Calendar

This Calendar lists all of the meetings which have been approved by the Council up to the date this issue of the c){otiuiJ 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 aasigned. 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

720 January 23-27, 1975 Washington, D.C. Nov. 6, 1974 (81st Annual Meeting) 721 March 20-21, 1975 Mobile, Alabama Jan. 28, 1975 722 March 23-26, 1975 New Y<>'rk, New York Jan. 28, 1975 723 April 11-12, 1975 St. Louis, Missouri Feb. 18, 1975 724 April 18-19, 1975 Monterey, California Feb. 18, 1975 725 June 20-21, 1975 Pullman, Washington Apr. 29, 1975 726 August 18-22, 1975 Kalamazoo,· Michigan June 17, 1975 (79th Summer Meeting) November 7-8, 1975 Blacksburg, Virginia November 15, 1975 Los Angeles, California January 22-26, 1976 San Antonio, Texas (82nd Annual Meeting) April 23-24, 1976 Reno, Nevada June 18-19, 1976 Portland, Oregon *Deadline for abstracts not presented at a meeting (by title). February 1\)75 issue: January 21 April 1975 issue: February 11 OTHER EVENTS January 29-30, 1975 Symposium on Some Mathematical Questions in Biology New York, New York November 6, 1974 January 31, 1975 Symposium on Theory vs. Practice in the Finite Element Method - New York, New York

Please aft"JX the peel-off label on these c){oliui) to correspondence with the Society concerning fiscal matters, changes of address, promotions, or when placing orders for books and journals. The c){otiui) of the American Mathematical Society is published by the American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940, in January, February, April, June, August, October, November, and December. Subscription per annual volume is $10. Member subscription of $5 is included in annual dues. 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 Island, and additional mailing offices.

Copyright© 1974 by the American Mathematical Society Printed in the United States of America OF THE AMERICAN MATHEMATICAL SOCIETY

Everett Pitcher and Gordon L. Walker, Editors Wendell H. Fleming, Associate Editor CONTENTS

MEETINGS Calendar of Meetings Inside Front Cover Program for the November Meeting in Nashville, Tennessee 312 Abstracts for the Meeting: A-599-A-622 Program for the November Meeting in Los Angeles, California • . • . • . • . • . . • • . • . • • • . • 318 Abstracts for the Meeting: A-623-A-630 Program for the November Meeting in Houston, Texas . . • . . . . • ...... • . . • • . • . . . . 321 Abstracts for the Meeting: A-631-A-642 PRELIMINARY ANNOUNCEMENTS OF MEETINGS. • • . . . . . • • . . . . • . • • . • . • . . • • • . . • . . . • . . . . 324 DOCTORATES AND JOBS, 1974 REPORT...... 335 INVITED SPEAKERS AT AMS MEETINGS...... 340 ORGANIZERS AND TOPICS OF SPECIAL SESSIONS . . . • • • • . • • . . . • . . • . . • . . • . . • . • • ...... 340 THE PH.D. ANCESTRY OF THE LEADING U.S. MATHEMATICS FACULTIES...... 341 CASE STUDIES, Some Mathematicians with Nonacademic Employment . . . . . • • . • . . . . . • . • . • • 346 AMS RESEARCH FELLOWSHIP FUND, Request for Contributions . • . • . . . . . • . • • . • . . • • . • • • . 348 PRELIMINARY REPORT OF THE MAA-AMS COMMITTEE ON EMPLOYMENT OF MATHEMATICIANS IN TWO-YEAR COLLEGES • • . . • . • . . . • • • • . . • . . • . . . . • • . . . . • • • . • 349 NEW AMS PUBLICATIONS • . • . . •. . . . • . . • • . . . . . • . • . • . . • . • • . . . . • . •• . • . . • . • . . . . . • . • . . • • . 350 VISITING MATHEMATICIANS, Supplementary List . • . • • . • • . • • . . . . • . . . . . • . • . . . . • • • • . • . • . 353 SPECIAL MEETINGS INFORMATION CENTER . • . . . .• .• • . •• •• . • . . • . . • . • . . • . • . . . . . • . • . . • 355 LETTERS TO THE EDITOR . • ...... • . . . • • • . • . . . . . • . . . • . . • . . • . . . • • • . • • . • . . . . • • . • . • • . 357 NEWS ITEMS AND ANNOUNCEMENTS . . • . . . . • . . . . • • . . • . • . . • . . . . • . . . . . • . . . • • . • • . • . . . . . 363 PERSONAL ITEMS . • . • • . • • • • . • . . • . • . . . • ...... • ...... • • . . • . . • . . • . • . . • • . • . . . • • • • • . . • • 364 ABSTRACTS . • . . • . . . . • . . . • . . . • • . • . . • . . . • . • ...... • • . . • . • . • . . . . . • . . • . . . • • . • . • • . . • . • . . A-587 SITUATIONS WANTED ..••...•..••.•..••...•..•....•.••.••..•.· .•.•....•••..•.....•.. A-643 RESERVATION FORMS ...•..•...... •....•....•..•..•....•....•...... ••..•... A-651 The Seven Hundred Seventeenth Meeting Vanderbilt University Nashville, Tennessee November8-9, 1974

The seven hundred seventeenth meeting of Nashville is on Interstates 24, 40, and 65, the American Mathematical Society will be held and is served by Amtrak, and the Greyhound and at Vanderbilt University in Nashville, Tennessee, Trailways Bus Lines. Many airlines have service on Friday and Saturday, November 8-9, 1974. to Nashville; limousine service from the airport By invitation of the Committee to Select to the Vanderbilt area is $3. 25. Cars may be Hour Speakers for the Southeastern Sectional rented at the airport through the Avis, Hertz, Meetings, there will be three one-hour addresses, and National agencies. the first being at 1:00 p.m. on Friday in the Three cafeterias on campus will be open auditorium of the Sarratt Commons. All other for all meals. Also a list of local restaurants sessions will be held in the Stevenson Center for will be available at the registration desk. the Natural Sciences. Professor Trevor Evans of Three motels near the campus are holding Emory University will give an address entitled blocks of rooms (see page 238 of the October "Word problems." An address entitled "Geom­ cJ/oticeiJ). Reservations should be made directly etry of submanifolds in Euclidean space" will be with them, with mention of this meeting included given by Professor Robert B. Gardner of the in that correspondence. The zip code for each of University of North Carolina, and Professor J.R. the following is 37203. Retherford of Louisiana State University will present an address entitled "Applications of HOLIDAY INN-VANDERBILT Banach ideals of operators." (100 rooms reserved) There will be three special sessions in ad­ 2613 West End Avenue dition to the regular sessions. A special session (five blocks from the Stevenson Center) entitled "Combinatorial Theory," organized by Phone: (615) 383-1147 Professor J. V. Brawley of Clemson University, Single: $14.00 up Clemson, South Carolina, will include as partici­ Double: $18.50 up (one bed) pants Professors J. V. Brawley, Kim Ki-Hang $20.50 up (two beds) Butler, Leonard Carlitz, John D. Fulton, Henry W. Gould, Bernard R. McDonald, S. Brent ALLEN MOTEL (15 rooms reserved) Morris, David P. Roselle, Cecil C. Rousseau, 2004 West End Avenue and Richard Scoville. Participants included in a (six blocks from the Stevenson Center) special session on "Number Theory" organized Phone: (615) 327-1841 by Professor Robert M. McConnel of the Uni­ Single: $12.00 versity of Tennessee will be Professors Jacob Double: $14.00 T. B. Beard, Jr. , Bruce Berndt, Ezra Brown, Richard Hudson, Gordon Pall, Carl G. Wagner, ANCHOR MOTEL (40 rooms reserved) and Charles R. Wall. Professor RichardS. 1921 West End Avenue Varga of Kent State University, Kent, Ohio, has (six blocks from the Stevenson Center) organized a special session on "Approximation Phone: (615) 327-4581 Theory." Included in this special session will be Single: $13.00 Professors Steven Demko, Thomas R. Lucas, Double: $16.00 (one bed) G. W. Reddien, Jr., John A. Boulier, Edward $18.00 (two beds) B. Saff, and A. K. Varma. An organizational meeting of the Associa­ tion for Women in Mathematics will take place at Accommodations are also available at: 10:15 a.m. in room 1415 of Stevenson Center on Saturday, November 9, Professor Lid a K. Barrett HOLIDAY INN-WEST END presiding. The agenda will include the role of 1800 West End Avenue women in mathematics, discrimination, etc. (eight blocks from the Stevenson Center) The registration desk will be located in the Phone: (615) 329-3711 lobby of the Mathematics Building in the Steven­ SHERATON-NASHVILLE HOTEL son Center. Registration hours will be from noon 920 Broadway to 5:00p.m., on Friday, November 8, and from (one mile from the Stevenson Center) 9:00a.m. to noon, on Saturday, November 9. Phone: (615) 244-0150 There will be a beer party from 8:00p.m. until 12:00 midnight on Friday at the University Club;· Emergency messages may be left for delivery details will be available at the registration desk. at (615) 322-6672.

312 PROGRAM OF THE SESSIONS

The time limit for each contributed paper in the general sessions is ten minutes and in the special sessions is twen­ ty minutes. To maintain this schedule, the time limits will be strictly enforced.

FRIDAY, 1:00 P.M.

Invited Address, Sarratt Commons Auditorium (1) Geometcy of submanifolds in Euclidean space. Professor ROBERT B. GARDNER, University of North Carolina at Chapel Hill (717-Dl)

FRIDAY, 2:15P.M.

Special Session on Combinatorial Theoi'ffi I, Room 5326, Stevenson Center for the Natural Sciences 2:15- 2:35 (2) Enumeration of -.fferent collections of subsets of ann-set admitting SDR. Professor KIM KI-HANG BUTLER, Alabama State University (717-A18) 2:45- 3:05 (3) Partitions with prescribed pattern. Professor L. CARLITZ, Duke University (717-A4) 3:15- 3:35 (4) Simple subrings of a matrix over a finite field. Professor BERNARD R. McDONALD, university of Oklahoma (717-A1) 3:45- 4:05 (5) A combinatorial problem involving q-Catalannumbers. Professor DAVID ROSELLE, Virginia Polytechnic Institute and State University (717-A39) 4:15- 4:35 (6) The number of polynomial functions which permute the matrices over a finite field. Professor J. V. BRAWLEY, Clemson University (717-A26)

FRIDAY, 2:15P.M.

Special Session on Approximation Theory, Room 4327, Stevenson Center for the Natural Sciences 2:15- 2:35 (7) Local mappings onto spline spaces. Professor STEPHEN DEMKO, Georgia Institute of Technology (717-B2) 2:45- 3:05 (8) Patch bases and singular splines. Preliminacy report. Professor G. W. REDDIEN, Vanderbilt University (717-C2) 3:15- 3:35 (9) Rational approximation in unbounded regions. Dr. E. B. SAFF*, University of South Florida, and Dr. R.S. VARGA, Kent State University (717-B19) 3:45- 4:05 (10) Some interpolation (characterization) type results for cubic (bicubic) splines with applications to boundacy value problems. Professor THOMAS R. LUCAS, University of North Carolina at Charlotte (717 -B12) 4:15- 4:35 (11) The approximation of functions by algebraic polynomials. Dr. T.M. MILlS and Dr. A.K. VARMA*, University of Florida (717-B15) 4:45- 5:05 (12) The degree of copositive approximation. Professor JOHN A. ROULIER, North Carolina State University (717-B4)

FRIDAY, 2:30P.M.

Session on Abstract Topolo~, Room 1206, Stevenson Center for the Natural Sciences 2:30- 2:40 (13) Weaky uniform bases. I. Preliminacy report. Professor R. W. HEATH, uni­ versity of Pittsburgh, and Professor W. F. LINDGREN*, Slippecy Rock State College (717-Gll) 2:45- 2:55 (14) Weakly uniform bases. II. Preliminacy report. Professor R. w. HEATH*, University of Pittsburgh, and Professor W. F. LINDGREN, Slippecy Rock State College (717 -G12) 3:00- 3:10 (15) Equivalence of almost continuous and Darboux functions under closure. Pre­ liminacy report. Dr. MAURICE HUGH MILLER, Jr., University of Missis­ sippi (717 -G10) 3:15- 3:25 (16) On C and C*-embedding. Preliminacy report. C. E. AULL, Virginia Poly­ technic Institute and State University (717 -G6)

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

313 3:30- 3:40 (17) Scattered compactification of N U {pl. Preliminary report. Mr. M. JAYACHANDRAN, Madurai College, India, and Professor M. RAJAGOPALAN*, Memphis State University (717-G1)

FRIDAY, 2:30 P.M. Session on Algebra, Miscellaneous, Room 1307, Stevenson Center for the Natural Sciences 2:30- 2:40 (18) Varieties of Steiner quasigroups and Steiner loops. Preliminary report. Pro- fessor ROBERT QUACKENBUSH, University of Manitoba (717-A3) 2:45- 2:55 (19) On projective representations of finite wreath products. Professor JOHN R. DURBIN and Professor K. BOLLING FARMER*, University of Texas at Austin (717-A5) 3:00- 3:10 (20) Matrix commutators over an algebraically closed field. Professor PETER M. GIDSON, University of Alabama in Huntsville (717-A6) 3:15- 3:25 (21) Remarks on topological algebra. WALTER TAYLOR, University of Colorado (717-A19) 3:30- 3:40 (22) Nilpotent .t-groups are representable. Preliminary report. Dr. JORGE MARTINEZ, University of Florida (717-A21) 3:45- 3:55 (23) Factors and roots of the van der Pol polynomials. Professor FREDRIC T. HOWARD, Wake Forest University (717-A34) 4:00- 4:10 (24) Uniform distribution of sequences of algebraic integers. Dr. HARALD NIEDERREITER, Institute for Advanced Study, and Mr. SIU KWONG LO*, Southern Illinois University (717 -A16) 4:15.- 4:25 (25) The second largest prime factor of an odd perfect number. Dr. CARL POMERANCE, University of Georgia (717-A14)

FRIDAY, 2:30P.M. Session on Abstract Analysis, Room 4309, Stevenson Center for the Natural Sciences 2:30- 2:40 (26) A ratlk theorem for infinite dimensional spaces. Dr. J.P. HOLMES, Auburn University (717-:02)

2:45- 2:55 (27) On the complementation of c0 • Mr. WILLIAM H. CHAPMAN and Professor DANIEL J. RANDTKE*, University of Georgia (717-B5) 3:00- 3:10 (28) Topological dynamics and C*-algebras. Dr. WILLIAM L. GREEN, Georgia Institute of Technology (717-B6) 3:15- 3:25 (29) A finiteness property of l.c.s. Preliminary report. Professor ROBERT KNOWLES, University of Connecticut, Waterbury (717-Bll) 3:30- 3:40 (30) Stability and control of hereditary systems. Preliminary report. Professor JAMES A. RENEKE, Clemson University (717-B16)

FRIDAY, 2:30P.M. Session on Lattices and Generalizations, Room 5212, Stevenson Center for the Natural Sciences 2:30- 2:40 (31) SM-semilattices. Mr. M. F. JANOWITZ, University of Massachusetts, Amherst (717-A36) 2:45- 2:55 (32) Semilattices do not have equationally compact hulls. Dr. EVELYN NELSON, McMaster University (717 -A2) 3:00- 3:10 (33) The lattice of varieties of de Morgan lattices. Preliminary report. Professor JOHN A. KALMAN, University of Auckland and Pennsylvania State University (717-A22) 3:15- 3:25 (34) Semilattices of bisimple orthodox semigroups. Dr. D. R. LATORRE, Clemson University (717 -A28) 3:30- 3:40 (35) Boolean algebras with ordered bases and the basis property for ultrafilters. Professor RICHARD A. SANERIB, Jr., Emory University (717-A35) 3:45- 3:55 (36) Homomorphisms on groupoids with local identities. Preliminary report. Dr. JAPHETH HALL, Jr., Stillman College (717-A37) 4:00- 4:10 (37) A note on Dilworth's embedding theorem. Dr. WILLIAM T. TROTTER, Jr., University of South Carolina, Columbia (717 -A38) FRIDAY, 4:15 P.M. Invited Address, Room 4309, Stevenson Center for the Natural Sciences (38) Word problems. Professor TREVOR EVANS, Emory University (717-A29)

314 SATURDAY 8:45A.M.

Special Session on Number Theory, Room 4327, Stevenson Center for the Natural Sciences 8:45- 9:05 (39) Perfect polynomials. Preliminary report. Professor JACOB T.B. BEARD, Jr.* and Mr. JAMES R. O'CONNELL, Jr., University of Texas at Arlington (717-AlO) 9:15- 9:35 (40) Arithmetic distribution functions. Preliminary report. Professor CHARLES R. WALL, University of South Carolina (717-All) 9:45-10:05 (41) Integral-valued polynomials over GF [q,x]. Preliminary report. Professor CARL G. WAGNER, University of Tennessee(717-A23) 10:15-10:35 (42) Integral positive quadratic forms of determinants less than 1. Professor M.S. CHEEMA, University of Arizona, and Professor GORDON PALL*, Louisiana State University (717 -A27) 10:45-11:05 (43) Diophantine equations of the form x2+ D = ynr Professor EZRA BROWN, Virginia Polytechnic Institute and State University (717 -A7) 11:15-11:35 (44) The least pair of consecutive kth power nonresidues (mod p). Professor RICHARD H. HUDSON, University of South Carolina (717-AB) 11:45-12:05 (45) Positive sums of the Legendre symbol. Preliminary report. Professor BRUCE C. BERNDT, University of Illinois (717-A20)

SATURDAY, 9:00A.M.

Invited Address, Room 4309, Stevenson Center for the Natural Sciences (46) Applications of Banach ideals of operators. Professor JAMES RETHERFORD, Louisiana State University, Baton Rouge (717 -B3)

SATURDAY, 10:15 A.M.

Special Session on Combinatorial Theory II, Room 5326, Stevenson Center for the Natural Sciences 10:15-10:35 (47) Some properties of the Catalan numbers. Professor RICHARD SCOVILLE, Duke University (717-Al2) 10:45-11:05 (48) Generalized Ramsey theory for multiple colors. Professor C. c. ROUSSEAU, Memphis State University (717-A24) 11:15-11:35 (49) Gauss sums and solutions to simultaneous equations over GF (2Y). Prelimi­ nary report. Professor JOHN D. FULTON, Clemson University (717-A25) 11:45-12:05 (50) Generalized inversions and.Dedeki.nd sums. Preliminary report. Dr. S. BRENT MORRIS, Duke University (717-A13) 12:15-12:35 (51) Some new inverse series relations and new formulas for Stirling numbers. Professor H. W. GOULD, West Virginia University (717-Al5)

SATURDAY, 10:15 A.M.

Session on Analysis, Room 1206, Stevenson Center for the Natural Sciences 10:15-10:25 (52) Matrix transformations on analytic sequence spaces. Preliminary report. Dr. R. T. JACOB, Jr., University of New Orleans (717-B9) 10:30-10:40 (53) On the uniqueness of solutions to boundary value problems for hyperbolic equations. Professor C. C. TRAVIS, University of Tennessee (717-BB) 10:45-10:55 (54) Oscillation criteria for elliptic differential equations. Preliminary report. Professor S.M. RANKIN III, Florida Institute of Technology (717 -B7) 11:00-11:10 (55) On domains of univalence for certain meromorphic functions. ProfessorS. H. LAMEIER*, Thomas More College, and Professor E. P. MERKES, University of Cincinnati (717 -Bl) 11:15-11:25 (56) Subordinating factor sequences for convex maps in en. Professor JAMES ERNEST MILLER, West Virginia University (717-B17) 11:30-11:40 (57) Three point boundary problems for second order differential systems. Pre­ liminary report. Professor G.J. ETGEN, University of Houston and Professor S.C. TEFTELLER*, University of Alabama in Birmingham (717-BlB) 11:45-11:55 (58) Then-dimensional Cauchy-Riemann equations. Professor MARIO 0. GONZALEZ, University of Alabama (717-B14)

315 SATURDAY, 10:15 A.M.

Session on AJ?Plied Mathematics and Analysis, Room 4309, Stevenson Center for the Natural Sciences 10:15-10:25 (59) Distortionless wave propagation in inhomogeneous media and transmission lines. Mr. VICTOR BURKE*, Professor D. HAZONY, Case Western Reserve University, and Professor R. J. DUFFIN, Carnegie-Mellon University (717 -C1) 10:30-10:40 (60) Adjacency graphs of transition probability spaces. Preliminary report. Pro- fessor JOHAN G. F. BELINFANTE, Georgia Institute of Technology (717-C3) 10:45-10:55 (61) Unconditional probabilities for the voter's paradox. Preliminary report. Dr. JAMES J. BUCKLEY, University of South Carolina (717-Fl) 11:00-11:10 (62) Multi-parameter Stieltjes transformations. Dr. R. SHONKWILER, Georgia Institute of Technology (717-B13) 11:15-11:25 (63) Oscillation in even order linear differential equations. DAVID LOWELL LOVE­ LADY, Florida State University (717-B20) 11:30-11:40 (64) Separable quotients of Banach spaces. Professor STEPHEN A. SAXON, Uni­ versity of Florida (717-B21)

SATURDAY, 10:15 A.M.

Session on Abstract Topology, Room 1307, Stevenson Center for the Natural Sciences 10:15-10:25 (65) Remarks on a property of 9-refinable spaces. Preliminary report. Professor JOHN M. WORRELL, Jr., and Professor HOWARD H. WICKE*, Ohio Uni­ versity (717 -G16) 10:30-10:40 (66) Some properties related to [a, b)-compactness. I. Professor JERRY E. VAUGHAN, University of North Carolina at Greensboro (717-G15) 10:45-10:55 (67) Metrization of spaces with large basis dimension. Preliminary report. GARY GRUENHAGE* and PIDLLIP ZENOR, Auburn University (717 -G13) 11:00-11:10 (68) Some topological properties determined by PN-operators. Preliminary report. Professor PIDLLIP ZENOR, Auburn University (717-G9) 11:15-11:25 (69) On Q-sets in normal first countable T2-spaces. Professor GEORGE M. REED, Ohio University (717-G7) 11:30-11:40 (70) Sectional mappings. Preliminary report. Professor J.D. LAWSON and Pro­ fessor B.L. MADISON*, Louisiana State University (717-G4) 11:45-11:55 (71) CompletableAronszajnspaces. Dr. THOMAS M. PIDLLIPS, Auburn University (717-G18)

SATURDAY, 10:15 A.M.

Session on Geometrical and Algebraic Topology, Room 5212, Stevenson Center for the Natural Sciences 10:15-10:25 (72) Some special decompositions of E3. Professor CHARLES D. BASS, Pembroke State University (717-G5) 10:30-10:40 (73) Sewings of closed n-cell complements. Preliminary report. Professor ROBERT J. DAVERMAN, University of Tennessee (717-G2) 10:45-10:55 (74) A factorization of the direct limit of Hilbert cubes. Dr. RICHARD E. HEISEY, Vanderbilt University (717 -G14) 11:00-11:10 (75) Six types of retracts. Preliminary report. Professor KENNETH R. KELLUM, Miles College (717-G17) 11:15-11:25 (76) On the structure of free z4-bordism. Preliminary report. Dr. R. PAUL BEEM, University of Pennsylvania (717-G3) 11:30-11:40 (77) Pontryagin duality for abelian k-groups. Preliminary report. w. F. LaMARTIN, University of New Orleans (717-G8)

SATURDAY, 10:15 A.M.

Session on Rings and Generalizations, Room 1307, Stevenson Center for the Natural Sciences 10:15-10:25 (78) Pure subfields and modular extensions. Preliminary report. Mr. JAMES K. DEVENEY, Virginia Commonwealth University (717-A9) 10:30-10:40 (79) Nilpotent elements of commutative semigroup rings. Preliminary report. Dr. TOM PARKER, Kirtland AFB, Albuquerque, New Mexico, and Pro­ fessor ROBERT GILMER*, Florida State University (717-A17)

316 10:45-10:55 (80) Finitely generated inseparable field extensions. Preliminary report. Mr. DAVID TUCKER, Florida State University (717 -A30) 11:00-11:10 (81) Finite embeddability in a class of infi.nitary algebras. Professor ALLAN B. CRUSE, University of San Francisco, and Professor M. F. NEFF*, Emory University (717 -A31) 11:15-11:25 (82) The existence of a normal basis in certain Galois extensions of commutative rings. Preliminary report. Professor H. F. KREIMER and Mr. PAUL M. COOK II*, Florida State University (717 -A32) 11:30-11:40 (83) Products of differential schemes. Preliminary report. Professor WILLIAM F. KEIGHER, University of Tennessee (717-A33)

0. G. Harrold, Jr. Associate Secretary

Tallahassee, Florida

PRESENTORS OF TEN-MINUTE PAPERS Following each name is the number corresponding to the speaker's position on the program • Invited one-hour lectures

Aull, C. E, #16 Heisey, Richard E. #74 Quackenbush, Robert #18 Bass, Charles D. #72 Holmes, J. P. #26 Rajagopalan, M. #17 Beard, Jacob T. B., Jr. #39 Howard, Fredric T. #23 Randtke, Daniel J. #27 Beem, R. Paul #76 Hudson, Richard H. #44 Rankin, S, M. III #54 Belinfante, Johan G. F. #60 Jacob, R. T., Jr. #52 Reddien, G. W. #8 Berndt, Bruce C. #45 Janowitz, M. F. #31 Reed, George M. #69 Brawley, J. V. #6 Kalman, John A. #33 Reneke, James A. #30 Brown, Ezra #43 Keigher, William F. #83 • Retherford, James #46 Buckley, James J, #61 Kellum, Kenneth R. #75 Roselle, David #5 Burke, Victor #59 Knowles, Robert #29 Roulier, John A. #12 Butler, Kim Ki-Hang #2 LaMartin, W. F. #77 Rousseau, C. C. #48 Carlitz, L, #3 Lameier, S. H. #55 Saff, E. B. #9 Cook, Paul M. II #82 Latorre, D. R. #34 Sanerib, Richard A., Jr. #35 Daverman, Robert J. #73 Lindgren, W, F. #13 Saxon, StephanA, #64 Demko, Stephen #7 Lo, Siu Kwong #24 Scoville, Richard #47 Deveney, James K. #78 Lovelady, David Lowell #63 Shonkwiler, R. #62 • Evans, Trevor #38 Lucas, Thomas R. #10 Taylor, Walter #21 Farmer, K, Bolling #19 McDonald, Bernard R. #4 Tefteller, S. C. #57 Fulton, John D. #49 Madison, B. L. #70 Travis, C. C. #53 • Gardner, Robert B. #1 Martinez, Jorge #22 Trotter, William T., Jr. #37 Gibson, Peter M. #20 Miller, James Ernest #56 Tucker, David #80 Gilmer, Robert #79 Miller, Maurice Hugh, Jr. #15 Varma, A, K. #11 Gonzalez, Mario 0. #58 Morris, S. Brent #50 Vaughan, Jerry E. #66 Gould, H. W. #51 Neff, M. F. #81 Wagner, Carl G. #41 Green, William L. #28 Nelson, Evelyn #32 Wall, Charles R. #40 Gruenhage, Gary #67 Pall, Gordon #42 Wicke, Howard H. #65 Hall, Japheth, Jr. #36 Phillips, Thomas M. #71 Zenor, Phillip #68 Heath, R. W. #14 Pomerance, Carl #25

317 The Seven Hundred Eighteenth Meeting University of Southern California Los Angeles, California November 23,1974

The seven hundred eighteenth meeting of HOLIDAY INN the American Mathematical Society will be held 1020 South Figueroa Street (90007) at the University of Southern California in Los Phone: (213) 748-1291 Angeles, California, on Saturday, November 23 Single $16,00 1974. ' Double $20,00 By invitation of the Committee to Select Hour Speakers for Far Western Sectional Meet­ LOS ANGELES HILTON HOTEL ings, there will be two invited hour addresses, 930 Wilshire Boulevard (90017) Professor C. Edmund Burgess of the University Phone: (213) 629-4321 of Utah will lecture on "Embeddings of surfaces Single $15, 00 in Euclidean three-space" at 11:00 a.m.; Profes­ Double $20.00 sor Paul R. Chernoff of the University of Cali­ OLYMPIAN MOTOR HOTEL fornia, Berkeley, will lecture at 3:30p.m. The 1903 West Olympic Boulevard (90006) title of his talk is "Quasi-analytic vectors and Phone: (213) 385-7141 quasi-analytic functions. 11 Both addresses will Single $13. 00 be given in the auditorium in Olin Hall of Engi­ Double $17,00 neering, Room 122. There will be sessions for contributed VAGABOND MOTOR HOTEL papers on Saturday, All sessions will be held in 3101 South Figueroa Street (90007) Olin Hall of Engineering. Blackboard space and Phone: (213) 746-1531 overhead projectors will both be available. Late Single $14. 00 papers will be accepted for presentation at the Double $19,00 meeting, but they will not be listed in the printed program. The noon meal can be taken on campus at Professor William A. Harris, Jr. of the the Commons and Faculty Club. Restaurants University of Southern California is organizing near campus are Carl's and Julie's. a special session of twenty-minute papers on The campus is located between Exposition Analytic Theory of Ordinary Differential Equa­ Boulevard and Jefferson Boulevard and between tions, These sessions will be held on Saturday Vermont Avenue and Figueroa Street. It can be morning and afternoon. The speakers will in­ reached via the Santa Monica Freeway or the clude Ivar Bakken, Louis J. Grimm, William A. Harbor Freeway. Persons driving to the meet­ Harris, Jr., FrederickA. Howes, James S. ing via the Harbor Freeway should enter the Muldowney, Robert E. 0 1Malley, Jr., Yasutaka campus at the main entrance on Exposition Sibuya, and Gilbert stengle. Boulevard and drive about three blocks on The registration desk will be located in the Hoover Street to Parking Lot K. The parking lounge of Olin Hall of Engineering. Registration fee is 75~ per day. Campus guards will be on hours will be from 8:30a.m. to 12:00 noon and duty if further help is needed, A football game from 1:00 p,m, to 2:30p.m. is scheduled near the campus at 1:15 p, m. so The following hotels and motels are located that the campus may be difficult to approach be­ in Los Angeles, The Vagabond Motor Hotel is tween 11:00 a.m. and 2:00p.m. the nearest to the campus; it is about a twenty­ Limousine service (Airportransit) is avail­ minute walk away. Reservations should be made able from Los Angeles International Airport to directly with the hotel or motel, The rates the Los Angeles Hilton Hotel. To get from the listed are special USC discounts. To obtain hotel to the campus, take a taxi or take bus #9 them, state that you are attending a conference from 7th and Grand to Jefferson and Hoover at USC and that you would like the USC rate. Street. Alternatively, one can take bus #51 from the airport to Jefferson and Hill and then AMBASSADOR HOTEL take bus #18 west on Jefferson to the campus. 3400 Wilshire Boulevard (90010) To reach the campus by car from the airport, Phone: (213) 387-7011 take Century Boulevard to the Harbor Freeway Single $15. 00 and proceed north on Harbor Freeway to Double $21,00 Exposition Boulevard.

318 PROGRAM OF THE SESSIONS

The time limit for each contributed paper in the general sessions is ten minutes and in the special sessions is twen­ ty minutes. To maintain this schedule, the time limits will be strictly enforced.

SATURDAY, 9:00A.M.

Special Session on Analytic Theory of Ordinary Differential Equations I, TV Studio B, Olin Hall of Engineering 9:00- 9:20 (1) A unified theory of asymptotic integration. Professor WILLIAM A. HARRIS, Jr.,* University of Southern California, Los Angeles and Professor DONALD A. LUTZ, University of Wisconsin-Milwaukee (718-B13) 9:30- 9:50 (2) An inequality of Caplygin and Polya. Dr. JAMES S. MULDOWNEY, University of Alberta (718-B3) 10:00-10:20 (3) Solvability of singular differential systems. Professor LOUIS J. GRIMM, Uni­ versity of Missouri-Rolla (718-B1) 10:30-10:50 (4) A nonlinear scattering problem. Professor GILBERT STENGLE*, Lehigh Uni­ versity and Professor JACK NARAYAN, State University of New York, College at Oswego (718-B5)

SATURDAY, 9:15A.M.

Session on Topology, TV Studio C, Olin Hall of Engineering 9:15- 9:25 (5) A van Kampen theorem. Preliminary report. Professor STEVEN C. ALTHOEN, Hofstra University (718-G7) 9:30- 9:40 (6) On convergence groups. Preliminary report. Dr. ROMAN FRIC, Zilina, Czechoslovakia (718-G4) (Introduced by Professor Edwin Hewitt) 9:45- 9:55 (7) Super-Cauchy sequences. Professor RAYMOND KILLGROVE, California State College, Bakersfield (718-G5) 10:00-10:10 (8) A note on Jones' function K. Mr. JOHNS. ROSASCO, University of California, Los Angeles (718-G2) (Introduced by Dr. C. L. Hagopian) 10:15-10:25 (9) Fixed point theorems for products and hyperspaces. Dr. CHARLES L. HAGOPIAN, California State University, Sacramento (718-G1) 10:30-10:40 (10) Monotone decompositions of continua. Professor ELDON VOUGHT, California State University, Chico (718-G6)

SATURDAY, 11:00 A.M.

Invited Address, Room 122, Olin Hall of Engineering (11) Embeddings of surfaces in Euclidean three-space. Professor c. E. BURGESS, University of Utah (718-G3)

SATURDAY, 1:15 P.M.

General Session, TV Studio C, Olin Hall of Engineering 1:15- 1:25 (12) Manifolds carrying bounded quasiharmonic but no bounded harmonic functions. Dr. LUNG OCK CHUNG, University of California, Los Angeles (718-Bll) (Introduced by Professor Leo Sario) 1:30- 1:40 (13) Stiff stability and its relation to Ao and A(O)-stability. Dr. ROLF JELTSCH, University of California, Los Angeles (718-C1) 1:45- 1:55 (14) The Cramer-Petrov large deviation theorem for triangular arrays. Prelimi­ nary report. Professor STEPHEN A. BOOK, California State College, Dominguez Hills (718-F1) 2:00- 2:10 (15) An index for convergence of sums of independent random variables. Prelim­ inary report. Mr. PATRICK L. BROCKETT, University of California, Irvine (718-F2) (Introduced by Professor Howard G. Tucker.)

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

319 2:15- 2:25 (16) Interpolation of localized Lipschitz spaces of functions with applications to multipliers. Dr. WILLIAM C. CONNETT and Dr. ALAN L. SCHWARTZ*, University of Missouri-st. Louis (718-B9) 2:30- 2:40 (17) H*-algebra valued measure on a locally compact space, Professor PARFENY P. SAWOROTNOW, Catholic University of America (718-B8) 2:45- 2:55 (18) Common fixed points of two multi-valued mappings. Preliminary report. Pro­ fessor L. S. DUBE, Sir George Williams University (718-B10) 3:00- 3:10 (19) Williamson type matrices of order 2q(q+1). Professor ALBERT LEON WIDTE­ MAN, University of Southern California (718-A1) 3:15- 3:25 (20) Forcing with tagged trees. Preliminary report. Mr. JOHN STEEL, University of California, Berkeley (718-E1)

SATURDAY, 1:45 P.M.

Special Session on Analytic Theory of Ordinary Differential Equations II, TV Studio B, Olin Hall of Engineering 1:45- 2:05 (21) A multiparameter eigenvalue problem in the complex plane. Preliminary re­ port. Dr. IVARBAKKEN, University ofMinnesota (718-B7) 2:10- 2:30 (22) Transitional problems in nonlinear singular perturbation theory. Preliminary report. Dr. FREDERICK A. HOWES, Courant Institute, New York University (718-B4) 2:35- 2:55 (23) Phase plane solutions to some singular perturbation problems. Professor R. E. O'MALLEY, Jr. , University of Arizona (718-B2) 3:00- 3:20 (24) On the problem of uniform simplification in a full neighborhood of a transition point. Preliminary report. Professor YASUTAKA SIBUYA, University of Minnesota (718-B6)

SATURDAY, 3:30P.M.

Invited Address, Room 122, Olin Hall of Engineering (25) Quasi-analytic vectors and quasi-analytic functions. Professor PAUL R. CHERNOFF, University of California, Berkeley (718-B12) Kenneth A. Ross Eugene, Oregon Associate Secretary

320 The Seven Hundred Nineteenth Meeting University of Houston Houston, Texas November 23,1974

The seven hundred nineteenth meeting of REGISTRATION the American Mathematical Society will be held The registration desk will be located in the at the University of Houston, Houston, Texas, lobby of the Continuing Education Center. It will on Saturday, November 23, 1974. All sessions be open from 9:00 a, m. to 4:00 p.m. on Friday will be held in the Continuing Education Center and from 8:00a.m. to 3:00p.m. on Saturday. of the university. The registration fee for the meeting will be $2. By invitation of the Committee to Select Hour Speakers for Western Sectional Meetings, ACCOMMODATIONS there will be two one-hour addresses. Professor Seymour V. Parter of the University of Wiscon­ The Continuing Education Center is locat­ sin will address the Society at 11:00 a.m.; his ed on the University of Houston campus in the topic will be Differential Equations with "Turning Conrad Hilton School of Hotel and Restaurant Points" and Numerical Methods. Professor Management. The School maintains an 80 room William A. Veech of Rice University will give an hotel for guests. Persons wishing to stay at the hour talk at 1:45 p.m. on the subject Topological School hotel should make their own reservations Dynamics. by writing: By invitation of the same committee there University of Houston Hotel will be two special sessions of selected twenty­ minute papers. Professor E. Ward Cheney of the Attention: Reservations Desk University of Texas has organized a special ses­ 4800 Calhoun sion on Approximation Theory; the speakers will Houston, Texas 77004 Telephone: (713) be Bill D. Anderson, Hermann G. Burchard, 741-2447 Charles K. Chui, Frank R. Deutsch, Oved The Ramada Inn at 3815 Gulf Freeway, Shisha, Philip W. Smith, Arun Kumar Varma, (800) 228-2828 (toll free), has reserved a block and Daniel E. Wulbert. Professor Richard D. of rooms for the meeting. It is located at the Sinkhorn of the University of Houston has organ­ Cullen exit on the Gulf Freeway (IH45) and is ized a special session on Matrix Theory; the about 6 blocks from the campus. speakers will be Peter M. Gibson, Darald J. There are four classes of rates for guest Hartfiel, Emilie V. Haynsworth, Mark Hedrick, rooms at the Continuing Education Center: Marvin Marcus, William L. Morris, Robert J. Standard (60) $14 single, $18 double Plemmons, and Tom E. Salazar. There will also Executive (10) $20 single, $25 double be sessions for contributed papers both morning Suites (5) $28 single, $32 double and afternoon. Presidential Suite $50 On Friday, November 22, 1974, the day before the meeting itself, the University of There is a $4 charge for each additional person Houston will sponsor a Symposium on Pure and in a double room. Food service is available in Applied Mathematics in Memory of Pasquale the Center's public dining room. Porcelli, formerly Professor of Mathematics at TRAVEL Louisiana State University. This symposium will include a fifty-minute talk by J. S, Mac Nerney of Persons coming to Houston by air are the University of Houston on "Iterations of non­ advised that some lines (Braniff, Texas Inter­ linear partial differential equations", a fifty­ national, and Southwest, at this writing) have minute talk by J. W. Neuberger of Emory Uni­ service to Hobby Airport, which is much closer versity on "The second dual of a space of nmc­ to the campus than Houston International. The tions of bounded variation", and twenty-five mi­ University of Houston is located off the Gulf nute talks by Robert M. Brooks, HeronS. Freeway (IH45) approximately 3 miles south of Collins, Richard M. Crownover, Richard B. downtown Houston. The Continuing Education Darst, James R. Dorroh, Ronald G. Douglas, Center is easily found by entering the campus at Charles F. Dunkl, John A. Dyer, Kenneth 0. entrance 1 on Calhoun Street. The Continuing Leland, Donald E. Ramirez, James R. Rether­ Education Center has underground parking ford, and Joseph L. Taylor. facilities for those attending the meeting.

321 PROGRAM OF THE SESSIONS

The time limits for each contributed paper in the general sessions is ten minutes and in the special sessions is twenty minutes. To maintain this schedule, the time limits will be strictly enforced.

SATURDAY, 8:15 A. M.

Session on Algebra, Topology, and Geometry, Room 185, Uranus Room 8:15- 8:25 (1) The group structure determined by permutations derived from card shuffling. Dr. S. BRENT MORRIS, Duke University (719-A6) 8:30- 8:40 (2) Nonnegative idempotent elements in a partially ordered linear algebra. Prelimi­ nary report. Professor RALPH DeMARR, University of New Mexico (719-A5) 8:45- 8:55 (3) Unitary post algebras. Preliminary report. Professor R. ARTHUR KNOEBEL, New Mexico State University (719-A16) 9:00- 9:10 (4) On the endomorphism monoids of complete Boolean algebras. Mrs. PONNAMMAL NATARAJAN* and Dr. CARLTON J. MAXSON, Texas A&M University (719-A9) 9:15- 9:25 (5) A representation theorem for multi-algebras and the axiom of choice. Professor HARTMUT HOFT* and Professor PAUL E. HOWARD, Eastern Michigan Uni­ versity (719-A14) 9:30- 9:40 (6) Group algebras and other constructions over near-rings. Preliminary report. Professor HENRY E. HEATHERLY, University of Southwestern Louisiana (719-All) 9:45- 9:55 (7) Congruences on certain subsemigroups of Baer-Levi semigroups. Preliminary report. Ms. DIANA LINDSEY* and Professor BERNARD MADISON, Louisiana State University (719-A10) 10:00-10:10 (8) A convenient category for [topological] algebraists. Preliminary report. Pro­ fessor J. H. CARRUTH, University of Tennessee (719-A15) 10:15-10:25 (9) Semilattices on Peano continua. Professor W. WILEY WILLIAMS, University of Louisville (719-A13) 10:30-10:40 (10) On separating open covers in Moore spaces. Professor GEORGE M. REED, Ohio University (719-G1) 10:45-10:55 (11) The Gauss-Codazzi equations. Preliminary report. Professor HOWARD J. JACOBOWITZ, Rice University (719-D1)

SATURDAY 8:15 A. M.

Session on Applied Mathematics, Room 183, Saturn Room 8:15- 8:25 (12) Necessary condition for a noncollision singularity in the four body problem. Preliminary report. Mr. ROBERT 0. SHELTON, Rice University (719-C3) 8:30- 8:40 (13) Automata and solvability. Preliminary report. Professor MARGARET M. LaSALLE, University of Southwestern Louisiana (719-C1) 8:45- 8:55 (14) An extension of linear programming duality to piecewise linear maps. Prelimi­ nary report. Dr. ROGERT BLEIER, University of Texas at Austin (719-C2)

SATURDAY 9:00 A. M.

Special Session on Matrix Theory, Room 183, Saturn Room 9:00- 9:20 (15) Regular splittings and the discrete Neumann problem. Professor R. J. PLEMMONS, University of Tennessee (719-A 7) 9:30- 9:50 (16) Matrix factorizations and their associated condition number. Professor WILLIAM L. MORRIS, University of Houston (719-A1) 10:00-10:20 (17) Products of symmetric doubly stochastic matrices. Professor PETER M. GIBSON, University of Alabama in Huntsville (719-A2) 10:30-10:50 (18) The permanent at a minimum on certain classes of doubly stochastic matrices. Dr. MARK BLONDEAU HEDRICK, Pasadena, Texas (719-A17) (Introduced by Professor Richard Sinkhorn)

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

322 SATURDAY 9:00 A. M. Special Session on Approximation Theory, Room 181, Jupiter Room 9:00- 9:20 (19) Extensions and approximations in function spaces. Preliminary report. Dr. B. D. ANDERSON, East Texas State University (719-B9) 9:30-9:50 (20) Degree of convergence of piecewise polynomial approximation on optimal meshes. II. Professor H. G. BURCHARD, Oklahoma State University (719-Bll) 10:00-10:20 (21) Some recent results on Pade approximation. Professor CHARLE-S K. CHUI, Texas A&M University (719-B8) 10:30-10:50 (22) Existence and continuity of best approximations. Professor FRANK DEUTSCH, Pennsylvania State University (719-B7) SATURDAY, 11:00 A.M. Invited Address, Room 261, Constellation Room (23) Differential equations with "Turning Points" and numerical methods. Professor SEYMOUR V. PARTER, University of Wisconsin (719-B15) SATURDAY, 1:45 P. M. Invited Address, Room 261, Constellation Room (24) Topological dynamics. Professor WILLIAM A. VEECH, Rice University (719-H1) SATURDAY, 3:00 P. M. Special Session on Matrix Theory, Room 183, Saturn Room 3:00- 3:20 (25) Results on measures of irreducibility and full indecomposability. Professor D.J. HARTFIEL, Texas A&M University (719-A4) 3:30- 3:50 (26) Aspects of the numerical range. Professor MARVIN MARCUS, University of California, Santa Barbara (719-A3) 4:00- 4:20 (27) Extreme operators on polyhedral cones. Preliminary report. Professor MffiOSLAV FIEDLER and Professor VLASTIMIL PTAK, CSAV Prague, Czechoslovakia, and Professor EMILIE HAYNSWORTH*, Auburn University (719-A8) 4:30- 4:50 (28) An abstract model for matrix theory. Column model. Professor RALPH DeMARR and Mr. TOM E. SALAZAR*, University of New Mexico (719-A12) SATURDAY, 3:00 P. M. Special Session on Approximation Theory, Room 181, Jupiter Room 3:00- 3:20 (29) Characterization of best relative approximation. Professor A. BACOPOULOS, University of Montreal, Professor 0. SIDSHA*, University of Rhode Island, and Professor G. D. TAYLOR, Colorado State University (719-B14) 3:30- 3:50 (30) Nonlinear approximation theory. Preliminary report. Professor PIDLIP W. SMITH, Texas A&M University (719-B10) 4:00- 4:20 (31) Korovkin approximations. Professor D. E. WULBERT, University of California at San Diego (719-B6) 4:30- 4:50 (32) A new proof of Timan approximation theorem. II. Dr. A. K. VARMA, University of Florida (719-B12) SATURDAY, 3:00 P. M.

Session on Analysis, Room 185, Uranus Room 3:00- 3:10 (33) Local evolution systems in general Banach spaces. Preliminary report. Dr. ALBAN ROQUES, Louisiana State University, Baton Rouge (719-B13) 3:15- 3:25 (34) Finitely additive set functions. II. Linear operations on a space of functions of bounded variation. Preliminary report. Professor J. S. Mac NERNEY, Uni­ veristy of Houston (719-B1) 3:30- 3:40 (35) Nonlinear integral equations and product integrals. Mr. ALVIN J. KAY, Texas A&I University (719-B2) 3:45- 3:55 (36) A linear Riemann-Stieltjes integral equation system. Preliminary report. Dr. WILLIAM F. DENNY, University of Oklahoma (719-B3) 4:00- 4:10 (37) The solution of a Volterra integral equation for rings. Professor BURRELL W. HELTON, Southwest Texas State University (719-B5) 4:15- 4:25 (38) A correspondence associated with Stieltjes-Volterra integral equations. Prelimi­ nary report. Dr. WILLIAM L. GIBSON, Houston, Texas (719-B4) Urbana, Illinois Paul T. Bateman Associate Secretary

323 PRELIMINARY ANNOUNCEMENTS OF MEETINGS The Eighty-First Annual Meeting Shoreham Hotel Washington, D. C. January 21-26,1975 SHORT COURSE ON MATHEMATICS IN OPERATIONS RESEARCH, January 21 and 22 On the recommendation of its Committee on Fleming, Calvin C. Moore, RichardS. Palais, Employment and Educational Policy, the Ameri­ and Martha Kathleen Smith. can Mathematical Society will present a one and The program will consist of lectures by the one-half day Short Course on Mathematics in following six distinguished mathematical scien­ Operations Research on Tuesday and Wednesday, tists: Dr. Ralph E. Gomory, Vice President and January 21 and 22, in the Palladian Room of the Director of Research, IBM Corporation, "Stock Shoreham Hotel. The course is designed to iden­ cutting and its ramifications: Mathematical op­ tify opportunities for mathematicians in opera­ erations research in industrY''; Dr. Carl M. tions research, both in employment and in intel­ Harris, Program Director, RMC Research Cor­ lectual content; to give substantial introductions poration (Professor on leave from George Wash­ to several of the important areas of operations ington University), "Queueing theory and applica­ research; and to introduce some of the mathe­ tions: Some mathematical frontiers"; Professor matically challenging aspects of the subject. This Frank Proschan, Florida State University, short course, which is open to all who wish to "Mathematical theory of reliability, with appli­ participate, will be similar in format to the short cations"; Dr. Gordon Raisbeck, Vice President courses given at the 1973 summer meeting in (and Head of Physical Systems Research), Missoula and the annual meeting in San Francisco Arthur D. Little, Inc., "Mathematicians in the last January. practice of operations research"; Professor The program is under the direction of Dr. Arthur F. Veinott, Jr., Stanford University, Alan J. Goldman, National Bureau of Standards. "Lattice programming and inventory theorY''; The members of the AMS Committee on Employ­ and Dr. Christoph Witzgall, Senior Mathemati­ ment and Educational Policy are Richard D. cian, National Bureau of Standards, "Linear Anderson (chairman), Michael Artin, Wendell H. programming and flows of networks".

