Notices of the American Mathematical

Home , Almost perfect number, Anil Nerode, Generic property, Isaac Namioka, The 1975

Calendar

This Calendar lists all of the meetings which have been approved by the Council up to the date this issue of the cNoliaiJ was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the American Mathematical Society. The meeting dates which fall rather far in the future are subject to change; this is particularly true of meetings to which no numbers have yet been assigned. Abstracts should be submitted on special forms which are available in most departments of mathematics; forms can also be obtained by writing to the headquarters of the Society. Abstracts to be presented at the meeting in person must be received at the headquarters of the Society in Providence, Rhode Island, on or before the deadline for the meeting.

Meeting Deadline for Abstracts* Number Date Place and News Items

725 June 20-21, 1975 Pullman, Washington Apr. 29, 1975 726 August 18-22, 1975 Kalamazoo, Michigan June 17, 1975 (79th Summer Meeting) 727 October 25, 1975 Cambridge, Massachusetts Sept. 2, 1975 728 November 1, 1975 Chicago, illinois Sept. 2, 1975 729 November 7-8, 1975 Blacks burg, Virginia Sept. 23, 1975 730 November 15, 1975 Los Angeles, California Sept. 23, 1975 731 January 22-26, 1976 San Antonio, Texas Nov. 5, 1975 (82nd Annual Meeting) March 4-5, 1976 Tallahassee, Florida March 15-20, 1976 Urbana, illinois April 23-24, 1976 Reno, Nevada June 18-19, 1976 Portland, Oregon November 19-20, 1976 Columbia, South Carolina November 26-27, 1976 Albuquerque, New Mexico January 27-31, 1977 St. Louis, Missouri (83rd Annual Meeting) *Deadline for abstracts not presented at a meeting (by title) June 1975 issue: April 22 August 1975 issue: June 10

Please affix the peel-off label on these cJYoliai) to correspondence with the Society concerning fiscal matters, changes of address, promotions, or when placing orders for books and journals. ThecJYoliaiJof 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. Orders for subscriptions or back numbers (back issues of the last two years only are available) should be sent to the Society at P. 0. Box 1571, Annex Station, Providence, Rhode Island 02901. All orders must be accompanied by payment. Other correspondence should be addressed to P. 0. Box 6248, Providence, Rhode Island 02940. Second class postage paid at Providence, Rhode Island, and additional mailing offices.

Everett Pitcher and Gordon L. Walker, Editors Hans Samelson, Associate Editor

CONTENTS

MEETINGS Calendar of Meetings . • ...... • ...... • . . . • . . Inside Front Cover Program for the April Meeting in St. Louis, Missouri ...... • ...... • • . . . 130 Abstracts for the Meeting: A-394-A-426 Program for the April Meeting in Monterey, California . • . . . . • . . • . . . • ...... • • . 139 Abstracts for the Meeting: A-426-A-431 PRELIMINARY ANNOUNCEMENTS OF MEETINGS . . . • • . . • • . • • . . • • . . • • • • • • . . • • . • 141 ORGANIZERS AND TOPICS OF SPECIAL SESSIONS . . . . . • . . • ...... • ...... • . . 151 INVITED SPEAKERS AT AMS MEETINGS ...... • . . . • . . . . • . . . . • . • . . • ...... • . . 151 NONACADEMIC EMPLOYMENT OF PH.D's IN THE MATHEMATICAL SCIENCES . • • . • • • • . 152 COMBINED MEMBERSHIP LIST 1975-1976 . . • ...... • . . . • . . . . . • . . • ...... • . . 156 NEW AMS PUBLICATIONS . . . • . . . . • . . . • . . • . . • . . • . • . . . . • • . . . . . • ...... 157 LETTERS TO THE EDITOR ...... • . . . . . • . . . . • . • • . . . . • ...... • . . . . • . • . . . • 158 NEWS ITEMS AND ANNOUNCEMENTS ....•...••...•...•..•.••.....•.••.. 155, 159 SPECIAL MEETINGS INFORMATION CENTER. . . • • . . . • . • ...... • ...... • . . . • 160 SUMMER GRADUATE COURSES (Supplementary List) . . • . . . • . . • . . . • . • . . . . • . . • . • • • 161 QUERIES ...... • . • . . • . . . . . • . . • • . . . . . • . . . • • . . . . • . . . . • . • • . • • • . . 162 PERSONAL ITEMS . • . . . . • • . • . • • . . • . . . . • . . . . • . . . • • . . . . • . . . . • • . . . . . • • • . 162 ABSTRACTS . • • • . • . • ...... • • . . . . • . . • ...... • ...... • . • • • . . A-377 SITUATIONS WANTED A-432 RESERVATION FORM ...... • • . • • ...... • • • . . . . • . . • . • A-438 The Seven Hundred Twenty-Third Meeting University of Missouri St. Louis, Missouri April11-12, 1975

The seven hundred twenty-third meeting of Charles P. Lanski, Donald S. Passman, Gary L. the American Mathematical Society will be held Peterson, Richard E. Phillips, Derek J. S. at the University of Missouri, st. Louis, Missouri, Robinson, Martha K. Smith, Keoneth W. Weston, on Friday and Saturday, April 11 and 12, 1975. Julian s. Williams, and Cleon R. Yohe. Profes The University of Missouri at st. Louis is lo sor Richard P. Jerrard of the University of illi cated about ten miles northwest of downtown st. nois at Urbana-Champaign has arranged a spe Louis and about four miles east of st. Louis Lam cial session on Geometric Topology, to be held bert Airport. All sessions will be held in the Friday morning; the speakers will be Joan S. J. C. Penney Building of the university. Birman, John E. Connett, Robert J. Daverman, By invitation of the Committee to Select Mary-Elizabeth Hamstrom, and Jan Jaworowski. Hour Speakers for Western Sectional Meetings, Professor Rangachary Kannan of the University there will be four one-hour addresses. Profes of Missouri at st. Louis and Michigan state Uni sor A. 0. L. Atkin of the University of illinois versity has arranged a special session on Non at Chicago Circle will speak on Friday, April11, linear Functional Analysis, to be held Saturday; at 11:00 a.m. ; the title of his talk will be "Super the speakers will be Lamberto Cesari, Michael singular games". Professor Kuo-Tsai Chen of G. Crandall, Jack K. Hale, Roger D. Nussbaum, the University of illinois at Urbana-Champaign Paul H. Rabinowitz, Duane P. Sather, Luc Tar will address the Society on Friday, April11, at tar, and Hans F. Weinberger. Professor Walter 1:45 p.m.; his subject will be "Iterated path in Leighton of the University of Missouri at Colum tegrals". Professor Kenneth Kunen of the Uni bia has arranged a special session on Ordinary versity of Wisconsin at Madison will speak on Differential Equations: Oscillation Theory, Saturday, April 12, at 11:00 a.m.; the title of Boundary Value Problems, to be held all day his talk will be "What good are ultrafilters ? " Friday and Saturday morning; the speakers will Professor Guido L. Weiss of Washington Univer be Sui-Sun Cheng, William J. Coles, Arlington sity at st. Louis will address the Society on M. Fink, Louis J. Grimm, Heinrich W. Guggen Saturday, April12, at 1:45 p.m.; his topic will heimer, Don B. Hinton, Lloyd K. Jackson, Mar be "The use of Hardy spaces and their generali vin S. Keener, Kurt Kreith, Alan C. Lazer, zations in harmonic analysis". All four hour William T. Reid, Jerry R. Ridenhour, Curtis C. talks will be held in the auditorium of the J. C. Travis, and W. Roy Utz, Jr. Professor Marian Penney Building. Boykan Pour-El of the University of Minnesota By invitation of the same committee there has arranged a special session on Recursion will be nine sessions of selected twenty-minute Theory, to be held Saturday; the speakers will be papers. Professor David Drasin of Purdue Uni Harvey Friedman, Carl G. Jockusch, Jr., versity has arranged a special session on Classi Manuel Lerman, Thomas G. McLaughlin, Anil cal Function Theory, to be held Friday afternoon Nerode, Hilary Putnam, Wayne H. Richter, and Saturday morning; the speakers will be Al Gerald E. Sacks, and Robert I. Soare. Profes bert Baernstein IT, Burgess J. Davis, Frederick sor Grant V. Weiland of the University of Mis W. Gehring, Simon Hellerstein, James A. Jen souri at St. Louis has arranged a special session kins, John L. Lewis, Richard H. Rochberg, on Harmonic Analysis and Related Topics, to be Donald C. Rung, Glenn E. Schober, Ted J. Suf held Friday; the speakers will be David R. Adams, fridge, and Allen W. Weitsman. Professor Richard J. Bagby, Richard A. Hunt, Alexander J. David L. Elliott of Washington University has ar Nagel, Victor L. Shapiro, Elias M. stein, ranged a special session on Differential Geomet Mitchell H. Taibleson, Alberto Torchinsky, S. ric Problems in Control Theory, to be held all Vagi, Wo-Sang Young, and William P. Ziemer. day Friday and Saturday morning; the speakers Professor David J. Winter of the University of will be John B. Baillieul, William M. Boothby, Michigan has arranged a special session on Roger w. Brockett, Jan M. Gronski, Robert Finite Dimensional Field Extensions, to be held Hermann, Henry G. Hermes, Ronald M. Hir Friday; the speakers will be stephen u. Chase, schorn, Velimir Jurdjevic, Arthur J. Krener, Lindsay N. Childs, James K. Deveney, Ray Deborah Rebhuhn, Jackson L. Sedwick, Jr., mond T. Hoobler, Herbert F. Kreimer, Jr., M. B.Suryanarayana, and Hector J. Sussman. Andy R. Magid, Moss E. Sweedler, and David J. Professor Franklin Haimo of Washington Univer Winter. There will also be five sessions of con sity has arranged a special session on Applica tributed ten-minute papers. tions of Ring Theory to Groups, to be held both On Thursday, April10, the day before the Friday and Saturday; the speakers will be Paul F. meeting itself, the University of Missouri at st. Conrad, Vance Faber, Burton I. Fein, Charles E. Louis and Washington University will sponsor a Ford, Brian Hartley, Jutta Hausen, Israel N. brief symposium on Harmonic Analysis and Re Herstein, Arun V. Jategaonkar, Everett L. Lady, lated Topics, which will supplement Professor

130 Weiland's special session. This symposium will Single $16 be held in the Auditorium of the J. C. Penney Twin/Double $22 Building according to the following schedule: Triple $28 9:30 a.m. , Ronald R. Coifman, "On multilinear Requests for room reservations at the Eight Days singular integrals" (Abstract 723-Bl8); 11:00 Inn should be made directly with the Inn using the a.m. , Charles L. Fefferman, "On several com room reservation form which appeared on the plex variables"; 1:30 p.m., Nestor M. Riviere, last page of the January c}(otit:tLJ. Reservations "Finite perimeter and rectifiability"; 3:00p.m., at other motels should be made directly, and stephen Wainger, "Hilbert transforms along mention should be made of this meeting in order curves". to obtain the quoted rates. COUNCIL MEETING FOOD SERVICE The Council will meet at 5:00 p.m. on Fri A fixed lunch will be available in Room 78 day, ApriJ 11, 1975 in the Champagne Room of of the J. C. Penney Building from 12:00 noon to stan Musial & Biggie's st. Louis Hilton Inn, 1:00 p.m. on Friday, Aprilll, at a cost of 10330 Natural Bridge Road, st. Louis 63134. The $2. 50. The cafeteria in the J. C. Penney Build Champagne Room is located on the first level of ing will also be open for lunch on Friday but is the Jnn. rather crowded until after 12:45 p.m. A fixed REGISTRATION lunch will be available in the cafeteria from 12:00 noon to 1:00 p.m. on Saturday, April 12, The registration desk will be located in the at a cost of $3. 00, but the cafeteria will not be lobby of the J. C. Penney Building. The desk open otherwise. Reservations for either or both will be open from 8:00 a.m. to 4:00 p.m. on Fri of the fixed lunches should be made at the regis day and Saturday, April 11 and 12, 1975. The tration desk, since the number of places is registration fee for the meeting will be $2. 00. limited. A list of nearby restaurants will also be ACCOMMODATIONS available. The following four inns are located between TRAVEL AND LOCAL INFORMATION St. Louis Lambert Airport and the University of The University of Missouri at St. Louis is Missouri at st. Louis. They are listed in the or located on Natural Bridge Road approximately der of increasing distance from the university. four miles east of St. Louis Lambert Airport The first two are essentially at Natural Bridge and about three miles east of the inns listed Road and Brown Road; the last two are essentially above. Natural Bridge Road is just south of and at Natural Bridge Road and Woodson Road. All roughly parallel to Interstate Route 70, the uni four are within a mile of the airport. versity being in the vicinity of mileage marker ROYAL INN* 240 on the Interstate 70. Automobile drivers 9600 Natural Bridge Road 63134 should leave Interstate 70 as follows: (1) Those (314) 428-9732 coming from the west who wish to go directly to Single $17 the University should use Exit 238 (Natural Double $20 Bridge Road and Brown Road) and then go three miles east on Natural Bridge Road to the Uni RAMADA INN versity. (2) Those coming from the west who 9636 Natural Bridge Road 63134 wish to go to one of the inns listed should use (314) 426-4700 Exit 237 southbound (Lambert Airport) and then Single $17 turn east on Natural Bridge Road, which is the Double $21 service road to the Interstate at that point. Twin $23 (3) Those coming from the east should use Exit Extra person $ 4 241 southbound (Lucas and Hunt Road), go one EIGHT DAYS INN* mile south on Lucas and Hunt Road (to the first 4545 Woodson Road 63134 (just south of 10232 traffic light), and then go one and one-half miles Natural Bridge Road) west on Natural Bridge Road to reach the Uni (314) 423-6770 versity. The inns listed are about three miles Single $ 9.88 farther west on Natural Bridge Road. Twin/Double $12.88 ADDRESS FOR MAIL Extra person $ 3.00 Mail for those attending the meeting may STAN MUSIAL AND BIGGIE•S ST. LOUIS HIL be addressed in care of American Mathematical TON INN Society Meeting, Department of Mathematics, 10330 Natural Bridge Road 63134 University of Missouri at st. Louis, 8001 Natur (314) 426-5500 al Bridge Road, st. Louis, Missouri 63121.

*N. B. The Eight Days Inn and the Royal Inn will provide free courtesy-van shuttle service between their respective motels and the campus on Aprilll and 12.

131 PROGRAM OF THE SESSIONS The time limit 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. FRIDAY, 8:30A.M. Special Session on Ordinary Differential Equations: Oscillation Theory, Boundary Value Problems I, Room 126 8:3o- 8:50 (1) Principal solutions of positive linear Hamiltonian systems. Professor DON HINTON, University of Tennessee (723-Bll) 9:00- 9:20 (2) Some positivity results for second order systems. SHAlR AHMAD, Oklahoma State University and ALAN C. LAZER*, University of Cincinnati (723-B16) 9:30- 9:50 (3) Related self-adjoint differential and integro-differential systems. Professor WILLIAM T. REID, University of Oklahoma (723-B44) 10:00-10:20 (4) Oscillation theory for nth-order linear differential equations. Professor MARVIN S. KEENER, Oklahoma State University (723-B43) 10:30-10:50 (5) Differential inequalities and k-point boundary problems. Mr. WILLIAM J. FITZPATRICK and Mr. LOUIS J. GRIMM*, University of Missouri-Rolla (723-B12) FRIDAY, 8:30A.M. Special Session on Harmonic Analysis and Related Topics I, Room 222 8:30- 8:50 (6) Lipschitz spaces and Fourier transforms on Lorentz spaces. Professor RICH ARD J. BAGBY, Washington University (723-B6) 9:00- 9:20 (7) L2 boundedness of Hilbert transforms along surfaces. Professors ALEXANDER NAGEL* and STEPHEN WAINGER, University of Wisconsin (723-B50) 9:30- 9:50 (8) Problems related to the a. e. convergence of Fourier series. Preliminary re port. Professor RICHARD A. HUNT, Purdue University (723-B64) 10:00-10:20 (9) Mean continuity and the stationary nonlinear Navier-Stokes equations. Prelimi nary report. Professor VICTOR L. SHAPIRO, University of California, Riverside (723-B21) 10:30-10:50 (10) Inner functions on bounded symmetric domains. Mr. ADAM KORANYI, Yeshiva University and Mr. STEPHEN VAGI*, DePaul University (723-B26) FRIDAY, 8:30A.M. Special Session on Geometric Topology, Room 225 8:30- 8:50 (11) Euclidean G-retracts. Preliminary report. JAN JAWOROWSKI, Indiana Univer sity (723-G2) 9:00- 9:20 (12) Cyclic group action and coincidence points. Professor JOHN CONNETT, Northern Ulinois University (723-G3) 9:30- 9:50 (13) Ambient isotopy classes in graph complements. Preliminary report. Professor MARY -ELIZABETH HAMSTROM, University of nlinois (723-GB) 10:00-10:20 (14) Heegaard splittings of prime 3-manifolds are not unique. Professor JOAN S. BIRMAN*, Columbia University, Professor FRANCISCO J. GONZALEZ-ACUNA, University of Mexico, and Professor J. M. MONTESINOS, Universidad Complu tense, Madrid, Spain (723-G4) 10:30-10:50 (15) Sewings of collared objects. Preliminary report. Professor ROBERT J. DAVERMAN, University of Tennessee (723-G7) FRIDAY, 9:00A.M. Special Session on Differential Geometric Problems in Control Theory I, Room 121 9:00- 9:20 (16) On the classification problem for the control vector fields. JAN M. GRONSKI, University of Missouri-St. Louis (723-C1) 9:30- 9:50 (17) On the closure of sets of attainability. Preliminary report. Ms. D. REBHUHN, Vassar College (723-C2) 10:00-10:20 (18) Upper semicontinuity of set valued functions. Preliminary report. Professor M. B. SURYANARAYANA, Eastern Michigan University (723-B17) 10:30-10:50 (19) Higher order controllability conditions. Preliminary report. Professor HENRY HERMES, University of Colorado (723-B9)

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

132 FRIDAY, 9:00A.M. Special Session on Finite Dimensional Field Extensions I, Room 75 9:00- 9:20 (20) Galois theory using higher derivations. Preliminary report. Professor N. HEEREMA, Florida state University, and Professor J. DEVENEY*, Virginia Commonwealth University (723-A23) 9:30- 9:50 (21) Inseparable Galois descent. Preliminary report. Professor STEPHEN CHASE, Cornell University (723-A15) (Introduced by Professor David J. Winter) 10:00-10:20 (22) Hopf algebras and Galois cohomology. Preliminary report. Professor H. F. KREIMER, Florida state University (723-A25) 10:30-10:50 (23) Galois corings. MOSS SWEEDLER, Cornell University (723-A8) (Introduced by Professor D. J. Winter) FRIDAY, 9:00A.M. Special Session on Applications of Ring Theory to Groups I, Room 72 9:00- 9:20 (24) A condition for complete reducibility. Preliminary report. Professor BRIAN HARTLEY, University of Wisconsin (723-All) (Introduced by Professor F. Haimo) 9:30- 9:50 (25) Centrality conditions in group algebras. Preliminary report. Professor MARTHA K. SMITH, University of Texas (723-A13) 10:00-10:20 (26) The hulls of semiprime rings. PAUL CONRAD, University of Kansas (723-A2) 10:30-10:50 (27) Morita duality and Noetherian rings. Preliminary report. Professor ARUN V. JATEGAONKAR, Fordham University (723-A19) FRIDAY, 9:30A.M. Session on Analysis, Room 229 9:30- 9:40 (28) Real-valued set functions defined on dense subsets of [a, b). CHARLES COPPIN, University of Dallas (723-B39) 9:45- 9:55 (29) Some necessary conditions for inner functions. T. L. McCOY, Michigan state University (723-B59) 10:00-10:10 (30) Integer coefficients of the theta function nome. Dr. H. R. P. FERGUSON, Brigham Young University (723-B29) 10:15-10:25 (31) On absolute high indices theorems. Preliminary report. Professor DANIEL WATERMAN, Syracuse University (723-B60) 10:30-10:40 (32) On certain nonlinear integral equations. RICHARD W. LEGGETT, University of Tennessee (723-B52) FRIDAY, 11:00 A.M. Invited Address, Room 101, Auditorium (33) Supersingular games. Professor A. 0. L. ATKIN, University of Illinois at Chicago Circle (723-A21) FRIDAY, 1:45 P.M. Invited Address, Room 101, Auditorium (34) Iterated path integrals. Professor KUO-TSAI CHEN, University of nlinois at Urbana-Champaign (723-Gll) FRIDAY, 3:00P.M. Special Session on Finite Dimensional Field Extensions II, Room 75 3:00- 3:20 (35) Normal field extensions and K/k-bialgebras. Professor DAVID J. WINTER, University of Michigan (723-A27) 3:30- 3:50 (36) An infinitesimal version of purely inseparable Galois theory. Professor RAY- MOND T. HOOBLER, Rice University (723-A24) 4:00- 4:20 (37) Informal Session 4:30- 4:50 (38) Abelian Galois extensions as modules over the group ring. Preliminary report. Professor LINDSAY N. CHILDS, state University of New York at Albally (723-A22) 5:00- 5:20 (39) The Galois theory of commutative rings. Dr. ANDY MAGID, University of Okla- homa (723-A26) FRIDAY, 3:00P.M. Special Session on Application of Ring Theory to Groups II, Room 72 3:00- 3:20 (40) Endomorphism rings and automorphism groups of abelian p-groups. JUTTA HAUSEN, University of Houston (723-A3)

133 3:30- 3:50 (41) Large abelian subgroups of the associated group of a nil ring. Preliminary re port. Professor VANCE FABER, University of Colorado, Denver (723-A12) 4:0o- 4:20 (42) Galois groups and division algebras. Professor BURTON FE1N*, Oregon State University, and Professor MURRAY SCHACHER, University of California, Los Angeles (723-Al) 4:30- 4:50 (43) Rings with involution whose symmetric units are Abelian. Preliminary report. Professor CHARLES LANSKI, University of Southern California (723-AlO) 5:00- 5:20 (44) Polynomials associated with groups of exponent four. M. F. NEWMAN, K. W. WESTON*, and TAH-ZEN YUAN, University of Wisconsin-Parkside (723-A14) 5:30- 5:50 (45) Representing the automorphism group of a noetherian module. Professor CLEON YOHE, Washington University (723-Al6) FRIDAY, 3:00P.M. Special Session on Classical Function Theory I, Room 229 3:00- 3:20 (46) Picard's theorem and Brownian motion. Dr. BURGESS DAVIS, Purdue Univer sity (723-B22) 3:30- 3:50 (47) Derivatives of entire functions and a conjecture of Polya. Professor SIMON HELLERSTE1N*, University of Wisconsin and National Science Foundation, Washington, D.C., and Professor JACK WILLIAMSON, University of Hawaii, (723-B24) 4:00- 4:20 (48) Examples of subharmonic functions in spaces. Dr. JOHN L. LEWIS, University of Kentucky (723-Bl4) · 4:30- 4:50 (49) Coefficients and means of functions omitting values. W. K. HAYMAN, Imperial College, London, England, and A. W. WEITSMAN*, Purdue University (723-B35) 5:0o- 5:20 (50) A variational method for families of K(z)-quasiconformal mappings. Professor GLENN SCHOBER, Indiana University (723-Bl3) 5:30- 5:50 (51) Meier type theorems for general boundary approach and a-porous exceptional sets. Professor D. C. RUNG, Carleton University (723-Bl9) FRIDAY, 3:00P.M. Special Session on Ordinary Differential Equations: Oscillation Theory, Boundary Value Problems II, Room 126 3:00- 3:20 (52) Periodic solutions of second order differential equations with nonlinear, non- differentiable damping. Professor W. R. UTZ, University of Missouri (723-B47) 3:30- 3:50 (53) Oscillation and disconjugacy in topological dynamics. Professor H. GUGGEN- HEIMER, Polytechnic Institute of New York (723-BS) 4:00- 4:20 (54) Upper bounds for conjugate points of nonselfadjoint differential equations. Dr. ALLAN EDELSON and Dr. KURT KREITH*, University of California, Davis (723-Bl) 4:30- 4:50 (55) Some sufficient conditions for nonlinear oscillation. Professor W. J. COLES, University of Utah (723-B40) 5:00- 5:20 (56) Some questions related to the existence of solutions of boundary value problems. Professor LLOYD JACKSON, University of Nebraska (723-B42) FRIDAY, 3:00P.M. Special Session on Harmonic Analysis and Related Topics II, Room 222 3:00- 3:20 (57) Wirtinger and Poincare inequalities for BV functions. Professor NORMAN C. MEYERS, University of Minnesota and Professor WILLIAM P. ZIEMER*, Indiana University (723-BlO) 3:30- 3:50 (58) Singular integrals on nilpotent groups. E. M. STE1N, Princeton University (723-B36) 4:00- 4:20 (59) Every n-dimensional space over a local field has a natural field structure. Pre liminary report. Professor MITCHELL TAIBLESON, Washington University (723-B4) 4:30- 4:50 (60) Parabolic HP spaces. Professor A. P. CALDERON, University of Chicago and Professor A. TORCH1NSKY*, Cornell University (723-B7) 5:00- 5:20 (61) Mean convergence of generalized Walsh-Fourier series. Dr. WO-SANG YOUNG, University of Chicago (723-Bl5) 5:30- 5:50 (62) Bessel potentials and Bochner-Riesz summation operators. Dr. DAVID R. ADAMS, University of Kentucky (723-B2)

134 FRIDAY, 3:00P.M. ecial Session on Differential Geometric Problems in Control Theo II, Room 121 3:00- 3:20 (63) On the determination of controllability of bi inear systems. Preliminary report. Professor WILLIAM M. BOOTHBY, Washington University (723-C10) 3:30- 3:50 (64) The geometry of the space of reduced rational functions. Preliminary report. Professor ROGER W. BROCKETT, Harvard University (723-C3) 4:00- 4:20 (65) Informal Session 4:30- 4:50 (66) Equivalence of nonlinear and bilinear control systems. Dr. J. L. SEDWICK* and Dr. D. L. ELLIOTT, Washington University (723-C4) 5:00- 5:20 (67) Topological groups and nonlinear control systems. Dr. RONALD HIRSCHORN, Queen's University (723-C7) 5:30- 5:50 (68) Equivalence of systems and infinite Lie groups. Professor ROBERT HERMANN, Rutgers University (723-C5) FRIDAY, 3:00P.M. Session on Topology, Room 225 3:00- 3:10 (69) Unknotting links in s3 by maps. Professor HOWARD LAMBERT, University of Iowa (723-G6) 3:15- 3:25 (70) Inverse limits and mappings of minimal topological spaces. LOUIS M. FRIED- LER* and DIX H. PETTEY, University of Missouri, Columbia (723-G5) 3:30- 3:40 (71) The hereditary Lindeltlf property, primitive structures, and separable metrizability. Professor H. H. WICKE* and Professor J. M. WORRELL, Jr., Ohio University (723-GlO) 3:45- 3:55 (72) Upper semicontinuous decompositions of convex metric spaces. Preliminary report. Dr. S. E. RODABAUGH, University of Missouri-Columbia (723-G9) 4:00- 4:10 (73) Fixed point theorems for mappings with a convexity condition. Preliminary report. Dr. S. P. SINGH* and Mrs. MARY VEITCH, Memorial University (723-G1) SATURDAY, 8:00A.M. Session on Algebra, Room 78 8:00- 8:10 (74) Generalized lattices. Preliminary report. Dr. LEONARD E. FULLER, Kansas state University (723-A20) 8:15- 8:25 (75) Graded modules bounded below are direct sums of cyclics. Preliminary report. Professor CARY WEBB, Chicago state University (723-A28) 8:30- 8:40 (76) Semi-perfect rings with quasi-projective left ideals. Mr. S. C. GOEL* and Professor S. K. JAIN, Ohio University (723-A29) SATURDAY, 8:30A.M. Special Session on Recursion Theory I, Room 75 8:30- 8:50 (77) Perspectives in algebraic recursion theory. Professor HARVEY FRIEDMAN, State University of New York at Buffalo (723-E4) 9:00- 9:20 (78) Inductive definitions and reflecting ordinals. Preliminary report. Professor WAYNE RICHTER, University of Minnesota (723-E10) 9:30- 9:50 (79) Recursion on inadmissible ordinals. Preliminary report. Professor GERALD E. SACKS, Harvard University and Massachusetts Institute of Technology (723-E8) 10:00-10:20 (80) Minimality and gap ordinals. Professor HILARY PUTNAM, Harvard University (723-E3) 10:30-10:50 (81) R.e. presented vector spaces. Preliminary report. Professor GEORGE META- KIDES, University of Rochester and Professor ANIL NERODE*, Cornell Univer sity (723-E6) SATURDAY, 8:30A.M. Special Session on Classical Function Theory II, Room 229 8:30- 8:50 (82) The * function for subharmonic functions inn-space. Professor ALBERT BAERNSTEIN II*, Washington University, and Professor B. A. TAYLOR, b;J.sti tute for Advanced Study (723-B53) 9:00- 9:20 (83) Degeneration of algebras of analytic functions. Preliminary report. Mr. RICH- ARD ROCHBERG, Washington University (723-B38) 9:30- 9:50 (84) Open problems for quasiconformal mappings. Professor F. W. GEHRING, Uni- versity of Michigan (723-B62) 10:00-10:20 (85) On univalent polynomials. Professor TED J. SUFFRIDGE, University of Kentucky, (723-B37) 135 10:30-10:50 (86) The mapping of strip domains. Professor JAMES A. JENKINS, Washington Uni versity (723-B55) SATURDAY, 9:00A.M. Special Session on Nonlinear Functional Analysis I, Room 222 9:00- 9:20 (87) Some generalizations of the Borsuk-Ulam theorem. Preliminary report. Professor ROGER NUSSBAUM, Rutgers University (723-B30) 9:30- 9:50 (88) Bifurcation and stability theory for nonlinear gradient operators. Preliminary report. Professor D. SATHER, University of Colorado (723-B20) 10:00-10:20 (89) Periodic solutions in the large of nonlinear ordinary differential equations. Professor LAMBERTO CESARI*, University of Michigan, and Professor RAN GACHARI KANNAN, University of Missouri, St. Louis (723-B58) 10:3D-10:50 (90) Birfurcation with several parameters. Professor JACK K. HALE, Brown Uni versity (723-B3) SATURDAY, 8:30A.M. Special Session on Differential Geometric Problems in Control Theory III, Room 121 8:30- 8:50 (91) Some optimal control problems on Lie groups. Preliminary report. Mr. JOHN BAILLIEUL, Harvard University (723-C6) 9:00- 9:20 (92) structural stability of control systems. Preliminary report. Professor ARTHUR J. KRENER, University of California, Davis (723-B25)(Jntroduced by D. L. Elliott) 9:30- 9:50 (93) Informal Session 10:00-10:20 (94) A sufficient condition for local controllability. Professor HECTOR J. SUSSMANN, Rutgers University (723-B57) 10:30-10:50 (95) Local controllability and the existence of stabilizing feedbacks. VELIMIR JURDJEVIC, University of Toronto (723-C8) SATURDAY, 9:00 A. M. Special Session on Awlications of Ring Theory to Groups III, Room 72 9:00- 9:20 (96) On the automorphism group of an integral group ring. II. Preliminary report. Dr. GARY L. PETERSON, Michigan state University (723-A9) 9:30- 9:50 (97) Cohomology of solvable groups of finite rank. Professor DEREK J. S. ROBIN SON, University of Illinois at Urbana-Champaign (723-A4) 10:00-10:20 (98) Endomorphism rings of finite rank torsion free abelian groups. Preliminary re port. Mr. EVERETT LEE LADY, University of Kansas (723-A7) 10:30-10:50 (99) Finite groups and division algebras. Professor CHARLES FORD, Washington University (723-A18) SATURDAY, 9:00A.M. Special Session on Ordinary Differential Equations: Oscillation Theory, Boundary Value Problems III, Room 126 9:00- 9:20 (100) On a comparison theorem for scalar Riccati equations. Professor C. c. TRA VIS, University of Tennessee (723-B46) 9:30- 9:50 (101) Open questions in quantitative estimates of disconjugary intervals A.M. FINK, University of Minnesota (723-B41) 10:00-10:20 (102) Boundary-value functions. Preliminary report. Professor JERRY RIDENHOUR, Northern nlinois University (723-B45) 10:30-10:50 (103) Systems conjugate and focal points of fourth order nonselfadjoint differential equations. Mr. SUI-SUN CHENG, University of California, Davis (723-B5) SATURDAY, 9:00A.M. Session on Awlied Mathematics, Analysis, and Probability, Room 225 9:00- 9:10 (104) An alternative method for approximating the potential and gradient at any point between an insulated cable to plane. T. J. LANOUE*, Westinghouse Electric Corporation, Muncie, Indiana, and Dr. MlR M. ALI, Ball state University (723-H1) 9:15- 9:25 (105) A boundary value problem for equations of mixed type. Professor JOHNS. PAPADAKIS, University of Rhode Island and Naval Underwater Systems Center, New London, Connecticut (723-B48) 9:30- 9:40 (106) Local field singular integrals with odd kernel. Preliminary report. Professor KEITH PHILLIPS, New Mexico State University (723-B63)

136 9:45- 9:55 (107) The invalidity of the Calderon-Zygmund inequality for singular integrals over local fields. Professor JAMES DALY, New Mexico State University (723-B33) 10:00-10:10 (108) Double commutators in R2• Mr. LONE YOUNG YEE, Washington University (723-B54) 10:15-10:25 (109) The method of moments for linear random initial value problems. Dr. MELVIN D. LAX*, Southern Dlinois University and Professor WILLIAM E. BOYCE, Rensselaer Polytechnic Institute (723-F1) 10:30-10:40 (110) Sufficient conditions for an operator valued Feynman-Kac formula. MICHAEL D. GRADY, University of Utah (723-F2) SATURDAY, 11:00 A.M. Invited Address, Room 101, Auditorium (111) What good are ultrafilters? Professor KENNETH KUNEN, University of Wis consin (723-E9) SATURDAY, 12:45 P.M. Special Session on Nonlinear Functional Analysis II, Room 222 12:45- 1:05 (112) A bifurcation theorem for potential operators. PAUL H. RABINOWITZ, Univer sity of Wisconsin (723-B28) 1:15- 1:35 (113) Conditions for a local Pareto optimum in a Banach space. Professor H. F. WEINBERGER, University of Minnesota (723-C9) SATURDAY, 1:45 P. M. Invited Address, Room 101, Auditorium (114) The use of Hardy spaces and their generalizations in harmonic analysis. Professor GUIDO L. WEISS, Washington University (723-B56) SATURDAY, 3:00P.M. ecial Session on Recursion Theo II, Room 75 3:00- 3:20 (115) Congruence r ations and definability in lattices of a-recursively enumerable sets. Professor MANUEL LERMAN, University of Connecticut (723-E2) 3:30- 3:50 (116) Degrees of generic sets. Preliminary report. Professor CARL G. JOCKUSCH, Jr. , University of lllinois (723-E5) 4:00- 4:20 (117) The infinite injury priority method. Professor ROBERT SOARE, University of nlinois at Chicago Circle (723-E7) 4:30- 4:50 (118) Finiteness predicates for .1\(Z). Professor THOMAS G. McLAUGHLIN, Texas Tech University (723-E1) SATURDAY, 3:00P.M. Special Session on Applications of Ring Theory to Groups IV, Room 72 3:00- 3:20 (119) Group rings. Preliminary report. Professor DONALDS. PASSMAN, University of Wisconsin (723-A5) 3:30- 3:50 (120) Zero divisors in rings and centrality on groups. Preliminary report. Professor RICHARD E. PHILLIPS, Michigan State University (723-A17) 4:00- 4:20 (121) Relation modules as generators. Preliminary report. Dr. JULIAN WILLIAMS, University of Wisconsin-Parkside (723-A6) 4:30- 4:50 (122) The relation between normal subgroups and ideals in rings. Professor ISRAEL N. HERSTEJN, University of Chicago (723-A30) SATURDAY, 3:00P.M. Special Session on Nonlinear Functional Analysis III, Room 222 3:00- 3:20 (123) A semilinear elliptic equation in L1(IRN). Preliminary report. Professor PH. BENILAN, Professor H. BREZIZ, University of Paris, and Professor M. G. CRANDALL*, University of Wisconsin (723-B61) 3:30- 3:50 (124) Global existence for some semilinear hyperbolic systems. Professor LUC TAR TAR, University of Wisconsin (723-B49) SATURDAY, 3:00 P. M. Session on Analysis and Geometry, Room 126 3:00- 3:10 (125) Distributed parameter penalization. Preliminary report. Professor RUSSELL D. RUPP, State University of New York at Albany (723-B34) 3:15- 3:25 (126) Harmonic analysis of harmonic functions. Professor LEE A. RUBEL, University of nlinois (723-B31)

137 3:30- 3:40 (127) A characterization ofthe cokernel of a singular Fredholm differential operator. Professor LEON M. HALL, University of Nebraska-Lincoln (723-B23) 3:45- 3:55 (128) Comparison theorems for ordinary differential equations. Professor ALLAN PETERSON, University of Nebraska-Lincoln (723-B27) 4:00- 4:10 (129) Oscillation of even order delay differential equations. Preliminary report. Dr. TSAI-SHENG LIU, University of Oklahoma (723-B32) 4:15- 4:25 (130) Nonoscillatory solutions of forced second order linear equations. Dr. RONALD C. GRIMMER and Dr. WILLIAM T. FATULA*, Southern Ulinois University (723-B51) 4:30- 4:40 (131) Spectra of the Laplace-Beltrami operator on compact semisimple Lie groups. Dr. BRIAN BEERS, Johns Hopkins University and Dr. RICHARD MILLMAN*, Southern Ulinois University (723-Dl) Paul T. Bateman Associate Secretary Urbana, Ulinois

PRESENTORS OF PAPERS Following each name is the number corresponding to the speaker's position on the program *Special Session speakers • Invited one-hour lectures *Adams, David R. #62 *Hermes, Henry #19 *Rebhuhn, D. #17 •Atkin, A. 0. L. #33 *Herstein, Israel N. #122 *Reid, William T. #3 *Baernstein, Albert, II #82 *Hinton, Don #1 *Richter, Wayne #78 *Bagby, Richard J. i6 *Hirschorn, Ronald #67 *Ridenhour, Jerry #102 *Baillieul, John #91 *Hoobler, Raymond T. #36 *Robinson, Derek J. S. #97 *Birman, Joan S. #14 *Hunt, Richard A. #8 *Hochberg, Richard #83 *Boothby, William M. #63 *Jackson, Lloyd #56 Rodabaugh, S. E. #72 *Brockett, Roger W. #64 *Jategaonkar, Arun V. #27 Rubel, Lee A. #126 *Cesari, Lamberto #89 *Jaworowski, Jan #11 *Rung, D. C. #51 *Chase, Stephen #21 *Jenkins, James A. #86 Rupp, Russell D. #125 eChen, Kuo-Tsai #34 *Jockusch, Carl G., Jr. #116 *Sacks, Gerald E. #79 *Cheng, Sui-Sun #103 *Jurdjevic, Velimir #95 *Sather, D. #88 *Childs, Lindsay N. #38 *Keener, Marvin s. #4 *Schober, Glenn #50 *Coles, W.J. #55 *Kreimer, H. F. #22 *Sedwick, J. L. #66 *Connett, John #12 *Kreith, Kurt #54 *Shapiro, Victor L. #9 *Conrad, Paul #26 *Krener, Arthur J. #92 Singh, S. P. #73 Coppin, Charles #28 •Kunen, Kenneth #111 *Smith, Martha K. #25 *Crandall, M. G. #123 *Lady, Everett Lee #98 *Soare, Robert #117 Daly, James #107 Lambert, Howard #69 *Stein, E. M. #58 *Daverman, Robert J. #15 Lanoue, T. J. #104 *Suffridge, Ted J. #85 *Davis, Burgess #46 *Lanski, Charles #43 *Su:ryanarayana, M. B. #18 *Deveney, J. #20 Lax, Melvin D. #109 *Sussmann, Hector J. #94 *Faber, Vance #41 *Lazer, Alan C. #2 *Sweedler, Moss #23 *Fein, Burton #42 Leggett, Richard W. #32 *Taibleson, Mitchell #59 Ferguson, H. R. P. #30 *Lerman, Manuel #115 *Tartar, Luc #124 *Fink, A. M. #101 *Lewis, John L. #48 *Torchinsky, A. #60 *Ford, Charles #99 Liu, Tsai -Sheng #129 *Travis, c. c. #100 Friedler, Louis M. #70 McCoy, T. L. #29 *utz, W. R. #52 *Friedman, Harvey #77 *McLaughlin, Thomas G. #118 *Vagi, stephen #lo Fuller, Leonard E. #74 *Magid, Andy #39 Waterman, Daniel #31 *Gehring, F. W. #84 Millman, Richard #131 Webb, Cary #75 Goel, s. c. #76 *Nagel, Alexander #7 *Weinberger, H. F. #113 Grady, Michael D. #110 *Nerode, Anil #81 •Weiss, Guido L. #114 *Grimm, Louis J. #5 *Nussbaum, Roger #87 *Weitsman, A. W. #49 *Gronski, Jan M. #16 Papadakis, JohnS. #105 *Weston, K. W. #44 *Guggenheimer, H. #53 *Passman, Donalds. #119 Wicke, H. H. #71 *Hale, Jack K. #90 Fatula, William T. #130 *Williams, Julian #121 Hall, Leon M. #127 Peterson, Allan #128 *Winter, David J. #35 *Ha.mstrom, Mary-Elizabeth #13 *Peterson, Gary L. #96 Yee, Lone Young #108 *Hartley, Brian #24 Phillips, Keith #106 *Yohe, Cleon #45 *Hausen, Jutta #40 *Phillips, Richard E. #120 Young, Wo-Sang #61 *Hellerstein, Simon #47 *Putnam, Hilary #80 *Ziemer, William P. #57 *Hermann, Robert #68 *Rabinowitz, Paul H. #112

138 The Seven Hundred Twenty-Fourth Meeting Naval Postgraduate School Monterey, California April19, 1975 The seven hundred twenty-fourth meeting STAGE COACH LODGE of the American Mathematical Society will be Single $15 ($20) held at the Naval Postgraduate School, Monterey, Double/Twin $18 ($22) California, on Saturday, April19, 1975. All functions will be held in Ingersoll Hall. Registra CALIFORNIAN tion will begin at 10:00 a.m. in the lobby. Single $12 ($19) By invitation of the Committee to Select Double/Twin $14 ($22) Hour Speakers for Far Western Sectional Meet ings, there will be two one-hour addresses. Pro Requests for reservations should be ad fessor Kennan T. Smith of Oregon State Univer dressed to Visitors and Convention Bureau, sity will lecture at 11:00 a.m. on "Practical and P. 0. Box 1770, Monterey, California 93940; mathematical aspects of the problem of recon phone (408) 375-2252. Requests should indicate structing objects from x-rays." Professor type of accommodation, arrival and departure Isaac Namioka of the University of Washington time, and order of preference of motels. They will lecture at 3:30 p.m. He will speak on "Some should refer to the American Mathematical So topological questions related to Banach spaces." ciety meeting, must be accompanied by one Both hour addresses will be in Room I-122 of In night's deposit, and it is suggested that you ask gersoll Hall. for a confirmation. Deadline for requests was There will be sessions for contributed pa March 24, 1975. pers. Abstracts should have been submitted to Less expensive accommodations are avail the American Mathematical Society so as to ar able at the Motel 6, 2124 Fremont A venue (old rive prior to the deadline of February 18, 1975. Highway 1), Monterey, California 93940; phone Late papers will be accepted for presentation at (408) 373-3500. Reservations at the Motel 6 the meeting, but will not appear in the printed must be made directly with the motel and should program of the meeting. be made well in advance of the meeting. Their The Visitors and Convention Bureau of rates range from $6. 95 to $12. 95. Monterey has blocked off rooms in the following mo Noon meals will not be available on campus, tels at the specified rates (rates in parentheses but several restaurants are within a ten are for Saturday night) . All motels except the minute walk of the meeting. There are over last are within a ten-minute walk of the meeting. fifty restaurants in the Monterey area. More DEL MONTE HYATT HOUSE detailed information will be available at the Single $24 meeting. The Monterey Twin $34 Airport is served by United and Hughes-AirWest. Greyhound buses have fre ROYAL INN quent service to Monterey, and Amtrak serves Single $20 nearby Salinas daily. By car, one should first Double $26 get on Highway 1. As you approach downtown, Twin $30 look for signs reading "Naval Postgraduate MONTEREY MOTOR LODGE School." Follow signs to the school, thence to Single $15 Ingersoll Hall. Parking will be available in Twin $22 three lots adjacent to Ingersoll Hall.

