OF THE

AMERICAN

MATHEMATICAL

SOCIETY

VOLUME 13, NUMBER 3 ISSUE NO. 89 APRIL, 1966

cNotiaiJ OF THE

AMERICAN MATHEMATICAL SOCIETY

Edited by John W. Green and Gordon L. \\' alker

CONTENTS

MEETINGS Calendar of Meetings • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 2 84 Program of the April Meeting in New York. • • • . • • • • • • • . • • • • • • • • 285 Abstracts for the Meeting- Pages 324-347 Program of the April Meeting in Honolulu • • • • • • • • • • • • • • • • • • • • • • 293 Abstracts for the Meeting- Pages 348-357 Program of the April Meeting in Chicago • • • • • • • • • • • • • • • • • . • • • • 297 Abstracts for the Meeting- Pages 358-375 PRELIMINARY ANNOUNCEMENTS OF MEETINGS.. • • • • • • • • • • • • • • • • • • 304 ACTIVITIES OF OTHER ASSOCIATIONS • • • • • • • • • • • • • • • • • . . • • • • • • . • 311 1966 INTERNATIONAL CONGRESS OF MATHEMATICIANS . • . • • • • • • • . • • • 312 COMMITTEE ON SUPPORT OF RESEARCH IN THE MATHEMATICAL SCIENCES...... 314 ASSISTANTSHIPS AND FELLOWSHIPS...... 316 GRADUATE COURSES...... 316 PERSONAL ITEMS...... 318 NEWS ITEMS AND ANNOUNCEMENTS. •••.•••••.•••••••••••• 313, 315, 320 SUPPLEMENTARY PROGRAM- Number 38...... 321 MEMORANDA TO MEMBERS The Mathematical Sciences Employment Register • • • • • • • • • • • • • • • • 323 Change of Address • • • • • • • • • • • . • • • • • • • • • • . • • • • • • • • • • • • • • 323 ABSTRACTS OF CONTRIBUTED PAPERS...... 324 INDEX TO ADVERTISERS • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 413 RESERVATION FORM • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 414 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 c}/otiai) 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*

635 june 18, 1966 Victoria, British Columbia May 4 August Z9 - September Z, 1966 (71st Summer Meeting) New Brunswick, New jersey july 8 January Z4-Z8, 1967 ( 73rd Annual Meeting) Houston, Texas August Z8 -September 1, 1967 (7Znd Summer Meeting) Toronto, Ontario, Canada january, 1968 (74th Annual Meeting) San Francisco, California

*The abstracts of papers to be presented in 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 llnes also apply to news items. The next two deadline dates for the by title abstracts are April Z8, and july 1, 1966.

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The cN'otiaiJ of the American Mathematical Society is published by the Society in January, February, April, June, August, October, November and December. Price per annual volume is $7.00. Price per copy $2.00. 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, J'.O. Box 6248, Providence, Rhode Island 02904. Second-class postage paid at Providence, Rhode Island, and additional mailing offices. Authorization is granted under the authority of the act of August 24, 1912, as amended by the act of August4, 1947 (Sec. 34,21, P. L. and R.). Accepted for mailing at the special rate of Postage provided for in section 34,40, paragraph (d).

Copyright«), 1966 by the American Mathematical Society Printed in the United States of America Six Hundred Thirty-Second Meeting Waldorf-Astoria Hotel New York, New York April 4-7, 1966

PROGRAM

The six hundred thirty-second meet­ four sessions on the afternoon of Tuesday, ing of the American Mathematical Society April 5, on Wednesday, April 6, and on the will be held at the Waldorf-Astoria Hotel morning of Thursday, April 7, as follows: in New York on April4-7, 1966. All sessions will be held in public rooms of the hotel. Session I. Computation w-ith symbolic By invitation of the Committee to and algebraic data. Select Hour Speakers for Eastern Sec­ Session II. Numerical methods for com­ tional Meetings, there will be two addresses. puters. Professor Walter Feit of Yale University Session III. Software systems; mechanical will speak on Monday, April 4, at 2:00P.M. linguistics, computer analy­ in the Sert Room. The title of his lecture sis of language. is "Modular representations of finite Session IV. Theory of automata; artificial groups" Professor Tsuneo Tamagawa of intelligence. Yale University will address the Society on Tuesday, April 5, at 11:00 A.M. in the Sert Room. His lecture is entitled" Z-func­ The subject of the Symposium was tions of simple algebras." The Sert Room chosen by the Committee on Applied Math­ is at the northwest corner of the main floor ematics which consisted of A. H. Taub (the corner nearest to Park Avenue and (Chairman), V. Bargmann, G. E. Forsythe, Fiftieth Street) and is near the registration C. C. Lin, Alfred Schild, and H. S. Wilf. area. Financial support comes from the Air There will be sessions for contrib­ Force Office of Scientific Research, the uted papers on Monday morning and after­ Institute for Defense Analyses, and the noon and on Tuesday morning, April 4 and U. S. Army Research Office--Durham. 5. There is room for a limited number of The Association for Computing Machinery late papers. and the Association for Symbolic Logic are co-sponsoring the Symposium. THE ASSOCIATION The Invitations Committee, respon­ FOR SYMBOLIC LOGIC sible for the planning of the program and the choice of speakers, consists of Jack The Association for Symbolic Logic Schwartz (Chairman), Courant Institute of will meet in the same hotel, also on April Mathematical Sciences; Martin Davis, 4. Their program includes an invited ad­ Courant Institute of Mathematical Sciences; dress by Professor Hilary Putnam of Har­ H. H. Goldstine, IBM Research Center; vard University. The complete program D. H. Lehmer, University of California; is reproduced on page 311 in the section on John Todd, California Institute of Tech­ activities of other associations. nology; H. S. Wilf, University of Pennsyl­ vania; Calvin C. Elgot, IBM Research Cen­ SYMPOSIUM ON MATHEMATICAL ter; Saul Gorn, University of Pennsylvania; ASPECTS OF COMPUTER SCIENCE Harry Huskey, University of California; and Anthony G. Oettinger, Harvard Univer­ The Symposium will be presented in sity.

285 REGISTRATION can take a shuttle bus to the west side ter­ minal and use the Independent Subway Sys­ The registration desk for attendance tem (E or F cars) to the 53rd Street stop. at the meeting will be in the Terrace Court Those arriving by car will find on the main floor near the west end of the many parking facilities in the neighborhood hotel (the Park Avenue entrance). It will be in addition to those at the hotel. Pickup and open Monday through Wednesday, April4-6, delivery service can be arranged through from 9:00 A.M. to 5:00 P.M. and on Thurs­ the hotel at a cost of $3.25 for a 24-hour day, April 7, from 9:00 A.M. till noon. period, plus $1.25 for each pickup-delivery.

TRAVEL ROOM RESERVATIONS

The Waldorf-Astoria occupies an Persons intending to stay at the entire city block on the east side of New Waldorf-Astoria should make their own York City, from 49th to 50th Street and reservations with the hotel. A reservation from Lexington to Park Avenues. blank and a listing of room rates was on Those arriving by train at Pennsyl­ page 278 of the February issue of the vania Station may take the Independent Notices. There was a deadline of March 21 Subway System (E or F cars) to the 53rd mentioned with the reservation form, a Street and Lexington Avenue stop, a short date which may well precede receipt of walk from the hotel. this program. From Grand Central Station one may Some persons may wish to be re­ take the I.R. T. Lexington Avenue local sub­ minded that Sloane House, a YMCA Hotel way to the 51st Street stop. for both men and women, is located at 356 Those arriving by bus may take the West 34th Street. They offer a variety of Independent Subway System (E or F cars) services. Some of their rooms are priced from the west side bus terminal. There is as low as $3.15. a shuttle bus service from La Guardia and Kennedy Airports to the East Side Terminal MAIL ADDRESS with a transfer bus to Grand Central ·sta­ tion. (It is suggested that those arriving in Registrants at the meeting may re­ a group of three or more may find it as ceive mail addressed in care of the Ameri­ economical to take a taxi directly to the can Mathematical Society, The Waldorf­ hotel.) Astoria, 301 Park Avenue, New York, New Those arriving at Newark Airport York 10022.

PROGRAM OF THE SESSIONS

The time limit for each contributed paper is ten minutes. The papers are scheduled at 15 minute intervals in order that listeners can circulate among sessions. 1:2. maintain the schedule, the time limits will be strictly en­ forced.

MONDAY, 10:00 A.M. General Session, Louis XVI Suite, Fourth Floor 10:00-10:10 ( 1) On ~ class of graphs with connectivity 2 Professor M. E. Watkins, University of North Carolina (632-5) 10:15-10:25 (2) Prime z-ideals in C(R). I Mr. M. W. Mandelker, University of Rochester (632-32) 10:30-10:40 (3) Radon and Helly theorems for finite dimensional unbounded sets Professor R. T. Ives, Harvey Mudd College (632-29)

286 10:45-10:55 (4) Positively expansive endomorphisms of compact groups Dr. Murray Eisenberg, University of Massachusetts (632-28) 11:00-11:10 ( 5) Ergodic automorphisms on compact groups Professor Ta-Sun Wu, University of Massachusetts (632-64) 11:15-11:25 ( 6) Oscillations of arithmetical functions Professor Emil Grosswald, University of Pennsylvania (632-21) 11:30-11:40 ( 7) Linear separability of switching functions Professor A. G. Azpeitia, University of Massachusetts 1632-43)

MONDAY, 10:00 A.M. First Session on Analysis, Jansen Salon, Fourth Floor 10:00-10:10 (8) On the analytic validity of formal simplifications of linear differential equa­ tions Professor Wolfgang Wasow, University of Wisconsin (632-63) 10:15-10:25 (9) The bifurcation of an equilibrium point into a closed orbit Mr. Nathaniel Chafee, Brown University (632-52) 10:30-10:40 ( 10) Two-parameter second order problems in singular perturbations Professor R. E. O'Malley, Jr., University of North Carolina {632-51) 10:45-10:55 { 11) Bounded solutions for the equation of the linear adiabatic oscillator Dr. R. B. Kelman and Mr. N. K. Madsen*, University of Maryland (632-47) 11:00-11:10 ( 12) Uniqueness theorem for the reduced wave equation under an nth order differ­ ential boundary condition Professor R. C. Morgan, St. John's University (632-59) (Introduced by Professor L. V. Quintas) 11:15-11:25 ( 13) A theorem on random Fourier series on noncommutative groups Dr. Alessandro Figa-Talamanca* and Dr. Daniel Rider, Massachusetts Institute of Technology (632-66) 11:30-11:40 (14) A front-ended proof of Lebesgue's theorem Professor Stephen Simons, University of California, Santa Barbara ( 632-7) (Introduced by Professor R. C. Thompson)

MONDAY, 10:00 A.M. Session on Logic and Foundations, West Foyer, Third Floor 10:00-10:10 ( 15) Composition of operations in a generalized algebraic system Professor D. E. Muller, University of Illinois {632-70) 10:15-10:25 { 16) A calculus for failure diagnosis Dr. J.P. Roth, IBM Watson Research Center, Yorktow~ Heights, New York (632-61)

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

287 10:30-10:40 ( 17) On incompactness of logics with infinitely long expressions. Preliminary re­ port ProfessorS. G. Mrowka, The Pennsylvania State University (632-56) 10:45-10:55 (18) Recursively enumerable sets not contained in any maximal set Mr. R. W. Robinson, Cornell University ( 632-4) 11:00-11:10 ( 19) Mathematical set theory Professor Hidegoro Nakano, Wayne State University (632-9) 11:15-11:25 (20) Full co-ordinals of recursive equivalence types Dr. A. B. Manaster, Massachusetts Institute of Technology (632-48) 11:30-11:40 (21) Elementary differences between the isols and the co-simple isols Professor Louise Hay, Mount Holyoke College (632-6) 11:45-11:55 (22) Neo-Newtonian dynamics. Preliminary report Professor A. C. Sugar, Northern Michigan University (632-15)

MONDAY, 2:00 P.M. Invited Address, Sert Room, Main Floor Modular representations of finite groups Professor Walter Feit, Yale University

MONDAY, 3:15P.M.

Second Session on Analysis, Jansen Salon, Fourth Floor 3:15-3:25 (23) Some inequalities for the Ahlfors-Shimizu characteristic and the area on the Riemann sphere Mr. Hari Shankar, Ohio University (632-68) 3:30-3:40 (24) Boundary behavior of conformal maps from simply connected domains onto half-planes Professor W. J. Schneider, Syracuse University (632-54) 3:45-3:55 (25) Analytic continuation of Laplace transforms by means of asymptotic series Dr. W. A. Beyer* and Mr. Leon Heller, Los Alamos Scientific Laboratory, University of California, Los Alamos (632-50) 4:00-4:10 (26) Interpolation with remainder of real analytic functions in several variables Professor J. M. Sloss, University of California, Santa Barbara (632-3) 4:15-4:25 (27) On division of tempered distributions by polynomials Professor Leon Nower, San Diego State College (632-26) 4:30-4:40 (28) A time domain characterization of linear scattering systems Dr. E. J. Beltrami, Grumman Aircraft Engineering Company, Bethpage, New York (632-17)

288 MONDAY, 3:15P.M. Third Session on Analysis, West Foyer, Third Floor 3:15-3:Z5 (Z9) Schauder decompositions in (m) Professor D. W. Dean, The Ohio State University (63Z-44) 3:30-3:40 (30) Quasi-reflexivity and weakly unconditional convergent series Professor J. R. Retherford, Louisiana State University, Baton Rouge (63Z-36) 3:4~-3:55 (31) A representation of bounded linear operators on Orlicz spaces. Preliminary report Mr. J. J. Uhl, Carnegie Institute of Technology (63Z-35) 4:00-4:10 (3Z) A necessary and sufficient condition for two volume spaces to generate the same space of Lebesgue-Bochner summable functions Professor W. M. Bogdanowicz, Catholic University of America (63Z-ll) 4:15-4:Z5 (33) An integral over complex Hilbert spaces. Preliminary report Mr. N. F. Rehner, University of Rochester (63Z-57) 4:30-4:40 (34.) A counterexample in the new maximum-minimum theory of eigenvalues Mr. W. G. Stenger, University of Maryland (63Z-1Z)

MONDAY, 3:15P.M. First Session on Algebra, Louis XVI Suite, Fourth Floor 3:15-3:Z5 (35) A class of semigroups and its finite quotients Professor J.D. McKnight and Mr. A. J. Storey*, University of Miami, Coral Gables (63Z-67) 3:30-3:40 (36) Ideals and automorphism of metabelian groups Professor Seymour Bachmuth and Professor H. Y. Mochizuki*, University of California, Santa Barbara (63Z-18) 3:45-3:55 (37) A note on solvable K-groups Professor H. F. Bechtell, Bucknell University (63Z-Z5) 4:00-4:10 ( 38) Subrings of finite index in finitely generated rings Professqr Jacques Lewin, Syracuse University (63Z-31) 4:15-4:Z5 (39) Weak radical of a ring Dr. Kwangil Koh, North Carolina State University (63Z-49) 4:30-4:40 (40) Prufer rings Professor W. W. Smith*, University of North Carolina and Professor H.S. Butts, Louisiana State University, Baton Rouge (63Z-34) 4:45-4:55 (41) On the spectrum of classes of algebras. Preliminary report Professor G. A. Gratzer, Pennsylvania State University (63Z-ZZ)

MONDAY, 3:15P.M. Session on Topology, Jansen Blue Room, Fourth Floor 3:15-3:Z5 (4Z) Homeotopy groups of fibered knots and links Dr. C. H. Giffen, The Institute for Advanced Study ( 63Z-1 O) 289 3:30-3:40 (43) Induced fibrations and cofibrations Professor Theodor Ganea, University of Washington (632-13) 3:45-3:55 (44) Semigroup structures for families of functions. II. Continuous functions Professor K. D. Magill, Jr., SUNY at Buffalo (632-19) 4:00-4:10 (45) Normal monotone mappings Mr. A. C. Connor, University of Georgia (632-55) 4:15-4:25 (46) Compact and product bases Mr.R.H.Prosl* and Professor Paul Slepian, Rensselaer Polytechnic In­ stitute (632-42) 4:30-4:40 (47) Function systems Miss Kazumi Nakano, Wayne State University (632-37) 4:45-4:55 (48) Cyclic and fine-cyclic elements Professor C. J. Houghton, SUNY at Binghamton (632-16)

TUESDAY, 9:00 A.M. Second Session on Algebra, Louis XVI Suite, Fourth Floor 9:00-9:10 (49) A characterization of v*-algebras Mr. G. H. Wenzel, Pennsylvania State University (632-23) 9:15-9:25 (50) Boolean extensions and normal subdirect powers of finite universal algebras Mr. M. I. Gould* and Professor G. A. Gratzer, Pennsylvania State Uni­ versity (632-27) 9:30-9:40 (51) Topological post coset theorem. Preliminary report Mr. R. L. Richardson, University of Florida (632-58) (Introduced by Professor A. D. Wallace) 9:45-9:55 (52) On the generalized (double) triple-systems with some applications Dr. Volodymyr Bohun-Chudyniv, Seton Hall University (632-39) 10:00-10:10 (53) A generalization of the van der Waerden conjecture Dr. D. W. Sasser and Dr. M. L. Slater*, Sandia Corporation, Albuquerque, New Mexico (632-24) 10:15-10:25 (54) Elementary divisor theorem for direct sum and tensor products of modules Professor Yeh-er Kuo, The University of Tennessee (632-30) 10:30-10:40 (55) Eigenvalue problems of a 2n X 2n matrix Professor Srisakdi Charmonman* and Mr. H. R. Baste!, McMaster Uni­ versity (632-38)

TUESDAY, 9:00 A.M. Fourth Session on Analysis, Jansen Salon, Fourth Floor 9:00-9:10 (56) On sigma-finite invariant measures. Preliminary report Mr. L. K. Arnold, Brown University (632-8)

290 9:15•9:25 (57) Representation of stationary spectral measures Dr. M. G. Nadkarni, Washington University (632-14) 9:30-9:40 (58) Some inbetween theorems for Darboux functions Professor J. G. Ceder* and Professor M. L. Weiss, University of Cali­ fornia, Santa Barbara (632-1) 9:45-9:55 {59) The E-entropy and E-capacity of a compact operator on a Hilbert space Dr. R. T. Prosser, Lincoln Laboratory, Massachusetts Institute of Tech­ nology (632-65) 10:00-10:10 (60) On localization and domains of uniqueness Dr. Roe Goodman, Massachusetts Institute of Technology (632-60) 10:15-10:25 (61) Summation of bounded divergent sequences and the second adjoint matrix Professor J.P. Crawford, Lafayette College (632-46)

TUESDAY, 9:00A.M. Session on Applied Mathematics and Probability, West Foyer, Third Floor 9:00-9:10 ( 62) Some results on transport theory and their application to Monte Carlo methods Dr. Jerome Spanier, Westinghouse Electric Corporation, Bettis Atomic Power Laboratory, West Mifflin, Pennsylvania (632-20) 9:15-9:25 (63) Ergodic properties of expectation matrices of a Markov branching process with countably many types Professor Shu-Teh Moy, Syracuse University (632-33) 9:30-9:40 {64) The linear search problem Professor Anatole Beck, University of Wisconsin (632-53) 9:45-9:55 ( 65) Invariant probability measures for temporally homogeneous Markov processes. Preliminary report Professor H. M. Schaer£, McGill University (632-40) 10:00-10:10 (66) Descent on the boundary of a sphere Professor A. A. Goldstein, University of Washington (632-69) 10:15-10:25 (67) On the Newton-Raphson method for systems of equations Professor Adi Ben-Israel, University of Illinois at Chicago Circle ( 632- 62) 10:30-10:40 (68) Two integration formulas of modified Runge-Kutta type Mr. C. R. Cassity, Roland F. Beers, Incorporated, Alexandria, Virginia (632-45)

TUESDAY, 9:00 A.M. General Session, followed by Late Papers, Jansen Blue Room, Fourth Floor 9:00-9:10 ( 69) An inclusion theorem for minimal surfaces Professor J.C.C. Nitsche, University of Minnesota (632-2) 9:15-9:25 (70) Conditions which imply that a 2-sphere in s3 is locally tame except at two points Professor C. E. Burgess, University of Utah (632-71)

291 TUESDAY, 11:00 A.M. Invited Address, Sert Room, Main Floor z-functions of simple algebras Professor Tsuneo Tamagawa, Yale University

SYMPOSIUM ON MATHEMATICAL ASPECTS OF COMPUTER SCIENCE

TUESDAY, 1:30 P.M. Session on Calculations with Symbolic and Algebraic Data, Sert Room, Fourth Floor A review of automatic theorem proving (one hour) Professor J. A. Robinson, Rice University Machine computation of a spectral sequence (one-half hour) Professor Mark Mahowald, Northwestern University Some illustrations of mathematical applications of on-line computation {one-half hour) Dr. Glenn Culler, University of California (Santa Barbara)

WEDNESDAY, 9:00A.M. Session on Numerical Methods, Sert Room, Fourth Floor The use of computers in the theory of numbers (one hour) Professor H. P. F. Swinnerton-Dyer, Department of Mathematics, Trinity College, Cambridge University The calculation of zeros of polynomial and analytic functions (one hour) Dr. J. F. Traub, Bell Telephone Laboratories and Stanford University Numerical hydrodynamics of the atmosphere (one hour) Professor C. E. Leith, Lawrence Radiation Laboratories

WEDNESDAY, 1:30 P.M. Session on Software Systems, Mechanical Linguistics, Sert Room, Fourth Floor Assigning meanings to programs (one hour) Professor R. W. Floyd, Carnegie Institute of Technology Computer analysis of national languages (one hour) Professor Susumu Kuno, Harvard University

THURSDAY, 9:00A.M. Session on Theory of Automata; Artificial Intelligence, Sert Room, Fourth Floor A mathematical theory of geometric pattern recognition (one hour) Professor Marvin Minsky and Dr. Seymour Papert, Massachusetts Insti­ tute of Technology Automata, computational complexity and decision problems (one-half hour) Professor Juris Hartmanis, Cornell University Mathematical theory of automata (one-half hour) Professor Michael Rabin, Hebrew University Proving the correctness of computers (one hour) Professor John McCarthy, Stanford University Everett Pitcher Bethlehem, Pennsylvania Associate Secretary Six Hundred Thirty-Third Meeting University of Hawaii Honolulu, Hawaii April9, 1966

PROGRAM

The six hundred thirty-third meet­ from members upon registration. ing of the American Mathematical Society Coffee will be served in the regis­ will be held on Saturday, April 9, 1966 at tration area at coffee breaks in the morning the University of Hawaii in Honolulu, Hawaii. and afternoon. Luncheon will be available By invitation of the Committee to in the cafeteria in jefferson Hall at the Select Hour Speakers for Far Western East-West Center near Kuykendall Hall. Sectional Meetings, there will be hour ad­ The University of Hawaii is located dresses by Professor Branko Grunbaum of about three miles from downtown Honolulu, Michigan State University and the Hebrew and about two miles from Waikiki. The University, Jerusalem, and by Professor campus can be reached by bus, but it is Jakob Korevaar of the University of Cali­ advisable to use taxi service, since the fornia, San Diego. Professor Grunbaum will bus trip requires two transfers. The cost speak at 11:00 A.M. on "Polytopes, graphs, of a taxi trip from Waikiki is about $1.7 5, and complexes." Professor Korevaar's with reduced rates if two or more persor..s lecture on "Distributions" will begin at share a cab. 1:30 P.M. Both of these addresses will be Honolulu is served by several air­ given in the. Kuykendall Hall Auditorium. lines, including Northwest, Pan American, There will be sessions for contrib­ and United Airlines. The round trip thrift uted papers at 9:30 A.M. and at 3:00 P.M. class fare from Los Angeles and San Fran­ in Kuykendall Hall. If necessary, there will cisco is approximately $200. From Portland be a session for late papers. Information and Seattle the fare is about $220. Group concerning late papers will be available at travel from the West Coast to this meeting the Registration Desk. will not be practical. However, one free The Registration Desk will be lo­ ticket is given with block reservations of cated on the Lanai of Kuykendall Hall, and sixteen or more. Limousine service is it will be open from 8:30 A.M. Since this available between the airport and hotels meeting is being held at the Conference in downtown Honolulu and Waikiki. Center of the University, the Council of the There are numerous hotels in Hono­ American Mathematical Society has autho­ lulu. All travel agents are happy to provide rized a registration fee to cover costs of the complete information on available accom­ meeting. A fee of $3.00 will be collected modations and rates.

PROGRAM OF THE SESSIONS The time limit for each contributed paper is ten minutes. To maintain the schedule, the time limit will be strictly enforced. SATURDAY, 9:30A.M. General Session, Room 203, Kuykendall Hall 9:30-9:40 { 1) A unified theory of types Professor M. D. Resnik, University of Hawaii (633-10) (Introduced by Professor Kenneth Rogers)

293 9:45-9:55 (2) Dominance semigroups l!lf the modular group Professor R. J. Wisner* and Mr. D. W. Hardy, New Mexico State Univer­ sity (633-31) 10:00-10:10 (3) The quadratic reciprocity law via the geometry of numbers Professor Kenneth Rogers, University of Hawaii (633-22) 10:15-10:25 (4) A refinement of Selberg's asymptotic equation Mr. Veikko Nevanlinna, University of Hawaii {633-4) (Introduced by Professor Kenneth Rogers) 10:30-10:40 ( 5) Asymptotic diophantine approximations to e Mr. W. W. Adams, University ·of California, Berkeley (633-9)

SATURDAY, 9:30A.M.

Session on Probability and Statistics, Room 204, Kuykendall Hall 9:30-9:40 (6) Monoidal families of discrete distributions Dr. R. B. Leipnik* and Mr. F. C. Reed, U.S. Naval Ordnance Test Station, China Lake, California (633-29) 9:45-9:55 ( 7) A stochastic process based on an alternating renewal process Dr. D. S. Newman, Boeing Airplane Company, Seattle, Washington (633-15) 10:00-10:10 (8) Some aspects of uniform recurrence Professor J. G. Baldwin, University of Houston (633-3) 10:15-10:25 (9) Existence of invariant measures for Markov operators Professor J. R. Brown, Oregon State University (633-Z5) 10:30-10:40 ( 1 O) Epsilon entropy of the Wiener process Dr. E. C. Posner* and Dr. Howard Rumsey, Jr., Jet Propulsion Labora­ tory, California Institute of Technology (633-5)

SATURDAY, 9:30A.M. Session on Analysis, Room 205, Kuykendall Hall 9:30-9:40 ( 11) A note on linear topological spaces Professor D. H. Hyers, University of Southern California (633-16) 9:45-9:55 (1Z) Invariant means and uniformly continuous retractions. Preliminary report Professor Aleksander Pe,lczynski, University of Washington (633-28) 10:00-10:10 ( 13) An integral formula for abstract harmonic or parabolic functions Professor H. S. Bear*, University of California, Santa Barbara, and Pro­ fessor A.M. Gleason, Harvard University (633-1) 10:15-10:25 ( 14) Spectral inner functions Professor M. J. Sherman, University of California, Los Angeles {633-7)

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

294 SATURDAY, 11:00 A.M. Invited Address, Kuykendall Hall Auditorium Polytopes, graphs, and complexes Professor Branko Grunbaum, Michigan State University and the Hebrew University, Jerusalem

SATURDAY, 1:30 P.M. Invited Address, Kuykendall Hall Auditorium Distributions Professor Jakob Korevaar, University of California, San Diego

SATURDAY, 3:00P.M. Session on Algebra, Room 203, Kuykendall Hall 3:00-3:10 (15) Primality-in-the-small and simplicity Professor R. A. Knoebel, New Mexico State University (633-6) 3:15-3:25 ( 16) Abelian groups which are determined by their group of mappings Professor R. A. Beaumont, University of Washington (633-12) 3:30-3:40 (17) On overrings of Prufer domains. II. Preliminary report Professor R. W. Gilmer, Jr., Florida State University (633-8) 3:45-3:55 (18) Nearly regular p-groups Professor C. R. Hobby, University of Washington (633-14) 4:00-4:10 { 19) Another proof of a theorem on rational cross sections Professor Maxwell Rosenlicht, University of California, Berkeley (633-11) 4:15-4:25 (20) Abelian curves of small conductor Professor A. P. Ogg, University of California, Berkeley {633-18)

SATURDAY, 3:00P.M.

Session on Analysis and Applied Mathematics, Room 204, Kuykendall Hall 3:00-3:10 (21) Weights of cyclic codes Dr. Gustave Solomon* and Mr. R. J. McEliece, Jet Propulsion Laboratory, California Institute of Technology ( 633-13) 3:15-3:25 (22) A self contained trajectory model Dr. Mark Lotkin, General Electric Company, Philadelphia, Pennsylvania (633-2) 3:30-3:40 (23) Confluent expansions Mr. J. L. Fields, Midwest Research Institute, Kansas City, Missouri (63_3-32) 3:45-3:55 (24) Applications of superpositions for nonlinear operators Mr. Alfred Inselberg, University of Illinois ( 633-30) 4:00-4:10 {25) A condition of halo type for the differentiation of certain classes of set func­ tions Professor C. A. Hayes, University of California, Davis (633-17)

295 4:15-4:25 (26) The chain rule in the transformation theory for measure space Professor Robin Chaney, Western Washington State College (633-21)

SATURDAY, 3:00P.M. Session on Geometry and Topology, Room 205, Kuykendall Hall 3:00-3:10 (27) Equivariant cohomology theory Professor G. E. Bredon, University of California, Berkeley (633-23) 3:15-3:25 (28) On the homogeneity of infinite products of manifolds Professor Theodor Ganea, University of Washington (633-26) 3:30-3:40 (29) E-Mappings and generalized manifolds Professor Sibe Marde~ic*, University of Zagreb, Yugoslavia, and Univer­ sity of Washington, and Professor Jack Segal, University of Washington (633-27) 3:45-3:55 (30) Proximate absolute extensors Professor A. L. Yandl, Western Washington State College (633-19) 4:00-4:10 ( 31) On a theorem of Hanner and Dowker Professor B. H. McCandless, Western Washington State College (633-20) 4:15-4:25 ( 32) Inflection hyperplanes of polygons Professor Douglas Derry, University of British Columbia (633-24) R. S. Pierce Seattle, Washington Associate Secretary

296 Six Hundred Thirty-Fourth Meeting University of Chicago Center for Continuing Education Chicago, Illinois April20-23, 1966

PROGRAM

The six hundred thirty-fourth meet­ Sessions for the presentation of con­ ing of the American Mathematical Society tributed papers will be held at 3:15 P.M. will be held at the University of Chicago on Friday and at 9:00 A.M. and 3:15P.M. on April 20-23, 1966. Registration and all on Saturday. sessions will be held at the University of Chicago Conference Center, officially en­ SYMPOSIUM ON SINGULAR titled "The Center for Continuing Educa­ INTEGRALS tion," and located at 1307 East 60th Street about one-half mile to the southeast of There will be a Symposium on Singu­ Eckhart Hall. No papers will be presented lar Integrals on the afternoon of Wednesday, at Eckhart Hall. April 20, on Thursday, April 21, and on the Rooms will be available at the Cen­ morning of Friday, April 22. The subject ter at the rate of $10 per single room and of the Symposium was chosen by the Com­ $7 per person in a twin-bedded double mittee to Select Hour Speakers for West­ room. In the event of an overflow, the ern Sectional Meetings. The Committee Center will undertake to place people at consisted of Seymour Sherman (Chairman). nearby hotels. The hotel in question will, Felix Browder, and Irving Reiner. Finan­ in that case, confirm the reservation. All cial support comes from the National Sci­ meals will be served at the Center and ence Foundation and the University of there is a bar which opens daily a 11:30 Chicago. A.M. The official at the Center in charge of The Invitations Committee respon­ the meeting is Mr. Barney M. Berlin. In­ sible for the planning of the program and quiries pertaining to the Center may be the choice of speakers consists of: Alberto directed to him. P. Calder6n (Chairman), University of By invitation of the Committee to Chicago; K. 0. Friedrichs, Courant Insti­ Select Hour Speakers for Western Sectional tute of Mathematical Sciences; Robert T. Meetings, Professor Glen Baxter ofPurdue Seeley, Brandeis University; and Antoni University, Professor Hans Grauert of Zygmund, University of Chicago. Gottingen and Notre Dame, and Professor R. G. Swan of the University of Chicago will present hour addresses. Professor Baxter COUNCIL MEETING will speak on "Some aspects of the Ising model;" Professor Grauert will speak on The Council will meet at 1:00 P.M. "N onarchimedean analysis," and Professor on Saturday, April 2 3, in Room 1-B of the Swan will speak on "Modules over finite Center. groups." Professor Baxter will speak at 2:00 P.M. on Friday, Professor Grauert will speak at 11:00 A.M. on Saturday, and REGISTRATION Professor Swan will speak at 2:00 P.M. on Saturday. All lectures will be in the Ass em­ There will be a registration fee of bly of the Conference Center. $1.50 (no charge for students).

297 ENTERTAINMENT

There will be a no-host cocktail P.M. to 7:00 P.M. Snacks will be furnished party on the second floor lobby of the Cen­ by the University of Chicago Mathematics ter on Friday, April 22, 1966, from 5:00 Department.

SYMPOSIUM ON SINGULAR INTEGRALS

WEDNESDAY, 2:00P.M.

The Assembly 2:00 P.M. Singular integrals, harmonic functions, and differentiability properties of functions of several variables. Professor Elias M. Stein, Princeton University 3:15P.M. A sharp estimate for pseudo-differential and difference operators Professor Louis Nirenberg, Courant Institute 3:45P.M. On the existence of singular integrals Professor Antoni Zygmund, University of Chicago

THURSDAY, 10:00 A.M.

The Assembly 10:00 A.M. L2 estimates for generalized pseudo-differential operators Professor Lars Hormander, Institute for Advanced Study 11:15 A.M. Remarks about L2 -theory of singular integral operators Professor H. O. Cordes, University of California at Berkeley 11:45 A.M. On symmetrizable differential operators Professor K. 0. Friedrichs and Professor Peter D. Lax, Courant Institute

THURSDAY, 2:00 P.M.

The Assembly 2:00P.M. Singular integrals on manifolds Professor Robert T. Seeley, Brandeis University 3:15P.M. Singular integral operators on orbit spaces Professor Isadore M. Singer, Massachusetts Institute of Technology 3:45P.M. Symbolic calculus for singular integral operators and applications Professor Alberto P. Calder6n, University of Chicago FRIDAY, 10:00 A.M.

The Assembly 10:00 A,M. Singular integrals and the boundary value problems for the heat equation Professor B. F. Jones, Jr., Rice University and Institute for Advanced Study

298 11:15 A.M. Systems of parabolic differential equations with uniformly continuous co­ efficients Professor Eugene B. Fabes, Rice University 11:45A.M. , . .,_ Une classes d integrales smgu1 ll::res Professor Paul Kree, Nice, France

PROGRAM OF THE SESSIONS

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

FRIDAY, 2:00 P.M.

Invited Address, The Assembly Some aspects of the Ising model Professor Glen Baxter, Purdue University

FRIDAY, 2:00P.M.

Session on Analysis I. The Assembly 3:15-3:25 ( 1) On the existence of non tangential limits of subharmonic functions Professor M.G. Arsove*, UniversityofWashington andProfessor Alfred Huber, Polytechnic Institute, Zurich {634-30) 3:30-3:40 (2) Elliptic regularization for symmetric positive systems. II Professor Leonard Sarason, University of Washington {634-42) 3:45-3:55 (3) An elementary proof and application of an extension of Aufgabe 160 Professor W. J. Schneider, Syracuse University ( 634-4 7) 4:00- 4:10 (4) Remarks on some convergence conditions for continued fractions Professor D. F. Dawson, North Texas State University (634-29) 4:15-4:25 ( 5) An expansion of the real-number system Professor L. P. Maher, Jr., Indiana University {634-34) 4:30-4:40 ( 6) Sums of powers of complex numbers Professor J.D. Buckholtz, University of Kentucky (634-45) 4:45-4:55 ( 7) Recurrence of set trajectories in a discrete dynamical system Professor B. K. Wong, Western Illinois University (634-11)

FRIDAY, 3:15P.M.

Session on Analysis II, Room 2-BC 3:15-3:25 (8) Topological groups and integral norms. Preliminary report Professor R. C. Hooper, Wichita State University ( 634-1)

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

299 3:30-3:40 (9) Analytic measures Professor Frank Forelli, University of Wisconsin (634-19) 3:45-3:55 (10) Representations of topologically simple Banach algebras Professor Jesus Gil de Lamadrid, University of Minnesota (634-24) 4:00-4:10 (1) Prime z-ideals in C(R). II Mr. M. W. Mandelker, University of Rochester (634-33) 4:15-4:25 ( 12) A Lipschitz characterization of analytic functions Professor K. 0. Leland, University of Virginia (634-37) 4:30-4:40 (13) Invariant subspaces Professor J. H. Wells and Professor C. N. Kellogg*, University of Ken­ tucky ( 634-41) 4:45-4:55 (14) Order convolution on L2 (a,b) Professor L. J. Lardy, Syracuse University (634-51) 5:00-5:10 ( 15) A representation for continuous linear operators on C Professor J. T. Darwin, Jr., Auburn University ( 634- 55)

FRIDAY, 3:15P.M. Session on Topology, Room 2-EF 3:15-3:25 ( 16) On extending homeomorphisms on the Hilbert cube Professor R. D. Anderson, Louisiana State University (634-56) 3:30-3:40 (17) On the interchangeability of 2-links, II Professor W. C. Whitten, Jr., University of Southwestern Louisiana, Lafayette (634-39) 3:45-3:55 (18) Some results on H-maps. Preliminary report Professor M.A. Arkowitz, Dartmouth College and Professor C. R. Curjel*, University of Washington (634-28) 4:00-4:10 ( 19) A note on Seifert circles Mr. R. E. Goodrick, University of Wisconsin (634-22) 4:15-4:25 (20) Some topological invariants of Seifert fiber spaces. Preliminary report Mr. P. P. Orlik, The University of Michigan ( 634-4) 4:30-4:40 ( 21) Topological significance of a certain class of cardinals. Preliminary report ProfessorS. G. Mrowka, Pennsylvania State University (634-49) 4:45-4:55 (22) On the axioms of topological structures Professor Heinrich Matzinger, University of Washington (634-38) (Introduced by Professor C. R. Curjel) 5:00-5:10 (23) Inverse star envelope of a subset of a L TS Professor J. E. Allen, North Texas State University (634-50)

300 SATURDAY, 10:00 A.M. Session on Analysis I, The Assembly 10:00-10:10 (24) Differentiation almost everywhere of a function of several variables Mr. G. V. Weiland, Purdue University (634-52) 10:15-10:25 (25) A generalization of the Wielandt inequality Professor T. L. Boullion*, University of Southwestern Louisiana and Mr. P. L. Odell, University of Texas (634-36) 10:30-10:40 (26) A singular eigenfunction expansion in anisotropic neutron transport theory Mr. D. H. Sattinger, University of California, Los Angeles (634-18) 10:45-10:55 (27) On using the Ilstow integral to solve a certain partial differential equation Mr. R. A. Kallman, University of Minnesota, Duluth (634-46)

SATURDAY, 10:00 A.M.