ANNUAL MEETING, January 23-26 The eighty-first annual meeting of the reservation form. Tickets will be on sale in the American Mathematical Society will be held at registration area at the meeting; further details the Shoreham Hotel in Washington, D. C., from will be announced in a later issue of these Thursday, January 23, through Sunday, Jan­ cNoficeiJ. uary 26, 1975. The meeting will be held in con­ The AMS Committee on Employment and junction with the annual meeting of the Associa­ Educational Policy is planning a panel discussion tion for Symbolic Logic (January 23-24) and the on Friday evening, January 24, at 8:30p.m. annual meeting of the Mathematical Association Professor Martha Kathleen Smith of the Univer­ of America (January 25-27). The National Coun­ sity of Texas, Austin, will serve as moderator; cil of Teachers of Mathematics will meet jointly the topic to be discussed is "Seeking employment with MAA on Saturday and Sunday, January 25 outside academia: Views from some who have and 26. The Association for Women in Mathe­ recently succeeded." Names of the additional matics will hold a session on Friday, January 24, members of the panel will be announced later. at 10:00 a.m. The Conference Board of the There will be a discussion of public science Mathematical Sciences will present a panel dis­ policy on Saturday, January 25, at 8:30 p.m. cussion on "Aspects of the uses of statistics" at This subject and the speakers were selected by 2:00p.m. on Saturday, January 25. There will a Society committee appointed by the Executive be a panel discussion sponsored by the AMS-MAA Committee and Board of Trustees. The commit­ Committee on the Training of Graduate students tee members are R. H. Bing (chairman), Paul to Teach, on Sunday evening, January 26, at Halmos, and A. H. Taub. TheDirectorof the Of­ 8:00p.m. fice of Technological Assessment, Emilio Q. Professor Henry L. Alder will retire at the Daddario, and H. Guyford Stever, Director of end of January 1975 as Secretary of the Associa­ the National Science Foundation, will be the tion following fifteen years of service in this speakers. position. A luncheon will be held in his honor on There will be one set of Colloquium Lec­ Saturday, January 25, at 12:00 noon. Partici­ tures which will consist of four one-hour talks. pants planning to attend the luncheon should check Professor H. Jerome Keisler of the University the appropriate space on the preregistration and of Wisconsin, Madison, will present these lee-

324 tures; the title of these talks will be "New direc­ Graph Theory, Professor Frank Harary, Uni­ tions in model theory," versity of Michigan; Interrelations between Com­ The Josiah Willard Gibbs Lecture will be putation and Number Theory, Dr. Morris New­ presented by Professor Fritz John of the Courant man, National Bureau of Standards; Structure Institute of Mathematical Sciences, New York and Representations of Lie Algebras over Gen­ University, on Thursday, January 23, 1975, at eral Fields, Professor George B. Seligman, 8:30p.m. in the Regency Ballroom. He will Yale University; Fourier Integral Operators, speak on 11 A priori bounds, geometrical effects Professor Francois Treves, Rutgers University; and asymptotic behavior. " and Mathematics and Games, Professor The Retiring Presidential Address will be Stanislaw M. illam, University of Colorado. given at 2:15 p.m. on Friday, January 24, by Professor Saunders Mac Lane of the University of Chicago. The title of his lecture will be "To­ pology and logic as sources of algebraic ideas." COUNCIL AND BUSINESS MEETING The Frank Nelson Cole Prize in Algebra The Council will meet on Wednesday, Jan­ and the 1974 Steele Prizes will be awarded at a uary 22, at 2:00 p.m. in the Diplomat Room of session at 3:15 p, m. on Friday, January 24. the Shoreham Hotel. Most of the meeting is open By invitation of the Committee to select to members of the Society as observers. The Hour Speakers for Annual and Summer Meetings, agenda will be posted. The Diplomat Room is lo­ there will be eight hour addresses. They will be cated on the lower lobby level of the hotel. given by Professor Donald W. Anderson, Uni­ The Business Meeting will be held on Fri­ versity of California, San Diego; Professor day, January 24, at 4:30 p, m. in the Regency Donald L. Burkholder, University of Illinois, Ballroom of the Shoreham Hotel which is located Urbana-Champaign; Professor Sigurdur Helgason, one level below the Diplomat Room. The Secre­ Massachusetts Institute of Technology; Professor tary notes the following resolution of the Council. Linda Keen, Graduate School and University Cen­ Each person who attends a Business Meeting of ter, City University of New York; Professor the Society shall be willing and able to identify Haskell P. Rosenthal, University of Illinois, himself as a member of the Society. In further Urbana-Champaign; Professor Rainer Sachs, explanation, it is noted that "each person who is University of California, Berkeley; Professor to vote at a meeting is thereby identifying him­ Wilfried Schmid, Columbia University; and Pro­ self as and claiming to be a member of the fessor Nolan R. Wallach, Rutgers University. American Mathematical Society. " The titles and times of these lectures can be found in the Summary of Activities which follows this announcement. There will be no limit on the number of MEETING PREREGISTRATION contributed ten-minute papers. No provision AND REGISTRATION will be made for late papers. Operations Research short course partici­ pants may register in SPECIAL SESSIONS at the desk the upper lobby level which is located directly adjacent to the Many special sessions of selected papers main lobby of the Shoreham. The desk will be of varying lengths will be held. The subjects of open from 11:00 a.m. to 3:00 p, m. on Tuesday, these special sessions and the names of the January 21, and from 8:30a.m. to 1:00 p.m. on mathematicians arranging them are as follows: Wednesday, January 22. Operator Theory, Professor William B. Arveson, The registration desk for the joint meeting University of California, Berkeley; Topological will be in the same location. The desk will be Dynamics, Professors Joseph Auslander and open from 2:00 p.m. to 8:00 p, m. on Wednesday, Nelson G. Markley, University of Maryland; January 22; from 8:00 a.m. to 5:00p.m. on Probabilistic Analysis, Professor A. T. Thursday, January 23; from 8:00 a.m. to 4:00 Bharucha-Reid, Wayne State University; Set­ p.m. on Friday through Sunday, January 24-26; theoretic and Combinatorial Methods in Topology, and from 8:30 a.m. to 2:30 p.m. on Monday, Professor W. Wistar Comfort, Wesleyan Uni­ January 27. versity; Hyperbolic Conservation Laws, Pro­ Participants who wish to preregister should fessor Constantine M. Dafermos, Brown Uni­ complete the Meeting Preregistration Form found versity; Commutative Algebra, Professor David on the last page of these c}fofice0. Those who pre­ Eisenbud, Brandeis University; Singular Cauchy register will pay a lower registration fee than Problems, Professor Bernard A. Fusaro, those who register at the meetings, as indicated Salisbury State College; Interpolation of Opera­ in the schedule below. Preregistrants will be tors and Applications, Professors John E. able to pick up their badges and programs when Gilbert and George G. Lorentz, University of they arrive at the meeting. Complete instructions Texas, Austin; Analytic Number Theory, Pro­ on procedure for making hotel reservations are fessor Emil Grosswald, Temple University; given in the section entitled ACCOMMODATIONS.

325 Checks for the preregistration fee should the session in order to become familiar with be mailed to arrive in Providence not later the details in advance. An informal question than December 20, 1974. It is necessary to and answer session will follow the meeting. complete the Meeting Pre:r:e_gistration Form This orientation session is designed to increase on the last page of these cNotJuiJ to take ad­ the efficiency of the Register and make it run vantage of the lower meeting registration more smoothly. fee even though the services of the housing bureau are not required. EXillBITS The book and educational media exhibits .Please note that separate registration is will be displayed in the Ambassador Room required for the short course and the joint meet­ of the Shoreham from Thursday through ings. Registration fees for the meetings Sunday, Jan­ are as uary 23-26. The exhibits will follows: be displayed from Preregistration At meeting noon to 5:00 p.m. on Thursday; from 9:00 a.m. (by mail prior to 5:00 p.m. on Friday, Saturday, and Sunday. to 12/20/74) All participants are encouraged to visit the ex­ hibits during the meeting. The Ambassador Operations Research Room is located adjacent to the Regency Ball­ Short Course room. All participants $ 9 $12 BOOK AND AUDIO TAPES SALE Joint Mathematics Books published by the Society and Meetings the As­ sociation and audio tapes of invited addresses Member 10 12 will be sold for cash prices somewhat below the Student or unemployed usual prices when these same books and tapes member 1 2 are sold by mail. Nonmember 16 20 One-day fee (Monday, January 27 only) $4 ACCOMMODATIONS There will be no extra charge for mem­ Forms for requesting accommodations will bers of the families of registered participants. be found on the last two pages of these c}/oticei). The unemployed status refers to any mem­ Please note that there are two separate and dis­ ber currently unemployed and actively seeking tinct preregistration forms-one for students and employment. It is not intended to include mem­ unemployed members who would like to reserve bers who have voluntarily resigned or retired accommodations in a room with triple occupancy from their latest position. and a second form for regular preregistration Students are considered to be only those and reservations. currently working toward a degree who do not The use of the housing services offered by receive an annual compensation totaling more the Washington Convention and Visitors Bureau than $7,000 from employment, fellowships, requires preregistration for the meeting. Per­ and scholarships. sons desiring accommodations should complete A fifty percent refund of the preregistra­ the appropriate form (or a reasonable facsimile) tion fee will be reimbursed for all cancellations and send it to the Mathematics Meetings Housing received prior to January 21. There will be no Bureau, P. 0. Box 6887, Providence, Rhode refunds granted for cancellations received after Island 02940, Reservations will be made in ac­ that date or to persons who do not attend the cordance with preferences indicated on the res­ meetings. ervation form, insofar as this is possible, and all reservations will be confirmed. Deposit EMPLOYMENT REGISTER re­ quirements vary from hotel to hotel, and par­ The Mathematical Sciences Employment ticipants will be informed of any such require­ Register will be maintained from 9:00 a m to ments at the time of confirmation. REQUESTS 4:00 p.m. on Friday, January 24, with fnt~r­ FOR RESERVATIONS SHOULD BE MAILED TO views scheduled from 9:00 a.m. to 5:40 p.m. on ARRIVE IN PROVIDENCE NO LATER THAN Saturday, Sunday, and Monday, January 25-27. DECEMBER 20, 1974. The addition of a third day of interviews contin­ BARBIZON TERRACE ues a previously adopted procedure recommended Singles $16 by the Joint Committee on Employment Oppor­ Twins 18 tunities in an attempt to expand the interview Extra person in room 4,50 schedule and to eliminate the necessity for eve­ ning interviews. The Register will be located in BRIGHTON HOTEL the Empire Room of the Shoreham Hotel which is Singles $12 adjacent to the lower lobby reception area. Doubles or twins 16 A short meeting of persons planning to Suites 24 participate in the Employment Register has Extra person in room 4 been called by the committee for Thursday, SHERATON-PARK January 23, at 4:30 p.m. in The Forum of the Singles $22,24,26,28,30 Shoreham Hotel. The purpose of this meeting Doubles or twins 26,28,30,32,34 is to explain Register procedures and the *Triples 27 various printed forms which are used. Both Suites 75 and up applicants and employers are urged to attend Extra person in room 9 *The triples are reserved for students and unemployed members; see special reservation form which may be found on penultimate page of these c){oticei) for criteria governing eligibility for these rooms.

326 SHOREHAM HOTEL (1) The U. S. Capitol. Capitol Hill at east Singles $20,21,22,24,26,28 end of the Mall. Visitor hours 9:00 a.m. to 4:30 Doubles or twins 26,27,28,30,32 p.m. ; guided tours leave from the upstairs ro­ *Triples 27 tunda daily every fifteen minutes from 9:00 a.m. Suites 60,70,80,90 to 3:45p.m. The tour takes 45 minutes. The Extra person in room 6 charge is 25~ for adults; children are free. (2) The White House. 1600 Pennsylvania WASHINGTON HILTON Avenue, N. W. Visitor hours 10:00 a.m. to noon Singles $23,25,27 Tuesday through Saturday. Doubles or twins 28,30,32 (3) Jefferson Memorial. South side of *Triples 33 Tidal Basin. Visitor hours 8:00a.m. to mid­ night. YMCA (4) The Washington Monument. For 10~ Singles or twins $ 5. 25 you can ride to the top in one minute. Visitor hours daily N. B. Space is available to accommodate only 35 9:00 a.m. to 5:00 p.m. (5) The Lincoln Memorial. male and 15 female participants; reservations Open around the clock. Fifteen will be accepted on a first-come-first-served minute tours are conducted at random daily by National Park service guides. basis. Reservations must be accompanied by a (6) The John F. Kennedy Center separate check for $5. 25 (made payable to YMCA for the Performing Arts. At Rock Creek of Metropolitan Washington), which will cover Park and New Hampshire the first night•s lodging, and sent to the Housing Avenue. The Center contains an Opera House, Concert Hall, Eisenhower Bureau in Providence along with the appropriate Theater preregistration form and fee. (for live drama), a Film Theater, an Exhibition Hall and three restaurants. It is worth a sight­ YWCA (Strong Residence) seei;g trip even if you do not attend one of the 101117th street, N. W. performances. The building is open for tours Washington, D. C. 20036 from 10:00 a.m. to 5:00 p.m. daily. (7) National Archives. 7th and 9th Streets, strong Residence (for women only) has N. W. Open 9:00a.m. to 10:00 p.m. weekdays limited accommodations. YWCA will not quote and holidays; 1:00 p.m. to 10:00 p.m. on Sun­ rates for 1975 until the latter part of November. days. Singles are currently $8. Reservations will be (8) The National Gallery of Art. Offers taken by mail starting in December; write general tours which convene in the rotunda at directly. 11:00 a.m., 3:00p.m., and 5:00p.m. Monday It should be noted that all rooms in the through Saturday, and at 5:00 p.m. on Sunday. Shoreham Hotel have two beds. Participants, who Special tours and concerts are also scheduled. are able to do so, are urged to share a room The Gallery is open to visitors from 10:00 a.m. whenever possible. This procedure will be eco­ to 5:00 p.m. Monday through Saturday, and nomically beneficial. For instance, two partici­ from 2:00 p.m. to 10:00 p.m. on Sunday. pants sharing a double or twin room will each (9) The Museum of History and Technology. pay only $13, $13. 50, $14, $15, or $16 (depend­ The newest of the Smithsonian buildings, 14th ing on category of room) daily as compared with and Constitution on the north side of the Mall. the single room rates of $20, $21, $22, $24, Visitor hours are 9:00 a.m. to 5:00 p.m. daily. $26, and $28. Lines 12 and 13 of the Room Res­ (10) Museum of Arts and Industries. South­ ervation Form should be completed to ensure side of Mall, Independence at 9th st., S. W. proper assignment of rooms. Each participant Visitor hours 9:00a.m. to 5:30p.m. daily. should complete a separate preregistration form; Apollo 11, Spirit of st. Louis, Wright Brothers it would be helpful (but it is not essential) to re­ plane on display. ceive the forms in Providence at the same time. Brochures on these and other attractions in The Shoreham Hotel has two dining rooms: the Washington, D. C. area will be available at The International Snack Bar (a coffee shop) which the Information Desk. It is anticipated that sev­ is open daily from 7:00a.m. to 11:00 p.m.; and eral tours will be organized for registrants at The Greenery (a more formal restaurant serving the Mathematics Meetings. liquor) which is open for breakfast, lunch, and dinner from 7:00 a.m. to 11:00 p.m. every day. TRAVEL AND LOCAL INFORMATION NATIONAL SCIENCE FOUNDATION The District of Columbia is served by two airports: Washington National (4 miles from NSF staff members will be available to city center) and Dulles International (26 miles provide counsel and information on NSF pro­ from city center). Allegheny, American, grams of interest to mathematicians from 9:00 Braniff Delta Eastern, National, Northwest, a.m. to 5:00 p.m. on January 24, 25, and 26, Piedmdnt TWA and United Airlines provide in the Committee Room. This room is located service t~ Washington National. Several com­ on the main lobby level of the Shoreham Hotel. muter airlines (i.e., Altair, Pennsylvania, etc.) ENTERTAINMENT also operate service into National. Transfers are available at variable rates and limousine Eighteen million people visit the nation• s service at $2. 50 is available from National capitol each year. The "Top Ten Tourist Attrac­ Airport directly to the hotel. tions" according to the Washington Convention Dulles International Airport is served by and visitors Bureau, are listed below in no American, Braniff, Delta, Eastern, Northwest, particular order.

327 Ozark, Pan American, Piedmont, Southern, inches of rain. Medium to heavy clothing is re­ TWA, United, and Viasa Airlines, as well as commended. Colgan Airway, a commuter line. Metered taxi service to the hotel will cost approximately $20, MAIL AND MESSAGE CENTER plus fifty cents for each additional passenger. All mail and telegrams for persons at­ Limousine-bus service for $3. 75 is available tending the meetings should be addressed in care from Dulles to the Washington Hilton, not quite of Mathematics Meetings, The Shoreham Hotel a mile south of the Shoreham. City buses & Motor Inn, Cal vert Street and Connecticut (Metrobus L2, L4, and L6) run on Connecticut Avenue, N. W., Washington, D. C. 20008. Mail Avenue past the Washington Hilton and the Shore­ and telegrams so addressed may be picked up at ham. the Mail and Information Desk located at the Amtrak service is available from many registration area in the upper lobby of the hotel. points to Union Station. The high-speed Metro­ A message center will be located in the liner runs between New York, Philadelphia and same area to receive incoming calls for all Washington. Metrobus numbers 96 and 98 go members in attendance. Messages may be left from Union Station past the Shoreham. The fare for registrants during the hours the registration anywhere within the city is 40~ in exact change. desk is open, cf. the section entitled MEETING From Exit 20 on the Capitol Beltway PREREGISTRATION AND REGISTRATION, (Interstate 495) south to Calvert Street is approx­ above. Messages will be taken down, and the imately 8-1/2 miles. The Shoreham is located name of any member for whom a message has at 2500 Calvert Street, N. W., near Connecticut been received will be posted until the message Avenue. There is a parking garage within the is picked up at the Message Center. Members hotel. The charge is $2. 50 for each 24-hour are advised to leave the following number with period; there is no extra· charge for in/out ser­ anyone who might want to reach them at the vice. Parking is available in this garage for meeting (202) 797-8356. commuters as well as hotel residents. LOCAL ARRANGEMENTS COMMITTEE WEATHER H. L. Alder (ex officio), Ruth A. Bari, Washington 'Zeathe:b in January varies from James A. Donaldson, Irving I. Glick, Walter brisk to balmy (10 to 60 ) but rareltprevents H. Gottschalk (ex officio), Hewitt Kenyon travel. The average daily high is 44 ; the (chairman), James C. Owings, William Swyter, average daily low is 29°; in January there are Choy-tak Taam, Juanita S. Tolson, and Gordon approximately eleven days with at least . 01 L. Walker (ex officio).

328 III = Shoreham Hotel 111 = Barbizon Terrace IKI = Washington Hilton Hotel l1J = Sheraton-Park Hotel 1!.1 = Brighton Hotel 1&:1 = Y. W.C.A. 121 = Y.M.C.A.

329 SUMMARY OF ACTIVITIES

The AMS Committee to Monitor Problems in Communications has recommended that a Summary of Activities appear in the issue of the c}/oficei) which contains a reservation form for either an annual or a summer meeting. The pur­ pose of this summary is to provide assistance to registrants in the selection of arrival and departure dates. The program, as outlined below, is based on information available at press time.

AMERICAN MATHEMATICAL SOCIETY SHORT COURSE ON MATHEMATICS IN OPERATIONS RESEARCH

TUESDAY, January 21 11:00 a.m. - 3:00 p.m. REGISTRATION 1:45 p.m. lhtroductory remarks Alan J. Goldman 2:00 p.m. - 3:15 p.m. Mathematicians in the practice of operations research Gordon Raisbeck 3:45 p.m. - 5:00 p.m. Linear programming and flows in networks Christoph Witzgall

WEDNESDAY, January 22 8:30 a.m. - 1:00 p.m. REGISTRATION 9:00 a.m. - 10:15 a.m. Stock cutting and its ramifications: Mathematical operations research in industry Ralph E. Gomory 10:45 a.m. - 12:00 noon Mathematical theory of reliability, with applications Frank Proschan 2:00 p.m. - 3:15 p.m. Lattice programming and Inventory theory Arthur F. Veinott, Jr. 3:45 p.m. - 5:00 p.m. Queueing theory and applications: Some mathematical frontiers Carl M. Harris

AMS - MAA ANNUAL MEETINGS

American Mathematical Society Other Organizations

WEDNESDAY, January 22 2:00p.m. AMS Council Meeting I 2:00p.m. - 8:00p.m. REGISTRATION

THURSDAY, January 23 8:00 a.m. - 5:00 p.m. REGISTRATION 9:00 a.m. - 10:00 a.m. INVITED ADDRESS I Lie groups and differential equations (title tentative) Sigurdur Helgason 9:00 a.m. - 11:30 a.m. Sessions of Contributed Papers and Special Sessions 9:00 a.m. - 12:00 noon Association for Symbolic Logic Sessions 10:15 a.m. - 11:15 a.m. INVITED ADDRESS Representations of semislmple Lie groups (title tentative) Wilfried Schmid 12:00 noon - 5:00 p.m. EXHIBITS 1:00 p.m. - 2:00p.m. COLLOQUIUM LECTURE I New directions in model theory H. Jerome Keisler 1:00 p.m. - 6:00 p.m. Sessions of Contributed Papers and Special Sessions 1:30 p.m. - 5:00 p.m. ASL Sessions 2:15 p.m. - 3:15 p.m. INVITED ADDRESS Riemann surfaces and the moduli problem Linda Keen 2:45 p.m. Mathematicians Action Oroup Business Meeting 3:30 p.m. - 4:30 p.m. INVITED ADDRESS 1 Some recent discoveries in the isomorphic theory of Banach spaces I Haskell P. Rosenthal

330 SUMMARY OF ACTIVITIES American Mathematical Society Other Organizations

THURSDAY, January 23 4:30p.m. Open Register orientation session

8:30p.m. JOSIAH WILLARD GIBBS LECTURE A priori bounds, geometrical effects, and asymptotic behavior Fritz John FRIDAY, January 24

8:00 a.m. - 4:00 p.m. REGISTRATION 9:00 a.m. - 10:00 a.m. INVITED ADDRESS A survey of current non-quantum general relativity (title tentative) Rainer K. Sachs 9:00a.m. - 11:30 a.m. Sessions of Contributed Papers and Special Sessions 9:00 a.m. - 12:00 noon ASL Sessions 9:00 a.m. - 4:00 p.m. EMPLOYMENT REGISTER 9:00 a.m. - 4:00 p.m. Mathematical Association of America IBoard of Governors Meeting 9:00 a.m. - 5:00 p.m. EXHIBITS 10:00 a.m. - 12:00 noon Association for Women in Mathematics Session Panel discussion: Action program Lenore Blum Edith H. Luchins Maita Levine Carolyn T. MacDonald 10:15 a.m. - 11:15 a.m. INVITED ADDRESS A representation theoretic proof of a formula of Max Noether Nolan R. Wallach 1:00 p.m. - 2:00p.m. COLLOQUIUM LECTURE II New directions in model theory H. Jerome Keisler 1:30 p.m. - 5:00 p.m. ASL Sessions 2i15 p.m. - 3:15 p.m. RETIRING PRESIDENTIAL ADDRESS Topology and logic as sources of algebraic ideas Saunders Mac Lane 3:15 p.m. - 4:30 p.m. Cole Prize Session Steele Prize Session 4:30 p.m. AMS Business Meeting 7:00 p.m. - 10:20 p.m. MAA - Film Program Allendoerfer Films 7:00p.m. - 7:25p.m. Gauss-Bonnet theorem 7:26 p.m. - 7:48p.m. Cycloidal curves or tales from the Wanklen­ burg Woods 8:00 p.m. - 8:22p.m. A film of the Topology Films Project: How to turn a sphere inside out 8:25p.m. - 8:50p.m. Topology - a B. B. C. Broadcast as part of the Open University Foundation Course in Mathe­ matics 8:55p.m. - 9:10p.m. Mathematics of the honeycomb 9:15 p.m. - 9:45 p.m. Numerical analysis - Excerpts from a video­ taped course produced by Ben Noble at Oberlin College 9:55 p.m. - 10:20 p.m. Statistics at a glance 7:30 p.m. - 9:00 p.m. Panel discussion: Computer oriented supple­ ments to the undergraduate mathematics curriculum (Report of an N.S. F. workshop group) Richard Alo Lyle Mauland Noal Harbertson Joseph Mayne Carl Leinbach Harold Weinstock J. C. Mathews

331 SUMMARY OF ACTIVITIES American Mathematical Society Other Organizations

FRIDAY, January 24 8:30p.m. AMS Committee on Employment and Educa- tional Policy Panel discussion: Seeking employment out­ side academia: Views from some who have recently succeeded Martha Kathleen Smith (moderator)

SATURDAY, January 25

8:00 a.m. - 4:00 p.m. REGISTRATION 9:00 a.m. - 5:00 p.m. EXHIBITS 9:00 a.m. - 5:40 p.m. EMPLOYMENT REGISTER 9:00 a.m. MAA Joint Sessions with National Council of Teachers of Mathematics 9:00 a.m. - 10:00 a.m. A personalized system of instruction B. A. Green, Jr. 10:15 a.m. - 10:45 a.m. The first U. S. participation in the Interna­ tional Mathematical Olympiad S. L. Greitzer 11:00 a.m. - 11:50 a.m. What is an automorphic form? L. J. Goldstein 12:00 noon LUNCHEON - Honoring Henry L. Alder 1:00 p.m. - 2:00p.m. COLLOQUIUM LECTURE ill New directions in model theory H. Jerome Keisler 1:00 p.m. - 6:00 p.m. Sessions of Contributed Papers and Special Sessions 2:15 p.m. - 3:15 p.m. INVITED ADDRESS Applications of homological algebra to topology Donald W. Anderson 2:00 p.m. - 4:00 p.m. Conference Board of the Mathematical Sciences Panel discussion: Aspects of the uses of statistics Joan R. Rosenblatt (moderator) 5:00 p.m. - 6:30 p.m. NQ-HOST COCKTAIL PARTY 7:00 p.m. MAA - Film Program 7:00 p.m. - 8:15 p.m. An application of computer graphics to teach­ ing mathematics: Exhibition of computer­ animated super 8-movies and slide sequences on calculus, statistics, pure and applied analysis, by Professors R. B. Kirchner and R. W. Nau of Carleton College 8:30 p.m. AMS Panel Public science policy R. H. Bing (moderator) Emlllo Q. Daddario H. Guyford Stever

SUNDAY, January 26

8:00 a.m. - 4:00 p.m. REGISTRATION 9:00 a.m. - 5:00 p.m. EXHIBITS 9:00 a.m. - 5:40 p.m. EMPLOYMENT REGISTER 9:00a.m. MAA-NCTM- Joint Session 9:00 a.m. - 9:50 a.m. Panel discussion: Teaching mathematics to the beginning undergraduate C. A. Lathan M. H. Protter Andrew sterrett, Jr. Benjamin Volker A. B. Willcox (moderator) 10:00 a.m. - 11:00 a.m. MAA Business Meeting 1:1:10 a.m. MAA-NCTM- Joint Session 11:10 a.m. - 12:00 noon The Open University in Great Britain­ a progress report R. J. Wilson 332 SUMMARY OF ACTIVITIES American Mathematical Society Other Organizations

SUNDAY, January 26

1:00 p.m. - 2:00 p.m. COLLOQUIUM LECTURE IV New directions in model theory H. Jerome Keisler 1:00 p.m. - 6:00 p.m. Sessions of Contributed Papers and Special Sessions

1:30 p.m. National Association of Mathematicians Panel discussion: Open admissions and the mathematics curriculum Theodore Sykes (moderator) 2:00 p.m. - 6:00 p.m. CBMS - Council Meeting 2:15 p.m. - 3:15 p.m. INVITED ADDRESS Martingale methods in analysis Donald L. Burkholder 7:30 p.m. - 10:30 p.m. CBMS - Council Meeting 8:00 p.m. - 10:00 p.m. AMS-MAA Panel discussion Teaching assistant training programs Anthony L. Peressini G. Thomas Sallee David I. Schneider Philip Treifman

MONDAY, January 27

8:30 a.m. - 2:30 p.m. REGISTRATION 9:00 a.m. - 5:40 p.m. EMPLOYMENT REGISTER 9:00a.m. Sessions of the MAA 9:00 a.m. - 9:50 a.m. The challenges to a discipline in an era of interdisciplinary emphasis L. J. Paige 10:00 a.m. - 10:50 a.m. The National Institute of Education J. M. Mays 11:00 a.m. - 11:50 a.m. Ordering of fields and prime ideals of Witt rings Alex Rosenberg 1:30 p.m. Sessions of the MAA 1:30 p.m. - 2:20 p.m. On measuring things, Eudoxus revisited A. M. Gleason 2:30 p.m. - 3:20 p.m. The early days of probability theory: reminiscences and reflections MarkKac 3:30 p.m. - 4:20 p.m. ! Intuitive geometry is alive and well I Branko Griinbaum '

Walter H. Gottschalk Associate Secretary Middletown, Connecticut

333 Symposium on Some Mathematical Questions in Biology New York, New York January 29-30, 1975

The ninth annual symposium on Some Cambridge, England; Brian Charlesworth, Mathematical Questions in Biology will be held "Natural selection in age-structured popula­ for one and one-half days on January 29-30, tions," University of Sussex, England; Hirsh G. 1975, in the LaLoire Room 2 and 3 of the Ameri­ Cohen, "Mathematical developments in Hodgkin­ cana Hotel in New York City, The symposium Huxley theory and its approximation, 11 T. J. is being held in conjunction with the annual meet­ Watson Research Center, IBM; Simon A. Levin, ing of the American Association for the Advance­ "Spatial heterogeneity in ecological systems," ment of Science, It will be cosponsored by the Cornell University; George F. Oster, "Dynamics American Mathematical Society and the Society of age-structured populations, 11 University of for Industrial and Applied Mathematics. The California, Berkeley; John Rinzel, "Simple support of the National Science Foundation is model equations for active nerve conduction and anticipated. Registration and local arrangements passive neuronal integration, " National Institute were announced in the September 27, 1974, issue of Health, of Science. The third session will be devoted to twenty­ The program has been arranged by the minute short papers selected and refereed in ad­ AM8-SIAM Committee on Mathematics in the Life vance by the committee. There will be no provi­ Sciences, whose members are Hans J. Bremer­ sion for late papers. A complete program of the mann, Jack D. Cowan, Murray Gerstenhaber, sessions will be included in the January 1975 is­ Alston s. Householder, Simon Levin (chairman), sue of these cJ{otiai). and Richard C. Lewontin. The symposium will be divided into three Simon A. Levin, Chairman half-day sessions; the main topics of the sym­ Organizing Committee posium will be ecology and evolutionary biology Ninth Annual AMS-SIAM and neurobiology. There will be six forty­ Symposium on Some Mathe­ minute lectures presented in two sessions by the matical Questions in Biology following: Horace B. Barlow, "Is neurobiology Ithaca, New York totally unmathematical ? 11 , Trinity College,

Symposium on Theory vs. Practice in the Finite Element Method New York, New York January 31, 1975 A symposium on Theory vs. Practice in nite element method is now widely studied math­ the Finite Element Method will be held on the ematically by numerical analysts. The goal of morning of January 31, 1975, at the Americana this symposium is to introduce its audience to Hotel in New York City, in conjunction with the the finite element method from both the engineer­ annual meeting of the American Association for ing and mathematical points of view, and hope­ the Advancement of Science. The symposium fully thereby stimulate new areas of application. will be sponsored by the American Mathematical The speakers and the titles of their lectures Society and will be supported by the Air Force are Professor Todd Dupont, The University of Office of Scientific Research. Registration and Chicago, "Modeling wave propagation with fi- local arrangements were announced in the Sep­ nite element methods"; Dr. Robert E. Nickell, tember 27, 1974, issue of SCIENCE. Sandia Laboratories, "Computer program con­ The program, which will be held in the La struction and maintenance-the future of central­ Loire Rooms 4 and 5 at the Americana Hotel, is ized finite element activitY''; Professor Ridgway being arranged by Professors Ridgway Scott and Scott, The University of Chicago, "Interplay be­ Gilbert Strang (co-chairmen). The finite element tween the mathematical and engineering approach­ method has been applied to such diverse prob­ es"; Professor Gilbert Strang, Massachusetts lems as the study of earthquakes, calculation of Institute of Technology, "Can mathematics be the modes of vibration of the human skull, de­ useful?"; and Professor Robert L. Taylor, Uni­ sign of nuclear reactors and airplanes, and guid­ versity of California, Berkeley, "The impact on ing secondary recovery techniques in oil wells. structural engineering. " First developed by structural engineers, the fi-

334 DOCTORATES AND JOBS, 1974 REPORT by R. D. Anderson*

The data for 1974 employment and for num­ courses, reversing the 2% decrease of a year bers and distribution of doctorates are generally previous, thus some pressure may develop for similar to that of a year ago with the most in­ increasing total faculty size. teresting changes noted below. Using compari­ The bad news is that we continue to head sons with data from the past several years, some rapidly toward great inflexibility of our national trends are now discernible, a number being en­ faculty, a condition where the percentage with couraging but others alarming. tenure or some form of "moral tenure" is very The good news is that, statistically, unem­ high. With a stable total faculty size, there will ployment was not worse than last year, the num­ soon be very few positions other than replace­ ber of new Ph. D. •s in pure mathematics contin­ ments for those retiring or dying, except for ued to decline (generally in all categories of de­ temporary positions in some Ph. D. producing de­ partments), and the percentage of the new doc­ partments. Thus there are limited employment torates who are not U.S. citizens went up mar­ opportunities in academia for those who have had kedly, from 21% a year ago to 28% this year. degrees for four or more years; the plight of From fall 1972 to fall 1973 there was some in­ nonretained people in this category is severe and crease, about 2%, in enrollment in mathematics probably becoming much worse.

TABLE 1 1974-1975 EMPLOYMENT STATUS OF NEW DOCTORATES IN THE MATHEMATICAL SCIENCES

PURE MATHEMATICS

~ ~ p ~ '<:r 0 'b q '<:r CZ> ""<: ~~ ~ fJ Ill ~ :tj §'51 .;:;. '1!§ fJ~ ~'i t>t;; !J '<:r 'II ~i; ';}~!? ~0 §ez> .;:;.e !].;:;. g~ ~'ll ~~ ;§ff 11 fri !l ~ !.~ (!): ~~ i Type of Employer ~~ "<::r! c!~ ~ Ut8 ot:J.~ ~$ $J ~ University 41 69 47 5 6 33 28 2 29 7 267 College 49 52 43 5 6 27 21 2 12 9 8 234 Two-year colleges and high schools 11 6 5 1 1 1 2 1 28 other academic de- partments and re- search institutes 1 3 7 2 1 14 13 7 11 4 63 Government 6 4 2 1 9 5 4 10 2 43 Business and in- dustry 9 14 4 4 15 30 13 18 10 117 Canada 8 12 4 3 2 6 10 3 8 5 61 Foreign 20 23 20 3 5 24 14 6 9 1 3 128 Not seeking employ- ment 1 1 3 1 6 Not yet employed 17 34 21 3 3 9 4 15 1 4 111 Unknown 12 17 14 3 8 8 17 8 3 90

Totals 169 237 172 26 36 146 143 37 121 13 48 1,148

* This report has been prepared by the author on behalf of the AMS Committee on Employment and Educational Policy whose other members are Michael Artin, Wendell H. Fleming, Calvin C. Moore, Richard s. Palais, and Martha K. Smith. The data in the report have been compiled by the AMS staff under the direction of Lincoln K. Durst.

335 This year's table on the Employment Status number, but only six had degrees in pure mathe­ of New Doctorates (Table 1) is based only on the matics. These 50 are grouped with thirteen po­ degrees listed in the October c/{oticti); last year's sitions in research institutions in Table 1. table included degrees listed in both the October (3) The number taking positions in two­ and November issues of cJVoticei) and was there­ year colleges and high schools was twenty-eight, fore more complete. A comparison with last up from nineteen a year ago but still a very small year's figures follows. percentage of the total . .(1) Most of the degree and employment pat­ (4) The percentage of the new doctorates terns were very similar to those of a year ago. going into business and industry was almost ex­ (2) This year a separate count was main­ actly the same as last year, of those going into tained of new Ph. D. 's taking positions in aca­ government was down about 10% and those going demic computer centers, research laboratories, abroad up about 5%. The percentage of those not or teaching in departments other than the mathe­ yet employed was up less than 5%. matical sciences; there were 50, an encouraging

FACULTY FLOW DIAGRAM 1973-1974 to 1974-1975 Full-Time Mathematics Faculty in Four-Year Colleges and Universities in the U.S.

institutions 380 rr·-~ 380 From disciplines other 60? 80 than math. sciences J Deaths and retirements I I

From outside U. S. 70 (visitors?) 60 J Left the U.S. I I OOCTORA TE-HOLDING From business, industry 30 FACUL'rY (12, 350) 110 To business , industry and government in 1973-1974 J I 'L and government I

Ph. D.'s not employed 50? 50 Other, e. g., left math.J full-time in 1973-1974 .I postdoc, ,dean, etc. +350 -, New math. Ph. D.'s Seeking employment not holding the same < r;,.l July, 1974 positions last year 200 I 30 Faculty getting doctorates and not moving (includes some math. ed.) 200 Deaths and retirements I ~ - 10 J Left the U.s. I From teaching assistant to NONDOCTORATE-HOLDING 60? To business, industry instructor (same school) FACULTY (4, 950) 70 J and government J in 1973-1974 I To faculty positions prior 70 Other, e. g., returned to receipt of doctorate 80 J to school -400 I Seeking employment 60 July, 1974 50 50 I To different institutions

336 The Faculty Flow Diagram which includes from fiscal 1968 through fiscal 1973 (i.e. through faculties in statistics and computer science is June 1973). It recorded a slight drop in pure generally quite similar to that of a year ago ex­ mathematics degrees from fiscal 1971 to fiscal cept that there was less mobility among the non­ 1972 and a rather sizable drop (about 10%) from doctorate faculty, and apparently the total num­ fiscal 1972 to fiscall973. The author has count­ ber of retirements and deaths was down from ed the numbers of pure mathematics degrees 240 a year ago to 160 this year. This reduction reported to the AMS by departmental chairmen may be a statistical abnormality or it may rep­ over the past three years. The pure mathematics resent a tendency to postpone retirement in the classification is not clear-cut since some degrees face of current and prospective inflation. As be­ in partial differential equations, numerical anal­ fore, the diagram represents projections to the ysis or probability, for example, could be clas­ total population based on reports from depart­ sified either as pure or applied. However, the ments involving about half the total faculty. The same classification scheme has generally been numbers with question marks are based on in­ used for the three years and thus the compara­ ferential rather than direct information. tive figures are probably reliable. Unfortunately, not all departments report (on time) every year THE REDUCTION OF NUMBERS OF to the AMS; in particular, some lower ranked PURE MATHEMATICS PH.D.'S departments do not report some years. Table 2 shows the number of pure mathe­ As cited in a CEEP report in the October matics doctorates by those departments which 1974 c}/otiui), there has been a reduction in the have reported~ each of the~ three years annual number of pure mathematics degrees in class I (the top twenty-seven ACE ranked over the past three years. That report dealt with mathematics departments), class II (the thirty­ numbers of degrees as compiled by the National eight other ACE rated departments) and class ill Research Council giving numbers of all degrees (the 91 ACE unrated mathematics departments).

TABLE 2 PURE MATHEMATICS DOCTORATES Percentage Drop Fiscal1972 Fiscal1973 Fiscal1974 1972 to 1974 Class I 291 281 219 25% Class II 212 180 169 20% Class ill 158 127 122 23% Total 661 588 510 23%

Jn addition, there were 65 pure mathemat­ per year by 1974. ics doctorates in 1972, 87 in 1973 and 62 in 1974 reported by departments (chiefly in class III) that NUMBERS OF GRADUATE STUDENTS did not report degrees each year. There are There was a drop of about 6% in the total known to be a few other pure mathematics de­ number of graduate students in Ph. D. producing grees that are (so far) not reported to the AMS. mathematics departments from 1972 to 1973 but It is encouraging that the reduction in pure the number in 1974 is expected to remain the mathematics degrees over the past two years has same as that in 1973. The number of first year generally occurred fairly uniformly in all classes students is expected to rise about 2% from the of departments and that the reduction has been so 1973 level after falling about 5% from 1972 to pronounced. Had the numbers not dropped about 1973. The number of teaching assistants in these 180-200 (or 25%) over the past three years, the departments fell 2% a year ago and is expected to present unemployment situation would be much drop another 2% this fall. more severe since almost every employer seek­ Jn master's level departments and in sta­ ing pure mathematics doctorates has found ac­ tistics and computer science departments, the ceptable and available candidates from among number of graduate students, of first year gradu­ those applying. With a prospective ten-year ate students, and of teaching assistants have steady state annual employment demand of per­ generally risen a total of 10% to 20% over the haps 200-300 pure mathematics doctorates (for past two years. long range retention), we should continue to re­ duce the numbers getting degrees until we are CITIZENSHIP OF NEW DOCTORATES much closer to equilibrium. However, as a pro­ Table 3 shows the percentages of U. S. doc­ fession we can take comfort that we have been torate recipients in the mathematical sciences reacting far more quickly than had been antici­ who are not U.S. citizens. The data are from the pated. Our total annual number of degrees in the annual National Research Council reports through mathematical sciences in the U. s. as counted by fiscal 1973 and from recent incomplete AMS fig­ the National Research Council has stabilized at ures for fiscal1974. Jn fiscal1973, the compa­ about 1, 200, far below the earlier projection of rable AMS figure was 21%, in agreement with the Office of Education for over 2, 000 degrees the NRC figure.

337 TABLE 3 doctorates on the national faculty and a decrease 90 women nondoctorates on the national DOCTORATES WITH FOREIGN CITIZENSHIP of about faculty. In Ph. D. producing mathematics depart­ Fiscal Years ments, women comprise about 4% of the doctor­ faculty; in masters' and bachelors• 1968 15.7% ate-holding level departments, they comprise between 9% 1969 15.3% and 10% of the doctorate-holding faculty. 1970 15.6% 1971 17.6% TEACHING LOAD PATTERNS 1972 18.2% This summer, the AMS collected several 1973 20.8% types of information relative to teaching loads. 1974 (28. O%) They show no increase in average teaching loads The geographical location of first employ­ from fall 1972 to fall 1973 and a probable slight ment of new doctorates in mathematics is also increase in teaching loads anticipated by depart­ recorded by the NRC. Of those for whom there­ ment chairmen in fall1974 as compared to fall gion of employment was known, the percentage 1973. The data for fall1972 to fall 1973 are with foreign employment went up from about 7% based on chairmen's reports of numbers of sec­ in 1970 to 13% in 1973. (The 1974 matrix (Table 1) tions, numbers of students enrolled and numbers of areas of degrees and types of initial employ­ of faculty members (full-time, part-time, and ment includes Canadians and is not directly com­ teaching assistants). Based upon reports from parable to NRC figures. ) 40%-50% of all department chairmen, repre­ more than 50% of the student body, the EMPLOYMENT OF THOSE WITH DEGREES IN senting offered in 1973 was al­ APPLIED AREAS total number of sections most exactly the same as that of 1972, down There was very little unemployment re­ about 0. 2% for mathematics departments as ported among computer science, statistics, and such and up about 0. 2% for all departments in operations research doctorates, either among the mathematical sciences. The total faculty was those previously employed in academia or among almost exactly stable for those two years, one those with 1974 degrees. There was indication of estimate showing, for example, the full-time a modest number of available positions still un­ faculty the same and another down less than 0. 3% filled in July. The publication "Employment In­ (i. e. about 50 out of 17, 500). It should be re­ formation for Mathematicians" also revealed sub­ marked that the data, which show very little if stantial demand for statisticians and computer any change, are generally consistent across scientists. several classifications of departments. While The percentage of new doctorates in applied the numbers of sections were stable, the course mathematics as such (presumably classical ap­ enrollments were up slightly, about 2% from plied mathematics) who were not yet employed in 1972 to 1973, offsetting a 2% drop from 1971 to July was almost the same, 12%, as the percent­ 1972. Thus, with a stable faculty size, there was age of new doctorates in pure mathematics who a slight deterioration in the student/faculty ratio, were not yet employed. reversing a favorable pattern of previous years. The numbers of new doctorates in computer But such a development should increase the science and statistics were apparently about the pressure for more faculty. same as last year (or down slightly), but because It was not possible, of course, to get en­ a substantial number of departments in these rollment or section figures for the fall of 1974 areas do not submit reports to the AMS, our fig­ so, in order to get information concerning the ures are not complete or necessarily accurate. changes from the fall of 1973 to the fall of 1974, The National Research Council figures, which the AMS asked all department chairmen to choose will be available next summer, should give es­ one of five alternatives to describe anticipated sentially complete information. changes in average or normal teaching load of WOMEN ON THE FACULTY full-time faculty in 1974-1975 as compared to 1973-1974. The responses are presented in There was an increase of about 50 women Table 4.

TABLE 4 ANTICIPATED CHANGES IN WEEKLY TEACHING LOAD 1973-1974 - 1974-1975 DECREASE ESSENTIALLY INCREASE Response More than Half-hour NO CHANGE Half-hour More than Rate Class Half-hour or less or less Half-hour 15/27 I 0 0 13 2 0 23/38 II 0 0 22 1 0 52/91 III 0 1 47 0 4 22/65 IV 0 0 17 2 3 18/106 v 0 1 15 0 2 11/30 VI 0 1 9 1 0 150/325 M 3 1 124 13 9 301/910 B 5 4 253 10 29

338 Thus a net of about 10% of chairmen expect some and others who were hired while writing their increase in teaching loads for 1974-1975, about dissertations. This replacement of nondoctorate half of those expecting increases of one-half hour faculty has been one of the important a venues of or less per week. Whether this will be borne out academic employment for doctorates. However, by figures available next summer remains to be two recent developments suggest strongly that seen. (For 1973, chairmen expected very minor except for self-replacement and natural attrition increases which statistically did not materialize. ) by retirement or death, we can expect very little The classifications used in Table 4 are: further reduction in nondoctorate level faculty. I the top 27 ACE ranked departments These developments are: II the other 38 ACE rated departments (1) only about two-thirds as many nondoc­ III the 91 unrated Ph. D. producing mathemat­ torates left faculty positions this summer as last ics departments summer and IV the 65 Ph. D. producing statistics, biosta­ (2) of all nondoctorate faculty above the rank tistics and biometry departments of instructor about 80% are tenured and at least V the 106 other U.S. Ph. D. producing de­ another 15% are individually expected to be re­ partments (64 computer science, 18 opera­ tained indefinitely. tions research and 24 miscellaneous, in­ cluding applied mathematics) TENURE VI the 30 Canadian Ph. D. producing depart­ The AMS collected data on changes in num­ ments bers of faculty with tenure from fall1973 to fall M the 325 masters' producing (but not Ph. D. 1974 on two different forms with slightly different producing) departments questions. The results showed rises of 3% to 4% B the 910 bachelors' producing (but not mas­ in the percentage of doctorate-holding faculty ters' producing) departments with tenure, the one set of data showing a rise under 3% and the other about 4%. A further ques­ THE REPLACEMENT OF tion was asked to show how many of the existing NONDOCTORATE FACULTY faculty without tenure were, as individuals, ex­ pected to be retained indefinitely. For the mathe­ For each of the past three years, from 300 matics departments in groups I, II, III, M and B to 400 nondoctorate faculty have been replaced by (all U.S. departments except Computer Science, doctorates, perhaps one-half being self-replace­ Statistics, Operations Research, and Applied ments, that is, faculty members who have re­ Mathematics) the results are summarized in ceived doctorates. Among the latter are some Table 5 for all full-time doctorate faculty at the who returned to graduate school for their degrees assistant professor level or above.

