OF THE
AMERICAN MATHEMATICAL SOCIETY
Edited by Everett Pitcher and Gordon L. Walker
CONTENTS
MEETINGS Calendar of Meetings ...... 490 Program for the April Meeting in Madison, Wisconsin...... 491 Abstracts for the Meeting- Pages 521-549 Program for the April Meeting in Davis, California...... 504 Abstracts for the Meeting- Pages 550-557
PRELIMINARY ANNOUNCEMENTS OF MEETINGS 508 SUMMER INSTITUTES AND GRADUATE COURSES 510 VISITING MATHEMATICIANS...... 512 ACTIVITIES OF OTHER ASSOCIATIONS...... 513 PERSONAL ITEMS ..... , ...... 513
NEW AMS PUBLICATIONS ...... , ...... 514 NEWS ITEMS AND ANNOUNCEMENTS ...... 503, 512, 514, 515 ABSTRACTS OF CONTRIBUTED PAPERS ...... 518 ERRATA ...... 585
INDEX TO ADVERTISERS ...... 600 MEETINGS Calendar of Meetings NOTE: This Calendar lists all of the meetings which have been approved by the Council up to the date at which this issue of the cJ{oti.t:a) 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 the meetings to which no numbers have yet been assigned.
Meet- Deadline ing Date Place for No. Abstracts•
676 June 20, 1970 Tacoma, Washington Apr. 30, 1970
677 August 24-28, 1970 Laramie, Wyoming June 30, 1970 (75th Summer Meeting)
678 October 31, 1970 Washington, D. C. Sept. 10, 1970 679 November 20-21, 1970 Athens, Georgia Oct. 6, 1970 680 November 21, 1970 Pas adena, California Oct. 6, 1970
681 November 28, 1970 Urbana, Illinois Oct. 6, 1970
682 January 21-25, 1971 Atlantic City, New Jersey Nov. 5, 1970 (77th Annual Meeting)
683 March 26-27, 1971 Chicago, Illinois April 7-10, 1971 New York, New York *The abstracts of papers to be presented 0 person at the meetings must be received in the Head quarters Offices of the Society in Providence, Rhode Island, on or before these deadlines. The dead lines also apply to news items. The next two deadliQEl§ for by-title abstracts will be April23, 1970 and June 23, 1970.
OTHER EVENTS
September 1- 10, 1970 International Congress of Mathematicians Nice, France
The cJioUceiJ of the American Mathematical Society is published by the American Mathematical Society, 321 South Main Street, P. 0. Bol< 6248, Providence, Rhode Island 02904 in January, February, April, June, August, October, November and December. Price per annual volume .is $10.00. Price per copy $3.00. Special price for copies sold at registration desks of meetings of the Society, $1.00 per copy. Subscriptions, orders for back numbers (back issues of the last two years only are available) and inquiries should be addressed to the American Mathematical Society, P. 0. Bol< 6248, Providence, Rhode Island 02904. Second class postage paid at Providence, Rhode Island, and additional mailing offices.
Copyright© 1970 by the American Mathematical Society Printed in the United States of America
490 Six Hundred Seventy-Fourth Meeting Wisconsin Center Madison, Wisconsin April14-18, 1970 The six hundred seventy-fourth on Saturday morning, April 18, will be on meeting of the American Mathematical the subject of Geometric Topology and has Society will be held at the Wisconsin been arranged by Professor James M. Center of the University of Wisconsin, Kister of the University of Michigan. The Madison, Wisconsin, on April14-l8, 1970, speakers will be James W. Cannon, in conjunction with the 19 70 spring meeting Jerome Dancis, Richard K. Lashof, Ewing of the Association for Symbolic Logic. An L. Lusk, RichardT. Miller, and Laurence announcement of the ASL program appears C. Siebenmann. The third special session, on page 513 of these c}/otiai). All sessions on the subject of Qualitative Theory for will be held in the meeting rooms of the Functional and Differential Equations, to Wisconsin Center. be held on both morning and afternoon of By invitation of the Committee to Saturday, April 18, has been arranged by Select Hour Speakers for Western Sec Professor John A. Nohel of the University tional Meetings there will be four one of Wisconsin and the speakers will be hour addresses. Professor Jack E. Mc Fred G. Brauer, Charles C. Conley, Neil Laughlin of the University of Michigan will H. Fenichel, Jacob J. Levin, Richard P. speak on Friday, April 17,at ll:OO a.m.; McGehee, Kenneth R. Meyer, Paul H. his topic will be "Finite groups generated Rabinowitz, Joel W. Robbin, Aaron S. by transvections." Professor Kurt Mahler Strauss, and James A. Yorke. of the Ohio State University will address There will be sessions for contribu the Society on Friday, April 17, at 1:45 p.m.; ted papers on April 17-18, 1970. Some his subject will be "Geometry of numbers provision will be made for late papers if of convex bodies." Professor Lee A. Rubel necessary. of the University of Illinois will speak on Saturday, April 18, at 11:00 a.m.; his talk SYMPOSIUM ON REPRESENTATION will be entitled "Bounded convergence of THEORY OF FINITE GROUPS analytic functions." Professor Morton L. In l)onor of Professor Richard Curtis of Rice University and the Univer Brauer of Harvard University there will sity of California, Berkeley, will address be a symposium on Representation Theory the Society on Saturday, April 18, at 1:45 of Finite Groups and Related Topics on p.m.; his topic will be "Finite-dimensional April 14-16. This Symposium has been H-spaces." All four addresses will be made possible by an anticipated grant presented in the auditorium of the Wiscon from the National Science Foundation. sin Center. The topic was chosen by the Committee By invitation of the same committee, to Select Hour Speakers for Western there will be three special sessions of se Sectional Meetings, which consists of lected twenty-minute papers.Oneof these, Paul T. Bateman (chairman), Roger C. to be held on both morning and afternoon Lyndon, and Mark E. Mahowald. The of Friday, April 17, has been arranged by Organizing Committee of the Symposium, Professor John H. Walter of the University responsible for the planning of the program of Illinois on the subject of Finite Simple and the choice of speakers, consists of Groups. The speakers will be Jonathan L. Irving Reiner (chairman), Richard Brauer, Alperin, Michael J. Collins, George Charles W. Curtis, Walter Feit, and Glauberman, Daniel Gorenstein, Koichiro James A. Green. The speakers at the Harada, William M.Kantor, Takeshi Kondo, Symposium will be Clark Benson, Richard Michael E. O'Nan, Ernest E. Shult, W. Brauer, P. Nicholas Burgoyne, Edward T. Brian Stewart, Michio Suzuki, and John H. Cline, Samuel B. Conlon, Charles W. Walter. Another special session, to be held
491 Curtis, Everett C. Dade, Larry L. Dorn MADISON INN, one block from the Wiscon hoff, Andreas W. M. Dress, Paul Fong, sin Center, Patrick X. Gallagher, David M. Gold 601 Langdon Street schmidt, James A. Green, Donald G. Madison, Wisconsin 53703 Higman, I. Martin Isaacs, Noboru Ito, Single $12.50-$20.00 Heinz Jacobinski, Gerald J. Janusz, Twin 15.00- 28.00 Robert W. Kilmoyer, Jr., Karl Kronstein, Tsit- Yuen Lam, John H. Lindsey II, EDGEWATER HOTEL, six blocks from Donald S. Passman, William F. Reynolds, the Wisconsin Center, Klaus W. Roggenkamp, Andrei V. Roiter, 666 Wisconsin Avenue Leonard L. Scott, Jr., Louis Solomon, Madison, Wisconsin 53703 Tonny A. Springer, Richard G. Swan, David Single $15.00-$19.00 B. Wales, Walter D. Wallis, Warren J. Twin 17.00- 25.00 Wong, and Hans J. Zassenhaus. In addition, the talks by Michael J. Collins, Koichiro FOOD SERVICE Harada, and Takeshi Kondo in the special The Wisconsin Center Coffee Shop session on Finite Simple Groups are to be will serve coffee, juice, and doughnuts regarded as part of the Symposium. from 9:30a.m. to ll:OOa.m.,andluncheon REGISTRATION from 11:30 a.m. to 12:30 p.m. The cafeteria in the Wisconsin The registration desk will be in the Union will serve meals at the following lobby of the Wisconsin Center. It will be times: open on Tuesday from 8:00 a.m. to 5:00 Breakfast 7:00 a.m. to 9:00a.m. p.m.; on Wednesday, Thursday, and Friday (except Saturday and Sunday) from 9:00 a.m. to 5:00 p.m.; and on Lunch 11:00 a.m. to 1:15 p.m. Saturday from 9:00 a.m. to 4:00 p.m. (from 11:45 on Saturday and ACCOMMODATIONS Sunday) Lowell Hall, a private dormitory Dinner 5:00p.m. to 6:45 p.m. one block from the Wisconsin Center, will The Wisconsin Union has other have accommodations in twin-bedded dining facilities as well. rooms for 150 of those attending the meet A list of nearby restaurants will be ing. The rates are $6.00 per person for a vailab1e at the registration desk in the double occupancy. A form which may be Wisconsin Center. used in mailing reservations at Lowell Hall can be found on page 488 of the TRAVEL February issue of these cNoticei). Alterna Madison is served by North Central, tively reservations may be made by calling Northwest Orient, and Ozark Air Lines. ( 608) 256-2621. In particular Northwest Orient Air Lines There are a number of motels and has direct flights to Madison from Chicago, hotels in the area, some of which are Detroit, Milwaukee, Minneapolis, New listed below. (A more extensive list may York, and Washington. The Milwaukee be found on page 675 of the August 1968 Road has several trains a day between issue of these cNoticei).) In making reserva Chicago and Madison. Badger Coach Lines tions with one of the hotels or motels has express bus service between Milwau listed below, participants should mention kee and Madison, while Greyhound offers that they are attending the mathematical express bus service between Chicago and meetings, since each of these places has Madison, and also a special connection reserved a block of rooms for those between O'Hare Field and Madison. attending the meeting. ENTERTAINMENT TOWN-CAMPUS MOTEL, two blocks from the Wisconsin Center, There will be a no-host cocktail 441 North Frances Street party in the Alumni Lounge of the Wiscon Madison, Wisconsin 53703 sin Center on Friday, April 17, at 5:30 Single $1 o.oo- $13 .oo p.m. An admission fee of $0.25 per person Twin 15.00- 17.00 will be levied in order to defray the cost
492 of snacks. Beer and mixed drinks will be Streets, which is one block from the Wis available at a cash bar. consin Center.
PARKING
Parking is extremely limited on campus. Those staying in Lowell Hall or ADDRESS FOR MAIL AND TELEGRAMS one of the other hotels listed above should plan on walking to the Wisconsin Center. Persons attending the meetings may Those coming from greater distances may be addressed at Mathematical Meetings, be able to park in the m unci pal parking Wisconsin Center, University ofWisconsin, facility at the corner of State and Lake Madison, Wisconsin 53706.
SYMPOSIUM ON REPRESENTATION THEORY OF FINITE GROUPS AND RELATED TOPICS (In honor of Professor Richard Brauer, Harvard University) TUESDAY, 9:00A.M.
First Session, Auditorium, Wisconsin Center 9:00-9:20 On the affine group over a finite field Professor Louis Solomon, University of Wisconsin
9:25-9:55 A construction of a 12-dimensional group projectively representing the Suzuki group Professor John H. Lindsey II, Northern Illinois University 1 0:00-10:20 Direct summands in representation algebras Dr. Walter D. Wallis, LaTrobe University, Bundocra, Victoria, Australia
10:25-10:55 Bass orders and the number of indecomposable lattices of orders Professor Klaus W. Roggenkamp, McGill University, Montreal, Canada 11:00-11:40 Symplectic action and the Schur index Professor I. Martin Isaacs, University of Wisconsin
Second Session, Auditorium and Room 210, Wisconsin Center 2:00-2:20 On the degrees and rationality of certain characters of finite Chevalley groups I Professor Charles W. Curtis, University of Oregon (Auditorium) 2:25-2:45 On the degrees and rationality of certain characters of finite Chevalley groups II Dr. Clark Benson, University of Oregon (Auditorium) 2:50-3:10 Jordan's theorem for solvable groups Professor Larry L. Dornhoff, University of Illinois (Auditorium) Group rings of infinite groups Professor DonaldS. Passman, University of Wisconsin and Institute for Defense Analyses (Room 210)
3:15-4:15 Cusp forms in the character theory of GLn(Fq) Dr. Tonny A. Springer, Math Institute, Umversity Centrum-De Lithof, Utrecht, The Netherlands (Auditorium) The modular theory of permutation representations Dr. Leonard L. Scott, Jr.,University of Chicago (Room 210}
493 4:20-4:50 The reflection character of a finite group with a EN-pair Professor Robert W. Kilmoyer, Jr., Clark University (Auditorium)
WEDNESDAY, 9:00A.M. Third Session, Auditorium, Wisconsin Center 9:00-9:20 Minimal vertices and irreducible characters Professor Edward T. Cline, University of Minnesota, Institute of Technology 9:25-10: 10 Defects and vertices Professor James A. Green, University of Warwick, Coventry, England 1 0:15-11:00 Decomposition and cancellation of lattices Professor Heinz Jacobinski, Chalmers University of Technology, Goteborg, Sweden 11: 05-ll:45 Operations in representation rings Professor Andreas W. M. Dress, University of Bielefeld, Bielefeld, Federal Republic of Germany
Fourth Session, Auditorium and Room 210, Wisconsin Center 2:00-2:30 Blocks of representations Professor Richard Brauer, Harvard University (Auditorium} 2:35-2:55 Extensions of blocks Professor Everett C. Dade, Universite de Strasbourg, Strasbourg, France (Auditorium) 3:00-3:20 Group characters and sets of primes Professor William F. Reynolds, Tufts University (Auditorium) Some decomposable Sylow 2-subgroups and a nonsimplicity condition Professor Paul Fong, University of Illinois at Chicago Circle (Room 210) 3:25-3:45 Faithful representations of p-groups over fields of characteristic p Professor Gerald J. Janusz, University of Illinois (Auditorium) Factorizable groups Professor Noboru Ito, University of Illinois at Chicago Circle (Room 21 0) 3:50-4:20 Character tables and the Schur index Professor Karl Kronstein, University of Notre Dame (Auditorium) Sylow 2- subgroups with nonelementary centers Professor David M. Goldschmidt, Yale University (Room 210) 4:25-5:15 The radical ideal Professor Hans J. Zassenhaus, Ohio State University (Auditorium)
THURSDAY, 9:00 A. M. Fifth Session, Auditorium and Room 210, Wisconsin Center 9:00-9:20 The number of conjugacy classes in a finite group ~ Professor Patrick X. Gallagher, Columbia University (Auditorium) 9:25-10:25 Operations inK-theory Professor Richard G. Swan, University of Chicago (Auditorium) 1 0:30-10:50 Representations of Chevalley groups in characteristic p Professor Warren J. Wong, University of Notre Dame (Auditorium)
494 10:55-11:10 Modular representations of some linear groups Professor p. Nicholas Burgoyne, University of California, Santa Cruz (Auditorium) Simple groups of order 7• 3a•2b Professor David B. Wales, California Institute of Technology (Room 210)
Sixth Session, Auditorium, Wisconsin Center 2:00-3:00 Matrix questions and the Brauer-Thrall conjectures Professor Andrei V. Roiter, Mathematics Institute, Academy of Sciences of Ukrainian SSR 3:05-3:25 Remarks on the Krull-Schmidt theorem Dr. Samuel B. Conlon, University of Sydney, Sydney, Australia 3:30-4:10 Restrictions of representations and Grothendieck groups Professor Tsit- Yuen Lam, University of California, Berkeley 4:15-4:45 Some characterization theorems for rank 3 groups Professor Donald G. Higman, University of Michigan It is expected that, in addition to the lectures listed above, there will be several supplementary talks, These will be announced at the beginning of the Symposium. In ad dition, the lectures of Collins, Harada, and Kondo in the Special Session on Finite Simple Groups are to be regarded as part of the Symposium,
PROGRAM OF THE SESSIONS The time limit for each contributed paper is 10 minutes. The contributed papers are scheduled at 15 minute intervals, To maintain this schedule, the time limit will be strictly enforced,
FRIDAY, 8:20 A, M. Special Session on Finite Sim;>le Groups, Auditorium 8:20-8:40 ( 1) A characterization of alternating groups Dr, Takeshi Kondo, Institute for Advanced Study (674-91) 8:45-9:05 (2) A characterization of unitary groups. Preliminary report Professor Michio Suzuki, University of Illinois (674-95) 9:10-9:30 (3) Prim~power factor groups of finite groups Professor George Glauberman, University of Chicago (674-86) 9:35-9:55 (4) Simple groups of small 2-rank. Preliminary report Professor Jonathan L. Alperin, University of Chicago (674-84) 10:00-10:20 (5) Centralizers of involutions in finite simple groups. I. Preliminary report Professor Daniel Gorenstein, Rutgers University, and Professor John H. Walter*, University of Illinois ( 674-87) 10:25-10:45 (6) Centralizers of involutions in finite simple groups. II. Preliminary report Professor Daniel Gorenstein*, Rutgers University, and Professor John H. Walter, University of Illinois (674-88) f*For papers with more than one author., an asterisk follows the name of the author who I !plans to present the paper at the meeting.
495 FRIDAY, 9:00A.M. Session on Logic and Algebra, Room 224 9:00-9:10 (7) Deduction in continuous logics Mr. Fred Halpern, Rider College (674-66)
9:15-9:25 (8) Reduced powers and limits Dr. Michael M. Richter, University of Texas (674-79) (Introduced by Professor Leonard Gillman)
9:30-9:40 (9) Compactification of relational structures, Preliminary report Mr. Walter F. Taylor, University of Colorado (674-80)
9:45-9:55 (IO) Implication in orthomodular posets. Preliminary report Professor Edwin L. Marsden, Kansas State University (674-71)
I 0:00-10:10 ( 11) Free profinite groups Professor Dian Gildenhuys and Mr. Chong-Keang Lim*, McGill University (674-56) 10:15-10:25 (12) Indecomposable modules for semisimple groups. Preliminary report Professor james E. Humphreys, Courant Institute, New York University (674-44)
10:30-10:40 (13) Solutions of pure equations in rational division algebras Professor Burton I. Fein* and Dr, Murray M. Schacher, University of California, Los Angeles (674-6)
FRIDAY, 9:00A.M. Session on Partial Differential Equations and Potential Theory, Room 210 9:00-9:10 (14) Nonexistence of a continuous right inverse for surjective linear partial dif ferential operators Dr, D. K. Cohoon, University of Wisconsin (674-30) 9:15-9:25 (15) Coercive inequalities for certain classes of bounded regions Professor james M. Newman, Florida Atlantic University (674-82)
9:30-9:40 (16) Viscosity matrices for two-dimensional nonlinear hyperbolic systems Professor C, C, Conley and Professor joel A. Smaller*, Courant Institute, New York University (674-28) 9:45-9:55 (17) Geometrical characterization of fluid flow Professor Robert H. Wasserman, Michigan State University (674-37) 10:00-10:10 (18) A potential-theoretic inequality and some of its applications Professor W. j. Schneider, Syracuse University (674-77) 10:15-10:25 (19) The field inside a charged hollow conductor, Preliminary report Professor jacob Korevaar* and Mr. Tune Geveci, University of California, San Diego (674-25) 496 FRIDAY, 9:00A.M.
Session on Approximation Theory, Room 227 9:00-9:10 (20) Simultaneous approximation Professor Alex C. Bacopoulos, Michigan State University (674-48) 9:15-9:25 (21) Best uniform approximations via annihilating measures Professor William R. Hintzman, San Diego State College (674-4) 9:30-9:40 (22) Entire functions and Miintz-Szasz type approximation Professor Wilhelmus A. J. Luxemburg*, California Institute of Technology, and Professor Jacob Korevaar, University of California, San Diego (674-17) 9:45-9:55 (23) Cardinal hermite spline interpolation, Preliminary report Mr. Peter R. Lipow, University of Wisconsin (674-54) 10:00-10:10 (24) A generalized minimum norm property for spline functions, Preliminary report Mr. Franklin B. Richards, University of Wisconsin (674-46) 10:15-10:25 (25) Some problems concerning generalized polyvibrating equations Professor Demetre John Mangeron* and Professor M. N. Oguztoreli, University of Alberta (674-53) 10:30-10:40 (26) Error formulas for analytic continued fractions Professor Evelyn Frank, University of Illinois at Chicago Circle (674-8)
FRIDAY, 11:00 A.M. Invited Address, Auditorium Finite groups generated by transvections Professor Jack E, Me Laughlin, University of Michigan
FRIDAY, 1:45 P. M, Invited Address, Auditorium Geometry of numbers of convex bodies Professor Kurt Mahler, Ohio State University
FRIDAY, 3:00 P. M. Special Session on Finite Simple Groups, Auditorium 3:00-3:20 (27) On some doubly transitive groups Dr, Koichiro Harada, Institute for Advanced Study (674-89) 3:25-3:45 (28) On the fusion of an involution in its centralizer, Preliminary report Professor Ernest Shult, Southern Illinois University ( 674-93) 3:50-4:10 (29) A characterization of u3(q), q odd, Preliminary report Dr.Michael E. O'Nan, Rutgers University (674-92) (Introduced by Professor John H. Walter)
497 4:15-4:35 (30) Finite groups with a split EN-pair of rank 1 Professor Christoph Hering, Professor William M. Kantor*, and Professor Gary M. Seitz, University of Illinois at Chicago Circle (674-90) 4:40-5:00 (31) Finite groups admitting almost fixed-point-free automorphisms Professor M. J. Collins, University of Illinois at Chicago Circle (674-85) (Introduced by Professor John H. Walter) 5:05-5:25 (32) Strongly self centralising 3-centralisers. Preliminary report Dr. W. B. Stewart, University of Illinois and University of Oxford, Oxford, England {674-94)
FRIDAY, 3:00P.M. Session on Fourier Analysis and Functions of a Complex Variable, Room 227 3:00-3:10 (33) On the almost (A) summability of the derived Fourier series Professor Narendra K. Govil*, Loyola of Montreal, and Professor Badri N. Sahney, University of Calgary (674-69) 3:15-3:25 (34) Everywhere convergence of Fourier series Professor Casper Goffman, Purdue University (674-64) 3:30-3:40 (35) Summability and Fourier analysis. Preliminary report Professor George U. Brauer, University of Minnesota (674-16) 3:45-3:55 (36) Radial Nth derivatives of Blaschke products Professor P.R. Ahern, University of Wisconsin, and Professor D. N. Clark*, University of California, Los Angeles (674-5) 4:00-4:10 (3 7) Successive derivatives of entire functions of exponential type Professor James D. Buckholtz, University of Kentucky (674-3) 4:15-4:25 (3 8) The range set of a meromorphic function Professor Leon Brown*, Wayne State University, and Professor Paul M. Gauthier, University of Montreal (674-36) 4:30-4:40 (39) A generating function for certain coefficients involving several complex variables. Preliminary report Professor Hari M. Srivastava, University of Victoria (674-27)
FRIDAY, 3:00P.M. Session on Probability, Integral Equations and Ordinary Differential Equations, Room 210 3:00-3:10 (40) On the infinite divisibility of mixtures of r-distributions Professor Fred W. Steutel, University of Texas ( 674-68) (Introduced by Professor Klaus R. Bichteler) 3:15-3:25 (41) Gaussian Markov expectations and related integral equations Professor John A. Beekman*, University of Iowa, and Professor Ralph A. Kallman, Ball State University (674-19)
498 3:30-3:40 ( 42) On an asymptotic property of a Volterra integral equation Professor Antonio F. lze, Brown University (674-52) (Introduced by Professor Jack K. Hale) 3:45-3:55 (43) Inequalities involving II f II and II f(n)llq for f with n zeros Mr. James E. Brink, Icfwa State University (674-61) 4:00-4:10 ( 44) Stability of differential equations with homogeneous right hand sides Professor Ronald C. Grimmer, Southern Illinois University (674-45) 4:15-4:25 (45) Liapunov functions and global existence without uniqueness Professor Stephen Bernfeld, University of Missouri at Columbia (674-14}
FRIDAY, 3:00P.M. Session on Geometry, Room 224 3:00-3:10 (46) Blocking sets in finite planes Mr. Aiden Bruen, University of Missouri-Columbia (674-78) (Introduced by Professor J. L. Zemmer) 3:15-3:25 ( 47) Additive loops, linearity, and distributivity in projective planes. Preliminary report Professor George P. Graham and Professor James W. Petticrew*, Indiana State University (674-75) 3:30-3:40 (48) Boolean 2-Geometry. Preliminary report Dr. Hubert J. Ludwig, Ball State University (674-10) 3:45-3:55 (49) Some generalizations of G32 • Preliminary report Mr. Gerald Schrag, Kansas State University (674-65) (Introduced by Professor Richard J. Greechie) 4:00-4:10 (50) Monotone norms. Preliminary report Professor G. P. Barker, University of Missouri-Kansas City (674-57) 4:15-4:25 (51) A characterization of certain L3 sets Professor Arthur G. Sparks, Georgia Southern College (674-43)
SATURDAY, 8:20A.M. Special Session on Geometric Topology, Auditorium 8:20-8:40 (52) New proofs of Bing's approximation theorem for surfaces Professor James W. Cannon, University of Wisconsin (674-81) 8:45-9:05 (53) Approximating cellular maps by homeomorphisms Dr. Laurence C. Siebenmann, Princeton University (674-40) 9:10-9:30 (54) Smoothing four manifolds Professor Richard K. Lashof*, University of Chicago. and Professor J. Shane son, Princeton University ( 674-83)
499 9:35-9:55 (55) Approximating codimension 3 embeddings Mr. RichardT. Miller, University of Chicago (674-22) 10:00-10:20 (56) An embedding theorem and a general position lemma for topological mani folds (metastable range) Dr, Jerome Dancis, University of Chicago (674-60) 10:25-10:45 (57) Homotopy groups of spaces of embeddings Mr. Ewing L. Lusk, University of Chicago (674-23)
SATURDAY, 8:45A.M. Special Session on QualitativeTheory for Functional and Differential Equations, Room 210 8:45-9:05 (58) On perturbations of asymptotically stable systems Professor Fred G. Brauer, University of Wisconsin (674-20) 9:10-9:30 (59) Differential inequalities and differential games Mr. A, Lasota, University of Maryland, and Professor Aaron S. Strauss* and Professor Wolfgang L. Walter, Mathematics Research Center, U, S, Army, University of Wisconsin (674-49) 9:35-9:55 (60) Controllability of linear oscillatory systems and of nonlinear systems Professor James A. Yorke, University of Maryland (674-96) 10:00-10:20 (61) On the asymptotic behavior of the bounded solutions of some integral equations Professor Jacob J. Levin* and Professor Daniel F. Shea, University of Wisconsin (674-21) 10:25-10:45 (62) Nonlinear Sturm-Liouville problems. Preliminary report Dr. Paul H. Rabinowitz, University of Wisconsin (674-58)
SATURDAY, 9:00A.M. Session on Group Theory, Room 227 9:00-9:10 ( 63) The annihilator of radical powers in the modular group ring of a p-group Dr. Edward T. Hill, Cornell College (674-63) 9:15-9:25 (64) Conjugate p-subgroups with maximal intersection Mr. John J. Currano, University of Chicago and Roosevelt University (674-31) 9:30-9:40 (65) Some results on p-groups of maximal class Mr. Raymond T, Shepherd, University of Chicago (674-38) 9:45-9:55 (66) On groups with chain conditions Professor Bernhard Amberg, University of Texas (674-67) 10:00-10:10 (67) On the existence of trivial intersection subgroups Dr. Mark P. Hale, Jr., Southern Illinois University (674-73)
500 10:15-10:25 (68) Large abelian subgroups of some infinite groups Mr. Vance Faber, Washington University (674-33) 10:30-10:40 (69) A class of abelian torsion groups characterized by their automorphism group Professor Jutta Hausen, University of Houston (674-35)
SATURDAY, 9:00A.M. Session on Functional Analysis, Room 224 9:00-9:10 (70) The lattice of barreled topologies. Preliminary report Dr. Forrest R. Miller, Kansas State University (674-59) 9:15-9:25 (71) Nonasymptoticity in projection lattices. Preliminary report Mr. Louis M. Herman, Plymouth State College (674-62) 9:30-9:40 (72) On the strong lifting property. Preliminary report Professor Klaus R. Bichteler, University of Texas (674-50) 9:45-9:55 (73) Absolutely continuous measures on a locally compact semigroup. Preliminary report Mr. Garry Hart, Kansas State University (674-70) (Introduced by Professor Karl R. Stromberg) 10:00-10:10 (74) On helixes in Hilbert space Professor Pesi R. Masani, Indiana University (674-72)
SATURDAY, 11:00 A.M. Invited Address, Auditorium Bounded convergence of analytic functions Professor Lee A. Rubel, University of Illinois
SATURDAY, 1:45 P.M. Invited Address, Auditorium Finite-dimensional H-spaces Professor Morton L. Curtis, Rice University and University of Cali fornia, Berkeley
SATURDAY, 3:00P.M. Special Session on Qualitative Theory for Functional and Differential Equations, Room 210 3:00-3:20 (75) A sufficient condition for structural stability Professor Joel W. Robbin, University of Wisconsin (674-2) 3:25-3:45 (76) Relations between a closed invariant set and its stable and unstable sets Professor Charles C. Conley, University of Wisconsin and Courant In stitute, New York University (674-97) 3:50-4:10 (77) Perturbation of invariant manifolds for vector fields. Preliminary report Mr. Neil H. Fenichel, Courant Institute, New York University (674-74) 501 4:15-4:35 (78) Generic bifurcation and stability properties of periodic points Professor Kenneth R. Meyer, University of Minnesota (674-7) 4:40-5:00 (79) Unbounded orbits in the restricted three body problem Professor Richard P. McGehee, Courant Institute, New York University (674-98)
SATURDAY, 3:00P.M. Session on Rings and Algebras, Room 227 3:00-3:10 (80) The Grothendieck ring of dihedral and quaternion groups. Preliminary repor Dr. John J. Santa Pietro, Stevens Institute of Technology (674-55) 3:15-3:25 {81) Isomorphic polynomial rings Professor Donald B. Coleman* and Professor Edgar E. Enochs, Uni- versity of Kentucky (674-11) 3:30-3:40 (82) Zero divisors and nilpotent elements in power series rings. Preliminary report Mr. David E. Fields, Stetson University (674-9) (Introduced by Professor William A. LaBach) 3:45-3:55 {83) Quasi-Frobenius rings Dr. Abraham Zaks, Northwestern University (674-32) 4:00-4:10 (84) Characters and orthogonality in Frobenius algebras Professor Timothy V. Fossum, University of Utah (674-42) 4:15-4:25 (85) On a class of partially stable noncommutative algebras Professor John D. Arrison, Monmouth College (674-26) 4:30-4:40 (86) On the types of functions which can serve as scalar productfj in a complex linear space Professor Konrad John Heuvers, Michigan Technological University ( 674-15)
SATURDAY, 3:00P.M. Session on Topology, Auditorium 3:00-3:10 (87) Isolated singularities defined by weighted homogeneous polynomials Professor John W. Milnor, Massachusetts Institute of Technology, and Professor Peter Orlik*, University of Wisconsin (674-18) 3:15-3:25 (88) Total spaces of circle bundles over lens spaces Professor Melvin C. Thornton, University of Nebraska (674-29) 3:30-3:40 (89) A characterization of universal crumpled cubes Professor Robert J. Daverman, University of Tennessee (674-41)
502 3:45-3:55 (90) Local finite cohesion. Preliminary report Mr. William C. Chewning, Iowa State University (674-34) 4:00-4:10 (91) J-compact spaces Professor Phillip L. Zenor, Auburn University (674-39) 4:15-4:25 (92) Quasi uniformities and quasi pseudo metrics Mr. Ivan L. Reilly, University of Illinois and Eastern Illinois University (674-51) (Introduced by Professor Mary-Elizabeth Hamstrom) 4:30-4:40 (93) Chains of simple closed curves and a dog bone space Professor Edmund H. Anderson, Mississippi State University (674-13) 4:45-4:55 (94) Some theorems on fixed points Professor Sankatha P. Singh, Memorial University of Newfoundland (674-76) Paul T. Bateman Urbana, Illinois Associate Secretary
NEWS ITEM
MENTOR PROGRAM The Council of the Society has ap to correspond with them and serve as pointed a special committee composed of R. mathematical mentors during the period Creighton Buck, Chandler Davis, and J. L. of service. In many cases, these mathe Kelley to concern itself with developing a maticians will be faculty members who means of helping advanced mathematics know the student and have a continuing students to maintain their mathematical interest in him. In cases where a student interests, identity, and competence during does not name a specific individual to act a period of service in the Armed Forces as mentor, the committee will attempt to or other enforced absence from the mathe select one. matical community such as a prison term In order that a list of possible men for draft resistance. tors can be compiled, the committee re Letters have been sent to all col quests that any person interested in be leges with an M. A. program, asking for coming a part of this program send his information about such students, so that the name and any relevant information to Dr. AMS can build a roster. When the list is Gordon L. Walker (Attention: Mentor Pro complete, the students will be questioned gram), American Mathematical Society, about their interest in the program, and the P. 0. Box 6248, Providence, Rhode Island committee will then try to find college or 02904. university mathematicians who are willing
503 Six Hundred Seventy-Fifth Meeting University of California, Davis Davis, California April25, 1970
The six hundred seventy-fifth VOYAGER INN meeting of the American Mathematical Interstate 80 (P. 0. Box 310) Society will be held at the University of Single $ 9.00 California, Davis, in Davis, California, Double 11.00 up on April 25, 1970. By invitation of the Committee to YOLANOINN Select Hour Speakers for Far Western 221 D Street Sectional Meetings, there will be two Single $ 9.00 hour addresses at this meeting. Professor Double 11.00 David Blackwell of the University of Twin 13.00 California at Berkeley will lecture at 11:00 a.m. on Saturday. The title of his UNIVERSITY LODGE address is" Ordinal solution of Gil games." 123 B Street Professor Ramesh Gangolli of the Univer Single $12.00 sity of Washington will speak at 2:00 p.m. Double 14,00 on "Aspects of the theory of spherical All reservations should be sent functions on symmetric spaces." There directly to the preferred motel. will be sessions of contributed papers on Saturday morning and afternoon. Late MEALS papers may be accepted for presentation at this meeting. Information concerning Buffet luncheon will be available on late papers and program changes will be the second floor of the Memorial Union. available at the registration desk. Tickets for the luncheon are priced at All sessions of the meeting will be $1. 75. They will be sold at the registration held in Young Hall. desk. Short orders can be purchased at the "Coop" in the Memorial Union. REGISTRATION TRAVEL Registration for the meeting will begin at 8:30 a.m. on Saturday. The re Davis is located on Interstate 80, gistration desk will be located outside ten miles west of Sacramento. The campus Room 198 in Young Hall. is in the southwest corner of the city, just north of Interstate 80. Davis is served ACCOMMODATIONS by the Southern Pacific Railroad and by the Greyhound Bus Lines ,Air transportation The following motels are located in is available to the Sacramento Metropolitan Davis, California 95616: Airport, with limousine service to Sacra DAVIS MOTEL, Highway 40 and Davis mento, and bus from Sacramento to Davis. Highway (P .0. Box 367) Taxi from the airport to Davis is not re Single (Commercial) $ 8.00 commended because of the high cost. How Double 9.00 ever, rental cars can be obtained at the Twin 11.00 airport.
504 PROGRAM OF THE SESSIONS The time limit for each contributed paper is 10 minutes. The papers are scheduled at 15 minute intervals in order that listeners can circulate among the sessions. To maintain the schedule, the time limit will be strictly enforced.
