .Book Company Norma-Jeanne Bruce 330 West 42nd Street College Division, 13 New York, New York 10036

Prices subject to change without notice. Offer good in Continental USA only.

A-491 DIFFERENTIAL GAMES 4 VOLUMES By Avner Friedman, Northwestern University This book provides the mathematical founda­ tions for the theory of differential games. The IN PURE author defines a game as a certain limiting process and explains concepts of value and AND APPLIED strategy. In the following chapters, it is proven that under general settings the game has value MATHEMATICS: and saddle points. Computational formulas de­ rived in previous works in a heuristic manner A Series of Texts and Monographs, are obtained by rigorous methods, and the notion of an 7)-saddle edited by R. Courant, point is introduced. 1971 360 pages illus. $17.50 L. Bers, and J. J. Stoker APPROXIMATION OF LINEAR OPERATORS ELLIPTIC BOUNDARY· Parts I, II, Ill VALUE PROBLEMS By Nelson Dunford, Former James E. English By Jean-Pierre Aubin, Universite de Paris Professor of Mathematics, Yale University, and IX-Dauphine Jacob T. Schwartz, Courant Institute of This is a rigorous mathematical approach to Mathematical Sciences, New York University the foundations of the numerical analysis of Part I: General Theory non-homogeneous boundary-value problems. "A tremendous enterprise ... The authors The author reformulates, at a highly sophisti­ deserve thanks for their labors and congrat­ cated and abstract level, various approximation ulations on their achievement." methods of boundary-value problems-con­ -Bulletin of the American Mathematical Society centrating on the Rayleigh-Ritz-Galerkin 1958 858 pages $24.95 method and the method of finite differences. Part II: Spectral Theory (Self Adjoint Within the framework of the language of func­ Operators in Hilbert Space) tional analysis, the interrelationships between "There can be no doubt that this book repre­ these methods are clarified, and developed sents a remarkable achievement ... into the variational finite element method. " ... it deals in depth with so many interest­ ing and highly non-trivial applications." 1972 368 pages illus. $17.95 -Bulletin of the London Mathematical Society CONCEPTS OF 1963 1072 pages $37.50 Part Ill: Spectral Operators COMMUNICATION Recently published, Part Three of Linear Interpersonal, lntrapersonal, and Operators covers spectral theory of non-self­ Mathematical adjoint operators and discusses the similarity Edited by Edwin F. Beckenbach and Charles theory of operators due to Friedrichs and Kato. B. Tompkins, both of the University of 1971 720 pages $32.50 California, Los Angeles A volume in the University of California Engineering and Physical Sciences Extension ALGEBRAIC NUMBERS Series By Paulo Ribenboim, Queen's University, In this volume, the editors have sought to Kingston, Ontario, Canada bring together the important contributions from Here is a successful, up-to-date approach to various disciplines so that workers in different the theory of algebraic numbers. Since the fields can understand and profit from the know­ theory of algebraic numbers requires results ledge, experience, and results of others. The from ring theory and also the structure of fi­ first part of the book is concerned with inter­ nitely generated modules over principal ideal personal communication systems which reports domains, the author chooses to develop these on communication among humans. The second theorems as the necessity arises, and in each part deals with intrapersonal communication case, he immediately derives applications of systems. It includes such topics as neural in­ the property in question. The material included formation processing and information storage in Algebraic Numbers was recently taught to in the brain. The third part covers the mathe­ senior undergraduate and graduate students at matical aspects of communication sciences, Queen's University, Ontario, and proved to be binding the material of the first two parts to­ a highly successful approach to the theory of gether through mathematical models and algebraic numbers. analysis. 1972 314 pages illus. $17.95 1971 446 pages 170 illus. $19.95