TABLE 5 I II III M B Tenured - Fall 1973 70.7% 68. O% 61.5% 61.1% 51.6%

Tenured - Fall1974 72.8% 70.5% 64.3% 63.9% 54. O% Nontenured but expected to be retained indefinitely, Fall 1974 3.5% 5.1% 12.2% 16.0% 26.0% TOTAL 1974 (Rows 2 and 3) 76.3% 75.6% 76.5% 79.9% 80. O%

The figures are from the AMS faculty mo­ THE EMPLOYMENT POSSIBILITIES bility survey and show a slightly lesser increase FOR NONRETAINED FACULTY in tenured faculty than do the annual salary sur­ vey figures. The Ph. D. producing departments As the Faculty Flow Diagram shows, there I, II, and III have about 5, 500 total faculty (5, 100 were about 170 nonretained doctorate-holding with doctorates), the masters' level departments faculty members not yet employed as of July. also have about 5, 500 (3, 900 with doctorates), This was an improvement over the comparable and the bachelors• level about 4, 500 (2, 700 with number, 210, for July 1973. This summer the doctorates) according to 1974 projections. The AMS collected specific further information on situation is probably more rigid than the num­ hiring patterns of new faculty. For example, all bers indicate since some (or many) new, or re­ 438 masters' and bachelors' level departments cent, department members will be retained even reporting hired not a single new faculty member though not yet identified as expected to be re­ with tenure. Indeed these departments hired only tained indefinitely. Also, as reported above, at thirteen people who had their degrees seven years least 95% of the nondoctorate faculty above the or longer and were from U.S. academia. In the instructor level in the masters' and bachelors' same departments only twenty-seven people who departments are tenured or are expected to be had their degrees from three to six years were retained indefinitely. hired from U.S. academia. Projections from the

339 sample to the total number of such departments now in academia and with many in positions that indicate that only about 80-90 mathematicians will not be permanent, we are clearly headed in­ who had their degrees more than three :years to a most difficult time for such nonretained were hired from U.s. academia for 1974. Fur­ people. As higher precentages of the national thermore, projections to all156 Ph.D. producing faculty become tenured, the situation for those mathematics departments (in classes I, II and who must move gets more and more severe. A III), show the total number of hirings from U. S. complicating factor is that most available posi­ academia of people out more than three years to tions appear to be at or near the beginning level, be about 45-50, of whom about fifteen were thus creating financial problems even for the hired with tenure. nonretained who do get academic positions. It would appear that for each 11, 4 ;;i; 11 ;;i; 7, The AMS and the Committee on Employ­ there were only about thirty doctorates out 11 ment and Educational Policy welcome suggestions years who left their academic positions and found as to how they may be helpful to those who are not jobs in four-year college and university mathe­ retained in academia and do not find positions in matics departments, and this figure includes academia. Efforts by CEEP to locate sources of those who moved voluntarily or got temporary funds for potential training of such people in con­ positions. temporary applications of mathematics have been With, for each such n, an average of per­ discouragingly unsuccessful so far. haps 500 pure mathematicians out n years and

INVITED SPEAKERS AT AMS MEETINGS

This section of these c/{otiaJ) lists regularly the individuals who have agreed to address the Society at the times and places listed below. For some future meetings, the lists of speakers are incomplete. Washington, D. C., January 1975

Donald W. Anderson H. Jerome Keisler (Colloquium Lecturer) Donald L. Burkholder Haskell P. Rosenthal Sigurdur Helgason Rainer K. Sachs Fritz John (Gibbs Lecturer) Wilfried Schmid Linda Keen Nolan Wallach

ORGANIZERS AND TOPICS OF SPECIAL SESSIONS Abstracts of contributed papers to be considered for possible inclusion in special sessions should be submitted to Providence by the deadlines given below and should be clearly marked "For consideration for special session on (title of special session)." Those papers not selected for special sessions will automatically be considered for regular sessions unless the author gives specific in­ structions to the contrary. Washington, D. C., January 1975 October 29, 1974

William B. Arveson, Operator Theory Joseph Auslander and Nelson G. Markley, Topological Dynamics A. T. Bharucha-Reid, Probabilistic Analysis W. Wistar Comfort, Set-theoretic and Combinatorial Methods in Topology Constantine M. Dafermos, Hyperbolic Conservation Laws David Eisenbud, Commutative Algebra Bernard A. Fusaro, Singular Cauchy Problems John E. Gilbert and George G. Lorentz, Interpolation of Operators and Applications Emil Grosswald, Analytic Number Theory Frank Harary, Graph Theory Morris Newman, Interrelations between Computation and Number Theory George B. Seligman, Structure and Representations of Lie Algebras over General Fields Francois Treves, Fourier Integral Operators Stanislaw M. illam, Mathematics and Games

340 THE PH.D. ANCESTRY OF THE LEADING U.S. MATHEMATICS FACULTIES By William H. Pell * The assessment of the quality of a depart­ strong department" or that "the department at ment of mathematics (or any discipline for that Bayshore Tech is not quite as good as that at matter) is a difficult task, and, indeed, ulti­ Siwash". On the other hand, almost all will read­ mately impossible, if it is required that the as­ ily admit that his or her quality measuring mech­ sessment consist in the assignment of a univer­ anism is not up to such fine-grained judgments sally agreed upon number to represent quality. as listing the top twenty-five departments in the The reason is simply that the concept of quality order of decreasing quality or (even more diffi­ is subjective-a value judgment-and as such can­ cult!) of assigning numbers measuring the quali­ not be tied to a number. ty of the department of mathematics at these in­ Despite this, almost all of us feel that we stitutions. have a rather good idea as to what quality of a Quality has many attributes (indices, indi­ department of mathematics really means, and cators, parameters, characteristics, or other we are generally right-in the sense that if a terms may be used), and each person arrives group of colleagues is polled we will usually at his or her definition of quality through some find a fair consensus that "Siwash has a fairly more or less dimly perceived integration over

Table 1 Composition of ACE Group 1 Tenured Mathematics Faculties: Contribution from Faculties of this Same Group

IZ1 0 < "'co ...."' INSTITUTION 1 California (Berk. ) 5 6 10 6 3 2 1 1 3 2 3 1 1 2 1 61 1 Harvard 4 3 1 12 3 Princeton 14 2 1 2 23

4 Chicago 12 87 1 11 1 1 29 5 MIT 4755 2 1322 2 37 6 Stanford 12 42 1 1 11 1 1 1 19

7 Yale 2 4 1 2 2 14 8 NYU 3 4 1 1 17 4 1 1 43 9 Wisconsin 24516236314 3 3 3 2 67

10 Columbia 2 1 1 2 9 10 Michigan 1221521 5 1 4 3 1 2 2 1 2 1 48 12 Cornell 146511 1 1 2 2 2 32

12 Illinois 43432611347272 1 2 1 1 4 1 1 81 14 California (LA) 7 3 5 2 3 1 1 4 1 1 1 2 4 2 1 1 1 49 15 Brandeis 1 2 2 4 2 12

15 Brown 3 4 3 1 2 2 1 16 15 Calif. Inst. Tech. 1 3 1 1 3 1 1 1 18 18 Minnesota 6 4 3432421 5 2 1 ·3 2 1 2 58

18 Pennsylvania 1 3 5 3 2 1 2 1 1 27 18 U. Washington 5 4 5 5 1 1 1 3 1 1 1 2 3 2 1 46 21 Purdue 3 5 243512 3 5 2 2 1 1 3 2 3 1 67

21 Rockefeller 1 1 1 1 1 1 7 23 Johns Hopkins 1 1 1 7 23 Northwestern 1412112 1 1 1 1 1 1 1 23

23 Virginia 2 1 1 3 4 1 1 15 26 California (SD) 2 2 4 1 2 3 17 26 Indiana 1 2 1 1 1 1 2 1 1 2 1 1 23

*This study arose out of the author's activities as an observer with the AMS Committee on Employ­ ment and Educational Policy. In no way does it reflect the views of either the American Mathemati­ cal Society or the National Science Foundation.

341 some set of these. Examples of such indices are: ly subjective ratings by individuals deemed to be publication record of the faculty, number of qualified. This is the procedure which has been members of the National Academy of Sciences, followed in two recent highly regarded studies adequacy of the mathematics library, number of assessing the quality of departments in a large NSF fellows, number of Guggenheim or Sloan number of disciplines, including mathematics, fellows [1], etc. On the face of it, all of those at a fairly large number of institutions. The named appear to be objective indices {except per- earlier of these was by Allan M. Cartter [2] in haps for adequacy of the library}, since they may 1964. The second, very much like the first ex- be assigned numerical values. But this is illu- cept for minor changes in technique, was by sory; except possibly for the publication cri- Kenneth D. Roose and Charles J. Andersen in terion, all are subjective measures of quality 1969 [3]. In both studies attention was focussed once removed, so to speak. For Guggenheim on two aspects of departmental quality of major Fellows and members of the National Academy concern alike to administrators, students, and are selected by peer groups exercising subjec- faculty members: {1} quality of the graduate fac- tive judgments, and the process by which a de- ulty and (2} effectiveness of the graduate pro- partment acquires a NSF fellow, though more gram. These, of course, will be recognized as complex, is also surely subjective. Many other quality attributes such as we mentioned earlier, factors influencing quality could be listed, some although very important ones which may them- subjective and some perhaps truly objective. selves be thought of as aggregates {in some But even though such a count be made, the sig- sense} of a congeries of lesser attributes. nificance of each characteristic must be deter- The Cartter and Roose-Andersen studies mined, i.e., a weight assigned to each, and a were opinion surveys, but were very carefully sum taken to determine the ultimate quality rat- done, and the results (at least in the case of [2]} ing. But these weights also are subjective, and analyzed very thoroughly. The faculty members cannot be quantified. asked to make evaluations were selected because Clearly the procedure just sketched is not they were widely regarded as knowledgeable and feasible. An alternative which makes no attempt statesmanlike, and who by virtue of scholarly to identify and measure the myriad of quality at- and professional work had earned the respect of tributes is simply that of tabulating the admitted- their colleagues and peers. The high degree of Table 2 Number of Ph.D. •s from Institutions Outside the Group 1 Departments {1969 ACE Study) Which Have Achieved Tenure in Group 1 Departments

ACE Group 2 Foreign Institution ---Number Institution Number Carnegie Mellon 3 Cambridge 13 Duke 3 Eidgen. Tech. Hochschule 6 North Carolina 4 Gtlttingen 6 Ohio State 6 Hebrew 4 Rice 7 Indian Stat. Institute 3 TUlane 6 Manchester 5 Nagoya 5 ACE Group 3 Paris 5 Tokyo 9 ~stitution Number Toronto 8 Warsaw 3 owa 5 Zurich 4 ~racuse 8 [rexas 5

Table 3 Ph. D. Source of Tenured Members of Mathematics Departments in ACE Group 1 by Institutional Group

Number Institutional Number of of Degrees Group Institutions in Group

661 ACE Group 1 27 37 ACE Group 2 17 31 ACE Group 3 21 9 ACE Dnranked 9 738 Total D. S. 74 10 Canadian 2 109 Other Non-D. S. 51 119 Total Non-D. S. 53 857 Grand Total 127

342 consensus among these expert witnesses makes torates in a department which are from depart­ it seem likely that these studies, despite the sub­ ments generally agreed to be of high quality, jectivity which must be admitted to in an individ­ then the higher the quality of the department. A ual's rating, have succeeded in establishing qual­ little reflection on this statement makes it clear ity ratings of substantial precision. that one should not assign too much significance Although, as we have seen, it is not feasi­ to this criterion, since it assumes at the outset ble to arrive at overall departmental quality rat­ some definition of high quality departments. ings by the summation of contributions of a mul­ Nevertheless, it is believed that the tabulation titude of attributes, the individual measurement of this data on Ph. D. origins has some interest of such in some instances can nevertheless give and provides information not hitherto available information of significance and interest regard­ regarding the faculties and graduate programs of ing departmental quality. One such indicator of some of our institutions of higher learning. no little interest, though of some complexity, The data on which this study is based was emerged from discussions between the writer, collected in the spring of 1972 by the author and his colleagues, and various members of the ac­ Dean Lowell J, Paige, of the University of Cali­ ademic community some time ago, Like many of fornia at Los Angeles. Only the top 27 depart­ the attributes of departmental.quality that have ments in the 1969 ACE study (Roose-Andersen) been proposed or defined (and sometimes mea­ were considered, and under the presumption that sured), it seems to be generally felt that the one the awarding of tenure indicates significant in question has some real measure of relevance achievement, only the tenured faculty of each to the quality of the graduate faculty. Beyond department has been considered. It must be this, admittedly, it is difficult to say much, ex­ pointed out that because of the time required to cept that our indicator reveals some fascinating achieve tenure, the results presented here rep­ information regarding our better departments of resent the situation at roughly the mid-nineteen mathematics, and in perhaps an instance or two sixties. By the same token, the data we present puts on a sound factual basis what has hitherto been represents an integration of degree production folklore. over the period from about 1930 to the time men­ The quality attribute that we have in mind tioned above. we shall call, for lack of a better term, the The results of our efforts may be exhibited Ph. D. ancestry pattern of a department. This in a 27 x 127 matrix. Considerations of space is a tabulation of the number of doctorates which make display of this matrix difficult, and since its members have received from various schools. our primary interest is with the 27 Group 1 de­ The argument that this constitutes a reasonable partments, we have shown in Table 1 only that attribute of quality simply resides in the as­ portion of it which relates to these. This is to sumption that the greater the proportion of doc- be interpreted as follows: the institutions on the Table 4 ACE Group 1 Departments Ranked by Number of Ph. D. •s on Tenured staff of Group 1 Departments

Number of Ph. D. •s on Institution Group 1 Faculties ACE Rank

1. Princeton 100 3 2. Harvard 68 1 3. Chicago 67 4 4. California (B) 50 1 5. MIT 42 5 6. Michigan 41 10 7. NYU 35 8 8. stanford 30 6 9. Yale 27 7 10. Wisconsin 24 9 11. Cornell 23 12 11. IDinois 23 12 13. Columbia 21 10 14. California (LA) 19 14 15. Cal. Tech. 17 15 16. Brown 15 15 17. Minnesota 14 18 18. Purdue 10 21 18. Johns Hopkins 10 23 20. Washington 6 18 20. Indiana 6 26 22 Pennsylvania 5 23 22. Virginia 5 23 24. Northwestern 2 23 24. Brandeis 2 15 26. California (SD) 0 26 26. Rockefeller 0 21

343 Table 5a

Number of Ph. D. 1s Contributed to ACE Groups 1A and 1B by the ACE Rated and Other Groups

ACE GROUPS Other Foreign u.s. Total 1A 1B 2 3 Canada Other

Group 1A 226 52 14 14 1 5 48 360 Group 1B 278 105 23 17 8 5 61 497 Group 1 504 157 37 31 9 10 109 857

Table 5b Data of Table 5a Reduced to Percentage of Total Tenured Faculty of Group 1 Institutions

ACE GROUPS Other Foreign u.s. 1A 1B 2 3 Canada Other

Group 1A 62.8% 14.4% 3.9% 3.9% 0.3% 1.4% 13.3% Group 1B 55.9% 21,1% 4.6% 3.4% 1.6% 1.0% 12.3% Group 1 58,8% 18.3% 4.3% 3.6% 1.1% 1.2% 12.8% left (defining the rows) are those of the Roose­ ments. In Table 4 the 27 departments of Group 1 Andersen ACE study on whose faculties we are again appear, but here they are ranked by the reporting. Across the top are the institutions number of Ph.D.'s they have contributed to the (defining the columns) whose doctorates comprise tenured staffs of the same 27 departments. The the mathematics faculties of those institutions on ACE Roose-Andersen rank of each is given in the left. Thus the entry N in a given row and col­ the right column (there were ties), and it will be umn means that there are N members of the seen that the correlation of the two rankings is mathematics faculty of the school associated with quite good, although in a few instances there are the row in which N appears who obtained the notable disparities. The first four departments Ph. D. at the school defining the column in which are the same as in the ACE 1969 (Roose-Ander­ N occurs. The institutions on the left are listed sen), ACE 1964 (Cartter), and 1957 Keniston in order of decreasing rank in the ACE study, studies, except for permutation; indeed, the while at the top we have the same order repeated same departments, except for Berkeley, were as one moves from left to right. The portion of the highest ranking departments in the earliest the matrix omitted is composed of columns which formal study of quality known to the writer [4]. represent the contributions to the faculties of the The dominance of Princeton is perhaps mildly ACE Group 1 departments by: (a) ACE Group 2; surprising. It may be conjectured that Berke­ (b) ACE Group 3; (c) unrated U.S. departments; ley's relatively low numerical score is due to and (d) non-U.S. institutions. Since the contri­ its fairly recent and rapid rise to the top. butions of most of the departments of (a) - (d) The quality ratings of the 1969 ACE Roose­ to the faculties of the Group 1 departments are Andersen report were determined by essentially not numerous, the omitted portion of the matrix the same methodology as in the 1964 report by has largely zero elements. Nevertheless, a sub­ Cartter, but the quality groupings were somewhat stantial number of these departments have made different. The departments forming Group 1 in significant contributions to the Group 1 depart­ the 1969 report were those with scores in the ments, and in Table 2 we have listed the insti­ interval 3, 0-5.0 (the exact meaning of these tutions of (a) - (d) which have contributed a to­ numbers is not material here), while in the Cart­ tal of 3 or more members to the Group 1 depart­ ter report the departments in this interval con­ ments.* stituted two groups: "distinguished" (4. 0-5. 0) The 27 departments surveyed had a total of and "strong" (3. 0-4. 0). Since it is of interest to 860 tenured faculty members. Of these, 857 had compare a relatively few departments of acknowl­ an earned Ph. D. degree (virtually all in mathe­ edged excellence with others, Group 1 of the matics), but 3 also held a second doctorate from Roose-Andersen study has been divided somewhat a foreign institution; the latter degrees are not arbitrarily (the exact procedure of 1964 could not considered here. Table 3 gives the number of de­ be followed since individual ratings were not giv­ grees arising from each of the five groups men­ en in 1969) into Groups 1A and lB. The former tioned above and the number of institutions in contains departments of rank 1 through 10 of each. Note that the top 27 departments drew 661 Group 1 and the latter, of course, the remaining (77%) of their tenured faculty from their own 16 departments of this Group. Table 5a is a ranks, 119 (14%) from without the United States, breakdown of the data in Table 2 to show the and only 77 (9%) from lower echelon U.S. depart- contribution of ACE Groups lA and 1B and the

*Copies of the complete matrix may be obtained from the author at the Mathematical Sciences Sec­ tion, National Science Foundation, 1800 G Street N. W., Washington, D. C. 20550. Since the number of copies is limited, the author would be grateful if only one copy per department 1s requested.

344 Table 6a Table 6b Ranking of Group 1 Departments by Number of Ranking of Group 1 Departments by Number of Ph, D. •s on the Tenured Staff of Group lA Ph. D. •s on the Tenured Staff of Group lB Departments Departments

Ph.D. •s in Ph.D. •s in Group lA Group lB Institution Departments Institution Departments

1. Princeton 62 1. Chicago 41 2. Harvard 29 2. Harvard 39 3. Chicago 26 3, California (B) 38 4. NYU 24 3. Princeton 38 5. MIT 19 5. Michigan 25 6, Michigan 15 6. MIT 23 7. California (B) 12 7. stanford 22 8. Columbia 11 8, Cornell 19 9. Brown 10 9. Yale 17 9. Wisconsin 10 ~0. California (LA) 15 9. Yale 10 ttO. Ulinois 15 12. illinois 8 ~2. Wisconsin 14 12. stanford 8 13. Cal Tech 11 14. Cal Tech 6 13. NYU 11 15, Johns Hopkins 5 15. Columbia 10 15. Minnesota 5 16. Minnesota 9 17. California (LA) 4 17. Purdue 8 17. Cornell 4 18. Brown 5 19. Brandeis 2 18. Johns Hopkins 5 19. Indiana 2 18. Pennsylvania 5 19. Purdue 2 21. Indiana 4 19. Virginia 2 21. Washington 4 19. Washington 2 23. Virginia 3 24. California (SD) 0 24. Northwestern 2 24. Northwestern 0 25. Brandeis 0 24. Pennsylvania 0 25. California (SD) 0 24. Rockefeller 0 25. Rockefeller 0 other categories of institutions to Groups lA and where the Ph. D. recipients of a given department lB, while Table 5b gives the same data in per­ have gone or at where the Ph. D. holders in a de­ centages. As might be expected, the contribu­ partment have come from. The entries in the tion of Group lA to itself is somewhat greater column associated with a given institution an­ than to Group lB, and that of Group 1B to itself swers the first question, those in the row asso­ considerably greater than to Group lA. It is in­ ciated with a particular institution the latter. teresting that the contribution of Group 1 to each More accurately, this dual point of view can be of the Groups lA and lB is almost exactly the taken only for the Class 1 departments. Since we same, 77%. This seems a bit surprising, for do not have information on the faculties of depart­ since Group lB is presumably of somewhat low­ ments in the other categories, we are unable to er quality than lA, it might be expected that the give the row entries for these. contribution to its faculties from Groups 2, 3, etc. would be such as to make the proportional con­ References tribution of Group 1 somewhat smaller than to 1. On the number of Sloan fellows produced by lA. The contribution of Groups 2 and 3, other various departments, see R. D. Anderson and U.S. and Canadian institutions to both groups is W. H. Pell, A Measure of Quality of Doctoral about the same, approximately 10%. The con­ Programs, these c){otiai] 21(1974),260-262. tribution of foreign institutions to Group lA is 2. Allan M. Cartter, An Assessment of Quality only a little larger than to lB, 13.3% as com­ in Graduate Educatfciii., American Council on pared with 12. 3%. This essential equality may Education, 1966 (cf. these cNotiaiJ 13(1966), seem a little surprising, since it might be ex­ 978-980). pected that the high quality of faculty with Euro­ 3. Kenneth D. Roose and Charles J. Andersen, pean and Japanese training would contribute more fl Rating of Graduate Programs, American strongly to Group lA than is the case. Tables 6a Council on Education, 1970 (cf. these c){otiai] and 6b are a breakdown of Table 4, in the sense 18(1971), 338-339). that we have here the departments of Group 1 4. By Raymond Hughes, a report to the annual ranked by the number of Ph.D's they have con­ meeting of the Association of American Col­ tributed to the departments of Groups lA and lB. leges. Reprinted in American Universities It should perhaps be pointed out in closing and Colleges, American Council on Education, that one can look at the data we have presented 1928. here from two points of view. One can look at

345 CASE STUDIES Some Mathematicians with Nonacademic Employment

Mathematicians, faced with the current undersupply of academic positions, are increasingly considering finding jobs in industry or other nonuniversity areas. Indeed, there are indications (e. g., concern prompted by the energy crisis; an increase in the number of classified advertisements for technological positions in some newspapers) that opportunities for highly trained scientific personnel in industry and government may be increasing. As an aid to persons considering such positions, the AMS Committee on Employment and Educational Policy has solicited case studies from mathemati­ cians currently working in nonacademic jobs. Some of these case histories are to be published from time to time in the cNoticeiJ. It is hoped that this will not only give mathematicians in general some idea of what they can do outside of academia, but will also suggest how to go about obtaining such a position. The Committee on Employment and Educational Policy seeks more case studies, including some involving somewhat less traditional employment than that represented by the three below. In addition, an evening panel discussion session is being arranged at the annual meeting in Washington, D. C. in January 1975 devoted to employment of mathematicians in positions not of the more tradi­ tional sort. Members who wish to participate by submitting case studies for consideration, or who have suggestions to make, are encouraged to communicate with Professor Martha K. Smith, Univer­ sity of Texas, Austin, Texas 78712.

LYNN 0. WILSON useful to me in one way or another. But also Mathematics and Statistics Research Center very important is the training I had outside the Bell Telephone Laboratories area of "pure" mathematics, particularly in physics. For someone interested in working at I received a Ph. D. in Applied Mathematics this type of job, a broad background is valuable, from the University of Wisconsin four years ago. including a solid understanding of physics or My thesis involved magnetohydrodynamics and some branch of engineering such as electrical the asymptotic solution of a system of nonlinear engineering, as well as a knowledge of mathe­ partial differential equations. Although most of matics. It would also be advantageous to have the courses which I took at the University of some experience with computer programming. Wisconsin were in analysis or applied mathemat­ ics, I also audited several courses and seminars C. R. GIARDINA in physics and engineering. Before going to Wis­ Singer-Kearfott Aerospace and Marine Division consin, I majored in physics at Oberlin College and did a year of graduate work in the Division For the past four years, I have been em­ of Applied Mathematics at Brown University. ployed by Singer-Kearfott Aerospace and Marine During some summers, I managed to get sum­ Division. I am involved in the design and devel­ mer jobs involving research in physics. opment of inertial and aided navigation systems I have been working at Bell Labs since the My principal tasks are: specifying and mechaniz­ time I got my Ph. D. I was recruited on campus ing the mathematical navigational equations by a member of the Bell Labs technical staff. which are subsequently put into hardware; simu­ Although many people here do research in a lating the potential software to guarantee ade­ specialized field, my problems tend to be quite quate performance and to make tradeoffs in varied. For example, I have helped an experi­ mechanizations; developing error budgets for mental physicist to develop the theory of optical instruments to guarantee adequate output per­ dielectric wave guides, worked with several of formance; determining output performance and my colleagues on a project to understand an reliabilities of present and future navigational electroplating process, worked on various prob­ systems; utilizing modern mathematical (func­ lems concerned with electromagnetic shielding, tional analytic) techniques to improve and de­ acoustic surface waves, and operations re­ velop state-of-the-art navigational systems. search-and even studied the mathematical I received my Ph. D. from Stevens Insti­ theory of epidemics. This variety is one of the tute of Technology in 1970. My mathematical things which makes my job fascinating. The academic training was highly nonpractical; to problems I work on frequently arise from dis­ compensate for this, I took numerous engineer­ cussions with other people here at Bell Labs. ing courses (beyond my mathematical Ph. D. re­ One part of my job, of course, is to transmit quirements). On the other hand, I have found the solution to the problems back to the people nonconventionally applied courses like abstract who posed them. The ability to communicate, algebra, topology, functional analysis and logic both verbally and on paper, is very important. very useful in creating and understanding inno­ Another aspect of my job is that of continuing vative industrial concepts. education. I have participated in a number of All my jobs were obtained by applying in courses and informal reading seminars. person at the company• s personnel office. With­ Much of my mathematical training has been out question, my "love of mathematics" and the

346 ability to converse mathematics in engineering psychologists, mathematicians and others. In my terms were major contributors to obtaining em­ two years with GM I have been afforded the op­ ployment. portunity to interact with colleagues on projects My positions at Singer-Kearfott have been both inside and outside the department as well both challenging and highly rewarding. Without as to further personal professional interests. going out of my way to use "fancy" mathematical Projects I have worked on include scheduling on techniques, I have found, on almost a daily basis, automated guideways and psychometric analysis the need to use quaternion differential equations, of public attitudes towards innovative transpor­ stochastic differential equations, constructive tation systems such as dial-a-bus, brief excur­ function theory, discrete state continuous time sions into engine geometries and the partial dif­ Markov chains, differential geometry, Fourier, ferential equations of diffusion processes. I have Laplace, Z and other transforms and so on. met with some of the consultants retained by GM, For example, in modeling the changing at­ and have had time to publish in professional titude of a vehicle by a unit quaternion differen­ journals, among them the Transactions of the tial equation and then mechanizing these equations AMS and Linear and Multilinear Algebra. I on a computer by way of difference equations, one have attended a number of mathematics confer­ must take into account the roundoff, truncation ences at company expense. and quantization errors involved in this mechani­ 2. Mathematics background zation. Because of these errors, the solution to the quaternion differential equation will not nec­ I received a B. A. in mathematics from the essarily be of unit length. Therefore, a normal­ University of California, Riverside, and a Ph. D. ization scheme based on some norm minimiza­ in mathematics from the University of California, tion procedure is performed. Seeing the actual Berkeley, in 1970, where my thesis title was software work in real life is very gratifying. "Representing Measures and Associated Function Another satisfying feature of this position Theory on Finite Bordered Riemann Surfaces." is that it is a continual learning experience. One My course background was mostly "pure," e. g., must always be aware of and understand such measure and integration, modern algebra, alge­ things as virtual memory computers, silicon on braic topology, differential geometry, functional sapphire transistors, ring laser gyros, strap­ analysis and Riemann surfaces, with a course in down navigation, Kalman filtering, digital filters differential equations as an undergraduate. and other modern or state-of-the-art concepts. 3. other background A person seeking employment in industry needs a broad mathematical background, in­ I completed an undergraduate minor in cluding operations research, statistics, proba­ physics; mechanics has proved to be especially bility, numerical analysis and computer science. useful. Courses in economic geography and These courses, along with stochastic processes, psychology also tied in with departmental activi­ differential geometry, classical mechanics and ties, but these relationships are largely fortu­ control theory, are very important in the aero­ itous and job-specific. space world. Before joining GM I held a variety of math­ Many concepts in theoretical mathematics ematics teaching positions: Acting Assistant have a physical interpretation. An instructor Professor at the University of Hawaii (1969- would need very little course time just to men­ 1970), Woodrow Wilson Teaching Intern, Virgin­ tion these applications, which can be found in ia State College, Petersburg (1970-1971) and engineering, biological, business or physics Lecturer, University of California, Berkeley, books. (1971). An individual with adequate mathematical 4. Job application background who researches a company and ex­ amines pertinent literature on the company• s The placement office at Berkeley provided technology will recognize the mathematics in­ information on an expansion at the GM Research volved and will only have to learn the names of Laboratories. I marketed myself on an ability the symbols in order to speak intelligently at the to relate my mathematics background to ongoing interview. research interests at GM, citing for example that some areas in spectral theory can be ap­ plied fruitfully in psychometrics and data analy­ DAVID NASH sis. I emphasized that the responsibilities in Transportation and Urban Analysis Department my last position at Berkeley included teaching General Motors Research Laboratories upper division li.Jld graduate level courses in real analysis, where more than half of the stu­ 1. Job description dents were not mathematics majors. This could I am employed as an Associate Senior Re­ be considered valuable experience for the multi­ search Mathematician by the Transportation and disciplinary setting at GM. Urban Analysis Department, General Motors Re­ 5. Personal traits search Laboratories, Warren, Michigan. A fun­ damental goal of the depart:ment is to advance Good communication skills and the ability analytical techniques for the design, analysis to work well with others are crucial, as are and evaluation of transportation systems. The dedication to work and achievement. Industry effort involves urban planners, engineers, com­ evaluates much more about a prospective em­ puter scientists, regional planners, economists, ployee than just his or her academic background.

347 6. Job rewards cians, operations researchers, computer scien­ My position is as interesting, challenging tists, etc. A member of this group whose train­ and rewarding as I make it. My job description ing was mostly in mathematics per se could con­ should give an idea of the potentials. Frankly, sider large portions of this training irrelevant I find being associated with the industry leader to his job. Such an attitude is too negative, un­ exciting in itself. On the more mundane side, creative and usually indicative of a failure to salary and fringe benefits at GM are very attrac­ grasp the potentialities of the situation. If one tive. sees one• s job as a challenge to relate his math­ ematics to concerns of colleagues and vice 7. Uses to employer versa, then both may profit. For this, I consider An industrial research mathematician can all of my mathematics training relevant. help formulate and devise practical methods for A final comment: Unfortunately, the gener­ solving problems of importance to his company• s al image of mathematicians to industry has often operation. He can seek new situations into which suffered because enough have shown aloof, pur~ mathematics can be infused profitably and serve er-than-thou attitudes towards applied profes­ as a problem solver and sounding board for other sions, considering them somewhat below their professionals trying to formulate their ideas rig­ dignity. orously. As a permanent employee, his day-to­ 9. Advice to job seekers day availability and knowledge of company pro­ jects are valuable adjuncts to the services of A resume prepared for a university posi­ outside consultants, with whom he may act as an tion generally would not be appropriate for in­ efficient liaison in mathematical matters. Con­ dustry; neither would any one resume necessari­ tributions to outside professional research ly attract a number of industries. The applicant journals and meetings can lead to validation of should first find out what a specific company corporate research efforts through exposure to wants or expects, and if he is interested, should a wide expert audience. address his application to these points. Part-time or summer industrial positions B. Relevance of training which are sometimes available to students and Training in such areas as applied statistics, faculty can provide opportunities to employee operations research and computer science would and employer for evaluation of career possibili­ be valuable to a mathematician seeking industrial ties. employment. Familiarity with a variety of ap­ Professors who are industrial consultants plied fields could certainly be an asset, but be­ can often advise on industrial positions in addi­ yond a certain stage there are dangers of dilet­ tion to those listed in placement offices. tantism and loss of identity as a mathematician The top industrial research laboratories in a reasonably strict sense. Indeed, probably are very selective on professional qualifications most industrial employees with the title of math­ and value research abilities and general academ­ ematician in fact practice primarily as statisti- ic excellence as highly as top universities.

AMS RESEARCH FELLOWSHIP FUND Request for Contributions

The AMS Research Fellowship Fund, estab­ strates the importance the Society and its mem­ lished by the Society in 1973, was created in re­ bers attach to research. sponse to the limited funds available for postdoc­ The survival of the Research Fellowship toral fellowships. The fellowships are intended to program depends on the contributions the Society support research fellows for one year and are receives. It is hoped that every tenured member awarded strictly on the basis of mathematical of the Society will be willing to contribute at merit. Serving on the Committee appointed by least $100 to the Fund, but any contribution is President Mac Lane to administer the Fund are welcome. The Society itself has pledged to con­ C. B. Bell, Walter Feit, Leonard Gillman, Peter tribute a minimum of $9, 000 and will match one­ J. Hilton, Mark Kac and Alice Schafer. The half the funds in excess of $18, 000 raised from Committee hopes to award for 1975-1976 several other sources. However, the Society can con­ partially tax exempt fellowships of approximately tribute no more than $20,000. Contributions are, $10, 000 each, a sum equivalent to the salary of course, tax deductible. Checks should be made which a research person with a recent Ph. D. in payable to the American Mathematical Society, mathematics might expect. Two such awards clearly marked "AMS Research Fellowship were made in August 1974. Although the number Fund" and sent to the American Mathematical of fellowships awarded may be small, the exis­ Society, P.O. Box 6248, Providence, Rhode tence of the Research Fellowship Fund demon- Island 02940.

348 PRELIMINARY REPORT OF THE MAA-AMS COMMITTEE ON EMPLOYMENT OF MATHEMATICIANS IN TWO-YEAR COLLEGES The Mathematical Association of America faculties should indicate a willingness to work and this Society established a Joint Committee on cooperatively with the universities in areas of the Employment of Mathematicians in Two-Year mutual concern. In the graduate schools there Colleges. The charge was composed in October of should be a focus on teaching as well as on re­ 1973 but the appointments were made in Novem­ search and a positive response to the requests of ber and December, The Chairman is Dr. Jerome the two-year colleges for good teachers with M. Sachs. other members are Louis Auslander, broad training including more than a superficial Betty J. Hinman, Henry J. Osner, and Edwin H. knowledge of applied mathematics. More gradu­ Spanier. Moreover, by agreement, Professor ate schools should encourage some of their stu­ John L. Kelley acted initially in place of Profes­ dents to observe and participate in two-year col­ sor Spanier. lege teaching. There should also be widespread The charge concerned the effective educa­ attention given to other practices found helpful tion of students in mathematics at two-year col­ such as record-keeping and evaluation reports on leges, the advantages and the disadvantages of teaching by teaching assistants, as well as on Ph. D. •s for teaching in such colleges, and the academic progress, supervision of teaching by possibility of training Ph. D. candidates in gradu­ experienced faculty members, observation of ate school to equip them better to teach in two­ graduate student teaching by peers and faculty, year colleges. The Committee was charged to informal discussions by graduate students on "study the situation, clarify it, and bring recom­ teaching problems at various levels, observation mendations to the Council and the Board of Gov­ of recognized faculty teaching experts and teach­ ernors which, when presented to two-year col­ ing seminars led by them. leges and to graduate schools, will contribute to The two-year college faculties, particularly better cooperation and improved teaching and in urban areas, should be consulted by the gradu­ learning." ate school faculties on the training needed to pro­ The Committee produced a preliminary re­ duce effective two-year college teachers. This is port. It was received with thanks and discussed part of the recognition of the skills a two-year by the Council of the Society. It was received college teacher needs to solve the difficult teach­ with thanks and discussed by the Executive and ing problems he encounters. If the graduate and Finance Committees of the Association. Each schools wish to place some of their students in group has communicated its reactions to the two-year colleges, their faculties should discuss, Committee and neither group has endorsed the with two-year college faculties, some serious report, each preferring to wait for a definitive questions. How do we produce Ph. D. 1s who can report before proceeding further. and will teach in two-year institutions ? How can In receiving the preliminary report, the we work together on common problems? What Council of the Society recommended its publica­ can the graduate schools do to help keep faculty tion, the intent being to secure public reaction to members of two-year colleges in touch with the expressed direction of ideas before taking ac­ developments in mathematics? Are exchange tion on the anticipated final report. The text of teaching programs possible? Are internship pro­ the preliminary report follows. grams possible? Are sabbatical interchanges possible? What kinds of cooperative programs The Committee agrees that it is essential will benefit both the two-year colleges and the that the communications gap between two-year universities? Out of discussions of questions and four-year institutions be closed. The mathe­ like these can come a spirit of camaraderie with­ matics faculties of two-year colleges should be in the local mathematics community and an at­ made to feel that they are an integral and impor­ titude that good teaching at all levels is a major tant part of the mathematical community. These concern of all.

349 NEW AMS PUBLICATIONS

MEMOIRS OF THE AMERICAN an introduction; notation and basic results; the MATHEMATICAL SOCIETY thermodynamic limit; a class of translation in­ variant states; and applications-time evolution EQUILIBRIUM STATES ON TffiN ENERGY in a lattice gas, Markov random fields, and SHELLS by Richard Leslie Thompson microscopic limits. It also includes an appendix, a section on open ques tiona and a bibliography. Number 150 110 pages; list price $3.30; member price $2.48; ON CLOSED 3-BRAIDS by Kunia Murasugi ISBN 0-8218-1850-3 To order, please specify MEM0/150 Number 151 114 pages; list price $3. 40; member price $2. 55; In this Memoir, the author considers the ISBN 0-8218-1851-1 probability measures on configurations of parti­ To order, please specify MEM0/151 cles in a finite lattice obtained by restricting the grand canonical ensemble to an energy shell, or This Memoir presents the first systematic set of particle configurations which share a com­ study of closed 3-braids from the algebraic point mon total energy with respect to a vector of po­ of view. It is proven in this book that certain tentials. The set of weak limits of these meas­ numerical invariants of the conjugate classes of ures as the finite lattice expands to zTI is used 3-braids are in fact invariants of the link types to define a class, C (cp; ~), of translation invari­ of associated closed 3-braids. Complete charac­ ant states on 'l71. This class of states is convex terizations of trivial link, torus links and split and it is shown that the extreme points of the links as closed 3-braids are also presented. The class are Gibbs' states. determination of product links as closed pure Just as the partition function for the grand 3-braids gives a partial solution to Birman's canonical ensemble may be used to define the conjecture. pressure for a finite lattice so the partition func­ This Memoir is divided into sections on tion for this ensemble restricted to an energy terminologies and notations, normal forms, shell may be used to define a kind of partial Alexander polynomials, split links, symmetric pressure corresponding to that shell. The polynomials.j>raid index, signature of links, thermodynamic limit for this partial pressure tol"l;gllinks, Oi n ~j. classification of link types function as the finite lattice tends to zTI is stud­ in Oi(i f. 6), link type invariant for ft6, partial ied and shown to be a Legendre transformation classification of link types in ftfb braid index of of the classical pressure function. This result is product links, product links in Ui(i f. 6), and used to analyze the structure of C (cp; ~). proofs. Appendices include the table of braid The analysis of C(cp; ~) is used to show index for classical knots. that a Markovian time evolution model for a lat­ tice gas introduced by F. Spitzer exhibits con­ ON THE GENERAL ROGERS-RAMANUJAN densation of liquid droplets from an initially THEOREM by George E. Andrews gaseous phase as time passes. It is also used to study translation invariant Markov random fields Number 152 and Markov chains. Ergodic Markov random 86 pages; list price $3 .10; member price $2. 33; fields are shown to be characterized by their ISBN 0-8218-1852-X property of invariance with respect to a certain To order, please specify MEM0/152 group, G. This G is essentially the group of In this Memoir, the author extends previ­ permutations of particles which preserve the ous generalizations of the Rogers-Ramanujan number of adjacent pair of particles of different identities. Special cases of the types. main theorem in­ clude B. Gordon's generalization of the Rogers­ Markov chains are given a global charac­ Ramanujan identities, the Gtnlnitz-Gordon iden­ terization. They are shown to be limits as n .... tities and their generalization, and I. J. Schur's oo of states on strips of length n which give second partition theorem and its generalization. equal weight to all configurations of particles of In the conclusion, two conjectures are made a certain "total energy" within the con­ strip and zero cerning further work in this area. to other configurations. The "total energy" is the The primary subject classification for sum of energies between adjacent events this in the Memoir is Number Theory-mainly chain, and these Partitions; energies depend on the stochas­ the secondary classifications are Combinatorics tic matrix of the chain. and Ordinary Differential Equations. The Memoir is divided into five sections:

350 TRANSLATIONS OF operators, 9. Elements of nonlinear analysis; MATHEMATICAL MONOGRAPHS Chapter II. The Linear Equation with a Constant Operator: 1. Solution of the homogeneous and STABILITY OF SOLUTIONS OF DIFFERENTIAL inhomogeneous equations, 2. The behavior of EQUATIONS IN BANACH SPACE by Ju. L. the solutions of the homogeneous equation at Dalecki1 and M.G. Krein infinity, 3. Boundedness of the solutions of the homogeneous equation, 4. Conditions for the ex­ Volume 43 istence of a bounded solution of the inhomoge­ 386 pages; list price $36. 40; member price neous equation; $27. 30; ISBN 0-8218-1593-8 Chapter III. The Nonstationary Linear Equation. To order, please specify MMON0/43 Bohl Exponents: 1. Evolution operator and for­ mulas for the This Monograph is concerned with differ­ solution of linear equations, 2. Integral ential equations for vector functions with values inequalities. Com paris ion of evolution in infinite-dimensional spaces. In the first chap­ operators, 3. Stability and bistability. Compara­ ter the authors recall the basic definitions and ble equations, 4. Ljapunov and Bohl exponents classical theorems of the theory of Banach of the homogeneous equation on [0 oo) 5. Con­ spaces. It is assumed, however, that the funda­ dition for boundedness on [0, oo) of th~ solutions mentals of the geometry of Hilbert spaces are of the inhomogeneous equation, 6. Equations known to the reader. A reader unfamiliar with with precompactly valued operator functions; functional analysis can read the book by assuming Chapter IV. Exponential Splitting of the Solutions everywhere that it is dealing with finite-dimen­ of the Linear Equation: 1. Conjugation operators sional spaces; the authors hope that even in this f?r projections, 2. Kinematically similar equa­ case a reader qualified in the theory of differen­ tions, 3. Exponentially dichotomic equations, 4. tial equations will find something new in the book. Exponential splitting, 5. Stability of the Bohl The easiest chapter to understand is the exponents of an exponential splitting, 6. Expo­ second chapter, in which stationary linear equa­ tential splitting in a Hilbert phase space; tions are considered. In the third and fourth Chapter V. The Equation with a Periodic Oper­ chapters the authors present the general theory ator Function: 1. Monodromy operator. Reduc­ of nonstationary linear equations; in particular, ibility, 2. Exponential dichotomy of the solutions they carry out a detailed investigation of the im­ of the periodic equation, 3. Localization theorems portant notion of an exponential splitting of the on the spectrum of the monodromy operator solutions of an equation. More restricted classes (!!I = 8':)), 4. Canonical differential equations, 5. of linear equations are studied in the fifth and Second order equations (infinite-dimensional sixth chapters, the former being devoted to analog of Hill's equation), 6. Expansion of the equations with periodic coefficients, especially logarithm of the monodromy operator in powers the strong stability of canonical equations, and of a small parameter; the latter to equations in the complex plane with Chapter VI. Linear Differential Equations in the a regular singularity. The seventh chapter is Complex Plane: 1. The equation with a regular devoted to the consideration of problems con­ singularity. Simplest case, 2. Case of integral nected with the behavior of the solutions of non­ differences of eigenvalues, 3. Finite-dimen­ linear equations. In the eighth chapter the authors sional case (dim !!I < oo), 4. Case of integral return to linear equations and consider special differences of points of the continuous spectrum asymptotic developments of the solutions with of A0; respect to a large parameter of the equation. Chapter VII. Nonlinear Equations: 1. Existence The exercises are of a diverse character of solutions. Basic stability concepts, 2. Stabil­ and pursue different ends. Some of them are ity and instability of the solutions of the nonlinear routine drill problems for the main text. Others equation with a stationary principal part, 3. The together with hints to their solution contain brief nonlinear equation with a nonstationary principal developments of important additional questions part, 4. The nonlinear equation with a principal which were not included in the main text in order part the spectrum of which does not intersect to avoid unduly increasing the size of the book. the imaginary axis, 5. Stable integral manifolds, Finally, many exercises contain additional his­ 6. Integral manifolds bounded for t ... + oo (or torical and bibliographical commentary and t ... - oo), 7. Averaging principle; formulations of the results of various authors Chapter VIII. Asymptotic Representation of the The headings of the chapters and their Solutions of a Linear Differential Equation with sections follow: a Large Parameter: 1. Approximate decomposi­ tion of the equation, 2. Estimate of the error, Chapter I. Some Information from the Theory of 3. The equation with a rapidly oscillating coef­ Bounded Operators in Banach Spaces: 1. General ficient. propositions on the geometry of Banach spaces and their linear mappings, 2. Functions of Each chapter includes as well a brief in­ bounded linear operators, 3. Solution of some troduction, a set of exercises and some biblio­ linear operator equations, 4. Exponential oper­ graphic notes . ator function, 5. Generalized Ljapunov theorem, The book also contains an extensive bibli­ 6. A theorem of B. Sz.-Nagy, 7. Hilbert space ography, and notation, author, and subject in­ with an indefinite metric, 8. Stable W-unitary dexes.

