Calendar
This Calendar lists all of the meetings which have been approved by the Council up to the date this issue of the cNotiaiJ was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the American Mathematical Society. The meeting dates which fall rather far in the future are subject to change; this is particularly true of meetings to which no numbers have yet been assigned. Abstracts should be submitted on special forms which are available in most departments of mathematics; forms can also be obtained by writing to the headquarters of the Society. Abstracts to be presented at the meeting in person must be received at the headquarters of the Society in Providence, Rhode Island, on or before the deadline for the meeting. Meeting Deadline for Abstracts* Number Date Place and News I terns
August 1974 No summer meeting; Inter- June 15, 1974 national Congress (see below) (News Items only) 716 October 26, 1974 Middletown, Connecticut Sept. 3, 1974 717 November 8-9, 1974 Nashville, Tennessee Sept. 24, 1974 718 November 22-23, 1974 Los Angeles, California Sept. 24, 1974 719 November 23, 1974 Houston, Texas Sept. 24, 1974 720 January 23-27, 1975 Washington, D. C. Nov. 5, 1974 (81st Annual Meeting) April 18-19, 1975 Monterey, California June 20-21, 1975 Pullman, Washington August 18-22, 1975 Kalamazoo, M!cb!gan January 22-26, 1976 San Antonio, Texas (82nd Annual Meeting)
*Deadline for abstracts not presented at a meeting (by title). August 1974 issue: June 8 October 1974 Issue: August 26
OTHER EVENTS
Angust 21-29, 1974 International Congress of Mathematicians Vancouver, B. C., Canada April 15, 1974
The zip code of the Post Office Box of the Society has been changed from 02904 to 02940. Correspondents are requested to note this change in their records.
Please affix the peel-off label on these cNofiaiJ to correspondence with the Society concerning fiscal matters, changes of address, promotions, or when placing orders for books and journals. The c)fofiaiJ of the American Mathematical Society is published by the Americao Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940, in Jaouary, February, April, June, August, October, November, aod December. Price per aonual volume is $10. Price per copy $3. Special price for copies sold at registration desks of meetings of the Society, $1 per copy. Subscriptions, orders for back numbers (back issues of the last two years only are available), and inquiries should be addressed to the American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940. Second class postage, paid at Providence, Rhode Islaod, aod additional mailing offices.
Copyright © 197 4 by the American Mathematical Society Printed in the United States of America OF THE AMERICAN MATHEMATICAL SOCIETY
Everett Pitcher and Gordon L. Walker, Editors Wend ell H. Fleming, Associate Editor CONTENTS
MEETINGS Calendar of Meetings ...... Inside Front Cover Program for the May Meeting in DeKalb ...... 172 Abstracts for the Meeting: A-453-A-464 INVITED SPEAKERS AT MEETINGS...... 178 PRELIMINARY ANNOUNCEMENTS OF MEETINGS ...... • ...... 179 NEW FORMAT FOR ABSTRACTS ...... 179 THE ROLE OF THE DISSERTATION IN THE PH. D. PROGRAM ...... • ...... 180 QUERIES •...... •...... •.....•.•.• , ...... , ...... • . . . . . • . . . . . 186 THE PRESERVATION OF ARCHIVAL MATERIALS 187 SPECIAL MEETINGS INFORMATION CENTER ...... • ...... 188 PERSONAL ITEMS . . . . . • ...... 190 NEW AMS PUBLICATIONS ...... •.••...... , . . . . . 191 NEWS ITEMS AND ANNOUNCEMENTS ...... 179, 189, 191 RELATIONS BETWEEN US-USSR SCIENTIFIC JOURNALS AND PUBLISHERS ...... 192 ABSTRACTS ...... •...... A-431 ERRATA TO ABSTRACTS ...... A-464 SITUATIONS WANTED ...... •. A-465 Seven Hundred Fifteenth Meeting N orthem Illinois UDiversity DeKalb, Illinois May 13-17,1974
The seven hundred fifteenth meeting of the John Milnor, David Mumford, Albrecht Ffister, American Mathematical Society will be held at Julia B. Robinson, Guido Stampacchia, John T. Northern nlinois University in DeKalb, nlinois, Tate, Robert Tijdeman, ArthurS. Wightman, from Monday, May 13, to Friday, May 17. The and Chung-Tao Yang. principal feature of the meeting will be a sym There will be five hour addresses per day posium on Mathematical Developments Arising in the symposium on Monday through Thursday. from the Hilbert Problems. In additon there will be four hour addresses for In 1900, David Hilbert delivered a lecture the Society meeting on Friday whose subject mat before the Second International Congress of ter will be an integral part of the symposium; Mathematicians at Paris. In this lecture, he they will be given by John H. Conway, 0. Timothy posed only ten of the twenty-three problems that O•Meara, Hugh L. Montgomery, and James B. appeared in the published reports. The paper Serrin. was first published in the Gottinger Nachrichten At the request of the Organizing Committee [{1900), 253-297], and Archiv der Mathematik of the Symposium on Mathematical Developments und Physik [3d ser. 1 (1901), 44-63, 213-237]. Arising from the Hilbert Problems, Professor The English translation by Dr. Mary Winston Jean Dieudonne has set up a committee of emi Newson appeared in the Bulletin of the American nent mathematicians to assemble a list of the Mathematical Society [50 (1902), 437-479] under most outstanding unsolved problems of present the title "Mathematical problems"; and a sum day mathematics. It is hoped that such a list mary in French of Hilbert• s talk and the list of might serve to focus the interest of young re the problems appeared in L' Enseignement Mathe search mathematicians on worthwhile projects matique [2 (1900), 349-355]. A substantial por and help them see their way through the con tion of the Bulletin article, with some additional siderable amount of mathematical publications. comments, was reproduced as Chapter 10, "The It is hoped that a preliminary version of this future of mathematics," in Hilbert by Constance collection of problems will be available at the Reid, Springer-Verlag, 19~ symposium and that a more definitive version Hilbert died in 1943 at the age of eighty-one. will be published in the proceedings of the sym An obituary by Hermann Weyl appeared in the posium. Bulletin of the American Mathematical Society There will be sessions for the presentation [50 (1944), 612-654], and in the Boletin da So of contributed ten-minute papers early Friday ciedade de Matematica de Sao Paulo [1 (1946), morning and late Friday afternoon, May 17. 76-104, and 2 (1947), 34-601. Professor Weyl All the addresses on the Mathematical De stated: "His Paris address on •Mathematical velopments Arising from the Hilbert Problems problems• .•. straddles all fields of our science. will be presented in the Carl Sandburg Audito Trying to unveil what the future would hold in rium of the University Center. The sessions of store for us, he posed and discussed twenty contributed papers will be held in Reavis West, three unsolved problems which have indeed, as a classroom building four-minutes walk from we can now state in retrospect, played an im the University Center. portant role during the following forty odd years. The Council of the Society will meet at A mathematician who had solved one of them 2:00 p.m. on Sunday, May 12, 1974, at the thereby passed on to the honors class of the Holiday Inn, 1212 W. Lincoln Highway in DeKalb. mathematical community." In the time that has REGISTRATION elapsed since Weyl wrote this, further develop ments have strongly endorsed his opinion. The registration desk will be located in the This symposium on Mathematical Develop lobby of the University Center. The desk will be ments Arising from the Hilbert Problems will be open from 8:30 a.m. to 4:30 p.m. Monday supported by a grant from the National Science through Friday. A message center will be main Foundation. This topic was chosen by the Com tained at the registration desk. The registration mittee to Select Hour Speakers for Western Sec fees for the meeting are as follows: members tional Meetings. The Organizing Committee of $5. 00, nonmembers $7. 50, and students and the symposium consists of Paul T. Bateman, unemployed mathematicians $1. 00. Felix E. Browder (chairman), R. Creighton ACCOMMODATIONS Buck, Donald J. Lewis, and Daniel Zelinsky. The list of speakers includes Lipman Bers, The following accommodations will be Enrico Bombieri, Herbert Busemann, Nicholas available on or near the Northern nlinois Uni M. Katz, Steven L. Kleiman, Georg Kreisel, versity campus for this meeting. R. Langlands, G. G. Lorentz, Donald A. Martin, A. Dormitory accommodations in Grant
172 South, a student dormitory complex ten-minutes From O•Hare to DeKalb walk from the University Center. These accom modations are not recommended for families with Lv. O•Hare Ar. DeKalb small children. Maid service is not provided. Room rates are $3. 50 per person per night on a Sun. May 12 3:35p.m. 5:20p.m. double occupancy basis, and $5. 50 per person 5:35p.m. 7:20p.m. per night in a single room. 7:35p.m. 9:20p.m. B. Accommodations in the University Plaza, a private dormitory type building five Mon. May 13 9:35a.m. 11:20 a.m. minutes walk from the University Center. Maid 11:35 a.m. 1:20 p.m. service is provided. Room rates are $7. 50 per 7:35p.m. 9:20p.m. person per night on a double occupancy basis, Tues. May 14 7:35p.m. 9:20p.m. and $8. 50 per person per night in a single room. All rooms have shared baths. Wed. May 15 7:35p.m. 9:20p.m. C. A limited number of rooms in the Uni Thurs. May 16 3:35p.m. 5:20p.m. versity Center. All of these rooms are for dou 5:35p.m. 7:20p.m. ble occupancy at the rate of $7. 50 per person per night, and all have private baths. Fri. May 17 5:35p.m. 7:20p.m. Requests for room reservations in Grant Buses will deliver passengers to their residences South, University Plaza, or the University Cen ter should have been sent to Professor Donald for the conference. Ostberg prior to April15. A form for making From DeKalb to O•Hare room reservations was included on the last page of the April c/{otiuiJ. DeKalb Ar. O•Hare Individuals arrving on campus without room Lv. reservations should contact the University Center Mon. May 13 4:30p.m. 6:30p.m. registration desk to arrange accommodations. There is also a Holiday Inn located within Tues. May 14 4:30p.m. 6:30p.m. ten-minutes walk of the University Center. Its 6:30p.m. address and room rates are: Wed. May 15 4:30p.m. Thurs. May 16 3:30p.m. 5:30p.m. HOLIDAY INN 4:30p.m. 6:30p.m. 1212 W. Lincoln Hwy DeKalb, illinois 60115 Fri. May 17 4:30p.m. 6:30p.m. Phone: (815) 758-8661 Sat. May 18 7:30a.m. 9:30a.m. Single: $13. 50 10:30 a.m. 12:30 p.m. Double: 16. 50 (2 persons) 12:30 p.m. 2:30p.m. Twin: 18. 50 (2 persons) Reservations should be made directly with the Buses will pick up passengers at their resi Inn, and mention should be made of this meeting dences for departure. in order to obtain the quoted rates. B. Direct limousine service is available FOOD SERVICE between O•Hare and DeKalb at a cost of $9. 50 Breakfast, lunch, and dinner will be avail per person each way. Reservations should have been made in able in the University Center throughout the con advance for transportation between Hare and ference. A snack bar will be open in Grant South o• DeKalb, using the form on the last page of the during the evenings, but meals will not be served April and mailed to Professor Donald in the dormitories. There are also several res CJ/oticriJ, Ostberg, Department of Mathematical Sciences taurants within walking distance of campus. A Northern Illinois University, DeKalb, Illinois ' list of these restaurants with directions for 60115, to arrive prior to May 1, 1974. reaching them will be available at the registra tion desk. ENTERTAINMENT TRAVEL A cash bar will be in operation from 5:30 p.m. to 6:30 p.m. on Monday through Friday DeKalb, illinois, is located on Illinois evenings. In addition, the Department of Mathe Route 38, approximately 70 miles west of matical Sciences at Northern Illinois University Chicago. The campus of Northern Illinois Uni will host a beer party for all those attending the versity is adjacent to Route 38 on the west side meeting on Wednesday evening, May 15. Details of DeKalb. concerning the location of the cash bar and the Travelers driving to DeKalb from the east time and place of the beer party will be available should take one of the routes shown on the map. at the registration desk. Travelers driving to DeKalb from the west should arrive on Illinois Route 38. PARKING Travelers arriving by air should fly to O•Hare Airport, Chicago, from which point Since the meeting will be held after the end ground transportation will be available to De of the spring semester at Northern Illinois Uni 'Kalb as follows: versity ample parking will be available on the A. Continental Air Transport Company and campus. Visitors are requested not to park in University Bus with a transfer at the Ramada Inn metered or reserved parking places, or in no in St. Charles, Illinois. Cost: $3. 25 each way. parking zones.
173 SUGGESTED ROUTES O'HARE-DEKALB
.tJ NORTHERN ILLINOIS UNIVERSITY
CRANE DR. GRANTN. + 0 LINCOLNN. 0 "'-l_, ~ "' LUCINDA AVE. ~ STADIUM DRIVE NORTH u > 0 ::;,z 0 ;,i COLLE E i Q ~ ::;; > < Q s.... ---~ -l -l < i ...< 0 0 Q Ul "'_, _, "' < ""'... CIRCLE 6 LINCOLN HIGHWAY US 30 REAVIS WEST 4 GRANT TOWERS RESIDENCE HALL 2 UNNERSITY CENTER & TOWER 5 UNIVERSITY PLAZA 3 ADAMS HALL (MATHEMATICS DEPARTMENT) 6 HOLIDAY INN 174 PROGRAM FOR THE SYMPOSIUM ON MATHEMATICAL DEVELOPMENTS ARISING FROM THE HILBERT PROBLEMS All the addresses in the symposium will be presented in the Carl Sandburg Auditorium of the University Center. MONDAY, MAY 13 9:00 a.m. - 10:00 a.m. Hilbert• s first problem: The continuum hypothesis. Professor DONALD A. MARTIN, Rockefeller University (715-E1) 10:30 a.m. - 11:30 a.m. What have we learnt from Hilbert• s second problem? Professor GEORG KREISEL, Stanford University (715-E2) 1:30 p.m. - 2:30p.m. Hilbert• s fourth problem: Desarguesian spaces. Professor HERBERT BUSEMANN, University of Southern California (715-D1) 3:00 p.m. - 4:00 p.m. Hilbert• s fifth problem and related problems on transformation groups. Professor CHUNG-TAO YANG, University of Pennsylvania 4:30 p.m. - 5:30 p.m. Hilbert• s sixth problem: Mathematical treatment of the axioms of physics. Professor ARTHUR S. WIGHTMAN, Princeton University (715-C3) TUESDAY, MAY 14 9:00 a.m. - 10:00 a.m. Hilbert• s seventh problem: The method of Gel• fond-Baker and its applications. Dr. ROBERT TIJDEMAN, University of Leiden, Netherlands 10:30 a.m. - 11:30 a.m. The Riemann hypothesis for curves. Professor ENRICO BOMBIERI, University of Pisa, Italy and Institute for Advanced Study 1:30 p.m. - 2:30 p.m. An overview of Deligne• s proof of the Weil conjectures and where it leaves us. Professor NICHOLAS M. KATZ, Princeton University 3:00 p.m. - 4:00p.m. Hilbert s ninth problem: Reciprocity laws. Professor JOHN T. TATE, Harvard University 4:30 p.m. - 5:30p.m. Hilbert• s tenth problem: Diophantine equations. Dr. YURI V. MATIJASEVIC, Steklov Institute of Mathematics, Leningrad, Professor MARTIN D. DAVIS, Courant Institute, New York University, and Dr. JULIA B. ROBINSON*, Berkeley, California WEDNESDAY, MAY 15 9:00 a.m. - 10:00 a.m. Hilbert• s twelfth problem: Class field theory. Professor R. LANG LANDS, Institute for Advanced Study 10:30 a.m. - 11:30 a.m. Hilbert• s thirteenth problem: Representation of functions in terms of functions of a smaller number of variables. Professor G. G. LORENTZ University of Texas and University of Stuttgart, Federal Republic of Germany 1:30 p.m. - 2:30 p.m. Hilbert• s fourteenth problem: Finiteness of rings of invariants. Professor DAVID MUMFORD, Harvard University 3:00 p.m. - 4:00 p.m. Hilbert• s fifteenth problem: Rigorous foundation of Schubert• s enumerative calculus. Professor STEVEN L. KLEIMAN, Massachusetts Institute of Technology and University of California, San Diego 4:30 p.m. - 5:30 p.m. Hilbert• s seventeenth problem and related problems on definite forms. Professor ALBRECHT PFISTER, Johannes Gutenberg University, Mainz, Federal Republic of Germany 175 THURSDAY, MAY 16 9:00 a.m. - 10:00 a.m. Hilbert• s eighteenth problem: Discrete lattices and fundamental domains. Professor JOHN MILNOR, Institute for Advanced Study 10:30 a.m. - 11:30 a.m. Hilbert's nineteenth problem: Minimal surfaces. Professor ENRICO BOMBIERI, University of Pisa, Italy and Institute for Advanced Study 1:30 p.m. - 2:30 p.m. Hilbert• s twenty-first problem: An overview of Deligne• s work on fuchsian equations and where it leaves us. Professor NICHOLAS M. KATZ, Princeton University 3:00 p.m. - 4:00 p, m. Hilbert• s twenty-second problem: Uniformization. Professor LIPMAN BERS, Columbia University (715-A21) 4:30 p.m. - 5:30 p, m. Hilbert• s twenty-third problem: Extensions of the calculus of variations. Professor GUIDO STAMPACCHIA, University of Pisa, Italy PROGRAM OF THE SESSIONS The time limit for each contributed paper in the general sessions is ten minutes. To maintain this schedule, the time limits will be strictly enforced, 'FRIDAY, 7:45 A. M. Session on Number Theory, 120 Reavis West 7:45- 7:55 (1) An iterative function of Euler's ~-function. Preliminary report. Dr. LARRY W. DAVIS, University of Wisconsin-Whitewater (715-A19) 8:00- 8:10 (2) Size of the integers involved in the solution of Fermat• s last theorem. Professor J. M. GANDHI* and Mr. B. G. PATEL, Western Illinois University (715-A16) 8:15- 8:25 (3) The two-dimensional Diophantine approximation constant. Dr. THOMAS W. CUSICK, State University of New York at Buffalo (715-A4) 8:30- 8:40 (4) Asymptotic independence of sequences of integers. Professor MELVYN BERNARD NATHANSON, Southern Illinois University (715-A1) FRIDAY, 8:00 A. M. Session on Algebra, 122 Reavis West 8:00- 8:10 (5) Generalized polynomial identities. II. Dr. LOUIS HALLE ROWEN, University of Chicago (715-A9) 8:15- 8:25 (6) Extensions of partial orders on semigroups. Professor FRANK A. SMITH, Kent State University (715-A2) 8:30- 8:40 (7) A universal mapping property for a lattice completion. Professor ALAN A. BISHOP, Western Illinois University (715-A3) FRIDAY, 9:00 A. M. Invited Address, Carl Sandburg Auditorium, University Center (8) Hilbert• s third problem: Dissection of polyhedra. Professor JOHN H. CONWAY, Cambridge University, England and University of Michigan (715-D2) FRIDAY, 10:30 A. M. Invited Address, Carl Sandburg Auditorium, University Center (9) Hilbert• s eighth problem: Problems about prime numbers. Professor HUGH L. MONTGOMERY, University of Michigan (715-A22) FRIDAY, 1:30 P. M. Invited Address, Carl Sandburg Auditorium, University Center (10) Hilbert• s eleventh problem: Quadratic forms with any algebraic numerical coefficients. Professor 0. TIMOTHY O•MEARA, University of Notre Dame (715-Al2) 176 FRIDAY, 3:00 P. M. Invited Address, Carl Sandburg Auditorium, University Center (11) Hilbert• s twentieth problem: Solvability of boundary value problems. Professor JAMES B. SERRIN, University of Minnesota (715-BB) FRIDAY, 4:30 P. M. Session on Category Theory and Group Theory, 120 Reavis West 4:30- 4:40 (12) Differential categories are cotripleable. II. Preliminary report. Dr. WILLIAM F. KEIGHER, Southern lllinois University, Carbondale (715-AlO) 4:45- 4:55 (13) Group completing monoidal categories. Preliminary report. Professor JAMES E. KETTNER, Northern lllinois University (715-A15) 5:00- 5:10 (14) The category of topological space objects in a topos E is E-initial, E-complete, and E-cocomplete. Mr. LAWRENCE NEFF-STOUT, University of lllinois (715-A18) 5:15- 5:25 (15) Non-Abelian simple groups are not B-groups. Preliminary report. Dr. LARRY E. KNOP, Southern lllinois University (715-A13) 5:30- 5:40 (16) A strengthening of the torsionless property in abelian groups. Professor CARY H. WEBB, Chicago State University (715-A5) 5:45- 5:55 (17) Reflexive subsemigroups in a regular semigroup. Preliminary report. Dr. FRANCIS E. MASAT, Glassboro State College (715-All) FRIDAY, 4:30 P. M. Session on Associative Rings, 122 Reavis West 4:30- 4:40 (18) Going down and open extensions. Preliminary report. Professor STEPHEN J. McADAM, University of Texas (715-AB) 4:45- 4:55 (19) Matrices and symmetric matrices with zero diagonal. Preliminary report. Professor FRANK W. OWENS, Ball State University (715-A20) 5:00- 5:10 (20) Idealizers in rings. Preliminary report. Professor E. P. ARMENDARIZ and Professor JOE W. FISHER*, University of Texas (715-A17) 5:15- 5:25 (21) When are proper cyclics quasi-injective? Preliminary report. ProfessorS. K. JAIN, Professor SURJEET SINGH, and Mr. ROBIN G. SYMONDS*, Ohio University (715-Al4) 5:30- 5:40 (22) Rings whose faithful left ideals are' cofaithful. Professor JOHN A. BEACHY and Professor WILLIAM D. BLAIR*, Northern lllinois University (715-A7) 5:45- 5:55 (23) On Morita• s localization. Professor JOHN A. BEACHY, Northern illinois University (715-A6) FRIDAY, 4:30 P. M. Session on Analysis, 204 Reavis West 4:30- 4:40 (24) Annular functions and residual sets. Preliminary report. Mr. RUSSELL W. HOWELL, Ohio State University (715-B1) 4:45- 4:55 (25) Cluster sets of arbitrary functions. Preliminary report. Professor PAUL D. HUMKE, Western lllinois University (715-B6) 5:00- 5:10 (26) On existence of asymptotic paths for subharmonic functions in Rm. Preliminary report. Professor M. N. M. TALPUR, University of Illinois (715-B5) 5:15- 5:25 (27) Applications of a universal measure. Preliminary report. Professor WILLIAM H. GRAVES* and Dr. SUZANNE MILLER MOLNAR, University of North Carolina at Chapel Hill (715-B4) 5:30- 5:40 (28) Square roots of sectorial operators. Preliminary report. Dr. ALAN G. R. MciNTOSH*, Institute for Advanced Study, and Mr. BRUCE EVANS, University of Pennsylvania (715-B7) 177 FRIDAY, 4:30 P. M. Session on Approximation Theory and Applied Mathematics, 320 Reavis West 4:30- 4:40 (29) On the error in the Pade approximants for a form of the incomplete gamma function including the exponential function. Professor YUDELL L. LUKE, University of Missouri-Kansas City (715-B2) 4:45- 4:55 (30) Comparison of rational approximations. Preliminary report. Professor YUDELL L. LUKE and Mr. RICK WHITAKER*, University of Missouri Kansas City (715-B3) 5:00- 5:10 (31) On generalized Gaussian quadrature. Professor YUDELL L. LUKE, Mr. BING Y. TING, and Ms. MARILYN J. KEMP*, University of Missouri-Kansas City (715-C2) 5:15- 5:25 (32) Association of Schroedinger equation (quantum) with von Karman-Howarth equation (deterministic, macroscopic). Professor MARIA Z. v. KRZYWOBLOCKI, Michigan State University (715-C1) FRIDAY, 4:30 P. M. Session on Topology, 328 Reavis West 4:30- 4:40 (33) K-theory of special normed algebras. Professor BARRY H. DAYTON, Northeastern Tilinois University (715-G1) 4:45- 4:55 (34) Configuration-like spaces and the Borsuk-Ulam theorem. Professor FREDERICK R. COHEN* and Professor EWING L. LUSK, Northern Tilinois University (715-G2) 5:00- 5:10 (35) The Mod p Smith index and a generalization of the Borsuk-Ulam theorem. Preliminary report. Professor EWING L. LUSK, Northern Illinois University (715-G3) 5:15- 5:25 (36) Divisibility of the index with applications to inertia groups and H*(G/0). Preliminary report. Mr. LAURENCE R. TAYLOR, University of Notre Dame (715-G4) 5:30- 5:40 (37) On the product of suborderable spaces. Dr. PETER J. NYIKOS, University of Chicago (715-G5) Paul T. Bateman Associate Secretary Urbana, Tilinois INVITED SPEAKERS AT AMS MEETINGS This section of these c/Votia?iJ lists regularly the individuals who have agreed to address the Society at the times and places noted below. For some future meetings, the lists of speakers are incomplete. DeKalb, Tilinois, May 1974 John Conway Hugh L. Montgomery 0. Timothy O•Meara James B. Serrin Nashville, Tennessee, November 1974 Trevor Evans J. R. Retherford R. B. Gardner Washington, D. C., January 1975 Linda Keen Wilfried Schmid 178 PRELIMINARY ANNOUNCEMENTS OF MEETINGS The Seven Hundred Sixteenth Meeting Wesleyan University Middletown, Connecticut October 26, 1974 The seven hundred sixteenth meeting of the submitted to the American Mathematical Society, American Mathematical Society will be held at P.O. Box 6248, Providence, Rhode Island 02940, Wesleyan University, Middletown, Connecticut, so as to arrive prior to the deadline of Septem on Saturday, October 26, 1974. ber 3, 1974. The preliminary announcement of By invitation of the Committee to Select the meeting will appear in the August issue and Hour Speakers for Eastern Sectional Meetings, the final program in the October issue of these there will be two one-hour addresses. Special cJ/otiuiJ. sessions are being planned. Sessions for con Walter H. Gottschalk tributed ten-minute papers will be scheduled in the morning and the afternoon. No provision will Associate Secretary be made for late papers. Abstracts should be Middletown, Connecticut NEW FORMAT FOR ABSTRACTS Members of the Council of the American in having their abstracts published (a require Mathematical Society voted at their January 1974 ment for contributed papers presented at meet meeting to have authors provide camera ready ings) may (1) type their own abstracts on the copy for abstracts submitted to these cJ/otiuiJ. form provided or (2) request that the Providence The abstract supplied by the author will be pho office type and proofread the abstract, a service tocopied at 73% reduction and will appear in these for which the author should include a check for cJ/oilai) exactly as typed by the author. This pro $7. 00 along with the abstract. cedure, which will be implemented with the Octo Authors electing to do their own typing ber 1974 issue of these c/'v~tiJ:eiJ, will save the So should take care that the instructions provided ciety an estimated $6, 000 a year. at the bottom of the new abstract form are fol In May 1974, copies of a new abstract form lowed exactly. Author-typed abstracts that can will be sent to all Departments of Mathematics not be satisfactorily reproduced will be typed and listed in the Administrative Directory. (Copies proofread at the Providence office, and the author of the new form are also available from the will be charged $7. 00 for this service. Providence office on request. ) Authors interested NEWS ITEMS AND ANNOUNCEMENTS POLICY ON RESEARCH ANNOUNCEMENTS privilege of submitting six (6) research announce The Council at its meeting of January 14, ments per year for publication in the Bulletin. Of 1974 effected a change of policy for research an these six research announcements, at most one nouncements in the Bulletin. In particular, it will may be a complete paper. The editors of the restrict both the number and the length of re Bulletin, in notifying a Council member of re search announcements that are published. The ceipt of a communication, will inform him, for text of the Council resolution is as follows: example, that this is paper number 2 of the six papers he is permitted to communicate during "Research announcements are intended for the current year. rapid communication of outstanding results that are to appear elsewhere with complete proofs. "Each research announcement is normally limited to five (5) typed pages. More precisely, It is only exceptionally that a paper containing complete proofs should appear as a research an no research announcement is normally accept nouncement. Generally, if a paper is acceptable able if it exceeds about 100 typed lines of 65 to the Proceedings or similar journal it should spaces each." not be accepted as a research -announcement. Everett Pitcher "Each Council member who is a member of Secretary one of the SocietY's editorial committees has the 179 THE ROLE OF THE DISSERTATION IN THE PH.D. PROGRAM At the Annual Meeting of the Society in San Francisco, January 17, 1974, a panel discussion on the Ph. D. thesis was sponsored by the AMS Committee on Employment and Educational Policy. What follows is an account of that session. The panel discussion at San Francisco was one of a series de voted to major questions of graduate education in mathematics. The first in the series, entitled Graduate education in mathematics in the coming decade, was held at the Dallas meeting in January 1973. An account of the panel session in Dallas was published in the April 1973 issue of these c/{otiui), volume 20, number 3, pages 123-130. The panel discussion in San Francisco was organized by P. Emery Thomas, University of California, Berkeley, who served as moderator. The members of the panel were William Browder, Princeton University; Karel de Leeuw, Stanford University; I. N. Herstein, University of Chicago; and Saunders Mac Lane, University of Chicago. The Committee on Employment and Educational Policy consists of Richard D. Anderson (Chairman), Louisiana State University; Michael Artin, Mas sachusetts Institute of Technology; Wendell H. Fleming, Brown University; Calvin C. Moore, Uni versity of California, Berkeley; RichardS. Palais, Brandeis University; and Martha K. Smith, Uni versity of Texas, Austin. teaching jobs to research mathematicians. For, William Browder as a profession, we owe a duty not only to our What does a Ph. D. degree in mathematics students and colleagues, but also to our subject, represent? It is not a mark on a continuous scale to further its development and progress. And of quality (say=. 001 Gauss), but rather denotes this progress will be aided more by raising the a change of type, or nature, like puberty. The standards of Ph. D. 's, and the dissertation, than Ph. D. degree represents the statement "This by changing its nature. We should produce fewer person is a mathematician." And the dissertation and better mathematicians and, at the same represents the proof that the person has created time, make them more qualified for the avail mathematics, which is what makes one a mathe able jobs by encouraging broader knowledge and matician. more attention to teaching. I don• t say that the statement is not some A Ph. D. degree without a research thesis times a lie. The quality of the mathematics rep is like a mock apple pie. It inay look the sall).e resented by a given thesis may not be very high, from the outside, but when you bite into it you or it may sometimes represent the creative taste crumbled crackers instead of fruit. efforts of the advisor, or both. But a faculty can lie on any subject, and would lie about another I. N. Herstein kind of thesis as easily. But that research thesis represents the statement that the "Doctor" has In September 1969, I published an article proved a theorem, has become a practitioner, a in the American Mathematical Monthly in which creator. No amount of watching from the side I discussed what I felt was wrong with the Ph. D. lines, of organizing, of teaching has the same training given in mathematics in America. Many quality. things have happened in American life, in gen If the quality of Ph. D. theses is low, make eral, and in academic life, in particular, since efforts to raise it, not do away with the thesis. then. Despite all these changes and events that The qualitative difference in a person who have transpired, my own thinking on the nature has created mathematics makes him a different of our Ph. D. has not undergone much change. kind of teacher from a person who has only wit So, I am afraid that what I will say today will be, nessed~ for the most part, a paraphrase of much of what Certainly the Ph. D. does not indicate I said then. teaching ability, nor does the thesis confer spe One vital factor that has changed, and cial pedagogical powers; but neither does it in drastically so, has been the availability of jobs dicate the lack of these. It does indicate a first for our fresh Ph. D.'s. Another, and perhaps hand experience, which changes the perspective even harsher, change has been in the opportuni and the possibilities. If Ph. D. •s don•t teach well ties to get tenure for junior faculty members. A enough, teach them more pedagogy, but not at third (and to my mind, this one represents a the expense of the essential. positive step) has been the significant role that If a department wants to give a degree in classroom performance now plays in the assess history, or psychology, or sociology, or expo ment of a faculty member. I believe that these sition of mathematics, OK, but don• t call it a changes argue for, rather than against, there Ph. D. in mathematics. And don•t consider it marks I had previously made. comparable, and don• t expect to make the job It seems obvious that the primary objective, crisis better-it will be made worse. in any training, should be that this training and In a time of overproduction of Ph. D. 