SATURDAY, 9:00A.M. First Session on Algebra, Room 194, Young Hall 9:00-9:10 (1) The number of one-sided identities, zero and semilattice-components of finite semigroups Professor Taka yuki Tamura, Mr. Howard B. Hamilton, and Mr. Yee Chung B. Ying*, University of California, Davis (675-21) 9:15-9:25 (2) Right zero unions of archimedean semigroups Mr. Robert P. Dickinson, Jr., Lawrence Radiation Laboratory, University of California, Livermore, ( 675-19) 9:30-9:40 (3) Medial &-semigroups. Preliminary report Mr. W. A. Etterbeek, Sacramento State College (675-25) 9:45-9:55 (4) The Grothendieck group of commutative semigroups Professor Takayuki Tamura and Mr. Howard B. Hamilton*, University of California, Davis (675-20) 10;00-10:10 ( 5) Tensor product of semilattices Professor Naoki Kimura, University of Arkansas (675-16) 10:15-10:25 ( 6) Attainability of system of identities on rings Professor Takayuki Tamura, University of California, Davis (675-8)
SATURDAY, 9:00A.M. General Session, Room 184, Young Hall 9:00-9:10 (7) Multiple points and Wallman-type compactifications Dr. Charles M. Biles, Humboldt State College (675-15) 9:15-9:25 (8) How to recognize homeomorphisms Professor Carlos R. Borges, University of California, Davis (675-24) 9:30-9:40 (9) Two-dimensional Miintz-Szasz type approximation. Preliminary report Professor Jacob Korevaar, University of California, San Diego (675-23) 9:45-9:55 ( 10) On a generalization of the Cauchy-Riemann equations. Preliminary report Dr. Heinz Leutwiler, University of WP.shington (675-18) (Introduced by Professor M. G. Arsove)
*For papers with more than one author, an asterisk follows the name of the author wh plans to present·the paper at the meetin •
505 10:00-10:10 ( 11) Reflection principle for system of first order elliptic equations with analytic coefficients. Preliminary report Professor Chung Ling Yu, Florida State University (675-12) (Introduced by Professor Eutiquio C. Young) 10:15-10:25 (12) Sturmian theorems for characteristic initial value problems Professor Kurt Kreith, University of California, Davis (675-1)
SATURDAY, 11:00 A.M. Invited Address, Room 198, Young Hall Ordinal solution of G6 games Professor David Blackwell, University of California, Berkeley
SATURDAY, 2:00P.M. Invited Address, Room 198, Young Hall Aspects of the theory of spherical functions on symmetric spaces Professor Ramesh Gangolli, University of Washington
SATURDAY, 3:15 P. M. Second Session on Algebra and Theory of Numbers, Room 194, Young Hall 3:15-3:25 ( 13) Subdirect decomposition of generalizations of distributive lattices. Prelimi nary report Professor John A. Kalman, University of Auckland, New Zealand, and University of California, Berkeley (675-14) 3:30-3:40 ( 14) Groups with free nonabelian subgroups Professor Melven R. Krom, University of California, Davis, and Professor Myren Krom*, Sacramento State College (675-9) 3:45-3:55 ( 15) Groups of sequences and direct products of countable groups Professor Thomas J. Head, University of Alaska (675-4) 4:00-4:10 ( 16) Determinantal ideals, identities, and the Wronskian Professor David G. Mead, University of California, Davis (675-11) 4:15-4:25 ( 17) Algebraic numbers for which the exponent operation is commutative Dr. Daihachiro Sa to, University of Saskatchewan, Regina Campus (675-1 0)
SATURDAY, 3:15P.M. Session on Analysis, Room 184, Young Hall 3:15-3:25 ( 18) Capacity theory on some Banach spaces of functions Mr. Peter A. Fowler, California State College, Hayward (675-3) 3:30-3:40 ( 19) Interpolation spaces and Banach algebras. I. Preliminary report Professor John E. Gilbert, University of Texas (675-13)
506 3:45-3:55 (20) Concerning the domains of generators of strongly continuous semigroups of linear transformations Professor John W. Spellmann, Texas A & M University (675-2)
SATURDAY, 3:15P.M. Session on Applied Mathematics and Analysis, Room 168, Young Hall 3:15-3:35 (21) Lattice structure of some linear recurrence pseudo-random points in Monte Carlo calculations Dr. William A. Beyer, Los Alamos Scientific Laboratory, Los Alamos, New Mexico (675-17) 3:30-3:40 (22) On self-contained numerical integration formulas for symmetric regions Dr. Frederich N. Fritsch, Lawrence Radiation Laboratory, University of California, Livermore (675-22) 3:45-3:55 (23) Differential and integral relations for two classes of hypergeometric functions Professor Moses E. Cohen, Fresno State College (675-7) 4:00-4:10 (24) On some properties of a class of polynomials unifying the generalized Hermite and Laguerre polynomials. Preliminary report Professor Chandra Mohan Joshi, Texas A & M University (675-5) 4:15-4:25 (25) Special functions related with generalized polyvibrating equations Professor Demetre John Mangeron * and Professor M. N. Oguztoreli, University of Alberta (675-6) R. S. Pierce Seattle, Washington Associate Secretary
507 PRELIMINARY ANNOUNCEMENTS OF MEETINGS The Six Hundred Seventy-Sixth Meeting Pacific Lutheran University Tacoma, Washington June 20, 1970 The six hundred seventy- sixth meet soap are not provided in the dormitories. ing of the American Mathematical Society Dormitory reservations should be sent to will beheld on Saturday, June 20, 1970, at Professor John Herzog, Department of Pacific Lutheran University in Tacoma, Mathematics, Pacific Lutheran University, Washington. The Mathematical Association Tacoma, Washington 98447. The room re of America and the Society for Industrial quests should include the following in and Applied Mathematics will hold North formation: name and address of the sender; west Sectional meetings in conjunction expected time of arrival; names of all with this meeting of the Society. The persons for whom space is required; type Association and SIAM will have sessions of accommodations desired. on Friday and Saturday, June 19 and 20. Pacific Lutheran University is lo By invitation of the Committee to cated in the Parkland suburb of Tacoma. Select Hour Speakers for Far Western The nearest motel is the Rodeway Inn on Sectional Meetings there will be two in Interstate Highway 5, about 20 minutes vited hour addresses atthis meeting. Pro driving time from the campus. There are fessor Robert C. James of the Claremont several hotels in downtown Tacoma, in Graduate School will address the Society cluding the Winthrop Hotel at 9th and at 11:00 a.m. on Saturday. The title of his Broadway, and the America West Tacoma lecture is "Geometry of normed linear Motor Hotel at 242 St. Helens. The rates spaces." At 2:00 p.m. on Saturday there at all of these hotels and motels are ap will be an address by Professor Charles proximately $11.00 to $12.00 for a single J. Stone of the University of California at room, and $14.00 to $16.00 for a double. Los Angeles. He will speak on "Infinitely Bus service is available from downtown divisible processes and their potential Tacoma to the campus. The ride takes theory." There will be sessions for con approximately 30 minutes. Anyone who tributed papers on Saturday morning and wishes to stay in a hotel or motel should afternoon. The deadline for contributed make his reservations through a travel papers is April 30, 1970. However, late agent, or write directly to the preferred papers may be accepted. It is requested hotel. that all abstracts be submitted on the new Meals will be available on Thurs forms containing a subject classification day and Friday at the Campus Dining scheme with 63 categories. These are Room. The prices are $1.10 for break available from most mathematics depart fast, $1.40 for lunch, and $2.2 5 for dinner. ment offices, or can be obtained by writing The Campus Coffee Shop will be open to the Society headquarters. daily. In addition, there are several res The registration desk for the meet taurants within walking distance of the ing will be located in Mordtvedt Library. campus. The Mathematical Association It will be open throughout the period of the of America will sponsor a banquet on meetings. All sessions of the meetingwill Friday evening. Tickets for the banquet take place in the Administration Building will cost about $4.00. and in Xavier Hall. Air transportation to Tacoma ar Dormitory space will be available rives at the Seattle-Tacoma International on Thursday, Friday, and Saturdaynights. airport. There is limousine service from The rates at the dormitory are $3.50 per the airport to downtown Tacoma. person per night on a double occupancy R. S. Pierce basis, and $5.00 per person per night in a Associate Secretary single room. Families can be accom modated in adjoining rooms. Towels and Seattle, Washington
508 The Seventy-Fifth Summer Meeting University of Wyoming August 25-28, 1970
The seventy-fifth summer meeting tion, will be given by Professor Harry of the American Mathematical Society will Kesten of Cornell University. The title of be held at the University of Wyoming, Professor Kesten's series of lectures will Laramie, Wyoming, from Tuesday, August be "Escapades of a Random Walk." The 25, through Friday, August 28, 1970. All Institute of Mathematical Statistics will sessions of the meeting will take place on meet from Tuesday, August 25, through the campus of the university. Friday, August 28. The Colloquium Lectures will be given by Professor R. H. Bing of tl:).e Uni COUNCIL AND BUSINESS MEETING versity of Wisconsin. They will be pre The Council of the Society will meet sented at 1:00 p.m. on Tuesday, August 25, at 5:00 p.m. on Tuesday, August 25 in the and at 9:00 a.m. on Wednesday, Thursday, Rendezvous Room at the Washakie Center. and Friday, August 26-28. The Business Meeting of the Society will There will be three invited hour be held in the Auditorium of the Arts and addresses at the meeting, and several Sciences Building at 4:00 p.m. on Thurs sessions for contributed ten minute papers. day, August 27. Abstracts of contributed papers should be sentto the American Mathematical Society, REGISTRATION Post Office Box 6248, Providence, Rhode Island 02904. It is requested that all ab The registration desk will be in the stracts be submitted on the new forms lobby of Washakie Center. It will be open containing a subject classification scheme on Sunday from 2:00p.m. to 8:00p.m.; on with 63 categories. These are available Monday from 8:00 a.m. to 5:00 p.m_,; on from most mathematics department of Tuesday, Wednesday, and Thursday frtYm fices, or can be obtained by writing to the 9:00 a.m. to 5:00 p.m.; and on Friday from Society headquarters at the above address. 9:00a.m. to 1:00 p.m. The deadline for papers contributed to The registration fees for the meet- this meeting is June 30, 19 70. There is no in&" are as follows: limit on the number of papers that will be Member $5.00 accepted for presentation at the 1970 sum Students $1 ;00 mer meeting. However, no provision will Nonmember $10.00 be made for late papers. There will be no employment re This meeting will be held in con gister at this meeting. junction with meetings of the Mathematical Association of America, the Institute of EXHIBITS Mathematical Statistics, the Society for Book exhibits and exhibits of educa Industrial· and Applied Mathematics, Pi Mu tional media will be displayed in the Wis Epsilon, and Mu Alpha Theta. The Mathe consin Room of the Washakie Center. matical Association of America will meet from Monday, August 24, through Wednes R •.S. Pierce day, August 26. The Earle Raymond Hed Associate Secretary rick Lectures, sponsored by the Associa- Seattle, Washington
509 SUMMER INSTITUTES AND GRADUATE COURSES
The following is a list of graduate courses, seminars, and institutes in mathematics being offered in the summer of 1970 for graduate students and college teachers of mathematics. This list is in addition to the list found ~n pages 363 through 368 of the February issue of these cJVoficu,). Graduate Courses GEORGIA June 8-July 17 311 - Introduction to Distribution Theory 703 - Characteristic Functions EMORY UNIVERSITY Atlanta, Georgia 30322 July 20-August 28 Application deadline: May 15 312 - Introduction to Statistical Inference Information: Professor Trevor Evans, 746 - Seminar in Design Chairman, Department of Mathematics SYRACUSE UNIVERSITY June 15-August 13 Syracuse, New York 13210 Math 325 - Set Theory Information: D. E. Kibbey, Math 420 - Fourier Series and Integrals Chairman, Department of Mathematics Math 431 - Dimension Theory Math 450 - Category Theory June 29-August 7 Math 607 - Fundamentals of Analysis KANSAS Math 635 - Modern Algebra Math 703 - Functions of a Complex Variable Math 706 - Functions of a Real Variable KANSAS STATE COLLEGE OF PITTSBURG Math 790 - Intermediate Seminar Pittsburg, Kansas 66762 Application deadline: May 15 July 20-August 28 Information: Helen Kriegsman, Math 790 - Intermediate Seminar Department of Mathematics June 8-July 31 NORTH CAROLINA Probability and Statistics UNIVERSITY OF NORTH CAROLINA AT Foundations of Euclidean Geometry CHARLOTTE Functions of Complex Variable Charlotte, North Carolina 28205 Application deadline: May 15 NEW JERSEY Information: Dr.j. F. Schell, Chairman,De- partment of Mathematics, P .O.Box 12665
TRENTON STATE COLLEGE June 10-August 3 Trenton, New Jersey 08625 Mat 600 - Foundations of Mathematics Application deadline: June 26 Mat 603 - Computer Techniques and Numeri Information: William Hausdoerffer, cal Methods Chairman, Mathematics Department July 22-August 28 June 29-August 6 Mat 550 - Analytic Functions Modern Statistics Foundations of Geometry PENNSYLVANIA Seminar in Mathematics Education UNIVERSITY OF PITTSBURGH Pittsburgh, Pennsylvania 15213 Application deadline: June 1 NEW YORK Information: Admissions Committee, Department of Mathematics STATE UNIVERSITY OF NEW YORK AT BUFFALO June 23-August 13 Amherst, New York 14226 Math 251 - Algebra II Information: Mrs. Josephine D.Wise, Room A1 A, Math 270 - Topology I 4230 Lea Road Math 280 - Geometry I
510 TEXAS Math 603 - Operations Theory & Partial Differential Equations TEXAS A & M UNIVERSITY Math 610 - Numerical Methods in Differential College Station, Texas 77843 Equations Application deadline: 4 weeks before semester Math 685 - Problems begins Math 691 - Research Information: The Registrar UNIVERSITY OF HOUSTON June 1 Houston, Texas 77004 Math 601 - Higher Math for Engineering & Application deadline: June Physics Information: D. R. Brown, Chairman, Math 602 - Higher Math for Engineering & Graduate Program, Department of Mathe- Physics matics Math 609 - Numerical Analysis May 28- August 15 Math 685 - Problems Measure and Integration Math 691 - Research Schauder Bases July 13 Harmonic Analysis Math 60 l - Higher Math for Engineering & Seminar on Compactifications Physics
Summer Institutes
COLORADO Requirements for Admission: Complete one year of graduate work in mathematics and UNIVERSITY OF COLORADO have some background in fluid mechanics Boulder, Colorado 80302 Information: Meeting Arrangements Depart ment, American Mathematical Society, Computer Science in Social and Behavioral P. 0. Box 6248, Providence, Rhode Island Science Research 02904. Dates: June 15- July 17 Subjects Covered: Computing Requirements for Admission: Ph. D. TURKEY Application deadline: April 1 NATO ADVANCED STUDY INSTITUTE ON Information: Professor Daniel E. Bailey, MATHEMATICAL PHYSICS Ketchum 106 Istanbul, Turkey Mathematical Physics NEW YORK Dates: August 10-August 21 Subjects Covered: Mathematical Problems of Statistical Mechanics, Problems in Non RENSSELAER POLYTECHNIC INSTITUTE linear Transformations,Generalized Spaces, Troy, New York 12181 Generalized Functions, Generalized Eigen Mathematical Problems in the Geophysical vectors, Generalized Algebras, Special Sciences Functions of Mathematical Physics and Dates: July 6-July 31 Group Representations Subjects Covered: Mathematical topics in the Information: A. 0. Barut, Department of atmospheric, oceanographic and earth Physics, University of Colorado, Boulder, sciences Colorado 80302.
511 VISITING MATHEMATICIANS
The list of visiting mathematicians now includes both foreign mathematicians visiting in the United States and Canada, and Am · ~ricans visiting abroad. Note that there are two separate sections.
FOREIGN MATHEMATICIANS VISITI NG IN THE UNITED STATES AND CANADA
Name and Home Country Host Institution Field of Special Interest Period of Visit Miranda, Guillermo (Chile) Courant Institute of Applied Mathematics 9/69-8/70 Mathematical Sciences, New York University
Novak, Josef (Czechoslovakia) Virginia Polytechnic General Topology 3/70-6/70 Institute
AMERICANS VISITING ABROAD
Harris, Bernard (U.S.A.) Technological University 2/70-9/70 Eindhoven, The Netherlands
Lehman, Eugene H.,Jr. (U.S.A.) Marlborough Girls' College, 3/70-6/70 Blenheim, New Zealand
Williams, R. F. (U.S.A.) Institute d e s Hautes Etudes 3/70-6/70 Scientifiques, Bures-sur- Yvette, France
NEWS ITEMS
MATHEMATICS IN THE GRADUATE PROGRAM IN STATISTICS ARCHEOLOGICAL AND UNIVERSITY OF MASSACHUSETTS HISTORICAL SCIENCES The graduate program in statistics at The Royal Society of London and the the University of Massachusetts, Amherst, Academy of the Socialist Republic of is undergoing great expansion. The new Romania have agreed in principle to program starts with a core course in pro arrange an Anglo-Romanian conference bability, from which the student may with international participation on Mathe branch out into various specialized areas matics in the Archaelogical and Historical of statistics or probability, or the student Sciences to be held in Bucharest in Sept may pursue the application of statistics in ember 1970; the opening session will be business, the social sciences, or computer on September 16. Excursions to archaeo and engineering sciences. In particular, logical sites near Bucharest and on or there is the unique opportunity to study near the Black Sea Littoral will be organi empirical logic, which bears on the zed. The principal themes of the confer foundations of inference. The statistics ence will be typology and taxonomy; chro faculty for the 1969-70 academic year nology and seriation; and the mathematical consists of I. Guttman, J. Horowitz, R. problems common to population genetics, Kleyle, R. Lew, G. B. Oakland, and M . historical demography, and the linkage of Skibinsky. Members of the empirical logic manuscripts, etc. For further information, group are: D. Catlin, D . Foulis, N. Hurt, please write to Dr. F. R. Hodson, Scie n R. Piziak, C . Randall, M. H. Stone, and tific Secretary, Mathematics intheArchaeo R. Weaver. For further information, please logical and HistoricalSciencesConfeFence, write to Professor Robert Kleyle, Depart Institute of Archaeology, 31-34 Gordon ment of Mathematics and Statistics, Uni- Square, London, W .C.l. , United Kingdom, versity of Massachusetts, Amherst, or Dr. P. Tautu, Organizing Secretary, Massachusetts 01002. Centre ofMathematicalStatistics, 21 Calea Grivitei, Bucharest 12, Romania.
512 ACTIVITIES OF OTHER ASSOCIATIONS
ASSOCIATION FOR SYMBOLIC LOGIC
The 1970 Spring Meeting of the sor Ronald B. Jensen of the Rockefeller Association for Symbolic Logic will be University will give a survey lecture on held on Wednesday and Thursday, Aprill5 "Using the axiom of constructibility." At and 16, 1970, at the Wisconsin Center of 2:00 p.m. on Thursday, April 16, Pro the University of Wisconsin, in conjunction fessor Jack H. Silver of the University of with a regularly scheduled meeting of the California, Berkeley, will give an hour American Mathematical Society. All ses address on a topic to be announced later. sions will be held in the Lake Shore Room There will be sessions for contributed of the Wisconsin Center. At 2:00 p.m. on papers at 9:30 a.m. on Wednesday, at 3:30 Wednesday, April 15, Professor Yiannis p.m. on Wednesday, at 3:30p.m. on Thurs N. Moschovakis of the University of Cali day, and also, if necessary, at 9:00 a.m. fornia, Los Angeles, will give an hour on Thursday. Detailed programs will be address on a subject to be announced. At mailed to members of the Association. 11:00 a.m. on Thursday, Aprill6, Profes-
PERSONAL ITEMS
Dame MARY L. CARTWRIGHT of To Associate Professor.Pennsylvania Claremont Graduate School has been ap- State University, King of Prussia Branch: pointed to a professorship at Case F. P. CALLAHAN. Western Reserve University. To Assistant Professor. Lehigh Uni Professor MELVIN D. GEORGE of versity: Dr. GARY B. LAISON. the University of Missouri, Columbia has been appointed Dean of the College of DEATHS Arts and Sciences at the University of Nebraska. Dr. G. M. CONWELL, formerlyMas ter at St. P au!' s School, died on October PROMOTIONS 30, 1969,attheageof86.Hewas a member of the Society for 56 years. To Assistant Dean. Yeshiva Univer sity: Dr. CHARLES R. P ATT. Professor WILLIAM FELLER of Princeton University died on January 14, To Professor. Herbert H. Lehman 1970, at the age of 63. He was a member College (CUNY): GODFREY L. ISAACS. of the Society for 30 years.
513 NEW AMS PUBLICATIONS
SELECTED TRANSLATIONS-SERIES II tial equations, by V. R. Petuhov; On an integral connected with symmetric Rie mann spaces of nonpositive curvature, by S. G. Gindikin and F. I. Karpelevi~. Volume 85
TWELVE PAPERS ON FUNCTIONAL MEMOIRS OF THE AMERICAN AI.:AL YSIS AND GEOMETRY MATHEMATICAL SOCIETY 262 pages; List Price $13.60; Member Number 94 Price $10.20 STABLE MODULE THEORY Inequalities for norms of deriva By Maurice Auslander and Mark Bridger tives in weighted Lp spaces, by V. P. Glusko and S. G. Kre1n; Some criteria 148 pages; List Price $2.25; Member for the completeness of a system of root Price $1.69 vectors of a linear operator in a Banach space, A. S. Markus; Plus-operators in a The first part of this Memoir is space with indefinite metric, M. G. Kre'ln devoted to the investigation of the vanish and Ju. L. Smul'jan; 3-polar representa ing of functors and their satellites on tion of plus-voperators, by M. G. Kre'I'n projective and injective objects. These and Ju. L. Smul'jan; On the theory of "stability" propefties, applied to wnctors linear operators in spaces with two norms, of the type Ext R (M,_) and Tor1 (M,_), by I. C. Gohberg and M. K. Zambicki1; yield information and natural generaliza Unitary representations of the Lorentz tions of the notions of torsion-freeness group in a space with indefinite metric, and reflexivity; the remainder of the by R. S. lsmagilov; Infinitesimal bendings Memoir is therefore concerned with these of convex surfaces with bush constraints, applications to module theory. The two by V. T. Fomenko; Almost-reducible and most productive cases--modules of finite symmetric almost-reducible affinely con projective dimension and modules over nected spaces, by N. M. Pisareva; Af Gorenstein rings- -are subsumed by the finely connected spaces admitting a transi introduction of the notion of Gorenstein' tive group of motions with a completely dimension, and most results are obtained reducible stationary linear subgroup, by when this is finite. Finally, modules N. M. Pisareva; The geometry of nested satisfying certain depth conditions are families with emptyintersection. Structure shown to be "approximable" by ones of of the unit sphere of a nonreflexive space, finite projective dimension, and also to by D. P. Mil'man and V. D. Mil'man; possess a homological "composition" Geometry of the Mobius strip and differen- series.
NEWS ITEM STEELE PRIZES FOR EXPOSITORY PAPERS
Through a bequest to the American year, beginning in 1970, at the summer Mathematical Society by Leroy P. Steele, meetings of the Society. Most favorable a prize has been established in honor of consideration will be given to papers George D. Birkhoff, William Fogg Osgood, distinguished for their exposition and and William Caspar Graustein for out covering broad areas of mathematics. It standing published mathematical research. is expected that each prize will be in the One or more prizes will be awarded each amount of $1,000 or more.
514 NEWS ITEMS AND ANNOUNCEMENTS
GRADUATE SUMMER SESSION 1971 IMS SUMMER RESEARCH IN STATISTICS INSTITUTE The 16th Graduate Summer Session The Institute of Mathematical Statis in Statistics, sponsored by the Committee tics is planning to hold the first in a new on Statistics of the Southern Regional series of summer research institutes Education Board, will be presented by the sometime during the summer of 1971. The Department of Statistics at Southern tentative plans are to invite some of the Methodist University from June 8 through distinguished experts who will survey some July 17, 1970. Graduate level courses in of the selected areas, disseminate recent probability, mathematical statistics, re research, discuss the major open problems gression, stochastic processes, multi and provide stimulation for further re variate analysis, Markov processes, search and scholarship through informal sampling, design of experiments, decision discussions. In order to stimulate com theory, and Bayesian statistics will be munication between the applied and mathe offered. A seminar series will also be matical statisticians, people will be in presented in nonparametric statistics and vited from universities as well as from other topics. The courses will be avail industries. Persons interested in partici able to a closed circuit television network pating are requested to send suggestions of colleges and industries in the North including the topic for the summer re Texas Region. Courses and seminars will search institute to Professor Madan L. be presented by visiting professors in Puri, Chairman, IMS Committee on Sum addition to local faculty. Tuition awards mer Research Institutes, Department of plus stipends of $400 each are available Mathematics, Indiana University, Bloom for a limited number of participants. Pre ington, Indiana 4 7401. ference will be given to college instructors. Deadline for receipt of applications is SYMPOSIUM ON GROUP THEORY May 1, 1970. Further information may be The Department of Mathematics of obtained by writing to Professor Paul D. the University of Cincinnati will sponsor a Minton, Southern Regional Graduate Sum Symposium on Group Theory on Friday mer Session in Statistics, Department of afternoon and Saturday, May 1-2, 1970. Statistics, Southern Methodist University, The program will include hour lectures by Dallas, Texas 75222. Professors Reinhold Baer, Norman Black burn, Zovnimir Janko, Wolfgang Kappe, IMS SUMMER INSTITUTES FOR and Hans Zassenhaus. These lectures will TEACHERS be held in the Executive Conference Room The Institute of Mathematical Statis of the Tangeman Center on the campus of tics has appointed a committee to study the University of Cincinnati. Further in the possibility of the Institute of Mathe formation may be obtained by writing to matical Statistics sponsoring summer in Professor Donald B. Parker, Chairman of stitutes for teachers. This committee will the Symposium Committee, Department of consider all problems related to the teach Mathematics, University of Cincinnati, ing of statistics. The committee will be Cincinnati, Ohio 45221. happy to receive any ideas that you have relative to these proposed summer insti SEMINAR ON COMMUTATIVE ALGEBRA tutes for teachers. Please write to Dr. AND ALGEBRAIC GEOMETRY Franklin A. Graybill, Chairman, IMS Com Under the sponsorship of the Canadian mittee on Summer Institutes for Teachers, Mathematical Congress, the Department of Statistical Laboratory, Colorado State Uni Mathematics of the Universite de Montreal versity, Fort Collins, Colorado 80521. is organizing the ninth session of its Inter-
515 national Seminaire de Mathematiques Supe lnstitut fur Mathematische Statistik der rieures which will be held from June 29 Universitat Munster, Federal Republic of to July 24, 1970. The subject of this semi Germany; Guus Zoutendijk, Centraal nar will be Commutative Algebra and Alge Reken-lnstituut der Rijksuniversiteit, The braic Geometry. The program will consist Netherlands. Requests for the detailed of four main courses given by the following program and information on registration invited speakers: Alexandre Grothendieck, and accommodations should be directed to Shreeram Abhyankar, Michael Artin, and Professor J. Ben Rosen, Mathematics Re Masayoshi Nagata. Additional lectures will search Center, The University of Wiscon be given by guest speakers. Registrants sin, Madison, Wisconsin 53706. may make application for financial assis tance to cover travelling and living ex ACM CONFERENCE ON THEORY penses. To obtain further information OF COMPUTING concerning the program and the invited The ACM special interest committee lecturers, and to obtain registration forms, on automata and computability theory please write to Seminaire de Mathemati (SIC ACT) is holding its second annual con ques Superieures, Universite de Montreal, ference on May 4-6, 1970, at the Colonial Case postale 6128, Montreal 250, Quebec, Statler Hilton Inn, Northampton, Massa Canada. chusetts. A total of 30 papers will be presented. Their subject matter includes SYMPOSIUM ON NONLINEAR computational complexity, theory of pars PROGRAMMING ing and optimization, abstract families of The Mathematics Research Center, languages, tree automata, logic and theo U. S. Army, The University of Wisconsin, rem proving, and algebraic structure of is holding a symposium on Nonlinear Pro automata. gramming on May 4-6, 1970. Nonlinear Information and advance registration programming has developed over the last forms can be obtained by writing to two decades in three, often independent, Professor Jeffrey D. Ullman, Department directions: theory, algorithms, and appli of Electrical Engineering, Princeton Uni cations. By means of this symposium, it is versity, Princeton, New Jersey 08540. hoped that a closer rapport between these three aspects of nonlinear programming will be established. Speakers have been PITTSBURGH CONFERENCE AND WORK encouraged to give priority to computation SESSIONS IN GENERAL TOPOLOGY ally promising methods. The following is the list of speakers: Jean M. Abadie, Uni During the second week of June 1970, versite de Paris and Electricite de France; the University of Pittsburgh and Carnegie Ian Barrodale, University of Victoria, Mellon University will sponsor a con British Columbia; James W. Daniel, The ference concentrating on dimension theory, University of Wisconsin; R. J. Duffin, generalizations of metric spaces, categori Carnegie-Mellon University; Roger Flet cal topology, and cardinal invariants. The cher, Atomic Energy Research Establish conference will be followed by two weeks ment, Harwell, England; Arthur Geoffrion, of informal work sessions and seminars. University of California, Los Angeles; The following mathematicians plan (some Gene H. Golub, Stanford University; Pierre tentatively) to participate in theconference: Huard, Electricite de France; Carlton E. A. Arhangel'ski1, Z. Frollk, H. Herrlick, Lemke, Rensselaer Polytechnic Institute; F. B. Jones, K. Kuratowski, E. Michael, Garth P. McCormick, Research Analysis K. Morita, J. Novak, A. Okuyama, A. H. Corporation and Mathematics Research Stone. Inquiries are invited from interested Center; R. R. Meyer, Shell Development topologists as well as from advanced Company; Lucien W. Neustadt, University graduate students in topology. For further of Southern California; M. J. D. Powell, information, please write to Professor J. Atomic Energy Research Establishment, Nagata, Department of Mathematics, Uni Harwell, England; R. Tyrrell Rockafellar, versity of Pittsburgh, Pittsburgh, Penn The University of Wisconsin; H. Witting, sylvania 15213.
516 CONFERENCE ON DIFFERENTIAL are made only upon nomination by an In GEOMETRY stitute Director, interest in such assis tance should be expressed promptly to the The Department of Mathematics of appropriate director, not to NSF. General Michigan State University will hold a information about these awards, and a list regional conference on Differential Geo of institutes, is available from the Ad- metry, June 15-19, 1970. Professor Yozo vanced Science Education Program, Matsushima will give a series of lectures National Science Foundation, Washington, on "Recent results on holomorphic vector D. C. 20550. fields." A number of contributed papers and small discussions will also be sche WORKSHOPS ON PROCESSING MODELS duled. Contingent upon a grant from the IN PERCEPTION AND PSYCHOPHYSICS National Science Foundation, travel and subsistence allowances will be available Several workshops in mathematical for a limited number of participants. psychology will be held in Miami, Florida, Further information may be obtained by on August 26 through September 1, 1970. writing to Professor G. D. Ludden, De This is the week just prior to the Mathe partment of Mathematics, Michigan State matical Psychology and American P sy University, East Lansing, Michigan 48823. chological Association meetings, both in Miami. The primary emphasis of these SUMMER INSTITUTE FOR TEACHERS workshops will be on information proces OF MATHEMATICS IN DEVELOPING sing and automaton-theoretic approaches COLLEGES to perceptual and psychophysical prob lems. The program will consist of six The University of Montana will con tutorial lectures on theory of automata, duct a summer institute for Teachers of with special emphasis on topics of rele Mathematics in Developing Colleges from vance to perception; six invited research June 22 to August 14, 1970. The program lectures; and either three or four indivi is supported by the Department of Health, dual workshops which will each meet two Education, and Welfare under Title V-E hours daily. Attendance will be by invi of the Education Professions Development tation only. Postdoctoral applicants should Act of 1967. The grant provides for sti prepare a brief statement of area(s) of pends for 30 college teachers, and these relevant research interest and list one or participants will be selected about May 1. two workshops in which they have partici For further information, please write to pated as contributors. In addition, they Professor Howard Reinhardt, Department should state whether or not they have other of Mathematics, University of Montana, sources of funds to cover transportation Missoula, Montana 59801. to Miami. A limited number of advanced (but not beginning) graduate students will NATO ADVANCED STUDY INSTITUTES be invited. They should not apply directly, The National Science Foundation has but should have a brief letter of recommen announced that it expects to award inter dation sent in by their faculty advisors. national travel grants to about 80 young Applications should be sent immediately U. S. scientists to attend 39 NATO Ad to Professor R. Duncan Luce, Institute vanced Study Institutes in Europe this for Advanced Study, Princeton, New Jersey are pro coming summer. Subjects covered by 08540. Funds for the workshops these institutes range broadly over the phy vided by the National Science Foundation sical, life, and social sciences. Advanced in a grant to the Mathematical Social graduate and postdoctoral students, and Science Board of the Center for Advanced junior faculty, who are U.S. citizens are Study in the Behavioral Sciences. Each generally eligible to apply. Since awards letter of invitation will specify the stipend.
517 ABSTRACTS OF CONTRIBUTED PAPERS The March Meeting in New York March 25-28, 1970
673 -lOS. J. BEE BEDNAR, Drexel University, Philadelphia, Pennsylvania 19104. Facial characterizations of ordered Banach sp3.ces which are abstract (L)-spJ.ces. Preliminary report.