351 INTRODUCTION TO THE THEORY OF ENTIRE Introduction: 1, Standard notation, 2, Semicon­ FUNCTIONS OF SEVERAL VARIABLES by L. I. tinuous functions, 3. Convex sets and convex Ronkin functions, 4. On the growth of functions, 5. Distributions, 6. Multiple power series; Volume 44 Chapter 1. Subharmonic Functions: 1. Elementa­ 274 pages; list price $28, 50; member price ry theory of subharmonic functions, 2. Integral $21. 38; ISBN 0-8218-1594-6 representation of a subharmonic function, 3. To order, please specify MMON0/44 Functions subharmonic in the entire space, 4. Capacity of sets and related concepts, 5. Regn­ This Monograph presents the general larization, 6. Hartogs functions (n = 1), 7. theory of entire functions of several variables. Phragmen-Lindeltlf principle; The first work in this direction appeared as Chapter 2. Plurisubharmonic Functions: 1. Gen­ early as the beginning of this century; however, eral theory of plurisubharmonic functions, 2. an intensive investigation of entire functions in r-projection and r-capacity, 3, The sets Tt) en' stemming from the general upsurge of in­ and stJ for plurisubharmonic functions, 4. terest in the theory of holomorphic functions of Representation of functions plurisubharmonic several variables, began 25 to 30 years later. An throughout en, 5. Hartogs functions ( n > 1), important part in this respect was played by 6. Functions of class m, 7. Functions of class Lelong1s work, particularly his studies of \!:\· plurisubharmonic functions. The methods of the Chapter 3. Growth of Entire Functions of Sever­ theory of plurisubharmonic functions provided a able Variables: 1, Associated orders, associated general approach to several problems in the types and related concepts, 2. Growth of an en­ theory of entire functions and led to substantial tire function in one of the variables, the others results. The first two chapters therefore present being fixed, 3. Growth of an entire function of the general theory of subharmonic and plurisub­ finite order and distribution of singnlarities of harmonic functions and a few related specific the associated function, 4. Fourier transform of subjects. In the next two chapters, which are entire functions of exponential type, 5. Radial devoted to the theory of entire functions proper, indicator, 6. Operator methods; particular attention is paid to questions of growth Chapter 4. Distribution of Zeros of Entire Func­ and distribution of zeros of entire functions of tions of Several Variables: 1. Integration over several variables. To make the book self­ an analytic set, 2. Characteristics of the dis­ contained, the author has added an introduction tribution of zeros of entire functions, 3. On presenting certain material from the theory of analogs of the Weierstrass canonical product. functions of a real variable, the theory of distri­ butions, etc,, not usually included in general The book also includes a bibliography and subject university courses. and notation indexes. The book is organized into chapters and sections as follows:

352 VISITING MATHEMATICIANS Supplementary List The list of visiting mathematicians includes both foreign mathematicians visiting in the United States and Canada, and Americans visiting abroad, Note that there are two separate lists. American and Canadian Mathematicians Visiting Abroad Name and Home Country Host Institution Field of Special Interest Period of Visit Gil de Lamadrid, J, (U. S,A,) University of Munich, West Functional Analysis 9/74 - 8/75 Germany Phillips, Ralph (U.S.A.) Mittag-Lefler Institute Functional Analysis 3/75 - 5/75 Senechal, Lester (U.S.A,) University of Groningen Numerical Analysis 8/74 - 8/75 Shahin, Jamal (U.S.A,) American University, Cairo Differential Geometry 1974 - 1975 Foreign Mathematicians Visiting in the United States Andler, Daniel M, (France) University of California, Logic-Model Theory 9/74 - 2/75 Berkeley Becker, Ronald I, (South University of California, Differential Equations on Banach 9/74 - 2/75 Africa) Berkeley Spaces Bhattacharga, C. G, (Gbana) University of Waterloo Probability and Distribution 9/74 - 8/75 Bicudo, lrineu (Brazil) University of California, Universal Algebra-Model Theory 9/74 - 8/75 Berkeley Caffarelli, Luis (Argentina) University of Minnesota Partial Differential Equations 9/74 - 6/75 Davison, S, G. (England) University of Waterloo Quantum Theory of Solids and 9/74 - 8/75 Their Surfaces Dhombres, J, G, (France) University of Waterloo Functional Analysis 9/74 - 12/74 Ghoshal, Sudhanshen Kumar Princeton University Numerical Analysis, Functional 9/74 - 8/75 (India) Analysis, Fluid Mechanics Genizi, A, (Israel) University of Waterloo Multivariate Analysis and Statistical 9/74 - 4/75 Consulting Gingold, Harry (Israel) University of Southern Ordinary Differential Equations 9/74 - 6/75 California Herzberger, J. (West University of California, Applied Mathematics, Integral 9/74 - 12/74 Germany) Berkeley Analysis Hughes, D. R, (England) University of Pittsburgh Algebra and Geometry 1/75 - 4/75 Johnson, Claes G. L, (Sweden) University of Chicago Numerical Analysis 10/74 - 10/76 Joergensen, Troels (Denmark) University of Minnesota Complex Function Theory 1/75 - 3/75 Klingenberg, Wilhelm University of California, Differential Geometry 9/74 - 1/75 (Germany) Berkeley Kluvanek, I, (Australia) University of Pittsburgh Functional Analysis 9/74 - 12/74 Landrok, Peter (Denmark) University of Chicago Group Theory 9/74 - 12/74 Lickorish, WJ.lliam B. R, University of California, Algebraic Topology 9/74 - 1/75 (England) Berkeley Lim, Teck-Cheong (Singapore) University of Chicago Analysia 10/74 - 10/76 McConnell, John (England) University of California, Ring Theory 9/74 - 6/75 Berkeley Okuyama, A, (Japan) University of Pittsburgh Topology 5/75 - 8/75 Olsson, Jorn (Denmark) University of Chicago Group Theory 7/74 - 6/75 Paris, Jeffrey B, (England) University of California, Logic 9/74 - 1/75 Berkeley Plesken, Wilhelm (Germany) California Institute of Group Theory, Representation 9/74 - 8/75 Technology Theory, Algebraic Number Theory Presutti, Errico (Italy) Stanford University Ergodic Theory 9/74 - 7/75 Raghunathan, M. S, (India) University of Chicago Algebraic groups 5/75 - 6/75 Regev, Amitai (Israel) University of Chicago Algebra 7/74 - 10/76 Rehm, Hans P. (Germany) California Institute of Ideal Theory, Algebraic Number 9/74 - 8/75 Technology Theory

353 Name and Home Count!:,! Host Institution Field of ~ecial Interest Period of Visit Ruchti, Rene (Switzerland) University of California, Gelfand-Fuks Cohomology, Analysis 9/74- 8/75 Berkeley on Manifolds Sagher, Yoram (Israel) University of Minnesota Real Analysis 9/74 - 12/74 Schmid, Jurg (Switzerland) California Institute of Model Theory and Algebra 10/74 - 9/75 Technology Shizuta, Yasushi (Japan) University of California, Operator Theory 9/74 - 9/75 Berkeley Tanaka, T. (Japan) University of Pittsburgh Topology 10/74 - 5/75 van Douwen, E.K. University of Pittsburgh Topology 5/75 - 8/75 (Netherlands) van Moerbeke, Pierre Stanford University Probability 1/75 - 8/75 (Belgium) Young, Wo-Sang (Hong Kong) University of Chicago Analysis 10/74 - 10/76

354 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, dotes or geographical area become apparent. A first onnouncelllent will be published in thec/Vol~Ce.D if it contains a call for papers, place, dote, and subject (when applicable); a second announcement must contain reasonably complete details of the meeting in order for it to be published. Information on the preliminary planning will be stored in the files, and will be available to anyone desiring information on prospective conferences. All communications on special meetings should be sent to the Special Meetings Information Center of the American Mathematical Society.

January 1- December 13, 1975 April27-May 3 MATHEMATISCHES FORSCHUNGSINSTITUT Methoden und Verfahren der mathematischen Physik OBERWOLFACH (Mathematics Research Institute of Chairmen: B. Brosowski, Gllttingen; E. Martensen, OberwoHach) Karlsruhe Information: Participation in the meetings is only possi­ May 4-10 ble by personal invitation. Interested persons may apply Gruppen und Geometrien to the Institute. Chairmen: D. Higman, Ann Arbor; H. Salzmann, TUbingen January 1-4 May 11-17 Arbeitstagung Ringe, Moduln und homologische Methoden Chairman: H. Salzmann, Tttbingen Chairmen: F. Kasch, MUnchen; A. Rosenberg, Ithaca January 5-11 May 18-24 Kontlnuumsmechanik der festen Kllrper Gruppentheorie Chairmen: W. GUnther, Karlsruhe; H. Lippmann, Chairmen: W. GaschUtz, Kiel; K. W. Gruenberg, London; Karlsruhe B. Huppert, Ma!nz January 12-18 May 25-31 Wahl'scheinlicbkeitsrechnung in der Schule Numerische Methoden der Approximationstheorie Chairmen: R. Borges, Giessen; H. Dinges, Frankfurt Chairmen: L. Collatz, Hamburg; G. Meinardus, Erlangen; January 19-25 H. Werner, MUnster Mengenlehre und Modelltheorie June 1-7 Chairmen: K. P. Podewski, Hannover; M. Richter, Mathematische Methoden in der Biologie Tttbingen; E.-J. Thiele, Berlin Chairman: K. P. Hadeler, TUbingen January 26-February 1 June 8-14 Stllrungstheorie und Operatorfunktionen Distributionen, Convolutionen und partielle Chairmen: B. Gramsch, Kaiserslautern; G. Neubauer, Differentialgleichungen Konstanz Chairmen: J. Wloka, Kiel! Z. Zielezny, New York February 2-8 June 15-21 EinhUllende Algebren von Lie-Algebren Masstheorie Chairmen: P. Grabriel, ZUrich; R. Rentschler, Orsay Chairmen: A. Ionescu-Tulcea, Evanston; D. Klllzow, February 9-15 Erlangen Komb!natorlk June 22-28 Chairmen: D. Foata, Strassburg; A. Kerber, Aachen Riesz-Spaces February 16-22 Chairmen: W. A, J. Luxemburg, Pasadena; H. H. Funktionentheorie Schaefer, TUbingen Chairmen: D. Gaier, Giessen; J. Winkler, Berlin; June 29-July 5 H. Wittich, Karlsruhe Kategorien February 23-March 1 Chairmen: J. W. Gray, Urbana; H. Schubert, DUsseldorf Medizinische Statistlk July 6-12 Chairmen: H. Klinger, DUsseldorf; R. Repges, Aachen Funktionalgleichungen March2-8 Chairmen: J. Aczel, Waterloo; o. Haupt, Erlangen; Partielle Differentialgleichungen A. M. Ostrowski, Basel Chairmen: E. Heinz, G!lttingen; G. Hellwig, Aachen July 13-19 March 9-15 Graphentheorie Mathematische Stochastik Chairman: G. Ringel, Santa Cruz Chairman: K. Krickeberg, Bielefeld July 20-26 March 16-22 Wahrscheinlichkeitstheorie in Banach-R!I.umen GeWilhnliche Differentialgleichungen und Anwendungen Chairmen: A. Beck, London; K. Jacobs, Erlangen Chairmen: H. W. Knobloch, Wllrzburg; R. Reissig, July 27-August 2 Bochum Numerische Methcden der Optimierung und March 30-April 5 Operationsforschung Kommutative Algebra und algebraische Geometrie Chairmen: W. Oettli, Mannheim; K. Ritter, Stuttgart Chairmen: E. Kunz, Regensburg; H.-J. Nastold, August 10-16 MUnster; L. Szpiro, Paris Algebraische Zahlentheorie April6-12 Chairmen: H. Hasse, Hamburg; P. Raquette, Heidelberg Arbeitstagung August 17-23 Chairmen: M. Kneser, Gllttingen; P. Raquette, Heidelberg Harmonische Analyse und Darstellungstheorie topologischer April13-19 Gruppen · ~adratische Formen Chairmen: H. Lept!n, Bielefeld; E. Thoma, MUnchen Chairmen: M. Knebusch, Regensburg; A. Ffister, Mainz; August 24-30 W. Scharlau, MUnster Himmelsmechanlk April20-26 Chairmen: E. Stiefel, ZUrich; V. Szebehely, Austin Mathematische Loglk Chairmen: K. SchUtte, MUnchen; E. Specker, ZUrich

355 August 31-September 6 Is hoped that proceedings of the seminar will be published. Topologie loformation: E. Mendelsohn, Department of Mathematics, Chairmen: D, B. A. Epstein, Coventry; T. tom Dieck, University of Toronto, Toronto, Ontario, Canada M5S1A1 Saarbrucken; C. B. Thomas, London September 7-13 February 13-14, 1975 Homotopietheorie EIGHTH ANNUAL COMPUTER SCIENCE AND STA­ Chairmen: T. tom Dieck, Saarbrucken; D. Puppe, TISTICS: A SYMPOSIUM ON THE INTERFACE Heidelberg University of California, Los Angeles September 17-20 Sponsors: Southern California chapter of the American Zufallsschwingungen und deren Stabllitll.t Statistical Association; Statistical Computing Section of Chairmen: W. Wedig, Karlsruhe; F. Weidenhammer, the American Statistical Association; Los Angeles chapters Karlsruhe of the Association for Computing Machinery and ACM September 21-27 Slgmetrics Geometrle Program: This meeting will provide a forum for persons Chairmen: P. Dombrowski, Kllln; K. Leichtweiss, interested in the interrelationship between computer Stuttgart science and statistics and will encourage continuing September 28-0ctober 4 dialogue among those working in these fields, Grundlagen der nichtlinearen Geometrie loformation: Avis Williams, Department of Biomathe­ Chairmen: A. Barlett!, Bologna; W. Benz, Hamburg; matics, University of California, Los Angeles, California H. Karzel, Munchen; R. Lingenberg, Karlsruhe 90024 October 5-11 Funktlonalanalys is March 28-31, 1975 Chairmen: H. Kllnlg, Saarbrucken; G. Kllthe, Frankfurt; SIXTH NATIONAL MATHEMATICS CONFERENCE OF H. H, Schaefer, Tttbingen; H. G, Tlllmann, Mainz !RAN October 12-18 Jundi Shapur University, Ahwaz, Iran Arbeitstagung Deadline for Abstracts: December 21, 1974. Abstracts Chairmen: M. Kneser, Gllttingen; P. Raquette, Heidelberg should be accompanied by vitae and sent to the Organiz­ October 19-25 ing Committee, Sixth National Mathematics Conference, Problemgeschichte der Mathematik Department of Mathematics, Jundi Shapur University, Chairmen: B. Bockstaele, Heverlee-LIIwen; C, J, Scriba, Ahwaz, Iran. Berlin loformatlon: Ali Amid!, Department of Mathematics, October 26-November 1 JuDdi Shapur University, Ahwaz, Iran Operatorungleichungen Chairmen: N. W. Bazley, Kllln; J. Schrllder, Kllln November 2-8 April 15-17, 1975 Zahlentheorie (insbesondere elementare und analytlsche COMPUTATIONAL METHODS IN NUCLEAR ENGINEERING Zahlentheorie) Mills Hyatt House, Charleston, South Carolina Chairmen: H, E. Richert, Ulm; W. Schwarz, Frankfurt; Information: Henry C. Honeck, Savannah River Labora­ E, Wirsing, U1m tory, Aiken, South Carolina 29801; Weston M. Stacey,Jr., November 9-15 Argonne National Laboratory, 9700 South Cass, Building Fortbildungslehrgang rur Studienr!tte 208, Argonne, Ulinois 60439 Chairman: Unknown November 16-22 Numerische Behandlung von Differentialgle!chungen, April 30-May 2, 1975 insbesondere mit der Methode der f!niten Elemente FORTY-SEVENTH NATIONAL ORSA MEETING Chairman: J. Albrecht, Clausthal-Zellerfeld TIMS 1975 NORTH AMERICAN MEETING November 23-29 Palmer House, Chicago, illinois Automatentheorie und Formale Sprachen Program: The program will consist of over 60 technical Chairmen: G. Hotz, Saarbrucken; H. Langmaack, sessions covering· a wide range of theoretical and applied Saarbrucken; H. Walter, Darmstadt topics. Several sessions will emphasize, in the theoreti­ November 30-December 6 cal area, the stochastic and programming methodology. Angewsndte Mathematische Statlstlk The application areas will range from sessions on ab­ Chairmen: D, Morgenstern, Hannover; H. Witting, stract modeling to sessions concerned with very practical Freiburg aspects of real-world problems. December 7-13 Speakers: Richard Bellman, Professor of Mathematics, Fragen zur Didaktlk der Mathematik Electrical Engineering, and Medicine, University of Chairman: H. Kunle, Karlsruhe Southern California; Herbert A, Simon, Professor of Computer Science and Psychology, Carnegie-Mellon January 6-10, 1975 University, REGIONAL RESEARCH CONFERENCE ON FOLIATIONS Deadline for abstracts: November 1, 1974, Address ab­ OF MANIFOLDS stracts to Brian Schaefer, Contributed Papers Chairman, Washington University, St, Louis, Missouri IE/MS Department, Northwestern University, Evanston, Program: Ten principal lectures by H. Blaine Lawson, of Illinois 60201, the University of California, Berkeley, There will also be Information: WJ.lliam P. Pierskalla, Meeting Chairman, provisions for informal working sessions after each IE/MS Department, Northwestern University, Evanston, lecture. Illinois 60201 Support: (anticipated) National Science Foundation Deadline for applications: There will be support for about May 20-22, 1975 25 invited participants. Inquiries should be made before TENTH NEW ZEALAND MATHEMATICS COLLOQUIUM December 1, 1974, University of Otago, Dunedin, New Zealand loformation: Gary Jensen, Department of Mathematics, loformation: Gloria Olive, Department of Mathematics, Washington University, St. Louis, Missouri 63130 University of Otago, Dunedin, New Zealand

January 6-10, 1975 May 20-24, 1975 SEMINAR ON ALGEBRAIC ASPECTS OF COMBINATORICS INTERNATIONAL SYMPOSIUM ON INTERVAL MATHE­ University of Toronto, Toronto, Ontario, Canada MATICS Registration: The $25 registration fee wlll be waived for Universitll.t Karlsruhe, Germany graduate students and unemployed, Information: Karl Nickel, Universitllt Karlsruhe, Institut Contributed papers: 20-minute presentations, 20 lecture ft1r praktlsche Mathematik, Postfach 6380, 75 Karlsruhe times to be filled on a first come, first served, basis. It 1, Federal Republic of Germany

356 LETTERS TO THE EDITOR

Editor, the cJ{oticei) Perhaps concerted action of the whole American scientific community In the publication might be noticed by the of the Mathematics Ac­ South African government, tion Group but certainly no such for July 30, 1974 I discovered that act of ours will the Council be. of the American Mathematical Soci­ The ety has AMS and the MAA did put pressure on cancelled the reciprocity agreement with its Southern the members by a firm policy of hold­ South African Mathematical Society. In the ing no segregated same publication meetings. That policy made I read about the assassination no moral judgement of Mr. Onkgopotse of us, and was not intended A. Tiro of the University of to influence state the North in South Africa governments. For not deseg­ after his expulsion regating faster, and flight to Botswana it imposed on us the rather in September 1973. minor penalty In view of of excessive travel time to meet­ this information I want to ap­ ings and it plaud the was a constant reminder that our action of the Council. I hope that the colleagues disapproved Society of which of the way we lived. That I have been a member for forty reminder was good years will for us, and the AMS and MAA recognize to the full its responsibili­ were better organizations ties to Mathematicians for insisting on what as well as to Mathemat­ was morally correct. I believe ics. After all, if there we should follow were no mathematicians that precedent in this case. there would be precious little mathematics. But that is not my principal reason for op­ Harry Goheen posing the agreement. Does anyone believe we Faculty Advisor have many Black members who would not be of­ African Students Association fended and hurt if this extension of reciprocity Oregon State University to South Africa had gone on? Would saving a few South African mathematicians a few dollars on Editor, the c#oticti) journals advance the cause of mathematics so greatly as to warrant the neglect of the convic­ A letter in this section made the case for tions of our Black members ? The answer is approval of a reciprocity agreement between the certainly no. (I say nothing of the effect on the South African Mathematical Society and our own many white mathematicians who share my views. ) Society. The Council has wisely (I believe) de­ It is easy to enter into such an agreement with­ cided against implementing that agreement, but out full thought as to its implications; but to con­ reasons to support their action have never ap­ tinue it would be a retreat from the firm policy peared in the CJ(oticei). Since a letter of mine of anti-discrimination in this country that the to the Council apparently initiated the discus­ Society has so successfully maintained for all sion that led to its decision, I would like to give these years. here my own reasons for opposing the agreement. Gail Young Professor Mostert• s letter [these CJioti.ceiJ 21(1974), 93-94] supporting the agreement de­ EDITORIAL COMMENT: scribes a situation Following a request that is very familiar to a from the South African Mathematical white Southern mathematician. Society, the Like Southern American Mathematical Society university faculties established a rec­ in the '40s and '50s, the iprocity agreement with the SAMS appears to be SAMS. The agree­ a group mostly liberal on ment was cancelled by the the race question, opposed Council of January 14, to apartheid, and 1974 and the cancellation was doing what little they safely reported to the can to ameliorate membership in the BULLETIN, it or change it. That is what most 80, 652-657 of us in the [July 1974, seep. 655]. When South did. We believed that our a member of the presence there SAMS received a routine notice of the helped change social and political cancella­ opinion, and tion, the Secretary of SAMS, N.J. certainly helped change the H. Heideman, universities. We on April 4, 1974 inquired into the could not see that our departure situation. The to the Northern major portion of the reply from universities would contribute the Secretary, to anything besides dated April18, 1974, was as follows: our comfort. But few of us ever did anything that put us in serious physical danger or that really Our Council in August 1972 authorized our imperiled our jobs. I suspect the South African Executive Director, Dr. Gordon Walker, mathematicians are much the same as we were. to establish a reciprocity agreement with I cannot possible say they are thereby disquali­ The South African Mathematical Society. fied from membership in the AMS unless I say The Council gave more serious considera­ that I and my colleagues were not fit to belong, tion than usual to this step because of the and that I do not believe. In fact, I would op­ concern of some of the members of the pose strongly any attempt to keep individual Council over the possible operation of a South African mathematicians from regular color bar to membership in the SAMS or membership in the Society. to effective participation by mathemati­ Nordoibelievethatthepower of the AMS cians in its meetings, irrespective of is such that tyrants cringe at our resolutions. their color. When the question first arose,

357 instead of authorizing the agreement, the cians who are members of our Society, Council commissioned one of its members, but all our members. Furthermore the Professor Paul Mostert, to report on the termination of the reciprocity agreement problem. On the basis of his report and will have no influence at all on the polit­ recommendation at the August 1972 meet­ ical situation in South Africa, since our ing, the agreement was established. Dur­ Society is a non-political organization ing public discussion subsequent to the open to all mathematicians. We endeavor meeting of August 1972, he published a as a matter of principle, not to allow po­ letter [ cNotietiJ, January 1974, p. 93-94] litical considerations to disturb the ami­ of which I have attached a copy. I assure able relations existing between members. you that it is an accurate account of what We have in the past bad to defend this he reported at the Council meeting when principle. We sincerely hope that the the initial decision was made. AMS will adhere to similar principles The fact of the establishment of the reci­ in this respect. procity agreement was published in the I shall appreciate it if you would pass c}/oticei) of June 1973. The page is attached. this information on to your committee Its existence aroused the indignation of a which has been charged with examining vociferous segment of our membership. the principles of reciprocity. I think it is their position that it is an of­ fense to the black members of the AMS Editor, the c}/oticei) for the AMS to have a mutually conve­ nient relationship with a Society in a coun­ I was very much interested in the report try where there is racial segregation and [these c}/otietiJ, June 1974, pp. 180-185] of the associated unequal opportunity. However, panel discussion on "The role of the dissertation there are many shades of opinion. There in the Ph. D. program", because it illuminated so is a segment of the membership which very clearly some attitudes and habits of thinking holds that the laws and customs of another that prevail in some parts of the mathematical com­ country are not a proper concern of the munity and that make it difficult for all of us to AMS as an organization, whatever the di­ adapt to the changed circumstances. I am writing verse social views of its individual mem­ this letter, from my vantage point as a mathe­ bers. matician interested in applications and in teaching applications, to call attention to some such at­ At the meeting of the Council of January titudes. 1974 "the Executive Director was instruct­ 1) "The Ph. D. dissertation represents the ed to cancel the reciprocity agreement with statement •This person is a mathematician'; the the South African Mathematical Society and dissertation represents the proof that the person the President was authorized to appoint a has created mathematics". The implication is committee, to include black mathematicians, that a person either is or is not a mathematician, to examine the principles of reciprocity and that he either does or does not create. To and to report to the Council." think in rigid categories such as these is typical The Committee is just now being appointed. for mathematics ("a theorem is either true or I have no idea what sort of principles or false; a mathematical object either does or does crit.eria they will advance or what kind of not exist"). However, when dealing with practical reaction to them another Council meeting situations (and the job crisis is an eminently will have. I shall inform you of develop­ practical situation) they are of little value, not ments but I expect many months to pass only because concepts like "mathematician" and before there is any action. "creativity" are not clearly defined, but also because even if they were defined, the practical Professor Heideman• s reply was dated problem would not be brought one inch nearer to April 26, 1974. Most of his letter, including a solution by any axiomatic reasoning about these the second and third paragraphs below in entire­ concepts. More important ingredients in finding ty at his request, is reproduced here. solutions for problems such as ours are common It was with shock and surprise that I sense, and political acumen. A mathematician read of the decision of the AMS to dis­ applying mathematical reasoning to an extra­ continue the Reciprocity agreement so mathematical problem should remain aware of soon after its inception. the limitations implicit in the rigidity of the model. Here, perhaps, the applied mathema­ I feel I must point out to you that since tician is at an advantage; he knows, at least, the founding of our Society, nobody was where not to apply mathematics. barred from membership, and the only 2)1•we should produce better Ph.D. •s." entrance qualification is an Honours de­ The present state of the mathematical community gree in Mathematics. Our Society was much resembles an industrial corporation manu­ established to advance mathematical know­ facturing an automobile being able to cruise at ledge and teaching at all levels and in all 200 mph, but only on very specially designed sections of the population, and this is still highways. Now the demand for such an automobile our sole aim. turns out to be smaller than anticipated. To in­ The denial of a reciprocity agreement is crease the quality of the Ph. D. in the sense of not only going to effect white mathemati- William Browder is much like deciding to in-

358 crease the speed of the automobile from 200 mph Princeton has been more successful in this re­ to 300 mph. gard than any other institution in the U.S. A. 3) "The best exposition is practically Clearly Princeton should continue to do what it always done by some of the most creative re­ can do best. search mathematicians." The best exposition of "Perhaps there are too many imitations of new and systematically developed mathematics Princeton which don1t yet succeed in adequate is quite frequently done by the innovator himself. measure. Some additional models of Ph. D. train­ But this is not the only kind of exposition we ing are in order as pointed out by Karel deLeeuw need. We need exposition which opens windows, during the panel discussion. For my part, I still for views of the extramathematicallandscape, urge that 1 some of our theses tackle some of the Take elementary linear algebra. This is tra­ relevant problems in the general structure of ditionally regarded as an introduction to mathe­ mathematical knowledge. 1 My attempts to do matical thinking, preparing for all the wonderful this at Chicago have hitherto not succeeded, but larger things to come. None of the currently it is still worth trying. Ideally, we should get used major texts emphasizes factor analysis, or creative Ph. D. 1 s in Mathematics proper who are the input-output matrix, two very important willing and eager to apply their knowledge on a applications of linear algebra in the outside broad front. If I were talking again on that panel, world, broadly used and applied by people at I would also emphasize the Ph. D. as a consul­ scientific meetings in numbers far exceeding tant. Sound applications of Mathematics are to those attending an annual meeting of the AMS, be emphasized-but not always easy to reach. No other professional group is taking a similarly Twice recently I have been excited to learn of a disdainful view of its own role in society. Could new application of some ideas which concerned you imagine a law school where students are me-only to discover that the applications were taught only to play an abstract game called "law", quite superficial. Good Mathematics is never without ever discussing any actual law cases as easy." presented in court? Saunders Mac Lane 4) "Would Chicago be interested in hiring someone who has not written a research thesis?" Editor, the cJiotiai) The quote comes close to fostering the image that I should like to add to the many published the sole purpose of the Ph.D. is to produce a pool views on employment and related issues in grad­ of potential professors of mathematics, The Uni­ uate education. I elaborate on these views and versity of Chicago will always be free to hire any describe operations research (OR) consulting in Ph, D. to its taste. But unless we consider mathe­ an article to appear in the Monthly, fall 1975, a .matical education a closed system we should also digest of which is in SIAM News, October 1974. try to create mathematicians who are able to OR itself has considerable long-range em­ serve society in more direct, if more mundane, ployment potential. It is still a young and grow­ ways, ing science, upon which new demands are made Let me add that the San Francisco dis­ as the world grows more complex. cussion brought to light some very good, practical It is feasible for a Ph. D. in basic (i.e."pure") ideas, I liked Herstein•s idea to accept lecture or applicable mathematics who obtains a position in notes for new courses (topic x from the point of OR to have a fruitful career compatible with his edu­ view of topic y) as dissertations, And I liked the cation. I assert this because it has been repeat­ remark of Mac Lane that "applied mathematicians edly demonstrated by my colleagues. The key ad­ have gradually made us recognize that (besides ditional requisites are (1) strength in research theorems) there are also methods of compu­ ability, preferably evidenced by a good thesis; tation", a point that I emphasized in a talk that I (2) a sincere interest in applications (not just a gave at Missoula last summer. flight from the academic job market); (3) ability Peter Henrici to model operational problems; (4) research ver­ satility, i. e. , facility in shifting from one type of application to another; (5) expository ability; Editor, the cJiotiai) (6) ability to communicate with scientists of other disciplines and non-scientists; and (7) a temper­ When a previous draft of Professor Henri­ ament which accepts inexactitudes. ci• s letter came originally addressed to me, I OR work can be satisfying to a mathemati­ responded on July 11, 1974 as follows: cian through importance of the applications, as "Many thanks for your stimulating letter opposed to the beauty of one's methods in ab­ about that panel discussion on the role of the dis­ straction. Most OR work makes few explLcit de­ sertation in the Ph. D. degree. The problem is a mands on graduate mathematics, althoug t ere tough one, and this panel (like most panels) are many counterexamples. However, many of a didn't settle it. The problem of getting creative mathematician's OR results which appear to be mathematicians is also a very subtle one. It within reach of non-mathematicians are not ob­ isn't comparable to the production of automobiles, tained by such others, or at least not as well. and, if it is, the production of Cadillacs and Hence, strength in mathematics makes a differ­ Mercedes in Mathematics is always going to be ence. Most mathematicians in OR have ample in order. This is why William Browder• s re­ opportunity for publishable research, e. g. , in marks are very much to the point. His institu­ OR-related applicable mathematics. tion at Princeton has been extraordinarily suc­ The catch regarding findin! employment is cessful in training creative mathematicians. Ac­ that most industrial employers o not hold the cording to some plausible numerical measures, views expressed here, e. g., see the survey by

359 Gaskell and Klamkin in the Monthly, 1974, p.699. ciplinary mutual respect for the presence and My advice to a Ph. D. in basic (or applicable) cultivation of ability to do original research. mathematics seeking industrial employment is as Daniel H. Wagner follows. "If your heart remains in academia, I doubt that you will succeed in industry. If your EDITOR'S NOTE: Dr. Wagner is founder and interest in applications is sincere and if you have president of Daniel H. Wagner, Associates, an done excellent work in your academic research, OR consulting firm, staffed primarily by thir­ then you should not lack confidence in performing teen Ph. D.'s in the mathematical sciences, well merely from possible lack of applied train­ mostly in basic mathematics. The firm does ing. However, to convince prospective employ­ extensive work on naval search problems among should acquire some applied trappings. ers, you other areas. Also in applications you will really need tools usually not taught in basic mathematics, many Editor, the c}/ot.iaiJ fairly easy to acquire. You have a better chance of employment by managers who are mathemati­ This letter is intended to acquaint American cians. If you obtain a position in an organization Mathematicians with developments in Mathematics active in complex quantitative scientific problems, Education and Research in the Philippines, the your graduate education, particularly your the­ third largest English speaking country. I was re­ sis research, should serve you well." cently privileged to lecture under the auspices It seems obvious to many that a graduate and support of the Mathematical Society of the student in mathematics should be learning things Philippines at the Ateneo de Manila University, he will use to earn his living. To me it is equally De La Salle College, and Mindanao State Univer­ obvious that his overriding objective is to learn sity. Up to now, graduate work in Mathematics mathematics. He initially seeks a good general beyond the Master's level has not been readily knowledge via courses in analysis, algebra, and available within the Philippines and Ph. D.'s have topology. Subsequent work is more specialized, been obtained abroad, many in the States. The but the accent must be on cultivation of original­ level of undergraduate and first year graduate ity. In the thesis research, probably for the first education is quite high in the selective institutions, time, he solves hard problems without knowing and many of these students have been successful in advance whether they have nice solutions. abroad (and many of the best have not returned). An applied practitioner needs this training. There is an intense need for scientifically and It is crucial that the thesis be well done and technically trained personnel throughout the Phil­ be in an area to the student's liking. If this ippines, and a genuine acute shortage in the choice is in applicable mathematics, employment mathematical and physical sciences. There is is within easier reach; however, excellence in a pronounced shortage and need for Filipino this climax of his formal education is paramount. Ph. D. •s in our disciplines, pure and applied. I strongly agree with W. Browder (these c}/of.iai), Three leading institutions-Ateneo de Ma­ 1974, p. 180) in opposing the granting of non­ nila University, University of the Philippines, research doctorates, certainly when industrial and De La Salle College-are in the process of employment is a goal. banding together in a consortium with sufficient Curricula in applicable mathematics must breadth to offer a high quality graduate program guard (this a caution, not an accusation) against in mathematics and the physical sciences. This emphasis on teaching students to solve problems will improve in large measure the whole scienti­ which have already been solved, rather than fic environment. Separately, President Marcos equipping them to devise tools to solve unknown has approved in principle a decree (widely pub­ new problems confronted in applied work. An lished in the press ; this much attention to mathe­ imbalance not favorable to the latter would sug­ matical sciences we do not get at home) estab­ gest augmenting the basics; these change slowly, lishing a Philippine Institute for Advanced Study while within a practitioner's career, applicable in Mathematics and Physical Sciences. This techniques will change drastically. An applied would for the first time provide a home for re­ mathematician should be educated in basic mathe­ search in mathematical sciences. This is still in matics well beyond the level of his applications. the planning stage. We know the profound influ­ I think the best single mathematics course to pre­ ence that such an institution can exercise. pare for an OR career is real analysis. An Institute, run directly by scientists of the It is a common misconception among mathe­ highest caliber without bureaucratic interference, matics professors that industry is a logical home is likely to be of real scientific and educational for their less talented but relatively personable impact. The major risk in the Philippines (since students. Although there are many non-demand­ the Institute will be government funded) is that ing uses of mathematics in industry, these are this direct administration of science by active not the roles for which Ph. D. •s should be pro­ scientists might be in the end thwarted by bureau­ duced. A Ph. D. should be capable of independent cracy. It would be heartening to our Filipino col­ research and this ability is the dominating value leagues if some of our knowledgeable prominent in industrial R & D as well as in academia. researchers expressed the importance to science In past generations the giants of mathemat­ of such an endeavor (and especially the impor­ ics moved freely between basic and applied re­ tance of scientific independence) directly to the search. It is unfortunate that there has been con­ President of the Mathematical Society of the siderable aloofness in both directions between Philippines, Professor Bienvenido F. Nebres, basic and applied mathematicians in recent de­ Dean of the Graduate School and College of Arts cades. What is greatly to be desired is interdis- and Sciences of the Ateneo de Manila University.

360 For our part, what can we do to help our Hilbert space, and category theory. Clearly a Filipino colleagues? First, under the present perfect balance will never be possible, since system, most of the faculty there will still have there are so many fields and so much freedom to go abroad for advanced degrees, and financing in the topics which invited speakers may choose. from Filipino sources is very limited. In the It is equally clear that a program which starts · United States, we have a larger Ph. D. production by subdividing mathematics into 20 fields is like­ program than We need, and many of our institu­ ly to miss things that fall between the fields, and tions have many graduate teaching assistantships is likely to miss other topics that may not be available. I strongly urge my colleagues to sup­ well known to the committees immediately in­ port the admission and financial support of these volved. qualified English speaking students of mathemat­ Second, there should be more expository ics, who are sorely needed with advanced train­ addresses, enough so that they can compete with ing at home, many of whom have solid college each other. At Vancouver there were 17 such ad­ teaching experience. dresses. They were in general excellent, but Second, there is a great need in the expand­ only one was presented at a time. There was ing higher education system here in the Phil­ simply no auditorium big enough to contain all ippines for scientific visitors at all levels-from those who wished to hear, while on the simulta­ fresh Ph. D. •s who may help improve offerings neous closed circuit TV versions the formulas and give seminars to the staff at the newer insti­ were often very difficult to read. More exposi­ tutions here, to well-known researchers and tory addresses could give a more balanced pic­ educators to give breadth to the faculty intellec­ ture of the current state of mathematics. If two tual life at the major institutions. In particular, or more such addresses came at one time, they the Mathematical Society of the Philippines will would assist the attendance problem. be hosting a Southeast Asian Summer Institute in Third, the sessions for short communica­ Graph Theory from 21 April to 16 May 1975 (de­ tions offered on the initiative of individuals were tails will be announced in the cJ{ofkti) ). successful. At the Congress in Nice no such in­ The Filipino funding available.is quite dividual communications were given in person, modest and on a person-to-person or institution­ perhaps because of a belief that it is possible for to-institution basis; but living costs are low. the organizers to select the "most important" It is hoped that once an lAS is set up, it will fields of mathematics and to cover all of these serve as a clearing house and center for such fields adequately by invited addresses. The Con­ visits. On the U.S. end, NSF has a new SEED gress at Nice demonstrated that not even the program (Scientists and Engineers in Ecqnomic French were really able to do this. Even if there Development-Office of International Programs) are fields which are unimportant their practi­ which can in principle finance travel and salary tioners may wish to meet, and the smaller ses­ for visits approved by a host institution. sions of communicated papers offer an excellent Our Filipino colleagues are infinitely over­ possibility for people from different countries worked because of shortages in personnel, and with common interests to get together. any help we can offer would certainly be appre­ Fourth, organized splinter groups or sem­ ciated. inars on special topics are effective. During the On a personal note, I would like to mention last three Congresses (Moscow, Nice, and Van- that in my visit to Marawi (Mindanao State Uni­ . couver) it has indeed been possible for energetic versity), I saw an inspiring example of a univer­ senior mathematicians to organize such splinter sity which is a spearhead of cultural and economic groups in their own fields. At Vancouver the ac­ development for a whole region, and in which the tual facilities for doing this were splendid. mathematicians are playing a great role. MSU Rooms were available, permission was granted has clearly become the repository and preserver easily, and the notice was spread by an effective of Maranao culture. closed TV network. What is needed is better ad­ Anil Nerode vance announcement of these possibilities. When people know that this can be done it is likely to Editor, the cJ{otira) attract them to a Congress. Finally, the planning of the program of a The 1974 International Congress of Mathe­ Congress should be made a more public matter maticians in Vancouver, B. C. is now over, leav­ and more open to initiatives from the Interna­ ing with all those who attended the memory of tional Mathematical Union (IMU) and other many excellent mathematical lectures, and of sources. In the recent past (since Stockholm in splendid and gracious hospitality, However, our 1962) this program has been planned by a nine­ pleasure and gratitude for these accomplishments member Consultative Committee with Chairman should not blind us to the fact that there may be and four members appointed by IMU, while four substantial questions about the arrangement of other members are appointed by the host coun­ the program, particularly of invited lecturers, try. This Committee then appoints subcommit­ for this Congress. tees for specified fields of mathematics and First, there was indeed an imbalance in oversees the choice of invited speakers. More­ the representation of certain fields of mathemat­ over, this whole rigid process is kept secret un­ ics. Some active topics such asK-theory and fo­ til the Congress. Unfortunately, the difficulties liations were extensively and vigorously repre­ inherent in this process were not in any way dis­ sented, while other topics (some almost equally cussed at the General Assembly of the Interna­ active) were completely omitted; for example, tional Mathematics Union just prior to the Van­ homotopy theory, infinite groups, operators on couver Congress, and the only change made in

361 the arrarigements for the next Congress is the (1) The steep rise in the median salary for new provision that the host country will be rep­ male Ph. D.'s with degrees from 1954 to 1962 resented by two to four members on the Consul­ and no such steep rise for females. tative Committee. (2) The large number of women with no spe­ This letter is to call for more open and ef­ cial field. fective discussion of the planning of the next (3) If the women working part-time who are Corigress. seeking full-time employment are added to the Saunders Mac Lane women unemployed, the percentage of women seeking work is 4. 8 percent. The corresponding Editor, the cfloticeiJ figure for men is 1. 5 percent, so perhaps the I should like to supplement the information headline of Science and Government Report, 15 supplied in the cJioticeiJ (August 1974) on the sta­ May 1974, is not misleading, as Professor An­ tistics of the Commission on Human Resources. derson suggests. Here I have used the fact that (These data are taken from the 1973 Comprehen­ O. 6 percent of male Ph. D. •s in mathematics are sive Roster of Doctoral Scientists which may be part-time employed. half of them seeking full­ obtained at cost from the Commission. ) The data time employment while 12 percent of female raise some questions which are not so easily an­ Ph.D.'s in mathematiqs are part-time, one swered (marginal differences should be ignored quarter of them seeking full-time employment. because of the small numbers involved). Some Finally, a comment on the third and fourth explanations: Percentages are the percentage of comments of Professor Anderson. The data is the male and the percentage of the female respon­ not just a reflection of unemployment but a record dents to a question. The total number of female of overlap among the sciences and I, for one, am respondents was approximately 500 and repre­ happy to see that 5. 3 percent of the mathemati­ sents a total population of approximately 860. The cians have branched off into other areas while male sample was approximately 2, 200 out of a about 10 percent of the body of the mathematically total population of 15,000. employed are bringing in some flavor of engineer­ I believe that these tables, like any tables, ing and physics. should speak for themselves but I would like to call attention to the following: Cathleen Morawetz

By Math. Field of Employment - Employed Mathematicians - 1973

Sex Total Alg. Anal/ Geo. Logic Nbr. Prob. Math. Top. Comp. Opern. App. Comb, Phys. Math. Math. Functl Theory Stat. Thry. Res. Math. App. Math. Gen. Other Anals. Pract. Math. Male 100.0 9. 2 14.1 2.? 1.6 2.0 2. 5 9.? 6. 5 19.? 5.4 10.4 1.5 1.2 10.9 2.?

Female 100.0 12.? 12.5 4. 0 1.6 2.1 1.2 8.8 5.4 10.0 1. 4 4.5 1.2 .5 28.9 5.5

Part-Time Employed - 1973 - seeking Full-Time Employment

sex Yes No No Report

Male 55.5% 4~. 2% 5. 5%

Female. 24.2% ?5. 5% 2. 5%

Occupation by Percentage

Sex Total Academic Professor Assoc. Asst. Instr. Lect. Other No Non- Not No % Prof. Prof. Rank Report Academic Employed Report

Male 100.0 ?4.? 25.3 22.1 24.0 .5 .3 1.1 1.4 19.? 4.? .9

Female 100.0 ?4.9 20.5 18.? 26.2 3.0 2.0 1.4 3.2 10,0 14.2 .8

Median Sal:!!!::i: b:t: Sex and Year of Ph.D., Academica112: Em,elo2:ed Mathematicians, ] 973 Academic Sala~ Fiscal Year of Total 29-35 36-41 42-45 46-49 50-53 54-57 58-61 62-63 64-65 66-67 68-69 70-71 72 Ph.D. Male 16565 25184 245?3 26122 22419 23580 23010 20881 1?92? 1?523 1661? 14095 13081 12658

Female 14160 18492, 20912 16280 16515 15935 14252 12998 12555 12450 Insufficient nwnbe>' of Pespondents

Median Salary by Sex and Age, Academically Emplo2:ed Mathematicians, ]973 Academic Salary

Aqe Total 25-29 30-34 .35-39 40-44 45-49 50-54 55-59 60-64 65-69

Male 16566 12751 10885 1o045 18826 21444 21989 2209? 21907 21901

Female 14122 12?20 12814 1401? 13745 16154 180?5 18144 1??00 •rnaufficient numbe!' of !'espondents.

362 NEWS ITEMS AND ANNOUNCEMENTS

SALEM PRIZE Institute will emphasize applied analysis and The Salem Prize for 1974 was awarded to numerical mathematics and related areas of Dr. Hugh L. Montgomery, of the University of mathematics and computer science. Michigan, for his work in number theory. The Plans are currently being formulated; how­ prize, established in 1968, is given every year ever, opportunities are anticipated for visitors who wish to participate in projects developing to a young mathematician who is judged to have and evaluating algorithms or software for linear done an outstanding work in the field of interest algebra, minimization, ordinary differential of Salem, primarily on Fourier series and re­ equations, special functions, quadrature, and lated topics. The recipient was Dr. Nicholas numerical differentiation. Projects involving the Varopoulos in 1968, Dr. Richard Hunt in 1969, utilization of algorithms and software in compu­ Dr. Yves Meyer in 1970, Dr. Charles Fefferman tational modeling are also planned. Activities in 1971, Dr. Thomas Kllrner in 1972 and Dr. E. aimed at the development of automated program­ M. Nikisin in 1973. The jury consisted of Pro­ ming aids to assist the generation and utilization fessor A.. Zygmund, Professor L. Carleson, of numerical software will provide additional op­ Professor J.-P. Kahane and Professor Ch. portunities. In addition to collaborating with Ar­ Pisot. gonne staff members on research projects, ap­ AMS APPOINTMENTS pointees will have the opportunity to present and attend lectures in an extensive seminar program. The President of the Society h3.s appointed Approximately twenty twelve-week appoint­ Harry Kesten as the Society's representative on ments for undergraduate and graduate students the Organizing Committee of the Seventh Berkeley and fifteen appointments for faculty and recent Symposium on Mathematical Statistics. Joseph B. doctoral recipients will be available. Additional Keller and Fr;1nk L. Spitzer have been nominated opportunities for brief visits to participate in the to be representatives of the Society to the Division seminar program are expected. of Mathematical Sciences of the National Research The response from interested faculty and Council. their students will influence the direction of this summer program. For further information in­ terested parties may write to: J. C. T. Pool, AMS COMMITTEES Associate Director, Applied Mathematics Divi­ sion, Argonne National Laboratory, Argonne, PROCEEDINGS EDITORIAL COMMITTEE. Illinois 60439. Chandler Davis has been elected a member of the Editorial Committee and of the Council to replace FEDERAL GOVERNMENT STAFFING Jacob Feldman, for the remainder of 1974. Richard K. Miller has been elected a member of The U.S. Civil Service Commission recent­ the Editorial Committee and of the Council to re­ ly received the annual estimates from Federal place Fred Brauer for the remainder of this year. agencies of the number of people agencies plan to W. Wistar Comfort has agreed to serve as the hire in the mathematics fields in fiscal year 1975, managing editor effective September 1, 1974. through next June 30, Anticipated job vacancies nationwide at grade GS-9 through 15 (all positions above entry level) number approximately 250, LOWELL J. PAIGE The Commission reports, however, that NAMED NSF ACTING DEPUTY DIRECTOR the supply of qualified applicants is more than Dr. H. Guyford Stever, Director of the adequate to meet current and anticipated Federal National Science Foundation, named Dr. Lowell needs through next June-the end of the fiscal J. Paige as Acting Deputy Director of the NSF. year in the Government. In the Washington, In addition to his new duties, Dr. Paige will con­ D. C., office alone, there are approximately tinue to serve as the Foundation• s Assistant Di­ 3, 200 persons who have eligibility for mathe­ rector for Education. Dr. Paige will be Acting matics jobs, including mathematicians, statis­ Deputy Director until the position is permanently ticians, and operations research analysts. In filled by Presidential appointment and Senate con­ addition to the list of eligibles in Washington, firmation. there are lists in tenotheroffices across the country. MATHEMATICAL AND COMPUTER SCIENCES Commission officials emphasized that, RESEARCH INSTITUTE: SUMMER 1975 while any qualified citizen is entitled to file for Federal positions, they want to avoid disappoint­ The Applied Mathematics Division of Ar­ ments by potentially interested applicants result­ gonne National Laboratory, in conjunction with ing from unrealistic expectations of Federal job the Argonne Center for Educational Affairs, will opportunities. conduct a Mathematical and Computer Sciences The Federal agencies are, however, short Research Institute during the summer of 1975. of qualified actuaries in the Federal service-an Expanding upon a program initiated in 1973, the occupation related to mathematics. Currently,

363 the Department of Labor and the Internal Revenue throughout the economy. Interested applicants Service have vacancies ranging from G8-9 should call collect AC202-254-6200 for further ($12, 841) to GB-15 ($29, 818). These agencies information. prefer a background in the pension field, since the positions will be involved in the new pension Dean D. Larrick, Director guarantee program. This is landmark legislation Office of Staffing Coordination to protect and insure the pensions of workers U, S. Civil Service Commission

PERSONAL ITEMS

RICHARD F. BASENER of Yale University PROMOTIONS has been appointed to an assistant professorship at Lehigh University. To Professor. Oakland University: RICHARD EWING of the University of Texas RONALD DeVORE; University of Tokyo: AKIO at Austin has appointed to al;l assistant professor­ HATTORI; To Professor and Acting Assistant ship at Oakland university. Vice President for Academic Affairs. Virginia JERROLD GROSSMAN of the Massachusetts Polytechnic Institute and State university: Institute of Technology has been appointed to an RAYMOND F. DICKMAN, Jr. assistant professorship at Oakland University. To Associate Professor. Central Michigan WALTER E. MAXEY of Purdue University University: EDWIN H. KAUFMAN, Jr.; Oakland has been appointed to an assistant professorship University: IRWIN SCHOCHETMAN. at Hartwick College. RICHARD MOLNAR of the University of INSTRUCTORSffiP North Carolina, Chapel Hill, has been appointed Hartwick College: GARY E. STEVENS. to an assistant professorship at Oakland Univer­ sity. DEATHS SUZANNE MOLNAR of the University of Professor MARY LOUISE BUCHANAN of North Carolina, Chapel Hill, has been appointed Adelphi University died on August 23, 1974, at a lecturer at Oakland University. the age of 43. She was a member of the Society L. J. NACHMAN of Oakland University has for 20 years. been appointed to a visiting assistant professor­ Professor MARY ANN LEE of Sweet Briar ship at the University of Maryland, College Park, College died on September 6, 1974, at the age DON PORTER of SUNY at Albany has been of 65. She was a member of the Society for 28 appointed a systems programmer at Rensselaer years. Polytechnic Institute Office of Computer Services. Professor CARLOS B. de LYRA of the RICHARD RUBIN of Washington University University of Sao Paulo, died on July 21, 1974, has been appointed to a visiting assistant pro­ at the age of 46. He was a member of the Society fessorship at Oakland University. for 17 years. J. SETHURAMAN of Florida state Uni­ Professor R. L. MOORE of the University ·versity has been appointed to a visiting profes­ of Texas, Austin, died on October 4; 1974, at sorship at the Department of statistics, Uni­ the age of 91. He was a member of the Society versity of Michigan "for the year 1974-1975. for 68 years. He served as Vice President ofthe Society in 1923, and as President 1936-1938.