's and its ultimate use be closely related. Does this a job shortage, efforts should be made to make hold true for the training our Ph. D.'s get today? the standards of theses higher and to open more It seems to me that it does not. I'm speaking of 180 the majority of our Ph. D.'s, not the relatively as far as the work effort is concerned. It cer small percentage that goes on to become highly tainly represents a real effort. Is it genuine and creative in research. intellectual? I have my doubts. The effort is The primary role that our Ph. D.'s will genuine; the end-product is, far too often, con assume will be that of teachers; the primary trived. Is it intellectual? Too often it is merely goal of our training is that of researchers. This mechanical, a going-through-motions of doing disparity between education and goal really be research. gins to manifest itself only after the first year 6. Is it evidence of independence and of of graduate work. The reason is clear. At this intellectual growth? Yes, it is evidence of inde stage the student starts thinking about, and look pendence. But whose? The student• s or the ad ing for, a possible thesis topic. At this point visor• s? Is it intellectual growth to be led by the comes a rather brusque shift from general math hand through a very well marked maze? ematical education to a more specialized one In going through this list of what I consider that can lead to a thesis. Because the avowed desirable goals for the Ph. D. dissertation, I purpose of this dissertation is to make an origi come to the conclusion that the thesis, as con nal contribution to mathematical research, the stituted now, does not reach these goals. At any student is now moulded and readied to try to do rate, not for the majority of our students. The this. It is here that we go astray for the large reason is simple. The thesis is designed to ac majority of our students. commodate people geared to going into research, In my opinion the Ph. D. dissertation should while most people seeking the Ph. D. are not serve several purposes. going to end up doing research. 1. It should be a culmination of one• s So, the first change needed in the disserta training. tion is to be realistic about it. We must give 2. Its writing should widen and educate recognition to the fact that it does not serve the the writer. needs of most of our Ph. D.'s. I feel that one 3. It should open up other possible roads can satisfy the criteria I outlined by means other to follow. than the traditional one of proving some new re 4. It should give the writer a sense of sults. participation in the field. First, we must recognize the obvious. The 5. It should represent a genuine intellec importance of the dissertation is to its writer, tual effort. not to the subject in the large. The contribution 6. It should be evidence of independence it makes must be to its creator. The thesis topic and of intellectual growth. should be chosen for the particular candidate Does the present day Ph. D. thesis satisfy with this in mind. these criteria, at l~ast as far as most of our If the candidate wants to go into research Ph. D. 's are concerned? I feel that it does not. and seems to have ability in this direction, then Let me take up these points one at a time. the kind of thesis we now use almost universally 1. Is some minor, highly specialized set may be the right one for him or her. If the can of results, of limited interest, in a narrow area didate wants to concentrate on teaching, several of research, a culmination of anything? These alternative kinds of theses should be available. words, unfortunately, all too accurately describe One kind of thesis might be to develop a new the average research-type of thesis. In fact I•ll course at the junior-senior level; the thesis go a little further. If this does represent a cul would be the text, or extensive notes, for the mination, then the situation is really sad, for it course. Another option is in a similar vein; in indicates that the student• s sense of taste and stead of creating a new course, re-work an old picture of what mathematical research is all course from a different point of view. Mac Lane about has been completely distorted. "Be wise, has suggested, for instance: develop the first generalize" may be a practical way of writing a course in algebra from the point of view of cate thesis, but is this what mathematics is about? gory theory. Here, too, the thesis would be the 2. Does it widen and educate? Well, it text or course notes for the course. These are certainly doesn• t widen; in fact it narrows the in the direction of the kind of things a faculty student• s perspective. Does it educate? Yes, but member will be doing: designing or re-designing one could easily question whether the education courses. Another kind of thesis might be a re gained, in something so special, is really worth working of some standard part of mathematics gaining. Couldn•t one use all this time and effort or an attempt to unify some pieces of mathemat to learn something more fundamental and useful? ics. These are cited as some of the possibilities. 3. Does it open up new roads to follow? In With time, many other options would evolve. general the answer is no. All too often the thesis Let me say that these would not be cheap topic represents a dead end. The evidence for theses. A great deal of creativity and intellectual this is clear. The number of Ph. D.'s that go on effort Would go into them. The kind of depart to write even a second paper after the thesis is ment capable of giving such Ph. D. 1 s would have very small. to be first-rate, one with great width in addition 4. Does it give one a sense of participation to depth. They could lead to our Ph. D. 1 s being in one• s field? Unquestionably yes. But here, better fitted to the positions they will fill for too, one can ask: Aren•t there other ways, pos most of their lives. sibly more fruitful, to give one a sense of in volvement with mathematics? I•ll come back to Saunders Mac Lane this later. Today there are many different views as 5. Does it represent a genuine intellectual to how we ought to treat or change our Ph. D. effort? Here, too, the answer is yes, at least degree. It is my considered opinion that we 181 should take clear steps to increarse the quality On the whole, we can nevertheless observe of the Ph. D. and the breadth of the Ph. D. dis that the number of Ph. D. 's in mathematics has sertation. There are basic and intrinsic reasons suddenly become very large. Many graduate for taking such steps, but they are also appro schools have increased the number of their stu priate to the present situation. dents and many new schools have been added. First, I speak to the situation. We have Just for example, the December 1965 c){otiai) too many Ph. D.'s who are hard put to find suit listed fellowships and assistantships from 256 able positions. This is partly because society departments of mathematical sciences. The De as a whole does not suitably value their talents cember 1973 c}/oticeiJ listed such positions from and partly because their talents are not always 449 such departments. This is a 75 per cent in such as to be well valued by society. It is im crease in eight years. Is this justified? portant that we explain why society should value No one can argue that the needs of mathe mathematical thought and expertise more effec matical research and discovery call for any such tively. Moreover, we should make our actions fantastic increase. The beauty of a mathematical fit with our explanation by seeing to it that the result is not improved by multiple discoveries I talents of mathematicians become better ad In the ill-fated report by COSRIMS, it was argued justed to their use. For example, if we wish that we needed many more Ph. D.'s because they society to value our teaching, then we, too, were required for teaching. This argument was should value it. based on very optimistic estimates of the m,J.m One recent proposal* holds that the Ph. D. ber of students who would need mathematics degree should drop the requirement of a disser courses for this or that purpose. These esti tation. This proposal is tragically and com mates were overinflated for several reasons: pletely wrong. Not only that, it is presented They were one-sided, on the needs of teaching, without a scintilla of evidence. The writer says, not of research; moreover, it is clear that they "As I see it, we must effectively eliminate the were written far too much as a piece of propa Ph. D. degree as we know it today." I don't see ganda, not as a hard-headed analysis of what it so. Today• s Ph. D. degree is fundamentally can happen. The mathematics community must good. The writer says of the last two years of take note of this and make sure that any new graduate work, which are typically spent strictly COSRIMS report-there is a prospect of such on the dissertation, "These last two years are does not depart from hard-headed analysis. irrelevant for the large majority of the recipi But, whatever the number of Ph. D. 's, their ents. " I do hope that they were not irrelevant for theses should be of high quality. those of us here; when I look in the c){otiai) at Heavy specialization is another aspect of other reports of the discussion in Dallas, I dis the Ph. D. With many more mathematicians ac cover that they were relevant for at least 52 per tive, many older fields have become more tech cent of the Ph. D.'s trained at Michigan. nical, and many newer fields have developed. The dissertation was, is, and should be the The bright young mathematician anxious to get center of the Ph. D. degree in mathematics. The to the frontier and to do something for himself, whole effectiveness of graduate education depends has often plunged himself too deeply into parti on the fact that here students are at last really cular technicalities about this or that. I submit asked to do something on their own. The only that the present Ph. D. thesis is by far too spe known way to accomplish this is to have them cialized, and that this has come about because really do something on their own. Any objection the idea of making a contribution to knowledge that some Ph. D. theses were actually written by has been emphasized at the expense of making the supervisor is beside the point and is a cri sense out of knowledge. In consequence, very tique only of that particular supervisor. Any ob many young Ph. D.'s will be heavily specialized jection that the student does no research beyond in fields which depend more on novelty than on his thesis again is beside the point-he has at any depth; in a few years• time, the field may be a rate learned what it is like to think, and if he de;id end and the young mathematician will be has learned that he doesn• t want to think, it is intellectually dormant. better that he learn that soon. All the evidence at It is high time that we broaden the Ph. D. hand indicates that writing a thesis matters to I don• t mean to clutter it up with the needless the student and that it will continue to matter, addition of terminology like "Doctor of Arts", whatever his future. but I do mean to allow theses really to deal with I now turn to a much more uncertain point. the advancement of analytical and historical It may be that there are too many theses and too knowledge on all sorts of fronts, not just the many Ph. D.'s in mathematics. This I say with latest high differentials in somebody• s spectral considerable hesitation, because I surely cannot sequence, or three-dimensional adjoint functor point to anyone and say, "This one Ph. D. is too theorems. It is high time that some of our theses much. " It may be that the best response to the tackle some of the relevant problems in the gen question, "Are there too many Ph. D. 's ?", is eral structure of mathematical knowledge. Here the scribbled note added to this title when it ap are some samples: The modernization of a num peared once on our departmental bulletin board: ber of classical fields, such as geometrical op "No, there is one too few; namely, the under tics. Second, the conceptual organization of signed." mathematics. Category theory by no means does *Calvin C. Moore, Graduate education in mathematics in the coming decade, these c}/otiui), Volume 20, Number 3, April 1973, pp. 123-125. 182 the whole job. It is high time that other people course. Other students had taken algebra and think on general or organizational ideas. Third, trigonometry in high school, but were helpless the applications of mathematics. The congeries in the basic calculus and probability course. of ideas which today pass for "systems analysis" Many students of both classes exhibited a common should be subjected to a systematic analysis. defect: they could not analyze even the simplest New points of view on old subjects are needed. problem. It was as though their previous mathe There are too many point set topologists, and matical education consisted solely of formula not enough who pay attention to what a topological memorization. They could apply the formulas space ought to be. For example, one ought to correctly only in problems which closely resem know how to describe a compactly generated bled those which they had memorized completely. space directly from suitable axioms and how to In short, these students had only the most super give a suitable course on this basis. ficial understanding of mathematics and were To summarize: The Ph. D. dissertation in totally unable to use it. To compound their dif mathematics is still much needed; perhaps not ficulty, they had poor self-images with regard so many dissertations are needed, but those that to their mathematical ability. are there should be both better and deeper. To help these students, we designed a course which emphasized understanding, visual ization, figuring things out and problem solving. Karel de Leeuw * Using our version of the Keller plan, the course American mathematical scholarship has provided a great deal of individual attention for been the envy of the mathematical world. It has students by using more advanced students as made great progress toward the solution of many tutors. We wrote much of the course material problems in pure and applied mathematics. The ourselves to emphasize visualization and con educators and administrators in mathematics ceptualization, and we revised the material in have more than succeeded in attaining the goal response to student feedback. The entire expe of producing many good mathematical researchers rience was unusual because the students were and much fine mathematical research. bright verbally, but backward mathematically. The anticipated depression in employment By exploiting their verbal facility, the course of untenured Ph. D. 's has caused many thoughtful materials and format enabled many of the stu educators to criticize the Ph. D. program for dents to understand and apply mathematics for what once seemed to be its greatest strength: its the first time. emphasis, by means of the dissertation, on re The course was personally rewarding to us search. In mathematics, as in most sciences, because it represented a challenge in mathemat research which advances our knowledge keeps ics education which we were able to meet. It was the subject alive and growing, so there can be gratifying to see students previously afraid of little room for doubting whether research is mathematics gain the confidence to take further essential. mathematics courses. Nevertheless, there are valid questions Potentially, our course could serve as a about what the nature of mathematical research laboratory for research on the learning of mathe ought to be. Graduate schools currently empha matics. As such, it could provide valuable train size research in traditional mathematics, where ing not only for would-be college teachers, but as job openings in the colleges and in industry also for interdisciplinary research students in seem to require varying degrees of interdisci mathematics and fields such as psychology, plinary training. There may be just one role physiology, engineering, and linguistics. The which all mathematicians and mathematics edu possibility of using the course in this way arose cators ought to play. However, until that role from our reading of recent research on the work becomes much clearer than it is now, it behooves ing of the brain ([1], [2] and [3]). us to experiment witl1in our laboratories of living In particular, research on the split brain mathematics. It's obvious that not every school conducted by Sperry at Caltech and by others can be a Princeton, and it may yet become ob allowed us to conceptualize what we were at vious that not every school should try to be a tempting to do in the course. Summarized brief Princeton. ly, this research suggests that although both As an example, but not a paradigm, of the halves of the brain are symmetric with respect kind of experimentation just advocated, we offer to sensory and motor functions, they are strik our experiences in establishing an experimental ingly asymmetric with respect to higher func course at Stanford. There we had the good for tions. It has been known for 100 years that the tune to enjoy the support of our department and left half of the brain contains the primary con the university administration. This point is es trols over the language skills. What has only pecially important because it shows a willingness recently become known is that the right half of to support a nonstandard activity in a field which the brain controls geometric and spatial percep allegedly rewards only standard research. tion, pattern recognition, gestalt synthesis, and It had become increasingly apparent during musical development. The left hemisphere func recent years that many students at Stanford who tions analytically, step-by-step, whereas the needed some work in mathematics had training right hemisphere functions synthetically, all-at inadequa:te to do our algebra and trigonometry once. The following quote from John Eccles in *These remarks are taken from a preliminary version of a paper written with Philip Faillace at Stanford. 183 The understanding of the brain [2] illustrates any level of mathematical study are those who these conclusions: can take material presented to the left hemi suitable for hemisphere programming the sphere and translate it into material "The right Then they must is greatly superior in all kinds of geo right hemisphere manipulation. left hand results of right hemi and perspective drawing. The left be able to translate the metrical into words which give a unable to carry out even simple sphere manipulation hemisphere is faithful left hemisphere representation. Con kind .... It is an arithmetical tasks of this versely, those who fail at any level of mathe hemisphere, but not a geometrical hemisphere. are those who do not use both It is quite surprising how sharply this distinc matical study hemispheres in their various appropriate func tion can be made. It could never have been pre tions, for example, by attempting to cope through dicted." memorization rather than understanding. In light of the above description of brain Our experience with the Stanford course hemisphere roles, consider how your own cre and our reading in split-brain research suggests ative work in mathematics involves a continual that the time may be right for good work here by shifting from analytic, sequential, discrete mathematicians as part of an interdisciplinary reasoning to synthetic, unordered, continuous team. A team, consisting of graduate students reasoning, and back again. This shifting, which in mathematics and other disciplines such as we now know to be from side to side, struck psychology, education, physiology, communica Einstein as fundamental to the creative process. tions and engineering could live with the course In response to Hadamard• s request for informa and manage on-going experiments, formulate tion on the internal or mental images used by hypotheses, write materials and, finally, pre mathematicians [4], Einstein observed, in part: pare not only a report of their work, but also for trig the language, as they are actual materials and tested procedures "The words or and creativ written or spoken, do not seem to play any role gering mathematical understanding as our recom my mechanism of thought. The psychical en ity. This should not be construed in in mathe tities which seem to serve as elements in thought mendation for a Ph. D. dissertation appropriately are certain signs and more or less clear images matics. Instead, it seems more which can be •voluntarily• reproduced and com the barest outline of a Ph. D. program in inter with specialization in bined. disciplinary research mathematics. "There is, of course, a certain connection Some of the fruits of research described between those elements and relevant logical con above might be applicable to mathematics educa cepts. It is also clear that the desire to arrive tion in general and even to the Ph. D. program finally at logically connected concepts is the in mathematics. If our students could learn to emotional basis of this rather vague play with the improve control of their creative processes, above-mentioned elements. But taken from a dissertations might come easier and be better. psychological viewpoint, this combinatory play The same could be said of traditional research seems to be the essential feature in productive after the Ph. D. With respect to some of the thought-before there is any connection with logi social problems facing our profession, this re cal construction in words or other kinds of signs search might help open the doors for greater which can be communicated to others." mathematical participation by women and under represented minorities. It might also produce Einstein• s statement constitutes a striking more employable Ph. D. holders. account of creative thought motivated by a left This paper began by remarking that Amer connected con hemisphere desire for logically ican mathematical scholarship has been the envy hemisphere, cepts, but taking place in the right of the world. In closing, it is appropriate to re in then followed by verbalization and checking mark that another enterprise has also been the us are not so suc the left hemisphere. Most of envy of the world-the automobUe industry. cessful as Einstein in the control of the creative Changes beyond the control of that powerful in great deal of hard process. There is always a dustry now threaten its continued success. Simi work by the left hemisphere which lays the larly, changes beyond the control of our pro a problem. Then groundwork for understanding fession also threaten its continued success. The of incubation during there seems to be a period response we make now will determine to a large after which the left hemisphere does nothing, but extent the judgment of our worth by future in far which insight flashes unexpectedly, usually tellectual historians, just as the auto industrY's from the chalkboard or desk. response will determine the judgment of their In retrospect, the bearing of these split worth by future political historians. brain discoveries upon our course is obvious. Finall~ we wish to acknowledge our debt The course materials which emphasized visual to George Polya. He has showed all of us how a ization and conceptualization provided mecha mathematician can make significant contributions nisms by which verbally bright students could beyond the pale of traditional research, as well learn to trigger their right hemisphere pro as within it. Our work is just a tiny contribution cesses, and then translate the products of that to a field in which his work is a model for all. process into words. Naturally, the course ma terial at this point is only a crude, almost acci BIBLIOGRAPHY dentally effective device in need of much refine ment based on knowledge not yet obtained. In *[1]. J. E. Bogen, The other side of the particular, it would be useful to refine and test brain, Bulletin of the Los Angeles Neurological the following hypothesis: students who do well at Societies, 34 (1939), 135. 184 [2]. J. C. Eccles, The understanding of [5]. R. Ornstein, The nature of human the brain, McGraw-Hill, 1973. consciousness, Freeman, 1973. *[3]. M. S. Gazzaniga, The split brain in man, Scientific American, 217 (1967), 24. [4]. J. Hadamard, An essay on the psy chology of invention in the mathematical field, *[1] and [3] are reprinted in [5]. Dover, 1954. In response to the question of how standards could be raised, Browder replied that the intrinsic quality of theses should be improved by increasing their significance, breadth, and relevance to other parts of mathematics. He said that he didn•t know what mechanism one might use to accomplish this. Thomas asked Mac Lane if Chicago would be interested in hiring someone who had not written a research thesis. Mac Lane said he thought one reason that recent students at Chicago have not chosen to write expository or organizational theses is that ambitious young people starting off in graduate work do believe that the way to make their mark in the world is to prove a better theorem. He said it is hard to persuade people that they ought to value organizational and deeper expository accomplish ments as much as proving theorems, and, although he hasn• t been able to persuade his colleagues at Chicago, he was trying to persuade everybody present. He asked Thomas if he had been persuaded: Would Berkeley hire such a person? Thomas said he thought it was possible, that his own views were shifting a bit, and that Berkeley has been changing its hiring pattern in recent years. Thus there are now more two-year appointments with very little, if any, prospect of re-appointment after those two years. In this context one might use criteria different from those that would apply in the case of a person one might want to keep on as an assistant professor. In particular, someone who had written a really splendid thesis of a broad and deep historical nature might make a very big contribution to the students during the two years he or she would be in the department. Mac Lane said he wanted to emphasize that they were really not talking just about Berkeley and Chicago. He said he thought they were talking about a state of mind in the mathematical community. The state of mind used to be that only theorems count; but applied mathematicians have gradually made us recognize that besides theorems, there are also methods of computation and there is understanding of other situations as well. He added that we have to loosen up still further, that he sees signs that this may happen, and that he• s an optimist. Browder stated that in his opinion, the best organization, and the best exposition, was practi cally always done by some of the most creative research mathematicians; he cited the examples of Serre, Milnor, and Hirzebruch. He recalled a quotation from Norman Steenrod who said that although many people say that exposition is difficult, it is the exposition of half-baked ideas that is difficult. The process of making a half-baked idea into a good, clear idea is research. And research is difficult. Browder concluded that it was very idealistic to expect someone who is not research oriented, and not very much in the line of research, actually to do a good job of organization. Mac Lane agreed with Browder• s examples and with Steenrod• s statement, but he added that there are also examples of notable research mathematicians who, though they were clear about the ideas, were not as clear in the expositions. And, in addition, there are others who were more con cerned with exposition than with research-he cited Herglotz, and also Edwin B. Wilson, as well as E. H. Moore in his later years. He said he agreed that good exposition was hard but that we should encourage people to do it. Calvin Moore said that he had been successful in at least one of his aims, which was to provoke discussion of the Ph. D. thesis. He said he concurred completely in the judgment of Mac Lane and Herstein that too many theses are very narrow and lack significance, This and the fact that the train ing has been oriented toward research mathematics where there are no jobs were his reasons for sug gesting that things ought to be changed radically. The Ph. D. program in mathematics is certainly effective in doing the things that were considered appropriate ten or fifteen years ago, but now things have changed and we have too many Ph. D.'s. His proposal is to use the degree to certify a certain level of competence, a familiarity with a large part of mathematics, as well as certain levels of dis tinction and achievement. Research training, in his scheme, would take place at the postdoctoral level, but otherwise in much the same way as at present. He observed that, even today, competence as a research mathematician is not measured by the fact that one has "Ph. D." after his name; it is measured by a bibliography of research papers. He added that he agreed with many of the panel mem bers and cited in particular Browder• s statement that there should be fewer and better research mathematicians. 185 QUERIES Edited by Wendell H. l<'leming The QUERIES column is published in each issue of these cJv"oliaiJ . This column welcomes questions from AMS members regarding mathematical matters such as details of, or references to, vaguely remembered theorems. sources of exposition of folk theorems, or the state of current knowledge concerning various conjectures. When appropriate, replies from readers will be edited into a definitive composite answer and published in a subsequent column. All answers received to QUERIES will ultimately be forwarded to the questioner. Consequently, all items submitted for consideration for possible publication in this column should include the name and complete mailing address of the person who is to receive the replies. The queries themselves, and responses to such queries, should be typewritten if at all possible and sent to Professor Wendell H. Fleming, American Mathematical Society, Post Office Box 6248, Providence, Rhode Island 02940. QUERIES max lu(x) 1~ i(b- a) J~ lu"(x) lctx 37, E. Mendelsohn (Department of Mathematics, where t (b-a) is the best constant. University of Toronto, Toronto, Canada M5S 1A1). I have been unable to find any literature on the following problem RESPONSES TO QUERIES and it has vast and important combinatorial implications. Replies have been ,r,.eceived to queries published in "Let G, H be small subgroups of Sn (say O(G), O(H) ~ recent issues of these cJVotiaiJ, as follows. The editor wishes to thank those who have responded, in!). When does there exist rJ E S such that G n ~ ~ n 26, (Small, October 1973), In addition to responses 1?" printed in the April 1974 c){oticti), the querier has pointed 38, Lawrence J, Dickson (Mathematics Department, out the following reference to a theorem of R. Lipschitz University of New South Wales, Kensington 2033, used in the proof: Australia). Let Hn be the space of n X n Hermitian R, Lipschitz, Transformation d•une somme de deux ou de trois carres, J, Mathematiques 4(1886), 374-439, matrices, H~ c Hn the set of positive definite matrices, The same theorem is utilized in a recent paper of B. F. Suppose ~: mm .. Hn is a linear map with the following Rice: Bart F. Rice, Rings of integral quaternions, J, properties: (a) F ~ ~(nlm) n H~ is nonempty; (b) k ~ Number Theory 5(1973), 524-536 dim ker ~(a) is constant for 0 'I ~(a) E ot(mm) n oH~; (c) 31, (Shelupsky, February 1974) This query concerns (z E ltn:z E ker ~(a) for some ~(a) E F- (O}} spans 186 3. J. Levine, Variable matrix substitution in 8, I. S, Reed, Information theory and privacy in algebraic cryptography, Amer. Math. Monthly 65(1958), data banks, Proc. AFIPS Nat. Comput, Conf. 42(1973), 170-179. 