Session on Analysis II, Room 2-BC. 10:00-10:10 (28) A Lane integral with no summability set which is everywhere dense Professor F. M. Wright and Mr. K. P. Smith*, Iowa State University (634-8) 10:15-10:25 (29) On the existence of Stieltjes integrals Mr. C. B. Murray, TRACOR, Incorporated, Austin, Texas (634-10) 10:30-10:40 (30) Denjoy integration in abstract spaces. III Mr. D. W. Solomon, Wayne State University (634-27) 10:45-10:55 (31) A Baire type characterization of Lebesgue-Bochner measurable functions Professor Witold Bogdanowicz, The Catholic University of America (634-21)

SATURDAY, 10:00 A.M. Session on Topology, Room 2-EF 10:00-10:10 (32) Locally-flat and locally-tame embeddings Mr. Jerome Dancis, University of Wisconsin, and University of Maryland ( 634-44) 10:15-10:25 (33) Locally cyclic continua. I Mr. A. C. Connor, University of Georgia (634-48) 10:30-10:40 (34) Cellular decompositions of 3-manifolds that yield 3-manifolds Professor Steve Armentrout, University of Iowa (634-53) 10:45-10:55 (35) Complete amonotonic decompositions of continua Professor H. L. Baker, Jr., University of Massachusetts (634-54)

SATURDAY, 11:00 A.M. Invited Address, The Assembly Nonarchimedian analysis Professor Hans Gruert, Gottingen and Notre Dame University

301 SATURDAY, 2:00 P.M. Invited Address, The Assembly Modules over finite groups Professor R, G. Swan, University of Chicago SATURDAY, 3:15P.M. Session on Analysis I, The Assembly 3:15-3:25 (36) On the summability of the differentiated Fourier series Professor Daniel Waterman, Wayne State University (634-16) 3:30-3:40 (37) Universal continuous functions ProfessorS, W. Young, University of Utah (634-35) 3:45-3:55 (38) On a fixed point theorem for nonlinear P-compact operators in Banach space Professor W. V. Petryshyn, The University of Chicago (634-31) 4:00-4:10 (39) An elementary proof of the uniqueness theorem in the theory of best L 1 -approximation Professor Ranko Bojanic, Ohio State University (634-5) 4:15-4:25 (40) Pointwise limits of quasi-continuous functions Mr. C. T. Tucker, University of Texas and TRACOR, Incorporated(634-14) (Introduced by S. H. Gould) 4:30-4:40 (41) Pointwise limits of sequences of quasi-continuous functions Professor C. S. Reed, The University of Texas (634-7)

SATURDAY, 3:15P.M. General Session, Room 2-BC 3:15-3:25 (42) Cohomology of coalgebras. Preliminary report Professor D. W. Jonah, Wayne State University (634-3) 3:30-3:40 {43) Nonnegative matrices diagonally equivalent to row stochastic matrices Professor R, A. Brualdi*, ProfessorS. V. Parter and Professor Hans Schneider, University of Wisconsin {634-12) 3:45-3:55 (44) A-transforms for Noether lattices. Preliminary report Mr. E. W. Johnson, University of California, Riverside (634-13) 4:00-4:10 (45) Cyclic extensions without relative integral bases Dr, L. R. McCulloh, University of Illinois {634-17) 4:15-4:25 (46) The Lie theory of monic induced group representations Mr. Willard Miller, Jr., University of Minnesota (634-32) 4:30-4:40 ( 47) Closed uniform subgroups of solvable Lie groups Dr. Clifford Perry, University of Minnesota (634-23) (Introduced by Dr. Leon Green) 4:45-4:55 (48) An extremal problem associated with a certain nonlinear integral equation Professor R. H. Rolwing*, University of Cincinnati and Professor I. A. Barnett, Fairleigh Dickinson University (634-2)

302 5:00-5:10 (49) Application of fixed point principles to solutions of P(x) = 0 Mr. J. E. Dennis, University of Utah (634-2.5) (Introduced by Dr. R. E. Barnhill)

SATURDAY, 3:15P.M. Session on Topology, Room 2.-EF 3:15-3:2.5 (50) Reversibly continuous bisensed transformations of an annulus into itself Mr. M.D. Seeker, University of Texas (634-6) 3~30-3:40 (51) Concerning the sides from which certain sequences of arcs converge to a compact irreducible continuum Mr. R. D. Davis, University of Texas (634-9) (Introduced by Professor R. L. Moore) 3:45-3:55 (52.~ A space whose regions are the. simple domains of another space Mr. J. W. Rogers, Jr., University of Texas (634-15) (Intr"oduced by Professor R. L. Moore) 4:00-4:10 (53) Concerning continuous collections of mutually exclusive continua Professor E. L. Bethel, Clemson University (634-2.0) 4:15-4:2.5 (54) Polyhedral approximation of spheres in E Mr. F. M. Lister, University of Utah ( 634-2.6) (Introduced by Professor C. E. Burgess) 4:30-4:40 (55) Isotopies of 2.-spheres in 3-manifolds Mr. R. F. Craggs, University of Wisconsin (634-40) 4:45-4:55 (56) Solution of a problem of R. D. Anderson Professor Howard Cook, University of Georgia (634-43) Seymour Sherman Bloomington, Indiana Associate Secretary

303 PRELIMINARY ANNOUNCEMENTS OF MEETINGS

Seventy-First Summer Meeting Rutgers, The State University New Brunswick, New Jersey August 30-September 2, 1966

The American Mathematical Society Mathematical Association of America, The will hold its seventh-first summer meet­ Institute of Mathematical Statistics, the ing at New Brunswick, New jersey from Society for Industrial and Applied Mathe­ Tuesday through Friday, August 30--Sep­ matics,'Pi Mu Epsilon, andMuAlpha Theta. tember 2, 1966. The Society for Industrial and Applied Math­ All sessions will be held in lecture ematics will present Professor Eugene P. rooms and classrooms of Rutgers, The Wigner of Princeton University as the john State University, on the College Avenue von Neumann Lecturer. The Mathematical Campus in New Brunswick. Association of America will present Pro­ There will be more than the usual fessor Nathan j. Fine of the Pennsylvania number of invited hour addresses but no State University as the Earle Raymond colloquium, following the established pat­ Hedrick Lecturer. tern for years in which there is an inter­ national congress. There will be sessions for contrib­ COUNCIL AND BUSINESS uted papers on Tuesday afternoon, August MEETINGS 30,, on Wednesday morning, August 31, and on Thursday and Friday, September 1 and2, The Council of the Society will meet both morning and afternoon. The number of at 5:00 P.M. on Thursday, September 1. contributed papers which can be accepted The Business Meeting of the Society is limited by action of the Council. The will be held on Friday, September 2 at limit for the Rutgers meeting is 180. The 11:00 A.M. number of papers contributed to a summer meeting has never exceeded the limit. However, if necessary, suitable contributed papers will be accepted for the program in ADVANCE REGISTRATION order of their receipt, with random choice among the final lot of papers simultaneously The advance registration procedure received. In any event, the deadline for the will be used. On Page 414 of this issue of receipt of papers to be placed on the pro­ the Notices is a registration form. The gram is july 8, 1966. Abstracts of con­ same form will appear in the june issue of tributed papers should be sent to the Amer­ the Notices, but it will not appear in the ican Mathematical Society, P .0. Box 6248, August issue. The form provides for regis­ Providence, Rhode Island 02904. Abstract tration and payment of registration fees, blanks can be obtained on request from the parking permits, and reservation of dor­ same address. mitory rooms. It allows for the preparation There will be no provision for late in advance of badges and an information papers. packet for. the registrant and his party. Several organizations will cooperate Copies of the advance registration in holding meetings or council meetings on form may also be obtained by writing to the the same campus and at approximately the Society at the address already given for same time as the Society. These include the the submission of abstracts.

304 REGISTRATION tional media will be displayed in Records Hall on Tuesday through Thursday. Records The registration desk will be in The Hall is across George Street from The Ledge on George Street in the River Dor­ Ledge. mitory area of the College Avenue Campus. It will be open on Sunday, August Z8, from BOOK SALE Z:OO P.M. till 8:00 P.M.; on Monday, August Z9, from 8:00 A.M. till 5:00 P.M.; Books published by the Society will on Tuesday through Thursday, August 30 be sold for cash prices somewhat below through September 1, from 9:00 A.M. the usual prices when these same books are through 5:00 P.M.; and on Friday, Septem­ sold by mail on invoice. ber z, from 9:00 A.M. till 1:00 P.M. The registration fees will be as fol- DORMITORY HOUSING lows: Member $3.00 Dormitory rooms will be available. Member's family $0.50 Guests with children will be assigned to the Bishop Campus Halls on George Street, one for the first such registration and no charge block from The Ledge. Others will be housed for additional registrations in the Riverside Halls on George Street. Student no fee These dormitories are provided with eleva­ Others $6.00 tors. They are not air-conditioned. Reservations for dormitory rooms Registration fees for Members and Others should be made in advance, using the form include $1.00 collected for Rutgers. provided. See the section on ADVANCE The preferred procedure is to re­ REGISTRATION. Advance reservations will gister in advance (see section ADVANCE be acknowledged by mail and a local map REGISTRATION). One then completes the will be enclosed. process by picking up a registration packet Although it is possible that rooms at the registration desk. It is desirable to will still be available to persons who have have one's local address already established not registered in advance, this is not when completing registration, as this in­ guaranteed. formation will be recorded at the registra­ Rooms will be available from 10:00 tion desk for the visual index. This will be A. M., Saturday, August Z7, to 10:00 A.M., easy for those occupying dormitory rooms Saturday, September 3. Bed linen, towels, since room registration will be accom­ soap, and room cleaning service are p:t.o­ plished at The Ledge also. vided. The rates are $5.00 per day, single, and $3.50 per day, per person; double. EMPLOYMENTREGmTER Extra beds will be furnished for older children (not adults) in double rooms at a The Mathematical Sciences Employ­ charge of $Z.OD per day. Extra beds will be ment Register will be in Frelinghuysen available by advance reservation only. Hall, Leval A, on George Street east of There are no special provisions or The Ledge. It will be open Tuesday through furniture available for babies or young Thursday, August 30 through September 1, children. Persons who wish to rent .cribs from 9:00 A.M. to 5:00 P.M. on each of the for babies or small children should com­ three days. Attention is invited to the municate directly with Bud Rentals, P .0. announcement of the Employment Register Box 595, Stelton Road and Washington Ave­ on page 3Z3 , in particular to the deadline nue, Piscataway, New Jersey, 08854 dates for application to the register and to (Phone: (ZOl) 75Z-8500) or Miller Rental the necessity for prompt registration at Service, Z 18 Sandford Street, New Bruns­ the Employment Register desk by both ap­ wick, New Jersey, 08903, (Phone: (ZOl) plicants and employers. Z47-8888). Cribs rent for $4.00 to $5.00 per week. EXHIBITS Upon arrival on campus all guests should check in at The Ledge on George Book exhibits and exhibits of educa- Street to register for dormitory rooms and

305 to secure keys and other housing informa­ September 3. The service will be cafeteria tion. Student bellhops will be available at style, on a cash basis, and the hours for The Ledge to guide guests to residence meals follow: halls and parking areas. They will accept Breakfast 7:00 A.M. - 9:00A.M. tips; however, it is not necessary to use Luncheon 11:00 A.M. - 1:30 P.M. their services. Dinner 5:00P.M.- 7:00P.M. Guests are urged to arrive on the campus during the normal registration The Rathskeller of the Commons will hours at The Ledge. Those who must arrive be open on Sunday through Friday from at unusual hours should go to The Ledge, 7:00A.M. to 7:30P.M. where a member of the Housing Staff will The Snack Bar in The Ledge will be be on duty 24 hours a day to issue dormi­ open on Saturday, August 27 from 8:30A.M. tory keys for advance reservations and to 11:00 P.M. and, if the service is re­ assign rooms for late arrivals. quired, the same hours will continue through Guests may pay for their rooms when Friday, September 2. On Saturday, Septem­ they arrive on the campus or when they ber 3, it will be open from 8:30 A.M. check out of their rooms. A member of the through 12:00 noon. University Cashier's Office will be on duty A list of nearby restaurants will be at the Registration Headquarters ln The available at the Registration Headquarters Ledge during the hours indicated in the in The Ledge. section titled REGISTRATION and on Satur­ day, September 3, from 9:30 A.M. through 12:00 noon. MOTELS AND HOTELS FOOD SERVICE There are a number of motels and The air-conditioned University Com­ hotels in the area. Some of them are listed mons will be open for breakfast on Sunday, below, with coded information which is ex­ August 28 through breakfast, Saturday, plained at the end of the list.

LIST OF HOTELS AND MOTELS Phone THE ARCH MOTEL {201) 722-3555 U.S. Highway 22, East bound, Somerville 42 Rooms (E) Single $9-$14; Double $10-$16 Extra Per. $2.00. Code: RT-FP-SP-TV-AC 25 minutes drive from The Ledge BRUNSWICK INN {201) 846-1400 Exit 9, N.J. Turnpike and Route 18, East Brunswick 200 Rooms (E) Single $10-$13; Double $14-$18 Code: RT-AC-TV-SP-CL 15-20 minutes drive from The Ledge EDISON MOTOR LODGE (201) 985-6000 U .S.Highway 1 and Wilson Ave. (P .0. Edison) 50 Rooms (E) Single $7-$9; Double $9-$12 Extra Per. $Z. Code: FP-TV-AC 20-25 minutes drive from The Ledge HOLIDAY INN (609) 452-9100 U.S. Highway 1 and Aqueduct Rd., Princeton 104 Rooms (E) Single $10- $12; Double $14-$18 Extra Per. $2. Code: RT-CL-FP-SP-TV-AC 30 minutes drive from The Ledge

306 HOST WAYS MOTEL (201) 257-8700 247 State Highway 18, East Brunswick 60 Rooms {E) Single $9; Double $11 Extra Per. $2. Code: RT-SP-TV-AC 15-20 minutes drive from The Ledge HOWARD JOHNSON'S MOTOR LODGE (201) 249-8000 U.S. Highway 1, New Brunswick 119 Rooms (E) Single $10-$12; Double $14-$20 Extra Per. $3, Code RT-CL-FP-SP-TV-AC 15-20 minutes drive from The Ledge LAKE-WOOD COURT MOTEL (201) 297-1036 U.S. Highway 130, Deans 14 Rooms (E) Single $7,5 0-$8.50 Double $9,50- $1.50 Extra Per. $2 Code: FP-TV-AC 25 minutes drive from The Ledge PARK HOTEL (201) 756-3400 123 W. Seventh St., Plainfield 104 Rooms (E) Single $9.50- $15; Double $12-$20 Extra Per. $3. Code: RT-CL-FP-TV-AC 25-30 minutes drive from The Ledge PRINCETON MOTOR LODGE (609) 452-2100 U.S. Highway 1 and Meadow Rd., Princeton 28 Rooms (E) Single $8- $10; Double $10-$16 Extra Per, $2. Code: FP-SP-TV-AC 30 minutes drive from The Ledge ROGER SMITH MOTOR HOTEL (201) 247-6000 10 Livingston Avenue, New Brunswick 100 Rooms (E) Single $7- $12,50; Double $10-$18,50 Extra Per. $3,Code: RT-CL-FP-TV-AC 5 minutes drive from The Ledge SILVER CREST MOTOR LODGE (201) 297-2100 U.S. Highway 130, North Brunswick 50 Rooms (E) Single $7- $9; Double $8-$12 Extra Per. $2 Code: FP-SP-TV-AC-RT 25 minutes drive from The Ledge CODE: R T - Restaurant SP - Swimming Pool CL - Cocktail Lounge TV - Television FP - Free Parking AC - Air Conditioned E - European Plan

CAMPING Tuesday evening - Chicken Fry followed There are no suitable camping sites by a Square Dance located near the Rutgers New Brunswick Wednesday afternoon - Presidential Tea campuses. Wednesday evening, following the von Neumann Lecture - IMS Mixer ENTERTAINMENT Thursday - nonmathematicians Beach The tentative plan calls for the fol­ Excursion lowing events: There will be free nonmathematical mov­ ies each evening, Monday through Fri­ Monday evening - SIAM Beer Party day.

307 The 18-hole Rutgers Golf Course, Air taxi service is available from located on the University Heights Campus, Kennedy Airport directly to New Brunswick. will be available. The greens fees are The air taxis leave Kennedy Airport every daily $3.00, and Saturday and Sunday, $5.00. hour on the hour from 8:00 A.M. to 11:00 Tennis courts are available without P.M. Arrangements can be made in ad­ charge on the first come, first served basis. vance for this service through American The Rutgers Swimming Pool in the Airlines. Gymnasium on College A venue will be available to the conferees and their fam­ PARKING ilies, providing planned repairs are com­ pleted prior to this Conference. Swimmers The campus Parking Security and must bring their own suits and towels. Transportation Office will issue, gratis, Lifeguards will be provided. There will be parking stickers which will permit the use no charge for this service. of campus facilities. The stickers will be johnson Park offers excellentfacili­ made available to those who indicate on ties for picnics: tables, benches, fire­ the advance registration form that they places, drinking water, sanitary and re­ will drive to the meeting. creation facilities, and a small wildlife exhibit. It is located on the opposite side BOOKSTORE of the Raritan River from the George Street Dormitories. The University Bookstore (air-con­ ditioned) is adjacent to the University Com­ TRAVEL mons and will be open Monday through Friday from 9:00 A.M. to 5:00 P.M. In ad­ Labor Day weekend follows this dition to textbooks, it sells University meeting. Persons who need return travel souvenirs and articles for personal use. reservations are advised to make them in ~dvance even though there will be a travel LIBRARY ~esk at The Ledge. New Brunswick, located in central The University Library, air-condi­ New jersey, is on the main line of the tioned, located on College Avenue, will be Pennsylvania Railroad, 30 miles southwest open on Monday through Friday from 9:00 of New York City, and 60 miles northeast A.M. to 5:00 P.M. of Philadelphia. The daily train service is very good. Express bus service (one hour) is MEDICAL SERVICES available from New York City at the Port Authority Bus Terminal, 40th Street and The Student Health Center located on 8th Avenue. The following long distance bus Bishop Place, Bishop Campus, will provide lines also stop in New Brunswick; Continen­ out-patient clinic service, Monday through tal Trailways, Eastern Greyhound Lines, Friday, from 9:00 A.M. to 5:00 P.M. After Safeway Trails, and Suburban Transit. these hours and on Saturday and Sunday, By automobile, New Brunswick is for emergency illnesses and accidents, one easily accessible from Exit 9 of the New may phone CHarter 7-1766, Extension 6211 jersey Turnpike and U.s. Highway 1. Other (Campus Parking Security and Transporta­ nearby main highways are the Garden State tion) for assistance. Parkway, and U.S. Highway 22, 130, 202, 206, 287 and N. j. Highway 18. ADDRESS FOR MAIL The nearest airport in Newark, N.j. AND TELEGRAMS (25 miles) provides facilities for air travel via most of the major airlines. The most Individuals may be addressed at economical way to travel from the airport to New Brunswick is by taxi or bus to the AMS - Summer Meeting - The Ledge Pennsylvania Railroad Station in Newark, Rutgers, The State University and by train from there to New Brunswick. New Brunswick, New jersey 08903

308 COMMITTEE Joshua Barlaz L. F. McAuley, John Bender Chairman The meeting arrangements have been Saul Blumenthal Mrs. Barbara L. made through the good offices of Mr. F. E. Clark Osofsky Robert Collett, Director of the Bureau of R. M. Cohn Everett Pitcher Special Services of Rutgers, The State Uni­ J. H. Griesmer M. S. Robertson versity. G. L. Walker The Committee consists of the fol­ K. G. Wolfson lowing persons: Everett Pitcher H. L. Alder Katherine E. Hazard Associate Secretary Bethlehem, Pennsylvania

Six Hundred Thirty-Fifth Meeting University ofVictoria and Canadian Services College, Royal Road June 18, 1966

The six hundred thirty-fifth meeting for residence accommodation should be of the American Mathematical Society will sent as soon as possible to Professor be held at the University of Victoria in Phoebe Noble, Department of Mathematics, Victoria, British Columbia on June 18, 1966, University of Victoria, Victoria, B. C. in conjunction with meetings of the Pacific Requests should include the number and Northwest Section of the Mathematical names of adults and children, dates and Association of America and the Society for times of arrival and departure, and check Industrial and Applied Mathematics. The or money order for pre-payment of room. Society will meet on Saturday, and the (If payment is made in American money, Association and SIAM will hold their ses­ rates are $4.75, $3.80, $0.95.) sions on Friday, June 17. As a major tourist center, Victoria By invitation of the Committee to abounds in motels and motor hotels. For a Select Hour Speakers for Far Western list of motels or further information con­ Sectional Meetings, the Society will be ad­ cerning motels or ferry timetables write dressed at 11:00 A.M. on Saturday by Pro­ to Phoebe Noble at the given address. fessor P. Emery Thomas of the University Many hotels are available in Victoria of California at Berkeley. The title of the although none are located close to the talk by Professor Thomas is "Vector Fields Gordon Head Campus of the University. on Manifolds.·" Below is a partial list of downtown hotels. Sessions for contributed papers will (Minimum rates are quoted). be held on Saturday at 9:00A.M. and 2:00 Anyone who wishes to stay in a P.M. The deadline for papers contributed hotel should write directly to the hotel for to this meeting is May 4. reservations. These hotels are approxi­ A limited amount of residence ac­ mately 3 miles from the campus. commodation will be available on campus There will be a complimentary lun­ for the nights of June 16, 17, 18. The daily cheon on Saturday. Other meals will be adult rates for rooms without meals are available in the University Cafeteria. $5.00 for single; $4.00 for double rooms; Victoria is served by Air ~anada Children under age 10, $1.00. Reservations from Seattle to Vancouver. There is also

309 Single Double Twin The Dominion Hotel 7 59 Yates Street $7,00-$10,00 $7,00-$11.00 $10.00-$15,00 The Douglas Hotel 1450 Douglas Street $6.00- $8.00 $8,00-$10.00 $10,00-$12,00 The Empress Hotel 721 Government Street $13,00 $19.00 $19.00 The Strathcona Hotel 919 Douglas Street $8,00 $9.50 $10,50 a direct bus service from downtown Van­ is an excellent car ferry service from couver to downtown Victoria via the B.C. Tsawwassen to Swartz Bay, twenty miles Government Ferries and a bus service north of Victoria; from Anacortes, Wash- from downtown Seattle to Port Angeles, ington to Sidney, British Columbia; and Washington, via the Hood Canal Bridge from Port Angeles, Washington to down­ which connects directly with the Port An­ town Victoria. geles-Victoria ferry. Members who are able to do so are advised to come to Victoria by private car since only in this R. S. Pierce way can one take full advantage of the Associate Secretary scenic beauties of the Victoria area. There Seattle, Washington

310 ACTIVITIES OF OTHER ASSOCIATIONS MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC New York City, April 4, 1966

The April 4 meeting of the Associa­ Monday morning and afternoon. This meet­ tion for Symbolic Logic will be held at the ing is being held in conjunction with the Waldorf-Astoria Hotel, in New York City. symposium on Mathematical Aspects of All sessions will be held in the Sert Room. Computer Science, for which the ASL is The program includes an invited hour ad­ one of the joint sponsors. Information re­ dress by Professor Hilary Putnam of Har­ garding the symposium is given und~r the vard University. His title is "Constructible program of the AMS April meeting in this sets and predicative hierarchies." There issue. will be sessions for contributed papers

PROGRAM OF THE SES·SIONS

MONDAY, 9:00 A.M. Morning Session, Sert Room 9:00-9:20 Hyperanalytic predicates Yiannis N. Moschovakis, University of California, Los Angeles 9:25-9:45 Probabilistic recursive functions Irwin Mann, New York University 9:50-10:10 Applications of recursive set functions to infinitary logics Carol Karp, University of Maryland 10:15-10:35 Analytic cut Raymond Smullyan, Belfer Graduate School of Science, Yeshiva Univer­ sity 10:40-11:00 Some consequences of normality of the space of models Hidemitau Sayeki, University of Montreal

MONDAY, 11:15 A.M. Invited Address, Sert Room Constructible sets and predicative hierarchies Hilary Putnam, Harvard University Papers to be presented by title How to make Boolean algebra post complete Storrs McCall, Ma,kerere University. College, Kampala-Uganda

Rem~rks on Hilbert's eight class problems Albert A. Mullin, Lawrence Radiation Laboratory On the semantical completeness of two systems of infinitary propositional calculus Richmond H. Thomason, Yale University New York Martin Davis

311 1966 INTERNATIONAL CONGRESS OF MATHEMATICIANS

The 1966 International Congress of activities of the Congress and to receive its Mathematicians is being held August 16-26, publications. Guests are entitled to the 1966 in Moscow. The scientific sessions of same privileges as members with the ex­ the Congress will take place in the auditoria ception of delivering brief communications of Moscow University on Lenin Hills. and of receiving the Congress publications. The scientific program of the Con­ Fees for the two classes of partici­ gress will consist of Plenary Sessions, pants are as follows: Members: up ·to during the course of which about twenty March 15, 1966--27 rubles (about 30 dol­ invited one-hour reports will be presented; lars); after March 15, 1966--36 rubles and Section Sessions, devoted for the most (about 40 dollars). Guests: up to March 15, part to brief scientific communications by 1966--22.5 rubles (about 25 dollars); after the participants of the Congress, but also Marchl5,1966--27rubles (about 30 dol­ including about 70 invited half-hour reports. lars). Listed below are the topics of the Congress Participants are advised to make Sections. accommodations and service arrangements through the U.S.S.R. Company for Foreign 1. Mathematical Logic and Foundations Travel, Intourist. This organization is also of Mathematics arranging tours and excursions (see sum­ 2. Algebra mary which follows) for the benefit of par­ 3. Theory of Numbers ticipants. 4. Classical Analysis All correspondence concerning the 5. Functional Analysis Congress should be addressed, in one of 6. Ordinary Differential Equations the working languages, to the Secretary 7. Partial Differential Equations General, V. G. Karmanov, Mathematical 8. Topology Congress, Moscow University, Moscow, 9. Geometry V 234. Cable address is MOSCOW MGU 10. Algebraic Geometry and Complex MATEMATIK. Manifolds 11. Probability and Statistics INTOURIST ARRANGEMENTS 12. Applied Mathematics and Mathematical Physics As mentioned earlier, Intourist, the 13. Mathematical Problems of Control U.S.S.R. Company of Foreign Travel, has Systems offered its services to participants of the 14. Numerical Mathematics Congress, to the extent that it will arrange 15. History and Pedagogical Questions travel in the Soviet Union, will provide various classes of accommodations, supply The working languages of the Congress will participants with meals and organize ex­ be English, French, German and Russian. cursions covering the sights of Moscow Reports may be delivered to the Congress and tours of the Soviet Union. in any of these languages. The program of Four classes of accommodation are the Congress Sessions, abstracts of brief available through Intourist. Deluxe Class, scientific communications and the hour which includes three meals a day, hotel and half-hour reports will be publ-ished by accommodation, daily excursions with a the Organizing Committee by the opening of private guide-interpreter with car (provided the Congress. for not more than 2 1/2 hours a day), is There are two classes of partici­ priced at $35 per day per person for sep­ pants in the Congress: (1) Congress Mem­ arate accommodation, and $25 per day per bers and (2) guests (persons accompanying person with "double" room accommodation. Congress Members). Members are entitled Classes A and B are also hotel ac­ to take part in the scientific and social commodatio

312 commodations, including two meals (break­ Theatre and the Kremlin Palace of Con­ fast and dinner) per day, and a guide-in­ gresses) if requested. terpreter available for groups of 2.5 per­ Several tours of the Soviet Union, sons 7 hours a day. Prices for these classes ranging from one to three days in length, range from $9.50 to $14 per day •. are available through Intourist after the Class C includes accommodation in Congress. There are separate tours to more the students' hostel of Moscow University, than ten different cities, including Lenin­ and two meals (breakfast and dinner) in grad, Kiev, Tbilisi, Vladimir, Suzdal', the students' canteen. The price is $6 dol­ Zagorsk, Yaroslavl', Yerevan and the beach. lars per day. resort Sukhumi. Persons desiring a more All classes of hotel accommodation extensive tour of the country may wish to includetransportation between the hotel and take advantage of a seven day tour which the Congress Sessions and to and from the includes visits to Tashkent, Samarkand, station or airport by car. and Bukhara. The three day tours range A, B, and C class arrangements in­ in price from $76 to $12.7. Shorter tours clude a tour of the sights of Moscow, a range from $5 to $66, depending on the dis­ visit to the Kremlin and its museums and tance traveled and the means of transporta­ a tour of the Exhibition of Achievements tion. The one seven day tour is $2.51. of the National Economy of the U.S.S.R., as Participants may purchase any of part of the services which are offered. The these tours in their native countries Company will also arrange to book parti­ through tourist firms having contractual cipants for theatre and concert perform­ relations vyith Intourist. The deadline for ances in Moscow (including the Bolshoi purchasing tours is july 15, 1966.

NEWS ITEMS AND ANNOUNCEMENTS

THIRD SOUTHEAS'li'ERN CONFERENCE ON THEORETICAL AND APPLIED MECHANICS

The Third Southeastern Conference viscoelasticity, plates and shells, wave on Theoretical and Applied Mechanics propagation, kinematic synthesis, vibra­ (SECTAM) will be held at the University of tions, nonlinear mechanics, boundary layer South Carolina, Columbia, South Carolina, phenomena, compressible flow, experi­ on March 31 and April 1, 1966. Three con­ mental mechanics, anisotropy, creep, sta­ current sessions will be held on each date bility, thermoelasticity, plasticity, viscous with the number of papers totaling approxi­ flow, vortex flow, and others. mately forty-five. Further information about the con­ Invited guest lecturers will include ference together with copies of the complete Eric Reissner, Massachusetts Institute of program may be obtained from J.D. Waugh, Technology, and Ian Sneddon, University University of South Carolina, Columbia, of Glasgow, Scotland. South Carolina. Topics of papers are as follows:

313 COMMITTEE ON SUPPORT OF RESEARCH IN THE MATHEMATICAL SCIENCES

The Committee on Support of Re­ this country to a preeminent world position. search in the Mathematical Sciences On the other hand the applications of mathe­ (COSRIMS), established by the· Division of matics have burst the traditional bound­ Mathematics of the National Academy of aries set by classical physics and have used Science--National Research Council, and recent mathematical results in a sophisti­ supported by the National Science Founda­ cated way. tion, was formed during the latter part of The consequences of these transfor­ 1965 to survey the bases for support of mations have sharply increased the burdens mathematics. Objectives of the Committee placed upon mathematical scientists and are threefold: ( 1) to study the current state mathematical institutions. A description of of both mathematical research, and of the nature and extent of these burdens is a mathematical education at the undergradu­ major purpose of the COSRIMS study. The ate, graduate, and postdoctoral levels; (Z) Committee is presently supported by a to study the current levels and forms of grant of $75,000 by the National Science support of mathematical research by fed­ Foundation to the National Academy of eral and private agencies; (3) to indicate Sciences--National Research Council. Sta­ the appropriate support needed in the im­ tistical material required by the study will, mediate future if a healthy state of mathe­ in large part, be made available to cos­ matical activity is to be maintained. RIMS by the Survey Committee of the Con­ In connection with the studies men­ ference Board of the Mathematical Sciences, tioned above, the Committee has been which is supported by the Ford Foundation. charged with preparing a statement of their Liaison between these two committees is findings, directed to those responsible for very cordial." federal science policy in the Congress and Members of the Committee are as the Executive Branch of the government, follows: Lipman Bers, Columbia University which will include an estimate of the cost (chairman); T. W. Anderson, Columbia Uni­ over the next five to ten years of main­ versity;. R. H. Bing, University of Wiscon­ taining a national research effort in the sin; H. W. Bode, Bell Telephone Labora­ mathematical sciences compatible with tories, Inc.; R. P. Dilworth, California In­ national goUs in science and technology. stitute of Technology; Mark Kac, Rocke­ COSRIMS will attempt to have this ready feller University; C. C. Lin, Massachusetts for release sometime in t967. Institute of Technology; j. W. Tukey, Mark Kac, Chairman of the Division Princeton University; F. j. Weyl, Office of Mathematics of NAS-NRC has made the of Naval Research; Hassler Whitney, In­ following comment on the importance of the stitute for Advanced Study; C. N. Yang, study: Institute for Advanced Study. "The COSRIMS study is of the great­ In addition, the Committee is served est importance in establishin~ communica­ by four panels. The four panels and chair­ tion within the mathematical community, men are: Levels and Forms of Support, between the mathematical community and Mina Rees, City University of New York; scientists in other disciplines, and between New Centers, Mark Kac, Rockefeller Uni­ the mathematical community and national versity; Graduate Education, R. P. Boas, policy makers. It is, perhaps, not an ex­ jr •• Northwestern University; Undergradu­ aggeration to say that the leading mathe­ ate Education, John Kemeny, Dartmouth. maticians in the United States have been The Committee held its first meet­ struggling with two distinct but related ing on October 8-9, 1965 at Columbia Uni­ rapid developments. On the one hand their versity, COSRIMS headquarters. research has raised pure mathematics in COS RIMS is being advised by the

314 Committee on Science and Public Policy, sity of California, Berkeley; S. S. Chern, of the National Academy of Sciences--Na­ University of California, Berkeley; George tional Research Council. Forsythe, Stanford University; Peter Hilton, The Committee recently announced Cornell University; Joseph Keller, New the names of newly appointed members to York University; Elias Stein, Princeton three of these four panels, Newly appointed University; Dorothy Stone, University of members are: Rochester. Support: Eleazer Bromberg, New Undergraduate: Grace Bates, Mount York University; D. E. Christie, Bowdoin Holyoke; Llayron Clarkson, Texas South­ College; Morton Curtis, Rice University; ern University; George Handelman, Ren­ Mahlon Day, University of Illinois; W. J, sselaer Polytechnic Institute; Frederick Dixon, University of California, Los Ange­ Mosteller, Harvard University; H. 0. Pollak, les; L. H. Farinholt, Sloan Foundation; Bell Telephone Laboratories, Inc.; Hartley Andrew Gleason, Harvard University; Jo­ Rogers, Jr., Massachusetts Institute of seph LaSalle, Brown University; F. J, Weyl, Technology; John Toll, State University of Office of Naval Research. New York; Robert Wisner, New Mexico Graduate: R. D. Anderson, Louisiana State University. State University; David Blackwell, Univer-

NEWS ITEMS AND ANNOUNCEMENTS

SREB OFFERS BROCHURE ON 1966 GRADUATE SUMMER COURSES IN STATISTICS

A brochure entitled "19 66 Graduate publication. Information regarding support­ Summer Courses in Statistics" has been ing funds for this session available for prepared by the Committee on Statistics predoctoral and postdoctoral students may of the Southern Regional Education Board. be obtained from A. C. Cohen, Department The brochure describes graduate courses of Statistics, University of Georgia, Athens, that will be available this coming summer Georgia 30601, at 22 major universities in the Southern Copies of the brochure may be or­ region. dered in reasonable quantity at no cost The tentative announcement of the from the Southern Regional Education Regional Graduate Summer Session in Board, 130 Sixth Street, N. W ., Atlanta, Statistics, to be held at the University of Georgia, or from any of the participating Georgia in 1966, is also included in the universities.

315 ASSISTANTSHIPS AND FELLOWSHIPS

The following were received too late for inclusion in the December issue.

Florida State University ENROLLMENT Undergraduates Graduates TALLAHASSEE, FLORIDA 32306 E.P. Miles, jr., Director Total University Computing Center Math. or Statistics 10 10 Applications must be filed by March 16, 1966.** majors subsidized Stipend Type of financial assistance (with Amount 9 or Tuition, if not Service Required number anticipated in 1966-1967) (dollars) 12 mo. included in stipend Hrs/Week Type Graduate Assistants (10) 3220-4000 12 $150 per trimester up to 20 "

*Compute-r programming, Instruction, Operation or consulting. **Applications should be filed jointly with the Director of the Computing Center and with the Head of the Department of Mathematics or the Department of Statistics in which the prospective student anticipates pursuing his graduate degree.

Loyola. University CHICAGO, ILLINOIS 60626 ENROLLMENT Undergraduates Graduates Robert B. Reise!, Acting Chairman Total University 11000 1500 Department of Mathematics Math. majors 300 30 Applications must be filed by April I, 1966.* (no. subsidized) none 7 Stipend Type of financial assistance (with Amount 9 or Tuition, if not Service Required number anticipated in 1966-1967) (dollars) 12 mo. included in stipend Hrs/Week Type

Teaching Fellowship (7) 1800 9 None 6 Teaching

*Late applications will be considered.

GRADUATE COURSES

SUPPLEMENTARY LIST

The following schools will offer graduate courses during the coming summer. This list is supplementarytothe one which appeared in these NOTICES, February, 1966, pp. 203-212.