364 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 IDy title. The abstracts are grouped according to subjects chosen by the author from categories listed on the abstract form. The miscellaneous group includes all abstracts for which the authors did not ifldicate a category. An individual may present only one abstract by title in any one issue of the cfloticei] 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-A232 ROLANDO CHUACUI, Universccac Cat6lica d•. Chile. C~sill2 114-0. Santisgo,Chi~a. A Classification of Sim~le CarcinBl ~lg-.bras

Simple Cardin~! AlgEbras ( sco.) were c~fined ~nd s tU''ic d by T'lrski in "C~rrlin8l ~.lgcbras", :Jxford U. Prase 1949 (9.31--9.39). Let OZ.=<. A,+,!.') bo an SCil.• It is "asy ':o :Jr:Jv" that- if Ot. is not isomorphic ':o '9 , :hr; SCA of nonnn for vary " e A, -"hor ·

is a b E A such thP.t 2·b ~a !5 ..O·b. H<=nc<;, using 3 of th2 aut.hor 1 s "Com;ol'l~r.ly Divisiblf' Cardinal Algebras", these Notices OctobP.r 1974, ~r;; obt'lin: Th•.. orc;m 1. If (fl. is em SCP. enc is not isomor"Jhic to ~ , rh>ln f!l, is com""J~'·.t

isom~rphic to j; ; and 4) 8lsabrns in uhich every ~lament belongs to R subalgabra isomorphic to n, to SCA of nonnegative r•cal numbers. It is still O'Jan wh

(1974) -~-288 W·l g•Jt '-hr· following: Theorem 2. ·A Card.inal Alg;:,br"l (/1.. !..s isomorphic to 1i if-;­ tha follo·.:~lng condirions nr·· s~tisL ·c: (i) 02. :.s s:.rnpl:; (ii) th,·r· :s no ic~mcompos'lbl:>

Glement; and (iii) thcr~ is q f;ni•3 R E A, ~ I 0, such that for qu8ry b E A, b f 0, there is an n (oO such that a ~ n•b. (Received July 16, 1974.)

*74T-A233 ALBERT A. MULLIN, 9213 Kristin Lane, Fairfax, Virginia 22030. Topological methods in number theory. Preliminary report. This note explores applications of elementary topology to number theory. Specifically, it investigates the modification of classical fixed­ point theorems for the study of lattice points in En in order to establish the existence of solutions to certain classes of diophRntine equations and integer programming problems. By analogy it is well known that fixed-point theorems for suitable (Banach) function spaces entail the non-constructive existence of solutions of certain classes of ODE's. Lemma 1· Let B(n, r) be an origin-centered n-dimensional ball of radius r in Euclidean space, En, n ~ 2. Then there exists a homeomorphism h of B(n, r) onto itself such that the only fixed-point of h is a lattice point distinct from the origin. Lemma £• Prove that for each n > 1, there exists B(n, r) which can be mapped homeomorphically onto itself such that the only fixed-point is a highly visible lattice point. Problem. Determine necessary and sufficient conditions on a continuous map f of B(n, r) into itself in order that f has a fixed lattice point distinct from the origin. (Received August 7, 1974.)

Ac587 *74T-A234 Earl S. Kramer and Dale H. I1esner, University of Nebraska, Lincoln, Nebraska 68508. Admissible ~arameters for Steiner systems S(t,k,v) with a table for all (v-t) < 98.

A Steiner system S(t,k,v) is a system~ of subsets of size k (called

blocks) from a v-set X, such that each t-subset from X is contained in pre-

cisely one block. Our primary concern is when 2 < t < k < v, but we do allow

t = 1 when discussing necessary conditions for an S(t,k,v). A necessary con­ dition for an S(t,k,v) is that bs = (~=:)~(~=:)must be an integer for 0 < s < t. Fisher's inequality further requires that (v-t+l) >

(k-t+2)(k-t+l). We say a triple (t,k,v), with 2 < t < k < v, is then admis-

sible if the above two conditions hold. J. Tits has shown that an

S(2+h,m+h,n+h) hash 2[n-3(m-l~/(m-2). Further results concerning admissible

parameters are presented along with a table of all admissible tri~les (t,k,v)

for (v-t) < 498. (Received August 15, 1974.)

74T-A235 RONAlD D. BAKER and RICHARD M. WILSON, The Ohio State University, Columbus, Ohio, 43210. The Whist-Tournament Problem of E. H. Moore. Preliminary report. E. H. Moore [Amer. J. Math. 18(1896), 290-303] defines a Whist-Tournament in v players Wh[v] as an arrangement of v-1 rounds of play such that every player meets every other player once as partner and twice as opponent, a round being a list of ~v tables exhausting the list of players. A table is a set of four players with a distinguished partition into pairs called partners, the other four pairs being known as opponents. Without the distinction between partners and opponents a Wh[v] becomes a resolvable block design with k=4, A=3. Without the assumption that the tables are arranged by rounds, several authors have cited particular cases of solutions to the "Bridge tournament problem". The main result is that Wh[ v] exist for all v=4n save possibly v=l32, 152, 168 or 264. Similar results can be obtained for: 1. v=4n+l, 2. "directed" Whist-Tournaments DTh[ v], 3. Moore's "triple" Whist-Tournaments TWh[ v], 4. other types of tables, e.g. "five handed poker" and 5. self-orthogonal latin squares with symmetric orthogonal mates (resolvable SAMDRR tournaments). (Received August 30, 1974.)

*74T-A236 KEN'!' R.. FIJILm, University of Iowa, Iowa City, Iowa 52242. Qn rings .2£ ~ ~ Ei:e2· Consider the followi..."lg conditions on a ring A: (a) Every left A-module is a direct sum ,r f.g. modules; (b) A is left artinian and every sequence of proper monanorphisms Ho -+-M:I_-+

~-"'••• between f.g. indecomposable left A-modules terminates; (c) A has Morita duality and every sequence of proper epimorphisms ••• ~~ 4--M:I_·~Mo between f'o.p,. indecanposable left A­ modules terminates; (d) A is a ring of finite module type. Auslander (Comm. in Algebra 1 (1974), 269-310) has proved that ((b) and (c)) implies (d), and that (d) implies (a}. THEOREM:

Conditions (a), (b), (c) ~ (d) ~ EilluivaJ.ent. The proof is an application of Jacobson ra.d-

A-588 ical theory to certain subrj.ngs o:f.' the endomorphism ring of a set of modules that consists of one isomorphic copy of each f.g. indecomposable left A-module. (Received September 4, 1974.)

74T-A237 JAVED AHSAN, University of Islamabad, Islamabad, Pakistan Modules over qc-Rings

A ring R is called a (right) qc-ring if each cyclic (right) R-module is quasi-injective

(see Ahsan [Proc. London Math. Soc. £L (3), 1973, 425-439] and Koehler [Quart. J. Math., Oxford (2), ~ (1974), 51-55]). The purpose of this ~ote is to make a brief study of certain properties of modules over qc-rings. Among other results, we prove that a qc-ring

R has the characterlzlns property that each submodule of a cyclic R-module is quasi-injective.

Endomorphism rings of certain modules over qc-rings have also been studied. (Received August 22, 1974.) (Author introduced by Professor Russell Marris.)

DAVID ZI:ITLDf,1650 Vucat !Tee lortb.,M1Daeapol18 1 MI. 55411. J"tptitiu t2l 1ntear J!C!!!IU!I igyol.yig Silt A'eateU ip1jgqr tgpctiga.VII.Pr.:J "•'"1'7 repor\.

N,Q = 1,2, ••• , W ,w ,w;,~ are illtegers; W are t110 eolutiou 0 1 8 ,w: ot wn+2 =MW 8 +1 + SQW8 , S =11. It W0 = o, w1= 1, thea 118 ; 1t W= W.: 0 2, w1= M, thea W8 !! v8 • Set B(.f)= w.fw; + w.f+1w: , AO() = lx2w3w:- IB(j)+ D(j)•(1+ V )Vk+j 1 O(j): W~+1 w;l, 2 Wj+1W~ -(JI/2)B(j)- SQW.fw: , rfl =x2+4SQ 1 Z:2 = O(j)/T, F = Vz'(1+V2), and (:r] the g.i. tunctioD. 2Hll• S = 1, with 1 !I Q(M+'I ud H ~ 1J P) 0 i8 the root ot x2~-Q =o • .Qu! 6• S =-1, with 1 !I Q(K-1 tUld H ~ 3; P)1 is the root ot z2..JCz+Q = o. Thtore 4(tor bo'th oau•)• It A(P) <(~)/(D(k)q2k), than tor n ~ k ~ o ad 8 intepr j, [~n+jW: t 2y'UJt(-sQ) + F]: WD-tk+jw:+Jt • IIIJINa 1o'FibPDncci ~~ with S=H=Q='I, trB S F8 , VB 5 ad P)O ~~ is the root ol x2-a~~oo1=0. X.\ W8 !! F111 w: !: r._, aad j=k =1.

For a ~1,[p2p.2nL2D- +(13 + = Let VB: "•' 1 21'5)/~ F2n+1~. w::: ~' ud j = o. It IPR0..S,I then [P2k:F + 2zrkFk(-1)B (("'5'·P2k)/(~2k), 8 S. +.7s] =F 11+.tBa+k torn~, where ~ =CS,-iHc,)/..J;'. !p'P]t 2.Wa+2 = 4Wa+1-W8 , W0=1,W1 =3,W:=O,W~=2; P)1 is the :roo\ ot z2-~=0.For 3=0, B = 1/t/3, and n ~ 1, [~( + 4B + (44/15)] =Wn+1w:., •Theorem 4 giTeS Theor• 3(•ee VI) a• llll'boue. (Received September 6, 1974.)

*74T-A239 DAVID SIBLEY, The Pennsylvania State University, University Park, Pa. 16802, Coherence in Finite Groups with a Frobenius Subgroup

Let G be a finite group and p an odd prime, Suppose a Sylow p-group P of G is a trivial intersection set and its normalizer NG(P) is a Frobenius group with Frobenius kernel

P. Let S be the set of all irreducible characters of NG(P) whose kernels do not contain P. THEOREM. Either S is coherent or P is elementary abelian and S contains only one character. (Received September 6, 1974.)

A-589 *74T-A240 G.J.RIEGER, Technical University, D-3 Hannover.On an theorem of Churchhouse concerning sums of fractional powers. Denote by [a] the integer part of the real number a.Using a method of Vinogradov, Churchhouse (Bull.London Math. Soc 5 (1973),111-117) proved: For real c with 1 < c < 4/3 , every sufficiently large natural number n can be written in the form n = [xc + yc] with natural numbers x,y .Using a theorem of Erdos and Turan (Indag. Math. 10 (1948), 370-378,406-413) and a well-known estimate of van der Corput (see e.g. Koksma, Diophantische G~ei­ chungen, Berlin 1936; p. 114) we prove the more general Theorem. For real num- bers c,d with c > 1, d > 1, c-1 + d-1 > 3/2 there exist real numbers c1(c,d) > 0, c2 (c,d) > 0 such that every natural number n > c1(c,d) can -1 -1 be written in more than c2 (c,d)n-1+c +d ways in the form n = [xc + yd] with natural numbers x,y. (Received September 9, 1974).

*74T-241 DUANE M. BROLINE, Department of Mathematics, University of Ibadan, Ibadan, Nigeria. Properties of Irreducible Character Kernels

Let G be a finite group and let Irr(G) be the set of complex irreducible characters of G.

Theorem I. Let k be the minimum of { IKerX I : X e Irr(G)}. If ljJ e Irr(G), then k divides IKerl!JI. Theorem z. If ljJ e Irr(G) and Kerl!J is not nilpotent, then there exists X e Irr(G) with Kerx. < Kerlj.l and X(l) > lj.l(l). We also examine those groups G such that IKerx.l= k for every X e Irt(G) with X(l) > 1, where k is defined as above. (Received September 13, 197~ (Author introduced by Dr. I. Martin Isaacs.)

*74T-A242 WILLIAM SHUTTERS, University of Iowa, Io~1a. City, Io~1a 52242. Exchange rings and P-exchange rings. Prelimine.ry report. The main results of this paper concern the relationship between lifting idempotents and various exchange properties for projective modules over Rand R/J. It is sho~lll that R is an exchange ring if R/J is an exchange ring and idempotents lift modulo the radical. Conversely, over a commutative ring R, R/J is an exchange ring and idempotents lift modulo the radical. It is also shown that projectiveleft R-modules have the exchange property in projective left R-modules if and only if R/J satisfies the same condition and J is left T-nilpotent. (Received September 16, 1974.)

74T-A243 G. OTIS KENNY; University of Kansas; Lawrence, Kansas 66045. Archimedean ~ .2f.!. Representable !,-Group, Preliminary Report.

Let G be a representable i -group. For g E G, let G(g) be the convex i -subgroup of G generated by g. We show A(G) = {g E G I G(!gl) is archimedean} is the (unique) maximal archimedean convex i-subgroup of G. A(G) has the following properties: 1) A(G) is characteristic in G; 2) If K is a large or a convex 1-subgroup of G, then A(G)nK = A(K); 3) If G is an X-group, so is A(G), for X= L,P,SP ot 0 (seeP. CONRAD,.!!!!. Hulls of Representable i-groups ~ £-rings, J. Aust. Math. Soc. 16(1973), 385-415); 4) A(G)X ~ A(G~ A-590 for X= L,P,SP or 0 but equality need not hold for X= P or SP even if G has a basis;

5) If G is completely distributive, then A(G)L = A(GL); 6) A(G) is closed in G. (Received September 16, 1974.)

*74T-A244 M. BHASKARAN, 9-18, Butterick Place, Girrawheen, W. Australia 6064. On the cyclic decomposition of polynomials and prime divisors of degree two •

This is a revised version of 'On the cyclic decomposition of polynomials' (See Abstract, these Notices

21 (1974) October issue) • The resulf~ of the earlier abstract except (iii), which is wrong, are valid only if P· I cyclic factors of the form (x-b) I + c mod q are considered where b does not q depend on the varying primes q. The idea of this paper is also used to prove: Theorem. Let(} be an algebraic integer with the defining polynomial not of the form g (r (x)) where r (x) is a quadratic polynomial. (This is true if e is

of odd degree). Let G be the Galois group of the normal closure of Q (~)and D (0<:) be the discriminant

of o< Let q be any rational prime not dividing D ( 0 )TI D (e + ~ e ). Then, if q has a prime 01 e G divisor of degree 2 in

*74T-A245 B. N. DATTA, Ahmadu Bello University, Zaria, Nigeria. A new Routh-tyPe canonical form of a nonderogatory matrix.

It is shown that a nonderogatory matrix A is either a stability matrix or has all its eigenvalues in right half plane if and only if A is similar to a special tridiagonal real matrix B of Routh-type. The matrix B = (bij) has the following structure:

b 11 ~o, bii=O for i=2, ... ,n and bii+1 =-bi+1 i fori=1,2, .•. ,n-l.

Eigenvalues are all in right or left half plane according as b11 is positive or negative. (Received September 17, 1974.)

T. Y. LAM, University of California, Berkeley, CALIF. 94720. *74T-A246 Series summation of stably free modules.

THEOREM 1. Let R be a right noetherian ring, and P~ (i ~ 1) be right m· n· ... R-modules such that PieR ~ ~ R 1 • Assume that ni > mi+l for all i. Then, for r sufficiently large, the partial sums P1 $ • · .e Pr are all free. (This says, in essence, that the •series• L!Pi •converges• to free modules.) THEOREU 2. Let R be a commutative ring, and P be a nonzero R-module such that PE9Rm ';;;; Rn. Then, r·P is free for r ~ m+m/(n-m). By definition, a (finitely generated) right module P over a ring R is called stably free of type m if P$Rm is free. The followine; are special cases of the theorems above. COROLLARY 1. Let R be right noetherian, and Pi ~ 0 be stably free right R-modules of type m. Then, L Pi converges to free modules. COROLhillY 2. Let R be commutative, and P be stably free of type m. Then, r·P is free for r ~ 3m/2. If' rank P :;;: m, then, r•P is free for r :;;:; m + 1. (For m = 1, Cor. 2 is first proved by l,i. R. Gabel). Most of these results generalize to abelian categories. (;l.eceived September 17, 1974.)

A-591 Analysis

*74T-B231 J. P. SINGH, Department of Mathematics, Kurukshetra University, Kurukshetra (India). Absolute summability factors of a Fourier series and series conjugate to it.

Let f'(x) be a function v:hicf} is inte<:rc.;':Jle (L) ,n.1 per-iodic wit'l period

2n, let its Fourier series be L nn(x) :mi L: -3n(x) t'1e correspondin£ con.jugate series. In this note we orove: If a J~l¢h(u)Jdu = O 9 then the series 1 An "'n (t) is sumoJ.':Jle I C ,h+l j .1t t~1e point t = x \vhere P·n! is convex sequence such th:1t 1 1-r/n < co. Our theorern inclt:.des some of the prev­ iously known results due to Hsiang [Comp.!'f.:J.th.l7(1965),156-1GO],Li

Sci. 22( 1968), 72-7.~] ,Gaut3.m[Ha th .Sduca tion H ( 197?.) ,90-99 J, .:lharma and Gupta

[Kodai Hath.Jam. ;lep. 22(Fl70),ol-64]. An an'llogus result for conjugate series is also given. (Received April 26, 1974.)

*74T-B232 Jon C. Helton, Dept. of Mathematics, Arizona State University, Tempe, AZ 85281, Existence and Approximation of Integrals. In the following, R denotes the set of real numbers and N denotes a ring which has a multiplica- tive identity element represented by 1 and a norm J·J with respect to which N is complete and

Jll = 1. Further, if r and s are positive integers, then Nrs denotes the set of r x s matrices with elements from N. Matrix addition, subtraction and multiplication are defined in the usual manner. If h is a function from R to Nrs' ~ach of G and K is a function from R x R to

Nrr' each of Hand Lis a function from R x R to Nss' his quasi-continuous on [a,b], each of

G, H, K- 1 and L- 1 has bounded variation on [a,b] and each of Jbc,JbH,Jb(K- 1) and b Jb aaa Ja (L - 1) exists, then (LR) aanu(1 + G)K dh L vnb(1 +H) exists. In addition, if E > 0, then there exists a subdivision D of [a,b] such that, if {x.}~ is refinement of D, then 1. ].= 0 b u b In i-1 n 11 II (LR) TI (1 + G)K dh L TI (1 +H) - . 1 [n. 1 (1 + G.)]K. dh.L.[n ..+ 1 (1 +H.)

74T-B233 WITHDRAWN

A-592 *74T-B234 J.M. Chadam and R.T. Glassey, Indiana University, Bloomington, Ind 47401. On the Maxwell-Dirac Equations with Zero Magnetic Field and Time-Dependent Hartree Equations. Under the assumption of a vanishing magnetic field, a transformation of variables is exhibited which uncouples the Maxwell-Dirac equations. It is then shown that the Cauchy problem has a unique global solution for c~ data in two space dimensions. In addition, the existence and properties of global solutions to the Cauchy problem for time-dependent Hartree equations for N electrons are established. (Received September 3, 1974.)

*74T-B235 KENNETH J. PRESKENIS, Newton College, Newton, Massachusetts 02159. Another view of the Stone-Weierstrass theorem. Preliminary report.

Let f be a complex valued continuous function on D = dzl ~ 1), Pf =the uniform closure on D of polynomials in z and f. Theorem. If f = u + iv satifies the conditions; (i) u and v are Gateaux differentiable in dzl < 1), (ii) Re(fz) ~ jfzj everywhere in, dzl < 1), (iii) C1(f(a)) is countable for each a in D, then Pf = C(D). Theorem. If f is a class C• function in a nbd. of D such that Re(L) jf l z ~ z everywhere in dzl < 1) and lfzl > lfzl almost everywhere, then Pf = C(D). Corollary. Let g be a class c• function in a nbd. of D which is quasi-conformal and which satisfies Re(gz) > jgzj everywhere.

If f = g, then Pf = C(D). The last two facts bring to mind the Stone-Weierstrass theorem and are related to the result which the author gave previously (see Abstract 70T-B148, these cf.fotiaiJ 17(1970), 665):

Theorem. If f is a class c• function in a nbd.of D and If zl > If zl everywhere (i.e., f is an orientation reversing immersion of D in the plane), then C(D) =the uniform closure on D of rational functions in z and f which are finite. (Received September 4, 1974.) (Author introduced by Professor Joseph A. Sullivan.)

*74T-B236 SIDNEY L. HANTLER, IBM, P. 0. Box 218, )!'orktown Heights, N.Y. 10598, Estimates for ~producing Kernels in Weighted Hilbert Spaces of Entire Functions

Let~ be a plurisubharmonic function on en and let A 2 (~) be the space of all entire fun­ ctions which are square integrable with respect to the measure e-~dA where dA denotes the Lebesgue measure on en. To each w E en there corresponds a unique reproducing kernel K in A 2 (~) with the property that_f(w) = ~ f Kw e-~dA for all f in A 2 (~). Estimates are der~ved for the Neumann operator of the 3-complex defined by L. Hormander (Acta. Math. 113 (1965), 89-152).

These estimates are employed to determine sufficient conditions on ~ to insure that there exist positive functions H so that the reproducing kernels of A 2 (~) belong to A 2 <~-H). These results can be applied to prove approximation theorems in A 2 (~) because the linear span of the repro­ ducing kernels in A 2 (~) is dense in A 2 (~). Example: If~ E c2 (e), A~~ c > 0 and d <~then Kw E A 2 (~-2djzj) for each wE en, and if max{jzj=r} ~(z) - min{lzl=r} ~(z) < 2dr+a, where a is an arbitrary constant, then the polynomials are dense in A 2 (~). These results are then used to prove results about polynomial approximation in spaces with measure jPj 2 e-~dA, where P is an exponential polynomial or an entire function of order less than one. (Received September 11, 1974.)

*74T-B237 H. M. SRIVASTAVA and REKHA PANDA, University of Victoria, Victoria, British Columbia, Canada V8W 2Y2, New generating functions involving several complex variables. Preliminary report. Recently, H. M. Srivastava [Nederl. Akad. Wetensch. Proc. Ser. A 74 = Indag. Math. 33 (1971), 483-486; see also Proc. Nat. Acad. Sci. U. S. A. 67 (1970), 1079-1080] gave a class of

A-593 generating functions, expressed as Taylor series, for certain multidimensional polynomial systems with arbitrary coefficients. The object of this paper is to present a new class of generating functions, in terms of Laurent series, for a certain sequence of functions of several complex variables. It is also shown how the main results can be suitably applied to yield the corresponding generating functions for the generalized Lauricella functions which were introduced and studied in an earlier paper by H. M. Srivastava and M. C. Daoust [Nederl. Akad. Wetensch. Proc. Ser. A 72 = Indag. Math. 31 (1969), 449-457; see also Math. Nachr. 53 (1972), 151-159]. The results obtained in this paper provide extensions, among others, of two theorems proved recently by David Zeitlin [Scripta Math. 29 (1973), 43-48; cf. Theorem 1, p. 44 and Theorem 2, p. 46]. (Received September 13, 1974.)

74T-B238 C.J, Mozzochi,Box 1315,Hartford,Conn~cticut,0610l. The Foundations of Analysis: Landau Revisited.

Landau's classic t~xt: Grundla~en der Analysis is reconstructed in its entirety from the naive set theoretic point of view, (Received September 13, 1974.)

*7.4T-B239 S.Zaidman,Universite de Montreal,Canada:Bohr-Neugebauer theorem for operators of finite rank in Hilbert spaces.

An elementary simple proof is given for the following

Theorem. Let T be an operator of finite rank in a Hilbert space;let be g(t) a H-valued almost periodic function;let u(t) be a strong solution of the ordinary differential equati_on

u'(t) =: T u(t) 4- g(t) and assume that sup II u(t) Jl ~ L < oO .Then u(t) isH-almost-periodic, --·ct<- (Received September 16, 1974.) Applied Mathematics

74T-c44 SHIH-LIANG WEN, Ohio University, Athens, Ohio 45701. On the existence theorem of an eigenvalue problem arising from hydrodynamics, Preliminary report.

The existe&ce of a denumerable set of eigenvalues and eigenfunctions are proved for the problem: g" + {}. 2 - h.>-1u)g = 0, h" + A. 2h = -g with h(O) = h' (0) "'h(l) = h' (1) • 0. The procedure involves a suitable transformation, asymptotic expansions and some complex variable techniques. (Received August 30, 1974.)

*74T-C45 THOMAS OTTMANN, University of Karlsruhe, Fed. Republik of Germany, 75 Karlsruhe, P.B. 638o. A 7~symhol 2-state universal Turing machine with a ~odimensional tape. A 7- symbol 2-state universal Turing machine with a twodimensional tape is constructed. The construction proceeds in several stages: Simulation of Turing machines with an onedimensional tape by infinite chains of finite automata, simulation of finite automata by networks constructed from simple components, and simulation of infinite networks of finite automata by Turing machines with a twodimensional tape. {Submitted for publication in ElK.) (Received September 3, 1974.)

A-594 *74T-c46 H. K. Verma, Punjab Agricultural University, Ludhiana, Punjab, India Evaluating film coefficient in a single-kernel drying.

In this paper we solve the equation of moisture diffusion for a homogeneous, isotropic,

symmetrical sphere (grain-kernel) that arises in the study of drying state of fully exposed

grain kernels using the classical method of 'Separation of Variables' and the solution is

obtained in the form of a convergent infinite series. (Received September 4, 1974.)

74T-C47 RINA COHEN, Technion-liT, Technion, Haifa, Israel and ARIE GOLD, Technion-liT, Haifa. The w Kleene Closure of families of languages. Preliminary report.

For earlier work see Abstract 74T-C40, these NOTICES 21(1974), A-549. Definition: For any finite alphabet E, let E00 denote the set of all w length sequences over E.

For any language L s E*, let Lw denote all w sequences in E00 obtained by concatenating infini- tely many words from L. Definition: For any family F of languages over alphabet E, define the k w Kleene closure ofF; wKC(F) = {L ~Ew ] L = U U.V~ for some Ui' Vie: F, i = 1,2, ••• ,k i=l 1 1 k = 1,2, ••• }. Theorem 1: Thew Kleene closure of the family of context free languages equals precisely Fw(type 2). Theorem 1 does not generalize to the upper families in the language hierarchy. Theorem 2: Thew Kleene closure of the family of context sensitive languages is properly contained in the w Kleene closure of the type 0 languages, which,in turn, is properly contained in F00 (type 1). For other known families of (finite string) languages, the position of their w Kleene closure in the w language hierarchy is established. The properties of the w Kleene closure of any family F are related to the properties of F. (Received July 1, 1974.) (Authors introduced by Igal Golan.)

Logic and Foundations

JEFFREY B. REMMEL, Cornell University, Ithaca, New York 14850. Co-R.E. Cohesive Vector Spaces. Let M be a recursively presented model of the theory of an infinite dimen­ sional vector space over a recursive field F. Given a subset A of M we write ct(A) for the subspace generated by A. Definition. An infinite dimen­ sional subspace V<;;; M is called cohesive if for any recursively enumerable subspace w either (i) vnw is finite dimensional or (ii) there is a finite set SSM such that c.f(WUS).s;;v. We note that condition (ii) is equivalent to saying that V-W generates a finite dimensional subspace. We construct cohesive subspaces V such that M-V is an r.e. set. Our notion of cohesive subspace seems to be a natural analogue of the notion of cohesive set. Our method of construction is a modification of the maximal set construction of Yates. We also produce co-r.e. cohesive subspaces that are complete or incomplete, analogous to the results of Yates (1965) and Sacks (1964). In the finite field case, we prove that a co-r.e. cohesive subspace Vis a co-hypersimple set, but that V is not a co-maximal set. (Received July 11, 1974.)

*74T-E91 HARVEY FRIEDMAN, State University of New York at Buffalo, Amherst, New York 14226. Primitive Recursive Arithmetic Without Induction.

The set of identities provable in primitive recursive arithmetic without induction (PRE),

with or without standard quantifier free successor axioms, is recursive. A finite number of

identities can be added to PRE such that the set of identities provable become complete r.e. A-595 (with or ,~ithout SIJCCE>ssor axioms). If the successor axiom y f. 0 .... :i!x(S(x) = y) is added to

PRE then the set of identities provable become complete r.e. (with or without 1 f. 0 ). If

PRE is augmented by definition by cases, then the set of identities provable become complete r.e. (with or without successor axioms). (Received August 5, 1974.) (Author introduced by Mr. Dallas Webster.)

74T-E92 WILLIAMS K. FORREST, Simon Fraser University, Burnaby, B. C. VSA 1S6 ~ Result .2!!. the Cardinality of Definable Sets Preliminary Report

Let T be an w-stable theory and suppose that m is a model of T prime over an uncountable set of indiscernibles. If ~(v0 ) is an L(m) formula and I~ w then l~

74T-E93 STEVE NEWBERRY, 1415 Bellevue Avenue, Burlingame, California 94010. A retraction of previously announced results.

The results announced in these c/Votiui), Abstract 73T-E37, The recursive unsolvability of Hilbert's lOth problem is not domain invariant, Page A-339; Abstract 73T-E42, Finite quantification theory of second-order is stronger than you think, Page A-443; Abstract 73T-E76, An algorithm for the decision problem for finite classes, Page A-500; and Abstract 74T-E66, Diagonal paradoxes. I. Preliminary report, Page A-502 were unqualifiedly incorrect and are hereby retracted, (Received September 9, 1974,)

Statistics and Probability

74T-F14 LAIF SWANSON, University of California, Berkeley, California, 94720, Two induced automornhisms. Preliminary report, Let (X, J- , L<--) be a probability space and T a measure automorphism of X. If xEA, AE J-, _...... (A)>O, the induced automorphism TA is defined on the nor­ malized probability space A by TA(x)= Tk(x) where k= k(x)= min(n>O•Tn(x)~A), By the cutting and stacking methods of N, A. Friedman, D. s. Orn­ stein, and P, Shields, an example of a Lebesgue space X, an er~odic automor­ phism T, and measurable sets A and B so that TA is a Bernoulli shift and TB is a K-automprphism which is not Bernoulli has been constructed, (Received September 16, 1974.)

Topology 74T-Gl32 C,J,M, RAO, Indian Institute of Technology, Kanpur-208016, India On a problem of Fletcher, Hoyle and Patty, Preliminary Report

In this note we prove that every connected, compact Hausdorff topological space has more than one compatible uniform convergence structures [Math. Ann, 173 (1967), 290- 306], This answers a problem raised by Fletcher, Hoyle and Patty [Duke Math, J,36 (1969),

325-331]. (Received May 20, 1974.) (Author introduced by Professor S.A. Naimpally.) A-596 *74T-G133 ROBERT A HERRMANN, U. s. Naval Academy, Annapolis, Maryland 21402 The point monad and topological generalizations, Preliminary report. In this paper, we use nonstandard set-theoretics to generalize continuous, open, perfect maps; open, closed sets; the closure operator, separation properties, compactness, local compactness and simple extension theory so that many of the results which hold for these topological concepts hold for these generalizations. The results obtained apply to continuity, weak and strong- B -continuity, e-continuity, neighborhood space continuity, c-continuity, semi-open continuity, almost-continuity (Singal, Husain); almost-open, e­ perfect maps; compact, quasi-H-closed, nearly compact, locally-quasi-H-closed, bounded, regular, T1 , T2, Urysohn, semiregular, almost-regular spaces etc., under the appropriate interpretations. 1'/e apply these results to our generalized simple extension theory, one point type compactifications and among other things extend a result of Rudolf on weak-G-continuity. Finally, we Show that numerous published and new results are simple interpretations of known results about such spaces as regular, T2, compact, locally compact etc. (Received July 22, 1974.)

74T-G134 FRANKLIN D. TALL, University of Toronto, Toronto, Canada MSS lAl and W.A.R. WEISS, University of Toronto, Toronto, Canada MSS lAl. eN-N may or may not be Blumberg. Preliminary report.

A space is Blumberg if every real-valued function is continuous on some dense set. H.E. White

proved that the continuum hypothesis implies eN-N is Blumberg. Theorem. There is a model of

set theory in which the continuum hypothesis fails and eN-N is Blumberg. Theorem. There is

a model of set theory in which eN-Nis not Blumberg. (Received July 30, 1974.)

*74T-G135 JAMES R. BOONE, Texas A&M University, College Station, Texas 77843 On Irreducible Spaces II.

A topological space is said to be irreducible if every open covering has an open refinement

that covers the space minimally. Irreducibility is a fundamental property related to

cardinality conditions for open coverings. In this paper, a constructive proof is presented

to establish that the weak e-refinable spaces of Smith are irreducible. Various results

concerning cardinality conditions for open coverings follow as corollaries. Some examples

are included. (Received August 9, 1974.)

*74T-G136 MARVIN ISRAEL, University of Illinois, Urbana, Illinois 61801. (TOP, CAT)- Local contractibility in spaces of homeomorphisms. Preliminary report.

Let CAT be one of TOP, PL or DIFF and let 'Jt CAT (X) be the space of CAT homeomorphisms of X (majorant topology). We say 'Jt TOP (X) is (TOP, CAT) - LC if there is a neighborhood U of id in 'Jt TOP (X) and a continuous function W : U __, 'Jt TOP(X X [0, 1]) satisfying: (1) for all f E 'Jt TOP(X), W(f) is an isotopy of X, W(f) 1 = f and w0 = id; and (2) if f E 'Jt CAT (X), then, for t E [0, 1), W(f) lx X [0, t) .... X X [0, t) is a CAT homeomorphism. Theorem. If M is a CAT manifold, then 'Jt CAT(M) is (TOP, CAT)­ LC, This theorem has relative versions and can be generalized to spaces of embeddings (with neighbor­ hoods). The proof utilizes Cernavskii's meshing techniques. In particular, a relative CAT-local lemma is proved. (Received August 12, 1974,)

A-597 *74T-Gl37 HAROLD BENNETT, Texas Tech University, Lubbock, Texas 79409 ~Note on LOTS, Preliminary report

Theorem. Let X be a LOTS with a weak G0-diagonal{Gn:nEZ1 such that if p and q are elements of X, then there is a natural

number N such that for all i greater than or equal to N noelement

of Gi contains both p and q . Definition. An open covering F of a space X is an F-cover if

p is point-separating and if p and q are in X, then there are only

finitely many elements of F that contain both p and q,

Theorem. An F-cover for' a LOTS is a point countable base for X. (Received September 16, 1974.) (Author introduced by Mr. Hal Martin)

74T-Gl38 R. GRANT WOODS, University of Manitoba, Winnipeg, Canada R3T 2N2. Topological extension properties. Let ~ denote a topological property of Tychonoff topological spaces. A space X is

P-regular if it is homeomorphic to a subspace of a product of spaces with P P is called

a topological extension property if P is closed-hereditary, productive, and each P-regular

space has a maximal P-regular compactification S~ . Herrlich and van der Slot [Indag.

Math. 29(1967), 524-529] showed that for such P, each ~-regular space X has a maximal

P-extension YpX , possessing P and containing X densely, such that if Y has P and

f: X ~ Y is continuous, then f extends continuously to y~ . A r-reguLar space X is

called P-pseudocompact if Y~ is compact. In this paper we undertake a systematic study

and classification of topological extension properties, using as the main tool the concept of P-pseudocompactness. A typical result is the following Theorem: If P and :J are

co-regular extension properties and if each space with 2 is ~-pseudocompact, then

Y.f- yPX is dense in ~X- y~X for each ~-regular space X . (Received September 16, 1974.)

*74T-Gl39 BARADA RAY, Regional Engineering College, Durgapur 713209, India. On common fixed points. Preliminary report.

Let (X, d) denote a complete metric space. Theorem 1, If T1 and T2 be two operators trans­ forming X into itself and if (i) d(Tl Ti x, Tl Ti y) :§ ad(x, y) + {3d(x, Ti Ti x) + yd(y, Tl Ti y) + l5[d(x, Ti Ti J + d (Y, T i Ti x)] where x, y E X, p, q > 0 are integers and a > 0, fJ > 0, 15 > 0, y > 0, a + fJ + y + 215 < 1.

(ii) T 1 T 2 = T 2 T 1 for all x E X then T 1 and T 2 have a unique common fixed point in X. Theorem 2. Let T 1 and T 2 be two self maps of X such that d(Tl Ti x, Ti Ti y) ;;; ad(x, y) + {Jd(x, Tl Ti x) + q p J: d q p d p Tq f E R yd(y,T2 T1 y)+u[ (x,T2 T1 y)+ (y,T1 2 x)] orallx,y X,a>O,">O,y>O, u>O,a+"+Y+2u

A-598 *74T-Gl40 ROBERT WELLS, Pennsylvania State University, University Park, Pennsylvania 16802. A combinatorial immersion theorem.

Suppose K is a linear cell complex in Rm, such that IKI is a closed n-manifold. For each vertex v of K define the excess e(v) to be the number of edges of K incident to v, minus n + 1. Define the excess e(K) of K to be the maximum of the e(v)•s. Theorem. There is an immer-sion of K in Rn+e(K)+1. The proof is an easy application of Whitney's compression technique. (Receive d Septem b er

3, 1974,)

*74T-Gl41 ROBERT CONNELLY, CORNELL UNIVERSITY, ITHACA, NEW YORK, 14850, AN ATTACK ON RIGIDITY. I. PRELIMINARY REPORT. We consider the problem of the continuous rigidity (not to be confused with infinitesi­ mal rigidity) of polyhedra mapped into R3. The outstanding conjecture is that all embed­ ded (or immersed) two dimensional surfaces in R3 are rigid. Except for convex surfaces (which are rigid, essentially) very little is known. We have developed some techniques which we hope will be u~eful in the more general situation, and we have applied them to the case of a polyhedral suspension. We first set up a system of polynomial equations which desc:r.•ibe the motion of the polyhedron around a vertex as it flexes. We then extend the domain of the variables involved and investigate the nature of the complex variety defined. This can be used to prove the following: Theorem: If a polyhedral suspension flexes with the distance between the suspension points changing, then the winding number of the equator about the line through the suspension points is zero. We also present an alternate approach to suspensions where the flexing about the suspension points is described in terms of an integral. This can be used to generalize the above theorem to when the suspension is taken over a piecewise-smooth curve (rather than a piecewise-linear curve). (Received September 17, 1974.)

TheN ovember Meeting inN ashville, Tennessee November8-9, 1974

Algebra & Theory of Numbers

*717-Al BERNARD R. McDONALD, The University of Oklahoma, Norman, Oklahoma 73069. Simple Subrings of~ Matrix Ring ~~Finite Field.

Let R denote an n-by-n matrix ring over a finite field k . Let S be a simple subring

of R having the same identity as R . This paper determines the number of isomorphism

classes of simple subrings of R and enumerates the simple subrings S in each class. The

approach is by way of the Hochschild Galois theory for simple rings which permits the simple

subrings of R to be identified with the regular subgroups of Aut(R) . (Received July 24, 1974.)

A-599 *717•A2 EVELYN NELSON, McMaster University, Hamilton, Ontario, Canada, L8S 4Kl. Semilattices do not have equationally compact hulls.

For each infinite cardinal ~. an example is given of a semilattice with ~

elements which has no pure, equationally compact, semilattice extension (and hence has no

pure equationally compact extension at all). As a bonus, all these semilattices turn out to

be pure-irreducible. (Received July 1, 1974.)

*717-A3 ROBERT QUACKENBUSH,University of Manitoba, Winnipeg, Manitoba,Canada R3T 2N2. Varieties of Steiner Quasigroups and Steiner Loops Preliminary report.

A Steiner quasigroup (squag) is a groupoid satisfying i) x• x=x, ii) x• y=y• x, iii) x• (x• y)=y; a Steiner loop (sloop) is a groupoid with identity l satisfying; a) 1 • x = x, b) x • x = 1, c) x • y = y • x, d) x • (x • y) = y. As is well known, these are two ways of associating algebras with Steiner triple systems. In this paper it is shown that the lattice of subvarieties of the variety of all squags (sloops) contains as a cover preserving sublattice a lattice isomorphic to the lattice of all finite subsets of a countably infinite

~et; the zero of this sublattir.e is the unique atom in the lattice of subvarieties of squags (sloops). (Received August 2, 1974.)

L. GARLITZ, Duke University, Durham, North Carolina 27706. ~­ *717-A4 tions with prescribed pattern. ·

Let r,k1 ,k2 ,k3 , ••• be arbitrary positive integers and let Pr(n;k1 , ••• ,kr)

denote the number of solutions in positive integers a 1 ,a2 , ••• ,am(m = k 1+ ••• +kr)

of the equation n = a 1+a 2+ •.• +am, where a 1~ ••• ~ak, ak +l~ ••• ~ak +k ••• and 1 1 1 2

Generating functions are obtained for

The results simplify considerably when the {ki} are periodic,

that is, for some s,kis+j = kj' where i = 0,1,2, •.• and j = 1,2, ••• ,s. In par-

ticular, for k 1 = ••• = kr = k, put Pr(n;k) = Pr(n;k, •.. ,k). It is proved that

E P (n;k)xnzrk = {E 00 (-l)sx~ks(s+l)zsk/(x) k}-1 , where (x)k = (l-x)(l-x2 ) n,r r 0 s (1-xk). This result may be compared with the corresponding result for permu-

tations (Math. Nachrichten, 58(1973), 31-53). (Received August 6, 1974.)

*717-A5 JOHN R. DURBIN and K. BOLLING FARMER, University of Texas at Austin, Austin Texas 78712. On Projective Representations of Finite Wreath Products.

The method for obtaining projective representations by induction is applied to finite wreath products. For finite Abelian and Abelian-wreath-cyclic groups, the factor sets are determined explicitly by establishing a one-to-one correspondence between certain lower triangular matrices and the inequivalent factor sets of these two classes of groups. This correspondence is used to determine the number and degrees of the inequivalent, irreducible projective representations. (Received August 15, 1974.)

A-600 717-A6 PETER M. GIBSON, University of Alabama in Huntsville, Huntsville, Alabama 35807.

Matrix commutators over an algebraically closed field.

Let A be ann-square matrix with zero trace over an algebraically closed field F, and let the characteristic of F not divide n. It is shown that A can be expressed as A = XY - YX where the eigenvalues of X andY may be arbitrarily specified as long as those of X are_distinct.

This generalizes a result obtained by C. R. Johnson [Proc. Amer. Math Soc. 42(1974), 351-353] for complex matrices. (Received August 22, 1974.)

*717-A7 EZRA BROWN, Virginia Polytechnic Institute & State £niversity, Blacksburg, Virginia 24061. Diophantine equations of the form x + D = yn.

The author proves, among others, the following theorems. THEOREM 1. The diophantine 2 n equation x + 3 = y has no solutions with n > 2. THEOREM 2. The diophantine equation x2 + 5 = pn, where p is a prime, has no solutions with n > 2. (Received August 23, 1974.)

*717-AB RICHARD H. HUDSON, University of South Carolina, Columbia, South Carolina 29208. The least pair of consecutive kth power nonresidues (mod p).

In addressing the 1972 Number Theory Conference in Boulder, Colorado, P.D. T.A. Elliott announced (1-e- 10/2)/4 + £ f that the least pair of consecutive quadratic nonresidues (mod p) was bounded above p or each (> 0 and p ~ 5. Elliott pointed out the "lack of a structural argument" which would make it possible to provide a more substantial improvement of his earlier bound p1/ 4 +£, £ > 0, p ~ 5 (Proc. London Math.

Soc. (3) 21(1970)). In a paper (J. Reine Angew. Math., to appear) we provide a structural argument which makes it possible to significantly improve Elliott•s result for all nonprincipal Dirichlet characters (mod p) of order k > 3. In particular, one obtains the bound p1/ 2uk + f for £ > 0, where uk is the (unique) solution of p(u) = 1/k and p(u) is Dickman's function defined by p(u) = 1 for 0 ""u"" 1, up(u) = -p(u -1) for u > 1. Thus, for example, the least pair of consecutive quintic nonresidues (for which Elliott obtained no bound at all since he only dealt with real characters (mod p)) is bounded above by p" 221. · · and, of course, the bound for large k is even more dramatically sharpened. (Received August 30, 1974.)

717-A9 JAMES K. DEVENEY, Virginia Commonwealth University, Richmond, Virginia, 23284, Pure Subfields and Modular Extensions, Prelim. Report

Let K be a purely inseparable modular extension of the field k. An intermediate n n field F is called pure if F and k(Kp ) are linearly disjoint over k(Fp ) for all n (Waterhouse). Pure subfields have many properties similar to pure sub­ groups. Theorem 1. If K/k is of bounded exponent, then F is pure iff

K = F G)k F' for some F'. Theorem 2. If F/k is of bounded exponent, then F

is pure iff K = F @ k F' for some F'. Theorem 3. If K is a tensor product of simple ex tens ions of F, then F is pure iff K = F G) k F' for some F'.

(Received September 5, 1974.)

A-601 717-AlO JACOB T.B. BEARD, JR. and JAMES R. OCONNELL, JR., University of Texas at Arlington, Arlington, Texas 76019. Perfect polynomials, preliminary report.

The polynomial A(x) e: GF[p,x] is perfect over GF{p) provided cr{A(x)) = A(x)(mod p), cr(A(x)) denoting the sum of the distinct monic divisors in GF[p,x] of A(x). Generalizing results of E.F. Canaday ["The sum of the divisors of a polynomial," Duke~· Math., vo1.7{1941), 721-737] over GF{2), the authors characterize all perfect polynomials over GF{p) having the form ~=~ (x-i)k and partially characterize those of the form ~=~ (x-i)a(i). Other results include the Theorem: If A(x) is perfect over GF{p) and there exists an e: GF(p) such that (x-i)IA(x), then ~=~ (x-i)IA(x). As an analog of the classical problem, the authors conjecture that xiA(x) whenever A(x) is perfect over GF(p).

{Received August 12, 1974.)

717-All CHARLES R. WALL, University of South Carolina, Columbia, South Carolina 29208. Arithmetic distribution functions, Preliminary report.

Several functions of the form F(t) = o{n: f(n)~t} are considered, where f is an arithmetic function and oA is the density of a set A of positive integers. We have computed upper and lower bounds which are close enough to provide good graph sketches if f(n) is one of the fol­ lowing: n/a(n), n/ljJ(n), n/a*(n), (n)/n, a(n)(n)/n2 or a*(n)/a(n). (Here a is the sum of divisors function,$ is Dedekind's function, a* is the sum.of unitary divisors function, and

(n)/n2 and (0~) for the others). Of special interest is the result that the density of the abundant numbers is between 0.24750 and 0.24893. Also con­ sidered is a distribution function of two variables which is related to Jordan's totient. {Received September 9, 1974.)

717-Al2 RICHARD SCOVILLE, Duke University, Durham, N.C., 277u6. Some Properties o( the Catalan numbers. (Invited paper for special session in combinatorial theory)

Generating functions for various sequences related to the Catalan numbers are

obtained.' For example, given statements p1, .•• , pn each with probability {3, and random1y joined by ---t (implies), the probability yn ({3) of a true statement and lim V'n ((3) are found. (Received September 12, 1974.) 717-Al3 S. BRENT MORRIS, Department of Mathematics, Duke University, 27706, Generalized Inversions and Dedekind Sums , Preliminary Report.

C. Meyer (J. Reine Angew. Math., 205 (1960) 186-196) has shown an interesting relationship

between I(h,k) and s(h,k), the Dedekind sum, where I(h,k) counts the number of inversions in

the sequence {ki-[ki/n]n}, i = 1, •.. , n. This note extends the concept of inversion to two

dimensions- by considering ordered pairs (x,y), l ~ x ~ m, 1_::; y _::; n, (m,n) = l, with the norm

i(x,y) I= x/n + y/m. Then, following Meyer's method, we are able to find a re~ationship

between the generalized inversion I(k;m, n) and s(km ,n), s(kn,m), s(k ,mn), and certain

other multiple. sums. {Received September 9, 1974.) A-602 *717-A14 CARL POMERANCE, University of Georgia, Athens, Georgia 30602 The second largest prime factor of an odd perfect number. Recently, Hagis and McDaniel have studied the largest prime factor of an odd perfect number, proving it is at least 100129. We begin the study here of the second largest prime factor, showing it is at least 139. We apply this result to show that any odd perfect number not divisible by at least 8 distinct primes, must be divisible by 15 or 21. The proofs depend on the mentioned result of Hagis and McDaniel and methods developed in Pomerance, Odd perfect numbers are divisible by at least seven distinct primes, Acta Arith. 25 (1974), 265-300. (Received September 11, 1974.)

717-Al5 H. W. GOULD, West Virginia University, Morgantown, W. Va. 26505 Some new inverse series relations and new formulas for Stirling numbers. \'n n \n n We prove the following Theorem: F(n) /. Ak(p)f(k) iff f(n) = L k=O Bk(p)F(k), k=O where: k n -k n! \' Ak(p) (p + 1) I (-1 )k-j ( :) ( (p:1 )j) k! L j=O

and k n n nl Bk(p) (p + 1) ") j/(:•' = k! I ,_,H (: ) ( l) j=O Corollary: The Stirling numbers of the first kind are given by s(n,k) = A~(-1) and the Stirling numbers of the second kind are given by S(n,k) = ~(-1). These results are found by a study of the equation DxU(x,a) is an extension of Truesdell's F-equation (corresponding to p = 0, h = 1). An extensive study of the extended equation has also been made. (Received September 13, 1974.) *717-A16 HARALD NIEDERREITER, The Institute for Advanced Study, Princeton, N. J. 08540 and SIU KWONG LO, Southern Illinois University, Carbondale, Ill. 62901 Uniform distribution of sequences of algebraic integers. Let 0 be the ring of algebraic integers in a fixed algebraic number field, and let I be a nontrivial ideal of 0. A sequence (an), n = 1,2, ... , of elements of 0 is called uniformly distributed (u.d.) mod I if every coset of I contains asymptotically the same number of an, and it is called u.d. in 0 if it is u.d. mod I for all I. Based on explicit character formulas for 0/I, we establish Weyl criteria for both types of uniform distribution. A Banach-Buck measure can be introduced similarly to the one in Z. Various connections between Banach-Buck measurability, relative density, and density of subsets of 0 and u.d. sequences in 0 are found. The discussion of sequences of polynomial values leads to a detailed study of permutation polynomials over 0. Furthermore, the relation between u.d. sequences in 0 and u.d. sequences in the localizations of 0 is investigated. The local method yields results of the following type: (i) there exists a sequence in 0 that is u.d. mod all primary ideals but not u.d. in 0; (ii) a subset A of 0 has outer Banach-Buck measure 1 iff its elements can be arranged into a u. d. sequence in 0. (Received September 13, 1974.)