Two faces F and G of a convex subset C of a linear space E are complimentary if F and G 673-106. JOHN W. HEIDEL, University of Tennessee, Knoxville, Tennessee 37916. Rate of £~_!.1:_~~920.!!~ory_~lutio~s of y~~q(t)yY ~-O!_o_::_y < 1. The equation to be considered is (1) y" t q(t)yy = 0 where q(t) is nonnegative and continuous on [O,oo), 0 < y < 1, andy = k/.l are odd positive integers. There are conditions, necessary and suf ficient, for the existence of nonoscillatory solutions y(t) of (1) satisfying either limt_ 00y(t) = a¥ 0 or limt_ 00 y(t)/t = a 1 0. Our purpose here is to discuss other possible nonoscillatory solutions, for example, solutions which grow like a !_ractional power of t. Criteria will be given for the existence of such solutions. Also it will be she>wn that when 0 ~q(t) "'ct (y+3 )/Z, c > 0, nonoscillatory solutions of (1) are "separated" in the same way as for (2) z" t p(t)z = 0 when 0 ~ p(t) ~ l/4t2 . (Received January 29, 1970.) 673-107. GEORGE G. WEILL, Polytechnic Institute of Brooklyn, Brooklyn, New York 11201. Harmonic forms on Riemannian spaces with positive Ricci curvature. Harmonic p-forms which imitate a given "singularity" p-form in.a neighborhood of the (ideal) boundary are constructed on complete orientable C. 00 Riemannian manifolds with positive Ricci curvature by generalizing Sario's method of normal operators. A particular attention is given to the Dirichlet operator. (Received January 29, 1970.) 673-108. W. L. HARWOOD, VICTOR LOVASS -NAGY, and DAVID L. POWERS, Clarkson College of Technology, Potsdam, New York 13676. A note on the generalized inverses of some partitioned ------matrices. A metho:i is developed for calculating the Mo:>re-Penrose generalized inverses of partitioned matrices expressible as a sum of Kronecker (tensor) products in which either all the left, or all the right, factors are powers of a single normal matrix. (Tne other factors need not be square.) The 518 generalized inverse of the partitioned matrix is given in terms of the generalized inverses of matrices of lower order. The results provide, for example, least-squares solutions of singular systems arising from the discretization of Poisso'l's eq:Jation with von Neumann's boundary conditions. (Received February 2, 1970.) 673-109. M AKOTO ITOH, North Carolina State University, Raleigh, North Carolina 27607. By introducing Kronecker 1i-op~rator into the finite m·valued function lattice Lm in which v(join) and A(meet)areddin.~clas a(t) v b(t) = max[a(t), b(t)], a(t) Ab(t) = min[a(t), b(t)) (the domain oft being arbitrary). one obtains a "functionally complete" m-valued 1i -lattice. In particular, when m " 2, this lattice becomes the we!l-kno;vn Boolean lattice. Next making use of the ordinary tensor notation, we develop a sort of finite discrete tensor calculus. On the basis of such m-valued func· tio'l.a.l calculus, oae can establish "Finite Automata Theory" as a theory of transform.1tion semigroups from Lm into Ln. (Received January 29, 1970.) 673-110. RICHARD B. LAKE!N, State University of New York, Buffalo, New York 14226. ~-<:_om..E_~i_:>9~_of three continued fr~~tion alg~rithms over the Gauss field. Preliminary report. Three continued fraction algorithmil over the Gauss field are considered. Two are classical (A. Hurwitz, Acta Math. I!; J. Hurwitz, Acta Math. 25) and one is due to the auth~H. The algorithms are compared for effectiveness in obtaining a fundamental solution to Pell's equation in the Gauss field: T 2 - 1i v 2 = 4, 4i. Relative efficiency is also discussed on the basis of machine computation. (Received January 30, 1970.) 673-111. VOLODYMYR BOHU:.J-CHUDYNlV, 1372 Deanwood Road, Baltimore, Maryland 21234 and Seton Hall University, South Orange, New Jers"'y 02079. On the G-loop problem. If the right (Rx) and left (Ly) binary operations of a nonabelian loop are isomorphic and iso topic (Rx-Ly, Rx!::. Ly), then operations Rx and Ly we shall call g·operations, designated g(Rx, Ly). A loop L with g-operations we shall call a g-operation loop, designated Lg or (S, g(Rx,Ly)). If in a set ( Lg}, i = T;n n ~ 2, all loops are isomnphic and isotopic (Lga.-Lg8, Lga. t. LgfJ)' then the set of loops ( Lgi) we shall call double G-loops, denoted G( Lgi}, i = T;ii n ~ 2. If Rx = Ly, then the set of isomarp!lic and isotopic abelian loops we shall call "the set of simple G-loops" or "G-loops", deaoted G( Li l , i = G n;; 2. A simHar terminology is adopted for quasiloops, using QLi instead of Li. One of th.~ unsolved problems in loop theory is the construction of G-loops of any order n;; 5. The aims of this paper are: (I) To establish algorithms for constructing sets of double and simple G-loops, giving illustrative examples. (II) To construct isomorphic and isotopic n-tuple systems which, with additional relations, can be used as binary composition for double and simple G-loops and G·quasiloops. (III) To determine double G-quasilo:Jps satisfying relations x • xy = yx and loops iso topic to them. (Received January 30, 1970.) 673-112. DAVID L. POWERS, Clarkson College of Te::hnology, Potsdam. New York 13676. Ap proximatioll__?f matrix functions. If the eigenvalues or eigenvectors of a Hermitian matrix Mare known exactly, then any matrix 519 function defined for M may be computed exactly by diagonalization or from the Lagrange interpolating polynomials, In this paper methods and error bounds are developed for approximating functions of Hermitian matrices by the use of approximate eigenvectors, approximate eigenvalues and the Frechet differential. (Received February 2, 1970,) 673-113. JACK M. SHAPIRO, Graduate Center, City University of New York, New York 10036. Algebraic properties of the representation ring of compact Lie groups. Let G be a compact connected Lie group, H a subgroup of maximal rank, and T a maximal torus contained in H. Let R(G), R(H) and R(T) be the complex representation rings of G, H and T, respectively. The inclusion maps induce monomorphisms, and we consider R(G) as contained in R(H) and R(H) as contained in R(T), The inclusion maps i: T ... H and j: H ... G also induce what Bott denotes ,A,. /". as i*R(T) ... R(H) and j*: R(H) .... R(G ). (R. Bott, "Homogeneous Differential Operators," Princeton University Press, Princeton, N. J,), Define an R(H)-linear form F with F(x,y) = i*(A(H)xy), where A(H) is the element of R(T) which generates the alternating elements, and x,y are in R(T), Similarly define G(z,w) = j*(A(G/H)zw), an R(G)-linear form. A(G/H) =A( G)/ A(H) an element of R(H), and z,w also in R(H), Theorem I. For suitable G, Hand T, R(T) is a free R(H) module of rank IW(H)j, where W(H) is the Weyl Group of H. R(H) is a free R(G) module of rank IW(G)/W(H)I. Theo~2. For G, Hand T as in Theorem 1, F and G are nonsingular. The first theorem is proven using informa tion about integrally closed rings, and the second uses induction on the subgroups of maximal rank contained in the given group G. (Received January 29, 1970,) 673-114. JOHN MEAKIN, University of Florida, Gainesville, Florida 32601. The kernel of a congruence on an orthodox sem igroup. The kernel of a congruence p on a regular semigroup S is defined to the set of p -classes which contain idempotents of S. G. B. Preston has proved that two congruences on a regular semi group coincide if and only if they have the same kernel, The kernel of a congruence on an orthodox semigroup S (a regular semigroup whose idempotents form a subsemigroup) is characterized as a "kernel normal system" of S and a construction of the unique congruence associated with such a kernel normal system is provided, Some results concerning idempotent- separating congruences and group congruences on orthodox semigroups are stated, (Received February 2, 1970.) 673-115. JAMES E. BRENNAN, University of Kentucky, Lexington, Kentucky 40506. Existence of invariant subspaces, Let X be a compact subset of the complex plane having positive two-dimensional Lebesgue measure, For each p, 1 !! p < oo, let HP(X, dx dy) denote the closure of the complex polynomials in LP(X, dx dy). If p f 2 it is known that HP(x, dx dy) possesses a nontrivial closed subspace invariant under multiplication by the complex identity function z. If X has "finite perim;:lter" the same is true for H2(X, dx dy). (See Abstract 68T -B28, these cAfoticeiJ 15(1968), 797 ,) Theorem. H2(X, dx dy) has a nontrivial closed z-invariant subspace for an arbitrary compact X. (Received February 3, 1970.) 673-116. WITHDRAWN. 520 673-117. WITHDRAWN. 673-118. WITHDRAWN. 673-119. J. M. GANDHI, Western Illinois University, Macomb, Illinois 61455. Two inequalities 2 4 We prove 1T (n) ~ 4/ TT ~~~; )/ ]G(4i t 2)/(4i t 2) and I; r=l G2(4i t 2) ~ 4/ TT I;r=l R(4i t 2)/(4i t 2) where [x) denotes the greatest integer ~ x and TT (n) denotes the number of primes ~ n, G(n) the num ber of ways an even number n can be expressed as a sum of two primes. The representations p 1 t p 2 and p2 + p 1 are considered to be different, R(n) denotes the number of ways an even number can be expressed as a sum of four primes. The results will appear in Mat. Vesnik, (Received February 9, 1970.) 673-120. WITHDRAWN. The April Meeting in Madison, Wisconsin Aprill4-18, 1970 674-l. WITHDRAWN. 674-2. JOEL W. ROBBIN, University of Wisconsin, Madison, Wisconsin 53706. A sufficient condition for structural stability. Let M be a compact smooth manifold without boundary and f: M - M a C 1 diffeomorphism. Let d be an admissible metric on M and define a new metric df on M by the formula dix,y) = sup( d(fn(x), fn(Ylll n E Z} • Note that the topology determined by df may be different from the topo logy of M; for example, iff is expansive, the former topology is discrete. Let Xf(M) denote the space of all continuous vector fields on M which are Lipschitz with respect to the metric df; i.e. if (a.,U) is a chart on M, there exists K > 0 such that 1117a.(x) - 17a.(Y)IJ ~ Kdf(x,y) for x, y E U where To. • 17 (x) = (.a(x), 17a.(x)). Xt(M) can be made into a Banach space, Define a continuous linear operator flr :xf(M)- Xf(M) by f'IF17 = Tf-l, 11 • f. Theor<:!tll, U l- f/F is split surjective, then f is structurally stable. The 521 hyp:>thesis of this theorem is satisfied by both Anosov and Marse-Smale diffeomorphisms. (Received November 26, 1969.) 674-3. JAMES D. BUCKHOLTZ, University of Kentucky, Lexington, Kentucky 4a5a6. Succes sive derivatives of entire functions of exponential type. The Whittaker constant W is the supremum of numbers t with the following property; if f is an entire function of exponential type less than t, and f and all of its derivatives have a zero in the closed unit disk U = {lz I ~ 1), then f a a. Theorem. Iff has exponential type less than W, then either f is a polynomial or there are infinitely many n such that the convex hull of f(n) (U) does not contain a. (In the other direction, it is known that there exists an entire F of exponential type W such that F(n\u) contains a for every nonnegative integer n.) Theorems of R. P. Boas [Duke M-ath. J. 6(194a), 719- 721] and G. A. Read [J. London Math. Soc. (2) 1(1969), 189-192] on univalent derivatives of entire functions are obtained as special cases by noting that the condition on f(n)(U) implies that f(n·l l is univalent in U. (Received December 6, 1969.) 674-4. WILLIAM R. HINTZMAN, San Diego State College, San Diego, California 92115. Best uniform approximations via annihilating measures. A general correspondence theorem on best approximations and annihilating measures is given for best approximations fa in the uniform norm from A, a subspace of C(K), to g, a continuous com plex valued function defined on the compact set K. The theorem states that g- fa= llgiiAIP-l a.e. diJ where llgiiA is the distance from g to A and tpd'JJ. is an annihilating measure of A of total variation 1 which gives the distance llgiiA = Jg.f,lld~ where I'PI= 1 a.e. diJ anti diJ ~ a. The theorem is then applied tog in C(K) where K is the closed unit disk and A is the subspace of all f in C(K) which are analytic in K. The first result is that each best approximation fa is unique since the support of the correspond ing annihilating measure !pdiJ is large enough to ensure the uniqueness of fa· Second, an algorithm is given for obtaining the best approximation fa to g, a polynomial in z, by constructing the annihila ting measure IP diJ and solving for fa in the above equation. Finally, the algorithm is extended to a larger class of functions g by utilizing a larger class of annihilating measures. {Received December 15, 1969.) 674-5. P. R. AHERN, University of Wisconsin, Madison, Wisconsin 537a6 and D. N. CLARK, University of California, Los Angeles, California 9aa24. Radial Nth derivatives of Blaschke products. Let B(z) = ~= 1 (ak/lakl)(ak- z)/(1- akz) be a Blaschke product, defined for lzl < 1 and --1 ( lz I> 1) \( ak ) . !heorem. (i) Let N > 0 be even. Necessary and sufficient that the Nth derivative of B and all its subproducts be bounded as z, lz I < 1, tends radially to eix is that (•) ~(1- lak I>! leix- ak IN+ 1 522 674-6. BURTON I. FEIN and MURRAY M. SCHACHER, University of California, Los Angeles, California 90025. Solutions of pure equations in rational division algebras. Let Q denote the rational field, If f(x) E Q [x], then f(x) is Q -adequate .. there is a finite dimensional division algebra D with center Q and A ED such that f(A) = 0. The Q-adequate pure polynomials are characterized by the following result, Theorem, Let b E Z and m E z+. (1) xm + b 2 is Q-adequate if any of the following conditions are satisfied: (i) b < 0, (ii) 4 {' m, (iii) b r/. Z • (2) x4 m + b is Q-adequate if and only if x4 + b is Q-adequate. (3) If bE 4Z4 , then x4 + b is Q- 4 2 r 2 adequate, (4) Let b E Z 2 , b r/_ 4Z , and suppose b = c where c = 2 de , r = 0 or l,d odd and square- free, d,c E Z, c f. 0. Then x4 + b is Q-adequate if and only if one of the following conditions is satis fied: (i) There are two distinct primes "' 3 (mod 4) dividing d, (ii) There is a prime ;; 3 (mod 4) dividing d and r = 0. (iii) There is a prime "' 3 (mod 4) dividing d,r = 1, and d ¥ 7 (mod 8). (5) Let u,v E Z, uv f. 0, v :fo- 1, (u,v) = 1. xm + u/v is Q-adequate if and only if xm + vm-lu is Q-adequate, (Received January 19, 1970.) 674-7. KENNETH R. MEYER, University of Minnesota, Minneapolis, Minnesota 55455, Generic bifurcation and stability properties of periodic points. A classification of the periodic points of a generic area preserving diffeomorphism which depends on a parameter is given, The stability properties and nature of the bifurcation of these periodic points are analyzed in detail, The classification can be carried over to periodic solutions of a Hamiltonian differential equation by the usual method of section maps. In this case the parameter is energy. (Received January 20, 1970.) 674-8. EVELYN FRANK, P.O.Box 361, Evanston, Illinois 60204, Error formulas for analy- tic continued fractions, New error formulas are developed for the computation of certain real and complex analytic continued fractions. (Received November 14, 1969.) 674-9. DAVID E. FIELDS, Box 1270, Stetson University, DeLand, Florida 32720. Zero divisors and nilpotent elements in power series rings. Preliminary report, It is well known that a polynomial f(X) over a commutative ring R with identity is nilpotent if and only if each coefficient of f(X) is nilpotent; and that f(X) is a zero divisor in R[X] if and only if f(X) is annihilated by a nonzero element of R. This paper considers the problem of determining when a power series g(X) over R is either nilpotent or a zero divisor in R[[X]]. If R is Noetherian, then g(X) is nilpotent if and only if each coefficient of g(X) is nilpotent; and g(X) is a zero divisor in R[(X]] if and only if g(X) is annihilated by a nonzero element of R. If R has positive characteristic, then g(X) is nilpotent if and only if each coefficient of g(X) is nilpotent and there is an upper bound on the orders of nilpotency of the coefficients of g(X). Examples illustrate, however, that in general g(X) need not be nilpotent if there is an upper bound on the orders of nilpotency of the coefficients of g(X ), and that g(X) may be a zero divisor in R([X]] while g(X) has a unit coefficient. (Received January 22, 1970.) 523 674-10. HUBERT J. LUDWIG, Ball State University, Muncie, Indiana 47306. Boolean 2- __ Boolean 2-Geometry is the study of the space (B.p ,a.), where B is a Boolean Algebra, p maps B X B into B and is defined by 0 (a,b) = (ab t a'b' )' and a. maps _B X B X B into B and is defined by a.(a,b,c) =(abc t a'b'c')'. L. M. Blumenthal [Rend. Circ. Mat. Palermo (2) 1(1952), 343-360] has shown that p has distance-like properties and has studied the space (B.p ). It is shown here that a. has area-like properties and is onto. Two equivalences for the between relation studied by Blumen thal are developed. An element d is said to be interior to elements a, b and c, I(a,b,c;d), if and only if a, b, c and dare distinct and the sum of a.(a,b,d),a.(a,d,c) and a.(b,c,d) is equal to a.(a,b,c). It is shown that given distinct elements a, band c of B there exists an element d of B such that I(a,b,c;d). Four equivalences for I(a,b,c;d) are developed and it is also shown that the interior relation has a transitive property in that if we have I(a,b,c;d) and I(a,b,d;e) we also have I(a,b,c;e). Some relations between I(a,b,c;d) and interior relations involving complements of these elements are developed. (Received January 26, 1970.) 674-ll. DONALD B. COLEMAN and EDGAR E. ENOCHS, University of Kentucky, Lexingtoa Kentucky 40506. Isomorphic polynomial rings. By" ring" we will mean" associative ring with identity element.'' We say that a ring A is _strongly invariant if whenever C1: A[X] - B [X] is a ring isomorphism of polynomial rings, then cr (X) is a B-algebra generator of B[X]. If A is a strongly invariant ring, then A is invariant in the sense that if B is a ring such that A[X]"' B[X], then A"' B. Let J(A) denote the Jacobson radical of A. Theorem. If J(A[X]) = J(A)[X] and if A/ J(A) is strongly invariant, then A is strongly invariant. Corollary. Every left or right perfect ring is strongly invariant. Corollary. Every left or right A Artinian ring is strongly invariant. (Received January 26, 1970.) 674-12. WITHDRAWN. 674-13. EDMUND H. ANDERSON, Mississippi State University, P. 0. Box 5140, State College, Mississippi 39762. Chains of simple closed curves and a dogbone space. A closed chain 4> is a collection L 1, •.. ,LN' N :;:; 3, of simple closed curves such that, with suitable numbering and subscripts modulo N, each Li links only Li-l and Litl" A dogbone space is an USC-decomposition of E 3 into points and tame arcs such that each nondegenerate element is expressible as the intersection of a tower of solid double tori. Theorem. Suppose If! is a closed chain in a topo logical 3-cube M and Dis a homeomorphic image of the two-sphere in E 3 Then, for each Li in 674-14. STEPHEN BE RNFE LD, University of Missouri, Columbia, Missouri 65201. Liapunov functions and global existence without uniqueness. We consider the ordinary differential equation x = f(t,x) where f: R X Rn- Rn is continuous and provide necessary and sufficient conditions for the global existence of solutions in terms of Liapunov functions which depend upon solution funnels. Theorem. All solutions of x = f(t,x) exist forever if and 524 only if there exists a function V: R x Rn - R such that (a) V(t,x) - oo as JxJ - oo uniformly fort in compact sets of Rand (b) for every point p = (t0,x0 ) c R X Rn and each solution x( ·, t 0 , x 0 ) we have V(t, x(t, t 0 , x 0)) ~ rp(t) where rp: R - R, satisfies rp(t0 ) = V(t0,x0 ), and is bounded above fort in compact sets of R. (Received january 30, 1970.) 674-15. KONRAD jOHN HEUVERS, Michigan Technological University, Houghton, Michigan 49931. On the types of functions which can serve as scalar products in a complex linear space. In this paper a generalized inner product (xJy) is defined as a binary function with complex values which satisfies the following: (i) for any nonzero vector y and any complex number C there existsavectorxsuchthat (xJy) = C,(ii) (x1 tx2 Jy 1 ty2) = (x 1 Jy 1 } + (x2 Jy 1 ) t(x1Jy 2 )t ( x 2 Jy 2 ), (iii) (YJx) = f[ (xJy}] where f is a continuous function and, (iv) (xJIJy} = g[IJ, ( xJy} J where g is a continuous function. These conditions induce several functional equations which are then solved. By making a linear combination of (xJy) with its complex conjugate a new function (xjy) is obtained which is either symmetric, antisymmetric, or Hermitian. The functions (xJy} and (xjy) have the same orthogonal vectors. (Received February 2, 1970.) 674-16. GEORGE U. BRAUER, University of Minnesota, Minneapolis, Minnesota 55455. ?ummability and Fourier analysis. Prelim:nary report. If s = (sn J is a sequence and ({J is a summation method which evaluates s to o we write Jsd tfJ = CT. We define the Fourier transformation <0 of a summation method rp as a linear functional on a space M of functions f(z) = r:;;o=of(n)zn analytic in the unit disc D; this functional is given by ca(f) = Jt(n)D m. Theorem !. If a linear functional L agrees with the Fourier transform of a regular matrix m:'!thod with the Borel property, then L(!/1 - z exp (ia))is zero or one according as a is or is not an integral multiple of 2rr. Let Mp denote the space of functions f(z) = ~ 0 ?(n)zn analytic in D such that JlfiJMp =lim supr-I(l- r)1/p'rfo 11 jf(rl/p'ei e)Jpd ej2rrJI/p is fi.1ite p' = p/(p- I). If Lis a well-defined linear functional on a space M , p :.; L(l/(1 - z)) = 1, then L agrees with the Fourier p transform of a summation method which includes strong Abel-p' -summability. A functional L on a normed space is said to be well-defined if L(f) = 0 whenever JlfiJ = 0. (Received February 2, 1970.) 674-17. WILHELMUS A. j. LUXEMBURG, California Institute of Technology, Pasadena, Cali fornia 91109 and j. KOREVAAR, University of California, Sau Diego, La jolla, California 92037. Entire functions and Miintz-Szasz type approximation. Let (An) be a sequence of distinct complex numbers such that JRe Ani '1!: o I A nJ' o '1!: 0, and let (ljt), t '1!: 0, be a nonnegative increasing function. Then there exists an entire function f(z) of arbi trary exponential type -r > 0 such that f(iAn) = 0, n = ! 12, ••• and jf(x)J 1! exp ( -W(JxJ J}, - oo < x E' 1/ J). n I 525 ~ k ~ Re~k Re~k inequalitiesdk=d(x ,Sc(x n,nfk}1!!< (b-E') (asRe~k~+oo),dk~(a+E') (as Re~k ~- oo), Inequalities of this form were known in special cases, but previous proofs were very complicated: they reduced the case of [a,b] to [0,1] (cf. L. Schwartz, "Etude des sommes d'exponentialles reelles," Hermann, Paris.J 1943), Applications. determination of the closed span Sc(x ~n); certain boundary value problems. (Received February 2., 1970,) 674-18, JOHN W. MILNOR, Massachusetts Institute of Technology, Cambridge, Massachusetts 02.139 and PETER ORLIK, University of Wisconsin, Madison, Wisconsin 53706, Isolated singularities defined by weighted homogeneous polynomials. Let f(z0, ... , zn) be a complex polynomi.al which is "weighted hom'Jgeneous ": that is f is a linear combination of monom:als z~O ... z~n for which i0;w0 + ... + in/w n = 1, where w0, ... , wn are fixed p:Jsitive ratio!lal numb·ers. Th.=orem. Iff has an isolated critical point at the origin, then the middle - 1 ·a homology group Hn F of the hypersurface F = f (e1 ) is free of rank equal to (w 0 - 1) ... (w n - 1). In particular this product is always an integer. (The hypersurface F has trivial homology in other di mensions. See Ml.lnor, "Singular points of complex hypersurfaces," Princeton Univ. Press, Princeton, N. J,,1968.) The characteristic polynomiall'>(t) of the m:>nodromy homomorphism HnF ~ HnF can be comp.tted as follows. To ea·~h moaic p:Jlynomia\ (t - a. 1) ... (t - a./J)with a. 1, ... , a.IJ. E C * = C - (0 l, assign the divisor (a. 1) + ... + {a.ll)' considered as an element in the integral group ring ZC *. The divisor of tk - 1 will be denoted by Ak. Theorem .. Expressing each wi as a fraction u/vi in lowest terms, the divisor of l'>(t) eq·.tals the product (v -l.A - 1) ... (v - 1 A - 1) in zc~. (Received February 4, 1970.) 0 UfJ n UT 674-19. JOHN A. BEEKMAN, Statistics Department, University of Iowa, Iowa City, Iowa 52.2.40 and RALPH A. KALLMAN, Ball State University, Muncie, Indiana 47306, Gaussian Markov expecta tions and related integral equations. Let (X(w), s ~ w ~ tl be a Gaussian Markov stochastic process with continuous sample functions. Examples of such processes are the Wiener, Ornstein-Uhlenbeck, and Doob-Kac processes. An operator valued function spaGe integral is defined for each process. This was done for the Wiener process by R. H. Cameron and D. A. Storwick in J, Math. Mech. 18 (1968), 517-552. For functionals of the form F(x) = exp ( J! 8(t - w, x(w))dw J where 8(t,u) is bounded and almost everywhere continuous, the special integrals satisfy integral equations related to the generalized Schroedinger equations studied by J, Beekman in J, MHh. Mo~ch. 14 (1965), 789-806. For the Wiener process, a "backwards time" equatio!l is coupled with the Cameron-Storwick equation to give a pair of integral equations. (Received February 2., 1970.) 674-20, FRED G. BRAUER, University of Wisconsin, Madison, Wisconsin 53706. On perturbati~~ of as~mptot!_cally stab_!~~stems. Uniform asymptotically stable systems of differential equations have certain perturbation prot~erties which are not shared by all asymptotically stable systems. For physical applications it is desirable to know as large a class as possible of systems with these perturbation properties. Here a form of stability is defined in terms of the variational system of a given system which reduces to uniform asymptotic stability for linear systems with constant coefficients, and results are obtained 526 on the effects of bounded perturbations arid of perturbations which tend to zero. The class of systems having this type of stability prop·~rty overlaps with the class of uniform asymptotically stable systems, but neither class contains the other. (Received Ja.:mary 23, 1970.) 674-21. JACOB J. LEVIN and DANIEL F. SHEA, University of Wisconsin, Madison, Wisconsin 53706. 0'1_!!1_:~~~~~,9tic behavior of the bounded solutions of so~_e_integral equations. We study the behavior as t-oo of the bounded solutions of the equations: (1) x'(t) + J~00 (x(t - ~)) dA(t) = f(t) (' = d/dt, - oo < t < oo), (2) x(t) + J~00 g(x(t - ~))dA(~) = f(t) (- oo < t < oo), where g, A, and fare prescribed functions. It is assumed that g E C, A E BV, and f(oo) exists. The analysis dep~nds on relating solutions of (l) and (2), respectively, to certain solutions of the "limit" equations: (l *) y'(t) + J~00 g(y(t- ~))dA(~) = f(oo) (- oo < t 674-22. RICHARDT. MILLER, U; Let k :! n - 3 and £ > 0. Theorem l. If H: Dk - R n is a topological embedding, there is a piecewiselinear embedding G: ok- Rn that £-approximates H. Corolla.!)' 2. Let Mk and Qn be piece wiselinear m,nifolds and let H: Ml: - Qa be a topological embedding. Then there is a piecewiselinear embedding G: Mk- Qn tha: £-approximates H. Cor,9llary 3. Let Mlc and Qn be topological manifolds and let H: Mk - Q" b·~ a topological embedding. Then there is a locally-flat embedding G: Mk: - Qn that £-approximates H. (Received February 5, 1970.) 674-23. EWING L. LUSK, University of Chicago, Chicago, Illinois 60637. Ho.!I.!_'Jtopy ~oups of spa:::e~~Lembeddings. Let M and Q be P L manifolds of dime~1sious m and q respectively, with M .:;ompact. Theorem 1. Let F: M X 8 5 - Q X Bs be a level-pre3erving, proper embedding which is level-preservingly locally flat. Let £ > 0 be given and suppose m + s ~ q - 3. Then there is a level-preserving ambient isotopy H:Q X8 5 XI- Q XBs XI such that Ho = l, H 1F is PL, and d(H,l) < £. Let TOP(M,Q) be the semi simplicial co:«plex of proper embeddings of Minto Q and let PL(M,Q) be the subcomplex of PL em beddings. Let Hom(M,Q) be the top:>logical space (compact-open top:>logy) of proper embeddings of M into Q and let HomPL(M,Q) be the subspace of PL emheddings. Theorem 2. If m + s ~ q- 3, then Hom(M,Q) and Homp L (M,Q) are locally s-connected. Theorem 3. If m + s ~ q- 4, then Trs(TOP(M,Q)) = Tl's(P L(M,Q)) = Trs(Hom(M,Q)) = Tl's(Homp L (M,Q)). The proofs of the theorems make use of theorems of Morlet on generalized concordances and isotopies and of Richard T. Miller on approximating embeddings by 527 P L embeddings, as well as techniques of Homma and Gluck and of Bryant and Seebeck which were used for tam:ng locally tame and locally nice embeddings. (Received February 6, 1970.) 674-24. WITHDRAWN. 674-25. JACOB KO"EVAAR and TUNE GEVECI, University of California, San Diego, La Jolla, California 92037. The f!~l_d ins!_d_~_<:_harged _t:_oll~ con~uctor. Preliminary report. Let D be the interior of a simple closed analytic curve C in the complex plane. For each n, let znl'"'' znn denote nth Fekete points on C, that is points z 1, ... , zn for which n j;tklzj- zkl is a maximum. What can one say about the corresponding sums sn (z) = 0k= 1 l /(z - znk)? The authors show that sn(z) tends to a lim~t function rl(z) in D as n ~oo, uniformly on every compact subset of D. Physical interpretation, For large n, the field in D due to an equilibrium distribution of n electrons on C is at most of the same order of magnitude as the field due to a single electron on C. The authors show also that rl(z) is identically zero if and only if C is a circle. (It is somewhat surprising that when C is not a circle, the field due to an equilibrium distribution is not as small as that due to elec trons placed at conformal images of nth roots of unity.) References. G. R. MacLane, Duke Math. J, 16 (1949), 461-477; J. Korevaar, Ann. of Math (2) 80 (1964), 403-410; C. Pommerenke, Math. Ann. 179 (1969), 212-218. The three-dimensional case appears to be much harder, although not hopeless. (Received February 5, 1970.) 674-26. JO:-IN D. ARRISOC\1, Monmouth College, Monmoath, Illinois 61462. Q!!.__a cla~~_o_f par~~!!Y stable_E-.:>~::£...~1!1-'!..t~!!_v_c::._~.!.gebJO._~~- Let A be a finite dimensional, strictly p;>wer-associative, partially stable algebra satisfying x(xy) t (yx)x = 2(xy)x over an algebraically closed field F of characteristic p not equal to 2 and 3. Let A+ be the algebra with the same vector space structure as A and multiplication, x • y, on A+ given in terms of multiplication in A by x • y = (xy t yx)/2. Theorem l. A is simple if and only if A+ is simple. Let A= A1 t A 112 t A0 be the Peirce decomposition of A relative to a stable idempotent 2 of A. It is known that there is an element win A112 with w = l. Let B be the set of all elements b in A 1 t A0 such that (w • b) • w = b. If the characteristic o: A is also not equal to 5, we may state the following structure theorem. Theorem 2. B is an associative commutative truncated polynomial algebra, B is isomorp:1ic to A 1 ac1d A0 , A 1 + A0 is an associative commutative algebra and A is flexible. (Received February 6, 1970,) 674-27. HARI M. SRIVASTAVA, University of Victoria, Victoria, British Columbia, Canada. A gen:_rati~~-~unction _!_:>_!_~rtain_~~gici~~s inv_olv!ng several ~~?_~plex ~~i:._ables. Preliminary rep;>rt. For the sake of brevity, let {k 1 denote the seq•1ence of r nonnegative integers k 1, ... , kr' and let ~mk = !j~ m.k .. Also let a and b be arbitrary complex numbers, and define a function of several I=! 1 1 complex variablea z 1, ... , zr by the eqilality (*) f[z 1, ... , zr] = !:Dc 1 =04>(k 1, ... ,kr; z 1, ... , zr), where r k. ~(k 1 , ... , k ; z 1, ... , z ) = C(k 1, ... , k ) n. (z.) 1/(k.)!, the C(k 1, ... , k ) are arbitrary constants, real r r r !=I .l L r a+ 1 - l or complex. In the present paper the author proves the generating function (**) (I tv) (l - bv) •fl(- )mlz (- vmrz II= ~oo (a+(bt!)n)tn 'I;Lmk~n {(- n)~ml•/(1 +a+ bn)~.m·' v 1' "·• r n=O n {k 1=0 ~ ' "-' k' 528 4>(k 1 , ... , kr; zl•···• zr), where vis a function oft defined by v(t) = t(l + v)b+l, v(O) = 0. For r = l, (**)would correspond to an earlier result of the author [Abstract 69T-B 198, these c)/oticei) 16 (1969), 975). Ind,~ed in the ganeral case, f[z 1, ... , zr] can be expressed fairly easily as the generalized La·.rrlcella function defined recently in a joint paper of the present author {Nederl. Akad. Wetensch. Proc. Ser. A 72 = Indag. Math. 31 (1969), 449-457], and the coefficients generated by(**) would then turn out to be a class o~ generalized hyp·ergeometric polynomials of several complex variables. (Received February 9, 1970.) 674-28, C. C. CONLEY and JOEL A. SMOLLER, Courant Institute, New York University, New York, New York 10012. Vis<:?sity ~trices_~r two-~imen~~n~-~onlinear hyperbolic systems. Consider the 2 X 2 hyperbolic system (1) Ut + F(U)x = 0, and the associated parabolic ("viscosity") system (2) Ut + F(U)x = (PUxx' where F satisfies the conditions in the paper (J. Smaller, "On the solution of the Riemann pro!:>lem with general step da~a for an extended class of hyperbolic systems~ Mich. Math. J. 16 (1969), 201-210), Pis a positive definite sym~etric matrix and ( > 0. We consider shock-wave solutio:~s o~ (1) and ask whether they can be obtained as limi.ts of standing wave solutions of (2) as ( ""'0. Such P are called admi.ssable. Using techniqlles for the qualitative analysis of planar flo·Ns we find criteria for admissability and inadmissability. These we apply to the above systems and show, for examp~e that the id·entity matrix is always admissable. For a more special class of systems, we show that all P with positive entries are admissable and find precisely the ways in which admissability fails. (Received February 9, 1970.) 674-29. MHLVIN C. THO:tNTON, University of Nebraska, Lincoln, Nebraska 68508. Total Using elementary fiber bllndle theory, well-known results on the topological type of the total spaces of principal S 1- bundles over s3 and s2 X S 1 are extended to principal S 1-bundles over an arbi trary lens space. This extension may also be done easily with Chern classes, but it has not appeared in the literature. The_?reE:l_. Let ffi' 0 ~ i ~ p - 1 J be representatives of the elements in [L(p,q),CP 00J. Then the total space of the principal S1 bundle determined by fi is homeomorphic to L(diq) XS 1 where di = gcd(i,p). Corollary 1. Orientable four manifolds which are the total space of a s 1-bundle (wi.th structure group all homeomorphisms of S 1) over a fixed lens space are topologically determined by their fundamental groups. Corollary 2. The number of times s3 X S 1 occurs as the total space of a S 1 -bundle (with structure group all homeomorphisms of S 1) over L(p,q) is the number of integers between 0 and p - l relatively prime top. (Received February 9, 1970.) 674-30. D. K. COHOON, University of Wisconsin, Madison, Wisconsin 53706. Nonexiste~ of a continuous r~ht inverse f~r surjective linear partial differential operators. The author extends results on the continuous right inverse question previously communicated in Abstracts 663-608, 665-28, these c)/oticei) 16 (1969), 265, 540, for linear partial differential operators with constant coefficients and n iii! 2 independent variables to the case where the dimension of the space spanned by the characteristics is n. We show in particular that if N is a noncharacteristic direction of P (D) and n = (x E ~n: a < (x,N) < b 1 then P (D) has a continuous right inverse in C 00 (fl) if and only if N is a hyperbolic direction for P(D). (Received February 9, 1970.) 529 674-31. JO:-IN ]. CURRANO, Uaive:cslty of Chicago, Chicago, Illinois 60637 and Roosevelt University, Chicago, Illinois 60605. Co:Ijug~e_p~_!J_?_£Eo:Ip3 with maximal intersection. Let P be a p-s:Ibgroup of a finite group, G, g E G, and assum~ that Q = P n Pg has index pin P. Assume that g normalizes no nonid·entity normal subgroup of P, e.g., that G is a simple group generated by P and g. In an unpublished paper, Glauberman has shown that P has nilp·:>tence class at most 3 if pis odd and at most 2 if p = 2. Let H be the subgroup of G generated by P and Pg. Assume (I): If N is aay normal subgroup of H containing either PCH(Q) or PgCH(Q), then N = H. Theorem I. If class P = 2, Pis isomorp:1ic to a dire::t product of a certain number of copies of E(p) times a central elementary abelian subgroup, where E(p) is the nonab.elian group of order p3 generated by elements of order p. Theorem 2., If class P = 3, P is is·omorphic to a direct product of a certain number o~ co;:>les of E•(p) times a ce:1tral elementary abelia'l subgroup, where E*(p) is a sp·ecified gro:1p of order p6. This generalizes a res:Ilt of Glauber man which proves Theorems I and 2 under the hypothesis that P aad pg are co;J.jugate in H. Theorem l remains tr:Ie if (I) is replaced by (2): V = [Z(Q), H] is not central in H, and H = PC = pgc where C = CH(Q/V). (Received February 6, 1970.) 674-32. ABRAHAM ZAKS, Northwestern University, Evanston, Illinois 60201. Quasi.::_ Frobeniu~.!ings. For an injective R- module, define the order of M, o(M;., to be the least integer for which the R-module I:~~~) M is free. If no such integer exists, set o(M) = oo. The global order of R is the least up;>er bound of the orders of the injective R-modules. Proposition. Let R be a quasi-Frobenius ring. A necessary and sufficient conditio'l for R to be of finite global order is that any two-simple R-modJles are isomorpl1ic. Corol.lary. A ring R is aa n Xn matrix algebra over a local quasi Frobenius ring if a~1d only if its global order is n. Theorem, An Artinian ring R is a direct sum of local quasi- Frobenius rings if and only if every minimal faithful injective module is free, and HomR (Q.Q') i 0 implies that Q is isomorphic to Q' whenever Q and Q' are indecomposable inje::tive modules. Theorem. Let Q be a projective and injective R- module, such that Q has a unique jordan Holder series. Then the ring of endomorphisms of Q is completely indecomposable. (Received February 12, 1970.) 674-33. VANCE FABER, Washington University, St. Louis, Missouri 63130. Large abelian subgroups of some infinite groups. We prove a generalization of a conjecture by W. R. Scott ("Group theory," Second printing, Englewood Cliffs, N. ]., 1964, p. 21). Theorem l. If U = (Aa) aEW is a totally ordered system of sub groups of a group G containing the whole group G = A11 and some A0 =naEWAa' then ~aEJ[Aatl: Aa] ~ jA . A I~ n [A . A J where J =(a E WjA and A I form a jump in U}. Using Theorem!, we IJ • 0 - aE J at I · a ' .. a at show that the members of certain classes of infinite groups are guaranteed to have abelian subgroups of large order. For example, define FCI*-groups by analogy with SI*-groups. Then Theorem 2. Every infinite FCl*-group G has an abelian subgroup A with exp expjAjl'; jGj. (Received February 12, 1970,) 674-34. WILLIAM C. CHEWNING, Iowa State University, Ames, Iowa 500!0, Local finite =ohesio~. Prelimi.nary report. A set X has local finite cohesion (I .f.c.) if each open set about a point x EX contains the closure 530 of some region R about x, such that for R = A+ B, A and B closed and connected sets, the set A n B has no more than k com?onents which miss Fr R. A region such as R is a k- canonical region; k is an integer depending on R. For locally compact metric X, the author has Theorem 1, If X is either finitely coherent, locally unicoherent, or locally cohesive, then X is l.f.c. Theor~m 2. If X c R 2 is com?act and l.f.c., then X is finitely coherent. Theore~ 3. If X is l.f.c., then X is locally !-cohesive, i.e. each point lies in an arbitrarily small !-canonical region. The_9_E~m 4. If X is locally !-cohesive and locally rim connected, then X is locally cohesive. _!.!_leore~ 5. If X and Y are nondegenerate generalized Peano continua, then X X Y is locally !-cohesive and locally rim connected. Theorem 6. If X is l.f.c. and D is a totally disconnected subset of X, then the quasi-components of X - D are con nected. !he~~~ 7. If X is l.f.c. and Y is regular and T 1, then any connectivity function f: X - Y is perip:1erally continuo•.1s. The~ 8. If X is l.f.c. and r: X -X is a connectivity retract then r(X) is a generalized Peano continuum .. (Received February 12, 1970.) 674-35. JUTTA HAUSEN, University of Houston, Houston, Texas 77004. A class of abelian !.~~s_!?_:t_.l~:£.~!:!.1:~-~-c_!~rized b_y their_~i~m<~_phi~~-~oue_. Let G denote an abelian torsion group and A(G) its group of automorphisms. Theorem 1. The following conditions are equivalent. (i) A(G) possesses a subgroup r which is isomnphic to the additive groap of rational numbers. (ii) The groupS W of all permatations on a countably infinite set is isomorphic to a subgroup of A(G). (iii) G possesses infinitely many pairwise isomorphic subgroups 0 ¥Hi, i E I, such that G = Z::iEIHi + C for a suitable subgroup C. Call G homogeneous, if G is the direct sum of pairwise isomorphic cyclic or quasi-cyclic subgroups. Theorem 2. Every subgroup r i 1 of A(G) possesses a finite epimorphic image i 1 if (and only if) all homogeneous direct sum mands of G have finite rank. (Received February 13, 1970.) 674-36. LEON BROWN, Wayne State University, Detroit, Michigan 48202, and PAUL M. GAUTHIER, University of Montreal, Montreal, Quebec, Canada. The rang_e set of a meromorphic functio:1. Let f be a function meromorphic in the unit disc of the complex plane. The (global) range of f is the set of values assumed infinitely often by f (similarly the (local) range off at a boundary point x is the set of values assumed infinitely often in every neighborhood of x). We call a set E a ll-set if E is a countable intersection of nested, connected open sets. The range off is a 6.