PROGRAM OF THE SESSIONS The time limit for each contributed paper is ten minutes. Tomaintainthis schedule, the time limit will be strictly enforced. SATURDAY, 11:00 A.M. Invited Address, Room I-122, Ingersoll Hall (1) Practical and mathematical aspects of the problem of reconstructing objects from x-rays. Professor KENNAN T. SMITH, Oregon state University (724-C4) SATURDAY, 1:30 P.M. Session on Analysis, Room I-271, Ingersoll Hall 1:30- 1:40 (2) Bifurcation from normal eigenvalues. Preliminary report. Professor JOHN ALAN MACBAIN, Air Force Institute of Technology, Wright-Patterson AFB, Dayton, Ohio (724-B5)

*For papers with more than one author, an asterisk follows the name of the author who plans to pre sent the paper at the meeting. 139 1:45- 1:55 (3) The similarity problem for representations of a B*-algebra. Professor BRUCE A. BARNES, University of Oregon (724-B1) 2:00- 2:10 (4) Hilbert problems-A distributional approach. Professor MARION ORTON, Uni versity of California, Irvine (724-B4) 2:15- 2:25 (5) Dual trigonometrical series associated with boundary conditions of the first and third kind. Professor ROBERT B. KELMAN* and Mr. J. TIMOTHY SIMPSON, Colorado state University (724-B6) 2:30- 2:40 (6) L2-boundedness of pseudo-differential operators. Preliminary report. Mr. ANDREW GARY CHILDS, University of California, Berkeley (724-B7) 2:45- 2:55 (7) An extension of the Bazilevic functions. Preliminary report. Professor DOUGLAS M. CAMPBELL* and Mr. KENT PEARCE, Brigham Young University (724-B3) 3:00- 3:10 (8) Flow from a generalized sluice gate under gravity. Dr. VURYL J. KLASSEN, California State University, Fullerton (724-C1) 3:15- 3:25 (9) Time-dependent solutions of the viscous incompressible flow past a circular cylinder by the method of series truncation. III. Preliminary report. Professor V.A. PATEL, California state University, Humboldt (724-C2) SATURDAY, 1:30 P.M. General Session, Room I-282, Ingersoll Hall 1:30- 1:40 (10) A hybrid algorithm for nonlinear programming problems. Professor R. W. CHANEY, Western Washington State College (724-C3) 1:45- 1:55 (11) The probability that p implies q. Preliminary report. Professor PHILIP G. CALABRESE, California state College, Bakersfield (724-E1) 2:00- 2:10 (12) Measure structure for function spaces. Preliminary report. Mr. EUGENE WESLEY, Northwestern University (724-F1) (Introduced by Stanley Reiter) 2:15- 2:25 (13) A 3-dimensional treatment of N-dimensional non-Euclidean geometries. NIELS T. SORENSEN, J. G. Enterprises, Riverside, California (724-D2) 2:30- 2:40· (14) On the number of reflections it takes to render a nonconvex plane polygon convex. Dr. RICHARD R. JOSS*, California state University, Long Beach, and Dr. ROBERT W. SHANNON, New Mexico state University (724-D1) 2:45- 2:55 (15) Mapping products of Xconnected continua into E2• Professor CHARLES L. HAGOPIAN, California state University, Sacramento (724-G2) 3:00- 3:10 (16) Discrete series representations. Professor THOMAS J. ENRIGHT* and Professor V. S. VARADARAJAN, University of California, Los Angeles (724-G1) SATURDAY, 1:45 P.M. Session on Algebra and Number Theory, Room I-260, Ingersoll Hall 1:45- 1:55 (17) Commutative cancellative semigroups without idempotent. Professor H. B. HAMILTON*, California state University, Sacramento, and Dr. T. E. NORDAHL and Professor T. TAMURA, University of California, Davis (724-A5) 2:00- 2:10 (18) Cancellative semigroups with nonempty center. Dr. T. E. NORDAHL, Univer sity of California, Davis (724-A6) 2:15- 2:25 (19) Decomposition matrices and Cartan matrices of some finite simple groups. Dr. ELAINE ZASLAWSKY, San Jose state University (724-A2) 2:30- 2:40 (20) Two problems involving Schur functions. Professor RUSSELL MERRIS, Califor nia state University, Hayward (724-A1) 2:45- 2:55 (21) A generalization of Eisenstein's irreducibility criterion. Dr. DANIEL DAVIS, Naval Postgraduate School (724-A3) 3:00- 3:10 (22) A lower bound for odd triperfect numbers. Preliminary report. Professor RUDOLPH NAJAR*, University of Wisconsin, Whitewater, and Professor WALTER BECK, Wartburg College (724-A4) SATURDAY, 3:30P.M. Invited Address, Room I-122, Ingersoll Hall (23) Some topological questions related to Banach spaces. Professor ISAAC NAMIOKA, University of Washington (724-B2) Kenneth A. Ross Eugene, Oregon Associate Secretary

140 PRELIMINARY ANNOUNCEMENTS OF MEETINGS The Seven Hundred Twenty-Fifth Meeting Washington State University Pullman, Washington June 21, 1975 The seven hundred twenty-fifth meeting of Department of Mathematics, Washington state the American Mathematical Society will be held University, Pullman, Washington 99163, prior to at Washington state University in Pullman, Wash May 16. Indicate clearly the accommodations de ington, on Saturday, June 21, 1975. The Mathe sired and the nights of desired occupancy. Dormi matical Association of America and the Society tory housing should be paid for at check-in time. for Industrial and Applied Mathematics will hold The following motels are located in Pullman Northwest Sectional Meetings in conjunction with (zip code 99163). Reservations should be made this meeting of the Society. The Association will directly with them. have sessions on Friday and Saturday, June 20 and 21; featured addresses will be given by Pro THUNDERBIRD LODGE fessor Roy Dubisch, University of Washington, SE 915 Main street, Box 255 and Professor Ivan Niven, University of Oregon. Phone (509) 332-2646 The main speaker for the SIAM meeting will be Single $ 13up Professor Victor Kl.ee of the University of Wash Double $ 18 up ington. Professor Sidney G. Hacker of Washing ton state University will be the speaker at a ban ROYAL MOTOR INN quet on Friday evening, W. 120 Main Washington State University has a proposal Phone (509) 564-1254 pending with the NSF to host an NSF Regional Con Single $ 14up ference the following week, June 23-27. Professor Double $ 17up Solomon W. Golomb ofthe University of Southern California would present a series of lectures on TRAVELODGE "Practical applications of finite mathematics. " S. 515 Grand By invitation of the Committee to Select Phone (509) 564-1143 Hour Speakers for Far Western Sectional Meet Single $ 13up ings, there will be two invited addresses. Pro Double $ 18up fessor Theodore E. Harris of the University of Southern California will lecture at 11:00 a.m. MANOR LODGE MOTEL on Saturday on "Some results on Markov inter Main and Paradise action processes." Professor David W. Barnette Phone (509) 564-1245 of the University of California at Davis will lec Single $10.50 up ture at 3:30 p.m. on Saturday. The title of his Double $12.50 up address is "From convex polytopes to manifolds." Both addresses will be given in Room 16 of the AL KIRCHER'S HILLTOP MOTEL AND STEAK Physical Sciences Building. Sessions of contri HOUSE buted papers will be scheduled on Saturday. P. 0. Box 296 Abstracts should be submitted to the Ameri Phone (509) 564-1195 can Mathematical Society, P.O. Box 6248, Single $ 8.40 up Providence, Rhode Island 02940 prior to the Double $ 9.98up deadline of April19, 1975. Late papers will be accepted for presentation at the meeting, but WILSON COMPTON UNION (on campus) they will not be listed in the printed program of Box 2100, College station the meeting. Phone (509) 335-3578 or 335-3548 The registration desk will be located in the Single $10. 50 auditorium foyer of the Physical Sciences Build Double $13.65 ing, and will be open during the following periods: 8:00a.m. to noon and 1:00 p.m. to 4:00p.m. Meals can be taken at the local Pullman on Friday; 8:00a.m. to noon and 1:00 p.m. to restaurants; a list will be provided at the regis 2:30p.m. on Saturday. There will be a registra tration desk. The Compton Union Building cafe tion fee of $2. 00. teria will be open Friday only, for breakfast and Dormitory housing will be available on lunch. The Association is sponsoring a banquet campus. The rates are $5. 00 per person per Friday evening at Austin's steakhouse. night on a double occupancy basis, $7. 00 per Pullman is located in southeast Washington person per night on a single occupancy basis, at the junction of U. s. Highway 195 and state and $3.50 per person for college students. Highway 270. Moscow, Idaho, is eight miles to Reservations should be made by writing to the the east on Highway 270. The main campus

141 entrance to Washington state University is also service into Pullman is available. on IDghway 270, with the dormitory complex on the immediate right at this entrance. The Pull Kenneth A. Ross man-Moscow Airport, three miles east of Pull Associate Secretary man, is served by Cascade Airways. Limousine Eugene, Oregon The Seventy-Ninth Summer Meeting Western Michigan University Kalamazoo, Michigan August 18-22, 1975

SHORT COURSE ON APPLIED COMBINATORICS, August 16 and 17 The American Mathematical Society will computing and operations research recently given present a two-day Short Course on Applied Com at Society meetings (August 1973, January 1974, binatorics on the campus of Western Michigan January 1975). University, Kalamazoo, Michigan, on Saturday, The program is under the direction of August 16, and Sunday, August 17, 1975. The D. R. Fulkerson, Operations Research Depart course is designed to give substantial introduc ment, Cornell University. This short course tions to several important areas of application of was recommended by the AMS Committee on combinatorics and graph theory. It is intended to Employment and Educational Policy (CEEP), present both mathematically challenging aspects whose members are Michael Artin, Charles W. and connections with problems encountered in Curtis, Wendell H. Fleming, Calvin c. Moore, other disciplines, industrial practice, or work Martha K. Smith, and Daniel H. Wagner. of government agencies. This short course, A detailed announcement of the program which is open to all who wish to participate, will will appear in the June issue of these CNotit:rJJ. be similar in format to the short courses on SEVENTY-NINTH SUMMER MEETING, August 18-22 The seventy-ninth summer meeting of the Two sets of Colloquium Lectures are sched American Mathematical Society will be held at uled. E. R. Kolchin of Columbia University will Western Michigan University, Kalamazoo, Mich lecture on "Differential Algebraic Groups." The igan, from Monday, August 18, through Friday, other set oflectures will be given by Elias M. August 22, 1975. All sessions of the meeting stein, Princeton University; the title of his lec will take place on the campus of the university. tures will be announced in a later issue of these The AMS Committee on Employment and c}/ofiefi). Both series of lectures will be given in Educational Policy (CEEP) will sponsor a panel the Miller Auditorium. The first lectures in each discussion on "The role of applications in Ph. D. series will be given Tuesday afternoon, August 19. programs in mathematics" on Thursday evening, It is expected that there will also be about August 21, at 8:00 p.m. Members of the panel six invited one-hour addresses at the meeting. will include Richard D. Anderson, former chair The 1975 Leroy P. steele Prizes in honor of man of CEEP; Lipman Bers, president of the George David Birkhoff and the Norbert Wiener American Mathematical Society; Henry 0. Pol Prize in Applied Mathematics will be awarded. lak, president of the Mathematical Association There will be sessions for contributed ten of America; and Wendell H. Fleming, current minute papers on Wednesday morning and all day chairman of CEEP, will serve as moderator. Thursday and Friday. Abstracts of contributed The same committee is planning an open papers should be sent to the American Mathemati meeting at 4:30p.m. on Monday, August 18, cal Society, P.O. Box 6248, Providence, Rhode consisting of a brief report on the state of the Island 02940; the deadline for receipt of abstracts job market, followed by an open 9fscussion with is June 17, 1975. There is no limit on the number comments and suggestions welcomed from the of papers that will be accepted for presentation. audience. Those individuals having time preferences for the This meeting will be held in conjunction presentation of their papers should so indicate with the annual meetings of the Mathematical clearly on their abstracts. There will be a ses Association of America and Pi Mu Epsilon. The sion for late papers if one is needed, but late pa Mathematical Association of America will meet pers will not be listed in the printed program of from Monday, August 18, through Wednesday, the meeting. August 20. The Earle Raymond Hedrick Lectures, sponsored by the Association, will be given by SPECIAL SESSIONS Frederick J. Almgren, Jr., Princeton University. There will be fourteen special sessions of The title of his lectures will be "Geometric selected twenty-minute papers. The subjects of Measure Theory and the Calculus of Variations." these special sessions and the names of the The Association for Women in Mathematics mathematicians arranging them are as follows: will hold a session on Tuesday, August 19, at Functional Equations, Janos D. Aczel, Univer 3:30p.m. The Mathematicians Action Group will sity of Waterloo; Affine Algebraic Groups, Amas hold a panel discussion on the employment crisis sa Fauntleroy and Andy R. Magid, University of at 4:30p.m. on Wednesday, August 20. illinois and University of Oklahoma, respectively; 142 Scientific Computing, George Fix, University of augmenting the already huge military budget, it Michigan; Aspects of Real Analysis, Casper Goff is the sense of this meeting that the federal man, Purdue University; Ordered Groups, W. government must: (1) Fund a national open ad Charles Holland, Bowling Green state University; missions program at institutions of. higher educa Banach Spaces with the Radon-Nikodym Property, tion (2) Fund a massive public works program Robert E. Huff, Pennsylvania state University; which will use the skills of the presently and soon Topological and Chromatic Graph Theory, Paul to-be unemployed (including mathematicians) for C. Kainen, Case Western Reserve University; sorely needed socially useful tasks (3) Transfer Geometry of Metric Spaces, Leroy M. Kelly, massive funds from the military budget to ac Michigan state University; Categorical Methods complish these aims. We call upon the officers in Algebraic Topology, Pierre J. Malraison,Jr., of the Society to work towards effecting the im Carleton College; Riemannian Geometry, Tilla K. plementation of the above. Milnor, Rutgers University; Theoretical Com Motion II. Resolved that the AMS will not puter Science, David E. Muller, University of cooperate with UNESCO until such time as the illinois; General Topology, Peter J. Nyikos, ruling removing Israel from any regional group University of illinois; Matrix Theory and Numeri ing is rescinded. cal Ranges, Hans Schneider, University of Wis Motion ill. That the officers of the Society consin; Efficient Algorithms for Exact Computa be requested to arrange that an early meeting of tion, Peter J. Weinberger, University of Michi the Society schedule an open session intended to gan. Most of the papers presented at these four discuss the possible establishment of an indepen teen sessions will be by invitation. dent ASSOCIATION OF MATHEMATICIANS FOR Contributors of abstracts for the meeting SOCIAL ACTION. who feel that their papers would be particularly This announcement, with its subsequent re appropriate for one of these special sessions appearance, constitutes the notice to the full should indicate this conspicuously on the abstract membership required in the bylaws. At the meet and submit it by May 27, 1975, three weeks ear ing in August, each motion is subject to substan lier than the above deadline, in order to allow tive changes, such as germane amendment or sub time for the additional handling necessary. stitution, and is subject to subsidiary resolutions POSTER SESSION concerning the disposition of the main motion. There will be a Poster Session for contri MEETING PREREGISTRATION AND buted papers from 11:00 a.m. to 1:00 p.m. on REGISTRATION Wednesday; this represents an alternative method Registration for the short course ~will for presenting papers. At the poster session in begin on Friday, August 15. General meeting re dividuals will display their papers either on an gistration will commence on Sunday, August 17, easel or a bulletin board, and remain in the room at 2:00 p.m. Participants who are not attending set aside for this purpose to expand on the ma the short course are advised that no general terial and answer questions during the two-hour meeting information (or registration material) session. Individuals who wish their papers con will be available prior to the time listed below sidered for poster sessions should so indicate on for Joint Mathematics Meetings registration. their abstracts, clearly in large block letters. Following are the hours that the desk will COUNCIL AND BUSINESS MEETING be open as well as the respective locations: The Council of the Society will meet at Applied Combinatorics Short Course 5:00p.m. on Tuesday, August 19, in the Univer Date and Time Location sity Student Center. Friday, August 15 The Business Meeting of the Society will be 4:30 p.m.-7:30p.m. Goldsworth Valley #3 held in Miller Auditorium at 4:00p.m. on Tlmrs Saturday, August 16 day, August 21. The secretary notes the follow 8:00 a.m.-4:00p.m. Rood Hall Lobby ing resolntion of the Council: Each person who Sunday, August 17 attends a Business Meeting of the Society shall 11:00 a.m. -1:00 p.m. Rood Hall Lobby be willing and able to identify himself as a mem ber of the Society. In further explanation, it is Joint Mathematics Meetings noted that "each person who is to vote at a meet Date and Time Location ing is thereby identifying himself as and claim Sunday, August 17 ing to be a member of the American Mathemati,-. 2:00 p.m.-8:00p.m. Miller Auditorium Lobby cal Society." Monday, August 18 In accord with Article X, Section 1, of the 8:00 a.m.-5:00p.m. MillerAuditoriumLobby bylaws of the Society, the Business Meeting of Tuesday, August 19 to January 24, 1975, in Washington, D. C. has di Tlmrsday, August 21 rected that each of the following motions be 8:30 a.m.-4:30p.m. MillerAuditoriumLobby placed on the agenda of the Business Meeting of Friday, August 22 August 21, 1975, in Kalamazoo: 8:30 a.m. -1:30 p.m. Miller Auditorium Lobby Motion I. In the face of the deepening eco nomic crisis, the rapidly rising unemployment Participants who wish to preregister for the from which mathematicians are not exempt, the meetings should complete the Meeting Preregis ominous nature of massive budget cuts for edu tration Form on the last page of these cJ{ofit:rA). cation and other social services, and the Presi Those who preregister will pay a lower registra dential request for an immediate increase of 300 tion fee than those who register at the meeting, millions for military aid to the Thieu regime as indicated in the schedule on the next page.

143 Pre:t:egistrants will be able to pick up their bad applicants and employers to display resumes and ges and programs when they arrive at the meet listings. Message boxes will be set up for indi ing, Complete instructions on procedures for viduals to leave messages for one another re making hotel, motel, or dormitory reservations questing interviews. Tables and chairs will be are given in the sections entitled RESIDENCE provided in the room for interviews. Employers HALL HOUSING and HOTELS AND MOTELS, are encouraged to attend the meetings and parti Please note that a separate registration fee cipate if possible. Applicants should recognize is required for each of the meetings, These fees that the Mathematical Sciences Employment are as follows: Register cannot guarantee that any employers will in fact attend the meeting ApPlied Combinatorics Short Course or be able to participate. The AMS-MAA-SIAM Committee on Preregistration At Employment Opportunities will, however, re (by mail prior to 8/1) meeting quest employers listing in the July 1975 issue of All participants $12 $15 Employment Information for Mathematicians to signify Joint Mathematics Meetings in their listing their intention of parti cipating in the open register at the summer meet Preregistration At ing. (bymailprior to 8/1) meeting EXHffiiTS Member $10 $12 student or unemployed The book and educational media exhibits will be displayed on the second lobby level of member $ 1 $ 2 Miller Auditorium at Nonoonember $16 $20 the following times: August 18 (Monday), noon to 4:30p.m.; August 19-20 There will be no extra charge for members of (Tuesday and Wednesday), 8:30 a.m. to 4:30 the families of registered participants except that p.m.; and August 21 (Thursday), 8:30a.m. to all professional mathematicians who wish to at noon. All participants are encouraged to visit the tend sessions must register independently. exhibits sometime during the meeting. The unemployed status refers to any mem RESIDENCE ber currently unemployed and actively seeking HALL HOUSING employment. It is not intended to include mem The Goldsworth Valley Residence Hall Com bers who have voluntarily resigned or retired plex has been set aside for the exclusive use of from their latest position. Students are consid the Mathematics Meetings participants and for ered to be only those currently working toward a participants of the Applied Combinatorics Short degree who do not receive an annual compensation Course. The dormitories are within a ten to fif totaling more than $7, 000 from employment, fel teen minute walk from the Miller Auditorium and lowships, and scholarships. the meeting rooms which will be used during the Checks for the preregistration fee(s) should be meetings. Frequent shuttle bus service will be provided between the mailed to arrive not later than August 1, 1975, dormitories and the meeting for the meetings. Participants may make their area. There is a twenty-five cent ($0. 25) fare for each one-way own reservations directly with any hotel or trip. The cash fare will be collected by the bus motel in the area if they wish. It is essential, driver; it would be appreciated if exact fares are paid however, to complete the Meeting Preregistra by participants to avoid the neces sity of having tion Form on the last page of these cNOtiuiJ to drivers carry excess money. take advantage of the lower meeting registra Most of the sleeping rooms are in suites of tion fee(s), two double rooms which share a bath. Linens, towels, and daily maid service (beds made only) A fifty percent refund of the preregistration are provided with all rooms. Each room will fee will be reimbursed for all cancellations re contain the following furniture: two twin beds, a ceived in Providence prior to August 16. There chest of drawers, a lounge chair, a desk, and will be no refunds granted for cancellations re two study chairs. Keys, curtains, glasses, and ceived after that date or to persons who do not soap are also provided. Each dormitory has a attend the meetings, fully equipped laundry room with coin-operated MATHEMATICAL SCIENCES washers and dryers. Ironing facilities are also EMPLOYMENTREGmTER available. A limited number of irons are avail able at dormitory desks. An experimental variant of the open register Residence hall rooms can be occupied from will be operated on a limited basis during the noon on Friday, August 15, to noon on Sunday, meeting, providing an opportunity for applicants August 17, for participants in the Applied Com and employers to arrange interviews at their binatorics Short Course; and from noon on Sun mutual convenience. day, August 17, to noon on Saturday, August 23, It is anticipated that the register headquar for Joint Meeting participants. Clerks will be ters, located in a room in or near the Miller available on call in each housing unit twenty-four Auditorium complex, will be open on Tuesday, hours a day. Parking for residents will be avail Wednesday, and Tlm.rsday (August 19 through 21) able free of charge in lots near the dormitories. from 8:30a.m. to 4:30p.m. The room will be The daily rate per person is as follows: closed from noon to 1:15 p.m. The exact location Singles will be announced in a future issue of these $8.00 per person per day c}(ofictJ]. Doubles $5. 75 per person per day · There will be no interviews scheduled by Children will be housed at the regular rates in the staff. Instead, facilities will be provided for rooms adjacent to their parents. Cribs and cots

144 are not available from the university and sleeping HOLIDAY INN- EXPRESSWAY (616) 381-7070 bags are not permitted. An infant may occupy the 3522 Sprinkle Road (Use Exit 80 at I-94, tum parents• room at no extra charge if the parents south). supply a crib and bedding. A limited number of 146 rooms cribs are available for rent by writing in advance Singles 1 Bed 1 Person $16.00 to United Rent-AU, 403 Balch street, Kalamazoo, 1 Bed 2 Persons 20.00 Michigan 49003. Pets are not permitted in the Doubles 2 Beds 2 Persons 21.00 dormitories. 2 Beds 3 Persons 25.00 To be assured of a room, guests should Twins 2 Beds 4 Persons 29.00 register in advance. Please use the Room Reser Suites 1 Person 20.00 vation Form provided on the last page of these (two available only) cNotitxiJ. Residence hall reservation requests 2 Persons 24.00 will be acknowledged by the University. Code: FP-SP-AC-TV-CL-RT HOTELS AND MOTELS 7 miles from campus HOLIDAY There are a number of hotels and motels in INN- WEST (616) 375-6000 2747 - 11th Street (Use the area which are listed below. Rates and other Oshtemo Exit at U. S.131, turn west) information are presently available for the first 118 rooms six listed only. Further information on the others Singles 1 person listed will appear in the· June issue of these $17.00 cNotitxiJ. All prices are subject to change without 2 persons 21.00 Doubles notice; a six percent tax should be added to the 1 person 17.00 2 persons room rates listed. It should be noted that" this 22.00 Twins area is a popular vacation resort area so that Same as a double Suites early reservations are recommended. Partici 1 person 18.00 pants should make their own reservations. The 2 persons 23.00 Extra person following codes apply: FP - Free Parking; SP - in room 4.00 Code: FP-SP-AC-TV-CL-RT Swimming Pool; AC- Air Conditioned; TV- Tele vision; CL - Cocktail Lounge; RT -Restaurant. 3 miles from campus

NORTH to Grand Rapids

MOSEL •"'.. GU~L~L;~ •.. M·89 >0: ..> i<

I M • 43 WEST MAIN

EAST to Detroit KILGORE

INTERSTATE RTE. 94

WEST to Chicago ai SOUTH g .. .. z.. to Indiana .." -' ...... z •..

145 KALAMAZOO CENTER INN (616) 381-2130 dinner on that date. The Cafeteria will continue 100 West Michigan (Downtown Kalamazoo) serving through lunch on Friday, August 22, 288 rooms with the exception of the evening meal on Wednes Singles $22,00 day, August 20. Hours of service and prices for Doubles 30,00 individual meals are: SUites 30.00-76.00 Hours Adults Children Breakfast Extra person in room 6.00 (12 years and older) 7:15a.m.- 8:30a.m. $2.00 $1.50 Code: FP-SP-AC-TV-CL-RT Lunch 2 miles from campus 11:30 a.m.-12:45 p.m. 2.50 2.00 Dinner HOWARD JOHNSON'S (616) 382-2303 5:00p.m.- 6:30p.m. 4.50 3.25 1912 East Kilgore (Exit 78 off 1-94; turn south) A package plan including all meals served 70 rooms from breakfast on Monday, August 18, to lunch Singles $15.00 on Friday, August 22, (with the exception of the Doubles 19.00 evening meal on Wednesday, August 20) will be Twins 1 person 16,00 available at a price of $32 for adults and $24. 75 2 persons 21.50 for children under 12 years of age. Extra person in room 4.00 Light snacks and beverages will be avail Code: FP-SP-AC-TV-CL-RT able in the University Student Center; the hours 6 miles from campus of operation will appear in the June issue of these cNotiaiJ. RAMADA INN (616) 382-1000 PARKING 5300 S. Westnedge (Exit 76 off 1-94; turn north) 102 rooms Parking permits will 'be required for park Singles $15,50 ing on all areas of the campus, with the excep Doubles 18.50 tion of metered lots. Parking will be free of Twins 20,50 charge in the lots near the dormitories for per SUites (one available only) 24,50 sons residing there, and in the lot near Miller Extra person in room 3.00 Auditorium. Maps showing the location of the Code: FP-SP-AC-TV-CL-RT various college parking lots will be available at 5 miles from campus the registration desk along with the permits.

VALLEY INN MOTEL (616) 349-9736 CAMPING 200 N. Park (Downtown Kalamazoo) The following campgrounds are located one 107 rooms half hour or less from Western Michigan Univer Singles $14.50 sity: Doubles 20,00 1, KUnes Resort, Route 2, Box 257, Three Twins 20.00 Rivers, 49093 SUites $25. 00 and 30.00 Telephone: 616-649-2514 Extra person in room 4,00 Special group rates available Travel Time: 35 minutes from W. M. U. Code: FP-SP-AC-TV-CL-RT Facilities: Electric hook-ups, bathrooms, 2 miles from campus lakeside swimming, boat rentals, 40 sites Cost: $4.00 per night, $24.00 per week, HOLIDAY INN - CROSSTOWN (616) 349-6711 reservations accepted 220 E. Crosstown 2. Oak Shores Resort, Route 3, 28th Street, Vicksburg, 49007 SOUTHGATE MOTOR INN (616) 343-6143 Telephone: 616-649-1310 5630 S, Westnedge Travel Time: 30-25 minutes from W. M. U. Facilities: 80 acre lake, swimming, boating, Y -MASTER MOTEL (616) 345-8603 water and electricity at sites, also sewer 2333 Helen hook-up available, club area, 93 sites avail able KALAMAZOO MOTOR INN (Telephone number Cost: Water and electricity $4. 00/day, not available) sewer hook-up $4. 50/day 5882 Stadium Drive 3. Schnable Lake Campground, 11th Street, RED ROOF INN (616) 382-6350 Martin, 49070 3701 E. Cork Telephone: 616-672-7524 Travel Time: 30 minutes from W.M. U. KALAMAZOO TRA VELODGE (616) 349-8711 Facilities: 45 acre lake, swimming, canoeing, 1211 S. Westnedge water and electricity are provided, 93 camp sites Cost: $4.00 per night, deposits required FOOD SERVICES 4. Shady Bend Park, 15320 Augusta Drive, The Goldsworth Valley #3 Residence Hall Augusta, 49012 Cafeteria will be open starting with breakfast on Telephone: 616-731-4503 Saturday, August 16. Participants in the short Travel Time: 25 minutes from W. M. U. course arriving on Friday, August 15, will have Facilities: The pavillion houses showers and to make private arrangements for lunch and/or toilets, spring fed pond, canoeing, 62 sites 146 available On Wednesday, August 20, there will b~ a Cost: $3. 50 per night picnic at 6:00p.m. in Goldsworth Valle~, adJa cent to the residence halls. The menu will be 5. Willow Lake Campground, Box 295, Three barbecued chicken and baked ham; no alcoholic Rivers, 49093 beverages will be served. The cost will be about Telephone: (Not available) $5. Travel Time: 20 minutes from W. M. U. A beer party is being arranged to follow the Facilities: Tent camping, water, campfires, picnic from 9:00p.m. to 12:00 midnight toilet facilities, private fishing lake on 110 at the Holiday Inn-West. Soft drinks in cans will also acres, 51 sites available be available for Cost: $3. 00 per day those who desire them. Potato chips and similar snacks will be served. The BOOKSTORES estimated cost is $2 per person. Bus service to and from the Holiday Inn will be available; the The Campus Bookstore is located in the fare will be $0. 25 each way. University student Center. Its hours of operation Tickets to both the picnic and beer party are from 8:00a.m. to 5:00p.m., Monday through will be sold in advance at the Meetings Registra Friday. The University Bookstore (private) at tion Desk, but can also be purchased 2529 W. Michigan is open from 9:00a.m. to at each event. 5:00p.m. Monday through Friday, and from A block of seats will be reserved 10:00 a.m. to 3:00p.m. on Saturday. The Book for the Thursday evening (August 21) performance Raft (private) at 2624 W. Michigan is open from of the nearby Augusta Barn Summer stock 10:00 a.m. to 7:30p.m. Monday through Thurs Theatre. (The title of the play has not yet been announced). day, fromlO:OOa.m. to 9:30p.m. on Friday and Reservations to attend the performance can be Saturday, and from 10:00 a.m. to 5:00p.m. on made in the Mathematics Meetings registration Sunday. The latter two are also situated near the area campus area. on Monday and Tuesday, August 18 and 19. Several tours of local business concerns LIBRARIES are being planned (e. g. Kellogg Cereals, Upjohn Pharmaceutical, The mathematics library, including current wineries, etc.), and a visit to the Kalamazoo mathematical journals and books, is part of the Nature Center. Further informa tion on these Physical Sciences Library on the third floor. of and other events being planned will appear in the June issue of these Rood Hall, and is open from 8:00a.m. to mid cNoli£eiJ. There will be daily supervised arts night Monday through Thursday, from 8:00 a.m. and crafts to 5:00p.m. on Friday, from 10:00 a.m. to for elementary age children at the dormi tory complex. clas~ 5:00p.m. on Saturday, and from 1:00 p.m. to Information about the hours and location midnight on Sunday. Information can also be ob will be available at the Meetings Registration tained in the mathematics library regarding the Desk. location of books in other areas. The main col There are also several commercial day lection of other books is in the Waldo Library, care centers in Kalamazoo. Listed below are two which which is open from 8:00a.m. to 11:00 p.m., will be in operation during the meetings. They Monday through Thursday, 8:00 a.m. to 5:00 will take children on a short-term basis. Interested parties should write directly p.m. on Friday, 10:00 a.m. to 5:00p.m. on for fur ther information Saturday, and from 1:00 p.m. to 11:00 p.m. on and/or registration forms. Sunday. Child Development Center - W. M. U. 1401 Cherry Street The Kalamazoo Public Library has its main branch located at 315 S. Rose. The hours of opera Kalamazoo, Michigan 49008 Telephone: (616) 383-4076 tion are from 9:00a.m. to 9:00p.m. Monday through Friday, from 9:00a.m. to 6:00p.m. on Will take children two and one-half years to five years Saturday, and from 2:00p.m. to 6:00p.m. on of age Sunday. Hours of operation: 6:30a.m. to 6:30p.m. MEDICAL SERVICES Michigan Young World 110 W. Cork Street Kalamazoo is serviced by Borgess Hospital Kalamazoo, Michigan 49001 and Bronson Methodist Hospital. The emergency Telephone: (616) 349-2445 rooms there are staffed around the clock. The Will take children two and one-half years to Kalamazoo Academy of Medicine can also make eight years of age referrals Monday through Friday, from 8:00a.m. Hours of operation: 6:30a.m. to 6:00p.m. to noon, and from 1:00 p.m. to 5:00p.m. (Tele phone: 342-8502). Referrals to dentists can be TRAVEL AND LOCAL INFORMATION made by calling 381-0400 during usual office Kalamazoo is situated on two major ex hours. For dental emergency service, the dentist pressways, I-94 (east-west), and US-131 (north on call may be reached through the Bronson south), and is approximately halfway between Methodist Hospital. Additional information will be Chicago and Detroit. It is also serviced by North available at the Mathematics Meetings Registra Central Airlines, Greyhound and Indian Trails tion Desk. Bus Lines, and Amtrak. Car rentals from A vis, ENTERTAINMENT Hertz and National are available at the airport, but prior reservations are advisable. The air Western Michigan University is planning a port is within the city limits, and regular cab program of recreation and entertainment for service is available for transportation between mathematicians and their families. the airport and campus. An information desk will

147 be maintained at the airport at the times of the A free copy of the Official Michigan Transporta most frequently used incoming flights to assist tion (Highway) Map can be obtained by writing arrivals. (During the summer, Michigan is on the State Highway Commission, Lansing, Michi Eastern Daylight Time). gan 48926. The second largest industry in Michigan is vacationing and tourism. If you are interested in a vacation in Michigan you may obtain free vaca WEATHER tion information by writing to: Michigan Tourist The normal daytime high temperature dur Council, Suite 102, 300 s. Capitol Avenue, Lan ing this period is 84° F. Normal nighttime low sing, Michigan 48926. More specific information is 60o F. Rainfall in August averages 2. 78 in is available free from the four Regional Tourist ches, with a twenty percent to thirty percent Centers, indicated on the map below. probability of precipitation each day. Humidity 1. Michigan's Upper Peninsula Travel andRe normally ranges from a daytime high of eighty creation Association two percent to a nighttime low of fifty-five per P. 0. Box 400 cent. The record high and low temperatures for Iron Mountain, Michigan 49801 August are 101° F and 41° F respectively. Light jackets and sweaters are advised for evening 2. West Michigan Tourist Association wear. 136 Fulton street East Grand Rapids, Michigan 49502 MAIL AND MESSAGE CENTER 3. East Michigan Tourist Association The Log Office All mail and telegrams for persons attend Bay City, Michigan 48706 ing the meetings should be addressed in care of Mathematics Meetings, Western Michigan Uni 4, Southeast Michigan Travel and Tourist Asso versity, Kalamazoo, Michigan 49003. Mail and ciation telegrams so addressed may be picked up at the 1200 6th Avenue Mail and Information Desk located at the regis Detroit, Michigan 48228 tration area in the lobby area of Miller Auditori- urn. A message center will be located in the same area to receive incoming calls for regis trants during the hours the registration desk is open, cf. the section entitled MEETING PRE REGISTRATION AND REGISTRATION, on a previous page. Messages will be taken down, and the name of any member for whom a mes sage has been received will be posted until the message is picked up at the Message Center. The telephone number of the Message Center will be listed in a later issue of these cNotiaJJ.

LOCAL ARRANGEMENTS COMMITTEE Youse£ Alavi (chairman), Paul T. Bate man (ex officio), Jean M. Calloway, Gary Chart rand, A. Bruce Clarke, Florence M. Clarke, S. F. Kapoor, Don R. Lick, John W. Petro, James H. Powell, David P. Roselle (ex officio), Gordon L. Walker (ex officio), and Alden H. Wright.

148 SUMMARY OF ACTIVITIES

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

AMERICAN MATHEMATICAL SOCIETY

FRIDAY, August 15 SHORT COURSE ON APPLICATIONS OF COMBINATORICS

4:30p.m. - 7:30p.m. REGISTRATION (Short Course Only)

SATURDAY, August 16 8:00a.m. - 4:00p.m. REGISTRATION (Short Course Only) morning and afternoon Short Course

SUNDAY, August 17 11:00 a.m. - 1:00 p.m. REGISTRATION (Short Course Only) afternoon Short Course

AMS - MAA SUMMER MEETINGS

SUNDAY, August 17 American Mathematical Society Other Organizations 9:00a.m. - 4:00p.m. Mathematical Association of America.Board o·f Governors 2:00p.m. - 8:00p.m. REGISTRATION evening I MAA Film Program MONDAY, August 18 8:00a.m. - 5:00p.m. REGISTRATION 9:00a.m. - 9:15a.m. John Bernhard, President Western Michigan University Welcome Address 9:15a.m. - 10:15 a.m. Frederick J. Almgren, Jr. I MAA Hedrick Lecture I 10:30 a.m. - 11:00 a.m. AMS-MAA Panel Discussion on Accreditation 11:30 a.m. - noon Panel Discussion Continued noon - 4:30p.m. EXHIBITS 1:30 p.m. - 2:30p.m. Frederick J. Almgren, Jr. MAA Hedrick Lecture II 3:00p.m. - 4:00p.m. MAA Invited Address I 4:30p.m. - 5:30p.m. AMS Committee on Employment and Educational Policy. Open Meeting on the State of Job Market evening MAA Film Program

TUESDAY, August 19

8:30a.m. - 4:30p.m. REGISTRATION 8:30a.m. - 4:30p.m. EXHIBITS 9:00a.m. - 10:00 a.m. Frederick W. Almgren, Jr. MAA Hedrick Lecture III 10:15 a.m. - 11:15 a.m. MAA Business Meeting 11:30 a.m. - 12:15 p.m. MAA Invited Address II noon Pi Mu Epsilon Luncheon 1:00 p.m. - 2:00p.m. E. R. Kolchin, Colloquium Lecture I 2:15p.m. - 3:15p.m. Elias M. Stein, Colloquium Lecture I 3:30p.m. - 5:00p.m. Association for Women in Mathematics Panel Discussion and Business Meeting 5:00p.m. Council Meeting 6:30p.m. liME Banquet 8:00p.m. liME Invited Lectures evening MAA Film Program

149 American Mathematical Society other Organizations

WEDNESDAY, August 20

8:00a.m. TIME Dutch Treat Breakfast 8:30a.m. - 4:30p.m. REGISTRATION 8:30a.m. - 4:30p.m. EXHIBITS 8:45a.m. - 9:45a.m. E. R. Kolchin Elias M. Stein Colloquium Lectures II 9:00a.m. - noon Contributed Papers 10:00 a.m. Mathematics Action Group Business Meeting 10:00 a.m. - noon TIME Contributed Papers 1:30 p.m. - 2:30p.m. MAA Invited Address III 2:45p.m. - 3:45p.m. MAA Panel Discussion 3:45p.m. - 4:15p.m. Panel Discussion Continued 4:30p.m. - 6:00p.m. MAG Panel Discussion on Employment Crisis 6:00p.m. Picnic 9:00p.m. - 12:00 Beer Party

THURSDAY, August 21 8:30a.m. - 4:30p.m. REGISTRATION 8:30a.m. - noon EXHIBITS 8:45a.m. - 9:45a.m. E. R. Kolchin Elias M. Stein Colloquium Lectures III 11:00 a.m. - 1:00 p.m. Poster Session morning and afternoon Contributed Papers 4:00p.m. Business Meeting 8:00p.m. AMS Committee on Employment and Educational Policy Panel Discussion on Applications in Ph. D. Programs

FRIDAY, August 22 8:30a.m. - 1:30 p.m. REGISTRATION 8:45a.m. - 9:45a.m. E. R. Kolchin Elias M, Stein Colloquium Lectures IV morning and afternoon Contributed Papers

Paul T. Bateman Associate Secretary Urbana, Illinois

150 ORGANIZERS AND TOPICS OF SPECIAL SESSIONS Abstracts of contributed papers to be oonsidered 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. Deadline st. Louis, Missouri, April1975 Expired David Drasin, Classical Function Theory David L. Elliott, Differential Geometric Problems in Control Theory Franklin Raimo, Applications of Ring Theory to Groups Richard P. Jerrard, Geometric Topology Rangachary Kannan, Nonlinear Functional Analysis Walter Leighton, Ordinary Differential Equations: Oscillation Theory, Boundary Value Problems Marian Boykan Pour-El, Recursion Theory Grant V. Weiland, Harmonic Analysis and Related Topics David J. Winter, Finite Dimensional Field Extensions

Kalamazoo, Michigan, August 1975 May 27, 1975 Janos D. Aczel, Functional Equations Amassa Fauntleroy and Andy R. Magid, Affine Algebraic Groups George Fix, Scientific Computing Casper Goffrnan, Aspects of Real Analysis W. Charles Holland, Ordered Groups Robert E. Huff, Banach Spaces with the Radon-Nikodym Property Paul C. Kainen, Topological and Chromatic Graph Theory Leroy M. Kelly, Geometry of Metric Spaces Pierre J. Malraison, Categorical Methods in Algebraic Topology Tilla K. Milnor, Riemannian Geometry David E. Muller, Theoretical Computer Science Peter J. Nyikos, General Topology Hans Schneider, Matrix Theory and Numerical Ranges Peter J. Weinberger, Efficient Algorithms for Exact Computation

Chicago, lllinois, November 1975 August 12, 1975 Saunders Mac Lane, Category Theory INVITED SPEAKERS AT AMS MEETINGS

This section of these cJ{oticeL) 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 in complete. st. Louis, Missouri, April 1975 A. 0. L. Atkin Kenneth Kunen K. T. Chen Guido L. Weiss Monterey, California, April1975 Isaac Namioka Kennan T. Smith Pullman, Washington, June 1975 David W. Barnette Theodore E. Harris Kalamazoo, Michigan, August 1975 E. R. Kolchin Elias M. Stein (Colloquium Lecturer) (Colloquium Lecturer) Chicago, lllinois, November 1975 Jonathan L. Alperin R. 0. Wells, Jr.