581-587. 4. ----, Some elementary cryptanalysis of 9, C. H. Meyer and W. L, Tuchman, Pseudo algebraic cryptography, Amer, Math, Monthly 68(1961), random codes can be cracked, Electronic Design 23(Nov. 411-418, 1972), 5. C. V. Krishnamurthy, Computer cr· ptographic 10, D, Kahn, The code breakers, MacMillan, N. Y,, techniques for processing and storage of confidential 1967, information, Internat. J. Control12(1970), 753-761. 11. C. Hammer, Signature simulation and certain 6, F, A. Stahl, A homophonic cipher for compu cryptographic codes, Carom. of the ACM (1) 14(1971), tational cryptography, Proc. AFIPS Nat, Comput. Conf. 3-14. 42(1973), 565-568, 12. J, M. Carrol and P. M. McLellan, "Fast 7. B, Tuckerman, A study of the Vigenere-Vernan •lnfinite-key' privacy transformation for resource-sharing single and multiple loop enciphering systems. Report systems", in Security and Privacy in Computer Systems, RC 2879, ffiM, T .J. Watson Research Lab., Yorktown L, Hoffman (ed,),Wiley, N.Y. Heights, N. Y,, 1970. THE PRESERVATION OF ARCHIVAL MATERIALS A Renewed Appeal by the CMBS Advisory Committee on History ·Mathematics as a specialty and mathemat National Union Catalogue of Manuscript Collec ical research are nearing their centenary in the tions, and the Center for the History of Physics United States -if we take as their beginning such maintains a National Catalog of Sources for the symbolic benchmarks as the founding of the Johns History of Physics and Astronomy,) Hopkins University (opened to students in 1876) 2. Organizations should be diligent in pre and the publication of The American Journal of serving their records. Mathematics (founded by Sylvester in 1878). Our 3. Mathematics departments of colleges major mathematical organizations (the AMS, and universities were also urged to take appro MAA, and NCTM) have all celebrated their semi priate action, We would like to extend this latter centennials. The beginnings of American mathe suggestion as follows: matics are rapidly passing out of the realm of (a) Have a department history written and personal knowledge and oral anecdote into writ deposited in the department office or in the col ten history. It is now more than proper, it is lege• s library and archives. The committee is urgent that original sources and archival mate aware of such materials at Dartmouth, Harvard, rials which are the basis of this history be pre and Michigan, but has found that most traces and served. This committee made an appeal for such memories of an important American mathemati preservation through notes in the mathematical cian, including a diary of studies abroad, have journals a few years ago (see cJVotiu.iJ, volume largely disappeared at another university which 16 (October 1961), p, 888). A renewed and ex formerly went so far as to establish a fellowship panded appeal seems appropriate today with the in his name. passage of time and of many funded projects and (b) Collect, date, and label photographs committees. Our previous appeal suggested that: of staff members, visiting lecturers, partici 1. Individual mathematicians and their pants in colloquia and conferences held on cam families should preserve such items as unpub pus. lished and semi-published materials, course (c) Collect and preserve the unpublished notes and photographs of important colleagues, minutes and proceedings of committees, con data on the organization and early days of so ferences, projects, and perhaps even staff meet cieties. College archives, local museums, aca ings. demies, and historical societies are potential (d) Urge staff members to prepare a depositories. For example, the Archive of Con skeleton key to their own publications. temporary History at the University of Wyoming The Committee suggests that persons with has been given materials by the Rudolph Langer historical interests would do the mathematical family and is interested in acquiring materials community a service by undertaking such tasks relating to mathematics and science; the Harvard as are listed above and by pursuing studies of University Archives have papers of Benjamin periods, people, or influences upon American Osgood Peirce and George D. Birkhoff, while mathematics, such as the influx of foreign math the Niels Bohr Library of the Center for the ematicians in the period centered about World History of Physics has copies of mathematical War II. notebooks of Herman Minkowski. (There is a Guide to Archives and Manuscripts in the United Carl B. Boyer Kenneth 0. May States by P. M. Hamer (Yale University Press, Churchill Eisenhart David Rosenblatt 1961), the Library of Congress maintains The PhillipS. Jones, Chairman Charles Weiner 187 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. An announcement will be published in these CJVoticeiJ if it contains a call for papers, place, date, subject (when applicable), and speakers; a second announc~t will be published only if changes to the original announcement are necessary, or if 1t appears that additional information should be announced. In general, SMIC announcements of meetings held in the United States and Canada carry only date, title of meeting, place of meeting, speakers (or sometimes general statement on the program), deadline dates for abstracts or contributed papers, and name of person to write for further in formation. Meetings held outside the North American area may carry slightly more detailed information. Information on the pre-preliminary planning will be stored in the files and will be available to anyone desiring information on prospective conferences. All ~ommunications on special meetings should be sent to the Special Meetings In formation Center of the American Mathematical Society. Deadlines for parti~l~ issues of the cNOticeiJ are the same as the deadlines for abstracts which ~ppell!.~~ !~~ inside front cover of each issue. May 3, 1974 JljD-e 17-July 12, 1974 , , SYMPOSIUM ON THE MATHEMATICAL TRAINING OF SEMINAIRE DE MATHEMATIQUES SUPERIEURES THE NONACADEMIC MATHEMATICIAN Universite de Montreal, Montreal, Quebec, canada Rensselaer Polytechnic Institute, Troy, New York Program: Categorical methods in mathematical logic and (Room 330, Communications Center) in the theory of automata Program: The symposium will feature brief presentations Speakers: Jean Benabou, Universite Paris-Nord: Logique by Burton Colvin, National Bureau of Standards; Murray categorique; Samuel Ellenberg, Columbia University: Sur Klamkin, Ford Motor Company; Edward Hart, General quelques problemas algebriques de Ia theorie des auto Electric Company; and Alan J. Hoffman, International mates; Andre Joyal, Universite du Quebec a Montreal: Business Machines. George Carrier, Harvard University, Theorie categorique des fonctions recursives; William will serve as moderator. The presentations will be fol Lawvere, University of Perugia: Logic of topoi inside lowed by a general discussion centering on the appro and outside; Gonzalo Reyes, Universite de Montreal: priate mathematical training-at B.S. , M.S. , and Ph. D. Methodes de theorie des modeles en theories des catego levels-for students who aspire to become professional ries; Dana Schlomiuk, Unlversite de Montreal: Introduc mathematicians in a nonacademic environment. tion a Ia theorie des topos Support: (partial) Office of Naval Research, Mathematics Support: National Research Council of Canada, the De Branch partment of Education of the Government of Quebec and Information: R. C. DiPrima, Department of Mathematical the Universite de Montreal. Sciences, Rensselaer Polytechnic Institute, Troy, New Information: Seminaire de mathematiques superieures, York 12181 Departement de mathematiques, Universite de Montreal, case postale 6128, Montreal 101, Quebec, Canada May 30-June 1, 1974 EIGHTEENTH MEETING OF THEORETICAL June 18-21, 1974 PHYSICISTS AND MATHEMATICIANS SYMPOSIUM ON QUASIGROUPS AND FUNCTIONAL Instltut de Recherche Mathematique Avancee EQUATIONS Information: Secretariat de Ia R. C. P. N° 25, Instltut de Belgrade, Yugoslavia Recherche Mathematique Avancee, 7, rue Rene Descartes, Information: Matematillki Institut, Knez Mihailova 35, 67084 - Strasbourg Cedex, France YU-11000 Belgrade, Yugoslavia June 12-18, 1974 June 29-July 2, 1974 FIRST LOS ALAMOS WORKSHOP ON MATHEMATICS A SURVEY OF HARMONIC ANALYSIS IN THE NATURAL SCIENCES DePaul University, Chicago, illinois Los Alamos Scientific Laboratory, University of Program: Eleven expository talks California Speakers: E. M. Stein (Princeton University), "Harmonic Invited lecturers: L. Bers, R. Bott, J. H. Bramble, analysis on Euclidean n-space;" A. Zygmund (University G. F. Carrier, J. D. Cole, B. Efron, J. G. Glimm, of Chicago), "The development of Fourier series 1900- L. N. Howard, D. J. Kleitman, R. H. Kraichman, 1935;" R. Hunt (Purdue University), "The development of C. C. Lin, N. Rostoker, P. G. Saffman, J. T. Schwartz, Fourier series 1936-1970;" E. Hewitt (University of F. L. Spitzer, H. F. Weinberger, K. G. Wilson Washington), "Harmonic analysis on locally compact Panel discussions: Hydrodynamic simulation, lasers, abelian groups;" C. Fefferman (University of Chicago), mathematics in nuclear physics, mathematics in the "Harmonic analysis and the theory of HP spaces;" G. Weiss biological sciences, numerical analysis, plasma physics, (Washington University of St. Louis), "Harmonic analysis statistical physics of compact groups;" J. M. Ash (DePaul University), Information: Robert Y. Porton, Public Relations Office, "Multiple trigonometric series;" S. Vagi (DePaul Univer Los Alamos Scientific Laboratory, Los Alamos, New sity), "Bounded symmetric domains;" P. Sally (Univer Mexico 87544 sity of Chicago), "Harmonic analysis and group repre sentations;" D. Burkholder (University of Illinois), 11 Har- 188 monic analysis and probability;" Y. Meyer (University August 16-17, 1974 of Paris-Sud, Orsay), "Harmonic analysis and number CONFERENCE ON MEASURES OF INFORMATION AND theory." THEIR APPLICATIONS Information: J. Marshall Ash, Department of Mathematics, Indian Institute of Technology, Bombay, India DePaul University, Chicago, illinois 60614 Topics: Entropy, directed divergence, inaccuracy, dis tance measures, affinity, Fisher• s measure, axiomatic July 8-13, 1974 foundations, applications to coding theory, psychology, illANNUAL GENERAL MEETING OF THE SOUTHEAST communication engineering, information theory, econom ASIAN MATHEMATICAL SOCIETY AND JOINT CON ics, probability and statistics, weather forecasting sta FERENCE ON NEW AREAS IN MATHEMATICS tistical physics, structure of languages, questio~lre Penang, Malaysia theory, decision theory, music, automata theory Progl'am: Topics of invited addresses will be applied al Deadline for abstracts and papers: June 30, 1974 gebra, combinatorics, mathematical logic, model theory, Information: P. N. Rathie, Convener of the Conference graph theory and mathematical models. Other items in De~artment of Mathematics, McGill University, Montr~al, the program include short talks on a wide variety of sub Quebec, Canada jects, meetings of splinter groups for the specialist or August 26-30, 1974 professional in given areas, and working parties in which issues of common interest to the entire mathematical REGIONAL CONFERENCE ON MATHEMATICAL community will be discussed. THEORIES OF POPULATIONS: DEMOGRAPHICS, GENETICS AND EPIDEMICS ~: University of Science, Malaysia, and the Re University of West giOOan:;entre for Education in Science and Mathematics Florida, Pensacola, Florida (RECSAM) Lecturer: Frank C. Hoppensteadt, Cour-ant Institute of Mathematical Information: R. F. Turner-Bmith, Secretary, Southeast Sciences, New York University ~: (anticipated) Asian Mathematical Society, c/o Department of Mathe National Science Foundation and matics, Chung Chi College, Chinese "Ciiiifilrence Board of the Mathematical Sciences University of Hong Information: Kong, Shatin, N. T. Shawky E. Shamma, Faculty of Mathematics, and Statistics, University of West Florida, Pensacola, July 22-26, 1974 Florida 32504 REGIONAL CONFERENCE ON PROJECTIVE MODULES October AND 23-25, 1974 RELATION MODULES OF FINITE GROUPS SIAM 1974 FALL MEETING University of Wisconsin-Parkside, Kenosha, Wisconsin Shoreham-Americana Hotel, Washington, D. C. Program: Karl W. Gruenberg, University of London at Program: The themes of the meeting will be on computing Queen Mary College, principal lecturer applications in science and engineering and some Contributed papers: To be scheduled topics in nonlinear waves. Some of the Support: (pending) specific areas to be pre National Science Foundation; travel sented may include optimization and subsistence for 25 participants and parameter fitting, numerical solutions of ordinary differential Information: R. W. Gatterdam or Kenneth equations W. Weston, finite element methods and nonlinear waves. University of Wisconsin-Parkside, Kenosha Wisconsin ' 53140 • Abstracts deadline: July 15, 1974 Information: Society for Industrial and Applied Mathe July 29-August 2, 1974 matics, 33 South 17th Street, Philadelphia, Pennsylvania SECOND COLLOQUIUM ON THE THEORY OF AUTO 19103 MATA, LANGUAGES, AND PROGRAMS Saarbriicken, West Germany August 27-September 2, 1975 Information: J. Loeckx, Computer Science Department, INTERNATIONAL CONGRESS OF LOGIC, METHOD University of Saarlandes, D-66 Saarbriicken, Federal OLOGY, AND PffiLOSOPHY OF SCIENCE Republic of Germany University of Western Ontario, London, Ontario, Canada Program: The objective of the congress is to provide a July 29-August 10, 1974 forum for the presentation and critical discussion of re A NATO ADVANCED STUDY INSTITUTE: MODERN cent and ongoing work by research workers from all over COURSE ON STATISTICAL DISTRIBUTIONS IN the world in the fields oflogic, methodology, and philos SCIENTIFIC WORK ophy of science. University of Calgary, Alberta, Canada Sponsors: The International Union of History and Philos Program: July 29-31: A modem theoretical seminar on ophy of Science distributions; August 1-3: A modem applied seminar on Support: National Research Council of Canada; The Uni related scientific problems; August 5-7: A modern course versity of Western Ontario on characterizations and applications; August 8-10: Re Contributed papers: Must be submitted in their entirety search communications and contributions. before January 1, 1975, to Jaakko Hintikka, Department Sponsors: NATO, National Research Council of Canada, of Philosophy, Stanford University, Stanford, California International Statistical Institute, Indian Statistical In 94305, stitute, University of Calgary, Pennsylvania State Uni Information: M. K. Ward, ;Executive Secretary, Inter versity national Congress of Logic, Methodology, and Philosophy Information: G. P. Patil, Distributions Project, Depart of Science, c/o National Research Council of Canada, ment of Statistics, Pennsylvania State University, Uni Ottawa, Canada K1A OR6 versity Park, Pennsylvania 16802 NEWS ITEMS AND ANNOUNCEMENTS SITUATIONS WANTED August issue ls June 15, 1974. Application forms On the recommendation of the Society• s are available from the editorial department of the Committee on Employment and Educational Poll Society, but lt is not necessary to use one pro ey, the editors of these c)l{otitxi) announce a tem vided all the necessary information is supplied. porary relaxation of the policy covering free sit Applicants who are not members of the Society uation wanted advertisements. Any person who are requested to have their application endorsed receives a doctoral degree in the mathematical by a member of the Society, Free advertisements sciences in the period July 1973-June 1974 wlll are limited to 50 words (equivalent to six 65- be entitled to a free situation wanted advertise space typed lines). Extra words are $0.15 each, ment in the August 1974 issue. See page A-465 of $1 minimum. For anonymous listings, there is this issue for instructions. The deadline for re a fee of $5. All fees must be prepaid. ceipt of situation wanted advertisements for the 189 PERSONAL ITEMS NAZAR ABDELAZIZ of the University of HONNAPPA SIDDALINGAIAH of the Univer Maryland, College Park, has been appointed to sity of Pittsburgh has been appointed a member an assistant professorship at the University of of the Technical Staff, System Sciences Division, Maryland, Baltimore. Computer Sciences Corp. , Silver Spring, Mary ANNIE L. ALEXANDER of Clarkson College land. of Technology has been appointed to an assistant ALLAN J. SILBERGER of the Institute for professorship at Hampden-Sydv,ey College. Advanced Study has been appointed to a visiting ROBERT E. BOZEMAN of Vanderbilt Uni professorship at the University of Bonn. versity has been appointed to an assistant pro VIVIAN E. SPENCER of the U. S. Bureau fessorship at Morehouse College. of Minas baa been appointed to a professorship RALPH CZERWINSKI of the University of at the U:lli.t, 190 Dr. HYMAN N. LADEN of Brooklandville 56. She was a member of the Society for 8 years. Maryland, died on December 21, 1973, at the ' Professor WILLIAM J. WALBESSER of age of 58. He was a member of the Society for SUNY at Buffalo died on October 22 1973 at the 34 years. age of 45. He was a member of the 'societ~ for Mrs. RUTH B. RINGEL of Maplewood New 13 years. Jersey, died on February 17, 1974, at the ;ge of NEW AMS PUBLICATIONS AUTHOR INDEX OF multiple entries and cross-references as ex plained below. Although no compilation of this MATHEMATICAL REVIEWS, 1965 -1972 kind can be free of error, a specific check has (Volumes 29- 44) been made to ensure that every review in Math ematical Reviews during the period in queStioil List price $200; institutional member $150; has been accounted for. individual member $100; reviewer• s price $50; The bulk of the index is an alphabetic list ISBN 0-8218-0027-2 ing by author. Items having several authors are To order, please specify MREVIN/65/72 listed in full under each author. All items by a This index has been prepared directly from given author are listed under a so-called to the the working files of Mathematical Reviews and •primary' name and cross-references from any other forms of lists all the reviews in Volumes 29 through 44 of primary form are given have been used in items Mathematical Reviews, which were published the author• s name that 29 through 44 of Mathemat from 1965 through 1972. In many respects it is reviewed in Volumes similar to the earlier cumulative indexes of ical Reviews. The entries under a given heading Mathematical Reviews, namely the 20 Year are listed in order of nominal publication dates. Author Index of Mathematical Reviews 1940- If a reviewed item is clearly connected with a 1959, and the Author Index of Mathem~tical Re person who is not its author (e. g., an editor) views, 1960-1964 (Volumes 21-28), published by then a cross-reference is given from the name the American Mathematical Society Providence of that person to the full entry. R. I. , in 1961 and 1966, respectiv~ly. However,' Collections, proceedings and other similar there have been a number of improvements. publications that may have editors but do not The index is in four volumes (3025 pages) have specific authors are given alphabetically by and covers about 127, 000 reviews (by way of title in a separate list (corresponding to the key comparison, note that the four volumes in the word index in the monthly issues of Mathematical two earlier cumulative indexes covered 25 years Reviews) following the list of authors. Such items and listed about 156, 000 reviews). The total are usually listed under several headings, with number of entries is about 200, 000, including cross-references, for ease of identification. NEWS ITEMS AND ANNOUNCEMENTS ARTHUR B. COBLE MEMORIAL LECTURES Sciences at Carnegie-Mellon University. The The Department of Mathematics of the Uni subject of the lectures will be optimization prob versity of Ulinois at Urbana-Champaign an lems of mathematical science; each of the three independent topic from nounces that the fifth annual series of Arthur B addresses will be on an lectures are intended Coble Memorial Lectures will be delivered Sep~ applied mathematics. The to appeal to those in the fields of engineering tember 3, 4, and 5, 1974, by Professor Richard science, and economics ' J. Duffin, University Professor of Mathematical 191 RELATIONS BETWEEN US-USSR SCIENTIFIC JOURNALS AND PUBLISHERS by William Browder* Soviet publishers have long made it a prac that normal practice would be for them to sub tice to reproduce photographically and distribute scribe in the normal way to Western journals, widely in the USSR and Eastern Bloc the leading as we subscribe to theirs. scientific and mathematical journals of the West. Many professional organizations, such as Now that the USSR has joined the International the American Mathematical Society, are engaged Copyright Convention, they are proposing to the in on-going programs of translation of Russian editors and publishers of the West that they journals and books. Normal practice would be should be authorized to continue this practice with for the translation rights to be given in exchange payment of nominal royalties to the copyright for a royalty, and negotiations to this end should owners. be pursued. A Russian offer to exchange rights I feel very strongly that the technical pub to reproduce our journals for rights to translate lishers of the West should deny them this privi theirs is obviously to exchange things of far dif lege, which no other country in the world asks ferent value, treated equally. A translation is a for, and invite them to subscribe in the normal major and expensive enterprise, involving new way to these journals at the usual rates, as West literary effort and totally new production from ern libraries do with Soviet journals. It is clear editing to typesetting. Photographic reproduction that the USSR would like to make the great eco gets all the benefits of the editorial work of oth nomic savings and savings of hard currency which ers, and yields a product indistinguishable from their practice yields. By the same token, this the original except perhaps by marginally lower practice costs Western scientific journals a very quality. large amount of income, and all these journals Let all these be treated as the separate are hard pressed financially. questions they are, and let the Soviet publishers As an illustration, let us consider the AN establish truly normal relations with their West NALS OF MATHEMATICS. According to Soviet ern counterparts on the basis of equality, with figures, their reproduced edition had approxi out special status. mately 180 subscribers last year. They propose It is sometimes suggested that the Russian to pay us royalties of 7 1/2 percent of the sub publishers can flout the copyright and be unac scription price of $24 per subscription, or a to countable except under Soviet law, and hence tal of $1. 80 per subscription. Hence, our reve they have the upper hand and we should take what-, nue last year on 180 subscriptions would have ever deal they offer. This would not be in keep been $324, instead of $4,320 if they had sub ing with their announced intentions, and should scribed in the usual way. Our marginal cost of certainly be fought if they attempted it. lf the producing 180 more issues would certainly have Western professional societies formed a common been less than half this amount, so that our loss policy toward this question, they could bring a of revenue under this plan is striking. great force of public opinion and publicity to Our journal has run at a substantial deficit bear, and could even make legal counterattacks for the last several years, and the USSR scienti in Western courts (perhaps by reciprocally at fic establishment has contributed nothing to our tacking Soviet copyrights here). It is not clear cost of production in that time. They propose to that the Soviets do have bad intentions in fact, continue to derive benefit from our journals, but it is clear that they will accept a bargain if it while contributing a tiny fraction of what all other is offered them. readers contribute. I propose that true reciprocity should guide The magnitude of this problem when extrap our relations, and the USSR should not be given olated over all the scientific, mathematical, special deals or bargains. medical and technical literature is enormous. I lf, in some cases, the Russians wish to estimate that Russian yearly savings in hard cur withhold translation rights from the professional rency under their proposal would be on the order societies, etc., presumably they will award of $40 million per year or more. This revenue is them elsewhere for royalties, and the transla coming out of the pockets of the publishers of tions will continue under different auspices, so these journals, and either directly or indirectly that there will be no scientific loss. lf they try (by higher prices) out of the pockets of the West not to sell translation rights to anyone in retalia ern scientific establishment. It is a subsidy of tion, then other measures of retaliation could be Soviet science by Western science. considered, such as refusing to sell translation The USSR wishes to normalize its trade rights to our works. However, this seems un and scientific relations with the world. It is clear likely in the present mood of the Soviets to try to normalize relations. *Professor of Mathematics, Princeton University, Member of the Editorial Board of the Annals of Mathematics 192 ABSTRACTS PRESENTED TO THE SOCIETY Preprints are available from the author in cases where the abstract number is starred. Invited addresses are indicated by • 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. Tbe 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 cffoliaiJ 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 74T-A!07. WILLIAMS K. FORREST, Simon Fraser University, Burnaby 2, British Columbia, Canada, A property of uncountable free algebras. Preliminary report. Suppose that A is an uncountable free algebra of countable order and Il(x) is a set of equations and inequations with parameters from A of cardinality less than lA I· We denote by ll(A) the set of simultaneous solutions of fl(x) in A. T'-len either III(All < IIl(xll + X 0 or lll(A)I = lA I. (Received January 30, 1974,) (Author introduced by Dr. Alistar H. Lachlan.) 74T-Al08, ANDREAS ZACHAR IOU, University of Athens, Athens 143, Greece. Reversible orthogonalization. Theorem. Let V be a real inner product space. Then V is of countable dimension iff every subspace of V admits an orthonormal basis. Sketch of proof. If dim V =' N 0 , it is well known that every subspace of V admits an orthonormal basis. For dim V > N 0 it is shown that V contains a subspace W which does not admit any orthogonal basis. In fact if V does not admit any orthonormal basis we are done, If V admits an orthonormal basis B, then fix e E R and take W as the subspace of V spanned by all x- e, x E B, x f e. (Received February 4, 1974.) 74T-Al09, KIM KI-HANG BUTLER, Instituto de Fisica e Matematica, Lisbon, Portugal and Route 1, Box 24B, Lumberton, North Carolina 28358. Connection between doubly stochastic matrices and equivalence relations. Let D n be the set of all n x n doubly stochastic matrices, En the set of all equivalence relations defined on n-set, E(D n) the set of all idempotents in D n' lXI the cardinality of a set X, and S(n, k) the Stirling's number of the second kind. Lemma 1. IE(D) I = ~~ = 1 S(n, r) = IE n I· Theorem 2. E(D n) is isomorphic to En' (Received February 15, 1974.) * 74T-Al10, N. E. AUSPITZ, University of Waterloo, Waterloo, Ontario, Canada. On Q·sheaves of Banach spaces. Preliminary report. Let Ban 1 be the category of (real or complex) Banach spaces (where the morphisms are linear contractions) and (X, :f) a topological space. We apply Fakir's process (C. R. Acad. Sci. Paris Ser. A-B 270(1970), A99 - AlOl) to the Godement "standard construction" derived from the category of Ban 1 pre sheaves on X, FtfoP, Ban 1) to obtain an idempotent monad (Q, 71) on FCJ 0 P, Ran 1) which yields a category called "Q-Sheaves of of Banach spaces on X." The functor Q serves as a "sheafifying" functor except, generally, Q does not preserve kernels. Characterization of our Q-sheaves is comparable to the definition of a j·sheaf in a topos where j is a modal operator, We construct an operator on subpresheaves of an idempotent, inflationary, meet-commuting ("up toE") presheaf of Banach spaces which defines (as in Lawvere-Tiemey) precisely the Q·sheaves, We relate our construction to the "fields of spaces" of Dauns A-431 and Hoffman (Mem. Amer. Math. Soc. No. 83(1968)) via an "espace etale" construction which yields Q-sheaves, and also, yields Dauns and Hoffman's construction for fields of Banach spaces. Applications of our constructions to results of Dauns and ~loHman are considered. (Received February 18, 1974.) (Author introduced by Professor K. A. Rowe.) 74T-Alll. ALAN E. ROSENBERG, Wesleyan University, Middletown, Connecticut 06457, On the simplicial radical of the group algebra. Preliminary report, The subject of when the simplicial radical of a group algebra is trivial has not been previously investigated. Let k be a field of characteristic zero, G a group, k[G] the group algebra of G over k. Lemma. If a P.I. algebra has trivial Jacobson radical then it has trivial simplicial radical. Proposition. If H is an Abelian subgroup of G s.t. [G:H] is finite, then the Jacobson radical of k[H] is trivial. Theorem. The simplicial radical of k[G] is trivial. (Received February 18, 1974,) • 74T-A112. COLIN L. MALLOWS and NEIL J. A. SLOANE, Bell Laboratories, Murray Hill, New Jersey 07974 and ANDREW M. ODLYZKO, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. Upper bounds for modular forms, lattices, and codes. Let W(z) = 1 + Ade 217 idz + Ad+ie 2rri(d+l)z +· · · be a modular form of weight n for the full modular group. Then for every constant c there exists an n 0 = n0 (c) such that if d::: n/6- c and n > n0 , then one of Ad' Ad+!' ••• has a negative real part. This implies that there is no even unimodular lattice in En, for n > n 0 , having minimum nonzero squared length 2: n/12- c. A similar argument shows that there is no binary self-dual code of length n > n0 having all weights divisible by 4 and minimum nonzero weight :::: n/6 - c. (Received February 22, 1974.) 