-set, and con versely, it is known that every 6.-set is the ra~1ge set of some function meromorphic in the unit disc. In this paper, we show that the intersection of open connected sets is a 6.-set. We also prove a theo rem on 1:!. -sets, one consequence of which is that if the cluster set of a function meromorphic in the unit disc has a nonem?ty interior, then the range set has infinite linear (Hausdorff) measure. We also present an example to show that this theorem is, in a certain sense, sharp. (Received February 16, 1970.) 674-37. ROBERT H. WASSERMAN, Michigan State University, East Lansing, Michigan 48823. Ge~~=~rical c_h_~r-~_!~!iza~5!E~~!luid !!9w. By introducing the unit vector t = vI lv I along the streamlines and the unit vector N = i7p/ lvp I normal to the constant pressure surfaces we write the partial differential equations of steady com- 531 pressible nondissipative fluid flow in intrinsic form. Certain purely geometrical conditions are de rived involving only the curvatures of the T andN congruences, div t, and the angle between the two families of curves. It is shown that, conversely, if one has two fami.lies of curves that satisfy these geometrical conditions, then one obtains solutions of the partial differential equations in which the streamlines are the curves of the o:1e family and the constant pressure surfaces are norm>l to the curves of the other family. The derived conditions simplify in various cases of special geometric hyp:>theses. In particular, considerab!e simplification is achieved when we assume the constant pressure surfaces are parallel surfaces, and under the further assumption of plane flow, actual solu- tions are obtained. (Received February 16, 1970.) 674-38. RAY MONO T. SHEPHERD, University of Chicago, Chicago, Illinois 6063 7. ~~me resalts on p-gro:Ips o: maximal class. Let P be a p-group o~ maximal class of order pn, p:.; 7, n !!!> 4. Denote the lower central series of P by P, P 2, P 3 , ... , P n = 1. Define P 1 = Cp(P 2/P 4). Let k ~ n- 3 be maximal such that for any i, j ~ l, [Pi' P j] ~ P itj+k' By a slight extension of Blackburn's methods (Acta. M3th. 100 (1958), 45-92) we can show (1) if k = 1 then n ~ 10 when p = 7 and n" p + 2 when p., 11. (2) If k"' 0 (mod p - 1) then n - 2k ~ p + 1. Corollaries of (2) are: (3) n - 2k ~ 4p - 5 and (4) if n ~ 12p-25 then the class of P 1 ~ 3. The best possible bounds in (3) and (4) are not yet determined. (Received February 16, 1970.) 674-39. PHILLIP L. ZENOR, Auburn University, Auburn, Alabama 36830. j-compac~ac~. Let J denote the integers. X is ]-compact if X is homeomorphic to a closed subspace of Jm for some cardinal number m. Theorem A. Suppose that X is a come>letely regular space such that either X has large inductive dimension zero or X is extremally disconnected, then X is realcom?act if and only if X is ]-compact. Theor._e_n:!, B. The com?letely regular space X is the image of a realcome>act space under a proper mapping if and only if X is the image of a ]-compact space under a proper mapping. (Received February 16, 1970.) 674-40. LAURENCE C. SIEBENMANN, Princeton University, Princeton, New Jersey 08540. A[Jp~ximatin~~}!_ll_l_a_r_~~E~b_L_il.£_m_eom~_p_his~. A proper map f: M ~N of topological n-manifolds without boundary is called cellular if, for each pointy E N, f- 1(y) has small neighborhoods homeomorphic to an open n-cell. For n '!; 5, it is know'l that f is cellular iff, for ally E N, the compactum f- 1(y)has a certain intrinsic property called cell-like. Theorem. Suppose f is cellular and n i 4. Then f can be approximated by homeomorphisms. In fact if r: N ~ (O,oo) is a given co:J.tin•Jous function, and d is a metric on N, one can find a proper homotopy ft: M~ N, 0 ~ t ~ l, off= fo such that ft is a hom•,omorphism fort> 0 and, for all x, d(ft(x), f(x)) < r(f(x)). The case n = 3 has been proved by Armentrout (see Bull. Amer. Math. Soc. 75 (1969), 453-456) using classical methods. A multi-parameter versio.ct of this theorem holds. It implies an extension theorem for cellular homotopies. It also implies that the two spaces, made up respectively of proper cellular maps and of homeomorphisms M to N, are weakly homotopy equivalent for the compact-open topology. Pro::>fs are based on the main diagram of Kirby and Siebenmann, Bull. Amer. Math. Soc. 75 (1969) 745, and a device to eliminate the Alexander isotopy. (Received February 18, 1970.) 532 674-41. ROBERT]. DAVERMAN, University of Teanessee, Knoxville, Tennessee 37916. ~haracterization of universal crumpled cubes. A crumpled cube C is said to be universal iff for each crumpled cube C* and homeomorphism h of Bd C to Bd C* the identification space C U h C* is homeomorphic to the 3-sphere s 3 . W. T. Eaton has shown that the following is a sufficient conditio'! for a crumpled cube C to be universal: (Eaton's Condition) for each Cantor set Kin Bd C, C- K is 1-ULC. The_ore_I!l;_l. A crumpled cube Cis uni versal iff Eaton's conditio:t is satisfied. The~ 2. There exists a particular crumpled cube T such that a crumpled cube C is universal iff for each homeomorphism h of Bd C to Bd T the space C Uh T is homeomorphic to s 3 . The exam ole T which tests for universality is a modification of a crumpled cube described by Stallings (A:m. of Math. (2) 71 (1960), 185-186). (Received February 19, 1970.) 674-42.. TIMOTHY V. FOSSUM, U.1iversity of Utah, Salt Lake City, Utah 841ll. Characters Le A be a Frob·enius algebra over a field K with nondegenerate, associative, bilinear form f. D·dine .p: A- A*= HomK(A,K) by A-modules M,N, respectively, and let (ai) and (bi) be K-bases for A which satisfy f(ai' b j) = liij (Kronecker delta). Then :0; x (a .)~ (b.)= 0 unless M ~Nand x = C. If P E A* is the character of the • 1 1 left A-module AA, then .p- 1(P) =:B.a.b., and it follows that x(0.a.b.) = m0.x(a.)x(b.), where m is 1 1 1 l 1 1 1 1 1 the multiplicity of M as a compositio:t factor of A A. In particular if :0 i a ib i = a for a E K, then a \1{1) = mLx(a.)\( (b.). The elementary orthogonality relations for gro<.~p-characters over the complex 1 1 field follow from this. (Recelved February 18, 1970.) 674-43. ARTHUR G. SPARKS, Georgia Southern College, Statesboro, Georgia 30458. Definition I. A compact set in E 2 is said to be simply-connected iff its complement is connected; De_f_igi_t:!~ 2. Let~ be a famHy of. non void sets in Em. The family~ is said to be an L0 family iff it is an intersecting famHy. If n l!, l, the,,~ is said to be an Ln fam!.ly iff for every A, B E ~.there exists a polygonal path P n' consis\:ing of no more than n segments and joining A to B, such that P n c U ~ . Theorem. Let S be a compact, simply- connected L 3 set in E 2 whose second order convex kernel is nonvoid. Then, there exists an L 1 family~ of co:J.vex sets such that U C!- = S. Remark_: This theorem and a result of Horn and Valentine suggest a possible characterization of certain Ln+2 sets in terms o: Ln families of convex sets. (Received February 19, 1970.) 674-44. JAMES E. HUMPHREYS:, Courant Institute, New York University, New York, New York 10012. ~<:_somposabl~o~ules_~o_!_~~misi~t?_~e-~roc~p~. Prelimi.nary report. Let K be an algebraically clos-ed field of prime characteristic p,G a semisim,ole algebraic group over K, W the Weyl groc~p of G. If ). ,IJ are dominant weights of G, related by the condition ). + p = (IJ + p )er (mod p) for som.~ cr E W(P =half-sum of positive ro·::>ts), call ). and 1J linked. Let h be the Coxeter number of W (=order at the prod.Ict of all simple reflectio'ls). Theorem .. Sup p::>se p >h. If M is an indesomposable ratio:1al G-module, then all highest weights of composition factors of Mare linked. The proof of the theorem (conjectured in ge:1eral by D.-N. Verma) involves an aaalogo•.1s result for Lie(G). (Received February 19, 1970.) 533 674-45. RONALD C. GRIMM3R, Southern Illinois University, CarboJ.d3.le, Illinois 62901. ~~~~~i_!¥ of d ff~E~~!i:_~-~q·~~09~~~~th hom?_§:~:!.~_right ~an·~_!des. Consider the system of first order scalar differential eqlations (l} x' = P(x,y)x, y' = Q(x,y}y, where P and Q are on R X R into R, are homogeaeo:1s of d·~gree 2m and satisfy a local Lipschitz conditioa. Assume (0,0) is the only singular point of (1) and if a is a cluster point of the zeros of P(cos 1'1, sin fl)Q(cos 1}, sin 8), either P(cos a, sin a) < 0 or Q(co.s a, sin a}< 0. Theorem, The zero solution of (1) is globally asymptotically stable if aad only if (a) P(l,O) < 0, Q(O,l) < 0, and (b) for any (x,y) where P(x,y) > 0 and Q(x,y) > 0, P(x,y) 'I Q(x,y). The proof is geometric and uses the properties of homogeneous fullctioas and a co:npa::ison theorem appUed to dy/dx = Q(x,y)y/P(x,y)x. This generalizes a result of Norman and Trench lJ. Differential Equations 5 (1969), 470-475, Theorem 1]. (Received February 19, 1970.) 674-46. FRANKLIN B. RICHARDS, University of Wis<::oasin, Madison, Wisconsin 53705. Let IT: 0 = x0 < x 1 < ... < x = l be so:nc subdivisioa of [9, 1] aad L m '= [f; f E Cm-l, f(m-l) is ' n p absol]Jtely continuous, and f(mJ E Lp[0,1]}, 1 ! p! oo. IfF E L2m and S is the natural spline of degree 2m- 1 interpolating to F on IT, the minimum ~orm ?roperry states that lls(m)ll2 ~ IIF(m)ll2' Similar remarks hold if S is p·eriodic or has mth order knots at the end,JDints. Co'ljecture. Supp·:>Se F E L;, 1 !! p ~ oo. Then there exists a constant K = K(m,p) independent of the subdivision IT such that IIS(m) lip ~ K IIF(m) lip· The conjecture is shown to be valid for the cubic case, the quintic case under certain very mild conditions on IT, and the case of a uniform subdivision (xi = i/n) if S is natural or periodic of degree 2m - I. (Received February 19, 1970.) 674-47. WITHDRAWN. 674-48. ALEX C. BACOPOCJLOS, Michigan State University, East Lansing, Michigan 48823, Simultaneous approximation of a given function f E C [a,b] by members p of an n-dimensional H.!l.ar subspace of C [a,b] is defined by llf - p lis = llw 1 (f - p) II + llw 2 (f - p) 11. where II • II is the supremum norm on [a,b] and w 1 and w 2 are two strictly positive continuous (weight) functions. Observe that, in case f does not change sign on [a,b], the special case w 1 " l and w2 = 1/f is of practical interest. Uniqueness of best simultaneous approximation fails, in general. Each best simultaneous approxima- tion satisfies a generalized oscillation which is not sufficient to characterize. Yet, in spite of these differences from Cebyshev theory, an efficient Remes- type algorithm has been developed which finds all the best simultaneous approximations. (Received February 20, 1970.) 674-49. A. LASOTA, University of Maryland, College Park, Maryland 20742, AARONS. STRAUSS and WOLFGANG L. WALTER, University of Wisconsin, Mathematics Research Center, Madison, Wisconsin 53706. Differential inequalities and differential games. For certain linear differential games (l) y' = A(t)y + ~;l Bj(t)uj with quadratic cost, there exists for each player a uniq•Je feedback equilibrium strategy u. = D .(t)y + r .(t). If, however, the J J J present state is not known, the players might choose to use the correct feedback formula on old data. 534 That is, in the nth play of the game, each could play u':(t) = D .(t) jt~- 12i-nyi(t) + rn(t)] + r .(t), where J J 1= 0 J lr (t) I ~ 0 as n ~ oo but is otherwise unknown. Now substitute uJ?-(t) into (I), call the solution yn(t), n J and let xn(t) denote the difference between yn(t) and the equilibrium solution. Then xn(t) satisfies a differential inequality in terms of xi(t), i = 0, I, ... , n - I. If xn(t) ~ 0 as n ~ oo, then uj(t) ~equilibrium strategy. This is the m~tivation behind our main result. Theore_!!!.. Let {xn(t)) be a sequence of absolutely continuous functions satisfying jx' (t)j ~ L'-.n A .(t)jx.(t)j + 1.1 (t) for almost all tin ro,TJ n 1= 0 m 1 n l! and all n ~ 0. Suppose that lxn(O)j + ll1.1nlll ~a as n ~ oo. Suppose also that, for some p >I, IIAnillp ~o. as n .... oo for each i 1" 0 and I: I_! IIA .11 is bounded as n ~ oo. Then lx (t) I ~ 0 uniformly on 0 :! t !. T as 1= 0 n1 p n n ~ oo. Here, II• II denotes the L norm. This result is false for p = 1. (Received February 19. 1970.) p p 674-50. KLAUS R. BICHTE LER, University of Texas, Austin, Texas 78 712. On the strong !_ifti:'~ proe,c:,£ry_. Prelimi.nary report. Let X be a locally compact Hausdorff space, M(X) the complete lattice of Radon measures on X, :!. 00 the algebra of bounded measurable functions, f, L 00 the algebra of their classes f. An alge bra homomorpilism T: L 00 ~ :!. 00 which is an inverse for f ~ f is an almost strong lifting if there is a locally negligible set N such that TiD= !I' on X\N for all continuous functions If) in :!.00 • Theorem 1, The set L(X) of Radon measures 1.1 on X such that 11.11 admits an almost strong lifting is a band in M(X). J_!leorem 2. In order that L(X) = M(X) for all locally compact spaces X it is sufficient that L(X) = M(X) for all products X of unit intervals. (Received February 20, 1970.) 674-51. IV AN L. REILLY, University of Illinois, Urbana, Illinois 6180 I and Eastern Illinois University, Charleston, Illinois 61920. Quasi uniformities and quasi pseudo metrics. This paper proves that the well-known Th~!:~~ "Any uniformity U on X is generated by the family of all pseudo metrics on X which are uniformly continuous on X X X" generalizes to the context of quasi uniform spaces. Q~finiti~~ 1. A real valued function f on a quasi uniform space (X,U) is quasi uniformly upper semi continuous if for each 15 > 0 there is a V E U such that (x,y) E V implies f(y) < f(x) + 15, Th«:?.~C:~ 1. Let (X,U) be a quasi uniform space, and p be a quasi pseudo metric on X. Then p is a quasi uniform~y upper semi continuous function on (X X X, u-l xU) if and only if Up, 15 E U for each 15 > 0, where Up, 15 = {(x,y): p(x,y) < 15). !?efi~tion 2. Let F be a family of quasi pseudo metrics on X. The quasi uniformity U on X which is the smallest such that each mem':Jer of F is quasi uniformly upper semi continuous on (X X X, u- 1 XU) is called the quasi uniformity generated by the fami.ly F. Theorem 2. If (X, U) is a quasi uniform space, then U is generated by the family of all q:~asi pseudo metrics on X which are quasi uniformly upper semi continuous on (X X X, U- 1 XU). (Received February 20, 1970.) ~ 674-52. ANTONIO F. IZE, Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912. On an~symDt_9ti<:_property of a Volterra integral equation. It is proved that if q(t,s) is bounded and f(t,s) is "small", the solutions of the integral equation x(t) = a(t) + J~q(t,s)f(s,x(s}}ds satisfies the condition x(t) = h(t) + P (t)a(t) limt~00 a(t) = a constant where p (t) is a nonsingular diagonal matrix chosen in such a way that P (t)h(t) is bounded. The results are extended to the m'.lre general integral equation x(t) = h(t) + J~F(t,x(s))ds and contain, some results 535 on the boundedness, asymptotic behavior and existence of nonoscillatory solution of differential equations. (Received February 20, 1970.) 674-53. DEMETRE JO!-IN MANGERO'\J and M. N. OGUZTORELI, University of Alberta, Edmonton 7, Alberta, Canada. Some problems concerning generalized polyv.ibrating equations. Starting from their own research work related with "polyvibrating equations (called by various scientists "Mangeron's equations"), of importance, for instance, in the automated design of free form surfaces and in the approximations theory using the spline functions, the authors consider in this paper various problems pertaining to the generalized polyvibrating equations having as a prototype the boundary value problem (A(x)u' + pB(x)u] + p[B(x)u' + C(x)u] = 0, x = (x 1, x2, ... , xm)R = n·-1 n.-1 [a J' '!! x.!! b.; j = 1,2, ... , m], (u = Ciu/CixJ. = ... = 21 J u/ox. J ) b = 0. Using their previous J J J xj=aj, j results related with a Dynamic Programming approach (Ist. Lombardo Accad, Sci. Lett, Rend. Al02 (1968), 250-256), the authors obtain new functional equations concerned with the boundary value problems taken into consideration. The novel aspects of the theorems lie in the interpretation of R as am-dimensional "rectangle" and the symbol ' as designating a generalized polivibrating deriva 1 ~ive asu/21x~ ox2 nz oxm nm, s = n 1 + n 2 + ... + nm' that reduces for n 1 = n2 = ... = nm = 1 to the first order total derivative in the Picone's sense.(See D. Mangeron, L. E. Krivosheine, Rend. Sem. Mat. Univ. Padova 23 (1963), 226-266.) (Received February 23, 1970.) 674-54. PETER R. LIP OW, University of Wisconsin, Madison, Wisconsin 53 706. Cardinal The problem of interpolating at all of the integers has come to be known as the Cardinal Inter polation Problem .. Here it is shown that Hermite interpolation (involving derivatives also) can be done for a fairly wide class of interpolating data when the interpolating functions are spline functions. By a doubly infinite sequence we mean a sequence y = ( ... , Y.z• y_ 1, y0 , y 1, y 2, ... ). ~~ Given r (r l!t 1) doubly infinite sequences each in lp (I ::, p < oo) which represent the values and first r - derivatives of a function at all the integers, there is a unique polynomial spline function of degree 2m - 1 (any m ~ r) and continuity class c 2 m- l-r which interpolates the data and is in L . A similar p theorem is true if 1 and L are replaced respectively by lvls) and O(lxls) (s ~ 0). Here the doubly p p o: infinite sequence y = (yv) is in 0( lv Is) if there is a constant K such that IYv I < K lv Is for all sufficiently large lvl. (Received February 23, 1970.) 674-55. JOHN J, SANTA PIETRO, Stevens Institute of Technology, Hoboken, New Jersey 07030. The Gro_£~endieck ri~g of dihedr~and qua~_EE:io~!l:_oups. Preliminary report. Let G be a finite group, ZG its integral group ring and K0 (ZG) the Grothendieck ring of ZG. In case G is cyclic formulas determi.ning multiplication in K0 (ZG) have been given by Stancl (J. Algebra 7 (1967), 77-90). Let F be the rational field, Dt the dihedral group of order 2t and Ht = (x). A full set of irreducible FDt modules is computed and is shown to determine a ring homomorphism K0 (ZHt) .... K0 (ZDt). A direct consequence is that the formulas given by Stancl serve to determine m'~ltiplication in K0 (ZDt). If Qt is the generalized quaternion group of order 4t it is shown that FQC ~ FDt e A, where A is a direct sum of quaternion algebras over real subfields of cyclotomic 536 fields. Cho-osing a maximal order in each quaternion algebra gives a linear map f: K0 (FQt) -K 0 (ZQt) necessary for determining multip.!ication in K0 (ZQt) (cf. Swan (Topology 2 (1963), 85-110)). Considera tion of special values oft indicates that m•1ltiplication form'llas for K0 (ZQt) could be derived in this way. (Received February 23, 1970.) 674-56. DION GILDENHUYS and CHONG-KEANG LIM, McGill University, Montreal, Quebec, Let C- be a full subcategory of the category of finite groups, closed under subobjects, quotient objects and finite products. Let(} be the full subcategory of G'FG, (the category of profinite groups) with the prop2rty that G E 8 iff G/N E C- for all open normal subgroups N of G. Objects of f' are called pro-C--groups, The_?re:E_ 1. The forgetful functors from (J to the categories of sets, groups, topological spaces, pointed topological spaces, compact Hausdorff spaces, totally disconnected spaces are tripleable. :!'heore~- 2. (J is equivalent to the category of models of the theory of the forgetful functor from::!- to the category of sets. Let F (respectively F') be the left adjoint of the forgetful functor from {i>FG (respectively (J) to the category of topological spaces. Theorem 3. For any dis- ~ 2card X ----- crete space X, card F(X) = 2 or 2 according as X is finite or infinite. :!'heorem 4. For any compact Hausdorff space X, the cano::tical map 7)X: X- F'(X) is injective iff X is totally disconnected. _!heore~ 5. The free pro-p-group on countably many generators (in the sease of]. P. Serre in Cohomologie Galoisienne) is an epimorphic image of some p-sylow subgroup of the free profinite group F (2). (Received F ebraary 23, 19 70 .) 674-57. G. P. BARKER, University of Missouri, Kansas City, Missouri 64110. MJnotone norms. Preliminary report. Let V be a real n-dim<:nsional vector space and let K be a cone in V (cone here means a closed, pointed and full cone). K determines a partial order in V by x ~ y iffy - x E K. A norm tJ is called monotone in the nonnegative orthant iff 0"' x"' y implies v(x) ~ v(y). Theorem I. If K is a cone, then there is a norm v monotone in K. The dual norm IJ need not be m:motone on the dual cone. However, if lub(A) is the matrix norm induced by IJ we have Theorem 2. v and 1J are monotone inK and K* iff lub is monoto::te oa the no:1negative matrices of rank I. Theorem 3. If Sis any set of nonnegative matrices for which lub(A) = P(A), the spectral radius, for all A E S, then lub is monotone on S. (Received February 23, 1970.) 674-58. PAUL H. RABINOWITZ, University of Wisconsin, Madison, Wisconsin 53 706. Nonlinear eigenvalue problems for(*) Lu"' -(pU')' + qu = \F(x,u,u'), 0 < x < TT, together with separated bo:.tndary conditions (B.C.) are studied. More general cases are also treated. Let r$ = IR X (C 1 [0, TT] n B.C.). A solution of (*) is a pair (\ ,u) E r$ and (). ,0) is a trivial solution. Associated with(*) is a linear equation Lv = Xav, v E B.C., with eigenvalues \k and eigenfunctions vk possessing the usual nodal properties. The existence of continua of nontrivial solutions ()., u) of(*) joining (\k, 0) to OJ in r$ with u having the same nodal properties as vk is proved. (Received February 23, 1970.) 537 674-59. FORREST R. MjLLER, Kansas State University, Manhattan, Kansas 66502. The lattice ~f barreled topologi:._e_!l_: Prelimi.nary report. Let E be a real or complex vector space, E* be its algebraic dual, L be the lattic·e of T2 locally convex topologies (adjoin a 0}, and B be the set of T2 barreled locally convex topologies with 0 adjoined. We define an idempotent mapping tfJ : L - B and show that B is a lattice which is a sublower semilattice of L. The tool for studying the mapping tfJ is an inductive limit topology onE*, and this topology is handled using the theory of convergence structures as defined by H. R. Fischer. This leads to results concerning topologies (such as the top;:, logy on E *) on a vector space with respect to which the vector operations are oaly separately continuous. It will be seen that closed graph theorems and completeness conditions can be obtained from these methods. Finally, the case of Banach topologies and open questions will be mentioned. (Received February 23, 1970.) 674-60. JEROME DANCIS, University of Chicago, Chicago, Illinois 60637. An em_!?eddi~g th~o_r~m a~~~neral positiO_:'llemma for topological manifolds (metastable ran~). Embedding theorem, Let f: M- Q be a (2m- g + I)-connected map of a compact m-manifold M into a q- manifold Q, m ~ (2/3}q - 1. Suppose aM = !il. Then f is homotopic to a locally flat embedding of M into Q. Our proof uses the following General position lemma for topological manifolds. Let f: M- Q be a continuous map from a compact m-manifold Minto a q-manifold Q, m ~ (2/3)q- 1 and let ( > 0 be given. Then there is a continuous map g: M- Q and finite complexes Km and Kq such that (i} Km is the singular set of g; (ii) dim Km ! 2m - q; (iii) g sends Km piecewise linearly onto Kq; (iv) g IM - Km is a locally flat embedding: and (v) d(g,f) < (. Furthermore for each x E Km there are locally flat m• and q-simplices t,m and l'!.q in M and Q resp,, x E 6 m, such that (vi) g sends l'!.m piecewise linearly into l'!.q and (vii) the P L structures of Km and Kq are com?atible with those of /';m and Aq resp. (Received February 23, 1970.) 674-61. JAM~S E. BRINK, Iowa State University, Ames, Iowa 50010. Inequalities involving lif and lif(n} :..:fo:..:rc...:f:_w.:..:..:.;it::.h...::nc...:z:..:e:..:r..:.o::s _ _.. -pII ----qII Let II II denote the L -norm. Let A be the set of all f E CnJll,l] satisfying llf(n)ll = 1, p p q (I) f(i)(O) = 0, i = 0, ... , a.- l, and (2) f(i)(l) = 0, i = 0, ... , n- a.- 1. Then the numbers K(n,p,q,a.)= sup( llfllp: f E A) exist and satisfy K(n,l,oo,a.} ~ K(n,p,q,a.) ~ K(n,oo,1,a.) for all extended real numbers p,q l!= 1. It is found that K(n,p,oo,a.) = llta.(t- l)n-a.llp/n! and that K(n,p,l,a.) = SU 674-62. LOUIS M. HERMAN, Plymouth State College, Plymouth, New Hampshire 03264. Nonasymptoticity in projection lattices. Preliminary report. Following S. Ma.eda (J. Sci. Hiroshima Univ. Ser. A.I Math. 24 (1960), 155- 161}, two left ideals E and F in the lattice L(A) of left ideals generated by idem?otents in a Rickart ring A, erstwhile Baer ring, are said to be sem:.orthogonal, E IF, if they are generated by orthogonal idempotents. Using an 538 algebraic characterization o: j. Feldman (D\ssertatio~, Univ. of Chicago, 1954), it is easy to verify that if A is a von Neumann algebra, then E IfF if and only if their uniqlle generating projections are no~asympo:otic. Theore~ l. If A is a Rickart matrix ring of order ~ 2 such that E n F = (0) implies E IfF for all E and F in L(A), then A is a regular ring. Using the structure theory of types, this theorem yields a nonspatial proof of a result of D. M. Topping (Pacific j. 1\hth. 20 (1967), 317-325): Theorem 2. Avo~ Neumann algebra contains no asymptotic pairs of projections if and only if it is the direct sum of an abelian algebra and a finite dimensional algebra. (Received February 24, 1970.) 674-63. EDWARD T. HILL, Cornell College, Mo"Jnt Vernon, Iowa 52314. The~!_hilator of radic~_po·Ner~n- th«:_~~ular JL:r::>_U.£. ring o: a p-group. The study of modlllar group rings of a p-group has been concerned with the properties of powers of the radical(S. A. jennings, "The structure of the group ring of a p-group over a modular field," Trans. Amer. Math. Soc. 50 (1941), 175-185). We use jennings' results to obtain the annihilator of radical powers and obtain as corollaries reslllts on the ideal structure of the group ring. Theore_~· Let G be a p-group and KG be the group ring of Gover K = GF(p), the field with p elements. If Lis the exponent of the radical N, of KG, then the annihilator of Nw is NL-w+l. Corollary_, KG has exactly one ideal of dimension one. _gor~!_ary. G cyclic implies KG has exactly one ideal of each dimension. (Received February 24, 1970.) 674-64. CASPER GO'? FMAN, Purdue University, Lafayette, Indiana 47907. Everywhere convergence of Fourier series_. A common simple genesis is given for results of Salem and L. C. Young and of Garsia and Sawyer on convergence o~ Fo,Hier series for functions of generalized bo"Jnded variation. (Received February 19, 1970.) 674-65. GERALD SCHRAG, Kansas State University, Manhattan, Kansas 66502. Some We observe that the dllal (in the geometric sense) of the qllasi-graph of the Greechie lattice G32 is the Petersen graph. This leads to a natural generalization of G32. Indeed, Watkins (j. Combinatqrial Theory 6 (1969), 15.2- 164] has defined the generalized Petersen graph G(n,k) for 1 < 2k < n to consist of the 2n vertices u0 , u 1, ... , un-l' v0 , v 1, ... , vn-l and the 3n edges [ui' uitl]' [ui, v iJ, and [vi, vitk1 where all subscripts are modulo n. The Petersen graph is G(5,2). Theorem 1. G(n,k) is the dual of the quasi-graph of an orthomodlllar poset (resp. orthomodular lattice) if and only if n/(n,k) "f 3 (resp. k "f 1 and n/(n,k) "f 3 or 4). Theorem 2. Fork "f j, G(n,k) ~G(n,j) if jk = ± 1 mod n. Furthermore if (n,k) = l and k 2 "f i 1 m:1d n there exists a positive integer j "f k such that G(n,j) !!!! G(n,k). (For jk = t 1 mod n, the first part was proved by Watkins (Ibid.).) Com bining the results of Sabidussi [Amer. j. Math. 76 (195·4), 447-487], Harary and Palmer [Acta Math. Acad. Sci. Hungar. 19 (1968), 263-269] and Watkins, Graver, and Frucht [Abstract 672-426, these c}/oticeiJ 17 (1970), 204], one ca. 539 674-66. FRED HALPERN, Rider College, Lawrenceville, New jersey 08602. Deductio~ Using a variant of Smullyan's meth::>d of tableaux (Raym:md Smallyan, "Logic") we construct a complete deductive system for the continuous logics of Chang-Keisler (C. C. Chang and H. jerome Keisler, "Continuous m.Jdel theory", Princeton Univ. Press, Princeton, N.j. 1966). The results o~ Mostowski (" Axiomatizability of some many valued predicate calculi", Fund. Math. 50 (1961/1962), 165-19J) are strengthened by actually producing th·e effective enumerating function. A strong version of the Craig interpolation theorem is proven, actual interp;)lation formulas are shown to exist. ~eceived February 25, 1970.) 674-67. BERNHARD AMBERG, University of Texas, Austin, Texas 78712. On groups with chain conditions. If n is any formation of groups we define for any ordinal B inductively the following group classes: n°; n, a group G is an n f3 -group if and only if every epimorphic image HI l of G possesses a normal subgroup N I 1 which is radical (in the sense of Plotkin) or such that H/ cHN is an n~' -group for som.e ~' < f3. Let n• denote the union of all classes nil. Theorem_. If G is an n*-group and the abelian subgroups of G are ~-minimax groups (in the sense of I3aer, Math. Ann. 175 (1968), 1-43) for some set of primes~. then G is an extension of a soluble poly-~-minimax group by ann-group. This theorem may for example be used to give characterizations of the following classes of groups: groups w\th maxim·.1m (respectively m!nimum) condition on (i) subgroups, (ii) subnormal subgroups, or (iii) normal subgroups. Also the class of (not necessarily soluble) polym:.nimax groups and many others may be characterized. (Received February 25, 1970.) 674-68. FRED W. STEUTEL, University of Texas, Austin, Texas 78712. On the infinite di visib~ity 0~-~i xwres of r-dis ~ributions. It is well known (Ann. Math. Statist 38 (1967), 1306) that mixtures of exponential (i.e. r(l)) distributions are infinitely divisible (i.d.). From this it follows that f(a.)-mixtures are i.d. for 0 674-69. NARENDRA K. GOVIL, Loyola College, Montreal, Quebec, Canada, and BADRI N. SAHNEY, University of Calgary, Calgary, Alberta, Canada. On the almost (!I) summability of the derived Fourier series. Let f(t) E. L(- 1r, 1r) periodic with periodic 2rr and 1/.(t); f(x t t)- f(x- t), g(t); 1/J(t)/4 sin t/2 -.£ If n; 0,1,2,3, ... ; k; 0,1,2, ... , n is a triangular matrix of real where.£ is a function of x. (A); (A n, k); or complex numbers, the (A) transforms of the sequence (sn) are defined by means of the sequence 540 fcr } where cr = b:kn X ksk. The seqJence [s } is said to be almost (II) summable to the sums, if n n = 0 n, n the seqJence [ O"nJ is a!m-:)St convergent to s [cf. ]. P. King, "Almost summable sequences," Proc. Amer. Math. Soc. 17 (1966), 1219-1225]. Here a necessary and sufficient condition for the (A) sum mability of the derived Fourier series is obtained. Theorem. Let g(t) be of bounded variation in [0,11] and g(t) - 0 as t- 0. If (1\) is almost regular then the derived Fo".lrier series of f(x) is almost (1\) summgble if and only if limp~00 (1/p)0~:~-lr;~=O It; X r,k I= 0 uniformly inn, where f;X r,k = .Xr,k -).. r,ktl. (Received February 26. 1970.) 674-70. Gi\RRY HART, Kansas State University, Manhattan, Kansas 66502. Ab~olu~ Let M(S) b-e the measure algebra of a locally compact sem'.group S, with convolution multiplica tio::~. By considering norm <::onvergence oi translates of an element of M(S), we identify a closed ideal M0 (S), which in th-e ca.3e oi a locally compact group is the ideal of all bo:Jnded measures absolutely with respect to Haar measure. Using Sreider's theory of generalized characters ~'The structure of maximal ideals in rings of measures with convolution," Mat. Sb 27 (69) (1950), 247-318; English transl. Amer. Math. Soc. transl. (l) 8 (1962),365-391] w·~ identify all multiplicative linear functionals on M,7 (S) with certain semi.characters, generalizing a well-know~ theorem for groups. (Received February 26, 1970.) 674-71. EDWIN L. MARSDEN, Kansas State University, Manhattan, Kansas 66502. Implicatiofl_ !_!1__9rth:>~od~l~_E_Evs.ets. Preliminary report. A state on an orthomodular poser (OMP) P is a mapping a: P- [0, 1} for which a(O) = 0, a(l) l, and x "'y' implies a(x Vy) = a(x) t a(y). The set Sp of all states on P is called strongly order determining (SOD) in case x ~ y iff for all a ESP' a(x) = l implies a(y) = l. A C-filter F in an OMP is an order filter (x E F and x ~ y im?lies y E F) for which x A y E F and x com mutes with Y (x = (x A y) v (x A y')). Let P be an OM? in which Sp is SOD, and let F be a C -filter in P. For elements x, y E P, x F-implies y, writteu. x F- y, if for all a E SP' [x} U F <;; a- 1(1) imo;>lies y I' a- 1(1). Elementary prop.,rties of F-implicatio::t are, for example, (l) if x F- y andy F- z, then x F- z, and (2) if x F- y, x F- z, andy commates with z, then x F- y A z. If B is a Boolean lattice, then SB is SOD and C-filters are (laaice) filters. The definition o£ F-implication generalizes B :Jolean implicatioct: if F is a filter in a Boolean lattice B and x, y E B, then x F - y iff x' v y E F. (Received February 26, 1970.) 674- 72. P ESI R. MAS AN!, lndiac Hilbe_r_£_spa~~· Let x be a continuous function on Ill. to a complex Hilbert space U and f(a,b,c,d) = (xb- x a' xd - x c), a,b,c,d E Ill.. We call x a helix iff. f(a t t, b t t, c t t, d t t) =f(a,b,c,d), 1f t E Ill.. Associated with every helix is a strongly continuous "shift group" (Ut' t E Ill.) of unitary operators on Uonto U such that Ut(xb- xa) = xb+t- xa+t. Now associate with every such Ut-group the operator-valued measure T u< •) defined by T U(a,bJ= b rub - ua- I a Utdt }/Jz, a 541 a,{! E 'II and a U -g1:oup the functio:J x, defined by xt = such that xb - xa T u {a,b](a). (b) Given [ /3 + Tu(O,t)(a) or P - T U (t,O](a) according as t ~ 0 or t ~ 0, is a helix with shift group (Ut' t E II!.). This theorem gives a (stro.>g) time-domain characterization of all helixes x in 'II, from which the (strong) spectral representation o: x due to Kolmogorov [C.R.(Doklady) Acad. Sci. URSS 26 (1940), 6-9} and Doob ["Stochastic pro~esses", p. 552], a;ld the (weak) spectral representations of f(a,b,c,d) and f(a,b,a,b) d:.~e to Kolmogorov ~oc. cit.], and Sch.C)enberg a.-td von N·~•lmann [Trans. Amer. M.1th. Soc. 50 (1941), 226-251] emerge as easy corollaries. (Received February 26, 1970.) 674-73. MARK P. HALE, JR., So:~thern Illinois U~ivzrsity, Carbondale, Illinois 6290 l. A subgroup H of G is a trivial interseo:tio:~ (t.i.) subgro:~p if H intersects any distinct conjugate of Hat the identity element oaly. For !T a set of primes, G is IT-isolated if IT-elements of G commate only with IT-elements. Theorem l. Let G be a finite group, transitive on a set X, and let H be the stabllizer of an element of X. If (i) H is a 11-Hall subgro:~p of G and G is 1T -isolated, and (ii) Every nonidentity element in H fixes b letters in X, then H is a t.i. subgroup of G, and if b > l, H is nilpotent. Theorem?... If the centralizer of each !T-element of G is a rr-Hall subgroup of G, then G has abelian t.i. !T-Hall subgroups. For the case b = l, Theorem l reduces to a classical remark due to Frobenius, and hypo!hesis (i) is unnecessary. Theorem 2 is a direct application of the first result. The proof of Theorem l def>ends oa the structure of partitioned groJ?s .. Th·~ so.'vabllity of groups of odd order, structure theocems on g:roJp'3 having fixed-p:Jint-free automorphisms, and classical permutation- theoretic techniques are also used, (Received February 26, 1970.) 674-74. NEIL H. FENICHEL, Courant Institute, New York University, New York, New York Let x0 be a cr vector field on R n, r ~ l. Let M= M U oM be a compact, connected cr sub manifold with boundary embedded in Rn, invariant in the sense that x0 1'M' lies tangent toM and points strictly outward on oM. Any metric on Rn splits the tangent space of Rn over Minto TM $ N, where N is the bundle of vectors normal to TM. Denote the normal projection by lT: TR n IM - N. Let F~ be the flow of x0 . For each m E M define !J(m) = lim t-oo IID(F (/M) (m) lll/t and v(m) = t(F t (m))lll/t • 1.1 measures the rate of ·approach of orbits in M to limit sets. limt_00 II!T • DF0 0 v m·=asures the rate of approach of nearby orbits to M. Both numbers are independent of the choice of the metric on R n. Theorem. Suppose v(m) ~ 1 and !ls (m) v (m) ~ l for all m E M. Then if X is 1 X. a cr vector field C - clpse to x0 there is a Cs manifold MX diffeomorphic to M and invariant under If oM = !il the asymptotic stability condition v ~ l may be replaced by a splitting of N into contracting and expanding bundles. MX is constructed as a section of a disc bundle over M. A section cor res ponding to an invariant manifold satisfies a functional equation. This equation is solved by a con traction schem·=. (Received February 26, 1970.) 542 674-75. GEORGE P. GRAHAM and JAMES W. PETTICREW, Inciana State University, Terre Haute, Indiana 47809. Additive loops, linearity, and distributivity in projective planes. Preliminary report. Let a coordinate system for a projective plane P be given with 0 as the origin, r the coordinate set, j the unit tim·~. Also let j' be a line on 0 with slope m. Let +be the usual addition on the vertical axis using j and+' adjition using j 1 , (Hall, "Theory of groups"). Theorem. Given m E r linearity of equations of all lines of slope m is equivalent to equality of + and +'. Theorem. Linearity and equality of + and +' implies that m distributes fr:>m the right. Let (oo), (0), (l) denote the points at infinity on the vertical axis, horizontal axis, and identity line. The~-· Addition is independent of (0) and (I) if and only if P is (L00 , (oo)) transitive. Theorem. If P is ((0), L 00) transitive then linearity of eq:.~atio:J.s of all lines of slope m is equivalent to right distributivity of m. (Received February 27, 1970.) 674-76. SANKATHA P. SINGH, Memorial University, St. John's, Newfoundland, Canada. So~e theo~em__:'3 on fixed___£oin~ The following theorems have been given: (I) Let X be a metric space and let T: X -X be a continuous mapping such that d(Tx, Ty) < l/2 (d(x,Tx) + d(y, Ty)} for xi y. If for some x0E X, the sequence (Tn x0 J has a subsequence (Tni x0 ) converging to z, then (Tn x 0 } converges to z and z is a unique fixed point of T. (2.) Let T be a continuous map of a metric space X into itself such that (a) d(Tx, Ty)! l/2 (d(x, Tx) + d(y, Ty)} , (b) if x f. Tx, then d(Tx, T 2x) < d(x, Tx), (c) for some x0 E X, the sequence (Tnx0 J has a subsequence (Tnix0 l converging to z. Then the sequence (Tnx0 } converges to z and z is a unique fixed point. (Received February 27, 1970.) 674-77. W. j. SCHNEIDER, Syracuse University, Syracuse, New York 13210. A potential theoretic inequality and some of its applications. Theorem. Let h(z) be subharm:>nic in the stripS : (z : x + iy IO < y < 1, - oo < x < oo }, continuous to the upper and lower boundaries with upper and lower boundary valves that are in L 1(- oo, co). In addition let h(z) be majorized by the harmonic function obtained by substituting the boundary valves in the Poisson- Green formula for the strip. Under these conditions J-: h(x, ~)dx is a convex function of ~- Corollary 1 (extended Fejer-Riesz theorem). Let J be a jordan domain bounded by a rectifiable curve and let a be a proper subarc of oj. Let t.o(z) be the harmonic measure of a with respect to J. Under these conditions the length of the curve (z lw (z) : ~1 is a convex function of ~ [this result follows by letting h(z): lf'(z) I (where f maps S onto J with the top boundary of S going onto a)]. Corollary 2.. Let J, f(z) and w(z) be as above, then area (z ll/3 < w(z) < 2/3} ~ (area J)/3 (this follows by letting h(z): lf'(z) 12). (Received February 27, 1970.) 674-78. AlDEN BRUEN, University of Missouri, Columbia, Missouri 65201. Blocking sets in finite planes. A blocking set S in a finite projective plane fT is a subset of the points of fr such that every line of fr contains at least one point in S and at least one point not in S. Denoting the number of points in S by IS I we have the Theorem. lf fr be of order n then n + Jn + l :!. IS I :!. n 2 - Jn. Both of these 543 bounds are best possible for many of the known planes of square order. We discuss some applications of the theorem to finite nets and to partial spreads of projective 3-space. (Received February 27, 1970.) 674-79. MICHAEL M. RICHTER, University of Texas, Austin, Texas 78712. Reduced powers and lim its. Let (I, ~ ) be a directed set and let C be the category of relational structures (with homomor- phisms as morphisms). Then functors •: C - C are considered which satisfy: (i) there is a natural transformation n: Identity -• s.t. n: A- A* is always an embedding; (ii) there is a transformation F which assigns to every directed system ~ = ((I, 1! ), (A., i E I), (f.., i 1! j)) a directed system F(~) = 1 1] ((I, ~ )*, (Bi, i E I*), (gij' i ~ill s.t. for all i, j E I BTJ(i) = A;, gTJ(i)TJ(j) = f;j and ~ F(~) "" (~ ~ )*. Theorem A. If D is a filter over I extending "~" and if G is a filter over I X I, then the (I,D)-reduced powers and the (I,D,G)-limit reduced powers are functors with (i) and (ii). Theorem B. If (I,~) is well ordered and if the morphisms in lll are embeddings, then for the functors in Theorem A holds:~ (A;/i E I)"" ll!!! 674-80. WALTER F. TAYLOR, University of Colorado, Boulder, Colorado 80302. Compactification of relational structures. Preliminary report. Say that ~ =E !!l for relational structures (which may have operations) if ~and !!l satisfy the same pr, !I)-sentences. For notions of purity, retract and (weak) atomic-compactness, consult Wrglorz (Fund. Math. 59 (1966), 289-Z98). Theorem. If~ is weakly atomic-compact, then there exists a (to within isomorphism) unique weakly atomic-compact IS: =E ~ such that IS: is a retract of every weakly atomic-compact !!l =E '11; this IS: is, moreover, atomic-compact. Corollary l. If the algebra !ll. has any closure (an algebra '1l containing the subalgebra ~, such that ('1.3 ,a) aE A is weakly atomic-compact), then '1! is a subalgebra of an atomic-compact algebra. This answers a question of W"glorz (Fund. Math. 60 (1967}, 92). Corol~r:y_2. If there exists a pure closure of '11, then there exists an atomic-compact pure extension of~- Related results and examples are given. The "pure compactification" whose existence is asserted in Corollary 2 generalizes to model theory Maranda's notion of pure-injective envelope, described by Warfield (Pacific]. Math. 28 (1969), 699-719). The main theorem is proved using the results of the author (Abstract 69T-E74, these cNof.i.cei) 16 (1969), 980). (Received February 27, 1970.) 