A-492 APPLIED FUNCTIONS OF APPLIED GRAPH THEORY A COMPLEX VARIABLE By Clifford W. Marshall, Polytechnic Institute By A. Kyrala, Arizona State University of Brooklyn This book can be used both as an intro­ Covering a wide range of material from the duction to and a review of complex variable field of linear graphs, Applied Graph Theory methods and applications. In addition to dis­ presents all the basic concepts and provides cussions on electrostatics, magnetostatics, hy­ a sound basis for utilizing the theory in several drodynamics, elasticity, Green's functions by applications. Illustrative applications are pre­ conformal mapping, and Laplace and Fourier sented from the social sciences, physics, opera­ transformations, the authors discuss topics tions research, and related fields. Concepts of generally not covered in previous books on random graphs not found in other texts are the subject. These include- discussed because of their increasing impor­ tance in theory and applications. • Dispersion relations • Hilbert Transforms 1971 336 pages 159111us. $17.50 • Plemelj formulae • Bandwidth-duration and uncertainty ELEMENTS OF APPLIED principles • Linear systems and stability STOCHASTIC PROCESSES • Filters By U. Narayan Bhat, Southern Methodist • Servomechanisms and feedback amplifiers University, Dallas, Texas Special features of this volume include a chart that shows interrelationships of theorems and A volume in the Wiley Series in Probability subject areas, a table of conformal mappings and Mathematical Statistics, edited by with illustrations, a short table of Laplace trans­ R. A. Bradley, J. S. Hunter, D. G. Kendall, forms, a short table of Fourier transforms, and and G. S. Watson a table of normalized Sturm-Liouville eigen­ Elements of Applied Stochastic Processes functions and eigenvalues. crosses the boundaries of a wide range of dis­ ciplines: operations research, traffic engineer­ 1972 374 pages illus. $17.95 ing, mathematical statistics, etc. The author's unified approach will guide the reader to a ALGEBRAIC METHODS IN firm understanding of stochastic elements and supply him with the necessary tools for modi­ STATISTICAL MECHANICS fying information to fit his particular need. AND QUANTUM FIELD 1972 400 pages illus. $17.50 THEORY By Gerard G. Emch, University of Rochester GRAPH THEORY A volume In lnterscience Monographs and By Wataru Mayeda, University of Illinois Texts in Physics and Astronomy, This book covers almost all the essential edited by R. E. Marshak properties of linear graphs. After the properties This volume offers a systematic algebraic are developed systematically, they are applied approach to problems which involve a large to electrical networks, switching networks, com­ number of degrees of freedom. This approach munication and transportation systems, and extends the traditional formalism of quantum system diagnosis. Numerous figures and mechanics, and it eliminates conceptual and examples supplement the text. mathematical difficulties that have plagued the 1972 608 pages 450 illus. $24.95 development of statistical mechanics and quantum field theory. By the same token, the Available from your bookstore or from Dept. 092 approach is linked to recent research in applied and pure mathematics. The dual interests of this approach are reflected in the interplay between a careful formulation of the physical lllill!l motivations and a self-contained description of the mathematical methods. These two features enable the reader to know the physical bases WILEY -INTERSCIENCE of the theory and to become fluent in the vo­ a division of JOHN WILEY & SONS, Inc. cabulary and grammar of this branch of mathe­ 605 Third Avenue, New York, N.Y. 10016 matical physics. 1972 352 pages $19.95 In Canada: 22 Worcester Road, Rexdale, Ontario

A-493 ------~

One ofthe glories ofthe human intellect, afounding document of our modern exact science.

ISAAC NEWTON'S PHILOSOPHIAE NATURALIS PRINCIPIA MATHEMATICA The Third Edition ( 1726) with variant readings Assembled and edited by I. Bernard Cohen and Alexandre Koyre. Assisted by Anne Whitman This is the first edition of the 'Principia' that enables the reader to see at a glance the stages of evolution of the work from the completion of the manuscript draft of the first edition in 1685 to the publication of the third edition in 1726. A photographic reprint of this final version, the present edition exhibits on the same page the variant readings from seven other texts. A series I of appendices provides additional material on the development I of the 'Principia'. Two Volume Set, $60.00 ~

INTRODUCTION TO i NEWTON'S 'PRINCIPIA' By I. Bernard Cohen This work serves both as a guide to Newton's 'Principia' and as a i biography of the work itself. It provides an analytic history of how Newton came to write the book, and-above all-how Newton continually corrected, revised, and enlarged the text over four decades. $30.00