UNIVERSITY OF ALABAMA, Tuscaloosa, Alabama Information: Dr. John P. Gill, Department of Business statistics June 6-July 11 July 14-August 19 BS 225 statistics for Business Decisions BS 101 Advanced Business statistics

DUKE UNIVERSITY, Durham, North Carolina 27706 June 8-July 20 Information: Dr. Marion R. Bryson Math 224 Mathematical statistics

UNIVERSITY OF FLORIDA, Gainesville, Florida 32601 April 27-June 18 Information: Dr. William Mendenhall, Department of statistics STA 320 Introduction to statistics STA 642 Theory of statistics STA 440 Mathematical statistics STA 670 Multivariate Analysis I STA 521 Methods of statistics STA 671 Multivariate Analysis IT STA 628 Problems in statistics STA 699 Master's Research

GEORGIA INSTITUTE OF TECHNOLOGY, Atlanta, Georgia 30332 June 27 -September 9 Math 415 Introduction to Probability Ind Eng 339 Evaluation of Engineering Data I Ind Eng 340 Evaluation of Engineering Data II Ind Eng 639 Experimental statistics Ind Eng 649 Design of Industrial Experiments (tentative)

UNIVERSITY OF KENTUCKY, Lexington, Kentucky 40500 June 13-August 5 Information: Dr. Dana G. Card Economics 207 statistical Methods Education 657 Educational Statistics

316 UNIVERSITY OF MARYLAND, College Park, Maryland 20742 June 22-August 13 Application deadline: June 1 Information: Professor R. Sedgewick, Mathematics Department 100 Vectors and Matrices 146 Fundamental Concepts of Mathematics 103 Introduction to Abstract Algebra 163 Analysis for Scientists and Engineers ll 128 Euclidean Geometry 181 Introduction to Number Theory 133 Applied Probability and Statistics 183 Introduction to Geometry

UNIVERSITY OF :MISSISSIPPI, University, Mississippi 38677 June 10-July 16 Application deadline: May 20 and June 28 .Information: T. A. Bickerstaff, Chairman, Department of Mathematics 513 Theory of Numbers 556 Adv. Calculus ll 525 Modern Algebra I 557 Theory of Integrals 537 Non-Euclidean Geometry 573 Probability 555 Advanced Calculus I 576 Mathematical statistics ll 575 Mathematical statistics I 631 Foundations of Geometry 655 Theory of Functions of Complex Variables I 656 Theory of Functions of Complex Variables n 519 Matrices

:MISSISSIPPI STATE UNIVERSITY, starkville, Mississippi 39759 Information: Dr. Charles N. Moore, Department of Business statistics and Data Processing June 7-July 15 July 18-August 20 BSD 753 Business statistics Using Computers BSD 963 Operations Research Problems BSD 953 statistics for Business Decisions BSD 983 Research Methods in Business and Industry

NORTH CAROLINA STATE UNIVERSITY, Raleigh, North Carolina 27607 Information: Jack Suberman, Director of the Summer Sessions, 1911 Building Registration Deadline: June 13 and July 25 June 7 -July 15 July 19-August 25 MA 511. Advanced Calculus I MA 512 Advanced Calculus ll MA 512 Advanced Calculus ll MA 5.24 Boundary Value Problems MA 513 Introduction to Complex Variable MA 527 Numerical Analysis I Theory MA 532 Theory of Ordinary Differential Equations MA 514 Methods of Applied Mathematics MA 541 Theory of Probability I MA 532 Theory of Ordinary Differential MA 622 Vector Spaces and Matrices Equations MA 625 Introduction to Differential Geometry MA 541 Theory of Probability I MA 6'32 Operational Mathematics I MA 622 Vector Spaces and Matrices MA 625 Introduction to Differential Geometry MA 632 Operational Mathematics I

OKLAHOMA STATE UNIVERSITY, stillwater, Oklahoma 74075 June 6-July 30 Information: Dr. Carl E. Marshall, Department of Mathematics and statistics 403 statistical Methods I 413 statistical Methods ll 343 Elementary Mathematical statistics I 503 Experimental Desigiis 513 Sample Survey Designs

SALEM STATE COLLEGE, Salem, Massachusetts 01970 July 5-August 12 Application deadline: July 5 Information: Dr. Robert S. Fishman, Department of Mathematics Abstract Algebra Fundamentals of Mathematics, I Analytic Geometry Fundamentals of Mathematics, ll Sets, Relations & Functions Introduction to Probability & statistics Linear Algebra Calculus, I Differential Equations Advanced Calculus Logic & Boolean Algebra Numerical Analysis College Algebra & Trigonometry Mappings and Cardinals

SOUTHERN METHODIST UNIVERSITY, Dallas, Texas 75222 June 4-July 15 Information: Dr. Paul D. Minton, Department of statistics 135 Theory of Probability

UNIVERSITY OF TEXAS, Austin, Texas 78712 June 7 -July 17 Information: Dr. Francis B. May Statistics 384 Business Dynamics (nine weeks) statistics 381K statistical Methods Applied to Business (six weeks)

TEXAS A&M UNIVERSITY, College station, Texas 77843 Information: Dr. H. 0. Hartley, Institute of statistics July 6-July 15 July 18-August 26 406 statistical Methods 622 Advanced Topics and Developments in 621 Advanced Topics and Developments statistical Methodology in statistical Theory 601 Statistical Analysis

317 PERSONAL ITEMS

Professor S. A. AFRIAT of Rice Uni­ search, Bombay, India has been appointed versity has been appointed to Professor of to a professorship at the Eidgenossische Economics at Purdue University. Technische Hochschschule, Zurich, Swit­ Miss M. V. AUMANN of San Francisco zerland. College for Women has been appointed to Dr. M. N. CHASE of Technical Opera­ an assistant professorship at the San Jose tions Research, Arlington, Virginia has State College. been reassigned to Vice President of the Dr. ADI BEN-ISRAEL of the Technion­ Washington Area Activities. Israel Institute of Technology, Haifa, Israel Dr. J. F. CHEW of the Virginia Poly­ has been appointed to an associate profes­ technic Institute has been appointed to an sorship at the University of Illinois at assistant professorship at Kent State Uni­ Chicago Circle. versity. Professor A. R. BERNSTEIN of the Mr. C. A. COMPTON of Malpar In­ University of California, Los Angeles has corporated, Alexandria, Virginia has ac­ been appointed to an assistant professor­ cepted a position as Mathematician and ship at the University of Wisconsin. Reliability Engi~eer with the Stanwick Cor­ Dr. B. J. BIRCH of the University of poration, Arlington, Virginia. Manchester, Manchester, England has been Professor C. H. CUNKLE of Clarkson appointed a Reader at the University of College of Technology has been appointed Oxford, England. to a professorship at Kansas State Univer­ Dr. W. W. BLEDSOE of Panoramic sity. Research Incorporated, Palo Alto, Cali­ Mr. R. B. DIFRANCO of Indiana Uni­ fornia has been appointed to a professor­ versity ha.s been appointed to an assistant ship at the University of Texas. professorship at Fordham University. Professor F. F. BONSALL of the Mr. H. R. DURHAM of the University University of Newcastle Upon Tyne, England of Georgia has been appointed to an assis­ has been appointed to a professorship at the tant professorship at the Appalachian State University of Edinburgh, Scotland. Teachers College. Professor G. R. BORGES of the Uni­ Mr. MARTIN DUSZYNSKY, JR. of the versity of Nevada has been appointed to an University of South Carolina has accepted assistant professorship at the University a position as Mathematician with Wilbur of California, Davis. Smith and Associates, Columbia, South Dr. T. F. BRIDGLAND, JR. of the Carolina. University of South Carolina has been ap­ Professor WADE ELLIS of Oberlin pointed to a professorship at the University College is on leave for the second semester of Alabama Research Institute. and will work as a consultant with the Ford Mr. R. A. BRUALDI of the National Foundation in work concerned with im­ Bureau of Standards, Washington, D. C. proving education in biology, chemistry, has been appointed to an assistant pro­ mathematics and physics in the Mexican fessorship at the University of Wisconsin. educational system, Mexico City, Mexico. Dr. R. S. BUCY of the University of Mr. J. E. FALK of the University of Maryland has been appointed to an Asso­ Michigan has accepted a position as a mem­ ciate Professor of the Aerospace Depart­ ber of the Technical Staff of the Research ment at the University of Colorado. Analysis Corporation, McLean, Virginia. Mr. D. E. CATLIN of the University The Reverend W. J.FEENEYofWeston of Florida has been appointed to an assis­ College has been named Acting Dean of the tant professorship at the University of Graduate School of Arts and Sciences at Massachusetts. Boston College. Professor K. CHANDRASEKHARAN Professor R. W. FELDMANN of the of the Tata Institute of Fundamental Re- State University of New York at Buffalo has

318 been appointed to an assistant professor­ associate professorship at the Illinois State ship at Lycoming College. University. Professor R. j. FLEMING of Florida Dr. W. C. HOFFMAN of the Boeing State University has been appointed to an Scientific Research Laboratories, Seattle, assistant professorship at the University Washington has been appointed to a pro­ of Missouri. fessorship at Oregon State University. Mr. N. E. FOLAND of Kansas State Dr. A. S. HOUSEHOLDER, Director University has been appointed to an asso­ of the Mathematics Division at the Oak ciate professorship at Southern Illinois Ridge National Laboratory, Oak Ridge, University. Tennessee has been awarded the degree Dr. H. L. GARABEDIAN of the General Doktor der Naturwissenschaften Ehren­ Motors Research Laboratories, Warren, halber by the Technische Hochschule, Mu­ Michigan has been appointed to a Professor nich, Germany. of Mathematics and Energy Engineering at Mr. ALEXANDER HURWITZ of the the University of illinois at Chicago Circle University of California at Los Angeles has effective January 1, 1967. accepted the position of a Systems Research Professor JESUS GIL DE LAMADRID Staff Member with the International Bus­ has returned to the School of Mathematics iness Machines Corporation, Los Angeles, at the University of Minnesota after a sab­ California. batical leave at the Centre Universitaire Professor G. M. KOETHE of the Uni­ International, Paris, France. versity of Heidelberg, Germany has been Mr. R. G. GILLESPIE of the Boeing appointed to a professorship at the Univer­ Company, Seattle, Washington has accepted sity of Frankfurt, Germany. a position as Manager of the Advanced Dr. FRANKLIN LOWENTHAL of New Development Department of the Control York University has beer. appointed to an Data Corporation, Palo Alto, California. assistant professorship at the University Mr. G. H. GLEISSNER of the U.S. of Oregon. Naval Weapons Laboratory, Dahlgren, Vir­ Professor L. P. MAHER, JR. of the ginia has accepted a position as Associate University of Texas has been appointed to Technical Director for Applied Mathematics an assistant professorship at Indiana Uni­ and Head of the Applied Mathematics Labo­ versity. ratory at the David Taylor Model Basin, Professor Emeritus MARSTON MORSE Washington, D. C. of the Institute for Advanced Study and a Professor E. L. GODFREY of the member of the National Academy of Science Wright-Patterson Air Force Base has been has been given a membership in the Aca­ appointed to an associate professorship at demy of The Romanian Socialist Republic. the Rose Polytechnic Institute. Professor HILARY PUTNAM of the Professor ABRAHAM GOETZ of the Massachusetts Institute of Technology has University of Warsaw, Poland has been ap­ been appointed a Professor in the Philo­ pointed to an associate professorship at the sophy Department at Harvard University. University of Notre Dame. Professor L. V. QUINTAS of St.John's Mr. J. D. HALPERN of the California University has been awarded a National Institute of Technology has been appointed Science Foundation Science Faculty Fellow­ a Member of the Institute for Advanced ship for a twelve month period of study in Study, Princeton, New jersey. Mathematics at The City University of New Professor E. H. HANSON of the U. S. York. Naval Postgraduate School has accepted a Dr. J. R. ROSENBLATT of the National position as Director of Advanced Analysis Bureau of Standards has received the annual with Data Dynamics Incorporated, Monterey, award for scientific achievement in math­ California. ematics from the Washington Academy of Professor F. S. HARPER of Georgia Sciences in recognition of her contributions State College has been appointed a Pro­ to systems reliability theory and other areas fessor of Actuarial Science at Drake Uni­ of statistics. versity. Dr. M. W. SHELLY of the U.S. Naval Professor T. L. HICKS of the Univer­ Ordinance Testing Station, China Lake, sity of Cincinnati has been appointed to an California has been appointed to Associate

319 Professor of Social Psychology at Kansas Deaths: University. Professor OSCAR ZARISKI of Harvard Professor Emeritus C. I. DAVISON University was awarded an honorary de­ of Wilson College died on July 23, 1965 gree at a convocation dedicating a new at the age of 88. She was a member of the science center at Brandeis University. Society for 32 years. SISTER MARIE GERTRUDE McNEIL The following promotions are announced: of Seton Hill College died on December 12, 1965 at the age of 72. She was a member To Consultant: of the Society for 30 years. Dean E. B. STOUFFER of the Univer­ Dr. BENJAMIN LEPSON has been promoted sity of Kansas died on November 24, 1965 to Consultant in Mathematics and Compu­ at the age of 81. He was Vice President of tation, Nucleonics Division, Naval Research the Society from 1936-1937 and Editor of Laboratory and has also been promoted to the Bulletin from 1945-1950. He was a adjunct Professor of Mathematics at the member of the Society for 54 years. Catholic University of America.

To Professor: Ohio State University: J. S. RUSTAGI.

To Associate Professor: University of Tennessee: D. R. BROWN.

The following appointments are announced: Errata

To Instructor: The following is a correction of an announce­ ment in the January issue of the Notices. Catholic University of America: J. W. BESSMAN; Pennsylvania State University: Dr. B. E. RHOADES, Executive Direc­ J. H. BIGGS; Rensselaer Polytechnic In­ tor of the Committee on the Undergraduate stitute: C. W. HAINES; Southern Illinois Program in Mathematics of the Mathemati­ University: F. D. PEDERSEN; State Uni­ cal Association of America, has been ap­ versity of New York at Buffalo: S. R. pointed to an associate professorship at GRACZYK; Trinity College: E. J. BOYER. Indiana University.

NEWS ITEMS AND ANNOUNCEMENTS

FINDING EMPLOYMENT IN THE MATHEMATICAL SCIENCES

The Mathematical Sciences Employ­ suited to his abilities and training. The ment Register, which is sponsored by the role of the mathematician in teaching, aca­ American Mathematical Society, the Mathe­ demic and industrial reserach, computing, matical Association of America, and the and government is discussed. The booklet, Society for Industrial and Applied Mathe­ also, lists the sources of information avail­ matics, has recently published a booklet able to the young mathematician who is entitled "Finding Employment in th.e Math­ seeking a position. There is no charge for ematical Sciences." This booklet gives in­ this booklet, and it may be obtained by formation to the young mathematician, who writing to the Employment Register, P .0. is just entering the professional field_. in­ Box 6248, Providence, Rhode Island 02904. formation on how to find employment best

320 SUPPLEMENTARY PROGRAM-Number38

During the interval from january 7, to February 11, 1966 the papers listed below were accepted by the American Mathematical Society for presentation by title. After each title on this program there is an identifying number. The abstracts of the papers will be found following the same number in the section on Abstracts of Contributed Papers in this issue of these cJ.foticrL)s. One abstract presented by title may be accepted per person per issue of the cJ.foticrL). 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. ( 1) Relation between different systems of connected domain modal logic Mr. R. Coifman, University of Chi­ Mr. Stal Aanderaa, Harvard Univer­ cago and Professor G. L. Weiss, sity (66T-227) Washington University (66T-205) (Introduced by Professor Hao Wang) ( 11) Rapid calculations by Turing machine (Z) On the bounds of the minimal length of Mr. S. A. Cook, Harvard University sequences representing simply ordered (66T-222) sets ( 12) The representation of biregular rings Professor Alexander Abian and Mr. by sheaves David Deever, The Ohio State Univer­ Professor john Dauns and Professor sity (66T-188) K. H. Hofmann, Tulane University (3) Examples of generalized Sheffer func- (66T-195) tions (13) Strength of certain set theories Professor Alexander Abian and Mr. Mr. H. M. Friedman, Massachusetts Samuel La Macchia, The Ohio State Institute of Technology ( 66T-t29) University ( 66T-226) ( 14) On the concept of a function cf> on a set (4) Concerning a certain linear integral S into the same set equation Dr. Baruch Germansky, Singel 270, Professor W. D. L. Appling, North Amsterdam C, Netherlands (66T-202) Texas State University (66T-209) ( 15) Homeotopy groups of 3-manifolds which ( 5) A Harnack inequality for nonlinear fiber over a circle parabolic equations Dr. C. H. Giffen, The Institute for Professor D. G. Aronson and Pro­ Advanced Study (66T-210) fessor james Serrin, University of ( 16) One-way stack automata. I Minnesota (66T-196) Dr. Seymour Ginsburg, Systems De­ (6) Some metrization theorems velopment Corporation, Santa Mon­ Professor C. E. Aull, Virginia Poly­ ica, California, Professor S. A. technic Institute (66T-197) Greibach, Harvard University and (7) On regular nonarchimedian Banach Professor M. A. Harrison, Univer­ algebras sity of California, Berkeley (66T- Mr. Edward Beckenstein, Polytechnic 199) Institute of Brooklyn (66T-220) ( 1 7) On the spectrum of classes of alge­ (B) On the univalence of an integral bras. Preliminary report Mr. W. M. Causey, University of Professor G. A. Gratzer, Pennsyl­ Kansas (66T-224) vania State University (66T-183) (9) Inequalities of Poincare type ( 18) Necessary conditions in the calculus Professor C. W. Clark, University of variations for extremal curves which of British Columbia and University are absolutely continuous but whose of California, Berkeley (66T-211) derivatives are not necessarily bounded ( 1 O) A factorization theorem for functions Professor Hubert Halkin, University in the Nevanlinna class of a multiply of California, San Diego (66T-193)

321 (19) WITHDRAWN. integrals (ZO) Some properties of integral closure. Mr. C. B. Murray, TRACOR, Inc., Preliminary report Austin, Texas ( 66T-189) Mr. W. J. Heinzer, Florida State ( 33) Analytic functions of polynomial growth University (66T-190) on a polycylinder. II (Zl) A trivial result in set theory Dr. D. E. Myers, University of Ari­ Dr. C. M. Howard, University of zona (66T-184) California, Los Angeles (66T-Zl8) (34) The homotopy type of some automor­ (ZZ) On relative regular neighborhoods. phism groups Preliminary report Dr. G. J. Neubauer, University of Mr. L. S. Husch, Florida State Uni­ Notre Dame (66T-Zl5) versity (66T-ZlZ) ( 35) On a semihereditary property of sets. (Z3) Relationships between reducibilities. Preliminary report Preliminary report Professor V. F. Pfeffer, George Mr. C. G. Jockusch, Jr., Massachu­ Washington University (66T-194) setts Institute of Technology (66T­ (36) Points in {3X associated with certain Z03) subalgebras of C(X) (Introduced by Professor Hartley Rogers, Mr. D. L. Plank, University of Ro­ Jr.) chester (66T-ZZ8) (Z4) Functions of bounded variation and (37) Consequences and generalizations of change of variable in a Lebesgue inte­ a lemma of Tong gral Professor Mary Powderly, Univer­ Mr. K. G. Johnson, Virginia Poly­ sity of Connecticut (66T-Z04) technic Institute (66T-ZZ3) (38) A number-theoretic estimate (ZS) The impossibility of effectively charac­ Professor Burton Randol, Yale Uni­ terizing weakly representable Boolean versity ( 66T-198) algebras (39) Quasigroupes demi-symthriques. Iso- Professor Carol Karp, University topies preservant la demi symetrie of Maryland (66T-187) Professor Albert Sade, 14, Bd du J. (Z6) Unions v s. joins, for disjoint r.e. sets Zoologique, Marseille 4°, BDR 13, Professor T. G. McLaughlin, Uni­ France (66T-Z17) versity of Illinois (66T-Zl6) (40) Ahlfors' conjecture concerning ex­ (Z7) Structure of direct factor sets treme Sario operators Professor Robert McLeod and Pro­ Professor N. W. Savage, Arizona fessor R. S. Spira, University of State University (66T-ZZ5) Tennessee (66T-ZOO) (41) On regularly accretive extensions (Z8) Continued binary exponentiations and Professor Martin Schechter, Insti­ a function obtained from them. Pre­ tute for Advanced Study (66T-Z19) liminary report (4Z) Denjoy integration in abstract spaces. Professor Laurence Maher, Indiana II University (66T-Z01) Mr. D. W. Solomon, Wayne State (Z9) A variation of the Tchebycheff quad- University (66T-ZZ1) rature formula (43) A sufficient condition for the Riemann Professor Amram Meir and Pro­ hypothesis fessor Ambikeshwar Sharma, Uni­ Professor Robert Spira, The Uni­ versity of Alberta, Canada (66T-Z08) versity of Tennessee (66T-Zl4) (30) Contraction theorems. Preliminary (44) The numerical range of an operator report Professor J. G. Stampfli, New York Mr. P. R. Meyers, National Bureau University (66T-19Z) of Standards, Washington, D. C. (45) Automorphism groups of Abelian p­ (66T-191) groups and generalized Boolean alge­ (31) Another estimate of the pi-function. bras Preliminary report Professor R. W. Stringall, Univer­ Mr. A. A. Mullin, University of sity of California, Davis (66T-Z30) California, Livermore (66T-18Z) (46) Integral inequalities for two functions (3Z) On a substitution theorem for Stieltjes Dr. B. A. Troesch, Aerospace Cor-

322 poration, El Segundo, California (66T- (49) An invariant formulation of the new 213) maximum-minimum theory of eigen­ (47) Differences, convolutions, primes. IV values Mr. Benjamin Volk, Yeshiva Univer­ Professor Alexander Weinstein, Uni­ sity (66T-186) versity of Maryland (66T-207) (48) Homomorphisms of I-bisimple semi­ (50) Extremal length and conformal capacity groups Professor W. P. Ziemer, Indiana Professor R. J. Warne, West Vir­ University (66T-206.) ginia University (66T-181)

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323 ABSTRACTS OF CONTRIBUTED PAPERS

The Meeting in New York City April4-7, 1966

632-1. J. G. CEDER and M. L. WEISS, University of California, Santa Barbara, California 93106. Some inbetween theorems for Darboux functions.

Given a class 5:f of r.v. functions on a real interval I we study the question of whether, given f, g c '£with g(x) < f(x) for all x E I, there is an h E 5:f such that g < h < f over I for the following classes (the Darboux functions), %-(the uniform closure of 9), 9~ (the Darboux, Borel n 5:f: 9 (1 functions), and %~11 • The main results obtained are: (1) There are two comparable §~2 functions having no 9J function between them; (2) There exists a reasonable condition for a pair of comparable functions to admit a g function in between them; (3) Between two comparable 991 (resp. %:,

, The problem of in­ resp. %' ~11 n ~ 2) func_tions there is a 9~2 (resp. %', resp. %'~ n+ 1) function. serting a ;f) ~l function between two comparable ;;2)~1 functions is unsolved. (Received November 23, 1965.)

632-2. J. C. C. NITSCHE, University of Minnesota, Minneapolis, Minnesota 55455, An inclusion theorem for minimal surfaces.

Consider two Jordan curves r 1 and r 2 in parallel planes of Euclidean (x,y,z)-space-- r 1 in the plane z: - h, interior to a circle K 1 in this plane; r 2 in the plane z: h, interior to a circle K2 in that plane. The following theorem is proved. If r 1 and r 2 bound a minimal surface of the type of the circular annulus, then also the circles K 1 and K2 bound such a minimal surface. This is a gen­ eralization of a result of P. M. Bailyn (Doubly-connected minimal surfaces, Dissertation, New York

University, October, 1964), where the circles K1 and K2 are supposed to be coaxial. In view of the fact that the totality of doubly-connected minimal surfaces, bounded by two circles in parallel planes, is known through the work of A. Enneper, B. Riemann, and M. Shiffman and can be represented in terms of elliptic integrals, the above theorem also contains a new necessary criterion for the capa­ bility of the curves r 1 and r 2 to span a minimal surface of the type of the circular annulus. In this sense it is in the spirit of the theorems proved in J. Math. Mech. 13 (1964), 659-666 and Rend. Circ. Mat. Palermo (2) 13 (1964), 192-198. (Received December 3, 1965.)

632-3. J. H. SLOSS, University of California, Santa Barbara, California. Interpolation with remainder of real analytic functions in several variables.

Let r 0, 1 ;;i; i ~ r such that if we expand r about u0 of C(L) then its radius of convergence about u6, with u6, ••. ,u~-l, u6+ 1, ••• ;uij" held fixed is > L/4 + e, f > 0, 1 ;;;; i ~ r. Let u~i : ai + L/2 + L/2 cos ~2ji + 1)/(m + 1}(1r/2)], 0 ;;;; ji ;;;; m, 1 ;;;; i ;;;; r. Theorem. If

324 r( 1 2 r we interpolate u) at (uh ,uj2 , ••• , ujr ) by Pm(u) then we have Pm converges uniformly to ron C(L) as m ---+ oo. Moreover : I r(u) - Pm (u) I ~ max j < < lr I (1 + (4L/1r(L + 4E))]r (1/(1 + 4E/L))m+ 1 t ECi=J=r •((3r- ·1)/2) where Cj are simple and well defined contours. Simple expressions of the remainder for interpolation of certain compact subsets of analytic r-dimensional surfaces in En can also be derived. (Received January 20, 1966.)

632-4. R. W. ROBINSON, Cornell University, Ithaca, New York. Recursively enumerable sets not contained in any maximal set.

For notation see Abstract 626-8, these cNotiaiJ 12 (1965), 692. Call a r.e. set atomless if it is coinfinite and is not contained in any maximal set. Theorem 1. There is an atomless r-maximal set. The proof is by a construction similar to those reported in the previous abstract. Theorem 2. The degree of unsolvability.!!. of any atomless set is such that.!!.' >Jr. The proof provides directly a maximal superset for any co infinite r.e. set of degree .Q. such that Jr = Q.'. This confirms a conjecture of D. A. Martin, but leaves open the exact classification of the degrees of atomless sets. (Received November 26, 1965.)

63.2-5. M. E. WATKINS, University of North Carolina, Chapel Hill, North Carolina 27515. On a class of graphs with connectivity 2.

G will denote a finite undirected graph with no loops or multiple edges. V will denote its vertex set and X will denote its connectivity. Define f' to be the largest integer not exceeding IV I such that given a set B c v with IB I = r. then there is a cycle in G through all the vertices of B. It

is known (G. A. Dirac, Math. Nachr. 22 (1960), 61-85) that if X !;:; 2, then r;;:; X. A result due jointly to D. M. Mesner and the author (Abstract 625-32, these cNOticeiJ 12 (1965), 553) characterizes G for r = X ;;:; 3, but fails for X= 2. Theorem. Given G, r= 2 if and only if X= 2 and there exists a set

S CV such that one of the following three sets of conditions holds: I (a) S = {u1,vpu2,v2,u3 ,v3J; (b) Each set {ui,vit separates G, i = 1,2,3; and (c) A pair of elements of Sis joined by a path in G having no interior vertex inS if and only if the pair is not of the form {ui,vjJ for some i i j;

II(a) S = tu,v1,v2,v3J; (b) Each set {u,vi} separates G; and (c) Each pair of elements of Sis joined by a path in G having no interior vertex in S; or III(a) S = {u, v J; and (b) The component index of S

is ~ 3. (Received January 3, 1966.)

632-6. LOUISE HAY, Mount Holyoke College, South Hadley, Massachusetts 01075. Elementary differences between the isols and the co-simple isols.

Let A= the isols = the collection of equivalence classes of finite or immune sets under 1-1 partial recursive mappings, Az = the co-simple isols =the collection of isols of sets with recursively enumerable complements. A class {So 1of sentences in a first-order functional calculus with+ and is exhibited which are true in (A,+, •) but false in (Az,+, ·). Let 0 be a function on the nonnegative integers and P n = the nth positive prime. The content of So is to assert the existence of an element X such that for each n, O(n) = the largest y such that pY divides X. This assertion is true in A for n every 0 [Dekker and Myhill, Math. z. 73 (1960), 127-133] but false in Az unless 0 is expressible in both 5-quantifier forms in the arithmetic hierarchy. To obtain So as a first- order sentence, a formula

325 is required which characterizes the finite isols in both (A,+, •) and (Az,+' •). This is provided by

showing that the finite !sols are the only elements of Az comparable (under ~) to all other elements of Az. The proof involves a priority construction. (Received January 6, 1966.)

632-7. STEPHEN SIMONS, University of California, Santa Barbara, California. A front-ended proof of Lebesgue's theorem.

We discuss a theorem on lattice-ordered groups from which Ptak's combinatorial theorem on the existence of convex means can be deduced, which can be used to give a short proof of Banach's criteria for the weak convergence of a sequence in the Banach space of all bounded, real functions on an abstract set, and which can be used to give a non measure-theoretic proof of Lebesgue's dominated convergence theorem for a sequence of continuous functions on a countably compact topological space. (Received January 7, 1966.)

632-8. L. K. ARNOLD, Brown University, Providence, Rhode Island. On sigma-finite Invariant

measures. Preliminary report.

Let (X,F ,m) be a finite measure space. Let T be a one to one transformation of X onto X. Sup­ pose (i) B is a measureable set implies T(B) and T- 1(B) are, too, and (ii) m(B) = 0 implies m(T(B)) =

m(T- 1(B)) = 0. A measure m' defined on F is equivalent (tom) if m and m' have the same sets of

zero measure, and m' is invariant If m'(B) = m'(T(B)) for every measureable set B. A decomposition is a finite or countable collection of pairwise disjoint, measureable sets whose union Is X. Theorem. The following are equivalent: (1) There exists a sigma-finite, equivalent, invariant measure m' de­ fined on F. (2) For every f > 0 there is a decomposition [Xili = 1,2, ..• J such that for each i, when­ ever B Is a measureable set, B C ~, and TP(B) C Xi for some integer p, then (1/1 + f)m(B)

~ m(TP(B)) ~ (1 + f)m(B). (3) There is a decomposition {Yili = 1,2, ••• } and positive constants ki, i = 1,2, ••• , such that whenever B is a measureable set, B C Y i' and TP(B) C Y i for some integer p, then kim(B) ~ m(TP(B)). The theorem can be used to show there is no sigma-finite invariant measure equivalent to Lebesgue measure for the members of a certain class of transformations of (almost all of) [0,1] onto itself. The class includes a transformation first constructed by A. Brunei. (Received

January 21, 1966.)

632-9. HIDEGORO NAKANO, Wayne State University, Detroit, Michigan 48202. Mathematical set theory.

A set class _yr of a space S is called a choice Ideal, if (1) Y 3 ¢, (2) Yo;; X -:::J Y -1 4> implies _yro:; Y, (3) Y 3 X,Y implies Y3 X UY, (4) Y 3 XC Y ~ S implies Y3Y, and (5) every X E Y satisfies the choice axiom, All finite sets of S form a choice ideal, which is called the finite ideal of S. A mapping T from another space R into S is said to be compatible to Y, if T(R) E Y. A set

A C S Is said to satisfy the choice axiom with respect to Y, if there is a choice mapping of A which is compatible toY. Then, a set A C S satisfies the choice axiom with respect to Yif and only If ~ .7 A E J; For two sets A, B C S we write A "- B, if there is a one-to-one mapping from A onto B, which is compatible to Y. S is said to be infinite with respect to Y, if we can find A C B C S such

326 that A t- B and A ~ B. We can prove that S is neither finite nor infinite with respect to Y if and only if S is not finite and Yis the finite ideal. For instance, the space of all natural numbers is neither finite nor infinite with respect to the finite ideal. The detail is found in the book: H. Nakano, Set theory, which will be published in the U.S.A. (Received January 20, 1966.)

632-10. C. H. GIFFEN, The Institute for Advanced Study, Princeton, New Jersey 08540. Homeotopy groups of f!bered knots and links.

Let G(X, Y) be the C - 0 topological group of autohomeomorphisms of the pair (X, Y) and 3 M(X,Y) = 1r0 (G(X,Y)) the homeotopy group of (X,Y). Let (S ,L) be a link whose complement fibers over a circle. Theorem. M'(S3 ,L) is naturally isomorphic to the subgroup of the outer automorphism group of the peripheral group syst~m of (S 3 ,L) generated by the automorphisms which send each meridian to some meridian or its inverse. Corollary. If they hold, invertibility, amphicheirality, and interchangeability of fibered knots and links can be verified algebraically. Let (S 3 ,K) be a fibered knot and 0 E K. The stable homeotopy group ~ (S3,K) is gotten by considering only auto­ homeomorphisms which are the identity near 0. Theorem. ~(S 3 ,K) is naturally isomorphic to the group of automorphisms of the peripheral group system of (S 3 ,K) which are the identity on some fixed peripheral subgroup. Examples. ~(torus knot)= z 2 , .M"(figure eight knot)= dihedral of order 8, M';,(torus knot)= Z, and ~(figure eight knot)= Z X z. (Received January 31, 1966.)

632-11. W. M. BOGDANOWICZ, Catholic University of America, Washington, D. C. 20017. A necessary and sufficient condition for two volume spaces to generate the same space of Lebesgue­ Bochner summable functions.

In the paper, A generalization of the Lebesgue-Bochner-Stieltjes integral and a new approach to the theory of integration, Proc. National Acad. Sci. U.S.A. 53 (1965), 492-498, has been presented an approach to the theory of integration generated by a volume space (X, V, v) (Cf. also Abstract 630-92,

these cNoticriJ 13 (1966), 84.) Using the notation and the notions of the paper let v 1, v 2 be two vol­

umes defined, respectively, on pre-rings V 1' V 2 of the space X. Let Y be a Banach space. ~

rem 1. We have L(vl'Y) C L(v2,Y) and lifllv 1 = lifllv 2 for all f E L(vl'Y) iff (a) XA E L(v2,R) and v 1 (A) = J~ dv 2 for all A E V 1. Theorem 2. If the condition (a) is satisfied then the family N 1 of null-sets generated by the volume v 1 is contained in the family N2 of null-sets generated by the volume v 2 and the integrals coincide, that is Jfdv1 = ]fdv2 for all f E L(v l'Y). In Theorem 1 by R we denote the space of reals. It follows from the above theorems that integration generated by any measure on the Borel ring in the Euclidean space Em can be obtained by integration with respect to

a volume on the segment pre-ring, that is on the pre-ring consisting of all sets A= J 1 X ..• X Jm' where Jj = (aj,bj), a 1 ~ bj" This result will appear in Math. Ann. (Received January 26, 1966.)

632-12. W. G. STENGER, University of Maryland, College Park, Maryland, Institute for Fluid Dynamics and Applied Mathematics. A counterexample in the new maximum-minimum theory

of eigenvalues.

Let A be a compact, negative definite, symmetric operator on a real Hilbert space H, with

eigenvalues X1 ~ X2 ~ ..• and eigenvectors ul' u 2, ...• Let Pp p 2, ..• ,pn-l be any n- 1 vectors in H

327 and let ,\(p 1, p 2 , ... ,pn-l) be the minimum of the Rayleigh'quotient on the subspace orthogonal to

P1• P2 .... ,pn-l' Let >.m-l < .\ m; .\m+l; ... = .\n ~ ,\n+l ~ .... 2 ~ m ~ n, A. Weinstein, J, Math, Mech, 12 (1963 ), 23 5-246 has given necessary and sufficient conditions for satisfying the equality

.\(p1, p 2 , .. .,pn-l); .\n. In the case m; n, a necessary and sufficient condition for the above equality is that the quadratic form with the Weinstein determinant wm-1(.\n- E); det{L~l(pi'uj)(pk,uj)/ (.\j- .\n + d}. i,k; 1,2, .. ,, m- 1, is negative definite for all sufficiently small E > 0, When m < n, this condition is sufficient but it is not necessary as is shown by the following counterexample, Let

A1 < A2 ; A3 so that m; 2 and n; 3, Let p 1 ; u 2 and Pi; p 2 ; u 1 + au2 where a is not zero but is otherwise arbitrary. It is clear by inspection that A(p 1,p2); A3. On the other hand, a simple calcu­ lation shows that both W 1 (A 3 - f) and Wi (A 3 - t) are positive, (Received January 26, 1966.)

632-13, THEODOR GANEA, University of Washington, Seattle, Washington 98105, Induced fibrations and cofibrations,

Let F ~ E ~ B be a fibration in which B is (m - !)-connected and 1Tq(F) 'I 0 only if n ;;;; q ~ n + 2m - 2, where m ;;;; 1 and n ;;;; 1, If there is a homotopy equivalence 0: F ~ !>IY such that the composite F • !liB ~IF ~ I!>!Y --> Y is nullhomotopic, then the fibration is induced by some map B --> Y; the first map in the composite is the Hopf construction associated with the operation p: F x!i'IB --> F, the second is '!:0, the third sends (s, w) into w(s). Dually, let A- X ----> B be a co­ fibration in which A is (n- I)-connected, B is m-connected, HN(B) is free and Hq(B); 0 for q > N, where N ; m + n + Min (m,n). If there is a homotopy equivalence fl: IY----> B such that the composite Y--> !>IIY--> !liB ----> !i'I(BbiA) is nullhomotopic, then the cofibration is induced by some map Y -->A; the last map in the composite is the Hopf invariant of the cofibration as defined by the author in Comment, Math. Helv, 39 (1965), 295-322; the other maps are the canonical inclusion and !>!0, There are applications to principal fibrations and principal cofibrations. (Received January 26, 1966,)

632-14. M. G. NADKARNI, Washington University, St. Louis, Missouri 63130, Representation of stationary spectral measures,

Let l:i be a (complex) Hilbert space, Assume that H is separable, Let E be a spectral measure on the family B of Borel subsets of R, the real line. The values of E are orthogonal projections on subspaces of H. Definition, A spectral measure E is called stationary if there exists a commutative group Tt, t E R, of unitary operators such that TtE(u)T-t ; E(u + t) for every measurable set u E B. E is called C-stationary if Tt is strongly continuous, Stationary spectral measures are studied and canonical representations for C-stationary spectral measures are obtained, Let L 2 (R) be the Hilbert space of all B-measurable functions f on R such that jfj2 is summable, Let N be a cardinal number 2 ~ N0 • Let L~(R); {

328 632-15. A. C. SUGAR, Northern Michigan University, 401 West College Avenue, Marquette, Michigan. Neo-Newtonian dynamics. Preliminary report.

Relativistic dynamics is incomplete in the sense that it is only a point dynamics. It does not possess a rigid dynamics. Among other complications this makes for a duality principle for time. Time is therefore both relativistic and Newtonian. At the (Newtonian) time (1879) of the Michelson­ Morley experiment, the wave theoretic concept of light was dominant. A particle theory of light would also have led to their negative result for the anticipated phenomenon of interference. Conse­ quently, the Michelson-Morley experiment could have been interpreted rather as a justification of the particle concept and hence could have been construed as an empirical predecessor for the founda­ tion of quantum dynamics rather than for relativistic dynamics. The author's theory of gravitation, which also predicts the advance of the perihelion, will in conjunction with a particle theory confirm the deflection of light. Furthermore, this theory will generate a new theory of the potential. A non­ relativistic electrodynamics based on that developed by Vannevar Bush when added to the prior in­ gredients would lay the foundations for the development of a neo-Newtonian dynamics. (Received January 26, 1966.)

632-16. C. ]. HOUGHTON, State University of New York at Binghamton, Binghamton, New York 13901. Cyclic and fine-cyclic elements.

At the 1'958 Summer Institute on Surface Area and related topics, ]. W. T. Youngs suggested the following relationship between the work of Neugebauer [Trans. Amer. Math. Soc. 88 (1958), 121-136; Ill. J. Math. 2 (1958), 386-401] on B-sets and fine-cyclic elements and classical cyclic element theory. Theorem 1. Y is a Peano space with a finite degree of multicoherence iff there exists a Peano space X with the property that every B-set of X is an A-set and a finite collection of

two point identifications f = f 1,f2, ... ,fk such that fkfk-l ••. f 1 (X)= f(X) = Y. Theorem 2. Let f(X) = Y be as above. Then C is a fine-cyclic element of Y iff there is a proper cyclic element D of X such that f(D) = C. Furthermore f(D) = C is a homeomorphism. The proofs use middle-space topology where the middle spaces are finite-coherent regular curves which have a fairly simple structure, (Received January 27, 1966.)

632-17. E. J. BELTRAMI, Grumman Aircraft Engineering Company, Bethpage, Long Island, New York. A time domain characterization of linear scattering systems.

A linear scattering system is one describable by a linear time invariant and continuous

operator S on the space of vector valued and real Schwartz distributions 1l' ® R u such that the domain of S contains distributions of compact support and the dissipative con?ition J~oo [(u,nl - (Su,Sn))dt ~ 0

holds for all t and all u E C~ ® Rn. One then proves the Theorem If S describes a linear scatter­ ing system then Su = w * u where w is a real matrix valued distributional Greens kernel satisfying

(i) w E 'DL 2 n 'I) .j. ® ~R u ,Rn), (ii) b TOn o - w* w T)b is Bochner positive for each b E R n , (iii) S can be extended as a mapping of L 2 ® R u into itself. Conversely if w is the kernel of an operator S for which (i), (ii) and (iii) hold then S describes a linear scattering system. This theorem extends some earlier results of Youla, Castriota, and Carlin (IRE Trans, CT-6, 1959) and Wohlers and Beltrami (IEEE Trans, CT-12, 1965.) (Received January 27, 1966.)

329 632-18. SEYMOUR BACHMUTH and H. Y. MOCHIZUKI, University of California, Santa Barbara, California 93106. Ideals and automorphism of metabelian groups.

e 1 er Let F be the free group of rank q ~ 2. Let m : p 1 ••• Pr , the Pi distinct primes, U(m): F /F "(F ')m and A(G) the automorphisms of G which induce the identity on G/G'. Theorem. A(U(m)) - e 1 er ~ A(U(p 1 )) X ••• X A(U(pr )). A matrix representation due toW. Magnus (Ann. of Math. 40 (1939), 764-768), is used in the proof. As an application, a complete solution to the problem of determining when an automorphism of U(m) mod the n'th term of the lower central series is induced by an auto­ morphism of U(m) is given. This continues the investigations begun by Bachmuth (Trans. Amer. Math. Soc., to appear). (Received February 7, 1966.)

632-19. K. D. MAGILL, JR., SUNY at Buffalo, Buffalo, New York 14214. Semigroups structures for families of functions. II. Continuous functions.

Let X and Y be topological spaces and f a continuous function from Y onto X. Let ~*(X, Y, f) denote the semigroup of all continuous functions from X into Y where the binary operation is given by fg: f 0 fog for all f, gin ~*(X,Y,f ). (!*(U,V,g) is defined similarly. It follows that if I) and tare homeomorphisms from X onto U and Y onto V respectively such that I) o f : g o t, then the mapping cb defined by cb(f): t<>foi)is an isomorphism from onto ~*(U,V,g). There are examples, however, of semigroups ~*(X,Y,f) and ~*(U,V,g) and isomorphisms from the former onto the latter for which no such homeomorphisms exist. Conditions are given which insure that for each isomor­ phism ¢from

~ and Y onto V respectively such that I) o f : go t and cb(f): to fol) for each f in ~*(X, Y, p. These results are used to show that for certain ~*(X, Y •f ), the automorphism group is isomorphic to a cer­ tain subgroup of the group of all homeomorphisms on Y. Applications are then made to near- rings of continuous functions from a topological space into a topological group. Finally, semigroups of continuous functions whose domains are subsets of X are discussed. (Received February 7, 1966.)

632-20. JEROME SPANIER, Westinghouse Electric Corporation, Bettis Atomic Power Laboratory, West Mifflin, Pennsylvania 15122. Some results on transport theory and their application to Monte Carlo methods.

In the Monte Carlo estimation of weighted integrals of the solution of the neutron transport equation, a correspondence is established between the physical model and a probability model upon which the sampling is based. The integral transport operator .5e' is called subcritical if it is a bounded operator on L 1 (f) for some euclidean space r, and if its spectrum is contained in the open unit disk. The probability model is called subcritical if the number of collisions suffered by every particle is finite with probability one. It is shown that if 5f' is subcritical, so is the analog probability model based on .5e' and the source S, and the integral transport equation has a unique solution 1/1 which is the L 1-limit of the Neumann series. In fact, under the same hypotheses, it may be shown that the expected number of collisions is finite. A more useful criterion for subcriticality is also established, namely iiJtnii 1 ~ 0. Some extensions are given to nonanalog probability models, i.e., models which do not necessarily mimic the physical behavior of the neutrons. (Received February 7, 1966.)

330 632-21. EMIL GROSSW ALD, University of Pennsylvania, Philadelphia, Pennsylvania 19104. Oscillations of arithmetical functions.

Landau's result (Math. Ann. 59 (1905), 527- 550) permits to prove oscillation theorems for functions f(x), provided that F(s) = f(x)x -sdx (s = u + it) is regular at Its abscissa of holomorphic 0• .(~ 0 A sharpening of Landau's result, dispensing with the requirement of regularity at 0 (Israel J, Math., to appear) permits to prove oscillation theorems for a large class of arithmetical functions f(x). Examples: (I) Let \(s) be Riemann's zeta function, set 0"' sup{uil(u+ it)= OJ; let W(n) = Lpln1 and " w(n) -2 r g(x) = L....n:5x2 - 611" x(log x +a) (a= 2')'- 1- 2~'(2)/1(2), 'Y= Euler constant). Then, for any f >0, - 0/2- f 0/2-f C > 0, each of the two inequalities g(x) > Cx , g(x) < - Cx holds on an Infinite set of Inte- gers W = {xn}; moreover, if W(y) = Lxn~Y1, then lim supW(y)/(logy) >0. Also, g(x) = n+(x 1/ 4 ). (II) Let cn(m) be Ramanujan's sum, Sm(x) =Ln,;xcn(m); then each of the inequalities Sm(x) > Cxli-f, Sm (x) < - c/1- f holds on a set W of integers as -under (I). (III) Many results of "comparative prime number theory" obtained by Knapowskl and Tur11'n, and some new ones also follow. (Received February 8, 1966,)

632-22. G. A. GRATZER, The Pennsylvania State University, University Park, Pennsylvania 16802. On the spectrum of classes of algebras. Preliminary report.