74T-All3. THOMAS C. BROWN, Simon Fraser University, Burnaby 2, British Columbia, Canada, Some new Van der Waerden numbers. Preliminary report. Let n(a 1, a 2, • • ·, ak) denote the smallest positive integer x such that if {1, 2, 3, • • •, x! is partitioned into k classes A 1, A 2, • • • , A k then for some j, the class A i contains an arithmetic progression of j terms. New values are n(2, 3, 4) = 21, n(2, 3, 5) = 32, n(2, 3, 6) = 40, n(2, 4, 4) = 40, n(2, 4, 5):::: 71, n(3, 3, 4) = ~1, n(2, 2, 3, 3) = 17, n(2, 2, 3, 4) = 25, n(2, 2, 3, 5) = 43, n(2, 2, 4, 4) =53, n(2, 3, 3, 3) = 40, n(4, 6)::: 70, n(5, 5)::: 177. These calculations were done in 1971, however to the author's knowledge the bound on n(5, 5) is still the best known. This list extends a list of V. Chvatal ("Combinatorial structures and their applications", Gordon and Breach, 1970), (Received February 25, 1974.) * 74T-A114. LAWRENCE JAMES RISMAN, Technion- Israel Institute of Technology, Haifa, Israel. Stability: Index and order in the Brauer group. A field F is stable if for every division algebra A in the Brauer group of F, order of A = index of A. In general, order divides index and they have the same prime factors. Any local or global field is stable by class field theory. In this paper, index and order in the Brauer group of a field F with discrete valuation and perfect residue class field K are calculated. Division algebras with specified order and index are constructed, giving another proof of Brauer's classic result and extending this result to power series fields and fields of rational functions in fewer variables. For F complete, necessary and sufficient conditions for the stability of F are given in terms of the Brauer group of K. Theorem. F is stable iff K is stable and for every cyclic extension L of K and every division algebra D in the Brauer group of K the index reduction factor of L and D = the g.c.d. of the index of D and the degree of L over K. Results. A finite extension of a stable field need not be stable. The power series field K((x)) is stable for K a local field. K((x)) and K(x) are not stable for K a global field. (Received February 27, 1974,) * 74T-A115. JON FROEMKE and ROBERT WILLIS QUACKENBUSH, University of Manitoba, Winnipeg, Manitoba R 3T 2N2, Canada. The spectrum of an equational class of groupoids. Preliminary report. The spectrum of an equational class K is the set S(K) = lnl3 V: E: K, IV: I '~ n!. Obviously 1 E: S(K) and x, y E S(K) = xy E: S(K) for any equational class K. G. Cratzer has shown that if S is any multiplicative A-432 monoid ,of positive integers then there is an equational class .K s.t. S(.K) c S. We prove that .K can be chosen to be an equational class of groupoids. (Received March I, 1974.) * 74T -Al16, DONALD R. JACOBS, Western New England College, Springfield, Massachusetts 01119. Lie derivations on the s kcw elements of simple rings with involutzon. This paper investigates the relationship between Lie derivations defined on the skew elements K of a simple ring with involution and ordinary derivations on the ring R. In particular, the following theorem is established. Let H. be a simple ring with involution, of characteristic not 2, having two symmetric orthogonal idempotents e 1, e 2 (/ 0, 1) with e 1 > e 2 f 1, and in addition, R not isomorphic to the 4 x 4 matrices over a field or the 3 x 3 matrices over a field of characteristic 3. Then if L is a Lie derivation of K into R, there exists an ordinary derivation D of H. into H. and an additive mapping T of R into the center of R such that T[R, R] = 0 and L = D + T on K. (Received March 1, 1974.) (Author introduced by Professor Wallace S. Martindale III.) • 74T-A 117. JOSEPH B. DENNIN, University of Connecticut, Storrs, Connecticut 06268. The genus of fields of elliptic modular functions. Let K(n) be the field of elliptic modular functions of level n. K(n) is a finite Galois extension of C(j) where j is the Weierstrass invariant. Theorem. Fix a genus g. The number of fields F of genus g s.t. C(j) <; F <; K(n) for some n is finite. Since [' f['(n) is isomorphic to the Galois group of K(n) over C(j), the theorem proves a conjecture of H. Rademacher chat the number of congruence subgroups of genus 0 of the modular group [' is finite. (Heceived March 4, 1974.) * 74T-A118, ARMANDO R. GINGRAS, IBM'Corporation, Boulder, Colorado 80302. Convergence lattices. Preliminary report. A complete lattice in which order convergence coincides with topological convergence w.r.t. the order topology () is termed a convergence lattice. Each complete chain, each arbitrary product of convergence lattices, and each lattice in which each chain is finite is a convergence lattice. We say that a complete lattice is locally intervaled if every 0-neighborhood of each point in the lattice contains an interval that is also a O-neighborh9od of the point, Theorem. A complete lattice is a convergence lattice iff it is locally-intervaled. (Received March 4, 1974.) *74T-Al19. MARTHA KATHLEEN SMITH, University of Texas, Austin, Texas 78712. Rings with an integral element whose centralizer satisfies a polynomial identity. The following generalizations of a theorem of S. Montgomery (Israel J. Mach., to appear) are obtained. Theorem I. Let R be an algebra over an integral domain 0, a E R integral over 0, D the derivation Dx = ax- xa. Suppose that D satisfies some polynomial f(A) E O[A] such that f'(O) is invertible in 0. Then if the centralizer of a in R satisfies a polynomial identity, R also satisfies a polynomial identity. (The condition on D is satisfied whenever 0 is a field and a is separable over 0.) Theorem 2. Let R be a prime ring and a E R integral over the centroid of R. If the centralizer of a in R satisfies a polynomial identity, then R satisfies a polynomial identity. (Received March 6, 1974.) *74T-Al20, MOSHE GOLDBERG, E. TADMOR and GIDEON ZWAS, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel. On the numerical radius of a matrix. Preliminary report. Let A be an n x n complex matrix with spectral radius p(A) and let r(A) =max li(Ax, x)l, lxl = 1l be the numerical radius of A. In general, p(A) :S r(A). A matrix A is said to be spectral if p(A) = r(A). We prove chat A is spectral iff for some integer m, m ::': n, rm(A) = r(A m). Additional properties and characterization are also investigated. (Received March 8, 1974.) 74T-A121. DAVID ZEITLIN, lli50 Vincent Avenue North, \.linneapolis, Minnesota 55411. Identities for integer sequences involving the greatest integer function. V. Preliminary report, M .and Q are positive integers; Wn is an integer sequence given by Wn +2 = MW n +i + SQW n' S = ± 1, where 11' 0 and 11' 1 are integers; if 11' 0 = 0, W1 = 1, then Wn oo Un, and if W0 = 2, 11' 1 = M, then Wn = Vn. Set A(x) = A-433 2 2 lxW 0 - W1i, F = V/(1 + V3), where V3 = 3SMQ + M2, D(j) ~ (1 + V 3)Uk.+;' T = M + 4SQ, Z 1 = 2 (l'i~ - MW 0 II' 1 - SQW 6)/(T ), and [x] the greatest integer function. Case 1. S = 1, with 1 ;: Q < M + 1 and M :; 1; and p > 0 is the root of x 2 - Mx- Q = 0, Case 2, S =- 1, with 1 ;S Q < M- 1 and M;;; 3; and P > 1 is the root 2 3 of x 2 - Mx + Q = 0, Theorem 2 (for both cases), Let Rk = PU 3k. + SQU 3k._ 1• If A3(P) < (T P ")/(D(2k)Q 3"), then for n;;; k ~ 0, we have [p 3 kw~ + 3Z 1(- SQ)n(RkWn- (- SQ)k.W n+k.) + F] = W~+k' Example 1. Fibonacci 2 3 3 sequences, with· S = M = Q = 1, U n =F n , V n =L n , and P > 0 is the root of x - x- 1 = 0, For n >= 1, (P F 2n + 3 (,6)(L 2n+l + {5F 2n) + ,8] = F~n+l' and (P L ~n+Z - 6PL Zn+Z - 3L 2n+ 4 + .8] = L ~n+ 3 ' Example 2, Wn+Z = 4W n+l 2 3 Wn' W0 = 1, W1 = 3, and P > 1 is the root of x - 4x + 1 = 0. For n;;; 1, [P W~ -\1(Wn-l + BWn) + ,8] = W~+l' where 8 = 22 + (15)vJ: and P 3 = 26 + (15)yJ. Theorem 1 (for squares W~) was given in IV (Abstract 74T-A95, these ~ 21(1974), A-367),(Received March 8, 1974.) 74T·A122, ROBERT J. PERRY, Worcester State College, Worcester, Massachusetts 01602, Relational models as generators of triples through reflective and coreflective functors. Preliminary report. If (f is a coreflective subcategory of e and 5\ is a reflective subcategory of t', let U: t' _, (j and F: e-> 5\ be the induced functors, then Vi5\: 5\-> {'[ and Fifi: (j _, 5\ are adjoint. We say (51, t') generates the resulting triple T on a. When (f = Ens, we extend the comparison functor ¢: ~B -> Ens T to H(P,, e): e -> A('f), where A(T) is the smallest subcategory of the T-relational models that accepts H(:B, t') for all generating pairs (5\, e). We develop characterizations for the objects of AcT). If i is the identity triple and B is the ultrafilter triple, then A(l) is sets with equivalence relations, while A(lJJ is the completely regular spaces. Furthermore, A(T) provides a right bicategory structure (1, P) on the T-premodels which leads to a series of P-reflective subcategories. (Received March 11, 1974.) * 74T-A123. BRIAN A. DAVEY, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada. Dualities for varieties of relative Stone algebras. Preliminary report. For 2 S n < w let en ce~) denote the n-element chain as a Heyting algebra (Brouwerian algebra), Let £ (S ) denote the variety, qua category, generated by e ce' ). Denote by X the category of Boolean n n n n n Iii spaces with a continuous action of the endomorphism monoid End(en), and by t)n the category of pointed , Foolean spaces with a continuous action of the endomorphism monoid End ce~ ) s .t. the constant endomorphism I acts as the retraction to the distinguished point. In both Xn and )&n the morphisms are the continuous maps preserving the endomorphism action. Note that en E xn and e~ E !Jn. Theorem. (i) Xn(-, en): x:p -> £n is right adjoint to £ (-, e ): £ _, Xop. Furthermore, the unit 71: Id., _,X (£~ (-, e ), e ) is a natural equivalence. n n n n .l..n nn n n (ii) Similarly for S and We may apply these dualities to characterize weak injectives. Theorem. I is a n ID n • , weak injective in £n (Sn) iff [Q/!8 x P {IQI!P), B a complete Boolean algebra and P a complete n·valued Post algebra. The dualities may also be applied to describe free products in £n and Sn. In particular, a new proof is obtained for the description of the finitely generated free algebras in £n and Sn (see P. Kohler, Dissertation, Math. Institut. Giessen, 1973). (Received March 11, 1974.) 74T·A124. THOMAS H. FOREGGER, University of Wisconsin, Madison, Wisconsin 53706. On facial minimizing matrices for the permanent function. Preliminary report. Let R be a set of ordered pairs (i, j), 1 S i, j S iz, such that a(O, 1) matrix with 1 's in the positions specified by R is fully indecomposable, Let F(R) =!XIX is n x n doubly stochastic and ";; > 0 implies (i, j) E R !. Theorem. Let A belong to F(R) and suppose Per{A) = min {Per (X) IX E F(R)l. Then A is fully indecomposable, Per(A) = Per(A(ilj}} for aii > 0, Per(A(ilj)) ~ Per(A) for aii = 0 if (i, j) E R. For n?: 2 let f(n) denote the minimum value of the permanent of an n x n nearly decomposable doubly stochastic matrix. Conjecture. f(n) is 1/2n- 1 and the only n x n nearly decomposable doubly stochastic matrix for which equality is attained is \1(1 + P), where P is a full cycle permutation matrix. (Of course permutations of \1(/ + P) also give equality.) The conjecture is true for n S 8. (Received March 11, 1974.) A-434 74T-Al25. VINCENT C. HARRIS, California State University, San Diego, California 92115 and M. V. SUBFARAO, University of Alberta, Edmonton, Alberta T6C 2Gl, Canada. Unitary multiperfec/ numbers. We define 11 to be unitary multi perfect if o*(11) kn for any integer k -- 2. (Here o*(n) denotes the sum of the unitary divisors of n.) We obtain here some properties of such integers n including the following: (i) 11 must be even; (ii) for any given integer m, the number of n's with m prime factors or for which 2mjln is finite; (iii) the density of the set of unitary multiperfect numbers is zero. (Received March 11, 1974.) 74T-Al26. JUTTA STRECKER, Vanderbilt University, Nashville, Tennessee 37235. Strong canc-ellation for free products in the rategory of bounded distributi1'e lattices. Preliminary report. Let L 1, L 2 and L~ be bounded distributive lattices and denote by * the free-product operation in the category of bounded distributive lattices. Consider the statements (E) L 1 * L 2 ~ L 1 * L~ => L 2 = L~ and (F) L 1 does not have a nontrivial complemented element. Comer and Dwinger, [Abstract 711-06-10, these 1lotice6 21(1974), A-44] proved that when IAut L 21 > 1' (E) =(F). Under the following finiteness conditions it can be shown that (F) = (F): (I) !Aut L 21 < w, (2) 3 an element in L 1 which is at the same time the finite join of join-irreducible elements and the finite meet of meet-irreducible elements, (3) L 1 has at most finitely many dual atoms and every element in L 1 is contained in one of them. Condition (2) generalizes a result from the abstract cited above. (Received March 11,' 1974.) (Author introduced by Professor Stephen D. Comer.) * 74T-A127. SOON-KIONG SIM, Universidad de Los Andes, M~rida, Venezuela. Right noetherian rings of which every prime kemel functor is symmetric. Preliminary report. Let R be an associative ring with unity. Following G. Krause [J. Algebra 23(1972), 88-99], R is fully right bounded if for each prime ideal P, every large right ideal of R/P contains a nonzero two-sided ideal, Theorem. If R is a right noetherian ring, then R is fully right bounded iff every prime kernel functor [Goldman, J. Algebra 13(1969), 10-47] is symmetric, in the sense of Murdoch and Van Oystaeyen (Abstract 73T-A216, these 1ktt~.-_ 20(1973), A-480). (Received March 12, 1974.) * 74T-Al28. SYLVIA MARGARET WIEGAND, University of Nebraska, Lincoln, Nebraska 68508, Semi local domains whose finitely generated modules are direct sums of cyclics. Preliminary report, An FGC-ring is a commutative ring such that every finitely generated module is a direct sum of cyclics, Theorem. Every nonzero prime ideal in an FGC-domain is contained in a unique maximal ideal. Corollary. Let R be a s~miprime ring with noetherian maximal ideal spectrum. Then R is FGC iff (1) R is Bezout, (2) every localization of R is an almost maximal valuation ring, and (3) R is a finite direct product of h-local domains, (Received March 14, 1974.) 74T-Al29. MICHAEL KEISLER, North Texas State University, Denton, Texas 76203. A characterization of the closed meet-ideals of the cone of a complete /-group. As a generalization of the C-sets investigated by Appling (Abstract 656-14, these :lktie""'- 15(1968), 506) the closed (order topology) meet-ideals of the positive cone P of a complete /-group G are characterized in terms of lattice endomorphisms. Theorems. 1. S is a closed meet-ideal of P iff S = a(P) for a unique a in g = laia: P-> P s.t. for x, y E P, (i) as;;:, (ii) a is isotone, and (iii) a(x + y) = a(a(x) + a(y))!. If a, f3 E g, then for X E p let (ae ,B)(x),, a(x) + fh -a(x)), (a v f)(x) = a(x)V,B(x), and (a l\f3)(x) = a(,B(x )). 2. If F = I e, v ' II I. then ·[g, F] is a (conditionally) complete l-monoid with maximal cone p* = laix = Vlxll(z- a(z))iz E Pl, for every x E Pl. 3. P is isomorphic to P= lal3 a E P s.t. a(x) = al\x, for x E PI, which is a subset of P* with the property that f3 = Vlf311ala E PI, for ,B d. 4. P is isomorphic to p* iff P= P*. 5. p* is the lateral completion of P (cf. Conrad, Proc, London Math. Soc. '(3) 19(1969), 444-480). 6. If a E g, then 3 a unique aE p*, and a unique a E ~ s.t. a is additive on P, a1\ a= 0, and aala= a. The technique of lateral completion apparently differs from that of Nakano and others. Additional C-set generalizations are established. (Received March 15, 1974,) A-435 74T-AI30. DAVID GEIGER, 3432 W. 66th Street, Chicago, Illinois 60629. Coherent algebras. An algebra A is coherent if every pair class be coherent is that, for some n, derived operators F 1, • • ·, F n and H exist which satisfy the identities F; *74T-AI31. F. THOMAS FARRELL, Pennsylvania State University, University Park, Pennsylvania I6802. The K-dimension of H 2(G, KG). Let Z denote the ring of integers and Q the field of rational numbers. Then we prove the following results. Theorem I. If G is a finitely presented group which contains at least one element of infinite order, and K is a field with nonzero characteristic; then any sub-G·module of H 2(G, KG) has dimension 0, I, or "" when considered as a K vector space. Corollary. If G is a finitely presented group which contains at least one element of infinite order, then the abelian group 11 2(G, ZG) either vanishes, is infinite cyclic, or is not finitely generated, Theorem 2. If G is a finitely presented, torsion-free group whose cohomological dimension is ""• then any sub·G·module of H 2(G, QG) has dimension O, I, or "" when considered as a Q vector space. (Received March I8, 1914.) *74T-AI32. CHARLES F. WELLS, Case Western Reserve University, Cleveland, Ohio 44I06. Centralizers of transitive semigroup actions and endomorphisms of trees. A tree is a poset in which every ancestor set is well-ordered. It is locally finite if the interval between any two points is finite. A local isomorphism of a tree with itself is a surjective endomorphism which is an isomorphism when restricted to any interval. Theorems. I, Let T be a nontrivial locally finite tree. Then the local isomorphism monoid of T is transitive iff T has no maximal or minimal elements. The automorphism group of T is transitive iff T has no maximal or minimal elements and the sibling sets of any two elements have the same cardinality. An example is given of a non-locally-finite tree with transitive automorphism group. 2. Let a: X-> X be a surjective function for which air. has no fixed points for all k > 0. Then there is a transitive monoid of transformations of X which commute with a. If the inverse images of any two points of X under a have the same cardinality, this monoid can be taken to be a group of permutations. If a is centralized by such a monoid, then a is surjective and, if a is not injective, ak has no fixed points for k > 0, Theorem 2 extends the result of Abstract 711-20-87, these~ 21(1914), A-103. (Received March I8, 1974.) * 74T -AI33. JAMES OSTERBUFG, University of Cincinnati, Cincinnati, Ohio 45221, Completely outer groups of automorphisms acting on R/J(R). Let R be a ring with unit, f(R) its Jacobson radical, and assume R/J(R) Artinian. Let G be a finite group of automorphisms of R that induces a completely outer group on R/J(R), Then R is G-Galois over the fixed ring S if R is projective over the usual crossed product 1'1, or if the order of G is invertible in R, or if R is Artinian. Moreover, R has a normal basis, Furtherm~re, the Jacobson radical of 1'1 is J(R)I'l, where f(R) is the Jacobson radical of R. (Received March 18, 1974.) (Author introduced by Professor Charles Groetsch.) * 74T-A134. KARL K. NORTON, Universit~ de Gen~ve, Geneve, Switzerland. On the number of restricted prime factors of an integer. I. Let E be a nonempty (possibly finite) set of primes, let p 1 < p 2 < · · · be the members of E. For each positive integer n, let O(n; E) denote the number of prime factors of n which lie in E, counted according to their multiplicities in the unique prime factorization of n. Define E(x) = ~~p- 1 : p .:S x, pEE}. Theorems. I, If x 2: 1 and I< z :5 pl' then ~n:<;xz'l(n;E) :5 xG 1(x, z)exp l(z- l)E(x) + 4z}, where G 1(x, z) = 1 1 min {I+ (logx)(log p 1)-1, p 1(p 1 -z)- }. (If z =Pp G 1(x, z)= I+ (log x)(logp 1)- .) If P, 74T-A135. OLGA TAUSSKY TODD, California Institute of Technology, Pasadena, California 91109, The determinant of the additiue CfJTfl'lllULator of two integral 2 x 2 matrices. If the matrices A, R have irrational characteristic roots in the fields Q(a), Q({3) respectively then - det (AR- HA) is a norm in either field. Also expressing (~ ~). m E Z, as a commutator of rational matrices leads to a result of related nature. (Recei.ved March 18, 1974,) * 74T-A131i. GERTRUDE ElfRLICH, University of Maryland, College Park, Maryland 20740. Units and one -oided units in regular rings. A ring R with identity is unit-regular if for each a E R there is a unit x E R such that axa =a. Theorems. 1, A ring R with identity is unit-regular iff, for each a E R, there is a unit u E R such that aR e u(Ra)' ~I~ (where (I~ a)' is the right annihilator of Ra in R), This ideal-theoretic characterization of unit-regularity is used to prove 2, A regular ring R with identity is unit-regular iff it is von Neumann finite (i.e., contains no strictly one-sided units), One-sided unit-regularity, defined in the obvious way, is also characterized in ideal-theoretic terms, and the ring of all linear operators on an arbitrary vector space over a division ring is proved to be one-sided unit-regular. (Received March 19, 1974,) Analysis * 74T-B106, HAROLD EXTON, University of Salford, Salford, United Kingdom, Incomplete hypergeometric functions. Preliminary report. Generalised hypergeometric functions in one or more variables may be represented by integral formulae in which the limits of integration are fixed. If these limits are made variable, the functions thus defined aie called incomplete hypergeometric functions. Some of the many possible types of incomplete hypergeometric functions are here introduced; this work extends the concepts of incomplete beta and gamma functions and incomplete elliptical integrals, (Received January 3, 1974,) * 74T-Rl07, JOSEPH LEHNER, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, The boundedness of integrable automorphic forms, Let A/G), B q(G) be the Banach spaces of automorphic forms of weight q 2: 1 on a fuchsian group G, 2 given by the norms JJ(1 - jzj )q lf(z)j dKdy, suplzl< 1 (1 -lzl 2)q lf(z)j, respectively, where G acts on !zl < 1 and R is a fundamental r~gion for G. Theorem. Let G satisfy: jtrace Aj 2: 2 + m for all hyperbolic A in G, where m > 0 depends only on G. Then Aq(G) C B q(G) and the inclusion map is continuous. The proof proceeds by constructing, for each elliptic vertex w and each parabolic cusp p, regions B"' and Kp s,t, if f E Aq(G) and z E BUK, B = U., R"" K~UPKP, then (1-jzj 2)q lf(z)j < m'll!lll' m' = m'(G, q), In the complement of BUK, A, Marden's results ("Universal properties of fucltsian groups in the Poincare metric", Proc. Conf. Univ. of Maryland, 1973) are used in an essential way, (Feceived \larch 7, 197-4,) *74T-Bl08, ETHELBERT N. CIIUKWU, Cleveland State University, Cleveland, Ohio 44115. On the boundedness of a certain fourth The equation has form (*) x< 4> + f(x, x', _,,:'', _,!")i" + g(x, x', x")x" + h(x, x') + i(x) = p(t, x, x', x", x"') where f, g, h, i, and p are functions which depend (at most) only on the arguments displayed explicitly. For similar treatment of simpler variants of (*) see Tejumola (Ann. Mat, Pura Appl. (IV), 80(1968), 177-196) and A-437 Ezeilo and Tejumola (Ann. Mat. Pura Appl. (IV), 95(1973), 131-145). Generalizations of the Routh-Hurwitz criteria and other sufficient conditions are formulated which guarantee that ultimately all solutions of (*) satisfy \x(t)l S K, \ ..:(t)'i S K, \x(tY'\ S K, \x(t)"' \ S K, where the magnitude of the constant K depends only on the constants associated with (*) as well as on f, g, h, and i. The result obtained extends known results in several directions. (Received January 16, 1974.) 74T-B109. ERWIN 0. KREYSZIG, University of Windsor, Windsor, Ontario, Canada. Representation of a Bergman kemel. 2 Let (u, v) € C (U) be a solution of (1) grad u = A grad v on a domain 0 C R 2 , where A= (ajk) € C'(O) is orthogonal, det A= +1, and the variable a in the usual expressions of the ajk's in terms of sin a and cos a is linear in x and y. Then from the integrability condition of (1) one obtains a p.d.e. for v which has a corresponding Bergman integral operator whose kernel can be represented as a power series in t (the variable of integration) s.t. the coefficients em of t 2m are polynomials in tan a of degree m. This representation is useful in certain problems of compressible fluid flow. It is also shown that that p.d.e. is not of class P, so that it does not have corresponding Bergman kernels which reduce to polynomials in t. (Received January 23, 1974.) * 74T-Bll0. CHUNG-LING YU, Florida State University, Tallahassee, Florida 32306. Solvability of the Cauchy problem for linear elliptic equations with analytic coefficients. Let L x be a mth order general linear analytic elliptic differential operator in a neighborhood N of the origin, x = (xl' · · ·, xn)' K(x, y) a symmetric fundamental solution of Lx inN (i.e., K(x, y) is also a fundamental solution of L~), and M a bilinear differential operator s.t. any solution u of L[u] = 0 can be 1 1 represented by u(x) = JBN M[u((), K(x, ()]d(, and let li; the functions li/x')- GJ(x') are analytic in a. Furthermore, the solution is given by u(x) = u0(x) + G0 (x), where u0(x) is a solution of L[u] = 0 with analytic Cauchy data lli;(x')- GJ(x')l. (Received January 28, 1974.) 74T-B111. MAXWELL 0. READE, University of Michigan, Ann Arbor, Michigan 48104. Generalizations of Bazilevic functions. I. Preliminary report. Let f(z) = z + · · · be al)alytic in the unit disc ti, with [f(z)/'(z)/z] /o 0 there, and U(w) a "suitable" function defined on f(tl}. If (aja(}) arg [z/'(z) U(f(z))] 2: O, z =rei 8 holds in ti, then f(z) is OST-convex in h. If J3td[arg(z['(z)U(f(z)))] > -11 holds for all 0::: 01 S 0 2 ::: 217, and all 0::: r < 1, then f(z) is OST close-to-convex (w.r.t. U(w)) in ti, Ogawa [J. Math. Soc. Japan 13(1961), 431-44I], Sakaguchi [ibid, I4(I%2), 3I2-321] and Takatsuka [Duke Math, J. 33(I966), 583-593] have shown OST-convex and OST-close-to-convex functions are univalent. We obtain (1) representation theorems for OST functions, using the technique of Reade [Pub!. Math. Debrecen 11(1964), 39-43], (2) geometric descriptions of OST functions that appear for special choices of U, and (3) some distortion theorems for special cases. These results are related to those of Bazilevic [Mat. Sb. 37(79)(1955), 471-476], the author, and Sheil-Smail [Quart. J. Math. Oxford Ser. (2)23(I972), 135-142], among others. (Received February IS, 1974.) *74T-Bll2. PAUL A. FUHRMANN, Harvard University, Cambridge, Massachusetts 02I38. !-symmetric restricted shifts. Preliminary report. Let J be a bounded operator in a Hilbert space H s.t. J = J* = J- 1• A bounded operator T in H is J -symmetric if T = JT* f. Let ¢ be inner and T the restricted right shift in I¢H21.l, i.e., T 1,! = PI factor of - 1, given by (JfXei1) = e-itcp(ei1)f(e-i1). Let K± =If € I¢H2 l.l\J/ = :!fl. Theorem 2. K+ (K _) is finite dimensional iff ¢ is a finite Blaschke product. If ¢ is a finite Blaschke product of n factors then dim K + = [(n + 1)/2] and dim K_ = [n/2]. Corollary. T ¢ is selfadjoint iff cp(z) = (z- .\)(I- .\z)- 1 for some real .\, \.\\ A-438 74T-Bll3. JA~lES LH.l!NC; \I'ANC, !lro"n Cniversity, Providence, Rhode Island 02912, Strong r<'gularity at 11<>11/H'I/k. points. Preliminary report. Suppose II is a uniform algebra and x is a point of the maximal ideal space, A is said to be strongly regular al x if the set of functions in II which vanish on some neighborhood of x is uniformly dense in the maximal ideal at x. \l'e }Construct a uniform algebra which is strongly regular at a nonpeak point. This answers a question of John Wermer (cf. D. Chalice, ".\'-algebras on sets in en", Proc, Amer. Math. Soc, 39(1973), 300- 304), (Received February 20, 197 4.) 74T-Rll4. ~l!CHAEL !J. RIC[, Ohio University, Athens, Ohio 45701, Uniformities in the descriptive theory of sets. II. '1'/wmcms. I, The complete metric spaces satisfying Lusin' s separation principle are precisely those which may be written as the union of a separable subspace and a a-discrete subspace. 2. Each (metric valued) Raire measurable map on a complete metric space pM is a map of class a, for some a < 0, iff m*pM is proximally fine. 3. Let M be a complete separable metric space and N any metric space. If Mi..."' is Baire measurable, then f(M) is separable. 4. Let pM be a complete, locally separable metric space. Then each (metric-valued) Raire measurable map on M is of class a, for some a < n, and hence m*pM is proximally fine. The last result is used to dispose of two unsolved problems. Example. Let M o- [0, 1] x D, where D is a discrete space with power I( 1• Then m*pM is a measurable proximally fine space which is not locally fine and the Baire sets of M is a proximally fine a-algebra that is not h-closed. (Received February 25, 1974.) (Author introduced by Professor Thomas James.) *74T-B115. JOHN J. F. FOURNIER, University of British Columbia, Vancouver, British Columbia V6T 1W5, Canada. Multipliers of weak type. Let G be an infinite, locally compact, abelian group, or an infinite, compact, nonabelian group. Let I < p < 2. It is shown that there exists a multiplier operator (convoluteur) on G that is of weak-type (p, p) but is not of strong-type (p, p). This generalizes results of Zafran (Abstract 73T-B26I, these :Jlo.tu. 74T -Bll6. J. P. SINGtf, Kurukshetra University, Kurukshetra, India. A characterisation of an entire function in terms of its mean values. Preliminary report. ·e Let f(z) be an entire function of order p and lower order >.. For 0 < 8 < .oo, and z = re' , let 17 8 8 1 8 17 8 8 118 M8(r, f) =[(I/.Z 17 )J; jf(rei )j d0] 1 and M8(r, /ml) ~[(l/2rr)J; jf(m)(rei )j d0] • Theorem. For every entire function, except a polynomial, we have (i) lim supy_.00 log r[M s(r, /m l)/M s(r, f)] 1/m /log r = P• (ii) lim infy_.,.log r[M 8(r. /ml)/M 8(r, f)] 11m /log r ~A, m = I, 2, • • •, and 0 < 8 proved (i) with ~ instead of ~. (Received February 2I, I974.) 74T-Bll7, HEINRICH W. GUGGENHEIMER, Polytechnic Institute of New York, Brooklyn, New York 1120I. Convexity arguments for differential equations. II. 2 4. If p(t, x, x'): R + x Rn x Rn ·~ R + satisfies 0 < p ~ M where M(t 1 - t 0 ) < 4 then for every x 0 there exists a neighborhood V(xo) for which x' + px = o, x(to) = Xo• x(tl) "'XI E V(xo) has a solution. 5. Specialization of the results improve known existence theorems of scalar equations. 6. Let J be the matrix of a rotation of angle rr/2. Let y:R+ x R x R 2 x R2 -. R 2 be continuous, sgn (xTJx') ·xTJy :::_ 0, y(O, t, x, x~ = 0, p = (x'Tfx)- 1 ' x'Tfy(>., t, x, x') 2: f(A) for bounded domains (t, x, x') with lim A·->"" f(>.) = ""• (xTJx')xTJy bounded. Statements I to 3 (see "Convexity arguments for differential equations. !")for x" + px = 0 remain true for x" + y(>., t, x, x') = 0, 7, Let F: R + x R x R 2 ... Hom (R 2, R 2) be continuous, x T JFx definite, the larger eigenvalue of the quadratic form x T J F(A, t, x)x tends to "" with A, F(O, t, x) is a constant operator of vanishing square. Then the problem x' = F(>., t, x)x, x(t 0 ) = x 0 , det (x(t 0 ), x(t 1)) = 0 has eigenvalues of all orders. (Received February 28, 1974.) A-439 * 74T -B118. ANTOINE P. BRUNEL, University of Paris \'1, 75005, Paris, France and HUMPHREY S. FONG and LOUIS SUCHESTON, Ohio State University, Columbus, Ohio 43210, An ergodic super-property of Banach spaces defined by a class of matricrs. (ani) is called an R-matrix if (A) !i"ni .,. 0, and (B) limn ani~ 0 for each i. A Banach space X is called R-ergodic if for each isometry T and each x EX, there is an R-matrix (ani) s.t. 'i.ian?ix-"!!t (converges weakly), Given two Banach spaces F and X, write F fr X if for each finite -dimensional subspace F' of F and E > 0, there is an isomorphism V from F' onto a subspace of X such that ll!xll - I!Vxlll < ' for each x € F' with llxll .:::; 1, X is called super-R-ergodic if F is R-ergodic for each F fr X. Theorem. X is super-R-ergodic iff X is super-reflexive. The proof is based on the following Theorem. Let T be a linear operator on X, (an) a matrix satisfying (A), x EX s.t. 'i.i aniTix~ x. Then there is a constant a s.t. (x- ax) € (I - T)X. (Received February 28, 1974.) 