Harvard University Press Cambridge, Massachusetts 02138 Cambridge University Press London N. W. 1, England i@!1 ~~~~~~~@~~~~~~~~~~~~~~~~m@~~~~~~j

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MATRICES AND LINEAR TRANSFORMA­ THEORY OF GROUPS TIONS, by Homer Bechtell, University of New Hamp­ SECOND EDITION shire by Charles G. Cullen, University of Pittsburgh This one-semester text bridges the gap between While remaining essentially the same in its over­ a graduate introductory algebra course and all coverage and content, this second edition more advanced work in group theory, and is has expanded its treatment of metric concepts suitable for independent study as well as course in Euclidean and Unitary spaces, Cauchy­ work. The central thread is extension theory Schwartz and triangle inequalities, and orthogo­ which is intertwined with other basic concepts nality. Along with a greater use of mapping and of group theory for application to the structure lattice diagrams, the text also includes many of a group. new examples and exercises. 144 pp (1971) $10.50 Aprill972 A SURVEY OF NUMERICAL SET THEORY MATHEMATICS, VOLUME I by Charles C. Pinter, Bucknell University by David M. Young and Robert Todd Gregory, A presentation of fundamental topics of set The University of Texas at Austin theory within the framework of an informal Written for the standard one-semester course in axiomatic system, this book begins with easy numerical mathematics at the advanced under­ concepts and simple proofs and slowly rises, graduate level, this text is designed to study through gradual stages, to the more difficult computer-oriented numerical algorithms for notions of set theory. A large number of signifi­ solving various types of mathematical problems. cant exercises are included. The book blends classical mathematics, numer­ 216 pp, 28 illus (1971) $8.95 ical mathematics, and computation. April1972 METHODS IN CLASSICAL AND FUNCTIONAL ANALYSIS NUMERICAL ANALYSIS AND COMPUT A­ by Einar Hille, University of New Mexico TION: THEORY AND PRACTICE Designed for courses in modern analysis, this by E. K. Blum, University of Southern Cali· advanced undergraduate and graduate level text fornia emphasizes methods of construction more than This first-year graduate text for students of general theory, and gives an historical perspec­ computer science, applied mathematics and sci­ tive on the subject. Illustrated examples are ence, and engineering is a self-contained treat­ worked out throughout the book, and each sec­ ment of the first principles of numerical analy­ tion contains exercises. sis. Functional analysis methods are used to 486 pp (1971) $13.95 unify the treatment of finite and infinite­ dimensional problems, while numerical analysis DIFFERENTIAL MANIFOLDS is viewed as a link between discrete and contin­ by Serge Lang uous mathematics. Superseding the author's Introduction to Dif· March 1972 ferential Manifolds, this expanded version fills the gap in existing literature between advanced calculus and the three great differential theories Addison-Wesley THE SIGN OF EXCELLENCE (differential topology, differential geometry, PUBLISHING COMPANY, INC. ordinary differential equations) and covers con­ Reading, Massachusetts 01867 cepts essential to all three. 230 pp (1972) $12.95

A-495 Summing up ...