151 NONACADEMIC EMPLOYMENT OF PH.D.'s IN THE MATHEMATICAL SCIENCES by Wendell H. Fleming

A recent NAS-NRC study showed that, of retirements, and resignations. Thus, it appears about 16,000 Ph.D. 1S in the U.S. identified as that an increasing number of Ph. D. 1s, trained mathematicians, roughly 20% were employed by in both pure mathematics and applied fields, will nonacademic organizations (see August 1974 seek nonacademic employment. c/{olicfi), pp. 212-214). The percentage is slight Manpowe1' projections by the NSF and the ly higher for recent Ph. D. 1s. Other data col Bureau of Labor Statistics predict for coming lected in the Annual AMS SUrvey for 1974 indi years a steady increase in the nonacademic em cate that roughly 300 mathematical science ployment of physical scientists, engineers, and Ph. D. 1s found nonacademic positions last year. mathematicians. The increase is seen to be This includes both new Ph. D. 1s and those pre partly attributable to more openings in traditional viously holding academic positions. The majori R and D types of work and partly to new kinds ty, but by no means all, were trained in an ap of jobs outside traditional R and D. Another plied field (operations research, statistics, com consideration is the gradual enrichment of the puter science, physical applications, etc.) work force as Ph. D. 1s undertake jobs previously Among all mathematical science Ph.D. 1S held by others. holding nonacademic employment in the U.S. , Obviously predictions are subject to uncer roughly 63 % are employed by business, 27% by tainties about the future state of economy and the government agencies, and 10% by nonprofit or pattern of federal research funding. Another un ganizations. Of new mathematical science Ph. D. 1s certainty in predicting the future number of non recently employed by business, around 60% were academic jobs for mathematical science Ph. D. 1s hired by larger companies in the areas of aero is the extent to which employers accept the idea space, defense, electrical manufacturing, com of hiring them. This will be conditioned by the munications, and data processing. Smaller com performance of mathematicians recently, or panies providing consulting, engineering, or soon to be, hired by nonacademic organizations. computer services employed another 20%. The It will also be affected by the success individual remaining 20% of new Ph. D. 1 s employed by mathematicians seeking jobs have in selling business were divided among the automotive, themselves to prospective employers. petroleum, and pharmaceutical industries, in At the time of writing, the current econom surance companies, and other organizations. The ic recession seems to have had a mixed effect on geographic distribution of nonacademic jobs is the market for scientific manpower. Some or uneven. Over 65% of nonacademic positions re ganizations which have traditionally employed cently obtained by new Ph.D. 1s were located in significant numbers in the mathematical sciences the Northeast, mid-Atlantic states, or Califor- are in a period of retrenchment. On the other nia. hand, estimates of hiring by U.S. Federal The Committee on Employment and Educa Government agencies for the current fiscal year tional Policy is interested in learning of nontra 1975 in mathematical science categories are ditional kinds of employment of mathematicians. somewhat higher than the corresponding esti There are a number of health-related positions, mates for fiscal 197 4. These estimates, broken often filled by statisticians. Some mathematical down by Federal agency, are contained in an science Ph. D. 1 s find university positions out article by Thomas Kramer expected to appear side mathematical science departments, either in Employment Information for Mathematicians in other academic departments or, for example, (April 1975). in the computing laboratory. We have learned of Future growth of opportunities in the a mathematician doing operations research-type mathematical sciences is likely to be selective. work for a guitar factory and of another employed Probable areas of growth include computing, by a winery. operations research, and energy-related or Future prospects of employment health care-related research. Most of those hired may expect to do applied work directly re The status of the academic job market was re lated to current projects. Openings in large ported by R. D. Anderson in these c/{otiQ.i) , Novem corporate laboratories devoted to long-range ber 1974, pp. 335-340. There is little prospect of basic research will be few. The job applicant significant growth in the total number of four must be prepared to convince the employer that year college and university teaching positions. he can fill a specific need of the organization. However, there continues to be some increase in For some jobs, preference will be given to those the number of faculty who are tenured or expected whose background identifies them with a parti to be retained indefinitely, with a corresponding cular field of application. shrinkage in the number of nontenured positions. We must expect soon to reach a situation in which Where job seekers can seek practical advice the number of tenured openings is no more than In addition to sources usually available on the number of replacements due to deaths, campus (faculty and the placement office), the

152 following publications of the AMS and other or tial nonacademic job opportunities? ganizations may be consulted. The pamphlet, (b) What role do you see for the U. S. Government, math Seeking Employment in the Mathematical Sciences, ematical organizations, or other agencies in this regard? may be obtained by writing the AMS office in (c) What barriers do you see to the increased nonacadem Providence. Emplo~nt Information for Mathe maticians lists a sm but growing number of ic employment of Ph. D. mathematicians? nonacademic positions. It also publishes from In responding to these questions, concern time to time articles dealing with various as was repeated]y expressed about the set of atti pects of the job market. Individuals who recently tudes and interests with which young mathemati found nonacademic employment are often an ex cians leave graduate school. One respondent cellent source of advice. A series of Case stud concluded that "there is no match. " Attitudes of ies written by such individuals is appearing in young people tend to be inherited from their pro these cJ{otica) (see November 1974, pp. 346-348 fessors. Mathematicians have sometimes indi and February 1975, pp. 100-102). The Case stud cated that work in industry is unsuitable (or even ies of the speakers at the panel discussion, demeaning) for their better Ph. D. •s. A role for "Seeking employment outside academia: Views of the AMS is seen in helping to modify such atti some who have recently succeeded," held at the tudes among graduate mathematics faculty, with recent Washington, D. C. meeting of the AMS, the idea that most Ph. D. -producing departments may be of particular interest in this regard. may expect some of their graduates to seek non These Case Studies are expected to appear in the academic careers. June 1975 cJ{otica). The Geographic Listing in the The following are excerpts from responses Combined Membership List, 1974-1975, lists to Question 1. The respondents have been indi organizations which employ two or more members cated by A, B, •.• rather than by name. of the AMS, MAA, or SIAM. A list of recent A. "The AMS cannot help by promoting the nonacademic employers of new mathematical hope that somewhere they can ply the mathe science Ph. D. •s appears in the February 1975 maticians• trade-only under slightly different issue of Employment Information for Mathemati circumstances. There are such jobs, but they cians. The Mathematical Sciences Administra have been filled long ago by specialist Ph. D. •s tiVebirectory lists heads of nonacademic re who now know the subject area as well as the search groups as well as academic department mathematical research tools. The AMS can pro chairmen. other sources which might prove use mote the true application attitude and ca.ilii-in.g ful include the College Placement Annual and ad in talented industrial leaders to talk about the vertisements in large metropolitan newspapers opportunities and the necessary change of view or trade journals. Finally, the publications listed point-and the pleasures of successful applica at the end of this article are good background tions." reading about the role of mathematicians outside B. "The new Ph. D. is immersed in an arti academe. ficial world; he feels, as indeed he should, that Views of some senior mathematical scientists his results are important; that they are still in complete; and that he, because of his intense con As part of an effort by the Committee on centration in his particular area, is the proper Employment and Educational Policy to encourage person to harvest them. All too frequently he more nonacademic employment of mathematicians, views his industrial job as a method of marking a questionnaire was sent to a number of senior time until he can get back into academia. " mathematical scientists with long experience in C. "There has been a common misconcep nonacademic organizations. The responses were tion among academic mathematicians that in thoughtful and frank. I have tried to summarize dustry is a logical home for their less talented below responses to some key parts of the ques but relatively personable students. (I frequently tionnaire, together with opinions obtained ver hear comments which reflect this attitude during bally from other people. I would like to thank reference checks. Does this carry an implication the following respondents to the questionnaire that personality characteristics are unimportant for their cooperation: J. Harold Ahlberg, John J. in classroom instruction?) Now there are many Anton, Thomas A. Brown, Lester R. Ford, Jr. , uses in industry of mathematics in nondemanding William J. Jameson, Jr., Henry J. Kelley, ways. IJowever, these are not the roles for which Joseph P. LaSalle, George W. Morgenthaler, Ph.D.'s should be produced. A ·Ph.D. is, by Raymond W. Rishel, Daniel H. Wagner. Thanks definition, supposed to be capable of independent are also due Alan J. Goldman and the speakers research and this ability is the dominating value at his Short Course on Mathematics in Operations in research and development activities in indus Research, Washington, D.C., January 21-22, try as well as in academia. " 1975, for thoughts expressed during a lively dis D. "Ph. D. mathematicians tend to be cussion. method-oriented rather than problem-oriented, For purposes of this report, several items and unresponsive to the needs and pressures of from the questionnaire have been grouped into a nonacademic environment. " two long questions. Some items on the question For u.S. Government or other agencies, a naire dealing with data collection and other mat role is seen supporting concentrated educational ters have been omitted. programs in applied mathematics for mathe Question 1. (a) What measures might the AMS Commit maticians trained in other areas, for example, short courses like ones recently given at annual tee on Employment and Educational Policy advocate toward AMS meetings. Another role is the support of dealing with the problem of matching highly trained mathemat internships through which practical experience ical talent to the needs of society, as reflected in actual or poten- in a nonacademic setting could be obtained.

153 Examples are the NSF Faculty Research Parti maticians,' in a very real sense." cipation Program (see news item p .155 ) and the H. "Most nonacademic organizations can SIMS transplant program. A suggestion was made afford to pay only for that education which is that a "Math. Corps" of volunteers in an applied pertinent to their activities. Moreover, normal context might be created. Finally, efforts should ly they refuse to hire Ph. D. •s for B.S. -level be continued to obtain good projections of the de (and paid) jobs." mand for Ph. D.'s in mathematics and related disciplines. Some personal observations by the author Competition with engineers, physicists, I received the Ph. D. in 1951, a year in and economists was mentioned as a barrier to which college teaching jobs were quite scarce. increased employment of mathematical science My first position was a nonacademic one, with Ph.D.'s. Another possible barrier is the rela the RAND Corporation. At that time pure mathe tive scarcity of senior mathematicians in hiring matics Ph. D. •s encountered little competition positions in industry and government agencies. from the few with degrees in applied areas. In This may be partly attributed to the fact that contrast, nearly half of 1974 mathematical sci many mathematicians who entered nonacademic ence Ph. D. •s were classified as applied (see employment soon after WWII were attracted dur November 1974 c}/oficeiJ, p. 335). ing the late 1950's and 1960's back to the (then) Many employers prefer persons with an green pastures of academe. applied degree, other things being equal. How Question 2. (a) To what extent are pure mathematicians ever, a case can (and should) be made for hiring at a disadvantage competing with those in applied fields, in a those trained in pure mathematics. Many per situation where the number of Ph. D. mathematicians seeking sons with such training have made significant nonacademic employment may reach twice the 300 or so who contributions in industry and government. others obtained it last year? will do so in the future, though rarely by proving theorems. The problems of nonacademic organi (b) Would a relatively limited experience in an applied zations are often as challenging as mathematical area (say, through a few graduate-level courses) be of much help research, but they are different. The Ph. D. in in this regard? mathematics is a significant accomplishment. Opinions regarding part (a) of the question Those who earn it must demonstrate both ability varied from "great disadvantage," to "some ad and self-discipline. Mathematicians have been vantage" for pure mathematics Ph.D.'s with a trained to keep absolutely clear the basic issues sincere interest in applications and the right per and assumptions in a problem. This ability is sonal attributes. Regarding (b), most respon extremely important in dealing with complex ap dents felt that such courses are useful to job plied problems, and is often lacking among indi seekers. At the least they help establish the ap viduals with other backgrounds. When a senior plicant's credibility with the prospective employ officer at RAND was once asked why it employed er. Several respondents felt that practical ex mathematicians, he answered "to keep the engi perience working in a nonacademic setting is of neers honest." more value in that regard than formal course The transition into industry or a govern work. ment lab is often difficult for a mathematician. The following are excerpts from responses Jt is not just that he may lack certain applied to Question 2: mathematical tools and has not been involved be E. "Training in the tools that have been fore with real applied problems. He will prob found useful in the past in one's area of applica ably wonder whether he is going to be a mathe tion is certainly desirable, but to a substantial matician in the sense he understood the term in extent such tools are fairly easily acquired by a graduate student days. My answer is "maybe." good mathematician. Give me an individual who However, I would suggest instead to change the can create his own tools to meet the new needs definition and use the term mathematician in the that he encounters. In preparation for the work wide sense. Jt is only by doing so that we shall of our firm, if one had to make a choice between find enough opportunities for productive use of good courses in real analysis and nonlinear pro our best mathematical talent. gramming, for example, I would certainly prefer Other sources of information. Below are the former, even though we seldom use directly listed reports and articles giving insight into the concepts that go beyond the elementary stages of role of mathematicians in industry and govern a real analysis course." ment. The reader may consult issues of these F. "A few graduate-level courses in an ap c}/otiai.J , as well as the American Mathematical plied area would not be as much help as several Monthly, SIAM News, and CBMS Newsletter for months of practical experience working for a other articles which will appear from time to nonacademic employer. Many so-called applied time. These publications also announce short courses are not very representative of how mathe courses, panels, symposia on new areas of ap matics is actually applied in the nonacademic plication, and the like. world." 1. Aspects of Professional Work in the G. "Very few grrups I've seen appear will Mathematical Sciences, by J. P. LaSalle, C.R. ing to train in the mathematics. They will train Phelps, and D. E. Richmond, CBMS Report of in the area of application, i.e., what's been the Survey Committee, vol. III, 1970. done, what are the specific problems of impor 2. The Future of Applied Mathematics, tance. Graduate-level courses are a minimum Proc. Sympos. (Brown University), Quart. for credibility. Realize that most electrical en Appl. Math. 30(1972), 1-125. gineering graduates are now 'Applied Mathe- 3. The Industrial Mathematician Views

154 his Profession, by R. E. Gaskell and M. s. Monthly, Nov. 1975. Klamkin, Amer. Math. Monthly 81(1974), 699- 5. The Mathematical Training of the Non 716. academic Mathematician, Proc. Sympos. (Rens 4. Mathematicians in Operations Research selaer Polytechnic Institute), May 1974. SIAM Consulting, by D. H. Wagner, Amer. Math. Rev. (To appear)

NEWS ITEMS AND ANNOUNCEMENTS

BERTRAND RUSSELL MEMORIAL NSF FACULTY RESEARCH LOGIC CONFERENCE: ESSAY PRIZE PARTICIPATION PROGRAM The Bertrand Russell Memorial Logic Con This program provides opportunities for ference is offering a prize of £200 for a pre college and university teachers to participate dur viously unpublished essay whose length does not ing the summer in the ongoing activities of labo exceed 20,000 words and which examines in de ratories engaged in research on problems of tail some aspect of the relationship between national interest and concern. Opportunities for mathematics and the development of social or summer 1975 are listed in the FaculttiiResearch economic conditions. The essay should be of Participation Directory published by e Division general interest to mathematicians, and prefer of Higher Education in Science, National Science ably to a wider readership, and should include a Foundation, Washington, D.C. 20550. Copies of consideration of current mathematical practice. this Directory have been sent to colleges and Essays may be written in English, Spanish, universities. The Directory may also be obtained French, Japanese, Polish or Russian and should by writing to the above address. A number of op be typewritten. To ensure objective judging, the portunities in computer science, operations re name of the author should not appear on any es search, and other applied mathematical special say, but instead should be given together with ties are listed. Requests for information about the author's address on a separate piece of paper specific projects and application forms should be enclosed with the essay. directed to the participating organizations listed Essays should be submitted by February 1, in the Directory. To be eligible, an individual 1976. Essays and inquiries about the prize should must: (1) Be a full-time faculty member of a U.S. be addressed to: Bertrand Memorial Essay Prize, academic institution; (2) Have had at least five c/o Dr. A. Slomson, School of Mathematics, The years of full-time teaching experience, at the University, Leeds LS2 9JT, England. rank of Instructor or higher; and (3) Hold an ad The organizing committee is: J. L. Bell vanced degree in a scientific discipline. (London School of Economics), M. A. Dickmann (CNRS, Paris), M. Machover (Chelsea College), G. Priest (st. Andrews University), A. B. Slom NEW AMS COMMITTEES son (Leeds University), Y. Suzuki (Sussex Uni versity, and G. M. Wilmers (Manchester Uni Special Committee on Graduate Study. versity). President Lipman Bers has appointed Lewis A. Coburn, Ronald G. Douglas, and Edgar R. Lorch POSTER SESSION to the newly formed Committee on Graduate Study. An innovation at the Summer Meeting in Kalamazoo will be the introduction of a poster Committee on Mathematical Models as session for contributed papers. Individuals will Used in Government Policy Decisions. In the display their papers either on an easel or a transition of presidency, Saunders Mac Lane bulletin board, and remain in the room set aside authorized the formation of the Committee on for this purpose to expand on the material and Mathematical Models as Used in Government answer questions during a two-hour session. In Policy Decisions. Mark Ka.c will serve as Chair dividuals who wish their papers considered for man, with David Gale as Vice Chairman. Its the poster session should so indicate on their ab members are: Harold Grad, Jack Kiefer and stracts. Halsey Royden. The magazine Science in its June 28 , 197 4 issue had a report on poster sessions. They were used at the BiochemistryI Biophysics 19 74 meet Committee on External Membership. As a ings sponsored by the American Society of Bio result of a report given to the Council by the logical Chemists and the Biophysical Society. Committee on Principles of Reciprocity, an Ad The poster session will allow communication Hoc Committee on External Membership has on a personal basis between presentor and those been appointed by President Lipman Bers. Its really interested in the subject of the paper. The members are: Arthur P. Mattuck (Chairman), display can be of graphs, tables, text, or any Tilla Klotz Milnor, and Murray H. Protter. other format that seems applicable.

155 COMBINED MEMBERSHIP LIST 1975 - 1976

CHANGE OF ADDRESS OR STATUS

The AMS computer file of the members of the Society contains several items of information in addition to mailing addresses, information on subscriptions and types of membership. Date of birth and date of election to membership, as well as name of employer and title of position are also in cluded. The last items of information have recently become of considerable interest to AMS com mittees concerned with the state of the profession, especially the employment patterns of mathe maticians. Members who move or change jobs would provide valuable assistance for such studies by making certain that their entries are accurate and current. Post Office Department change of address forms are inadequate, unless they are supplemented by the information requested below.

If there have been any changes in address, institution, or position, and the Society has not been notified, please fill in the appropriate portions of the form below and return it to the American Mathe matical Society, P. 0. Box 6248, Providence, R.I. 02940, rio later than May 31, 1975. Since the CML appears in the fall, changes should reflect the location or position you anticipate will be in effect beginning in September.

Address changes submitted by members throughout the course of the year result in the automatic deletion of existing CML information unless the Society is instructed otherwise. Thus, if no institution is provided either as part of the mailing address, or as additional information, a name will appear in the geographic section under city and state only, and not institution. If an address change is not accompanied by information concerning position, no position will appear in the CML entry. Members are also reminded that names are entered in the CML in the form in which they are provided to the Society.

Members who have not submitted address changes since the appearance of the 1974-1975 CML should consult that edition to check the accuracy of our data. Those who have submitted address changes, but who did not provide new information concerning position, or the name of an institution either as part of the mailing address or as additional information, should fill in the pertinent portions of the form below.

Peel off the c}(otiai) mailing label and place in the space provided. The mailing label contains the member code, and with this information clerks are able to process address or other changes more efficiently. If your mailing label is lost, damaged, or being used for other purposes, please print your complete name, address, and member code in the space provided.

Name______Permanent position=------Permanent institution or business Location of permanent institution ·------or business (city and state only).______Temporary position=------ Temporary institution or business'------ Address for mail ~------

Date the above change in address for mail becomes effective'------

156 NEW AMS PUBLICATIONS MEMOIRS OF THE AMERICAN morphisms of the domain). The method used is MATHEMATICAL SOCIETY the Selberg trace formula and involves the detaUed analysis of the conjugacy classes in the arithmetic EQUIV ARIANT SINGULAR HOMOLOGY AND group, as well as the computation of integrals COHOMOLOGY I by Stlren Illman associated to each conjugacy class. The volume of a fundamental domain for the arithmetic group Number 156 is also obtained by a general method due toR. P. 74 pages; list price $3.10; member price $2.33 Langlands involving Eisenstein series and pre ISBN 0-8218-1856-2 viously applied only in the case of Chevalley To order, please specify MEM0/156 groups. In this Memoir the author constructs an equivariant homology and equivariant cohomology PROCEEDINGS OF THE theory, defined on the category of all G-pairs STEKLOV INSTITUTE and G-maps (G is a topological group), which BOUNDARY both satisfy all seven equivariant Eilenberg VALUE PROBLEMS FOR DIFFER Steenrod axioms ENTIAL EQUATIONS. ill edited by V. P. and have a given covariant and MihaD.ov contravariant, respectively, coefficient system as coefficients. He also establishes some further Number 126 (1973) properties of these equivariant singular homology 256 pages; list price $30. 60; member price and cohomology theories, such as a naturality $22.95 property in the transformation group, transfer ISBN 0-8218-3026-0 homomorphisms, and a cup-product in equivariant To order, please specify STEKL0/126 singular cohomology with coefficients in a com mutative ring coefficient system. This volume contains papers in which the behavior of the solutions of boundary value prob TWO PAPERS: J/-COEXTENSIONS OF MONOIDS, lems in unbounded regions and the Cauchy prob AND THE STRUCTURE OF A BAND OF GROUPS lem for nonstationa:ry equations are investigated. by Jonathan E. Leech The majority of them are devoted to a study of the stabilization of the solutions for t --. oo. Number 157 The titles of the articles follow: "On esti 97 pages; list price $3. 30; member price $2.48 mates of the solutions of boundary value prob ISBN 0-8218-1857-0 lems for a parabolic equation of second order" To order, please specify MEM0/157 by A. K. Guscin, "On the rate of decay of a vor tex in a viscous by This Memoir consists of two papers. In fluid" V. N. Maslennikova, the first paper, the J/-coextension problem for "Asymptotic behavior for large values of time of the solutions mono ids is studied in full generality. This of the second and third exterior is done boundary value by techniques which generalize many of the problems for the wave equation with two spatial methods of group extension theory. The variables" by L. A. Muravei', main "On the stabilization tool in carrying out these generalizations is an of the solution of a boundary value problem for a parabolic equ.atjon" by order isomorphism which exists between the D. sub-J/ congruences on a semigroup and the sub V. Proka, and "A mixed problem with a dis continuous boundary: operator for functors of a certain group valued functor. The a hyperbolic equation" by V. I. Cehlov. paper ends with a discussion of the relationships between the J/-coextension problem and a gen eralized cohomology theory for mono ids. In the TRANSACTIONS OF THE second paper, the structure of bands of groups is MOSCOW MATHEMATICAL studied by means of techniques of the first paper. SOCIETY Volume 27 (1972) THE DIMENSION OF SPACES OF AUTOMORPHIC 272 pages; list price $30. 70; member price FORMS ON A CERTAIN TWO-DIMENSIONAL $23. 03; ISBN 0-8218-1627-6 COMPLEX DOMAIN by Leslie Cohn To order, please specify MOSCOW/27 This translation of Transactions Number of the 158 Moscow Mathematical Society for the Year 96 pages; 1972 list price $3.30; member price $2.48 was prepared jointly by the American Mathemat ISBN 0-8218-1858-9 ical Society and the London Mathematical To order, Society. please specify MEM0/158 The volume contains five papers in differential In this Memoir holomorphic automorphic equations, one in dynamical systems, and one forms on the complex domain of pairs (z, u) in in topology. .:2 such that 2 Im z - lu 12 > 0 are studied. The The titles of the papers follow: "Spectral goal is to determine the explicit formula for the asymptotics of nonsmooth elliptic operators. 1." dimensions of such forms (defined with respect by M.S. Birman and M. Z. Solomjak, "On the to a certain specific arithmetic group of auto- decomposition of a discontinuity for a system of

157 two quasilinear equations" by V. A. Borovikov, functional analysis, one in statistical mechanics, "Vibro-stable differential equations with contin and one in probabilistic potential theory. uous right-hand side" by M.A. Krasnosel• ski1 The titles of the papers follow: "Spectral and A. v. Pokrovskii, "Smooth ergodic flows on asymptotics of nonsmooth elliptic operators. II" surfaces" by A. A. Blohin, "Projection spectra" by M. s: Birman and M. Z. Solomjak, "On the by v I Zaicev "A quasidifferentiation operator plane problem of the motion of a body immersed and bo~dary v~lue problems connected with it" in a fluid" and "On the problem of the steady by v. Ja. Sikirjavyr, and "Ge11eralized functions state oscillations of a fluid layer of variable and differential equations in linear spaces" by depth" by B. R. Varnberg and V. G. Maz•ja, v. I. Averbuh, 0. G. Smoljanov, and S. V. Fomin. "Boundary value problems and the parametrix for systems of ~lliptic pseudodifferential equa Volume 28 (1973) tions" by G. I. Eskin, "Development of a plane 256 pages; list price $37. 50; member price parallel symmetric boundary layer in a sudden $28.13; ISBN 0-8218-1628-4 generation of motion" by V. N. Samohin, "Gen To order, please specify MOSCOW /28 eralized measures or distributions on Hilbert This translation of Transactions of the space" by D. N. Dudin, "Example of a Martin Moscow Mathematical Society for the Year 1973 compactum with a nonnegligible irregular was prepared jointly by the American Mathemat boundary point" by M.G. 1fu.r, "On critical se!_s ical Society and the London Mathematical So of Morse-Smale systems" by V. S. Afraimovic ciety. The volume contains three papers in fluid and L. P. ~il'nikov, and "Limiting equivalence mechanics two papers in partial differential of various ensembles for one-dimensional sta tistical equations, 'one in dynamical systems, one in systems" by A. Haitov.

LETTERS TO THE EDITOR Editor, the cNotiaiJ hers of which one was Mexican-American while the second university had 35 Spanish surnamed On page 308 of the October 1974 issue of the facu1ty members of which again one was Mexican we see that last year in this country _22 . c/V'OtiaiJ American. These other Spanish surnamed faculty Doctorates in Mathematics were awarded to mdi members were not even American citizens and viduals with a Spanish surname, yet only two of consisted entirely of upper-class individuals from these were American citizens. It is unfortunate Central America, South America and Spain. that HEW and others use this category when dis There is no way that these faculty members can cussing issues related to Equal Opportunity and serve as a role model for the Mexican-American Affirmative Action. The merits of Equal Oppor students and the Mexican-American community; tunity and Affirmative Action need not be argued worse yet, they show contempt for our culture here, since both are now federal law. However and have no sympathy for, nor do they understand, the above statistic is yet another example of a our problems. This kind of compliance with problem which as a Mexican-American has been HEW Affirmative Action guidelines is actually a a source of great frustration to me. Namely the giant step backwards for the Mexican-American. injustices which have resulted from the indiscri Of course these universities are playing a game minate lumping of all Spanish surnamed individ and I can in no way fault these Latin American uals either residing or going to school in this aristocrats for allowing the universities to satis country into one category. What does Affirmative fy the letter of the law instead of the spirit of the Action have to do with a foreign aristocrat? law. I have been told by friends inside HEW that Affirmative Action is an effort to correct what we have been describing is a recognized the injustices that our society has committed by technique for allowing universities to "get off denying a particular group its civil rights and the hook" and as such there has been considerable civil liberties. With respect to groups within the pressure exerted against any change, Spanish Surname category which have been eco There are 10 million Mexican-Americans nomically, educationally or socially deprived we. in this country, there are approximately 10 are primarily speaking about the Mexican-Amen Mexican-Americans with the Ph. D. degree in can in the West, the Puerto Rican in the East and mathematics and about 100 Mexican-Americans to a lesser degree the Cuban in the Southeast. with the Ph. D. degree in science. We have little Then why don't we say so? The Ford Foundation or in its graduate fellowship program is perhaps the no representation on policy setting committees only organization which recognizes this distinc either at the governmental level or at the level tion. of the professional society. Our representation in science and particu1arly in mathematics is ridi Jn the Southwest the Mexican-American con stitutes a huge majority of the Spanish Surname culously small. There is considerable evidence to show that foreign students are category, yet we represent an extremely small receiving more benefits from science training part of this category when we restrict our atten than our native born minorities. There is also substantial tion to graduate students, holders of advanced de evi dence to show that while Spanish surnamed citi grees and university faculty members. In the state of Texas the Mexican-American is the zens of foreign countries constitute an extremely small fraction of our Spanish surnamed popula dominant minority comprising nearly 20 per cent tion they constitute the large of the total population. A recent survey disclosed majority of the graduate students and faculty members that at two typical Texas universities, the first at Ameri can universities. The category Spanish Surnamed university had 15 Spanish surnamed faculty mem-

158 as it is presently used by HEW and other organi is great and the supply is small efforts are made zations including our own math society not only to increase the supply. Since the government is distorts the truth but is an extreme detriment to pushing affirmative action programs and the num the Mexican-American in his fight for Equal Op ber of available Blacks and Women is so portunity small, and equitable representation as graduate the government should be called upon to help solve students and faculty members of American uni the problem. Under Title III, there is a provision versities. for faculty development in developing institutions. Richard Tapia Under this provision, an institution may receive federal funds to grant to its faculty members for the purpose of earning an advanced degree. This Editor, the c){oticei) provision has been used by nearly all Black in stitutions Here is my suggestion to help those faculty members without for a way to increase Ph. D. •s to earn one. the number of black mathematicians. Provisions are also made According whereby the faculty member receiving such a to information given in the Octo grant ber, 1974 must return to the granting institution for a issue of c){oticei), of the new doctorates stated in mathematics period of time. conferred during the period July It is 1973-June 1974 not inconceivable that a white institu only 20 were Black. This is only tion could recruit 2% of the total qualified Blacks with Masters number of doctorates conferred degrees, hire during this period. them as instructors, and receive This is not surprising since grants from the federal there is presently a government to finance gradual decline in the num their education. As ber of Blacks majoring with the Black institutions, in mathematics. Affir there could be mative action pressure some arrangements made with on university and college the person involved whereby administrations from the federal he must return to government has the institution when he completes led to many mathematics departments his study. This actively has been done for years by the seeking qualified Blacks for employment. Black colleges and Although there is no reason why it couldn't be done at the the demand for Blacks and Women in white colleges. mathematics is great This is also a method of getting at the present time, the the government involved supply is not. Many mathematics in the solutions to some department of the affirmative action problems. chairpersons follow, to the letter, the affirma The mathematics department chairperson tive action guidelines in filling vacancies. Some has other means of helping Blacks to get the because they have to and others because they necessary sincerely funds for study. These include scholar want to do what is right. Some go be ships and yond these guidelines fellowships granted by the university. to recruit qualified Blacks. And of course there They view the Black Ph. is always the graduate as D. in the department as sistantship or teaching an asset not only where assistantship. Jn many affirmative action is cases, these are already concerned, but also being used by many de in the area of recruitment of partments to train Black students. It is sometimes qualified Blacks. useless to at This is in no way tempt to induce Black students reverse discrimination to come to a nor an attempt to get persons mathematics department with employed who are an all white faculty. not qualified to teach mathematics. The desirability of having a Black It is merely Ph. D. in a possible solution to one of the problems a mathematics department is one thing, but the involved in the implementation of affirmative action availability is another. The Black Ph. pro D. is sought grams. There is a large number of Blacks with after by business, industry and government. He masters degrees in mathematics, teaching or is also sought after by university and college ad holding non-teaching ministrations jobs, who are qualified and for administrative appointments. want to earn But with a Ph. D. in mathematics, but can only 2% of the new doctorates black, not because of financial there just isn•t enough considerations. to go around. This creates There are not many problems a problem for a lot of people. that mathe But there is a pos maticians working together cannot solve. sible solution to this problem. When the demand Louis Dale NEWS ITEMS AND ANNOUNCEMENTS

ABSTRACTS FOR SPECIAL SESSIONS LILLIAN R. CASEY At the Meeting of January 22, 1975 the The Council approved the following resolution. "The Executive Committee of the Council Council records its position that an individual adopted the following resolution in a mail ballot who presents a twenty-minute paper at a meeting dated February 21, 1975: of the Society, either by contributing it or by in The Executive Committee records with vitation, shall present an acceptable abstract sorrow the untimely death on February 20, 1975, for publication in the c){otiai) by the date of the of Lillian R. Casey, head of the section on published deadline for receipt of abstracts for Meeting Arrangements. Mrs. Casey was a tal that meeting. As with contributed ten-minute ented and effective worker in smoothing the paths papers, the Associate Secretary will ordinarily of the officers and members of the Society· For not schedule a twenty-minute paper for which no this and for her friendly and open manner abstract is at hand. " she will be long remembered.

159 SPECIAL MEETINGS INFORMATION CENTER The purpose of this center is to maintain a file on prospective symposia, colloquia, institutes, seminars, special years, meetings of other associations, and to notify the organizers if conflicts in subject matter, dates, or geographical area become apparent. An announcement will be published in these CJVotit:ri) if it contains a call for papers, place, date, subject (when applicable), and speakers; a second announcement will be published only if changes to the original announcement are necessary, or if it appears that additional information should be announced. In general, SMIC announcements of meetings held in the United States and Canada carry only date, title of meeting, place of meeting, speakers (or sometimes general statement on the program), deadline dates for abstracts or contributed papers, and name of person to write for further information. Meetings held outside the North American area may carry slightly more detailed information. Information on the pre-preliminary planning will be stored in the files, and will be available to anyone desiring information on prospective conferences. All communications on special meetings should be sent to the Special M!'~~ngs Information Center of the American Mathematical Society. Deadlines for particular issues of the cNotia,U are the same as the deadlines for abstracts which appear on the inside front cover of each issue.

May 5-7' 1975 June 16-July 11, 1975 SEVENTH ANNUAL ACM SYMPOSIUM ON THEORY OF FOURTEENTH SESSION OF THE SEMINAIRE DE COMPUTING MATHEMATIQUES SUPERIEURES Hotel Plaza, Albuquerque, New Mexico Universite de Montreal, Montreal, Quebec, Canada Program: More than 30 talks in such areas as analysis Program: Numerical methods in applied mathematics. of algorithms, computational complexity, formal languages Sponsors: National Research Council of Canada, the and automata, theory of computation, and theory of pro Department of Education of the Government of Quebec, gramming, and the Universite de Montreal Information: J. W. Carlyle, Department of System Sci Invited speakers: P. Arminjon, G. Birkoff, A. Chorin, ence, 4531 Boelter Hall, University of California, Los P,G, Ciarlet, M. Fortin, R. Glowinski, H,-0, Kreiss, Angeles, California 90024 R. Varga Information: Aubert Daigneault, Department of Mathe May 21-August 8, 1975 matics, Universite de Montreal, c. P, 6128, Montreal, SUMMER COURSE IN COMPLEX ANALYSIS Quebec, Canada H3C 3J7 International Centre for Theoretical Physics, Miramare Trieste, Italy July 7-11, 1975 Program: Geometrical Aspects of the Theory of Functions REGIONAL RESEARCH CONFERENCE ON NUMERICAL of Complex Variables (differential geometry with special SOLUTION OF TWO-POINT BOUNDARY-VALUE PROB reference to the complex domain; conformal mapping; LEMS Riemann surfaces; complex manifolds), Analytical Aspects Texas Tech University, Lubbock, Texas of the Theory of Complex Variables (Nevanlinna theory of Program: Herbert B. Keller, California Institute of meromorphic functions; analytical aspects of the theory Technology, wlll be the principal lecturer, There will of several complex variables; potential theory in the plane also be Informal discussions and seminars open to all and on complex manifolds). participants. Support: United Nations Development Programme and Support: (anticipated) National Science Foundation; United Nations Educational, Scientific and Cultural . travel and subsistence allowances for 25 invited partici Organization pants, Information: International Centre for Theoretical Physics, Deadline: Inquiries should be received by May 15, 1975, P, 0, Box 586, I-34100 Trieste, Italy Every effort will be made to notify the selected partici pants by June 1, 1975. June 2-6, 1975 Information: Paul Nelson, Department of Mathematics, SYMPOSIUM ON COMBINATORIAL OPTIMIZATION Texas Tech University, Lubbock, Texas 79409 The Johns Hopkins University, Baltimore, Maryland Program: Jack Edmonds, University of Waterloo, and July 7-18, 1975 William Pulleyblank will give a series of instructional SEMINAR ON MODERN MODELLING OF CONTINUUM lectures on optimal matchings. Other speakers are ex PHENOMENA pected, Rensselaer Polytechnic Institute, Troy, New York Information: Jack Elzinga, Department of Mathematical Appllcation deadline: March 17, 1975 Sciences, The Johns Hopkins University, Baltimore, Information: American Mathematical Society, Editorial Maryland 21218 Department, P. 0. Box 6248, Providence, Rhode Island 02940 June 16-20, 1975 GORDON RESEARCH CONFERENCE ON THEORETICAL July 14-20, 1975 BIOLOGY AND BIOMATHEMATICS INTERNATIONAL CONFERENCE OF MATHEMATICAL Tllton School, Tilton, New Hampshire SCIENCES Program: The general theme of the conference is the Karachi, Pakistan mathematical modelling of biological populations at Sponsor: The Karachi Mathematical Association various levels of organization, The complete program Program: The objectives of the conference are to help for the 1975 Gordon Research Conferences is published scholars in Pakistan to acquaint themselves with the in Science, March 14, 1975, (Reprints are available on latest advancements in the field of mathematics and thus request.) to encourage them in their own research work; to study Information: Alexander M. Cruickshank, Director, the history and progress of mathematics throughout the Gordon Research Conferences, Pastore Chemical centuries; and to focus attention on the works of scholars Laboratory, University of Rhode Island, Kingston, from Central Asia during the period from the ninth to the Rhode Island 02881 thirteenth century in mathematics,

160 Information: Hakim Mohammed Said, President, Hamdard Sponsors: The International Federation of Automatic National Foundation, Hamdard Centre, Nazimabad Control (IFAC) Systems Engineering Committee, Appli Karachi - 18, Pakistan cations Committee, and Theory Committee Program: The aim of the symposium is to determine the September 22-24, 1975 state of the art in the application of the large-scale SYMPOSIUM ON CALCULUS OF VARIATIONS AND systems approach to the solution of either practical or CONTROL THEORY theoretical decision and control problems. Mathematics Research Center, University of Wisconsin Deadline for papers: July 1, 1975 (in triplicate) Madison Information: G. Guardabassi, lstituto dl Elettrotecnlca Program: Fifteen invited lectures dealing with classical ed Elettronica-Politecnlco dl Milano, P.zza L. Da Vinci calculus of variations, optimal control theory and control 32 - 20133 Milano, Italy ' theory in general, The symposium will honor L. C. Young on the occasion of his retirement. June 28-July 2, 1976 Information: D, L, Russell, Mathematics Research COLING 76: INTERli!ATIONAL CONFERENCE ON Center, University of Wisconsin-Madison, 610 Walnut COMPUTATIONAL LINGUISTICS Street, Madison, Wisconsin 53706 University of Ottawa, Ottawa, Canada Information: Guy Rondeau, Department of Linguistics June 16-20, 1976 and Modern Languages, University of ottawa, ottawa, IFAC SYMPOSIUM ON LARGE SCALE SYSTEMS THEORY Canada AND APPLICATIONS Udine, Italy

SUMMER GRADUATE COURSES Supplementary List

The following is a list of graduate courses being offered in the mathematical sciences during the summer of 1975, MINNESOTA VIRGINIA MANKATO STATE COLLEGE VillGINIA COMMONWEALTH UNIVERSITY Mankato, Minnesota 56001 Richmond, Vll"glnla 23284 Information: Howard L, Prouse, Department of Information: R, W. Farley, Acting Chairman, Mathematics Department of Mathematical Sciences June 9 - July 12 June 16 - July 16 Mathematical Analysis I Analysis I History of Mathematics Methods of Applied Mathematics I Modern Algebra Mathematical Statistics I Linear Algebra Abstract Algebra I In Service: Models and Modeling Research and Thesis In Service: Patterns in Problem Solving ·June 16 - July 25 July 14 - August 15 Operational Methods Mathematical Analysis II Nonparametric Statistics Probability Topics in Topology: Uniform Spaces Modern Geometry In Service: Materials for Secondary School Mathematics July 17 - August 15 Analysis II Methods of Applied Mathematics II PENNSYLVANIA Mathematical statistics II UNIVERSITY OF PITTSBURGH Abstract Algebra II Pittsburgh, Pennsylvania 15260 Application deadline: June 23 Information: E, Kovacs, Department of Mathematics July 7 - July 18 Group-testing: Theory and Applications Statistical Methods in Biological Assay Reliability Theory

161 QUERIES Edited by Hans Samelson

This column welcomes questions from AMS members regarding mathematical matters such as details of, or references to, vaguely remembered theorems, sources of exposition of folk theorems, or the state of current knowledge concerning published conjectures. When appropriate, replies from readers will be edited into a composite answer and published in a subsequent column. All answers received to questions will ultimately be forwarded to the questioner. The queries them· selves, and responses to such queries, should be typewritten if at all possible and sent to Professor Hans Samelson, American Mathematical Society, Post Office Box 6248, Providence, Rhode Island 02940.