674-81. JAMES W. CANNON, University of Wisconsin, Madison, Wisconsin 53706. New proofs of Bing's approximation theorems for surfaces. Bing's Side Approximation Theorem (Approximating surfaces from the side," Ann. of Math. 77 (1963), 145-192) is proved on the basis of Dehn's Lemma, elementary homology linking theory, standard cut and paste techniques of three-dimensional topology, and a minimum of plane topology. The key to this new proof is that of finding, without the aid of the Side Approximation Theorem, those arcs in a 2-sphere S in E 3 whose complements in E 3 have the homotopy properties of the complements 544 of tame arcs in E 3 . Around a !-skeleton of such arcs inS a very nice polyhedral handlebody can be built; polyhedral disks can be spanned across the "holes" in the handlebody; and polyhedral 2- spheres which side approximate S from each side of S can be identified in the union of those spanning disks and the surface of the handle body. Bing's Side Approximation Theorem (for open subsets) is, in turn, used to give a proof, considerably simpler than the original, of another of Bing's fundamental theorems: A 2-sphere S in E 3 is tame if, for each £ > 0, there are £-homeomorphisms h: S - Int S and h' :.S- Ext S ("Conditions under which a surface in E 3 is tame," Fund. Math. 47 (1959), 105-139). (Received February 27, 1970.) 674-82. JAMES M. NEWMAN, Florida Atlantic University, Boca Raton, Florida 33432. Coercive inequalities for certain classes of bounded regions. In this paper, sufficient conditions are given for the coerciveness of form ally positive integra differential forms over complex-valued functions satisfying zero boundary conditions in certain bounded domains in R 2; the boundaries need not be smooth. This work extends results given by the author in Comm. Pure Appl. Math., December, 1969. In addition, sufficient conditions and partial necessity conditions are given for coercive-type inequalities involving differential operators in the Holder norm; here the results hold for complex-valued functions with no boundary conditions; the regions are bounded subdomains of Rn having the cone property. (Received February 27, 1970.) 674-83. RICHARD K. LASHOF, University of Chicago, Chicago, Illinois 6063 7, and J. SH ANESON, Princeton University, Princeton, New Jersey 08540, Smoothing four manifolds. LetS= k(S 2 X S 2) = S2 X S2 # ... # S 2 X s2 some k. Theorem 1. If X4 is a simply connected Poincarl complex and f a vector bundle over X, reducing the Spivak normal fibration, then X # S has the homotopy type of a smooth manifold if cr(X,~) = 0. Theorem 2. ( 1) If M =closed top manifold whose (unstable) tangent bundle has P L reduction, then M # S is smooth able with tangent bundle the pull-back of the given reduction via the natural map M #S-M. (Call such M S-smoothable.) (2) Two S-smoothings of M corresponding to the same reduction areS-equivalent. Theorem 3. If M4 is a simply connected top manifold and the stable tangent bundle reduces, then M # S is h-cobordant to a sm.~oth manifold. (Received February 27, 1970.) 674-84. JONATHAN L. ALPERIN, University of Chicago, Chicago, Illinois 60637. Simple groups of small 2-rank. Preliminary report. In classifying simple groups by the structure of their local subgroups some general methods fail when the 2-rank is not sufficiently large. In particular, description of all simple groups of 2-rank at most three would be useful. The author surveys the classification of simple groups of 2-rank two. The possible Sylow 2-subgroups are easily found and each case must then be dealt with. Preliminary results on the structure of the Sylow 2-subgroups of simple groups of 2-rank three are also given. These are applicable to the case of solvable 2-local subgroups. (Received February 27, 1970.) 545 674-85. M. ]. COLLINS, University of Illinois at Chicago Circle, Chicago, Illinois 60680. Finite groups admitting almost fixed-point-free automorphisms. The solubility of certain finite groups admitting an automorphism a with few fixed points is shown. In a minimal counterexample, which is simple, information is available directly only for proper a-invariant subgroups. Theorems involving the Thompson subgroup are used to extend this information to a larger class of local subgroups. (Received February 27, 1970.) 784-86. GEORGE GLAUBERMAN, University of Chicago, Chicago, Illinois 60637. Prime- power factor groups of finite groups. Let p be a prime and S be a Sylow p-subgroup of an arbitrary finite group G. Let oP(G) be the subgroup of G generated by the p'-elements of G. In a previous paper (Math. Z. 107 (1968}, 159-172}, several results about G/OP(G) are proved for the case in which p > 5. By defining two characteristic subgroups K (S) and K00 (S}, both of which contain Z(S}, analogous results are obtained for p = 5. 00 00 Theorem. Suppose p > 3. Let N1 = N(K00(S)) and N2 = N(K (S)). Then G/Op(G) is isomorphic to N. ;oP(N.) fori= 1,2. Corollary 1. If p >3, Sf. 1, and N(S)/C(S) is a p-group, then G/Op(G) f. 1. 1 1 - Corollary 2. Suppose that, for every prime q and every Sylow q-subgroup Q of G, N(Q)/C(Q) is a q-group. Then G is a q-group for some prime q. (Received February 27, 1970.) 674-87. DANIEL GORENSTEIN, Rutgers University, New Brunswick, New jersey 08903, and JOHN H. WALTER, University of Illinois, Urbana, Illinois 61801. Centralizers of involutions in finite simple groups. I. Preliminary report. The semisimplicity of the components of the centralizer of involutions in finite simple groups is discussed. In particular, results are obtained on the basis of assumptions made on those components Lin centralizer of an involution t for which O(CL(t)) P: O(L}. The main theorem is expected to be useful as a crucial step in the characterization of Chevalley group over fields of odd characteristic and alternating groups. (Received February 27, 1970.) 674-88. DANIEL GORENSTEIN, Rutgers University, New Brunswick, New Jersey 08903, and JOHN H. WALTER, University of Illinois, Urbana, Illinois 61801. Centralizers of involutions in finite simp~g_:oups_, II. Preliminary report. The authors will discuss a sufficient condition on the structure of the centralizers of involutions in a simple group G for the centralizer H of some involution x of G to be in standard form. By definition H is in ~andard form !lrovided H possesses a normal subgroup L such that (a) L is a perfect central extension of a nonabelian simple group; and (b) CH(L) has cyclic or generalized quaternion Sylow 2-subgroups. (Received February 27, 1970.) 546 674-89. KOICHIRO HARADA, Institute for Advanced Study, Princeton, New jersey 08540. On som•o doub1y,transitive groups. Let G be a finite group satisfying (•): (1) G is a doubly transitive group on S1 = (1,2, ... , n}; (2) if K is the stabilizer of two different points, then K has even order; and (3) K n Kg has odd order if g E G does not normalize K. Theorem l. Let G be a group satisfying(*). If the degree n of G is odd, then (1) G has a regular normal subgroup R and an involution z such that G = C G(z) • R, or (2) a Sylow 2-subgroup of G is dihedral, quasidihedral, wreathed product z 2nJ z 2 or z 2n X ZZ.n' n ~ ~ Theorem 2.. Let G be a group satisfying (*). If the degree n of G is even, then (1) G has a normal subgroup isomorphic to PSL(2,q) where q 1 is odd, or (2.) G is isomorphic to an automorphism group of AG(d,qz.) where d = 11 2. and q 2 is even, or (3) G;;, A6 and n = 6, or G"" PrL(2,8) and n = 2.8. (Received February 27, 1970.) 674-90. CHRISTOPH HERING, WILLIAM M. KANTOR, and GARY M. SEITZ, University of Illinois at Chicago Circle, Chicago, Illinois 60680. Finite groups with a split BN-pair of rank l. A group G is said to have a faithful split BN-pair of rank l if G has a faithful 2.-transitive permutation representation on a set S1 such that, for a E rl, G a has a normal subgroup regular on S1 - a. E. Shult has classified all such groups G when 1>~1 is odd. Theorem. If 1>~1 is even, then G has a normal subgroup M such that M ::> G '! Aut M and M acts on S1 as one of the following groups in its usual permutation representation: a sharply 2-transitive group, PSL(2,q) PSU(3,q), or a group of Ree type. The proof involves a detailed study of the 2-Sylow subgroups of G based on the given permutation structure. (Received February 2.7, 1970.) 674-91. TAKESHI KONDO, Institute for Advanced Study, Princeton, New jersey 08540. Theorem, Let Am be the alternating group of degree m and zbe an arbitrary involution of Am Let G be a finite group with the following properties: (l) G has no subgroup of index 2 and (2.) G con tains an involution z such that CG(z) is isomorphic to C A \z). Then if m = 2 or 3 mod 4 and m ~ 10, m G is isomorphic to Am. Remark. This is a generalization of my previous work in which I treated the case where z is an involution of Am (m; an arbitrary positive integer ~ 8) with the cycle decompo sition z = (1,2)(3,4) ... (4n- 1, 4n) where n is the largest integer not exceeding m/4 (cf. "On the alternating groups. III," j. Algebra 14 (l970)).However I have not obtained any such generalization in the case m = 0 or l mod 4. The reason lies in the point that we cannot find out any method to examine the fusion of involutions in the case m = 0 or l mod 4. (Received February 27, 1970.) 674-92. MICHAEL E. O'NAN, Rutgers University, New Brunswick, New jersey 08903. A characterization of U 3(q), q odd. Preliminary report. The followi.ng theorem is proved: Let X be a set with 1 + q3 points, where q is the power of some odd prime number. Let G be a doubly transitive group on X and oo, 0 points of X. Let H be the subgroup of G fixing oo and K the subgroup of G fixing 0 and oo. Suppose, in addition (1) H has a nor mal subgroup Q, regular on the points of X - oo. (2) K is cyclic of order (q2 - 1)/(q + 1,3). Then G 547 is isomorphic to U 3 (q), the three-dimensional projective unitary group over the field with q elements. The proof is obtained by associating with such a group G a block design on X preserved by G. This block design is proved to be isomorphic to the unitary block design of u3 (q). The automorphism group of the unitary block design of u3(q) is determi.ned to be PrU(3,q). From this the theorem follows. (Received February 27, 1970.) 674-93. ERNEST SHUL T, Southern Illinois University, Carbondale, Illinois 62901. On the The following has been proved: Theore!!!_. Let t be an involution in a finite group G, let T be the weak closure oft in its centralizer in G, and suppose (a) Tis abelian, and (b) that T is strongly closed in its normalizer in G. Then the smallest normal subgroup of G containing t has the form (*) (tG)/Z((tG )) = X 1 X ... X Xn X Y where Y is 2-nilpotent with elementary 2-Sylow group and a· a· a· Xi:.. SL(2,2 1 ), Sz(2 1), or U(3,2 1). Corollary 1. The theorem holds if condition (b) is replaced by (b'): t lies in the center of some 2-Sylow subgroup of G. Corollary 2. Let Q be a 2-subgroup of a finite group G such that (i) Q is strongly closed in some 2-Sylow subgroupS of G, (ii) Q is c-closed in NG(Q), (iii) CG(t) ~ NG(Q) for some involution tin the center of Q. Then the normal closure oft in G has the form (*) of the theorem. 9__£rolla_Ey 3. Let G be a transitive permutation group on a finite set of letters 0. Suppose the stabilizer G a of a point, a, contains a normal nontrivial 2- subgroup A which is sem!.regular on 0 - (a). Then either G contains a normal regular subgroup or else G is 2 transitive on 0, 101 = 1 + q, 1 + q2 or 1 + q 3, q = 2n, and G is an extension of SL(2,q), Sz(q) or U(3,q) (respectively) by a subgroup of its outer automorphism group. (Received February 27, 1970.) 674-94. W. B. STEWART, University of Illinois, Urbana, Illinois '61801 and University of Oxford, Oxford, England. Strongly self centralising 3-centralisers. Preliminary report. Theorem 1. Let G be a finite simple group containing a subgroup A of order divisible by 3 which is the centraliser of each of its nontrivial elements, and with IN(A)/ AI = 2, Then G =< PSL(2,q). Theorem 2. Let G be a finite simple group containing a cyclic subgroup of order 42 a- 1, the centralisers of whose proper subgroups are is·Jmorphic to the centralisers of the corresponding sub groups in PSU(3,42 a). Then G ~ PSU(3,42 a). In each theorem character theory shows G has but one class of involutions; a study of 2-groups normalised by an element of order 3 reduces the problem to a known characterisation of finite simple groups in terms of involution centralisers of Sylow-2-sub groups. (Received February 27, 1970.) 674-95. MlCHIO SUZUKI, University of Illinois, Urbana, Illinois 61801. A characterization of ~nitary groups. Preliminary report. The unitary groups in characteristic 2 are characterized by the structure of the centralizer of involutions. Let Un be then-dimensional (projective special) unitary group over a finite field of characteristic 2, and let Cn be the centralizer (in Un) of an element corresponding to a transvection. Theorem. Let G be a simple group such that G contains a subgroup H which satisfies the following two properties: (i) His isomorphic to Cn and (ii) His the centralizer CG(t) of any nonidentity element t of the center of H. Then G is isomorphic to Un. It has been known that the Theorem holds 548 in case n ~ 5 (G. Thomas), Proof is similar to the case of linear fractional groups (Bull, Amer, Math. Soc. 75 (1969}, 1043-1091) and depends on a construction of a building of type Cm (2m= nor n- 1) on which G operates. A simi.lar characterization of simplectic groups over a finite field of charac- teristic 2 has been obtained also. (Received February 27, 1970.) 674-96. JAMES A. YORKE, Department of Fluid Dynamics, University of Maryland, College Park, Maryland 20740. Controllability of linear oscillatory systems and of nonlinear systems. A linear autonomous control system x' = Ax + bu is considered where A is a constant matrix and b is a vector in R n and u E [9,1]. Under what conditions (using only u(t) 1!; 0) is the reachable set (of solutions starting from the origin) a neighborhood of the origin. Necessary and sufficient conditions are that the controllability matrix must have rank n and no eigenvalue of A can be real. The above theorem was discovered with S. Saperstone. The second problem concerns the controllability at x = 0 of the nonlinear system (C) x' = f(x,u), u E r!. The usual approach is to linearize the system in x and u. That is, y = fx(O,O)y + fu(O,O)u. However, there is no justification for linearizing in u since r! might even be a discrete set. Theorems are given for controllability of (C) which include the case in which r! is discrete. (Received January i6, 1970.) 674-97. CHARLES C. CON LEY, University of Wisconsin, Madison, Wisconsin 53706, and Courant Institute, New York University, New York, New York 10012. Relations between a closed invariant set and its stable and unstable sets. The theorem to be proved concerns isolated invariant sets which carry a one form. Let I be a closed invariant set of a smooth flow on a smooth manifold and let w be a one-form. Suppose that I carries w; namely that the integral of w on long enough orbit segments in I is positive. Suppose also that w is a closed form when restricted to some open neighborhood, U, of I. Then w must represent nontrivial cohomology in U as well as in I; in the latter case, via the inclusion induced homomorphism from H'(U)- H'(I). Finally suppose I is an isolated invariant set in the sense that there is neighbor hood U of I such that I is the largest invariant set in ci(U). For such a U, let a+(a-) be the set of points P in CIU such that the positive (negative) half orbit of p stays in U. The set a+ is then in the stable set of I and a- is in the unstable set; since I is isolated, a+ U a· is not empty. Theorem. With notation as above: For small enough U, w represents nontrivial cohomology in each of Ci(U), I, a+ and a provided the relevant set is nonempty. (Received February 27, 1970.) 674-9_8. RICHARD P. McGEHEE, Courant Institute, New York University, New York, New York 10012. Unbounded orbits in the restricted three body problems. The behavior of unbounded orbits is examined for the planar restricted three body problem. A transformation is made which enables one to consider infinity as a hyperbolic fixed point of a map. It is proved that the stable and unstable manifolds of the fixed point are analytic and that they intersect. These conditions imply an alternative: either the stable and unstable manifolds are the same, or they have a nontangential intersection. In either. case there exists infinitely many periodic orbits passing near infinity. (Received February 27, 1970.) 549 The April Meeting in Davis, California April25, 1970 675-1. KURT KREITH, University of California, Davis, California 95616. Sturmian theorems for characteristic initial value problems. Let u(x,y) and v(x,y) be sufficiently regular nontrivial solutions of characteristic initial value problems for uxy + pu = 0, vxy + qv = 0 in the first quadrant of the x,y-plane. Suppose that u(x,y) = 0 on the boundary of a domain D contained between two nonintersecting curves C 1 and C 2 which are asymptotic to the x and y axes. Theorem I. If q(x,y) ~ p(x,y) in D, then v(x,y) has a zero in D. Theorem 2. If q(x,y) ~X (xy)u for constants \ > 0 and u > - I, then v(x,y) is oscillatory at oo (in the sense that v(x,y) has zeroes arbitrarily far from the origin). (Received December I, 1969.) 675-2. JOHN W. SPELLMANN, Texas A & M University, College Station, Texas 77843. Concerning the domains of generators of strongly continuous semigroups of linear transformations. Suppose T: ( T(x) lx ~ 0} is a strongly continuous semigroup of bounded linear transformations on a real Banach space S. Let A denote the infinitesimal genera tor of T. Let 0 l!t a < b, p E S and f be a continuous nonnegative function defined on [a,b]. Sufficient conditions on f and p are given for b Ja T(f(x))p dx to be in the domain of A. Both a change of variable technique and a closed operator technique are considered. (Received January 15, 1970.) 675-3. PETER A. FOWLER, California State College, Hayward, California 94542. Capacity theory on some Banach spaces of functions. Let X be locally compact Hausdorff, C the space of continuous scalar-valued compact support functions on X. Let D = D(X, t> be a B-space of (equivalence classes of) scalar-valued functions on X, locally integrable with respect to a dense Radon measure ~ ~ 0, and verifying the Dirichlet axioms (Beurling and Deny, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 208-215). Capacity of an open w c:: X is cap w= inf( llulllu ~ l a.e. ~on w} exterior capacity for any E ex is capeE = inf(capt<>IE cw open}. a set E c:: X is capacitable iff capeE = sup {capeK IE ::::l K compact}, f ED' is a pure potential if there exists a Radon measure 1.1"" 0 such that cp ED n C implies (tp,f) = Jtpd!J (write f = u#.l). Theorem 1. Ex terior capacity is a true capacity in the sense of Brelot (Lectures on potential theory, Tata Inst. Fund. Res., Bombay, 1960). Theorem 2. For any capacitable E c:: X, capeE = 0 iff jJE = 0 for all 1.1 such that u#.l E D' is a pure potential. (Received January 23, 1970.) 675-4. THOMAS J. HEAD, University of Alaska, College, Alaska 99701. Groups of sequences and direct products of countable groups. Let G be a group. Let GN be the group of functions from the set N, of positive integers, into the group G with pointwise multiplication as operation. Theorem. GN is the restricted direct product of countable (finite or countably infinite) subgroups if and only if G is abelian and is the 550 direct product of a bo'.lnded group and a divisible group. This theorem provides another context in which to see the well-known fact that zN is not a free abelian group (where Z is the additive group of integers). (Received February 16, 1970.) 675-5. CHANDRA MOHAN JOSHI, Texas A & M University, College Station, Texas 77843. On some properties of a class of polynomials unifying the generalized Hermite and Laguerre polynomials. Preliminary report. In a recent communication, we defined a class of polynomials, by means of J~a) (x,r,p,q) = c(q,n)x- a exp(pxr)Dn ( xu+qnexp(- pxr)) where for the sake of brevity, c(q,n) = (-I)n(q-l)(q- 2)/2 /2nq(q-l)(t/2nq(q - 2) and q is a nonnegative integer, which unify not only the gen eralized Hermite polynomials (Gould and Hopper, Duke Math. j. 29 (1962), 51-64) and the generalized Laguerre polynomials (Singh and Srivastava, Ricerca (Napoli) (2) 14 (1963), 11-21; see also Chatter jea, Rend. Sem. Mat. Univ. Padova 34 (1964), 181-190) but also the generalized Bessel polynomials of Krall and Frink (Trans. Am~r. Math. Soc. 65 (1949), 100-115). The primary object of this paper is to study certain properties, particularly the generating and bilinear generating relations, the explicit form·~la and the determinant representation. In addition, num"!rous scattered results have been exhibited as interesting special cases of our formulas. (Received February 19, 1970.) 675-6. DEMETRE JOHN MANGERON and M. N. OGUZTORELI, University of Alberta, Edmonton, Alberta, Canada. Special functions related with _g:eneralized polyvibrating equations. In a set of their papers, partially in print, the authors studied using various methods the existence, the unicity, the approximate determination of solutions, the dynamic programming approach as well as various optimal control problems concerning the generalized polyvibrating equations having as a prototype the following variational system. (*) D[f] = minft 1 ... [:m A(x)f' 2 (x)dx = 0, b b a 1 am 2 x = (x 1, x 2 , ... , xm), (**) H[f] = J al ... Jam [2B(x)f(x)f'(x) + C(x)f (x)]dx = + 1 (or- l),(f =of/ox.= -- 1 1 m J - ., n j -If/ n j- - 0 - r "' "" . . - ] s n 1 n 2 nn - ••• - o ox. ) _ ·b - , R- La . " x. =b., J- 1,2, ...• m, o f/ox ox2 ... ox , s- J x faj' j 1 J J 1 m n 1 + n 2 + ... + nm· In the present note various classes of special functions and polynomials, orthogonal or not in R or in some other domains and corresponding to different weight functions, using the authors' polyvibrating and generalized polyvibrating systems are given and a number of related theorems is exposed. The main topics and the bibliography related with the authors' previous studies devoted to the polyvibrating equations, called by various scientists "Mangeron's equations", are to be found in the authors' sets of papers published in the last years in Atti. Accad. Naz. Lincei Rend. Cl,Sci, Fis.Mat. Natur., C. R. Acad. Sci. Paris Ser. A-Band Bul. Inst. Politehn. Iasi. (Received February 23, 1970.) 675-7. MOSES E. COHEN. Fresno State College, Fresno, California 93726. Differential and integral relations for two classes of hypergeometric functions. [c~ (x)]2 , the square of the Gegenbauer polynomial has been investigated as a special case of a class of hypergeometric polynomials in [Math. Z. 108 (1969), 121). This has prompted the study of two classes of hypergeometric functions with resulting differential and integral relations presented here. 551 These relations unify many recorded results and lead to new formulas for several well-known func tions, such as the product of two, as well as single, Bessel, Jacobi, Associated Legendre of the second kind,. Whittaker and other functions. (Received February 23, 1970.) 675-8. TAKAYUKI TAMURA, University of California, Davis, California 95616, Attainability of system of identities on rings. Let A be an algebraic system with binary operations and (J be a class of algebraic systems A. Let!'= (f~ (x).l'"'' x).n) = g).(x). 1, ••• ,x).n):). E A) be a system of identities related to((. f).' g). are words involving the binary op·erations in (J as symbol. !'is called attainable on (J if, in the greatest !'-decomposition of each A of (J, every congruence class if it is a sub-algebraic-system of A is !'-indecomposable. The greatest !'-decomposition of A is a decomposition induced by a smallest congruence p such that A/P satisfies!'. The author proved in 1965 that only semilattice ( x2 = x, xy = yx} is a nontrivial attainable system of identities on the class of all semigroups [J, Algebra 3 (1966), 261-276]; and the author and F. M. Yaqub proved that there is no nontrivial attainable system of identities on the class of all distributive lattices (Boolean Algebras). [Math. Japon. 10 (1965), 35-39). Is there any nontrivial attainable system of identities on the class of all rings? The answer is negative, It is proved by using certain polynomial rings. (Received February 23, 1970.) 675-9, MELVEN R. KROM, University of California, Davis, California 95616 and MYREN KROM, Sacramento State College, Sacramento, California 95819. Groups with free nonabelian sub- Theorem 1, A group G has a free subgroup on two free generators iff there are two subsets M, N of G and two elements a,b of G such that: (!) M UN= G, (2) aM 1'1 bN = j;J, (3) aM U bN c M n N. '!:!:eo~o:_~ 2, A group G is not amenable (F. P. Greenleaf, "Invariant means on topological groups and their applications," Van Nostrand, New York, 1966, p. 4) iff there is a finite sequence (a 1M 1), ... , (an, M nl of (not necessarily distinct) ordered pairs where ai E G and Mi c G for i = 1, ... , n and such that for every x E G the number of terms with x E M i is strictly greater than the number of terms with x E a M.. The conditions in Theorem 2 when restricted to two termed sequences are equivalent to i 1 the conditions in Theorem 1. Theorem 3. Any ACt, class of groups (A. Tarski, "Contributions to the theory ofmodels. I," Koninklijke Nederl. Akad, Wetensch,, Proc. Ser. A,. 57 (1954), 577) which contains all finite groups also contains a group with a free subgroup on tw;> free generators. (Received February 23, 1970.) 675-10, DAIHACHIRO SATO, University of Saskatchewan, Regina, Saskatchewan, Canada. ~eb!:.~ic nu_:nbers for w_!lich the exponent operation is commutative. It is well known that there is only one integer solution of xY = yx (0 < x < y), namely 2 4 = 42 and it is not difficult to show that there are infinitely many rational solutions of the equation. With respect to the real algebraic solution of the equation xY = yx (0 < x < y), we have Theorem. The real algebraic solution of the equation xY = y x (0 < x < y) is parametrized by x = s l/(s-l) andy= ss/(s-l) 552 where s > 1 is a rational number. Corollary. Let 0 < x < y be an algebraic solution of the equation xY = yx, then the algebraic degree of xis the same as the algebraic degree of y, actually y/x =sis always a rational number. Corollary. The rational solution of xY = y x (0 < x < y) is parametrized by ----- X= (n + l)n/nn, y = (n + l)n+ 1;nn+l, where n = 1,2,3, .... Corollary. The quadratic solution of xY = yx (0 < x < y) is parametrized by x = ((2n + l)/(2n - l))n-(l/2)and y = ((2n + 1)/(2n -1 ))n+( 1/ 2) where n = 1,2,3, .... Behavior of some solutio!l of the equation in higher degree, especially those which d~viate from the usual parametrization is also discussed. (Received February 23, 1970.) 675-11. DAVID G. MEAD, University of California, Davis, California 95616. Determinantal ideals, identities, and the Wronskian. Let Y..• i ~ N = (1, ... , nJ, j E P = (0,1,2, ... }, be indeterminants over a field F, and let R = l,J F[yi,j]. With S = (a 1, ... , ak1 c:N, ud fr 1, ... , rkl c: P, let (S; r 1, ... , rk) represent the k Xk deter- minant with Yr.• a. in the ith row and jth colum!l. (If F is a differential field of characteristic zero, I J and Yi,j is the jth derivative of yi then W = (N; 0,1, ... , n- l) is the Wronskian of y 1, ...• yn.) Consider ideals It = (W, W 1, ... , W t) where Wj is any fixed linear comblnatio!l with nonzero coefficients in F, of all determi.nants (N;r , ... , r ) where ~r. = n(n - l)/2 + j. Ordering both subsets S c: N and subscripts 1 n 1 ( r 1 , ... , r k 1, consider the collectio!l B of all products of determinants for which the corresponding S 's and subscripts both form chains. Using certain determinantal identities and a grading of R in which every direct summand is of finite dimension, one shows B is a basis of R, as a vector space over F, and that the deletion of certain elements of B yields a basis of R/I where I = I0 U I 1 U ... This provides a constructive test which can be used to determine whether or not any particular element of R belongs to the ideal I and generalizes the corresponding result for n = 2 of B. D. McLemore and the author. (Received February 23, 1970.) 675-12. CHUNG LING YU, Florida State University, Tallahassee, Florida 32306. Reflection principle .!5!.!._ syst~~f first order elliptic equations with analytic coefficients. Preliminary report. Let D be a simply connected domain D of the z = x + iy plane, whose boundary contains a portion cr of the x-axis. Also let A(z,C ), B(z,C), F(z,CJ,a(z), (J (z) and 6(z) be analytic functions for z, C E D U crUD, with a(z) - iP (z) f. 0 and a(z) + it6(z) f. 0. Our reflection principle is that for any solution w = u + iv of an equation of the type Clw/Ciz = A(z,z)w + B(z,z)w + F(z,z) in D under the boundary condition a(x)u + IJ(x)v = 6(x) on cr, w can be continued analytically across the x-axis, onto the entire mirror image D. (Received February 23,1970.) 675-13. JOHN E. GILBERT, University of Texas, Austin, Texas 78712. Interpolation spaces and Banach algebras. I.' Preliminary rep;:,rt. Let I be a Banach space of functions or distributions on R n for which the. translation operators {Tj (tj): tj E R 1), l ~ j ~ n, form strongly continuous groups of isometries; let I (r) be the domain of definition (suitably normed) of powers of the infinitesimal generators. It is known that in certain examples, e.g. I = Lp or ~HLP), interpolation spaces generated by I and I(r) are Banach algebra~ under pointwise multiplication. Interpolation theorem'3 associated with these spaces are used to dis cuss multipliers on the closed ideals whose zero-set is a point or a coset of a closed subgroup. The 553 Strong Ditkin Condition (and hence the Ditkin Condition) follows. There are applications to embedding theorems for Besov spaces as well as to the various Beurling algebras. (Received February 24 1970.) 675- 14. JOHN A. KALMAN, University of Auckland, Auckland, New Zealand and University of California, Berkeley, California 94720. Subdirect decomposition of generalizations of distributive lattices. Prelim:nary report. Given an algebra T, let SP(Ti be the class of all isomorphic images of subalgebras of direct powers ofT. This paper treats extensions of the theorem that the class of distributive lattices is SP(L), where L is the two-element lattice, to certain classes of algebras (A) with two binary distribu- tively related idempotent associative operations II and v but which need not be lattices, or (B) which are distributive lattices with additional structure. (AI) If we require the operations to be commuta- tive without necessarily obeying any absorption laws (cf. P)onka, Fund. Math. 60 (1967), 191-200) we obtain SP(X), where X= L U f oo}, Lis a subalgebra of X, and x II oo = oo II x = x V oo = oo V x = oo for all x in X. (A2) If the operations may be noncommutative but obey certain absorption laws, we obtain SP(L X N), where N is the two-element "nest" (cf. Abstract 564-66, these c}fotiai) 6(1959), 796; Gerhardts, Math. Ann. 161 (1965), 231-240). (B) The distributive lattices with an additional operation x f--4 x' which is a dual automorphism of period two are the algebras in SP(D), where D is a four elem~nt lattice with dual automorphism x ,_. x' tkaving two elements fixed (cf. Trans. Amer. Math. Soc. 87 (1958), 485-491). (Received February 25, 1970). 675-15. CHARLES M. BILES, Humboldt State College, Arcata, California 95521. Multiple points and Wallman-type compactifications. Let a.X and -yX be Hausdorff compactifications of the Tychonoff space X with yX ~ aX. Suppose a.X is a Wallman-type compactification of X. We inquire into conditions for which yX is also a WaH man-type compactification of X utilizing the multiple points of yX with respect to a.X. In particular, if aX= wZ[E(X, a.X)), where E(X, a.X) is the ring of continuously extendable functions from X to a.X, and yX has a countable number of multiple points with respect to a.X, then a.X is a z- compactification of X. We conclude by showing that every compact space is regular Wallman iff j3X is regular Wallman for each (noncompact) locally compact Hausdorff space X. (Received February 26, 1970.) 675-16. NAOKI KIMURA, University of Arkansas, Fayetteville, Arkansas 72701. Tensor product of semilattices. Let Ai be a semilattice for each i E I. A subset F of the product P = OiA i is called a filter, if it is dual hereditary and for each element a E F and for each i E I the subset { x E F: a and x have the same j-coordinate for every j ¥ i 1 is a semilattice. For any subsetS of P, there exists the smallest filter containing S which is said to be generated by S and is denoted by [S]. The set T of all filters generated by nonempty finite subsets of P forms a semilattice under the operation v, where F v G = [F U G). It will be shown that Tis isomorphic with the tensor product of Ai's. This construc tion of the tensor product will make it easy to observe that of two semilattices, expecially, when one of them is a semilattice of a special type, for example, a chain. (Received February 26, 1970.) 554 675-17. WILLIAM A. BEYER, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87544. Lattice structure of some linear recurrence pseudo-random points in Monte Carlo calculations. Generators of the form xi+ 1 = >..xi + l.l (mod 2P) where x0 , >.., l.l• fJ are given integers are used as pseudo-random number generators on binary computers. For sampling of points in the n-dimen sional cube, each side of length 2$, one can use either the sequence A = ((xi' x it 1, ... ,xi +n _ 1); i = 0,1, ... ) orB= (xin'xin+l''"'xi(n+l)-l; i = 0,1, ... }. Conditions under which A orB form a lattice are given, it being understood that A orB is extended by periodicity to all of n-space En. A set G c En is a lattice if there exist..l.i. vectors It'"= (y1, ... , yn} (called a basis of G) and y 0E En such that a E G iff :B: integers g. such that a= y + !;n g .y.. ft is called reduced if every m-tuple - 1 0 1= 1 1 1 (2 ~ m ;;; n) of "?n: Yi , ... , Yi satisfies IYi.l ~ ldk I for l ~ j ~ m, l ~ k ~2m, where dk are the 2m l m J body diagonals of the parallelepiped with the set Yi. as adjacent edges. An algorithm is given for J finding a reduced basis. If Rn is reduced, then q = min n IY .\/max ., n IY ij is a measure of the qual- n YiER 1 YiE" ity of the pseudo-raadom points. W. W. Wood (J. Chern. Phys. 48 (1968), 415) has reported on the case n = 2. Some aspects of the reduction algorithm are due toR. B. Roof. (Received February 26, 1970.) 675-18. HEINZ LEUTWILER, University of Washington, Seattle, Washington 98105. On a generalization of the Cauchy-Riemann equations. Preliminary report. We fix a,~ (~ 2 < 4a) and generalize the product of two complex numbers by z 1 * z 2 = r l [L(z l )/(z z)], where 1-: z = X + iy ~ i(z) = X - .. * z = >..z (>..real) and i * i =-(a+ ifJ ). The inverse element 1//z of z (f 0) is equal to (x - ~ y - iy);d2 with d2 = x2 - f3 xy + al > 0. Let now a,~ ($ 2 < 4a) be C 1-functions, defined on a region rl. The existence of limh~((f(z +h) - f(z))jh) (divided with respect to a(z), fJ (z)) leads to the partial differential equations (CR): ux = ~ vx + vy, uy = - avx. We require now that the function z ~ i * i = - (a(z) + i~ (z)) satisfies (CR): ax= ~fJx + ~ Y' ay = - a~x and call then the solutions f = u + iv of (CR) "(a,tl )-holomorphic" functions. Note that the identity : z ~ z = f(z) is for any a,~ (a,fJ )-holomorphic. Theorem. The sum f 1 + f2, >..£ 1 (A real constant), the product f 1 * f2 and the quotient f2,Jf1 (f1 f 0) oftwo (a, 8 )-holomorphic functions are again (a, B)-holomorphic. The imaginary part v of an (a,p)-Jiolomorphic c 2 -function solves the elliptic differential equation av + (3v + v t (a + ~ y)vx = 0 which will be transformed XX xy yy X into its canonical form by the function .l*: z - i-t(z). An example: a= (ax2 + bx + c)/(ay2 + dy + e), p = (2axy + dx +by + f)/(ay2 + dy +e) satisfies the above conditions for r2 = (z l$2 (z) < 4a(z)} f 0, which is bounded by a conic section. (Received February 26, 1970 .) 675-19. ROBERT P. DICKINSON, JR., Lawrence Radiation Laboratory, University of California, Livermore, California 94550. Right zero unions of archimedean semigroups. Let F = (Sa: a E A) be a disjoint family of semigroups. Then F has a Right Zero Union (R ZU) if there exists a semigroup S which is a disjoint union of the Sa' where each Sa is a left ideal of S. There are four types of commutative archimedean semigroups: Type l. nil-semigroups; Type 2, An ideal extension of a group G (IGI > l) by a semigroup N of Type l (written (G,N)); ~3. cancellative without idempotents (~-semigroup); ~ 4. noncancellative without idempotents. Theorem l. Let F = [Sa: a E A} be adisjoint family of commutative archimedean semigroups. Assume that some 555 S 110 is Type 1. Then F has a RZU if and only if each S 11 is Type 1. Theo~2. Let F = {S 11 : a E A} be a disjoint fam~ly of comm•ltative archimedean semigroups. Assume that some S 110 ~(GaO' N 110 ) is of Type 2. Then F has a RZU if and only if each S 11 is Type 2, (Ga• Na), with G11 ~Gao· ]'_!leo~-~ 3. Let F = {S 11 : a E A) be a disjoint family of Types 3 and 4 having a RZU. Then {Sa/c a: a E A 1 has a RZU where c 11 is the smallest cancellative congruence on S 11 (Sa/ca is always Type 3). (Received February 27, 1970.) 675-20. TAKAYUKI TAMURA and HOWARD B. HAMILTON, University of California, Davis, California 95616. The_Grothendieck group of com_-nutative semigroups. Let S be a commutative semigroup and G(S) be the Grothendieck group of S. (See the definition inS. Lang, Algebra, Addison-Wesley.) Let ybe the homomorphismS- G(S) with the required universal property. Theorem. LetS be a comm".ltative archimedean semi.group. (l) S is an abelian group if and only if y is injective and surjective. (2) S is not a group, and has an idempotent if and only if y is not injective, but surjective. (3) S is cancellative, and has no idempotent if and only if y is injective, but not surjective. (4) S is not cancellative, and has no idempotent if and only if y is neither injective nor surjective. !~eorem. LetS be a semigroup. The following are equivalent: (l) S has a maximal group h:>m·:>morphic image. (2) Each proper ideal I of S has a maximal group homomorphic image. (3) Some proper ideal I of S has a maximal group homomorphic image. (Received February 2 7, 1970 .) 675-21. TAKAYUKI TAMURA, HOWARD B. HAMILTON and YEE-CHUNG B. YING, University of California, Davis, California 95616. !~~~be_!_~~ one-sided identities, zero and semilattice components of finite semig:roup~. Let n be the order of a finite semigroup S and let m..li' mri' nlz' nrz' s denote respectively the number of left, right identity elements, left, right zeros of S and the number of components of the greatest semilattice decomposition of S. The ordered sextuplet (mli' m ri' niz' nrz' s, n) is called the proper system of S. Let (x 1,x 2,x3,x4,x5,x6) be an ordered sextuplet of nonnegative integers. This paper answers the following question: Under what conditions between xis (i = I, ... , 6), can (x1 ,x 2,x3'x4,x5,x6) be the prop~r system of some finite semigroup? (Received February 27, 1970.) 675-22. FREDERICK N. FRITSCH, Lawrence Radiation Laboratory, University of California, Livermore, California 94550. On self-contained numerical integration formulas for symmetric regions. Preliminary report. An approximate N-point integration formula of the form JRf(x)dx = '!;~= 1 Akf(lk) + E(f) with nodes Xk and weights Ak for a region R in Euclide;m n-space En is said to be of degree m if E(f) = 0 whenever f is a polynomial of degree at most m in the n variables 1 = (x 1, ... , xn). Such a formula is ,said to be self-contained if xk E R fork= !, ... , N. A region R!:: En is said to be simplicially_::.(fJ!!Jcl symmetric if there exists ann-simplex t,n (n-cube Cn) such that TR = R whenever Tis an affine,trans formation such that T~ = ~ (TCn = Cn). Theorem. Let n = 2. If R is connected and sirnplicially- (fully-) 556 symmetric, then R possesses an equally-weighted 3-point (4-point} formula of degree two (three) that is self-contained. These results are the tirst to guarantee self-contained formulas for reasonably large classes of regions. This work was performed under the auspices of the U. S. Atomic Energy Commission. (Received February 27, 1970.) 