Oscar Zariski: Collected Papers The Collected Works of Leo Szilard: Volume 1: Foundations of Algebraic Geometry and Scientific Papers Resolution of Singularities edited by Bernard T. Feld and Gertrud Weiss Szilard edited by H. Hironaka and D. Mumford with Kathleen R. Winsor By introducing ideas from abstract algebra into alge­ This volume contains the full scientific output of one braic geometry, Zariski undertook to rewrite its of the greatest and most fertile minds of this century. foundations completely, taking an approach that The subtitle, Scientific Papers, is exact, for Leo made no use whatever of topological or convergent Szilard's work ranged from physics to biology; he power series methods and that made no appeal to crossed over with ease from the world of neutrons to vague geometric intuition. The most important char­ that of neurons, from the nuclei of atoms to the acteristic of this approach toward algebraic geom­ nuclei of cells. etry, and in particular toward the problem of resolu­ Further volumes will present Szilard's social and tion of singularities, is that it uses the available power political writings, and biographical material. of modern algebra as fully as possible not only as a $15.00 through April 30, 1972; source of techniques in each step of solving a spe­ $17.50 thereafter cific problem but also in reformulating the problem Differentiable Dynamics: at a fundamental level. Professor Hironaka writes An Introduction to the Orbit Structure of that "By this type of fundamental approach (not to Diffeomorphisms mention specific techniques he invented to over­ by Zbigniew Nitecki come specific difficulties in the problem), he made it much easier for other mathematicians in later works Specialists teaching courses in this field as well as to follow the tracks and make further progress." nonexperts interested in a comprehensive introduc­ Oscar Zariski: Collected Papers is part of a new tion should welcome Differentiable Dynamics, the series, Mathematicians of Our Time, edited by first work to collect and explain in detail a wide selec­ Gian-Carlo Rota. tion of results and techniques which have formerly $15.00 been scattered in the primary literature. Approaching this literature directly can moreover Abstract Lie Algebras be somewhat treacherous since a number of obso­ by David J. Winter lete results are embedded within it. In this regard, it One purpose of this book is to give a solid but com­ is worth noting that one expert in this area who pact account of the theory of Lie algebras over fields examined the book in manuscript wrote that "Nitecki of characteristic 0, with emphasis on the basic sim­ has it seems to me made an admirable choice of plicity of the theory and on new approaches to the material -both in what he has put in and what he major theorems. Another is to give a general and has left out." Some of this material is so recent that extensive treatment of Cartan and related subalge­ at present it exists only in preprint form. bras of Lie algebras over arbitrary fields. $12.50, hardcover $12.00 $6.50, paperback The MIT Press Massachusetts Institute of Technology r..,mhrirln<> M""""~"'h"""tt" n?1.4?

A-496 More Math Texts From W A. Benjamin, Inc.

THE GREEN BOOK OF CALCULUS PERSPECTIVES IN MATHEMATICS by Joshua Chover, University of Wisconsin by David E. Penney, The University of Georgia This new text offers a fresh, student-oriented This text is designed for use in courses for approach for the introductory calculus course. nonscience majors at the freshman/sophomore Distinguishing features include: level who do not plan to go on to calculus but • Problems and worked-out examples from who want to be exposed to some college level such diverse fields as economics, biology, mathematics. Much of the material will be new and physics. to the student since the topics were deliberately • A lively and informal writing style chosen for their intrinsic interest in order to through which the author engages the avoid the typical rehash of high school algebra. reader in exploratory discussions to give The ten chapters in the book, which together him the feeling of the interplay of cover the whole spectrum of modern mathe­ idealized concepts and practical applica­ matics, are wholly independent, allowing the tions. instructor to select whatever material he thinks · "Byways" in green type that offer appropriate. optional material for deeper study by 349 pp, 110 illus (1972) $9.95 the motivated student. · A continuing (but optional) sequence of C.E.M. (COMMITTEE ON EDUCATIONAL probabilistic discussions interwoven in MEDIA) the text. A PROGRAMED COURSE IN CALCULUS • A brief, intuitive introduction to multi­ Vol. I: Functions, Limits, variable calculus providing a transition to and the Derivative 296 pp, illus further courses in multivariable calculus, Vol. II: The Definite Integral linear algebra, and differential equations 130 pp, illus for physical science majors. Vol. III: February Transcendental Functions 1972 130 pp, illus Vol. IV: Applications STUDY GUIDE FOR THE GREEN and Techniques BOOK OF of Integration 214 pp, illus CALCULUS Vol. by Norman Locksley, University V: Infinite Sequences and Series of Maryland 122 pp, illus Following the chapters of the text, this study All volumes paperbound (1968) $3.95 each guide reviews major aspects of each chapter. It Complete set, $16.50 also reviews one important and typical example These programed volumes are equally effective from the text, working through the example as main texts in introductory calculus courses, exhaustively, both verbally and mathematically. as Sample test questions are included. supplements to other traditional texts, or as refresher material. Detailed answers and explana­ Spring 1972 tions are provided in the summaries, self-tests, reviews, and exercises. The set is the result of four years of preparation, classtesting, and revision by the Programed Learning Project of the Committee on Educational Media, an affil­ iate of the Mathematical Association of ..,~+\A.t1~ America. Directed by Brewster Gere, of Hamil­ < ·- -·;:::: zn ton College, this committee of educators repre­ 1- t fiJ W. A. BENJAMIN, INC. sented schools and other educational institu­ ....8 lis"-"' Reading, Massachusetts 01867 tions throughout the United States.