A-603 717-A1.7 TOM PARKER, Air Force Weapons Lab, Kirtland AFB, Albuquerque, New Mexico 87117 and ROBERT GILMER, Florida State University, Tallahassee, Florida 32306. Nilpotent elements of commutative semigroup rings. Preliminary report. Assume that R is a commutative ring and (S,+) is a commutative semigroup with zero. We have determined the nilradical of the semigroup ring R[X;S] of S over R; the following are some special cases of the results. If N is the nilradical of R, then N[X;S] is contained in the nilradical of R[X;S]; if R has positive characteristic, then equality holds if and only if S is p-torsion-free for each prime divisor p of the characteristic of R. If R is an integral domain of characteristic 0, then R[X;S] has nonzero nilradical if and only if there exist distinct asymptotically equivalent elements of S. (Received September 16, 1974.)

*717-A1.8 KIM KI -HANG BUI'LER, Alabama State University, ~bntgomery, Alabama 36101 Enumeration of different collections of subsets of an n-set admitting SDR In this paper we shall answer the following open problem: Given an n-set, how many different collections of subsets of an n-set admits SDR ? The formula is too long and complicated and so it lrill appear elsewhere. (Received September 5, 1974.)

*717-A1.9 WALTER TAYLOR, University of Colorado, Boulder, Colorado 80302. Remarks~ topological algebra.

Continuing the results in Abstract 74T- A224 Theorem 5. If 1( has modular or k- permutable congruence relations, then the fundamental group of every topological algebra in V is commutative. Theorem 6. If 1( has factorable congruences, then a· topological algebra in 1( has all homotopy groups zero; conversely, if X is an acyclic cell complex, then X admits a continuous majority function. Theorem 7. If K is a Mal 1cev- definable class of varieties not containing unary algebras,then no topological algebra in 1f E K has simple fundamental group. (Received September 20, 1974.)

*717-A20 BRUCE C. BERNDT, University of Illinois, Urbana, Illinois 61801. Positive sums of the Legendre symbol. Preliminary report.

If p is a prime with p = 3 (mod 4), it follows from one of Dirichlet's class number formulas that there are more quadratic residues than nonresidues in (0, p/2). The proofs in the literature that are independent of class number considerations use Fourier series. Two completely new proofs are given, but, like the others, neither is elementary. One of the proofs can be generalized to give other results om positive sums of the Legendre symbol. For example, there are more quadratic residues than nonresidues in (O,p/3). Lastly, from another of Dirichlet's class number formulas, it follows that if p =1 (mod 4), the number of residues exceeds the number of nonresidues in (0, p/4). Two new proofs, independent of class number formulas, are given. (Received September 20, 1974.)

*717-A21 Jorge Martinez, University of Florida, Gainesville, Florida 32611. Nilpotent ~-groups are representable. Preliminary report.

A group G is said to admit an ascending central chain if

there is a ·chain of subgroups 0 = G0 <;; G1 <; .•. <; Gcr <;; ••• , where cr is an or-

dinal number, so that [G, Gcr+ll ~ Gcr' for each ordinal cr, and G GT, for a

suitable ordinal T. The main results are: 1) if G is an ~-group and admits an

ascending central chain then G is representable, and 2) if G is also finite valued then each convex ~-subgroup is normal in G. (Received September 20, 1974.)

A-604 717-A22 JOHN A. KALMAN, University of Auckland, New Zeala~d and Pe~nsvlvanis State University, University Park, Pennsylvanis 1~~02. · The lattice of varieties of de Xorgan lattic~~· Freli~:nary report A "de ~or~an lattice" (or "d1~tributive i-lattice") is an alRebra with a distributive latti~e and xI-+ x' an involution (dual automorphism of period two) of . rh3 author !Trans ..L\!'!'!er. Math. Soc. R7 (lg5R} p. 4R6) has determined all subdirectlv irre~u~ible de MorP'an lattices. It is not hard to dedu::e that the lattice of' varieties of de Morp.an lattices is a chain with rour ele~ents. This result is closely related to Theorem 4 of the paper cited above. (Received September 20, 1974.)

717-A23 CARL G. WAGNER, University of Tennessee, Knoxville, Tennessee 37916. Integral­ valued polynomials over GF[q,x] • Preliminary Report.

Let GF[q,x] be the ring of polynomials over the finite field GF(q) , and let GF(q,x)

be the quotient field of GF[q,x] A polynomial f(t) with coefficients in GF(q,x) is

called integral-valued if f(M) E GF[q,x] for all ME GF[q,x] • Garlitz [Duke Math. J. 6

(1940), 486-504] has constructed an ordered basis for the GF[q,x] ~module of integral-

valued polynomials. We use this basis to construct bases over GF[q,x] for modules of

integral-valued polynomials with integral-valued differences of various orders.

(Received September 20, 1974.)

*717-A24 C. C. ROUSSEAU, Memphis State University, Memphis, TN 38152, Generalized Ramsey Theory .for Multiple Colors.

In recent work of Erdos, Faudree, Rousseau, and Schelp, the generalized

Ramsey number r(G1 ,G2 , .. ,Gk) is studied for the case where G2 ,G3 , •. ,Gk consist of complete graphs, complete bipartite graphs, paths, and cycles

of fixed order and where G1 is either a cycle or a path of order n. This Ramsey number, denoted r(CPn,[K], [B], [C]), is determined exactly for all

sufficiently large n. Two of the several corollaries yield r(Cn,Ct,Ck)

and r(Cn,ct,Ck,Cm), again for t, k, and m fixed and n sufficiently

large. (Received September 20, 1974.)

*717-A25 JOHN D. FULTON, Clemson University, Clemson, South carolina 29631, Gauss sums and solutions to simultaneous equations over GF(2Y), Preliminary report. Let q = 2Y and let F = GF(q) denote the finite field of order q. If Flxs

is the vector space over F consisting of vectors X= (x1 , x 2 , ••• , x 5 ) and if Q is a full-rank quadratic form on Flxs with associated bilinear form g, the num­

ber of pairs of solutions (~, xl e Flxs x Flxs to the simultaneous equations

Q{~) = t, Q(x) = u, and g(~, xl = v, for arbitrary, but specified, u,v,t in F, is determined by using multiple Gauss sums. (Received September 23, 1974.)

A-605 *717-A26 J. v. BRAWLEY, Clemson University, Clemson, S. C. 29631. The number of polynomial functions which permute the matrices over a finite field.

Let F = GF(q) denote the .finite field of q elements, and let Fnxn denote the algebra of

nxn matrices over F. A function f:Fnxn + Fnxn is called a scalar polynomial function in case

there exists a polynomial f(x) £ F[x] which, under substitution, represents f. In this paper

a formula is obtained for the number of permutations of Fnxn which are scalar polynomial

functions. (Received September 23, 1974.)

*717-A27 M.S.CHEEMA, University of Arizona, Tucson, Arizona 85721, and G.PALL, Louisiana State University, Baton Rouge, Louisiana 70803. Integra1 positive quadratic forms of determinants less than 1 • Let fn denote such an n-ary form of the least possible determinant k/2n, where k is 1, 2, 3, or 4 according as nor -n is congruent to 0, 1, 2, or either 3 or 4 mod 8. Up ton= 10 there is only one class at this lowest level. Upton= 8, fn is also a minimal form, giving the highest possible ratio of its minimum to the n-th root of its determinant. 2 2 2 2 2 2 2 2 We can take f 8 to be x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x1x6 + x 1 ~ + x1x8 - x2x5 + x2x7 - x 2x8 - x3x6 - x3~ + x3x8 - x4x5 + x4x6 - x4~, and can then obtain f 7, f 6, ••• , t 1 by successively setting x1, x8, x2, ~· x3, x6, x4 equal to zero. It would be interesting to determine the number of classes in each order and genus of determinant less than 1 up to n = 8. It may turn out that all of these contain only one class, This is true at any rate when n = 8 for the determinants 1/4, 1/16, 1/64, and 1/256; and this fact is of importance in establishing unique factorization in the rings of integral elements in the Cayley algebra which contain the naive ring (corresponding to a sum of eight squares)as a subring, It is also true for the positive integral forms of determinants 1/2h when n = 3, 4, 5, 6. Some remarks are also made about the minimal form f 24 associated with Leech's simple group. (Received September 23, 1974.)

*717-A28 D. R. LATORRE, Clemson University, Clemson, S. C. 29631. Semilattices of Bisimple Orthodox Semigroups.

A construction of all bisimple orthodox semigroups was given by Clifford [Semigroup Forum

5(1972), 127-136]. This article determines all homomorphisms of a certain type from one bi-

simple orthodox semigroup into another, and then applies the result to give a structure theorem

for any semilattice of bisimple orthodox semigroups with identity, in which the set of iden-

tities forms a subsemigroup. The methods depend heavily upon Clifford's construction. (Received September 23, 1974.)

*717-A29 TREVOR EVANS, Emory University, Atlanta, Georgia 30322, Word problems.

• This is a survey of both solvability and unsolvability results for decision problems for various classes of algebraic systems (excluding groups). Our main interest is on the relationship between algebraic structural properties of a class of algebras and the solvability or unsolvability of the word prob­ lem and related decision problems for the algebras in the class. The topics covered include (i) normal form theorems, (ii) finite separability properties, (iii) the connection between embedding partial algebras and the word and isomorphism problems, (iv) some recent results on the relationship between the word problem and embedding algebras in simple algebras. (Received September 23, 1974.) A-606 717-A30 DAVID TUCKER, Florida State University, Tallahassee, Florida 32306. Finitely generated inseparable field extensions, Preliminary report.

Let k be a field of characteristic pI 0 and K be a finitely generated ex-

tension of k. Theorem 1. There exists a unique minimal finite purely inseparable extension

ko of k such that K(k0)/k0 is separable. The relationship between separability and mod- ularity allows us to prove: Theorem 2. There is a unique minimal finite purely inseparable

extension k1 of k, containing k0 , such that K(k1)/k is modular. Also we find that: Theorem 3. ko is the unique minimal finite purely inseparable extension of k such that:

K ~ F ®k k0 for any distinguished subfield F of K/k. (Received September 23, 1974.)

*717-A31 ALLAN B. CRUSE, University of San Francisco, San Francisco, California 94117 and M. F. NEFF, Emory University, Atlanta, Georgia 30322. Finite embeddability in a class of infinitary algebras. The variety of w-skeins, infinitary analogues of quasigroups, is shown to contain models of every finite order and to satisfy the strong finite embeddability property, i.e., any finite partial w-skein is embeddable in some finite w-skein. These results generalize recent work on quasigroups and latin squares and their finite-dimensional analogues. (Received September 23, 1974.)

*717-A32 H. F. KREIMER and PAUL M. COOK II, Florida State University, Tallahassee, Fla. 32306. The existence of a normal basis in certain Galois extensions of commutative rings. Preliminary Report. For R a subring of a commutative ring S, assume that the maximal ideal spectrum of R with the Zariski topology has a basis of open/closed sets. Theorem 1. If S is Galois over R with Galois group G (Chase, Harrison, Rosenberg; "Galois Theory and Galois Cohomology of Commutative Rings", Memoirs A.M.S. 52, 1965), then S ~ R(G) as left-R(G)-modules where R(G) denotes the group ring; i.e. S has a normal basis over R. This includes the previously known case where R is semi-local; the theorem also applies if R is von Neumann regular. More generally we obtain Theorem 2. Let S be an H-Galois object for H a finite Hopf algebra over R. (Chase, Sweedler; "Hopf Algebras and Galois Theory" 97 Lecture Notes in Mathematics, Springer-Verlag), then S-:::::: H as left-H'"-modules where H'' denotes the dual of H. (Received September 23, 1974.)

*717-A33 WILLIAM F. KEIGHER, University of Tennessee, Knoxville, Tenn. 37916 Products of differential schemes. Preliminary report,

A differential scheme X is an LDR(= local differential ringed)-space which admits

an open covering (X~) by differential affine open sets, i.e., open sets isomorphic to

Spec 0 (A~) for some differential ring A~, where Spec0 (A~) denotes the LDR-space consisting of the differential primes of A with an appropriate sheaf of differential ~ rings. A family of morphisms (1jii : Xi + S\E! of differential sch.emes is realizable

if s has a covering of differential affine open s·ets such that each X, "'1jJ.-1 (S) l.~ l. C4 has a covering by differential affine open sets such that 1Ji.Ju u . + s l. yuv.. yl.a a is induced by a family of differential ring homomorphisms cjl • : A + B . yl.~ C4 yl.ct (Xi) iE! is realizable over S if there is a realizable family (1jii : xi+ s)iei · Theorem. Let X,Y and S be differential schemes such that X andY are realizable over S .

Then X x 5 Y exists and is a realizable differential scheme over S . (Received September 23, 1974.)

A-607 *717-A34 FREDRIC T. HOWARD, Wake Forest University, Winston-Salem, North Carolina 27109, Factors and roots of the van der Pol polynomials. 3 xa x x -1 The van der Pol polynomials Vn (a) are defined by means of x e [6x(e + 1) - 12 (e - 1)] =

L:;00 v (a) xn/n!. Related to the zeros of the Bessel function of the first kind of index 3/2, n= 0 n these poly­ nomials were defined and examined to some extent by the author in a previous paper ("Properties of the van der Pol numbers and polynomials," J. Reine Angew. Math. 234(1969), 45-64). By considering the polynomials modulo 5, we can now show that neither v2n(a) nor v2n+l (a)/(a- 1/2) has rational roots. The question of irreducibility is difficult, but by using Eisenstein's irreducibility criterion we can show that if n = 2 • 3m, m;;; 0, or n =3m+ 3t, m > t > 0, or n = m(p- 3), p a prime number, then V n(a) and v (a)/ (a - 1/2) are both irreducible over the rational field. It is also true that V (a) is irreduci- ~· n ble if n = 2k, k;;; 0. Some results concerning possible factors of the van der Pol polynomials can be obtained by considering Vn(a) modulo 5, For example V2n(a) has no factor of the form c0 + c1a + •.• + ckak, k > 0, where c0, ••• , ck are integers and ck ¢ 0 (mod 5). (Received September 23, 1974.) *717-A35 RICHARD A. SANERlli, JR., Emory University, Atlanta, Georgia 30322. Boolean algebras with ordered bases and the basis property for ultrafilters.

J.,, in these NOTICES 73T-Al36 we say a filter F in a Boolean algebra (BA) C~has a basis (weakly independent set of generators) {a.}.l l8 I ifF= <{a.}.J.. J..C I> (i.e,,b8F iff there exist i 1, ... ,i such that a.•. ·a.~ b) and for i 1, ... ,i 1 distinct,a.·· •. •a. ~ n 2 1 ln n+ 2 1 J..n a. If G and P are filters in a l3A m with Gcp, then {ai} i 8IC:: G-F is a basis for Q over E 2 n+l if G = and for il' ... ,i l distinct, -a. +... + -a. +a. I F. Let Ol be a BA l l8 n+ 2 1 ln ln+l with an ordered base X. Theorem l CR has the basis property for ultrafilters (i.e., every ultrafilter in Ot has a basis) if and only if every initial segment of X has cofinali ty ~ K o and every tail of X has coini tiali ty ~ 1'-{~. Corollary If ar has the basis property for ultra­ filters, then the cardinality of ~is ~2~~. Theorem 2 An ultrafilter Fin OT has a basis if and only if F has a basis over G for every proper filter G::>F. Corollary or has the basis prop.,rty for ul trafil ters if and only if every homomorphic image of Ol has the basis property for ultrafilters. (Received September 25, 1974.) *717-.A36 M. F. Janowitz, University of MassachusP.tts, Amherst, Mass. 01002 SM-semilattices.

A mapping p on a bounded semilattice is called semimultiplicative if ab ,£ 0 implies (ab)p = (aP)(bp). An SM-semilattice is one that can be coordinatized by a left Baer semigroup of semimultiplicative residuated maps. The concept initially arose because of the observation that every bounded implicative semilattice is an SM-semilattice, These semilattices are characterized, and the characterization used to prove that the following classes of bounded semilattices are SM-semilattices iff they are implicative& (1) atomless modular semilattices; ( 2) pseudocomplemented modular semilatticesa (3) dis­ tributive semilattices in which p an atom implies p vx exists for all x. Finally, the structure of orthomodular SM-lattices is obtained, as well as that of modular SM-semilattices having the property that p an atom implies p 1/ x exists for all x. (Received September 25, 1974.) A-608 *717-A37 JAPHETH HALL, JR., Stillman College, Tuscaloosa, Alabama 35401, Homomorphisms on groupoids with local identities. Preliminary report.

Let (V; +) be a groupoid. If (W, +1 ) is a groupoid, then associated with each homomorphism g 1 from (V,+) to (W,+1) isagroupoid (Vg,+g) definedasfollows: Vg= jg- (lg(x)!):xEV!;if xEV and y E V, then g-1 (!g(x)l) +g g-1 (lg(y)!) = g-1(lg(x+y)!). Let B be a subalgebra of (V,+). V is right B­ unitary if for each x E V, x + b = x for some b E B. B is left-absorbing if (x +b) + B = x + B whenever x E V and bE B. B is left skew-normal if (x +B)+ (y+B) = (x+y) + (B+B) whenever x E V and y E V.

Let TB = !x+B:xEVl. If x E V and y E V, let (x+B) +B (y+B) = (x+y) +B. Proposition 1, if V is right B-unitary, then TB is a partition on V if and only if B is left-absorbing. Proposition 2. If V is right B-unitary while B is left-absorbing and left skew-normal, then (T B' +B) is a groupoid.

Theorem. If V is right B-unitary, then the following are equivalent: (i) There is a groupoid (W, +1 ) 1 havingarightidentity 0 andahomomorphism g from (V,+) to (W,+1) suchthat g- dolJ=B and +B = +g; (ii) B is left-absorbing and left skew-normal. (Received September 25, 1974,)

*717-A38 Dr. William T. Trotter, Jr., University of South Carolina, Columbia, South Carolina 29208. A Note ~ Dilworth's Embedding Theorem.

For a finite poset X, the width of X is the smallest positive integer t such that

X= c1 u c2 u •.• u Ct where Ci is a chain. Also, the dimension of X is the smallest

positive integers for which X is a subposet of D1 x D2 x .•. x Ds where each Di is a chain. R. P, Dilworth proved that for a distributive lattice L = £x, the dimension of

L equals the width of X. In this paper, we prove that for each k ~ 2, the smallest

positive integers such that L~ D1 x D2 x ••. x Ds where each Di is a chain of at most

k points equals the least positive integer t such that X = c1 u c2 u u Ct where each

c1 is a chain of at most k - 1 points. (Received September 24, 1974.)

717-A39 DAVID ROSELLE, Virginia Polytechnic Institute and State University, Blacksburg, Va. 24061. A combinatorial problem involving q-Catalan numbers.

A combinatorial meaning is assigned to the coefficients appearing in the expansion of the q-Catalan numbers. This answers a question raised by Professor Carlitz at the Conference on Eulerian Series and Applications, May 1974, The Pennsylvania State University. (Received September 25, 1974.)

Analysis

*717-Bl S. H. LAMEIER, Thomas More College, Covington, Kentucky 41017 and E. P. MERKES, University of Cincinnati, Cincinnati, Ohio 45221. On domains of univalence for certain ~orphic functions. - --

Let K be a set of ~ojnts in the complex plane and let F(K) denote the family of meromorphic functions f(z) = :0. A./(z- k.), A.> 0, k. e K(i=l,2, ... ,n). A l=1 l l l l domain of univalence of the class F(K) is an open connected set U such that each f e F(K) is univalent in U and, if R is a region properly containing u, there is a function f e F(K) that is not univalent in R. Distler (Proc. Amer. Math. Soc. 15(1964), 923-928) identified one do-

A-609 main of univalence U(K) for F(K) when K is any closed set in the complex plane. We so~ve completely the problem of determining all the domains of univalence of F(K) when K = V i=lKi is the union of n pairwise disjoint, closed, convex sets K., l ~ i ~ n. The couple (A,B) of sets A and B is called a partition of K if A and B are uniBns of the sets K. such that A f ¢, B f ¢, An B = ¢, and K = A L) B. Let c8 , S 8 I, be the family of all open disks, each of which has a diameter with one endpoint in A and the other in B. Finally, let L(A,B) denote the interior of the intersection r-\C . Theorem. The (only) domains of univalence of F(K) S8I S are (l) U(K), provided it is non empty, and (2) W(K) = L(A,B){) U(A)(\ U(B), provided it is nonempty, where (A,B) is any partition of K. (Received July 8, 1974.)

*717-B2 STEPHEN DEMKO, Georgia Institute of Technology, Atlanta, Ga. 30318. Local mappings onto spline spaces.

The approximation theoretic properties of a class of linear mappings onto spaces of polynomial spines, those which are locally defined, are investigated. It is shown that these mappings behave somewhat like projections. This fact is used to obtain sharp L , 1 < p < ro , p - - . error estimates for these mappings. It is also shown that in all of the interesting cases there exist local mappings of minimal norm. (Received September 3, 1974.)

*717-B3 JAMES RETHERFORD, Louisiana State University, Baton Rouge, Louisiana 70803. Applications of Banach ideals of operators.

• The address will be a (hopefully) general interest survey of applications of ideals of operators (in the sense of A. Pietsch) to the structure theory of Banach spaces (in particular the Lp-spaces). Beginning with the work of Grothendieek-we will try to make a comprehensive survey, including the most recent (in some cases, unpublished) work in the area. Topics in the written version of the lecture and in the address

(as time permits) include: (1) operator characterizations of Lp-spaces; (2) parameters induced by ideal norms; (3) the L. Schwartz duality theorem; (4) LUST; (5) the class GL and some factoring problems of

Gothendieck; (6) the subspace structure of ideals of operators on Hilbert space; and, (7) subspaces of Lp. (Received September 9, 1974.)

*717-B4 JOHN A. ROULIER, North Carolina State University, Raleigh, North Carolina 27607. The Degree of Coposi ti ve Approximation.

We say that two functions f and g are copositive on [a,b] if f(x)·g(x) ~ 0 for all x in [a,b]. Let En(f) be the degree of best uniform polynomial approximation to f on [a,b], and let E (f) be the degree of n best approximation by polynomials copositive with f on [a,b]. This paper shows that for certain functions f and for n sufficiently large E (f) < C(w(6E (f)/o) + En(f)) n - n where w is the modulus of continuity of f. (Received September 9, 1974.)

*717-B5 WILLIAM H. CHAPMAN and DANIEL J. RANDTKE, University of Georgia, Athens, Georgia 30602. On the complementation of c 0 • C(X) denotes the usual Banach space of continuous scalar valued functions on a compact Hausdorff space X and c 0 denotes the usual Banach space of zero-convergent scalar valued sequences. Every linearly homeomorphic copy of c 0 in C(X) is complemented in C(X) whenever X

A-610 is the continuous image of an arbitrary product of any of the following types of spaces: (a) compact metric spaces, (b) closed ordinal spaces or (c) weakly compact subsets of Banach spaces. Every linearly homeomorphic copy of

c 0 in a locally convex topological vector space E is complemented in E whenever E is the continuous linear image of an arbitrary product of weakly compactly generated Banach spaces. These two results extend certain known results (cf. Mich. Math. J. 20 (1973), 39-44; Proc. Amer. Math. Soc. 28 (1971), 627-628; and Colloq. Math. 11 (1963), 55-63 ) . (Received September 16, 1974.)

*717-B6 WILLIAM L. GREEN, Georgia Institute of Technology, Atlanta, Georgia 30332. Topological dynamics and C*-algebras.

Let A be a C*-algebra with identity and G a group of automorphisms of A. Let S(A) denote the state space of A with the weak* topology. Then composition of mappings defines an action of G on S (A), and the transformation group (S (A) ,G) is uniformly almost periodic if and only if G has compact pointwise closure in the space of all maps of A into A. Consideration of the enveloping semigroup of (S(A),G) shows that in this case, this pointwise closure G is a compact topological group consisting of automorphisms of A. The Haar measure on G is used to define an analogue of the center-valued trace in a finite von Neumann

algebra. If A possesses a sufficiently large group G0 of inner automorphisms such that (S(A),G) is uniformly almost periodic, then the primitive ideal 0 space of A is Hausdorff. The notion of a uniquely ergodic system is applied to give necessary and sufficient conditions that an approximately finite dimensional C*-algebra possess exactly one finite trace. (Received September 18, 1974.)

*717-B7 S.M. RANKIN, III, Florida IP~titute of Technology, Melbourne, Florida 32901, Oscillation Criteria ~ Elliptic Differential ~ations. Preliminary report.

An oscillation criteria is given for ~ 2 u + cu = 0 in terms of the eigenvalues of the system

2u + cu " 0 on G .. { x E'. < = v ro En : 0 a < I x\ < b}, u 0 on S11 = tx E:. En : \ x \ = a} and Ju;c:J.n = 0 on Sb = l x E'. Ef : lx\ .. b). Also a strong comparison theorem is given. (Received September 19, 1974.)

717-B8 C. C. Travis, University of Tennessee, Knoxville, Tennessee 37916, On the unique­ ness of solutions to boundary value problems for hyperbolic equations.

It is well known that the Dirichlet problem for hyperbolic equations does not in general con­ stitute a well-posed problem. In the case of the two-dimensional wave equation utt - uxx = O, for instance, Bourgin and Duff have shown that for a rectangle with sides parallel to the coordinate axes, uniqueness of solutions of the Dirichlet problem holds if and only if the ratio of the sides of the rectangle is an irrational number. This basic result has been

A-611 extended in many directions. We shall present a proof of these known results which yields new information when applied to the singular hyperbolic boundary value problem. k n utt + tut- L Di(a .. (x)Dju) + C(x)u = f(x,t) i,j=l l.J

u(x,O) = u(x,a) for x E G

avau + u~(x)u o o n aG x [0 , a] ,

where G is a bounded regular domain in En and k is a real parameter, -m < k < ® • (Received September 20, 1974.)

717-B9 R. T. JACOB, JR., University of New Orleans, New Orleans, La. 70122. Matrix Transformations on Analytic Sequence Spaces, Preliminary report.

Let A denote the space of all complex sequences a such that if z is a complex number and I z I < 1 then I:a zn converges, and let B denote the space of all complex sequences b for which n there is a complex number z such that lzl > 1 and I:b zn converges. Theorem 1. If p > 1, n q = p/(p-1), and M is an infinite matrix, then the following are equivalent: (i) M transforms ip into A; (ii) M' (the transpose of M) transforms B into iq; (iii) lim supj (I:kiMjklq)l/j ~ 1.

Theorem 2. Under the hypotheses of Theorem 1, the following are equivalent: (i) M transforms ~P lnlu B; (ii) M' transforms A into iq; (iii) there exist numbers t and r such that 0 < r < 1 and I:k1Mjklq5t rj. Theorem 3. If M is an infinite matrix, then the following are

equivalent: (i) M transforms A into B; (ii) each row and each column of M is in B, and there exist numbers t and r such that 0 < r < 1 and IMjkl 5 t rj+k whenever each of j and k is a nonnegative integer. Theorem 4. If M is an infinite matrix, then the following are equivalene (i) M transforms B into A; (ii) each row and each column of M is in A, and if E > 0, there is a positive number m such that IMjkll/(j+k) < 1 + E whenever each of j and k is a nonnegative integer and j + k ~ m. (Received September 20, 1974.)

717-BlO. WITHDRAWN.

*717-Bll ROBERT KNOWLES, The University of Connecticut, Waterbury, Connecticut. A finiteness property of 1. c. s. Preliminary report.

Let E and F be locally convex spaces in duality. Following Amemiya and Komura [Math. Ann. 177(1968), 273], we say that F is finite with respect ta E if every O'(F,E)-bounded subset of F is finite dimensional. In this note we point out some of the properties of this concept and discuss some examples. (Received September 23, 1974.)

717-B12 THOMAS R. LUCAS, University of North Carolina, Charlotte, North Carolina 28223. Some interpolation (characterization) type results for cubic (bicubic) splines with applications to boundary value problems. Preliminary report.

Recent results of the author concerning certain superconvergence results for smooth cubic splines over uniform meshes are extended to give a new result which leads to a 4th order subdomain method for solving two point boundary value problems. A characterization theorem for smooth bicubic splines over a rectangle is shown to give the result that bicubic spline collocation applied to the Dirichlet problem over a rectangle (tlu = f, ul r = 0) with variable mesh spacing gives O(n2) convergence to the solution and its derivatives provided that fir= 0. The proof gives some insight into the strengths and weaknesses of collocation methods as compared with Galerkin methods. (Received September 23, 1974.) A-612 717-B13 R. SHONKWILER, Georgia Institute of Technology, Atlanta, Georgia 30332. Multi-parameter Stieltj es transformations.

Fix n = 1, 2, .•. , let W = (0, m)n, and let f(x) be a real-valued function defined on the set - W = (- w, O)n. The difference kernel K(x2, x1) of f for the points xp = (x~, ... , xJ, p = 1, 2, is defined to be (n? (~ - ~ ))- 1· L(-1)I;pif(x1 , .•. xn ) where the summation extends over all choices p. = 1, 2 for J=1 2 1 P1 ' Pn J j = 1, .•. , n. If f has continuous nth partial derivatives, then K can be defined as a limit in case certain factors in the denominator vanish. Theorem. In order that a real-valued function f(x) admit a represen- . . . -1 tation in the form f(x) =Jwnf= 1 ~(1-xltl) da(t), x E -W, where a is a nonnegative bounded measure on W it is necessary and sufficient that: (1) f have continuous nth partial derivatives, (2) for each subset

)i,j, ... , Pl c \1, ... , nl the limit of f(x)/xi ... xP exists as xi, ... ,xP tends to zero, and (3) the difference kernel is of positive type, i.e. L~. 1a.a.K(x.,x.);;; 0 for all finite choices a1, .•• ,am complex and 1, J= J 1 J 1 xl' .•. ,xm in -W. (Received September 23, 1974.)

*717-Bl4 MARIO 0. GONZALEZ, University of Alabama, University, Alabama 35486 ..The n-dimensional Cauchy-Riemann equations.

The Cauchy-Riemann equations in n-dimensional vector spaces are obtained by imposing on the mapping function the conformality property. An alternative form of these equations is expressed in terms of the gradients of the components of the mapping function. Also, a generalized form of the Laplace equation is derived. (Received September 23, 1974.)

*717-Bl5 T. M. MILLS and A. K. VARMA, University of Florida, Gainesville, Florida 32601 The approximation of functions by algebraic polynomials.

The object of this paper is to give a new proof

S. A. Teljakovskii and I. E. G. Gopengauz concerning the approximation of

functions by algebraic polynomials (see for i.e. S.A. Teljakovskii, two

theorems on the approximation of functions by algebraic polynomials.

(Russian) Mit. Sb. vol. 70, (112) 1966, 252 - 265). It is of considerable

advantage that our polynomial is of degree ~ n - 1 and depends on n + 2 values of f (x) only. (Received September 23, 1974.)

*717-Bl6 JAMES A. RENEKE, Clemson University, Clemson, S. c. 29631, Stability and control of hereditary systems, Preliminary report.

Suppose that X is a linear space, G is a linear space of functions from the reals S into X, Nis a nondecreasing function from S into the pseudonorms on G such that Nu(f)=O only in case f(x)=O for x.::_u and {G,N} is complete, P is a function from S into the projections on G such that

Nv(Puf)=Nu(f) for u.::_v, B is the set of operators on G to which B belongs only in case B con­ tains {0,0}, [Bf] (u)=O for u.::_O, and there is a nondecreasing real function k on S such that rv N (P Bf-P Bg-P Bf+P Bg) < (L) J N (f-g) dk (x) for u _< _< w v v u v - ux v w, and A is the set of operators on G to which A belongs only in case A-1 is in B. See Abstract 712-BlO, these Notices 21(1974), A-339.

A-613 If each of A and o is in A, A is linear, and C is in B then the following are equivalent:

I) D= [l+AC]-lA; II) there are integrals K1 and K2 on G defining systems with input-output func­ tions A and o, and [Cf](u) = [K1 (0,u)f](u)-[K2 (0,u)f](u) if O~u; III) there are transition functions w1 and w2 on G defining systems with input-output functions A and D, and w 2 (u,O)~ = w1 (u,O)f+(L,R)J~w1 [u, ]VW2 [ ,O)f if O~u, where V(v,u)=PvC-PuC if O

*717-B17 JAMES ERNEST MILLER, West Virginia University, Morgantown, West Virginia 26506. Subordinating factor sequences for convex maps in en.

Let Kn be the class of holomorphic and univalent maps of the unit disc in en. An extension of subordinating factor sequences (H. S. Wilf, "Subordinating factor sequences for convex maps of the unit circle", Proc. Amer. Math. Soc. 12(1961), 689-693) to the class Kn is made, From this characterization of subordinating factor sequences, the distortion theorem and the coefficient bounds for Kn are found. Also the Polya-Schoenberg conjecture for Kn is proved. (Received September 25, 1974.)

*717-B18 G. J. Etgen, University of Houston, Houston, Texas 77004 and S, C, Tefteller, University of Alabama in Birmingham, Birmingham, Alabama 35294, Three Point Boundary Problems for Second Order Differential Systems, Preliminary Report.

Consider the second order differential system (1) y' = k(x)z, z' = -g(x;A;~)y, x EX= [a,c],

A E L = (L1,L2), ~ EM= (M1,M2), together with the boundary conditions (a) ~(A,~)y(a) -

13(A,~)z(a) = 0; (b) y(A,~)y(b) - 6(A,~)z(b) = 0; (c) e(~)y(c) - cp(~)z(c) = 0; a< b 0, 13 > 0, 8 > 0 and cp > 0, In addition, it is assumed that each of the functions g, (~/13) and (y/6) are increasing in A, with lim g(x;A;~) = + 00 and lim g(x;A;~) = ~ for each x,~ E XxM, Under these hypo- A~Lz A~L 1 theses, .there exists a sequence of continuous functions lAn (~)J: on M such that for each

~ E M and each nonnegative integer n, the solution ly(x;A (~) ;~), z (x;A ~) ;~)j of (1) sat- n n isfies (a) and (b), and has exactly n zeros on (a,b). Sets of eigenvalues A,~ for the three point boundary problem are then obtained by varying ~ on M. (Received September 25, 1974.)

*717-B19 E.B. SAFE', University of South Florida, Tampa, Florida 33620 and R.S. VAOOA, Kent State University, Kent, Ohio 44242. Rational App:rox:irnation in unbounded Regions.

Let f(x) be defined and finite on I 0,-+<><>) and assume there exists a sequence of rational functions {rn}~=l of respective degrees n such that for the sup nonn -1. II II 1/n 1 ~ { f-rn L.. [0,+..)} = q < 1. Suppose further that the rn {z) have no poles in an unbounded region H of the z-plane symnetric about [ 0 ,-ioo) , General results are obtained which :irrply geanetric oonvergence of the r n {z) in an unbounded symmetric subregion H* of H. For exarrple, if H is an infinite sector of the fonn Iarg z I < 4> , then H* is also an infinite sector of the fonn larg z I < 4>*, where *= 4>*(4> ,q). If

A-614 H = {z = x+iy : X ::.. 0, IYI ~ h(x)}, where for all x, h(x) > 0, h' (x) ::_ 0, and lim h' (x} = o, then H* = {z = x+iy : x ~ 0, 1y 1 .s_ s h (x) } , where s = s (q} • Applications of £h:Se -z overconvergence results are given; in particular when f (z) e and the rn (z) are PacM approxirnants. (Received September 25, 1974.)

*717-B20 DAVID LOWELL LOVELADY, Florida State University, Tallahassee, Florida 32306, Oscillation in even order linear differential equations

Let q be a positive continuous function on [0, ®), let n be an integer, n ~ 2, (2n) 1 and consider (*) u + qu = 0 on [0' ®). THEOREM: If w"(t) + (( 2n-3)! r (s - t/n-3 t­ q(s)ds) w(t) = 0 is oscillatory then every solution of (*) is oscillatory, and if

t2n-2 w"(t) + ( 2n-2)! q(t)w(t) = 0 is nonoscillatory then there exists a nonoscillatory solution of ( *). (Received September 25, 1974. )

*717-B21 STEPHEN A. SAXON, University of Florida, Gainsville, Fla. 32611 Separable quotients of Banach spaces •

Whether every (infinite-dimensional) Banach space admits a separable (infinite-dimensional) quotient by a closed subspace is a long-standing open question, considered recently by such noted Banach space specialists as Johnson and Rosenthall. Locally convex space theorists may well ponder the question, as a joint effort by the author and Professor Wilansky suggests:

~ Banach space has ~ separable quotient if and only if it has ~ dense, non-barrelled subspace· (Received September 25, 1974.)

Applied Mathematics

*717-Cl DISTORTIONLESS WAVE PROPAGATION IN INHOMOGENEOUS MEDIA AND TRANSMISSION LINES Victor Burket, R. J. Duffintt & D. Hazonyt, tease Western Reserve Univ.,Cleveland,Ohio & ttCarnegie-Mellon Univ. Pittsburgh,PA Of concern are mechanical or electrical waves in a media which may be nonuniform and dissipativs The problem posed is to find conditions for the undistorted propagation of signals. The elec­ trical transmission line is chosen as the general model. Along the length of the transmission line there are four functions which may be prescribed essentially arbitrarily. These are series resistance, series inductance, shunt conductance, and shunt capacitance. A different.ial equa­ tion is derived, relating these functions, which gives a necessary and sufficient requisite for distortionless transmission of a voltage wave. Various corollaries of this theorem are developed. For instance it is shown that simultaneous voltage and current waves can be transmitted without distortion if and only if the characteristic impedance of the transmission line is positive at each point. (Received August 30, 1974.)

717-C2 G. W. REDDIEN, Vanderbilt University, Nashville, Tennessee, 37235. Patch Bases and Singular Splines. Preliminary Report.

The use of a class of singular splines to approximate the solution to singular two-point

boundary value problems by means of collocation and other projection methods was studied in

Numer. Math. 21 (1973), 193-205 and these Notices 20 (1973), 709-C6. The existence of patch

A-615 bases for the singular splines used in these methods was left open, but has now been settled.

Explicit formulas and properties of these bases will be given along with numerical examples of their application to collocation. (Received September 11, 1974.)

717-C3 JOHAN G. F. BELINFANTE, Georgia Institute of Technology, At1anta, Georgia 30332. Adjacency Graphs of Transition Probability Spaces. Preliminary Report, J. von Neumann added axioms about transition probability to the lattice axioms for filters in quantum mechanics. (J.v.N., Collected Works, Pergamon Press, Oxford (1961), vol. 4, pp. 191- 194.) Certain poset properties already follow from the transition probability axioms if we identify filters with orthoclosed subspace& of a transition probability space. In particular, the poset of orthoclosed subspace& is orthomodular. (Abstract 712-C4, these Notices 21(1974), A-344.) A~ is an element of a transition probability space. States x and y are ad1acent if they are distinct, but not orthogonal. The valence of a state is the number of states adjacent to it. A transition probability space is irreducible if it is not the union of two nonempty orthogonal subspacea. Theorems. The adjacency graph of an irreducible transition probability space has no articulation points. Any state of valence two belongs to exactly one four-element irreducible subspace, and this subspace is orthoclosed. An irreducible transition probability space having two adjacent states of valence two has exactly four states. There are no irreducible transition probability spaces with two, three or five states. An irreducible transition probability space in which all states have the same valence can not have seven states. (Received September 20, 1974.)

Geometry

717-D1 ROBERT B. GARDNER, University of North Carolina, Chapel Hill, North Carolina 27514. Submanifolds of Euclidean space.

8 In this report we survey recent results and open problems in the most classical area of differential geometry, the theory of submanifolds in JRn. In particular we will present selective results on the follow- ing questions. (1) The Isometric Immersion Problem: (Given a m-dimensional Riemannian manifold M, when does there exist an isometric immersion into Euclidean JRm+p ?) Here the report will center on refinements of the problem which are obtainable when M is assumed compact. (2) The Rigidity Problem: (When do two submanifolds of JRn differ by a Euclidean motion?) Here the report will describe the af­ firmative solution of a conjecture of H. Hopf concerning convex hypersurfaces, and recent results in high codimension, (3) The Uniqueness Problem: (What differential geometric conditions characterize a par­ ticular submanifold up to Euclidean notions ?) Here the report will concentrate on underlying geometric ideas leading to recent characterizations of spheres. (Received September 17, 1974.)

*717-D2 J, P. HOUIES, Auburn University, Auburn, Alabama 36830. A Rank Theorem for Infinite Dimensional 8paces

Suppose X is a Banach space, U is an open set of X containing 0, and f is a continuously differentiable function from U to X satisfying f(O) = 0 and f'(o)2 f'(O). Condition I: There is a neighborhood 1'1 of 0 in X sothat (f'(O) [ f(W)) is one to one. Condition II: 'There is a neighborhood W of 0 in X so that if V is a neighborhood of 0 in f'(O)(X) then f(V) is a neighborhood of 0 in f(W). f satisfies condition I if and only if f satisfies condition II. If X is finite dimensional then

A-616 f satisfies condition I if and only if there is a neighborhood W of 0 in X so that (rank o f' lvl) is constant. If f satisfies condition I there are reversibly continuously differentiable homeomorphisms h and k, each with domain a neighborhood of 0 in X, so that (fldom (h))= k o f'(O) o h. If f satisfies condition I there is a neighborhood W of 0 in X and a continuously differentiable function q with domain W satisfying q o q = q, im(q) is a neighborhood of 0 in f-1( { o) ), and f x q is a homeomorphism onto a neighborhood of 0 in f(W) x f-1( { 0} ) . (Received September 19, 1974.)

Statistics and Probability

*717-Fl JAMES J. BUCKLEY, Department of Mathematics and Computer Science, University of South Carolina, Columbia, South Carolina 29208. Unconditional probabilities for the voter's paradox. Preliminary report.

Let n be the total number of voters, m the total number of issues, and n/2 < k. Let Ri' 1;;;; i;;;;

J, = m! be all rankings of the issues, pi= (p~, ••• , p~) a (culture) probability vector for the ith voter (i selects Rj with probability p~, and P = (p1, .•• , ifl>. One issue I is preferred to another J by the committee if and only if k voters prefer I to J. Let A be all probability vectors (universe of cultures),

F is a probability distribution on An, and E is an event (no majority winner, a cycle, etc.) obtained by the committee voting on all pairs of issues. P[E], the (unconditional) probability of E for a random sample of n cultures, is calculated and shown to equal the (conditional) probability of E under an im­ partial culture (pi = (J.-l, .•• , t-1), all i) for a large class of F. This class includes the uniform,

(truncated) normal, and symmetric discrete distributions, Necessary and sufficient conditions for the two probabilities to be equal are given. Examples where they are not equal are also cited. (Received

September 23, 1974.)

Topology

*717-Gl M. JAYACHANDRAN, Madurai College, Madurai, INDIA and M. RAJ.A.GOP.ALAN, Memphis State University, Memphis, Tennessee 38152. Scattered cmnp-actifica·tion of N U W Preliminary report.

Let P be a point of pz-z where Z is the discrete space of integers. Let p be a p-point

of the derived set X of a countable set~ Pl, p2 .. Pn. -~ of p-points p1 ,p2 .• Pn·· in pz-z. The N \J\P1 admits a scattered compactification. (Received August 12, 1974.)

717-G2 ROB~T J. DAVERMAN, University of Tennessee, Knoxville, Tennessee 37916. SewJ.ngs of closed n-cell complements. Preliminary Report.

A closed n-cell complement is a space homeomorphic to the closure of the complement of an n-cell topologically embedded in then-sphere Sn. A sewing h of closed n-cell complements Cl and C2 (i.e., a homeomorphism h: Bd c1 ~ Bd c2) satisfies the Mismatch Property if there exist sets Fi in Bd Ci such that FiU Int Ci is 1-ULC(i =1,2) and h(F1)(1 F2 = 0. Theorem. For n ~ 5 satisfaction of the Mismatch Property is a sufficient but not a necessary condition for a sewing h of C and C to · ld sn (th · f c U c 1 2 y1e at J.s, or 1 h 2 to be

A-617 homeomorphic to sn). Theorem. If c1 and c2 are closed n-cell complements such that there exist sets X. in Bd C. such that X. lJ Int C. is 1-ULC and X. is a countable union of Cantor l. l. l. l. l. n sets that are tame in Bd Ci (i = 1,2), then a sewing h : Bd c1 + Bd c2 yields S (n ~ 5) iff h satisfies the Mismatch Property. Theorem. If C is the suspension of a closed (n-1)-cell complement (n ~ 5), then, for any closed n-cell complement D, every sewing of C and D yields Sn. (Received August 19, 1974.)

*717-G3 R. PAUL BEEM, University of Pennsylvania, Philadelphia, Pa. 19174 On the structure of free z4-bordism, Preliminary report.

Let N*(z4) denote the bordism algebra of free z4-actions on closed unoriented 1 A manifolds, a = [ S ,i] denote the class in N1 (Z4) of the circle with the generator of z4 acting as multiplication by v:r and J* denote the N*-subalgebra of N*(Z4)

generated by a • Let K* denote the image of the extension homomorphism from free z 2 A bordism to N*(z4). "z 4 ~ 'J'hPOTPm: N* = J* ®N* K* , as algebras.

A Z4 Theorem: There are free J*-module generators of N* b2i in K2i , such that if 2 [s j+l,i] denotes the class of the 2j+l sphere with standard free z4 action, then ,[j/21 a~ _ CP(2n)b . n-0 2J-4n modulo the ideal of N*-decomposables. (Received August 23, 1974.)

717-G4 J.D. LAWSON and B. L. MADISON, Louisiana State University, Baton Rouge, Louisiana 70803. Sectional mappings. Preliminary report.

A continuous (onto) function f:X ~ Y is ~ctional if a subset U of Y is open in Y if and only if for each section (not necessarily continuous) s:Y ~X of f ' f-1 (u) n sfYT is open in STYT . Clearly a sectional map is a quotient map, but there are quotient maps that are not sectional. In each of the following results the situation is that f:X ~ Y is a continuous (onto) function. Then f is a sectional map provided 1) f is a quotient map, X is a k-space, and Y is k-Hausdorff; 2) f is a compact-covering map and Y is a k-Hausdorff k-space; 3) f is a perfect map and Y is Hausdorff; 4) f has continuous local sections. For an application of this notion, denote by q(X) a space X with the topology defined by saying U is open in X if and only if for each compact Hausdorff space K and continuous function g:K ~ X the set g-l(u) is open in X [See R. M. Vogt, Arch. Math. 22(1971), 545-555.] Using the above re­ sults one can show that if X is a k-Hausdorff space, then q(X) is X with the k-topology. These and Vogt•s results yield results concerning quotients of topological algebras. (Received September 16, 1974.)

*717-G5 CHARLES D, BASS, Pembroke State University, Pembroke, North Carolina, 28372. Some special decompositions of EJ.

A condition, suggested by August Lau, is shown to be sufficient for an upper semicontinuous decomposition of EJ to yield a space homeomorphic to EJ.

A-618 For p, q € EJ, p = (xl,x2,xJ)' q = (yl,y2,yJ)' p~ q if and only if xi ::S y i for each i. Define• R(p,q) = [P' E: E3\P ~ p' ~ q). Let X be a compact subset of EJ. Suppose that X contains a point p such that, for every p £X, p ~ p and R(p,q) c X. Then X is a universally monotone set, If G is a monotone upper semicontinuous decomposition of EJ. into universally monotone sets, then EJ/G is homeomorphic to E3• (Received September 19, 1974.)

*717-G6 c. E. AULL, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061. On C and C*-embedding, Preliminary report.