675-23. jACOB KOREVAAR, University of California at San Diego, La jolla, California 92037. Two-dimensional Miintz -Sza'sz type approximation. Preliminary report. Let I = [0,1} and let 0 < nk 1 oo. The sequence (t nk 1 spans Lp (I) if and only if E 1/nk = oo (Muntz, Sz.l:sz); the latter is the condition under which (nk 1 is a set of uniqueness for B(H}, the bounded holomorphic f(z) on H (Re z > 0). What are the spanning properties of sets A(n) = ( s mktn k 1 (mk' n k > 0) in spaces LP(I2)? Under what conditions is !1'" ((mk,nk) l a set of uniqueness for B(H2}, the bounded holom.nphic f(z,w) on H2 ? Let 0 l:! a< 1/p < ~. If ((mk + a, nk + a) 1 is a set of uniqueness, A(S1) spans LP(I2}, and if A(S1) sp.ns, ({ink+ f3, nk +~)}is a set of uniqueness. Examples starting with n• = ((y + m, mn4}, m,n = 1,2., ... ) show that translation may spoil a set of uniq.Ieness. Thus if 1 '!! p < r, there exist A(S1) which span LP(I2) but not Lr. Such examples--and a quite different one discovered independently by S. Hellerstein-- represent boundary effects. For sets S1 in an angle AX ~ u l:! !JX (A .> 0) one has the following exte11sion of joint work with Hellerstein. Theorem. Distinct points (p .. , q.) in an angle and such that q. - q. !!': 6 > 0,1: 1/q. = oo, ~ .1/p .. !!' £ > 0 form a set of lJ 1 1 + 1 1 1 J lJ uniq·~eness. Conjecture. A set of lattice points S1 = ((mk' n k)} "in an angle" is a set of uniqueness (and the correspo!!ding A(S1) a spanning set) if and only if~ 1/mk n k = oo. There are similar results for n variables. (Received February 2. 7, 1970 .) 675-2.4. CARLOS R. BORGES, University of California, Davis, California 95516. We prove, among others, the following results: (1) Let f: X- X be a continuous onto function with X a compact Hausd:>rff space. If the fami.ly of (composition) iterates (f,f2, f3 , ... 1 is evenly continuous, then f is a homeomorphism, (2.} Let (X,d} be a com[Jact metric space and f: X - X a nonexpansive onto map. Then f is a homeomorphism. (Received February 27, 1970.) 675-2.5, W. A. ETTERBEEK, Sacramento State College, Sacramento, California 95819. ~·edial Ll-sem~roup~~ Preliminary report. A medial Ll-semigroup S is a semigroup whose lattice of congruences is a chain and in addition satisfies the identity xyuz = xuyz for all x,y,u,z E S. We establish the following classification for medial Ll-semigroap3. Theorem. Every medial Ll-semigroup is one of tlw following types: (1} A quasicyclic group, (2.} A cyclic group of order pn where p is a prime, (3) Type 1 or 2. with zero adjoined. (4) A left or right zero semi.group with no more than two elements. (5) Type 4 with zero adjoined. (6) A commutative nil-sem!.group satisfying the divisibility chain condition. (7} Type 6 with identity adjoined. (Received February 2.7, 1970.) 675-2.6, WITHDRAWN. 557 ABSTRACTS PRESENTED TO THE SOCIETY The next deadline for Abstracts will be April 23, 1970. 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 miscel laneous group includes all abstracts for which the authors did not indicate a category. One abstract presented by title may be accepted per person per issue of these c/{oliai). Joint authors are treated as a separate category; thus, in addition to abstracts from two authors individually, one joint abstract by them may be accepted for a particular issue. Algebra & Theory of Numbers 70T-A66. GEORGE A. GRATZER and H. LAKSER, University of Manitoba, Winnipeg 19, Manitoba, Canada. Congruences and injectives in equational classes of pseudocomplemented distributive lattices. For notations and terminology see G. Gratzer and H. Lakser, "Absolute retracts in equational classes of pseudocomplemented distributive lattices," Abstract 70T- A48 these cNoticeiJ 17 (1970), 429. A class of algebras has the congruence extension property if, given any algebra A, any sub algebra B, and any congruence 0 on B, there is a congruenceS on A whose restriction to B is e. Theorem l. Any class of pseudocomplemented distributive lattices has the congruence extension as property. Theorem 2. In 8 1 and 132 the injective algebras are exactly the absolute retracts, characterized in G. Gratzer and H. Lakser, loc. cit. R. A. Day observed that in /3n, 2 < n o!< oo, the only injective algebras are the complete Boolean algebras. For 8 1, the class of Stone algebras, Theorem 2 is equivalent to the characterization in R. Balbes and G. Gr"atzer, Abstract 69T-A33 these cNoticeiJ 16 (1969), 407. In 82 injective hulls are characterized exactly as for 8 1 in H. Lakser, Abstract 69T-Al05, these cNoticeiJ 16 (1969), 814-815. (Received December 15, 1969.) 70T-A67. JOEL L. BRENNER, University of Arizona, Tucson, Arizona 85721. The fundamental theorems of matrix theory. Preliminary report. This expository paper presents many new or novel, greatly shortened, definitions and proofs; the base domain is the real field of the complex field as a rule. (The treatment could be expanded.) The notions "linear dependence", "vector space", are not required. The fundamental theorems include: (l) If a square matrix has a right inverse, it has an inverse. (2) Rank AA* =rank A, and more details on rank. (3) Sylvester's law of inertia. (4) If A,B are positive definite, det(A +B) > det A. (5) Existence and uniqueness theorems for systems of linear equations. (6) Abbreviated discussion of pseudo-inverses. (7) Characterizations of the determinant function. (8) The orthogonal group. (9) Computability. (Received December 29, 1969.) 70T- A68. VLASTIMIL DLAB, Carleton University, Ottawa 1, Ontario, Canada. Structure of Schur rings. Definition. A subring A of the full (m X m) matrix ring .M(n,D) over a division ring D is said to be a D-matrix ring if, for every (a.ij) E A and (Pij) E M(n,D), (Pij • a.ij) EA. A ring (with unity) which is isomorphic to a finite direct sum of Di·matrix rings is called a Schur ring. Theorem. A ring R is a Schur ring if and only if R is a (right) perfect ring which is strongly (left) torsion-free 558 and both left and right locally semiuniserial. Here, a right perfect ring R is called left locally semi uniserial if every factor of the left socle sequence of each of the (left indecomposable) components of R is a direct sum of simple mutually nonisomorphic R-modules; and, R is said to be strongly left torsion-free if, for every pER and every nonzero homogeneous IT-component SIT of the left socle of R, SITP = 0 implies P= 0, The above characterization of Schur rings is a consequence of a more general result on matrix representation of left locally semiuniserial rings which are strongly left torsion-free. (Received January 15, 1970 .) 70T-A69. PHILIP KELENSON, Israel Institute of Technology, Haifa, Israel, Semiprimality and structure lattices. Preliminary report. We follow the terminology of Hu, "On the fundamental subdirect factorization theorems of primal algebra theory," Math. Z. 112 (1969). For any algebra II, c 0(11 )denotes the lattice of congruence relations over !I . Theore_l!!. Let K be a locally semiprima1 cluster. If !& satisfies the identities of K in the small and e E c 0 (!8) then f)= n iE:'Ie i where each 8i is maximal in c 0 (1i). Moreover if !ll is finitely generated then c 0(lll) is a finite Boolean algebra. This generalizes results of Foster, "Ideals and their structure in classes of operational algebras," Math. Z. 65 (1956). (Received January 12, 1970.) (Author introduced by Profess'n Stephan R. Cavior.) 70T- A70. ALEX ANDER ABIAN, Iowa State University, Ames, Iowa 50010. On isomorphism of partially well ordered sets. Let P as well as Q be a partially ordered set with (i) no infinite descending chain and (ii) no infinite diverse subset (i.e,, P as well as Q is a partially well ordered set). Theore!!!_. If there exists a one-to-one order preserving mapping f from P onto Q and a one-to-one order preserving mapping g from Q onto P then P is similar to Q (in fact f is a similarity mapping). There are examples to show that the "onto" requirement in the theorem cannot be relaxed to "into". Similarly, there are examples to show that the conclusion of the theorem does not hold if either of conditions (i) or (ii) is dropped from its hypothesis. (Received January 23, 1970.) 70T- A 71, STEPHEN J. McADAM, University of Chicago, Chicago, Illinois 60615. Going down. Preliminary report. Let all rings be commutative with one. Recall going down between R c:: T means whenever P c:: M are primes of R and N is prime in T with N n R = M, then there is a prime Q c:: N with Q n R = P. Definition. If R c:: T and P is prime in R, say P is unibr_a.gched in T if there is a unique prime Q of T with Q n R = P. Theore_!!!_. If R is a Noetherian domain and R c:: T c:: R' = the integral closure of R, then going down holds between R and T if and only if every prime of R of rank greater than one is unibranched in T. Without the Noetherian assumption, but with going down between the polynomial rings over R and T, the following holds. Theorel!!.: R c:: T domains, the quotient field of T an algebraic purely inseparable extension of the quotient field of R, if going down holds for R [X] c:: T[Xl then for any prime P of R, either P is unibranched in Tor PT = T. (Received February 2, 1970.) 559 70T-A72. JUDITH D. SALLY, University of Chicago, Chicago, Illinois 60637. Failure of the saturated chain condition in an integrally closed domain. Preliminary report. Among Nagata's examples of "bad" Noetherian rings is a Noetherian domain (not integrally closed) which fails the saturated chain condition for prime ideals (SCC). It is an open question whether integrally closed Noetherian domains satisfy this condition. The following is an example of an integrally closed (non-Noetherian) domain which fails SCC. Let k be a field and x,y,z,w indeter minates. Let R denote the restricted polynomial ring k + k[y,z]x + k[y,z]x 2 + .... Then R [w] is an integrally closed domain which fails SCC. T=k[x,y,z] is an algebraic extension of R. Let N denote the kernel of the homomorphism of R [w] onto T. Let I denote the contraction of an ideal I from T to R, and J* the expansion of an ideal J from R to R [w]. Then the chains 0 c:: yT* c:: '\y";Z)T'* c:: xT* and 0 c::Nc::xT* exhibit the failure of SCC. (Received January 28, 1970.) 70T- A 73. MICHAEL RICH, Temple University, Philadelphia, Pennsylvania 19122. A characterization of the prime radical in Jordan rings. Preliminary report. C. Tsai (Proc. Amer. Math. Soc. 19 (1968), 1171-1175) has defined the prime radical R(J) of a Jordan ring J and has shown it to be equal to the intersection of all the prime ideals of J. In any ring J let P(J) be the Baer Lower Radical of J (defined in a manner analogous to the definition for asso ciative rings). Then we have the following results. Theorem I. In any Jordan ring J, P(J) is the intersection of all ideals Qi of J such that J/Qi has no nonzero nilpotent ideals. Theorem 2. In any Jordan ring J, P(J) = R(J). (Received January 23, 1970.) 70T-A74. LARRY R. NYHOFF, Calvin College, Grand Rapids, Michigan 49506. The influence on a finite group of the cofactors and subcofactors of its subgroups. The effect on a finite group G of a condition X imposed on its subgroups has been studied by Schmidt, Iwasawa, Ito, Huppert, Rose and others. The following are examples of results obtained by imposing X on only the cofactors H/cor GH' or subcofactors H/scor GH' of the subgroups H of G (where cor GH and scor GH are the normal core and subnormal core respectively of H in G). Theorem I. If H/scor GH is (a) p-nilpotent for all self-normalizing H < G, or (b) p-nilpotent for all abnormal H < G and p is odd or the p-Sylows of G are abelian, then, in either case, G has a normal p-subgroup P 0 such that G/P 0 is p-nilpotent. Theorem 2. For ria fixed ordering of 1T(G), if H/scor GH is (a) ri~Sylow towered for all self-normalizing H < G, or (b) CI-Sylow-towered for all abnormal H < G and the 2-Sylows of G are abelian, then, in either case G is solvable with G/F(G) cr-Sylow-towered. Analogous results are obtained for X= nilpotent, nilpotent of class ~ n, solvable of derived length ~ n, super solvable. Also, additional structure in G is determined if X is imposed not only on these "worst" parts of the "bad" subgroups (from the viewpoint of normality), but also on the "good" subgroups, those which are normal in G or are close to being normal in the sense that their cofactors are small. (Received February 2, 1970.) 560 70T-A75. AHMAD SHAFAAT, Carleton University, Ottawa, Ontario, Canada. Residual finiteness in commutative monoids. Preliminary report. Let Ki,j be the variety of commutative monoids defined by the identity xi = xi+j The following results are obtained: (i) Every m:moid in K 1,j, j > 0, is isomorphic to a submonoid of a cartesian power of the monoid obtained from the free monogenic monoid of Kl,j by adding a zero. (ii) If i > 1 then Ki,j contains m:moids which are not residually finite. (iii) In addition to the subvarieties there 2 2 is only 'one subquasivariety of K 1, 2 which is defined by the implication V x,y((xy = x y)- x = x ). (iv) For i > 1 every maximal chain in the lattice of a subquasivarieties of Ki,j is infinite. (Received February 2, 1970.) 70T- A 76. ROBERT C. SHOCK, Southern Illinois University, Carbondale, Illinois 6290 l. Right orders in self-injective rings. Preliminary report. Let R be a ring with unity, Q the complete ring of right quotients of R. Call an annihilator of the injective hull of R a rational closed right ideal; an internal characterization of such right ideals has been given. R has the dense extension property if each R -map from a right ideal into R can be extended to a dense right ideal of R. Theorem. Q is self-injective if and only if R has the dense extension property. R is a right order in a quasi-Frobenius ring if and only if R has the dense extension property, has the max. condition on rational closed right ideals and the prime radical is the right singular ideal of R. (Received February 2, 1970.) 70T- A 77. MICHAEL SLATER, University of Bristol, Bristol, England. Structure of alternative All rings are alternative. (1) Suppose R is prime but not assoc. Then R is a Cayley-Dickson (CD) ring if either (a) 3R ¥ (0) or (b) R is free of loc. nilp. ideals. Hence (2). If either (a) R is semiprime and 3x = 0 implies x = 0, or (b) R is free of loc. nilp. ideals, then R ., a s·ubdirect sum of CD rings and prime assoc. rings. (3) R is also purely alt. (no nuclear ideals) iff the assoc. sum mands are redundant. Applications. (4) The Smtley and Jacobson radicals coincide. Zevlakov (Algebra i Logika Sem. 8 (1969), 309) proves this if R = 6R; our proof uses the (new?) identity 0 = h(a,b,x) = - (a,b)2(2x - t(x)) + (a,b)(ab,x) + (b, ab)(a,x) + (ab, a)(b,x) - (a,b)(a,b,x), valid in any quadratic alt. algebra. (5) Any nil p.i. ring is loc. nilp. Sirltov has proved this if 2x = 0 implies x = 0, but our proof implicitly uses his result, also a result of Zwier. (6) We can produce explicitly loc. nilp. ideals in any free alt. algebra on~ 4 generators. (Received February 5, 1970.) 70T-A78. D. z. DJOKOVIC and C. Y. TANG, University of Waterloo, Waterloo, Ontario, Canada. On Frattini subgroups. The following theorems on Frattini subgroups have been proved. Theorem 1. Let G be the generalized free product of Ai, i E I, amalgamating a finite subgroup H. If Y is the family of all G normal subgroups MinH such that ifcr·: G- G/M is the canonical epimorphism the abelian component of the socle of cr Ghas complements Ci in cr Ai such that Ci n CT H, i E I, are identical, then .P(G ), the Frattini subgroup of G, IS the umque minimal element of Y, Y being ordered by inclusion. Theorem 2. Let N be a finite, abelian, normal, .p -free subgroup of a group G. If M is a norma1 sub- 561 group of G contained inN, then CJ (¢ (G)n N) = ~(Cl G) naN where a is the canonical epimorphism of G onto G/ M. (Received February 9, 1970.) 70T-A79. HYMIE LONDON, McGill University, Montreall!O, Quebec, Canada. On the Dio- phantine equation y2 + 372I = x3• Preliminary report. The Diophantlne equation y2 + 3721 = x3 has no integer solutions x, y. (Received February !2, 1970.) 70T-A80. j. R. SENFT, University of Notre Dame, Notre Dame, Indiana 46556. Endomorphism semigroups of free algebras. Preliminary report. If 21 = (A, F) is a finitary algebra with free generating set X c A, then the semigroup of endo morphisms of !ll , E(2! ), by definition must satisfy the condition: for every map T : X -A there is a unique element rp E E(al) with tp /X = T. Theorem. If E is a submonoid of the monoid of all mappings on A, and if for some finite subset X c A, E satisfies the condition above, then there is a finitary algebra IU =(A, F) with free generating set X and E = E('l.l ). The proof is a trivial verification that the required algebra is ~I =(A, (fa/a E A}) where if X= ( I •••• ,n} then fa(!, ••• ,n) =a and, in general, fa(a 1, ••• ,an) = a(jl where cp is the unique element of E S< tisfying i~Q = ai fori = l, ••• ,n. By adding an additional requirement to the above condition, a similar characterization for infinite X is obtained. (Received February !6, 1970.) 70T-A8!. SURJEET SINGH, Department of Mathematics and Statistics, Aligarh Muslim Uni versity, Aligarh (U .P .),India. Principal ideals and multiplication rings. All rings R considered here are commutative, containing at least two elements, but may not contain unity. A ring R is said to satisfy (K)-property if, for each nonzero ideal A of R there exists an ideal A' of R such that AA' is a principal nonzero ideal. A ring R is said to satisfy (KH)-property if every (nonzero) homomorphic image of R satisfies (K)-property. A ring R is said to satisfy (KPH) property if for each semiprimary ideal A (F R) whose radical is a nonzero prime ideal R/ A satisfies (K)-property. Following are the main results proved. (I) A quasi-local ring R satisfies (KH) property if and only if (i) R is a special primary ring or (ii) R is a disc rete valuation ring of rank on one. (II) A ring R satisfies (KPH)-property if and only R is au-ring such that each semiprimary ideal of R is a prime power and for each nonzero proper prime P, R/ P is a Dedekind domain. (III) A subdirectly irreducible ring satisfies (KPH)-property if and only if R is a special primary ring. Three examples are also given. One example is of a local ring R which is not a multiplication ring, but it satisfies (K)-property. (Received February 16, 1970.) (Author introduced by Professor Paul V. Reichelderfer.) 70T-A82. KARL K. NORTON, University of Michigan, Ann Arbor, Michigan 48104. On the distribution of primitive roots. Preliminary report. Let p be an odd prime, a = cp (p - 1 ), and let g 0 < g 1 < ••• < ga-l be the a primitive roots mod p in the interval [l,p]. Write ga = p + g 0 and ~(p,8) =L;j= 1(gj- gj_ 1 )8 for real 8 ~ 1. Theorem 1. The maximum number of consecutive primitive roots mod p is 0 (pl/ 4+£ ) for each £ > 0, the implied ( constant depending at most on (. Theorem 2. Let 0 ~ k < j ~a, and let 6 > 0. If j - k ~ pl/ 4+ ~. then 562 . . . . 1/4+6 1/4t6H gj - gk = { p/tp (p- 1)} (J- k) ( 1 t o(1)} , wh1le 1f J- k < p , then gj- gk = o 6 '£ (p ) for each£> 0. Theorem 3, If ( > 0 and p > p 0(f ), then there is some j, 1 ~ j ;~;a, with gj- gj_ 1 < (e'Yt£) •loglogp, where y is Euler's constant. (Note: Turan has shown that ga- ga-l> clog p for infi nitely many p and some absolute constant c > 0.) Theorem 4. For each real fJ ~ 1, 5(p, B) ii; p( p/tp(p- 1)}tl-1 • Also, for each f > 0, 5(p,B) =o£ (p1H) if 1 ~ 8;~;3, and 5(p,fJ) = 0 s,( (p( fJ + 1 l I 4+ () for fJ > 3. (The upper and lower bounds in Theorem 4 are essentially best pos- sible for 1 ;~; 8 ;~; 3.) These theorems can be generalized to the case of primitive roots mod n, where n has the form pa or 2pa, a being any positive integer. Certain generalizations can also be obtained for integers of (multiplicative) order b mod n, where b < ~ (n), but it seems to be necessary to require that b have magnitude almost as large as rp (n). (Received February 16, 1970.) 70T-A83. EUGENE W. JOHNSON and JOHN P. LEDIAEV, University of Iowa, Iowa City, Iowa 52240. Join-principal elements and the principal-ideal theorem. Theorem. Let £ be a Noether lattice in which 0 is prime and every maximal element is join principal. Then every element in .! is principal and .t is representable as the lattice of ideals of a Noetherian ring. This result is a consequence of the following theorem which extends R. P. Dilworth's generalization of the Krull Principal Ideal Theorem [Pacific J. Math. 12(1962), 481-498]. Theorem. Let iL be a local Noether lattice and let E be an element of £ which is either meet-or join-principal. Then any minimal prime of E has rank at most one. (Received February 16, 1970.) 70T-A84. DAVID E. FIELDS, Box 1270, Stetson University, DeLand, Florida 32720. Some ideal theoretic properties of pow·er series rings. Preliminary report. It is well known that if R is a com mutative ring with identity which has dimension k, then the dimension of R [X] is at least k + l and at most 2k + l. This paper considers the problem of deter mining the dimension of V [[X]] where V is a valuation ring of finite rank n. If V is discrete, then the dimension of V [[x]) is n + l. If V is rank one nondiscrete, then the dimension of Vi[X]] is at least 3; if Vis of finite rank nand if Vis not discrete, then the dimension of v[[x]] is at least n t k + l where k is the number of idempotent proper prime ideals of V. The following conditions are equivalent: (l) If Vis a rank one nondiscrete valuation ring, then v[[x]] has finite dimension. (2) If V is a valuation ring of finite rank, then V [[X]] has finite dimension. (Received February 18, 1970.) (Author introduced by Professor Gene W. Medlin.) 70T-A85. ALBERT A. MULLIN, USATACOM, Warren, Michigan 48090. On the logic of classes of arithmetic functions. Preliminary report. This note studies maximal sets of arithmetic functions which "factor" if certain hypotheses are satisfied and maximal sets of arithm~tic functions which "partition io if certain hypotheses are satisfied. Definitions. Let H~ 2 ), i E I, be a family of binary arithmetic hypotheses (e.g. (a,b) = l; 1 the natural num';)ers a and b have the same set of prim~ factors; the mosaics of a and b have no prime in common). A H~-multiplicative function f(·) satisfies f(a• b)= f(a)• f(b} provided ViHi hold (disjunction of hypotheses). An H~-multiplicative function f( ·)satisfies f(a ·b)= f(a) . f(b) provided 1\ .H. hold (conjunction of hypotheses). A K~-additive function g( ·)satisfies g(a · b)= g(a) + g(b} 1 1 pr·Hided V .K. hold. A K~additive function g( · ) satisfies g(a · b) = g(a) + g(b) pro\'ided /\ .K. hold. 1 1 1 I 563 If the index set I is empty, f( ·) is com?letely multiplicative and g( ·) is completely additive. Clearly, ·au the classes of arithmetic functions described above are. nonempty. Attention is given to con structing "towers" of classes of arithmetic functions. (Received February 19, 1970.) 70T-A86. STANLEY N. BURRIS, University of Waterloo, Waterloo, Ontario, Canada. Results on the equational theory of unary algebras. Let !:; denote a system of equations for a variety of unary algebras. Theorem. There is a (constructive) procedure for deciding if !:; has a nontrivial finite model (based upon an examination of one-one functions). Theor~m. Given an arbitrary cardinal m there is a system!:;, containing only equations of the form figih(x) = x and figjh(x) = constant, such that any nontrivial model of !:; has cardinality greater than m. Theorem. There are a continuum of equationally complete theories of type ( 1,1} • Theorem. L (( 1,1)) satisfies no special lattice laws. Theorem. The consistency problem for recursive!:; (of type ( 1,1)) is recursively unsolvable. (Received February 19, 1970.) 70T-A87. SURJEET SINGH, Aligarh Muslim University, Aligarh (U.P.), India and RAVINDER KUMAR, University of Delhi, Delhi, India. Principal ideals and prime powers. In the following R is a commutative ring having unity 1 'I 0. Following concepts are introduced: (i) A ring R is said to be a (p) -ring if every principal ideal is power of a prime ideal, (ii) a ring R is a restricted (p)-ring if each nonzero principal ideal is power of a prime ideal and (iii) a ring R is called an almost (p)-ring if Rp is a (p)-ring l P E Spec (R). Following are the main results proved: Theorem l. Let R be a ring satisfying a.c.c. on prime ideals. Then R is a (p)-ring iff (i) R is a special primary ring, or (ii) R is a discrete valuation ring of rank 1. Theorem 2, Let R be a (p) ring. If R is none of the rings of Theorem 1, then for every P 1 and P E Spec(R) for which there is no P' E Spec(R) such that P 1 < P' < P, P 1 is the union of all prime ideals of R which are strictly con tained in P 1. Theorem 3. Let R be a Noetherian ring. Then R is a restricted (p)-ring iff (i) R is a special primary ring, or (ii) R is a discrete valuation ring of rank 1 or (iii) R = F E9 T, where F is a field and T is a local Dedekind domain. Theorem 4. A ring R is an almost multiplication ring iff R is an almost (p)-ring with a.c.c. on prime ideals. (Received February 20, 1970.) (Author intro duced by Professor Klaus E. Eldridge.) 564 Analysis 70T-B72. C. J. MOZZOCHI, Yale University, New Haven, Connecticut 06520. An application of a theorem of Stein and Weiss. In [l] R. A. Hunt, "On the convergence of Fourier series," Orthogonal expansions and their continuous analogues (Proc. Conf. Edwardsville, Ill. 1967) Southern Illinois Univ. Press, Carbon dale, Ill'., 1968, pp. 235-255. It is first shown that the M* operator is of restricted weak type (p,p) for I < p < oo. Then using the theory of interpolation in Lorentz spaces it is shown that the M operator is bounded from LP(- TT, TT) into LP(- TT, 1T) for I < p < oo. In this paper we show (in detail) that it is possible to go directly from the former result to the latter without the use of Lorentz space techniques by means of the interpolation theorem found in [2] E. M. Stein and G. Weiss, "An extension of a theorem of Marcinkiewicz and some of its applications," J. Math. Mech. 8 (1959), 263-2.84. This approach is mentioned in [1], and an outline of the proof was communicated to me by the first author in (?]. (Received October 31, 1969.) 70T- B 73. SAMUEL ZAIDMAN, Universite de Montreal, Montreal, Quebec, Canada. On integrals of almost- periodic functions. Let G(t), - oo < t < oo -IS: (X,X) be a strongly almost-periodic group of operators in the Banach space X. Suppose that U xEXcr(G(t)x) is a countable set (IJ.n)~ of real numbers, where cr(h(t)) means the spectrum of the almost-periodic function h(t). Let now f(t) be an almost-periodic function, 00 - oo < t < + oo- X, rr(f(t)) = (\n) 1 and assume:>! a> 0 such that [>..n- llm [>a> 0 'if n,m = 1,2, .... Then F(t) = fa G(t- 1))f(1))d1) is almost-periodic, - oo < t < + oo- X. This extends a classical result by Favard. (Received November 5, 1969.) 70T- B 74. WILLIAM D. L. APPLING, North Texas State University, Denton, Texas 76203. A yet more elementary proof of part of the Bochner-Radon-Nikodym Theorem. U, F, R A' R +A and the notion of integral are as in previous abstracts. Suppose m is in R ~ Theorem H (an elementary known fact about Hellinger integrals). If g is in R A and for some K '!; 0, Km - J[g I is in R ~· then JU ji2 /m] exists. Iheor:.~ l. If h is in R A and for some T '!; 0, Tm - J[h [ is in R ~· then Theorem H and the following very easy assertions establish the Lipschitz condition portion of the I3ochner-Radon-Nikodym Theorem: If Dis a subdivision of U and for each V in D and I in F included in V, B(I) = h(I)- [.h(V)/m(V)]m(I), then !:DJvlBI = ~ofv[<[B[/[.m 1 / 2 J)m 1 / 2] ~!:DrJv[B2/m11l/2m(V)l/2 ~t..EDfvCB2/m1}l/2m(U)l/2 = ~[ •m(U)l/2. = Uu[h2/m}- !:D[}!(V)2/m(V)]} l/Zm(U) 1/ 2 . (Received November 24,1969.) 70T-B75. HENRY E. FETTIS, Aerospace Research Laboratories, Wright-Patterson AFB, Dayton, Ohio 45433. A new method of computing toroidal harmonics based on quadratic transforma tions of the argument. Toroidal harmonics (Legendre functions of integer order and half- odd degree) are usually com puted by employing a backward recurrence method similar to that used by J. C. P. Miller in com puting Bessel functions. The present method, which permits cal~ulation in the forward direction with- 565 out appreciable loss of significant digits, is based on transforming the argument until it is suf ficiently large for the functions to be expressed by elementary functions, and then working backward from these values to obtain values corresponding to the given argument. It is a generalization of Gauss' transformation for Elliptic integrals. (Received October 31, 1969.) (Author introduced by Dr. Charles L. Keller.) 70T-B76. ALAN L. LAMBERT, University of Michigan, Ann Arbor, Michigan 48104. Strictly cyclic operator algebras. Preliminary report. Let';/ be a complex Hilbert space and let .e.(';/) be the algebra of all bounded linear operators on';/. A (not necessarily selfadjoint) sub algebra a of .e.(';/) is strictly cyclic if ax=';/ for some x in';/ . Theorem. Let a be an abelian subalgebra of .t (';/) and let x be cyclic for a (i.e. ax is dense in';/). Then a is strictly cyclic if and only if a is maximal abelian and the dual space of a consists entirely of the maps A- (Ax,y), y in"· If a is strictly cyclic, then the strong and uniform operator topologies on a are identical. A subalgebra 3" of .e.(';/) is transitive if the only closed subs paces of ';/invariant for 3" are ( 0) and';/. Theorem. Let 3" be a transitive subalgebra of .e.(';/) and suppose 3" contains an abelian strictly cyclic algebra. Then 3" is weakly dense in .e.(';/). This theorem is proved by using techniques developed by W. Arveson in (Duke Math. J. 34(1967), 635-647), and gener alizes a result of E. Nordgren, H. Radjavi, and P. Rosenthal concerning Donoghue shifts ("On density of transitive algebras," to appear). (Received December 1, 1969.) 70T-B77. SALVATORE D. BERNARDI, New York University, Bronx, New York 10453. Univalent convex maps of the unit circle. II. This paper is a continuation of an earlier paper with the same title (Abstract, 70T- B 54 these c){oticei) 17(1970),441. Let (S) denote the class of univalent functions f(z) = z + ... analytic in the unit disk E ( z: lz I < 1) with ranges D(f); T(f) the complement of D(f); A(f) the set of fixed points z 1 E E of f(z). Let (K) and S • (y) (0 ;!!! y < 1) denote the subclasses of (S) with convex, and starlike of order y ranges, respectively. Let 4f(z,w) = [f(z) - f(w)J + (z - w) for z 'I w, llf(z,z) = f'(z), for all z, w in E. Theorem. Let f = f(z) = z + ~Cf anzn E (K); z 1 E A(f); k E T(f); a. real; w,z in E; lzl = r; lwl =a. Then we have (A) Re[w/f(w)] llf(z,w) > 1/2, (B) F(z) = [w/f(w)]zllf(z,w) E S*(l/2), (C) H(z) = (k/2) ·[{ k/(k- fh - 1] e (S), (D) L(z) = [wzllf(z,w)/f(w)] exp (-I o[rP (t)/(1 + ttp (t))]dt) E S*(O), where rp (z) is analytic and ltp (z) I !1 1 for z in E, (E) (1/4) < [1/(1 + a)(l + r)] ~ lllf(z,w)l ~ (1/(1- a)(l- r)], (F) Re [w(f - z)/zf] < (1/2)(1 + a 2), (G) Re ( w [(1/z) - (1 - r 2)f' /f]) < (1/2)(1 + a 2), 2 . 2 (H) Re fiy(l - wz)f(z)/zf(w)] > (1/2)(1 - a ), (I) cos a. - r ~ Re (e 1 a.z/f] ;!!! cos a.+ r, (J) (cos a. - r)/(1- r ) 2 ;!!! Re[ei 70T-B78. ANDRE DE KORVIN and RICHARD J. EASTON, Indiana State University, Terre Haute, Indiana 47809. A representation theorem. We consider a vector valued setting similar to that of D. J. Uherka C'Generalized Stieltjes integrals," Math. Ann. 182(1969), 60-67]. The main theorem is as follows: Theorem. Let H be a normal topological space satisfying the condition that whenever F and 0 and subsets of H, F closed, 566 0 open, with F c 0, then there exists a closed a 6 set C such that FcC c 0. If Tis a continuous linear operator from c 8 (H,X) toY, then there exists a unique, weakly regular, finitely additive, B (X, Y**). valued, Gowurin set function K, defined on the field F generated by the closed subsets of H, such that every totally bounded fin c8 (H,X) in integrable with respect to K. Moreover, if T** is continuous on SF(H,X) U CT8 (H,X), then T**(f) = J HdK • f for all fin CTB(H,X). This is ana,l~ ogous to the theorem given on p, 262, "Linear operators," Vol. I, N. Dunford and J, Schwartz, generalized to the vector valued setting. It includes the results of D. J, Uherka, and the techniques• would also yield generalizations of the result of K. Goodrich and K. Swong in the locally convex setting. The results are related to those obtained by J, Edwards and S. Wayment, [Abstract 672-221, these cJVoticeiJ 17(1970), 146). (Received January 22, 1970.) 70T-B79. GUILlERMO MIRANDA, Courant Institute, New York University, New York, New York 10012. Regularization of singular systems of integral equations with kernels of finite double- norm on L 00 • There are known examples of linear integral transformations T of finite double-norm on L 00 such that neither the transformation nor any of its iterates is compact, so that Fredholm 1 s alter native does not hold unrestrictedly for the equation (I - A. T) g = f (A. is a complex number, g, f E L 00). It is also known that the alternative holds true for lA. I less than the Fredholm radius of T. A kernel decomposition permits the introduction of a quantity w such that the alternative holds for lA. I < w·. This is shown by reduction to a regular Fredholm system. In contrast to the Fredholm radius, w can be easily computed in terms of the kernels of the system. While w is not as sharp as the Fred holm radius, it is good enough for many applications. (Received January 22, 1970,) 70T-B80, SYED M. MAZHAR, Department of Applied Science, Aligarh Muslim University, Aligarh, India. Some theorems on absolute Nor lund summability. Let Pn ~ 0, Pn = L:;~=Opk, qk > 0 and Qn = L:;k=Oqk. The following theorems have been proved. Theorem 1. Let (N,pn) be an absolutely regular method and let (i) pkL);11Pn+l-k/Pn+l- Pn-k/Pnl< C, k = 1,2, ... , (ii) Pnqn = O(Qn). Then the necessary and sufficient conditions for L; an£ n to be sum mabie IN,pnl whenever L:;an is summable IN,qnl• are (a) E'n = O(Pnqn/~) and (b) I:Jt.n = O(q/Qn_ 1). Theorem 2, (For notations see Mazhar, Math. Scand. 21(1967), 90-104.) If Jg t- 1 1111a(t)ldt < oo, 0 ;!! a < 1, then the series L; (sn - s)/n is summable IN,pn I, where ( Pn} is a monotonic sequence such that ( (n + l)pn/Pn) E BV and (Pk/ka)L:;~k (na/(n + l)Pn) ~ C, k = 1,2, .... Theorem 1 includes a number of well known theorems while Theorem 2 includes a theorem of the author for 0 ~ a < 1 (see the paper cited above). (Received January 22, 1970 .) 70T-B8l. A. H. SIDDIQI, Aligarh Muslim University, Aligarh, India. On the degree of approxi-, mation for Lip j(t) class. In the present paper a number of theorems concerning Lip j(t) class (see Masaka and Izumi: J, Math. Mech. 18(1969), 857-880) have been proved. A few typical results are as follows. Thea !~ l. Let j(t) be a positive and nondecreasing function defined on the interval (0, 1), satisfying the following conditions (1) s;j(u)u- 2du;!! Aj(t)t-l as t- 0, (2) J~j(u)u- 1 du >! Aj(t) as t- 0, then f Lip j(t) if and only if En(f) ~ Aj(l/n). Theorem 2. A necessary and sufficient condition that a 567 trigonometric series T(x) be the Fourier Series of f(x) belonging to Lip j(t) is that C7n(x) - C7m(x) = O(j(l/n)) uniformly in x for all m > n, where j(t) satisfies the condition of Theo rem 1 and j(t) ... 0 as t ... 0. These theorems, include, as particular cases, the results of Natanson, ("Constructive theory offunctions," 1961, p.97) and Duplessis (Proc. Edinburgh Math. Soc. 10(1954), 100). (Received January 22, 1970.) (Author introduced by Dr .. Syed M. Mazhar.) 70T-B82. STERLING K. BERBERIAN, University of Texas, Austin, Texas 78712. A note on the algebra of measurable operators of an AW*-algebra. Let A be an AW*-algebra, and let M be the algebra of measurable operators affiliated with A [K. Said\, T&hoku Math. J. 21(1969), 249-270). Theorem. If every order-bounded increasing sequence (or directed family) in A has a least upper bound, then (1) M has the same property, and (2) M has the property that 0 i!! xi!! y implies x 1/ 2 i!! y 1/ 2. (Received January 23, 1970.) 70T-B83. DOMINGO ANTONIO HERRERO, University of Chicago, Chicago, Illinois 60637. Full-range invariant subspaces of Hi. I. Let K be a complex separable Hilbert space and let H~ be the Hilbert space of all (classes of equivalence of) analytic square integrable (on the unit circle with respect to the Lebesgue measure) functions with values inK, as defined in H. Helson's book "Lectures on invariant subspaces". Ac cording to that reference, a full-range invariant (under multiplication by eix) subspace of H~ can be written as ?11 = UHi:, where U is an inner function-operator. ?11 is said to be an IN-subspace if there is a scalar inner function q such that qH~ c ?1! • The first part of this paper studies the properties of a large family of analytic operators- -which include the inner function- operators-- and the structure of the operator U when ?11 is an IN-subspace, from the viewpoint of the theory of in variant subspaces in the sense of Helson. The results include a matricial version of a theorem due to 0. Frostman about the density of Blascke products in the set of all inner functions. The second part is devoted to analytic continuation of inner function-operators and it also contains several partial answers to the following conjecture (M. J. Sherman): "If ?11 is an invariant subspace and for every F E Hi there exists a nonidentically zero scalar function f such that fF E ?11, then ?11 is an IN-subspace." It is proved, e.g., that Sherman's conjecture is true whenever the set of singu larities of U has measure zero. (Received January 30, 1970.) 70T-B84. LEONARD SARASON, University of Washington, Seattle, Washington 98105. Existence of solutions for some separable problems. Preliminary report. Let G = (R+)n be the subset of R n with all coordinates positive. For each i = 1, ... ,n, let Li be a selfadjoint semipositive operator determined by a linear symmetric ordinary differential operator Pi (C!l I 3 xi) with selfadjoint homogeneous boundary conditions u E Bi (with constant coeffi cients) at xi= 0. Let L denote the collection (L1, ... ,Ln). Let p(A) be a polynomial inn complex variables A = (A 1, ... ,A n>· Then iff E crJ (G), the boundary value problem p(L)u = f has a solution u E C 00 (G). The proof imitates Malgrange's proof for the equation P(D)u =fin Rn, i.e., the spectral representation off with respect to L is used; where the generalized eigenfunctions of L are analytic in A , an inverse is constructed by contour integrals. Near branch points in A one writes p as a product p 1p2 such that P1 can be inverted by the above method and p2 is a monomial. Thus Pz can be inverted by using variation of parameters. (Received January 28, 1970.) 568 70T-B85. S. j. POREDA, Franklin and Marshall College, Lancaster, Pennsylvania 17604. An example of best uniform approximation in the complex plane. Let n > l, p(z) a polynomial of degree m :;:; l, E the lemniscate lp(z) I= l, and f(z) = (ap(z) t b)/(p(z) - n) (l · np(z)). Theorem. The polynomial of degree N, km ~ N < (k + l)m, of best uniform approximation to f(z) on E is given by: PN(z) = 2 2 2 (ap(z) + b- K 1p(z)k (l - ap(z)) -K2 (p(z) - a) )/(l - ap(z))(p(z) - a), where K 1 =(aa t b)/ak(l - a )l and, where K2 = a(a t ba)/(1 - a2)2 . The proof requires the followingLem~~ If F(z) is continuous onE and F(z 1) = F(z2) whenever p(z 1) = p(z2), then the polynomial of best uniform approximation to F(z) onE is a polynomial in p(z). (Received january 30, 1970.) 70T-B86. FRANCIS E. SULLIVAN, Department of Computer Science, University of Pitts burgh, Pittsburgh, Pennsylvania 15213. Projective automorphisms on Boolean algebras. Let B be a complete Boolean algebra and let H: B ~ B be an automorphism such that H2 = IdE. Let A,;; B consist of all e E B such that H(e) = e and K !;; B consists of all e E B such that e A H(e) = 0. Lemma l. If e E B then e = x 6 y where x E A, y E K and x is a maximal element of A bounded above by e and x V y V H(y) is a minimal element of A bounded below by e. Theorem 2. A 2 is the range of a complete homomorphism P on B such that P = P. Theorem 3. If B is a measure algebra and j is an arbitrary automorphism on B, then there exists an automorphism H J such that HJ = H J and H J has the same fixed points as j. (Received February 2, 1970.) (Author introduced by Professor H. B. Cohen.) 70T-B87. J. R. EDWARDS and STAN LEY G. W AYMENT, Utah State University, Logan, Utah 84321. Variation represented by an integral for vector valued functions. Iff is an absolutely continuous function from the real interval [a,b] into an arbitrary normed space X, then the absolute derivative exists almost everywhere, is Lebesgue measurable, and the variation fro:n a to b off is given by the Lebesgue integral of the absolute derivative. The result extends immediately to the setting where X is replaced by an arbitrary metric space, and the abso lute derivative is seen to be an extension of the notion of the spread of a transformation. (Received February 2, 1970.) 