A-497 The University of Minnesota Morris, a small undergraduate liberal arts college of approximately 1800 students, invites AN ENGINEERING mathematicians or statisticians to apply APPROACH TO for the Chairmanship of the Division of Science and Mathematics, appointment LINEAR ALGEBRA beginning in the fall of 1972. Rank and W. W. SAWYER salary are open. Applicants should hold Professor of Mathematics, the Ph.D. degree and an associate or full University of Toronto professorship. Half-time teaching is re· quired, and administrative experience is A freshman text for non-mathematics ma­ desirable. Persons having special expertise jors in engineering and other fields. Empha­ in statistics or applied mathematics are sis is placed on the significance of the preferred. Others will be considered. mathematical techniques and the concepts Interested candidates should apply to: embodied in the formal calculations are clearly shown. Long and rigorous treat­ ments are eliminated. Many physical analogies and worked and unworked examples are integrated into the Professor W. D. Spring, Chairman text. There are 150 line illustrations. $11.50 Search and Screening CommiHee University of Minnesota, Morris A COURSE OF GEOMETRY Morris, Minnesota 56267 FOR COLLEGES AND UNIVERSITIES D. PEDOE Professor of Mathematics, University of Minnesota A stimulating textbook covering all the geometry normally required in undergrad­ INDEX uate and first-year graduate courses, except for the theory of convex sets. The author TO develops Euclidean and projective geometry in 2, 3 and n dimensions by means of map­ ADVERTISERS pings, transformations, and geometrical representations. The book is self-contained Academic Press ...... A-486, A-487, cover 3 and the material may be used over several Addison-Wesley Publishing Co...... A-495 semesters. American Mathematical Society ...... A-485 The text, which has been successfully tried W. A. Benjamin, Inc ...... A-497 out in a number of universities, contains Cambridge University Press ...... A-498 many examples and over 500 exercises. The author is an internationally knoWn. geom· Harvard University Press ...... A-494 . eter with many years of experience teaching The Macmillan Co ...... A-488 in colleges and universities. McGraw-Hill Book Co... A-489, A-490, A-491 "A heartening sign of a long-needed UniversityofMinnesota ...... A-498 revival in the teaching of geometry."­ TheM.I. T. Press ...... A-496 American Mathematical Monthly $11.50 Springer-Verlag New York, Inc ...... cover 4 John Wiley & Sons, Inc ... A-484, A-492, A-493 Cambridge University Press 32 East 57th Street New York, N.Y. 10022