Let K be a class of algebras, N the set of positive integers. The spectrum of K, Sp(K) is the set of orders of finite algebras InK (see T. Evans, Abstract 627-40, these cNotiaiJ 12 (1965), 798). Theorem 1. S <:;;:; N is the spectrum of an equational class K (defined by an arbitrary set of identities) if and only If 1 E S and S • S <:;;:; S. Theorem 2. Every S <:;;:; N is the spectrum of a universal class K (defined by an arbitrary set of universal sentences). Theorem 3. Let K be an equational class defined

by a finite set of identities. Then there exists an equational class K 1 defined by four identities such

that Sp(K) = Sp(K 1). Theorem 4. There exists an equational class K such that Sp(K) = Sp(K1) for no equational class K 1 defined by a finite set of identities. (Received February 8, 1966.)

632-23. G. H. WENZEL, Pennsylvania State University, University Park, Pennsylvania 16802. A characterization of v*-algebras.

A universal algebra ~ = (AiF) is called a v*-algebra if it satisfies the following conditions: (I) If a E A Is not a polynomial constant in 0£. then a is self-independent. (II) If {al'"''anJ is an

independent subset of A and [a 1, ... ,an• an+ 1J is not an independent subset of A, then an+ 1 E [al' ... ,anJ· Independence is in the sense of Marczewski (Bull. Acad. Polan. Sci. 12 (1958), 733). {}[: (AiF) is a v**-algebra if the Sf-independence of a nonempty subset B <:;;:;A (i.e. b r/:c [B- {b}] for all bE B) implies the independence of B in Oi. Theorem. Let cJi-be a v**-algebra. Then CtL is a v*-algebra

if and only if every maximally Independent subset of Oi is a basis. This solves a problem of W. Narkiewicz (Fund. Math. 54 (1964), 124). (Received February 8, 1966.)

632-24. D. W. SASSER and M. L. SLATER, Sandia Corporation, Albuquerque, New Mexico 87115. A generalization of the ver der Waerden conjecture.

For any m X n matrix A and 1 ~ k ~ min(m,n), let Pk(A) be the sum of the permanents of all

331 k-square submatrices. Let nn be the class of n-square doubly stochastic matrices and let Jn be the matrix in Un with all entries equal. Conjecture. If A E rln' At Jn and 1 < k ~ n, then Pk(A) > F}c (Jn). The van der Waerden conjecture is k = n. Theorem 1. If A E nn, A t Jn and IIA - Jn II < fn then

P k (A) > P k (Jn)' 1 < k ~ n. Theorem 2. If 1 < k ~ n, and for some A E int(Un), Pk(A) :,; Pk(S) for all S E nn w~th IIA - S II < fn' then A= J 11" Theorem 3. If A E rln' At Jn and A is symmetric positive semi-definite, then Pk(A) > Pk(Jn), 1 < k ~ n. Theorem 4. If k = 2 or 3, n ?; k, A E nn, At Jn' then Pk(A) > Pk(Jn). These theorems generalize results due to M. Marcus, H. Mine, M. Newman and others (see Permanents, Amer. Math. Monthly 72 (1965), 577-591). Theorem 1 is a consequence of 2 2 fk (A) = L Pk (Ak)P 1 (Ak) - (k /n ) Pk (A) P 1 (A) > 0 for 1 ~ k ~ n - 1 and~ (not only doubly stochastic) A in 0 < IIA- En II < fn' where the sum is over all k-square minors of A, and En= njn. For A non­ negative it is known that fk(A) ~ 0 for (a) k = 1, n ?, 1, (b) k = 2, n = 3, and (c) IIA- En II < fn; fk(A) ?; 0 for all n ?; k <; 1 would easily imply the conjecture. (Received February 8, 1966.)

632-25. H. F. BECHTELL, Bucknell University, Lewisburg, Pennsylvania 17837. A note on solvable K-groups.

A finite K-group is a group having a complemented subgroup lattice. It is established that with respect to the general linear group GL(n,P), P a field of p elements, that a primitive solvable K-group is a semidirect product of a direct product of elementary Abelian groups by a cyclic group of square-free order. Furthermore a K-group G that is a maximal solvable irreducible subgroup of GL(n,P) iii a semidirect product of a subgroup A by a subgroup B such that (1) A is the direct product of k copies of a group H, H a semidirect product of a cyclic group of square-free order by a cyclic group of square-free order and (2) B is a K-group that is isomorphic to a certain maximal transitive subgroup of the symmetric group of degree k. The investigation is motivated by the fact that each center less solvable K-group G is a subgroup of a direct product of solvable K-groups that are isomor­ phic to subgroups of G and that are also primitive permutation groups. (Received February 8, 1966.)

632-26. LEON NOWER, San Diego State College, San Diego, California 92115. On division of tempered distributions by polynomials.

Let

2 by induction. Assuming

Ek' E 02, 0 ~ k' ;;; p', and k = jk'l = k 1 + k 2, which reduces the problem to one of defining E • 1/rk for E E 0 2• Assuming f to be monic inx, one has the global factorization f(x,y) = TI~= 1 (x- a)y)), ai E 01" 1 1 Let E = E 0 ; it is shown that if Ei_ 1 = D~i- )(k+ )Hi_ 1 , with Hi_ 1 E 0 2, 1 ~ i ~ n, then 3 a polynomial 1r.(y) and an HiE 0 3 E. • 1r./(x- a.)k = pi(k+ 1>H. = Ei. With T = JJ? 1ri' this gives E • r;rk = E 1 2 1- 1 1 1 x 1 1= 1 n as a tempered distribution of order ~ n (k + 1). Finally, E • 1/fk is obtained when En • 1/r is constructed and shown to be a tempered distribution of order§. n(k + 1) + ZN, where N is the degree of r. (Received February 9, 1966.)

332 632-27. M. I. GOULD and G. A. GRATZER, Pennsylvania State University, University Park, Pennsylvania. Boolean extensions and normal subdirect powers of finite universal algebras.

A. L. Foster showed the equality, up to isomorphism, of the class of Boolean extensions and the class of normal subdirect powers, of a finite f- algebra. We establish this result for arbitrary finite algebras, using the following definition (which coincides with Foster's when rJi is a finite f-algebra);

(;\: contains the diagonal and is closed under the normal transform T, defined by: T(a.,/J,-y ,o} = t means t(i) = 'Y(i) if a(i) = ,B(i) and t(i) = o(i) if a.(i) i ,B(i). We show that if di[\B] is the extension of (}£by the Boolean algebra ~. and I is the set of atoms of an atomic Boolean algebra containing \B, then a normal subdirect power isomorphic to Of[~] is the subalgebra of rJ£1 consisting of the maps 'Cr'defined, for a E A[B], by Ci'(i) = a iff i ~a a. For the converse, we choose the Boolean algebra of sets of the form [ija.(i) = aJ where a E A 1 and a EA. We also establish, without using f-algebras, Foster's theorems on extension of algebraic identities and existence (and uniqueness) of subdirect representation, (Received February 10, 19"66.)

632-28. MURRAY EISENBERG, University of Massachusetts, Amherst, Massachusetts 01003. Positively expansive endomorphisms of compact groups.

An unpublished result of T. S. Wu asserts that a compact connected finite~dimensional group admitting an expansive automorphism is abelian. Call an endomorphism f of a topological group G positively expansive if there is some neighborhood U of the identity e of G such that, given x E G with xi e, then xfi ¢ U for some nonnegative integer i. Theorem. If a compact connected finite­ dimensional group G admits a positively expansive continuous surjective endomorphism, then G is abelian. To prove the theorem, an expansive shift transformation of an inverse limit of copies of G is constructed, and Wu's result is applied, (Received February 10, 1966.)

632-29. R. T. IVES, Harvey Mudd College, Claremont, California 91711. Radon and Helly theorems for finite dimensional unbounded sets.

The affine span of a set A, (flat A), is the smallest flat that contains A. The set A is affine independent iff VxEA x ~ flat (A~ x). Theorem. In an arbitrary vector space, a set is affine depen­ dent iff it can be partitioned into two disjoint subsets with overlapping convex hulls. Suppose c/1- is a family of convex subsets of an n-dimensional space and C is a k-dimensional cone such that each set in $contains a translate of C. Theorem. If cJf contains at least n - k + 2 members, then subfamilies

_<;81 and ~2 exist such that .9!11 n ~2 is empty and conv( U ~ 1 ) hits conv( U ~ ~· Theorem. If cf1 is finite with at least n - k + l members and each subfamily of n - k + l sets has a nonempty inter­ section, then ell has a nonempty intersection. (Received February 11, 1966.)

632-30. Y. KUO, The University of Tennessee, Knoxville, Tennessee 37916. Elementary divisor theorem for direct sum and tensor products of modules,

Let R be a commutative Euclidean ring. The matrix of diagonal form in the Elementary Divisor Theorem [van der Waerden, Modern algebra, Vol. II, 106-109] is abbreviated as the matrix of

333 ED. Let dkk be the nonzero entry at (k,k)-position of the matrix of ED. Each individual matrix in the direct sum or the tensor products of matrices is called a component matrix. Theore-m. Any nonzero entry dkk in the matrix of ED, for the direct sum or for the tensor products of a finite number of modules of linear forms with respect to R, can be written in the form hk/hk-l (h0 is defined to be the identity of R). For the direct sum, hk is the g.c.d. of all possible products of k number of non­ zero entries, where the nonzero entries are chosen from the component matrices of ED. For the tensor products, let fj be a product of nonzero entries, where the nonzero entries are chosen the way that exactly one nonzero entry comes from each component matrix of ED. Then hk is the g.c.d. of all possible products of k number of~ (j = l, .•• ,k). (Received February 11, 1966.)

632-31. JACQUES LEWIN, Syracuse University, Syracuse, New York. Subrings of finite index in finitely generated rings.

Let S be a subring of the finitely generated ring R. The index of S is the order of the quotient group R/S. The following analogues of well known theorems of group theory are proved: Lemma. If S has finite index then S contains an ideal of R of finite index. Theorem. If S has finite index then S is finitely generated. Theorem. R contains only finitely many subrings whose index is a given positive integer. (Received February 11, 1966.)

632-32. M. W. MANDELKER, University of Rochester, Rochester, New York. Prime z-ideals in C (R). I.

9 denotes the family of all prime z-ideals in the ring C(R) of all real-valued continuous func­ tions on R. Theorem 1. (Extending a result of Kohls.) A prime z-ideal is minimal if and only if the zero-set of each member has nonvoid interior. Theorem 2. The intersection of any countable chain in 9 is nonminimal. Theorem 3. There is a decreasing w1-sequence in _9 with nonminimal intersection. Theorem 4. For every countable ordinal J.l, there is an increasing p-sequence in 9. Theorem 5. Each nonminimal prime z-ideal has a family of predecessors that is order-isomorphic with the family of~prime z-ideals. Theorem 6. Let p E 13 R. The following are equivalent. (a) p is a remote point. (The continuum hypothesis implies the existence of remote points [Fine and Gill­ man, Proc. Amer. Math. Soc. 13 (1962), 29-36].) (b) The maximal ideal MP of C(R) contains no other prime ideal. (c) The z-ideal oP is prime. Theorem 7. Assume the continuum hypothesis and let a i 0 be a countable ordinal. There is a maximal chain in 9 of type a • if and only if a is not of the form ,\ or ,\ + 1 for a limit ordinal ,\. The structure of 9 in the vicinity of a given prime z-ldeal will be considered in more detail in part II, see Abstract 634-33. (Received February 11, 1966.)

632-33. S. -T. C. MOY, Syracuse University, Syracuse, New York 13210. Ergodic properties of expectation matrices of a Markov branchin& process with countably many types.

Let I be a set containing countably many elements and let (Mij), i, j E I be an infinite matrix with nonnegative entries. Let (M~j)> = (Mij)n. (Mij) is assumed to be irreducible. Let<;!. be the period or the cyclic Index of (M1j). In the following are some main theorems. (1) There is a number !. such that for all i, j E I L:: 1 Mfj>sn converges for Is I < r and diverges for Is I > r and there are "oo (n) n "oo (n) n . only two possible cases. Case I. L...n= 1M ij r < oo for all i,j. Case II. L...n= 1Mij r = oo for all i,J.

334 (nd) nd (2) For Case II. Mjj r __, 1r(J) as n __, oo and either 1r(j) > 0 for all j or 1r(j) = 0 for all j. There are positive functions u(i), v(i) such that (1) r L;JMiju(j) = u(i) for all i, (2) r Li v(i) Mij = v(j) for all j, and any nonnegative solution of either (1) or (2) is a constant multiple of u or v. We also have Li u(i)v(i) < oo or L;iu(i)v(i) = oo according as tr{j) > 0 or 1r(j) = 0. (3) For Case II with 1r(j) > 0 and d = 1, we have rni:j M}j)f(j)----> -yu(i) for all i iff satisfies ~ v(i)jf(i)J < oo and rni:i f(i) M~J)----> ov(j) for all j iff satisfies Li u(i) jf(!) J < oo. -y, o are constants. (Received November 22, 1965.)

632-34. W. W. SMITH, University of North Carolina, Chapel Hill, North Carolina and H. S. BUTTS, Louisiana State University, Baton Rouge, Louisiana. P;riifer rings.

Let R be a commutative ring with unity. R is a Priifer ring if the ideals are linearly ordered in the quotient ring 11> for each proper prime ideal P of R. jensen (Proc. Amer. Math. Soc. 15 (1964), 951) has shown a Priifer ring is an arithmetical ring. We consider the following nine conditions, where A, B, and C represent ideals of R: (I) R is a Prufer ring. (II) If A C B and B is finitely gen­ erated then there exists an ideal C such that A= BC. (Ill) If C is finitely-generated then (A + B): C = A: C + B: C for any A and B. (IV) If A and B are finitely generated then C: (A riB)= C: A+ C: B for any C. (V) A = () Aec for each A where the intersection is taken over all quotient rings of R which have linearly ordered ideal systems, and e and c denote ideal extension and contraction, re- spectively. (VI) If one of B or Cis regular then A(B nC) = AB nAC. (VII) If one of A orB is regu­ lar then (A+ B)(A nB) = AB. (VIII) Finitely-generated regular ideals are invertible. (IX) If A is finitely-generated and regular and AB = AC then B = C. Theorem. Properties I through V are equivalent and VI through IX are equivalent. Any ring satisfying I through V will satisfy VI through IX. (Received February 14, 1966.)

632-35. j. j. UHL, Carnegie Institute of Technology, Pittsburgh, Pennsylvania. A representation of bounded linear operators on Orlicz spaces. Preliminary report.

Let (n.~.ll) be the triple, ~ a ring of subsets of n,ll a finitely additive measure on ~. and ~O C ~ the ring of sets of finite 11-measute. A partition ll. is a finite disjoint collection [Ev} in ~ 0 •

ll.1 ;:;; ll. 2 iff each E in ll.1 is a union of sets of ll. 2• Let X andY be B-spaces; B(X,Y) be the B-space of bounded linear operators from X toY. If is a Young's Function, the space W(B(x,y)) is the collection of finitely additive B(X,Y)-valued set functions on ~O satisfying sup)inf[k > 0:

supll. ~ll.((Jiy *F(Ev)Jix*/kll(Ev)) i!(Ev));:;; 1], y• E Y*, Jly•JI;:;; 1] = IJFIJw< oo. II II is a norm). Theorem. If (IF) are complementary Young's functions, continuous, M C L(n.~.ll.X) is the closed subspace determined by wsimple functions, then B(Mlr(B(X,Y)).

If T E B(M, Y) there is G E w>lr satisfying (*)T(f) = limll.~ll. {G(Ev>EJE/di!J/i!(Ev)J. Conversely every G EW>Ir (B(X,Y)) defines aT in (B(X,Y)) through(*). Corollary. If(2x);:;; M(x), then Wi'(B(X,Y)) is isometrically isomorphic to B(L, Y). Spe_fialized results are obtained. (Received February 14, 1966.)

632-36. j. R. RETHERFORD, Louisiana State Unitersity, Baton Rouge, Louisiana. Quasi­

reflexivity and weakly unconditionally convergent series.

Write ( X,P) if a Banach space X possesses a property P. We say that X has (1) property Vn if there is a total normed closed subspace F C X •• such that co dim F = n and if K C X • and satisfies

335 (*)limp supx*EK x*(xp) = 0 for every w.u.c. series LpXp in X then K is w(X*,F)-sequentially com­ pact; (2) property v; if there is a total norm closed subspace G C X* such that codimx•G = n and if K C X satisfies(**) limpsupxEKxf,(x) = 0 for every w.u.c. series LpXp in X* then K is w(X,G)­ sequentially compact; property wen if there is a Gas in (2) such that every w(X,X*) Cauchy sequence in X is w(X,G) convergent to some element of X. In all three cases n is the smallest nonnegative integer yielding a subspace with the desired properties. Finally X has property 0 if w.u.c. series are u.c. Motivated by previous work of A. Pelczynski we prove the following: (A) (X, Vn ) ~ (x•,v~); (X,V~) ~ (X,Wcn) =>(X,O); (B) (X,V~) and(X*,O) ~Xisquasi-reflexiveoforder n; (C) (X, Vn) and (X,O) 9 X is quasi-reflexive of order n; (D) X is quasi-reflexive of order n ~(X,Vn) and (x,v; ). (Received February 14, 1966.)

632-37. KAZUMI NAKANO, Wayne State University, Detroit, Michigan 48202. Function systems.

A system of bounded functions on a spaceS is called a unit if [cf>(x); cf> E <1>} is a bounded set for each xES. A collection of units is called a unity. For two unities 6£, £,we define ;!; -< ~ if for any E dS and for any f. > 0, we can find o > 0 and a finite number of units >¥1, 'l-2, ••• ,'{In E M such that I ~(x) - ~(y) I < ofor all ~ E U ~= 1 'l'i implies ltf>(x) - cf>(y) I < f for all cf> E <1>. We define L9f ~ Z if (Ji-< ;G and tJt >- ;6 simultaneously. A unity (}£is called simple if every unit of tYt consists of a single function. A directed system a0 E S (o E A) is called a Cauchy system if for any E rJt­ and f > 0, there exists o0 E .:l such that lcf>(a 0 )- cj>(a0 )1 < f for all t/> E <1>, and for all o,o' ~ o0• We can prove a unity (J[ is equivalent to a simple unity if and only if any directed system contains a partial Cauchy system. (Received February 14, 1966.)

632-38. SRISAKDI CHARMONMAN and H. R. BASTEL, McMaster University, Hamilton,

Ontario, Canada. Eigenvalue problems of a 2n X 2n~.

B. Friedman proved in Eigenvalues of compound matrices [New York University, Mathematics Research Group, Research Rept. No. TW-16 (1951)] that if A and B are real square matrices of order nand S = CAB, BA' then A(S), the eigenvalues of S, is the set of 2n numbers A(P) and A(Q) where P = A + B, and Q = A - B. In the present paper we give a simple proof and three extensions: (I). An algorithm to find the eigenvectors of S in terms of the eigenvectors of P and Q. (II). If S is a cyclic permutation matrix of order 2in, where i is a positive integer, then .\(S) is the set of 2in eigenvalues of matrices of the type Q of orders 2i- 1n, 2i- 2n, ••• ,2n,n, and a matrix of the type P of order n. (III). If R = CAB,BTA, then A(R) is the set of 2n numbers A(A +C) and A(A- C), where c 2 = B T B. For the coefficient matrix of the five-point difference equation approximating Laplace's equation in a rectangular domain, C is obtained by inspection. We found that the use of the smaller matrices P and Q is superior to the use of the original matrix in view of the effect of round-off error as well as the computer storage and the number of computational operations required. (Received February 14, 1966.)

336 632-39. VOLODYMYR BOHUN-CHUDYNIV, Seton Hall University, South Orange, New Jersey. On the generalized (double) triple-systems with some applications.

In Abstract 622-61, these cNOticeiJ 12 (1965), 344 was established a general method of con­ structing triple-systems of any order n = 6q + r (r = 1,2) (a) without exception, and triple-systems of 25 and 85 orders were constructed, which cannot be constructed by Netto's and others' particular methods, including the author's particular methods (1961, 1962, 1964). In this paper the author gen­ eralizes triple-systems by including ordering, and establishes a method to construct them, finding some applications to loop theory. It is proved: (1) The order of the generalized triple-system (GTS) has to satisfy relation (a) and has to be a prime number. Each triple is one-sided, (2) The GTS con­ sists of two sets (double) of triple-systems of the same order. (3) In case n = 7 each of two triple­ systems with additional relations can be used to represent binary composition of Cayley algebra and Moufang loop. (4) Each double t:rip1e-system of order n with additional relations can represent binary composition of the loop of order n + 1 and idempotent quasi-loop of order n. (5) Multiplication of any three elements belonging to the same triple is completely semiassociative. Generalized triple­ systems of orders 7, 13, 19, and 31 are constructed. The double triple-system was obtained as a separate case of the n-tuple system. The paper on methods of constructing n-tuples with some ap­ plications will be presented to the Canadian Mathematical Congress in June 1966. (Received February 14, 1966.)

632-40. H. M. SCHAERF, McGill University, Montreal, Canada, Invariant probability measures for temporally homogeneous Markov processes. Preliminary report.

For each member g of a semigroup G let Pg be a finitely additive transition probability of a temporally homogeneous Markov process on a algebra S of subsets of a set X. Theorem. If G is left amenable (in particular: abelian), then there is for every finitely additive probability measure 1.1 invariant under all P (i.e, such that (x,A)JL(dx) = l.l(A) for m on S a like probability measure g fxP g all A E S, g E G) and such that JL(A) is between the extreme bounds of the set of all numbers fxp g(x,A) m(dx) with g E G for all A E: S provided JX ph (y ,A)P g(x,dy) = Phg(x,A) for each x E: X, A E S, g, h E G. This result, whose specialization to the semigroup G of positive integers is related to one of Y. Ito (Trans. Amer. Math. Soc. 110 (1964), 152-184), is proved by a method similar to the one in the author's paper on the existence of measures invariant under set mappings (Bull. Amer. Math. Soc. 67 (1961), 504-507). (Received February 15, 1966.)

337 632-41. WITHDRAWN.

632-42. R. H. PROSL and PAUL SLEPIAN, Rensselaer Polytechnic Institute, Troy, New York. Compact and product bases.

This paper strengthens the conjecture that topological bases constitute adequate structures in which to study the topics of "point set topology". Bentley and Slepian (see Abstract 622-12, these c#oticei) 12 (1965), 328) studied continuity and limits in the basis setting. The present paper extends these investigations, defines compact bases and product bases, and shows that, where these topics are concerned, it is possible to prove analogues of established topological theorems including the Alexander Theorem and the Tychonov Theorem. Analogues of the usual topological separation axioms are presented, definitions of ~\. regular, completely regular, normal, and separable bases are given, and the operation of forming product bases from the sundry types of bases defined is shown to yield results which are formally the same as the ones obtained in the analogous topological situation. (Received February 15, 1966.)

632-43. A. G. AZPEITIA, 650 Huntington Avenue, Apartment 20J, Boston, Massachusetts. Linear separability of switching functions.

Let rn be the unit cube in En. A switching function f(x): rn ____. {o,Ij is said to be linearly separable or realizable iff there is a polynomial p(x) = L~=OCmxm: En ____. R' such that for all x Ern p(x) ~ 1 iff f(x) = 1. A minimal weight realization of f(x) is any realization p(x) which minimizes L~=O JCm J. The Crn may or may not be required to be integers. Theorem: _!!_f(x) and f 0 (x) are switching functions on rn such that f(x) is not realizable and that fx jf(x) = f 0 (x) = 0 J i IJ, then there are two switching functions f 1(x) and f 2 (x) such that f(x) = (f0 (x)vf1(x))Alz (x) and that each h -1 -1 -1 one oft e sets f 1 (1} and f 2 (1) has more elements than f (1). An algoritl!.m is established that (i) Supplies a realization (minimal if required) of any realizable function, and (ii) By applying the

Theorem and solving a finite sequence of linear programs, gives a representation of any switching function as a Boolean polynomial of realizable functions each of which may be required to satisfy additional minimality conditions. (Received February 16, 1966.)

632-44. D. W. DEAN, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43'210. Schauder decompositions in (rn).

A decomposition of the Banach space E is a sequence of subspaces Bi such that each b in B has a unique representation b = :Lfb·i' bi in Bi' and the convergence is in norm. It is a Schauder de­ composition if the associated projections Pib = bi' for each bin B, are continuous. Let B have the following properties: (1) If limnfn = f weakly* where (~) C B ', the conjugate space of B, then lirnnfn = f

338 weakly (weak* convergence of a sequence to f implies weak convergence of the sequence to f). (2) If X andY are Banach spaces and if X ----+ Us ----+ Vy where U and V are weakly compact opera­ tors, then VU is compact. Theorem. If X has properties (1) and (2) then it does not have a Schauder decomposition. Corollary. If H is compact, Hausdorff, and extremally disconnected then C (H) does not have a Schauder decomposition. ln particular (m) has no such decomposition. (Received February 16, 1966.)

632-.45. C. R. CASSITY, Roland F. Beers, Incorporated, P. 0. Box 264, Alexandria, Virginia 22313. Two integration formulas of modified Runge-Kutta type.

Two formulas of modified Runge-Kutta type are developed. The modification consists of the addition of terms involving the finite difference Y 0 - Y _ 1 and the function value f_ 1 to each Runge­ Kutta estimate of the increment. A fifth-order formula requires three evaluations of f; a sixth­ order formula requires four evaluations. Each form11la contains two free parameters. (Received November 26, 1965.)

632-46. J.P. CRAWFORD, Lafayette College, Easton, Pennsylvania. Summation of bounded divergent sequences and the second adjoint matrix.

Let B(X, Y) be the set of bounded linear transformations from the Banach space X to the Banach space Y. Consider those TE B(X, Y) such that (•): T**-l(Y) = X, where Xindicates the natural embedding of X in x**. (•) resembles the characterization of weakly compact operators given in (Dunford and Schwartz, Linear operators, Part 1, Interscience, New York, 1964, Theorem VI.4.2). Theorem 1. If T has closed range and finite dimensional kernel, then T satisfies (•). Theorem 2. If X= Y = c and Tis a conservative matrix, then (•) is satisfied if and only if T sums no bounded divergent sequence. Theorem 3. Let T be a conservative matrix. In order that T sum no bounded divergent sequence it is necessary and sufficient that T have closed range and finite di­ mensional kernal. Sufficiency follows immediately from Theorems 1 and 2. (Received February 16,

1966.)

632-47. R. B. KELMAN and N. K. MADSEN. University of Maryland, College Park, Maryland 20742. Bounded solutions for the equation of the linear adiabatic osc!llator.

The equation (*)y" + (1 + f +h cos 211x)y = 0 is studied. Theorem. Let h be a function of bounded variation on [O,oo) such that h = o(l) and let c = inf [niJg>lhl"dx < oo, n > 0} < oo. Let f be absolutely integrable on [O,oo]. Let 11 satisfy one of the following conditions: (i) 11 is irrational; (!1) 11 = p/q where p and q are coprime positive integers such that p > c or q > c; (iii) 11 = p/q where p and q are coprime positive integers such that p > q, p + q = 1 (mod 2), and p > c - 1. Then each solution of equation (*) and its first derivative are bounded as x ----+oo. ln fact there is a unique correspondence between all pairs of real numbers (kl'k2) and all solutions such that each solution y satisfies the 2 relation y = k 1 sin (x + :E1:SJ

339 632-48. A. B. MANASTER, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. Full co-ordinals of recursive equivalence types.

For basic definitions see j.C.E. Dekker and j. Myhill, Recursive equivalence types [University of California, N. S. vol. 3, No.3, pp. 67-214] and j. N. Crossley, Constructive order types. I [Formal systems and recursive functions, Amsterdam, 1963]. A co-ordinal is called full if every initial seg­ ment of it is a predecessor of it. (cf. Crossley, Theorem IV. 5.4.) The field of a co-ordinal is, in a natural sense, an R.E. T. Associate with each R.E. T., X, the set, FO(X), of ordinals of full co-ordinals with field X. For infinite X, either FO(X) "' [w, w1) where w1 is the first uncountable ordinal, or FO(X)"' [w,w" • n) for some countable n > 0 and finite n > 1. If X is not an isol, FO(X)"' [w, w1). For each countable n > 0 and each finite n > 1, there are !sols X for which FO(X) = [w,w" • n). (Received February 16, 1966.)

632-49. KWANGIL KOH, North Carolina State University, Raleigh, North Carolina 27607. Weak radical of a ring.

If I is a proper right ideal of a ring R then we say I is almost maximal provided that it is irreducible and (l) if a E R and a - 1I > I then a E I (!!) if N(I) is the normalizer of I in R and J is a right ideal which contains I properly then N (I) n J > I and if a - 1 J ~ I then a- 1J > I for any a E R. It is known that a ring R is weakly transitive (See Abstract 622-78, these c}/oticei) 12 (1965), 447) if and only if R contains a faithful almost maximal right ideal. Let ~ R be the family of all almost maximal right ideal in a ring R for each In E l:R, let S(In) = fx E R IRx ;:;; InS· We define W(R) =

nrnE!R S(In). In case ~R is a vacuous then we define W(R)"' R. Theorem A. W(R/W(R))"' 0. Theorem B. If U(R) is the upper radical or F and J(R) is the jacobson radical of R then U(R) ;:;; W(R) ;;; J(R) and there exists a ring R such that W(R) < J(R). Theorem C. If a E R, In E ~R then either a- 11 "'R or a- 11 E ~Rand W(R) = n 1 E}'_l • Conjecture. U(R)"' W(R). (Received February 16, a ~ a -K a 1966.)

632-50. W. A. BEYER and LEON HELLER, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87544. Analytic continuation of Laplace transforms by means of asymptotic series.

Conditions under which a Laplace transform L fF(t)} = f(s) may be analytically continued, by means of an asymptotic expansion of F, outside the half plane of convergence of the Laplace transform integral are investigated. It is supposed that F(t) = t-il{L,~,. 0 ant-n + RN(t)j for some fixed iJ and that F is integrable on (O,k) for some k > 0. First it is shown that if RN(t) "' 0 fN! (u/t)N -l} uniformly in N and t E (k,oo) for some u > 0, then the analytic nature off at s "' 0 can be dete'imlned in terms of a 1• If iJ is a negative integer or zero, then in some neighborhood of s "' 0 1 it is shown that L [F (t)} =s iJ- L;!0ai -iJI'(l - 1 - /))s i + (log s)g(s) + h(s) where g and h are analytic at s"' 0 and g(s)"' LF: 1<- 1)1aisi-l/(1- 1)! If (3is not an integer, in some neighborhood of s"' 0 it is shown that LfF(t)} "'s(l-lg(s)+ h(s) where g(s)"' L~Oaif(l-!- iJ)si. Second, if the estimate on

~ (t) holds uniformly in N and in the complex t plane in the region (It I > k) n (larg t I < 7r/2 + A) for some A > 0, then the analytic continuation off to the entire s plane except s "' 0 can be determined

340 k' -st in terms of the ai. For any k' > k we have L£F(t)J = J0 e F(t)dt + f~ [a(t} f(2 - {3. k' (s + t})/(s + t}z- {3)dt, where r is the incomplete gamma function and

a(t} = L~oaiti /i!. (Received February 16, 1966.)

632-51. R. E. O'MALLEY, JR., University of North Carolina, Chapel Hill, North Carolina. Two- parameter second order problems in singular perturbations.

Generalizing questions discussed previously by Wasow, Latta, Visik and Lyusternik, et al.,

consider boundary value problems for equation (•) E y" + 1-!a(x)y' + b(x)y = 0 on 0 ~ x ~ 1 where a(x) i 0 'I b(x) and E and ll are small interrelated parameters which simultaneously approach zero. (•) has a pair of fundamental solutions which, in general, feature exponential decay determined primarily by the singular (as 1-1 and E ----> 0) roots of the auxiliary polynomial E 0 2 + 1-1a(x)D + b(x) = 0. Behavior is considerably different in the cases (1) E/1-1 2 _, 0 as 1-1-> 0 and (2) 1-1 2/E----> 0 as E ---->0. In case (1), e.g., the solutions of the initial-value problem when a(x) and b(x)/a(x) are positive, the terminal value problem when a(x) and b(x)/a(x) are negative, and any problem which prescribes at each endpoint either y or y' when a(x) and b(x)/a(x) have oppposite signs will all converge non uniformly (as E, ll ____, 0) to zero (the solution of the "reduced equation"). Complete asymptotic solutions including "boundary layer terms" for representative problems are obtained and formal results are carefully proved valid. Extensions to the nonhomogeneous problems, higher order equations, and to partial differential equations have been achieved. (Received February 17, 1966.)

632-52. NATHANIEL CHAFEE, Brown University, Providence, Rhode Island 02912. The bifurcation of an equilibrium point into a closed orbit.

In this paper one considers an autonomous ordinary differential equation in En for which the origin is an equilibrium point. Corresponding to the linear part of this equation there are n eigen­ values two of which one assumes are simple and complex conjugate. It is also assumed that as a certain parameter varies these two eigenvalues cross the imaginary axis from left to right and the other eigenvalues remain to the left of the axis. Thus, the equation being considered changes its state of genericity as the parameter varies. At the point where the equation is nongeneric, i.e., where the two eigenvalues cross the imaginary axis, one assumes that the origin is asymptotically stable. Under these conditions one investigates the qualitative behavior of solutions near the origin. The fundamental result is that as the two complex conjugate eigenvalues move into the right half complex plane the origin bifurcates into a closed orbit. This orbit lies within a two dimensional exponentially stable local integral manifold and is asymptotically orbitally stable from the inside with respect to solutions in this manifold. A similar problem is considered for a retarded functional differential equation with finite time lag and analogous results are obtained. (Received February 17, 19n6.)

632-53. ANATOLE BECK, University of Wisconsin, Madison, Wisconsin. The linear search

problem.

I am located at the point 0 of the real line and I wish to locate an object which is placed some­ where on the real line according to a certain probability distribution. I can move in either direction,

341 with a fixed speed, and turn around wherever I please, How should I search so as to minimize the expected time spent? The foregoing question, which generalizes a common experience, is known as the linear search problem, This problem is still unsolved, but something is known about the exist­ ence of best strategies, and estimates exist on the size of the minimum (or infimum) of the expected time losses. Treated as a game between the seeker and the originator of the probability distribution, the problem has some surprising aspects. (Received February 17, 1966.)

632-54. W. ]. SCHNEIDER, Syracuse University, Syracuse, New York 13210, Boundary behavior of conformal maps from simply connected domains onto half planes.

Theorem 1 (Extension of Caratheorory's theorem on behavior of conformal maps at corners). Let D be a simply connected domain in the UHP bounded by two continuums C 1 and C 2 each joining the origin to the point at infinity. Furthermore for any f > 0 let C 1 (C 2) be eventually contained in

[ z lu1 - f < arg z < n2 + ~ ( [z ln5 - f < arg z < u6 + f}) w.bere 0 ::;; n 1 ;;;; n2 < n 3 ::;; n 4 < n 5 ::;; n 6 ::;; 1r, Let A be an arc in D terminating at the origin which for any f > 0 eventually remains within

{ z ln3 - f < arg z < n4 + f J. Let f(z) be a conformal map of D onto the UHP, continuous at 0, with f(O) = 0, Then for any f > 0, f(A) is eventually contained in {w l((n3 - n2)/(n6 - n2)) 1r- f < arg w <

((n4 - n1)/(n5 - n 1))7r + f }. The proof follows from just a simple comparison of certain harmonic measures on D, D 1 ( = { z ln1 < arg z < n5J) and D2 ( = { z ln2 < arg z < u6J). This approach enables us to avoid the exceedingly complicated argumants involving NE geometry in Caratheodory's original proof. Theorem 2, Let f(z) be an analytic map from lz I< 1 into itself and let C(f,eili) be the cluster set of f(z) at eill. If there exists an n such that the union over IJ < IJ < n (- n < IJ < 0) of

C(f,eili) C [wl7r/2 ::;; arg(w- 1) ::;; 37r/4j <{wl57r/4::;; arg(w- 1) ::;; 37r/2j) then the image under f of the radius terminating at one is eventually contained in [w I larg(w - 1)- 1rl < 7r/8}. (Received February 17, 1966.)

632-55, A. C. CONNOR, University of Georgia, Athens, Georgia 30601. Normal monotone mappings.

G. T. Whyburn has raised the question of whether every monotone mapping of En onto En is a compact mapping. The following partial results are obtained here, A mapping f of En onto Em is called a normal mapping If there exist a continuum K in En and a point p in En - K such that f IK is an essential map of K into Em - f(p). It is proved that every normal monotone mapping of En onto En is compact; a corollary to this theorem is that every monotone mapping of En onto En which is one-to-one on some open set is compact. (Received February 17, 1966,)

632-56, S. G. MROWKA, Pennsylvania State University, 227 McAllister Hall, University Park, Pennsylvania 16802, On incompactness of logics with infinitely long expressions. Preliminary report.

We follow the terminology of S. Mrowka, Abstract 625-159, these cNoticeiJ 12 (1965), 592,

Let M be the class of all cardinals m satisfying: N-defect of Xm ~ m. It is shown that the (first order predicate) logic LN (containing expressions of the length < N1) has the following property 1 (which can be called "very strong incompactness "): if m E M, then LN 1 contains a set 2; of sentences

342 such that card ::!: ; m, every subset::!:' of ::!: with card::!:' < m has a model, while the whole set ::!: does not have one. The complete result following from considering cardinals in the class M is technically more complicated to formulate but gives the strongest type of incompactness of infinite logics. (Received February 17, 1966.)

632-57. N. F. REHNER, University of Rochester, Rochester, New York 14627. An Integral over complex Hilbert spaces. Preliminary report.

Banach's integral [S. Banach, The Lebesque integral in abstract spaces, note to S. Saks, Theory of the integral {Warsaw, 1933)] over the real Hilbert space 1 2(R) is modified to obtain an integral over the unit ballS of the complex Hilbert space 12(C); [z; (z0 ,z 1, ••• >1I~Izn1 2 < oo]. On identifying I 2{C) 00z wn(n!)-1/2 ; f (w) it turns out with the Segal-Bargmann space 5'i of entire functions by z <----> .2:: 0 n z that for almost all z E S fz is of growth (2,{2ef 1), where e is the base of natural logarithms. Under the isomorphism z <---> I:g:'znhn{x), where hn is the nth normalized harmonic oscillator function of quantum mechanics, S maps into the family of initial states. The series E(z); L~(n + l/2)lznl2, q(z) ; 2 1 / 2 l:~n 112Re(zn-l zn)' and p(z); 2 1 / 2 I:~n 112Im(zn-l zn)' representing the expected energy, position and momentum respectively of the initial state of the oscillator, each converge for almost all z E S. The mean and variance of each of the random variables E, p, q on S are computed. (Received February 17, 1966,)

632-58. R. L. RICHARDSON, University of Florida, Gainesville, Florida 32601. Topological Post coset theorem. Preliminary report.

An m-group may be defined as an algebraic structure {A, [ ]) with an m-ary operation [ ]:

Am--; A satisfying them-associative law [[x 1x 2 ••• xm]xm+l ••• Xzm-l]; [x 1 [x 2x 3 ••• xm+l]

••. x 2m_ 1]; ••• ; [?c: 1x 2 ••• xm_ 1[xmxm+l ••• x2m_ 1]) and the condition that for any set of m- 2 elements e 1, e 2, ••• , em_ 2 E A there exists uniquely an element em _1 E A (also denoted by (e 1,e2, •• .,em_ 2)- 1) such that [e 1e 2 .•• em_ 1x]; x; [xe 1 •.• em_ 1] for all x EA. When both the functions [ ] and (el'e2, ••• ,em_ 2) --7 (el'e2, ... ,em-z)-l are continuous under a topology T, then {A, [ J, T) is called a topological m-.&!.2.':!P.· Theorem. Any topological m-group is a coset of some ordinary topological group. (Received February 17, 1966.)

632-59. R. C. MORGAN, 8 Pequa Lane, Commack, New York. Uniqueness theorem for the reduced wave equation under an Nth order differential boundary condition.

Uniqueness is demonstrated for the solution to a mathematical problem that arises in the phenomenological theory of multi-mode surface wave propagation. The unusual aspect is that a boundary value problem for the reduced wave equation is posed using an Nth order differential bound- ary condition. Furthermore, the solution is assumed to be decomposed into three terms; a source, excited surface wave modes, and a radiating term. This last term satisfies a Sommerfield radiation condition and certain regularity conditions at infinity. (Received February 17, 1966.)

343 632-60. ROE GOODMAN, Massachusetts Institute of Technology, Cambridge, Massachusetts. On localization and domains of uniqueness.