QUERIES 14, Spain). I would like to get an up-to-date bibliography for Modern Mathematics of Economics, 63, Paul Schweitzer (Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598). A RESPONSES TO QUERillS fOU:rth order linear difference equation (LDE) with non constant coefficients is given, and numerical values of Replies have been received to queries published in the first few dozen fxn} are desired, The LDE has recent issues of these cJVotioi). The editor would like to thank those who have replied, four linearly-independent solutions with distinct asymptotic rates of growth xn+l/xn. The solution of interest is 53, (vol. 22, p. 71, Jan. 1975, Edgar). The following singled out by a set of constraints 2::0 anxn = b, The reference may be what is desired: ~elasko, W., Metric. slowest-growing (fastest-growing) solution can be numer generalizations of Banach algebras. Roz. Mat. 47(1965), ically computed by downward (upward) recursion, like Lemma 7. 5. Communicated by R. J. Loy. that used for generating Bessel (modified Bessel) functions from second-order LDEs. Unfortunately, the two 56. (vol. 22, p. 71, Jan. 1975, Zwillinger\. The largest prime I know of not of the form 2k ± 1 is 2 • 3 • 5 • 7 • solutions with intermediate rates of growth are desired. 11 • 13 • • • • 73 • 79 • 83 • 89 - 1 = ls there a convenient way to numerically. generate them? 23768741896345550770650537601358309 2,377 o1034, An attempt was made to numerically compute the fastest ~ This prime was discovered by Alan Borning, and details growing solution and then deflate the given LDE to lower may be found in his paper Some results for k I ± 1 and order, but this proved numerically unstable. 2 • 3 • 5 • • • • • p ± 1 in Mathematics of Computation- 64. Julio G. Villalon (lnstituto Financiero Actuarial, 26(1972), pp. 567-570, Communicated by D. E. Penney. Facultad de Economicas, Universidad de Bilbao, Bilbao-

PERSONAL ITEMS

NEIL E. BERGER of the University of PROMOTION lllinois at Chicago Circle has been awarded the first Monroe Martin Prize by the Institute for To Professor. Maharaja Sayajirao Uni Fluid Dynamics and Applied Mathematics, Uni versity of Baroda, India: CHANDULAL C. SHAH. versity of Maryland. MUNffiUR R. CHOWDHURY of Jahangirnagar To Associate Professor. John Jay College, University, Dacca, Bangladesh, has been ap CUNY: NACHMAN N. ESHEL. pointed a Commonwealth Academic staff Fellow for the academic year 1974-1975 at the Depart DEATHS ment of Pure Mathematics and Mathematical Statistics, University of Cambridge, England. Professor Emeritus FWRENCE BLACK ROBERT TODD GREGORY of the University of the University of Kansas died on September of Texas at Austin has been appointed to a pro 13, 1974. She was a member of the Society for fessorship in Mathematics and to the head of the 54 years, Computer Science Department. Professor Emeritus RAY E. GILMAN of MANFRED KOCHEN of the University of Brown University died in January, 1975, at the Michigan, Mental Health Research Institute, has age of 87. He was a member of the Society for received the Award of Merit, given by the Ameri 54 years. can Society for Information Science at its annual Professor ffiVIN L. LYNN of the Cooper meeting in Atlanta, Union died on January 19, 1975, at the age of 53, MARGARITA MATAS of Herbert H. Lehman He was a member of the Society for 15 years, College, CUNY, has been appointed to an as Professor Emeritus JAMES B. SCAR sistant professorship at Miami-Dade Community BOROUGH of the US. Naval Academy died on College. December 29, 1974, at the age of 89, He was a WILLIAM T. REID of the University of member of the Society for 58 years. Oklahoma received a Distinguished Alumnus Professor Emeritus CHARLES E, Award given by Hardin-Simmons University, WEATHERBURN of the University of Western GRAHAM H. TOOMER of Cornell University Australia died on October 18, 1974, at the age of has been appointed to an assistant professorship 90, He was a member of the Society for 46 at Ohio State University, Columbus. years,

162 ABSTRACTS PRESENTED TO THE SOCIETY Preprints are available from the author in cases where the abstract number is starred. Invited addresses are indicated by • The papers printed below were accepted by the American Mathematical Society for presentation by title. The 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 indicate a category. An individual may present only one abstract by title in any one issue of the c}/ofiai) but joint authors are treated as a separate category. Thus, in addition to abstracts from two individual authors, one joint abstract by them moy also be accepted for an issue. Algebra & Theory of Numbers 75T-A82 Daihachiro Sato, University of Regina, Saskatchewan, Canada. Ex ansion of the STAR OF DAVID THEOREM of H.W. Gould and David Singmaster. Prel1m1nary report I). The greatest common divisor property of the binomial coefficients, namely { n-1 n n+l { n-1 n n+l . GCD ( k ) , (k-l), (k+l) }=GCD (k-l), (k+l), ( k ) } was conJectured and named as THE STAR OF DAVID PROPERTY by H.W. Gould in 1972. [Notices AMS, 19(1972),A-685,72T A248J. David Singmaster who, among others, proved the equality and added that n-1 n-1 n-1 n-1 the said GCD's are also equal to GCD{(k_2 ), (k_1 ), ( k ),(k+l)}. We obtained many generalizations of this type of equalities by an algorithmic method named pennant closure process. We give here only one of the application which generalizes the STAR OF DAVID PROPERTY of H.W. Gould and David Singmaster, e.g., n n-1 n-2 n n+2 n+l } GCD { (k+2),( k ),(k-2),(k-l),( k ),(k+l) { n-2 n-1 n n+l n+2 n } =GCD ( k ) I (k-1), (k-2) I ( k ) I (k+2) I (k+l) =GCD{(k-S+j)n-2 I.J=l,2,3,4,5,6,7} See figures with (~)=mel' Further generalization of this theorem will follow.

(Received December 4, 1974.)

75T-A83 SIN HITOTUMATU, Kyoto University, Japan and DAIHACHIRO SATO, University of Regina, Regina, Saskatchewan, Expansion of the STAR OF DAVID THEOREM. Preliminary report (II). We name the sets of binomial coefficients as follows: { n-p+l I . Mp= (k-2p+j+l) J=l,2, •.. ,3p-2}, (p~l)

Ap--{ (k+p-l)n-p+j I·-J-l,2, ... ,3p-2}, (p~l) -{ n-p+j I . n Rp- (k_2p+j+l) J=l,2, ... ,3p-2}, (p~l), 61=v1=(k), -{ n-p+2t+l n+p-t-1 n-t I liP- ( k-p+t+l),( k+t ),(k+p-2t-l) t=l, 2 •····P-l}, (p~2) -{ ( n-t n-p+2t+l n+p-t-1 I vp- k-p+t+l) I ( k.+t ) I (k+p-2t-l) t=l,2, • • • ,p-1}, (p>2) •

UP=~Mr, VP=~Ar' WP=QRr' D =1..£.;6 , N =~v , B =M u Au R , S =~B r=l r=l r=l P r=l r P r=l r P P P P p r=l r =UPUVPUWP. Then each of the twelve sets of binomial coefficients, M ,A ,R , p p p llp,V ,u ,v ,w ,D ,N ,B and s has the equal greatest common divisor. The PPPPPPP P case p=2 is the original STAR OF DAVID THEOREH of H.W. Gould, and the case p=3 is reported above. See figures in the preliminary report (I). (Received December 4, 1974.) *75'r-.A84 MENDELSOHN, ERIC, Department of Mathematics, University of Toronto. On the Groups of Automorphism of Steiner Systems

If G is a FINITE group then there is a FINITE Stiener triple system and a FINITE Stiener quadruple system with G as full automorphism group. (Received January 15, 1975.) A-377 75T-A85 JOYCE LONGMAN, Villanova University, Villanova, Pa. 19085 On ~Alternative Algebras. Preliminary Report

An algebra is called almost alternative if A satisfies (1) (x, y, x) = 0

(2) (x2 , y, x) 0 and ( 3) (x, x [y, zJ)= 0. Block (Amer. J. Math. 94 (1972) pp 389-412) has shown that a simple finite dimensional almost alternative algebra is Jordan or alternative. We extend this to the infinite dimensional case in the following: Theorem. If A is a simple almost alternative algebra with idempotent e f 1, then A is Jordan or alternative.

The case in which the algebra A is prime is also investigated. (Received January 20, 1975.)

*75T-A86 KENNETH C. LOUDEN, McGill University, P.O. Box 6070, Montreal, Quebec H3C 3Gl. Torsion theories, ring extensions, and group rings. Preliminary report.

Let 13: R-t S be a ring homomorphism, let 13*: Mod-R-+ Mod-S be the obvious functor, and

let ~ be a hereditary torsion class for R with associated quotient functor Q. Then ~* = { M E Mod-S I 13*(M) E ~ J is clearly a hereditary torsion class for S. Let its associated

quotient functor be Q*. Let ! be the idempotent filter of right ideals of R associated to ~· and set ! 1 = f D ~ S I DnR E ! J • Call ~ "sufficiently nice" w.r. t. 13 if ! 1 is an idempotent filter. If ~ is "sufficiently nice", then Q(13*(M)) is an S-module for all ME Mod-S, and Q*(M) c Q(I3*(M)) functorially in M, with equality if S is left R-flat. There are many situa

tions where ~ is "sufficiently nice". In particular for group rings, if A is a ring, G is a group, H4G, R = AH, and S = AG, then AG fiAHQ(AH)CQ*(AG) when~ is G-invariant: g-lDg E F for all g E G and for all DE!· Equality holds if G:H

all free sums. The Lambek torsion theory is always G-invariant, and we obtain AG 8AH~x(AH)C ~ax(AG), with equality if G:H

*75T-A87 C~RLES WELLS, Dept. of Mathematics and Statistics, Case Western Reserve Univer sity, Cleveland, OH 44106. An extension of the Krohn-Rhodes Theorem.

Notation: Let C and D be categories. C divides D if there is a subcategory E < D and a sur-

jective functor H: E~ C. A finite category Cis irreducible if for any finite categories D

and E and any functor A: E ~ Sets, C I D wr E"' C I D or C I E, where 11 0 I D" means C A divides D. Theorem: The irreducible categories are all the finite simple groups, the dis-

crete category with two objects, and Krohn-Rhodes' u1 , u2, and u3• Any firiite category C divides a finite iterated wreath product of irreducible categories, each of which either

divides C or is u3• An analogous but weaker theorem is proved for functors to sets. {Received February 7, 1975.)

75T-A88 STUART A. STEINBERG, University of Toledo, Toledo, Ohio 43606. On lattice ordered rings in which the square of every element is positive.

An f-ring is a lattice-ordered ring (2-ring) that is a subring and a sublattice of a product of totally ordered rings. Theorem 1. A unital 2-ring in which the square of every

element is positive is an f-ring if it is either archimedean, semiperfect, ~-regular, or left ~-regular. An 2-domain is an 2-ring R whose positive cone R+ has no zero divisors.

A-378 Theorem 2. An ~-ring with squares positive and no nonzero right ~-quasi-regular right

~-ideals is a subdirect product of ~-domains. Let ~-~(R) denote the lower ~-radical of

the ~-ring R, and let Dd be the class of those ~-rings R for which R/~-~(R) satisfies the identity x+y+/\ x-y+ = 0. Denote the Johnson radical of the ~-ring R by J(R). As a generalization of a result of D.G. Johnson's we have Theorem 3. J is a special radical in

Dd. If R e Dd has the minimum condition on ~-ideals and a left identity element, then J(R)

is nilpotent, and R/J(R) is a direct sum of totally ordered unital ~-simple f-rings. (Received January 24, 1975.) 75T-A89 BONALD STATTON, Northwestern Univarsity, :&vanston, Illinois 60201. A characterization of commutative, separable algebras, Preliminary report.

Let R be a commutative ring with identity. An R-algebra, s, is separable if the (S,S)-bimodule epimorphism of SGhS onto S, which is determined by the ring multiplication in S, splits. Let S be a commutative R-algebra, finitely generated, projective and faithful as R-module.

Then S is separable if and only if there is a commutative, separable R-algebra, T, finitely

generated, projective and faithful as R-module such that Hom~(S,T) is generated overT by R-algebra homomorphisms from S to T. (Received February 10, 1975.) *75T-A90 BENJAMIN VOLK, 13-15 Dickens Street, Far Rockawa,, N.Y. 11691. On the Riemann Hyppthes1 g.

For the Riemann zeta function, it is found that the disc of radius 1/10 centered about the point (.7, 6) is zero-free. This is shown by applying a real-variable technique to a standard line-integral representation of said function, to derive an apparently' new approximation (with error tel'Jil) of the zeta function in the critical strip. On the other hand, a zero of the third partial sum of the aeries representat ion of the zeta function is found to be near the point (.4, 3.5). This is shown by obtaining seperate equations for the real and imaginary parts ot the zero, and using the Reglll.a Falsi. This latter result contradicts, it memory serves, a conjecture in Saaty: Lectures in Modern Mathematics. (Received January 27, 1975.) *75T-A91 S. BURRIS, Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario. Congruence lattices of subdirect products. Preliminary report.

Let ei be a congruence of A; = , 1 ~ i ~ n. Define ITe 1 on ITA; by € ITei iff € ei' 1 ~ i ~ n. A congruence e of ITA; which is not one of the ITe 1•s is a skew congruence. Con A denotes the lattice of congruences of A. Theorem 1. If Con(ITl~i~nAi) is modular and, for 1 ~ i < j ~ n, A; x Aj does not have any skew congruences, then ITl~isnAi does not have any skew congruences, hence Con(ITlsisnAi) ~ ITlsisnConA1• D Theor.em 2. Suppose Con(A) is modular. Then Con(A) is a Boolean lattice iff (i) A is semi-simple, (ii) any two members of A are identified by all but finitely many maximal congruences of A, and (iii) Con(A/~At 2 ) is distributive for all maximal congruences ~l and t 2. D Theorem 3. Suppose A has permutable congruences. Then Con(A) is a Boolean lattice iff (i) A is isomorphic to a direct sum of simple algebras {Ai}iEI' and (ii)for i,j E I, i~j. AiXAj does not have skew congruences. 0 Application: A group G has a Boolean lattice of congruences iff G ~ L;E 1G1, where each Gi is simple, and for any prime number p the simple group (Z,+)/(p) is equal to at most one Gi. (Received January 28, 1975.) A-379 75T-A92 MASAO KISHORE, University of Toledo, Toledo, Ohio 43606, Odd almost perfect numbers, Preliminary report.

A positive integer N is called quasiperfect if o(N) = 2N + 1 and almost perfect if o(N) = 2N- 1. H. L. Abbot, C. E. Aull, E. Brown, and D. Suryanarayana have shown (Acta

Arithmetica, XXII, 1g73, 439-447) that if N is quasiperfect, then (1) N > 1020 ,

(2) w(N) ~ 5 if 3\N and (3) w(N) ~ 8 if 3fN, w(N) being the number of different prime factors of N. Using computer, we prove that if N is an odd almost perfect number, then (1) N > 1030 , (2) w(N) ~ 6 if 3jN, (3) w(N) ~ 9 if 3fN and (4) for some

pjN pN is primitive abundant. (Received January 30, 1975.)

*75T-A93 G. GRATZER, Dept. of Mathematics, University of Manitoba, Winnipeg, Manitoba Canada 1 R3T 2N2 A characterization of the characteristic of an equational class. Prelim~nary Report. ' The characteristic of an (infinitary) algebra is the smallest infinite regular cardinal m such that every operation is of arity < m. The characteristic of an equational class K of algebras is the characteristic of any/all member of K. For an algebra !ll, Sub ij is the subalgebra lattice of !ll and Con !ll is the congruence lattice of !ll. A complete lattice L is m-algebraic if every element is the join of m-compact elements; an element a E L is m-compact if a :s: Vx (X,;; L) implies that a :s: Vx1, where x1 ,;; X and I xlj < m. Theorem. Let K be a nontrivial equational class of algebras and let m be an infinite regular cardinal. The following conditions are equivalent: (i) K is of characteristic m; (ii) m is the smallest infinite regular cardinal such that every Sub l!l, l!l E K, ism-algebraic; (iii) m is the smallest infinite regular cardinal such that every Con fU, !ll E K, is m-algebraic. (Received February 3, 1975.)

*75T-A94 G. GRATZER & H. LAKSER, Dept. of Mathematics, University of Manitoba, Winnipeg, Man.,Canada, Finitely presented lattices I. Preliminary Report.

Theorem. A finitely presented lattice L can be written as a finite disjoint union of convex sub lattices, Ai, each of which satisfies the following conditions: (SDi\) : xi\ y =xi\ z = u implies that xi\ (yV z) = u; (SD): xVy=xV z=u implies that xV (yi\ z) =u; (W) xi\y:s:uVv implies that x:S:uVv or y:S:uVv or xi\y:s:u or xi\y:S:v; the lattice Ai can be generated by J(Ai) U M(Ai), where J(Ai) and M(Ai) is the set of join-irreducibles and meet-irreducibles in Ai 1 respectively. As an easy corollary we obtain a result of T. Evans and D. Y. Hong, Algebra Universalis 2(1972), 284-285, stating that FM(4) is not finitely presented. (Received February 3, 1975.) *75T-A95 E.FRIED & G.GRATZER,Univeristy of Manitoba,Winnipeg,Canada,R3T 2N2. On auto mar hisms of the s~bal ebra lattice induced b automor hisms of the al ebra. Pre iminary report.

Let !ll be an algebra and Cl an automorphism of !ll. Then 'Ei : B.,. Bet= [ xct \ xE B} defines an auto-· morphism of Sub!ll 1 the subalgebra lattice of!ll and cp : Cl +Cf is a homomorphism of Aut!ll (the auto morphism group of!ll ) into Aut Sub!ll (the automorphism group of the subalgebra lattice of l!l). We characterize the triple (Aut!ll, Sub!ll,cp). Theorem. Let G be a group, let L be an algebraic lattice with more than one element and let cp be a homomorphism of G into Aut L. Then there exist a (finitary) algebra !ll and isomorphisms a 1: G _,. Aut !ll and ct 2: Sub !!!._,. L such that for aE G, ar:r1 = a 2 (acp)a;1. In other words, if we identify G with Aut!ll and Sub!ll with L, then for Cl E Aut!ll , Cf = Cl cp . The special case when cp collapses G onto the identity element of Aut L was represented by E. T. Schmidt, Acta Sci. Math. (Szeged) 24(1963) ,251-254. - Our result generalizes to m-algebraic lattices and infinitary algebras of characteristic m. There is a similar map * :Cl +ct * of Aut!ll into Aut Con!ll , the automorphism group of the congruence lattice of !ll. It would be interesting to characterize ( Aut!ll 1 Sub!ll 1 Con!ll ,cp ,*) . The first three components were characterized in W.A.Lampe, Algebra Universalis 2(1972), 99-112, 270-283, and 286-302. (Received February 3, 1975.)

A-380 75T-A96 N. BURGOYNE, University of California, Santa Cruz, Ca. 95064, R. L. GRIESS, JR. and R. LYONS, Rutgers University, New Bnmswick, N. J. 08903. Field automorphiSIIIS and maximal subgroups of finite Chevalley ~· Preliminary 'RijiOrt.

If G is a group, pis a prime, and A E End(G), write f!' (G) for the subgroup of G generated by all p-elements, and write GA for CG(A). Theorem 1. Let G be a simple algebraic group over an algebraically closed field of characteristic p;IO. Let A be an endomorphism of G onto itself such that GA is finite. Exclu~ the exceptional cases GA = ~ (2), ~ (3), or 2c2 (2). If M is a finite group such that oP (GA) ~ M ~ G, then there exists a positive integer n such that oP' C!V ~ M ~ ~· where Jl=An. Theorem 2. Let G = sL(q) be a finite simple Steinberg variation defined over the field GF(qs) of characteristic p. Let A be a standard automorphism of G obtained from an automorphism of GF(qs). Exclude the exceptional cases - 2 ...m 2 2 _pi G = Azn (.: ~), A2 (3), or A2 (5). Let M be a group such that u- (GA) s M s G. I£ p r IM:oP' (GA) I. then M ~ GA. If A has orders, then either M ~ GA or M = G. Corollazy. Let G be a finite simple Chevalley group, Steinberg variation, Suzuki group, or Ree group defined over a field k. Let A be an automorphism of G obtained from an automorphism of k of prime 2 order r. Exclude G ; ~ (2r) , ~ (3r) , 2c2 (Zr) , 2Az (3), ZAz (5) , and (if r=;) Azn (2m) • Then GA is a maximal subgroup of G and is the only maximal subgroup containing aP (GA). (The possibilities forM in the exceptional cases of Theorems 1 and 2 are tabulated.) (Received February 3, 1975.)

*75T-A97 Dr. B. N. DATTA, Ahmadu Bello University, Zaria- Nigeria. On the Routh-Hilrwitz Fujiwara and Schur-Cohn-Fujiwara theorems.

Simple and elementary proofs are given in this paper of the two classical results of Fujiwara [M. Fujiwara, Uber die algebraischen Gleichungen, dren Wurzdn in einem Kreise oder in einer Halbebene eiegen, Math. Z. 24 (1926) 160-169] on the solution of the well-known Routh-Hurwitz and Scbur-cohn problems. :re is shown that the Fujiwara matrix in each case satisfies a Lyapunov type equation and Fujiwara's result then follows froJI~. the recent inertia theorems due to Chen [C. T. Chen, A generaliza tion of the Inertia theorem, SIAM J. Appl. Math., 25 (1973), 158-161] and Wimmer [H. K. Wimmer, Remarks on inertia theorems for matrices, (to appear)]. These alternative proofs of the Fujiwara's results thus establish a link between the two different approaches of the solution of the root separation problem: one, the classical methods of solution using hermitian forms and the other, the solution via matrix equations. (Received February 4, 1975.)

*75T-A98 ZOLTAN PAPP, George Mason University, Fairfax, Virginia 22030. A note on mixed abelian groups.

The following theorems are proved. Theorem 1. If B is a bounded abelian group and G is an extension of B, then G splits iff G/B splits. Theorem 2. Let H be a pure torsion-free subgroup of G and assume that G/H splits. Then G splits too. Theorem 3. Every pure extension G of the group A has the property that the splitting of G/A implies that G splits iff A= B(:i) F E)D, where B is bounded, F is reduced torsion-free, and D is a divisible group. Theorem 1 is a direct generalization of Proposition 100.2 of L. Fuchs. (Infinite Abelian Groups, Vol. 2. Academic Press, N.Y., 1973.) (Received February 5, 1975)

FRANK HARARY, The University of Michigan, Ann Arbor, Michigan 48104 *75T-A99 and GARY CHARTRAND and DON R LICK, Western Michigan University, Kalamazoo, Michigan 49001 Point Arboricity Anticritical Graphs. The point arboricity p(G) of a graph G is the minimum number of colors assignable to the points of G so that no cycle is monochromatic. A non complete graph G is point arboricity anticritical with p(G) = m if for every pair of nonadjacent points in G , the line e joining them has the property that p(G +e) ~ m. Denote the join of graphs G1 and G2 by

G1 + G 2 . Further, the notation G1 + · • • + Gn denotes (G1 + • • • + Gn_1.> + Gn• A-381 It is shown that a graph G is point arboricity anticritical with p(G) = m if and only if (1) G = T1+ ••. + Tm (m ~ 1), where each Ti

is a tree of order at lea.st three, or (2) G = s 1 + .•. + sm•l+ K2 (m ~ 2) where each Si is a star and at least one si has order at least three. (Received February 6, 1975.)

*75T-A100 Harold Finkelstein, Emory University, Atlanta, Ga. 30322 and Kenneth Mandelberg, Emory University, Atlanta, Ga. 30322. On Solutions of Equations in Synnnetric Groups. Preliminary Report.

Let x E S , the symmetric group on n symbols. Let e E Aut(S ) and let the automorphism order of X withnrespect toe be defined by Ye(x) = min{k: X xe xe~ ••• xek = l} where xe is the image of x under e. Let ag E Aut(Sn) denote conjugation by the element g E Sn. Let b(g; s, k : n) _ i{s E Sn: k/ya (x)/sk}i where sand k are positive integers and a/b denotes a divides b. Further let h(§, k: n) = b(l; s, k: n). If g E S moves k symbols, let k' = f(g, m) denote - • n the number of symbols in g which are in cycles of length not dividing the integer m, and let gm denote the product of all cycles in g whose lengths do not divide m. Then ~ moves k' symbols. The main results proved are: (l) recursion; If n L k + m and s = n- k 1 - l then b(g; m, 1: n) = i1mb(g; m, 1:. n- i)(i~ 1 )(i- l)! (2) reduction; b(g;m, 1: k')h(m, 1: i) = b(g; m, 1: i + k'). (3) distribution; Let D(e, n) = {(k, b): k E z+ and b = b(6; l,k:n) I o}. Then D(6, n) = D(¢, m)V'm L N = N(e, ¢)iff e is conjugate to¢. (4) evaluation; the number of cycles in~ of any given length is smaller than the smallest prime in miff b(~; m, 1: k')=L If g = (l2 ••• pm)t and sk/pm then b(g; s, k: pm) = {±~(mod p). (Received February 6, 1975.)

75T-A101 mvm m,16SO vtaoe• m. lol"ttl,~ua.-. 55411. I"'""1p f.tE Daar ucmt1U1 Hisll!!Ds .b !!'I'MMD ~ ,..,.....,,.ILI'.relill1au7 repo;n. K.Q • 1,2, •••• W0 ,V, an 1Dtepn; ~2 • Mila+1 +-. 1 S • ,t1. It (W0 ,w1) • (O,t),(21K),tlaa. w. =u •• w• .:. v•• sn A(:x:) =I xVO-w,l. D(J)•(1+Vr)Uk+J ' il = Il-+ 4SQ , F(r)• V/(1+ Vr), aa4 (J'] the g.i.t• .Qaa 1. S • 1, M ~ 1, with 11€ Q(M+1; P)O ill a roo\ of ~-Jb:-Q=O. !!Ill 2. 8 • -1..,

II~ 3, with 1~ Q• Let. r • 11 21 31 1 •••• It Ar(P)((f"" t'k)/(D((r-1)k)tfk),there u:lats a flmc•ion I\.(S,K,Q,W0 ,W1,a,k)!! \- auch that [PrkV: + Br + F(rU • w:.fk torn~ t ~ o. Jaul.I'Gr r • 1,2,3,4, see III,IV,v,mz. 8 28 l"a2ua 7. s5 = sz, (-BQ) (Jkw!-(-sQ)J&:w!.x> + S~Q (.y_ - Q2k~),.11hen i'-z,• /,..aw0w1...sq( .1 and Jk .. Pl15k + SCJJ5k-1" J1.1."MJe=t ,aq., with ~.u. ~ r.,v.!! '.• and 'a .. 1 +Yf •

For D ~ 11 [P'r'2a +(SP+3),'2a + r'28+1 + PF2n + .21.2n-1 +(11/12>] • ,S2a+1 , and

~'L~2 +(2SP+1S)(~ • ~) - S(~) + L:...3) +(11/12)] • xi.+) • Add1t10118l repone are tortheGiing. (Received February 7, 1975.) *75T-A102 J. N. McNAMARA, State University College, Brockport, New York 14420. Compact Categories

Let T be a faithful covariant functor from a category A to the category of sets, S. A morphism f of A is a T-quotient iff T(f) is surjective and each function g with Dom(g) = Ran(T(f)) lifts to A whenever gT(f) lifts to A. The category A is compact iff (1) it contains finite products and coproducts, images, and pullbacks (with each preserved by T); (2) it contains a point-like object; (3) each epimorphism in A is aT-quotient;- (4) certain monomorphisms in A fit into a pushout-pullback diagram.

Various properties of A and an associated category A are developed. The results are similar to those obtained by Spanier (Duke Math. J. 30 (1963)) when A is the category of compact Hausdorff spaces and A is the category of quasi-topological spaces. (Received February 7, 1975.)

A-382 GUSTAV I. LEHRER, University of Sydney, N.S.W. 2006, Australia Classical Groups and Incidence Structures. Preliminary Report.

Let H0K be a pair of subgroups of the finite group r. The incidence structure s(r; H,K) whose points and blocks are the (left) cosets of H,K respectively is said to have~ !:!:!!!t if its matrix has rank [r: H] or [r: K]. It is shown that if r = W, a Weyl group of type~· B1, c1 or n1, then s(r; H,K) often has maximal rank if Hand K are maximal parabolic sUbgroups. When interpreted representation - theoretically, deformation theory may be used to translate this into a result about Chevalley groups of corresponding types, i.e. classical groups. Using the geometric connotation of parabolic subgroups one obtains the following geometric statement: Theorem:- Let V be a finite vector space equipped with a non-singular (sesquilinear or bilinear) form <,> of index v. For integers e,f such that 1 ~ e ~ f ~ v - e let Mef be the incidence matrix of totally isotropic e-subspaces vs totally isotropic f-subspac;es. Then Mef has maximal rank (equal to the number of totally isotropic e-subspaces). (Received February 10, 1975.) (Author introduced by Professor G. M. Kelly.) Analysis ~. l!ANQA,, University of Victoria, 'Victoria, British Columbia, Canada V8W 2Y2 an4~~enshaw College, Cuttack-3, India. An asymptotic confluent expansion for .cenain.. fnnct i nns-'lf ..sevexal .variables. Preliminary. report. J:n,tliher pftsent·. paper some recent results on asymptotic confluent expansions for functions of one and two variables, given by J. L. Fields [Math. Camp. 21 (1967), 189-197, especially p. 195, Theorem 3] and V. L. Deshpande [Math. Camp. 28 (1974), 605-611, especially p. 605, Theorem 1], are extenqed to hold for certain functions of several variables. The main results obtained in this paper would apply, for instance, to the four Lauricella hyper- geometric functions F(r) F(r) F(r) and F(r) of r variables [G. Lauricella, A ' B ' C D Rend. Circ. Mat. Palermo. 7 (1893), 111-158] and to their generalization which was defined recently by H. M. Srivastava and M. c. Daoust [Nederl. Akad. Wetensch. Proc. Ser. A 72 = Indag. Math. 31 (1969), 449-457, especially p. 454 et seq.; see also Math. Nachr. 53(1972), 151-159, especially &ect~on 5]. The paper concludes by showing how asymptotic confluent expansions for these special functions of several variables can be deduced from the results presented. (Received August 7, 1975.)

*7'5T~Jff7 1'.!<.KAfiilTHANand S,K.SINGH GAUTAM ,Department of Mathematics , I. I.T. l<:allJlu:r:..2o8016 ,India.Restricted double automorphisms of the spaces of . analytic Dirichlet functions. In this paper we consider two spaces X and Y where X is the space of all Dirichlet ftthctions analytic in the half-plane,eq1i.i_£Ped wilh a Frecnet Topolo- gy F,determined by the family fll··ll= I If II = ~lan leO' n , f(s) = QO s A cr cr n=r L.,ane n ,s=G"'+it E:C,cr<:.A of sel"'i-norms andY is a subspace with a topology n=1 G generated by the family f llfllv :i =1, 2, 3,... of semi-norrr.s such that G is finer than the induced topology on Y from F.The main results of this paper are: Theorem 1:--Let ~otn\ andf~} be bases in X for which the function~n= ~n- ~n belongs toY {n=l,2, ••• ) and satisfies sup f lim sup }< A. v~ (1. n-?oo Then for t ~n t to be proper , it i.s necessary and sufficient be proper. T be the endo~orph- Theorem 2:-- Let f«n} and {~;~ be proper bases in X ann let ontot{3n} • If the function qn= Pn -~n belongs to Y (n=1,2, .... ) and satisfies the condition log 11 ~nil lim sup A, ( =1,2,, •• ), n400i:l /\n then T is a ~estricted double automornhism on X and Y. (Received November 4, 1974.) {Au:thm:a..in.tr.odu.c.ed by. Dr. s. K, Bajpai.) *75T-B78 SHIMSHON ZIMERING, Ohio State University, Columbus, Ohio 43210. A remark about slowly varying seguences. We have shown the following relationship between slowly varying sequences (in the sense of Karamata) 8.nd slowly oscillating sequences (in the sense of Schmidt-Zimering): Theorem 1. Let (s ) be a slowly A-383 n varying sequence, i.e., sp.n/sn- 1 (n- ro) for any 0

We consider the weak Cauchy problem in arbitrary Banach space for equations (D-A) u: 0 ,as were defined by Kato-Tanabe.After proving some elementary relationships,we obtain a result which shows how uniqueness for strong solutions in the bidual space implies unique-

ness of the weak Cauchy problem.Then we give a partial extension of a result by Ljubi~- Krein(they have weakened solutions),and also an extension to non-reflexive Banach spaces of a result by Barbu-Zaidman,using,for the last result,Phillips theorem on adjoint semigroups in Banach spaces. (Received December 20, 1975.) 75T-B80 O.P. JUNEJA and M.L. I«>GRA, Department of Mathematics, Indian Institute of Technology, Kanpur-208016, India, On starlike functions of order a and trpe s. Preliminary report. ·

In this paper the concept of 'type' has been introduced fbr starlike

functions. Thus, let f(z) = z + L au zn be analytic in the unit disc D(izl < 1); then n=2 f(z) is said to be starlike of order a(O ~a < 1) and type 8(0 < B ~ 1) in D if the inequality

ICzf'(z)/f(z) - 1)/(zf'(z)/f(z) + (1- 2a))i < B holds for given a, 8(0 ~a< 1, 0 < S ~ 1)

and fbr all z E D. The class of starlike functions of order a and type S is denoted by sa,S'

We obtain a representation formula, distortion theorems, coefficient bounds and the radius of

convexity for functions in sa,S' A sufficient condition for a function to belong to sa,S has also been obtained. The results obtained generalize the corresponding results of Schild

(Amer. J. Math. ~ (1965), 65-70), Padmanabhan (J, Indian Math. Soc, ~ (1968), 89-103) and

Zmorovic UMat, Sb, ~ (110) (1965), 518-526; English transl., Amer. Math.' Soc, Transl. (2)

~ (1969), 203-213), (Received January 13, 1975.) (Author introduced by Professor J. N, Kapur.) *75T-B81 ELEMER E. ROSINGER, Technion-Israel Institute of Technology, Haifa, Israel 32000. A Modified Distribution Multiplication Theory.

In the author's previous paper: "A Distribution Multiplication Theory", associative,

commutative algebras with unit element, containing the distributions in D'(R1), were constructed. These algebras possessed linear mappings, extending the distribution derivative and satisfying the rule of the product derivative. Relations, such as: 1 1 (x-x0 )r•o(q) (x-x0 ), with variable x E R , given x0 E R , r,q = 0,1, ... ,r > q, and o the Dirac distribution, were proved. In the present paper, an other family of algebras, with stronger properties, is constructed. For instance, relations, such as: rv ( qv) 1 E (x-xv) •o (x-x\1), with (xv I v EN) locally finite in R , are proved. VEN (Received January 21, 1975.) (Author introduced by Francois Treves.)

*75T-B82 G. Bennett, V. Goodman and c. Newman, Indiana Umiversity, Bloomington, Indiana 47401. Random matrices and absolutely summing operators.

We obtain estimates for the distribution of the norm of a matrix transformation

A-384 2 to (2 p < ro) whose entries are independent, bounded, mean- from ~ n m ~p ~ zero, random variables. Asymptotically, the expected norm turns out to be re- markably small, and this enables us to produce an interesting class of bounded linear operators from ~ 2 to ~p • As an application we complete the descrip tion of the class of (p,q) - absolutely summing operators on Hilbert space by shvwing that, for 2 < q < p < oo, it coincides with the ideal generated by the Lorentz sequence space ~ 2p/q,p • (Received January 22, 1975.)

75T-B83 THOMAS KEAGY, North Texas State University, Denton, Texas, 76203. A Tauberian Theorem for Rearrangements, Preliminary report.

R. C. Buck (Bul. A.M. S. 49 (1943), 898-899) characterized convergent sequences by showing that a sequence x is convergent if there exists a regular summability method which sums every subsequence of x. I. J. Maddox (Bull. London Math. Soc. 2 (1970), 63-65) improved this result by showing that if A is a matrix method which sums every subsequence of some non- convergent sequence x, then A is a Schur matrix. J. A. Fridy (these Notices 22 (1975) Abstract 720-40-4, A-158) has obtained an analogue to Buck's theorem in which "subsequence" is replaced by "rearrangement." In this note an analogue to Maddox 1 theorem is proved in which "subsequence" is replaced by 11 rearrangement. 11 (Received January 23, 1975.)

75T-B84 JOE HOWARD, Portales, New Mexico 88130 and EDWARD OSTLING, HOFSTP~ University, Hempstead, New York 11550, THE TOPOLOGY OF UNIFORM. CONVERGENCE ON WEAK STAR NULL SEQUENCES. Let E and E' denote a Banach space and its dual and v(E,E') the topology on E of uniform convergence on the weak star null sequence of E! Theorem 1. v(E,E') and the norm topology coincide if and only if E is topol

ogically isomorphic to a closed subspace of c 0 • It is not generally true that v(E,E') and the norm topology have the same compact sets, the space m of bounded sequences providing a counterexample. Theorem 2. If E is weakly com-

pactly generated, then v(E,E') and the norm topolo~~ have the same compact sets. (Received November 8, 1975.)

*75T-B85 B.C.BURCH, Tulane University, New Orleans, Lousiana 70118. A Semi group Treatment of the Hamilton-Jacobi Equation. Preliminary report. We study the Hamilton-Jacobi equation, ut+F(Vu)=O, t>O, xcRn, Vu=gradxu' F convex, from the point of view of nonlinear semigroup theory. We define B(k)={ucL"'(Rn):iuloo2k; for every x,ycRn, lu(x+y)-u(x)l.::_kiYI and u(x+y)-2u(x)+u(x-y).::_klyl 2 }. We show, for an operator Au=F(Vu) with domain in B(k), that A satisfies the hypothesis of the Crandall-Liggett Theorem. In studying the operator A, we obtain existence and uniqueness results for h in B(k) for the equation u+AF(Vu)=h, A>O. This equation is studied via the reg ularized equation u+AF(Vu)-c~u=h, c>O. Existence for the regularized equation is obtained through Schauder type estimates. A maximum principle is used to study the solutions of the regularized equations. We show that the operator, defined as the union of the above A over k>O, generates a contraction semigroup on the space of bounded, uniformly continuous functions on Rn. We show that the semigroup leaves every B{k) invariant and that for initial data in any B(k), the semigroup satisfies the Hamilton-Jacobi equation almost everywhere. (Received January 31, 1975.)

A-385 *75T-B86 JON C. HELTON, Arizona State University, Tempe, Arizona 85281. Product Integrals and Linear Integral Equations.

In the following, R denotes the set of real numbers and N denotes a ring which has a multiplicative identity element 1 and a norm 1·1 with respect to which N is complete and llJ = 1. Further, if r and s are positive integers, then Nrs denotes the set of rxs matrices with elements from N. Matrix addition, subtraction and multiplication are defined in the usual manner. For convenience, 1 is also used to denote the multiplicative identity element in Nrr and Nss' If f and h are functions from R to Nrs' G, H, A and B are functions fromR>

Nss' (1 - H)-l exists and is bounded on [a,b], B = (1- H)-land A= (1 + G)(l - H)-l on (a,b], his quasi-continuous on [a,b], G and H have bounded variation on [a,b], fb G exists and fb H exists, then the following statements are equivalent: (1) f is boun~ed on [a,b] and (~R)fx(fG + fH) exists and is f(x)- h(x) for a ~x ~ b, and (2) (R)fx[dhB ~A] exists a a v and is f(x) - h(a) arrx A for a~ x ~b. Solutions are also obtained for integral equations

of the form f(x) = h(x) + (LR)fx(Gf + Hf), f(x) = h(x) + (LR)fx(fG + Hf) and

f (x) = h(x) + (LR) r(Gf + fH) f~r a < X ~ b. (Received Feb~y 3, 1975.) a *75T-B87, R. s. DAHIYA, Iowa State University, Ames, Iowa 50010. on zeros of solutions of the second order delay differential equation.