A-498 DIFFERENTIAL EQUATIONS IN ABSTRACT SPACES by G. E. LADAS and V. LAKSHMIKANTHAM, Department of Mathematics University of Rhode Island, Kingston The theory of differential equations in abstract literature; and 3) it provides the specialist with spaces is a fascinating field with important ap­ an excellent and wide-ranging reference book. plications to many areas of analysis and other Special features of the book: treats the calculus branches of mathematics. Nevertheless, until of abstract functions and the theory of semi­ now, no single book has been self-contained groups of operators used in connection with and simple enough to appeal to the beginner, differential equations in Banach spaces; in­ yet comprehensive enough to be of value to cludes numerous examples and applications to the expert. Here is a book, however, which ac­ functional and partial differential equations to complishes three main objectives: 1) it intro­ illustrate the abstract results developed; and duces the beginner and nonspecialist to this covers the theory of linear evolution equations. elegant theory and its powerful techniques; 2) it prepares the reader for the current published April 1972, 218 pp., $14.50 MEASURE AND INTEGRATION THEORY ON INFINITE-DIMENSIONAL SPACES: Abstract Harmonic Analysis by XIA DAO-XING, Fudan University, Shanghai, China Translated by ELMER J. BRODY, Chinese University of Hong Kong Introducing the projective limit concept for results along with those of other Chinese re- measure spaces, the author generalizes the searchers. Subsequently, the theory is special- classical Kolmogorov theorem, and uses infi- ized to Gaussian measures, including Wiener nita-dimensional measure techniques to present measure in particular. The book ends with a a novel and illuminating approach to the repre- lucid and concise account of the applications of sentation of positive functionals on topological quasi-invariant measures to representations of algebras. He then develops a very general the commutation relations in quantum field theory of quasi-invariant measures in consider- theory. able detail, and presents many of his own recent May 1972, about 420 pp., in preparation NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN ENGINEERING, Volume 2 by WILLIAM F. AMES, University of Iowa FROM A REVIEW OF VOLUME 1: "The erudition throughout a wide range of technical journals. that has gone into the compilation is matched It features a wealth of examples from a variety by the skill of the presentation .... a most valu­ of scientific and engineering disciplines, includ­ able addition to the mathematical literature .... " ing wave propagation, vibrations, boundary lay- J. Gillis in PHYSICS TODAY ers, elasticity, and diffusion. · This volume consolidates information scattered May 1972, about 300 pp., $16.50 INTEGRAL MATRICES by MORRIS NEWMAN, U.S. Department of Commerce National Bureau of Standards, Washington, D.C. This book develops the theory of matrices over CONTENTS: Background Material on Rings. rings-in particular, principal ideal rings, and Equivalence. Similarity. Congruence. Combined includes many applications. It provides the nec­ Similarity and Congruence. The Geometry of essary background material, discusses various Numbers. Groups of Matrices. The Classical kinds of equivalence relationships that may be Modular Group and Related Groups. Finite defined for matrices, and develops the classical Matrix Groups. Circulants. Quadratic Forms. theory of matrices over a field. The book is References. Index. eminently suitable as a first-year graduate text. Apri/1972, about 225 pp., $14.00 METHODS OF INTERMEDIATE PROBLEMS FOR EIGENVALUES: Theory and Ramifications by ALEXANDER WEINSTEIN, Department of Mathematics, Georgetown Univ., Washington, D.C. and WILLIAM STENGER, Department of Mathematics, Ambassador College, Pasadena, Calif. This book presents in detail two complementary gives on one hand several contributions to methods for the computation of upper and lower functional analysis, but it also includes for the bounds for eigenvalues, namely the Poincare- first time in book form some new devices for Rayleigh-Ritz Method and the Weinstein Method high precision computation of lower bounds, of Intermediate Problems and its variants. It such as for vibrations of cantilever plates, for begins with classical and modern variational the Mathieu equation, and for energy levels in principles and ends with a new discussion of quantum mechanics. perturbations of finite rank. The monograph 1972, 234 pp., $14.50