Localization and domains of uniqueness are considered for weak solutions u to an abstract hyperbolic equation du/dt = iAu, where for each t, u(t) belongs to a Hilbert space H of distributions on R n, and A is a self-adjoint operator on H. Under the hypothesis that A has no homogeneous Lebesgue spectrum (e.g. if the spectrum of A is a proper subset of R 1), it is proved that if K ~ R n,

then Supp(u(t)) .~ K fort < 0, implies that Supp(u(t)) ~ K for all t. As a corollary, if C is an open set

in R n+ 1 whose t-sections increase to R n as t -> - oo (e.g. if C is a cone), then u vanishing on C implies u = 0 (this generalizes previous results of the author). In the special case of the Klein­ Gordon equation oq, = m 2q,, m > 0 (where the initial data 1/l(O) and 1/lt(O) have Fourier transforms of at

most polynomial growth), if U ~ R n is ..!.!!X.. nonempty open set, then rP(t,x) = 0 for x E U and all t < 0 implies tP= 0. (Received February 17, 1966.)

632-61. j. P. ROTH, IBM Watson Research Center, Yorktown Heights, New York. The calculus for failure diagnosis.

One of the central aspects in the theory of design and organization of reliable automata is the calculation of tests to detect and locate failures of these automata. This paper is a description of a calculus for such calculation which has a range of effectiveness considerably in excess of previous techniques. An algorithm based on this calculus is described. (Received February'18, 1966.)

632-62. ADI BEN-ISRAEL, University of Illinois at Chicago Circle, P.O. Box 4348, Chicago, Illinois. On the Newton-Raphson method for systems of equations.

The Newton-Raphson method is extended to rectangular systems and singular Jacobians by

using generalized inverses. The convergence of the sequences: .J!;p+l = !p- j+(JS,0)1(!p) (p = 0,1, ••• ) ,l!;p+l = .Ap- J+(!p)!(!p)' (p = 0,1, ••• ) is studied, where .1!. E rfl, f: En---+ Em and J{,l!;) = (afi/

632-63. WOLFGANG WASOW, University of Wisconsin, Madison, Wisconsin. On the analytic validity of formal simplifications of linear differential equations.

Let dy/dx = (L:_hAr(x)trJy, h > 0, (1), be an analytic, vectorial, linear differential equation with a small parameter t. In the study of such systems near the singularity in t, at t = 0, one often changes them into a form dz/dx = l~-hBr(x)tr]z, (2), with a simpler coefficient matrix, by means of a transformation y = lL~oPr(x)t:r:)z, (3) The series in (1) and (2) are assumed to be asymptotic expansions of analytic matrix functions. The first part of such a simplification always consists in the construction of a formal power series "'coL...r=Op r(x)t r that achieves the desired transformation in the formal sense. In the special case that the function B (x, t) t-h represented asymptotically by L:-hBr(x)t r is a polynomial in x. Sibuya [Funkcial. Ekvac. 4 (1962), 83-113] has given conditions which guarantee that (3) defines asymptotically an actual transformation of (1) into (Z). In the present paper Sibuya's result is generalized to a fairly wide class of functions B(x,t) that are not polynomials. (Received February 18, 1966.)

344 632-64. TA-SUN WU, University of Massachusetts, Amherst, Massachusetts. Ergodic auto­ morphisms on compact groups.

Let G be a compact connected finite dimensional group. If G admits a continuous ergodic automorphism, then G Is abelian. (Received February 18, 1966.)

632-65. R. T. PROSSER, Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts 02173. The t-entropy and t -capacity of a compact operator on a Hilbert space.

The t-entropy H(t) and t-capaclty C(l) of a compact set in a Hilbert space were defined and examln.ed by Kolmogorov in his studies in information theory. Here we consider the case where the compact set is the image of the unit sphere under a compact operator. For this case we show that the order of growth and logarithmic order of growth of H(E) and C(t) as E -->0 are related to the ex­ ponent of convergence and logarithmic exponent of convergence of an eigenvalue sequence associated with the operator. This result permits us to calculate the asymptotic behavior of H(E) and C(t) from the asymptotic behavior of the associated eigenvalue sequence. We present the calculations for various examples of interest in information theory. (Received February 18, 1966.)

632-66. ALESSANDRO FIG2\.-TALAMANCA and DANIEL RIDER, Massachusetts Institute of Technology, Cambridge, Massachusetts. A theorem on random Fourier series on noncommutative groups.

Let G be an arbitrary compact group. We consider for x E G,the formal Fourier series (*)

I;diTr(UiAiD1(x)), where Tr is the ordinary trace, the D1 are distinct (nonequlvalent) Irreducible unitary representations of G of degree di' the Ai are linear operators and the Ui unitary operators on the Hilbert space of dimension d1• Keeping the A1 fixed we let fU 1J vary in the group H = TI %'(di) (the product of groups %'(di) of unitary operators on the dcdlmenslonal space). We prove that if there exists a set M ~ H of positive Haar measure such that for all U = {u1} E M the series (*) represents a function In Li(G), then I;d1Tr(AiAi) < oo. This Is a generalization of a known theorem on Fourier series with random coefficients defined on the unit circle. (Received February 18, 1966.)

632-67. j. D. McKNIGHT and A. j. STOREY, University of Miami, Coral Gables, Florida. A class of semigroups and its finite quotients.

Let a semigroup S be called equldivisible if ab = cd Implies either a = c and b = d or there exists e E S such that a = ce and eb = d or such that ae = c and ed = b. S is such a semigroup if and only if a 1 • a 2 • ... • an= b 1 • ... • bm Implies there exist ci (i = 1, ... ,p) InS such that a 1 = c1 • c2• ... • ci(a1)' a2 = ci(a1)+1 • ... • cl(a2)'"''an = ci(an-1)+1 ..... cp and b1 = c1 .... • ci(b1)' ... , b = c.(b • ... • c • Two of the extended Kleene Converse Theorems hold for equidlvlsible m 1 m-1)+1 p semigroups [Pacific j. Math. Vol. 14, no. 4, 1350]. Every free semigroup is equidivisible and an equidivlslble semigroup In which each element has a maximal length factorization is free. Any completely simple semigroup is equidivisible. (Received February 18, 1966,)

345 632-68, HARI SHANKAR, Ohio University, Athens, Ohio 45701, Some inequalities for the Ahlfors-Shimizu characteristic and the area on the Riemann sphere.

Let f(z) be a nonconstant meromorphic function in the open plane, of finite nonzero order P and positive lower order A, Let T*(r,f) denote the Ahlfors-Shimuzu characteristic function of f(z), and

A(r,f) denote the area of the image of iz I< r by f(z) on the Riemann sphere divided by 1r, the area of the sphere, Set D[d] =lim sup[inf]r ____, 00 T*(r,f)/A(r,f) and A[a] =lim sup[inf]r ____, 00T*(r,f)/rP, It is well-known that for a meromorphic function of finite order A(r,f) = O)T*(r,f)J for some arbitrarily larger, Also unless f(z) has zero order it can not have A(r,f) = o{T*(r,f)j, Theorem 1, For any positive real number k < 1, lim supr ____, 00\T*(r,f)jk/ A(r,f) = 0; then lim infr ___, 00 [T*(r,f)jk/ A(r,f) = oo if k > p/A > 1. From the inequalities established earlier by the author [see Abstract 621-40, these cJI;;tiuiJ 12 (1965), 315) it easily follows that exp{(a/A)- 1} ~ p,D ~A/a and d ~ 1/{J ~D. Theorem 2, exp [(a/ A) - lJ ~ (a/A) exp [(A/a) - lj ~ p,D ~ [1 + log(A/a)j ~A/a. Theorem 3, p, A.d ~A~ P ~ p,A,D, It is remarked that these and the earlier results, quoted above, will hold even if one uses, as comparison functions rpL(r) or rp(r) instead of rP, where L(r) is a slowly in­ creasing function in the sense of Karamata and P(r) is the Lindelo('s Proximate order of f(z) with respect to the function R *(r,f). (Received February 18, 1966.)

632-69, A, A, GOLDSTEIN, University of Washington, Seattle, Washington 98105, Descent on the boundary of a sphere,

Let S denote the boundary of the unit sphere in En, Assume f is in C 1 on S, Take x0 arbitrarily ins. Choose 3J.L ~min {IIY'f(x)f1 : X E. s}. For X E. s set x'(-y) = (x -'Y 'V'f(x))/llx- 'Y'V'f(x)ll. ~ (x,')') = f(x) - f(x' (y)) and g(x, 'Y) = - ~(x, 'Y)/'Y [W(x),x' (')') - x]. Here [x,y] denotes the inner product of points x andy in En. Theorem. Choose o, 0 < o& J.L, Take 'Yk ~ l/3j['V'f(xk), xk]l-l =Ilk such that o ~ g(xk, 'Y ) < 1 - oif g(xk,ll ) < o or 'Y = II otherwise, Set xk+l = (xk- yk'V'f(xk))/llxk- yk'V'f(xk)ll. k k k k . Then (a) Every cluster point z of the sequence [xk} satisfies 'V'f(z): ll'V'f(z)llz, [f(xk)J converges downward to a limit, and fxk+ 1 - xkj converges to 0, (b) Assume that for x in S the roots of the equation 'V'f(x) = ll'V'f(x) Ux, are finite in number, Then £xkJ converges, If this equation has only one root inS then lim £xk} is a minimizer off on S, (Receive

632-70. D. E. MULLER, University of Illinois, Urbana, Illinois 61805, Composition of operations in a generalized algebraic system,

Forms are expressions possibly containing the blank symbol x and/or free variables z from some alphabet z. If Q is a set of forms, R a set, and H: Z ---> 2R, let S(R,Q) consist of all forms obtainable by replacing each appearance of x in any form of Q by some element of R, and let T(H,Q) consist of all forms obtainable by replacing each appearance of every element of z of Z in any form of Q by some element of H(z). An algebraic system a. on a set A and a set P of forms is an implicit mapping such that each element of S(2A ,P) denotes an element of 2 A, and such that the form (x Ux) representing set union commutes with every form of P. That is, if G and H are mappings from Z to

2A and q is in S(Z,P), then T(G,q) UT(H,q) = T(GUH,q), Let P* = {(z0 ,K)jz0 E. Z,K E. (S(Z U fxj,P))~. Each (zoJ) in S(2A,p•)denotes HJ(z0 ) in 2A, where H J is the unique minimum of all mappings H from z to 2A satisfying: Vz E. Z, H(z) 2T(H,j(z)). Theorem, This implicit mapping from S(2A,P*) to 2A

346 is an algebraic system a• on A,P*. Theorem. (a*)* is the same as a• except for redundant repre­ sentations. Theorem. If P : f (xx), (xU x)} contains forms for concatenation and union, and A is the set of all strings on some alphabet, then P • generates the context free languages from the singleton sets of A. (Received February 18, 1965.)

632-71. C. E. BURGESS, University of Utah, Salt Lake City, Utah. Conditions which imply that a 2-sphere in s3 is locally tame except at two points.

In the following theorem and its corollary, letS be a 2-sphere in a 3-sphere s3 such that each component of s3 - S is an open 3-cell. Theorem. If for each x E S and for each t > 0 there exist a disk D and an open annulus A such that x E Int DC S, diam D < <, A ns: Bd D, and A intersects both components of s3 - S, then S is locally tame except possibly at two points. Corollary. If the set W of all wild points of S is 0-dimensional, then W contains at most two points. (Received February 15, 1966.)

347 The Meeting in Honolulu April 9, 1966

633" 1. H. S. BEAR, University of California, San Diego, La Jolla, California and A.M. GLEASON, Harvard University, Cambridge, Massachusetts. An integral formula for abstract harmonic or parabolic functions.

We define abstractly a linear space of "quasi-harmonic" real functions on a locally compact space X, and prove a Cauchy-type integral formula for these functions. Examples include harmonic functions and solutions to the heat equation. We poseulate the existence of local kernels of the follow­ ing sort on X: for each neighborhood N of each p E X there is an open set U with p E U C N, a com­ pact set Y C N, a Borel probability measure >. on Y, and a Borel measureable function G(p,y) on

U X X such that p ,---> G(p, •) is a continuous map of U into L 00 (X). We assume there are a countable number of such maps G. Define the space AN' for any open n C X, to be all continuous functions f on N

such that f(p) = Jyf(y)G(p,y)dX(y) whenever p E U C N and Y C N. Theorem. If B is all functions in AN which are continuous on N, for a relatively compact open set N, and if r is the Silov boundary of B inN, then there is a measure J.L on rand a function Q(p,b) on N X I' such that Q(•,b) E AN for all bE I' and f(p) = frf(b)Q(p,b)cW,(b) for all pEN. (Received November 26, 1965.)

633-2, MARK LOTKIN, 3198 Chestnut Street, Philadelphia, Pennsylvania. A self-contained trajectory model.

This paper describes a mathematical trajectory model which Is self-contained in the sense that it permits the determination of the ballistic parameter W /C0 A directly while the differential equations of C. G. motion are being integrated. The variable weight W of the vehicle, its variable cross- sectional area A, and its variable total drag coefficient c 0 may be generated directly from the ambient conditions of vehicle speed, sound speed, and atmospheric density only. Trajectory models of this type, provided they possess a satisfactory degree of accuracy, may serve a variety of practical purposes. Since such self-contained trajectory models permit the approximate calculation of weight, area, drag, and position, they obviate the need for lengthy and costly thermodynamic and aerodynamic calculations, including possibly the calculation of six degrees of freedom trajectories. Models of this type are currently being used for a variety of practical applications. (Received December 8, 1965.)

633-3. J. G. BALDWIN, University of Houston, Cullen Bouvelard, Houston 4, Texas. Some aspects of uniform recurrence.

A subset E of a topological space is said to be uniformly recurrent with respect to a mapping

T if there exists a positive integer ko such that wTs(w) E E for some s(w) ::£ k 0 , for each wE E. If T is continuous and E is compact, the fact that the forward orbit, { w Tk lk = 1, 2,. .. J, of each w E E intersects E does not insure uniform recurrence, However if E is "strongly" recurrent in the sense

348 that all forward orbits of points of E intersect the interior of E, then the pre-compactness of E in- sures uniform recurrence to the interior of E. One may consider the case where a set is not recur- rent but possesses the weaker property ("weak" recurrence) that each of its points returns to every neighborhood of the set (or equivalently, each of its points returns to the closure of the set). The con­ cept of uniform weak recurrence is then meaningful, but if E is recurrent and compact, it need not be uniformly weakly recurrent. However if E is weakly recurrent and compact, it is uniformly recur­ rent to any neighborhood of E, but the set of maximum uniform return times is not necessarily bounded above. (Received December 2, 1965.)

633-4, VEIKKO NEVANLINNA, University of Hawaii, 2565 The Mall Room 101, Honolulu, Hawaii, A refinement of Selberg's asymptotic equation.

Writing R(x) ~ x - \f!(x), where \f!(x) is the numbertheoretic function of Cheby!fev, using the Lemma of Tatuzawa and Iseki [Proc. Jap, Acad. 27 (1950), 340-342] and assuming the prime number

theorem it is proved that R(x)log X + f~R(x/t)d\f!(t) ~ - (C + 1)x + o(x), or \f!(x)log X + n'}l(x/t)d\f!{t) ~ - (2C + 1)x + o(x), where C is Euler's constant, (Received January 13, 1966,)

633-5, E. C. POSNER and HOWARD RUMSEY, JR., Jet Propulsion Laboratory,48000ak Grove Drive, Building 238, Room 420, Pasadena, California 91103, Epsilon entropy of the Wiener process,

The concept of probabilistic metric space X is defined as a metric space with a probability

measure in which every open set is measurable and has positive measure. Given f, o > 0, the

f; oentropy of X, H f;o (X), is defined as the infinium of the entropies of all f; o partitions of X. An f; o partition is a partition of X except for a set of probability oor less into measurable sets of

diameter less than f, The entropy of such a partition is defined as the entropy of the discrete dis­

tribution whose probabilities are the probabilities of the sets in the partition. The f- entropy of X is

the supremum of H f ;o(X) over all positive o. This definition in a certain sense extends the concept

of f -entropy of a compact metric space. An example of a probabilistic metric space is the space of

continuous functions on the unit interval vanishing at 0, under the L 2 -norm. The probability measure is taken to be Wiener measure, the measure induced by the Wiener process, the Gaussian process

x(t) with E(x(t)) ~ 0, E(x(s)x(t)) ~ min(s,t). It is shown that as E -> 0, the E-entropy of this probabilis­ tic metric space times E2 lies between two positive constants. (Received January 25, 1966.)

633-6, R. A. KNOEBEL, New Mexico State University, Box AM, University Park, New Mexico, Primality-in-the-small and simplicity,

Let iJt ~ (A, f 0 ,f 1,. .. ,f11 , ... ) be a universal algebra. (%#is the set of all finitary functions

which can be generated by composition of the f0 ,f1, ... ,f11 ,... I% is primal-in-the-small (primal/sm) iff for every finite subset F of A and for every finitary function g on A there is an hE tJt# which

agrees with g on F. It is easy to show that any framal/sm algebra is simple. We state three converse results out of several possible. First some definitions. ()tis !ramal iff there are 0, 1 E A and a

binary function x E at# s.t. 0 X a~ 0 ~ a X 0 and l X a~ a~ a X l for all a EA. rJt is 1-fold transit­ ive iff there is a set P ~ CJtf of permutations which, under composition of functions, forms a group

349 transitive on A. Theorem. Otis primal/am if cJt is simple, framal and 1-fold transitive and every two elements of A can be interchanged by some unary function in tJt#. Corollary. A ring with unit in which the additive inverse and all constant functions are adjoined is primal/sm iff it is simple, Proposition. If rJt is finite, then rJt is primal/sm iff c,t is simple, framal and 1-fold transitive. (Received December 6, 196 5,)

633-7. M. J, SHERMAN, University of California, Los Angeles, California 90024. Spectral inner functions.

If U is a finite dimensional inner function (for basic definitions and notations see H. Helson, Lectures on invariant subspaces, Academic Press, New York, 1964) we say that a scalar inner func­ tion q is an eigenfunction of U if det(U- qi) = 0 a. e. It is shown first that there are inner functions U for which no such q exists. An inner function which has eigenfunctions the sum of whose multiplici­ ties in an appropriate sense is the dimension of the underlying Hilbert space is called a spectral inner function, Theorem. U is spectral iff U = \8~'!3*, where '!l is a diagonal inner function whose diagonal entries are the eigenfunctions and Q3 is an arbitrary unitary operator valued function (which can be chosen most advantageously to have outer columns). Infinite dimensional extensions are dis­ cussed, as are special results when U arises from an operator on S) using constructions due to Rota and Potopov. (Received November 19, 1965.)

633-8. R. W. GILMER, JR., Florida State University, Tallahassee, Florida, On overrings of Prufer domains, II. Preliminary report.

Let D be a Prufer domain and let [Mx} be the set of maximal ideals of D. D has property(#) if for any A, DM -;p_ n ~ADM , One-dimensional Prufer domains with property (#)have been pre- A a _A( Viously investigated (see Abstract 627-2, these c;;voticelJ 12 (1965), 786). Theorem 1, In D, these conditions are equivalent: (a) D has property (#), (b) for each A, there exists a finitely generated ideal AA such that MA is the only maximal ideal containing Ax, (c) Dis uniquely expressible as an intersection of independent valuation rings. Theorem 2, For any D, these conditions are equivalent: (a) each domain between D and its quotient field satisfies (#), (b) if P is a proper prime ideal of D, there exists a finitely generated ideal A of D such that each maximal ideal of D containing A contains P. Theorem 3, If the ascending chain condition for prime ideals holds in D, the conditions of Theorem 1 are equivalent to: each maximal ideal of D is the radical of a finitely generated ideal, and the conditions of Theorem 2 are equivalent to: each prime ideal of D is the radical of a finitely generated ideal, (Received January 28, 1966.)

633-9. W. W. ADAMS, University of California, Berkeley, California, Asymptotic diophantine approximations to e,

We consider e, the base for the natural logarithms, Let w be a monotone function and let B be a positive real number, Let X(B,w) be the number of solutions in integers p,q of the inequalities

Jqe- pj < w(q)/q and 1 ~ q ~B. The result gives the asymptotic formula for A(B,W) as B-+ oo. Be­ fore stating the theorem we make some restrictions on w. The function 4x r(x + 3/2) is a strictly

350 monotone increasing function so let G(x) denote its inverse function (one sees that G(x) ~ log x/log log x). Now let

oo. We suppose w(G- 1(x)) = xn4J(x) for some integer n. We note that this restricts us to w's satisfying w(x)/G(x)k---> 0 as x ---> oo for some Integer k. For faster growing w Lang [Proc. Nat. Acad. Sci. U.S.A., Jan. 1966] has a general theorem giving the asymptotic formula. Theorem. (a) Suppose w(x)G(x) ---> oo. Then A(B,w) ~ 2 J?(B)w(G- 1(x))log x dx + J?(B)(2x w(G-l(x)))l/2dx. (b) If w(x)G(x) ---> c then for

0 ~ c ~ 1/2 there are only a finite number of solutions and for 1/2 ~ c < oo we have A(B,w) ~ [(2c)112]/(2c)112 f?(B)(2x w(G- 1(x))) 1/ 2dx. We note that In (a) if w(x)/G(x)/loif G(x) ---> 0 the second term dominates and if c~(x)/G(x)/log 2 G(x) ---> oo then the first term dominates. Also the first term Is asymptotic to 2 J~ w(x)/x dx. Explicit error terms have been computed. (Received January 31, 1966.)

633-10. M. D. RESNIK, University of Hawaii, 2560 Campus Road, Honolulu, Hawaii 96822. A unified theory of types.

The system ET Is a one-sorted formulation of the simple theory of types and a partial formu­ lization of the type concept. Unlike Quine's UT [Cf. Set theory and Its logic, pp. 266-272], ET dis­ penses with schematic type indices in axiom schemata and contains as theorems (xi), (xU) and (xiii) [Ibid., p. 271] in addition to all theorems of UT. ET has a model in Zermelo's set theory and is consistent relative to UT. The axioms and key definitions of ET follow: "PTxy" for "(Ez)(x E 2z. y E z)"; "STxy" for "(z)(PTzx = PTzy)"; "x = y" for "(z)(x E z ::J y E z. z Ex ::J z E y)"; "Tx" for "(Ey)(z)(z Ex = STzy)"; Ax. 1. x E y, z E w. STxz. =-.> STyw; Ax. 2, x E y. z E w. STyw. ::J STxz; Ax. 3, (Ey)(z)(zE: y = STxz); Ax. 4. x E y::::ly f. x; Ax. 5. (Ex)(y)- PTyx; Ax. 6. (Ez)PTzx.STxy. (z)(z E x = z E y). ::J x = y; Ax. 7. Tw. - (Eu)(u E w. (z) - PTzu) ,::J (Ey)(y E w. (x)(PTxy ::J, x C. y

=< Fx)). An axiom of infinity may also be added to ET without altering the results mentioned above. (Received February 3, 1966.)

633-11. MAXWELL ROSENLICHT, University of California, Berkeley, California. Another proof of a theorem on rational cross sections.

The extant proofs of the existence of a rational cross section for a transformation space for a connected solvable linear algebraic group either use a certain amount of algebraic curve theory or restrict themselves to the case of a principal space, where galois cohomology can be used. The present proof of the general result may be considered more elementary in that it depends only on the standard facts on fields of rationality of algebraic sets, (Received February 7, 1966.)

633-12. R. A. BEAUMONT, University of Washington, Seattle, Washington 98105, Abelian groups which are determined by their group of mappings.

lf G and H are abelian groups, then Map(G,H), the set of all mappings of G into H, is an abelian group. It is natural to look for classes of abelian groups G for which Map(G,G) determines G. lf G is finite and H is a group such that Map(H,H) ""Map(G, G), then H ""G. If G and H are torsion free rank one groups, G is of nonnil type, and Map(H,H) ""Map(G,G), then H ""G. If G is an uncount­ able torsion free divisible group and H is a group such that Map(H,H) "" Map(G,G), then H ""G. On

351 the other hand, if G is an infinite group and K is a direct summand of G, then Map(G,G) ;;:,; Map(G Ell K, G Ell K). This result provides many examples of groups which are not determined by their group of mappings. (Received February 7, 1966.)

633-13. GUSTAVE SOLOMON and R. ]. McELIECE, Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California. Weights of cyclic codes.

A new formula is presented for the weight, or number of ones, in a binary cyclic code. The formula allows the quick computation of the error-correcting properties of a large class of codes. The weight of a vector a in A, any (n,k) group code, is given by w(a) = 2.:~ r i(a)2i, when N = 1- 0 2 [log w ], and r i = C i mod 2 are expressed as a function of the parameters in the general repre- 2 2 w,2 a entation of cyclic codes in (H. F. Mattson, and G. Solomon, A new treatment of Bose-Chaudhuri codes, J. Soc. Ind. Appl. Math. 9 (1961), 654-669). The coefficients of the formula are obtained by using elementary field operations in the cyclic group of the nth roots of unity. Using the expressions for r 2 and r4, we obtain general results for weights occurring in certain cyclic codes. A general formula for higher-order f•s is given. (Received February 10, 1966.)

633-14. CHARLES HOBBY, University of Washington, Seattle, Washington 98105. Nearly regular p-groups.

A p-group G is said to be nearly regular if (l) there is a central subgroup Z of order p such that G/Z is regular, and (2) the subgroup generated by x,y is regular whenever x E Gandy E G', the derived group of G. Theorem. Suppose G is a nearly regular p-group where p is an odd prime. If x,y E G then there are elements a,b in G such that ( x,y) = ( a,b) and (ab)P = aPbPcP where c is an element of the derived group of the group (a, b). Thus every two generator subgroup is generated by elements which satisfy a regularity condition. (Received February 10, 1966.)

633-15. D. S. NEWMAN, Boeing Company, P. 0. Box 3707, Seattle, Washington 98124. A stochastic process based on an alternating renewal process.

Stochastic processes of the form Y(t) = 2.:: 1 A(t - Sk) where Sk is the kth partial sum of a re­ newal process, and A(u) = 0 for u < 0, arise in the theory of the shot effect, particle counters, and in communication theory. In this paper a process of the form Y(t) = 2.:: 1 (- l)k A(t - Sk) is considered, where the Sk are partial sums of an alternating renewal process, and A(u) again vanishes for u < 0. This problem arises in communication theory in connection with the random telegraph signal ~. 0. Ri:e, Selected papers in noise and stochastic processes, (N. Wax, ed.), 133-294]. A linear differen­ tial equation in t for the characteristic function .P(u,t) = E [exp iuY(t)) is obtained. The limiting dis­ tribution of Y(t) as t-> oo is obtainable from the asymptotic behavior of solutions to the differential equation. (Received February 10, 1966.)

352 633-16. D. H. HYERS, University of Southern California, Los Angeles, California 90007. A note on linear topological spaces.

Since Bourbaki, the study of linear topological spaces has been confined solely to those which are locally convex. In this note, an attempt is made to begin a study of the general case. (Received February 14, 1966.)

633-17. C. A. HAYES, University of California, Davis, California. A condition of halo type for the differentiation of certain classes of set functions.

Let 1 denote a u-algebra of subsets of a space S; ll, a measure on 1; p.*, the outer measure agreeing with ll on ..It, defined in the usual manner; B, a derivation basis with spread .f) C _,/{. For any M E _ff, 0 <

Let ¢and u denote functions nonnegative for 0 ~ t :::i oo, strictly increasing for large t, such that ::L:Nxn;u(Xn) < oo for some Nand some X> 1. Let Ybe the class of p.-integrals of all functions f satisfying JM¢(u(jf(x)j))dp(x) < oo and JMjf(x)jdp.(x) < oo whenever ME A', u(M) < oo. Theorem.

Suppose (1) B differentiates the JL-integrals of all bounded p-measurable functions; (2) if M0 E 1, ll(M0 ) < oo, there exists C > 1 such that p.*(S(ct,M)} ~ Cd>(l/ct)ll(M) whenever Mo :::J ME 1 and p.(M) and 0 are sufficiently small. Then B differentiates all members of .5£. (Received February 14, 1966.)

633-18. A. P. OGG, University of California, Berkeley, California 94720. Abelian curves of small conductor.

An abelian curve defined over the rationals with conductor 2X ,3 • 2 X, or 9 • 2X has a rational point of order 2; using this fact, it is easy to find all such curves. The results are consistent with the conjectures based on the recent work of Weil and Shimura on zeta-functions. (Received February 14, 1966.)

633-19. A. L. Y ANDL, Western Washington State College, Bellingham, Washington 98225. Proximate absolute extensors.

Let I be an index set, 3 = {(Xi, pi) I i E IJ a class of metric spaces, (X,p) a metric space and X ~ 0. Definitions: (1) X is called a weak ")..-proximate absolute extensor for 3 (abbreviated wX -PAE for 3) if and only if whenever i E I, A C Xi with A closed, f: A -->X is X- continuous, and

f > 0 we can find a (X + f)-continuous mapping g: Xi ----->X for which p(g(x), f(x)) < f for each x E X. (2) X is called a strong >.-proximate absolute extensor for 3 (abbreviated S X-PAE for J) if and

only if whenever i E I, A C Xi with A closed and f: A -->X is >.-continuous, the following is true. For

each f > 0, there exists a li > 0 so that given any p. > 0 there is a (A+ p.)-continuous mapping g: Xi__, X for which x E: Xi' a E. A and pi(x,a) < 1i implies P(g(x), f(a)) < ~. (See Abstract 630-134, these cNotiaiJ 13 (1965), 97). Theorem. A compact metric space X is a WPAR if and only if it is a WO-PAE for the class of compact metric spaces. Theorem. A compact metric space X is a SPAR if and only if it is a SO-PAE for the class of compact metric spaces. (Received February 14, 1966.)

353 633-20. B. H. McCANDLESS, Western Washington State College, Bellingham, Washington 98225. On a theorem of Hanner and Dowker.

Hanner and Dowker have shown that every metric ANR(normal) is separable and an absolute G5. In this paper we point out that the only property of an ANR (normal) needed for this result is that of being a 0-ANR (normal). This allows us to prove that a metric spaceY is an n-ANR (normal) if and only if Y is ann- ANR (metric), separable, and an absolute G0• This is analogous to the result of Hanner and Dowker that one gets by deleting the n's in the above statement. (Received February 14, 1966.)

633-21. R. W. CHANEY, Western Washington State College, Bellingham, Washington. The chain rule in the transformation theory for measure space.

Let [S,M,u} and [X,_N, p} be measure spaces and assume Tis a function from S onto X. Assume

Q C M such that TQ C N. A nonnegative extended real valued function 1 W with domain X X .M. is a weighting function for T if 1 W( •, G) = 0 a. e. p off TG for each G in Q, each function 1 W( •, M) is

Ji-measurable, and if 'W( •,M) = L 1 W( • ,Mi) a.e. p on X whenever Min M is the union of countably many disjoint sets Mi in M· Given 1 W we say that fin Lt(u) is a glbfW in case Jx'W( • ,M) dp = JMf du for each M. Now assume under appropriate conditions that T is a function from S onto X as above, T' is a function from X onto Y, 1 W is a weighting function forT, 1V is a weighting function for T', f is a glbfW, and that g is a glbfV. The following "chain rule" is then established. There is a weighting function 'U for T' o T such that f(g o T) is a glbfU iff f(g o T) is in L I (u); in this case we have 1 U(y,M) = limn'L:kk2-n1 V(y,E(k,n,M)) a.e. on Y for each M, where E(k,n,M) = [x EX: k2-n ~ 'W(x,M) < (k + 1)2-nj. This theorem can be applied to obtain certain versions of the "chain rule" in the transformation theory for measure space (see Abstracts 61 T-267 and 624-12, these cNotiaiJ 8 (1961), 518 and 12 (1965), 452). (Received February 14, 1966.)

633-22. KENNETH ROGERS, University of Hawaii, 2565 The Mall, Honolulu, Hawa!! 96821. The quadratic reciprocity law via the geometry of numbers.

In Cassels' An introduction to the geometry of numbers, pp. 100-102, Minkowski's convex-body theorem is used to prove that if (1) a l'a2,a3 are squarefree integers, relatively ·prime in pairs, (2) - aiaj is a quadratic residue mod ak for all permutations (i,j,k) of (1,2,3), and (3) either all a 1 are odd but not all mutually congruent mod 4; or one is even, say a 1, and then either a 2 + a3 or a 1 + a 2 + a 3 is a multiple of 8, then there exist integers ui not all zero, such that z=aiuf = 0. Hence, if the ai are all positive and satisfy (1) and (3), then (2) is false. By applying this result to triples such as (l,l,p), (1,2,p), (l,p,q), (p,q,r), much of the quadratic reciprocity law can be deduced. lf we assume Dirichlet's Theorem and the fact that for p of the form 4n + 1, - 1 is a quadratic residue, then all of the reciprocity law follows. (Received February 15, 1966.)

633-23. G. E. BREDON, University of California, Berkeley, California, Equivariant cohomology theory.

Let G be a finite group. Consider the category of CW complexes on which G acts cellularly

354 and of equivariant maps. A cohomology theory is defined on this category and is used to obtain an obstruction theory for extending equivariant maps. Analogues of Ellenberg- Mac Lane spaces are constructed and used to prove the representability of the cohomology theory. An analogue of the Eilenberg-Steenrod uniqueness theorem is also proved. This cohomology theory has a more elaborate structure than classical equivariant cohomology theory and, in particular, the coefficients used are not G-modules but members of a certain, more elaborate, abelian category. Generalized equivariant cohomology theories (especially stable equivariant cohomotopy theory) are also studied, and, in fact, our equivariant "classical" cohomology theory was primarily designed to aid this study. (Received February 14, 1966.)

633-24. DOUGLAS DERRY, University of British Columbia, Vancouver 8, B. C. Inflection hyperplanes of polygons.

Al' A2, •.• ,Am, m > n, are points of real projective space L n' n > 1, in general position. 1r: A1A2 ••• Am is a closed polygon in Ln with vertices Al' A2, ••. ,Am. [Ap Ai+ 1, ... ,Al+n-lJ = Ln-l is the hyperplane which contains the points Ai' Al+ 1, .•• ,Ai+n-l (Aj+m = Aj). If q is the number of points of the segment Al+lAi+2 ••• Ai+m-l of 1rin Ln-l' then Ln-l is·called an Inflection hyperplane of ?rif n + 1 + q is even or odd according as 1r is even or odd. For n = 2, L 1 is a straight line which contains

no vertex of 1r. If n > 2, L 1 is a straight line for which L 1 n [Ai'Ai+1, ... ,Al+n- 2J= (.J, l ~ i ~ m. For n = 2, r is the number of points of L 1 n 1r. If n > 2, r is the number of osculating n- 2-spaces L(Al+lAi+2 ••• Ai+n- 2) of ?rwhich intersect L 1. If k is the number of inflection hyperplanes of 1r, it is shown that r + k is even. This result is applied to show that every n + 3 points of Ln in general position are the vertices of exactly one polygon of real order n in Ln. (Received February 16, 1966.)

633-25. J. R. BROWN, Oregon State University, Corvallis, Oregon 97331. Existence of invariant measures for Markov operators.

Let (X~ ,JJ.) be a finite measure space, and let P(x,B) be the transition function for a Markov process with state space X. The process is nonsingular if P(x,B) = 0 a.e. iff JJ.(B) = 0 and ergodic

if P(x,X ~B)= 0 a. e. on B iff JJ.(B)u(X ~B)= 0. Au-finite measure a is invariant (excessive) if JP(x,B)a(dx) = ( ~) a(B) (B E .'1&). A family of jJ.-continuous measures [Qi: i E I] is almost equi­ continuous if ' 0 3Xf E !1& and o> 0 such that JJ.(X ~X f) < f and JJ.(A) < o, A E !1& nxf = Qi(A)

355 633-26, THEODOR GANEA, University of Washington, Seattle, Washington 98105. On the homogeneity of infinite products of manifolds.

M. K. Fort (?acific J. Math. 12 (1962), 879-884] has proved that a countable product of mani­ folds with boundary is homogeneous. It is proved here that a countable product of manifolds of which only finitely many have a boundary is never homogeneous. In fact the product has both stable and unstable points. (Received February 16, 1966.)

633-27. SIBE MARDESIC and JACK SEGAL, University of Washington, Seattle, Washington 98105. E-mappings and generalized manifolds.

Let X andY be metric compacta. A map f: X-> Y onto Y is an f-mapping, f > 0, provided the diameter diam { 1 (y) < f, for each y ~ Y. We say that X is n-manifold-like (Mn-like) provided for each f > 0 there exists an f-mapping of X onto some connected closed orientable triangulable n-manifold. Theorem. If X is an ANR of dimension n and is Mn-like, then X is an orientable n-dimensional generalized closed manifold (over any module of coefficients). Corollary. If X is an n-dimensional polyhedron and is Mn-like, then X is an orientable homology manifold (as defined in P. S. Aleksandrov, Combinatorial topology, III). Remark. T. Ganes has exhibited a 3-dimensional manifold-like ANR which is not a manifold (Michigan Math. J, 9 (1962), 213-215). (Received February 16, 1966.)

633-28. ALEKSANDER PE~CZYNSKI, University of Washington, Seattle, Washington 98105. Invariant means and uniformly continuous retractions. Preliminary report.

Let Y be a closed linear subspace of a Banach space X. Let Cu(X) (resp. Cu(Y)) denote the linear topological space of all uniformly continuous real valued functions on X (resp. on Y). The existence of invariant means on abelian groups is used to prove that each of the conditions (a) there is a uniformly continuous retraction from X onto Y, (b) there is an operator of simultaneous extension from Cu (Y) into Cu (X) implies (c) there is a bounded linear projection from the dual space X* onto its subspace Y.L = { x• E X*: x*y = 0 for y E Y }. The implication (a) ---->(c) was proved by J. Lindenstrauss, Michigan Math. J. 11 (1964), 268-287. The implication (a) ---->(c) is related to a theorem of Dugundji, Pacific J. Math. 1 (1951), 353-367. (Received February 16, 1966.)

633-29, R. B. LEIPNIK and F. C. REED, U. S. Naval Ordnance Test Station, China Lake, California, Monoidal families of discrete distributions.

The concept of a set of selections from a discrete multivariate distribution is defined In terms both of distributions and of generating functions. The concept of a self-associated parametric family of distributions is introduced, With the help of theorems of Aczel, the class of parametric families self-associated under a given associative set of selections Is characterized as to form. In the case where the monoid Is the group of positive reals under multiplication, the above class Is characterized In terms of representabllity as mixtures of stuttering Poisson distributions. (Received February 17, 1966.)

356 633-30, ALFRED INSELBERG, University of Illinois, Department of Electrical Engineering, Urbana, Illinois, Applications of superpositions for nonlinear operators.

Let T: D --> D be an operator, we say that T has a superposition if 3 a pair of functions F: DxD --> D, G: DxD--> D and an element e E E such that: T[F(x,y)) = G(x,y) Vx, y ED and G(x,y) = e when T(x) = T(y) = e. Many nonlinear operators, among which is the Riccati operator R (R(u) = u + p(t)u + q(t)u2 + r(t)), have such a superposition. By way of an illustration, a superposition is given for R from which the general solution of the Riccati equation (R(u) = 0) can be obtained in terms of two particular solutions, while a well-known result, the crosa-ratio formula, expresses the general solution in terms of three particular solutions of the Riccati equation, Furthermore, the generalized nth order Riccati equation is obtained, in a natural manner, from a canonical representation of R which is suggested from the superposition that is given, Further generalizations are also possible. Finally a result of E. Vessiot, obtained in 1893, is presented in conjunction with this notion of super­ position to provide information about the properties of certain nonlinear differential operators. (Received February 17, 1966.)

633-31. R. J. WISNER and D. W. HARDY, New Mexico State University, Box AM, University Park, New Mexico. Dominance semigroups of the modular group.

The notation and terminology are as in a previous paper (see Abstract 600-4, these c}/oticei) 10 (1963}, 265), Some special subsemigroups of ug (the semigroup of 2 X 2 unimodular matrics with nonnegative integral entries) are st.udied. One of them, called here L~, is suggested by the 0 array of Farey fractions, and some typical results are: (1) L 2 admits an ordering which is full, positive, archimedean, cancellative, and well; (2) the infinitude of primes in Lg can be described explicitly, and factorization is unique; (3} a prime number theorem is computed; (4} L~ is small in the sense that an infinite chain of semigroups lies between it and U~; (5} L~ is large in the sense that the full modular group in the smallest group containing it, (Received November 26, 1965.)

633-32. J. L. FIELDS, Midwest Research Institute, 425 Volker Boulevard, Kansas City, Missouri 64110. Confluent expansions.

The confluent hypergeometric function is a limiting case of the Gaussian hypergeometric

function, i.e., Limb~oo 2F l (a l b;c;z/b} = 1 F l (a;c;z). If a = c, this reduces to the familiar Limb (1 - z/b(b = ez. We will refer to this sort of limit process, as a confluence with respect to ->QO b. More generally, we will refer to any limit process of the form Lim b~ooL~Ofk(b) as a confluence with respect to b, if the functions fk(b}, up to a multiplicative constant dependent on k, are composed -k of a finite number of multiplicative factors of the form (± b + w1)k(b + w2) or their reciprocals,

where w1 and w2 are constants independent of band k, and (u)~ = r(u+~)jf(u). The value of the limit, if it exists, will be called the confluent limit with respect to b, In the situation where a conflu­ ence with respect to b is possible, it is of interest to consider what happens when b is large but finite. This leads in a natural way to expansions in inverse powers of b or a related variable. Such expan­ sions may be either analytic or asymptotic in nature, and will be referred to as confluence expansions with respect to b. Our results are concerned with characterizing several different types of confluent expansions. The results have special significance for the generalized hypergeometric functions. (Received February 16, 1966.)