A space is defined to be functionally Hausdorff (regular) [normal] if there exists a continuous function f to be reals such that f(A) n f(B) = 0 if A n B = ¢where A and B are points (A is a point, B is a closed set)[A and Bare closed sets]. D'Aristotle has shown F.H. closed sets (sets closed in every functionally Hausdorff space) are C-embedded in every functionally Hausdorff space they are embedded in. This is also an easy consequence of some earlier work of Stephenson. We prove that if a set is C*-embedded in every functionally Hausdorff space it is embedded in then it is F. H. closed. Lemma: If a pseudocompact space M is embedded in a functionally regular space X then it is C-embedded in X iff M with its weak topology is C-embedded in Xwith its weak topology. Note: If M is pseudocompact its weak topology is necessarily the relative topology for the weak topology for X. Theorem. A space is C*-embedded or C-embedded in every functionally regular space that it is embedded in iff it is almost F. H. closed. A closed pseudocompact subset with pseudocompact zero sets is C-embedded in a functionally regular space X if the weak topology for X is functionally normal. (Received March 25, 1974.)

'11717-G7 GEORGE M, RSED Ohio University Athens, Ohio 45701 On Q-sets in normal first countable T2-spaces. The e

717-G8 W. F. LaMartin, University of New Orleans, New Orleans, La. 70122. Pontryagin duality for abelian k-groups. Preliminary report.

The dual GA of a T 2 abelian k-group G is the set of all k-group morphisms from G into the circle group provided with the k-refinement of the compact-open topology. A k-group G is said to satisfy k-group duality if G is isomorphic to G~ via the canonical morphism. Obviously, locally compact abelian groups satisfy k-group duality. In analogy with results for Pontryagin duality, it is shown for any collection of k-groups, each satisfying k-group duality, that their categorical product and coproduct both satisfy k-group duality, where the dual of the product is the coproduct of the respective duals, and the dual of the coproduct is the product of the respective duals. In the course of this, it is

A-619 observed that any group satisfying k-group duality is in fact the k-refinement of a topological group which satisfies Pontryagin duality. The converse of this is still open. (Received September 20, 1974.)

*717-G9 PHILLIP ZEllOR, iluburn Unive.rsity, Auburn, Alabama 36830. Some Topological Properties Determined by P:I-operators, Preliminary report.

FX will denote the collection of closed subsets of X. A function q:> : X X FX ~ [0, 1] is a PN-operator (perfect normality-operator) if, for each H € l<'X, H =tx € X I F(x, H) =ot A sequence g1 ,g2 , ... of functions from FX into the collection of open subsets of X is a G -sequence for FX 0 if, for each H € FX, we have H =() i=l gi(H). If g1 ,~, ••• is a Gb -sequence such that, for each H in FX, H ={) :l gi (H), then g1 ,~, ·•. will be called a regular Gb -sequence. Theorem 1: The following conditions are equivalent for the T -space X: 1. X is developable' 1 2. There is a G6 -sequence g1 ,~, ••• for FX such that if for each x € g.(H) there are open sets J. U containing x and V intersecting H so that if Kn V f ¢, then U C gi(K). 3· There is a PN-operator which is upper semi-continuous when FX is endowed with the lower semi-finite topology. 'i'heorem 2: If X is a ·r1-space, then the following are equivalent: 1. X is metrizable 2. There is a regular G& -sequence for FX so that if H € FX and x € g. (H) , then J. there are open sets U containing x and V · intersecting H such that if K (') V f

*717-GlO MAURICE HUGH MILLER, JR., University of Mississippi, University, Mississippi 38677. Equivalence of almost continuous and Darboux functions under closure. Preliminary report. ------Denote by f. (i = 1, 2) the graph J. of a real function with domain and range the interval

I [0, 1]. A function fi is said to be a Darboux function if fi (C) is connected whenever C is a connected subset of [0, 1] and f. is said to be a connected function if f. is connecte~ J. J. A function fi is said to be almost continuous if every open set in I 2 which contains fi also contains a continuous function g with domain I, [see Stallings,~· ~· 47(1959), 249- 263]. Theorem 1. Denote by f 1 a Darboux (connected) function which is not almost continuous. Then there exists an almost continuous function f 2 such that ~ = 12 • Theorem 2. A sufficient condition that a Darboux (connected) function f. be almost continuous is that J. K meets f whenever K is a perfect set in I 2 such that the X-projection of Kllfi is uncountable. (Received September 20, 1974.)

717-Gll R. w. HEATH, University of Pittsburgh, Pittsburgh, Pennsylvania 15213 and W. F. LINDGREN, Slippery Rock State College, Slippery Rock, Pennsylvania 16057. Weakly uniform ~ !,, Preliminary report.

A base E for a space X is said to be weakly uniform if, for each x E X and each infinite subfamily A of ~ each member of which contains x, n A= {x). The Michael line, the pointed irrational extension of 'R and metacompact developable spaces have weakly uniform bases. A separable space which has a weakly uniform base is second countable. If' X has a weakly uniform

A-620 base, then it has a G0-diagonal. Hence any linearly ordered space or compact space which has a weakly uniform base is metrizable. If X is a locally connected k-space which has a weakly uniform base, then X has a point-countable base. (Received September 23, 1974.)

717-Gl2 R. w. HEATH, University of Pittsburgh, Pittsburgh, Pennsylvania 15213 and w. F. LINDGREN, Slippery Rock State College, Slippery Rock, Pennsylvania 16057. Weakly uniform bases II, Preliminary Report.

For the basic definition and other results, see the abstract Weakly uniform bases I, these

Notices. Further results are presented which delineate the position of spaces which have a weakly uniform base with respect to spaces having a point-countable base, uniform base, etc. An example of a non-first countable stratifiable space which has a weakly uniform base is given.

Nearly all these results can be proved under the weaker assumption of the existence of a point­ separating weakly uniform open cover. (Received September 23, 1974.)

*717-G13 GARY GRUENHAGE and PHILLIP ZENOR, Auburn University, Auburn, Alabama 36830. Metrization of spaces with large basis dimension. Preliminary report.

A collection r of subsets of a set X is said to have rank 1 if whenever g1, g2 E r and g1 n g2 t fJ, then g1 c g 2 or g2 c g 1. A topological space X is said to have large basis dimension :;;;; n (or a base of big rank;§ n + 1) if there exists a basis for X which is the union of n + 1 rank 1 collections of open sets. X has countable large basis dimension if it has a basis which is the union of countably many rank 1 collections, and each point has a base belonging to one of the collections. The following result extends results obtained by A. V. Arhangel•skii and P. Nyikos for nonarchimedian spaces (i.e., large basis dimension= 0). Theorem. A normal (normal and countably paracompact) space of finite (countable) large basis dimension is metrizable if and only if it is a L:-space or a wll-space. (Received September 23, 1974.)

*717-G14 RICHARD E. HEISEY, Vanderbilt University, Nashville, Tennessee 37235. A Factorization of the Direct Limit of Hilbert Cubes.

00 Let R = dir lim Rn, Q00 = dir lim Qn where R denotes the reals, Q the Hilbert cube.

It is known that if B is any separable, infinite-dimensional Banach space, then its

conjugate B* with the bounded weak-* topology is homeomorphic to Q00 Theorem 1.

00 00 Q ~ R00 X Q. Corollary. Let U,V be open subsets of R. Then U and V have the same

homotopy type iff U x Q ~ V x Q. Combined with theorem of D. W. Henderson, Theorem 1

implies the following. Theorem 2. (a) If X is a locally compact, separable AR (metric),

00 00 then X x Q x R ~ Q x R • (b) If X is a locally compact, separable ANR (metric), then

00 00 X x Q x R embedds as an open subset of Q x R • (Received September 23, 1974.)

*717-Gl5 JERRY E. VAUGHAN, University of North Carolina, Greensboro, North Carolina 27412. Some properties related to [a, b)-compactness. I.

We consider three properties which are related to [a, b)-compactness in the sense of open covers ([a, b]-.compactness) and [a, b)-compactness in the sense of complete accumulation points ([a, b]-compact­ nessr). Definitions. A space X has property N[a, b] if each net in X, whose domain is any S (m) = a lHJHc m and !HI< a) where a;§ m ;§ b (directed by inclusion), has a cluster point in X. A space X

A-621 has property S[a, b] if every open cover of X of regular cardinality :"§ b has a subcover of cardinality

< a. A space X has property G[a, b] if every net in X, whose domain D is

717-Gl7 KENNETH R. KELLUM , Miles College, Birmingham, Alabama 35208 Six ~ of Retracts Preliminary report.

Suppose f: X~ Y. That f is almost continuous (resp. r- almost contin-

ueus means that each open set in X x Y (rest. each open set of the form \ n I '-/ (X x Y) - (Ai x Bi), where Ai and Bi are closed) containing f i=l also contains a continuous function g:X ~ Y. That f is a connectivity

function means'that whenever C is a connected subset of X, fie is a

connected subset of X x Y.

The following types of retracts are considered: &-retracts, almost continuous retracts, r-almost continuous retracts, &-almost continuous

retracts, &-r-almost continuous retracts, and &-connectivity retracts. We show that under certain conditions all six are equivalent. (Received September 24, 1974.)

*717-G18 THOMAS M. PHILLIPS, Auburn University, Auburn, Alabama 36830. Completable Aronszajn spaces.

The concepts of a primitive sequence and a primitive representative are used to give an internal characterization of those topological spaces which are dense subspaces of complete Aronszajn spaces. Some general properties of arbitrary Aronszajn completions are examined by considering first completions formed in this primitive fashion. Three other sufficient conditions, one of which is also shown to be necessary, under which a space is a completable Aronszajn space are given. The remaining two conditions are shown to be necessary if the spaces considered are metacompact Moore spaces. (Received September 25, 1974.)

A-622 TheN ovember Meeting in Los Angeles, California November23,1974

Algebra& Theory ofNumbers

718-Al Albert Leon Whiteman, University of Southern California, University Park, Los Angeles, California 90007.Williamson type matrices of _or

Let q be a prime power - l(mod 4). It is proved in this paper that there exists an Hadamard matrix of Williamson type of order Zq(q + 1). The four matrices A,B,C,D in the Williamson form are circulant only when q is a prime, and multicirculant otherwise. (Received August 19, 1914.)

Analysis

*718-Bl LOUIS J. GRIMM, University of Missouri-Rolla, Rolla, Missouri 65401. Solvability of singular differential systems. The classical solvability theorem of F. Lettenmeyer proves the existence of nontrivial holomorphic solutions at singular points of linear homogeneous differential systems. This talk will discuss some recent results, including work of L. M. Hall, W. A. Harris, Y. Sibuya, L. Weinberg, and others, which have simplified the proof and extended the theorem in several directions. (Received August 8, 1974.)

*718-B2 R. E. 0 1MALLEY,JR, University of Arizona, Tucson, Arizona 85721. Phase Plane Solutions to Some Singular Perturbation Problems. We seek asymptotic solutions of the two-point problem e 2 ~ + f{x) = 0, 0 ~ t ~ 1 ,

x(O), x(l) prescribed, as the small parameter E tends to zero. In the (x,ex) phase-plane, solutions generally tend to remain at rest points (x,O) corresponding to maxima of the potential energy V(x) = Jxf(s)ds. Typical results are (a) If V has a finite number of

maxima between x(O) and x(l) 1 there is a unique monotonic solution which remains at these

maximum values for E + 0, and (b) if x(O) and x(l) are contained between successive

maximum points d1 and d2 of V with V(d1) = V(d2), there are denumerably many solutions which switch between the values d1 and d2 for E + 0. Examples of such behavior are of physical interest in addition to their value in furthering our knowledge of asymptotic solu- tion techniques for differential equations. (Received September 3, 1974.)

*718-B3 JAMES S. MULDOWNEY, University of Alberta, Edmonton, Alberta T6G-2Gl. An inequaZity q_f_ CapZygin and PoZya.

A theorem of Polya (1924) may be stated as follows: If L is a real linear nth n order differential operator disconjugate on [a,S] and tiE [a,S] , i O, ••• ,m ,

f{j)(ti) = 0, j = O, ••• ,ri-1, r 0 + ... + rm = k < n then Lnf ~ 0 => pk Ln-kf ~ 0

A-623 r. where pk(t) = IT(t-t.) ~, L f = f, L kf = W(~ , •.•• ~ k'f)/W(~ , ••• ,~ k) and {~ 4 } is ~ o n- 1 n- 1 n- • a basis for the set {~ : Ln~ = 0 , Z~(ti) = ri, i = O, ..• ,m} . This theorem is extended to allow a,S to be singular points of Ln As an application it is shown that if 2 f E c (-oo,oo) and llfll = sup If I < 00 then llfll 2_ f I If" - Afll , I If' II 2_ i/2 llf"-A£11 A 2 for all A> 0; also 11£'11 < llfll [2 llf"- Afll- Allflll for all A E (-"",00) (A= 0 is Landau's Inequality). A similar extension when k = n for one and two point problems is due to Willett. (Received September 3, 1974.)

718-B4 FREDERICK A. HOWES, Courant Institute, New York University, New York, New York 10012. Transitional problems in nonlinear singular perturbation theorv. Preliminary report.

Consider the nonlinear boundary value problem (1) l'Y" = f(t,y,Y', t'), y(O, t') = A(t'), y(l, t') = B(t'), 0 < ( << 1, and the corresponding reduced problem (2) 0 = f(t,u,u•,O), u(l) = B(O). Under appropriate as­ sumptions, the existence of solutions of the "transitional" problem (3) t'Z" = F(t,z•, t') = f(t, a(t), z•, t'), z(O, t') = A(t'), z(1, t') = B(t'), a E C[0,1], is used to deduce the existence and asymptotic behavior (as t' .... 0+) of solutions of (1). In addition, the existence of a solution u of (2) is also proved. Finally, a solution of (3) with a(t) = u(t) is shown to be a uniform O(t')-approximation of a solution of (1). As an example, for quasilinear functions f, i.e., f , = 0(1), (3) has a unique solution for each t' > 0; hence, (1), (2) have y -kt(-1 solutions y(t, t'), u(t), resp., which satisfy the estimate y- u = O(t') + O(e ),Y' - u• = O(t')+ O(t' -1e-ktcl), o ~ t ~ 1. For the solution z of (3) with a(t) =u(t), the sharper estimates y- z = O(t'), y' _ z• = O(t') + O(e-kt(2E"- 1), 0 ~ t ~ 1, also hold. (Received September 6, 1974.)

*718-B5 Gilbert Stengle, Lehigh U., Bethelehem, Pa. 18015 and Jack Narayan, State College of N.Y. at Oswego,~ nonlinear scattering problem. We consider neutral nearly diagonal n-dimentional systems of the form e 2y 1 = ixA(x)y + g{x,e,y). We study the propagation of solutions from x = - ~ to x = + ~ past the complete degeneracy of the linearized problem at _x = 0. Under several conditions on A and g we show that for small c in ~n there exists a global solution having the form 2 y = exp e - Jx sA (s) ds c near x = - oo and y = exp e - 2 J x sf\ (s)ds S(e, c) ' 0 0 near x = + ~. Here S{e,c) e ¢n is the scattering function. Our main result is an asymptotic formula for S{e,c). We show that if g(o,o,y) = 0 and g = Zgjk{x,e)y.yk + O{lyl 3) then

J { 1 S{e,c) = c + Zcjck(2Ti fi{O)-(Aj(O)+Ak(O))I})-2 gjke(O,O)+O(Icl 3)+0(e). To establish this formula we use the Kolmogorov-Arnold-Moser method and the Moser-Jacobowitz approximation method to obtain a priori estimates for solutions, These a priori estimates provide a rigorous justification for our calculation of explicit asymptotic formulas by a technique of matched asymptotic expansions. (Received September 5, 1974.) *718-B6 YASUTAKA SIBUYA, School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455. On the problem of uniform siiii.Plification in a full neighborhood of a transition point. Preliminary report. The problem of uniform siiii.Plification of a linear differential equation y"- A2 [xm+A -~(x, A) ]y = 0 in a full neighborhood of ·the transition point x = 0 was studied by w. Wasow (m = 1) , R. Y. Lee (m = 2) , andY. Sibuya (m ~ 3) • In this report, (i) the problem for a system of linear differential equations, and (ii) some refinements of the existing results will be discussed. (Received September 11, 1974.) A-624 IVAR Schoo~ 718-B7 • BAKKEN, or Mathematics, University or Mlnnesota, Mlnneapo~is, Mn. 55455. A multiparameter eigenvalue problem in the COIII.Plex plane. Preliminary report.

Consider a differential equation of the form y"- [xm + where m is an arbitrary positive integer and the at are arbitrary COIII.Plex numbers. Divide the COIII.Plex x­ plane into m+2 congruent sectors S0 , ••• ,Sm+l with the origin as common vertex. These sectors are Stokes regions for solutions of the differential equation. Consider the following multiparameter eigenvalue problem: Given extended-valued COIII.Plex numbers c0 , ••• ,cm+l, is there a differential equation of the form above with linearly independent solutions y1 (x) and y2 (x) such that y1 (x)/y2 (x) ~ ~ as x ~ = in Sk, for all k? Necessary and suffi- cient conditions for an affirmative answer follow from research by R. Nevanlinna on covering surfaces of the Riemann sphere with finitely many logarithmic branch points.· These conditions are: 1) ck ~ ck+l for all k, and 2) the set {c0 , ••• ,cm+l} contains at least 3 distinct points. It turns out that there are countably many differential equations, that is, countably many vectors (a2 , ••• ,am) , corresponding to each given vector (c0 , ••• ,cm+l) satisfYing the conditions above. We prove a local uniqueness property of the correspondence (c0 , ••• ,cm+l) ~ (a2 , • • • ,am) • (Received September 11, 1974.)

*718-B8 PARFENY P. SAWOROTNOW, The Catholic University of America, l·lashine;t~m, D.C.20064 H*-algebra valued r:1ecsure ::J•t a locall:/ C;)!.llJact spae:e.

This is an e~~tensi::m of the author's previ:>'-IS result [Titeorer:~ 1 in "Integral as a certa.i.a ty;?e of a :)ositive det'in.ite fw1ction", ~· AI,ffi, 3;! (1972) 93-95]. Let S be a locally compact !!ausdori'f space, let A be an H*-algeb1'r: nnd let p be all A-valued positive definite func·~ion defined on the cla.ss c (s) :Jf all continuous c:Jmplex 0 valued functions with compact support. Then there exists an A-valued measl.Lre r on a class of Borel sets of S suet tha. C (Received September 16, 1974.)

*718-B9 WILLIAM C. CONNETT and ALAN L. SCHWARTZ, University of Missouri -St. Louis, 63121. Interpolation of Localized Lipschitz Spaces of Functions with Applications to Mu.l t i pI i ers Let B(q,a) stand either for~q or Aqq (see Ch. V in E.M. Stein, (l (l Singular Integrals and Differentiability Properties of Functions, Princeton University Press (1970)); let ~(x) be a function in Coo which is supported in (-1,1) and define c V(q,al by I lfl lv( q,a ) = supll~ (x-klt(zX>I Is( ) • These are the spaces that arise k q,a in the study of multipliers, for if a? I is an integer, then fEV(q,a) iff f has a Lq-derivatives and s~pjf(xlj+ sup[f2n\xaf(a)(x) lq dx]l/q < Cllfll n n x V(q,a1

(l,~) = (1-s)(l a l + s(l a ) then M. Taibleson (On the theory of Lipschitz spaces of q qo' 0 qt' 1 distributions on Euclidean n-space, II, J. Math Mech 14, (1965), 821-840) showed B(q~) OBJs where[, ]s denotes complex interpolation. The same relation may fail for V(q,~l. but the following is true Theorem: V(q,a)c [VJs where V(q,a)C is the functions of compact support in V(q,a). Applications are given to the interpolation of multi pi ier theorems for ultraspherical expansions and Hankel transforms. (Received September 23, 1974.)

A-625 *718-BlO L. S. Dube, Sir George Williams University, Montreal, P.Q. H3GlM8. Common Fixed Points of Two Multi-valued Mappings, Preliminary Report.

Let X be a complete metric space with metric d and let CB(X) be the collection of all non-empty closed bounded subsets of X . Let H be the Hausdorff metric on CB (X) induced by the metric d. The main theorem states that if F1 and F2 are two multi-valued mappings of X into CB(X) satisfying

for all x,y e: X, where are non-negative real numbers such that

fixed point in X •

Some other related results are also established. (Received September 25, 1974.)

*718-Bll LUNG OCK CHUNG, University of California, Los Angeles 90024. MANIFOLDS CARRYING BOUNDED QUASI!fARM)NIC BUT NO BOUNDED HARMONIC FUNCTIONS.

The main unsolved problem in the harmonic and quasiharmonic classification of Riemannian manifolds is the existence of manifolds carrying quasiharmonic fUnctions with various boundedness properties but not carrying nonconstant harmonic fUnctions with similar bounded- ness properties. Only a few scattered results have thus far been obtained in this direction.

In the present paper, we will introduce a manifold which completely solves all these problems for dimension N ~ 3 and combines earlier results in one construction. (Received September 25, 1974.)(Author introduced by Professor Leo Sario).

*718-Bl2 PAUL R. CHERNOFF, University of California, Berkeley, California 94720. Quasi-analytic vectors and quasi-analytic functions.

• If all the derivatives of an analytic function vanish at a point, the function vanishes identically.

Hadamard asked whether other simply-described classes of functions share this property; such classes are termed "quasi-analytic". An elegant response is provided by the Denjoy-Carleman theorem: Let {Mnl be a logarithmically convex sequence. Let C(Mn) be the class of C00 functions f of one variable such that

llf(n)ll anM for some constant a. Then C(M ) is· a quasi-analytic class if and only if the series ~ n n L:M-l/n diverges. There is a related notion in operator theory. Generalizing E. Nelson's notion of n analytic vector, A. E. Nussbaum made the following definition. Let A be a symmetric operator on Hilbert space. A vector x is quasi-analytic (for A) if the series L:IIAnx!r1/n diverges. Theorem. A is essentially selfadjoint if it has a total set of quasi-analytic vectors. The interaction between function

theory and operator theory leads to an interesting circle of ideas. First, Nussbaum's theorem is really a

corollary of the Denjoy-Carleman theorem. But the latter, in turn, is merely Nussbaum's theorem applied

to the specific operator i d/dx. There are two routes to generalization. Purely operator-theoretic

techniques lead to further abstract theorems like that of Nussbaum. Aside from other applications, when

A-626 these theorems are specialized to suitable ordinary or partial differential operators they lead to new quasi­

analytic function classes. Moreover, these "classical" theorems can be applied to prove additional results

in operator theory, e. g. in the study of semigroup generators. (Received September 25, 1974.)

718-Bl3 WILLIAM A. HARRIS, JR. University of Southern California, Los Angeles, CA 90007 DONALD A. LUTZ, University of Wisconsin-Milwaukee, Milwaukee, WI 53211 A unified theory of asymptotic integration.

We are ~oncerned with the asymptotic integration of the nonautonomous linear differential system x' = A(t)x . We show how to obtain and extend to a wider class of differential systems the fundamental results of Levinson and Hartman-Wintner through a reduction to L-diagonal form. This approach yields a unified treatment of the non-resonant and resonant (adiabatic oscillator) cases and provides the asymptotic integration in terms of computable functions. (Received September 25, 1974.)

Applied Mathematics

*718-Cl ROLF JELTSCH, University of California, Los Angeles, CA 90024 Stiff Stability and its Relation to A0 and A(o)-stability

Necessary and sufficient conditions for A(O)-stability and stiff stability of convergent

multistep methods are established. Using these results it can be shown that Cryer's k-step method (Cryer, C. W., "A new class of highly-stable methods: A0-stable methods," BIT 13 (1973)) is A(O)-stable, stiffly stable and of order k. In addition it is shown that to

any k and any stiff stability parameter D there exists a stiffly stable k-step method

of order k with the prescribed parameter. The stability regions of Cryer's method are

plotted for k = 1(1)7. The angle of A(a)-stability and D of stiff stability of Cryer's method are listed for k = 1(1)16. (Received September 13, 1974.)

Logic and Foundations

718-El JOHN STEEL, University of California, Berkeley, Ca. 94720 Forcing With Tagged Trees, Preliminary Report

Theorem l. There 0 is a real X and II1 (x) singletons {y} and {z} so that y and z are arithmetically-in-x incomparable. Theorem 2. There is a real 0 x and a countable rr 1 Cx) set of reals A and YEA so that {y} is not arithmetic in x. These theorems answer the rela-

tivised versions of questions 56 (in its corrected form, with "arithmetic" for

"Turing") and 57 of Harvey Friedman's forthcoming paper, "Ninety-four Problems in Mathematical Logic". The method of proof is forcing over < Hw, E > . The con- ditions are finite trees, some of whose nodes are distinguished as initial segments of branches, while the rest are given ordinal tags. Such forcing over settles wl question 59 and the relativised version of 60 on Friedman's

A-627 list. That is, Theorem 3. There is an w-model of ~~-CA where fails. Theorem 4. There is an w-model of I 1-Dc and "(JX) u, 1 (X) t Hyp(X))". -1 l (Received September 9, 1974.)

Statistics and Probability

718-F1 STEPHEN A. BOOK, California State College, Dominguez Hills, California 90747. The Gram~r-Petrov Large Deviation Theorem for Triangular Arrgrs. Prelimin~ Repo~

For a triangular array {xnk: 1 :S k :S n, 1 :S n < oo} of row-wise independent random variables with moment-generating functions, the large deviation probability P( Sn 2:: zn), where S = ,,ll X_,_, and z ... oo and n-112z ... 0 as n ... oo, is studied· Under some uniformity n '"k=; ;11\. n n conditions, it is shown that, as n ... oo,

z P(S > z) = {1- -:h(z )} {1 + 0(..!!)} • n- n ::1:::" n fn where (j2 (x) = (2rr)-112 J x exp(-u2/2)du is a power series convergent for -oo all sufficiently small values of t, uniformly for all n. As a special case, the theorem yields a large deviation estimate for weighted sums of independent, identically distributed random variables, which can then be used to extend a version of the Erdos-~nyi law of large numbers to weighted sums. (Received September 13, 1974.)

*718-F2 Mr. Patrick L. Brockett, University of California at Irvine, Irvine, California 92664. An Index for Convergence of Sums of Independent Random Variables. (Preliminary Report). Let S = [ [X(n, j)} } be an infinitesimal system of random variables whose centered sums converge in law to a distribution function F determined by the triple

( 01, a 2 , M(x) ) . The indices i3 (S) and i3 (M) are defined respectively by the in£ of

[ 6>0: Jxj 0 ~P [ jx(n,j)J:z: jx!] -tO as (x,n) --t (0, CD) }and in£ (6>0: fo

Denoting M(n, x) = ~ P[X(n, j),;; x] if x < 0 and= - ~p [X(n, j)~ x] if x > 0 , it is shown that if 0 i3 (M): f~b I xj 6 M(n, dx) --t f~b jxj 6 M(dx) } . For rather general functions g satisfying g(x) = 0( Jxj 6 ) (x --t 0) for some 6 > i3(S), it is shown that 0 E g(X (n, j) ) ;: fg(x) M(dx). In case S is obtained from increments of a stochastically continuous process with (not necessarily stationary) independent increments, i3 (S) is explicitly calculated, and convergence in probability and first mean of the usual variational sums ~ g(X (n, j) ) is obtained. (Received September 16, 1974.) (Author introduced by Professor Howard G. Tucker.)

Topology

*718-Gl CHARLES L. HAGOPIAN, California State University, Sacramento, California 95819, ~ point theorems for products ~ hyperspaces,

A continuum is a nondegenerate compact connected metric space. If for each continuous function f of a continuum X into itself, there is a point x of X such that f(x) = x, then X is said to have the ~point property, A continuum X is A. connected if each two of its points can be joined by a hereditarily decomposable subcontinuum of X, Let X

A-628 and Y be )\ connected continua, Theorem l· If the topological product X x Y is disk­ like, then X and Y are arc-like and X, Y, and X x Y each have the fixed point

property. Theorem~· Both X and Y are circle-like when X x Y is torus-like, Theorem 3, If for each c > o, C(X), the hyperspace of compact connected subsets of X with the Hausdorff metric, can be £-mapped into the plane, then X is arc-like or circle-like and C(X) has the fixed point property. Fixed point theorems of 0, H. Hamilton, E. Dyer, J, Segal, and J, T, Rogers, Jr. are used in this paper, (Received September 6, 1974.)

*718-G2 JOliN S. ROSASCO, University of California, Los Angeles, California

90024, ~ ~ ~ ~ function !_. For each point x of a continuum H, F. B. Jones defines K(x) to be the closed set con·· sisting of all points y of ~I such that ~! is not aposyndetic at x with respect to y, Suppose ~! is a plane continuum and for any positive real number e there are at most a finite numher of complementary domains of H of diameter greater than £ . In this paper it is proved that for each point x of M, the set K(x) is connected. (Received September 9, 1974.) (Author introduced by Dr. C. 1. Hagopian.)

*718-G3 C. E, BURGESS, University of Utah, Salt Lake City, Utah 84112. Embeddings of surfaces in Euclidean three-space.

• Criteria for surfaces to be tamely embedded in Euclidean three-space, or in 3-manifolds, will be discussed. Some fundamental developments, such as Dehn•s Lemma, Bing's side approximation theorem, and triangulation of 3-manifolds, during the 1950's have led to a substantial amount of work in the last fifteen years on tame embeddings of surfaces, and subsets of them, in 3-manifolds. The discussion will, to some extent, be similar to a recent survey of these topics by the lecturer and J. W. Cannon ["Embed­ dings of surfaces in E3", Rocky Mountain J. Math. 1(1971), 259-344]. There will be a brief mention of similar work for higher dimensional spaces, recently summarized by R. J. Daverman ["A summary of results and problems concerning flatness of codimension one spheres in En", Proceedings of the geometric topology conference at Park City, Utah, February 19-22, 1974; Springer-Verlag, edited by L. C. Glaser and T. B. Rushing]. (Received September 9, 1974.)

*718-G4 ROMAN FRIC, Vysoka skola dopravna, 010 88 Zilina, Czechoslovakia. On convergence groups. Preliminary report.

We solve a problem of B.V. Hearsey (Portugal. Math. 30(1971),201-213) con­ cerning topological modification of a filter convergence group. Theorem. Let (G,l,A,+) be a sequential convergence group in the sense of J. Novak (General Topology and its Relations to Modern Analysis and Algebra III, pp.335-340 (Proc. Third Prague Topological Sympos., 1971), Academia, Prague, 1972). Then there is a convergence structure g for G such that (G,g,+) is a filter convergence group and kg = A (i.e. the pretopology kg corresponding so g is the pretopology A corresponding to the sequential convergence£ ). We solve Hearsey's problem as follows. Consider Novak's minimal completion (G,l,A,+) of the group of rational numbers. Since A =kg is an irregular topology for G, (G,kg,+) is not a topological group. (Received September 16, 1974.) (Author introduced by Professor Edwin Hewitt.)

*718-G5 RAYMOND KILLGROVE, California State College, Bakersfield, California 93309. Super-Cauchy sequences.

Def. A sequence in any topological space is Super-Cauchy iff for any given base for the

topology there is a positive integer t1 so that for any n, m, exceeding !1, there is a

A-629 member of the base with an and am as elements. Thm 1: ~.!ll space, every Super-Cauchy

sequence~_! cluster point. Thm £: ~_!regular space, _!cluster point of.! Super­

Cauchy sequence i.?_ _!limit point. Thm 1_: ~~metric space Super-Cauchy sequences

converge, while in~ topological space convergent sequences are Super-Cauchy. Consider the power set of the positive integers as a point set, and define a set U to be open iff

for each point p of U, all subsets of p are points of U. Then i....,. { i} is Super- Cauchy with the evens as a cluster point which is not a limit point. Now restrict to all finite subsets of integers, and the same (still Super-Cauchy), sequence has no cluster point.(Received September 20, 1974.)

*718-G6 ELDON VOUGHT, California State University, Chico, Calti'ornia, 9.5926. Monotone decompositions of continua

Let M be a compact, metric continuum and let \1 be a class of subcontinua of M. Define a monotone, upper semi-continuous decomposition ~ of M to be admissible with respect :t2,)! it given Y E t;. D E ~ such that Y n D ~ ~. then Y c D. It is proved that the continuum M has a unique minimal (in the sense of refinement) admissible decomposition ~ with respect to \1• For certain classes of continua \1 the quotient spaces (M,~) of ~ have nice properties, e.g., if J is the class of indecomposable subcontinua of M then (M .~_,) is hereditarily decomposable 1 if C is the class of continua of convergence of M then (M,~C) is hereditarilY locally connected; if ! is the class of layers of irreducible subcontinua or M then (M,~1 ) is hereditarily arcwise connected. Furthermore ~<9 , t.C' t.S. are the unique minimal monotone upper semi-continuous decompositions such that (M,t._,), (M,t.C)' (M,~S.) are hereditarily decomposable, hereditarily locally connected, hereditarily arcwise connected, respectively, if M i.s a certain type of continuum, e.g. 1 hereditarily unicoherent, atriodic,. irreducible. For these types of continua the following implications are valid in the sense or refinement ~J ::s:~ (Received September 25, 1974.) 718-G7 STEVEN C. ALTHOEN, Hofstra University, Hempstead, New York 11550. A van Kampen Theorem. Preliminary report.

The covering space construction referred to in Abstract 716-G4 these

Notices 21(1974), A- S70and Lemma 1.1 of "Some Small Aspherical Spaces" by E. Dyer and

A. Vasquez [J. Australian Math. Soc. XVI part 3(1973) p 333] are used to represent the fundamental group of a space, X, as the semi-direct product of the fundamental group of the covering space and the fundamental group of a particular graph associated to the cover. The fundamental group of the covering space is computable as a colimit of the fundamental groups of the intersections of the elements of the cover assembled according to the universal cover of the graph. As before, X is assumed to be path connected and locally path connected with a locally finite open cover. A technical condition is also imposed on the generalized nerve of of the cover. (Received September 23, 1974.)

A-630 TheN ovember Meeting in Houston, Texas November 23, 197 4 Algebra & Theory of Numbers

*719-Al WILLIAM L. MORRIS, University of Houston, Houston, Texas 77004. ~factorizations and their associated condition number.

Let A be an n x n real matrix, and let B belong to the set of matrices of the form PAQ, where P and Q are permutation matrices. The factorization B=GT, where T is upper triangular, can be realized in various ways. For each of these factorizations the spectral condition numbers of A and T are compared over all choices of P and Q. Applications of these results to the problem of solving linear equations are considered. (Received August 12, 1974.)

719-A2 PETER M. GIBSON, University of Alabama in Huntsville, Huntsville, Alabama 35~0/. Products of symmetric doubly stochastic matrices.

Let F be a field with unity 1, and let A be an n-square matrix over F. If all row sums

and all column sums of A are 1, then A is a doubly stochastic matrix. For n > 2, it is shown

that every n-square doubly stochastic matrix over F can be expressed as a product of two

symmetric doubly stochastic matrices if and only if the characteristic of F does not divide n.

Then it is shown that every doubly stochastic matrix over F is a product of three symmetric

doubly stochastic matrices. (Received August 22, 1974.)

719-A3 MARCUS, MARVIN, University of California, Santa Barbara, California, 93106, Aspects of the Numerical Range.

Let A be a linear map on a finite dimensional unitary space V with inner product (x,y). The numerical range (or field of values), W(A), is the image of the unit sphere I lxl I = 1 under the mapping XH (Ax,x). The classical Toeplitz-Hausdorff result states that W(A) is convex in the plane. Various generalizations of W(A) have been suggested and investigated by F. L. Bauer, c. A. Berger, P. R. Ha lrnos, J. de Pillis, R. Westwick, and many others. Some of this work can be placed in the following setting. Let l.i (X) Let X be a complex valued function defined for all m-square matrices x. xl, .• 0' m be an orthonormal set of vectors in V and set X= [(Ax.,x.)], i,j = 1, .•. , m. Then l. J define W (A), the I)-numerical range, to be the set of all complex numbers /.i(X) obtained l.i as the vectors ••• J X vary over all orthonormal sets. For example, if /.i(X) = tr(X) m

then W0 (A) specializes to them-numerical range defined by Halmos and proved to be convex by Berger. Various other choices of l.i will be discussed together with a generalization of the notion of a convexoid map, i.e., a map A for which W(A) is the convex polygon in the plane spanned by the eigenvalues of A. (Received August 30, 1974.)

*719-A4 D, J. HARTFIEL, Texas A&M University, College Station, Texas 77843. Results on measures of irreducibility and full indecomposability.

Let n > 1 be an integer, For a given nxn nonnegative matrix A and each integer

A-631 K, 0 ~ K ~ n-2, we define UK(A) = min max aij called the Kth measure of full IRI+Icl=n-K i£R jE:C indecomposability of A, and ~(A) min max a .. called the Kth measure of IRI+Icl=n-K iE:R 1 J R(JC=W jE:C irreducibility of A, where R and C denote nonempty subsets of row and column indices respectively with lsi being the number of elements in set s. A study of the algebraic behavior of UK and uK on products of matrices is given.

This study is then applied as follows: (1) We provide bounds involving UK and ~ on eigenvalues and eigenvectors for nonnegative matrices. (2) For a given sequence of non­ negative matrices, say A1 ,A2, ••. ,~, •.• we provide conditions which establish the existence of i!: A1A2 .•• ~, i!: ~- 1 ••. A1 and others. (Received September 3, 1974.)

*719-A5 RALPH DEMARR, University of New Mexico, Nonnegative idempotent elements in a partially ordered linear algebra. Preliminary report.

Let A be a partially ordered linear algebra which is Dedekind a-complete (dsc-pola). Let

u E A be a nonnegative idempotent element; that is, 0 < u = u2 We characterize such an

idempotent in terms of its relationship to elements x E A such that 0 < x < u • In

particular, we study an idempotent u having the property: if 0 ~ x ~ u , then ux = x

By using only abstract techniques, we show that if u is a matrix, then u must be in

modified reduced row-echelon form. (Received September 11, 1974.)

*719-A6 S. BRENT MORRIS, Department of Mathematics, Duke University, Durham, N. C. 2 7706, The Group Structure Determined by Permutations Derived from Card Shuffling. In this paper, we consider two operations which arise naturally in considering card shuffling,

and then determine the structure of the permutation group generated by them. For p elements,

the operation C = (p,p-l,p-2, ... ,1) and corresponds to cutting one card from the top of a

deck of p cards to the bottom of the deck. The other operation, 0 , is equivalent to n dividing the deck into n portions and shuffling these together. If the size of the deck is a

multiple of n less l, then 0 C = CnO , and the order of the group is the product of the n n order of C and the order of 0 . In all other cases, at least the alternating group A is n P generated, and if On or C are odd permutations, then the symmetric group SP is

generated. (Received September 11, 1974.)

*719-A7 R. J. PLEMMONS, University of Tennessee, Knoxville, TN 37916. Regular Splittings and the Discrete Neumann Problem

Iterative methods are discussed for approximating a solution to a singular but consis­ tent square linear system Ax = b . The methods are based upon splittings A = M - N with M nonsingular. Monotonicity, positive semidefiniteness, and the concept of regular splittings, introduced by Varga, are used to determine some necessary and some

A-632 sufficient conditions in order that the iteration xi+l = M-lNxi + M- 1b converge to a solution to the linear system. Applications are then given to solving the discrete Neumann problem by iteration which are based upon the inherent monotonicity in the formulation. Simple projection methods before and after the iteration enable one to compute the best least squares solution to the problem. (Received September 16, 1974.)

/ *719-A8 MffiOSLAV FIEDLER, VLASTIMIL PTAK, CSAV Prague, Czechoslovakia and EMILIE HAYNSWORTH, Auburn University, Auburn, Alabama 36830, Extreme operators on polyhedral cones, Preliminary report.

Suppose K1 and K2 are proper polyhedral cones in vector spaces E1 and E2 respectively. Then

'II'(K1,K2), the set of operators mapping K1 into K2, is also a proper polyhedral cone. Properties of the extremal operators in this cone are examined in the first part of this paper. In later sections decomposable cones are considered. A cone is decomposable if K = K1 + K2 , where spanK1 n spanK2 = \o! [Notation:

K = K1 (9 K2]. Otherwise K is indecomposable, Theorem. Let A be extreme in 'II'(Kl' K2). Then the image AK1 is indecomposable, As a corollary it follows that no operator of rank 2 can be extreme. Theorem. If cones C and K can be decomposed as C = L::~=l E9 Ci' K = L::;=l E9 Kj, with projections

P. (i = 1, ••• , r), Q. (j = 1, ••• , s) on the corresponding subspaces, then the statement that A is extreme 1 J in 'II'(C, K) is equivalent to the statement that for some pair of indices i, j, Q.AP. is extreme in tr(C., K.) 1 1 1 J and ~APk = 0 if t +j or k +i. (Received September 16, 1974.) 719-A9 ¥UNNAMMAL NATARAJAN and CARLTON J. MAXSON, Texas A&M University, College Station, Texas 77843. On The Endomorphism Monoids of Complete Boolean Algebras.

If every endomorphism of a Boolean algebra U =: < U, V, A , 1 , 0, 1 > is complete we

say that the semigroup of Boolean algebra endomorphisms, End U, is complete. In this

paper we investigate Boolean algebras U such that End U is complete. We say a maximal

ideal M of a Boolean algebra U is complete if for every infinite set I, V Ai exists ie:I in M, whenever each Ai is in M. Theorem. For a Boolean algebra U, the following are

equivalent: (1) End U is complete. (2) every two valued endomorphism of U is complete.

If U is also complete the above conditions are equivalent to: (3) every maximal ideal of

U is principal (4) every maximal ideal of U is complete. Theorem. If a Boolean algebra

U is (1) complete and atomless or (2) complete, non-atomic, non-atomless or (3) atomic,

then End U is not complete. Corollary. If U is complete (atomic), then End U is

complete if and only if U is finite. (Received September 16, 1974.)

719-AlO DIANA LINDSEY ar:d BERNARD ~lADISON, Louisiana State lT':iversity, Baton Rouge, Louisiana 70803. Congruences e>n certain subsemigroups of Ba~!:-fo~.Y::!:. ~~igro1!_E_~. Prelirunary repor"f:------

Let T be the Baer-Le-.ri s~:nigroup o;l a cou··ltable s~t A , i.e., 'l' is the semi group :::>:t' o~le-to-one funct lo,Js ¢ from A to A so tl--Jat; A\ (A )_0 is in!'ini-ce. Let a= [A lo,;:: A} be a L'O.Llec:tion of infin:ite snl,seta o:t" A Cl' satisfying the condition tha"t for each o.,~ e 1\ theYe is y e 1\ so tnat

A-633 A'J u As -.::. AY and AY \Aa J A13 is inrinite. Let s(a) be tf1e subsemigroup of T defined by ¢ e s(a) if there exists a e A such that (A)¢ c A . CL The semigroups s(a) are basic examples of idempotent-free semigroups that are right simple, left reductive, and have the property that any two elements have a common right identity. (See D. Lindsey, Semigroup Forum 8(1974), 298-311.) Numerous congruences on s(a) are exhibited and part of the structure of the lattice of congruences on s(a) is given. Basically there is a largest proper congruence f and a smallest proper congruence 6 with an an uncountable number of congruences between 6 and f • (Received September 16, 1974.)

*719-All HENRY E. HEATH~R~Y, University of Southwestern Louisiana, Lafayette, Lou~s~ana 70501. Group Algebras and Other Constructions Over Near-Rings. Preliminary report. ------

If N is an arbitrary (left) near-ring, then the standard constructions for the group algebra, n x n matrices, quaternions, polynomial forms, or formal power series over N yield algebraic systems which may not be associative and in the former case may not even be well-defined. There is a wide class of near-rings, properly including the class of all rings, for which each of these constructions yield an associative system. Theorem. A necessary and sufficient condition for any one of the aforementioned systems to be a near-ring is that N be product distributive, i.e. (ab + cd)x = abx + cdx. If N is a distributive near-ring (cf H. E. Heatherly, Distributive Near-Rings, Quart J. Math (2) 24 (1973), 63-70) then each of the systems is a distributive near-ring. Product distributive near-rings which are not distributively generated are exhibited. (Received September 18, 1974.)

719-A12 RALPH DeMARR and TOM E. SALAZAR, University of New Mexico, Albuquerque, New Mexico 87131. An abstract model for matrix theory. Column model.

Let A be a dsc-pola with 1 ~ 0. Put I= fy E A: y ~ 1 and y -1 ~ ol and let D = I- I. D is the set of "diagonal" matrices (note that D = A1 in T.-Y. Dai's, "On some special classes of partially ordered linear algebras", J. Math. Anal. Appl. 40(1972), 649-682). Column model. Let u1, ••• , um be elements from A such that u. ~ 0, u.u. = u. for all i,j = 1, ••• ,m. We assume that if x E A, then there 1 1 J J exist a. ED such that x = :B~ a.u. and that if :B~ b.u. ~ 0, then b.~ 0 for all i (b. ED). Example. 1 1= 1 1 1 1= 1 1 1 1 1 The m X m matrices with real entries, the usual matrix operations, and partially ordered entry by entry is a dsc-pola. In this case u. is the matrix consisting of l's in the jth column and zeros everywhere else. J We prove various theorems which justify this model. Many standard theorems are simple consequences of the assumptions made in the model. As examples consider the following: 1. Proposition. Suppose dE D is such that du1 = u1d then dx = xd for all x EA. 2. x E A, x~ 0, will be called "stochastic" if x = :B~ c.u., where :B~ c. = 1. Proposition. The product of two "stochastic" matrices is "stochastic". 3, 1= 1 1 1 1= 1 1 - Proposition. A commutative implies u1 = 1. (Received September 18, 1974.)

*719-Al3 W. WILEY WILLIAMS, University of Louisville, Louisville, Kentucky 40208. Semilattices on Peano Continua

A Peano continuum is cell-cyclic if every cyclic element is an n-cell for some n.

A-634 Theorem Every cell-cyclic Peano continuum admits a topological semilattice.

Corollary Every retract of r 2 admits a topological semilattice.

(Received September 23, 1974.)

*719-Al4 HARTMUT HOFT and PAUL E. HOWARD, Eastern Michigan University, Ypsilanti, Mich. !:_ representation theorem for multi-algebr·as and the Axiom of Choice.

Multi-algebras sre E:ets with multi-v<:lued opcr

719-Al5 J.H. CARRUTH, University of Tennessee, Knoxville, Tenn. A convenient category for [topological] algebraists. Preliminary Report.

A topological partial semigroup is such a triple (P,A,m) that P is a Hausdorff space,

A is a subspace of pxp , and m is a continuous function from A into P such that if

{(a,b) , (b,c) , (m(a,b),c) , (a,m(b,c))}C A, then m(in(:a,,b),c} = m (a,m(b,cJ). A con-

tinuous partial homomorphism from one topological partial semigroup (P,A.m) to another

(Q,B,n) is such a continuous function from P to Q that fXf(A] C B and f(!ll(a,b))

n(f(a),f(b)) for all (a,b) e A . It is shown that a large number of categories are re- flective subcategories of the category P of topological partial semigroups and continuous partial homomorphisms (that is, the inclusion functor admits a left adjoint). Among these

are the categories of [topological, compact] semigroups, groups, semilattices, and inverse

semigroups. One obtains as corollaries the existence of most of the classical "free" objects

of [topological] algebra as well as most of the compactifications. Moreover, the existence

of coproducts in each category is obtained as a corollary to the result. (Received September 25, 1974.)

*719-Al6 R.~THUR KNOEBEL, New Mexico State University, Las Cruces, New Mexico Dn~tary Post algebras. Preliminary report. A unitary Post algebra A is an algebra of type <2,2,1,0,0> such that is an Abelian group, is a semilattice with zero 0 and unit -1,

= (x-xy)(y-xy) 0, x(-y) ~ -(xy), xyz ~ x-xy+xyz ~ x, (x+y-xy)z = xz+yz-xyz,

[x(-x) + y(-y)]z = x(-x)z + y(-y)z. Let Pn = <{O,l, ..• ,n-1}; +,A,-,0,1> be the basia Post

algebra on n elements (n~2) where xAy = min{x,y}. Proposition: For a unitary Post algebra A the following are equivalent: (1) A is simple; (2) A is subdirectly irreducible; (3) A is a

nontrivial chain; (4) A ~ G/[g] where G is an Abelian, fully ordered, lattice-ordered group

with atom 1 and g>l. Theorem: A is a unitary Post algebra iff A satisfies the identities

A-635 common toP ,P , ... ,P , .... Corollary: The identities of each P are generated by those for 2 3 n n times n unitary Post algebras together withlx+x+ ... + x 1= 0, and d-1 < d for d a maximal divisor of n. (Received September 25, 1974.)

*719-A17 MARK BLONDEAU HEDRICK, 222 Kalmer, Pasadena, Texas 77502. The permanent at a minimum on certain classes of doubly stochastic matrices.

The author proves the following: Let A be a doubly stochastic matrix and let X be a set of doubly stochastic matrices with the same (0, I)-pattern as A in some neighborhood of A. If A is a critical point of the permanent relative to X, then per A= per A(ijj) for each positive aij. (Received September 25, 1974.) (Author introduced by Professor Richard Sinkhorn.)