~e purpose of this paper is to study the oscillatory behavior for the solutions of second order non-homogeneous equation

(1) (r(t)y1 (t)) 1 + pl (t)y(t) = f(t)

in relation to the solution of delay equation

(2) (r(t)x 1 (t)) 1 + P2 (t)x(t) = -P3 (t)x'f (t) where x (t) = x(t-'f(t)). 'f

~e following type of results are obtained under various sets of conditions: If all solutions of the equation (1) are oscillatory or nonoscillatory, then all solutions of the equation (2) are oscillatory or nonoscillatory respectively. (Received February 3, 1975.) *75T-B88 MAURICE H. MILLER, JR., University of Mississip~i~ University, Mississippi 38677 Limits of Seguences of Darboux Functions. Prel~~nary Report.

Let f be a function from I = [0,1] into I. A function f j,s said to be a Darboux function

if f(C) is connected whenever Cis a connected subset of [0,1]. We investigate the following

question due to Marcus: "What is a natural convergence "+" such that the following is true:

if {fn}n=l is a sequence of Darboux functions converging pointwise to a function f, then f

is a Darboux function if and only if fn + f?". Several results related to this question are obtained such as a characterization of the Darboux property in terms of certain types of

convergence. Also several theorems are proved concerning sequences of Darboux functions which

converge either pointwise or uniformly. The following theorem gives the flavor of these 00 results. ~· Let {fn}n=l be a sequence of Darboux functions converging uniformly to a function g. Then there exists a Darboux function h such that the closure of h is the closure

of g. (Received February 5, 1975.)

*75T-B89 AUTHOR: Andre de Kervin, Indiana State University, Terre Haute, Indiana 47809 Some Applications of Schauder Bases. Pre I iminary Report.

Proposition: If y is a finitely additive bounded function from an algebra of sets E into the space X with a Schauder basis then y is equivalent to a finite positive finitely additive

A-386 function A and A satisfies the countable chain condition. Let ~n be a sequence of positive

measures on E we say y<O there exists 4>0 such that ~n(Al<4 for alI

n imp! les I IY

y = Eynxn then y is extendable to the smallest a-algebra generated by E if and only if

Nikodym property then if y<<~ where ~ is positive and finitely additive then $~~ converges to

Y In the variation norm where $~ = E ~!~J XE as EE~ where ~ is a partition. (Received February 7, 1975.) *75T-B90 AUTHOR: Andre de Korvin, Indiana State University, Terre Haute, Indiana 47809 and Mr. Vo Van Tho Indiana State University, Terre Haute Indiana 47809 Non Linear lntegrafion of Totally Measurable Functions. ~reliminary Report.

If E and Fare two Banach spaces, let U(E,Fl denote alI functions from E into F which are

bounded on bounded sets of E and are moreover uniformly continuous on such sets. A finitely

additive set function m from E into U(E,Fl wi I I be cal led a Nemytskii measure if it satisfies

some continuity conditions. If f is a function with values in E then the integral of f

respectively tom is defined and it is shown that every totally measurable function has an

integral respectively tom. Let sv(mal be the semi variation of m restricted to thea-balI

of E, call m v.s.r. if An\,$ imp! ies sv ma

is countably subadditive a convergence Theorem holds when fn+f a.e. (Received February 7,1975)

75T-B91 L. M. GEARHART, University of illinois, Chicago, illinois 60680. On the invertibility of continuous shifts. Preliminary report. The results of Moeller (J. Math. Anal. Appl. 4(1962), 276-296; Duke Math. J. 31(1964), 99-108) are generalized to the case of countable multiplicity. For K a separable Hilbert space and H2 (0+;'(") the Hardy space of square summable K-valued functions defined on the upper half-plane, we have the following result for the semigroup fJ of multiplications f(cr) .... exp(itO')f(O'), f E ~(O+;M) and t;;; 0: Theorem. If !Dl = H2 (0+;'<") 111 G~ (0+;'<") where G is an inner function, then the compression of fJ onto !Dl can be imbedded in a group iff G(0')-1 exists and is bounded for all 0' in some half-plane, Im(O') > y > 0, and there is a t> 0 such that infflexp(ita)l + IIG(O')xll :0' E 0+ and X EK with llxll = 1}> o. The presentation is greatly simplified by invoking some results in interpolation theory. (Received February 10, 1975.) (Author introduced by Professor Jeff Lewis). 75T-B92 S.M. SHAH, University of Kentucky, Lexington, Kentucky 40506 Polynomial Approximation and Entire Functions

Let f(x) be complex-valued continuous function on [-1, 1] and let En(f) = inf sup I f(x) - p(x) I where '~~'n pEJrn -1~<;;;1 denotes the set of all polynomials p of degree at most n. Let a(x) and {3(x) be positive strictly increasing and differentiable on [a,oo) and tend to oo as x-> oo and suppose that {3(x log x)/f3(ex) -> 0, {3( ( 1 + o( 1 ))x)/f3(x) -> 1 and a(cx)/a(x)-> 1 as x-> oo for every c > 0. Theorem. Let (En(f)) 1/n-> 0 as n-> oo. Then f(x) is the restriction to [-1,1] of an entire function f(z) and (i)

liminf a(log M(r.f)) ;;;. liminf a(n)/{3 (1tog __j__( ) . r->oo {3(1og r) n->oo n En f 1

(ii) Suppose further that F(x,c) = {3- 1(ca(x)) and dF(x,c)/d(log x) = 0(1), x-+ oo for every c > 0. Then

limsup a(log M(r.f)) = limsup a(n)/f3(1tog - 1-) . r->oo f3(1og r) n->oo n En(f)

(iii) Assume further that En(f)/En+l (f) is ultimately a non-decreasing function of n. Then there is equality sign in (i). Part (ii) gives an extension of a theorem of R. S. Varga and Parts (i) and (iii) give extensions of two theorems of

A. R. Reddy. (Received February 10, 1975.)

A-387 *75T-B93 John Gill, Southern Colorado State College, Pueblo, Co. 81001 A Genera1ization of Certain Corresponding Continued lractiopa

0 Given f 0 (z) = 1 + ai_ >z + ••• , and two sequences X= f•k1; , D = tdk1:. where the l!Tk's are positive integers and the dk's are non-zero complex numbers, define a continued fraction recursively by f (z) = 1 + ai•>z + ••• + cai•>- ~)zl!T• + l!T m m ~z m/fm+l (z) , m=0,1,2,... The reaultiDg expansion is called an l!TD-fractiop and corresponds to f 0 • Certain conditions on 1!T and D produce C-fractions, !-fractions, and P-fractions. If f 0 is holomorPhic in jz 15 R and jf0(z) - 115M in j z j5 R for R ~ 2, then there are uncountably ~ l!TD-fractions converging uniformly to f 0 in neighborhoods of the origin. Given 0 < c 51, lfk = l!T, if Id 0 I > (lf+l)M/al'(l-(lf+l)R-c) > O, ldk I~ Rc-l!T, dk+ d, 1d 1 = Rc-lf, then the resulting l!TD-expansion of f 0 converges uniformly to :t0 on compact subsets o:r 1z 1 < :al-c/l!T. Improvements in both radiue and speed of - (" l!T}"' convergence occur if the converging :!actors 1. ~z 1 are employed. This work was motivated by Waadeland 'a results concerniDg !-fractions ( Xgl. l!Toreke Videnak. Selsk. Skr., l!Tr. 8 (1964) ) • (Received February l.O, l.975.) *75T-B94 RICHARD FRANKFUR·l', University of Minnesota, Minneapolis, Minn. 55455. Subnormal weighted shifts ~related function spaces. Let V be a finite positive Borel measure on [0,1], with V(lO}) • 0 and tr( [a. 1]) > 0 for all 0 ~ a< 1. Let D denote the open unit disk. The measure ,IlL defined on D by d~(r,e) = (1/2~)dv-(r)d9 is called a symmetric measure. If O< p< oo, EP(.,u.) is defined as the class of all analytic !unctions f in D such that sup J'n.lf(rw)IPd~(w)

The notation and definitions we use can be seen in M. de Guzman, Differentiation of integrals in Rn (Universidad C. de Madrid; Madrid, 1974). In this paper we show some characterizations of a Busemann - Feller basis that differen tiates L1 , and we prove the "halo conjecture" in this particular cas. THEOREM.- Let B a Busemann - Feller basis, invariant by homothecies and translations. Let ~(B) = {R e B(O) I IRI ~ 1}. Then the following statements are equivalent: (i) B satisfies the (snong) Vitali property; (ii) B diffei•entiates Jf for each f e L 1(Rn); (iii) ~(B) is of finite measure; (iv) K(B) is a bounded set; (v) the halo function $(u) at infinity behaves like u (i.e. c1 u ~ $(u) ~ c2u). (Rece~ved February l.l., 1975.) (Author introduced by Miguel. de GuzmB:n.)

*75T-B96 Lawrence Fialkow, western Michigan University, ~ similarity orbit of ~ normal operator. This notice revises Abstract 7ll-47-23(these Notices 21(1974), A-195)). If N is a bounded normal operator on a separable complex Hilbert space AI, let ~(N) denote the similarity orbit of N in ~(~) and let ~K(N) denote the set of all A-388 compact perturbations of elements of 4(N). It is proved that ./ (N) ( ..JK (N)) is norm closed in ~(~) if and only if the spectrum (essential spectrum) of N is finite. If the essential spectrum of N is infinite and M is a normal opera tor whose spectrum is connected and contains that of N, then M is in the closure of~(N). If the spectrum of N is connected, this result characterizes the normal elements of the closure of ~(N). A hyponormal operator is similar to a non-quasidiagonal operator if and only if its essential spectrum contains more than two points. (Received February 10, 1975.) Applied Mathematics *75T-C24 ROBERT JEROSLOW, Carnegie-Mellon University, Pittsburgh, Pa. 15213. Relaxations from Logical Conditions Unexplained terminology is from our previous abstract, Cuts from Logical Conditions. To every proposition A, assign a matrix R(A), the matrix of the inequalities below transposed to the form R(A)v~ 01 by induction on the number of connectives in A. For P atomic and of the form F(t)). f, write inequalities v(P,j)). F(ej)y, j = l, ••• ,n, and v(P,O).:;; yf. For A= Bt\D, atld v(A,j) ). v(B,j)+v(D,j), j = 1, ••• ,n and v(A,O) ;;;: v(B,O)+v(D,O), to the inequalities of R(B)UR(D). For A= BVD, add v(A,j) ~ v(B,j) ,v(A,j) ?v(D,j), j = 1, ... ,nand v(A,O) ..'( v(B,O), v(A,O) ..'(. v(D,O) to the inequalities of R(B)UR(D). The v' s are the variables of these inequalities, and are introduced by the above construction. Let Ej be a unit vector, with unit in the position of variable v(A,j) , j = O,l, ••• ,n. Theorem: (tl, ••• ,tn) e cp(A) iff

there is au~ 0 with uR(A) +tlEl +•••• +tnEn =EO. Comments: Combined with our previous abstract, we now have a method of assigning to a proposition A in only "and" and "or," a relaxation cp(A) for which we have just given a linear programming formulation. The relaxations can be successively tightened, and when disjunctive normal form is reached, bounded linear logical problems will have been solved. Note: As in our previous abstract, t ;;;. 0 and F is conical. (Received November 22, 1974.) STEPHEN GROSSBERG and SAMUEL ELLIASr M.I.T., Cambridge, Mass.t 02139 75T-C25 Pattern Formation and Oscillations ~n the Short Term Memory ot Shunt ing On-Center Off Surround Networks. Preliminary Report. The transformation of spatial patterns and their storage in STM by shunt ing neural networks of the form xi -Axi + (B-xi) [Ek~lf(xk)Cki + Ii]- xi Ek~lg(yk)Dki n yi -Eyi + !k=lh(xk)Fki i=l,2, ••• ,n are studied. xi (yi) is the average potential of the ith ex citatory (inhibitory) population of network cells. Mechanisms are described for real-time regulation of the amount of contrast with which a pattern is stor ed. Mechanisms for removing peak splits and other disinhibitory responses are described. Periodic responses (stable and unstable) corresponding to the time scale of slow cortical waves can be generated if a tonic input is set between two threshold activity levels. Their frequency as a function of tonic input $ize is unimodal. Order-preserving limit cycles are never found in STM1 hence sustained slow oscillations as a mechanism for storing a pattern in STM are ruled out in favor of steady states (i.e., fast oscillations) with spatially graded activity levels. (Received January 29, 1975.) 7 6 STEPHEN GROSSBERG and DANIEL LEVINEr M,I.T.,, Cambridge, Mass., 01239 5T-C2 Developmental and Attentional Biases in Contrast Enhancement and Short Term Meroorv of Neural Networks. Preljmjnary Report Systems of the form xi= -Axi + (Bi-xi)f(xi) -xi kfi f(xk) + Ii ,

i=l,2, ••• n, are studied, given A>O, Bi>O, all xi(O)~ 0 and all Ii~O and continuous. They describe neural networks with an on-center off-surround anatomy undergoing shunting interactions. The constants Bi embody certain developmental, attentional, or statistical biases in the net. The nets can

A-389 contrast enhance significant inputs, sustain this data in short term memory, adapt the total network activity x = Ek~lxk at multistable equilibrium points, suppress noise, prevent saturation of network response to patterns of high intensity, and bias the processing of data in favor of states with large Bi values. The choice of signal function f(w) regulates the distribution -1 of the probabilities Xi = xi x as t + "" • (Received January 30, 1975.) *75T-C27 STEPHEN GROSSBERG, M. I. T. , Cambridge, Massachusetts 02139 Development of Specificity in Striate Cortex~ Applications to Learn ing and Reaction-Diffus1on. Prel1minary Report. Recently suggested developmental mechanisms for tuning of striate cortex are recaste as adult learning mechanisms. One mechanism (conservation of syn aptic strength) is replaced by an adaptational property of shunting on-center off-surround networks that facilitates parallel processing of patterns in the presence of noise. The revised developmental model uses only shunting on-cen ter off-surround networks and cross-correlational synapses. The adaptational mechanism can also account for some data on spatial frequency adaptation. Shunting network mechanisms and properties mimic those of certain reaction diffusion (RD) systems. For example, positional information due to regulation in RD systems is analogous to constancies due to network adaptation, firing of a RD gradient is like contrast enhancement of network potentials, maintanence of RD morphagen concentrations is like network short-term memory, morphagen source density is like network inputs, etc. (Received February 10, 1975.)

75T-C28 MICHAEL J. FISCHER, M.I.T. Project MAC, 545 Technology Square, Cambridge, Massachusetts 02139, H.F. DE GROOTE and A. SCHONHAGE, Fachbereich Mathematik der Universitat, D7400 Tlibingen, Auf der Morgenstelle 10, West Germany. On quaternion multiplication.

Let U = (u1, ... ,u4), V = (v1, ... ,v4) be quaternions over R. The quaternion product UV can be computed by a straight line algorithm from inputs u1, •.. , u 4, v 1, ••. , v 4 using the real operations: +, -, x. A real multiplication is active if neither operand is constant. Theorem. Eight active real multiplications are sufficient to compute the quaternion product. This improves a previous bound of ten [Fiduccia, "Fast matrix multiplication," Pro c. 3rd ACM Symp. on Theory of Computing, 1971, 45-49] and a recently announced bound of nine by J.C. Lafon. A computer program using numerical approximation techniques played an essential role in the discovery of the theorem. (Received February 10, 1975.) 75T-C29 W. JOHN WILBUR, 11209 Rosarita drive, Loma Linda, California 923.54. Complete Projective Logics and Normalizations. Preliminary report.

This is an extension of results announced in [Abstract 706-81-2, these ~ 20(1973)A-

543). For terminology and some standard results we refer to [Geometry of ~Theory. I, D. Van Nostrand, Princeton, New Jersey, 1968]. Let P be a complete projective logic and. let D be the division algebra and 8 the involu tive anti-automorphism corresponding toP. Let J be the fixed set under e and let C denote the center of D. Then'Phas normalizations if and only if for each a E J either~ or.j=a exists and is in c. Theorem. If"P has normalizations, J has a natural ordering and D has one 2 of the following three forms: (1) D=J; (2) D=J(I) where I is negative in J; (3) D=J(i,j) where i 2=j2=-1 and ij=-ji(the quaternions over J). In case(2) &(I)=-I and in case(3) e(i)=-i and 9(j)=-j. Theorem. If '!J is infinite dimensional and has normalizations then J is isomorphic to the real numbers. One corollary is that every complete projective logic corresponding to the complex field is isomorphic to the logic of closed subspaces of a complex Hilbert space. (Received February 10, 1975.)

A-390 Logic and Foundations

75T-E31 STEVEN GARAVAGLIA, Yale University, New Haven, Connecticut 06520. Exactness for equationally compact groups. Preliminary report. Definition. An abelian group G has the exactness property if the Cech" homology theory with coefficients G satisfies the exactness axiom on the category of compact pairs. Theorem. (i) An abelian group G is equationally compact iff G has the exactness property; (ii) if T is a complete theory of abelian groups then T is w-stable iff every model of T has the exactness property; (iii) there is an enlargement such that *(Zw) does not have the exactness property, where Z is the integers. (Received November 21, 1974.)

75T~E32 WILLIAMS FORREST, Mathematics Department, Simon Fraser University, Burnaby, B. C. V5A 1S6. ~ Representation o:f Open Formulas Over Integral Domains

Let· A be an integral domain with unit . An open :formula 1jJ ( x1 , ... ,xn) over A is a

quantifier :free :formula in the first order language .I,(A). The open formula 1jJ(x1 , ... ,xn) over A is algebraic i:f for all integral extensions B of A the solution set o:f 1jJ

in B is :finite. Theorem 1 Suppose ljJ(x) is an open formula over A. There is a

polynomial p(x) over A such that ljJ(x) := (p(x) = 0) or ljJ(x) :=-, (p(x) = 0).

Theorem 2 Let 1jJ(x1 , ... ,xn) be an open algebraic formula over A. There are

polynomials :r1 , ... ,:fn such that 1jJ(x1 , ... ,xn) := :r1 (x1 , ... ,xn) = 0 A, ... , A fn

(x1 , ... ,xn) = 0. (Received December 23, 1974. )

*75T-E33 A.H. LACHLAN, Simon Fraser University, Burnahy, B.C. Canada VSA 156 A trichotomy of W-stahle theories. Preliminary Report.

Let T be a countable complete w-stable theory and I (a) be the number of

nonisomorphic models of T in power ~a then one of the three following possibilities

holds: (i) for all a , I(a) :0 max(lal, w) with equality when a~ w , (ii) for all

a~ l , I(a) = ia+llw, (iii) for all a , I(a) ~ la+WI Ia!. The spectra of those theories

for which I(a) < w for some a , l

Let X be a separable, complete metric space, y the set of 2-element subsets of X, A C Y ls

~!!!!!!!.if i

A simplicity sentence is a first-order sentence all of whose models are simple. Similarly define a subdirect irreducibility sentence. Let A = be an algebra, B a Boolean algebra (B complete if A is infinite), and U an ultrafilter on B. The relation eu(A) = { E I A[BJI 2 : v a(a) A s(a) E U} is a congruence on the Boolean aEA

A-391 pawer A[B]. Denote the quotient algebra A[B]/eu(A) by A[B]/U and call it a Boolean ultrapower of A. Assume that the language of A is countable. Theorem. A[B]/U is simple (subdirectly irreducible) iff either U is w-complete and A is simple (subdirectly irreducible), or A satisfies a simplicity (subdirect irreducibility) sentence. A similar result can be stated for ultraproducts. (Received January 27, 1975.) Topology *75T-G36 MARK E. KIDWELL, Department of Mathematics, Yale University, New Haven, Conn.06520 ! Non-Interchangeable Link.

3 Let L be a link in s with two components K1 and K2 • L is called 3 3 interchangeable if there is an autohomeomorphism f: s ~ s such that f(K1) = K2 and

Theorem: There exists a link L with two unkriotted components which is not interchangeable.

Proof: Let 6(x,y) be the Alexander polynomial of L. If L is interchangeable, then

6(x,y) = 6(y,x). I have constructed a link with ten crossings and linking number 2

whose Alexander polynomial is

+ 2x3 - Sx2 + 9x - 2

Since 6(x,y) + 6(y,x), this link is not interchangeable. (Received January 22, 1975.) (Author introduced by Dr. Ronnie Lee.) *75T-G37 PAUL BANKSTON, McMaster University, Hamilton, Ontario LBS-4Kl. Ultraproducts In Topology. Preliminary Report.

(See 74T-G42 Feb. 74 Notices) The ~o~ ultraproduct construction in model theory provides motivation for defining a similar concept in topology. This construction is seen to preserve many popular topological properties (eg. Hausdorffness, complete regularity, non compactness, linear orderability) and to fail to preserve many others (eg. non-(complete regularity), normality, non-normality, compactness). In this paper we focus on preservation and develop several techniques which have fairly wide-reaching applications outside of the preservation question. Among the results are: Theorem 1: Every ultraproduct via a countably incomplete ultrafilter was a topology which is closed under countable intersections. Theorem 2. (GCH): Every regular space has an ultraparacompact (open covers refine to clopen partitions) ultrapower. Theorem 3: The spaces 2K, for K>c, fail to be linearly orderable, to be non-Archimedian, and to be uniformizable with a linearly ordered basis. Theorem 4: Any two regular perfect spaces have homeomorphic ultrapowers.

A proof of Theorem 2 obtains from Theorem 4 without recourse to the GCH. (Received January 24, 1975.) 75T·G38 HEELil!iiA SHRIKlL'l.NDE, University of Wisconsin, Madison, ';;j_sconsin. CliTrent Address: Pahlavi University, Shiraz, Irru1. ?reli~inary Report. :·::or:otopy properties of Dea:omposition Spaces. iet x, Y be co::J.tinua in En. Theorem 1. Shape(:T) = Bhape(Y) implies :;:PmodX has the same homotopy type as EnmodY. Tlleo,:ero 2. E~od X is

locally simply connected if and only if X is ne~o.:;:l;T 1-movable. We define

_,._ 21ecessary condition for En mod X to be sLr[>l;;r c Jm-:tected. This is also

A-392 a necessary condition for X to be arcwise com1.ec·i:;ct1.. 'l'heorem 3. If X is a solenoid of VanDantzig (not necessarily di:·.c~.~-c~ -~::·en En mod X is not simply connected, Theorem 4. If X is stronc;l;r ;,·.c. ·cl:·. -::, non-1UV continuum in E3 then E3 mod X is not simply connected. (Received February 4, 1975.) (Author introduced by M. s. Shrikhande,)

75T·G39 l?.T. CHURCH, Syracuse University, Syracuse, New York 13210. Discrete maps on manifolds. Prel:iroinary Report.

ret fll and rfl be second countable manifolds, and let f :fll ->- rf1 be a map. The

branch set Bf is the set of points at which f fails to be a local har!ec:m:Jrphism. Theorem. If f is discrete (i.e. each f-1 (y) consists of isolated points), then dim Bf =dim f (Bf) ~

n- 1. Specifically, f is open if and only if dim Bf =dim f(Bf) ~ n - 2, and f is not

open if and iilnly if dim Bf = dim f (Bf) = n - 1. That discrete open implies dim Bf =

dim f(Bf) = n - 2 was proved by Cernavskii and Vaisala. If, more generally, f is countable

(i.e. each f-1 (y) is countable), Vaisala proved (Duke Math. J. 1974) for n ~ 3 that

int Bf = !if, i.e. dim Bf .::_ n- 1. Here the restriction on n is removed. (For a countable

map f, int f (Bf) may be nonatpty. ) (Received February 6, 1975. )

*75T-G4o STEPHEN LEON LIPSCOMB, Naval Surface Weapons Center, KGR, Dahlgren, Virginia 22448. An imbedding theorem for metric spaces.

A simple solution to the imbedding problem for the class of separable metric spaces has been known for a long time: (Urysohn's Imbedding Theorem) A topological space is separable metric if, and only if, it can be imbedded in the topological product of countably many unit intervals. We see that products of the unit interval make an especially informative type of imbedding space since the finite (Lebesgue) dimensional separable metric spaces are those that can be imbedded in a finite product of intervals. The author has recently shown that this result concerning the finite case could be extended to arbitrary metric spaces if we use a topological generalization of the unit interval. This present paper shows that if we use this same generalization of the interval, then we can obtain an analogue to Urysohn's Imbedding

Theorem, Besides presenting the first unified results which simultaneously generalize both separable cases, this paper contains comparisons with existing imbedding theorems and imbedding spaces, (Received January 31, 1975.)

*75T-G41 Robert w. Button, Carnegie-Mellon University, Pittsburgh, Pa., 15213 When Do *Continuous Extensions Exist? In this paper we show that there is a large class of pairs of topological spaces (X,Y) such that every f : X~ Y has a *continuous extension. These pairs are particularly interesting because they satisfy the additional con dition given in the following theorem. Theorem. Let X be a Urysohn space, let Y be a pathwise connected space

and let f : X ~ Y be an arbitrary function. Then there is a ·*continuous g : *X~ *Y such that glx = f. Moreover, if X is not a Urysohn space, then

there is a f 1 :X ~m with no *continuous extension. If Y is not pathwise

connected, then there is a f 2 : m ~ Y with no *continuous extension.

A-393 Corollary. A topological space X is Urysohn iff each function f has a *continuous extension g : *X ~ *m. Corollary. A topological space Y is pathwise connected iff each function f : m .... Y has a *continuous extension g : *X .. *lR. (Received February ll, 1975.)

Miscellaneous Fields *75T-H2 ALBERT A, l't~ULLIN', 1500 Ronstan Drive, Killeen, TX. ?6,541 Historical remarks concerning priree-number theory.

This note oresents alternative logical views and historical addenda on the evolution of the so-called Fundamental Theorem of Arithmetic (unique prime factorization) to those by s. Bochner, Amer. ~Monthly 81 (1974) 827-852. It su~~ests that Euclid's three arithmetical books were never subject to the "logical scandals" of his geometric books, although there are logical flaws in Book VIII unlike Books VII and IX, It proposes that L, E. Dickson's brilliant History be up-dated to include, among other things, a chapter on unioue prime factorization (UPF) and its numerous generalizations; Dickson's History does not mention, let alone devote special attention to, UPF. The author defends the following theses• Proposition 14 of Book IX, a uniqueness theorem on prime factorization, contains the essence of UPF; Euclid states, but does not orove, the infinitude of the primes in Proposition 20 of Book IX; and Gauss did not rigorously prove UPF for all natural numbers in .§ 16 of Disquisitiones. Finally, it is oroposed that comprehensive historical biograohies be preoared for four creative mathematicians trained in America• H. F. Blichfeldt, O, H. ll"itchell, R. D. Carmichael, and E. 1. Post. (Received January 28, 1975.)

The April Meeting in St. Louis, Missouri April11-12, 1975

Algebra & Theory ofNumbers *723-Al BURTON FEIN~ Oregon State University, CorvallisJ Oregon 97331 and MURRAY oCHACHER~ University of California, Los Angeles, California 90024. ualois groups and division algebras.

Let L be a field and G a finite group. Define (L,G) to have Property ~ if every division ring of index IGI with center L is a crossed product for G. For K a field, let UD(K,n,m) denote the division ring of index n over its center which arises as the quotient ring of the domain generated over K by m generic nxn matrices where m ~ 2. Amitsur (Israel J. Math 12(1972), 408-420) proved that if UD(K,n,m) is a crossed product for G, then (L,G) has Property A for every L ~ K. Theorem 1. Let K be a global field of characteristic p ~ 0. For p ~ IGI, (K,G) has Property A if and only if G ;;; c x c where n ~ 1, nlm, n m and n,11 E K. If PI IGI and (K,G) has Property A, then G has a normal Sylow p-subgroup with factor group Cn x Cm where n ~ 1, nlm and nlf E K. Theorem 2. Assume char K = p > 0 and let q be prime, q # p. Then UD(K,q3r,m) is not a crossed product for any r. If K is, in addition, global and q is odd with qlf f K, then UD (K,q2r ,m) is not a crossed product. (Received January 20, 1975.) *723-A2 PAUL CONRAD, University of Kansas, Lawrence, Kansas 66045. 'Ihe hulls of semiprime rings.

Elements a,b in a semiprime ring G are disjoint if aGb = 0. Define a~ b if agb = !Jgb A-394 for all g £ G. Then this is a partial order for G. Denote the annihilator ideal of a subset X of G by X'. G is a P-ring if G = g" (!) g' for each g £ G; an SP-ring if G = X'' (!) X' for each subset X of G; an 1-ring if each pairwise disjoint subset of G has a l.u.b.; an 0-ring if G is an 1-ring and an SP-ring. We prove the following theorems for X = P, SP, L or 0. Theorem A. Let G be a semiprime ring and let H be a left essential extension of G that is an X-ring. Then the intersection of all the subrings of H that contain G and are X-rings is a minimal left essential extension of G that is an X-ring; called an X-hull of G. Theorem B. Each semiprime ring admits a unique X-hull Gx. Moreover GX is semiprime and GX is reduced (commutative) iff G is reduced (commutative). If G has a unit 1, then 1 is also the unit for GX . Finally, GX is the minimal right essential extension of G that is an X-ring. Almost all of the theory for X-hulls of lattice-ordered groups has a counterpart for semiprime rings. There is also a feedback from the rings to the groups. (Received January 20, 1975.) *723-A3 JUTTA HAUSEN, University of Houston, Houston, Texas 77004. Endomorphism rings and automorphism groups of abelian p-groups.

Let R be the full ring of endomorphisms of an abelian p-group A, where p ~ 5. Relations between the ideal structure of R and the normal structure of its group of units, the automorphism group of A, are discussed. Suppose that A is not locally

cyclic. ~· R contains minimal two-sided ideals if and only if its group of units contains minimal non-central normal subgroups. These conditions are equivalent to A being a direct sum of a bounded and a divisible group. (Received January 20, 1975.) *723-A4 DEREK J. s. ROBINSON, University of illinois, Urbana, illinois 61801. Cohomology of solvable groups of finite rank. A proof is given of the following conjecture. If G is a finitely generated solvable group in which every elementary abelian p-section is finite, G is a minimax group, i.e. G has a finite series whose factors have either Max or Min. This is deduced from theorems of P. Hall and A. I. Mal•cev on infinite soluble groups and from the following cohomological theorem. Let G be a solvable minimax group and A a G-module. Assume that ~ additive group A is torsion and its primary components have finite rank. Assume also that for no prime p does G have a pro-section and A a pro-subgroup. Then the cohomology groups !f-(G,A) are torsion and have their primary components of finite rank. In the proof of this theorem essential use is made of the Lyndon-Hochschild-Serre spectral sequence. (Received January 23, 1975.) 723-A5 DONALDs. PASSMAN, University of Wisconsin, Madison, Wisconsin 53706, Group Rings, Preliminary report.

The study of the semisimplicity problem for group rings of infinite groups has given rise to a number of interesting group theoretic structures. In particular, the characteristic subgroups D.+(G), A +(G). 3(G) (for G solvable) and J (G) (for G locally finite) will be discussed in this context. (Received January 23, 1975.) *723-A6 JULIAN WlLLIAMS, UNIVERSITY OF WISCONSIN-PARKSIDE, KENOSHA, WISCONSIN 53140 RELATION MODULES AS GENERATORS. PRELIMINARY REPORT.

Let G be a fixed finite group and consider a presentation of G l+R/[R,R]+F/[R,R]~ G+l. Where R, F are free groups and [R,R] denotes the commutator subgroup. The group R/[R,R]=R is a ZG-module via conjugation in F/[R,R] and is called a relation module. If the rank of F is equal to the minimal number of generators of G (d(G)) then R is called a minimal relation module. A ZG-module M is called a generator if ZG is a direct summand of a direct sum of copies of M (i.e. ZGIM" for some n). The problem I wish to consider is the following "classify all those finite groups G for which a minimal relation module is a generator". In particular the following propositions can be proved. Proposition 1 If G is a nilpotent group then a minimal relation module is not a generator. Proposition 2 If G ~ PSL(2,pr) then a minimal relation module is a generator.

(Received January 29, 1975.) A-395 723-A7 EVERETT LEE LADY, University of Kansas, Lawrence, KS 66045. Endomorphism rings of finite rank torsion free abelian~· Preliminary report.

Many properties of direct sum decompositions of finite rank torsion free abelian groups can be tmderstood by looking at the endomorphism rings of the groups involved. For instance, we can prove the following cancellation theorems: 1) If A@ H = B @ C and no quasi-sunnnand of A is quasi-isomorphic to a quasi-sunnnand of H , then

there exist decompositions B = B1 @ B2 , C = c1 @ c2 such that A @ H = Bl ffi c1 ffi H = A ~ Bz @ c2 . 2) If A ffi H = B @ C and no quasi-sunnnand of A is isomorphic to a quasi-sunnnand of C , then there exists a decomposition

B = B1 ffi B2 such that A@ H = B1 @ H =A@ B2 @ C . (Received January 24, ~975.)

*723-AB MOSS SWEEDLER, University of Cornell, Ithaca, New York 14853· Galois corings.

A K-coring is a K-bimodule M with suitable maps 6: M ~ M®KM and E: M ~ K. If M is a K-coring then the left (or right) K-dual to M is a ring (with unit E) to which K maps. The image of K in the dual need not be central; thus, the dual need not be a K-algebra. Birings and Hopf rings are defined in analogy with the development coalgebra, bialgebra, Hopf algebra. Example: sup pose K is a field extension of k then M = K ®k K is a K Hopf ring. The K-dual to M is naturally isomorphic to EndkK and the XK/k bialgebra struc ture on EndkK -- independently studies by Winter and Sweedler -- naturally arises from the biring structure of M. Galois Hopf rings are analogous to the duals of Galois Hopf algebras K 0k K is a Galois Hopf ring for extension K/k. There is a one-one correspondence between the fields intermediate-between-k-and K and the quotient Hopf rings of K 0k K. This result does not depend upon K being a finite extension of k and may even be generalized to the case where K and k are division rings. In the latter case K ®k K is merely a coring, and the one-one correspondence is between the fields intermediate between-k-and-K and the quotient corings of K ®k K. The Jacobson-Bourbaki correspondence is dual to this quotient coring correspondence. (Received February 3, 1975.) (Author introduced by Professor D. J. Winter.)

723-A9 GARY L. PETERSON, Mich. State Univ., E. Lansing, MI 48824. On the Automorphism Group of an Integral Group Ring II. Preliminary report. Let G be a finite group, Z(Gl denote the integral group ring of G, and NA(Gl denote the group of normalized automorphisms of Z(G) (see Notices,Janu ary 1975, 700-20-15, Al04 for definition). For a normal subgroup N of G, let 6(Nl denote the kernel of the natural map from Z(G) to Z(G/N). It was previously known that Aut(G) (the automorphism group of G) has a normal complement in NA(G) when G is metabelian. This result can be gener alized by setting W(G,N) = (f ENA(G) \f(g) = gmod6(N)6(G) }. Then, if N is abelian, W(G,N) is a complement for Aut(G) in NA(G) if and only if G/N is either abelian or a Hamiltonian 2-group. For N ~ G and S ~ NA(Gl, we say that N isS-admissible if f(6(N)) = 6(Nl for all f ES. N is said to be NA-characteristic if N is NA(G)-ad missible. If N is NA-characteristic, it follows that N ·is a characteristic subgroup of G. Further, the converse is shown to hold if G is either an E. R. group (see above reference for definition) or if G contains an abelian normal subgroup A such that W(G,A) is a complement for Aut(G) in NA(Gl. Admissiblity plays a role in studying NA(G) when G = G1x ..• xGn. It is shown that if (!Gil' IGj ll = 1 for i I j, then NA(G1 )x ... xNA(Gn) has a nor mal complement in NA(G). The proof of this result depends on the fact that each Gi is NA-characteristic. (Received February 3, 1975.) 723-AlO CHARLES LANSKI, University of Southern California, Los Angeles, California 90007 Rings with involution whose symmetric units are Abelian, Preliminary report Assuming that all symmetric units are central, one observes the existence of a polar decompo sition for the group of units, if norms have symmetric square roots~ Semi-prime rings whose symmetric units are central are characterized as subdirect products of domains and orders in 4-dimensional simple algebras. An example shows that in a domain, the symmetric units can A-396 scheme, which includes semi-linear representations of group schemes and descent data for modules as special cases. (Received February 17, 1975·) (Author introduced by Professor David J. Winter.) *723-Al6 CLEON YOHE, Washington Univerij~tY., St. Louis, Mo. 63130 Representing the automorphism group of a noetherian module. Let R be a commutative noetherian ring, A be a finitely generated

R-module, and G the group of R-automorphisms of A. A method is discussed whereby G may be represented as a subgroup of a finite direct product of general linear groups over fields. Conditions on A ensuring that the representation be faithful or epic are discussed. (Received February 17, 1975.)

723-Al7 RICHARD E. PHILLIPS, Michigan State University, East Lansing, Michigan 48824. Zero divisors in Rings and Centrality on Groups. Preliminary Report. The notion of a residually central group was introduced by Durbin (J.

Alg. 9, 408-413 (1968)) and Ayoub (J. Austral. Math. Soc. 9, 218-227 (1969)). Utilizing properties of the integral group rings ZG, where G is either finite or nilpotent, we are able to show that residually central abelian by finite groups possess a central system. This work is being done jointly with J. E. Roseblade. (Received February 17, 1975)

*723-AlB CHARLES FORD, Washington University, St. Louis, MO 63130 Finite groups and division algebras.

Let G be a finite group with an irreducible complex character with va1u~in an algebraic number field. For each prime p dividing the Schur index one can find a section. A central element is found in such a section with order pn equal to the p-part of the Schur index. The section has an irreducible character with Schur index pn As a character on the appropriate subgroup of G , it induces the given character on G above. (Received February 17, 1975)

123-A19 ARUN V. JATEGAONKAR, Fordham University, Bronx, N.Y. 10458. Morita duality and Noetherian rings, Preliminary report. Let R be a Noetherian semi-local ring which is complete in its J(R)-adic topology. Our main result shows that R has a self-duality (in the sense of Morita) iff each R/Jn(R) has a self-duality and these dualities satisfy a 'compatibility condition' which is too technical to state here. If R is commutative, this immediately yields a well-known result of Matlis. As another consequence of our result, we show that various complete semi local rings which arise from integral group rings of polycyclic-by-finite groups have self-duality. (Received February 17, 1975)

723-A20 Leonard E. Fuller, Kansas State University, Manhattan, Ks 66506 Generalized Lattices, Preliminary report

The concept of partial orderin~ is extenden to an n-ply partial ordered set s. Some properties of the ordering are proven. From these a set of bounds on a subset A of S are defined. These bounds are shown to have four bRsic properties. An L~ lattice is defined to be an n-ply partially ordered set S in which every subset of n elements have A set of n bounds defined with the prescriben pro- A-398 be Abelian without being central; also, the domain satisfies no standard chain condition or polynomial identity. In a prime ring with idempotent, the group generated by the norms cannot be solvable unless it is central. (Received February 3, 1975.)

723-All BRIAN HARTLEY, University of Wisconsin, Madison, Wisconsin 5 3706. A condition for complete reducibility. Preliminary report. v Let G be a Cernikov group with A its unique minimal subgroup of finite index. Let k be a field of characteristic p 2: 0, with prime field k 0 and algebraic closure k, and assume that k 0(A) n k is finite dimensional over k 0, where k 0(A) is the subfield of k generated by all n-th roots of 1 as n runs over the orders of the elements of A. Let V be a kG-module, and assume that the set of centralizers in V of p'-subgroups of A satisfies the minimal condition. Theorem. If either G is a p'-group or the Frattini submodule

Theorem. Let R be an infinite nil ring. Let G = (R,o) with a o b = a + b + ab for

a, b E G G has an abelian subgroup A such that 21AI :>: !G! . (Received February 3, 1975)

723-Al.3 Y~THA K. SMITH, University of Texas, Austin, Texas 78712. Centrality conditions in group algebras, Preliminary report.

The study of growth in groups and algebras prompts consideration of the following prop-

erty for a finitely generated ring R:

(*) For each homomorphic image S of R, every non-zero ideal of S intersects the

center of S nontrivially.

Roseblade (Bull. London Math, Soc. 3 (1971), 351-355) has shown that if R is the in-

tegral group ring of a finitely generated group G, (*) implies G is nilpotent. However, if

R is the group algebra of G over a field of characteristic zero, it is only necessary that

the ascending central series of G terminate in a subgroup of finite index. In characteris-

tic not zero, the situation is more delicate. (Received February 13, 1975.)

M. _f. NeWffiil!l, K. W. W~ston*, and Tah-Zen Yuan. University of Wisconsin-Parkside, *723-Al.4 Kenosha, W~scons1n 5jl40 Polynoffilals associated with groups of exponent four

Complicated groups of exponent four have been constructed from the ring of polynomials in associating non-commuting indeterminates with coefficients from the field of two elements. The justification of these constructions depends on a computational reductio result. In this note a further reduction is obtained. The expressions involved seem to have an interesting combinatorial structure. (Received February 14, 1975.) *723-Al5 S.U. Chase, Cornell University, Ithaca, N.Y. 14853· Inseparable Galois descent, Preliminary report. We prove a descent theorem for semi-linear representations of truncated group schemes which is an analogue, for purely inseparable field extensions, of the non-commutative version of Hilbert's theorem 90 [Serre, Corps locaux, p. 159], and generalizes to arbitrary exponent a theorem of Cartier on semi-linear representations of restricted Lie algebras [Bull. Soc. Math. France 86 (1958), pp. 282-283]. The result is closely related to the author's inseparable Galois theory [Am. J. Math., to appear]. A principal ingredient of the proof is the concept of a representation of a groupoid

A-397 perties. This is the ordinary lattice when n = 2. !c set of n n-ary operBtions n are defined on S to obtain Rn equivalent definition of 11 • Th8se concepts are n then generalized to a set of mxl column vectors over 11 to obtain another lattice which is denoted as 1~. (Received February 17, 1975)

723-A21 A. o. L. ATKIN, University of Illinois, Chicago, Illinois 60680. Supersingular games. e The superslngular values of the modular invariant j in characteristic q are related to recent theorems and conjectures of Serre, Swinnerton-Dyer, and the author. In addition an unusual connexion with Berlekamp•s factorization algorithm for polynomials in characteristic q is established. (Received February 17, 1975.)