357 The Meeting in Chicago April20-23, 1966

634-1. R. C. HOOPER, Wichita State University, Wichita, Kansas. Topological groups and integral norms. Preliminary report.

If K is an abelian group, then an integral norm will be a function llx II: K ___, Z satisfying llx + y II ~ llx II + IIY II. llnx II = In I llnx II for n E Z, and llx II = 0 if and only if X is the identity element of K. Now if K is countable, then for any k' E K there exists a homomorphism f: K ---> Z with f(k') = Ilk' II and lf(x) I ;;; llxll for all x E K. It follows from this that if K is a subgroup of a separable normed space E with the norm restricted to K integer valued, then all the points of E/K are separated by characters from the Identity except for a subgroup R where R is discrete and decomposes into the product of finite cyclic groups. In particular there exists a normed space E with a discrete sub­ group K such that for E/K, R is a cyclic subgroup of order 2. A partial converse to the preceding is the theorem that if K is a discrete subgroup of a normed space E such that E/K has a basis of neighborhoods that are the intersection of weak character neighborhoods, then E can be renormed so that the norm restricted to K is an integral norm. It is to be noted that E/K can be bad since there is a discrete subgroup K 1 of £2 such that i 2/K has only the trivial character. (Received November 22, 1965.)

634-2. R. H. ROLWING, University of Cincinnati, Cincinnati, Ohio 45221, and I. A. BARNETT, Fairleigh Dickinson University, Teaneck, New Jersey. An extremal problem associated with a certain nonlinear integral equation.

In a paper On a system of quadratic equations and its integral analogue [An. Sti. Univ. "Al. I. Cuza" Iasi. Sect. I, 1965) the following integral equation y 2(x) = fy(x)- K(x)JJ6y(s)ds is discussed. The solution y(x) depends upon the equationfo.t(x) (1- ck(x)) 112dx = 1 in the unknown C for given K(x). Here f(x) is a step function assuming values + 1. The subject of this paper is to determine the interval on which c must fall as the functions k(x) vary over a certain class for a prescribed f function. This gives rise to the following problem. Find the maxlrrmm c-ror which Jaf(x) (1- ck(x))112dx = 1 and J6 [K(x)]rdx = 1 where r is any positive integer and f(x) is a prescribed step function. For the case r = 1, it is found that the maximum C is given by the expression - (1 - m)/(1 + m),m =J6f(x)dx. If one lets cP(x) =- Ck(x), the problem becomes that of minimizing Ja[(x))rdx subject to the condi­ tions (x) <;- 1, Jaf(x) (1 +(x)) 1/ 2dx = 1. For the case r odd, r > 1 the minimum is p(l/p2-1)- n if p 2 ,1;; 1/1 +X and- 1 + (Xr+1)/ (1 + X)1/2 if p2 > 1/(1 +X) where pis the measure of the set where f(x) "' 1, n is the measure of the set where f(x) = ~ 1, and where X is the unique root of the equation (2r - 1)xr t 2rxr- 1 - 1 = 0. For the case r even, results analogous to the previous are found but the results are much more complicated. (Received November 17, 1965.)

358 634-3. D. W. JONAH, Wayne State University, Detroit, Michigan 48202. Cohomology of coalgebras. Preliminary report.

We set up a cohomology theory for coalgebras over a category with multiplication and carry out the Eilenberg-Mac Lane-Hochschild program for interpreting the low dimensional cohomology groups. The proofs are categorical and yield a generalization of Hochschild's original work for linear associative algebras. (Received December 1, 1965.)

634-4. P. P. ORLI~. 804 Berkshire, Ann Arbor, Michigan 48104. Some topological invariants of Seifert fiber spaces. Preliminary report.

Let(O,o,glb;11 1, fJ1; ••• ; 11:r' fJ.), (O,n,klb;11 1, {31; ••• ; ar,flr),(N, o,glb; 111, /31; ••• ;11r•.Br},(N, ri.I,kl b; a 1, fJ1; ••• ; ar' f3r)' (N,nii,klb; <11' fJ1; ••• ; <1r' flr) and (N,nlll,k lb; <11' {31; ••• ; ar' {3r) be the 3-manifolds considered in Seifert: Topologie dreidimensionaler gefaserter Raume (Acta Math. 60 (1933), 147-23.8). The symbols left of the vertical bar denote: the orientability of the space; the orientability of the decomposition space; and the genus of the decomposition space. To the right of the bar are integers such that (11i' {3i) = 1 and 0 < {3i < ai all i. Assuming g > 0 the following results have been obtained toward a topological classification of these manifolds. Theorem 1. Except for the manifolds (N,o,ljO), (N,nl,2IO), (N,o,111) and (N, nl,2l1) the symbols left of the bar are topological invariants.

Theorem 2. For manifolds of type (O,o,glb; <11' f3r; .•• ; ar' {3r) and (N,nl,k lb; <11' /31; ••• ; ar' {3r) with r > 1 the set of integers [al'a2 , ... ,arJ is a topological invariant. (Received November 26, 1965.)

634-5. RANKO BOjANIC, Ohio State University, Columbus, Ohio 43210. An elementary proof of the uniqueness theorem in the theory of best L 1-approximation.

Let f be a continuous function on (a,b]. As it is well known the polynomial of best L 1-approxi­ mation to f is unique. D. jackson's proof of this result, as well as all subsequent proofs, depends essentially on measure theoretic arguments. The basic idea in most of these proofs is to show that f-P must either change sign n + 1 times or vanish on a set of positive measure. The aim of this note is to give an elementary proof of jackson's theorem based on the following results:__!! P and Q ~ polynomials of best L 1-approximation to f, then (i) (l/2)(P + Q) is also such a polynomial, (ii) If- (l/2}(P + Q)l = (1/2)1 f- PI+ (l/2)lf- Ql, (iii) l!_P is a polynomial of best L 1-approximation of degree ~ n~ f, then f - P has at least n + 1 different zeros in [!!.,b). The first two results are well known. The third result is intuitively clear, for if the number of zeros off - P were :>· n, it would be possible to construct a polynomial of degree ~ n which would approximate fin L 1 better than P. To prove the uniqueness theorem, suppose that P and Q are polynomials of best L 1-approximation of degree ~ n to f. Then by (i) and (iii) f- (1/2}(P + Q) has at least n + 1 zeros x 1 < ••• < xn+1 in [a,b] and by (ii) f(x11 )- P(x 11 ) = 0 and f(x 11 )- Q(x 11 ) = 0 i.e. P(x 11 ) = Q(x 11 }, v= 1, ••• ,n + 1 and the uniqueness theorem is proved. (Received December 3, 1965.)

634-6. M. D. SECKER, University of Texas, Austin, Texas. Reversibly continuous bisensed transformations of an annulus into itself.

If in the plane the simple closed curve J 1 encloses the simple closed curve J 2 and F is the

359 set of all points that lie within J 1 and without J 2 then (1) a collection G of mutually exclusive arcs filling up F is called an 11- collection if and only 1f it is true that if g is an arc of the collection G then continuous trans­ g • J1 is one end point of g and g • J 2 is the other one, and (2) if T is a reversibly formation ofF into itself such that T(J 1) = J 1 and T(J2) = J2, then a meaning is given to the statement that T is bisensed. It is shown that the subset K of F is closed and separates J 1 from J 2 if and only if there exist an 11-collection G and a reversibly continuous bisensed transformation T of F into itself such that a point belongs to K if and only if it is common to some arc of the collection G and the image of that arc under T. (Received November 19, 1965.)

634-7. C. S. REED, The University of Texas, Austin, Texas 78712, Pointwise limits of sequences of quasicontinuous functions.

The work previously outlined in Abstract 625-96, these cflotiai.J 12 (1965), 573 is continued, Suppose g is the graph of the real valued function f over the interval [a,b]. The following two state­ ments are equivalent. (1) f is the pointwise limit of a sequence of quasicontinuous functions each continuous on the right and (2) if p is a point of g, there exist two vertical lines h and k with p between them such that if u and v are two horizontal lines and Tis a subset of [a,b] having the property that each point of g with abscissa in T is between h and k and above u and v and B is a subset of Ja,b I having the property that each point of g with abcissa in B is between h and k and below u and v, then there is a point of T which is not a limit point of B from the right or there is a point of B which is not a limit point of T from the right, (Received November 22, 1965.)

634-8. F. M. WRIGHT and K. P. SMITH, Iowa State University, Ames, Iowa. A Lane integral with no summability set which is everywhere dense.

R. C. Bzoch (J. Indian Math. Soc., 1959) showed that 1f g is a real-valued function of bounded variation on a nondegenerate closed interval [a,b] of the real axis and f is a bounded real-valued function on [a,b] such that the Lane integral L J~f(x)dg(x) exists, and if G is a summability set for f and gin [a,b], then there is a real-valued function u on (a,b] such that u(x) = f(x) for all x in G, and such that the Stieltjes mean sigma integral Fmf~u(x)dg(x) exists. A less involved proof of this result is presented based on results for a weighted Stieltjes mean sigma integral discussed in earlier ab­ stracts by F. M. Wright. Bzoch also showed that if f,g are real-valued functions on a nondegenerate closed interval [a,b] of the real axis such that the Lane integral Lf~f(x)dg(x) exists, if G is a sum­ mability set for f and gin (a,b], and if Tis in [a,b], then G + [rl is a summability set for f and gin (a,b]; an example is presented of such an f,g for which there is no summability set for f and gin [a,b] which is everywhere dense on [a,b]. (Received November 24, 1965.)

634-9. R. D. DAVIS, 900A Spring Valley Plaza, Richardson, Texas. Concerning the sides from which certain sequences of arcs converge to a compact irreducible continuum.

Let S be the set of all points of a space :2: that satisfies Axioms 0 and 1 through 5 of R. L. Moore [Foundations of point set theory, Amer. Math. Soc. Colloq. Publ., vol. 13, Rev. Ed., 1962]. lf A and B are two points of the compact continuum M, a meaning is given to the statement that two

360 sequences of points converge to the point P of M from opposite sides with respect to A and B. Theorem. Suppose G is an uncountable collection of mutually exclusive compact conti'l.ua, no one of is compact and for each element g of G, A , B and P are three which separates S, such that G* g g g points of g such that g is irreducible from Ag to B g' Ag and B g are accessible from S - g and for

each point P 'g of g different from P g' there exists a subcontinuum of g that contains P g and P ~ and lies in g - (Ag + B g>· Then there exists a countable subcollection G' of G such that if g is an element

of G- G', there exist two sequences a. 1,a.2,a.3, ••• and (31, {32 , {33' ... of elements of G such that these­ quences P , P , P , ••• and P fJ ,P (3 ,P(3 , ••• converge to Pg from opposite sides of g with respect a.1 a.2 a.3 1 2 3 to A and B • (Received January 3, 1966.) g g

634-10. C. B. MURRAY, TRACOR, Inc., Box 7053 University Station, Austin, Texas 78712. On the existence of Stieltjes integrals.

Here the integrals are "refinement" limits, real-valued f is bounded on the interval [a,b) of real numbers, real-valued g is of bounded variation on [a,b). Theorems. (A) The left Cauchy­ Stieltjes integral (L)J~f dg exists if and only if for each w > 0 and s :::> 0 there is a subdivision a= t < t < ••• < t = b such that (l) Jbldgl- ""n ig(t ) - g(t < s, and (2) if Q is a set of 0 1 n a L...p= 1 p p- 1)1 positive integer(s) q ~ n such that the segment (tq-l'tq) has numbers x and y with f(y) - f(x) ;?; w then I:Qig(tq)- g(tq-l +)I < s. (B) The mean Stieltjes integral (M)J~f dg exists if and only if (L)J~f dg exists and for each tin [a,b) either f(t +)exists or g(t +) = g(t). (Compare G. B. Price, Cauchy­ Stieltjes and Riemann-Stieltjes integrals, Bull. Amer. Math. Soc. 49 (1943), 625-630.) (C) Suppose

that if a ~ u < v ~ b then either f(u +) exists or g(u +) = g(u), and either f(v -) exists or g(v -) = g(v). Then the author's condition, Theorem 1, Abstract 615-5, these cJioticeiJ 11 (1964), 663, is necessary

and suffici~nt for existence of F. M. Wright's weighted mean Stieltjes integral defined in Abstract 618-12, these cJ{oticei) 11 (1964), 762. These theorems fall if the condition that f be bounded on [a,b] is omitted. Methods of proof resemble those in the author's dissertation, University of Texas, Austin, 1964. (Received January 10, 1966.)

634-11. B. K. WONG, Western illinois University, Macomb, illinois 61455. Recurrence of set trajectories in a discrete dynamical system.

In Probl~mes dans la thc!orie des syst~mes dynamiques, Acta. Math. 95 (1956), to be referred

to as [T], Trjitzinsky proposed a generalization of Denjoy's recurrence theorems by replacing the perfect sets with closed sets with nonvoid perfect kernels. See [T] for background and notations. It is shown that to obtain the analogue of Denjoy' s results for Trjitzinsky' s classes (C) and (C), (9. 7) and (9.14) in [T],p. 246 and 248, as well as the theorems to follow, an additional condition, viz,

N(E) = E, is needed. With respect to the class (C0 ), Trjitzinsky proved that for any perfect set

P E: (C 0) such that all of its portions are also in (C0), the set E*(P) is everywhere-dense on P. Using an argument modelled after Denjoy's, we proved: For any perfect set P E (C 0 ), E*(P) is a residual of P. The last theorem answers the open question raised in [T] p. 248, and together with Trjitzinsky's cited above give the analogues of Denjoy's theorems in a more general setting. (Received January 10, 1966.)

361 634-12.. R. A. BRUALDI, S. V. PARTER and HANS SCHNEIDER, University of Wisconsin, Van Vleck Hall, Madison, Wisconsin 53706. Nonnegative matrices diagonally equivalent to row sto­ chastic matrices.

Theorem. Let A be an n X n nonnegative matrix with no zero rows. Then for every nonnegative B having zeros in exactly the same positions as A there exists a diagonal matrix D with positive diagonal (dependent on B) such that DB D is row stochastic if and only if the following condition is satisfied: Suppose the rows and columns of A have been permuted simultaneously so that aij : 0 for 1 ;;:; i, j ;;:; r, then there exists k and £with r < k ;;; n and 1 ;;; £ ~ r such that akj: 0 for j: l, •.• ,r and a £k > 0. Moreover, if A satisfies the above condition, then the diagonal matrix D for which DBD is row stochastic is unique. Corollary. If, A is an n X n nonnegative matrix with a positive main diagonal, then there exists a unique diagonal matrix D with positive diagonal such that DAD is row stochastic. (Received January 5, 1966.)

634-13. E. W. JOHNSON, University of California, Riverside, California. A-transforms for Noether lattices. Preliminary report.

Let .sf denote a Noether lattice as defined by R. P. Dilworth (Pacific J. Math. 12. (1962.), 481-498). Let I denote the greatest element of Yand let A be an arbitrary element. Set Ai: I for i ~ 0 and consider the collection of all formal sums L~-ooBi of elements of .st'which satisfy there­ lation Ai ~ Bi ~ Bi+l ~ ABi for all i. For sums B: LBi and C: LCi' define B ;;; C if Bi;;: Ci for all i, and B· C: Li(Vr+s=iBrCs). Let .9f(...5t;A) denote the resulting system. Then it is seen that .9f(...5t,'A) is a modular lattice. Moreover, one has the following Theorem 1 • .9f(Y,A) is a Noether lattice. If Yis local (i.e., Yhas a unique maximal element), then the lattice .9f(Y,A) can be used to establish the following Theorem 2.. J!. A is primary for the maximal element of Y, then there exists a numerical polynomial p(x) such that for n sufficiently large p(n) is the dimension of the finite dimensional quotient I/ An. A theory of minimal bases for elements. is obtained for local No ether lattices and the lattice !Jf(Y,A) is used to establish Theorem 3. _!!.A is an element of .st'(local) then there exists a numerical polynomial p'(x) such that for all sufficiently large n,p' (n) is the least number of principal elements with join An. (Received November 17, 1966.)

634-14. C. T. TUCKER, 2.704 B Salado, Austin, Texas 78705. Pointwise limits of quasi­ continuous functions.

Kempisty (Fund. Math. 2 (192.1), 131-135] showed that a necessary and sufficient condition for a real-valued function f with domain the set of real numbers to be in either Batre's class 0 or is that it be the uniform limit of a sequence each term of which is the difference of two upper semi­ continuous functions. The author extends this theorem to the case where the domain off is any subset M of the real numbers, and this result is used to prove the following Theorem. Suppose f is a real valued function with domain M. Each two of the following three statements are equivalent: (1) f is the pointwise limit of a sequence of quasi-continuous functions over m. (2.) f is the pointwise limit of a sequence of functions over M which are either conti11uous or else have at most countably many points of discontinuity. (3) There exists a countable proper subset T of M and a function g over M belonging to either Baire's class 0 or 1 such that g: f over M- T. (Received January 10,

1966.)

362 634-15, J. W. ROGERS, JR., 604 Harris Avenue, Austin, Texas 78705, A space whose regions are the simple domains of another space,

If 2: is a space that satisfies Axioms 0-5 of R. L. Moore's Foundations of point set theory, let 2:' denote a space whose "points" are the points of 2: and whose "regions" are the simple domains of 2:, It is shown that, for every 2:, 2:' satisfies Axioms 2,3,4, and 5, An example is given of a space

~satisfying Axioms 0-5 but such that 2:' does not satisfy Axiom 13, and a condition on 2: is given which is necessary and sufficient in order that 2:' satisfy Axiom 13• If 2: satisfies Axiom 13 so does 2:'. (Received January 13, 1966.)

634-16. DANIEL WATERMAN, Wayne State University, Detroit, Michigan 48202. On the summability of the differentiated Fourier series,

Suppose f E L(0,211") with Fourier series S(f). Fatou's theorem asserts that if the symmetric derivative f~{x 0 ) exists, then S'(f) is Abel summable to sum f~{x 0 ). We will say that y is the a-approximate symmetric derivative off at x0 if, on letting Ef = ftl(f(x0 + t) - f(x0 - t)/2t) - y I > f J for each f > 0, we have m(Ef n (O,Il)) = o(lln) as ll--+ 0+. Then iff is essentially bounded in a neigh­ borhood of x0 and this generalized derivative exists for some n ~ 2, the differentiated Fourier series is Abel summable to the value of this derivative. This is "best possible" in two senses; first, 2 can­ not be replaced by a smaller value, and second, essentially bounded cannot be replaced by integrable, (Received November 17, 1965.)

634-17. L. R. McCULLOH, University of Illinois, Urbana, Illinois 61801. Cyclic extensions without relative integral bases,

Finitely generated, torsion free modules M over a Dedekind ring R are characterized by their rank and an invariant C(M) which is an ideal class of R. Let R be the ring of integers in an algebraic number field K containing the nth roots of 1. The ring of integers S in a finite extension L/K is an R-module, We determine the possible invariants C(S) for cyclic extensions of fixed degree n. They are precisely those classes in the subgroup of dth powers of the ideal class groups of R, where d = if n is even or d = g,c,d. { (p - 1)/2; p a prime divisor of nj if n is odd, In particular, if d is not an exponent of the class group, there is a cyclic extension of degree n without a relative integral basis, (Received January 20, 1966.)

634-18. D. H. SATTINGER, University of California at Los Angeles, 405 Hilgard Avenue, Los Angeles, California, A singular eigenfunction expansion in anisotropic neutron transport theory,

Let K be a compact symmetric integral operator on L 2 (- 1,1) with kernel k{u,u') which is real and analytic - l ;;; u, u' ;?; 1, Consider the integral equation (s - u)c/Js = sKc/ls' where s is a complex number. If f{u) is Holder continuous on - l < u < l then f(u) can be expanded in the eigen­ functions of this integral equation provided one admits generalized functions {distributions) as singular eigenfunctions corresponding to the continuous spectrum - 1 < s < 1. This completeness theorem can be used in solving one-velocity, one-dimensional transport problems with boundary conditions, (Received January 17, 1966, ).

363 634-19. FRANK FORELLI, University of Wisconsin, Madison, Wisconsin. Analytic measures.

Let the real line R act as a topological transformation group on the locally compact Hausdorff space S, and let M(S) be the Banach space of bounded complex Balre measures on S. The action of R on S can be used to define the convolution of a measure in M(S) with a function in the group algebra L 1 (R), and convolution In turn can be used to associate with a measure in M(S) a closed subset of R called the spectrum of the measure. !lin M(S) is called analytic if its spectrum is contained in the nonnegative reals, and !lis called quasi-invariant if the collection of ll null sets Is carried onto itself by the action of R on S, Theorem. Analytic measures are quasi-invariant. When Sis a compact abelian group and the action of R on S is given by translating with the elements of a one-parameter subgroup of S, the theorem is the deLeeuw-Glicksberg generalization of the F. and M. Riesz theorems (Acta Math. 109 {1963), 179-205). (Received january 26, 1966.)

634-20. E. L. BETHEL, Clemson University, Clemson, South Carolina 29631, Concerning continuous collections of mutually exclusive continua.

M.-E. Hamstrom has shown that If G is a continuous collection of mutually exclusive arcs filling up a compact continuous curve M in the plane such that M/G is an arc, then G* is a simple closed curve plus Its Interior. The main purpose of this paper is to generalize Ham strom's result in the following Theorem. If S is a space satisfying Axioms 0 - 5 of R. L. Moore's Foundations of point set theory, and M C S such that (1) M has one and only one complementary domain, and (2) there exists a nondegenerate collection of mutually exclusive nondegenerate continua filling up M, then M is a simple closed curve J plus one of the complementary domains of j. As one of the con­ sequences of this theorem, we have the following Corollary. A necessary and sufficient condition that a bounded continuous curve M in the plane be a simple closed curve plus its interior is that there exist a nondegenerate collection of mutually exclusive nondegenerate continua filling up M. Some other consequences of this theorem shall be stated, {Received February 4, 1966.)

634-21. WITOLD BOGDANOWICZ, Catholic University, Washington, D. C. 20017. A Baire type characterization of Lebesgue-Bochner measurable functions.

Let Y ,R be a fixed Banach space and the space of reals. A family V of subsets of a space X is called a pre-ring if A,B E V implies An B E V and A'\B = C 1 U ... U Ck, where Ci E V are disjoint sets. If v is a nonnegative countably additive function on V then the triplet (X, V, v) is called a volume space, In the paper: Bogdanowicz, A generalization of the Lebesgue-Bochner-Stieltjes integral and a new approach to the theory of integration, Proc, Nat, Acad, Sci. U.S.A. 53 (1965), 492-498, has been presented a direct approach to the theory of the space L(Y) of Lebesgue-Bochner summable functions generated by a volume space {X, V, v). The space M(Y) of Lebesgue-Bochner measurable functions consists of all functions f which have V - u-finite support and, for every set A E V, the function

XA(x)a(f(x)) belongs to the space L(Y), where a(y) = (1 + IY j)- 1y for y E Y. Let s0 {Y) consist of sums of functions of the form XAy' where A EV, y Ey, Define the set Sn(Y) by induction to consist of limits under convergence everywhere of sequences of functions from Sn-l (Y). Theorem. A function f belongs to the space M(Y) iff there exists a function g E S 3(Y) such that f{x) = g(x) v-almost every­ where, This result will appear in Math, Ann. in the volume dedicated to Professor Gottfried Koethe.

(Received February 4, 1966.)

364 634-22. R. E. GOODRICK, University of Wisconsin, Madison, Wisconsin. A note on Seifert

1t is known that a polygonal knot In E 3 bounds an or!entable surface. R. H. Fox (A quick trip through knot theory, Topology of 3-Man!folds) obtains this surface by attaching twisted rectangles at crossings, constructing a collection of disjoint simple closed curves, called Seifert circles, and at­ taching disjoint disks to these simple closed curves. The Seifert circles will be called nested If, in a projection on a plane, some pair is nested. Fox noted, that in particular cases, the knot could be moved by a space isotopy to a position in which the Seifert circles do not nest. Theorem. Any poly­ gonal knot can be moved by a space isotopy to a position in which its Seifert circles do not nest. (Received February 3, 1966.)

634-23. CLIFFORD PERRY, University of Minnesota, Minneapolis, Minnesota 55455. Closed uniform subgroups of solvable Lie groups.

A characterization of those fundamental groups of compact solvman!folds which can be Im­ bedded as discrete uniform subgroups of connected, simply connected solvable Lie groups was given by L. Auslander (Trans. Amer. Math. Soc.99 (1961), 398-402). It is the purpose here to give sufficient conditions for a group r to be Imbedded as a closed uniform subgroup In a connected, simply connected solvable Lie group, and conversely, to characterize certain closed uniform subgroups r. Theorem. If r Is an algebraic strongly torsion free S group, then r can be imbedded as a closed uniform sub­ group of a connected, simply connected solvable Lie group. Theorem. If a closed uniform subgroup r of a connected, simply connected solvable Lie group G contains no proper normal analytic sub­ groups of G, then r Is an algebraic strongly torsion free S group. (Received February 3, 1966.)

; 634-24. JESUS GIL DE LAMADRID, University of Minnesota, Minneapolis, Minnesota. Representations of topologically simple Banach algebras.

For terminology and background material refer to the book of C. E. R!ckart, General theory of Banach algebras, Van Nostrand, Princeton, 1960. Theorem. Let B be a Banach algebra of dimension higher than 1. Then B is topologically simple if and only if the set B2 of all products of elements of B is total In B and there exists a continuous faithful representation of B as a dense two- sided ideal of a topologically simple Banach algebra of operators (with the operator norm) on some Banach space. The representation is the left regular representation on the entire algebra B and the proof Is completed with the help of the following Lemma. Let A be a Banach algebra whose left annihilator reduces to the element 0, and B a dense two-sided Ideal of A of dimension higher than l, itself a Banach algebra under some norm which majorizes the norm of A on B, and with respect to which B2 is total in B. Then A is topologically simple If and only if the Banach algebra B Is topologically simple. (Received February 7, 1966.)

634-25. ]. E. DENNIS, University of Utah, Salt Lake City, Utah 84112. Application of fixed point principles to solutions of P(x) = 0.

Let X and Y be Banach spaces and let P be a nonlinear operator Into Y from a neighborhood

U of a point x0 EX. Kantorovich and Akilov, in Functional analysis in normed spaces, present some

365 theorems due to Kantorovich insuring the convergence of Newton's Method, both original and modified, to a solution s of P(x): 0. These theorems locate s in a neighborhood V CU. If U is sufficiently large, then initial guesses close to x0 may be used in place of x0 in the iteration process. The present work applies fixed point techniques to the Modified Newton's Method under Kantorovich's hypotheses and obtains a much wider choice of initial guess without the additional assumptions on U. In addition, s is located in an annulus inside V. These results yield a convergence theorem for a modification of the Generalized Newton's Method of Greenspan, Lieberstein, and Yohe. (Received February 7, 1966.)

634-26. F. M. LISTER, University of Utah, Salt Lake City, Utah 84l12. Polyhedral approximation of spheres in E 3.

Bing's Side Approximation Theorem for 2-spheres in E 3 [Ann. of Math. 77 (1963), 145-192,

Theorem 16] is strengthened as follows. Theorem 1. If S is a 2:sphere In E 3 and l > 0, then there Is a homeomorphism h of S into E 3 and a finite collection of mutually exclusive f-disks [Ei1 ~on S such that (1) h moves no point a distance more than f, (2) h(S) is polyhedral, (3) h(S - L:mE.): 1 1 h(S) - L~(EI) c Int s, (4) s - I:~EI c Ext h(s), and (5) h(Ei) ns : Ei n h(S) : h(Ei) n Ei (i = 1, ••• ,m). This may be generalized to open subsets of S. Theorem 1, along with theorems proved by Bing [Mich. Math. J. 11 (1964), 35-45, Theorem 8.4) and Loveland [Tame subsets of spheres in E 3 , Pacific j. Math. (to appear)], Is used to obtain the following, Theorem 2. If F is a finite sum of tame finite graphs and tame Sierpinski curves on a 2-sphere Sin E 3, and his a homeomorphism of S + Int S into E 3 such that h(S) is tame from Ext h(S), then h(F) Is a subset of a tame 2-sphere in E 3• (Received

February 7, 1966.)

634-27. D. W. SOLOMON, Wayne State University, Detroit, Michigan 48202. Denjoy integration in abstract spaces. III.

Let X be a second countable, locally compact metric space which has a base cfl satisfying Romanovski's ten axioms [Math. Sbornik 9 (51) (1941)), andY be a Banach space with a countable determining set. This paper presents descriptive definitions of two integrals, DP • and DP, of point functions f, defined on a member of c:l/; and with range contained in Y, and a constructive definition of DP*, and studies some of the properties of these integrals. DP generalizes DP*, and DP* general­ izes the Pettis integral [Trans. Amer. Math. Soc. 44 (1938)) and two integrals defined by the author (Abstract 630-45, these cJ.fotiai). 13 (1966), 69, and Abstract 66T-221). In caseY is finite dimen- sional, all four of the author's integrals are equivalent, although this is not the case for infinite di­ mensional Y. The constructive process presented for DP* can be compared with those presented for the author's other two integrals. Any DP*-integral can be attained in at most countably many appli­ catiQns of the construction process. (Received February 7, 1966.)

634-28. M. A. ARKOWITZ, Dartmouth College, Hanover, New Hampshire and C. R. CURJEL, University of Washington, Seattle, Washington. Some results on H-maps. Preliminary report.

Let X and Y be H -spaces. Denote by [X, Y] H the set of all homotopy classes of maps X ----+ Y

which are H-maps (i.e., homomorphisms "up to homotopy"); [X, Y] His a subset of the set [X, Y] of

366 homotopy classes of maps X --> Y. Under the assumption that X and Y are homotopy associative H-spaces having the homotopy type of finite connected CW-complexes the following results are established: (1) [X, Y]H is infinite if and only if rank 1rn (X) • rank 1rn (Y) i 0 for some n. (2) Let~ be the element M • ld = Id + ••• + Id in the group [X,X] (the homotopy class of the power map x --> xM in case X is a topological group). Then for any X there exists an integer N > 0 such that 1s,t< is an H-map for all integers k. Furthermore [liP, Sq]H is completely determined if p,q are in the set­

{1,3, 1] (for example, _tl <":: 71'7(S 7) is an H-map if and only if N(N- I)"' 0 (mod 240). (Received February 7, 1966.)

634-29. D. F. DAWSON, North Texas State University, Denton, Texas. Remarks on some convergence conditions for continued fractions.

If a= { apf is a complex number sequence, let f(a) denote the continued fraction

l/1 + a 1/I + a 2/1 Theorem 1. If II+ an+ ant1 1 ~ 2lail' i = n, n + 1; n = 1,2,3, •.. , then f(a) converges. Theorem 2. If frp) is a sequence of nonnegative numbers such that r 1 11 + a 1 1

~ la 11, r 2 P + a 1 + a 2 1 ~ la2 1, rpp + ap_ 1 + apl ~ rprp_ 2 1ap_ 1 1 + lapl' p = 3,4,5, .•. , ft; r 2t ~ rJ is an infinite set , and Lll - r 2p I= oo, then f(a) converges at least in the wider sense, and if 1 extends a result of r 1 1I + a 1 1 > la 1 1 or r 2 P + a 1 + a 2 1 > la 2 1, thenf(a) converges. Theorem Farinha [Portugal. Math. 13 (1954), !45-148) and Theorem 2 extends a result of the author [Proc, Amer. Math. Soc. l3 (1962), 698-701). A result similar to Theorem 2 is obtained if the roles of even and odd indices are interchanged. (Received February 7, 1966.)

634-30. M. G. ARSOVE University of Washington, Seattle, Washington 98105 and ALFRED HUBER, Polytechnic Institute, Zurich, Switzerland. On the existence of nontangential limits of sub­ harmonic functions.

Suppose that u is a subharmonic function on the open unit disc having bounded mean modulus. Although Littlewood has shown that u has finite radial limits for almost all directions, Zygmund has pointed out that nontangential limits may fail to exist. Each of the following conditions, fundamentally different from earlier conditions of Tolsted, is sufficient for the existence of nontangential limits at almost all boundary points: (1) the mass distribution for u is given by a density function A for which

A(rei8) = O[(l - r)-2) (the exponent- 2 here is best possible); (2) the mass distribution for u is given by a density function A such that J6J~1r(l - r)2p- 11A(rei 8) IPrdlidr < + oo for some real number p > 1 (the condition p >I here cannot be replaced by p ~ 1). (Received February 7, !966.)

634-31. W. V. PETRYSHYN, University of Chicago, Chicago, Illinois. On a fixed point theorem for nonlinearP-compact operators in Banach space.

Let X be a real Banach space such that there exists a sequence fxnl of finite dimensional sub­ spaces Xn of X, a sequence of linear projections P non X, and a constant K > 0 such that Pn X= Xn,

Xn C Xn+ 1 and liP n II ~ K for n = 1,2,3, ... , and Unxn = X. Let Br denote the closed ball in X of radius r > 0 about 0 with boundary Sr Definition. A nonlinear operator A in X is called projectionally­ compact (P-compact) if Pn A is continuous in Xn for all large n and if for any constant p > 0 and any

bounded sequence fxn} with xn E Xn the strong convergence of fgn} = [P nAxn - pxn\ implies the

367 existence of a strongly convergent subsequence ~x I and an element x in X such that limn.x = x ( ni) 1 ni Let A be P compact. Suppose that for given and llmn1P niAxni = Ax. The main result is: Theorem. r > 0 and J.l. > 0 the set A(Sr) is bounded and A satisfies the condition (il~): If for some x in Sr the equation Ax = ax holds then n < J.L. Then there exists an element u in (Br - Sr) such that Au - J.I.U = 0. We deduce from our theorem the fixed point theorems of Schauder, Rothe, Altman and Kaniel, and some results on monotone operators in Hilbert space of Minty, Browder, and Shinbrot. Further extensions to unbounded operators will be indicated. (Received February 11, 1966.)

634-32. WILLARD MILLER, JR., University of Minnesota, Minneapolis, Minnesota. The Lie theory of monic induced group representations.

In order to provide the foundation for a general theory of special functions the classical Lie theory of local transformation groups is extended to monic induced representations of local groups., These representations are classified up to a suitably defined equivalence and the equivalence classes are shown to be isomorphic to certain cohomology spaces. Computational aspects of the theory are stressed throughout and the results are applied to special function theory. (Received February 11, 1966.)

634-33. M. W. MANDELKER, University of Rochester, Rochester, New York. ~ z-ideals in C (R). II.

For part I, see Abstract 632-32. Let Q be a prime z-ideal in C(R). We say that Q is~ if the z-filter Z [Q] contains the derived set of each member. An ultrafilter o/ on R is associated with Q if Z [Q] = o/ n Z(R). Theorem 8. Q is nonclosed if and only if some associated ultrafilter has a discrete member. In this case, Q has just one associated ultrafilter, and Q is not the union of its predecessors. Theorem 9. If Q is the intersection of its successors, then Q is closed. Theorem 10. If Q is closed and contains a function with countable zero-set, then Q is the union of its predecessors. Theorem 11. If Q is nonminimal and some associated ultrafilter has a countable member, then Q has an immediate predecessor, and, assuming the continuum hypothesis, a minimal immediate predecessor. Examples are given for all the above types of prime z-!deals. Among other examples, there is a prime z-ideal that is both the intersection of its successors and the union of its predeces­ sors, and one that has an immediate predecessor but also a predecessor that is contained in no im­ mediate predecessor. Other results and extensions to metric spaces are discussed, In particular, each result here and in part I holds in Rn. (Received February 11, 1966.)

634-34. L. P. MAHER, JR., Indiana University, Bloomington, Indiana 47401. An expansion of the real-number system.

The set of real-number axioms is modified and expanded so that there are uncountably many nontrivial operations. A geometric example is given which satisfies the new axiom set. The set of operations is linearly ordered so that it is topologically equivalent to the X axis and, for each real number x, there is an operation P x· For each x, Px is distributive over Px-l, and Px and P x- 1 have all the properties which the real-number axioms ascribe to multiplication and addition. If, for each x, Dx denotes the set of elements operated upon by P x• then /Dxl is monotonic and Dx C Dw, if

368 w < x. As a by-product, the set of functions •••• 1b - 1• 1b0 • 1b1, lb2, ••• (i.e., •.•• 2x, X, log2x, 1b t. If P denotes log2 (log2x), ..• ) is expanded so that, for each real number t, there exists a function 1 1 1 multiplication and Po denotes+ and D1 denotes the real-number set, then xP 2y means 2( bx)( by)

and xP _ 1y means 1b(2x + 2Y). There is a meaning for such things as 1og2 (- 5) and 1bt(- lxi). (Received February 14, 1966.)

634-35. S. W. YOUNG, University of Utah, 204 Mathematics Building, Salt Lake City, Utah. Universal continuous functions,

Let c 0 denote the space of real continuous functions g on [0,1] with g(O) = 0. llgll is the uniform norm. Theorem. There exists an f E c 0 such that fnt(x/n)J is dense in c 0• Iff E c 0 and there exists a sequence )an,bnf such that [anf(bnx)j is dense in c 0 , then f is called a universal continuous function. (u.c,f,). The differentiability of u.c.f.'s is examined, For example: (1) A u.c.f, of the type in the theorem stated above is not differentiable at 0. (2) Iff is a u.c.f. having nth derivative on a region of 0, then fn(O) = 0. (3) Iff is a u.c.f., then g(x) = J~f(t)dt is a u.c.f. (4) There exists a u.c.f. which is infinitely differentiable on [0,1). (Received February 14, 1966.)

634-36. T. L. BOULLION, University of Texas, Austin, Texas and P. L. ODELL, University of Southwestern Louisiana, Lafayette, Louisiana. A generalization of the Wielandt inequality.

The Scroggs-Odell pseudo inverse, A+, which inherits an important spectral property possessed by the inverse of a nonsingular matrix, is used to define a generalized spectral condition number for square (even singular) matrices using the spectral norm. i.e. K(A) = IIAII·IIA + 11. The generalized spectral condition number is used to establish the main result which is: Theorem 1. For any square matrix A and any pair of orthonormal vectors x and y in the orthogonal complement

of the null space of A, lx*My I ~ IIAy II II Ax II cos 01 where cot (0/2) = K(A), M = A* A and 11•11 is the Euclidean norm. The Kantorovich inequality is also generalized to square, singular matrices and follows from Theorem 1, Corollary, For any square matrix A and unit vector x in the orthogonal

complement of the null space of A, (x*x) 2 i'; x*Mx x*M+ x sin2 0. An indication of the usefulness of these generalizations to solving systems of simultaneous linear equations by iterative processes is established. (Received February 14, 1966.)

634-37. K. 0. LELAND, University of Virginia, Charlottesville, Virginia. A Lipschitz charac­ terization of analytic functions.

Let B and C be Banach spaces, N > 0, and F a family of locally bounded functions on open subsets of B into C, closed under the operations of addition, multiplication by scalars, and linear translation. Then F is called an LN family if for all f E F, M > 0, ll > 0, x E dom f, such that V = [y E B; IIY- xll < o] C domf, and llf(y)ll;;; M for ally E V, one has lif(y)- f(x)ll ~ NMiiy- xiW 1 for ally E V. The elements of an LN family are (real Frechet) differentiable, indeed infinitely differentiable, and may be locally expanded in power series. Examples include complex analytic

functions and harmonic functions. (Received February 14, 1966~)

369 634-38, HEINRICH MATZINGER, University of Washington, Seattle, Washington 98105, On the axioms of topological structures,

Let E be a set and t = f lJ ,,®-, ... j the set of all filters on E. We consider a map h: t --> t, such that (1) h( lJ) « IJ; (l) lJ >> ® =h((J') > >h( ~); (3) h(i{ A~)= h(ct)Ah( ®); (4) h o h = h; (5) h(VF) = Vh(F), where the union is taken for all F E f1. Topological spaces, proximity spaces and other generalizations of topological and metric spaces can be defined by such a map h. For topo­ logical spaces condition (4) turns out to be the usual fact, that any neighborhood contains an open neighborhood, whereas for metric like structures (proximity spaces etc.) the same condition (4) re­ presents the axiom corresponding to the triangle inequality. (Received February 16, 1966,)

634-39, W. C. WHITTEN, JR,, University of Southwestern Louisiana, Lafayette, Louisiana,

On the interchangeability of l-links, II,

Consider an oriented link, L = K 1 + Kz, of two components tamely imbedded in s 3, If L is not prime and if its hub is interchangeable, the conditions (l) and (3) of Theorem 1 of Abstract 6U-27, these c)/otiai) ll (1965), 333, can be applied in many instances as effective invariants of inter­ changeability of L. In this note several other interchangeability invariants are given, one of which is noted here, (L need not be nonprime,) Let M denote the gth cyclic covering space of s3 branched ...... ,...... g ...... about L. Let L = K 1 + Kz C M lie over L, where Ki lies over Ki, (i = l,l). Theorem. H 1 (M - K 1) g ~ g is isomorphic to H 1 (Mg - Kz). Corollary. The group H 1 (M - L) is isomorphic to the direct sum of ~ g H 1 (Mg - Ki)' (i = 1 or l), and the infinite cyclic group. (In the theorem K 1 and Kl may well be of different knot types,) If Lis interchangeable, then Mh- K1 is homeomorphic to Mg- Kz. which in turn implies that there is a 1-1 correspondence between the 1-dim homology groups of the r-fold unbranched cyclic covering spaces of Mg - K1 and those of Mg - K'2 such that corresponding groups are isomorphic, Using this method a certain prime link is shown to be noninterchangeable, (Received February 16, 1966.)