74T-Bll9. JOHN WERMER, Brown University, Providence, Rhode Island 02912. Polynomially conv<'x hulls of some sets. Preliminary report. X is a compact space, A a uniform algebra on X, M the spectrum of A. Let f € A and lJ be an open subset of C\j(X). Put r 1(V) = {p € Mlf(p) € VI, r 1(()" lp € Mlf(p) = (1. Theorrm 1. Fix g in A with g 1- 0 on r 1(V), For ( € lJ put Z(Q = sup lgl over r 1(Q, Then log Z is subharmonic in U. Let now X be a compact set in C 2 invariant under the map: (z, w)-> (ei8z, e-iB w), 0 :S 0 :S 2rr. Put h(X) ~ polynomially convex hull of X and I== zw. Let lJ be an open set contained in c\j(X) with 0 f. u. Theorem 2. Put r 1(U) -= l(z, w) € h(X)Izw E VI. If r 1(V),;, ¢. it contains an analytic disk. Consider the following example: a is a continuous complex-valued function on 0 :S r :S 1 with r- 1a(r) continuous at 0 and a(O) = a(1) = 0, s.t. a maps 0 .:S r :S 1 onto a simple closed curve with interior U, Let X be the disk {(z, w)llzJ :S 1, w = a(jzl)/zl. Under mild regularity conditions on a one has Theorem 3. Put D = {(z, O)llzl ::; 11. 3 a function ¢ analytic in V s.t. h(X) = (XuD)u{(z, w)lzw E U and Jzl = j¢(zw)l1- Theorem 1 is proved by a method of Oka (J, Sci. Hiroshima Univ. Ser. A-I 7(1937)); Theorem 2 makes contact with J. E. Bjork's manuscript "Compact groups operating on Banach algebras", (Received March 1, 1974.) 74T-B120, DAVID F. DAWSON, North Texas State University, Denton, Texas 76203. Continued fractions and totally monotone sequences. For notation and background see Abstract 711-40-3, these Jl.,~<.ce6 21(1974), A-155. Let BV 00 = n ;dBV;. Theorem. The continued fraction f(a) E BVoo in case r 1 j1 1- a 11;;; ja 11 and r 2 11 + a 1 + a 2 1;:; ja2 j, with strict inequality in one of these two relations, rpJ1 + ap-l + apl;;; rprp_ 2 Jap-ll +lap I for p > 2, and 1 < r ;$ r q and Jaql < k for q;;; 1, Theorem. Suppose R is any ray from the origin. (1) If R lies in the half-plane z + z f;: 0, M is a bounded subset of R, and every ap EM, then f(a) € BV oo' (2) If R intersects z + z = ~ at Q, M is any subset of R lying in the half-plane z + z >- ~ and bounded away from Q, and every ap EM, then f(a) E BV00 • Theorem. The sequence lzp I E BV.,. and is not a null sequence iff lzp I is bounded and 3 N 3 'i.; =N(1~kl- 2 )j(~kzp)/zp I < .,., k = 1, 2, 3, • • •. Theorem. If z = x + iy and z = -- ~. or x >- %, or -% < x ;S-% and y 2 > (- %)(4x t 1)(2x t 1)2{4x + 3)- 1, and every ap = z, then the even and odd pans of f(a) are in BV.,.· (Received March 1, 1974.) * 74T-B121. JON C. HELTON, Arizona State University, Tempe, Arizona 85281, Solution of integral equations by product integration. Consult B. W, Helton [Pacific J. Math. 16(1966), 297-322] for background and definitions. Iff and g are functions from R to N, G is a function from R >< R to N, h is quasi-continuous on [a, b] and G E 08° on [a, b], then the following are equivalent: (1) f is bounded on [a, b], G E OA 0 on [a, b], [(u)G(u, v) E OA 0 on [a, b] and f(x) = h(x) + J~f(u)G(u, v) for a :S x :S b, and (2) G E OM 0 on [a, b] and J~h(u)G(u, v)vrrx(l +G) exists and is f(x)- h(x) for a < x A-440 74T-B122. ROBERT E. JAMISO'I 11, University of Washington, Seattle, Washington 9fll9'\ and RICHARD CHRISTOPHER O'BRIEN and PETER D. TAYLOR, Queen's Universitv, Kingston, Ontario, Canada. Rmbedd111~ com·c·x sc•ts in locally com•c•,· linear spac<'s. Preliminary report. Theorem. If X is a compact convex set in a T 2 topological vector space (t.v.s.) and the relative topology on .\ has a basis of open convex sets, then X is affinely homeomorphic to a compact convex subset of a locally convex t.v.s. (This result can be proved in a slightly more general setting which includes convex subsets of certain topologized vector spaces which do not satisfy all of the t.v.s. axioms.) The proof proceeds by showing that an ample supply of lower semicontinuous affine functions exists on X. These are then used to define a continunus barycenter map on X- that is, a map which assigns to each probability measure on X a resultant point of X (d. R. Phelps, "Lectures on Choquet's theorem", Van Nostrand, N. ]., 1966), The barycenter map then provides the machinery to produce continuous affine functions with which to accomplish the embedding. (Received March 4, 1974.) 74T-B123. SWARUPCHAND M. SHA'I, University of Kentucky, Lexington, Kentucky 405010, Probability distributi This abstract is a continuation of Abstract 711-30-21, these lL:die.,.,_ 21(1974), A-123. Theorem 1. Let ¢ satisfy a linear differential equation (*) P k(t)¢(kl(t) + · · · t P 0 (t) ch(t) = Q(t), Q and P's polynomials and de g. P k :':,de g. P 7 (j- 1, • • •, k). Suppose there is an interval [a, b] on which ¢(t) is nonnegative and integrable L. Then the characteristic function (**) f(z) ~ ~'.' 1p. exp (ia .z) + Jb eizt cp(t)dt, p. > 0, a. all real and f(O) = 1, 1 ' 1 1 a 1 ·- 1 is an entire function of bounded index and bounded value distribution. Furthermore, if 0 Sa, <··· replace Q(t) in (*) by ~:~-!Qp(t) exp (apt), ap E C. Theorem 2, Suppose f is defined as in (**) and pk ~ 0, 0 Sale~ m, b > 0, 0 _:::a :S b and p(t) is nonnegative and integrable L on [a, b]. Assume that f does not reduce to a constant, Then {(::;) is an entire function of order one, exponential type and f and each /k) are univalent in lzl < P where p --min (77/2m, 71/2b), (Received March 8, 1974.) * 74T-B124. JINFU FENG and THOMAS H. MacGREGOR, State University of New York, Albany, New York 12222. lntewal mean estimates for derivatives of functions wzth a positive real part. Preliminary report, This paper considers the functions which are analytic and have a positive real part in the unit disk and are normalized by /(0) = I. By properties of subharmonic functions, the sharp bounds for the integral means, (1/277) f~ 71 [Re f(re;e)] "dO, are determined. Through a method of E. Landau [Jber. Deutsch. Math. - Verein. 34(1926), 239-243] the sharp bound i/nl(z)i S 2 · n! Re f(z);'[(l- lzi}n(l + lzi)] is found for n = 1, 2, • • •, These two results yield the upper bound (2 ·n!ft'(l - r)nA (1 + r) A for the integral means of lf * 74T-B125. ATHANASSIOS G. KARTSATOS and MANOUG N. MANOUGIAN, University of South Florida, Tampa, Florida 33620. Further results on oscillation of functional differential equations. Preliminary report. Consider the equation (*) x lim g(l) = +"" as I-> t-"" and Q E C[R+' R]. Theorem 1, Assume that uH(t, u) 2: 0, H is increasing in u, and 1 lim1"""'inf J~ sn-l[_ H(s, k) + Q(s )] ds =- ""• lim1_,00sup J~ sn- [- H(s, - k) ; Q(s)] ds = + oc for every k > 0, Let JQ(t)dt = m(l) + C, Jm(t)dt = m1(t) + C, where C is an arbitrary constant and m, m1 are continuous and bounded with m oscillatory. Then every bounded solution of (*) is oscillatory. Theorem 2. In addition to the hypotheses of Theorem 1, assume that g(t)"' t, H(t, u) = P(t)G(u), P(t) 2: 0, continuous, G(u) increasing, continuous, uG(u) > 0 for every u .J, 0, J;"[1/G(s)]ds <+ ""• J:::;'[1/G(s)]ds <+""for every f > 0, and J~tn-lp(t)dt = + oe. Then every solution (*) is oscillatory. Example. The hypotheses of Theorem 1 are satisfied by x< 4 l + [1/(1 + t6 )]x 113 = sin t. It can be shown that this equation has in fact some solutions which are bounded. (Received March 11, 1974.) A-441 .,, 74T-R126. REKHA PANDA, University of Victoria, Victoria, British Columbia VSW 2Y2, Canada and Ravenshaw College, Cuttack - 3, Orissa, India. Som<' polynomials of severalt;ariables. Preliminary report. This paper discusses a polynomial set !g:f'•qil(h; x 1, • • ·, x,)! generated by exp (ht 11 ) • G(x 1tq 1, ••• , X/q'), where h ,J. 0, the exponents p, q i' i ~ 1, • • •, r, are positive integers, and G(x 1, · • ·, x,) is a multiple power series with arbitrary coefficients. The properties given here include classes of generating functions, multiplication theorems, and differential recurrence relations for these polynomials and extend certain results of H. M. Srivastava [Illinois J. Math. 15(1971), 64- 72], and others. (Received March 11, 1974.) *74T-B127. DOUGLAS MOREMAN, Emory University, Atlanta, Georgia 30322. ContJexity specific functionals in convex topology. Preliminary report, This note is a sequel to a recent abstract in these ikti.c.u, Definition. S is regular relatively to convexity specific functionals means that if M is a convex point set and P is a point that does not belong to M then there exists a continuous convexity specific functional T from S such that T"(M) does not contain T(P), (Every locally convex linear topological space is regular relatively to convexity specific functionals.) Theorems. If S is regular relatively to convexity specific functionals then (1) a sequence a of points converges convexly to the point P iff, for each convexity specific functional T from S, the sequence T(a) converges to T(P), (2) if M is a convex point set then M is convex, (3) the convex topology 0 of S, B, C is the common part of all topologies for S relatively to which each B-continuous convexity specific functional from S is continuous, (4) 0 is a locally convex topology, (5) S is convexly perfectly compact iff, for each monotonic collection H of convex point sets, there is a point P such that if T is a continuous convexity specific functional from S and h is in Jl then T(h) contains T(P), and (6) if S is metric and a is a sequence of points that closes convexly then for each Lipschitz continuous convexity specific functional T from S, T(a) converges. (Received March I5, 1974.) 74T-B128. MYRON S. HENRY and KENNETH L. WIGGINS, Montana State University, Bozeman, Montana 59715. Approximate solutions of nonlinear functional differential equations. Define L(x)(t) = x"(t)- I.j =la/t)[x(g(t))]i- h(t) for appropriate functions h, g and a1, and consider the i.v.p. (*} L(x) = 0, x(O) = a 0 , x'(O) = a 1 on I= [- r, r] ..Set Pu(t) = a 0 + a 1t and, for n 2:2, choose P~n as the polynomial of degree at most k- 2 that solves the minimization problem (m.p.) inf11 EQA\_ 21\p- Ij=lajt{n-1 og- hll, where Qk_ 2 is the set of polynomials of degree at most k- 2 and 11·11 = sup1 l·l· Then Pkn(t) = a 0 + a 1t + f~ (t - s )p; n (s) ds. There is a positive number r 0 such that for r ~ r 0 the sequence !p kn 1;;=1 has a cluster point pk. Furthermore, if pk is the set of all polynomials of degree at most k satisfying the initial conditions of (*), then there is a positive number ri S r0 such that r ~ ri implies that pk is the unique polynomial of degree k or less that solves the nonlinear m.p. infpE!j>kiiL(p)ll· The sequence {pk!;=I converges uniformly to the solution of (*). The second algorithm of Remes may be employed to compute the sequence !pk 1. The above analysis extends the work of G. D. Allinger, D. E. Olson and the first named author, (Received March 18, 1974.) 74T-BI29. WILLIAM D. L. APPLING, North Texas State University, Denton, Texas 76203, A right hand integral representation theorem for a class of quasi-continuous functions. Suppose a< b and Q. C. 0 [a, b], B. V.[a, b] and S[a, b] are, respectively, the set of all functions f from [a, b] into the plane that are quasi-continuous, and 0 at a, of bounded variation, and such that {I-0 lf(q )I: D a subdivision of [a, b]! is bounded. For a~ y ~ b, J L (y) is the element of Q. C. 0[a, b] such that J L (y)(x} = 0 if a~ x ~ y and JL (y)(x) = I if y < x ~ b; and JR(y) is the element of Q. C. 0 [a, b] such that JR(a) = ~. and if a< y S b, then J R(y)(x) = 0 if a~ x < y and JR(y)(x) = I if y ~ x ~b. II II denotes sup norm. Theorem. Suppose f is in Q. C. 0 [a, b] and T is in the dual of Q. C. 0 [a, b], continuous with respect to II II· Then {(x, T(-J L (x))): A-442 a :S x ~ b I, denoted by I, is in fl. V.[a, b], {(x, T{J f~(x) - J L (.·>:))): a :S x :S b I, denoted by u, is in S[a, b], and llr- 2nf(q-H-J L(q)-- fr.(p)lll iII/- r -1/)[f(q)- f(q-H[JR(q)- h(q)]ll + IT(f-) -· ~nf(q-)[t(q)- t(p)]l + IT(f- r)- 2D]f(q)- f(q- l]u(q)l • 0 for refinements of subdivisions. Uniqueness-type assertions for - j L' j R ] L' t and u, with respect to the above representation, are established. (Received April 4, 1974,) 74T-B130. M. R. PARAMESWARAN, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada. Some Tauberian theorems for the Sa-method of summability. Making use of the fundamental paper on Sa ·methods due to Meyer-Konig [Math, Z. 52(1949), 275-304], it is proved that if a sequence Is I is S -summable to A (for some 0 < a < 1) and s ~ O(n>i), then Is I is n Cl n n summable to A by the Cesaro method C 1. Corollary. In the above, one may replace "Sa" by the Taylor method T a or Euler method E a or the P.orel method B. For E a and B the result is due to Knopp [Math. Z. 18(1923), 125-156] and Lord [Proc. London Math. Soc. (2) 38(1935), 241-256] respectively. (Received March 18, 1974.) (Author introduced by Professor Henry C. Finlayson.) 74T-B131. WILLIAM R. TRANSUE, State University of New York, Binghamton, New York 13901 and MICHAEL W. ROYD, State University College of New York, Fredonia, New York 14063. Properties of ultraproducts. Preliminary report. J. Rretagnolle D. Dacunha-Castelle and J. L. Krivine [Ann. Inst. H. Poincar,; Sect. B (N. S.) 2(1%5;156), 231-259.1 introduced the notion of ultraproduct of an indexed family of normed linear spaces and lattices, and Dacunha Castelle and Krivine later expanded this study [Studia Math. 41(1972), 315-334]. These ultraproducts are here investigated systematically. In particular, it is shown that if the spaces in the family are complete, the completion which has been included in the definition is unnecessary. Conditions for separability, a-completeness, semicontinuity of the norm and other properties of ultraproducts are obtained under appropriate hypotheses on the factor spaces. (Feceived March 18, 1974.) 74T-Rl32. SIAMAK KliALILI, University of Pittsburgh, Pittsburgh, Pennsylvania 15260. A sufficient condition for boundedness of the infinitesimal generator of a strongly continuous group of bounded linear transformations on Lp. Preliminary report. Let (i) n be a set, 93 a a-algebra over 0, and p. a countably additive, nonnegative, bounded measure on ~B. (ii) V p 2: 1, LP = LP((}, 93, fl.• R), (iii) X be any Banach space over R. Theorem 1. Let V n 2: 1 T n E CL(X, LP) where 1 P <"" and D =Domain of slim T =If: f and lim T f exists in L Then D is --< n EX n-+oen p 1. closed if ~ a positive decreasing function c(A) defined for A> 0 and tending to zero as A-> oe such that V f E X, E & > (i) p.{w:w 0 supn_ 1 1(T n f)(w)l > AJI/111 S c(A). Theorem 2. Let :Ill be a closed subspace of L p , where 1 --< p < ""• (ii) (U 1: I E H) be a strongly continuous group of bounded linear transformations on 'lll, and A be its infinitesimal generator, (iii) V f E lll, limh_.,0 I(Uh -0/hlf exist a,e, (p.). Then A E CL(lR, lll). (Received March 20, 1974.) 74T-Bl33. BERTRAM WALSf-1, Rutgers University, New Brunswick, New Jersey 08903. Positivity of the trace form on nuclear operators on A(S) characterizes S as a Choquet simplex. Preliminary report. Let E be a real Banach space ordered by a closed, normal generating cone K. Let K' denote the dual cone in E' and order f.(E) by the cone IT E f.(E): T[K] s; Kl, so T 2: 0 means T[K] s; K. For each tensor t € E'EDE let T1 denote the corresponding element of f.(E). Proposition. The following conditions are equivalent: (1) For each t € E'iE, T1 2:0 =>tr(i} 2:0. (2) E has the positive approximation property (P.A.P,): the identity I E .S'!.(E) is the limit uniformly on compacta of operators x-+ ~~=l A-443 (2) for any Banach space F each K-absolutely summing T: E-> F with \\T\\L .S 1 is integral with 1\T\1 1 .S 1; (3) E is (isomorphic as an ordered Banach space to) the space of affine functions on a Choquet simplex (equivalimtly, the base of k' determined by the order unit of K is a simplex). A dual characterization of AL-spaces is given. (Received March 20, 1974.) 74T-B134. TSANG-HAI KUO, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213. Weak convergence of vector measures on F -spaces. Preliminary repon. A Banach space X is said to be a Grothendieck spa,ce if every weak* convergent sequence in the dual space X* of X is weakly convergent. The original result of A. Grothendieck shows that if K is a compact Stonian space then C(K) is a Grothendieck space. Later this result was improved by G. Seever for K being a compact F-space. Let S be a compact F-space-. Our result allows the following generalization: if X is a reflexive Banach space then C(S, X) is a Grothendieck space, Suppose Y is a Ranach space for which both Y and y* have the Radon-Nikody,n propeny, and (un) a bounded sequence in M(S, Y), the Banach space of all regular vector measures with finite variation, such that (un(E)) converges weakly for every clopen subset E of S; then (un) converges weakly in M(S, Y). From this, an analogue of the Vitali·Hahn-Saks Theorem is established: if ~ is a Floolean algebra with the Interpolation property, v a positive finitely additive function on ~. and (un) a bounded sequence in the Banach space fa(~, Y) of all finitely additive Y -valued function on ~ with finite variation such that un are v-continuous and (un(A)) converges weakly for every A in ~. then (un) is uniformly v·continuous. (Received March 20, 1974.) *74T-Bl35. CHUNG-CHUN YANG, Naval Research Laboratory, Washington, D. C. 20375. A problem on polynomials. Preliminary report. The following results are proved. Theorem. Let p(z)- c 1(z- a 1)m1 •• • (z- ak)mk and q(z) ~ c 2(z- a 1)nl. · • (z- ak)nk be two nonconstant polynomials where c 1 and c 2 are nonzero constants, mi and ni are positive integers, and al' a 2, • • ·, ak are distinct complex numbers. Suppose that p'(z) ~ 0-=q'(z) = 0, Then (i) mtfn 1 ~ m/n2 ~· .. ~ mk/nk ~ m/n; (ii) p(z) = cq(z)mln (c is a constant). Corollary. Let p(z) and q(z) be two nonconstant polynomials such that p(z), q(z) have the same set of one points and p'(z), q'(z) have the same set of zeros. Then p(z) = q(z) if on.e of the following two conditions is satisfied: (i) both p(z) and q(z) are monic polynomials with degree p ~degree q, or (ii) the multiplicities for some preimage of 1, say a, of p and q are the same. (Received March 20, 1974.) (Author introduced by Dr. Charles F. Osgood.) * 74T-B136. EUGENE M. NORRIS, University of South Carolina, Columbia, South Carolina 29208, Relative act ideals. This paper generalizes to semigroup actions the well-known notions of relative semigroup ideals due to and studied by A. D. Wallace. An act (S, X) comprises a topological semigroup S acting as a semigroup of transformations on a Hausdorff space X. A subact (T, Y) of (S, X) consists of a subsemigroup T of S, a subset Y of X for which TY ~ Y. A left (respectively, right) (T, Y)-ideal of (S, X) is a pair (A, B) s.t. A is a T-ideal, B ~ Y, TB ~ B (A Y ~B). (A, B) is a (T, Y)-ideal if it is both a left and a right (T, Y)-ideal. (All terms are defined in the paper.) We investigate some of the elementary properties of (T, ¥)-ideals here. We determine, inter alia, the structure of "minimal" (T, ¥)-ideals, we give conditions guaranteeing the existence of minimal and "maximal proper" (T, ¥)-ideals. For example, if S and X are compact, T and Y ace closed, then each proper (T, Y)-ideal is contained in a maximal such. By way of illustration of the application of these ideas, we give a generalization of a theorem of Koch and Wallace: If (T, Y) is a closed subact of a compact act (S, X), if (M, N) is a maximal proper (7', Y)-ideal s,t, M contains idempotents of T then TY ~ N. This paper, dedicated to A. D. Wallace, will appear in the Czech journal Mat, Casopis Sloven. Akad. Vied, (Received March 20, 1974.) A-444 Applied Mathematics 74T-CI9. SUDHANSHU KUMAR GHOSHAL, Jadavpur University, Calcutta 32, India. lterati1w solutions of abstract multinomial equations. Preliminary report. The abstract multinomial equation has been defined with the help of multilinear operators. In the particular case this reduces to an abstract polynomial equation. The Banach-Caccioppoli fixed point theorem is used to obtain the iterative method in Banach space. An estimate of the rate of convergence is presented with the help of the theorem. Other alternative methods based on the Newton-Kantorovic and modified Newton Kantorovic methods have been developed. Some applications of our theory have been indicated. (Received March 6, 1974.) 74T-C20, SUDHANSHU KUMAR GJIOSHAL and ABHA GHOSHAL, Jadavpur University, Calcutta 32, India. A generalized method of solution of unsteady compressible boundary layer equations. Preliminary report. A theory of compressible boundary layers in which a two-dimensional body is accelerating through a compressible fluid at rest at infinity is developed, The flight velocity has form U(x, t) ~ X(x)N(t), which has important applications in aerodynamics and initial stages of rocket flight. N(t) is assumed to be square-integrable and X(x) E the class C"". A general solution of the velocity variable, with a solution for the temperature variable, when Prandtl's number = 1 is presented. Expressions for the general boundary layer characteristics, viz skin friction, displacement thickness, momentum thickness, etc., are found. We also discuss the method of determining the point of separation at different moments of time. (Received March 6, 1974.) *74T-C2I, ROBERT P. DALEY, University of Chicago, Chicago, Illinois 60637. Noncomplex sequences: Examples and characterizations. Preliminary report. A nonrecursive recursively enumerable sequence is constructed having extremely low bounded minimal-program complexity. Recursive function-theoretic characterizations, not involving notions from complexity theory, are given for all sequences of extremely low minimal-program complexity. Let j= = lf\f is unbounded total recursive!; lcf>;l be an acceptable Godel numbering of partial recursive functions; 1([1;1 a Blum complexity measure for lcf>;l; p. a Blum program size measure; x A the characteristic sequence of the set A; whenever cf>; is total, R. = ln\cf>.(n) =II; W. = dom cf> .• All variables denoting functions range over j', C:[f] = lx E IO, II"'\ fJn3j. 1 1 l z p.(j) s f(n) and Vm s n. cf>/m) = x(m)l; e[f\t] = lx E IO, II"'\ f;Jn3j. p.(j) .s f(n) and Vm s n[cf>/m) = x(m) and ([1/m) S t(n)]l. Theorems. 1, 3 A. A is r,e, and 3 Nf. x E C:[fJt]. 2. VA. (Vf. x A E C:[f]) =(Vr f!Jn. 3i S n. 3j n. [D ( )nW. = D ( )nA and D ( )nW. = D ( )nA]). Corollary. VA. If A is r.e. then (V/. xA E C:[j]) = -< rn t rn rn 1 rn (Vr n3j n. [D ( )nw. = D ( )nA]). 3. VA. (V/3 t. XA E e[fJt]) (Vr3s n3j. s(j) n and D ( )nR ( ') = YJ -< rn rn = 9 -< rn SJ D r(n)nA) 4. VA. (3tVf. x A E t'[[\t])-..•(3sVr Vn3j. s(j) S n and Dr(n)nR s(j) = D r(n)nA). {Received March 11, I974.) *74T-C22, LOKENATH DEBNATH, East Carolina University, Greenville, North Carolina 27834. Dissipation of wave energy in an axisymmetric ocean model. An unsteady investigation is made to determine the effects of the frictional bottom on the axisymmetric dispersive free surface flo111. s generated in an inviscid shallow rotating ocean by arbitrary surface forces, The integral representation of the free surface elevation is obtained 111.hich is then evaluated asymptotically for large time, The steady-state solution for the free surface elevation is explicitly determined for all values of the· frequency parameter w/f. It is shown that the free surface elevation represents the exponentially decaying progressive circular waves travelling with the phase velocity wc/B which is modified by the bottom friction. It is found that the inclusion of the bottom friction introduces an exponential damping factor into the amplitude of the axisymmetric waves which eventually decay in the far field. A comparison is made between the wave motions in the shallow ocean with or without bottom friction. (Received March I4, I974.) A-445 *74T-C23. OTOMAR f!AJEl<, Case Western Reserve University, Cleveland, Ohio 44106. The actually used part of target. This treats the differential game x = Ax - p + q in Euclidean space; the control constraint sets are nonvoid, compact, convex; the termination condition is f(x) :S 0 with real-valued f € C 1• For a point x on f(x) = 0 set up the test expression () = minpmaxq Df(x) • (Ax- p + q) (formally, this is min max df/dt). Theorem. If () < 0 then x is, and if () > 0 then x is not, a limit of points x k' outside the target, which can be forced to target within time intervals [0, tk] with tk-> 0, by nonanticipatory pursuer strategies. (Received \larch 18, 1974.) *74T-C24. PRABHA GAIHA and S. K. GUPTA, Indian Institute of Technology, Kanpur, 208016, India. Adjacent vertices on a permutohedron. The convex hull S of S = {a17(1)' a 17(2)' • • ·, a17(n)117 is a permutation of 1, 2, • • •, nl where a 1, a 2, • • ·, an are integers, is defined as a permutohedron. In this paper, the following result, concerning the vertices adjacent to a given vertex of a permutohedron, has been proved: If a 1 < a 2 < · · · n- 1 adjacent vertices, In fact, for any vertex (a 17( 1), a17( 2,,. • • , a17(n)) of S, the vertices adjacent to it are: (a17;0l' a 17;( 21 , · · ·, a17;(n)) for i = 1, 2, • • •, n- 1 where 17;(j) = 17(j) if 17(j) .f, i, i + 1; 17; (j) = i if 17(j) = i + 1, and 17 i(j) = i + 1 if 17(j) = i. The problem considered above has also been solved in case the given integers are not distinct. (Received March 19, 1974,) Geometry 74T-Dl4. CHANDAN S. VORA, Jundi Shapur University, Ahwaz, Iran. A version of Minty's theorem for a vector space with two Hilbert inner products. Preliminary report, Recently the author introduced the concept of comparability of two pairs of norms, and showed, in general, that if the pair of norms are not comparable then there is no Lipschitz extension preserving the two Lipschitz conditions. Here, the author shows again, under the same hypothesis, that, in general, for two inner products there exists no solution to a certain system of linear inequalities simultaneously as follows: Theorem. Let M be a vector space with real or complex scalars and two Hilbert inner products ( , >c and ( , )E, respectively. Let xl' • • ·, xm and Yp · • ·, ym be given s.t. Re (xi- xi' Y;- yi)C::: 0 and Re (xi- xi' yi- yi)E::: O, i, j = 1, • • •, m, and let x be any point of M. Then (a) if the pair of norms (C, C) and (E, E) are comparable then there exists a point y in M satisfying Re (xi- x, yi- y)c ::: 0 and Re (xi- x, y i- y)E ::: 0, i = 1, • • •, m; and (b) if the pair of norms (C, C) and (E, E) are not comparable then there exists a counterexample, (Received February 13, 1974.) 74T-Dl5, JOHN J. O'SULLIVAN, Institute for Advanced Study, Princeton, New Jersey 08540, Manifolds without focal points. Preliminary report. A riemannian manifold is said to have no focal points if for any nontrivial Jacobi field Y with Y(O) = O, d./dt ( Y, Y ) > 0 for t > 0, Theorem. Let M be compact, without focal points and of dimension ::: 2, Let G be an abelian subgroup of 17 1(M), Then G is finitely generated and free. Further, there exists an isometric and totally geodesic immersion of a flat torus T in M where dim T = rank G. Corollary. If 17 1 (M) is solvable, M is a flat manifold. Consider u € G as a deck transformation on Nt, the universal covering space of M. In proving the theorem it is shown that each pair of distinct a-invariant geodesics gives rise to an isometric and totally geodesic imbedding in Nf of a flat rectangular strip of surface. (Received February 18, 1974,) 74T-Dl6, MICHAEL GOLDBERG, 5823 Potomac Avenue, N. W., Washington, D. C. 20016, Covering a square by equal circles. What is the smallest radius of n equal circles that can cover a square? This equation is answered for n S 5. Conjectured solutions for most of the values of n < 40, and several larger values of n, are given. (Received March 4, 1974,) A-446 Logic and Foundations 74T-E49. DAN MAULDIN, University of Florida, (;ainesville, Florida 32611. On the measurability of the continuum. Let E be the reals and R the family of sets of the form A x B in the plane E 2• We record the following theorem under a model of set theory for Zermelo-Fraenkel together with Axiom of Choice 1. 21'(0 ~ K 1• 2. Martin's Axiom A. 3. Every subset of E 2 is in the family I<.ao. 4. Every subset of E 2 is in the a-algebra generated by R. 5. There is a countable ordinal a, s.t. if If is a family of 2~'<0 subsets of E, 3 a countable family G of subsets of E s.t. fl S G a· 6. If II is a family of 2~'<0 subsets of E 2 3 a countable family G of subsets of E s.t. H is a subcollection of the a-algebra generated by G. 7. If H is a family of 21'( 0 subsets of E, and each member of H has cardinality < 2~'<0, 3 a countable family G of subsets of E s.t. H is a subcollection of the a-algebra generated by G. 8. 2~' 74T-E50. AL!STAR 1-L LACHLAN, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada. The number of models of" an w-stable theory. Let T be a countable complete w-stable theory which is neither w-categorical nor w 1-categorical. T has at least w models in each uncountable power. The proof breaks up into two cases (1) when in the prime model there is an w-categorical strongly minimal set and (2) otherwise. (Received February 25, 1974.) 74T-E51. JAMES H. SC~IMER L, University of Connecticut, Storrs, Connecticut 06268. On K-like structures which embed stationary sets. Preliminary report. A structure £( = (A, <, · · . ) is K-like iff < is a linear ordering of A of cardinality K yet every proper initial segment has smaller cardinality. If X :::; K then £t embeds X iff there is a continuous, order-preserving function f: X-> A. For ordinals a and X:::; K define: X is [strongly] 0-stationary iff X is a stationary subset of K and K is [strongly] inaccessible; X is [strongly] a .,stationary iff for each {3 < a, {v < K: Xnv is a [strongly] {3-stationary subset of vi is a stationary subset of K. Theorem 1. If a is a first-order sentence s.t. for each n < w, a has a K-like model embedding a strongly n-stationary set, then whenever cf(A) > w, a has a A-like model embedding a stationary set. Corollary. If a has a K-like model which embeds a stationary set and K is weakly compact, then whenever cf(A) > w, a has a A-like model embedding a stationary set. Theorem 2. For each n < w there is a first-order sentence an which has the property that whenever X :::; K, then an has a K-like model which embeds X iff X is not n-stationary. Corollary. (V = L) There is a sentence a s.t. whenever cf(K) > w, then a has a K-like model embedding a stationary set iff K is not weakly compact. All of the above results have natural extensions to admissible fragments of L""w" (Received February 28, 1974.) 74T-E52. ROBERT I. SOARE, University of Illinois, Chicago, Illinois 60680. Degrees and structure of levelable sets. Preliminary report. Let {¢i: i E Nl be an acceptable numbering of the partial recursive functions and {c!li: i E Nl step counting functions which constitute a complexity measure in the sense of Blum. Let r.e. set Wi =domain ¢r Definition. An r.e. set A is levelable if there is recursive function r such that for all i with W i =A and for all recursive functions h, there exists Wj =A and there exist infinitely many x such that cll/x) > h(x) and ¢/x) S r(x). Definition. A sequence of recursive sets {Rn: n E Nl is a strong array of recursive sets if there is a recursive function f such that for all n, ¢/(n) is the characteristic function of Rn. Theorem. An r.e. set A is nonlevelable if and only if for every strong array {Rn: n EN I of recursive sets, {n: R 11 :::; A I Sr¢'. Corollary 1 (Paul Morris). An r.e. set A is speedable iff A x N is levelable iff A x N is speedable. Corollary 2. Every r.e. degree a contains a nonlevelable set, and an r.e. degree a contains a levelable set iff a'> O'. Corollary 3. Every finitely strongly hypersimple set is levelable. Corollary 4. Every nonspeedable r.e. set is effectively nonlevelable. (The theorem uses a result of Blum and Marques.) (Received March 14, 1974.) *74T-E53. JOACHIM REINEKE, Yale University, New Haven, Connecticut 06520. Reduced products of commutative rings. Preliminary report. We are studying under which conditions on the filter some important classes of rings are preserved A-447 by reduced products, We have the same notions as in Rell-Slomson, "Models and ultraproducts", and in ring-theory as Lambek, "Lectures on rings and modules". R means commutative ring with identity, F a proper filter on I and R 1/F denotes the reduced product. Main theorems. A. R 1/F is a local ring (or integral domain) iff F is an ultrafilter and R a local ring (or integral domain). B. R1/F is artinian iff F is a finite intersection of ultrafilters and R an artinian ring. C. If R 1/F is a principal ideal ring then F is a finite intersection of ultrafilters. D. Suppose F is an w-incomplete ultrafilter. Then either R is artinian or R1/F is not noetherian. Then by C and D: E. Assume card I= w or assume there exists no uncountable measurable cardinal number. Then R 1/F is a principal ideal ring iff F is a principal filter generated by a finite set X s; I and R is a principal ideal ring, .or R is an artinian principal ideal ring and F a finite intersection of ultrafilters. (Received March 8, 1974.) * 74T-E54. RUDOLF v. B. RUCKER, State University College of New York, Geneseo, New York 14454. Martin's axiom, scales, and the rich continuum. GBC is Godel-Bernays class-set theory with the axiom of global choice and without the power set axiom. 