723-A22 LINDSAY N. CHILDS, State University of New York, Albany, New York, 12222. Abelian Galois extensions as modules over the ~ring. Preliminary report. Number theorists have long sought to understand the RG-module structure of rings of integers S of Galois extensions L ~ K of number fields with Gal(L/K) = G, R = SG In case L/K is unramified at all finite primes, S is a Galois extension of R with group G in the sense of Chase Harrison, Rosenberg, and such S can thus be studied using techniques of Galois theory of rings. From this point of view when G is abelian it has been shown that any Galois extension of R with group G is a primitive element of Pic(RG). We survey what results are available towards a possible converse: any primitive element of Pic(RG) is represented by a Galois extension of R with group G. (Received February 18, 1975.)

723-A23 N. HEEREMA, Florida State U. , Tallahassee, Florida, 3 23 06 and J. Deveney, Virginia Commonwealth U., Richmond, Virginia 23284. Galois theory using higher derivations. Preliminary report. Let K be a finitely generated extension of a field k of characteristic p f 0. This report is concerned with Galois type correspondence especially between sub groups of the group of rank t, higher derivations on Kover k, and intermediate fields. If t < p, the fields of constants are intermediate fields over which K is purely inseparable modular. Moreover, every such field is a field of con stants. However, two distinct groups can have the same fields of constants. Thus, the main problem is to determine when a given group with field of con stants k 0 is the full group of all higher derivations trivial on k 0 • These groups are characterized in terms of certain types of generating sets. Although this approach to a purely inseparable Galois theory has certain disadvantages, namely the non-uniqueness of a group with a given field of constants and the fact that not all intermediate fields correspond to subgroups, it has some ad vantages. First, there is an intermediate theory related to the ciassical case. Secondly, this theory can be extended to inseparable extensions by allowing d 0 to be an automorphism rather than the identity map. If t = 00 , relationships are established between purely inseparable modular extensions and separable ones. (Received February 18, 1975.) 723-A24 RAYMOND T. HOOBLER, Mathematics Department, Rice University, Houston, Texas 77001. An Infinitesmal Version of Purely Inseparable Galois Theory. . A survey of the Gerstenhaber-Zaromp purely inseparable Galois theory W1th a sketch of some proofs will be given. An outline of how to pass between Chase's purely inseparable Galois theory and the G-Z theory will be presented. The relationship with Winter's approach may also be discussed. (Received February 18, 1975.) A-399 723-A25 H. F. KREIMER, Florida State University, Tallahassee, Florida, 32306. Hopf algebras and Galois cohomology, Preliminary report.

Let R be a commutative ring with identity element, and let H be a Hopf algebra with antipode over R. Assume that H is a finitely generated, projective R-module, and let H* be the dual of H. The definition of Galois H-object (Chase-Sweedler, "Hopf algebras and Galois theory", Lecture Notes in Mathematics 97, Springer Verlag; Definition 7.3) is easily extended from commutative R-algebras to non-commutative R-algebras, and this extended definition is used below. A Galois H-algebra S is said to have a normal basis if S and H are isomorphic as modules over the ring H*. Galois H-algebras S and S' are said to be isomorphic if there exists an R-algebra isomorphism of S onto S' which is also an H* module homomorphism. Theorem: If H is either commutative or cocommutative and there is a basis of open and closed sets for the structure space of maximal idea~s of R, then every Galois H-algebra has a normal basis. If H is cocommutative, i.e. H is commutative, a cohomological description of the set of isomorphism classes of Galois H-algebras which have normal bases is given. For any commutative Galois H*-algebra T, a mapping of the set of isomorphism classes of Galois H-algebras which have normal bases into the two dimensional Amitsur cohomology group H2(T/R) of T over R is described.(Received February 18, 1975.) 723-A26 ANDY MAGID, University of Oklahoma, Norman, Oklahoma 73069. The Galois Theory of Commutative Rings.

The goal of the Galois theory of commutative rings is to provide an elementary

description of the category of commutative, separable, projective algebras over a commutative base ring. If the base ring is a field, then the category in question is well-known to be anti-equivalent to the category of finite sets on which the (profinite) Galois group of the separable closure of the base field acts continuously.

In the general case, the category turns out to be anti-equivalent to the category of zero-dimensional compact topological spaces on which a certain profinite groupoid acts continuously. The groupoid in question is a (suitably topological) groupoid of partial automorphisms of a separable closure of the base ring. (Received February 18, 1975.)

723-A27 David J. Winter, University of Michigan, Ann Arbor, Michigan 48104 Normal Field Extensions and K/k - Bialgebras

The structure of a finite dimensional field extension K/k is closely related to the structure of a corresponding "K/k - bialgebra H(K/k) " (which is En~K as k-algebra) • The structure of H(K/k) can be studied in terms of "tori" (much as a Lie algebra is studied in terms of Cartan subalgebras). This talk is concerned with the relationship between K/k and H(K/k) and the role of tori in studying them, primarily when K/k is normal or purely inseparable. (Received February 18, 1975.)

723-A28 CARY WEBB, Chicago State University, Chicago, illinois 60628. Graded modules bounded below are direct sums of cyclics. Preliminary report. Suppose k is a field and M is a module over k[x] graded by the positive and negative integers. Theorem. If the degrees of the nonzero homogeneous components of M are bounded below, then M is a direct sum of cyclic (homogeneous) submodules. (Received February 18, 1975.) *723-A29 S. C. Goel and S. K. Jain, Ohio University, Athens, Ohio-45701. Semi-perfect rings with quasi-projective left ideals. R is called a left qp-ring if each left ideal of R is quasi-projective. The structure of right and left perfect left qp-rings has been investigated by Jain and Singh (Rings with quasi-proj ective left ideals, to appear in Pacific J. of Math.). In this paper we have studied left per fect left qp-rings and semi-perfect left qp-rings with nil radical. Some of the results proved are: (1) The following are equivalent: (a) R is a left perfect ring having all its finitely generated left ideals quasi-projective (b) R is a semi-perfect ring with nil radical, a.c.c. on principal left ideals, and all its finitely generated left ideals quasi-projective (c) R is a semi-primary ring having all its finitely generated left ideals quasi-projective. (2) Let R be a local ring with radical N. Suppose N is nil and R has a.c.c. on principal left ideals. Then the following are equivalent: (a) All left ideals generated by at most two elements are quasi-projective. (b) Either N2~(o) or R is a principal left ideal ring with d.c.c. (c) R is a left qp-ring. (3) Let R be a local ring with radical N. If N is nil and finitely generated then

A-400 the statements (a),(b), and (c) in (2) are equivalent. (4) Let R be a semi-perfect ring with nil radical N and a.c.c. on principal left ideals. If all the left ideals generated by at most two elements are quasi-projective then for each primitive idempotent e of R, eRe is a left qp ring. If we drop the ascending chain condition on the principal left ideals but assume that N is finitely generated then eRe is left artinian left qp-ring. Results concerning the global dimension for such rings have also been studied. (Received February 18, 1975.) Israel l'l. Herste:i.n, University of Chicago, Chicago, Ill:i.nois 60637 !he rel.at.ion between normal subgroups and ideaJ.s in rings.

The relationship between the normal subgroups and the group of units of a ring and the ideal structure of that ring is examined. (Received February 18, 1975.) Analysis 723-Bl ALLAN EDELSON and KURT KREITH, University of California, Davis, California 95616 Upper bounds for conjugate points of nonselfadjoint differential equations

The general real fourth order linear differential equation Ly = (p2 (t)y"- q2 (t)y 1 )"- (p1 (t)y 1 - q1 (t)y) 1 +Po (t)y = 0 with Pz (t) > 0 can be transformed into a second order system y" =a (t)y + b (t)x; x" = c (t)y + d (t) x = 0 which allows an interpretation in terms of a particle of unit mass moving in the (x,y)-plane. This representation has been used to show the existence of conjugate points ~(a) determined by the smallest B >a such that y(a) = y 1 (a) = 0 = y (~) = y 1 (II) is satisfied by a nontrivial solution of Ly = 0. By estimating the duration of various components of tl1e trajectory x(t),y(t) corresponding to this solution,

it becomes possible to obtain specific upper bounds for ~(a). The corresponding problem for equations of even order higher than four appears to open and of considerable interest. (Received January 8, 1975.) 723-B2 David R. Adams, University of Kentucky, Lexington, Kentucky 40506 Bessel Potentials and Bochner-Riesz Summation Operators

LetT~ denote the operator corresponding to the Fourier multiplier !1-J~ J2/R2)~. X>- 1, ~ERn, and R > 0. The idea is to determine both the values of Xand the size of the exceptional set of x's in Rn for which T~f(x) converges pointwise to f(x), as R tends to infinity, for all fin a "smooth" class of functions. For example iff= Pg, g E LP, P = Bessel potential operator of order a, it is sufficient to establish a so called capacitary weak type inequality for the

associated maximal operators: {*) Ba P [T~ o JC'lg > 1] .;;; c JJg JJP . B denotes a Bessel capacity restricted to an arbitrary ' p but fixed ball in Rn. Theorem: (*)holds whenever a+ X;;;.. n21, 1< p < oo. This improves an earlier result of Weiland (Studia Math. 44 (1972), 229-238). Conjecture:(*) holds whenever a+ X> (n-1) J _1.- iJ. It might be expected p 2 that(*) is much easier to establish than obtaining the usual LP estimates forT~ or even Tl. However, Theorem: Tl o Ja is a bounded operator on LP if and only if Tl is bounded on LP. And it is not difficult to see that(*) implies that TI o J01 is bounded on some Lq, q * 2 when p * 2. (Received January 9, 1975.) *723-B3 JACK K. HALE, Brown University, Providence, Rhode Island 02912 Bifurcation with several parameters

Let X,Z be Banach spaces, S:XxRn + Z be a Cr-mapping. If

T = S(·,O), TO= 0, then the problem is to describe the number of solutions

x of the equation S(x,A) = 0 for the vector A small by making some

generic hypotheses about the behavior of T on the null space of T 1 (0). The

theory and applications are given for the dimension of this null space equal to 0,1 and 2. (Received January 10, 1975.)

*723-B4 MITCHELL TAIBLESON, Washington University, Saint Louis, Missouri 63130. Every n-dimensional space over a local field has a natural field structure. Preliminary report. Let K be a local field (commutative p-field). Let Kn be the n-dimensional vector

A-401 space over K, The usual norm on Kn is given by \x\Kn = maxk \xk\K' wherex•(xl'x2, ... ,xn) and \xk\K = modK(~), Let K[T] be an algebraic extension of K of degree n. Write n-1 ) u E K['l'] as u = x1 + x2T + • , , + xnT , then u ~> (x1,x2, ••• ,xn establishes an isomorphism of K[T] and Kn as additive groups and modKn(•) is equivalent to \•\Kn' as is well known. Proposition: Let K[T] be the unique (up to isomorphism) unramified extension of K of degree n. Then modK[T](·) = (\ ·\Kn)n. Applications are given to the theory of Hp-spaces over Kn and to rP.lated topics in harmonic analysis on Kn, (Received January· 10, 1975.) 723-B5 SUI-SUN CHENG, University of California, Davis, California 95616. Systems conjugate and focal points of fourth order nonselfadjoint differential equations.

With reference to a system of second order differential equations y" = -a(x)y + b(x)z, z" = c(x)y - d(x)z, where a(x), d(x) 2:: 0 and b(x), c(x) > 0 are continuous on [x0 ,"'), ~ (x) is defined as the smallest ~ E (x ,"') such that (y,z)(x) = 0 = (y,z)(p) is satisfied 1 0 0 0 by a nontrivial solution of the system. Necessary conditions and sufficient conditions for the existence of ~ 1 (x0 ) are obtained. These lead to extensions of Barrett's results (Proc. Amer. Math. Soc. 12(1961), 205-213) on systems-conjugate and focal points to fourth order differential equations which are not selfadjoint. (Received January 13, 1975.)

*723-B6 RICHARD J. BAGBY, Washington University, St. Louis, Missouri 63130, Lipschitz Spaces and Fourier Transforms on Lorentz Spaces

By elementary means we obtain a real interpolation theorem for functions on Rn which satisfy Lipschitz-type smoothness conditions but no explicit global constraints. Using this we prove a version of Bernstein's theorem giving sufficient conditions for a function in the Lorentz space L(r,q) to be the Fourier transform of a function in L(r' ,q); namely that 1\L:>~ fliP\ Y\-n(l/p - l/r)E Lq(\ yrndy) where 1 :5 p :52 :5 r < "', p .; r . We then find sufficient conditions for a function f E L(r,"') to be an Lp,Lq multiplier. (Received January 15, 1975.)

*723-B7 A.P. CALDERON, The University of Chicago, Chicago, Illinois, 60637 and A. TORCHINSKY, Cornell University, Ithaca, New York 14853. Parabolic HP spaces.

Let F(x,t) = (f*~t)(x), where f is a tempered distribution in Rn and is an approximate identity of 1 ~t the form ~t(x) = (det At)- ~(At 1x), ~ in S, J~(x)dx ,£ o. We say that f E Hp if sup IF ( x, t) I E Lp' 0 < p. We t)o introduce the so called Lusin and Littlewood-Paley functions associated

with f and prove equivalences of the corresponding p norms involved.

Applications to multipliers, fractional integration, interpolation and to singular integrals with mixed homogeneity are discussed. (Received January 16, 1975.)

*723-B8 H. GUGGEN!lliiMER, Polytechnic Institute of New York, Brooklyn NY 11201, Oscillation and Disconjugacy in Topological Dynamics. creometric definitions (quite different from those derived from variational analysis) are given for conjugate and focal points of curves and flows. Applications are to the theory of oscillation and disconjugacy of x" f(t,x,x') and x' F(t,x,x 1 )x where x ~ R2 and F: RXR2x R2 2 2 --l>Hom(R ,R ), The paper will discuss problems connected with possible generalizations to higner dimensiom;. (Received January 16, 1975.)

A-402 723-B9 HENRY HERMES, University of Colorado, Boulder, Colorado 80302. High Order Controllability Conditions, Preliminary report.

Let M be an analytic n- manifold and lQ: dx/dt =X(x) + )' m u. (t)Yi(x) , ~ 1=2 1 be an analytic control system on M . For a collection, C , of vector fields on M , L(C) denotes the Lie algebra generated by C L(C)(p) the elements of L(C) evaluated at p

a 0 but this need not hold if Jui (t) I ,;; 1 If L{Y2, ••• , ym} defines an involutive distribution of dim. k < n , we give high order sufficient conditions that TX(t)p belong to the boundary

of a(t,p,~) for small t > 0 . These extend the authors results which appear in pro ceedings of the Vancouver congress. Next let ,.11 = { (adjX, yi): j ~ 0 , i = 2, ••. , m} . If L~) is solvable we give some results on when TX(t)p E int. a

723-BlO NORMAN C. MEYERS, University of Minnesota, Mpls., Minn. 55455, WILLIAM P. ZIEMER, Indiana University, Bloomington, Ind. 47401, Wirtinger and Poincare Inequalities for BV Functions. We consider bounded domains G C Rn that satisfy a regularity condition. For example,

Lipschitz domains are suitable. Let BV(G) be the space of functions whose distribution par

tial derivatives are totally finite measures on G • For f E BV(G) it is possible to define

f pointwise on closure G except possibly on a set of zero n-1 dimensional measure. Lemma: A positive measure ~ is in the dual of BV(G) iff for some k > 0 y(Br) ~ K rn-l where Br is any ball of radius r •

Theorem: There is a constant K such that if y is as in lemma and f E BV(G) , then

JG \f-0\ ~ K \lxll \IVf\I(G) where C = J fa~ , and y(G) = 1 Setting ~ equal to normalized Lebesgue measure or n-1 measure restricted to oG yields familiar Wirtinger (or Sobolev) type

inequalities, Also, if f ~ 0 on a set N of positive n-1 measure, there is a measure y carried by N such that \lxll is the reciprocal of a certain capacity of N , thus leading to a Poincare inequality. The capacity is the one used to determine the exceptional sets for BV f:mctions. (Received January 20, 1975. )

*723-Bll DON HINTON, University of Tennessee, Knoxville, Tennessee 37916. Principal solutions of positive linear Hamiltonian systems.

The Hamiltonian system Y' BY + cz Z' = -AY B*Z is considered where the co-

efficients are continuous on I [a,oo) c = C* > 0 and A = A* < 0 ' A solution (Y ,Z) satisfying Y*Z = Z*Y is said to be !-principal (2-principal) provided that (i) y-1 exists t t (Z-1 f Y-lcy*-ll-1 on I exists on I) and (ii) [ + 0 as t + 00 ( [f - z-lAz*-ll-1 + 0 a a as t + oo) Three conditions are given which are separately equivalent to the property

that a solution is !-principal iff it is 2-principal. For a self-adjoint scalar differential

operator L of order 2n , this problem is related to the deficiency index problem and to a

result of L. R. Anderson and A. C. Lazer (Trans. Amer. Math. Soc., 152 (1970), 519-530). (Received January 20, 1975.) 723-Bl2 WILLIAM J. FITZPATRICK and LOUIS J. GRIMM, University of Missouri-Rolla, Rolla, Missouri 65401. Differential Inequalities and k-point Boundary Problems.

Recently, L.J. Grimm and L.M. Hall (Springer-Verlag Lecture Notes in Mathematics 312 (1973), 41-53) have employed a scheme of monotone iteration of solutions of differential

A-403 inequalities to study solutions of two-point boundary value problems. In this paper we extend this procedure to k-point boundary value problems for nth order differential equations. (Received January 20, 1975.) *723-B13 GLENN SCHOBER, Indiana University, Bloomington, Indiana 47401. A variational method for families of K(z)-guasiconformal mappings. Let F be a family of plane quasiconformal mappings whose dilatation quotients (measures of local distortion) are bounded a.e. by a fixed function K E L~. Suppose f E F maximizes a real functional A over F • Under appropriate conditions on F and A , but not on K , variational consider ations lead to a functional differential equation satisfied by f • Some applications are given. (Received January 22, 1975.)

*723-Bl4 John L. Lewis, University of Kentucky, Lexington, Kentucky 40506 Examples of subharmonic functions in space

Let x = (x1,x2• • • ·, xnl be a point in Euclidean space, Rn, n;;;. 3, with r = 1x ~ x 1 = r cos 8, ()..; 8 .;;; 11'. Let Hm, m > 0, be Hausdorff measure in Rn. PutS= {x : jxj = 1} and C(B) = {xES: x 1 ;;;. cos 8} • If 0 < 'Y < oo, let lji'Y be the function satisfying: (sin8)2-n ~8 [(sinB)n-2 ~ (8)] = --'Y h + n- 2) lji'Y (8), 0<8 < 11', and lji'Y(O) = 1. Let A('y) M(r)} • Given e0, 0 < e0 .;;; 0, for which Hn-1 {yES: u(ry) > lji'Y(B0)M(r)};;;. Hn-1 [C(Boll. Theorem 1. I[O< p<'Y

tern y" + ~(t)y 0 where P is a continuous nxn matrix and the boundary value problem y(a) = y(b) 0, y(t) f 0, a < b. Under certain assumptions on~ and the assumption that b is the first conjugate point of a, the authors prove the existence of solutions of the problem whose components have certain properties. Stronger results are given for the case n = 2. Conversely, a sufficient con dition that b be the first point conjugate to a is given in terms of the exis· tence of such solutions is given. (Received January 27, 1975.)

A-404 *723-Bl7 M.B.SURYANARAYANA, Eastern Michigan University, Ypsilanti,Michigan 48197. Upper semicontinuity of set valued functions, Preliminary Report.

The purpose of this paper is to study various upper semicontinuity properties of set valued mappings used in control theory. Using the concept of semiclosure we present here the theory in a rather general setting, which brings about the underlying unity. We formulate certain relatively new semicontinuity properties, as yet unexploited in the literature. Among the several examples of semicontinuous set valued mappings, noteworthy is the class of maximal monotone operators on a Hilbert space. Indeed, we show here that maximal elements of certain classes of operators on Hilbert spaces are automatically semicontinuous in the sense of Kuratowski or Cesari. (Received January 27, 1975,)

723-B18 R,R.COIFMAN, Washington University, St. Louis, Missouri 63130 ~ Multilinear Singular Integrals

A general method developed by Yves Meyer and the author is applied to study multilinear singular integrals generalizing operators of the type

rra(x) - a(y))[b(x) - b(y)) f( )d ~ 3 y y where a', b', f (x - y) are in appropriate Lp spaces. Another application of the method yields estimates in Lr for 6-1(Vf • Vg) where V is the gradient 6-l the inverse of the Laplace operator and f c Lp (Rn) gc L q(Rn).! ~ .! + .! and where f(y) ,g(y) are supported in the quadrant y > 0 r p q 1 i • l ••• n (Received January 27, 1975.)

*723-Bl9 D.C. RUNG, Carleton University, Ottawa, Canada KlS SB6. Meier TYpe Theorems ~or General Boundary Approach and a-Porous Exceptional Sets.

Let {hn} be continuous fUnctions on [-1,1), (i) h(x) = h(-x); (ii) h(x) > o, x > 0, h(O) = 0 ;

9 (iii) h(x1 ) < h(x2 ), 0 S ~ < x2; (iv) hn(x) > hn+l(x) • Put h(~) = hc(x). At ei E c set

0 (a, h~) = {r eicpl r S 1 - h~cp-9)1; ~or 0 < a< b, RA(a, a, b,hn) = {reicpll-h~(cp-9) < r b i9 < 1-hn (cp-9), cp;::: 9} and LA(9, a, b,hn) with cpS 9 • If ~: D--> w, e is an hn-Picard point of ~ if the values taken in~initely often by f in each RA(A, a, b, hn) and LA(9,a, b,hn) i9 are all 0~ w except possibly two values; e is an hn-normal point o~ ~ i~ for each c > and seg_uences {z'l, z E 0 (9,hc), p (zn' implies X(~(z ),~(z' )) -->0 0 ' {zn}' n n n z~) --> 0, n n Main results: except ~or a a-porous set on C (i) c(~,O (9,h~)) = C(~, RA(9,a,b,hn)) =

C(f,LA(e,a,b,h )), any c > o, 0

723-B20 D. SATHER, University of Colorado, Boulder, Colorado, 80302. Bifurcation and stability theory for nonlinear gradient operators. Preliminary report.

A rigorous mathematical treatment of the general nonlinear theory of stability for thin elastic shells due to Keiter (e.g., see "Nonlinear Problems", edited by R. Langer, univ. of Wisconsin Press, Madison, 1963) leads in a number of cases to the study of an operator equation of the form (*) w- AAw + 0'2A2w + G(w ,0') = 0 , w EU' . Here U" is an appropriate real Hilbert space, A:U'-+U" is a positive, compact, self-adjoint operator and G:U'X R-+U' is a nonlinear gradient operator. Such equations include a number of well-known shell buckling problems in which A is a load parameter and 0' is a geometric parameter of the shell, e.g., buckling problems which involve uniformly compressed complete spherical shells and shallow spherical caps, and axially compressed cylindrical panels and shells. The approach used to study equation (*) involves a preliminary branching analysis of equation (*) by means of the Lyapunov-Schmidt method and the development of some new stability theorems for the resultant nontrivial branches. The indicated approach yields in some problems results which are not only completely rigorous but also more complete than earlier results obtained by various formal methods. For example, as part of some joint work with G. Knightly on buckled states of axially compressed "narrow", cylindrical panels and axisymmetric-buckled states of uniformly compressed complete spherical shells, it is shown that for A near the buckling A-405 load A one obtains a description of all stable and unstable buckled states which depend continugusly on the load parameter A ; in addition, the solutions obtained exhibit some of the important features of equilibrium states derived by more formal methods such as the exchange of stabilities at the buckling load, the formation of "dimples" in the case of a spherical shell, and "outward" buckling at the ends in the case of cylindrical panels. (Received January 31, 19'75. ) 723-B21 SHAPIRO, VICTOR L. Dept. of Mathematics, University or t:al.!fornia, Riverside, CA 92502, Mean Continuity and the Stationary Nonlinear Navier-Stokes Equations, Preliminary report.

Let the pair (y,p) be a classical solution of the stationary nonlinear Navier~

Stokes equations with zero external force in the punctured disc D(O,r0 )-(0l where y Suppose (i) i e > 0 such that v. is in L2+c [D(O,r J . J 0 j = 1, 2 such that JD\O,r)lv1 (x)-al 2dx = o(r2) as r-+0. Then y and p can be defined at 0 so that (y,p) is a classical solution in the whole disc D(O,r0 ) • A counterexample exists to show that the conclusion is false if "o" is replaced by "O" in (ii). Harmonic analysis is used in several places in the proof of the above result. (Rece'ived January 24, 1975.)

*723-B22 BURGESS DAVIS, Purdue University, West Lafayette, Indiana 47907. Picard•s theorem and Brownian motion. Properties of the paths of two dimensional Brownian motion are used as the basis of a proof of Picard's little theorem and its analog for complex valued functions defined on simply connected n di mensional manifolds which map certain diffusions into Brownian motion. (Received February 3, 1975,)

723-B23 LEON M. HALL, University of Nebraska-Lincoln, Lincoln, Nebraska, sesos. A c¥aracterization of the cokernel of a singular Fredholm dlfferentla operator. Let the functions b and u be analytic in the open unit disc and continuous in the closed disc, and define the operator L by Ly(z) = z 2 y ' (z) + b(z)y(z). Grimm and Hall (J. Diff. Equations, to appear) have found necessary and sufficient conditions for Ly = u to have a solution analytic in the open disc and continuously differentiable once on the closed disc. The ap plication of this result, however, requires knowledge of the cokernel of L, and involves solving a system of infinitely many equations in infinitely many unknowns. In this paper we find conditions which guarantee that coker(L) consists of functions in the Hardy space H2, and use the alternative problem techniques described by Hale (Lecture Notes 71-1, Division of Applied Math., Brown University, 1971) to solve the infinite system. (Received February 3, 1975.) 723-B24 SIMON HELLERSTEIN, University of Wisconsin and National Science Foundation, Washington, D.C. 20550 and JACK WILLIAMSON, University of Hawaii, Honolulu, HI 96822. Derivatives of Entire Functions and! Conjecture of Polya.

For each integer p .:. 0 denote by v2p the class on entire functions of 2p+2 the form f(z) = exp (-az )g(z) where a > 0 and g is of genus < 2p + 1, real on the real axis with only real zeroes. Set uo vo and for p > 1 only real put u2p = v 2p - v2p-2· Theorem 1. Let f e u2p' If f' has -----zeroes then f" has exactlr 2p non-real~· Theorem 2. Let F be !!!!_ entire function of finite order. If F, F' and F" have onlr real~

A-406 then F has one of the forms (i) F(z) = af(z), f E u0 (ii) F(z) = a exp (bz) (iii) F(z) = a{exp(icz) - exp(id)}, where a, b, c and d are constants

with c and d real. Theorem 1 affirms an old conjecture of Wiman. Theorem 2

verifies a long-standing conjecture of Polya in the case of finite order. (Received February 3, 1975.)

723-B25 ARTHUR J. KRENER, Department of Mathematics, University of California, Davis, California 95616. Structural Stability of Control Systems. Preliminary report. 00 Consider the control system on a C manifold M given in local coordinates by x=X0 (x)+uX1(x), lui ~ 1 where X. are c"" vector fields on M. Each measurable input u(t) generates a ].

trajectory x(t) defined for t ~ 0 and the points of M so covered are called the accessible points of the control system. If we restrict to a neighborhood N of x0 the points of N so covered are called locally accessible. In many examples it can be shown that boundary of this set is a stratified set, the disjoint union of a finite number of smooth submanifolds of N, each generated by a family of broken extremals. Such systems are called locally stratified, however their characterization is an open question. We give the space of systems the Whitney c"" topology. A locally stratified system is said to be locally structurally stable if all nearby systems are locally stratified and have boundaries generated by the corresponding families of extremals. Examples and some results in this direction are given. (Received February 3, 1975.) (Author introduced by D. L. Elliott.)

*723-B26 ADAM KORANYI, Yeshiva University, New York, N.Y., 10033, STEPHEN VAGI, DePaul University, Chicago, Ill. 60614 Inner functions on bounded symmetric domains.

Let DC[n be a bounded symmetric domain in Harish Chandra realization. Let S be its Silov boundary, De its Cayley transform. A bounded holomorphic function on D is inner if its boundary values on S almost everywhere have modulus one. The main results (for brevity stated only for irreducible D) are: (A) If D is of tube type then (1) The symmetry o at the base point of De is o = N-l grad N. N is an irreducible polynomial invariant under the stability group of the base point of De. The gradient is relative to the usual Euclidean metric of Cn(=ll.2n). The restriction of o to S cojncic!P.s with the jnvnluti.on z+-z of tn In the Jordan algebra realization o£ De, q(z) is the Jordan inver$e of z. (2) Every entire inner function on D is a power of N. (3) Every inner function continuous on the closure D of D is a rational function whose denominator has no zero in D. (4) Every rational inner function on D is of the form R(z) = Nk(z).Q(o(z))/Q(z), where Q is a polynomial with no zero in D; Q(z)=Q(Z). (B) If D is not of tube type, then every rational inner function on D is constant. Statements (1) and (2) extend results of Bojanic and Stoll (Proc. A.M.S. 13(1962), 115-116) and Bochner (ibid,, 117-120); (3) and (4) generalize results of Rudin for the polydisc (Function Theory on Polydiscs, W. A. Benjamin, New York, 1969). (Received February 5, 1975.)

723-B27 ALLAN PETERSON, University of Nebraska-Lincoln, Lincoln, Nebraska 68508. Comparison Theorems for Ordinary Differential Equations.

Let L be the classical nth order linear differential operator. At the outset we will be concerned with proving comparison theorems for the equation

Ly = k(x)y and the two comparison equations Ly = k 1 (x)y and Ly = k 2 (x)y where k1 (x) ~ k(x) ~ k 2 (x), a~ x

*723-B30 ROGER NUSSBAUM, Rutgers University, New Brunswick, New Jersey 08903 Some Generalizations of the Borsuk-Ulam Theorem, Preliminary report.

We prove the following result. Theorem. Suppose that X is a finite

dimensional metric ANR and that h: X ~X is a periodic homeomorphism of

period p (p a positive integer). Assume also that A {x e: X: h(x) x} is an ANR and that A {x e: X: hi(x) = x} for 1 ~ i ~ p-1. If W is an

open subset of X such that h (W) C W and if f: W ---7 X is a continuous

map such that fh = hf, f(W) is compact and {x e: W: f(x) x} is compact,

then it follows that ix(f,W) = iA(f,WnA) (Mod p). (The symbol i 2(g,U) denotes the fixed point index of a continuous map g: U- Z, U an open subset of Z). We prove that the above theorem implies some previous generalizations of the Borsuk-Ulam theorem and also gives new results concerning the topological degree. (Received February 10, 1975.)

*723-B31 LEE A. RUBEL, University of Illinois, Urbana, IL 61801 Harmonic analysis of harmonic functions. Let p be the complex plane and let E be the space of all entire functions. Let C(~) be the space of all complex valued continuous functions on p in the topology of uniform convergence on compact sets. Definition. A variety is a closed linear subspace V of C(P) s.t. if f E v and ~ E E then f o ~ E v. The main result is the six-varieties theorem that there are exactly six varieties; (0], ~' E, ~. H (all harmonic functions), and C(~). This completely resolves the problems of spectral analysis and spectral synthesis in this context. It includes the approximation-theoretic result that if f is a non-harmonic continuous function, then the span of the compositions f o ~ with ~ E E is dense in C(p). (Received February 10, 1975.)

723-B32 TSAI-SHENG LIU, University of Oklahoma, Norman, Oklahoma 73069. Oscillation of even order delay differential equations. Preliminary report.

The purpose of this paper is to consider the equations y(n)(t) + p(t)f(y(t),y(g(t))) = 0 and

y(n)(t) + f(y(t),y(g(t))) 0 (n ~ 2 an even integer) A-408 to find conditions which will ensure that every solution is either oscillatory or y(t)y•(t) > 0 for all sufficiently large t . This generalizes results of Chiou [Oscilla-

tion and nonoscillation theorems for second order functional differential equations, J. Math. Anal. Appl. 45(1974), 382-403] to higher order equations. In particular, it follows that every bounded solution is oscillatory under the given conditions. An example for the

occurrence of the case y(t)y•(t) > 0 is given. (Received February 10, 1975.)

*723-B33 JAMES DALY, New Mexico State University, Las Cruces, NM, 88003. The Invalidity of the Calderon-Zygmund Inequality for Singular Integrals over Local Fields.

We will show that the Calderon-Zygmund inequality II ,; C(p,rlllwP liT w p 'r is not valid in the local field setting, where i) Tw is the singular integral operator determined by the kernel

w, ii) is the L -operator norm, and iii) is the usual L -norm restricted 11-11 p p 11·11 r r to the unit ring (jxj = 1). (Received February 10, 1975.)

723-B34 RUSSELL D. RUPP, SUNY at Albany, Albany, N.Y. 12203. Distributed Parameter Penalization. Preliminary report.

An existence and convergence result is presented for a distributed parameter penalized Lagrange optimal control problem whose constrained form is to minimize JGL(t,~) (t),u(t))d~(t) subject to

(Dx) (t) = f(t,(Mx)(t),u(t)), and other control and state constraints,

all of which hold ~-almost everywhere. Here the control and state spaces

are arbitrary sets, and ~ is a countably additive measure on a a-algebra of subsets of G. This permits optimal, but non-measurable controls in cases where an Implicit Functions Lemma is not valid. Application is made to a problem arising in biochemistry, a minimum time problem in Banach space, and a standard multiple integral problem. (Received February 10, 1975.)

*723-B35 W. K. HAY~~. Imperial College, London and A. W. WEITSMAN, Purdue University, W. Lafayette, IN 47907. Coefficients and Means of Functions Omitting Values

Let f(z) = Eanzn be analytic in the disk U = {jz! < 1} and

IA(r,f) = ~TI J:Tijf(rei6)jd6 (0 (Zp)' ~d IA(r,f) ~ A(p,A)(~p)A(O < r < 1) if A< (Z~); further jan! < A(p)~pnZp-l (n > p). Here ~ = max Ia I and A(p,q, ••• ) are constants depending on P O~n~ n p,q, ••••

The proofs rest on an elementary potential theoretic inequality, and estimates on harmonic measure. (Received February 13, 1975.)

723-B36 • _E. M. Stein1 Princeton University P:J.nceton, N. ~ 08540. SJ.ngular Integrals on NJ.lpotent Groups. Further results in the application of singular integrals on nilpotent groups are described. (Received February 13, 1975.) A-409 *723·B37 TED J. SUFFRIDGE, University of Kentucky, Lexington, Kentucky 40506 On Univalent Polynomials

Let Sn denote the class of polynomials of the form z + a2z2 + · · · + _1. zn that are univalent in Iz I < 1. These n polynomials have an interesting geometric property which can be used to obtain considerable information about the

polynomial which maximizes Re F(P), P E Sn, where F is a given linear functional. Further, USn is dense in the class S of normalized univalent functions so that it is sufficient to study extremal problems for the class S over the restricted classes Sn. Also, for a.;; 1, a class Sn *(a) of polynomials z + a2z2 + · · · + _1.. zn can be defined such that n Sn *(a) c Sn when 0.;; a.;; 1 and such that f(z) = z + a2z2 + · · · , 1z 1 < 1, is starlike of order a if and only if there is a sequence {P nk} in S~k(a) satisfying~ Pnk (z) = f(z) (uniformly on compact subsets of 1z I < 1 ). These ideas can be used to prove a generalization of the Polya..Schoenberg conjecture. A similar characterization of the close-to-convex functions is possible. (Received February 10, 1975.)

*723-B38 RICHARD ROCHBERG, Washington University, St. Louis, MO 63130. Degeneration of Algebras of Analytic Functions, Preliminary report

For finite bordered Riemann surfaces s 1 and s 2 , let A(Si) be the

Banach algebras of continuous boundary value analytic functions on Si and let d(S1,s2)

= log inf[!iT!!!!T-lil; T an invertible linear map of A(s1) onto A(S 2)} • d induces a metric on R(S) , the Riemann space of the finite bordered Riemann surface S • Although the

topology induced by d is the Teichmuller topology, the metric space (R(S),d) is not

complete. Examples .of non-convergent Caughy sequences in (R(S),d) are discussed. These

examples are obtained by geometric degeneration of elements of R(S) • The corresponding

points in the Cauchy completion of (R(S),d) can be realized as algebras of analytic

functions on degenerate Riemann surfaces. (Received February 14, 1975.)

*723-B39 CHARLES COPPIN, University of Dallas, Irving, Texas 75061. Real-valued Set Functions Defined on Dense Subsets of [a,b]. The definitions of f is g-integrable on M and 8 are as in Abstract 663-8, these Notices 16 (1969), 89. aibfdg(M) = !fdg for each member M of 8 where M f is g-integrable on M. A real-valued function g with domain [a,b] is said to be quasi-continuous on [a,b] if and only if g has left-hand limit at each c in

(a,b] and g has right-hand limit at each c in [a,b). Theorem. Iff and g are real-valued functions with domain including [a,b], f and g have no common dis continuities from the left nor common discontinuities from the right and g is quasi-continuous on [a,b], then the range of aibfdg is dense in itself and between each two points of the range of aibfdg is another point of the range. (Received February 14, 1975.)

723-B40 W. J. COLES, University of Utah, Salt Lake City, UT 84112 "Some sufficient conditions for nonlinear oscillation:'

The equation y" + q(t,y) = 0, where q(t,y) compares with a ftmction of the form a(t)f(y) (a(t) not necessarily one-signed; yf(y) > 0 for y f O) is considered. Sufficient conditions for oscillation, in terms of weighted integrals of a(t), are discussed. (Received February 17, 1975.)

A-410 A.M. FINK, University of Minnesota, Minneapolis, Minnesota 55455. 723-B4l Open gueptions in quantitative estimates of disconjugary intervals.

We are interested in disconjugary interval theorems for linear differential operators of the form:

n-1 s. i~l \lai\\pi (b-a) ~Ci ~ 1 implies that. is disconjugate on

I = (a,b} • The problem is to find good constants c.• These constants are to be independent ~ of (b-a) and depend only on pi E [l,co] and n • Two methods for arriving at such estimates are discussed and the obvious places for improvement are pointed out. (Received February 17, 1975,) LLOYD JACKSON, University of Nebraska, Lincoln, Nebraska 68508. *723-~ Some questions related to the exist·ence of solutions of boundary value problems. Existence theorems for solutions of boundary value problems for equations of the form (1) y(n) = f(x,y,y', ..• ,y

, - ~) distribution of zeros on an interval I provided there exist points (i. - 1) = y J (t ) = 0 for • < tk in I such that y(tj) = y'(tj) =. j ., k. The oscillatory behavior of solutions of (E) is discussed under the

assumptions that no nontrivial solution of (E) possesses certain zero distributions on an

interval [a, ""), The results can be considered extensions of the well known results of

Leighton and Nehari for the equation [r (x)y") " - p (x)y = 0, r (x), p(x) positive and continuous on [a, ""), as well as Hanan's results for third order equations. (Received February 17, 1975.) 723-B44 WILLIAM T. REID, University of Oklahoma, Norman, Oklahoma 73069 Related self-adjoint differential and integra-differential systems.

The considered systems involve n-dimensional vector equations L1[u,v) = -v' + C(t)u

A*(t)v + Jb K(t,s)u(s)ds = 0, L2[u,v) = u' - A(t)u- B(t)v = 0, and two-point boundary con ditions al,j, q{l + Dvel'-, where: (i) on a given compact interval [a,b) on the real line the n X n matrix functions A,B,C belong to Lm, B* = B ~ 0, c* = c, while K is of class Lm on [a,b] x [a,b] with K(t,s) = [K(s,t)]*; (ii) A is a given linear subspace of R2n' Q is an hermitian 2n X 2n matrix, D = diag {-En,En}, and a, v are the respective 2n-dimensional boundary vectors (u(a);u(b)), (v(a);v(b)). Such systems and related boundary problems are shown to possess comparison and oscillation properties analogous to those of the ordinary differential system to which they reduce when K = 0, although some of the basic techniques A-411 for differential systems are no longer available. Particular attention is devoted to systems that are equivalent to related ordinary differential systems, or the composition of such related systems, together with the discussion of some unanswered questions. The presented results generalize those of a much earlier paper of the author [Amer. J. Math., 60 (1938), 257-292]. (Received February 17, 1975.)

723-B45 JERRY RIDENHOUR, Northern Illinois University, DeKalb, Il. 60115. Boundary-value functions, Preliminary report,

Let Ly =y(n) + Pn_1(x)y(n-l) + ··· + p0 (x)y = 0 be an n-th order linear differential equation with continuous coefficients on an interval I. For

positive integers i 1 , ••• ,ik with i 1 + ·•· + ik ~ n, the boundary-value function r. . (t) is defined for t in I by taking r. . (t) to be the 1.1 •• ,l.k 1.1 •• ,l.k infimum of the numbers r greater than t such that there is a nontrivial

solution of Ly = 0 with (i1 , ••• ,ik)-zeros on the interval [t,r]. Various inequalities among boundary-value functions are discussed and related open

questions are pointed out. (Received February 17, 1975.)

723-B46 C. c. TRAVIS, University of Tennessee, Knoxville, Tennessee 37916 On a Comparison Theorem for Scalar Riccati Equations.

A comparison theorem for the scalar Riccati equations

(1) 0

(2) s'(t) + s2(t) + q2(t); 0 will be stated which has as a consequence the following corollary: If (1) has a solution

r(t) existing on [a,b) such that ar(a) < 1 and

then (2) has a solution existing on [a,b) such that 0 < 1- ts(t) ~ 1- tr(t). The relationship of this theorem to several areas of oscillation theory will then be discussed. (Received February 17, 1975.)

723-B47 w. R. UTZ, University of Missouri, Columbia, Missouri 65201. Periodic solutions £f second ~differential equations ~~-linear, .!!2!l-differentiable damping.