634-40, R. F. CRAGGS, University of Wisconsin, Madison, Wisconsin 53706, Isotopies of l-spheres in 3-manifolds,

Bing's technique of building fences, (Conditions under which a surface in E 3 is tame, Fund, Math, 47 (1959), 105-139), is modified to obtain what follows. Results are stated for l-spheres; they generalize considerably. Let M be any 3-manifold without boundary whose distance function is p,

LetS, s 1, and Sl be l-spheres in M, and let hi (i = l,l) be a homeomorphism of S onto Si' Theorem 1, For any t > 0 there is a o > 0 such that if p(h i,I) < o (i = l,l) then there is an isotopy gt (0 ~ t ~ 1) of S into M where g 0 = h 1, g 1 = hl' and diameter g(p X I)< t where pES and g is the map given by g(p,t) = gt(p). If in addition s 1 and s 2 are disjoint tame l-spheres, then the map g may be taken to be an embedding of the annulus S X I. Two applications are: Theorem l, lsotopies between tame l-spheres in a 3-manifold can be approximated by tame isotopies, (A tame isotopy g tis one where the image under gt is tame for each t,) Theorem 3, Any l-sphere in a 3-manifold is isotopic to a tame l-sphere. (Received February 16, 1966,)

370 634-41. J. H. WELLS and C. N. KELLOGG, University of Kentucky, Lexington, Kentucky 40506. Invariant subspaces.

Let X be a compact Hausdorff space, A a Dirichlet algebra on X, and m a probability measure on X that is multiplicative on A. Denote by Am the set of all functions f in A such that Jfdm = 0. A closed subspace M of LP(dm) (l ::> p < oo) is simply invariant if the closure of Am M in LP(dm) is a proper subspace of M, and doubly invariant if (A U A)M is contained in M. We give new proofs of the structure of such subspaces. The innovation arises through an application of a double extremal pro­ cedure for analytic functions introduced by Rogosinski and Shapiro [Acta Math. 90 (1953), 287-318]. A similar procedure for function algebras was described by E. Bishop in Abstract 619-218, these cNotiaiJ 12 (1965), 123. For 1 < p < oo the characterization of simply invariant subspaces depends only on the reflexivity of LP(dm) and, for 1 = 1, on a partial generalization of a result of D. J. Newman to the effect that H 1(dll) is pseudo-uniformly convex [?roc. Amer. Math. Soc. 14 (1963), 676-679). This paper will appear in the Illinois J. Mathematics. (Received February 16, 1966.)

634-42. LEONARD SARASON, University of Washington, Seattle, Washington 98105. Elliptic regularization for symmetric positive systems. II.

The results announced in Abstract 65T-416, these oVotiaiJ 12 (1965), 84 are extended to mani­ folds with boundary; corners and edges are again permitted. Also, analogous results are derived for cylindrical problems, in which some form of 'weak equals strong' is assumed for the boundary value problem on a cross section, but no differentiability estimate. Assuming that after multiplication by a nonsingular matrix S(x) the reduced operator is independent of the cylindrical coordinates, cylin­ drical derivatives of the solution are estimated, and the method of elliptic regularization is shown to apply if the perturbing 'elliptic' operator involves differentiation only in the cylindrical directions. (Received February 16, 1966.)

634-43. HOWARD COOK, University of Georgia, Athens, Georgia. Solution of a problem of R. D. Anderson.

In answer to a problem of R. D. Anderson (Pathological continua and decompositions, Summer Institute on Set Theoretic Topology, Summary of lectures and seminars, University of Wisconsin, Revised 1958, 81-82) an example is. given of a compact metric continuum M such that, if His a sub­ continuum of M and f is a continuous mapping of H onto a nondegenerate subcontinuum of M, then f is the identity mapping of H onto itself. (Received February 16, 1966.)

634-44. JEROME DANCIS, University of Maryland, College Park, Maryland. Locally-flat and locally-tame embeddings.

Theorem 1 (Locally-flat approximations). Let g be a map of a compact, topologicalk-man!fold- k n with-boundary M into a topological n-manifold-without-boundary M , n ;;:; 2k + 2. Then given an k n k . f > 0, there exists a homeomorphism h of M into M such that d(g,f) ::> f and h(M ) IS locally flat. Corollary (Locally-flat embeddings). Every compact, topological k-manifold-with-boundary may be embedded in E 2kt2 as a locally-flat set. Theorem 2 (Locally-tame approximations). Let g be a map

371 of a k-complex K into a topological n-manifold-without-boundary Mn, n ~ 2k + 2, Then given an l > 0, there is a locally-tame embedding h of K into Mn such that d(g,h) ;;;; l, Theorem 3, Suppose that f and g are two embedding& of a compact topological k-manifold-with-boundary Mk into En, n ~ 2k t 2, such that f(Mk) and g(Mk) are locally flat, Then there is a homeomorphism h of En onto itself such that h • f : g and h is the identity of a compact set, (Received February 17, 1966.)

634-45, J, D. BUCKHOLTZ, University of Kentucky, Lexington, Kentucky. Sums of powers of complex numbers.

Suppose n is a positive integer and z l'z2, ... ,zn are complex numbers. Theorem. If rr~lzfl;;;; n, k: 1,2, ... , n, then all of the complex numbers zl'"''zn lie in the disc lzl < 2 + 2(2)112. Furthermore, the constant 2 + 2(2)112 is best possible. The method of proof is similar to a technique introduced by F. V. Atkinson (Acta Math. Acad, Sci, Hungar. 12 (1961), 185-188). The constant is shown to be best possible by taking z l'"''zn to be the zeros of the polynomial Lp:OCn+p-l,pzn-p and investigating the distribution of these zeros for large values of n. The above theorem can also be interpreted as an estimate for the zeros of a polynomial, and in this setting Is equivalent to the following. Let P(z) be a nonconstant polynomial of degree nor less with P(O) f. 0. Set zP'(z)/P(z): 00 k 1~ z : 0 Lk= 1ck z , and let B : min Jn/ck I , k : 1,2,.,.,n. Then the zero of P(z) which is nearest to lies in the annulus B/(2 + 2(2)1/ 2 )< Jzl;;;; B. (Received February 17, 1966,)

634-46. R. A. KALLMAN, University of Minnesota, Duluth, Minnesota 55812. On using the llstow Integral to solve a certain partial differential equation.

The Ilstow integral, f(s,t,ff), of a certain functional, F(t,; ,x; 9,u), (where x is a continuous function on [0, 1] vanishing at 0) Is used to express a solution of the equation f ~ ~- afts + O(t, ~)fs'

(s,t, ~) E (O,oo) <8l (O,t0 ) <8l (- oo,oo), with boundary conditions f(s,o+, $) : u(s) and aft(O+,t, ~) : O(t,f)f(O+,t,t). The Wiener and Feynman integrals of the same functional F were used by R. H. Cameron to express solutions for, respectively, the generalized heat flow and the Schroedinger equations. Our proof Involves extending certain theorems of Cameron dealing with the existence of certain llstow Integrals and estimates of their norms. We also establish differentiability properties for these Integrals, (Received February 17, 1966.)

634-47. W. J, SCHNEIDER, Syracuse University, Syracuse, New York 13210. An elementary proof and application of an extension of Aufgabe 160,

Aufgabe 160 in Pc:5lya and Szego's book states that there exists a nonconstant entire function Which is bounded outside a semi-Infinite strip, Much more can be said, Theorem 1, Let C : £zlz: teiJc(t), (j>k(t) real and continuous, 0 ;;;; t < oo} (k: 1, 2) be two curves bounding a pla:e Jordan domain J, then there exists a nonconstant entire function f(z) both bounded and univalent outside of J, The proof follows from a continued application and reapplication of Rung's theorem obtaining at each step a rational function Rn (z) with a single pole at zn E J, Each Rn approximates the previous Rn_ 1 on a simply connected domain on the Riemann sphere not containing zn or zn-l• The zn's and Rn's can be chosen so that limn~oozn: oo and limn---- 00 Rn(z) has the desired properties. Theorem 2,

372 (answer to a generalization of a question of Monte!, Le3ons sur les families normales, p. 85) Let C = [z lz = teic/>(t), c/>(t) real and continuous] and S be a closed subset of lz I= 1, then there exists an entire function f(z) with the property eiO • Cis a Julia curve for f(z) if and only if eiO E S. For C a half ray one can make a Judicious choice of constants an and linear transformations Tn(z) such that f(z) = L~ 1 ang(T n(z)) where g(z) is ,a function obtained from Aufgabe 160. Theorem 2 follows in somewhat the same manner from Theorem 1. (Received February 17, 1966.)

634-48. A. C. CONNOR, University of Georgia, Athens, Georgia. Locally cyclic continua, I.

A continuum M is said to be locally cyclic provided that for every positive number f, there exists a positive number o such that every two points of M at a distance apart less than o lie on a

simple closed curve in M of diameter less than f, The following results concerning locally cyclic continua are proved in this paper, which will be submitted to Fund. Math. Theorem 1, A necessary and sufficient condition for a continuum M to be locally cyclic is that for any two points p and q of M there exists a monotone open mapping f of M onto [0, 1] such that f- 1(0) = p and f- 1(1) = q. Theorem 2. A necessary and sufficient condition for a continuum M to be locally cyclic is that for any three points p, q and r of M there exist a monotone open mapping f of M onto a simple triad T such that the inverse under f of the set of noncut points of T is the set f p,q,r J, Corollary. Every locally cyclic continuum M contains a subcontinuum K such that M - K is the union of three mutually exclusive con­ nected domains such that K is the boundary of each of these domains. (Received February 17, 1966.)

634-49. S. G. MROWKA, Pennsylvania State University, University Park, Pennsylvania 16801. Topological significance of a certain class of cardinals. Preliminary report.

We follow the terminology of S. Mrowka, [Abstract 625-159, these cNOtiaiJ 12 (1965), 592]. The following are equivalent: (l) N-defect of Xm ;;; m; (2) Xm is the intersection of mFu -subsets of tJx ; (3) there exists a compactif!cation eX of X such that weight of eX = m and X is Q-closed m m- m m ------in eX • If m is of the form m = 2n, then each of the above conditions is equivalent to the following - m one: (4) Xm is a closed subspace of anN-compact space X~ such that X~ has a dense subset of cardinality n. (Received February 17, 1966.)

634-50. J. E. ALLEN, North Texas State University, Denton, Texas, Inverse star envelope of a subset of a L TS.

A subset S of a L TS L is called inverse starlike from a E L if for every x E S, ooxa C S (where

ooxa = {ax + (l - a.) a: a. !?; 1} ). The intersection of any collection of sets which are each inverse star­ like from a common point a is also inverse starlike from a, which leads to the definition of the inverse star envelope of any subset A of L from a point a E L: UxE..flxa denoted by ooAa. Theorem 1. (a) If A is closed, then ooAa is closed; (b) oo(A U B )a = (ooAa) U (ooBa). Theorem 2, Let K be closed, inverse starlike from a, a t/:- K, and B be the boundary of K. Then K = ooBa. Theorem 3. Let K be connected, inverse starlike from a, and a tf K. Then the boundary of K is connected. (Received February 17, 1966.)

373 634-51. L. J. LARDY, Syracuse University, Syracuse, New York 13210. Order convolution on L 1(a,b).

Let M denote the complex Banach space of all bounded regular Borel measures on an interval of real numbers with end-points a and b. By results of Hewitt and Zuckerman [_Pacific J. Math 7 (1957), 913-941] M is a commutative Banach algebra with convolution product defined by means of the Riesz representation and the requirement that I~ f(x)d(,u * v)(x) = J~I~f(max [x, y ])d.U(x)dv(y) for all continuous functions f which vanish at infinity. Assume that L 1 = L 1(a,b) is embedded in M in the usual way. Theorem. L 1 is a closed subalgebra but not an ideal in M. For f and gin L 1 we have (f * g)(x) = f(x)I~ g(y)dy + g(x) I~f(y)dy almost everywhere. Theorem, Every homomorphism h of L 1 onto the complex numbers is of the form h(f) =I~ f(y)dy for some unique x, a < x ~ b. The Gelfand topology is the interval topology. Thus the Gelfand transform is the indefinite integral. Theorem. L 1 is regular and self-adjoint. If the interval is finite then L 1 is singly generated. (Received

Febn'"ry 17, 1966.)

634-52. G. V. WELL AND, Purdue University, West Lafayette, Indiana 47906. Differentiation almost everywhere of a function of several variables.

E. M. Stein and A. Zygmund give a characterization for the differentiation of a function of a single variable in terms of a "Marcinkiewicz type" integral in On the differentiability of functions, Studia Math. 23 (1964), 247-283. R. L. Wheedon extends some of these results to the case of a func­ tion of several variables in On the n-dimensional integral of Marcinkiewicz, J, Math. Mech. 14 (1964), 61-70, C.]. Neugebauer uses the results of the first paper to give necessary and sufficient conditions for a function to be equivalent (i.eo equal almost everywhere) to a function which Is differentiable al­ most everywhere on a set in Proc. Amer. Math. Soc. (to appear). Here the results of the second paper are used to extend these results to the case of a function of several variables. To see the main result let Q0 be the unit cube in En and f be a measurable real valued function on ~· Let x and t be in En and let f{x,t) = {f(x + t) + f(x- t)- 2f(x)}(2tf 1• Let r/>(x) = 1- lxl if lxl < 1 and 0 elsewhere, where lxl = (xi+ ... + x~) 1 1 2 . Theorem. A measurable function, f, is equivalent to one which is dif­ ferentiable almost everywhere on a measurable set E if and only if for almost all x in E f(x,t)2 /ltlnr/>(fx(t)) is integrable with respect tot over some sphere with center at the origin. (Received February 14, 1966.)

634-53. STEVE ARMENTROUT, University of Iowa, Iowa City, Iowa. Cellular decompositions of 3-manifolds that yield 3-manifolds.

Suppose that M is a 3- manifold (without boundary). A subset X of M is cellular in M if and only if there exists a sequence c 1, c 2 , c 3, ... of 3-cells in M such that (1) for each n, Cn+ 1 Cint en, and (2) X n~ 1 Cn. If M is a 3-manifold, then by a cellular decomposition of M is meant an upper semi-continuous decomposition G of M such that each element of G is a cellular subset of M, Bing raised the question as to whether each cellular decomposition of E 3 that yields a 3-manifold neces­ sarily yields E 3 (Fund. Math. 50 (1962), 431-453 ). In this paper, we establish the following result. Theorem. If M is a 3-manifold, G is a cellular decomposition of M, and the associated decomposi­ tion space M/G is a 3-manifold, then M/G is homeomorphic toM. (Received February 18, 1966.)

374 634-54, H. L. BAKER, JR., University of Massachusetts, Amherst, Massachusetts. Complete amonatonic decompositions of continua,

In previous abstracts (619-119 and 626-24, these c}/oticei) 12 (1965), 91 and 697) the notions of an amonotonic collection of sets and of a complete amonotonic subcollection of a collection of sets were introduced. A complete amonotonic decomposition of a continuum M is a nondegenerate collec­ tion G of subcontinua of M such that no element of G is a subset of another one, but such that every sub-continuum of M either contains or is a subset of an element of G. In the second abstract a theo- rem is stated which is a corollary to the following result. Theorem. !!._K is a connected finite graph with no end point then there· exists a countable a monotonic decomposition of IK 1. In the first abstract mentioned above it is stated that a certain theorem implies that a large number of continua possess countable complete amonotonic decompositions. The same theorem implies that if K is a connected complex in which every 1 dimensional simplex is the face of a 2-dimensional simplex, then IK I has a countable complete amonotonic decomposition. Theorem, .!f_K is a connected (finite) complex and IKI is not a dendron (i.e. K is not a tree) then there exists a countable complete amonotonic decompo­ sition of IK 1. (Received February 18, 1966.)

634-55. J, T. DAR WIN, JR., Auburn University, Auburn, Alabama. A representation for continuous linear operators on C.

Let C denote the Banach space of real valued continuous functions from the interval [0, 1] with the least upper bound norm. A necessary and sufficient condition is given for a class of functions A, from [0, 1] X [0,1], in order that a bounded linear operator, T, can be represented by a Stieltjes integral with respect to a function in A. (Received February 18, 1966.)

634-56, R. D. ANDERSON, Louisiana State University, Baton Rouge, Louisiana 70803, On extending homeomorphisms on the Hilbert cube,

A closed subset K of the Hilbert cube I00 has property Z provided for each open nonempty homotopically trivial subset U of I 00 the set U\K is nonempty and homotopically trivial. An endslice of I00 is the set of all points of I00 for which some coordinate is 0 (or 1). Theorem. Any homeomor­ phism of a closed set K C I00 into an endslice can be extended to a homeomorphism of I00 onto itself iff K has property Z. Corollary. Every homeomorphism of I00 onto itself is stable (in the sense of Brown- Gluck). Corollary. The annulus conjecture is true for the Hilbert cube, Corollary (of this and other results). The group of all homeomorphisms of I00 onto itself is algebraically simple. (Received February 18, 1966.)

375 ABSTRACTS PRESENTED BY TITLE

66T-181. R. J. WARNE, 428 Cedar Street, Morgantown, West Virginia 26505. Homomorphisms of l-bisimp1e semigroups.

Let N denote the set of natural numbers. Theorem. LetS= (G,a) and S* = (G*,~) be 1-bisimple semigroups (Abstract 66T-81, these c}/oticei) 13 (1966), 230). Let (fi: i E 1\N) be a collection of homomorphisms of G into G*, (Xi: i E I\N) be a collection of nondecreasing functions of 1 into I, a be an element of 1°, and (zi: i E I\N) be a collection of elements of G* such that (1) f.~ac = afi -1 a 1 zi where xCz = zixzi for x E G*. (2) fi+ 1c =f .• (3) z.~ = z • (4) X = X +a for each element i zi 1 1 i+1 i+1 i EN, i E 1\N ofS, define (g,n + i, m + i)/1 = [zi-1pa(n-1> ••• z~ 1 pa.z~ 1 gf.zi""" z.,Ba(m-1), (g,n+i, m+i), n,m 1 1 1 1 Xi+ an, Xi + am). If n(m) = 0, the left (right) multiplier of gfi is identity. Then II is a homomorphism of S into s•. Conversely, every homomorphism of S into s• is obtained in this fashion. For the isomorphism theorem (Abstract 66T-81) let a= 1, Xi= i, and fi be an isomorphism of G onto G*. (Received October 27, 1965.)

66T-182. A. A. MULLIN, University of California, Box 808, Livermore, California 94551. Another estimate of the pi-function. Preliminary report.

This note considers an analogue for exponential number-theory of the classical number­ theoretic function D defined as follows: if natural number m ?; 1 and p is prime, D(pm) = 1/m; and D(n) = 0, otherwise. Definition. Let D* be defined as follows: if m ?; 1, D*(p(m)) = 1/m, where p(m) is the m-fold exponentiation of the prime p (e.g., p(3) is pPP); and D*(n) = 0, otherwise. Lemma.

D*(n) ~ D(n) for almost all n, but not all n. Further, 1r(x) ~ Lm~xD*(m) ~ Lm~xD(m) for every positive real number x. Theorem. :Em~xD*(m) ~ 1r(x). Finally, improved estimates are given for M*(x) = Lm~xJL*(m) and L*(x) = Lm~x A*(m). Based upon "empirical" evidence with an IBM 1 2 7094 as a "scratch pad", the author conjectures that M*(x) = O(x 1/ 2) and L*(x) = O(x / ). It may be that each of these claims entails the ordinary R. H. for the ordinary zeta-function. (Received November 22, 1965.)

66T-183. G. A. GRATZER, The Pennsylvania State University, McAllister Building, University Park, Pennsylvania 16802. On the spectrum of classes of algebras. Preliminary report.

Let K be a class of algebras, N the set of positive integers. The spectrum of K, Sp(K) is the set of orders of finite algebras inK (see T. Evans, Abstract 627-40, these c}/oticei) 12 (1965), 79P ).

Theorem 1. S ~ N is the spectrum of an equational class K (defined by an arbitrary set of identities)

if and only if 1 E S and S • S ~ S. Theorem 2. Every S ~ N is the spectrum of a universal class K (defined by an arbitrary set of universal sentences). Theorem 3. Let K be an elementary class (defined by a single first order sentence). Then Sp(K) is a (primitive) recursive set; but not every (primitive) recursive set can be so-represented. The following problems remain unsolved: Charac­ terize Sp(K) if (1) K is defined by k identities; (2) K is defined by a single universal sentence;

376 (3) K is defined by a single first order sentence. Some results concerning these problems: Theorem 4. Let K be an equational class defined by a finite set of Identities. Then there exists an equational class K 1 defined by four Identities such that Sp(K) = Sp(K 1). Thus we can assume In Problem 1 that k ~ 4. Theorem 5. There exists an equational class K such that Sp(K) = Sp(K 1) for no equational class K 1 defined by a finite set of identities. (Received December 6, 1965).

66T-184. D. E. MYERS, University of Arizona, Tucson, Arizona 85721. Analytic functions of polynomial growth on a polycylinder. II.

This Is a continuation of Abstract 629-6, these cNOticeiJ 12 (1965), 807 . Let Si =

[ziTi< R(z) < 1/iZ E C}, S = S1 X ••• X Sn and for Z E Cn, z = (zl'"""'zn). If {fj(z)j Is a sequence of functions analytic in S then [fj(z)} is said to be of uniform polynomial In growth In S If there exists

a polynomial P such that ifj(z)l < P(i(z)i) for all j and z E S, i(z)i = (iz 1 !, ... ,1zn1>. Theorem. If {fj ~ ----+ f(z) uniformly on compa::t subsets of Sand !fj(z)} is of uniform polynomial growth then f(z) is of polynomial growth In S. In the aforementioned abstract a representation was obtained for func­

tions of polynomial growth In terms of the Laplace Transform of ;i function g. Theorem. If ffj(z)J----+ f(z) uniformly on compact subsets of S, {fj(z)} of uniform polynomial growth InS and gj(t) the function whose transform determines fj(z), likewise g(t) for f(z) then gj(t) __,g(t) uniformly on bounded subsets of Rn. (Received January 10, 1966.)

66T-185. R. J. HANSON, University of Southern California, University Park, Los Angeles, California 90007. Reduction of those second order turning point problems of type T(h,l,O) (h <; 1).

Using the classification scheme given in [1], those 2nd order turning point problems of type

T(h,l,O) have the form (*)fhy' = [zA0 + fA(z,f))y. A0 is a constant diagonal matrix with eigenvalues ± 1. A(z,f) has a uniform asymptotic expansion as f ----+ 0, larg f I < fo with coefficient matrices

which are holomorphlc for lz I< o0• There exists a formal series P(z,f ), nonsingular at z = f = 0 and with coefficient matrices holomorphlc for iz I < o0 , such that y = P(z, f )w reduces (*) to (**)f hw' =

[zA0 + fB(f))w. The matrix B(f) depends only on f. This reduction to (**)Is analytically valid in re­ stricted sectors of the (z, f) space if A(z,f) satisfies certain restrictive hypotheses. [1] R. J. Hanson and D. L. Russell, MRC Tech. Summary Report No. 557, U. S. Army Mathematics Research Center, Madison, Wisconsin. (Received December 8, 1965.)

66T-186. BENJAMIN YOLK, 1315 Dickens Street, Far Rockaway, New York 11691. Differences, convolutions, primes. IV.

Let n be a given natural number, p = p(x) be a polynomial of degree n with integer coefficients which is Irreducible over the rationals, z be a complex.zero of p, E be the field consisting of the set of all polynomials with rational coefficients modulo p, R be the ring consisting of the set of all poly­

nomials with Integer coefficients modulo p, ~ , be the multiplicative semlnorm on E defined by

lie II = jk(z)j· where e is any element of E and k is any m'ember of the set e, f be a function from E to E defined on all of E, g be a function from R to E defined on all of R, Q be an n dimensional polyhedron with all of its vertices In E, where E Is now to be considered as a field on the set of n-tuples with

377 rational coordinates and R as the subring of E consisting of the set of n-tuples with integer coordin­ ates. Definition 1. f is analytic in E with derivative f' iff lif(e,e') - f'(e) II tends to zero whenever lie' - e II tends to zero, where e and e' are any points in E which are distinct. Theorem 4. If lif(a,b,c, ••• ,t) II .is bounded on every closed ball of E then f is analytic in E. Here f(a,b,c, ••• ,t) denotes the nth order difference quotient off taken at the n + 1 distinct points of E: a,b,c, ... ,t. Conjecture 4. f' need not be analytic in E. (Received October 6, 1965.)

66T-187. CAROL KARP, University of Maryland, College Park, Maryland 20742. The impossibility of effectively characterizing weakly representable Boolean algebras.

The negative result reported as the main theorem of Abstract 66T- 174, these cJiotiaiJ 13 (1966), 260 proved in a stronger form than stated. It is well known that for infinite cardinals K, the

K- WRB A's can be characterized by equations, but the theorem of this abstract shows that for certain K, there are difficulties in describing such sets of equations. A K-Boolean algebraic equation is built from variables by forming complements and J.l-placed joins and meets, J.l < K+. Let EK be the set of all such equations in a fixed set of K+ variables and let HK be the set of all sets hereditarily of power ~ K. An equation of EK may be thought of as an element of HK by means of a suitable coding. There is strong evidence to support the contention that if K <;;;; EK is K-decidable (in the sense that there is a uniform effective procedure using only elements of HK which will decide in K steps whether or not a given equation is in K), then there is !! E HK and there are ~ 1 -formulas A,B, in two variables such that A(y 0 ,~) defines K and B (Yo•!!) defines - K relative to EK. (For };1-formulas see Ll!vy, A hierarchy of formulas in set theory, Mem. Amer. Math. Soc. no. 57.] Such sets are called l 1(HK)n No 1r of K-Boolean 1r1 (HK)-definable. Theorem. If K : K, then there is no l:1 (HK) n 1(HK)-definable set algebraic equations that characterizes the K- WRBA's. (Received December 21, 1965.)

66T-188. ALEXANDER ABIAN and DAVID DEEVER, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210. On the bounds of the minimal length of sequences representing simply ordered sets.

Let sep S represent the smallest cardinal such that a simply ordered set (S, ~ ) has a weakly dense (see Abstract 65T-428, these c)/oticeiJ 12 (1965), 718) subset of power sep S. Moreover, let dy S represent the smallest cardinal N[J such that S has a representation by sequences of 0 and 1 of type w(3 ordered by first differences. Finally, let wo S represent the least upper bound of the powers of well ordered subsets of S. Theorem. For every infinite simply ordered set (S, ~ ) it is the case that wo S ~cry S ~ sep S. Furthermore, there exist simply ordered sets W, P, Q and R such that (with appropriate assumptions concerning the Generalized Continuum Hypothesis) (i) wo W = dy W : sep W, (!!) wo P < dy P : sep P, (iii) W Q < dY Q < sep Q and (iv) wo R : dy R < sep R. (Received January 7, 1966.)

66T-189. C. B. MURRAY, Box 7053, University Station, Austin, Texas 78712. On a substitution theorem for Stieltjes integrals.

Suppose the real-valued function f and the derivative g' of the real-valued function g are defined on the interval [a,bJ of real numbers, and I is the identity function on [a,b). In the following the inte-

378 grals .are either "norm" or "refinement" limits, J denotes either the mean Stieltjes integral, the left (or right) Cauchy Stieltjes integral, or the weighted mean Stieltjes integral of F. M. Wright, Abstract 618-12, these cJvOticei) 11 (1964), 762, which includes the others; and f Is bounded In the latter two cases. Theorem. If J~f dg exists and J~ fg'dl exists then J~f dg = J~ fg' dl. For the mean integral this improves a theorem in the author's dissertation, Univ. of Tex., Austin, 1964, where it was required In addition that for g be of bounded variation on [a,b], The result is well known (and easier) for the interior and Riemann-Stieltjes Integrals. Proofs here are straightforward from

the definitions of the Integrals and the following Lemma. If c > 0, d > 0, and [u, v) is a subinterval of

[a,b] then there is a partition u = t0 < t 1 < ••• < tn = v such ·that if Q is a set of positive integer(s) q ~ nand for each interval [tq_ 1, tq], q In Q, the difference quotient .:lg/.:lt differs from g'(tq_ 1) or g'(t ) by cor more thanL:QJh(t ) - h(t )J < d, h = I,g. (Received January 10, 1966.) q q q- 1

66T-190. W. J. HEINZER, Florida State University, Tallahassee, Florida, Some properties

of integral closure. Preliminary report.

Let D be an integrally closed integral domain with identity having quotient field K. Let L be an algebraic extension field of K, and let 5 be the integral closure of D in L. Theorem 1. If 5 is a Priifer domain, D is also Priifer. Corollary 1. If Dis, respectively, Dedekind or almost Dedekind, then Dis, respectively, Dedekind or almost Dedeklnd, Theorem 2. If 5 has the QR-property (Abstract 63T-l06, these cJvOticei) 10 (1963), 203), then D has the QR-property. Theorem 3. If D has property (#) [Abstract 627-2, these cJvOticei) 12 (1965), 786), then D has property (#). Corollary 2. If each averring of 5 has property (11'}, then each averring of D has property (#).

(Received January 10, 1966.)

66T-191. P.R. MEYERS, National Bureau of Standards, Washington, D. C. Contraction

theorems. Preliminary report.

Theorem. Let f be a continuous self-mapping of a metric space (X,d). Suppose further that f(O) = () for some 0 EX. Iff satisfies (a) fn(x) _, 0 for all x EX; (b) fn(U)---> {0} for some neigh­ borhood U of 0, then there is a metric d equivalent to d under which f is a contraction. (This weakens the hypotheses cited in Abstract 65T-267, these c/'.foticei) 12 (1965), 476.) Theorem. Iff is a continuous selfmapping of (X,d) and if fn is a contraction on (X,d) then there is a metric d under

which f and fn are contractions. Theorem, If Tt, t ~ 0 is a family of continuous maps of a metric

space satisfying Tt • Tt = Tt +t and if limt t T t = Tt i.e. limt~ t supx E xd(Ttx, Tt x) = 0, then 1 2 r 2 ~o o o o a necessary and sufficient condition for each T t to be a contraction (with respect to some metric dt

equivalent to d) is that some one Tt be a contraction. (Received January 10, 1966.)

66T-192. J. G. STAMPFLI, New York University, University Heights, Bronx, New York 10453.

The numerical range of an operator.

Theorem: Let T be an invertible operator on a Hilbert space. If the numerical range of both

T and T-1 are contained in the unit disc; then Tis unitary. (Received January 11, 1966.)

379 66T-193. HUBERT HALKIN, University of California, San Diego, California. Calculus of variations for extremal curves which are absolutely continuous but whose derivatives are not neces­ sarily bounded.

Professor Lamberto Cesar! suggested recently that I study the classical and previously un­ solved problem stated in the title of this note with the help of the machinery which I developed in the paper: On the necessary condition for optimal control of nonlinear systems (J. d' Analyse Math. 12

(1964),1-82). This study has been successfully completed. Theorem I, stated below, is the prototype of the results which I have obtained. The necessary conditions of the classical calculus of variations

(Euler-Lagrange, Weierstrass, etc.) are immediate corollaries of Theorem I. Similar results hold for the Boza problem and for the more general optimal control problem. Theorem I. We are given a function L(x,u,t) from E 0 X E 0 X [0, 1] into E 1. We assume that for any f > 0 there exists a o(f) > 0 that for all absolutely continuous function x(t) from (0,1] into En with maxtE[0, 1]jx(t)j;:;; o(f) and all integrable function u(t) from [0,1] into E 0 with fci ju(t) jdt ;:;; o(f) we have (!) L(x(t}, u(t}, t) and Lx(x(t}, u(t}, t) are defined and integrable in t over [0,1]. (ii) J61L(x(t), u(t), t}jdt and J61Lx(x(t), u(t}, t}jdt ;:;;f. (iii} J6L

66T-194. V. F. PFEFFER, George Washington University, Washington, D. C. 20006. On a semihered!tary property of sets. Preliminary report.

Let X be a compact Hausdorff topological space and let u cexp X be such that every x EX has a local base rx Cu. A property P of subsets of X is said to be semihereditary with respect to u if and only if whenever A C X, and A possesses P, it follows that either A nB or A- B inherits P for every B E u. Theorem. Let ¢f. A C X possess a property P, semihereditary with respect to u. Then there is an x E A such that every neighborhood U of x contains a set Bu which also possesses P. Let, in addition, u be closed under tl}e formation of set differences. Then the set Bu possessing P can be chosen such that x E Bu or even such that x E Bu provided A is closed. (Received January 19, 1966.)

66T-195. JOHN DAUNS and K. H. HOFMANN, Tulane University, New Orleans, Louisiana

70118. The representation of biregular rings by sheaves.

Theorem I. A necessary and sufficient condition that a ring A be b!regular (every principal ideal is generated by a central idempotent) is that it be isomorphic to the ring f' 0 (S} of sections with compact supports in a sheaf (S,1r,M). The base space M is homeomorphic to the maximal ideal space of A in the hull-kernel topology; the stalks '1T- 1(x} are A/x, where x is a maximal ideal of A.

Remark. R. Arens and I. Kaplansky (Trans. Amer. Math. Soc. 63 (1949), 461; Thm. 2.3) obtained necessary and sufficient conditions for A ""c0 (M,R), the ring of all continuous functions on M with compact supports into a discrete simple ring R(l E R). If S is the constant sheaf S = M X R, then c0 (M,R} <1:; r0 (S). Theorem II. (i) Let G be the automorphism group of a biregular ring and N the normal subgroup leaving every maximal ideal invariant as a set. N is isomorphic to the group of all global sections in the sheaf of automorphisms of (S, 1T,M ). (ii} H "' G/N is a subgroup of the full

380 group of homeomorphisms of M. Theorem III. G = H X N splits if A contains a certain kind of a subring. The above will appear in the Math, Z. (1966), (Received january 10, 1966,)

66T-196. D. G. ARONSON and jAMES SERRIN, University of Minnesota, Minneapolis, Minnesota 55455, A Harnack inequality for nonlinear parabolic equations.

Let ll be an open cube in En and let Q = ll X (0, T) for fixed T > 0. In Q we consider the equa­ tion (*)ut = divvo/(x,t,u,ux) + ~(x,t,u,ux>• where :/1 is a given vector function,~ a given scalar, ux = (Bu;Bxp .... au;Bxn>• and div.Yl= :L~= 1 (Bv~\;Bxi). We assume that p·.5¢l ~ clpl + dlul + g and IJfl ~ alpl + elul + h, where a, a > o are constants, b,c,e,f,h are functions of (x,t) in L Zq[O, T; L 2P(Il)], and d,g E L q [o, T; LP(Il)] with n/2p + 1/q < 1. Let n• be a closed proper subcube of ll and consider any two subcubes Q± = ll' X [r±, 11±_l, where Theorem, Let u be a nonnegative solution of (*) in Q. Then there exist constants K ~ 0, 'Y ~ 1 such that maxQ_(u + K) ~ 'Y minQ+(u + K). Here ')'depends on a, a and the norms of b,c,d,e and K depends on the norms of f,g,h; both constants also depend on the domains Q and Q±. This result generalizes the Harnack inequality proved by Moser for the equation ut = (a.ju ). 1 xi xj (Comm. Pure App. Math, 17 (1964), 101-134). In case everything is independent oft it reduces to the corresponding result for quasilinear elliptic equations proved by Serrin (Acta Math. 111 (1964), 247-302). (Received November 18, 1965,)

66T-197. C. E. AULL, Virginia Polytechnic Institute, Blacksburg, Virginia. Some m¢rizat!on theorems,

The following are proved. A T 3 space is compact and metrizable iff it has a countable base

~for the topology such that for each closed set f there is a base for the open sets containing f which is a subfamily of ~. A T 2 space is metrizable iff it is locally countably para compact and has a u-locally finite base, Definition. A topological space is locally countably paracompact if every point of the space has a countably paracompact neighborhood. (The neighborhood is a countably paracompact subset) In defining a countably paracompact subset, the topology for the space is used rather than the relative topology for the subset and the open refinement of a countable cover of the subset is locally finite with respect to all points of the space. (Received january 17, 1966.)

66T- 198. 8 URTON RANDOL, Yale University, New Haven, Connecticut, A number-theoretic

estimate,

Suppose k is an integer ~ 1, and denote by V the volume of the region in m-space defined by x2k + ... + x 2k ~ 1. Define numbers A,B, and C by setting A= (2k - l)(m - l)/4k2, 8 = m(m - 1)/ 1 m 2k(m + 1), C = max(A,B). Let N(x) be the number of integral lattice-points (n 1, ... ,nm)' satisfying nik + ... + n;; ;;; x, Theorem. N(x) = Vxm/Zk + O(xc). Moreover, if A > B, then this estimate is the best possible. I.e., in this case, N(x) = Vxm/Zk + ll(xc). (Received january 18, 1966,)

381 66T-199. SEYMOUR GINSBURG, Systems Development Corporation, Santa Monica, California, S. A. GREIBACH, Harvard University, Cambridge, Massachusetts and M. A. HARRISON, University of California, Berkeley, California. One-way stack automata. I.

A 1-sa A is a 9-tuple (K.~.¢.$, r,o,q0,z0 ,F) where (i) K.~.r are finite sets; (ii) ¢,$are symbols tj_ ~;(iii) (q0,z0 ) E K X I'; (iv) F ~ K; (v) o maps K X(~ u[¢,$}) X r into the finite subsets of [0,1} X K X f- 1,0,1} X I'*, * the Kleene closure; (vi) (d,q' ,e,w) E o(q,a, Z), with e = 0 if w f Z. Let 1 and be new symbols. Define the relation f- as follows. Let k, i i;: 1; a l'""" ,ak E 2: Uf ¢ , $}, ak+ 1 the unit word; z 1, .•. ,zf, Z E I'; andy E I'*. (l) If (d,q',e,Zj) E o(q,ai,Zj) where 1 ~ i ~ k and 1 ~ j ~ £ satisfy (a.)e;:;: 0 if j = 1, and ({3)e ~ 0 if j =£:write (q,a 1 .•• ~ai"""ak, z 1.•. Zj1 ••. Z£>f-(q',a 1 ... fai+d"""ak+l'

z 1 •.• zj+e1 ••• Z.£). (2) If (d,q',O,w) E o(q,ai'Z), write (q,a1••. ~ai" .• ak,yZ1)f-(q1,a 1•.• tai+d"""ak, yw1). Let f-* be the reflexive, transitive closure of f-. Let T(A) = fw/(q0 ,f¢w$, z 01>r*(q,¢w$f, y 11y2) for some (q,y l y 2) E F X I'* X I'•}. Call L a language if L = T(A) for some 1 - sa A. Languages are closed under limited substitution, union, product, •, intersection with regular set, gsm mappings, and reversal. Languages are not closed under complementation. Whether T(A) = IJ' is solvable. Whether T(A) is context free or regular is unsolvable. Each language is context sensitive. (Received January 17, 1966.)

66T-200. ROBERT McLEOD and R. S. SPIRA, The University of Tennessee, Knoxville, Tennessee 37916. Structure of direct factor sets.

Let Z be the set of positive integers. P and Q, nonempty subsets of Z, are said to form a

conjugate pair of direct factor sets (a notion due to Eckford Cohen) if (n1 ,n2) = 1 implies (n 1• n2 E P

iff n 1 and n 1 E P) and similarly for Q, and also if z E Z implies z has a unique factorization, z = a· b, a E P, b E Q. The authors determine all possible factor sets by finding all pairs of subsets

A, B C z 0 (= z U {0} ~. such that z E z0 implies there is a unique representation z = a + b, a E A, bE B. The generalization to several direct factor sets is under investigation. (Received January 25, 1966.)