Analysis

*719-B1 J. S. MAC NERNEY, University of Houston, Houston, Texas 77004. Finitely additive set functions. (II) Linear operations~~ space of functions of bounded variation. Let S be the space of all functions of bounded variation on [0,1] which are anchored at o, s+ be the set of all real nondecreasing functions ins, and, for each t in [0,1] and f ins, P f be t the function h in S such that h(u) is f(u) or f(t) accordingly as uS t or u 2 t. The equations A(A)(a)(t) A(Pta), for A in the dual D of Sand a ins+ and tin [0,1], define a linear iso­ morphism A from D onto the set of all functions g from S+ into S such that (1) there is a b 2 0 such that if a is inS+ and 0 S u < v S 1 then [g(a)(v)-g(a)(u) [ S b[a(v)-a(u)] {the least such b is the norm of the member A- 1(g) of D} and (2) if a and ~ are in s+ and there is a c > 0 such that la(v)-a(u)] S c[~(v)-~(u)] for 0 S u < v S 1 then g(a)(t) = j 0t(dg(~)da]/d~ for each tin (0,1]. If g = A(A) and f is inS then A(f) is an integral in this sense: for each a ins+ such that Hellinger's (subdivision-refinement) 1 2 1 integral j 0 [df[ /da exists, A(f) = j 0 (dg(a)df]/da. All this remains true in case, from the beginning, all the functions in S are further required to be right-continuous at each number between 0 and 1. These, and related results about repre­ sentation of linear operations, are presented in the somewhat more general context wherein S is a space of finitely additive set functions from a pre-ring R to a complete inner product space Y and the norm of a function h inS is the total variation of h relatively to the usual norm on Y. There are also., then, representations of the space E of continuous linear functions from S to Y. (Received July 5, 1974.)

*719-B2 ALVIN J. KAY, Texas A&I University in Kingsville, Kingsville, Texas 78363. Nonlinear integral equations and product integrals. Let S be a linearly ordered set, {G,+, II II } a complete normed abelian group, H the set

of functions from G to G that take 0 to 0, OA and OM classes of functions from SXS to H that

are order-additive and order-multiplicative respectively and satisfy a Lipschitz-type

condition, E be J. S. Mac Nerney's reversible mapping from OA onto OM ( see J. S. Mac Nerney

[Illinois J. Math. 8 (1963),621-696] for further details). Definitions. If each of K and M is

a function from SXS to H, K is differentially equivalent to M means there is a function k from

SXS to the real numbers such that for each {x,y,P} in SXSXG Iyk = 0 and jiK(x,y)P - M(x,y)Pii X ,;, k(x,y) liP II. ! and ]!_ are mappings such that if {V,W} is in OAXOM, each of

A-636 is the set of all functions that are differentially equivalent to V and W-1 respectively.

Theorem. $[E] = ~. This analysis is used to prove existence theorems for product integrals c -1 of the form W(x,c)P = IT [1-M] [l+K]P ( where each of K and M is in ~(OA) and M is linear X which we show solves a nonlinear integral equation of the form f(x) = P + (RL)/~(Kf + Mf). (Received September 6, 1974.)

719-B3 WILLIAM F. DENNY, The University of Oklahoma, Norman, Oklahoma, 73069. ~Linear Riemann-Stieltjes Integral Equation System, Preliminary report.

The system treated is of the form du = [dN]v , dv = [dM]u , where M and N are n x n matrix functions of bounded variation with N continuous and u and v are n x 1 vector functions. A sufficient condition for existence and uniqueness is given, and some related basic properties are studied. The Morse Quadratic Form of the system is then developed and is used to extend to this system oscillation, separation, and comparison theorems occurring in the generalization of the classical Sturmian theory due to Morse. (Received September 16, 1974.)

719-B4 WILLIAM L. GIBSON, 943 Beachcomber Ln., Houston, Texas 77058. A correspondence associated with Stieltjes-Volterra integral equations. Preliminary report. Suppose S is an interval, R is a complete normed ring with identity, {G,N} is the complete normed group of quasi-continuous functions from S into R normed with the supremum norm N, and H is the class of all functions from G into G to which {0,0} belongs. Let OA and OM be classes of functions from SXS to H that are order-additive and order-multiplicative respectively and satisfy a Lipschitz-type condition. Let E be J. S. Mac Nerney's reversible mapping from OA onto OM. Theorem. If {V,W} is in E, the following are equivalent: (i) V(x,y)g = (R)fYVl·g for each {x,y} in SXS and g in G, and (ii) W{x,y)g - g = {R)fYd{W[x,I]D·g foxr each { x,y} in SXS and g in G. Theorem. If {V,W} is in E, (i) abo~e holds, and each of A and B is a function D from SXS into R such that for each y in S, D[I,y] is in G, then the following are equivalent: {1) B{x,y) = A{x,y) + {R).[Y{Vl}{x)·B[I,y] for each {x,y} in SXS, and (2) A{x,y) = B(x,y) X - {R)..( d{W[x,I]l}{x)·A[I,y] for each {x,y} in SXS. Several fundamental results about Stieltjes-Volterra integral equations are corollary to these theorems. (Received September 18, 1974.)

*719-B5 BURRELL W. HELTON, Southwest Texas State University, San Marcos, Texas 78666. The solution of a Volterra integral equation for rings.

Theorem. If f, h and K are bounded and g has bounded variation on [a, b], then the following statements are equivalent: (1) on [a,b], (f,K,dg) E OA* and f(x) =h(x) + (L) J~f(t)K(x,t)dg(t); (2) on

[a,b], (h,K,dg) ~OM* and f(x) = V(a,x;h,K,dg). Definitions. For each subdivision D = ltil~ of [a,b],

V(D, h, K,llg) denotes the n X n determinant A= la .. l such that a .. = 0 for j > i + 1, a .. = -1, a. = 1J 1J 1, l+ 1 11 h(t.) + h(a)K(t., t0)llg1 and, if 1 < j ;:!§ i ;:!§ n, then a .. = K(t., t. )t,g.; V(a, b;h, K, dg) denotes the subdivision- 1 1 1J 1 J-1 J refinement-type limit of the set jV(D,h,K,l',g)l. The triple (f,K,dg) E OA* [(h,K,dg) E OM*] means if

A-637 e > 0, then there is a subdivision D of [a, b] such that, if It i!~ is a refinement of D and 0 < p ~ n and x = xp' then I( L) J~f(t)K(x, t) dg(t) - L:if(ti_1)K(x, ti_1)tlgil < e rlv(a, x;h, K, dg) - V(D', h, K,Ag)l < e, where

D' = It i~]. Also, sufficient conditions for (f, K, dg) E OA *, (h, K, dg) f OM* and the existence of

V(a,b;h,K,dg) are proved. (Received September 12, 1974.)

*719-B6 D.E.WUiiBERT, University of CaJ.ifornia at San Diego, La Jolla, CaJ.ifornia, KOrovkin Approximations Let £ be a class of operators on a Banach space E. Let K c E. The £-shadow of K is the largest subset, s, of E such that if (Li) is a bounded sequence in

;l and Lik converges to k for all k in K then Lis converges to s for all s in S.

Theorem. Let ;ll denote the contractive operators on a locally uniformly convex reflexive space. The :l1-shadow of a set is the range of a contractive projection.

Theorem. There is a subspace K of a reflexive space for which :l1-shadow K is not the range of a contractive projection. A set K cE is an ;l-Korovkin set if ;f.-shadow K =E.

Theorem. (with W. Kitto) Let ;l+ be the positive operators on J, • If v c J, p "- p i~ finite dimensional and contains a positive function then K is ;l+-Korovkin in J, p if and only if K is ;l+-Korovkin in c0 . (Received September 23, 1974.)

*719-~ FRANK DEUTSCH, Department of Mathematics, The Pennsylvania State University, Existence and Continuity of Best Approximations

A general existence theorem for best approximations is established which includes the majority of known results (e.g. every weak* closed subset of a dual space is proximinal, a closed sub- set of a finite-dimensional subspace is proximinal, the rational functions and exponential sums in C[a,b] are proximinal, every weak operator closed set in the space of bounded operators on a reflexive space is proximinal, etc.). The same theorem also establishes a certain kind of upper semicontinuity for the (set-valued) metric projection. (Received September 23, 1974.)

719-BB CHARLES K. CHUI, Texas A&M University, College Station, Texas 77843 Some Recent Results on Pade Approximation.

The [m/n] Pade approximants of a formal power series or of a function in Cm+n+l[O,~], o > O, are studied. If the series is a series of Stieltjes or if the function is an

R-function, then the Pade approximants are some Gaussian quadrature and convergence along the upper diagonals of the Pade table follows. (See also Abstract 74T-B32, these Notices

21(1974).) For a function f in Cm+n+l[O,o] satisfying some non-singularity condition,

J. Walsh (J. Approximation Theory 11 (1974) ) proved that the [m/n] Pade approximant of f is the limit, as E ~ 0, of the best (m,n) rational approximant of f on [O,E]. Using an

A-638 approximation theoretic proof, o. Shisha, P. W.· Smith and the author recently showed that

in the real case the hypothesis on the non-singularity of f is not needed. We also study

what we now call "local best approximation" by generalized rational functions. (Received September 23, 1974.)

719-B9 B. D. ANDERSON, East Texas State University, Commerce, Texas 75428. Extensions and Approximations in Function Spaces, Preliminary Report.

Let T be a topological measure space with a regular measure ~. and let Q be a

closed subset which is also a topological measure space. If there exists a closed neighborhood

U of Q and a retraction r from U onto Q such that ~(r-lA- Q) ~ k ~(A) for all Borel

sets A in r(U- Q), then there is an extension map E from Ct (Q) to Ct (T) such that

a) E is linear and II Ell~ 1 + k, b) II (Ef)(Eg)lf ~IIE(fg)ll, c) fg ~ 0 implies E(fg) >

(Ef)(Eg), d) fg = 0 implies E(fg) = (Ef)(Eg) and conversely if T is perfectly normal and

r(U - Q) = Q. Part a) was proven by Cheney and Anderson [Abhandl. Math. Univ. Hamburg, 36

(1971) 26-34]. Extension maps are related to the approximation of functions.

(Received September 23, 1974.)

719-BlO PHILIP W. SMITH, Texas A&M University, College Station, Texas 77843, Nonlinear Approximation Theory, Preliminary report.

The concepts of metric curvature and folding of a c1-representable manifold M in a normed linear space will be used to obtain sets which have unique best approximations from M. Some of the results of the Chui-Rozema-Smith-Ward paper "Metric curvature, folding, and best approximation" will be presented including the following two theorems. Let cr(m) and fld(m) denote the metric curvature and folding at m E M respectively. Theorem 1. Let M be a connected boundedly compact c1-representable manifold with cr(m) 0 for all m E M. Then M is Chebyshev. A partial converse to Theorem 1 is contained in: Theorem 2. If M is a boundedly compact Chebyshev c1-representable manifold, then for all m E M we have fld(m) = ® and either cr(m) = 0 or cr(m) = ®, We note that there does exist M satisfying the hypothesis of Theorem 2 with m1 ,m2 E M and cr(m1) = 0, cr(m2) = ®, Piecewise polynomials, exponentials, and other nonlinear families will be introduced to illustrate these as well as other results. (Received September 23, 1974.)

*719-Bll H.G. BURCHARD, Oklahoma State University, Degree of convergence of piecewise polynomial approximation on optimal meshes II.

Let Pkn(Skn,l) d enote h f t e am1"1 y o f p1ecew1se· · po 1ynomia 1 functions ( splines with simple knots) of degree n-1 with k segments on [0,1]. H.G. Burchard and

D.F. Hale (r. Appr. Th., to appear; also these Notices 20(1973) A-277] obtained (k +co) results for E (f,k) p,n dis~p(f,Pk n), 1 ~ p ~ oo; these imply similar though less sharp results for dis\p(f, s~· 1). They introduced F-spaces

(the closure of the smooth functions in ~p,n). In this paper the analysis is carried further, and additional degree-of-convergence results are

A-639 -n For example, l."f f € Np,n This obtained. we show Ep,n +l(f,k) = O(k ) ;;o . implies a considerable improvement of a result of Korneichuk and Freud-Papov, since we

show f € €"'()Np,n if Dn-l f BV (for n+p-l > 1). (Received September 23, 1974.)

*719-B12 A. K. VARMA, University of Florida, Gainesville, Florida 32601. A. new proof of Timan approximation theorem. II.

Let x. kf-:1, 2 .. , n be the zeros of T {·X) = Kn n cos n e, 1 2 n-1 2 cos9 = X . Define ~ (x) = 'Kn n n + n rlh "-rn Tr.(~n) Tr (x), "-rn = cos r=O, l, ..• n-1 Rn[f,x] =kjh f (~n) ~n (x). Main object of this paper is to give extremely simple proof of Timans funda- mental theorem of approximation with the help of Rn[f,x] compare these results with earlier results of the author and T.M. Mills [to appear in Israel Journal of Math] . (Received September 23, 1974.)

719-Bl3 ALBAN ROQUES, J.nui~d.ana State University, Baton Rouge, La. 7~03. Local Evolution Systems in General Banach Spaces, Preliminary Report.

Let { A(t) : o J. t J. T) be a family of W-dissipative, multi-valued operators in a real Banach space. Under certain assumptions on the family {A(t)), Crandall and Pazy, and A. T. Plant have constructed an evolution system {U(t,s)) ty means of a product integral,

t ))-1 () U(t,s) = f 8 (I-drA(r . We relax the conditions on {At } and obtain a local evolution

system by means of a similar product integral. (Received September 23, 1974.)

*719-Bl4 A. BACOPOULOS, University of Montreal, Montreal, Canada, o. SHISHA, University of Rhode Island, Kingston, Rhode Island, and G.D. TAYLOR, Colorado State University, Fort Collins, Colorado. Characterization of best relative approximation. Let f be a real function, continuous in [-1,1]-{0}, and let k be a positive integer. Suppose 0< inf I xk/f(x) I< suo lxk/f(x~O and pEnn be given" Call an x*E[-1,1] an

extreme point iff either x*~O and E(x*)=l lEI I, in which case set cr(x*)=sgn E(x*), or x*=O and at most one of the relations

lim E(x)=( !Ell, lim E(x)=-IIEII holds; if the first holds, set x+O x+O cr(O)=l; if the second holds, set cr(0)=-1. Theorem. Suppose either lim xk/f(x)O. Then p minimizes I lEI I iff either x+O x+O 0 is not an extreme point, or there are n+2 extreme points

xk:l~X1

A-640 719-Bl5 SEYMOUR V. PARTER, University of Wisconsin, Madison, Wisconsin 53706, Differential equations with "Turning Points" and numerical methods.

• Differential equations which depend upon a parameter ( > 0 have been studied extensively, A case of great difficulty and of great mathematical interest is the case where there is a decrease in the order of the equation at ( = 0; i, e., the singular perturbation case. Generally speaking, the theory is very well developed when the "reduced equation " (the problem with ( = 0) is a regular problem. However, when the reduced equation becomes singular at certain isolated points, called "Turning Points" or "Transition Points," the general theory is still not well understood, Asymptotic theories for such problems have been studied by many people with early fundamental work having been done by R. Langer, K. 0. Friedrichs, F. Olver, W. Wasow. In his celebrated "Gibbs Lecture" K. 0, Friedrichs discussed a large number of important applications in which such problems appear. When one turns to the problem of obtaining numerical approximations to the solutions y(x, f) with £ << 1, these difficulties appear in new forms, e.g,, "Stiff equations." In order to study numerical methods for these problems, one needs qualitative insight into the analytic theory. This lecture is concerned with such results. Particular emphasis is placed on the well-studied second order case. (Received September 25, 1974.)

Applied Mathematics 719-Cl MARGARET M. LaSALLE, University of Southwestern Louisiana, Lafayette, Louisiana 70501, Automata and solvability, Preliminary report.

A sequential automaton is defined and an automaton transformation is considered as a mapping of the set of all omega sequences. The solvability problem questions the following: if given a formula does there exist an automaton transformation and is it possible to construct an algorithm? The Theorem proved will show that the necessary and sufficient condition for the existence of such a transformation is the existence of an R-tree which leads to solvability, (Received July 24, 1974.)

*719-C2 ROGER BLEIER, University of Texas, Austin, Texas 78712. An extension of linear programming duality to piecewise linear maps. Preliminary report.

Consider the pair of optimization problems : (1) Maximize f(x) subject to the m constraints gi(x) ~ bi (i = l, ••• ,m), and (2) Minimize L rib. subject to r 1 , ••• ,r ~ 0 i=l 1 m and L rigi = f. The first problem is the primal, the second the dual. Here f and the gi are real-valued functions on some set X, and the ri and bi are rea! numbers. An

important duality thoerem in linear programming asserts that, for the case where f and

the gi are linear functions IRn+IR, if both primal and dual possess feasible solutions,

then both possess optimal solutions and their optimal values coincide. The author has

proved this theorem for piecewise-linear maps IRn+IR, subject to the restriction that the

vector sublattice of C(IRn) generated by f and the gi is a free vector lattice.

(Received September 23, 1974.)

719-C3 ROBERT 0. SHELTON, Rice University, Houston, Texas 77001. Necessary condition for a noncollision singularity in the four body problem. Preliminary report.

It is shown that if there is a singularity in a solution of the four body problem which is not a collision

A-641 then the motion of the bodies near the singularity is nearly one dimensional. This is established by grouping the bodies into natural clusters and showing the angular momentum of each cluster with respect to its center of mass tends to zero near the singularity. This is related to Sperling's proof of von Zeipel•s theorem (J. Reine Angew. Math. 245(1970), 15-40). (Received September 28, 1974.)

Geometry 719-Dl HOWARD J. JACOBOWITZ, Rice University, Houston, Texas 77001. The Gauss-Codazzi equations. Preliminary report.

It is shown that the Gauss-Codazzi equations for an isometric embedding of a Riemannian manifold M into a given Euclidean space are equivalent to the assertion that a certain bundle over M admits a flat connection compatible with the metric of M. This leads to a new and simple proof that these equations are necessary and sufficient conditions for isometric embeddings. (Received September 23, 1974.)

Topology

*719-G1 GEORGE M. REED Ohio University Athens, Ohio urJ701 On senarating open covers in Moore spaces. An open cover H of the space S is said to he separating provided that for each two points p and q in S there exists a member of H which contains p but does not contain q. This concept has recently proverl extreMely useful in the theory of metrization. It follows from the work of the author and P. L. Zenor in fMetrization of Hoore spaces and generalized manifolds, Fund, ~fath., to aopear] that each Moore srace of' card. $ c has a countable separatinp, open cover. This result raises the question as to whether each

Moore space, regardless of' cardinality, has a ~-discrete separating open cover. Theorem For each cardinal m > c , there exists a DCCC, locally separable Moore space of card. mHo which does not have a ~-discrete separating open cover, (Received September 20, 1974.)

Miscellaneous Fields

719-H1 WILLIAM A. VEECH, Rice University, Houston, Texas 77001. Topological dynamics.

• This will be a survey of recent results in topological dynamics. Topics will be selected from generalizations of almost periodic functions, the structure of point distal flows, generic points and uniform distribution, and horospherical flows. (Received September 24, 1974.)

A-642 SITUATIONS WANTED 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 quality. 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. Free advertisements may not exceed fifty words {not more than six 65-space lines), including address of advertiser; excess words are charged at the rate of $0.15 per word {minimum charge $1). Anonymous listings are carried for an additional fee of $5; correspondence for such applicants will be forwarded to them. Employed members of the Society may advertise at the rate of $0. 15 per word; nonmembers, currently employed, will be charged $0.50 per word {minimum charge $15). Deadline for receipt of advertisements is the same as that for abstracts; date appears on the inside front cover of each issue of the c}/oticei). Application forms may be obtained from, and all correspondence should be directed to, the Editorial Department, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940. 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.

MATHEMATICS TEACIDNG OR TEACIDNG AND RE­ MATHEMATICS TEACIDNG-UNDERGRADUATE, SERVICE SEARCH. Ph. D. 1961, Wisconsin, Algebra. Female, AREAS. Ph. D. 1967, Illinois, Algebra. M.S. in Opera­ age 37, 13 years experience. 4 papers on Representa­ tions Research expected 1975. Male, age 34, 7 years tions of Finite Groups. Prefer west, midwest. Available experience. 6 papers. Prefer west, midwest. Available Fall1975. Patricia Tucker Montague, Mathematics De­ Fall1975. J, Stephen Montague, 514 Rockingham Drive, partment, University of Tennessee, Knoxville, Tennessee Knoxville, Tennessee 37919. 37916.

A-643 MOST OF MATHEMATICS Norbert Wiener: Collected Works Discrete Multivariate Analysis: Volume 1: Mathematical Philosophy and Founda­ Theory and Practice tions; Potential Theory; Brownian Movement, by Yvonne M. M. Bishop, Stephen E. Fienberg, Wiener Integrals, Ergodic and Chaos Theories, and Paul W. Holland with the collaboration of Turbulence and Statistical Mechanics Richard J. Light and Frederick Mosteller edited by P. Masani $27.50 Norbert Wiener was one of the most versatile as Ordinary Differential Equations well as one of the greatest mathematicians of our by V. I. Arnold time. This first volume of his collected papers, translated from the Russian by while restricted to only three classes of his Richard A. Silverman mathematical work, nevertheless ranges widely. $17.50 In order to place the papers into the context of modern research, commentaries have been Elementary Real and Complex Analysis provided by 17 contemporary scholars, writing by Georgi E. Shilov on individual papers or groups of related ones. $19.95 These commentaries, long or short, review the Elementary Functional Analysis papers and trace both their genesis and their by Georgi E. Shilov impact on later research. They are authored by $19.95 E. J. Akutowicz, G. Deem, J. Feldman, A. Fine, The above two texts by Shilov are respectively L. Gross, F. E. Hohn, K. Ito, J.P. Kahane, H. E. Volumes 1 and 2 of his series Mathematical Kyburg, R. J. Levit, L. Lumer, P. Masani, B. Analysis. These revised English editions were McMillan, W. V. Quine, E. M. Stein, W. A. Veech, translated from the Russian and edited by and R. L. Wilder. Richard A. Silverman. The availability of further volumes will be announced at a later time. THE MIT PRESS Massachusetts Institute of Technology $25.00 Cambridge, Massachusetts 02142 Women in Mathematics by Lynn M. Osen $7.50 Mathematics Education in China Its Growth and Development by Frank Swetz $15.00 Fundamentals of Mathematics Volume I: Foundations of Mathematics/The Real Number System and Algebra Volume II: Geometry Volume Ill: Analysis edited by H. Behnke, F. Bachmann, K. Fladt, W. Suss, and H. Kunle translated from the German by S. H. Gould $45.00 the set ($15.95 per volume)

A-644 Count on Marcel Dekker, Inc. to offer a 25°/o discount on these new and outstanding books in mathematics

Calkin Algebras and Algebras of Operators on Banach Spaces (Lecture Notes in Pure and Applied Mathematics Series, Volume 9) by S. R. Caradus, W. E. Pfaffenberger, and Bertram Yood. For the graduate student and for the research mathematician-here is the ideal introduction to the interaction between Banach algebra theory and the theory of linear operators on Banach spaces. Contents: Introduction. Banach Algebras. Riesz Operators. Semi-Fredholm Operators. Ideal Theory for B(X). Generalizations of Fredholm Theory. 160 pages, 1974. Prepaid price: $13.75

Value-Distribution Theory (Pure and Applied Mathematics Series, Volume 25) edited by Robert 0. Kujala and Albert L. Vitter Ill, in two parts. PART A: Presents the results achieved by numerous mathematicians, working together to clarify the most important problems and stimulate further research in various areas of value-distribution theory. Contents: Some Remarks on Nevanlinna Theory, P. Griffiths. Nonstandard Analysis in a Nutshell, J. Hirschfelder. Moisezon Spaces and the Kodaira Embedding Theorem, R. Wells, Jr. Positive (p,p) Forms, Wirtinger's Inequality, and Currents, R. Harvey and A. Knapp. Applications of Geometric Measure Theory to Value Distribution Theory for Meromorphic Maps, B. Shiffman. Holomorphic Extension Theorems, P. Kiernan. Extending Holomorphic Functions with Bounded Growth from Certain Graphs, I. Cnop. The Complement of the Dual of a Plane Curve and Some New Hyperbolic Manifolds, M. Green. A Remark on the Transcendental Bezout Problem, J. Carlson. A Theorem in Complex Geometric Function Theory, R. Greene and H. Wu. Refined Residues, Chern Forms, and Intersections, J. King. Frenet Frames Along Holomorphic Curves, S. Chern, M. Cowen, and A. Vitter Ill. The Kobayashi Metric on IPn- (2n+ 1) Hyperplanes, M. Cowen. The Order Functions for Entire Holomorphic Mappings, J. Carlson and P. Griffiths. Generalized Blaschke Conditions on the Unit Ball inC", R. Kujala. 288 pages, 1974. Prepaid price: $11.63 PART B: (Deficit and Bezout Estimates) by Wilhelm Stoll. A fascinating collection of investigations, all directed toward a Bezout estimate. Contents: Introduction. Grassmann Algebra in Hermitian Vector Spaces. Stokes Theorem on Complex Spaces. Meromorphic Maps. The First Main Theorem. An Integral Average. Associated Maps. The Plucker Formulas. The Ahlfors Estimates. Pseudo-Convex Ex­ haustion. The First Deficit Estimates. The Second Deficit Estimates. Steady Maps. The Bezout Estimates. 288 pages, 1973. Prepaid price: $12.94

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A-645 COMMUTATIVE RINGS Revised Edition Irving Kaplansky Commutative ring theory is one of the major branches of today's mathematics. From a tool in algebraic number theory and algebraic geometry, it has grown rapidly into an independent discipline. This work by Irving Kaplansky takes the reader to the frontiers of research in the field, but only requires a very modest familiarity with abstract algebra. An unusual feature of the book is its collection of nearly 300 excercises-ranging from simple variations on a theme in the text to challeng­ ing problems of considerable interest in themselves. 1974 192 pages Cloth $9.75 POPULAR LECTURES IN MATHEMATICS SERIES Izaak Wirszup, Editor The Popular Lectures in Mathematics series, translated and adapted from the Russian, makes available to English-speaking teachers and students some of the best mathematical literature of the Soviet Union. A significant feature is the authors' use of new approaches to make complex mathematical topics accessible to both high school and college students.

INVERSIONS WHAT IS DISTANCE? I. Ya. Bakel'man Yu. A. Shreider 82 pages Paper $2.95 82 pages Paper $2.95

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A-646 THE LINEAR ALGEBRA TEXT THAT GIVES YOU MORE- Wwenthal If you're teaching the first course in linear algebra, there's quite a few texts you could choose from. But Lowenthal is one that gives you more. Here's what you'll find in Lowenthal's linear algebra- • Lowenthal gives you a fresh, logical approach to linear algebra-approach­ ing the topic through vector spaces and linear transformations, rather than linear equations and matrix algebra. And to complement the theory of finite dimensional vector spaces, Lowenthal uses function spaces and linear dif­ ferential operators throughout. • You'll also discover that Lowenthal gives your students more exercises and examples than any other text. Every new concept is illustrated with examples, carefully designed to combine a minimum of tedious computa­ tion with a maximum of insight. There's also both numerical and theoretical exercises to help your students move through the book with efficiency and understanding. • Lowenthal gives you unusually complete coverage-like the intensive treatment of inner product spaces and spectral theory. You'll find that the finite dimensional versions of such classical topics as the projection the­ orem, the Fredholm alternative, and the spectral theorem are thoroughly discussed. • Finally, Lowenthal gives you a special category, Comments, that examines all aspects and implications of both definitions and theorems. These are just four of the reasons why we think you'll like Lowenthal. Once you take a look at it yourself, we're sure you'll find even more! LINEAR ALGEBRA with Linear Differential Equations By Franklin Lowenthal, University of Wisconsin, Parks ide 1974 approx. 304 pages $12.95

For your complimentary examination copy, write to Robert McConnin, Dept. 232, N.Y. office. Please include your course title, enrollment, and present text.

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A-647 YOU CAN TRAVEL TO THE AMS ANNUAL MEETING IN WASHINGTON, D.C. AT A CONSIDERABLE SAVINGS (FROM MOST CITIES) Garber Travel of Brookline, Massachusetts, is pleased to offer transporta­ tion arrangements for all participants. Flights have already been reserved from major cities. Since there are many people traveling into the Washington area, many times it is difficult to secure passage. By utilizing the services of Garber, all participants will be able to take advantage of the reserved seats. We caution, however, that the seats are in limited supply and we urge you to make your reservations with Garber as soon as possible. For full information on this program, call Garber Travel's special AMS Hotline, (617) 738-0100, or write to Bob DeStefano, Garber Travel, 1406 Beacon Street, Brookline, MA 02146. The fare chart below indicates the startling savings realized on the special "IT X" and "Group" fares. These fares require that travel must be on the same airline in both directions. LOOK AT THESE SAVINGS SAMPLE PRICE COMPARISONS

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SPECIAL FARE INCLUDES: * Round-trip air via American Airlines and other scheduled airlines. * Transfers from airport to your hotel and return. NOTE: In order to avail yourself of the special package fares, you must prepay $65.00 worth of land arrangements. This land cost may include payment for your hotel, in which case payment must be made through Garber Travel, who in turn will transmit it on your behalf to your hotel.

Return To: GARBER TRAVEL 1406 Beacon Street, Brookline, MA02146 ATTN: Convention Department Yes, we plan to attend the AMERICAN MATHEMATICAL SOCIETY MEET­ ING in Washington, DC, January 21-27, 1975. Enclosed is my deposit of $65.00 per person. Please reserve for us the following preferred accommodations at the Shore- ham Hotel: Single ___ Double __ Suite __. Date of Arrival Date of Departure ______, Please arrange transportation from my home city of ------

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A-648 THE HEBREW UNIVERSITY OF JERUSALEM THE FACULTY OF SCIENCE APPLICATIONS ARE INVITED FOR POSTDOCTORAL FELLOWSHIPS FOR THE ACADEMIC YEAR 1975-1976 IN THE FIELDS OF MATHEMATICS, PHYSICS, CHEMISTRY, LIFE AND EARTH SCIENCES The applicant is required to submit his application (in letter form) before December 1, 197 4, together with his list of publications and curriculum vitae, containing details of his marital status, to the chairman of the Institute, in which he is interested to carry out his research. Applications for post-doctoral research in Earth Sciences should be submitted to the appropriate Head of Department, i.e., Geolog)-, Geography or Atmospheric Sciences. All applicants are requested to address copies of their correspondence to the Dean, Faculty of Science. The applicant is further required to arrange for at least two letters of recommenda­ tion. from persons well-acquainted with his personal and academic record, to be directed to the relevant Chairman, or Head of DepartmPnt: these recommendations to reach their destination by December l. l97·t. The fellowship provides for a tax-free salary ranging from approximate!)- IL. H,OOO to IL.l6,000 (dependent on the candidate's marital status). which is pa)·able in eleven monthly instalmt>nts. Tht> facult) usually allows for a singlP air-tieket to Israel, and a similar allowance is made if the eandidate leaves Israel on eompletion of his tenure as postdoctoral fellow. The fellowship is intended primarily for persons who attained their degrPe in very reeent years, and for those who will eompletP all requiremPnts for thP Ph.D. degrPe before the autumn of 1975.

CALIFORNIA STATE UNIVERSITY, FULLERTON THE DEPARTMENT OF MATHEMATICS Academic Administrative Opening OF CARNEGIE-MELLON UNIVERSITY DEAN School of Mathematics, Science and EngiHering invites applications and nominations for The Dean, as the Chief Academic Officer of the School, the position of Department Head. We are shall be responsible for the overall quality of the edu­ cational and research programs offered by the seven looking for a person with a distinguished academic units in the disciplines of Biology, Chemistry, research record and broad mathematical Earth Sciences, Engineering, Mathematics, Physics and interests. Science Education. Applicants should possess a doctorate and qualifica­ We are exploring new opportunities for tions for appointment at the rank af Professor in one mathematics, and the Department Head of the school disciplines. University level teaching will play a major part in redefining the experience and a record of scholarly achievement in aims and restructuring the program of basic or applied research is essential. Evidence of the Department. administrative ability is required and preference will be given to candidates with experience in administering Communications should be addressed to: engineering or applied science research programs. The University is a non-discriminatory, affirmative Professor Zeev Nehari action, equal opportunity employer. Chairman, Search Committee Anticipated Salary: $26,900-$32,700 Department of Mathematics Applicants should send a letter of intent plus a brief Carnegie-MeDon University resume of professional and academic experience to: Pittsburgh, Pennsylvania 15213 P.A.Wegner Department of Chemistry Carnegie-Mellon University Chair, Dean Search Committee is an Equal Opportunity Employer. California State University, Fullerton Fullerton, California 92634

A-649 Conference Board of the Mathematical Sciences Volume 22 CHAIRPERSOl\ "lectures on linear Groups," by 0. T. O'Meara DEPARTME!'iT OF CO'.\IPlTER SCIE!\CE The notes in this volume evolved from lectures at the THE UNIVERSITY OF KANSAS California Institute of Technology during the spring of The Department seeks a computer scientist 1968, from ten survey lectures on classical and Chevalley groups at an NSF Regional Conference at Arizona with scholarly reputation and experience in State University in March 1973, and from lectures on university administration to lead graduate linear groups at the University of Notre Dame in the programs (M.S. and Ph.D.) and under­ fall of 1973. graduate program. The department has The author's goal in these expository lectures is to fourteen faculty members. Applications explain the isomorphism theory of linear groups over integral domains as illustrated by the theorem should include a concise summary of ex­

PSL.(O) ~ PSL.,(o1) ~ n = n1 and o ~ 01 perience, education, honors, publications, for dimensions ~ 3. The theory that follows is typical of and other qualifications. A list of references much of the research of the last decade on the iso­ is required. Salary negotiable. Begin June morphisms of the classical groups over rings. The author 1, 1975. Detailed position description is a starts fr0m scratch, assuming only basic facts from available from the address below. Send first course in algebra. The classical theorem on the simplicity of PSL.(F) is proved, and whatever is needed applications promptly to: from projective geometry is developed. Since the primary Professor Francis D. Tuggle, Chairman interest is in integral domains, the treatment is commu­ tative throughout. In reorganizing the literature for Chairperson Search Committee these lectures the author extends the known theory from Computer Science Department groups of linear transformations to groups of collinear THE l"NIVERSITY OF KANSAS transformations, and also improves the isomorphism theory from dimensions ~ 5 to dimensions ~ 3. Lawrence, Kansas 66045 ISBN: 0-82!8-1672-J BOOK CODE: CBMS/22 AN EQUAL OPPORTUNITY EMPLOYER PAGES: 88 PRICE: Us! $4. J 0, Individual $3.08

u.s POSTAt.S~RVICE . ~EE INSTRUCTIONS i ST ATEM~~,T,?!u.e~~~-~~~,H1!.;h~~6~-~~~~~;~.3,~~r?t<~!,~.t~'UlATION ON PAGE 2 !REVERSE) 1"'TrT1.t"'l:irl"uiiTTCAYTON- - -· ------DA'i'E---o-Ffli:!No-

1 Notic:e.s cf the American Mat_h~tical 3ociet;r _ ___J_ ~~!::! .l9 ~ ~~:;~;,~~~:~:;:~~: oi~~~~-~i~~~~?.~~;.i:1·,~;~;JJ ------·- THE UNIVERSITY OF KANSAS ~f.ri.;&!l.l'~kf·~-if~~f,i'A-Jf.-0~~~m.;A~~s\'f~c~f~~4-,.r;;~o~•--- a · em 1 6 NAMES AND ADDBESSES 01· PUHI.ISHEH, EDI fOR. AND MANAGING EDI OR Lawrence, Kansas 66045 ~n:rJr(l'>.l..,,anJ.iir.l'f~>ir------·- -·------·-·---·- ~aa~~~:?..;';l.-;;~~jp,:.Vj-a't>;io;!a).,_ Sqc;i~t_y~_ 1-'...Q, §.!~_6gl~8~ ..Pr.QY;i1_en~_. __lYW.9&J~aM_Qgg_~Q__ -~ CHAIR IN MATHEMATICS I"Ji.a-d""~~~v~td~J}i1?~ Ql.:.....~~ti_g,_,_]C_ill,g_IL_TJniY.?nm~lbb..~h_g_l!l~.~!!.J,~ ~- -ft*~~~.~A:~~t;~~,,~~~~;!~nS~J,~.;:;:~~ 'sJ~~%J~\ti::.:d)·~~~~'n'~:.:.,~; alia~~ The Stouffer Professorship in Mathematics j :~:;~:·:~·:~.:::~·;, ~~~J;~,:.:. ";,':::.~' ;;·•:• ;~~:~.~;~,~~: ~~~rsr~·~::.~;,P::: ~;,~:.-"1 :·:::~::n:~::.::• =~~ ::~::;;~;:; will become vacant Fall 1975. Candidates with outstanding research credentials are sought. A person with broad mathematical interests and accomplishments is preferred. The criteria will be the candidate's likely total contribution in teaching and research to the Department and the University. Any preference in the area of expertise will only be insofar as it bears on that contribu­ tion. Appointment may be made on either a /lfch4ng•J,puMbh•rrniUI I !~.:~'~;~.~.':· ~~~,~·~·~·· ::';',.',;:':·•;;~:~~ ·:~:"·~:~:~~: !{] ~':;,~·:~:.O':.~~~d U ~.:::~~""9"~~~!~~·,-;,~. f•"M"JJ' permanent or visiting basis. The University '"'onoo to• pu

A-650 SPECIAL HILTON, SHERATON-PARK, OR SHOREHAM HOTEL ACCOMMODATIONS FOR STUDENTS AND UNEMPLOYED MEMBERS PREREGISTRATION AND HOTEL RESERVATION FORM Washington, D. C. Operations Research Short Course Joint Mathematics Meetings January 21-22, 1975 January 23-27, 1975 MUST BE RECEIVED NO LATER THAN DECEMBER 20, 1974 Please complete these forms and return with your payment to Mathematics Meetings Housing Bureau P. 0. Box 6887 Providence, Rhode Island 02940 The Hilton, Sheraton-Park, and Shoreham Hotels have set aside a special block of rooms, at a reduced rate, for stu­ dents and unemployed mathematicians. Rooms are available for triple occupancy only, at a daily rate of $9 per person (Shoreham and Sheraton-Park) or $11 per person (Hilton). The rooms must be shared by three students or unemployed members (of one sex), all of whom will be required to preregister and to sign an appropriate status verification state­ ment which is given below. Registrants should make every attempt to find two attendees with whom they would like to share a room (it would be helpful to receive the three preregistrations at the same time), or they may request that the Mathematics Meetings Housing Bureau make these arrangements. If a registrant cannot be matched with two compatible registrants by the Housing Bureau within three weeks of receipt of the form, it will be returned to the registrant with a request that al­ ternate housing arrangements be made. Do not make reservations directly with the hotel. All reservations will be confirmed. A deposit may be required. At the time of confirmation, registrant will be informed of deposit requirement which should be sent directly to the hotel as requested. In case cancellation is necessary after having been given a confirmed reservation please advise the Housing Bureau up to 15 days prior to the start of the meeting; within last 15 days, make cancellation directly with the hotel. All other changes (i.e. changes in arrival or departure dates) are to be made directly with the hotel. In ac­ cordance with common practice, reservations will be held until 6:00 p.m. on the day of arrival unless a later hour is specified below. Please note that all rates are subject to taxes totaling 6%. Please note that a separate registration fee is required for each of the two meetings. Joint Mathematics Meetings Preregistration At meeting (by mail prior to 12/20) Student or unemployed member $ 1 $ 2 Operations Research Short Course Preregistration At meeting (by mail prior to 12/20) $ 9 $12 ______~~ :h~':_k~ ~~b~e _t~ ~~R~~A_!'I_~A_!!_l~~T~~A_L_8~9~~ ______MEETING PREREGISTRATION FORM MEETING(S) (Please check): Joint Mathematics Meetings O; Operations Research Short Course D

1)~~~~--~~--~~------~~------=~~------NAME (please print) last first middle 2) ADDRESS (forcon~fi'-r~ma~t"-ion~)--~s~tr~e~e~t~an~ar.n=um~b~e=r~------~cnuy~------~s"fu~t~e------.zT.ip~c~o~de~--- 3) I am a student at or unemployed member D Male D Female D 4) Member of AMS 0 ASL CJ MAA CJ NCTM 0 AMOUNT ENCLOSED $___ (check or money order only) Charge my BankAmericard No.______Expiration date------Master Charge No.______Expiration date,______Signature.______Date,______

ROOM RESERVATION FORM 5) I am a student 0 or unemployed member 0 and would like a room from the special Hilton D Sheraton-Park 0 Shoreham 0 block; (preferred hotel). 6) I will arrive (date) at (hour) a.m./p.m. 7) I will depart (date) at (hour) a.m./p.m. Please list names of those persons with whom you plan to share a room. (Each participant should complete a sepa­ rate preregistration form and sign an individual status verification statement.) a. ______b. ______

0 I will share a room with person(s) assigned by housing bureau. STUDENT VERIFICATION I am currently a student working toward a degree and do not receive annual compensation in excess of $7, 000 from employment, fellowships, and scholarships. Signature'------UNEMPLOYED MEMBER VERIFICATION I am currently unemployed and actively seeking employment. My unemployed status is not the result of voluntary resignation or retirement from my last position. Signature 8) 0 I am interested in attending the luncheon in honor of H. L. ""A"l-;d-er------A-651 PREREGISTRATION AND HOTEL RESERVATION FORM FOR ALL PARTICIPANTS EXCEPT THOSE QUALIFYING FOR SPECIAL ACCOMMODATIONS Washington, D. C. Operations Research Short Course Joint Mathematics Meetings January 21-22, 1975 January 23-27, 1975

MUST BE RECEIVED NO LATER THAN DECEMBER 20, 1974 Please complete these forms and return with your payment to Mathematics Meetings Housing Bureau P. 0. Box 6887 Providence, Rhode Island 02940 Housing Bureau Services Do not make reservations directly with hotels. All reservations will be confirmed. A deposit may be required by some hotels. At time of confirmation, registrant will be informed of deposit requirement which should be sent di­ rectly to the hotel as requested. In case cancellation is necessary after you have been given confirmed reservations at a particular hotel, please advise the housing bureau up to 15 days prior to start of meeting; within last 15 days, make cancellation directly with hotel. All other changes (i.e. changes in arrival or departure dates) are to be made directly with hotel. In accordance with common practice, reservations will be held until 6:00p.m. on the day of arrival unless a later hour is specified below. Please note that all rates are subject to taxes totaling 6%. If re­ quested rate is not available, the next available rate will be assigned. Preregistration Only Those participants who desire to PREREGISTER ONLY should complete the preregistration section exclusively on the form below. Please note that a separate registration fee is required for each of the two meetings.

Joint Mathematics Meetings Preregistration At meeting (by mail prior to 12/20) Member $10 $12 * Student or unemployed member 1 2 Nonmember 16 20 * For definition of student and unemployed member, see section on Meeting Preregistration and Registration.

Operations Research Short Course Preregistration At meeting (by mail prior to 12/20) All participants $ 9 $12

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1),~~~~----~~--~~------~~~------~~------NAME (please print) last first middle

2)~~~~~--~~~~--~~~~~--~~------~~------~~----ADDRESS (for confirmation) number and street city state zip code 3)Employing institution'------or unemployed D 4) I am a student at 5) Accompanied by spouse (first name),______6) Accompanying children (number)___ 7) Names------8) Member of AMS CJ ASL 0 MAA CJ NCTM CJ AMOUNT ENCLOSED $___ Iebeck or money order only) Charge my BankAmericard No. ______Expiration date ------­ Master Charge No. ______Expiration date ------__ !~~~~~---_-_-_-_-______~~~---_-_-______ROOM RESERVATION FORM 9) I would like hotel accommodations at the following: (1st choice),______(4th choice)------(2nd choice),______(5th choice)------(3rd choice),______(6th choice)------10) Type of accommodations: ___ Single(s) at $___ Twin(s) at$ ___ Double(s) at$____ Suite(s) at$ 11) I will arrive (date)------'at (hour) a.m./p.m. I will depart (date)______at (hour) a.m./p.m. 12) Persons for whom this reservation is made. Please list names and type of room for each (bracket the names of those persons sharing a room). Each participant should complete separate preregistration form. a. ______c. ______b. ______d. ______

13) I will D I will not D share a room, N. B. Participants planning to stay at the Shoreham Hotel are reminded that all rooms have two beds; double occupancy is considerably less expensive (see paragraph on accommodations for further information). 14) CJ I am interested in attending the luncheon in honor of H. L. Alder A-652 NINTH EDITION Handbook of Mathematical Formulas by HANS-JOCHEN BARTSCH translation by HERBERT LIEBSCHER This compilation of formulas covers the whole Arithmetic. Equations. Functions. Vectors. field of mathematics-from the fundamental Geometry. Analytical Geometry_ Differential rules of arithmetic through geometry and in- Calculus. Differential Geometry. Integral Cal- finitesimal calculus, to the Fourier series and culus. Differential Equations. Infinite Series, the fundamentals of probability calculus- Fourier Series, Fourier Integral, Laplace Trans- and includes subjects of topical interests forms. Theory of Probability; Statistics; Error such as matrices, statistics, linear optimiza- Calculation; Mathematical Analysis of Obser- tion, Boolean algebra, and Laplace transforms. vations. Linear Optimization. Algebra of Logic CONTENTS: Mathematical Signs and Symbols. (Boolean Algebra). 1974, 528 pp., $9.50/£4.55

Elastodvnamics voLuME 1: FINITE MoTioNs by A. CEMAL ERINGEN and ERDOGAN S. SUHUBI Here is the first volume of a two-volume covers basic theory, propagation of singular treatise which provides perhaps the first com- surfaces, finite motion of elastic bodies, and prehensive and rigorous study of the mathe- small motions superimposed on large static matical theory of elastodynamics. Volume 1 deformations. 1974, 358 pp., $38.50/£18.50 Essentials of Pade Approximants by GEORGE A. BAKER, Jr. This book assembles the material of several ring classes of functions for which not only major areas. It first presents the classical convergence, but converging upper and lower material on the algebraic properties (the part bounds, can be obtained. The book contains forming the underpinning for computer alga- examples of several sorts. There are nu- rithms for the calculation of Pade approxi- merical examples which show how well Pade mants and many special functions) as well approximants can converge; general exam- as their relationship to continued fractions pies which serve to illustrate the discussion and orthogonal polynomials. Then, it covers throughout; and sample scientific applications the convergence theory (both point-by-point from such fields as statistical mechanics, and in measure, as appropriate). The book scattering physics, and electrical circuit pays special attention to some widely occur- theory. 1974, 328 pp., $26.00/£12.50 Sieve Methods by H. HALBERSTAM and H. E. RICHERT London Mathematical Society Monographs No. 4 There are many problems in the theory of gant account of Brun's sieve and a sketch of numbers which can be formulated in terms of Rosser's sieve where attention is directed sieve theory-among them are many of the towards aspects of the method which should oldest problems in classical prime number receive further attention. The rest of the book thoery, such as the prime twins and binary is a comprehensive description of Selberg Goldbach conjectures. The object of this book sieve theory, presented in a general yet highly is to show how one may approach these practical form which makes the theory readily questions with the powerful methods of mod­ applicable to a wide range of problems. ern sieve theory: in particular there is an ele- 1974, 378 pp., $26.00/£9.80

Prices subject to change without notice N Lecture Notes in Mathematics A Crash Course on Kleinian Groups San Francisco 1974 Lectures given at a special session at the January 1974 meeting 400 of the American Mathematical Society at San Francisco. Edited by L. Bers and /. Kra. This crash course is a result of a "special session," held at the 1974 Winter Meeting of the AMS, and is addressed to N nonspecialists. It gives the reader an introductory survey of some topics which are important in the modern theory of Klein ian groups. The references to the literature should enable anyone interested to obtain further information. N 1974. vii, 130p. paperj$7.40 Previously published ... Volume 391 Formal Category Theory: Adjointness for 2-Categories ByJ. W. Grey N 1974. xii, 282p. paper /$9.90 Volume 393 Stability of Unfolding& By G. Wassermann 1974. ix, 164p. paper/$8.20 N Volume 394 Iterative Methods for the Solution of a Linear Operator Equation In Hilbert Space-A Survey By W. M. Patterson 1974. iii, 183p. paper/$8.20 Volume 396 N The Pontryagln Duality of Compact 0-Dimenslonal Simllattlces and Its Applications By K. H. Hofmann, M. Mislove, and A. Stra/ka 1974. xvi, 122p. paper/$7.40 Volume 397 The Schur Subgroup of the Brauer Group N By T. Yamada 1974. v, 159p. paper/$7.40 Volume399 Functional Analysis and Its Applications International Conference, Madras, 1973 Edited by H. G. Garnir, K. R. Unni, and J. H. Williamson N 1974. xvii, 569p. paper/$18.10

Also of interest ... Volume401 Elliptic Operators and Compact Groups N By M. F. Atiyah 1974. v, 93p. paper/$7.40