Periodic solutions of x" + Q (x') + f (x) = 0 are determined with emphasis on cases where

Q is Lipschitzian but not differentiable. In case Q(x') is an odd function, with further

appropriate properties, a comparison theorem is given that applies to known useful 2 equations. In case Q is a function of x1 , conditions are determined to guarantee

a continuum of periodic solutions. (Received February 17, 1975) *723-B48 JOHNs. PAPADAKIS, University of Rhode Island, Kingston, Rhode Island 02881 and New London Laboratory, Naval Underwater Systems Center, New London, Connecticut 06320, A boundary value problem for equations of mixed type, The equation of elliptic-hyperbolic type utt - L; Ki (t) ux.x. = f(t, x1, x2), where Ki (t) :;: 0 if t:;: 0 is 1 1 considered in a domain D c R3. D is a generalization of the Frankl domain for the two dimensional Tricomi type equation studied by Morawetz and others. Under the same boundary conditions as in the two dimensional case the solution is shown to be unique. Also given that f(t,x1,x2) E H_1 it is proved that there exists a weak L2 (D) solution, The proof uses a priori estimates derived by the energy method. (Received February 17, 1975.) A-412 723-B49 LUC TARTAR, University of Wisconsin, Madison, Wisconsin 53706 Global existence for some semilinear hyperbolic systems.

We consider the solutions of the Broadwell's model:

u + u + uv - w2 0 With nonnegative intital data U(X,O) = u (X) t X 0 v - v + uv - w2 0 V(X,O) = VO(X) t X - 2UV + 2W2 0 W(X,O) = WO(X)

We prove that the solution exists for all t > 0 for every bounded initial data. We investigate also the case of data in LP (R). OReceived February 17, 1975)

•723-B50 ALEXANDER NAGEL and STEPHEN WAINGER, University of Wisconsin, Madison, Wisconsin 53706. L2 Boundedness of Hilbert Transforms along surfaces. Let a : lRk ..... JRn have the form:

Under certain restrictions on a, we show that the "Hilbert Transform along a"

00 dt dtn f(x1 , , • xn) ..... p, v. I I f(x - a(t)) ~ ••• -t- -oo -oo 1 n is bounded from L2(JRn) to L2(Rn). Using these transforms, we show that there is a large class of kernels on !Rn, which are homogeneous with respect to a multiple parameter group of dilations, which give rise to bounded convolution transforms on L2(JRn). OReceived February 17, 1975) *723-B51 Ronald C. Grimmer and William T. Patula, Southern Illinois Univer sity, Carbondale, Ill. 62901. Nonoscillatory solutions of forced ~order linear equations. Sufficient conditions are given for a forced second order linear equation to have at most one nonoscillatory solution. It is also shown that if the forcing function is of one sign and if the unforced equation is nonoscillatory, then every solution of the forced equation is nonoscilla tory. (Received February 17, 1975)

723-B52 RICHARD w. LEGGETT, University of Tennessee, Knoxville, TN 37916, On~ nonlinear integral equations. Preliminary Report.

By C(O) we mean the space of continuous real-valued functions on the compact set

Q c: Rn . A fixed point result is established for certain nonlinear, noncompact operators

T: C(O) ~ C(O) of the form Tx = x0 + (Lx)(Kx) , where x0 is a fixed element of C(O) ,

L (: C(O) ~ C(O)) satisfies a Lipschitz condition, and K (: C(O) ~ C{O)) is completely continuous. This result is applied to establish the existence of solutions of some nonlinear integral equations of the form

x(t) = x (t) + x(t) J K(t,s)f(s,x(s)) ds • (Received February 17, 1975.) 0 Q *723-B53 ALBERT BAERNSTEIN II, Washington University, St. Louis, Missouri 63130 and B. A. TAYLOR, Institute for Advanced Study, Princeton, New Jersey 08540, The * function for subharmonic functions in n-space. Let u be subharmonic in the spherical shell A: r 1 < lxl < r 2 in Rn The * function of u is the function u*(r1 9) "' sup { r u(ry)da{y} : a (E) "'a (K(B))} where a E •E denotes Lebesgue measure on the unit sphere sn-1 and K(~) denotes the spherical cap A-413 { y E Sn-1 : y1 2:, cos 9 J (y1 denotes the first coordinate of y }. The domain of u* is the plane semi-annulus A+,.{rei9:r1 < r < r 2 , 0 :;e ::>:~t} For n • 2 , the first author proved (Acta Math. 133 (1974}, 139-169} THEOREM A. If u is subharmonic in A , then u* is subharmonic in the interior of A+ • Applications were given in that paper to integral means problems for univalent functions. Earlier (Proc. London Math. Soc. (3} 26 (1973}, 418- 434} the first author had used a special case of Theorem A to give a proof of Edrei's "spread conjecture". In the present work we consider n 2:, 2 It is shown that u* satisfies, in a weak sense, the elliptic differential inequality * -1 * -2 * * u rr + (n-l}r u r + r (u 99 - (n-2} (cot 9 }u 9 } 2:, 0 This has also been proved (implicitly} by R. Gariepy and J. Lewis (Ark. Mat., to appear} using surface area theory. We give here a different proof, using ideas from harmonic analysis, which,we feel, sheds new light on the intuitive content of this inequality, even in the case n • 2 • (Received February 17, 1975)

723-B54 LONE YOUNG YEE, ~ashington University, St. Louis, Missouri 63130 ~ Commutators _!!! !!._

This paper is an extension to Rn of results obtained by Coifman, Meyer, and Calderon (to appear TAMS 1975}. Theorem: Let h(x} be homogeneous of degree -n -2, satisfying r • • + Jr; h(x'}dx' = ~ x;_h(x'}dx' = ~E h(x'}xi_xjdx' = 0 and JEl h(x'}jlog I h(x'}dx'

M(h,Va,Vb,f}(x) = cs>'o ljCa(x}- a(y})(b(x)-b(y)}h(x-y}f(y)dyl If Pl'P 2 ,P 3 ?::l,~•;t i2+ i3 lx-yl>c and 1:5 q

723-B55 JAMES A. JENKINS, Washington University, St. Louis, Missouri 63130. The mapping of strip domains.

A substantial part of the material for this talk is drawn from my joint work with ~taro Oikawa. The conformal mapping of strip domains was first treated in a unified manner by Ahlfors in his thesis. He derived two basic results which essentially relate to distortion under the mapping, the first of which is now called the Ahlfors Distortion Theorem. They were applied to two problems, first to give a solution of the Denjoy Conjecture, second to the problem of the existence of angular derivatives for which only partial results were obtained. A few years ago the speaker and Oikawa undertook a program to reexamine such results in terms of recent developments in the method of the extremal metric. (Illinois Journal of Mathematics, vol. 15, 1971, pp. 664-671}. In this context the Ahlfors Distortion Theorem becomes almost trivial. Further we gave a more complete version of Ahlfors "Second Fundamental Inequality". It is now observed that the restriction in this result that the strip be bounded by curves of bounded variation can be weakened. The Denjoy Conjecture has been treated in the same context (Canadian Journal of Mathematics, vol. 10, 1958, pp. 527-531, Mathematical Essays Dedicated to A. J. Macintyre, 1970, pp. 183-200}. Now we have given a solution of the problem of angular derivatives, giving necessary· and sufficient conditions for their existence and also such conditions in the case of unrestricted approach, A substantial part of the results does not require the assumption of univalence. (Received Februaryl7, 1975) *723-B56 GUIDO L. WEISS, Washington University, st. Louis, Missouri 63130. The use of Hardy spaces and their generalizations in harmonic analysis. • Motivated by the real variable characterization of classical Hp spaces obtained by R.R. Colfman in terms of "atoms," one can define the analogues of the Hardy spaces in a very general setting. Specifically, Hp spaces, p > 0, can be defined on spaces of homogeneous type (see Coifman-Weiss, Lecture Notes in Math. , vol. 242, Springer-Verlag). These general spaces include several classical Lie groups, boundaries of strictly pseudo-convex domains as well as many other situations in which one studies harmonic analysis. One can establish basic properties of these spaces that enable one to extend the theory of Calder6n-Zygmund singular integrals, study multiplier problems and obtain results on interpolation of operators. (Received February17,1975.)

A-414 *723-B57 HECfOR J. SUSSMANN, Rutgers University, New Bnmswick, NJ 08903 ~sufficient condition for~ controllability

Let S be·a finite set of vector fields in a neighborhood of p e~n. Call S locally controllable (L.C.) from p if every neig!Jborhood U of p contains a neighborhood V with the property that every point of V can be reached from p by a continuous curve in U which is a finite concatenation of integral trajectories of elements of S. If T is a set of vector fields, let T(p) = {X(p):X e T}, T(p)A =the convex hull" of T(p), and LT(p) =the largestAlinear subspace L of Hf with the property that 0 belongs to the I-interior of I~ T(p). (If no such L exists, put LT(p) = 0.) It is well known, and easy to prove, that S is L.C. from p if L = ~n, and that S is not L.C. from p if L = 0. Between these two extreme cases there is a gap which, presumably, can be filled by means of conditions involving the values at p of a suitable set of Lie brackets. (This should be possible when S is "finitely determined at p" in some reasonable sense.) Using A. Kraner's method of high order variations we have proved a theorem which constitutes a first step in this direction: Let s1 be obtained by adjoining to S all the brackets [X,Y], for all XeS, YeS such that X(p) and Y(p) belong to Ls(p). Then, if Is (p) = Bf, it follows that S is L.C. at p. (Received February 18, 1975.) 1

723-B58 LA.l®ERTO CESARI, University of Hichigan, Ann Arbor, Mich. 4810h and RANGACHARI KANNAN, University of Missouri at St. Louis, St. Louis, Mo 63121.

Periodic solutions in the large of nonlinear ordinary differential ~qua~ions •

On the basis of functional analysis and alternative methods the authors discuss the existence of periodic solutions of systems of Lienard·type nonlinear equations with periodic forcing terms under sole qualitative conditions • (Received February 18, 1975.) *723-B59 T. L. McCoy, Michigan State University, East Lansing, MI 48824 Some Necessary Conditions for Inner Functions.

Let D be a domain in the space of two complex variables (z,w). We say f is an inner function for D if f is holomorphic in D, JfJ ~ 1 in D, and f has radial limits of unit modulus almost everywhere on the boundary of D. Theorem; let B be the unit ball, and let f be inner for B. Then for every complex number c the slice function fc(z) = f(z,cz) is an inner function of z in the disc where B intersects the complex line w = cz. This theorem extends to several variables. Our method also proves a result of H. Alexander, to the effect that the only inner functions for the polydisc are constants, and yields a kind of localized version of Alexander's result. (Received February 18, 1975.)

723-BSO DANIEL WATERMAN, Syracuse University, Syracuse, New York 13210. On absolute high indices theorems, Preliminary report.

The absolute high indices theorem of Zygmund [T.A.M.S. 55(1944), 170-204] states that if

00 f(s) = E a e-Ans for s > 0, A1 > 0 and An+l/An ~ q > 1 for n > 1, then Lja J

468-4 78]. It has now been shown that for a> 0, Eja jA -a< A / 00 (1-e-s)ajf'(s)Jds. The 1 nn -qo theorem of Zygmund has been generalized by Levinson [Duke Math.J.31(1964),241-245] by replacing e-s by a function N(s). This result had severe restrictions on N(s) which have now been substantially reduced. (Received February 18, 1975.)

A-415 723-B61 Ph. BENILAN and H. BREZIS, University of Paris, and M. G. CRANDALL, University of Wisconsin, Madison, Wisconsin, 53705, A semilinear elliptic equation in L1(JRN), Preliminary report. Attempting to associate an m-accretive operator A in L1(lRN) with the expression Av = -.a.}..}

723-B62 F. W. GEHRING, University of Michigan, Ann Arbor, MI 48104 Open problems for quasiconformal mappings

This talk will be concerned with some open problems for quasiconformal mappings in the plane and for quasiconformal and quasiregular mappings in higher dimensions. (Received February 18, 1975.) 723-B63 KEITH PHILLIPS, New Mexico State University, Las Cruces, New Mexico 88003. Local field singular integrals with odd kernel. Preliminary report,

I Let K be a nondiscrete zero dimensional local field, R the ring of integers of K, D the units of R, P the maximal ideal of R, ~ a prime generating P,

Pk = ~~ for k E ~. ;k the characteristic function of (Pk)'. Let m be the modular function for K and w a function on D extended to K\{0} by homogeneity. If fnw = 0 and w satisties modulus of continuity and integrability conditions then the singular integral lim (;kw/m) * f exis~a.e. and in Lr norm (r > 1) (Phillips and Taibleson). k->oo If w is odd, the defining integrals over (Pk)' n P-n can be written in the form k . . ~JDw(y)Lj=-n[f(x - ~y) - f(x + ~y)]dy. The transform H(f;n,k) defined by the inner sum appears as a result of the fact that the value group of K is ~. The corresponding decomposition in Rm yields a Hilbert transform in each variable because the value group of

Rm is R+. Study of H(f;n,k) yields improved existence theorems. (Received February 18,1975)

723-B64 RICHARD A. HUNT, Purdue University, West Lafayette, IN 47907. Problems related to the a.e. convergence of Fourier series. Prel~minary repor~ ------The problem of determining those Orlicz spaces L~ for which

snf ~ f a.e. for all f e; L~ is related to the rate of growth of snf

for f e; L 1 and to the size of the Fourier coefficients en (f) for

n>l are considered. Known results and the nature of the relationship

between these results will be discussed. Also, the role of the conjugate

function in the proofs will be discussed. (Received February 18, 1975.)

A-416 Applied Mathematics

723-CJ. JAN·M. GRONSKI, Dept. of Math., U.M.S.L., St. Louis, MO 63121 ON ~ classification problem for the control vector fields

Let A(x) denote the set of attainability of a polydynamical system (PDS) B from x e M. Let G be the set of those x such that A(x) is open, and dim t(x) be the dimension of Lie algebra generated by~ at x. Theorem 1: If dim t(x) = n for all x e G, then G is open and the connected component of G containing xis the set C(x) = (yeG : A(y) = A(x)}. Theorem 2: Let x e JR 2 with A(x) closed and A(x) -/: JR 2 • If dim t(y) = 2 for y e ilA(x) then ilA{x) is either a simple arc separating the plane or a closed Jordan curve.

De·finition: We say that two PDS, ~ and B 1 , are equivalent if there is a homeomorphism H M-----} M such that H(A(x)) = A 1 (H(x)) for x e M. In case M = JR 2 and A(x) = ATXJ, author obtains necessary and sufficient conditions for two PDS to be equivalent. These conditions are basically expressed in terms of the boundaries of the sets of attainability. (Received November 11, 1974.)

*723-C2 D. REBHUHN, Vassar College, Poughkeepsie, N.Y. 12601, On the closure of sets of attainability, Preliminary report.------

There is an open set of control systems in the plane in which controls appear linearly and for which, generically, any set attainable from a point in positive time is a closed Whitney prestratified set. For the analagous systems in three space, closure of sets attainable in positive time cannot be a generic property unless "controls" depend on position as well as time. In three space,we can show that generically bad behavior appears on a closed two dimensional manifold. For any point off that manifold and for a sufficiently small positive time depending on the given point, the set attainable in the given positive time is a closed Whitney prestratifi~d set. (Received January 17, 1975.) 723-C3 ROGER W. BROCKETT, Division of Engineering and Applied Physics, Harvard University, Cambridge, Mass. 02138. The Geometry of the Space of Reduced Rational Functions. Preliminary Report.

Let .~ be either the real or complex field and let n be a positive integer. The reduced proper rational functions of degree n over §will be denoted by Rat(.~,n) = {(q sn-l+ ... +q s+q )/ n-1 1 o n n-1 I s +pn_1s + •.. +p 0 ) qi,pi £ .~. q(s), p(s) relatively prime}. We consider Rat(.~,n) as a metric space with p(q(s)/p(s), n(s)/m(s)) = Eilqi-nil+lpi-mil. Certain problems arising in control theory lead to the question of determining, as specifically as possible, the geometry of these spaces. We show that Rat(~,n) has n+l connected components, a particular component being characterized as those elements of Rat(J,f,n) having the same Cauchy index on (-00 , 00). On the other hand Rat(€. ,n) is connected. A sample of more specific results:

Rat(hf ,2) "' JR4 U (s1x :ni!') U TR 4, Rat(a:' ,2) = tk3xs1xTs 2 where "' indicates topological equi valence and TS 2 is the tangent bundle of the 2-sphere. The geometry of certain components of Rat(lf,n) are known for all n but others are still to be determined. (Received January 28,1975.)

723-C4 J. L. SEDWICK, Washington University, St. Louis, Missouri 63130 and D. L. ELLIOTT, Washington University, St. Louis, Missouri 63130. Equivalence of Nonlinear and Bilinear Control Systems.

Given a nonlinear control system, linear in the controls, all of whose terms have a common critical point, Lie algebraic conditions are established for the existence of a real

A-417 analytic transformation to coordinates in which the system is bilinear (that is, of type i =Ax+ Eui(t)Bix). The transformation is constructed locally by algebraic methods. The hypotheses used are analyticity, transitivity of the Lie algebra L associated with the bilinear system, and isomorphism of L to the Lie algebra of nonlinear vector fields associated with the original nonlinear system. (Received February 5, 1975.)

723-C5 ROBERT HERMANN, Rutgers University, New Brunswick, New Jersey 08903. Equivalence of systems and lnfinlte Lie groups. Consider a system of the form x= f(x, u, t), with x in Rn, u in Rm, t in R. Dlffeomorphisms which map systems into systems form an infinite Lie group. A good deal of contemporary systems theory can be unified as the study of equivalence of systems with respect to various subgroups of this group. As examples, classical work by Halphen, Picard, Vessiot and Wilczynski can be applied to classify certain types of linear, time-varying systems. Lie's theory of differential invariants plays a key role in the class ification of nonlinear systems by differential geometric methods. (Received January 20, 1975.)

723-C6 JOHN BAILLIEUL, Division of Engineering and Applied Physics, Harvard University, Cambridge, Mass. 02138. Some Optimal Control Problems on Lie Groups. Preliminary report. m Consider the controlled dynamical system X{t) = (A+ L u.(t)B)X(t) which we assume is controll i=l ~ able on a matrix Lie group G. Defining the trace form = tr{CtD) on the corresponding matrix Lie algebra~· we may formulate the following optimization problem: Find piecewise continuous controls ui(·) which steer the sy~tem from X0 £Gat time t = 0 to x1 £Gat time t = 1 so as to minimize the functional n = J: dt where Q: ~ + ~ is a self-adjoint linear mapping. In a number of interesting cases we shall find controls ui(•) which solve this problem and which allow us to write down the minimum return function n explicitly. These results generalize some of the classical theory of geodesics in that the constant drift term A is present and the Bi's are not required to form a basis for the Lie algebra~· (Received January 22,1975.)

*723-C7 RONALD HIRSCHORN, Queen's University at Kingston, Ont., Canada Topological Groups and Nonlinear Control Systems.

We consider the nonlinear control system ~ = f(x,u) which evolves on a real analytic manifold M The vector fields f(·,u) corres ponding to constant controls u generate a Lie algebra 'J.. of vector fields on M • In the case where i consists of complete real analytic vector fields the one-parameter groups of the vector fields in ~ generate a transformation group G of M We show that G can be given the structure of a topological transformation group of M in a natural way. The relationship between the Lie algebraic structure of ~ , the action of G on M , and the controllability of the system ~~ = f(x,u) is examined. We obtain an algebraic criteria for controllability and discuss applications. For example the series interconnection of two bilinear systems can be studied using this approach. (Received February 14, 1975.) *723-C8 VELIMIR JURDJEVIC, University of Toronto, Toronto, Canada, M5S lAl. Local controllability and the existence of stabilizing feedbacks,

Let (*)~~ = F(x ,u) be a control system defined on an n -manifold M , with admissible (J controls u taking values in c :IRm For u E fl , let X(x) = F(x , u) and let Yi (x) = ,aF (x , u) i =1 , 2 , ••• ,m Define ad0 X(Y) = Y , and aUi then define inductively adKX(Y) = [X , adK-lX(Y)] fol' all K > 0 , where [• , •] is the Lie bracket of vector fields. Let A-418 Adu(x) be the linear hull spanned by the vectors of the form

(1) If dim Adu (x0 ) = dim M for UE ints:l, then the corresponding solution x(x0 ,u,t) of (*) which originates at x0 at t = 0 belongs to the interior of the attainable set of from x0 for each t > 0 • Hence, is locally controllable.

(2°) If in addition to the preceding assumption, F(x0 ,u) = 0 for some u E int s:l , then there exists a function f defined in a neighborhood of x0 which takes values in s:l such that : = F(x , f(x)) is asymptotically locally stable at x0 • In general, local controllability does not imply local stabilizability. A possible explanation for this phenomenon is provided by the generic point of view: dim Adu(x0 ) = dimM is a generic condition. (Received February 14, 1975.)(Author introduced by Professor David L. Elliott.) *723-C9 H. F. WEINBERGER, Uhiversity of Minnesota, Minneapolis, Mn. 55455 Conditions for a local Pareto optimum in a Banach space.

Let u, c, and e be mappings with continuous first and second Frechet derivatives from a Banach space X to Banach spaces U, C, and E, respectively. U and E are partially ordered by positive cones with interiors. Let K = [x EX: c(x) = 0, e(x) ~0} • z EK is said to be a local Pareto optimum if there is a neighborhood N of z in K such that if x EN and u(x)~u(z), then x=z. * * * For fixed z E K let M=[[t,m,n} EU XC XE : t~o, n>O, ne(z) =0, t5u(z;x) +m5c(z;x) + n8e(z;x) = o-\1-x EX} It is shown that if lim inf min - [ ta2u(z;x ) +m52c(z,x ) + n -+co [t,m,n} EM n n n52e(z;x )] O , n n n * n - lim 5c(z;xn) =0, and liminfv5e(z,xn) ~0 whenever vEE , v ~0, and ve(z) =0 , then z is a local Pareto optimum. If 8c(z,•) is onto and has a bounded pseudoinverse, this condition with < replaced by < is necessary for a local Pareto optimum. (Received February 17, 1975)

723-ClO WILLIAM M. BOOTHBY, Hashington University, St. Louis, MO 63130 On the determination of controllability of bilinear systems. (Preliminary Report)

Let A1, ... ,Ar ben X n matrices determining a homogeneous bilinear system dx/dt = (u1A1 + u2A2 +· .. + urAr)x on the state space Rn-(0} with piecewise constant controls u(t) '"(u1(t), ... ,un(t)). Let A1, ....,Am be a basis of the Lie algebra..§. in

!!(n,R) generated by A1, .•. ,Ar' then it is known that the system is controllable if and only if them X n matrix with columns A1x, ... ,Amx has rank n for every x E Rn-(0), i.e. if and only if the(~ minors, which are n-forms in the variables x 1, ... ,xn do not have simultaneous zeroes. Controllable bilinear systems have been classified by the author (A transitivity problem from control theory, Jour. Diff Eqns, to appear). The problem considered here is the following: using this classification is it possible to determine in a finite number of steps involving only rational operations on the :Jatrices A1, ..• ,Ar l

Let APM be space of p-forms on the Riemannian manifold M and ~:APM + APM be the usual Laplace-Beltrami operator. Let SpecP(M) = {A E Rl There is 0 'f w E APM for which ~w = AW} be the spectrum of the Laplace-Beltrami operator. If A e SpecPM then the multiplicity of A is the dimension of {w e APMI ~w = AWL There has been a great deal of interest in the spectrum of a Riemannian mani- A-419 fold arising from both mathematics (e.g. Minakshisunderam-Pleijel type expan sions) and physics (e.g. expansions of forms in terms of eigenforms of 8). Unfortunately SpecP(M) has only been computed in very isolated cases. In this note we have reduced the problem of computing SpecP(G) for an arbitrary compact semisimple Lie group (with the Killing form metric) to a completely algebraic problem. The solution to this algebraic problem is known in general and can be readily computed in any specific case. Our methods also give the multiplici ties. We shall give some examples. (Received February 7, 1975.)

Logic and Foundations *723-E1 THOMAS G. McLAUGHLIN, Texas Tech University, Lubbock, Texas 79409 Finiteness Predicates for J\(jj~;,l

In their 1960 monograph on Recursive Equivalence Types, Dekker and Myhill pointed out that the natural numbers are characterized among the isols by sat isfaction of the first-order condition "is comparable with every entity". In 1967, Louise Hay published a proof that this same condition works for/\(~),the co-simple isols. In 1972, Ellentuck remarked that a much simpler argument gives a version of Hay's result for the modified condition "has all its pre decessors comparable with every entity", and that moreover this modified con dition is valid also in the co-hypersimple isols. He asked, then, whether the original Dekker-Myhill condition is good for the co-hypersimple case. The answer is yes. In fact, by means of a couple of purely recursion-theoretic lemmas (in the same general spirit as §3 of Hay's 1967 TAMS paper) we obtain the correctness inA(e)of each of the following two first-order finiteness conditions: I. "Is comparable with every highly decomposable entity". II. "Is comparable with every indecomposable entit¥". \ The proof of I extends- Hay's result so that it applies to the class /\teRJ of co-simple regressive isols; while II corresponds to what Dekker and Myhill actually proved. (Received January10, 1975.)

*723-E2 MANUEL L~urn, University of Connecticut, Storrs, Connecticut 06268, C?n~ence Relations ~ definability is lattices g! d.-recurs1ve y enumerable sets. Let oc. be an admissible ordinal, and let £ ( o..) denote the latt:l.oe of ~-recursively enumerable sets. Theorem: There is a largest congruence relation definable over E.( oc. ) in a language sui table for latt:l.oe theory. Similar results are obtained for filters and ideals replacing congruence relations. Some structure results for various quotient lattices of ~ ( "'-) are obtained. (Received January 22, 1975.) 723-E3 HILARY PUTNAM, Harvard University, Cambridge, Massachusetts 02138. ·Minimality and~ ordinals.

The extent to which a system of notations for ordinals is "minimal" depends on the presence of "gap ordinals" and the length of the "gaps". (A "gap" ordinal is an ordinal ')(. such that, for some y, there is no "real" in L (+d..- L y·) There are close connections between this fact and the behavior of intrinsic degree hierarchies. There are also connections with recent work of Jockusch and Simpson and also with recent work of Kechris. An un solved problem will be described. (Received January 29, 1975.) 723-E4 HARVEY FRIEDMAN, State University of New York at Buffalo, Amherst, NY 14226. Perspectives in algebraic recursion theory.

In ordinary recursion theory (generalized recursion theories) one has a set A with a partial binary operation F(x,y) on A , given by the Kleene enumeration theorem (generalized

Kleene enumeration theorems}. I.e., F(x,y) ~ ~x(y) . It is convenient to let = represent the undefined element, and consider algebras (B,•) , where B =AU{=}, and x• y=F(x,y) for x,y E A, and =• x=x • ==x. We discuss work of Wagner and Strong on such algebras; A-420 theorems and questions about their structure; decision problems associated with them; and their relation to the untyped (self-referential) A-calculus. This is a very promising underdeveloped area of recursion theory. (Received January 30, 1975.)

723-E5 CARL G. JOCKUSCH, Jr., University of illinois, Urbana, illinois 61801. Degrees of generic sets. Preliminary report. A degree of unsolvabllity ! is called generic if it contains a set which is generic for arithmetic (with respect to finite forcing conditions). If .! is generic, then the initial segment of degrees with greatest element !_: (i) is not a lattice and (ii) is not dense. (The proof of result (ii) uses an unpublished result of D. A. Martin's.) If .! is generic, the following also hold: (iii) ! is the supremum of the two nongeneric degrees, (iv) for any degree ,2 >! there is a generic degree £ incomparable with ,! such that ,! U £ = _2, and (v) if .Q. < £ < ,2 ~ ~ there is a degree .!! incomparable with £ such that £ U .!! = £. (Result (v) follows from (iv) and another result of Martin's.) For each result (vi)-(viii) it can be shown by category arguments (without use of the priority method) that there is a degree £ such that the result holds when relativized in the standard way to the degrees ~ £: (vi) there exist incomparable r. e. degrees, (vii) there is a nonzero r, e. degree ,!!; such that .!' = Q', and (viii) there is an r. e. degree £ < ,2' such that £' = Q". In some sense this shows, without using the priority method, that known recursion theoretic techniques (if consistent) could not refute (vi)-(viii). (Received February 3, 1975,)

*723-E6 GEORGE METAKIDES, University of Rochester, Rochester, New York 14627 and ANIL NERODE, Cornell University, Ithaca, New York 14850, R.e. presented vector spaces. Preliminary report. Recursively presenting one aspect of a model does not mean that other aspects of the model are also recursively presented. We study some cases of •recursive independence• of such model theoretic notions.

Example. Let V00 be the infinite dimensional vector space over a recursive field F consisting of all finitely nonzero infinite sequences of elements of F under pointwise operations of vector addition and scalar multiplication, A subspace W of V 00 is complemented (in £ (V00 ), the lattice of r. e. subspaces ofV~) if W isr,e.andthereisanr,e. subspace W• of V with wnw•=i.O} and WEllW•= V. w (X) (X) The r. e. (recursively) presented vector spaces are precisely those recursively isomorphic to V00 mod W for some space W which is an r.e. (recursive) subset of V00 , V has a dependence algorithm if V is r. e presented and there is a uniform effective procedure which applied to any n-tuple V 0, V 1, •.• , Vn _ 1 from V determines whether or not v0, V1 , ••• , Vn-l are linearly dependent. Porism. V has a depen dence algorithm if and only if W is complemented in J!.(V00 ). Definition. V E J!.(V00 ) is decidable if V mod V has a dependence algorithm. 00 Theorem,(F infinite) Thereexistsubspaces V of V00 which are recursive subsets of V 00 but not decidable. Corollary. There exist recursively presented vector spaces with no dependence algorithm. We apply the above to the study of the effective content of Morley rank, (Received February 3, 1975.)

723-E7 ROBERT SOARE, University of Illinois at Chicago Circle, Box 4348, Chicago, Illinois 60680. The infinite injury priority method One of the most distinctive tools in recursion theory has been the priority method whereby a recursively enumerable (r.e.) set is constructed to satisfy a sequence of conditions called requirements. Friedberg and Muchnik solved Post's problem by a method in which each requirement may be "injured" finitely often. This "finite injury" method is easily understood and has been widely applied. Sacks invented an apparently much more complicated method where requirements may be injured infinitely often. Using a key idea of Lachlan together with a further simplification the infinite injury method is reduced to a form whose construction and proof are very similar to the finite injury case. The method then yields those theorems of Sacks on r.e. degrees from his monograph (Degrees of unsolvability, Annals of Mathematics Studies, no. 55, 1963) and his density theorem (The recursively enumerable degrees !!!!_dense, Annals of mathematics, 80 (1964), pp 300-312). (Received February 4, 1975.)

A-421 *723-EB GERALD E. SACKS, Harvard University and Massachusetts Institute of Technology, Cambridge, MA 02138. Recursion ~ inadmissible ordinals. Preliminary Report,

Let L(S) be the set of all sets constructible in the sense of GBdel via ordinals less

than S· Various theorems concerning the S-degrees of subsets of S E1 over L(S),

already known to be true when S is E1 admissible, continue to hold when L(S) is primitive recursively closed. (Received February 6, 1975.)

723-E9 KENNETH KUNEN, University of Wisconsin, Madison, Wisconsin 53706, What good are ultrafilters ? • We review some of the current work on ultrafilters (ufs). Ufs are widely studied by diverse groups of people: model-theorists, combinatorists, and topologists. The methods used in these areas often overlap, as we point out by several examples. For the purpose of exposition, we (perhaps artifically) divide the subject into three areas: ufs on countable sets, ufs on small uncountable sets, and ufs on sets of measurable cardinality. Ufs on countable sets are used by model theorists, but have not been classified in detail by them, as (under the continuum hypothesis) they all behave alike regarding ultrapowers of countable models. To a combinatorist or a topologist, however, these ufs have much structure, and one studies selective ufs, P..points, Q-points, etc. Under CH (or MA) most reasonable questions have been answered, although we shall list a few exceptions. Without MA, there is still much room for in dependence proofs. The same techniques used for countable sets may be used to construct ufs on larger sets with various combinatorial properties. However, a large number of open questions remain regarding regularity and related questions on the cardinality of ultraproducts. If X is a measurable cardinal, most of the basic definitions regarding ufs generalize in an obvious way to x-complete ufs, Unfortunately, we lose our major tool for constructing ufs-transfinite induction. In fact, it is known that mere measurability of lt is not enough to imply very much of interest about x-complete ufs, so one usually considers strongly compact or super-compact x when dealing with these questions. (Received February 10, 1975,) *723-ElO WAYNE RICHTER, University of Minnesota, Minneapolis, Minnesota, 55455. Inductive definitions and reflecting ordinals. Preliminary report.

An operator ~:P(w) ~ P(w) determines a transfinite sequence (~S: s E ON) of subsets of w where ~a = U{Ug: g < a} J~J is the least a such that ~a+l =~a . For c a set of operators, JcJ = sup{j!PJ: t E C} . Nice characterizations of JcJ in terms of reflecting ordinals are known in the cases where C is the set of operators definable in IT 0 form, as n well as the cases where C is the set of operators second order definable in IT~ , or in ~ , form, We obtain similar characterizations for certain other sets c of second order definable operators. (Received February 12, 1975,)

Statistics and Probability 723-Fl MELVIN D. LAX, Southern Illinois University, Carbondale, Ill. 62901 and WILLIAM E. BOYCE, Rensselaer Polytechnic Inst., Troy, N.Y. 12181. The Method of Moments for Linear Random Initial Value Problems.

Theoretical justification is given for the use of the method of moments of functional analysis to find the mean and autocorrelation of the solution of a random initial value problem of the form

y(n) + q1 (t)Y(n-l) + ••• + Qn(t)Y = G(t) ,

Y(O) = Y'(O) = ... = y(n-l)(O) = 0 where Q1 , ••. ,Qn,G belong to a certain class of stochastic processes. The use of random Taylor series in conjunction with the method of moments for such problems is discussed. Three examples are given to illustrate the application of the method to particular problems. (Received January 16, 1975.) A-422 *723·F2 MICHAEL D, GRADY, University of Utah, Salt Lake City, Utah 84112 Sufficient conditions for an operator valued Feynman-Kac formula

Sufficient conditions are given for the existence of a solution to au/at = Au + Bu, u(O) = f where A is the generator of a c0 semigroup and Bu(x) = V(x)u(x) where each V(x) is the generator of a c0 semigroup and u(x) is an element of the domain of V(x) for almost every x in a second countable locally compact, Hausdorff space E. (Received February 13, 1975·) Topology *723-Gl S.P. SINGH &MARY VEITCH, Memorial University, St. John's, Nfld., Canada, Fixed Point Theorems for Mappings with a Convexity Condition, Preliminary report.

Theorem 1.· Let T : K + K be a demicontinuous mapping defined on a non empty, convex, weakly compact subset of a Banach space X. Suppose 1 - T is quasi convex on K, and inf{l lx- Txl l/x E K} = 0. Then T has a fixed point. Moreover, if 1- T is strictly quasi-convex, then T has a unique fixed point. A few known results due to Belluce and Kirk, Montagnana and Vignoli, and, Singh and Holden become corollaries to this theorem. Theorem 2. Let K be a closed, bounded, convex subset of a reflexive Banach space X and let F = {TA} be a commuting family of 1-set contractions of K to itself such that is quasi convex for each TA in F. Then the family F has a common fixed point in Theorem 3. Let S and T : X +X be functions on a complete metric space X. Let S and T satisfy (Sx,Ty) s q.max{d(x,y),d(x,Sx),d(y,Ty),d(x,Ty),d(y,Sx)}, where 0 S q < 1. Then S and T have a unique common fixed point. This theorem generalizes the results due to Kannan, Reich,and Hardy and Rogers. (Received January 27, 1975.)

*723-G2 JAN J.AWOROWSKI, Indiana University, Bloomington, Indiana 47401. Euclidean G-retracts, Preliminary report.

This paper is a characterization of euclidean neighborhood retracts in the category of G-spaces (G-ENR's), Where G is a compact Lie group. If X is a locally compact G-space with a finite nlllliber of orbit types and if G is embeddable in a euclidean space then X is a G-ENR if and only if the fixed point set of every isotropy subgroup of G is a topological ENR. An analo gous condition characterizes G-ER's. (Received January 22, 1975.)

*723-G3 JOHN CONNETT, Northern Illinois University, DeKalb, Illinois 60115 Cyclic Group Action and Coincidence Points

Recent theorems of Cohen, Lusk and the author are summarized. These are generalizations of the Borsuk-Ulam theorem and results of Munkholm, Bourgin, and Yang. The method of proof

involves analysis of the cohomology of configuration spaces and related spaces. The following are typical results:

Theorem A. If X is an (n- l)(p- 1) + 1- connected Hausdorff space and p; X +X has prime period p and f: X+ Rn is continuous, then there exists x E X such that f(x) f(~(x)).

Theorem B. If n is odd and p ~ 3 is prime, and~: Sn(p- l) + 1 + Sn(p- l) + 1 has period p, and f: sm X sn(p- l) + l + Rn is continuous, then the set

A = {(t, x) E Sm x Sn(p - l) + l/ f(t, x) = f(t, ~i (x)O for all i, l~ i ~ p - l} has cohomological dimension at least m. (Received February 3, 1975,)

A-423 *723-G4 JOAN S.BIRMAN,Columbia Univ.,N.Y.,N.Y.l0027,& FRANCISCO J.GONZALEZ ACUNA,Univ.of Mexcio,& J.M.MONTESINOS,Universidad Complutense,Madrid, Spain.Heegaard splittings of prime 3-manifolds are not unique. Let M be a closed orientable 3-manifold, and suppose that M = U U V U'U v' are two Heegaard splittings of genus g of M These splittings are equivalent if there is a homeomorphism h: M • M such that h(U) = U' or v•, and h(V) = v' or u' • Examples are given of an infinite family of prime, closed, orientable 3-manifolds (each being a homology sphere of Heegaard genus 2 ) each of which exhibits at least two equivalence classes of Beegaard splittings of genus 2. These manifolds also exhibit a second "bad" property: each may be represented as the 2-fold covering space of s 3 branched over a prime knot K , and also over a second prime knot K' , where K and K' are inequivalent knot types. The nature of the examples is such that one eXpects the phenomena is not an isolated one, and probably happens repeatedly for 3-manifolds which are

sufficiently complicated in structure. (Received February ~8, 1975.)

723-G5 LOUIS M. FRIEDLER and DIX H. PETTEY, University of Missouri, Columbia, Missouri 65201. ~ ~ !!!2. Mappings ~~Topological Spaces.

A conjecture of S. Willard that every H-closed space is the continuous image of a minimal Hausdorff space is verified. It is also shown that every R-closed space is the continuous image of a minimal regular space. Using the proof of this last result and an example due toR. Stephenson, Jr., an R(ii) space is constructed whose product with itself is not R(ii). Both mapping theorems use inverse limits. The inverse limit of non-empty H-closed spaces is proved to be a non-empty H-closed space. Sufficient conditions are given for an H-closed space to be the finite-to-one continuous image of a minimal Hausdorff space. (Received February 13, 1975.)

*723-G6 HOWARD LAMBERT, University of Iowa, Iowa City, Iowa 52242. Unknotting links in s3 by maps.

Let L be a (PL) link in the Euclidean 3-sphere S3 (i.e. L = U~=lLi' where each Li is a

polygonal simple closed curve in s3 and Li nLj = 0, i 1 j). Call a continuous (PL) map

f:X+X' strongly 1-1 on YCX if flY is a homeomorphism onto f(Y), f(X-Y) n f(Y) = 0 and f is locally 1-1 at each point of Y. Is it possible to find a map of s3 onto itself which is strongly 1-1 on Land each f(Li), i 1,2,···,n, is unknotted? If n = 1, the answer is yes (John Hempel, A surface in s3 is tame if it can be deformed into each complementary domain, Trans. Amer. Math. Soc., Vol. 111(1964), pp. 273-287). In this paper we show the answer is no

for n ~ 2. (Received February ~4, 1975.)

723-G7 ROBERT J. DAVERMAN, University of Tennes~e, Knoxville, Tenn. 37916 Sewings of collared objects. Preliminary Report.

This is an expansion of work announced in Abstract 717-G2, these Notices 21 (1974), A-617. A collared object C in Sn is a compact subset of Sn such that the frontier of c,

relative to sn, is collared from sn-c A sewing of two collared objects is a homeo-

morphism of their frontiers. A sewing h of collared objects cl and c2 is said to satisfy the Mismatch Property if there exists sets F. in Frontier such that ]. ci

Fi Int ci is 1-ULC (i=l,2) and h(F1)r! F2 = 0 Theorem. If cl and c2 are sn collared objects in (n ~ 5) and h is a sewing of cl and c2 that satisfies the A-424 Mismatch Property, then c1 1 th c2 is homeomorphic to Sn • The corollaries to this result describe unusual involutions on sn and decompositions of sn that yield sn

(Received February 17, 1975)

723-GS MARY-ELIZABETH HAMSTROM, University of Illinois at Urbana-Champaign, Illinois 61801. Ambient Isotopy Classes in Graph Complements, Preliminary repor In mimeographed notes (University of Liverpool, 1970), G. P. Scott

outlined a proof that (in the P. L. category) the space H(M,~M) of a solid torus M onto itself leaving its boundary pointwise fixed is homotopically trivial. Here M denotes the complement in s3 of a tubular neighborhood

of a connected graph. The space H(M,~M) need not be connected, but a method for "almost'proving this is described that in many instances is a real proof and which seems to open the way for an application of Scott's

techniques to a study of the PL homotopy type of such H(M,~M).

(Received February 17, 1975)

*'723-G9 s. E. Rodabaugh, University of Missouri-Columbia, Columbia, Missouri 65201. Upper semicontinous decompositions of~ metric spaces, Preliminary report.