70T-B88. KENNETH 0. LELAND, Illinois Institute of Technology, Chicago, Illinois 60616. A polynomial approach to topological analysis. III In this paper, employing properties of complex polynomials, a circulation index is constructed. In previous papers ~ee Abstract 69T- B84, these c){ofice0 16(1969), 664, and Compositio Math. 17 (1967), 291-298] a theory of complex variables based upon polynomial methods was developed for the case of continuously complex differentiable functions. The creation of the index makes possible the absorption into the polynomial approach of the methods of G. T. Whyburn r•Topological analysis," 2nd ed., Princeton Univ. Press, Princeton, N. j., 1964], allowing the handling of the general case when no conditions of continuity are placed on the derivative. Let c 0 (B) be the family of all con tinuous functions on B = ( z; lz I= 1) which do not vanish on B, and let T0 be the family of all elements of C 0 (B) of the form f(z) = E~nakzk For f E T, set L(f) = a0 , and let n(f) be the constant term of zf'(z)/f(z). Iff is a polynomial, n(f) is the number of zeros off inside B. From the Stone-Weierstrass 569 theorem, the elements of c0 (B) may be uniformly approximated by elements of T0 . The continuity of L, and the fact that n(f) takes only integer values allows the extension of n(f) to c 0 (B). (Received February 3, 1970 .) 70T-B89, JOHN J. BENEDETTO, University of Maryland, College Park, Maryland 20742. Sets of strong spectral resolution. Let closed E ~ r "'R/277 Z have measure zero and let A(r) be the Banach algebra of abso lutely convergent Fourier series. A'(E) is the set ofT in the dual of A(r) with supp T ~E. The compact open subsets of E form an algebra of sets and T can be considered as a finitely additive set function on this algebra. AE is the collection of elements in A(r) which have a finite number of values on a neighborhood of E. Theorem l, T E A'(E) is a measure if and only if its variation (as a finitely additive set function) is finite. Theorem 2. AE= A(r) if every T E A'(E) has finite variation. (Received February 4, 1970.) 70T-B90. ANDRE DE KORVIN, Indiana State University, Terre Haute, Indiana 47809. On the product of two Gowurin measures. I Let H be a compact space and !: the rr -field generated by closed subsets of H. Let X and Y be two normed linear spaces. A finitely additive set function A is called a Gowurin measure if sup I!: X (ei)xil ~ W suplxil where the sup is taken over all finite!: partitions of H, xi's are elements of X, A(ei) is a bounded linear operator from X toY**, and W is some positive constant. Now let k 1 and k2 be two Gowurin measures on !: and let X, Y, Z denote linear spaces. Let k 1 map !: into B(X, Y"*) and k2 map!: into B(Y**, Z**). On a measurable rectangle A X B define (k 1 X k2)(A X B)(x) = k 2(B)(k 1(A)x) for all x in X. Thus (k 1 xk2)(AXB) is in B(X,Z**). Let (J be the field generated by measurable rectangles. Lemma. k 1 X k2 extends uniquely to 9 , moreover k 1 X k2 is Gowurin on 9. Now let f be a scalar valued continuous function on H X H. For each x in X, f • x(s,t) = f(s,t)x defines a function f • x from H X H to X. Theorem. For every continuous function f on H X H and x in X, J Hdk2(t) [J Hdk1 (s)f(s,t) • x] = J HXI-Id(k 1 X k2)(s,t)f(s,t) • x. (Received February 5,1970.) 70T-B9l, ANDRE DE KORVIN, Indiana State University, Terre Haute, Indiana 47809 and RICHARD A. ALO, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, On the product of two Gowurin measures. II Let K 1 and K2 be two Gowurin measures defined on the Borel field !:of the compact space H. K 1 takes values in B(X,Y**) and K2 takes values in B(Y**,Z**). Definition. K2 is linearly nonzero respectively to K if for every relation E .K (e.)x. = 0 with lx.l ~ l and every C E !: whose Gowurin 1 1 1 1 1 1 constant respectively to K is positive there exists a partition ( cl·) of C and vectors x .. E X, 2 1,] lxi J·l ~I such that E . J.K (c.)K (e.)x .. = 0, Lemma. Assume that K is linearly nonzero respectively • 1• 2 J 1 1 1,] 2 to K1. If E has zero measure respectively to K 1 X K2 then the section Et has zero measure respec- tively to K1 for almost all t. TheOJ;:!UJl. If convergence a.e. implies convergence in measure (such is the case if the measure is weakly countably additive) and if K2 is linearly nonzero respectively to K 1 then if a Fubini theorem holds for fn and if fn~f then a Fubini theorem holds for f. (Received February 13, 1970.) 570 70T-B92. JAMES WARD BROWN, 1014 Church Street, Apt. B-2, Ann Arbor, Michigan 48104 and University of Michigan, Dearborn, Michigan 48128. Series transformations associated with cer tain Chebyshev inverse relations. Preliminary report. In another paper the author discusses the Chebyshev inverse relations 1/J n = L; [n/p] a t{3n L; [n/p]- n -{3nt{3pk k=O ( k ) tp n-pk and 70T-B93. RICHARD KRAFT, U.S. Department of Commerce, National Bureau of Standards, Washington, D. C. 20234. Incoming, outgoing type boundary conditions for analytic continuation. Preliminary report. For notation and background see [!; "Analyticity and reflexivity for first order systems of elliptic type in two independent variables," J. Math. Anal. Appl. 29(1970), 1-18]. Let R(Uk), I(Uk) k = l, ... ,N be the real and imaginary parts of the solutions of our elliptic systems. Let Bk(<\. 1,CIY!'R(Uk), I(Uk)) = 0 be the boundary conditions. In Bk(•, ·, •, •) replace R(Uk), I(Uk) by (l/2)(U•k t Uk), (l/2f(u*k- Uk) respectively; replace all x 1-derivatives by y 1 and y2 derivatives by using the equations of the elliptic system [1, equation (3.6)]. The outgoing data in the transformed boundary conditions can be expressed in terms of the incoming data. Thus a systematic procedure for solving the hyperbolic boundary initial problems in [1] can be given. (Received February !3, 1970.) 70T-B94. HARI M. SRIVASTAVA, University of Victoria, Victoria, British Columbia, Canada. A note on the evaluation of a definite integral. Preliminary report. In a recent paper to appear in J. Indian M.3th. Soc. use is made of a number of long and involved techniques including an application of the finite difference operator EIP f('ll) = f(\0 + I) to 6 evaluate the Eulerian integral (*) J6tP-l(l- t).8-I 2 F 1[.\. ,jJ; {3; 1- t] 11 Fcr~t'Y}c[xtm;o,ytm/ ]dt,where m, 6 are positive integers, and G[X, Y] denotes a (modified) G-function of two variables discussed earlier in a couple of joint papers of the present author [cf., e.g., Proc. Nat. Inst. Sci. India Part A 35(1969), 64-69; see also Proc. Cambridge Philos. Soc. 65(1969), 471-477]. In the present note it is first shown how rapidly one can evaluate the integral (*) under less restrictive assumptions by merely using Gauss's summation theorem. The form of the resulting formula and the aforementioned method of its derivation would readily suggest the existence of its obvious extensions with the hypergeometric 11 F cr function replaced by the generalized Lauricella function of several complex variables defined and studied in another joint paper of the present author [N ederl. Akad. W etens ch. P roc. Ser. A 7 2 = Indag. Math. 31(1969), 449-457]. (Received February 16, 1970.) 70T-B95. BENJAMIN VOLK, 13-15 Dickens Street, Far Rockaway, New York 11691. O.D.E.s Preliminary report. One solution is via quantified calculus existence theorems. Another solution is sequence of expansions obtained from sequence of overlapping-circular meshes imposed on desired-domain- 571 boundary as used for Cauchy integral-formula. Mesh is gotten as follows: One first changes by linear transformation all given data to be either zero or one- -except for possibility of some letter coefficients. Then expand unknown in power series about origin --obtaining recursion for coefficients which solve at end. Determine its convergence radius. Then locate its singularities on circle by taking midpoints of arcs left uncovered when convergence center is shifted and above process iterated to cover most of boundary of first disc. Expand about singularities. Continue process until desired domain is completely covered by mesh. Finally, recursion is made set of linear recursions by re placing one of its initial conditions by letter, writing nth coefficient as suitable power series in letter, focusing on its leading coefficient, next, next, ... , using induction; and obtaining convergent form of asymptotic series for coefficients by expanding in Faber polynomials of domain of convergence. (Received February 16, 1970.) 70T-B96. ALDO ]. LAZAR, Louisiana State University, Baton Rouge, Louisiana 70803. }"he unit ball in conjugate L 1 spaces. Theorem. Let K be a compact absolutely convex subset of a locally convex space. The fol lowing properties of K are equivalent: {1) There is a Banach space X such that X*= L 1(m) for some measure m and an affine homeomorphism of K onto the closed unit ball of X* considered with its w* topology; (2) If IJ. 1• #J. 2 are maximal probability measures in Choquet's ordering on K having the same barycenter and M is an arbitrary Borel subset of K then (!J. 1 - !J. 2){M) = (#J. 1 - ,., 2)(-M) where -M = { -k; k EM). Corollary. Let X be a Banach space such that X*= L 1(m) for some measure m. If E c:ext{ x* EX*: llx*ll ~I) is w*-closed and En (-E)= Ill then the w*-closed convex hull of E is a face of { x* EX*: llx*ll !!i l) . (Received February 16, 1970.) 70T-B 97. ERWIN 0. KREYS ZIG, Mathematisches Institut, University of Dusseldorf, Dussel dorf, Germany. Polynomials as generating functions of Bergman ~?P·erators. If {1) c(z, z*) = -m(m + l)kh'(z*)(kz + h(z*))- 2 (k arbitrary, h(z*) an arbitrary analytic func tion of z•, m a positive integer), then the solutions of (2) uzz• + c(z, z*)u = 0 can be generated by a Bergman integral operator whose generating function is particularly simple (a polynomial in the variable of integration, with coefficients depending on z and z *). This permits an easy translation of results from complex analysis into theorems about those solutions. The class defined by (I), (2) is a subclass of the class P and includes equations investigated by S. Bergman, H. A. Schwarz, I. N. Vekua, K. W. Bauer, E. Pesch!, and W. Watzlawek. (Received February 17, 1970.) 70T-B98. PAUL C. ROSENBLOOM, Columbia University, New York, New York 10027. Sobolev' s inequality. Let f be a function of class C 1 on R n, with compact support. Sobolev's inequalities are con-· cerned with bounds on llfllq/llv flip= S(f). If g is spherically symmetrical and equimeasurable with f, then llfll = llgll and, by a theorem of P6lya and Szego, llvgllp ~ lfv fll so that S(f) !!i S(g). It is q q p easy to get an upper bound of S(g) by elementary methods for those p, q, and n for which sup S(g) < + oo. We can also obtain the best bound of S(g) by t the variational method of E. Schmidt. This yields simple proofs of Sobolev's inequalities with explicit constants. (Received January 29, 1970.) 572 70T-B99. ANDERS LUNDBERG, Vendelso High School, Grenviigen 22, 13050 Vendelso, Sweden. A theorem on continuous solutions of the generalized associativity equation. Definition. A function F of a finite number of variables is said to be strictly dependent with respect to the variable x, if F has no constancy intervals with respect to x for any fixed values of the other variables. Theorem. Let~ be a real valued function of three real variables. Assume that ~ is strictly dependent with respect to each variable in the set M = X X Y X Z, where X, Y, and Z are intervals, and representable in the two forms (1) (x,y,z) = F(G(x,y),z) = H(x,K(y,z)) where F, G, H, and K are real and continuous. Then, in the set M, ~ is also representable in the form (2) ell (x,y,z) = rp (f(x) + g(y) + h(z)) where tf>, f, g, and h are continuous and strictly increasing. This theorem has been known before in the case when ell is strictly increasing in each variable. It follows by the method, used by M, Hosszt!, "On local solutions of the generalized functional equation of as sociativity," Ann. Univ. Sci. Budapest. Eotvl:is Sect. Math. 7(1964), 129-132. (Received February 19, 1970.) 70T- B 100. W. JOHN WILBUR, Pacific Union College, Angwin, California 94508. A class of Caratheodory outer measures in topological spaces. Given a topological space P, a collection .I of subsets of P, and any nonnegative real valued function ~ on .I; a procedure is given for defining a class of outer measures on P. Let 7f 1 stand for the collection of all families of open subsets of P and let 7f 0 stand for the subcollection of 7f 1 con sisting of all countable families of open sets. Let ')f be any subcollection of 7f1 containing 7f0 , closed under countable unions, and such that if A1 and A2 are in 7f then { U n V IU E A1• V E A2 J is in 7f • If A E ')f let .I(A) denote the collection of all finite pairwise disjoint subfamilies A of .I such that for each A E a there is aU E A with A c: U. Then set v(A) =sup {:E ~(A) (A E O)IO E .J(.A)). For any subset A of P let ')f (A) be the collection of all elements of ')f which cover A. Define 1J. (A) = inf{ IJ (A) lA E7f (A)}. We call IJ. a ')f -outer measure. It is shown that all 7f -outer measures measure the Baire sets while if Pis regular all ')f 1-outer measures measure the Borel sets, (Received February 20, 1970.) 70T-Bl0l. ALESSANDRO FIGA-TALAMANCA and GARTH I. GAUDRY, Yale University, New Haven, Connecticut 06520. Multipliers of LP. A bounded measurable function tp on Ill is called a multiplier of LP(lll) (l !i p ~ 2) if, for every f E Lp(ll!.) there exists g E Lp(lll) such that !p :J f = :J g (3t is the Fourier transform). The norm of a multiplier tfJ is IIIP II= sup { llgllp: !p 573 Applied Mathematics 70T-Cl7. K. B. RANGER and SUDHANSHU KUMAR GHOSHAL, University of Toronto, Toronto- In connection with the study of earth's magnetic field, the toroidal part of this field plays a dominant part. This gives rise to the importance of studying the effect of a toroidal magnetic field on flows past different bodies of revolution, especially past spheres, spheroids, and ellipsoids. In the present note inviscid flows past a sphere and a spheroid, are considered, for the case of a toroidal magnetic field originating in the fluid. In the case of a sphere the field inside consists of an electric dipole directed along the axis. In the case of the spheroid, the field inside it is due to an electric dip:>le and a quadrupole directed along the axis of symmetry, together with a uniform electric field which produces a uniform current along the axis. The nonlinear governing equations are solved by expanding the magnetic field inside Ui and outside U and the stream function ¢ in the form Ui = UiO + RmUil+... , U = 1/Jo + RmUl + ... +¢=¢ 0 + Rm¢ 1 + ... on the assumption Rm (mag netic Raynolds number) is small. An alternative procedure for large Alfven number is also formulated. (Received December 15, 1969.) 70T-Cl8. SUDHANSHU KUMAR GHOSHAL, University of Toronto, Toronto-181, Ontario, Canada. On unified solu!_!.on of the velocity variable for incompressible and compressible boundary layer flows. The first part of the paper presents unified solution of the boundary layer equations for velocity variable for compressible and incompressible flows past a body with a velocity distribution of the form U(x,t) = X(x)N(t) in the mainstream. By a transformation which is a generalization of Howarth Dorodnitsin transformation, the boundary layer equations for a compressible flow can be reduced to the corresponding equations for the incompressible flow,_ if Pellcx/Poo = const. (where oo and e de notes the values of the variables outside the boundary layer and at the edge r·~spectivdy). In this 2 case if N(t) be square integrable in (O,oo] and c 2 = J~N 2 (t)dt, and t = t 1/c, v~ 11/ct = (l/c ) rtl 2 - I- I - 'J 0 N (t1)dt1 and under the change of variables from x,y,t to x,t, '17 = (N(t 1)/c"' II) X (1/"' 3t) and the substitution¢ (x,y,t) = F(x,y,t) (../ 3V t)X(x), and expanding F(x,y,t) in powers •f X'(x) and X" and their products, the problem may be solved by analytical methods. If Pell00/p00 is >t a constant, it can be expanded also in powers of X'(x) and X"(x), and their products, where their~ efficients are known and the above method may be employed. (Received january 9, 1970.) 70T-C 19. BORO DOERING, Mathematisches Institut Technische Hochschule, Darmstadt 61, Darmstadt, Germany. On_c_~rtain classes of iteration methods. Let X,Y be arbitrary Banach spaces, k > 2 a given natural number and F: X=> XF ... Y a non linear operator having Frt!'chet derivatives (in S defined below) up to the order k. Consider the following two classes of iteration methods (containing, e.g., Halley's and Ceby~ev's methods) (J.llll 0 : = (0} UJ.llll; Fn: = F(xn); F~>: = F(j)(xn); fn: = F'(xn) \fnEJ.llll0): Forfixedk, xn+l: -1 "'m I -lj) (i-1) ., xn + dk-l,n \f n E lllllo where dl,n: = -fn Fn and Fn + F~dm,n + "-'j=2 (l j~)l 70T-D9. JOHN DO:JGLAS MOORE, University of California, Santa Barbara, California 93106 . .!_some~-::~c:.__!__I!lmersions of product manifolds. If M is a riemannian manifold, let F(M) be the interior of the set of points in M at which M is flat. '!:heorer_:::_ I. For I "! i ~ p, let Mi be a connected riemannian man if old of dimension ni such that F(Mi) = lil, and let M0 be a connected flat manifold of dimension n0 . Let Em be euclidean space of dimP.nsion m=(I:f=oni) + p. Thenanyisometricimmersionoftheriemannianproductf: Mo XMJ X ... xtvt, -Em is a product immersion. Theorem 2.. For l l! i ~ p, let Mi be a complete connected riemannian manifold of dimension n. ~ 2, and let Em be euclidean space of dimension m = (I:P n.) + p. Then any I I= I I isometric immersion f: M 1 X M 2 X ... XMP- Em is either a product immersion or else it carries a complete geodesic onto a straight line. Corollary_. If each Mi in Theorem 2 is compact, and has non negative sectional curvatures, then any isometric immersion f: M 1 X M2 X ... X Mp- Em is rigid. In particular, if S 2 isthetwo-sphereofconstantcurvatureone, then sZ xS 2 X ... XS2. (p times) is rigid in E 3P. (Received January 7, 1970.) Logic and Foundations 70T-EZ5. SAHARO"i SHELAH, Princeton University, Princeton, New Jersey 08540. On the number of nonisomorphic m-:>dels of a theory in a cardinality. Preliminary report. LetT be an infinite consistent first order fixed theory. Definit~ I. IT(A) is the maximal number of nonisomorphic models of T in the cardinality A . Definition 2.. S(A) (A c M) is the set of complete consistent types on A; T is stable if for every A, IS(A) I ~ lA liT I, T is supers table if for every A, IS(A)Il! IAI +ziTI (see Abstract 68T·El7, these cJ.fotiai) 15(1968), 930, and my paper in Israel J, Math. 7(1969)). Theorem I. If Tis unstable, 2.). > zziTI, then ITO.)= Z).. Theorem 2.. If T . ~0 IS not superstable, ). 1 ~ A = A ~IT I, then IT(), 1) >A. Corollary. If T is categorical in A > IT I, j.l. ~A then Tis categorical in j.l.. Tneorem 3. If Tis denumP.rable, T has a model omitting a type p in ::lw but not in :JWl' then for every a, IT(~ a)~ Ia +II. Theorem 4. If T has the f. c.p (see Keisler, 1.8 -a I J, Symbolic Logic 32(1967), 23-47) ITI = ~a' then IT(~ p> ~ Z . Those theorems have some gener- alization, and unwrittea strengthening. (Received October 31, 1969.) (Author introduced by Professor Elias M. Stein.) 70T-EZ6. STEPHEN H. HECHLER, Case Western Reserve University, Cleveland, Ohio 44106. Q_:r the indep:_nd~~! the existence of small, infinite, maximal, almost disjoint, families. Prelimi nary report. We define an alm-:>st disjoint famil~_(ADF) to be a family of infinite subsets of w such that any 576 two distinct members of the family have a finite intersection, and a maximal almost disjoint family (MADF) to be an ADF which is not properly contained in any other ADF. Lusin credits Cantor with proving that there cannot exist MADFs of cardinality ~ 0 • but notes that for any n E W the partitioning of W into n different infinite pieces results in an MADF of cardinality n. It is also known that there ~ must exist ADFs, and therefore MADFs, of cardinality 2 0. We prove: Theorem 1,. Martin's axiom ~ implies that all infinite MADFs have cardinality 2 0. Theorem II. It is consistent with ZF + AC that there e,xist, simultaneously, infinite MADFs of various different cardinalities. (Received February 9, 1970.) 70T-E27. DOUGLASS B. MORRIS, University of Wisconsin, Madison, Wisconsin 53703 . .:A_mode~!_E whic~~no!~~xtended to a model of ZFC without adding ordinals. Preliminary re port. Let NE be "Va. :l!x (xis a countable union of countable sets and the power set of x can be parti tioned into ~a. nonempty sets)." Let R(a.) = (x\x is of rank< a.) . Theorem l. Let M be any countable model of ZFC. There is a Cohen extension N of M such that cardinalities and cofinalities are pre served, for each ordinal a. of M&f M \= "R(a.) F ZF" then N \= "R(a.) \= ZF + NE," and N \= ZF + NE. The pro~f uses a tower of Dedekind finite sets (which by itself gives a generalization of the result announced in my abstract "Existence of models of ZF with involutions," Abstract 69T-E72, these c){oticei) 16(1969), 980). The countable unions of countable sets are on side branches; their generic subsets pick one element from each countable set and any two are forced to pick different elements after a certain point. Symmetry arguments are used to prove power set and replacement in N. Thea ~ 2. Let N \= ZF + NE, N be a submodel of N*, N* \= ZFC, Va,b((a EN and N* \= b E a).., bE N). Then there is an element ("set") a. of N* such that N* \= fJ E a whenever fJ is an ordinal of N. Proof. Let a= 2~0 inN*. N is the model referred to in the title. Observation. Strong inacces sibility in a certain sense is preserved in the passage from M to N. (Received February 9, 1970.) 70T-E28. JOHN K. TRUSS, The University, Leeds LS2 9JT, Yorks, England. Finite versions ~f th~,-~~!_11 of choice. For nEW, let \!11 denote this version of the axiom of choice:--There is a choice function for any set X such that x EX implies \x\ = n. If Z 1::: W, let [z] denote (1f z E Z) [z]. Consider the follow ing condition (M) on Z and n: --For every expression for n as a sum of (not necessarily distinct) primes, n = L:~pi there are nonnegative integers s 1, ... ,sr such that L:~sipi E Z. Mostowski showed (Fund. M,~th. 33(1945)) that (M) is necessary for [z} ... [n], and sufficient in these cases: --n l!! 14, n prime, n = 16, 18. It is also known to be sufficient in these cases:--n = 20, 24, 30, 42. Th_:~~2!:· (M) is sufficient for the implication [z] ... [n] if n = 15. In particular it is proved that [3,5,13] ... U5J. The method of proof is to show first that one can choose a proper nonempty subset of any given 15-e1ement set. If this subset has 8 elements, co:!lsider Y, the set of its 2-element subsets. Y has 28 members, and by first choosing a proper nonempty subset of Y, and then showing how to use this subset to choose a member of the original.set, all cases are covered. Whether or not this result generalizes to the case n = 21 is open. (Received February 18, 1970.) (Author introduced by Professor Frank R.Drake.) 577 Statistics and Probability 70T-F7. S. Cl-IANDRASEKHARAN, c/0 J, R. GEIGY AG, Wissenschaftliches Rechnenzentrum, Postfach 71, 4003 Basel 21, Switzerland. A recursive formula for the steady state probabilities of the queueing systemc M/D/C. For background and definitions see Crom m~lin (P .0. Elect. Eng. J. 25(1932}, 41-49}, who has shown that the steady state probabilities of this system are given by the generating function P(z) = -(c- ~)n(z- z.)/n(l- z.) (1- zce"-0-z)) where cis the number of channels,>. the poisson arrival 1 1 rate, and z. the ro'Jts of (1 - zce).(l-z) = 0). Theorem. Let P(z) be the generating function of the 1 ----·- system. Then for n < c p(n)(z) = (n - 1)! Ek:6p(k)(z) [(-1)(n-l- k) • An-k(z)/k! + Rn-k(z~ where Ar(z) = Ej0:6(-l)r+ 1 /(z - zj)r and R,(O) = 0 for r 70T-F8. JOH:N S. KALME, U.S. Naval Academy, Annapolis, Maryland 21402, §.!_~tiona!Y_.¥"_aUS ~ian P_!:~~_:;~~h trajectories in generalized Sobo.lev spaces. Let c 21Tbe the Banach space of real continuous functions of period 21T, with norm llfll = r, ] - 1J 211 2 r· 2 - 1 - supo~ x~ 2 11' if(x) 1. J..et LV(r)f (x) = (21T) 0 f(u}(l - r ) I} - 2r cos (x - u) + r ] du for 0 ~ r < 1, f(x) = -1imt L0(21T)- 1J ~ fr Topology 70T- G59. DAVID J. LUTZER, University o: Washingto!l, Seattle, Washington 98105. On gener alized or!iered spaces. Preliminary rep:>rt. Generalized ordered spaces (GO spaces) were introduced in Cech's"' "Topological spaces" where it was pointed out that they are precisely the subspaces of linearly ordered topological spaces (LOTS). The Sorgenfrey line is one example o.f a GO space which is not a LOTS. We prove, however, that if X is a GO space, then there is a LOTS Y(X) cotltaining X as a closed subspace. Mureover if X is metrizable or paracompact, so is Y(X). If X has a a -locally countable base then X is metrizable and if every open cover of X has a point finite open refinement thea X is paracompact. These results geaeralize two theorems due to Fedor~uk. If a GO space is semi.stratifiable, then it is metrizable. 578 For LOTS, a better theorem is known--a LOTS with a Gil diagonal is metrizable--but this theorem is false for arbitrary G·) spaces (consider the Sorgenfrey line). Among GO spaces, the existence of a Gil diagonal insures little more than hereditary paracompactness. (Received january 22, 1970,) 70T-G60. DiX H. P;,:TTEY, University of Misso•Jri, Columbia, Missouri 65201. One-to-one _Etap_pings. Let Y be a locally connected., locally compact metric space having the following property: for each simple closed curve J in Y, there is a 2-cell Kin Y such that J = Bd K and such that Int K is an open set in Y. Theorem. If X is a connected, locally connected, locally compact topological space and f is a 1- I mapping of X o~to Y, thea f is a homeomorphism. (This theorem is a generalization of a result which the author announced in Abstract 68T-H34, these c){otiai] 15(1968), 947,) (Received january 22, 197(),) 70T- G61. GLENN P. WELLER. University of Illinois at Chicago Circle, Chicago, Illinois 60680. Locally flat imheddings of topological m.1nifolds in codimension three. The nthor has improved the results anno:mced in his Abstract 68T- G20, these c){otiai] 15(1968). 94·). Let Mn and Qq be top.:>logical manifolds with q 5; n + 3 and M11 compact. Denote the boundaries of M and Q by M and Q respectively. If M is 2n - q connected and Q is 2n - q + 1 connected, then any . . map f: (M1 M) ~ (Q, Q) such that fiM is a locally flat imbedding is homotopic relative toM to a pro- per locally flat imbedding g: M -+ Q, From this result it easily follows that if M is closed and 2n - q + 1 connected and Q is 2n - q + 2 connected, then any two locally flat imbeddings f, g: M ~ Q which are homotopic are also concordant via a locally flat proper imbedding H: M X I -+ Q X I with H(x, 0) = (fx, 0) and H(x, 1) = (gx, 1). The codimension three requirement improves the previous dimension requirement of 2q > 3(n + 1). This improvement results from the piecewise linear approxi mation theorem of R. T. Miller announced in Abstract 69T-G42, these c){oticei) 16(1969), 583. (Received january 23, 1970.) 70T-G62, jAMES W. CANNON, University of Wisconsin, Madison, Wisconsin 53706, Hierarchy ~ wild~ets o~~-sph~~· LetS denote a 2-sphere in E 3. Consider the following properties: (I) Every Cantor set on S lies on a tame arc on S. (2) Every arc on S is tame. (3) Every chainable continuum on S is tame. (4) Every treelike continuum on S is tame. (5) Every !-dimensional continuum on S is tame. (6) S is tame. Clearly each property implies those which precede it. Examples of Bing [Duke Math. J. 28 (1961), 1-16] and Gillman [Duke Math. j. 31(1964), 247-254] show that (5) does not imply (6). The following theorem, which depends on modifications of Gillman's construction, shows that none of the conditions listed implies any of those following it. Theorem. Let M be a nondegenerate planar con tinuum, Then there is a 2- sphere S in E 3 and an embedding h: M -+ S such that (i) no open subset of h(M) lies on a tame sphere, and (ii) if M 1 is any closed subset of S such that M 1 n h(M) is a dimensional, then M 1 lies on a tame sphere in E 3. (Received September 22, 1969.) 579 70T-G63. RONNIE LEE, Institute for Advanced Study, Princeton, New Jersey 08540. Spherical Let p and q be two distinct odd primes such that p - I is divisible by q. Let &::p,q be the non abelian gro.1p of order pq. I_!leore~ l. Let L~kt 3 (&::p,q) be the surgery obstruction group of Wall. h - Then L4k+ 3(&::p,q)- 0. The_::>re~ 2. There exists a free differentiable action of the group &::p,q on a homr)topy sphere of dimension 2kq2 - I (k > 1). (Received January 29, 1970.) 70T-G64. GORDON BERG, Department of Mathematical Sciences, New Mexico State University, Las Cruces, New M·oxico 8800 I. Incompatibility of the metrical properties completeness and SC- WR. Two properties which metrics can possess are said to be compatible if for every space which has two metrics, each possessing one of the properties, there is a metric for the space having both properties. The space consisting of the open unit disk in the plane along with a convergent sequence of points on the boundary of the disk and its lim\t point has a complete metric because the space is a G0 subset of the plane. The ordinary plane metric restricted to the space is strongly convex and without ramifications (SC-WR). However, the space admits no metric which is both SC-WR and com plete. (Received February 2, 1970.) 70T-G65. JOHN L. BRYANT, Florida State University, Tallahassee, Florida 32306. Approxi mations of embeddings of polyhedra. Theo~~m. Suppose that f is a closed embedding of a p-dimensional polyhedron P into a P L q manifold Q (without boundary) with q - p ~ 3. Then for each positive continuous function ( on P there exists a closed P L embedding g: P - Q such that d(g(x), f(x)) < ( (x) for each x in P. The proof of this theorem depends upon a recently anno:.mced theorem of Cernavski!" which states that the above theorem is true if Pis a p-eel! and Q = Eq together with a result of Bryant and Seebeck in [Quart. J. Math. Oxford Ser. (to appear)]. (Received February 4, 1970.) 70T-G66. WERTHEN N. HUNSAKER, Southern Illinois University, Carbondale, Illinois 62901. Quasi-proximi.ty classes of quasi-uniformities. A quasi-uniformity '1.< on X is compatible with a quasi-proximity a if a is given by Aa B if and only if UnA X B 'I ill for every U in '1.<. !heorem. Let (X, a) be a quasi-proximity space. The collec tion of all sets of the form X X X - A X B, where AillB is a subbase for a totally bounded quasi uniformity which is compatible with a. Moreover, '1.< ~ is the coarsest quasi-uniformity compatible with a and is the only totally bounded quasi-uniformity compatible with a. Cor~lla~ A base for 'Up consists of all sets of the form U( Bi X Ai: I "'i "'n), where [ Bi) is a finite cover of X and Bi ill X- Ai for every i, I~ i ~ n. (Received February 5, 1970.) 70T-G67. LUDVIK JANOS, University of Florida, Gainesville, Florida 32601. On extensions of LetS 1 and s2 be topological semigroups, s2 is said to be an extension of S 1 iff there is an iseomorphism i: S 1 - s2 of S 1 into s2 . The semigroup S is monothetic iff there is X E S such that the closure of the family (X n In !!; I) is S. Theorem. If S is compact and monothetic then S has an exten- 580 sion of the form s 1 X G where s 1 is at most countable monothetic compact semigroup with zero element z E S 1 such that yn ~ z for all y E S 1 and G is a compact monothetic group. (Received February 5, 1970.} 70T- G68. DOUG W. CUR TIS, Louisiana State University, Baton Rouge, Louisiana 70803 and ROBERT A. McCOY, Virginia Polytechnic Institute, Blacksburg, Virginia 24061. Stable homeomor- ph isms -~~-i~!0_i!e -dimensional normed linear spaces. Let E be an infinite-dimensional normed linear space. For a topological space X, H(X} will denote the group of homeomorphisms of E onto itself, and SH(X} is the normal subgroup of stable homeomorphisms (i.e., the subgroup of H(X} generated by those homeomorphisms which agree with the identity map on a nonempty open set}. S Ann(E} will denote the strong annulus conjecture for E, which says that if C is a collared cell in Int B 2 (Br and Sr are the ball and sphere in E of radius r, centered at 0} and f is a homeomorphism from S 1 onto Bd C, then there exists a homeomorphism h from (B 2/Int B 1; s 1,S2} onto (B 2/Int C; Bd C, S2} such that h!S 1 = f and h!S 2 =identity. Theorem l. SH(E} = H(E} iff SAnn(E} is true. Corollary. If M is a connected manifold modeled onE and SH(E} = H(E}, then SH(M) = H(M). Theorem 2. If E is homeomorphic to EW, then SH(E} = H(E}. It might be noted that E is homeom0rphic to E"' for all infinite-dimensional Hilbert spaces, for all infinite-dimensional reflexive Banach spaces, and for all infinite-dimensional separable Fr~chet spaces. (Received February 6, 1970.} 70T- G69. L. DUANE LOVELAND, Utah State Uaiversity, Logan, Utah 84321. The boundary of a linearly connected c_:_~p.!~ cube ~..!!!_me. We define a subset X of E 3 to be linearly ~~nected if the intersection of each vertical line with X is a connected set. The following question, stated here in terms of the above definition, ap pears in §9.3 of the survey article by C. E. Burgess and J. W. Cannon r•Embeddings of surfaces in E 3," manuscript]: Is the boundary of a linearly connected crumpl·~d cube in E 3 a tame 2-sphere? In this note we present an affirmative answer to the question. The fact that a linearly connected crum pled cube C is a 3 -cell is an easy application of Theorem l of [J. W. Cannon, "*-taming sets for crumpled cubes, horizo:1ta.l sections in closed sets," manuscript]. We show that C is tame by ex hibiting C* = cl(E3 - C) as a countable union of *-taming sets. (Received February 11, 1970.} 70T-G70. T. BENNY RUSHING, University of Utah, Salt Lake City, Utah 84112. Everywhere wild cells and spheres. A topo.log.ica.l manifold M in En is said to be locally tame at x E M if there is a neighborhood U of x in En and a homeomorphism of U onto En which carries U n M onto a subpolyhedron of En. M is ~_!:ld at x E M if it is not locally tame at x, and M is everywhere wild if it is wild at every point. Jheorem. In En, n !!; 3, there are cellular, everywhere wild cells and everywhere wild spheres of all codimensions between 0 and n. It is immediate that there are closed, everywhere wild strings and half-strings of all codimensions. Noncellular everywhere cells are also obtained. Finally, every where wild arcs are exhibited in En, n!!: 3, which pierce a locally flat (n - i)-sphere at a single point. (Received February 11, 1970.} 581 70T- 071. THOMAS E. ELSNER, Michigan State University, East Lansing, Michigan 48823. _!.rbitrar~ pr_c:>duc~~ T 0 -~ac.:'.:__~':e not :!_' 0 • On page 108 of Thron's book "Topological structures" a question is asked and is settled partially by this example. Let 0 and 1 be m~de a Sierpinski space with 0 open. Take the enumerable product of copies of these and call the space Z. The derived of the point wtth all 0 coordinates is just the comrlem"'nt of this point and is not closed. Note that the factors of Z are T 0 . (Received February 4, 1970.) (Author introduced by Professor P. H. Doyle.) 70T- G72. DOUG W. CUR TIS, Louisiana State University, Baton Rouge, Louisiana 70803. All spaces are m"'trizable. Y is .!_~ally ~9,!:1_ico~~ec~_9 if there exists a map >..: U X I- Y, U a neighborhood of th~ diagonal in Y X Y, such that A (y0 ,y l'i) = Yi• i = 0, l, and A (y,y,t) = y. A c Y is >..-~.£~~~if A(A x A X I) cA. Y is .!_ocally con~ locally eq"..Iiconnecte_? if it has A -convex local neighborhoo:l bases. Every ANR is I.e.; the converse is not known. Y is I.e. if it has an open cover by I.e. spaces. Every l.c.l.e.space is an ANR; the converse is not true in general, but does hold for locally finite complexes. A ~table.!..£~ conve~ structure on Y is an open cover '4 and a sequence of maps y = >..n (t1, ... ,tn; y 1, ... ,yn)' n > 0, defined forti E I, 2:Jti = l, (y 1 , ... ,yn) cUE '1.{, such that: 1 (i) >.. (l,y) = y; (ii) >.. n(t 1, ... ,tn; y 1, ... ,yn) = >.. n-l(t 1, ... , )!'i•···•tn; y 1, ...• ~\····•Yn) if ti = 0; (iii) every open cover ?r has a refinement 'Jf such that ( An(t 1, ... ,tn; y 1, ... ,yn): 2:Jti = l) cV E ?!if (y 1, ... ,yn} c W E 'Jf, n > 0. Y admits a s.l.c.s. iff it is an A!'-IR. A retraction r; Y- A is a '4 -straight deformation retraction if for every open cover 'U of A, there exists a deformation F; Y X I- Y with F(y, l) = y, F(y,O) = r(y), r(F(y X I)) cUE '4. The countable inverse limit of ANR's, with '4 -s.d.r.'s as bonding maps, is an ANR, and the projestions are U -s.d.r.'s. (Re::eived February 17, 1970.) 70T- 073. ERNEST A. MICHAEL, University of Washington, Seattle, Washington 98 10 5. Para ~om2_~ctness and the Lindelof property for Xn and X W. It is well known that there are paracompact (in fact, hereditarily Lindelof) spaces X for which x 2 is not even normal. The following examples disprove some plausible conjectures by showing that, even if x 2 is paracompact, the behavior of the higher powers xn and Xw can be quite unpredictable. All spaces are regular, and (CH) indicates that the continuum hypothesis is assumed. (l) There ex ists a space X such that xn is paracompact for all n EN, but Xw is not normal. (2) (CH) There exists a s pa::e X such tha~ Xn is Lindelof for all n E N, but X"-' is not normal. (3) (CH) For all k E N, there exists a space X such that Xk is Undelof, Xk+ 1 is not LindelOf, and Xn is paracompact for all n E N. (4) (CH) For all k EN, there exists a space X such that xk is hereditarily Lindelof but xk+l is not normal. (5) (CH) There exist semimetrizable spaces X a:td Y such that x41 and yW are hereditarily Lindelof, but X X Y is not normal. (Received February 18, 1970.) 70T-G74. WILBUR WHITTEN, University of Southwestern Louisiana, Lafayette, Louisiana 7050 l. On noninvertible links with i~vertible proper sublinks. In 1963, H. F. Trotter gave a solution to a difficult problem of classical knot theory by ex hibiting a collection of noninvertible knots (H. F. Trotter, "Noninvertible knots exist," Topology 2(1964), 275-280). More recently (th·e author, "A pair of noninvertible links," Duke Math. j. 36(1969), 695-698) 582 examples of noninvertible links were given, each link consisting of two components each of which is invertible. An orie11ted, ordered link L of jJ comp:ments in the oriented 3-sphere s3 is invertible if there is an orientation -preserving autohomeomorphism of s3 taking each component of L onto itself with reversal of orieatation. In this paper we solve a "Brunnian type" problem on invertibility of links by presenting for each integer jJ 1!!: 3 a noninvertible link of jJ components with the feature that each proper sublink is invertible. (Received February 18, 1970.) 70T- G75. PAUL R. MEYER, Lehman College, City University of New York, Bronx, New York l046S. On_the cardinality of dense sets. It is well known that there exist separable sequential spaces of cardinality =·c. However, the following theorem shows that, under the generalized continuum hypothesis (GCH), such cardinals are rather special. Theorem l. (GCH) If X is a sequential space with unique sequential limits and 6 X< lXI, then lXI = ~a+l and 6 X= ~a' where a is 0 or is a limit ordinal. Conversely, if ~a .is a sequential cardinal, there exists a sequential Hausdorff space X such that ~a= 6X < IX I = ~a+ 1. (6X =least cardinal of a dense subset of X.) This and other results in this paper are proved more generally for nonsequential spaces by using the following definition: For an arbitrary topological space X, a X = sequential character of X = least cardinal m such that X is m -sequential. (For defini tion of m-sequential see Abstract 648-116, these cJVoticei) 14(1967), 664; Compositio Math. 21(1969), 102-106.) Known upper bounds on the cardinality of a Hausdorff space can be improved as follows: The or~ 2. If X is a Hausdorff space, then IX I ;§ 6 XO' X and IX I ;§ exp( 6 X • a X). Other properties of a X and the special case of Theorem 2 for sequential spaces (a X = ~ 0 ) were announced in Abstract 672-117, these cJVoticei) 17(1970), 116. (Received February 19, 1970.) 70T-G76. WITHDRAWN. Miscellaneous Fields 70T-Hl8. RENU LASKAR, Clemson University, Clemson, South Carolina 29631. An example of Let H be a net of dimension three (see previous abstract) with pa.rameters (k,n,b,r*). If two points of H are called first associates if they are incident with a line, second associates if incident with a plane but not incident with a line, and third associates if not incident with a plane then it can be shown by the following theorem that points of H yield a 3-class association scheme and all the parameters p~k can be calculated. Theorem. If a set S of treatments are such that any two treatments J l l l l are either first, seco;J.d, or third asso.::iates then the constancy of n 1, n2, n3, p 11, p1z = P2 1• P22• 2 2 2 2 3 3 3 3 ff' . h h 3 l . t' h P 11• P12 = Pzl' P22, P11• p 12 = p21• p22 are su 1cJ.ent to s ow t at s 1s. a -c ass assoc1a wn sc erne. (Received January 12, 1970 .) 583 70T--Hl9. R·)BERT L. CONSTABLE, Computer Science Department, Cornell University, Ithaca, New York 14850 and ALLAN B. BORODIN, University of Toro:~to, Toronto 181, Ontario, Canada. Let G be a programmi.ng language capable of computing all recursive functions, R, i.e. G could be ALGOL or PL/1. Let L be the Loop 1ang:~age of Meyer and Ritchie [?roc. twenty-second National ACM Conf., 1967, pp. 465-470]. This language computes only 1( 1, the primitive recursive functions, and is called a ~ub~-:_ursi~«:._ programming language. Let (!pi} , ( fi} be recursive enumerations of the programs of G and L respectively. The functions computed are denoted by tpi() an::! fi() a.:ld the r11n· times of the programs are respectively ~ .