1 is GBC + ('v'a)(3{3)[;; < jiJ ~ ('v'K)[MAK], MAK the Martin axiom asserting existence of Z-generic subsets of c,a.c. partial orderings whenever Z _:S K. The proof of the consistency of Martin's axiom can be adapted to give the consistency of 1 rei ZF. 1 yields that Pw is a proper class. We are interested as to how 1 models the viewpoint that Pw should be a proper class because it contains ''too much information''. It would be nice to prove r 1-V = L[Pw], which seems impossi.ble, since Pw seems only to have information about the width (rather than length) of V. Theorem. fi-(3W s; Pw)[V = L[W]). Proof. Let W code Pw and a subclass S of "'w s.t. the order typ:_of ~ith the Hausdorff by-end-pieces ordering is On. Remarks (i) Since MAK--. 2K·= 2 11 0, we can prove rr-i! = Pw, hence rH3R s; Pw X Pw)[ < Pw, R >""' < v. E >], and hence our theorem all without using scales. (ii) The scale technique is better, for it does not require the adjunction of as much information to Pw to obtain V. (iii) In the context of r, the continuum problem remains as: "Is V =On?", (iv) We answer no (see "A new approach to the Continuum Problem", to appear). (Received March 11, 1974.) * 74T-E55. J. A. PAULOS, Temple University, Philadelphia, Pennsylvania 19122. Truth-maximality and !1-closed logics. Preliminary report. Feferman in "Applications of many-sorted interpolation theorems" (to appear) proves that a logic is truth-maximal iff it is !1-closed. Thus, we prove Theorem. If a logic L is truth-complete, then no structure in which the syntax is represented is L-characterizable. This is proved by showing that from the L -characterizability of a structure in which the L-syntax is represented one can implicitly (with extra predicates), but not explicitly, define the satisfaction relation. That Henkin logic, L(Q), Q the chang quantifier, and Lk.+k+ fail to be !1-closed, satisfy interpolation or Beth's theorem, follows. (Received March 14, 1974.) (Author introduced by Professor Louis Rayman.) Statistics and Probability 74T-F9. SIAMAK KHALIL!, University of Pittsburgh, Pittsburgh, Pennsylvania 15260. On the oscillation of the Brownian motion random me.asure. Preliminary report, Let (il, Cf, P) be a probability space, and £ 2 = L /il, Ci', P, R). Definition. Let (i) G be a locally compact abelian group, :13 the collection of Borel subsets of G, and m a Haar measure for G, (ii) G0 E :13, and :13 0 = !B: B E :BnG 0 and m(B) < oe }. Then we say that .(; is a Brownian motion random measure (BMRM) over (0, Cf, P) with parameter domain G 0 iff (a) 'v' B E :13 0 , {;(B) is a random variable which is normally distributed with mean zero and variance m(B), (b) .(; is finitely additive, (c) .(; is independently scattered, i.e, if B I' B 2• · · ·, B n E :13 0 are pairwise disjoint, then ,f;(B 1 ), ,f;(B 2 ), • • ·, ,f;(B n) are independent random variables, Theorem. Let (i) .(;be a BMRM over (il, d, P) with parameter domain Rq, q 2: 1, (ii) 'v't =(II' t 2, • • ·, tq) E Rq, 'v'h = (hI' h2' . .. ' h ) E Rq 3 h. > 0 for i = 1, 2, •.• , q, (t, t + h) = nql (t ., t. + h ]. Then 'v' a > Y,, 3 n~ E (f 3 P(il~) = q z l z l 1 and 'v' t E Rq, w Eil~, lim suph-•O \.f;(t, t + h](w)\/!Leb(t, t + h]}a.= + oo, where Lebdenotes Lebesgue measure in Hq, (Received February 22, 1974.) A-448 74T-F10, THOMAS H. SAVITS, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, Supercritical age-dependent birth and death pwcess. Preliminary report. Let X1 be an age-dependent birth and death process given by the following description, A parent object alive at age x has probability Adx of giving births to offsprings and probability fLdx of dying before it reaches age x f dx (,\ and fl are constants), The probability generating function for offsprings is given by rr(s) -· ~; ~o pksk. Theorem 1, The process does not explode in finite time iff for every sufficiently small f > 0, J i _,!tM-- I) - A,;[ rr(.;') - 1W 1 d.; diverges. Define h(l) implicitly by J~(t) {fL(,;- 1) - A,;[,(,;) - 1] }- 1 d.;'"" t, where q <: c " 1 and q is the extinction probability. Then in the supercritical case, corresponding to m = 1 AfL- rr '(l) > 1, we have the analogous Seneta-Heyde result. Theorem 2, If 1 < m <~,then Z 1 log h(t) _. W a,s, as I • oo, where W is a nontrivial random variable and Z 1 is the number of particles alive at time t, (Received February 22, 1974.) Topology * 74T-G63. RORERT A. HEFRMANN, U.S. Naval Academy, Annapolis, Maryland 21402. H(i) semiregular, U(i) and J<(i) extensions. I. We continue our study of extensions for a space (X, r), where we assume no separation properties for X. Definition. An extension (Y, T) of (X, r) has the RO-intersection property if for each p E Y- X 3 a local base W(p) 3 for each IV E W(p), WnX is regular-r-open, Theorems. If (X, r) is a non-H(i) space, then 3 an extension (Y, T) of X 3 Y C X and (i) (Y, T) is H(i) and X E T, (ii) Y is T 2 except for X, (iii) Y has the 1<0-intersection property and each p E Y- X is semiregular, (iv) X is OCE m Y. If (X, r) is a non-H(i) space, then 3 an extension (Y, T 1) 3 Y C X and (i) (Y, T 1) is H(i), X E T 1, Y is T 2 except for X with the RO intersection property and X is OC E in Y, (ii) (Y, T 1) is a projective maximum in the class of all H(i) extensions of X which are T 2 except for X and have the RO-intersection property, (iii) (Y, T 1) is essentially unique, In a natural sense, (Y, T 1) is the smallest such extension which is a subset of X. (Received January 31, 1974.) * 74T -G64. PHILLIP L. ZENOR, Auburn University, Auburn, Alabama 36830. Hereditary separability and the hereditary Lindeli5f property in function spaces. If X and Y are spaces, then C(X, Y) denotes the space of continuous functions from X to Y with the point-wise topology. Theorem. If X is completely regular and m is a cardinal number, then the following conditions are equivalent: 1. For each integer n, xn is hereditarily m-separable (hereditarily m-Lindeliif; resp.). 2, xm is hereditarily m-separable (hereditarily m-Lindeliif; resp.). 3. For each n, C(X, En) is hereditarily m-Lindeliif (hereditarily m-separable; resp.) (En = Euclidean n-space), 4. C(X, fl) is hereditarily m-Lindeliif (hereditarily m-separable; resp.) (H = Hilbert cube). 5. C(X, Y) is hereditarily m-Lindeliif (hereditarily m-separable; resp.) for every space Y with wY :':: m, (Received February 7, 1974.) *74T-G65. V. KANNAN, Madurai University, Madurai 21, India. A note on countably generated spaces. Answering a question of Moore and Mrowka (Abstract 614-88, these :Jkiic-e.>- 11(1964), 554), S. P. Franklin proved (Proc, Amer, Mach. Soc. 21(1969), 597-599) that there exists a Hausdorff space determined by countable sets, but not determined by countable closed sets. He asked whether such a space can be regular. Here we prove that such a space can even be completely normal. The proof uses some elementary properties of the Lebesgue measure on the real line, (Received February 14, 1974.) (Author introduced by Professor Stanley P, Franklin.) *74T-G66. R. VASUDEVAN, Institute of Advanced Studies, Meerut University, Meerut 250001, India. Pairwise compactuess in bitopolugical hyperspaccs. Preliminary report. For a bitopological space (X, ~- 1 , :r 2) denote by 2X the collection of all nonemptv :f 1-closed or ~- 2-closed sets, Sc(X) the collection of all nonempty ;( 1 V ~- 2-closed subsets of X and the natural bitopological A-449 hyperspaces (2x, 2~\ 2~2), (2x, £ 11, ~-5"2), (Sc(X), 2:1\ 212' and (Sc(X), ~ 1 1, £:1"2), where 21i is the finite topology, ~5"; is the lower semifinite topology or "A-topology", 2r;is the upper semifinite topology or "K·topology" have been studied. Main results. (i) (X, :f 1, 5 2) is Kim (K·)·compact if (2X, 2 ~\ 2 12) is 11 12 K·compact and conversely if (X, 5 1, 5 2 ) is bicompact and (2X, £ , ~ ) is K·compact then (X, 5 1, 5 2) is K-compact, (ii) (X, 5 1, 5 2) is Fletcher-Hoyle-Patty (FHP-)-compact iff (Sc(X), 2~\ £ 3"2) or (Sc(X), ~ 1 1, 212) 1 is FHP-compact. (iii) (X, :r 1, 5 2) is Swart compact iff (Sc(X), 2~1, 2 2) is Swart compact. (iv) (X, 5 1, 5 2) is Birsan (B-) compact iff (2x, ~ 1 1, ~5"2) is B-compact. (F.eceived February 4, 1974,) (Author introduced by Professor J. N, Kapur,) *74T-G67, PETER J. NYIKOS, University of Chicago, Chicago, Illinois 60637. Basically screenable and countable-dimensional spaces. Preliminary report, A Hausdorff space is of large basis dimension :S a (a a cardinal number) if it has a base consisting of no more than a+ 1 rank 1 collections of open sets, s.t. every point has a local base belonging to one of the collections. (A collection ~ of subsets of a set X is of rank 1 if any two members of :13 are either comparable or disjoint.) A Hausdorff space X is basically screenable if it has a base consisting of countably many rank 1 trees of open sets. (A collection ~ is' a tree if the members of ~ containing any given member are well-ordered by reverse inclusion.) Theorem. Every metric space, every space of countable large basis dimension, and every proto-metrizable space [Abstract 74T-G34, these :naUc~-21(1974), A-328] is basically screenable, Theorem. Every basically screenable space is (hereditarily) screenable, Theorem. A space is separable and metrizable iff it is separable, regular, and basically screenable, Theorem. A metric space is countable-dimensional (a union of countably many strongly zero-dimensional subspaces) iff it is of countable large basis dimension. Theorem. If a space X of countable large basis dimension is perfectly normal, and a a-locally countable union of metric spaces, it is metrizable, Corollary. Every semistratifiable space of countable large basis dimension is metrizable, (Received February 18, 1974.) *74T-G68, DAVID B. GAULD, University of Warwick, Coventry CV74 7AL, England and University of Auckland, New Zealand. Local contractibility of spaces of homeomorphisms. Let K be a finite simplicial complex, H(K) the group of homeomorphisms of K with the C- 0 topology, and PL(K) the subgroup of PL homeomorphisms, Theorem. H(K), PL(K)) is locally contractible in the sense that paths starting in PL(K) remain in PL(K), In particular, PL(K) is locally contractible. The proof involves repeated applications of a truncated Alexander-type isotopy and an adaptation of the technique in § 8 of Edwards and Kirby [Ann. of Math (1) 93(1971), 63-88], (Received February 18, 1974.) *74T-G69. SIDNEY A. MORRIS, University of Wales, Bangor, Wales LL572UW, United Kingdom and DAVID CHRISTOPHER HUNT and ALFRED J. Vander POORTEN, University of New South Wales, Kensington, New South Wales 2033, Australia, Closed subgroups of products of reals. It is well known that any closed subgroup of a finite product Rn of reals is topologically isomorphic to Ra x zb, where Z denotes the discrete group of integers. For some time S, A. Morris has conjectured that an analogous result holds for infinite products of reals, To be precise, the conjecture is that any closed subgroup of an infinite product of reals is topologically isomorphic to a product of copies of R and Z. We prove that any closed connected subgroup of a product of reals is topologically isomorphic to a product of reals, Theorem. Let G be a closed subgroup of a product fla.E/Ra. of reals. Then the following conditions are equivalent: (i) G is connected; (ii) G is divisible; (iii) G is a real vector subspace of fla.e 1 R "'; (iv) G is topologically isomorphic to a product of reals. (Received February 21, 1974.) *74T-G70. JOHN J. WALSH, Institute for Advanced Study, Princeton, New Jersey 08540. Open mappings on manifolds. Let Mm be a compact, connected p.l. manifold, Q a compact, connected polyhedron, Y a compact, connected metric space. Theorems. (A), Let m 2: 3 and f a mapping(= continuous function) of M to Q; then / A-450 is homotopic to an open mapping of M onto !J iff the index of /*(rr 1(M)) in rr 1(Q) < ""· (B). If we assume, in addition, that !J is a p.l. manifold of dimension .:: m, then f is homotopic to a light open mapping of M onto Q iff the index of /*(rr 1(M)) in rr 1(Q) < ~. (C). If f is a monotone mapping of Mm onto Y (m;.:: 3), then f can be uniformly approximated by monotone open mappings of M onto Y. This last theorem is (in essence) one announced by R. D. Anderson in 1956; however, a proof was never published. Remark. It is also shown that Theorem (A) and some earlier theorems of the author (but not Theorem (B)) remain true under the assumption that Q is a compact, connected ANH. (Eeceived ~Aarch 1, 1974.) * 74T-G71. RICHARD H. WARREN, Aerospace Research Laboratories, ARL/LR, Wright-Patterson AFR, Ohio 45433. Continuity of mappings of fuzzy topological spaces. We introduce the n~tion of a neighborhood of a point in a fuzzy topological space, and show that this concept is fruitful in characterizing fuzzy continuous maps. For motivation, definitions and earlier results, see J. A. Goguen (J, Math. Anal. Appl. 43(1973), 734-742) and C. L. Chang (J. Math. Anal, Appl. 24(1968), 182-190). Definition 1, Let X be a set. A fuzzy set in X is a function from X into [0, 1]. A fuzzy topology T on X is a family of fuzzy sets in X 3 fl¢• flx E T; if gi E T, then V gi E T; and if g, h E T, then g flh E T. Definition 2, A fuzzy set n in a fuzzy topological space (X, T) is a neighborhood of a point x E X iff 3 g E T 3 g :; n and n(x) ~ g(x) > 0, Theorem. Let (X, T) and (Y, S) be fuzzy topological spaces and let f: X_, Y. The following are equivalent: (a) if b E S, then r 1(b) E T; (b) the inverse image of every element of a subbasis for S is in T; 1 (c) for every X EX and every neighborhood n of /(x), r (n) is a neighborhood of x; (d) for every X EX and every neighborhood n of /(x), 3 a neighborhood TTl of X 3 /(m) S n and m(x) = r 1(n)(x); (e) for every fuzzy set a in X, ((a) S /(a); (f) for every fuzzy set b in Y, r 1(b) S r 1(b), (Received March 1, 1974.) * 74T-G72. RORERT MILLER GEIST III, University of Notre Dame, Notre Dame, Indiana 46556. Semicharacterislic detection of obstructions to rational homotopy equivalence. The groups L7,(Qrr; cu) do indeed measure the obstructions to cobording normal maps of unit (rational) degree to rational homotopy equivalences; further, all obstructions can be realized, The accessibility of these obstructions is certainly improved by: Theorem. The semicharacteristic of Lee (Topology 12(1973), 183) can be reinterpreted as a monomorphism from L;k+i(Qrr; cu) to a quotient group of representations of rr; if cu = 1, it is an isomorphism. Using this semicharacteristic, we find that, for 11 finite abelian, the natural map L~k+!(Zrr; 1) _, L~.l:+i(Qrr; 1) is identically zero. Extension of this result to certain nonabelian and unoriented cases follows easily. (Received March 4, 1974.) 74T-G73. HARVEY ROSEN, University of Alabama, University, Alabama 35486, Wild universally pierced arcs. Preliminary report. A subset X of E 3 is a pierced set if it lies on a 2-sphere S in E 3 which can be pierced by a tame arc at each point of X. A subset X of E 3 is called universally pierced if each 2-sphere in E 3 which contains X can be pierced by a tame arc at each point of X. A point p in a set X is called a shrinking point of X if for every open set U in E 3 which contains X- {p}, there is an open set V which contains X- {p} s,t. each loop in V- X can be shrunk to a point in U -X. The following corollary answers a conjecture of L. D. Loveland and the author. Theorem 1. If an arc X in E 3 has a shrinking end point and if X is a cellular pierced set, then X is universally pierced. Corollary. Alford's wild arc is universally pierced, Theorem 2. There exists uncountably many nonequivalently embedded universally pierced arcs in E 3. (Received March 4, 1974.) * 74T -G74, ROBERT M. HARDT, University of Minnesota, Minneapolis, Minnesota 55455. Topological properties of subanalytic sets. A subset of a real analytic manifold is a subanalytic set (or semianalytic shadow) if locally it is the proper real analytic image of a semianalytic set, The stratification of either semianalytic sets or subanalytic sets (established in J.Iironaka [ "Subanalytic sets, " N·1mber Theory, Algebraic Geometry, and Commutative Algebra, A-451 Kinokuniya, Tokyo, 1973, 453-493] or Hardt ["Stratification of real analytic mappings and images," Preprint]) here leads easily, by consecutive projections in Euclidean space, to a CW decomposition, In the category of subanalytic se;:s and continuous maps with subanalytic graphs, theories of slicing, intersection, and homology result through use of the topological chains defined by subanalytic sets. (Received March 6, 1974.) 74T-G75. LUDVIK JANOS, University of Newcastle, Newcastle, New South Wales, Australia 2308, On representations of selfmappings by contractions. Under a selfmapping (s,m,) we understand a pair (X, f) consisting of an abstract set X and a mapping f: X -> X. If (Y, g) is another s.m., (X, f) can be represented in (Y, g) if there is an injection i: X -> Y s.t. i 0 g = f 0 i. If (X, f) is a s.m. and if the set X admits a metric for which f is a contraction, (X, f) is an abstract contraction. The cartesian product (X, f) x (Y, g) and the disjoint sum (X, f)u(Y, g) are defined in an obvious way. Theorem 1. Any s.m. (X, f) can be represented in (X, f)u(X 1, f 1) x (Y, g), (X 1, f 1) and (X 2, f 2) abstract contractions and g a bijection. Theorem 2, If the cardinality of X does not exceed c, the metrics in X 1 and X 2 can be chosen to be separable, (Received March 11, 1974.) 14T-G76, K. ALSTER and TEODOR C. PRZYMUSINSKI, Polish Academy of Sciences, 00-950, Warszawa, Poland. Normality and Martin's axiom. Martin's axiom (AM) plus the negation of the Continuum Hypothesis ( K 1 < 2 No)=> the existence of a whole variety of normal nonmetrizable Moore spaces, e.g. complete separable ones, countable chain condition metacompact ones, complete metacompact ones, It also yields a Lindelof space T such that T 2 is normal but not paracompact. Definition. Let 5" and m be two topologies on the set X. 5" is regular w.r.t. m, if 5" ::>m and for every U E 5 and x E U 3 a V E 5, 3 x E V c Vll cU. Theorem 1 (AM). If lXI < 2 No and ~ is regular w.r.t. a certain metric separable topology on X, then T"' is perfectly normal, T = (x, 5"), The existence of the above mentioned spaces (and even better ones) can be easily deduced from Theorem 1. Moreover, we have Corollary 1 (AM + N 1 < 2 No), 3 a (separable, first countable) Lindelof space T 3 T"' is normal but not paracompact. Corollary 2 (MA + N 1 < 2 No), 3 a (separable first countable) nonparacompact space T 3 T"' is perfectly normal. We also prove as a corollary to Fleissner's result Theorem 2 (Axiom of constructibility). Every normal locally countable chain condition Moore space is metrizable. (Received March 11, 1974.) *14T-G77. KRYSTYNA KUPERBERG, Rice University, Houston, Texas 77001, A note on the Hurewicz isomorphism theorem in Borsuk's theory of shape. In shape theory, the role of the homotopy groups "n (n 2: 1) is played by the so called fundamental groups .J!.n' introduced by K, Borsuk, and the homology groups which are useful there are of the Vietoris-Cech type. The classical Hurewicz isomorphism theorem gives a connection between the homotopy groups "n and the singular homology groups Hn with integral coefficients, An example of a compacturn X is constructed showing that there is no exact analogue of the Hurewicz theorem in shape theory. The example is simple: X is the double suspe,1sion of the 3-adic solenoid. The compactum X is arcwise connected and it has the following properties: (i) .[q(X) =0, for q = 1, 2, 3, and (ii) .rr4(X) and H4(X) are not isomorphic. (Received March 14, 1974.) (Author introduced by Professor William Jaco.) * 74T-G78. ED HUTCHINGS, University of British Columbia, Vancouver V6T 1W5, British Columbia, Canada. Local homotopy in Bing's dogbane space. Bing's dogbane space, called ~ here is a decomposition space of B 3 which is a nonmanifold although the associated decomposition is usc and pointlike. A small neighbourhood of a nonmanifold point of ~ is called a bad neighbourhood. We show that for an open bad neighbourhood V of ~. there are at least two freely related generators of rr 1 (V). This behaviour distinguishes T from all toroidal decomposition spaces. There are examples of bad neighbourhoods V of ~ s.t. (a) V is open and 17 /V) is free on two generators, (b) V is an open set plus a zero-dimensional set of boundary points and rr 1(V) is infinite cyclic. (Received March 15, 1974,) (Author introduced by Professor Dale Rolfsen.) A-452 * 74T -G79, RICHARD E. HODEL, Duke University, Durham, North Carolina 2770(), Metrizability of topological spaces. Conditions are given under which a topological space is metrizable or has a countable base. A quasi G s·diagonal for a space X is a sequence ~; 1, ~ 2 • • •• of open collections in X such that, given any two distinct points p and q in X, there is some 11 such that st(fJ. ~n) f, ¢ and q f- st(p, f!:jn), The following theorem extends work of Bennett on quasi-developments [Proc. Japan Acad, 45(1969), 6-9]. Theorem. Every regular 0-refinable {1-space with a qua si-S 0-diagonal is semistratifiable. Corollary. Every para compact wL\-space with a quasi-~ s· diagonal is metrizable. Theorem. Let X be a regular space with a point-countable separating open cover which hereditarily satisfies the CCC. Then X has a countable base if X is hereditarily a wL\-space or is a Baire space and a wL\-space. Theorem. Every locally connected, locally peripherally separable meta-Lindelof 'Aoore space is metrizable. Theorems concerning the metrizability of spaces having a weak base in the sense of Arhangel'skii are obtained, Finally, generalizations of results by Caban, Carson-Michel, Smirnov, and Stone on the metrizability of spaces which are the union of countably many metrizable subsets are given. (Received March 18, 1974.) Miscellaneous Fields * 74T-HI. FRANCIS E. MASAT, Glassboro State College, Glassboro, New Jersey 08028 and WALTER E. MIENTKA and ROBERT A. MEYER, University of Nebraska, Lincoln, Nebraska 68508. A semisemester approach to introductory mathematics. Prior to the fall of 1971, precalculus mathematics at the University of Nebraska served an average of 1500 students per year. The results of extensive surveys showed that the precalculus class had a significant numbers of failures, over-prepared and under-prepared students, and topics no longer relevant. To meet these problems and prepare for possible future change, a system of semisemester courses, called modules, was developed, A system of placement exams and an advising procedure were also adopted. The modules were taught in 8-week periods, with two periods per semester. The original modules corresponded approximately to elementary algebra, college algebra, and trigonometry. Modules could be taken concurrently and repeated. The program thus allowed flexibility in timing, content, and placement. Two years of the program gave ample data to support its continuation. In particular, the number of students failing or withdrawing was more than halved. The consensus of students and staff was that the ·modular approach was advantageous over other courses because of its extreme flexibility, shortness of modules, and the positive change of student attitude and morale towards introductory mathematics, (Received March 7, 1974.) The April Meeting in Santa Barbara, California April27, 1974 714-A7, ROBERT C. THOMPSON, University of California, Santa Barbara, California 93106. The eigenvalues of a partitioned Hermitian matrix involving a parameter. Let H(p) = [~Q .kQ] be a partitioned Hermitian matrix, where P, Q, R are blocks, P and - R pos1t1ve definite, and p is a real parameter. In a recent paper (Linear Algebra and Appl. 3(1970), 263-273), H. van Kempen observed that the positive (resp. negative) eigenvalues of H(p) are nondecreasing (nonincreasing) functions of the increasing parameter p. In the present paper (to appear in Linear Algebra and Appl.) these results are reproved using the classical inequalities for eigenvalues of Hermitian matrices, and precise asymptotic estimates for the growth of the eigenvalues of H(p) as p-> oo are obtained, (Received March 1, 1974.) The May Meeting in DeKalb, Illinois May 13-17,1974 Algebra & Theory of Numbers * 715-A1, MELVYN BERNARD NATHANSON, Southern Illinois University, Carbondale, Illinois 62901, Asymptotic independence of sequences of integers. Let A= la;l~=i be a se.quence of integers. For rn::: 2, let A(N, r, m) denote the number of a; with A-453 iS N such that ai "'r (mod m). If !imN_,~A(N, r, m)/N =a, exists for r = 0, 1, · · ·, m- I, then the sequence A has asymptotic distribution (a0 , a 1, • • • , am_ 1) modulo m. Suppose that the sequences A = Ia i I~= 1 and B = lb;l;'= 1 have asymptotic distributions (a0 , a 1 , • • ·, am_ 1) and ({3 0 , {3 1, ···,{3m- 1) modulo m, respectively. Let A x B(N, r, s, m) denote the number of pairs (ai, bi) with iS N such that ai = r (mod m) and bi "'s (mod m), If limN'"" A x B(N, r, s, m)/N = a 7 {3 5 for r, s = 0, 1, · · · , m - 1, then A and B are asymptotically independent modulo m. In this paper we show how to construct a sequence of integers that has a given asymptotic distribution and that is asymptotically independent of a given sequence of integers. (Received March 11, 1974.) * 715-A2. FRANK A. SMITH, Kent State University, Kent, Ohio 44242. Extensions of partial orders on semigroups. We give necessary and sufficient conditions to extend a partial order on an abelian semigroup so that any finite set can be linearly ordered, In addition, we prove the following theorem: If (S, ·, S) is an idempotent semigroup with neutral element 1, S can be extended to a linear order if and only if S = S 1 uS 2 where S 1, S 2 are abelian subsemigroups both of which have a minimal neutral element and pq £ {p, q I for all p, q £ S. (Received March 15, 1974.) * 715-A3. ALAN A. BISHOP, Western Illinois University, Macomb, Illinois 61455. A universal mapping property for a lattice completion. Theorems. 1. A mapping ¢: L -> M of lattices is a complete join homomorphism iff the inverse image of every complete ideal is a complete ideal, Let K(L) denote the completion by complete ideals of the lattice L and i L: L -> K(L) denote the natural embedding (x)i L = {y E L \y S x 1. 2. If ¢: L -> M is a complete join homomorphism, then there exists a unique complete join homomorphism K(¢): K(L) -> K(M) such that iLK(¢) = ¢iM. The pair (K(L), iL) has the following universal mapping property: 3. If ¢ is a complete join homomorphism of L into a complete lattice M, then there exists a unique complete join homomorphism p.: K(L) -> M such that iLp. = ¢. This characterizes the K(L) completion, (Received March 18, 1974.) * 715-A4. THOMAS W. CUSICK, State University of New York at Buffalo, Amherst, New York 14226. The two dimensional Diophantine approximation constant. Given two real numbers a, {3, there is a well-known Diophantine approximation constant, denoted by 2 2 c 1 (a, f3) in the paper, defined to be the infimum of those c > 0 such that the inequality \x + ay + f3z\ max (y , z ) < c has infinitely many solutions in integers x, y, z with y and z not both zero. It is a well-known unsolved problem to evaluate C = sup c 1(a, {3), where the supremum is taken over all pairs of real numbers a, f3. It is known that C 2: 2/7, and it is tempting to conjecture that equality holds, It is reasonable to conjecture further that if C' =sup c 1(a, (3), where the supremum is taken over all a, f3 such that 1, a, f3 is a basis of a real cubic number field, then also C' = 2/7, The theorem of the paper is that sup c 1(a, {3), where the supremum is taken over all a, f3 such that 1, a, f3 is an integral basis for the totally real cubic field of smallest possible discriminant, is equal to 2/7 iff a certain plausible but very deep condition on the continued fraction expansions of certain cubic irrationals holds, This hypothetical condition implies the existence of cubic irrationals whose continued fractions have unbounded partial quotients -another well-known conjecture. (Received March 20, 1974.) * 715-A5. CARY H. WEBB, Chicago State University, Chicago, Illinois 60628, A strengthening of the torsionless property in abelian groups. Suppose A, B, C, and C i' i E I, are abelian groups. Let the natural transformations B®llC i -> ll(B®Ci), Hom (A, B)®C _,Hom (A, B®C), and Hom (A, B)®C _,Hom (Hom (C, A), B) be denoted f, S and T, respectively. Theorems. l, If Ci is torsion-free for all i £I then f is a monomorphism. 2, If B is torsion-free then S is a monomorphism. C is called torsionless if T is a monomorphism whenever A = B = Z. It is well known that such groups are precisely subgroups of IIZ (arbitrary product). 3. Suppose A is a reduced rational group, B is torsion-free, CiS: A for every i = 1, 2, 3, ... and C is a subgroup of liCi' Then T is a monomorphism. (Received \1arch 27, 1974,) A-454 71'1-i\6. JOH~ A. llFACIIY, \lorthern Illinois University, J).~Kalb, lllinois (,OlJ~. On Morita's localio:ation. For a monad ,·T, '7 ·fl ·. of U-\lod, let (! be the monad defined by fJ ker ('IT · 7'7/). For the ring Q(R), let !': (_1{1~)-\lod • /~-~hd be the functor induced by the ring homomorphism 7JW): R ·• QW). (T is idempotent iff Q ,_. T, and if T is idempotent nod left exact, tl1en T(R) is a ring of quotients of R, and the category of '/'-algebras is the associated quotient category.) Theorem. !f l/: Q(I~)-Mod • R-Mod is left exact, then the category of Q-a!gebras is a full, Crothendieck subcategorv of /~-\lod. This yields several results of K. Morita (Quotient Rings, Rill!;!, Theory, ed. R. Gordon, 1972, pp. 2~7-28'i), To particular, consider T ,. HomE(Hom 11 (-, V), V) for certain modules 11 I' E, where E ·· End ( 11 1'). (Received March 22, 1974.) 715-A7. JOHN A. REACHY and lt'ILLIA:\1 D. BLAIR, Northern Illinois University, DeKalb, Illinois 60115. Rin!;!,s ll'hosc faithful left ideals arc cofaithful. A left ideal is said to be cofaithful if it contains a finite subset whose left annihilator is zero. A ring is said to contain enough uniforms if every nonzero left ideal contains a uniform left ideal. Theorem. A ring is an order in a semisimple Artinian ring if and only if it is semiprime, has enough uniforms and every faithful left ideal is cofaithful. The conditions that R have enough uniforms and that every faithful left ideal is cofaithful are both Morita invariants; the former condition goes up to polynomial rings while the latter does if the ring is commutative. (Received March 22, 1974,) * 715-AB. STEPHEN J. \lcADAM, University of Texas, Austin, Texas 78712. Going down and open extensions. Preliminary report. Let R C T be commutative rings, with R such that every ideal of R has only finitely many primes minimal over it. If R C T is either integral, or finitely generated and algebraic, and if the extension satisfies .going ·down, then the spec map from spec T to spec R is open. (Received March 22, 1974.) 