All decompositions in this paper are upper semicontinuous. Theorem 1. If G is a locally null decomposition of a locally compact, strongly convex without ramifications metric space X having closed edges and G has Property N, then G yields x. Corollary. If G is a locally null, starlike equivalent decomposition of En, then G yields En. Theorem 2. If G is a star-0-dimensional decomposition of a locally compact, strongly convex without ramifications metric space X having closed edges, then G yields x. Theorem 1 and an example given in this paper answer a question of J. W. Lamoreaux [Thesis, University of Utah(l967)]. The corollary strengthens a theorem of R. J. Bean [Illinois J. Math. 11(1967), 21-23]. Theorem 2 strengthens a theorem ofT. M. Price [Trans. Amer. Math. Soc. 122(1966), 427-435].(Received February 18, 1975.) *'723-GlO H. H. WICKE and J. M. WORRELL,JR., Ohio University, Athens, Ohio 45701. The hered ~ Lindeltlf property, primitive structures, and separable metrizability. ---

Theorem 1. Let X be a regular T0 space having primitive bases locally. Then the following conditions on X are equivalent: (a) Separable metrizability. (b) Hereditarily weakly oS-ref inable and hereditarily separable. (c) Hereditarily weakly 58-refinable and hereditarily jt1-compact. (d) Her.editarily Lindeltlf. (e) Closed sets are locally sets of interior con densation and Lindeltlf. (f) Weak o8-refinability,~1-compactness, and closed sets are G0 's. ~ 2, Let X be a regular T0 space which is locally primitively quasi-complete. Then conditions (c), (d), (e), and (f) of Theorem 1 are equivalent to X being an hereditarily Lind elBf p-space, If X also has a diagonal which is locally a primitive set of interior conden sation, then all conditions of Theorem 1 are equivalent for X. The following simple theorem, used in the proofs of 1 and 2, has a forerunner in a theorem of C. Aull on aS-bases [Preprint].

~ 3. A topological space is hereditarily Lindeltlf if and only if it is hereditarily weakly 58-refinable and hereditarily!tt-compact. Remarks.l.Theorem 2.5 of H. Bennett[Gen.Top. Appl.l(l971) 256] may be deduced from Theorem 1. His Example 2.6,loc.cit., shows that the word hereditarily may not be validly omitted from the conditions of Theorem 1. 2. Ostaszewski's example [M, E. Rudin, Notes for the 1974 Univ. of Wyoming Topology Conf.] shows that in ZFC the condition of hereditary weak o8-refinability is nonsuperfluous in the above theorems. (Received February 18, 1975.) A-425 723-G11 KUO-TSAI CHEN, University of illinois, Urbana, illinois 61801. Iterated path integrals.

• Instead of using a single integrand, we extend the usual notion of integration by taking a number of differential forms and integrating them repeatedly. SUch iterated integrals are integrals in the usual sense on path spaces and provide a link between analysis and topology of manifolds (or more general dif ferentiable spaces). (Received February 18, 1975.)

Miscellaneous Fields *723-Hl T. J. LANOUE, Westinghouse Electric Corp., MIR M. ALI, Ball State University, Muncie, Indiana 47305. An Alternative Method For Approximating the Potential and Gradient At Any Point Between An Insulated Cable to Plane. In a previous paper (Abstract 720-99-4, these Notices 22 (1975)) a mathematical method

for determining the approximate potential and gradient at any point in the insulating mediums separating a charged conducting cylinder and a conducting plane was presented. The method assumed there was that the interface of the two insulating mediums lie on an equipotential contour. With this assumption, only one analytic transformation was necessary for obtaining the solution to the problem. In this paper an alternative approach to this problem is made by transforming each medium independently with an appropriate analytical transformation for each medium and with the assumption that the interface between the two mediums is an equipotential surface. The transformations are such that the transformed systems have uniform field conditions. The transformed systems are then mathematically related by the condition that the electric flux density must be continuous across the boundary joining the two mediums. (Received February 10, 1975.)

The April Meeting in Monterey, California April19, 1975 Algebra & Theory of Numbers *724-Al RUSSELL MERRIS, California State University, Hayward, California 94542. Two Problems Involving Schur Functions.

A scalar valued function d of the m-square matrices is defined. It depends on a subgroup

G of Sm and an irreducible representation of G. Two questions are discussed: (1) What can be said about the roots of d(z1-A)1 (2) If A::: B and d(A) = d(B) f. 0, must A= B?

(The notation A ::: B means A, B, and A - B are positive semidefinite hermitian.) Some partial answers are obtained. (Received January 16, 1975.) *724-A2 ELAINE ZASLAWSKY, San Jose State University, San Jose, California 95192. Decomposition matrices and Cartan matrices of some finite simple groups. The author has obtained the decomposition and Cartan matrices of the groups SL(2,q), PSU(3,q), PSL(3,q), and Sz(q), for many values of q. These tables were obtained with the aid of the computer. Reprints of these tables are available on request. (Received January 31, 1975.)

*724-A3 DANIEL DAVIS, Naval Postgraduate School, Monterey, CA, 93940. A Generalization of Eisenstein's Irreducibility Criterion. Let p denote a rational prime, and f(X) a monic polynomial with (p-adic) integer coefficients. Theorem. Suppose that f(X) = [q(X)]n mod p, where q(X) is irreducible mod p. Suppose also that there exists a root a mod p of q(X) and a positive integer k such that f(a) - 0 mod pk+l. Then f(X) is the product of at most k factors.

A-426 The classical Eisenstein criterion is given by q(X) = X, a = 0, and k = 1. Corollary. If f(X) satisfies the classical Eisenstein criterion relative to p, if q(X)

is irreducible mod p, and if m is any positive integer, then f(q(X)m) is (p-adically) irreducible. (Received February 6, 1975.) *724-A4 RUDOLPH NAJAR, U. of Wisconsin-Whitewater, Whitewater, WI 53190 and WALTER BECK, Wartburg College, Waverly, Iowa 50677. A Lower Bound for Odd Triperfect Numbers. Preliminary report.

Let a be the sum-of-divisors function. A positive integer N is called triperfect if

cr(N) = 3N. Theorem: If N is odd triperfect, then N > 105°. Discussion of the method

of proof and a corollary. (Received February 10; 1975.)

*724-A5 H. B. HAMILTON, California State University, Sacramento, Calif. 95821 T, E, NORDAHL and T. TAMURA, University of California, Davis, Calif. 95616. Commutative Cancellative Semigroups ~ Idempotent. A commutative cancellative semigroup without (with) idempotent is called a CCIF-(CCI-)

semigroup. A CCIF-semigroup S is represented in terms of a CCI-semigroup C and a function

of C X C into integers. In particular, consider the case where C is an abelian group. A

CCIF-semigroup S is called an m-semigroup if it is subarchimedean, i.e., there is a E S

such that for each x E S, am= xy for some positive integer m and y E s. m.semigroups are

characterized in several ways, for example: Theorem 1. S is an m-semigroup iff some archi-

medean component of S is an ideal of s. Theorem 2. If S is a CCIF-semigroup of finite

rank, S has nontrivial homomorphisms into the semigroup of non-negative rational numbers under addition. Let S be a CCIF-semigroup. If Hom(s,JR:) .fo {0} where m: is the semi group of non-negative real numbers, then nanS = ~for some a E S. The converse is still n=l open. (Received February 17, 1975.)

*724-116 T. E. Nordahl, University of California, Davis, California 95616. Cancellative Semigroups with Non Empty Center.

Let S be a semigroup and L the greatest semilattice image of s. We call S pivoted if L has a "0" and if S is pivoted the semilattice component of S corresponding to the "0" of L is called the pivot of Sand is denoted Piv(S). The center of S, C(S), is defined as usual. Theorem. S is a cancellative pivoted semigroup satisfying Piv(S)(\C(S) + 0 if and only if S is a group or S is determined up to isomorphism by a triple (G,H,I) where G is a group,

H is either empty or a subgroup of G and I is a function mapping GxG into z+,O satisfying:

(i) I(a,b)+I(ab,c) = I(b,c)+I(a,bc) for all a,b,c in G, (ii) I(e,e) = 1 where e is the identity of G, (iii) I(a,h)>O and l(h,a)>O for all a in G and for all h in H and (iv) I(g,h) 1 for all g,h in H. Setwise S = z+• 0xGUH with product given as follows: (m,a)(n,b) =

(m+n+I(a,b),ab), h(m,a) = (m+I(h,a)-l,ha), (m,a)h = (m+I(a,h)-l,ah) and product as usual in H for all (m,a), (n,b) in z+• 0xG and for all h in H. (Received February 18, 1975.) Analysis *724-Bl BRUCE A. BARNES, University of Oregon, Eugene, Oregon 97403. The similarity problem for representations of a B*-algebra. It is an open question whether a continuous representation rr of a B*-algebra A on a Hilbert space H is necessarily similar to a *-represen tation of A on H. Assuming reasonable hypotheses on rr, we show that A-427 there exists a (perhaps unbounded) selfadjoint operator U with domain in H and a •-representation y of A on H such that TT u-1 y U on the domain of u. Furthermore, the •-representation v has the property that if TT is similar to some •-representation of A on H (via some bounded operator on H with bounded inverse), then TT is similar to y. (Received February 10, 1975.)

724-B2 I, NAMIOKA, University of Washington, Seattle, Washington 98195, Some topological questions related to Banach spaces. • Let K be a weak-compact (resp. weak*-compact) subset of a Banach space (resp, dual Banach space). The talk will be centered around the question: Does there exist a point in K where the identity map: (K, weak(resp, weak*)topology) .. (K, norm topology) is continuous? The origin of the questions, some related problems, and the known answers will be discussed, (Received February 14, 1975.)

724-B3 DOUGLAS M. CAMPBELL and KENT PEARCE, Brigham Young University, Provo, UT 84602. An Extension of the Bazilevil: Functions. Preliminary Report. 2 The authors examine analytic, locally univalent solutions f(z) = z + a 2 z + • • • of the differential equation

zf"(z) . f'(z) _ ~ ~ 1 + f' (z) + (a.+lj3-l) z f (z) - a.zg (z) + zh (z) + ij3

2 where g(z) = z+b2z + • • · is an analytic locally univalent function vanishing only at 0, h(z) 1 + • • · is analytic and nonvanishing, and a.+ij3 is an arbitrary complex number. For a.;;;,O, g starlike and h of positive real part, the solutions were proved by Sheil-Small

to define the univalent Bazilevi~ functions. The authors report on the significant differences of the results in the left half plane including nonexistence, nonuniqueness and nonunivalence. Relations between the geometry of f and its domain of analyticity are discussed as well as some positive results. (Received February 17, 1975.)

*724-B4 MARION ORTON, University of California, Irvine, California 92664. Hilbert problems-a distributional approach, Let U = R- U:!,1 ~} and let G(x) be infinitely differentiable and nonzero on U. Let hE b'(R) be known on U. We seek functions f(z) defined and analytic for Im z -i' 0 whose limits f(x .± iO) = lim 0)(x + iy) exist in the sense of convergence in b 1 (R) and satisfy the equation f(x + iO)- G(x)f(x- iO) ~- A = h(x) on the open set U. It is shown that for a large class of coefficients G factorizations G(x)K(x + iO) = K(x- iO) can be obtained from which solutions of the distributional Hilbert problem are constructed analogous to the classical case, Even if G has zeros or discontinuities at the points ~ solutions are seen to exist for arbitrary data h as above, The results are applied to mixed boundary value problems for Laplace's equations, a set of dual integral equations and certain singular integral equations, all posed as distributional problems. (Received February 18, 1975,) 724-B5 JOHN ALAN MACBAIN, Air Force Institute of Technology, Wright-Patterson AFB, Dayton, Ohio 45433, Bifurcation from normal eigenvalues, Preliminary report. In a real Banach space B, consider the operator equation Lu =Au + H~, u) where L is linear (bounded or unbounded), A is real, and H is continuous and uniformly o{llull) on bounded A intervals, Let A0 be an isolated normal eigenvalue of L having odd algebraic multiplicity. Then ~ 0 , 0) E R X B is a bifurcation point for our equation which possesses a maximal continuous branch of solutions in R X B. This maximal branch either is unbounded, meets a special surface S (dependent upon H), or is bounded and contains other isolated normal eigenvalues of L. In case three, the sum of the algebraic multiplicities of the eigenvalues met is even, (Received February 13, 1975,)

724-~ ROB&HT il. KELMAN and J. TD!OTHI SIMPSON ,Dept.o of Computer Science, Color~ Stat.e University, Fto Collins, CO 80521. lllal trigonometrical series associated w.ith o~ conditions of the first and third ld.nd. --- - Consider the d.Uatiigonometric series!_ "u/(ii;"hJ Sl.Il iii";" f(x), 0 c;. X c;. C_, and £ an sin nx : f(x), c .c: x ""- ir. Here equality may hold in a generalized sense, h is a

A-428 nonnegative constant, and f is a given function of bounded variation. We prove: tl~ is a

unique solution (~,~··••) neh t.hat an: o(n~) and this estimate is best possible for uniqueness) in general, the second series converges on no set of positive measure in the ordi-

nar.r sense and converges everywhere in the sense of Abel-Poisson. The proof i."!Volves per

turbation about h:::. 0 and at critical moments uses a reflection theorem of Fo Wolf(Duke

Math. J. 14(1947),877) and a regularity theorem of' N, lli.gley' (Math. Z.ll5(1970),33). The results extend to all other classic trigonometrical kernels. (Received February 18, 1975. )

724-B7 ANDREW GARY CHILDS, University of California, Berkeley, California 94720. L2- boundedness of pseudo-differential operators. Preliminary report.

It is proved that an operator defined by Au(x) = J eix •Za(x, z)u(z) dz for u E.,I'(RI) can be extended to Rn a bounded operator on L2(RI) when a is a bounded complex-valued function on Rn which is jointly, uniformly Hoelder continuous of order i + 6, 6 > 0, in all of its n variables. The proof is in two steps. The 2i+( 2 2ltt' 2 2t+( 2 2t+( first is the proof that b(x, z) = (1- d/dx1) ••• (1- d /dx ) (1- d /dz1) ..• (1- d /dz ) a(x, z) is n ~~£ 2 bounded on +(Rn X Rn for some f > 0. The second is the substitution of a(x, z) = (1- d /dxi) (1- d2/dz 2) b(x, z) into the expression for Au(x) n and the proof that l

Utilizing ·conformal mapping techniques the flow from a general sluice gate is mapped

onto a known flow. It is shown that a necessary condition for this mapping to exist is that

a nonlinear integral equation has a solution. Flows with a finite number of stagnation points

are considered. It is assumed that the pressure is constant on the free surface and that

gravity acts on the system. It is also assumed that the motion is steady, two dimensional and irrotational and that the fluid is inviscid and incompressible. (Received February 13, 1975)

*724-C2 V.A. PATEL, California State University, Humboldt, Arcata, California 95521. Time dependent solutions of the viscous incompressible flow past a circular cylinder by tiieiii.ethod of series truncation. III. Preliminary report. Semianalytic solutions of the Navier-Stokes equations are calculated for two-dimensional, sym- metrical, incompressible flow past a circular cylinder. The stream and vorticity functions are expanded in a finite Fourier series and then substituted in the Navier-Stokes equations. This led to a system of coupled parabolic partial differential equations which are solved numerically. More terms of the series are required as Reynolds number increases and the present calculations were terminated at Reynolds number 600 with 60 terms of Fourier series. The results are compared with similar calculations and experimental data for Reynolds numbers 100, 200, 500, 550 and 600. A secondary vortex appeared on the surface of the cylinder in the case of Reynolds numbers 500,550 and 600. The kink in the streamlines, vorticity and pressure distributions around the surface of the cylinder also indicates the existence of secondary vortex. (Received February 17, 1975,)

A-429 *724-C3 R. w. CHANEY, Western Washington State College, Bellingham, WA 98225 ~ Hybrid Algorithm for Nonlinear Programming Problems

We present a first-order algorithm, designed to solve finite-dimensional nonlinear programming problems having both equality and inequality constraints. The algorithm is built up from the method of exterior penalty functions, the Pironneau-Polak method of centers, and a quasi-Newton method of Luenberger. The algorithm is shown to generate sequences which, under appropriate hypoth eses, will converge linearly at a rate "asymptotically" independent of the penalty coefficient. The principal hypothesis is the assumption that second order sufficiency conditions hold at the proposed solution. The main technique employed in the convergence analysis is the finite-dimensional version of the Hestenes indirect method. (Received February 18, 1975.)

724-C4 KENNAN T. SMITH, Oregon State University, Corvallis, Oregon 97331. Practical and mathematical aspects of the problem of reconstructing objects from x-rays. • A very considerable amount of work is being done these days on the problem of reconstructing 3- dimensional objects from radiographs, neutron radiographs, electron micrographs, etc. The aim of this talk is to describe that part of it with which the speaker has been associated. The work divides in two parts. (l) The practical part involves the location of brain pathologies from ordinary hospital radiographs without the use of contrast materials. The results that have been obtained on phantoms (models) and on human brain tumor patients will be described. (2) The mathematical part involves mainly a study of the Radon transform. Various mathematical results have been uncovered through the practical point of view (e.g. lower dimensional integrabUity of L2 functions, improvements on the Paley-Wiener theorem, the indeterminacy of a finite number of x-rays, etc.). These also will be described. (Received Febrnary 18, 1975.) Geometry *724-Dl RICHARD R. JOSS, California State U., Long :Beach, Long lleaoh, CA 90840 ad ROBERT w. SHA!INON, New Mexico st. u., Las Cruces, lf.X. 88001. On the neber o~ reflections it takes to render a nonconve;z: plane polzgon canft%.

Let P be a nonoonvex n-gon. Let ab be a segment of the convex hull H o~ P such that P n(ab) .. aUb. A reflection operation r on P reflects that piece of P - (aUb) on the interior of H in ab as a mirror. r(P) :is a k-gon (n-2 :S. lt ~ n) with sides that are congruent to corresponding sides Gf P. Consider the sequence f~m(P)} where r 0{P) • P, rm(P) • r(r-1 (P)) (m> 0) and r is any arbitrarily chosen reflection operation. IaBar inoff and Bing (Abstract 564-155, these Notices 6(1959), 834) proved that the sequence £rm(P3 is finite and that the last member is convex. 'l'he present paper is to refute the conjecture of Kazarinoff and Bing that the sequence {rm(P)J contains no more than 2n elements by shoring that no such bound can exist. 'l'he paper alae proves that if r* is a reflection operation an P that reflects that piece of P - (ai.Jb) on the interior of H through the midpoint of ab so that this point acts as a center of symmetry for the pol;rgaa formed by the reflected piece and its original image, then£ r:(P)J is finite and contains no more than (n-1)1 elements. 'l'he authors conjecture that (n-l)J can be replaced by (l/4)n2• (Received December 9, 1975.)

724-D2 NIELS T. SORENSEN, J. G. Enterprises, 8607 Mission Boulevard, Riverside, California 92509. A 3-dimensional treatment of N-dimensional non-Euclidean geometries. A revised definition of orthogonals shows N-dimensional non-Euclidean geometries to be simply contained in Euclidean R3. (Received February 14, 1975.) Logic and Foundations 724-El PHILIP G. CALABRESE, California State College, Bakersfield, Ca. 93309. The Probability that p implies g. Preliminary report. Let (n,B,P) be a probability space such that under the correspondence n +-+II, U +-+V, complement(')+-+ negation (~),and n ++ 1, B is both a sigma-algebra of events and a logic of propositions. As usual, define p + q = ~p v q but define P(qjp) = { ~(q 1\ p)/P(p): ~~~l ~ ~~

A-430 Fact: P(~p V q) ~ P(qlp) with equality if and only if P(p) = 1 or P(~p V q) = 1. Thus in the 2-Valued Logic, where every proposition p has truth-value (probability) 0 or 1, P(qlp) is the truth-value of p + q. But in the logic of propositional functions or the class calculus, where p and q are sometimes true and sometimes false, we cannot define p + q ~P V q if P(qlp) is to be the truth-value of p + q. Can elements (qlp), "q given p" (or p + q, "p implies q") be adjoined to B, and the operations V & ~ extended to a system B*, closed under v. ~. and (1). such that B* is both a class of "conditional events" and an intuitively acceptable logic with probabilistic truth-values where the probability (truth-value) of the proposition "p implies q" is the conditional probability of q given p? (Received February 18, 1975.) Statistics and Probability

*724-Fl EUGENE WESLEY, Center for Mathematical Studies in Economics and Management Science, Northwestern University, Evanston, Illinois, 60201, Measure Stru£tu~~ for Function Spaces, Preliminary Report Let F be a set of Borel functions from X to Y, where X, Y are copies of the unit interval. A probability distribution (mixed strategy) A over F is said to be admissible if it has the following attribute: for any Borel probability measure ~ over X and any Borel subset E c Y, the probability of f(x) appearing in E when f E F, x f X are chosen at random is uniquely determined by ~' ~· We consider the following question: Let a be a countable Boolean algebra of subsets of F, where each element in 0 is definable by a formula in the language of set theory. Let !0 be the minimal cr-ring containing o. Given an arbitrary probability measure ~ over ~· can ~ be extended to an admissible distribu tion? An affirmative answer is shown to be consistent with the Zermelo-Fraenkel axioms of set theory. (Received February 18, 1975.) (Author introduced by Stanley Reiter.) Topology *724-Gl Thomas J. Enright and V. S. Va.radarajan, University of California at Los Angeles, California, Los Angeles, Calif. 90024- Discrete series representations

An infinitesimal. characterization is given for "almost all" discrete series representations

of a connected real semisimple Lie group with finite center. Let g be a real semisimple Lie algebra with Cartan decomposition g = l + a. Let b be a Cartan subalgebra of and assume

that b is a Cartan subalgebra of g also. Let @ (resp. R) denote the universal envel

oping algebra of g (resp. !). For certain weights A in b* the Verma modules with highest

weight A are used to construct irreducible R-finite ®-modules. These modules possess a

canonical R-type which occurs with multiplicity one and where t.1e scalar action of the central-

izer of R in @ is given by a simple formula. These @-modules are shown to be infinitesim-

ally equivalent to the discrete series modules for sufficiently regular A. (Received November 11, 1974.)

*724-G2 CHARLES L. HAGOPIAN, California State University, ~acramento, California 95819. Mapping products of A connected continua .!!!!2. E2•

We call a nondegenerate compact connected metric space a continuum. A continuum X is A. connected if every pair of points in X can be joined by a hereditarily decomposable continuum in X, Suppose that X and Y are 1\. connected continua and that for each t > 0, there exists an e-map of the topological product X X Y into the plane. Then X is either arc-like or circle-like. Furthermore, if X is circle-like, then Y is

arc-like. (Received February 12, 1975.)

A-431 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 postdodaral positions; however, veterans recently released from service will qualify. Applicants must provide (1) name of institution where last employed; (2) date of termination of service; (3) highest degree; (4) field. Applications from nonmembers must carry the signature of a member. 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 cJ{otiai). 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 direded 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.

UNIVERSITY TEACHER. Age 34. Dr. rer. nat. 1967, MATHEMATICS TEACHER AND RESEARCHER. Age 29. G6ttingen; varied teaching experience, some publications. Ph. D. University of Southern California, 1973, in Combi Seeks any visiting/temporary position in U. S. A. /Canada; natorial Mathematics. Excellent publications. Well versed available Fall 1975. M. R. Chowdhury, 30 Causewayside, in the applications of my work to the theory of design of sci Cambridge, U.K. entific experiments, coding theory,andstatistics. Experi ence in applying combinatorialtechniquestoproblems In R,ESEARCH AND/OR TEACHING MATHEMATICIAN. business. Two years professional experience at Louisiana Ph.D. 1971. Age 29. Specialty Algebra. 3 Papers. Have state University, teaching wide range ofundergraduate experience teaching in Spanish in Mexico and Venezuela. courses and lecturing at graduate seminars.References and Nationalized U. S. citizen. References and resume upon Resume on request. K. B. Gross, Department of Mathematics, request. Raul Machuca, 351 Warwick Avenue, Teaneck, Louisiana state University, Baton Rouge, Louisiana 70803. New Jersey 07666.

A-432 Structural Stability and Monte Carlo Principles and Morphogenesis Neutron Transport Problems An Outline of a General Theory of ". . . the book represents a well thought out text. It Models is 'divided rather sharply' into two halves. The first Translated hy D. H. FOWLER from the French half attempts an introduction to 'Monte Carlo,· pre edition, updated hy the author (With Forewords to sented from a unified point of view and directed to the original French and updated English editions by the novice. The second half is directed to readers C. H. WADDINGTON> who have a special interest in reactors and have had "This long awaited hmk i> important and will contact with reactor theory. The last two chapters probably he widely read. It summarizes the recent are entirely devoted to the computation of thermal biomathematical work of Thorn, Fields medal neutron fluxes and resonance escape probabilities. winning topologist. It applies recent work on dif

Here the famous tricks and techniques are developed 00 ferentiable (C ) mappings to a wide variety of which one needs to make the adjoint and superposi questions of general biology, among them embryolo tion methods practical. What makes this book very gy from which the name in part derives .... The text, valuable is a set of carefully selected references for in French, is relatively easy to follow for those who further study, both in mathematical and practical know the language and who are not put off by the aspects. This unusual book is both mathematically author's easy references to advanced mathematics. rigorous and useful for the practitioner.·· Nonetheless a translation would be useful and one -Mathematical Rel'ie•vs hopes it will soon appear." From review of 1972 /969. xiv. 234 pp .. ill us. French ed. in Bulletin of' Mathematical Biology hardbd. ISBN 0-201-07089-8, $17.50 1975, about400 pp .. illus. IUirdbd.ISBN 0-805-39276-9, $22.50 paperbd./SBN0-805-39277-7, $13.50

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A-433 ~You loved our Act in Washington ...

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Group Representation Theory "i::r (Pure and Applied Mathematics Series, Volume 7) by Larry Dornhoff A comprehensive two-part textbook that is directed toward graduate students in mathematics as well as to research mathematicians. Part A: Ordinary Representation Theory Contents: Introduction. Theory of Semisimple Rings. Semisimple Group Algebras. Splitting Fields and Absolutely Irreducible Modules. Characters. Burnside's paq• Theorem. Multiplicities, Generalized Char acters, Character Tables. Representations of Abelian Groups. Induced Characters. Representations of Direct Products. Permutation Groups. T. I. Sets and Exceptional Characters. Frobenius Groups. Clifford's Theorem. M-Groups. Brauer's Characterization of Characters. Brauer's Theorem on Splitting Fields. Normal p-Complements and the Transfer. Generalized Quaternion Sylow 2-Subgroups. A Theorem of Tate. Mackey Decomposition. lt6's Theorem on Character Degrees. Algebraically Conjugate Characters. The Schur Index. Projective Representations. The Finite Two-Dimensional Linear Groups. Special Conjugacy Classes. A Characterization via Centralizers of Involutions. Primitive Complex Linear- Groups. Jordan's Theorem a Ia Blichfeldt. Extra-Special p-Groups. Normal p-Subgroups of Primitive Linear Groups. The Frobenius-Schur Count of Involutions. Primitive Solvable Linear Groups. Simplicity of PSL(n,F) and PSp(2m,F). Jordan's Theorem for Solvable Groups. lt6's Theorem on Characters of Solvable Groups. Characters of SL(2,pn). 264 pages, 1971 Prepaid price: $14.62 Part B: Modular Representation Theory Contents: Indecomposable Modules and Chain Conditions. The Radical of a Ring. ldempotents. Com pleteness. Unique Decomposition Theorems. Lifting ldempotents. Principal Indecomposable Modules. Carlan Invariants. The Number of Irreducible Modules. Decomposition Numbers. Finite Extensions of Complete Local Domains. Existence of a Suitable Ring: p-Adic Integers. Relatively Projective RG-Modules. Green's Theorem. Vertices and Sources. Defect Groups. Central Characters. The Brauer Homomorphism. The Brauer Correspondence. Brauer's First Main Theorem. Brauer Characters. Orthogonality Relations. Characters in Blocks. Blocks of Defect Zero. Higher Decomposition Numbers; Brauer's Second Main Theorem. Extension of the First Main Theorem to DCa(D)/D. The Principal Block. Quaternion Sylow 2-Subgroups. The Z*-Theorem of Glauberman. Blocks with Cyclic Defect Group. Brauer: Groups of Degree <(p-1)/2. Fait and Thompson: Groups of Degree <(p-1)/2. p-Biocks of SL(2,p). p-Biocks of p-Solvable Groups. 270 pages, 1972 Prepaid price: $17.06

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A-435 The Sociedad Matematica Mexicana is starting a new ser~es: MATHEMATICAL NOTES AND SYMPOSIA The purpose of these series is to provide rapid and informal publication of Lecture Notes and Proceedings of meetings in the different areas of mathematics. The first two issues will be available in the second quarter of 1975. Volume 1. "CONFERENCE ON HOMOTOPY THEORY", Northwestern University, August 1974. Approximately 200 pages. $5.00 US. Contributors: E. H. Brown, D. M. Davis, H. H. Glover, D. C. Johnson, D. S. Kahn, A. Liulevicius, M. Mahowald, ). P. May, H. Toda, H. R. Miller, C. Mislin, ). C. Moore, F. P. Peterson, D. C. Ravenel, L Smith, V. P. Snaith, M. Tangora, W. Wilson. Volume 2. "THIRD MEXICO-U.S. SYMPOSIUM ON DIFFERENTIAL EQUATIONS", Centro de lnvestigacion del lPN, January 1975. Approximately 300 pages. $7.00 US. Contributors: D. Sanchez, S. Hahn-Goldberg, J. Adem, S. Bernfeld, G. B. Gustafson, R. Flores, A. Montalvo Robles, R. O'Malley, R. A. Rogers, K. Bohmer, R. Sacker, J. K. Hale, A. G. Kartsatos, R. Suarez, 1. E. Arellano Roig, ). P. Hennart, J. C. Lillo, G. A. Bogar, K. Schmitt,). R. Graef, P. W. Spikes, W. E. Fitzgibbon, A. Plls, D. R. Snow, Z. Vorel, M. Garanc;on, 'T. A. Burton, C. P. Tsokos, A. N. U. Roo, ). R. Haddock, M. Manongian, K. Gustafson, M. S. Henry. Standing orders or separate issues will be available from: BOLETIN DE LA SOCIEDAD MA TEMA TICA MEXICAN A APARTADO POSTAL 14-140 MEXICO 14, D.F. MEXICO

A NEW KONINKLIJKE VLAAMSE INGENIEURSVERENIGING PUBLICATION JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS The Journal of Computational and Applied Mathematics is a new quarterly journal that will initially be published in volumes of approximately 300 pages. Each volume consists of four issues of approximately 75 pages each. Volume I will be published in 1975. First issue, March 1975. The journal will take up contributions in fields where applied mathematics play a fundamental part. The journal offers the possibility to publish algorithms. The main purpose of the journal consists in providing the possibility for the reader to become acquainted with techniques of computational and applied mathematics in different fields, enabling him to apply them to his own field of research. The subscription price of a volume is $36, postage included. Orders for subscriptions should be sent to the publisher Koninklijke Vlaamse Ingenieursvereniging Jan van Rijswijcklaan 58 B -2000-Antwerp, Belgium Editorial Board: S. Amelinckx, J.P. Brans, P. Breesch, F. Broeckx, R. Butstraen, J. De Kerf (R. De Meersman), E. De Wolf, L. D'Hooge, P. Dingens, R. Gevers, H. Glejser, M. J. Goovaerts, C. C. Grosjean, C. Joachain, J. Meinguet, R. Piessens, H. Steyaert, A. Strub, P. Van Leuven, B. Van Styvendaele, L. Wuytack (all in Belgium) Board of Associate Editors: 0. Axelsson, Sweden; R. Bellman, U.S.A.; A. Bjorck, U.S.A.; L. Collatz, Fed. Rep. Germany; K. L. Cooke, U.S.A.; B. Einarsson, Sweden; I. Hansen, France; P. Henrici, Switzerland; J. Paelinck, Netherlands; R. Storer, Australia; W. Vandaele, U.S.A.; A. Van der Sluis, Netherlands; H. J. Zimmerman, Fed. Rep. Germany.

A-436 The University of Western Ontario NONARCHIMEDEAN FIELDS AND Department of Mathematics ASYMPTOTIC EXPANSIONS Applications are invited for an anticipated by A. H. LIGHTSTONE, Queen's University, Professor or Kingston, Ontario, Canada, and A. ROBINSON, position at the rank of Assistant Yale University, New Haven, Connecticut, Associate Professor. Candidates should have U.S.A. a Ph.D., and an A.S.A. or equivalent. Duties 1975. 213 pages. will include teaching and research in actuarial US$ 23.25/Dfl. 60.00 mathematics. Salary will be dependent on For many years mathematicians have been aware of a close connection between nonarchi qualifications and experience. The present medean systems and the orders of infinity and salary minima for the ranks of Assistant of smallness, associated with the asymptotic behaviour of a function. In examining the whole Professor and Associate Professor are background of this relationship from the view $13,025 and $15,725. point of nonstandard analysis, the authors de monstrate that infinitesimals and infinitely large Applications should be sent to: numbers form a natural background to asymp D. Borwein, Head totics. Department of Mathematics MODERN GENERAL TOPOLOGY The University of Western Ontario by Jun-lti NAGATA, Professor of Mathematics, London, Ontario N6A 589 University of Pittsburgh. 1968. 2nd rev. ed. 1974. 375 pages. US$ 26.95/Dfl. 70.00 Intended as an introductory text as well as an advanced reference book on general topology, Assistant or Associate Professor this book combines the material found in an ordinary text, with recent theories, especially of Mathematics on metric spaces, paracompact space, mapping, etc. An extensive bibliography and a large num Starting Fall 1975. PhD in Computer Sci ber of exercises which can be solved without too much difficulty are also included. ence or in a closely related area preferred. Candi dates should have the expertize to provide leader MODERN MATHEMATICAL ship in developing an undergraduate computer METHODS IN TECHNOLOGY, Vol. 2 Science curriculum. Demonstrated excellence in by S. FENYO, Technical University, Budapest teaching, and future professional growth expected. 1975. about 350 pages. Practical experience in the Computer Science field US$ 36.50/ Dfl. 95.00 Teaching load 12 hours per week. important. The first volume of the work Modern Mathema Boston State College is an affirmative tical Methods in Technology deals with con action/equal opportunity employer and encourages tinuous methods. This, the second volume, treats the so-called discrete methods. Two women and minority group members to apply. chapters cover extensively the theory of linear algebra and the theory of linear and convex Seymour Kass, Chairman optimization while a third introduces the theory of graphs. Department of Mathematics Boston State College :~~~:~d~!11 625 Huntington Avenue north-holland The Netherlands Boston, Massachusetts 02115 Distributors in the U.S.A. and Canada: American Elsevier Publishing Company, Inc., 52 Vanderbilt Avenue, New York, N.Y. 10017 0106

A-437 PREREGISTRATION AND RESERVATION FORM Western Michigan University Kalamazoo, Michigan

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A-438 Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations edited by ABDUL K. AZIZ This book contains the proceedings of the L. Lapidus, The Occurrence and Solution of Symposium on Numerical Solutions of Boundary Boundary Value Reaction Systems. G. H. Meyer, Value Problems for Ordinary Differential Equa- Invariant Imbedding for Fixed and Free Two tions held June 10-12, 1974, at the University Point Boundary Value Problems. S. V. Parter, of Maryland, Baltimore County Campus. A Posteriori Error Estimates. M. Lentini and TENTATIVE CONTENTS: T. E. Hull, Numerical V. Pereyra, Boundary Problem Solvers for First Solutions of Initial Value Problems for Ordinary Order Systems Based on Deferred Corrections. Differential Equations. H. B. Keller, Numerical J. Tauter, Numerical Solution of Boundary Solution of Boundary Value Problems for Ordi- Value Problems by Stable Methods Based on nary Differential Equations: Survey and Some the Transfer of Conditions. J. M. Varah, A Tale Recent Results on Difference Methods. M. R. of Two Methods for Solving Two-Point Boundary Scott, On the Conversion of Boundary-Value Value Problems. T. I. Seidman, Some Consid Problems into Stable Initial-Value Problems Via erations in the Numerical Simulation of a Several Invariant Imbedding Algorithms. /. Ba- Semi-Conductor Device. S. M. Davenport, L. F. buska, The Connection between the Finite Dif- Shampine, and H. A. Watts, Comparison of terence Like Methods and the Methods Based Some Codes for the Initial Value Problem for on Initial Value Problems for ODE. C. de Boor, Ordinary Differential Equations. J. N. Shoo A Smooth and Local Interpolant with "Small" smith, A High-Order Finite-Difference Method K-th Derivative. H.-0. Kreiss, Difference Ap- for the Solution of Two-Point Boundary-Value proximations for Singular Perturbation Problems. Problems on a Uniform Mesh. 1975, 384 pp., $18.50/£9.00 Arrows, Structures, and Functors The Categorical Imperative by MICHAEL A. ARBIB and ERNEST G. MANES This book is divided into two parts. In the The second part of the book introduces first part-devoted to arrows and structures- functors. Building upon the many examples of the authors present the most fundamental as- Part I, the authors introduce natural transforma pects of the point of view that a category tions and adjoint functors and show that reach models a system of "structures" generalizing ability and observability in automata theory specific domains of discourse such as sets and correspond to free and cofree constructions functions and vector spaces and linear maps. respectively. They study monoidal categories as They introduce the reader to the art of "chas- a framework for extending set-theoretic con ing" commutative diagrams as a way of constructions to other domains of discourse, noting structing or verifying equalities between chains very briefly that topos theory-part of the cur of arrows; present structures such as monoids, rent research frontier of category theory-offers metric spaces, and topological spaces from exciting new vantage points for problems in the first principles, and use categories induced by foundations of mathematics. Finally, they show these structures to demonstrate the diversity that universal algebra can itself be captured in of possibilities inherent in generalizing con- category theory language, using the concept of cepts from set theory to category theory. a monad. 1975, 200 pp., $8.951£4.35 Sieve Methods by H. HALBERSTAM and H. E. RICHERT London Mathematical Society Monographs No.4 Series editors: P. M. Cohn and G. E. H. Reuter There are many problems in the theory of in a general yet highly practical form which numbers which can be formulated in terms of makes the theory readily applicable to a wide sieve theory-among them are many of the old- range of problems. est problems in classical prime number theory, CONTENTS: The Sieve of Eratosthenes: Formu such as the prime twins and binary Goldbach lation of the General Sieve Problem. The Com conjectures. The object of this book is to show binatorial Sieve. The Simplest Selberg Upper how one may approach these questions with Bound Method. The Selberg Upper Bound the powerful methods of modern sieve theory: Method (Continued): 0-Results. The Selberg in particular there is an elegant account of Upper Bound Method: Explicit Estimates. An Brun's sieve and a sketch of Rosser's sieve Extension of Selberg's Upper Bound Method. where attention is directed towards aspects of Selberg's Sieve Method (Continued): A First the method which should receive further atten- Lower Bound. The Linear Sieve. A Weighted lion. The rest of the book is a comprehensive Sieve: The Linear Case. Weighted Sieves: The description of Selberg sieve theory, presented General Case. Chen's Theorem. 1974, 378 pp., $26.00/£9.80 Prices sub;ect to change without notice. ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers 111 Fifth Avenue, New York, N.Y. 10003 24-28 Oval Road, London NW1 7DX Ill NEW BOOKS 3 ~ ATHEMATIC Ill GRUNDLEHREN DER MATHEMATISCHEN WISSENSCHAFTEN ... ~ Volume 144 -~ Basic Number Theory 3rd Edition By A. Weil This is a "modern" account of algebraic number mathematically rigorous exposition of the theory by the adele method, followed by a principles of quantum mechanics. self-contained account (involving no 1974. viii,·211p. cohomology) of class field theory. In this third 10 illus. cloth/$23.00 edition in new appendix "Examples of "L-Functions" was added. Volume 215 1974. xviii, 325p. cloth/$19.60 Banach Lattices and Volume 211 Positive Operators The Theory of Ultrafilters By H. H. Schaefer By W. W. Comfort and S. Negrepontis This is a comprehensive account of Banach This work is a comprehensive and unified lattices and positive linear operators (including treatment of the significant results and applica- positive finite matrices) with main emphasis on tions of ultrafilters, including the most recent their relevance for and applications to measure developments. It deals with structural, top- theory, mean ergodic theory, operators be- ological, and model-theoretic problems in the tween LP-spaces and spectral theory. Examples theory of ultrafilters, including a detailed are given in each of the fifty sections, and treatment of the Rudin-Keisler minimal and the main text of each chapter is supplemented the good ultrafilters. by bibliographical notes and exercises. 1974. x, 482p. cloth/$40.20 1974. xi, 376p. cloth/$40.20

Volume 212 Volume 216 Algebraic Topology-Homotopy Problems and Theorems and Homology in Analysis II By R. Switzer By G. P6Jya and G. Szegoe This volume is an advanced text for students This collection of problems is characterized with an elementary knowledge of algebraic by its emphasis on furthering the understanding topology, covering such topics as homotopy of the mathematics of the problem rather than theory, generalized homology theories, spectra, on calculation of solutions. The student who K-theory, cobordism, products, duality, spectral works through these problems will develop his > sequences, characteristic classes, cohomology ability to carry out independent and creative i&'; operations and homology cooperations, and the mathematical research. ,_. Adams spectral sequence. 1975. approx. 400p. cloth/$45.10 1975. xiii, 526p. cloth/$50.10 Volume 214 ~ ~ Group Theory and § ~~ ~~ Quantum Mechanics ..,. By B. L. van der Waerden ~ This is the English edition of an earlier work in ~ !- t,) Gennao(meG,uppenth.,,•••cheMethodein Springer-Verlag < 00 • der Quantenmechanik. Grundlehren der Ill ::S "'" oe mathematischen Wissenschaften, Vol. 36) New York Inc. Z ~ - I and updated by the author. It presents the 175 F'fth A ~ S 8 P fundamentals of group theory and their appli- I venue ;::j llCI c:: - cations to quantum mechanics, with a New York, NY 10010 ~ 0. ~ S ..______,j~o;~ i