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Applied Mathematical Sciences Vol.234: Andreotti, A., W.Stoll: Vol. 4: Parcus, J.: Analytic and Algebraic > CI::ICI::t Dependance of Maromorphlc Combinatorial Methods 58 fig. IX, 194 pages. 1971 a Functions. Ill, 390 pages 1971. OM 26,-; US 8.10 w OM 24,-; US$ 7.50 S Vol. 5: Mlses, R. von, 1 Vol. 235: Global Dlflarentlable =-=- Dynamics K. 0. Friedrichs: Fluid Dynamics t ~ 216 fig. IX, 353 pages. 1971 1 $ Proceedings of the Conference, OM 24,-; US$ 7.50 held at Case Western ~ Reserve University, Cleveland, - Vol. 6: Freiberger, W., Ohio, June 2-6, 1969 U. Grenander: A Course In 6 h h Ed. by 0. Hajek, A. J. Lohwater, Computational Probability and A. McCann Statistics. 35 fig. XII, 155 pages 17 fig. X, 140 pages. 1971 1971. OM 24,-: US S 7.50 OM 16,-; US S 5.00 Distribution ri~hts for U.K., Commonwealt , and the Traditional eritish Vol.236: Barr, M., P.A.Grlllet, Market (excluding Canada): Allen & nwin Ltd., London D. H. van Oldol: Exact Categories and Categories of Sheaves I I I VII, 239 pages. 1971 Ergebnisse der Mathematik OM 20,-; US$ 6.30 Band 65: Browder, W.: Surgery • Vol. 237: Stenstr6m, B.: Rings on Simply-Connected Manifolds and Modules of Quotients Approx. 2 fig. Approx. 140 pages VII, 136 pages. 1971 1972. OM 42,-; US $13.10 OM 16,-; US$ 5.00 Vol. 238: Der kanonlsche Modul Graduate Texts in Mathematics elnes Cohan-Macaulay-Ringe Vol. 5: Mac Lane, S.: Categories --=· In German. Ed. by J. Herzog, For the Working Mathematician E. Kunz. VI, 103 pages. 1971 IX, 262 pages. 1972 OM 16,-; US$ 5.00 DM 31,50; US$ 9.80 Vol. 239: lllusle, L.: Complexe Cotangent et Deformations I Die Grundlehren der mathe· In French. XV, 355 pages. 1971 matischen Wissenschaften OM 26,-; US$ 8.10 Band 65: Fejes T6th, L.: Vol. 240: Kerber, A.: Lagerungen In der Ebene auf Re{re•entatlons of der Kugel und lm Raum Heidelberger TaschenbOcher Permutation Groups In German. 138fig. XII, 238 pages Part 1: VII, 192 pages. 1971 1972. OM 66,-; US$ 20.60 Band 103: Diederich, K., OM 18,-; US S 5.60 R. Remmert: Funktlonentheorle I Band 181 : Lions, J. L., In German. 9fig. XIII, 246 pages • Please ask for prospectus E. Magenes: Non Homogeneous 1972. OM 14,80; US$ 4.60 material Boundary Value Problems and Appllcetlone I Translated from the French Lecture Notes in Mathematics by K. Kenneth. XVI, 357 pages Vol. 230: Waelbroeck, L.: ;;... 1972. OM 78,-; US$ 24.30 Topological Vector Spaces and E-< Band 182: Lions, J. L, Algebras. VII, 158 pages i:il E. Magenes: Non-Homogeneous 1971. DM 16,-; US$ 5.00 u Boundary Value Problems and Vol. 231 : Reller, H.: - 0 Applications 11 L 1 -Algebras and Segal Algebras rn Translated from the French XI, 113 pages.1971 by P. Kenneth OM 16,-; US$ 5.00 .....:l XI, 242 pages < 1972. OM 58,-; US $18.10 Vol. 232: Ganellus, T. H.: u ~ Tauberlan Remainder Theorems 0 I E-< ~ Band 165: Rublnowlcz, A.: VI, 75 pages. 1971 - '='! Sommerleldsche Polynom- DM 16,-; US$ 5.00 Springer-Verlag < 0 methode ::a "0 In German. Approx. 300 pages Vol. 233: Tsokos, C. P., New York Inc. i:il 1 =~ d 1972. OM 56,-; US $17.50 W.J. Padgett: Random Integral 175 Fifth Avenue • New York X ril Equations with Applications New York 10010 E-< Ql G Band 194: lgusa, J.: to Stochastic Systems ::1 Theta Functions. X, 232 pages VII, 174 pages. 1971 Published by Springer-Verlag < -"0 0 e,:, 1972. OM 64,-; US S 19.90 OM 18,-; US$ 5.60 Berlin · Heidelberg · New York ::a .J: 00 ~ z -q< Cl:: '='! gj .s u~< u ! Ql ll. -Cl:: 0>C "0= s:: i:il~ ·;:; ... 0 .a ;:aq .. Q) <~~ ~