715-A9. LOUIS HALLE ROWEN, University of Chicago, Chicago, Illinois 60637. Generalized polynomial identities. II. Let R be a ring with 1. Informally speaking, R satisfies a generalized identity (GI) if some polynomial f(X 1, • • ·, Xm) with coefficients in R vanishes for all substitutions of elements of R for the indeterminates Xi' Assume f is linear in each Xi' Write f as a sum of monomials of the form w 1X771 w 2X772 • • • w,.X17mwm +!' 77 a permutation of (1, · · ·, m), although such an expression need not be unique. A generalized monomial f 17 of f is the sum of the monomials corresponding to the permutation 77, Let I ,(R) be the ideal generated by all evaluations in R of all generalized monomials of /; f is R-proper if 1/R) f, 0, and f is R-strong if 1/R) = R. Let P be a primitive ring. A GI of P is P-proper iff it is nontrivial on the central closure of P, so Amitsur's theorem says: If P has a proper ('I then soc P f, 0, Theorems. 1. If f is a proper GI of P then f is (P /soc F) improper, Amitsur's theorem follows. 2, lf R satisfies an R-strong Gl then R is a PI-algebra. (Received \larch 25, 1974.) 715-A10. WILLIAM F. KEIGHER, Southern Illinois University, Carbondale, Illinois 62901. Differential categories are cotripleable. II. Preliminary report. In a recent paper (Abstract 73T-A154, these :lt,tic.,.,_ 20(1973), A-419), the author has shown that the functors which forget the single derivation operator of an ordinary differential ring, module or algebra are both cotripleable and tripleable, that for an ordinary differential ring (A, d) where A contains the rationals Q, there is a natural open retraction p A: Spec A -+ SpecdA. We now show that the results of that paper are valid when one considers differential rings, modules and algebras with a family of commuting derivation operators indexed by an arbitrary set X. In particular, if Diff denotes the category of differential rings with an X-indexed set of derivation operators and tl. = (D, l>, f) the cotriple on Rings such that Diff ~ Pings4 , we have the following. Theorems. 1. For any A E Rings, there is a 1-1 correspondence between X-indexed families of derivation operators on A and ring homomorphisms a: A -+ D(A) which are ,\-co structures. 2, There is a natural ring homomorphism ¢A: A-455 A!LX]J ~ D(A), and if Q C A, ¢A is an isomorphism. 3. For any (A, (d)) E Diff with Q C A, there is a natural open retraction pA: Spec A -> Spec 715-All. FRANCIS E. \1ASAT, Glassboro State College, Glassboro, New Jersey 08028. Reflexive subsemigroups in a regular semigrvup. Preliminary report. A subset B of a semigroup S is called reflexive if ab E B -> ba E B for a, b E S. Let S be a regular semigroup and denote the set of idempotents of S by E. The reflexive subsemigroup, B, generated by E can be used to construct the kernel, N, of the minimum group congruence on S. (See Abstract 703-A2, these :lta..tiee<>- 20(1973), A-356.) The author expands those previous results, and presents some additional results in the direction of characterizing B for inverse [orthodox, conventional, regular] semigroups. For simple w-semigroups, B and N are described; some corollaries are given for bisimple w-semigroups, bicyclic, and inverse Clifford semigroups. For inverse semigroups, a counter example shows that in general E ,P B, but that Ew = N. In conventional semigroups, E = N iff E is left unitary. (Received March 25, 1974.) 715-Al2. 0. TIMOTHY O'MEARA, University of Notre Dame, Notre Dame, Indiana 46556. Hilbert's Eleventh Problem: Quadratic forms with any algebraic numerical coefficients. • A survey talk, suited to a general mathematical audience, emphasizing major contributions to the theory of quadratic forms with coefficients in algebraic number fields, related developments, and recent research. (Received March 26, 1974.) 715-Al3. LARRY E. KNOP, Southern Ulinois University, Carbondale, Illinois 62901. Non-Abelian simple groups are not B-groups. Preliminary report. A B-group, or Burnside-group, is a finite group B such that if B is a regular subgroup of a primitive group G, then G must be doubly transitive. It is well known that if there are no nontrivial S-rings over B then B is a B-group. A finite non-Abelian simple group G has a nontrivial primitive S-ring over G whose basis consists of all conjugate classes of G, but the converse of the theorem above is an open question. In this particular case however, the direct product G x G has a faithful, primitive but not doubly transitive permutation representation in which G is regularly represented, and the orbits of a subgroup fixing a letter are the conjugate classes of G. (Received March 26, 1974.) 715-A14. S. K. JAIN, SURJEET SINGH and ROBIN G. SYMONDS, Ohio University, Athens, Ohio 45701. When are proper cyclics quasi-injective? Preliminary report. Carl Faith (Pacific J. Math. 45(1973), 97-112) has characterized rings R such that every proper cyclic right R-module is injective (called right PCI-rings). In an attempt to generalize the theory of Dedekind domains, Faith proposes to study the class of rings R such that every proper cyclic right R-module R/A is injective modulo its annihilator ideal (of which right PCI-rings form a special class). Call a ring R a right PCQI-ring if each proper cyclic right R-module is quasi-injective. It is shown that (i) a right PCQI-domain is a right Ore-domain and (ii) a nonprime right PCQI-ring is semiperfect such that for every proper ideal A, R/A is a finite direct sum of rings which are either semisimple artinian or rank zero maximal valuation duo rings. (Received March 26, 1974.) * 715-A15. JAMES E. KETTNER, Northern Illinois University, De Kalb, Illinois 60515. Group completing monoidal categories. Preliminary report. The Grothendieck group of a category with product (i.e. a small symmetric (strict) monoidal category) is defined to be the group completion of the monoid of isomorphism classes of objects under the product operation. A particular form of this construction is imitated to obtain a group completion functor for the category of small monoidal groupoids, and any skeleton of the group completion of a symmetric monoidal groupoid M is shown to have objects isomorphic to the Grothendieck group K0M with morphisms forming a group isomorphic to K0M eK 1M. (Received \larch 27, 1974.) A-456 *71'5-Al6, J. \1. GANDHI and R. G. PATEL, Western lllinois University, Macomb, Illinois 61455. Size of the integer ..,· itn'ol1'cd in the> solution of Fermat's last theorem. Let (1) xP + yP = zP, p an odd prime and 0 < x < y < z Inkeri proved [Ann. Univ. Turku. Ser. A XVI (1953), 9] that for the first case of FLT i.e. when (xy;::, p) = 1, x > (2p 3 + p/log 3p)P while for the second case when one of x, y or z is divisible by p, x > p 3P- 4 , z > Y,p 3P-I. Improving Inkeri's result we prove: Theorem. If (1) has an integral solution for the second case then either (a) z > Y,p 3P-l with 3P- 1 = 1 (Mod p 2) or (b) z>Y,3P p 3P-I with p> 25000, Also either (a) z>Y,p 3P-I v.ith 2p-l 1 (\lodp 2) or (b) z>2P- 1p 3p-! with p > 25000, Using these results we calculate the values zP for p > 3 x 109 for the first case and 25000 for the second case, Suppose a computer has to print the least integers involved in the solution of (1), assuming that it can print 1 · 3() x 10 5 digits per minute, it will take more than 3 · '5 billion years to print the integers involved in the first case, and 22 days to print the integers involved for the second case, (Received \larch 27, 1974.) *715-Al7. E. P. ARMENDARIZ and JOE W. FIS!-JER, University of Texas, Austin, Texas 78712. Idealizers in rings. Preliminary report. For a ring S with right ideal A, let R be the idealizer of A in S. Various relationships between the ring S and R are investigated. When A is semimaximal, then all of the following properties of R transfer to S: prime, semiprime, Artinian, Noetherian, perfect, semiperfect, semiprim':'ry, regular, and self-injective. In general, properties of S do not transfer to R; however, when S is semiprime (with A not necessarily semimaximal), then various (but not all) of the above mentioned properties do transfer, (Received March 27, 1974.) 715-A18, LAWRENCE NEFF STOUT, University of Illinois, Urbana, Illinois 61801, The category of topological space objects in a tapas JI is §-initial, §.-complete, and §-cocomplete. Let §. be an elementary topos. A topological space object in § is defined to be a pair (A, T A) where A is an object of .f and T A is a subobject of the powerobject PA which contains the global sections corresponding to A and 0 as subobjects of A and which is closed under the formation of pairwise internal intersections and arbitrary internal unions, A morphism from A to B in §. is called continuous if the image of T B under the internalization of the functor "pull back along f" is contained in T A. Topological space objects and continuous morphisms form a category Top (JD enriched in g, Top (JD is .§-initial in that any .§_indexed family of morphisms induces an initial structure, Top 715-A19. LARRY W. DAVIS, University of Wisconsin, Whitewater, Wisconsin 53190. An iterative function of Euler's ¢-function. Preliminary report. For each positive integer n, f(n) is defined to be the least positive integer such that ¢f(n)(n) = 1, where ¢ is Euler's ¢-function. Theorems. 1. If m and n are both even, chen f(mn) = f(m) + £(n). Otherwise, £(mn) = f(m) + f(n) - 1, 'l:'hus f(n) can be written as a sum involving terms of the form £(p- 1), where p is a prime divisor of n. 2, If 2 · 3i-l < n _::; 21, then i < f(n) _::; j. It then follows that f(n)-->"" as n--. ""· 3. For each positive integer n, f(n) .:S ¢(n). (Received March 27, 1974,) 715-A20, FR AliJK W. OWENS, Eall State University, Muncie, Indiana 47306, Matrices and symmetric matrices with zero diagonal. Preliminary report. Let A 1, • • • , A k be n x n matrices with zero diagonal over a commutative ring with identity. Results on the standard polynomial [A 1, •.• , Ak] will be presented and discussed graph theoretically. (Received March 27, 1974.) * 715-A21. LIP\lAN BERS, Columbia University, Nev. York, New York 10027. On Hilbert's 22nd Problem. • Hilbert's 22nd Problem deals with uniformization of analytic curves, The calk will describe the classical and the modern methods for achieving this, the theorems on simultaneous uniformization of all algebraic A-457 curves of a given genus and the application (due to P. A. Griffiths) of these theorems to the uniformization of higher dimensional algebraic varieties. Open problems will also be discussed. (Received March 27, 1974.) 715-A22. HUGH L. MONTGOMERY, University of ~1ichigan, Ann Arbor, Michigan 48104. Problems of prime numbers. • A survey of developments since 1900 in prime number theory and a discussion of recent results and of new directions in which the subject is moving. (Received April 4, 1974.) Analysis * 715-Bl. RUSSELL W. HOWELL, Ohio State University, Columbus, Ohio 43210. Annular functions and residual sets. Preliminary report. Let U be the open unit disk and H(U) = 1/: f is analytic in Ul. H(U) has the (complete, metric) topology of almost uniform convergence. [A basic open neighborhood of f is V(f, K, £) where, for K a compact subset of U and f > 0, V(f, K, £) = lg € H(U):\f(z)- g(z)\ < f for all z € Kl]. Set E =If= "I';;= 0 anzn: an= ± 1 for all nl. E is a complete metric space with the relative topology. Definitions. f € H(U) is annular if there exists a sequence of closed Jordan curves Jn about 0 converging outward to the boundary T of U such that the minimum of \!\ on Jn approaches infinity as n -> oo. If the Jn are circles concentric with T, f is strongly annular. Using a method of Bonar and Carroll (Abstract 696-30-2, these il..li.c.e.6. 19(1972), 636), we let Sn =If € E: there exists s = s(f) with 1 - 1hz < s < 1 such that \f(z )\ > n for all \z\ = s I and s: = It € [0, 1]: there exists s = s(t) with 1- 1/n < s < 1 such that \/1(z)\ = \I;=Ork(t)zk\ > n for all \z\ = sl (rk is the kth Rademacher function). We show that Sn is open and dense in E, s: contains an open dense set of [0, 1], and hence we obtain the following connection between the topologies of E and [0, 1]. Theorems. 1. If € E: f is strongly annular} is residual in E. 2. It € [0, 1]: / 1(z) = I;=Ork(t)zk is strongly annular} is residual in [0, 1]. (Received March 18, 1974.) * 715-B2. YUDELL L. LUKE, University of Missouri, Kansas City, Missouri 64110. On the error in the Pade approximants for a form of the incomplete gamma function including the exponential function. Closed form expressions for all entries of the Pad~ matrix table and their errors are derived for the incomplete gamma function H(c, z) = cz-ce-zJ~ettc- 1 dt, R(c) > 0, H(O, z) = e-z. Asymptotic estimates for the error are developed. Let (p., v) be a position in the Pad~ matrix table where p. and v are the degrees of the denominator and numerator polynomials respectively which define the Pad~ approximant. Our error representations hold for c and z fixed, c not a negative integer, with p. or v or both p. and v approaching infinity. Under these conditions the Pad~ approximants converge along all rows, columns and diagonals. The asymptotic representations are remarkable in that they give very easy to apply and very realistic estimates when the parameters p. and v are rather small. In the case c = 0, i.e., the exponential function, uniform asymptotic estimates are developed for the main diagonal and first and second subdiagonal entries of the PaM matrix. (Received March 25, 1974.) 715-B3. YUDELL L. LUKE and RICK WHITAKER, University of Missouri, Kansas City, Missouri 64110, Comparison of rational approximations. Preliminary report. Let /(z) be represented by a power series (at least formally) near z = 0, Let A, B be polynomials in z of degree p., v resp. If Bf(z)- A = O(z~-'+"+ 1 ), then A/B is the Pad~ approximation for f(z) in the (p., v) position of the Pad~ matrix. The case p. = v = n is called the main diagonal Pad~ approximation of order n, In the volumes by Y. L. Luke, "The Special Functions and Their Approximations, " Vols. 1, 2, Academic Press, New York, 1969, there are developed rational approximations C/D for f(z), where C, D are polynomials in z of degree n. These approximations are such that Df(z)- C = O(zn), In the absence of complete asymptotic results to gauge errors, it is of interest to compare heuristically the latter approximations with the main diagonal Pad~ approximations of the same order. Both approximations are remarkable in that they may converge in the full finite cut complex domain where the corresponding power series for {(z) is divergent. In particular this is true for z- 1 ln (1 + z), \arg (1 + z)\ < 11. ln the unit disc, computations show that the main diagonal Pad~ approximations A-458 1 for z- In (1 + z) are vastly superior. As \z\ • oo, \arg (1 + z)\ < 11, this superiority diminishes and for z sufficiently large, say z > 10, there is very little difference. Similar comparisons are made for other functions. (Received March 25, 1974,) *715-B4. WILLIAM H. GRAVES and SUZA '\INE MILLER MOLNAR, University of North Carolina, Chapel Hill, North Carolina 27514, Applications of a unit1ersal measure. Preliminary report. Let R be a ring of subsets of a set X For any real, locally convex, Hausdorff, topological vector space W, p.: R-. W is a measure if p.(¢)- 0, p. is additive, and for any sequence IE;l of mutally disjoint members of R, the sequence, ~~~/l(E)l, of partial sums is Cauchy and converges to p.(UE,) if, in addition, UEi E R. There is a locally convex, Hausdorff, topological vector space M(R) and a measure (the universal measure for R) v: R ·> M(R) such that for any other measure p.: R -> W, W arbitrary as above, p. = (;: o v for a unique, continuous linear map (;:: M(R) -. W. We characterize the equicontinuous and weak compact subsets of M(R)', From this will follow the fact that M(R) is a Mackey space but is neither normable, metrizable, barreled, nor bornilogical. The Orlicz-Pettis theorem also follows from these characterizations. (Received March 26, 1974,) 715-BS. M. N. M. TALPUR, University of Illinois, Urbana, Illinois 61801, On existence of asymptotic paths for subharmonic functions in Rm. Preliminary report. It has been shown by the author that if u{z) is a nonconstant subharmonic function in the plane, then there exists an asymptotic polygonal path r such that u(z) -. + oo as z -> "" on r. For an extension of this result to Rm, results on growth of subharmonic functions u(x) are proved in terms of geometrical properties of the set lx\u(x) > 01. In case of a general subharmonic function in Rm for m::: 3, a weaker result is obtained in which the desired path is replaced by a chain of continua each intersecting the preceding one, The continua can be replaced by a polygonal path in case of continuous functions. Recently Hayman has shown that for bounded subharmonic functions, u(x)-> M as x-> oo on almost all st. lines, where M is the l.u.b. of u(x) in Rm. For unbounded functions, Fuglede has shown existence of a path by methods of fine topology. However the path shown by Fuglede is a trajectory of Brownian motion, The question of existence of polygonal path in Rm is still open and will be discussed, (Received March 26, 1974.) *715-B6. PAUL D. HUMKE, Western Illinois University, Macomb, Illinois 61455. Cluster sets of arbitrary functions. Preliminary report. In this paper we investigate the stability of ambiguous points of arbitrary functions from the Riemann sphere into itself. In particular we have found: Theorem 1, If z is an ambiguous point of a function f and there are two pairs of arcs of ambiguity, (a, {3 1) and (a, {3 2), at z then except for a set of the first Baire category the cluster set of f along {3 1 is identical with the cluster set of f along {3 2• This theorem generalizes a result due to \lcMillan ["Arbitrary functions defined on plane sets ",Michigan Math. J, 14(1967), 445-447], In addition, an example is presented to show that, in a sense, Theorem 1 is a best possible result. (Received March 27, 1974.) *715-B7. ALAN G. R. MciNTOSH, Institute for Advanced Study, Princeton, New Jersey 08540 and BRUCE EVANS, University of Pennsylvania, Philadelphia, Pennsylvania 19104, Square roots of sectorial operators. Preliminary report. There exists a maximal accretive operator A in a Hilbert space satisfying \Im (Au, u)\ ::: c Re (Au, u) for all u E ~(A), for which 5J(Al1) -f, ~(A* 11 ). However, if the additional assumption is made that \Im (Au, u)\m::;:: c Re (Au, u). \\u\\ 2(m- 1) where m > 2, then ~(A 11 ) = P.(A *11), lt is conjectured, but not proved, that this holds for all m > 1, (Received \larch 27, 1974.) 715-BB. JAMES B. SERRIN, University of Minnesota, Minneapolis, Minnesota 55455. Solvability of boundary value problems. (Hilbert's Twentieth Problem,) • The Twentieth Problem set by David Hilbert conjectured the existence of a general principle according to which boundary value problems for (elliptic) partial differential equations can be presumed to have solutions, Seventy-five years of work have made it seem likely that no such principle is available if one strays beyond A-459 rather definitely prescribed situations, and, in particular, if the equation is nonlinear. At the same time Hilbert's statement of the Twentieth Problem can alternately be interpreted simply as calling to the attention of mathematicians the important and profound problem of boundary values for p.d.e. In this sense Hilbert's insight has proved to be little less than spectacular, for a field which was then only in its infancy has now become a major area of mathematical endeavor. It would be almost impossible to summarize the many directions of research in this area today. Instead, the gradual accumulation of techniques which have enabled mathematicians finally to come to ·grips with the most concrete of the problems which Hilbert specified, namely the boundary value problem for quasilinear elliptic equations of second order including in particular the equations for minimal surfaces, equations for surfaces of constant Gaussian curvature and first order regular variational problems. We shall indicate how the answer to Hilbert's final query "has not every regular variational problem a solution provided certain assumptions regarding the (smoothness of the) boundary condition are satisfied?" must be no. At the same time we shall show that it is possible to distinguish and reasonably classify equations, domains, and data for which the answer is positive. (Received April 4, 1974.) Applied Mathematics 715-Cl. MARIA z·. v. KRZYWOBLOCKI, Michigan State University, East Lansing, Michigan 48824. Association of Schroedinger equation (quantum) with von Karman-Howarth equation (deterministic, macroscopic), Quantum theoretic methods (linear) emerge as the strongest methods available today in the field of mechanics, both solids and fluids. The macroscopic approach to turbulence through Navier-Stokes system (nonlinear) cannot claim to be very successful. The approach to quantum by J. von Neumann is accepted by mathematicians jointly with his hidden variables, The author starts from Schroedinger equation, follows Made lung's idea in "Quantum theory in hydrodynamic form" (1926) with electron, generalized to cluster of mass particles around electron and ends in macroscopic domain; viscosity phenomena are taken into account through hidden variables in form of diabatic flow (NASA, 1946), The author demonstrates that the well-known von Karman- l.Jowarth equation in isotropic turbulence is only a special case of quantum approach: Schroedinger-Madelung- John von Neumann (linear) vs. Euler-Navier-Stokes-Prandtl-Blasius-von Karman-Howarth (nonlinear). (Received March 14, 1974.) * 715-C2, YUDELL L. LUKE, BI\IG Y. TING and MARILYN J. KEMP, University of Missouri, Kansas City, Missouri 64110, On generalized Gaussian quadrature. Y, L, Luke has shown that if a set of polynomials satisfies an orthogonality relation with respect to integration, the set also satisfies an orthogonality relation with respect to summation. The orthogonality relations give rise to two representations of an arbitrary function f(x) in series of orthogonal polynomials qn(x). One is infinite with coefficients ck; the other is finite with coefficients dle,n and nodes are the zeros of qn+ 1(x), A connection between these coefficients is established. Integration of the finite interpolation formula leads to Gaussian type quadrature formulas, The error in the interpolation formula and in the quadrature formula is characterized in terms of the coefficients ck. The purpose of this paper is to extend the results noted above by developing a generalized interpolation formula from which we derive a generalized Gaussian quadrature scheme to include an arbitrary number of preassigned nodes, If (a, b) is the interval of integration and if the preassigned node(s) is at a or at both a and b then the integration formulas go by the names of Radau and Lobatto respectively. The error analyses noted above are applied to the generalized case, Several numerical examples are presented to manifest the efficiency of the error formulas for Gaussian, Radau and Lobatto quadrature schemes. (Received \ * 715-C3. ARTHUR S. WIGHTMAN, Department of Physics, Princeton University, Princeton, New Jersey 08540, Hilbert's Sixth Problem: Mathematical treatment of the axioms of physics. • Hilbert proposed as his Sixth Problem: "To treat in the same manner( ... as investigations on the foundations of geometry···), by means of axioms, those physical sciences in which mathematics plays an A-460 important part; in the first rank are the theory of probability and mechanics." As far as the general theory of probability is concerned, Hilbert's problem can be regarded as solved by the modern theory that evolved in the first three decades of this century. The theory of dynamical systems can be regarded as providing a precise axiomatic foundation for the mechanics of systems of a finite number of degrees of freedom. For svstems of an infinite number of degrees of freedom both in the context of classical and quantum mechanics the situation is less clear. Axiom systems have been proposed, but the study of their consequences has not been carried far enough to provide decisive evidence that the axioms have captured the essence of the heuristic ideas involved. Examples show that such requirements as stability, existence, and unicity of global solutions provide only partial guides to the adequacy of the axioms, It can and will be argued that axiomatization has a limited but essential role in the development of theoretical physics. The internal consistency of a physical theory depends, in general, not on agreement with nature but on mathematical consistency as examples in hydrodynamics, statistical mechanics, and quantum field theory show. (Received April I, 1974,) Geometry 715-Dl. HERBERT BUSEMANN, University of Southern California, Los Angeles, California 90007, Problem IV: Desarguesian spaces. • Hilbert's Problem IV calls for the study of those geometries in which the ordinary lines, i.e. the lines in n-dimensional projective space pn or parts thereof, are the shortest connections, in particular for the construction of all and the study of the individual metrics of this type. The distance need not be symmetric. The problem is related primarily to three fields: the foundations of geometry, the calculus of variations, and differential geometry. Hilbert's main interest was on the foundations of geometry, but the first contributor to the problem, G. Hamel, wrote a thesis in 1901 under Hilbert's guidance emphasizing the calculus of variations, The differential geometric aspect predominated around 1930 led by the then German University at Prague. In the lecture the problem will be formulated in modern terms. The significance of Desargues' Theorem (and hence the title) will be explained. According to a theorem, which in the case of symmetric distances goes back to Hamel, the range of the metric is either all of pn or leaves out at least one hyperplane and can therefore be considered as a convex set in affine space An. Examples will be given to show that there are interesting (non-Riemannian) Desarguesian spaces, but most are completely uninteresting. This means that the second aspect of Problem IV is, in its original form, not a well posed question, but the subject will probably continue to develop by the discovery of interesting special metrics. The first part was solved by Hamel in 1901 for n = 2, 3 under strong regularity and at that time necessarily vague completeness conditions. For the case of symmetric distances a complete solution for general n was sketched recently (1973) by Pogorelov in a Doklady note, His approach is based on an idea of integral geometry introduced by the author. For nonsymmetric distance Hamel's construction remains so far the only one. The foundations of geometry aspect enters partly through various characterization theorems; a simple example: just two Desarguesian spaces (Minkowski's and Funk's) have homothetic spheres. (Received March 29, 1974,) 715-D2, JOHN H. CONWAY, University of Michigan, Ann Arbor, Michigan 48104 and Cambridge University, Cambridge, England. Hilbert's Third Problem. • Hilbert's Third Problem was solved by Dehn even before the published version of the Hilbert problems appeared, In the simplest case this problem asks whether a regular tetrahedron can be dissected into a finite number of pieces which can be reassembled to form a cube. Dehn showed that the answer is no. More recently an elaborate theory of dissection problems has been created. Sydler finally showed that Dehn's necessary condition for the solubility of a dissection problem is also sufficient. This talk will discuss the algebraic theory of dissection problems developed after Dehn's work. (Received April 4, 1974.) A-461 Logic and Foundations 715-E1. DONALD A. MARTIN, Rockefeller University, New York, New York 10021, Hilbert's First Problem. • Hilbert's First Problem is identical with the continuum problem of Cantor: Is the cardinal number of the continuum the least uncountable cardinal number? Cantor's continuum hypothesis (CH) asserts that the answer is yes. The generalized continuum hypothesis (GCH) asserts that the least cardinal greater than any given in finite cardinal m is the cardinal zm. In 1938, Kurt Godel proved that the GCH could not be refuted in the usual formal axiomatic set theory (ZF). In 1963, Paul Cohen showed that the continuum hypothesis could not be proved in ZF, Cohen's methods yielded an almost complete solution as to what could be shown about the GCH in ZF. Many set theorists regard these developments as a complete solution of the continuum problem, insofar as it has mathematical meaning. Others hope for a solution of the continuum problem based on new axioms for set theory. The main candidates for new axioms are large cardinal axioms-axioms asserting that very large cardinal numbers exist. Such axioms have been used to settle certain cases of the GCH. Unfortunately, every known large cardinal axiom has been shown not to settle the continuum hypothesis itself. Although there is no immediate hope of settling the continuum problem, one can make plausible guesses as to the answer. If Cl:! is false, there is a map of the continuum onto w2 (i.e., onto the ordinals < w 2). One can define a very simple map of the continuum into w 2, and it is consistent with ZF that this map is onto. One can further define a map of the continuum which large cardinal axioms imply is into w 3, and might well be onto w 3• Using further axioms one gets a simple map into w4• The definitions of these maps fall into a known hierarchy and suggest a definite way in which the answer to the continuum problem might be negative. (Received April 1, 1974.) 