( ) and F. ( ) . We prove: If IP. E G and rp - ( ) E R 1, then th-ere is -- 1 1 1 1 an f. ELand n E ml such that FJ·(X) ~ n. ~i(x) for all x E Nl, and !p.( ) =f.( ). The result contrasts with J 1 J Blum's ~nformgtion and Control 11(1967), 257-265] theorem that for every recursive functions( ) there is an f 1- E L and !pJ· E G such that f.( ) = fP .( ), the size of f., lfd, is the shortest among all L 1 J 1 programs for fi( ) and yet s(lfP jl> < lfil. (Received February 5, 1970.) 70T- H20. JOEL H. SPENCER; The Rand Corporation, Santa Monica, California 90406. Maximal consisteat sets of----·------arcs in a tournament. A (round-robin) tournament T n consists of n nodes 1,2, ... ,n such that each pair of distinct nodes i and j is joined by exactly one of the oriented arcs ij or ji. The arcs in a setS are said to be consistent if it is possible to relabel the nodes of Tn so that if ij is inS, i > j. Let f(Tn) denote the maximum n~mb·~:c of consistent arcs that can b·a fo'.lnd in the tournament T 11 • It is shown that for all Tn' f(Tn) ~ n2/2 + cn312 where cis an absolute positive constant. The following result uses the same proof. Color the edges o.f the complete n-graph Kn red and blue. For some subsetS of the ~ 3/2 vertices l1r red edges - # blue edges I (in the complete graphS) is at least c'n where c' is an absolute pvsitive constant. (Received February 19, 1970.) 584 ERRATA Volume 16 MOHAMED A. AMER and WILLIAM P. HANF. Boolean~-~Ea__!;_o_!_ _!o_g!~1,l-~~igher order, Abstract 670-7, Page 1059. The third line from the bottom of page 1059 reads: If the nonexistence of a set-theoretically definable well-ordering of"' is assumed, LT(TL) is not atomistic. It should read: If the nonexistence of a set-theoretically definable well-ordering of the power set of"' is assum-~d, L T(T L) is not atomistic. R. ALAN DAY. Injectives in equational cl_13:sses in Heyting algebras. Preliminary report, Abstract 69T-Al85, Page 964. Line 7: Replace "some 1 ~ n < u• II with "n = 1,2". GEORGE A. GRATZER and J. P.(.ONKA. On the number of polynomials of an idempotent algebra. II, Abstract 69T-Al90, Page 1076. In Theorem 1, "pn('lt) i 0" should read "pn(~l) i 0 and pn+l(!l) i 0". In Theorem 2, second sentence, "Then" should read: "If Pm(tl) i !, then". The two occurrences of "Pm('l) < Pm+l(tr)" should read "Pm(~J) + 1 ~ Pm+l('J)". MITSUYOSHI KATO. Classificl!~ionof _co~p_act lllani~c_>l~ho~<)tg_p~g~ivalent to the sphere, Abstract 69T-Gl49, Pages 991-992. Page 992, line 10. "FTOPC~ ~ FPLC~ e Z 2" is incorrect and should be replaced by "there is an exact sequence 0 ~ FPLC~ ~ FTOPC~ ~ z 2 ~ 0 which does not split". CRAIG R. PLATT. Endomor'phism sem~~u_p_!l__ l!:nd lattices of subalgebras and congruences, Abstract 69T-A79, Page 657. The proofs of Theorem 1 (b) and of Theorem 2 were incorrect. If in Theorem 2 "join-preserving" is replaced by "order-preserving", then the result is true. Volume 17 SHELDON M. EISENBERG and BRUCE WOOD. Approximation of analytic functions by Bernstein-type operators, Abstract 672-35, Pages 94-95. The addresses of the authors should read as follows: SHELDON M. EISENBERG, University of Hartford, West Hartford, Connecticut 06117 and BRUCE WOOD, University of Arizona, Tucson, Arizona 85721. LEO:o.JARD D. LIPNER. Generalized quantifiers and___ ~_I!-.E CHARLES P. LUEHR and MARCOS ROSENBAUM. A derivation of the already-unified field equations by intrinsic methods, Abstract 672-528, Page 235. In the title as originally published, replace "deviation" by "derivation". In line 6, replace "J. Math. and Phys." by "J. Math. Phys." EVELYN M. NELSON. Educational classes of comm".ltative semigroups. II, Abstract 70T-Al2, Page 279. The title should read: Equational classes of comm".ltative semigroups. II. M. ROSENBLATT-ROTH. Stability of some limit theorems of nonhomogeneous Markov chains, Abstract 673-79, Page 414. Line 7: Replace the word "compact" by the word "complect" as originally submitted by the author. WILLIAM F. TRENCH. Discrete minim".lm V1!_Eia_l!£e smoothing of a polynomial plus correlated noise. Abstract 672-661, Page 273. Statement (b) should be amended to read as follows: "(b) Let C (A)= 'B~=-n W seis). ; it is shown that jC().) I has 2n - 2k relative maxima in 0 -< lA I ~ n." 585 Saunders Mathematics Textbooks Tightly-written, lucid texts that cultivate Munroe: an overview of both integral and differ CALCULUS ential calculus in order to help students immediately apply calculus to any con Outstanding for a three semester se current introductory physics course. 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Infinite-Dimensional Problems This brief introductory text shows how By BERNARD EPSTEIN, Univ. of New Mex functional analysis developed by the ex- ico. 229 pp. illus. $9.50. January, 1970. Ballard: level. It is particularly well suited for ele GEOMETRY mentary education majors, students pur suing programs in general education, and The "core" of Euclidean geometry - prospective secondary school teachers. plane geometry to the Pythagorean theo By WILLIAM R. BALLARD, Univ. of Mon rem - is thoroughly presented in this tana. 238 pp. 178 illus. $9.00. January, ideal text for a first course at the college 1970. Embry, Schell & Thomas: higher dimensional calculus using affine CALCULUS AND LINEAR approximations. ALGEBRA: AN INTEGRATED By MARY R. EMBRY and JOSEPH SCHELL, APPROACH Two Volumes Univ. of N. C.; and J. PELHAM THOMAS, Western Carolina Univ. Vol. I: About 640 Here is a text- suitable for college fresh pp. 124 illus. Ready june, 1970. 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Washington Sq., Phila., Pa.19105 587 Our fine mathematics list has acquired two outstanding additions CALCULUS WITH AN INTRODUCTION TO LINEAR ALGEBRA appeals directly to students in all disciplines by using motivation and examples from many areas of application, and by providing problems and exercises that are among the best ever developed for and employed by any calculus text. There is a felicitous balance between intuition and rigor, with each new concept being properly motivated. To this end, much of the traditional material is drastically rearranged with the early introduction and use of finite sequences; specific isolation of metric properties of the plane; and "flexible rigor" built into the presenta tions on limits and those on the definite integral. The author develops the calculus heuristically, thus enabling the student to intuitively grasp the particular subject matter in point, and anticipate or prepare for the progressively more formal mathematics which follow. In many instances, the proof of a result is preceded by an argument which outlines the proof, establishes the reasonableness of the approach, and isolates the chief difficulties implicit in the concept in exposition. Elements of linear algebra are presented in approximately 100 pages of textual discourse. There are over 2500 problems and exercises designed for the Mathematics, Engineering, and Science major, as well as a plentitude of problems in the Social Sciences, with their emphasis being the applicability or calculus to a wide number of disciplines. Instructor's Manual CALCULUS WITH AN INTRODUCTION TO LINEAR ALGEBRA John G. Hocking, Michigan State University April1970 I 962 pages I $12.95 (tent.) 588 INTRODUCTION TO LINEAR ALGEBRA presents the concepts of elementary linear algebra in a logical sequence: fundamental mathematical concepts, real vector spaces, linear transformations and matrices, linear equations and determinants, change of basis and similarity, Euclidean spaces. Examples, problems, figures, geometric illustrations, and schematic diagrams are employed to reinforce the theoretical development. An unusually complete and detailed first chapter leads the student from the computational world of beginning calculus to the conceptual world of higher mathematics. The Instructor's Manual contains solutions to all proplems not answered in the text, and 'provides teaching suggestions. INTRODUCTION TO LINEAR ALGEBRA Eugene Krause, University of Michigan May 1970 I 320 pages I $8.95 (tent.) WE'RE EXPONIBLE AND WE KNOW IT AND WE'RE PROUD OF IT Expose our text a little further. Write me, Copywriter Stacey Weilandt, for more information. Ill] HOI~ Rlnelllrl ~B~~!1~~!~~:!~~·New York 10017 589 Published and Forthcoming 1970 Titles MATHEMATICS FOR ELEMENTARY TEACHERS BY JOHN L. KELLEY AND DONALD B. RICHERT, UNIVERSITY OF CALIFOR NIA, BERKELEY. This text is written for the teachers and prospective teachers of elementary mathematics, especially the mathematics for the first six years of school. Most of the material is organized in the order in which it appears in the elementary school curriculum, and each new idea is introduced within the conceptual background prossessed by the school child at the time he is introduced to the idea. Intended for the more or less standard one or two quarter or one semester sequence in Mathematics for Elementary School Teachers this text would fit the requirements of a one-semester course in Number Systems that is a segment of a two semester sequence. April 1970, 400 pages, $9.50 (est.) INTRODUCTION TO COMPLEX VARIABLES BY NORMAN LEVINSON, MASSACHUSETTS INSTITUTE OF TECHNOLOGY. AND RAYMOND REDHEFFER, UNIVERSITY OF CALIFORNIA, LOS ANGELES. This book is an outgrowth of independent but parallel experience by two well-known mathematicians. The objective is twofold: to give a clean devel opment of the theory at an appropriate level of generality, and to provide a variety of non-trivial applications. The book contains more than 100 worked out examples and over 600 problems, many of which consist of several parts. This textbook is intended for use in both "pure" and "applied" courses of one or two semesters at the senior or early graduate levels. April 1970, 256 pages, $11.50 (est.) INTRODUCTION TO LINEAR ALGEBRA AND DIFFERENTIAL EQUATIONS BY SAMUEL WOLFENSTEIN, UNIVERSITY OF PARIS. This new text contains all of the material recommended by the CUPM for a first course in linear algebra and the two most important topics for a first course in differential equations: existence-uniqueness theorems and the general theory of linear differential equations. 1969, 254 pages, $10.95. ELEMENTARY CALCULUS BY GEORGE HADLEY, UNIVERSITY OF HAWAII. This text is intended for shorter one semester or one or two quarter courses for liberal arts, business and economics, biology, and psychology students, as well as for use in teacher training colleges and in high schools. The level of presentation makes the work suitable for students of very limited mathematical background. 1968, 430 pages, $10.50 LECTURES ON APPLICATIONS ORIENTED MATHEMATICS BY THE LATE BERNARD FRIEDMAN, EDITED AND INTRODUCED BY VICTOR TWERSKY, UNIVERSITY OF ILLINOIS, CHICAGO. This book consists of a program of courses to distill the essentials of various mathematical topics for presentation to an applications-oriented audience. Intended primarily for self study by applied mathematics majors, physicists, engineers, or graduate students. The book will be a useful supplementary text for upper division and first year graduate courses taken by math majors.1969, 272 pages, $13.95 STATISTICAL METHODS IN STRUCTURAL MECHANICS BY V. V. BOLOTIN, SOVIET ACADEMY OF SCIENCES TRANSLATED BY SAMUEL ARONI, AMERICAN CEMENT TECHNICAL CENTER. This is the only monograph available in English which developes from first principles the statistical methods and concepts applicable to the description of the mechan ical structures. It incorporates the methods of the leading researchers in this field and covers many original results developed by Russian Scientists. Suitable for graduate courses dealing with either the general question of statistical methods in structural mechanics or individual topics, also useful II!IJI for self-study. 1969, 264 pages, $13.95 111!1HOLDEN-DAV, INC. 500 Sansome Street, San Francisco, California 94111 590 BASIC CONCEPTS OF PROBABILITY AND STATISTICS SECOND EDITION, BY J. L. HODGES AND E. L. LEHMANN, UNIVERSITY OF CALIFORNIA, BERKELEY. Directed to students in all disciplines, this text explains the fundamental concepts of probability and statistics assuming only a knowledge of high school algebra. This revised edition of the successful text includes the addition of 300 problems, mostly elementary in nature; two new sections of finite probability; a rewritten section dealing with conditional probability; revision of the section dealing with conditional probability; revi sion of the section treating the Wilcoxon test; a new section which treats !-test. Selected answers are now included in the text; separate Answer Book also available. 1970, 470 pages, $9.50 INTRODUCTION TO PROBABILITY AND STATISTICS BY ROGER CARLSON, UNIVERSITY OF MISSOURI, KANSAS CITY. The most unique feature of this text is an attempt to give the mathematical and philoso phical background of statistics and to show the need for an understanding of the background of statistical inference. Directed to the non-mathematical reader, in particular the serious student of social science, the purpose of the book is to present the basic concepts of elementary statistics. April 1970, 512 pages, $9.50 (est.) ELEMENTARY STATISTICS BY GEORGE HADLEY, UNIVERSITY OF HAWAII. This text is intended for a one-semester introductory course in statistics for students in both the social and biological sciences, or for use as a text in some of the new general courses taught to students with considerable diversity of interest. The level of presenta tion is elementary; no calculus is used, however, probability theory is developed in more depth than is typical for texts at this level. 1969, 467 pages $10.95 FOUNDATIONS OF PROBABILITY BY THE LATE ALFRED RENYI, UNIVERSITY OF BUDAPEST. Introducing many innovations both as regards contents and methods this book deals with the foundations, basic concepts, and fundamental results or probability theory in several respects from a novel point of view. Written for students and mathe maticians who want to obtain a firm basis for the study of special topics of probability of mathematical statistics or of information theory. March 1970, 384 pages, $15.00 (est.) NONPARAMETRIC STATISTICS BY JAROSLAV HAJEK, CHARLES UNIVERSITY, PRAGUE. Designed primarily for graduate courses and for researchers who want to apply nonparametric methods in their fields. "A masterful consolidation and unification of rank procedures, Professor Hajek's exposition is clear and flawless". -Professor Robert Berk, Columbia University, 1969, 192 pages, $10.95. APPLIED PROBABILITY MODELS WITH OPTIMIZATION APPLICATIONS BY SHELDON ROSS, UNIVERSITY OF CALIFORNIA, BERKELEY. A natural follow-up to basic courses in Operation Research or Probability, this graduate level text provides a clear and precise exposition of the theoretical foundations underlying most of the models of applied probability. March 1970, 200 pages, $12.95 (est.) MARKOV PROCESSES BY DAVID A. FREEDMAN, UNIVERSITY OF CALIFORNIA, BERKELEY. De signed for second-year graduate courses, this book provides a readable, constructive treatment of Markov chains, Brownian motion, and diffusion. The book contains some of the author's own research; most of the proofs are new and many of the results are original. June 1970, 512 pages, $15.00 (est.) TIME SERIES ANALYSIS, FORECASTING AND CONTROL BY GEORGE E. P. BOX, UNIVERSITY OF WISCONSIN, AND GWYIL YM M. JENKINS, UNIVERSITY OF LANCASTER, ENGLAND. Concerned with dis crete time series and dynamic models and their use forecasting and control, the techniques are all illustrated with real examples using economic and indus- -- trial data. June 1970, 500 pages, $18.95 (est.) 111;1HOLDEN-DAV, INC. 500 Sansome Street, San Francisco, California 94111 591 Ergebnisse der Mathematik Vol. 111: K. H. Mayer, und ihrer Grenzgebiete Relationen zwischen charak Vol. 11: H.-H. Ostmann, Addi teristischen Zahlen. 102 pp. tive Zahlentheorie. Part 2: In German. 1969 Spezielle Zahlenmengen: OM 8,-; US $ 2.20 Reprint. 138 pp. In German Vol. 112: Colloquium on 1969. OM 28,-; US $ 7.70 Methods of Optimization. Vol. 27: R. Schatten, Norm Held in Novosibirsk/USSR, Ideals of Completely Con June 1968. Ed. by tinuous Operators. N. N. Moiseev. 298 pp. 1970 2nd printing. 88 pp. 1970 OM 18,-; US$ 5.00 OM 26,-; US $ 7.20 Vol. 114: H. Jacquet, R. P. Lang lands, Automorphic Forms on GL (2). 554 pp. 1970 * Lecture Notes OM 24,-; US $ 6.60 in Mathematics Vol. 115: K. W. Roggenkamp, Vol. 94: M. Machover, V. Huber Dyson, Lattices J. Hirschfeld, Lectures on Over Orders I. 309 pp. 1970 Non-Standard Analysis. 85 pp. DM 18,-; US $ 5.00 1969. OM 6,-; US $1.70 Vol. 116: Seminaire Pierre Vol. 96: H.-B. Brinkmann, Lelong (Analyse). An nee 1969 D. Puppe, Abelsche und lnstitut Henri Poincare, Paris. exakte Kategorien, Korre 200 pp. In French. 1970 spondenzen. 146 pp. In Ger OM 14,-; US $ 3.90 man. 1969. OM 10,-; US $ 2.80 Vol. 117: Y. Meyer, Nombres Vol. 104: G. H. Pimbley, Jr., de Pisot, Nombres de Salem Eigenfunction Branches of et Analyse Harmonique Nonlinear Operators, and Cours Peccot donne au their Bifurcations. 130 pp. College de France en avril 1969. OM 10,-; US $ 2.80 mai 1969. 63 pp. In French 1970. OM 6,- Vol. 105: R. Larsen, The Multi plier Problem. 291 pp. 1969 OM 18,-; US $ 5.00 Die Grundlehren der mathe· Vol. 106: Reports ofthe matischen Wissenschaften Midwest Category Seminar Vol. 43: 0. Neugebauer, Ill. Ed. by S. Mac Lane. 250pp. Vorlesungen iiberGeschichte 1969. OM 16,-; US $ 4.40 der antiken mathematischen Vol. 107: A. Peyerimhoff, Wissenschaften. Part 1: Vor Lectures on Summability griechische Mathematik 114pp.1969. OMS,-; US $2.20 2nd edition. 61 fig. 224 pp. Vol. 108: Algebraic K-Theory In German. 1969 and its Geometric Applica OM 48,-; US $ 13.20 tions. Ed. by R. M. F. Moss Vol. 68: G. Aumann, Reelle and C. B. Thomas. 90 pp. Funktionen. 2nd edition. 22fig. Springer-Verlag 1969. OM 6,-; US$ 1.70 426 pp. In German. 1969 Vol. 109: Conference on the OM 68,-; US $ 18.70 New York Numerical Solution of Differ Vol. 140: Mathematische Heidelberg ential Equations. Ed. by Hilfsmittel des lngenieurs. J. Ll. Morris. 281 pp. 1969 Ed. by R. Sauer and I. Szabo. Berlin OM 18,-; US$ 5.00 Part 2: Authored by L. Collatz, 115 FIFTH IVEIUE Vol. 110: The Many Facets of R. Nicolovius, W. Tornig 148 fig. 704 pp. In German lEW YORK. U.tOOtO Graph Theory. Proceedings of the Conference held at 1969. OM 136,-; US$ 37.40 Western Michigan University, * Distribution rights for this Kalamazoo/MI, October 31- series in Japan held by November 2, 1968. Ed. by Heidelberger Taschenbiicher Maruzen Co. Ltd., Tokyo G. Chartrand and S. F. Kapoor Vol. 64: F. Rehbock, Darstel 298 pp. 1969 lende Geometrie. 3rd edition • Please ask for prospectus OM 18,-; US $ 5.00 111 fig. 250 pp. In German material 1969. OM 12,80; US $ 3.60 592 ELEMENTARY COMBINATORIAL ANALYSIS By Martin Eisen, Temple University 1970 240 pp. Cloth $14.50/ Prepaid $11.60 Paper $ 7.50 f Prepaid $ 6.00 This book was written in a manner understandable by high school honors students or under graduates. No mathematical background beyond high school algebra is required. The wide applicability of combinatorial methods makes it useful for engineers, physical and social scientists, operations researchers and actuarial scientists. Topics covered include combina tions, permutations, recusion, generating functions, principle of inclusion and exclusion, probability problems, hit and rook polynomials, and current research. CODE NUMBER: 0226 LINEAR FUNCTIONAL ANALYSIS By Robert A. Bonic, University of California at Santa Cruz 1970 138 pp. Cloth $12.50/Prepaid $9.50 Paper$ 6.00/Prepaid $ 4.80 This book is intended for the student of pure mathematics and contains the basic prerequisites for global analysis which is currently one of the most active areas of mathematical research. The book covers important topics not usually treated in other works, including nuclear spaces and operators, chains of Banach spaces, Fredholm operators, and Holder continuous functions. CODE NUMBER: 0205 TOPOLOGICAL VECTOR SPACES By A. Grothendieck 1970 approx. 260 pp. Cloth $17.50/Prepaid $14.50 Paper$ 9.50/Prepaid $ 7.60 CONTENTS: Introduction to Topology: General Properties; General. Theorems of Duality in Localized Convexed Spaces; the Hahn-Banach Theorem and Its First Consequences; Spaces of Linear Applications; Study of Some Special Kinds of Spaces; CoQ'lpactness in the ELC. Northeastern University CODE NUMBER: 3002 AN INTRODUCTION TO REAL AND COMPLEX MANIFOLDS By Guiliano Sorani, Northeastern University 1970 approx. 220 pp. Cloth $12.50/Prepaid $9.50 Paper$ 7.50/Prepaid $6.00 The lecture notes of a course taught at Northeastern University during the winter and spring terms of 1967-68, this volume contains topics chosen in order to stimulate interest in students. Many examples and problems have been included. The principle aim of the course was to give the greatest amount of information of the subject, assuming the least possible background. CODE NUMBER: 6215 G ------ORDER FORM------8 GORDON AND BREACH, SCIENCE PUBLISHERS, INC. 150 FIFTH AVENUE • NEW YORK, NEW YORK 10011 Code # Author Edition Quantity Payment Enclosed D Send new catalog Total $ Prepaid orders: All orders from Individuals must be prepaid. We pay all Tax, if applicable $ _____ postage and handling charges. U.S. residents add applicable sales tax. Amount Remitted $ NAME, ______ ADDRESS CITYfSTATEJZIP 593 c LINEAR Linear algebra and calculus gain in depth and significance when inter related. Linear algebra simplifies L the theory and application of calculus while calculus illustrates and clarifies the study of linear c algebra. u ALGEBRA CALCULUS AND LINEAR ALGEBRA Volume I Vectors in the Plane and u One-Variable Calculus CACULUS AND LINEAR ALGEBRA s Volume II Vector Spaces, Many-Variable Calculus and Differential Equations By WILFRED KAPLAN and DONALD J. LEWIS, both of the University of Michigan This two-volume text fully integrates linear algebra with calculus and recognizes the geometric significance of the theory of linear algebra. Volume 1: reviews precalculus topics, introduces vectors in the plane and elementary transcendental functions early in the text, moves rapidly to differential and integral calculus, and concludes with a chapter on infinite series. Available March 1970 for the first year. Approx. 640 pages $9.95 Volume II: treats the theory of vector spaces, matrices and determinants, Euclidean geometry in space of 3 or n dimensions, many-variable calculus of scalar and vector functions, and differential equations (with emphasis on the linear case). Available January 1971 for the second year. JOHN WILEY & SONS, Inc. 605 Third Avenue New York, N.Y. 10016 llliiiiQ In Canada: 22 Worcester Road, Rexdale, Ontario 594 HARPER FILMS- READY IN MAY Calculus Films: Computer Generated HARRY M. SCHEY AND JUDAH L. SCHWARTZ 1817 Education Research Cent_er of the Massachusetts Institute of Technology -Under the Consulting Editorship of Gian-Carlo Rota - CALCULUS FILMS is a set of ten computer generated film loops illustrating fundamental principles of analytic geometry and calculus. Each film deals with a single concept. So that the films wi II appeal to as wide an audience as possible, detailed exposition is left to the accompanying notes and to references to some of the more widely used modern calculus texts. Thus the films can be used successfully with a wide range of students from those in high school encountering calculus for the first time, to college upper classmen who seek to deepen their intuitive understanding of mathematics. Available in both cartridges and reels, each film runs approximately four minutes. The films: Taylor Series I, Taylor Series II, Newton's Method, Mean Value Theorems, Conic Sections, Conic Sections - Ellipse, Conic Sections - Hyperbola, Conic Sections - Parabola, Indeterminate Forms, The Function of xn sin (1/x). $30 per film; $270 per set HARPER TEXTS- JUST PUBLISHED Numerical Methods That Work FORMAN S. ACTON, Princeton University This text deals with numerical solutions of problems in terms of available computer meth ods. Practical rather than theoretical, the book develops a feeling for the expedient use of numerical methods and teaches the student to recognize and handle the trouble spots that plague numerical computation. Problems, graphs, diagrams. Tentative: 540 pp.; $12.95 College Algebra CARL H. DENBOW, Ohio University Prepares students for modern courses in calculus, linear algebra, and probability. Gives a solid background in algebra and an introduction to functions (including domain and range analyses), composites, and the inverses of functions and relations, using sets and models. Two years of high school algebra and one year of high school geometry are prerequisite. 900 problems with answers to the odd-numbered ones; exercises; solved problems; illus trations; examples. Instructor's Manual. 433 pp.; $8.95.,Student Guide will be available. Algebra and Trigonometry THOMAS J. ROBINSON, University of North Dakota This logical approach to traditional and modern aspects of algebra and trigonometry pre pares the student - even the student who has had only one unit of high school algebra -for calculus. The book moves rapidly from elementary topics to more complicated ones. The author blends theory and application by intermingling theorems, examples, and drill. Problems, with selected answers; graphs and illustrations. Teacher's Manual. 407 pp.; $8.95. Student Guide will be available. A NOTABLE TEXT FROM HARPER & ROW The Calcu!us with Analytic Geometry LOUIS LEITHOLD California State College at Los Angeles A full treatment of elementary calculus, presenting theory in a well-motivated and rigor ous fashion, but with concern for computational aspects. Thoroughly class-tested; num erous problems and exercises. Answer Booklet. 976 pp.; $13.50 Also in a two-part edition: I. FUNCTIONS OF ONE VARIABLE AND PLANE ANALYTIC GEOMETRY (Chapters 1 through 17)- 648 pp.; $9.95; II. VECTORS, FUNCTIONS OF SEV ERAL VARIABLES, AND INFINITE SERIES (Chapters 18 through 24) - 348 pp.; $7.95 Send for our 1970 cata/ogue/#36-00426 HARPER & ROW, PUBLISHERS, 49 E. 330 STREET, NEW YORK 10016 595 Macmillan Mathematics Texts for 1970 ALGEBRA By JACOB K. GOLDHABER and GERTRUDE EHRLICH, both of the University of Maryland This book provides the algebraic background needed for specialization in algebra and num ber theory, as well as in other branches of mathematics. It is intended as a text for the mathematically mature beginner in algebra. The book points the way towards the modern spirit in algebra through its emphasis on maps and its frequent use of universal mapping properties in detining algebraic entities. 1970,418 pages, $11.95 INTRODUCTION TO MATHEMATICAL STATISTICS, Third Edition By ROBERT Y. HOGG and ALLEN T. CRAIG, both of the University of Iowa The Third Edition of this highly successful text features: a new chapter on non parametric methods, an expanded discussion of random variables, an earlier introduction to the proof of the independence of X and S2 (when sampling from a normal distribution), and an ex panded and strengthened supply of exercises. The text is designed for a one-year course in mathematical statistics at the junior-senior level and requires at least one year of a substan tial course in calculus. A manual will be available for use with the text. 1970,415 pages, $10.95 ELEMENTARY LINEAR ALGEBRA By BERNARD KOLMAN, Drexel Institute of Technology Designed for the student who has completed a course in single-variable calculus, Elemen tary Linear Al!febra provides a gradual and firmly-based introduction to postulational and axiomatic mathematics-while giving due attention to computational aspects of the subject. Exercises are closely integrated with the text, and CUPM recommendations are taken into consideration. 1970, 255 pages, $8.95 INTRODUCTORY COMPUTER METHODS AND NUMERICAL ANALYSIS Second Edition By RALPH H. PENNINGTON This new edition of a successful text for computer oriented courses in introductory numer ical methods and analysis is now geared to third generation computers. Requiring only a background in integral calculus. the text stresses methods of proven value in the computer solution of science and engineering problems. The Second Edition includes ASA standards for FORTRAN, a remote terminal language, expanded treatment of accuracy and error control, new methods, and more problems. 1970,497 pages, $10.95 INTERMEDIATE ANALYSIS By M.S. RAMANUJAN, The University of Michigan, a·nd EDWARDS. THOMAS, Jr., State University of New York, Albany This rigorous but informally written text is designed to bridge the gap between elementary calculus sequences and more advanced analysis courses. It helps build mathematical ma turity by exposing the student to rigorous proofs, abstraction, and up-to-date terminology. Numerous exercises give practice in supplying proofs. 1970, approx. 208 pages, Prob. $8.95 \\'rite to the Faculty Service Desk for examination copies. THE MACMILLAN COMPANY, 866 Third .-henue, New York, ~ew York 10022 In Canada, write to Collier-~larmillan Canada, Ltd., 1125B Leslie Street, Don Mills, Ontario 596 CALCULUS: A First Course By Louis Auslander, The City University of New York © 1970,550 pages, illus., hardbound $10.50 (tent.) CALCULUS OF ONE VARIABLE By Robert T. Seeley, Brandeis University © 1968,532 pages, illus., hardbound $10.95 CALCULUS OF SEVERAL VARIABLES By Robert T. Seeley, Brandeis University © 1970,288 pages, il/us., softbound $3.25 (tent.) COLLEGE TRIGONOMETRY By Fred Richman, Carol Walker, and Elbert A. Walker, New Mexico State University © 1970,208 pages, hardbound $5.95 BASIC MATHEMATICS FOR MANAGEMENT AND ECONOMICS By Lyman C. Peck, Miami University © 1970, 323 pages, il/us., hardbound $7.95 CORE MATHEMATICS By Larry L. Whitworth and John F. Leslie, Community College of Allegheny County © 1970, 350 pages, il/us., softbound $5.95 (tent.) ELEMENTARY MATHEMATICS: Number Systems and Algebra By Anthony J. Pettofrezzo, Florida Technological University Donald W. Hight, Kansas State College of Pittsburg © 1970, 500 pages, illus., hardbound $9.75 (tent.) ~Scotti' ~ lCD ..3 Division!! Scott, Foresman College Division Glenview, II. Palo Alto, Ca. Tucker, Ga. Oakland, N.J. Dallas, Tx. 597 ~~···································································· • r• • • • rnr• • or• • • • or• • • • orw • • or• • r• • • • • • e • • r• • • • • or• • • • .,. • • • • • • .,. • or• • •1r• or• • A Comprehensive I nlroduction to DIFFERENTIAL GEOMETRY M. Spivak, Brandeis University ~ /cloth $18.00 /Available now Volume 1. Theory of Differentiable Manifolds. 635 pages/ paper $9.00 Volume 2. Connections and Riemannian Geometry. Approx. 500 pages/Available July ~ These notes provide a coherent and thorough presentation of the foundations of differential geometry, which prepares the student to read classical works, as well as the diverse modern treatments of the subject. These different formulations are all presented and compared, but, although a modern spirit prevails, the classical language and viewpoint is also explained in detail; the second volume includes ~ which can be read concurrently. expositions of the foundational papers of Gauss and Riemann, Approximately 200 hand drawn figures are included, and about one-fourth of the material consists of Problems. Prerequisites: Calculus of several variables, using linear algebra, and basic properties of metric spaces. Contents (Volume 1.) 1. Manifolds. 2. Differentiable Structures. 3. The Tangent Bundle. 4. Tensors. 5. Vector Fields and Differential Equations. 6. Integral Manifolds. 7. Differential Forms. 8. Integration (Stokes' Theorems and de Rham cohomology). 9. Riemannian Metrics. 10. lie Groups. 11. In the Realm of Algebraic Topology (Exact Sequences, Poincare' Duality, Euler Class, Poincare'-Hopf Theorem). Appendix A. Non-Metrizable Manifolds. ~ Send orders prepaid to: Spivak Notes, Dept. of Math., Brandeis University, Waltham, Mass. 02154. Make checks payable to: Publish or Perish ;JI .:.~·································································~~• • • ra a • •~ • • a a rra a a a a a a a a a • a .. • a • a • • .,. a • a •~,. a • rra a a •~ .,. a~r• • • .,.~ a • .--.--.-. ~ MATHEMATICS OF THE USSR Mathematics of the USSR-Sbornik and Mathematics of the USSR Izvestija are two important translation journals of the American Mathe matical Society devoted to Russian mathematics. Izvestija is the English translation of the journal of the Academy of Sciences of the USSR, and Sbornik is the English translation of the journal of the Moscow Mathe matical Society. Both journals publish current research in all fields of pure mathematics. Mathematics of the USSR-Sbornik is published in three volumes of four issues annually. We are now accepting orders for the 1970 issues. The list price for a yearly subscription is $160.00 ($80.00 to members of the AMS). Mathematics of the USSR-Izvestija is published in one volume of six issues annually. We are now accepting orders for the 1969 and 1970 volumes. The list price for a yearly subscription is $110.00 ($55.00 to members of the AMS). To place your order, to request additional information, or to inquire about back Issues, please write: American Mathematical Society P. 0. Box 6248 Providence, Rhode Island 02904 598 CALCULUS AND ANALYTIC GEOMETRY by Douglas F. Riddle, Utah State University This very student-oriented text provides exceptionally detailed and numerous explanations which permit the student to read mathematics. The limit concept is presented three times: intuitively, geometrically, and by epsilon/delh definition. The mathematics is consistently careful. Over 4000 problems, graded for difficulty and variety, are included. A detailed Solutions Manual vouches for the effectiveness of the exercises. 1969. 832 pages. 7 x 10. Clothbound. GEOMETRY, AN INTRODUCTION by Gunter Ewald, Ruhr Universitat This general survey of modern geometries for mathematics majors is based on Klein's Erlangen Programm. The text emphasizes mappings as a central idea, and thoroughly covers the foundations of metric, affine, and projective geometries. New in 1970. 372 pages. 71/2 x 83/a. Clothbound. STUDY GUIDE for WOOTON AND DROOYAN'S INTERMEDIATE ALGEBRA, 2nd Alternate Edition by Bernard Feldman, Los Angeles Pierce College For those instructors who wish their students to be exposed to a programmed approach, as well as to a conventional text, this semi-programmed learning tool complements pedagogi cally the leading intermediate algebra text. 1969. 320 pages. 81/2 x 11. Paperbound. ELEMENTARY ALGEBRA by Lee A. Stevens, Foothill Junior College District Smoothly written and carefully tested, this beginning algebra text for college students em phasizes the solution of equations as the central theme and purpose. Numerous worked-out examples and graded exercises are included. Set terminology is used throughout. A semi programmed Study Guide is available. 1969. 288 pages. 61/2 x 91/4. Clothbound. ORDINARY DIFFERENTIAL EQUATIONS, 3rd Edition by Walter Leighton, University of Missouri at Columbia This revision features an expanded and especially careful presentation of stability theory, numerous pedagogical improvements, and increased exercise sets. A strong presentation of oscillation theory is provided. March 1970. 288 pages. 6 x 9. Clothbound. Write to Box NAMS-3A, Wadsworth Publishing Company, Inc. Belmont, California 94002 599 CUSHING-MALLOY, Overseas opportunities in a new science and engineering college for faculty and INC. department heads in mathematics, physics, 1350 North Main St., P. 0. Box 632 chemistry, geology and English, as well as civil, chemical, electrical and mechanical engineering. Challenging assignments in Ann Arbor, Michi~an 48107 research and undergraduate teaching LITHOPRINTERS growing into the graduate level. Minimum appointments of two years. Competitive salaries, free air-conditioned furnished • housing and unparalleled opportunities Known for for travel in Europe, Africa, and the East. Reply with resume to QUALITY- ECONOMY SERVICE College of Petroleum and Minerals Saudi Arabian Educational Mission • 17th Floor 880Third Avenue We specialize in reproducing out-of New York, N.Y. 10022 print books, as well as typewriter com position. Let us quote on your next printing. INDEX TO ADVERTISERS Academic Press...... cover IV Addison-Wesley Publishing Company, Inc...... cover III American Mathematical Society ...... 598 College of Petroleum & Minerals ...... 600 Cushing-Malloy, Inc...... 600 Gordon and Breach, Science Publishers, Inc...... 593 Harper & Row, Publishers ...... 595 Holden-Day, Inc...... 590, 591 Holt, Rinehart & Winston, Inc ...... 588, 589 The Macmillan Company ...... 596 Publish or Perish ...... 598 W. B. Saunders Company ...... 586, 587 Scott, Foresman and Company ...... 597 Springer-Verlag New York Inc...... 592 Wadsworth Publishing Company, Inc...... 599 John Wiley & Sons, Inc...... 594 600 Addison -Wesley TT• Math is Our Middle Name Calculus with Analytic Geometry: A First College Calculus with Analytic Geom Course, Second Edition etry, Second Edition by Murray H. Protter and Charles B. by Murray H. Protter and Charles B. Morrey, Jr., University of California Morrey, Jr., University of California Revised to fit in a modern course while The second edition of one of the most maintaining the successful approach of successful calculus texts ever pub the first edition in tone and emphasis. lished, designed for a three-term course. 548 pp, 374 illus, $9.95 (1970) 900 pp, 553 il/us, $13.95 (1970) Introduction to Partial Differential General Topology Equations: From Fourier Series to by Stephen Wi liard, University of Alberta Boundary Value Problems by Arne Broman, Chalmers University of This balanced introduction to both Technology, Goteborg, Sweden "geometry" and "topology for analysts" provides the basic material from which Calculus, Linear Algebra, Ordinary any of several one or two-semester Differential Equations, and Complex courses in general topology can be Analysis are prerequisites for this self taught. contained text on Fourier Analysis and In press (1970) applications thereof. In press (1970) Calculus: A Short Course Calculus and Statistics by Donald E. Richmond, Williams College by Michael C. Gemignani, Smith College A revision of the author's Introductory An integrated and thorough treatment Calculus, this text is designed for a one of calculus, statistics and probability semester course and presupposes no for a one-year course, this text is de knowledge of ana lytic geometry or signed primarily for the behavioral and trigonometry. social science major. In press (1970) In press (1970) Addison-Wesley THE siGN oF PUBUSHING COMPANY, INC. I EXCELLENCE Reading, MaaaachuaeHs 01867 AMERICAN MATHEMATICAL SOCIETY SECOND-CLASS POSTAGE PAID AT P.O. Box 6248 PROVIDENCE, RHODE ISLAND Providence, Rhode Island 02904 AND ADDITIONAL MAILING OFFICES Return Requested TIME VALUE CONTRIBUTIONS TO EXTENSION THEORY Of TOPOLOGICAL STRUCTURES Proceedings of the Symposium held in Berlin, August 14-18, 1867 Scientific editors: J. FLACHMEYER, H. POPPE, and F. TERPE. This book surveys important recent contributions to topological extension theory. It emphasizes the internal topological problems of extension theory. A number of papers discusses the colinection between topological extension theory and uniform structures, while others treat the algebraic and order theoretical problems of extension theory. Applications of extension in functional analysis, function theory, and potential theory are also discussed. 1969, 279 pp., $17.50. PROBABILITY ALGEBRAS AND STOCHASTIC SPACES A Volume of PROBABILITY AND MATHEMATICAL STATISTICS edited by Z. W. BIRNBAUM and EUGENE LUKACS by DEMETRIOS A. KAPPOS, University of Athens, Athens 144, Greece This monograph evolves the basic notions of the subject through the use of an abstr~ct measure and integration theory which has been developed in Boolean algebras. The funda mental concept of a probability algebra-an algebra of events associated with their proba bility-is discussed in detail together with the various possible structures of probability algebras in general and their representations by the probability spaces of classical theory. A general treatment of the concept of random variables, taking values on spaces endowed with any algebraic of topological structure, is given, emphasizing cases in which the space of the values is a lattice group, or vector lattice. 1969, 268 pp., $12.50. COHOMOLOGY OF GROUPS Volume 34 of PURE AND APPLIED MATHEMATICS edited by PAUL A. SMITH and SAMUEL ElLENBERG by EDWIN WEISS, Department of Mathematics, Boston University, Boston, Massachusetts In order to understand the subject of class field theory, the student must know certain basic arithematic results and be well versed in topics from homological algebra, centering upon the cohomology of groups. This book provides the student of mathematics with direct ac cess to this specialized material. The classical approach to cohomoly is taken because it is historically valid, relatively easy to understand, and it constitutes a shorter path to the desired results. The material is detailed, expansive, and essentially self-contained. Prob lems and exercises of varying degrees of difficulty are presented at the end of the books major sections. 1969, 274 pp., $15.00.