715-E2, GEORG KREISEL, Department of Philosophy, Stanford University, Stanford, California 94305. What have we learnt from Hilbert's Second Problem? • Hilbert asked for a proof of the consistency of formal arithmetic by 'elementary' means; or else an impossibiliry proof when 'elementary' is made precise in an appropriate sense; cf. the problem of proving number theoretic theorems stated in 'elementary' terms by means of 'elementary' methods, Hilbert's principal interest came from the relevance, which he had established, of his (mathematical) consistency problem to a familiar (philosophical) conception of mathematical rigor; based on the assumption or 'theory' that 'elementary' methods are particularly secure. The principal direct progress so far consists in correcting false early impressions of the nature of the problem (what is general and easy, what is problematic); the principal by-product of work on the consistency problem is, probably, recursion theory (the theory of arbitrary formal or mechanical rules), The latter theory is needed for a proper general formulation of Godel's incompleteness theorems which provide negative solutions to Hilbert's second problem under mild assumptions (on the meanings of 'formal arithmetic' and 'elementary methods') which will be explained in the talk. For special formal systems of arithmetic and certain elementary methods (of which Fermat's use of the methode de d~scente is typical), Gentzen gave a positive solution to the consistency problem. His analysis of the structure of formal proofs is of mathematical interest quite independently of the epistemological problem of the validity of the principles considered, His analysis is also of interest for judging the philosophical assumption (concerning rigor) mentioned above. In the last 25 years lfilbert's consistency problem has been extended to the functional interpretation problem (where 'elementary' methods include the ordinary mathematical language of higher types, but only weak axioms about them). The bulk of mathematical practice can be formulated in systems admitting such functional interpretations, and so can, automatically, be constructively reinterpretated (as illustrated by reference to Roth's result on rational approximations to algebraic numbers and its reinterpretation in terms of a bound on the number of (close) approximations). This discovery corrects the false impressions 50 years ago (and still current) concerning the nature of so-called constructive foundations. Though, apart from a few doctrinaires, the silent majority of mathematicians had no illusions about such foundations, they also had no precise tools to put those false impressions in their proper place and use them positively, for example, for Hilbert's 17th Problem. It should perhaps be added that, unlike the other problems (including the first), the consistency problem A-462 was formulated without the benefit of earlier related work of a mathematical character. Its basis was a mere case studv (establishing formalization). To be precise, the talk concerns not Hilbert's original formulation (which did not refer to 'elementary' methods), but the later version used by Hilbert after 1904. (Received April 2, 1974.) Topology * 715-G1. BARRY H. DAYTON, Northeastern Illinois University, Chicago, Illinois 60625. K-theory of special normed algebras. A real commutative unitary algebra A will be called special if it has a norm 1\ ·11 satisfying (i) llrll = lrl for real numbers r; (ii) the group of units A* is open, (iii) a 1-+a- 1 is continuous. Let it be the category of special normed algebras (and continuous homomorphisms) and no the category of special algebras. 1 For A E n let A be the algebra of continuous functions I: I -+ A and for A E no let A :J"(, n0 , and on the groups GL(n, A) in the obvious manner, The author shows that the algebraic K-theory functors K0, SK 1 are homotopy invariants with respect to these homotopy notions. This is used to define functors f<= on n and n0 and to derive exact sequences connecting them. Among the results obtained are (1) for a compact Hausdorff space X, f<~(CRX) = f * 715-G2. FREDERICK R. COHEN and EWING L. LUSK, Northern Illinois University, DeKalb, Illinois 60115. Configuration-like spaces and the Borsuk-Ulam theorem. Let "P be the cyclic group of prim•e order p, X be a path connected Hausdorff space which supports a free ITP action, and f: X-+ Rn be continuous. Define A(f, q) = lx E Xlx is an orbit in which at least q points get mapped to the same point by fl. Theorems. 1, If Hi(X; ZP) = 0 for 0 < i < (n- 1)(p- 1) + q- 1, then A(f, q) f, ¢. 2. If X is a closed ZP -orientable manifold of dimension m, and Hi(X; ZP) = 0 for 0 < i < (n- l)(p- 1) + q- 1, then dim A?: m- (n- 1)(p - 1) + q- 1. Remarks. Special cases of the above are: 1. The classical Borsuk-Ulam Theorem, Theorem 1 with q = p = 2 and X= sn; 2. The Bourgin-Yang Theorem, Theorem 2 with q = p = 2 and X= sn; 3. Munkholm's "Mod p Bourgin-Yang Theorem," Theorem 2 with q = p and X a homology m-sphere; 4. The case q = 2 of Theorem 1 (F. Cohen and J, E. Connett, to appear in Proc, Amer, Math, Soc.). Similar results can be obtained by our methods when Rn is replaced by other connected manifolds of dimension at least two, but we have not calculated the appropriate replacement for the expression (n - 1)(p - 1) + q - 1 in the theorems, (Received March 22, 1974.) *715-G3. EWTNG L. LUSK, Northern Illinois University, DeKalb, Illinois 60115. The Mod p Smith index and a generalization of the Borsuk.Vlam theorem. Preliminary report. Let ZP act freely on a Hausdorff space X, where p is prime, If z is an invariant cycle in X, an index v(z} in ZP is defined by means of the classifying map c: X/lp-+ BZP, It is shown that this index has properties analogous to those of the index described in the case p = 2 by Wang (Ann. of Math. 60(1954), 262-282]. This is used to prove the following generalization of a theorem of Fenn [Proc. Cambridge Philos, Soc. 14(1973), 251-256]. Theorem. Let a generate a free piecewise linear ZP -action on a ZP -cohomology n-sphere X and let g: X -+ Rm be any map. Let f: X-+ X be any map of degree not a multiple of p. Then if n ?: (m - l)(p - 1) + 1 q - 1, there are q points x 1' • • ·, x q in X such that a 1j(x 1) = •• • = aiqf(xq) and g(x 1) = ••• = g(x q). In some cases Rm can be replaced with other manifolds. Applications are given to finding homotopy classes of maps containing no embeddings, no maps without triple points, ere. (Received March 25, 1974.) A-463 * 715-G4. LAURE\ICE R. TAYLOR, University of Notre Dame, Notre Dame, Indiana 46)56. Dinsibillty of the index with applications to inertia groups and H*(GIO), Preliminary report. The theorem of Frank (Topology 11(1972), 245-252) on the divisibility of the index of a smooth manifold by a power of 2 depending on the number of sections of the stable tangent bundle is generalized to permit fibre-homotopy sections. This can be used to prove that iff: M4n _, G/0 is a surgery problem with M 2 oriented and with index surgery obstruction s(f), s(f) "'f*k 2w 4 n- 2(M)[M] (mod 2), .where k 2 E H (G/O; Z 2) is 4 1 the nonzero class. One application is that if l E bl' 4n is a generator; if M n- is orientable; and if h: M # l -> M is the obvious PL homeomorphism, h is never homotopic to a diffeomorphism. Brumfiel-Madsen-Hilgram (Ann. 4 of 'vlath. 97(1973), 82-159) define classes k 4 ; E H i(G/rOP; Z 2). We give a proof of their result that in 2 H*(G/0; Z 2), k 4 i goes to 0 unless i = 2i, If i = 2i, k 4 i goes to (k 2) i. (Received \larch 25, 1974.) *715-G5. PETER J. NYIKOS, University of Chicago, Chicago, Illinois 60637, On the product of suborderable spaces. Subspaces of linearly orderable topological spaces form an important class of monotonically normal spaces. Various necessary and sufficient conditions are here given on a suborderable space X for X x Y to be monotonically normal for some nondiscrete suborderable space Y. The strongest condition, implying all the others for general topological spaces, is that of wl-'·metrizability, equivalent to X admitting a uniformity with a totally ordered base. When combined with total disconnectedness, these conditions are shown to characterize those suborderable spaces X such that X2 is suborderable. In the case where X is not metrizable, the conditions imply total disconnectedness, and wl-'-metrizability implies suborderability as well. (Received March 27, 1974.) ERRATA Volume21 THOMAS C. GARD, Uniqueness criteria for Ito's equation. Preliminary report, Abstract 712-Fl, Page A-344. Line 7 should read, "(av ;ax)f(t, x) + (av ;ay)f(t, y) ..• " DAVID ZEITLIN, A conjecture on the general coefficients of the chromatic polynomial of a complete bipartite graph KP,q' Abstract 74T-A71, Page A-298. On line 11, the value of Cd- 2 is wrong. Let U, = ~i:f "i.~=k+tPkPt For each j, j = 1, 2, ••• , (;), let Pj "'TI~= 1 pj~>, as a product of three numbers, be one of the (;)"' R combinations obtained by choosing 3 numbers from the r numbers p 1' p 2, • • • , p ,- Then the correct value of C d- 2 is given by cd-2 = <~'>- ~f=l ~'r A-464 SITUATIONS WANTED SEE SPECIAL ANNOUNCEMENT, page 189 Unemployed mathematicians, or those under notice of involuntary unemployment, are allowed two free advertisements during the calendar year; retired mathematicians, one advertisement. The service is not available to professionals in other disciplines, nor to graduate students seeking their first postdoctoral positions;* however, veterans recently released from service will qualify. Applicants must provide (1) name of institution where last employed; (2) date of termination of service; (3) highest degree; (4) field. Applications from nonmembers must carry the signature of a member. 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Correspondence to applicants listed anonymously should be directed to the Editorial Department; the code which is printed at the end of the listing should appear on an inside envelope in order that correspondence can be forwarded. *Note the temporary exception announced on page 189 MATHEMATICS RESEARCH, INDUSTRIAL, GOVERN TEACffiNG AND/OR RESEARCH. Ph. D. 1967. Age 29. MENT, ACADEMIC. Ph.D. 1963, Male, age 36, Com Specialty: (numerical) analysis. Two articles accepted plex variables and scientific and interactive computing. for publication in the AMS BULLETIN. Available imme Fifteen papers in print or accepted (179 pages). Eight diately, Benjamin Volk, 13-15 Dickens Street, Far years teaching experience, Currently senior postdoctoral Rockaway, New York 11691. associate at Air Force lab. Resume and references avail able. K. 0. Leland, 619 Omar Circle, Yellow Springs, Ohio 45387. MATHEMATICS PROFESSOR. Ph. D. Cornell 1970. Age ANONYMOUS 30. Specialties: combinatorics, logic, applied mathemat MATHEMATICS PROFESSOR, TEACffiNG, RESEARCH ics, computers. Three papers published, one accepted, at a University/College, Ph. D. 1962. Speciality: Contin two submitted, three in preparation. Four years teaching uum Mechanics, F1uid Dynamics, Partial Differential experience, Popular teacher. Competent for all under Equations, Methods of Applied Mathematics, Engineering graduate mathematics including combinatorics, prob Mathematics. Thirteen published papers, two submitted, ability, statistics, computer science, logic. Also many Thirteen years of experience in industrial research in graduate courses. Available June or September 1974. U.S. , nine in teaching. Vita and references upon request. L. Taylor Ollmann, Department of Mathematics, Louisi Available September 1974. SW35, ana State University, Baton Rouge, Louisiana 70803, A-465 A series of paperbacks in pure and applied mathematics NORTH-HOLLAND M:ATHEM:ATICS STUDIES Vol. 1: Holomorphic Functions, Domains of Holomorphy and Local Properties By L. NACHBIN 1972, 2nd repr., 130 pages, Dfl. 18.00 (about US$ 6.60) Vol. 2: Degrees of Unsolvability By J. R. SHOENFIELD 1971, 120 pages, Dfl. 20.00 {about US$ 7.30) Vol. 3: Pseudo-convexite, Convexite Polynomiale et Domaines d'Holomorphie en Dimension lnfinie By Ph. NOVERRAZ 1973, 122 pages, Dfl. 20.00 (about US$ 7.30) Vol. 4: Spectral Theory and Complex Analysis By J.-P. FERRIER 1973, 108 pages, Dfl. 20.00 (about US$ 7.30) Vol. 5: Operateurs Maximaux Monotones et semi-groupes de contractions dans les espaces de Hilbert By H. BREZIS 1973, 184 pages, Dfl. 25.00 (about US$ 9.1 0) Vol. 6: Matched Asymptotic Expansions and Singular Perturbations By W. ECKHAUS 1973, 148 pages, Dfl. 22.50 (about US$ 8.20) Vol. 7: Finite Groups '72 By T. GAGEN, M. HALE and E. 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The central topic of integration in the complex Practical, concrete approach to linear algebra. Intro plane is introduced early in the text. Special attention is duces and illustrates topics with real problems from given to complex series and power series, facilitating science, business and industry. Proves all theorems later treatment of Taylor and Laurent Series. constructively and prepares readers to solve applied 1/74, 304 pp., cloth $12.50 problems in industry with techniques that translate eas ily into computer programs. Instructor's Guide available. VARIATIONAL METHODS IN OPTIMIZATION 2/74, 432 pp., cloth $12.50 Donald R. Smith, University of California, La Jolla Requring only elementary calculus as a prerequisite, ABSTRACT ALGEBRA: A FIRST COURSE this book shows how to solve many different classes of Larry Joel Goldstein, University of Maryland optimization problems using the simpler Euler This concise new book is designed for a full year Lagrange multiplier technique alone. Considers the course in algebra and presumes no prior background in classical problems of calculus of variations, including set theory or algebra. The historical approach used in problems with fixed and variable endpoints, isoperimet the book expounds upon contemporary algebra by care ric and certain types of global inequality constraints, and fully examining some of the important questions whose other optimization problems that have customarily been solutions gave birth to the subject of abstract algebra. handled by the more difficult methods of optimal control 1973,349 pp., Cloth $11.95 theory. 1/74, 400 pp., cloth $16.95 DIFFERENTIAL TOPOLOGY LIE GROUPS AND REPRESENTATION Victor Gulllemin, Massachusetts Institute of Technol V. S. Varadarajan, University of California, Los Angeles ogy and Alan Pollack, Harvard University. An advanced work which summarizes the techniques Intuitive, visualizable approach-much attention is and results found in Lie groups and Lie algebras. In a paid to pedagogical features such as pictures, exer well-organized format, the book reviews and explains cises, etc. Emphasis on elementary and intuitive ap thoroughly all aspects of Lie algebras. Exercises appear proach to fairly deep, substantive results. An annotated at the end of each chapter. bibliography at end of book provides a guide to related 6/74, 496 pp., price forthcoming. literature. Contains well developed exercises; essential MULTIVARIABLE MATHEMATICS: LINEAR adjuncts to book. 8/74, approx. 415 pp., cloth $14.95 ALGEBRA, DIFFERENTIAL EQUATIONS, CAL CALCULUS WITH THE COMPUTER: A LABORAT· CULUS ORY MANUAL Richard E. Williamson, Dartmouth College and Hale L. Carl Leinbach, Gettysburg College F. 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An Introduction to the Theory of Distributions (Pure and Applied Mathematics Series, Volume 14) by Jose Barros-Neto 240 pages, 1973 Usually: $16.00 Prepaid price: $12.00 Introduces the student to the theory of distributions, and stimulates him to do further study and research on the theories of topogolical vector spaces, distributions, and kernels, as well as their applications to analysis. Contents: Locally Convex Spaces. Distributions. Convolutions. Tempered Distributions and Their Fourier Transforms. Sobolev Spaces. On Some Spaces of Distributions. Applications. Tangent and Cotangent Bundles: Differential Geometry (Pure and Applied Mathematics Series, Volume 16) by Kentaro Yano and Shigeru Ishihara 432 pages, 1973 Usually:$27 .00 Prepaid price: $20.25 Offers a clear insight into classical results on tangent and cotangent bunales and provides many new problems in the study of modern differential geometry. Contents: Vertical and Complete Lifts from a Manifold to Its Tangent Bundle. Horizontal Litts from a Manifold to Its Tangent Bundle. Cross-Sections in the Tangent Bundle. Tangent Bundles of Riemannian Manifolds. Prolongations of G-Structures to Tangent Bundles. Non-Linear Con nections in Tangent Bundles. Vertical and Complete Litts from a Manifold to Its Cotangent Bundle. Horizontal Litts from a Manifold to Its Cotangent Bundle. Tensor Fields and Connections on Cross-Sections in the Cotangent Bundle. Prolongations of Tensor Fields and Connections to the Tangent Bundle of Order 2. Prolongations of Tensor Fields, Connections, and G-Structures to the Tangent Bundle of Higher Order. Rings with Polynomial Identities (Pure and Applied Mathematics Series, Volume 17) by Claudio Procesi 202 pages, 1973 Usually: $17.25 Prepaid price: $12.94 Provides a compact, systematic presentation of the results obtained over the last twenty-five years for one particular area of noncommutative algebra-the theory of rings with polynomial identities (PI-rings). Contents: Polynomial Identities in Algebras. Structure Theorems. The Identities of Matrix Alge bras. Representations and Their Invariants. Finitely Generated Algebras and Extensions. Fin iteness Theorems. Intrinsic Characterization of Azumaya Algebras. The Center of a PI-Ring. Harmonic Analysis on Homogeneous Spaces (Pure and Applied Mathematics Series, Volume 19) by Nolan Wallach 384 pages, 1973 Usually: $28.50 Prepaid price:$21.38 Examines the subjects of representation theory and nonabelian harmonic analysis. Contents: Vector Bundles. Elementary Representation Theory. Basic Structure Theory of Com pact Lie Groups and Semisimple Lie Algebras. The Topology and Representation Theory of Compact Lie Groups. Harmonic Analysis on a Homogeneous Vector Bundle. Holomorphic Vector Bundles over Flag Manifolds. Analysis on Semisimple Lie Groups. Representations of Semi simple Lie Groups. I am enclosing a check/money order for $ , including $.50 per book shipping and handling charge. 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Some features to look for in this new edition are: more emphasis on the definite integral concept; the logarithm is now defined as a definite integral and the exponential function as its inverse function; new material on the triple scalar product and the triple vector product; an explanation of Kepler's Laws of Plan etary Motion; addition of proof of the Implicit Function Theorem; new matter on the center of gravity, and a new appendix article on sum notation. Almost all the exercises in each chapter are new and present new ideas and concepts. A detailed Solutions Manual will be available for instructors use. Spring, 1974/1150 pages (tent.)/$15.00 (tent.) CALCULUS FOR THE NONPHYSICAL SCIENCES With Matrices, Probability and Statistics by Simeon M. Berman, New York University The aim of this outstanding new textbook is to teach mathematical methods to the non-math, non-science major. Expressly written for the student in social sciences, busi ness, economics, or life sciences, the book may also be used as a supplement in such courses as genetics, mathematical sociology, or operations research. Over 600 illustra tions and numerous applications and exercises from all areas of interest aid in making abstract theory accessible to students of the humanistic professions. The four main areas covered in the text are analytic geometry, matrix algebra, calculus of one and two variables, and probability/statistics. Each section is followed by an initial set of exercises devoted to drill techniques and then by more specialized problems and applications. Answers to the odd numbered exercises may be found in the back of the book; others are located in the comprehensive Instructor's Manual. Spring 1974/672 pages/$13.00 (tent.) FINITE MATHEMATICS by john M. Peterson, Brigham Young University This textbook is designed for a one semester or a one quarter course for non-mathematics majors with introductory algebra as its only pre-requisite. The approach is intuitive and does not require or involve calculus. The chapters are written independently to allow for flexible arrangement and ordering of topics. Emphasis is on examples and applica tions to provide today's student with manipulative resources. Basic ideas, purposes, and uses of statistics are covered without excessive use of formulas and abstract theory. Concise chapter summaries and review exercises help make this new textbook a per fect guide for the professor and his student. A workbook and Solutions Manual accom pany the text. Spring 1974/350 pages/$12 .00 ALGEBRA by Thomas W. Hungerford, University of Washington This is a basic text for an algebra course at the beginning graduate level. The major areas of algebra are treated in sufficient breadth and depth for this level, including sets and cardinal arithmetic; groups; rings; modules and vector spaces; fields and Galois theory; linear algebra; commutative algebra; structure of rings; categories and functors. There are over 800 exercises of varying difficulty. December 1973/528 pages/$17.00 For complimentary copies please send your course title and approximate enrollment to: Kaye A. Widmayer, Advertising Editor HOLT, RINEHART AND WINSTON, INC. 383 Madison Avenue New York, New York 10017 A-470 HONGKONG POLYTECHNIC Applications are invited from suitable candidates for the post of DIRECTOR the salary for which will be HK$13,500 per month, or at current exchange rates about US$32,400 per year. The successful applicant will be offered an appointment on a 4-year contract with a gratuity of 25% of total salary payable on satisfactory completion. 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Am lnstitut fur Mathematische Statistik der Universitiit GoHingen ist die Stelle eines TEACHING POSITIONS AVAILABLE 1974-75 Abteilungsvorstehers und Professors (associate professor) (Abteilung fiir Stochostik-Arbeits Assistant Professor - at lima Regional gebiet: Wohrscheinlichkeitstheorie oder Mothe Campus of The Ohio State University, motische Stotistik) voroussichtlich zum Herbst Ph.D. in Math required. 1974 zu besetzen. lecturer - (temporary position, maximum Bewerbungen mit Lebenslauf, Schriftenverzeichnis of 3 years) at Mansfield Regional Campus und Angoben iiber bisherige Lehrtatigkeit werden of The Ohio State University, M.S. in moglichst bold, spatestens bis 15.5.1974, erbeten Math required. on den Vorsitzenden des Fochbereichs Mathematik, Address inquiries with vita to: Professor Dr. R. Kress Dr. John Riner, Vice Chairman MATHEMATISCHES INSTITUT THE OHIO STATE UNIVE;RSITY DER UNIVERSIT AT GOTTINGEN Department of Mathematics 34 Gottingen, Bunsenstr. 3 I 5 231 West 18th Avenue Columbus, Ohio 43210 Die Anstellung schliesst eine Beomtung noch Niedersachsischem Beomtenrecht ein (Besoldung, Application Deadline: Versorgung noch AH 3). Anfrogen konnen on June 15, 1974 die oben genonnte Adresse gerichtet werden. A-471 LATTICE THEORY ILLINOIS INSTITUTE of PROCEEDINGS OF THE TECHNOLOGY UNIVERSITY OF HOUSTON Invites Nominations, Inquiries CONFERENCE and Applications for the Position of MARCH, 1973 Chairman 632 pages of articles by more than DEPARTMENT OF MATHl-iMA.'J'ICS 30 leaders in the field. Includes papers Position open September, 1975 byK.BAKER,G.GRATZER,K.H. Send Communications to: HOFMANN, J. LAWSON, R. Mc KENZIE, G. C. ROTA, R. WILLE. PROFESSOR LOUIS A. KOKORIS CHAIRMAN, SEARCH COMMITTEE Price $15 prepaid. Add $1 for mailing outside North America. ILLINOIS INSTITUTE OF TECHNOLOGY Send orders to: CHICAGO, ILLINOIS, 60616 Mathematics Department UNIVERSITY OF HOUSTON Houston, Texas 77004 Equal Opportunity Employer PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS MATHEMATICS DEPARTMENT HARMONIC ANALYSIS ON HOMOGENEOUS SPACES, edited by Calvin C. Moore CHAIRMAN Volume XXVI Nominations and applications for the 480 pages; list price $43. 50; member price $32. 63; ISBN 0-8218-1426-5 position of Chairman of the Department To order, please specify PSPUM/26 of Mathematics of Brooklyn College are This volume constitutes the proceedings of being solicited. the nineteenth Summer Research Institute of the The department currently has a full American Mathematical Society which was held at Williams College on July 31-August 18, 1972, time staff of fifty-three including thirty and was supported by a grant from the National in the professorial ranks. It offers bachelor's Science Foundation. The scientific program of the institute consisted first of all of six major and master's degrees and opportunities lecture series, each devoted to relatively broad for the doctorate through the CUNY areas within the subject and consisting of from doctoral program. four to six lectures. The lecturers were C. C. Moore, V. S. Varadarajan, Barish-Chandra, Nominations, applications and inquiries H. Furstenberg, s. Helgason, and B. Kostant should be addressed to: (jointiy with R. Blattner). The first part of this volume contains articles prepared by the lec Mordecai L. Gabriel turers and based on these lecture series. These are intended to be surveys of various aspects of Dean, School of Science the subject of the institute, and it is hoped that they will provide material not only for advanced BROOKLYN COLLEGE graduate students and postdoctoral mathemati Brooklyn, New York 11210 cians who want to get into the subject, but also for more senior mathematicians who work in re CUNY is an lated areas and who need a survey of what is Action Employer. known in the subject, and what techniques are Equal Opportunity/Affirmative available. American Mathematical Society A-472 Third Edition THE SPECTRAL ANALYSIS SOLUTION OF EQUATIONS OF TIME SERIES IN EUCLIDEAN AND by L. H. KOOPMANS This book presents the theory and methods BANACH SPACES of the spectral analysis of time series at a by ALEXANDER M. OSTROWSKI mathematical level that makes the material A Volume in the PURE AND APPLIED accessible to readers in a variety of scientific MATHEMATICS Series disciplines. The author writes in a discursive From A Review of the First Edition: "This is style and illustrates the methods by actual certainly a remarkable book which will be applications from such diverse fields as eco welcome to everybody who wants suggestions nomics, engineering, geophysics, and medical for original work or for the preparation of research. Thus he provides the knowledge lectures in numerical analysis with stress on necessary to construct time series models, to mathematical rigour."-Mathematical Reviews design and interpret spectral analysis experi In the thoroughly revised Third Edition, the ments, and to understand the workings of the author provides treatment of operator equa methods for processing the time series in tions in a Banach space. This replaces the cluded in the text. special treatment of systems of equations in 1974, 378 pp., $26.00/£12.50 the preceeding editions and is accompanied by a discussion of the elements of functional analysis. The author also develops a fully AN INTRODUCTION TO automatic method of solution of polynomial equations, and has added several new ap NONLINEAR BOUNDARY pendices-the most important of which deals VALUE PROBLEMS extensively with a feedback method for errors by STEPHEN A. BERNFELD a posteriori in iterative procedures. A Volume in the MATHEMATICS IN SCIENCE 1973, 432 pp., $34.00/£16.00 AND ENGINEERING Series This book, on an advanced level, exposes the MATHEMATICAL METHODS reader to the field of differential equations and OF STATISTICAL QUALITY provides a ready access to an up-to-date state of this art. The authors provide a variety of CONTROL techniques employed in the theory of non by K. SARKADI and I. VINCZE linear boundary value problems, including CONTENTS: INTRODUCTION: Statistical Meth methods that involve differential inequalities; ods of Quality Control. Probability Theory shooting and angular function techniques; and Its Role in the Methods of Statistical functional analytic approaches; and topological Quality Control. THEORETICAL FOUNDA methods. They have also included a chapter TIONS: Elements of Probability Theory. Ran on nonlinear boundary value problems for dom Variables. Fundamentals of Sampling. functional differential equations and a chapter Fundamentals of Mathematical Statistics. covering special topics of interest. Of value METHODS OF STATISTICAL QUALITY CON to research workers and students in the fields TROL: Statistical Methods in the Control of of mathematics, physics, life sciences, eco Production Processes. Acceptance Sampling. nomics, and engineering. Reliability Theory. APPENDIX: Special Tables; 1974, 400 pp., $18.501£9.00 Bibliography; Index. 1974, 415 pp., $22.00/£10.55 ALMOST SURE CONVERGENCE NON-EUCLIDEAN by WILLIAM F. STOUT TESSELATIONS AND This book presents a balanced and unified THEIR GROUPS treatment of the almost sure behavior of par by WILHELM MAGNUS tial sums of random variables. It comprehen Mathematical works produced around the sively traces the subject from its elementary turn of the century-particularly those of Felix roots to current research horizons and in Klein and Robert Fricke-contain a large num cludes such helpful features as an index of ber of beautiful drawings representing various symbols and conventions; exercises that pro tesselations of the hyperbolic plane. This duce a working familiarity with the subject; monograph presents reproductions of about chapter independence, i.e., later chapters may fifty of the most interesting and revealing of be used with only occasional reference to these drawings, together with comments on results of earlier chapters; an extensive bibli the algebraic relationships the drawings dem ography to be referred to for further study. onstrate. The text avoids lengthy and difficult Care has been taken to make the book ac presentations, giving new proofs and new re cessible to those with a relatively modest sults in an elementary manner. Of interest to mathematical background (basic real analysis, senior undergraduate and first-year graduate basic measure theory, and basic measure students, this book will be a useful aid to theoretical probability are the only prerequi courses in geometry and group theory, Rie sites). It can be used either as a book for self mann surfaces, and foundations of geometry. study or as a text for graduate courses in 1974, in preparation probability. 1974, about 375 pp., in preparation Price subject to change without notice. ~!:~~o~~o~u~,!~!!!~u!/~~· @ 111 FIFTH AVENUE, NEW YORK, NEW YORK 10003 24-28 OVAL ROAD, LONDON NW1 7DX Introducing ... Ill 3 Applietl Jla"'e•ali~s ~ Ill aM O,Ci.Uzalita ~ An International Journal -~ A. V. Balakrishnan (Managing Editor), Systems Science Department, UCLA, Los Angeles, California J. L. Lions (Managing Editor), -c IRIA-Laboria, Rocquencourt, France G. I. Marchuk, Computer Center, Academy of Sciences, ·-..• Novosibirsk, USSR - L. S. Pontryagin, Steklov Institute of Mathematics, ·-..iliili Academy of Sciences, Moscow, USSR The primary aim of this journal is to publish papers treating applied (practical) problems of optimization without compromising mathematical precision. The problems them selves may embrace a wide diversity of areas, e.g.: Physical Systems Chemical and Biochemical Systems Environmental Systems (Air Pollution, Water Quality Management, etc.) Problems of Optimum Design Biomedical Systems Aerospace Systems Public Service Systems t.iil Socio-Economic Systems I- Papers that treat both the theoretical (including compu- ~ tational) aspects and the applications of a particular ~ problem are welcome. From the Table of Contents .. On the Evasion Process in Differential Games ~ By L. Pontryagin, Steklov Institute of Mathematics, Academy of Sciences, Moscow, USSR .. Synthesis of Multivariable Regulators: The Internal Model Principle t.iil t By B. Francis, 0. A. Sebakhy, W. M. Wonham, University of Toronto, Toronto, Canada ~ • Optimal Estuary Aeration: An Application of Distributed Parameter ~ • Control Theory ~ g .:- By W. Hullett, The Singer Company, Glendale, California ~ ~ 'I Send your order or request to ! Vol. 1 (4 issues) 1974 ""'~..,.11:1~ ~ ~ Springer-Verlag New York Inc. $49.50 (including postage and handling) _ 175 Fifth Avenue, New York, NY 10010 Sample copies upon request. ~ or < oo (,!) Vol. 1 (4 issues) 1974 ::S • ~ Springer-Verlag/Werbeabteilung OM 122.-plus postage and handling z ~ a 1 Berlin 33, Heidelberger Platz 3 Sample copies upon request. ~ >e § a;. ;&-8 e ~~:~ o ·g a ~~.t .; Polygons, Perimeter and area of polygons, Circles and their measure, Trigonometry of the right angle. 1973 I 165 pages, paperback I $4.95 By WILLIAM ZLOT, MATTHEW GRABER, and HARRY E. RAUCH ARITHMETIC CONTENTS: Basic ideas about whole numbers, The whole number line and numeration, Computational methods, Exponents and primes, The idea of a fraction, Addition and sub traction of fractions, Multiplication and division of fractions, Decimal notation. 1973 I 205 pages, paperback I $4.95 By WILLIAM ZLOT, SAMUEL GALE, LOUISE S. GRINSTEIN, JUDITH E. JACOBS, HANNAH MASTERSON, and HARRY E. RAUCH