66T-201. LAURENCE MAHER, Indiana University, Bloomington, Indiana 47401. Continued binary exponentiations and a function obtained from them. Preliminary report.

Let B(x) = 2x. If, for some sequence s 1, s 2,s3 , ••. , each Si is 1 or- 1 and s 12, s 1B(S22),

S 1B(S2B(S32)), ••• -> x, then xis said to be a continued binary e~ponentiation. Example. ±. 1, - 1, 1,1,1, ••• are 1sJ 's for the continued binary exponentiation 0; 1,1,- 1,±1, - 1, 1,1, 1, •.• are fsi~'s for the c.b.e. 2 1/ 2 . Theorem. If some term, not S, of [si} is- 1, {si} has a c.b.e.; each real number xis a c.b.e. with only one fsi}, or with only two {siJ 'sending as ±1,- 1,1,1,1, •••• Let Z denote the function such that Z(x) is I:: 1 (l/2)n-liT~=lsi" (If x has 2 fsi!'s, they give the same Z(x).) E.g. Z(O) = ± 1 + 1/2 + 1/4 + 1/8 + ... = 0. Z(21/ 2 ) = 1 + 1/2- 1/4 t 1/8 ±. 1/16 ± 1/32 ± ••• = 5/4. Theorem. Z exists everywhere, has range of values (- 2,2), is increasing and continuous,

Z(x) = - Z(- x), limx 4 00Z(x) = 2, 2 - Z(x) = Z(l/x) if x >0, and 2 - Z(x) = 2(2 - Z(2x)). The func­ tions Z and z -l can be used to define uncountably many nontrivial real-number operations, each with properties paralleling all the axiomatic properties of multiplication. (Received January 21, 1966.)

382 66T-ZOZ, BARUCH GERMANSKY, Singel Z70, Amsterdam C, Netherlands. On the concept of a function

The second axiom PZ: (Vx)(x E S =:- 1/Jx E S) of Peano considered alone expresses the property of tP described in the title. The "fundamental" models of this axiom may be given by sequences of the form x 1, x 2, ••• , and ••• , x_ 2 , x_ 1, x0 , x.l'''"' where xtt1= 1' T" ~ ""12" n6Fk. Then t~eneral solution of PZ 1s the vector n 1A + n2B 1 + n2Bk + ••• + n3 11 + n3 c 12 + ••• + ----+ """"1" z T'"' T n4D + n5E 1 + n 5E 2 + ... + n 6F 1 + n 6F z + ... • These vectors constitute a semilinear space, (Received January 17, 1966.)

66T-Z03. C. G. JOCKUSCH, JR., Apartment 307, Westgate, Cambridge, Massachusetts OZ139. Relationships between reducibilities. Preliminary report,

Let A be a set of integers, and suppose either that A is simple and nonhypersimple, or that A (the complement- of A) is immune, nonhyperimmune and retraced by a total recursive function, Then the many-one degrees of A, A X A, A X A X A, ... are all distinct, It is a corollary that every r,e, nonrecursive Turing degree contains infinitely many r.e. many-one degrees and every non­ recursive truth-table degree contains infinitely many many-one degrees, The main tools used in proving the above are the recursive enumerability of the logical consequences of a recursively enumerable set of axioms, the existence (shown by Yates) of a•simple but not hypersimple set in every nonrecursive r,e, Turing degree and the existence of an immune, nonhyperimmune set retraced by a total function in every nonrecursive truth-table degree. By similar methods it may

be shown that every r.e. Turing degree contains an r.e. many-one degree consisting of a s~ngle one-one degree. This last result answers a question raised by Young. (Received January 21, 1966.)

66T-204. MARY POWOERL Y, University of Connecticut, Storrs, Connecticut. Consequences and generalizations of a lemma of Tong.

Recently H. Tong proved the following Lemma. Let g be a mapping from a subset of P(X) (the set of all subsets of X) into X. Let A C X be a well-ordered set with the property that g(S A (x)) = x for every x E A (where SA(x) is the (possibly empty) segment in A consisting of the elements of A which are< x). Then the union U of all A satisfying the above conditions can be well-ordered such that g(SU(x)) = x for every x E U. Moreover, if g(U) exists, then g(U) E U. From this lemma (the proof of which is short) he showed that Zorn's Lemma, the Well-Ordering Theorem, and other results follow quickly as corollaries, We show that Hausdorff's Maximal Chain Theorem and the Transfinite

383 Recursion Theorem are also immediate corollaries of Tong's lemma. We then give several generali­ zations of the above lemma. (Received January 19, 1966.)

66T-205. R. COIFMAN, University of Chicago, Chicago, Illinois and G. L. WEISS, Washington University, St. Louis, Missouri. A factorization theorem for functions in the Nevanlinna class of a multiply connected domain.

Let !?J be a bounded multiply connected domain in the complex plane whose boundary r consists of n disjoint Jordan arcs. A function, F, analytic in f» is said to belong to the Nevanlinna class W(!?J) if log+ IF I is majorized by a harmonic function. We show that such a function has the factoriza­ tion F = B G where B has precisely the same zeroes { ai} of F, is the uniformly convergent (on com­ pact subsets of !?J) product of conformal maps of f» onto a canonical "slit disc" domain mapping the a 1• s onto 0 and G is an exponential. This is shown by making use of a kernel 9(z,n, continuous on !J1 X r, analytic in z for r E r fixed, having the property that when F = u + iv is analytic in !?J,

F(z) = fr9(z,r>uds(f) + iv(z0) (here ds = ds(r> is the element of arc length and z 0 E f» is a fixed point). In terms of this kernel, a conformal map, B (z;a), mapping f» onto the unit disc with concentric circular slits (about 0) removed, such that B (a,a) = 0 is given by the formula B (z,a) = (z - a) .expf- Jr9(z,n log lr- alds(n}. Many other properties, including the basic results of classical HP-space theory, are obtainable by these methods. (Received January 25, 1966.)

66T-206. W. P. ZIEMER, Indiana University, Bloomington, Indiana 47401. Extremal length and conformal capacity.

Let R C E 3 be a. ring with nondegenerate boundary components B0 and B 1 and let u be that unique admissible ACT function for which JR j\7u 13dm = r{R), the conformal capacity of R c.f., [F. Gehring, Trans. Amer. Math. Soc. 103 (1961), 353-393]. Let F be any class of closed separating 3 sets, i.e. sets ~ C R for which B0 and B 1 lie in different components of E - 2:. For nonnegative Borel functions f in R, f A F will mean that jif2dH 2 <; 1 for all ~ E F, where H2 is 2-dimensional

Hausdorff measure. Let M3(F) = inf[ JRf 3dm: f 1\ F}. Theorem 1. For almost all s E [0,1], 2 2 J _1 I u 12dH2 = r(R). Moreover, Jl: IV'ul dH <; r(R) for all separating sets S except those that u (s) - l/2 are in some class N for which M3 (N) = 0. Theorem 2. M3 (A) = r(R) , where A is the class of all closed separating sets. This was first proved by Gehring when A was assumed to be the class of piecewise smooth separating sets, [F. Gehring, Michigan Math. J. 9 (1962), 13 7-150]. The results of Gehring needed to establish the above theorems as well as the theorems themselves generalize to ~. (Received January 26, 1966.)

66T-207. ALEXANDER WEINSTEIN, University of Maryland, College Park, Maryland. An invariant formulation of the new maximum-minimum theory of eigenvalues.

Let A be a compact, negative definite, symmetric operator on a real Hilbert space H, with eigenvalues ·\ ~ Az ~ ••• and eigenvectors u 1, u 2, .... Let p 1, p 2,. • .,pn-l be any n- l vectors in H and let A(p 1,p2, ... ,pn-l) be the minimum of the Rayleigh quotient on the subspace orthogonal to pl'p2, ... ,pn-l' Let Am- 1 < Am= Xm+l = ... = An~ An+l ~ .... 2 ~ m ~ n. Let W r(X) =

384 det fL~ 1 (pi,uj)(p k'ui )/(Aj - A)}, i, k = 1,2,. •. , r. Then the main result of a previous paper (A. Wein­ stein, j. Math. Mech. 12 (1963), 235-246) is reformulated in the following way. Theorem. For any choice of pl'p2, ... ,pn_ 1 one has the inequality A(p l'p2, ... ,pn_ 1) ::0 An. A necessary condition for the equality A(pl'p2, •.• ,pn_ 1) = An is that the dimension r of the subspace S generated by p 1,p2, ••• ,pn_ 1 satisfies m - 1 :;; r :;; n- 1. Assuming this condition and letting p 1,p2, ... ,pr be a basis for S one has the following necessary and sufficient condition for the above equality: The quadratic form with the symmetric matrix of the determinant Wr(An- E) has in normal coordinates x 1,x2, ••• ,xr the form- x~- x~- ... - x~_ 1 + x~ + x!+i + .•. + x; forE> 0 and sufficiently small. (Received january 26, 1966.)

66T-208. AMRAM MEIR and AMBIKESHUAR SHARMA, University of Alberta, Edmonton, Canada. A variation of the Tchebycheff quadrature formula.

Given a weight function p(x) > 0 and a fixed integer k, n :;: k + 2, we wish to determine the largest integer m = m(n) and real constants A~nl, y~n) (i = 1,2, .•. ,k), x1n) (j = 1, ••• ,n - k) and B (n) such that the formula (1) f_\ f(x)p(x)dx = :Lf= 1 A~n)f(y~n)) + B (n) Lj~-1kf(xt)) is valid for all polynomials of degree :;; m(n). If p(x) = 1, it was shown [Erdos and Sharma, Canad. j. Math. 17 (1965), 652-658] that m = m(n):;; Ck(n)112 , where Ck depends only on k. We proved that if k = 1, p(x) = (1 - x 2)a., 2 2 a.>- 1/2, then m < c 1n 1/ a.+ • In particular if (l) is valid for all polynomials of degree:;; n + 1, then n < n0 (a.), a.>- 1/2. In case a.=- 1/2, a formula of the form (l) is well known as Hermite quadrature formula. (Received January 26, 1966.)

66T-209. W. D. L. APPLING, North Texas State University, Denton, Texas. Concerning a certain linear integral equation.

Suppose F is a field of subsets of a set U, R is the set of all real-valued functions on F, R 8 is the set of all bounded elements of R, R A is the set of all finitely additive elements of RB, for each q in R A' A(q) is the set of all elements of R A absolutely continuous with respect to q, and for

each Pin Rand each I in F, Tp(I) = 1 if 0 ~P(I) and Tp(I) =- 1 if P(I) < 0. All integrals considered are Hellinger type limits of the appropriate sums. Suppose Q is in RB and h is in R A" Theorem. The equation, JyQ(I)g(I) = h(V) for all V in F, has a solution, g, in R A iff JUT Q(I)h(I) exists and there is a number M such that if 0 < K, then Ju[lh(I) 1/max [IQ(I) I,K J) exists and does not exceed M, in which case A(h) contains one and only one solution, r, of the equation. Furthermore, iff is in R A and is a solution of the equation, then fvlr(I)I:;; fvlf(I)I for all V in F. (Received january 27, 1966.)

66T-210. C. H. GIFFEN, The Institute for Advanced Study, Princeton, New jersey 08540. Homeotopy groups of 3-manifolds which fiber over a circle.

Let G(X) denote the C - 0 topological group of autohomeomorphisms of X and M(X) = 1r0 (G(X)) the homeotopy group of X. Let M be a connected, closed aspherical 3- manifold which fibers over a cirde. Theorem. l'l«(M) is naturally isomorphic to the outer automorphism group of 7ri (M). The homeotopy groups of the nonaspherical 3-manifolds which fiber· over a circle are as follows: 2 1 if(S2 X S 1) = Zz X z 2 X z 2 [H. R. Gluck], M'(p X S ) = z 2 *z 2 , and M{N) = z2 X z 2 where N is the

385 nonorientable bundle over S 1 with fiber s 2. For 3- manifolds with boundary which fiber over a circle, there is an analogous result involving automorphisms of the peripheral group system of the manifold. (Received January 27, 1966.)

66T-2ll. C. W. CLARK, University of California, Berkeley, California 94720. Inequalities of Poincare' type.

Let G be an open set in En' uniformly regular of class em. For o > 0 let G0 = {x E G: dist(x, 8G) < oj. Notation. lu lf.s.P = Ljal= j fs P(x) ID"u(x) !2dx, where P(x) !!: 0 on s c En.

Theorem. Let 0 ~ j ~ m and 'Y < 2(m - j). Then there Is a constant C'Y such that for all u In the Sobolev space H~(G) and all sufficiently small o > 0 we have ju j. G p ~ C'Y ju I G' where P y(x) = J, o• 1' m, [dist(x,8G)T'Y. This Is a generalization of "Polncare''s Inequality" (cf. S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand, Princeton, N. J. (1965). Next let B [u,v] =

I:~o(Bk [u, vJ + Bk [u, v ]) be a Dirichlet form of order m on G, where Bkjp' v] = Ljal= li3l=k(D"u,a a/3ol3v)0 ,G and where BjcJu,v] is obtained by replacing 1131 by 1131 + 1 in this sum; B0 = 0. Assume that Bm(u,v] is uniformly strongly elliptic over G, that aa/3(x) with Ia! = m and 1-131 = m or m - 1 are uniformly continuous and bounded over G, whereas aa/3(x) with In!= k and 1/31 = k or k- 1 (k < m) are bounded 1 on each set G - G0 and satisfy aa/3(x) = 0 [(dlst(x, dG)f j. x E G 0, for some o > 0 and 'Y < 2(m - k). Theorem. B [u,v] satisfies "Gardlng's inequality" ReB [u,u] <; c 1 11ull~- c 2 11ull~. u E H~(G),~ c 1 > 0. Other related Inequalities and applications are also obtained. (Received January 31, 1966.)

66T-212. L. S. HUSCH, Florida State University, Tallahassee, Florida. On relative regular neighborhoods. Preliminary report.

Hudson and Zeeman define the concept of "relative regular neighborhood" In their paper On regular neighborhoods (Proc. London Math. Soc. (3) 14 (1964), 719-45) and proceed to prove an existence theorem and two uniqueness theorems. Ralph Tindell has given a counterexample to the uniqueness theorems in his paper A counterexample on relative regular neighborhoods (submitted for publication). The purpose of this paper is to prove some uniqueness theorems under various additional hypotheses one of which is the following. Let M be a combinatorial manifold, X, Y be finite polyhedra in M such that X Is link-collapsible on Y and let N be a regular neighborhood of X mod YIn M. Let J,K,L,G be a triangulation of M,X,Y,N. We say N satisfies conditions /3if lk(u, G) \lk(u, Kit) for each u E L"l. (Received January 31, 1966.)

66T-213. B. A. TROESCH, Aerospace Corporation, 2350 E. El Segundo Boulevard, El Segundo, California. Integral inequalities for two functions.

In this paper a proof is given for a conjecture puqlished in Comm. Pure Appl. Math 18 (1965), 377. Let f(x) be an arbitrary continuous, piecewise smooth function with f(O) = 0, and let h(x) be a positive concave function with piecewise smooth hx(i.e., hxx ;'i 0 where it exists) satisfying hx(O) ;:;,; 0, 2 2 for 0 & x ;;l1. Then the inequality, L(h,f) = J0l h(x)fxdx/ 2 ffl0 h(x)dx JlOf dxJ ?, 11' /4 holds. Furthermore, equality holds for h(x) ""' 1, f(x) = sin(n/2). When h(x) "' 1, this inequality reduces to the Wirtinger inequality. The proof proceeds in ratchet fashion on the string of the four inequalities

386 L(h,f) ;:; L(h,f[h]);;; L(h,f[}l}) ~ L(h,f[liJ) :S L (h = 1, sin(7rx/2)) = 11" 2/4, where f[h] denotes the fundamen­ tal eigenfunction f belonging to the variational problem oL = 0 for given h, and ii denotes a concave function consisting of one or two straight line segments. Related inequalities are also proved, e.g.,

L(h,f) :S j 2 /2 ""7.34 ""3(11"2 /4) for L defined as above and j the first positive root of J 1• In this case, admissible functions are: f(x) continuous, piecewise smooth with mean zero, and h(x) positive concave, 2 1 2 with piecewise smooth hx. Equality occurs for h = x, f = J 0 (J(x) l/ ) and for h = 1 - x, f = Jo(j(l - x) 1 ). (Received January 31, 1966.)

66T-214. ROBERT SPIRA, University of Tennessee, Knoxville, Tennessee. A sufficient condition for the Riemann hypothesis.

Lets= u +it. Previously, the author conjectured that f(s) f 0 for u ~ 1/2. In this paper,

numerical evidence is given supporting the conjecture (8/au)logj§(s)J ~- (log(t/211"))/2 + 0(1/t) for

0 ~ u < 1/2. This conjecture implies the above conjecture as well as the Riemann hypothesis. Using the formulas developed, it is shown that on u = 1/2, t' (s) i 0 except possibly at the zeros of $(s). (Received February 3, 1966.)

66T-215. G. j. NEUBAUER, University of Notre Dame, Notre Dame, Indiana 46556. The homotopy type of some automorphism groups.

The groups of continuous automorphisms of the Banach spaces 1p (1 ~ p < oo) and c 0 (of any infinite dimension) are contractible. The proof is similar to that for Hilbert space given by N. H. Kuiper (Topology, 3, (1965), 19-30). (Received February 3, 1966.)

66T-216. T. G. McLAUGHLIN, University of Illinois, Urbana, Illinois. Unions vs. joins, for disjoint r.e. sets.

Let a.,{J be disjoint sets of natural numbers. By 11 j(a.,fJ) 11 is meant {2xJx E a.J Uf2x + ljx E fJJ. Everybody knows that a. U fJ and J(a., fJ) have the same Turing degree, provided a., fJ are"separable by r.e. sets. (J(a.,fJ) is, of course, a least upper bound for the sets a.,fJ with respect to most of the stand-

ard reducibilities.) Let 11 ~ m 11 , 11 ~tt 11 denote (as usual) the relations of many-one less than and truth-table less than, respectively. Theorem. For each of the conditions which follow, there exists a pair a.,fJ of disjoint r.e. sets satisfying that condition: (1) a. U fJ ~mj(a.,{J) but J(a.,{J) ~tta. U{1; (2) a. U {3 i J(a.,{J) while J(a.,/3) ~ a.U{J; (3) a. U fJ ~ J(a.,fJ) and J(a.,fJ) .1- a.UfJ. The proofs of m m 'I'm 'f tt (1) and (3) involve an ad hoc priority construction of the classical variety (finitely many injuries per requirement). The proof of (2) depends on a few generalities about creative sets. (Received February 4, 1966.)

66T-217. ALBERT SADE, 14, Bd du j. Zoologique, Marseille 4 BDR 13. France. Quasi­

groupes demi-sym~triques. Isotopies pr~servant la demi-sym~trie.

Suitede65T-319, c_Na/icei)12 (1965), 609. Pour qu'il existe une isotopie vraie (p,q,r) projetant un quasigroupe 1/2-sym. sur un 1/2-sym., il faut et il suffit que le 1 er poss~de une autotopie vraie -1 rc':= (X,Y,Z) avec XYZ = 1; et alors p = T, q =X T, r = ZT, T arbitraire dans -oE. Pour que l'isotopie

387 principale (p,q,l) pn!serve la demi-symt!trie de Q il faut et il suffit que p et q satisfassent Vx,.1xt = p~q, oil t = pq- 1 et oil let .1 sont les translations de Q. Pour que les isotopies (X- 1, Z,l) forment un groupe il faut et il suffit que les X soient 2-a-2 permutables; tous les 1/2-sym. isotopies de Q ont alors meme groupe d'autotopie. Si Q = E( ) est 1/2-sym., toute distorsion, (l,l,r), prt!servant cette demi-symt!trie ales proprit!tt!s suivantes: (i), r est transformt! en son inverse par toute translation de Q et rt!ciproquement, (ii), r appartient au centralisateur du complexe relatif aux translations de Q, (C), dans @SE' (iii), rest dans chaque sous-complexe de (C), (iv), rest rt!gulii!lre. Le sons-ensemble des distorsions 2-a-2 permutables qui respectent la 1/2-sym. d'un quasi­ groupe fini Q est un groupe. (Received February 4, 1966.)

66T-218. C. M. HOWARD, University of California, Los Angeles 24, California. A trivial result in set theory.

Let A ""' B mean A and B are equinumerous sets (there is a biunique function with domain A and range B). Theorem. If A ""B and C ""'D and also An C ""B n D, then A UC ""'BUD, where n and U are set theoretic intersection and union respectively. The theorem is interesting because it has a trivial proof, does not require complicated lemmas, and does not require the axiom of choice. In particular it does not seem to imply anything about infinite cardinal addition and leaves the question of whether or not the fact that a + a = a, for infinite cardinals a, requires the axiom of choice, open. (Received February 4, 1966.)

66T-219. MARTIN SCHECHTER, Institute for Advanced Study, Princeton, New jersey. On regularly accretive extensions.

Let A be a densely defined linear operator in a Hilbert space H. According to Kato [J. Math. Soc. japan 13 (1961), 246-274) it is called regularly accretive if there is a bilinear form a(u,v) such that D(a) 2 D(A), Re a is closed, there is a constant ')' > 0 such that Re a(u,u) ~ "Y lim a(u,u) 1. u E D(a), and f_or u E D(a), f E H, one has a(u, v) = (f, v) for all v E D(a) iff u E D(A) and Au = f.

Theorem. A densely defined operator A0 in H has a regularly accretive extension Iff (1) Re(A0u,u) ;;:; "Y 1Im(A0u,u) 1. u E D(A0), for some y > 0. There is an extension satisfying (1) with the same constant"(. Corollary. 1!_A0 is maximal accretive and satisfies (1), it is regularly accretive. This was proved by Kato [ibid.] for 'Y > 1. (Received February 4, 1966.)

66T-220. EDWARD BECKENSTEIN, Polytechnic Institute of Brooklyn, Brooklyn New York 1123 5. On regular nonarchimedian Banach algebras.

In this paper the algebraic and topological properties of commutative Banach algebras X with identity over fields with nonarchimedian rank one valuation are studied. The Gelfand topology (when definable) on the space of maximal ideals ..L Is found to be always zero dimensional and totally dis­ connected but not always compact as in the classical case. The Gelfand topology is found to be com­ pact if the field is complete with respect to its valuation induced metric topology, the valuation is discrete, and the residue class field is finite. Regular algebras are defined in the usual manner and it is found that if X is a Noetherian ring that It Is regular if and only if ..L is a finite set. This result Is shown to be true In the classical setting. The following results are shown to be true In the classical

388 and nonarchimedian settings. Theorem 1. If X is Noetherian then every prime ideal is maximal if and only if X is jacobson and the only irreducible closed sets in the hull kernel topology are one point sets. A set is irreducible if it cannot be written as a union of two closed sets where neither of the sets is the whole set. Theorem 2, Let X be Noetherian semisimple but not a field. Then every non­ zero prime ideal is maximal and .Lis infinite if and only if X is Jacobson and the hull kernel top­ ology is cofinite, Theorem 3, A regular Noetherian semisimple Banach algebra is an algebraic algebra. (Received February 7, 1966,)

66T-221. D. W. SOLOMON, Wayne State University, Detroit, Michigan 48202, Denjoy integration in abstract spaces, II.

Let X be a second countable, locally compact metric space which has a base Jf- satisfying

Romanovski's ten axioms [Math. Sbornik 9 (51) (1941}], and Y be a Banach space with a countable de­ termining set. This paper presents a descriptive and constructive definition of an integral, G, of point functions f, defined on a member of Jl1; and with range contained in Y, and studies some of the properties of this integral. The integral generalizes one previously presented by the author [Abstract 630-45, these c}/oticei) 13 (196.6}, 69], and which will be referred to as DB. The descriptive definition given differs from that given for DB. Several conditions are presented which involve local pseudo derivability of a set function F defined on Jfl-, and local DB-integrability of the pseudo derivative, and which are necessary and sufficient in order that F be a G-integral. The constructive process given for G can be compared with that presented for DB and, in fact, they can be shown to be equi­ valent in case Y is finite dimensional. As in the case of the DB-integral, any G-integral can be attained in at most countably many applications of the construction process. (Received February 7, 1966.)

66T-222, S. A. COOK, Harvard University, Cambridge, Massachusetts. Rapid calculation by Turing machine. 6(log n) 112 1 Let T(n) = n2 2 (Notice that T(n) = o(n tf) for any f >0). The machines considered here are Turing machines with two tapes; one tape called the input tape. Some of the finitely many internal states of a machine are labelled with output symbols. A sequence of these output symbols is designated as the machine assumes various labelled states during the course of its operation. A base-b multiplier is a machine whose designated output is the base-b notation (starting with the low order digit) of the product of any pair of integers superposed on its input tape in base-b notation

(with the head initially scanning the low order digits). Theorem. For every b ~ 2 there 1s a base-b multiplier which designates the first n product digits within T(n) operations for all n and all pairs of input integers. The proof uses ideas of A. L. Toom (Soviet Math. Dokl. 4 (1963}, 714-716) and F. C. Hennie and R. E. Stearns (G. E. Report No. 65-RL-4020E, Aug. 1965), As examples of applications, Newton's method can be used to prove a similar theorem on division, and to prove a strengthened version of Theorem 11 of Hartmannis and Stearns (Trans. Amer. Math. Soc. 17 (1965}, 285-306) (concerning the rate of calculation of algebraic numbers), in which our T(n) replaces n • (Received February 7, 1966.)

389 66T-223. WITHDRAWN.

66T-224. W. M. CAUSEY, 933 Kentucky, Lawrence, Kansas. On the univalence of an integral.

Let S be the class of functions f regular and univalent in lz I < 1 and normalized by f(O) = 0, f'(O) = 0. Let g(z) = J~(f(t)/t)Edt. Then for any E, 0 ;> E ~ (5 1/ 2 - 2)/4, and any f E S, the function g belongs to S. (Received February 9, 1966.)

66T-225. N. w. SAVAGE, Arizona State University, Tempe, Arizona. Ahlfors' conjecture concerning extreme Sarto operators.

Ahlfors conjectured in 1953 that the extreme (in the sense of convexity) Sarto operators are all of the form Tf = f(g) where g is a measure preserving transformation. It is shown that the con­ jecture is true if and only if every extreme Sario operator is multiplicative. An example due to Ryff is then given of an extreme operator which is not multiplicative, thus settling the conjecture in the negative. (Received February 9, 1966.)

66T-226. ALEXANDER ASIAN and SAMUEL LaMACCHIA, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210. Examples of generalized Sheffer functions.

For a natural number n ~ 3, let gn(x) denote a permutation on the set n = [0,1,. • ., n- 1} which together with the cyclic permutation (O,l, .. .,n - 1) generate the symmetric group Sn. Let such a permutation gn(x) be called an Sn -cogenerating permutation. Based on the definition of an n-Sheffer function, as introduced in Abstract 66T-88, these c){otiu.i) 13 (1966), 232 it is established: Theorem 1. For every n '<: 3 and every S -cogenerating permutation g (x) each of the n - 1 functions n n p~(x,y) from n X n onto n is an n-Sheffer function where 0 ~ m ;> n - 2 and (i) p~(x,x + k) = x + k + 1 with 0 ;> k ;> m, (ii) p:(x,x + m + 1) = gn(x) and (iii) p:(x,y) = 0 otherwise. Theorem 2, Same state­ ment as in Theorem 1 with (i), (il) being replaced respectively by (i') p:(x + k, x) = x + k + 1 with 0 ;> k ;> m, (ii') p:(x + m + 1, x) = gn(x). Theorem 3. For every n ~ 3 and every Sn-cogenerating permutation there are at least 2 (nn(n-l) - 1)/(nn - 1) n-Sheffer functions. (Received February 9, 1966.)

390 66T-227. STAL AANDERAA, The Aiken Computation Laboratory, Harvard University, Cambridge, Massachusetts. Relation between different systems of modal logic.

Let S2, S3, and S4 be Lewis' systems of modal logic. Let M be von Wright's system M, (see Georg H. von Wright, An essay in modal logic, Amsterdam, 1951) and LM be Lukasiewicz's modal logic (A system of modal logic, Journal of Computing Systems, Vol. 1, July 1953, pp. 111-149). Let

TM be the trivial modal system obtained by adding p ::J o p as axiom to S4. (We shall use 1:1, ::J, & for necessity, material implication and conjunction, respectively). Let p be the first propositional variable and a be the first propositional variable not occuring in a formula A. For each part B of A, we define o+B and o-B as an abbreviation of O(p ::Jp) ::J DB and oB&a, respectively. Let A+ and A- be

the result of replacing every occurrence of o by o+ and o- respectively. Theorem. I-S 2A iff

\-Ma ::JA-; \-M A iff l-s 2 A +; l-s 3 A iff I-S 4 a ::JA-; \-S 4 A iff \-S 3 A+; \-LM A iff 1-L T A-; 1-L T A iff 1-LM A+. (Received February 10, 1966.)

66T-228. D. L. PLANK, University of Rochester, Rochester, New York. Points in {JX associated with certain subalgebras of C(X).

Let C (X) denote the real or complex continuous functions on a completely regular space X. If A is a subalgebra of C(X) and if p E {JX, define MP = {_f E A: (fg)*(p) = 0 for all g E A}, an ideal in A. We shall consider only subalgebras A for which p-> MP is a homeomorphism of {JX onto the maximal ideal space of A. A set of sufficient conditions for this to hold is (i) 1 E A, (11) [z /3X (f*): f E A] is a base for the closed sets in {JX, and (iii) f E A, If I ~ 1 =:> f-l E A. Examples of such sub­ algebras are those mentioned below. Let X • = {JX\X; with A as above, we define p E X • to be an A -point if p ¢ aX • fq E X • : f E M~ for all f .E A. Theorem 1. Assume the continuum hypothesis; if X = R or N, then X • has a dense set of 2c A-points for all A. Theorem 2. lf X = R or N, then p is

a C*(X)-point ..;:>pis aP-point of x•. Theorem 3. pis a C(R)-point ~pis a remote point of R (see Fine and Gillman, Proc. Amer. Math. Soc. 13 (1962), 29-36). Let B = {f E C(N): lim supn~oof(n) ~ 1}, where f(n) = lf(n)ll/n for n EN (see Brooks, Studia Math. 24 (1964), 191-210). Theorem 4. pis a B-point ~Mp =[fEB :f{J(p) < 1]; Every B-point is aP-point of N*. Theorems 1,2 and 3 generalize to wider classes of spaces. (Received February 10, 1966.)

66T- 229. H. M. FRIEDMAN, 50 Massachusetts Avenue, Cambridge, Massachusetts 02139. Strength of certain set theories.

ZF* is a form of set theory described as follows: ZF* has the same axiom of extensionality and infinity as ZF. The axiom of foundation in ZF* asserts that every nonempty set has an t-least member and that the t-relation is antisymmetric. All other axioms of ZF* are closed comprehen­ sion axioms, asserting the existence of extensions to certain properties with only one free variable. All the other axioms of ZF can be written as the universal closures of comprehension axioms assert­ ing the existence of extensions to certain properties with one or more free variables. The rest of the axioms of ZF* are inductively given by eliminating the initial universal quantifiers in the rest of the axioms of ZF and substituting for them the definitions of the sets given by other axioms of ZF*. Thus power set, in ZF •, takes the form of a schema asserting the existence of a power set of all sets given by the axioms of ZF*. Sum set takes on a similar form, and replacement can be seen to take

391 on the form of a schema of schemas, asserting that usual replacement holds if the domain is given by an axiom of ZF* and if the function contains no free parameters, In addition, an axiom asserting the existence of w is added, It is shown as a theorem of number theory that ZF • is consistent if and only if ZF is consistent. The proof uses ideas of Skolem and Geidel. (Received February 10, 1966.)

66T-230, R. W. STRINGALL, University of California, Davis, California 95616, Automorphism groups of Abelian p-groups and generalized Boolean algebras,

Let E(G) denote the endomorphism ring of the Abelian p-group G, and define H(G) =

[a E E (G): x E G, px = 0 and height x < oo imply height a(x) > height x J. Then H(G) is an ideal in E(G) which contains the jacobson radical of E(G). Theorem A. The ring E(G) is additively generated by its units (automorphisms) if and only if the following conditions are valid: (i) E(G)/H(G) is gener­ ated by its units, (ii) every isomorphism of G into G is a sum of automorphisms of G. If G has a standard basic subgroup, the rings E (G)/H(G) are simply sub rings of nN0 Zp with identity. Theorem B. If I is any indexing set, then there is a one-to-one isomorphism preserving correspon­ dence between the subrings of n1 Zp and the Boolean rings contained in P(I). Moreover, if pi 2, then all such rings with identity are generated by their units, (Received February 11, 1966.)

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Designed 1348. 100 GREAT PROBLEMS OF ELEMENTARY MATHEMATICS: even for those with no calculus. 22 worked examples, Their History & Solution, Heinrich Dorrie. Archimedes' "Prob­ 60 exercises, answers. 7 figs. vii + 56pp. Paperbd. $1.00 lema Bovinum," Omar Khayyam's binomial expansion, the tangency problem of Apiollonius, Steiner's ellipse, circle, 1460. PARTIAL DERIVATIVES, P. J. Hilton. Differentiation, sphere problems, many more in a book of wide appeal. First implicit func~;ons, maxima and minima, etc. 18 worked trans. of "Triumph der Mathematik." 112 figs. x + 393pp. examples. SO exercises, answers. 3 figs. viii + S4pp. Paperbd. $2.00 Paperbd. $1.00 1241. ARITHMETIC REFRESHER FOR TECHNICAL MEN, A. A. Klaf. 1461. ELECTRICAL & MECHANICAL OSCILLATIONS, D•. S. Streamline methods, resharpen calculation skills, introduce Jones. IS worked examples. SO exercises, answers. 53 arithmetic via everyday applications. A Dover original. 809 figs. viii + t04pp. Paperbd. $1.00 problems. 937 questions, answers. Appendices: 4-place logs, il•bles, etc. vii + 438pp. Paperbd. $2.00 1462. COMPLEX NUMBERS, W. Ledermann. Clears away much of the mystery for students. 43 worked examples. All books 5% x 81/2 unless noted. 41 exercises, answers. 14 figs. 68pp. Paperbd. $1.00 1465. ELEMENTARY DIFFERENTIAL EQUATIONS & OPERA­ r------, TORS, G. E. H. Reuter. 28 examples, fully worked. 37 exer­ Dept. 5:3,7, DOVER PUBLICATIONS, INC., 180 Varick St., cises, answers. viii + 67pp. Paperbd. $1.00 N.Y., N.Y. 10014. Please send me the following books: 1466. FOURIER SERIES, I. N. Snedden. Both theory & application. 14 worked examples. 29 exercises, answers. 10 figs. viii + 69pp, Paperbd. $1.00

1011. OPERATIONAL METHODS IN APPLIED MATHEMATICS, H. S. I am enclosing $ ...... In full payment. (Payment in Carslaw & J. C. Jaeger. Easy, quickly learned application of full must accompany all orders, except those from libraries Laplace Transform to differential equations. 153 problems, or other public institutions, who may be billed.) Please many solved. 22 figs. xvi + 359pp. Paperbd. $2.25 add 10¢ per book to orders less than $5.00, for postage and handling. Please print 1238. INTRODUCTION TO HIGHER ALGEBRA, M. Bticher. Back in print in its first low-cost ed. 212 exercises. xi + 32\pp. Paperbd. $2.00 Name ...... 1239-40. FOUNDATIONS OF THE THEORY OF ALGEBRAIC NUMBERS, Address ...... Harris Hancock. Both analyzes theory's origins, clarifies its elements, applications. Total of !viii + 1256pp. City ...... State ...... Zip# ... . Two vols., Paperbd. $5.50 GUARANTEE: All Dover books unconditionally guaranteed. 1364. FUN WITH FIGURES, J, A. H. Hunter. First book collec­ Return an~ book within 10 days for full cash refund if you tion! 150 original, maddening puzzles by a brain-teaser par are dissatisfied. No questions asked. excellence. 15 typical solutions. xii + 109pp. Paperbd. $1.00 ~------

395 from ADDISON-WESLEY. . . NUMBER SYSTEMS OF ELEMENTARY MATHEMATICS BY G. CuTHBERT WEBBER, University of Delaware The system of complex numbers and its subsystems form the subject of discussion of this book. The primary purpose of the text is to develop a sense of proof and mathematical development early in the student's program, and to introduce him to basic ideas of modern mathematics. In terms of well­ defined content areas, the book develops the natural numbers, integers, rational numbers, and com­ plex numbers. In Press INTRODUCTION TO NUMERICAL ANALYSIS BY CARL-ERIK FROBERG, University of Lund, Sweden Requiring a background of elementary calculus and differential equations, this text is intended for introductory courses in numerical analysis, In the selection of topics, stress is placed on modern and efficient methods, and to a large extent, the book's content reflects the important role played by electronic computers in recent years. A brief account of the theory of matrices is presented, while applications of matrix methods have been treated in considerable detail. 340 pp, 22 illus. $8.95 TOPOLOGICAL VECTOR SPACES AND DISTRIBUTIONS, Volume I BY JoHN HoRVATH, University of Maryland The present text is an elementary introduction to topological vector spaces and their most important application: the theory of distributions of Laurent Schwartz. All the necessary definitions and results from algebra and topology, giving complete proof for all results which are not immediate consequences of the relevant definitions, are included. In Press INTRODUCTORY PROBABILITY AND STATISTICAL APPLICATIONS BY PAUL L. MEYER, Washington State University Assuming a calculus prerequisite, this one-semester text presents the basic ideas of probability and statistics emphasizing random variables. Employing an axiomatic approach, the emphasis is on practical applications, thus making the book of prime interest to engineers, physicists, those interested in the quantitative social sciences, and, in fact,. anyone needing the subject as a tool. 339 pp, 130 illus. $8.75 THE APPROXIMATION OF FUNCTIONS Volume I: Linear Theory BY JoHN R. RICE, General Moton; Research Laboratories The central problem and theme of this book is the approximation of a real continuous function by an approximating function depending on a fixed finite number of parameters. It is aimed at developing an intuitive feeling for the results and techniques of approximation theory and concentrates attention on the theoretical probleiDS motivated by the advent of computers. 203 pp, 61 illus. $9.75 FUNDAMENTALS OF ABSTRACT ANALYSIS BY ANDREW M. GLEASON, Haroard University This new text is designed for use in the first course in real variable theory at the advanced undergraduate or graduate level. The detailed explanation of how abstract mathematics works, and the explicit carrying out of the construction of the real numbers, will be of special interest to the reader. Other outstanding features of the text include the explanation of the axiom of choice and abstract set theory. In Press LINEAR GEOMETRY BY RAFAEL ARTzy, Rutgers, The State University Written at the advanced undergraduate level, this book is intended for courses in higher geometry. The emphasis is generally theoretical and will be of prime interest to students majoring in mathe­ matics. However, prospective teachers and students of the physical sciences will also find the book valuable. 273 pp, 147 illus. $9.75 WRITE FOR APPROVAL COPIES

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396 New books for today's up-Io-date mathematician ...

Numerical Solution of New volumes of Mathematics in Partial Differential Science and Engineering: Equations A Series of Monographs and Textbooks edited by James H. Bramble Includes works of many active in­ VOLUME 19 ternational investigators currently doing research in both the theoret­ ical and practical aspects of the field of numerical solution of par­ Lectures on Functional Equations tial differential equations. The topics treated cover a wide range and include many important sub­ and Their Applications fields. by Janos Aczel March 1966, 315 pp., $16.50

The first systematic treatise on the theory of functional equations, this book gives an account of the present state Topology of the theory in addition to descriptions of general meth­ by G. Choquet A VOLUME OF PURE AND APPLIED ods for solving functional equations. MATHEMATICS: A SERIES OF MONO­ GRAPHS AND TEXTBOOKS March 1966, 510 pp., $19.50 Provides an elegant and very modern treatment of topological VOLUME 23 spaces, mappings between them, and topological vector spaces. Both real and complex-valued functions are given. Differential Equations: February 1966, 337 pp., $12.50 Stability, Oscillations, Time Lags by A. Halanay An Introduction to Nonassociative Algebras Emphasis throughout this work is placed on the develop­ by Richard D. Schafer ment of a broad theoretical foundation, and on the clari­ A VOLUME OF PURE AND APPLIED fication of important concepts and techniques. MATHEMATICS: A SERIES OF MONO­ GRAPHS AND TEXTBOOKS 1965, 528 pp., $19.50 June 1966, about 125 pp., approx. $6.50

VOLUME 24 Handbook of Series Time-Lag Control Systems for Scientists and by M. Nam1k Oguztiireli Engineers by Visvaldis Mangulis A detailed study of ordinary delay-differential equations Summarizes some properties of and control systems involving time delay. Consists of two series; tabulates various expan­ sions of the most common func­ parts and an extensive list of references, and is designed tions (in power series, trigonomet­ as a reference work for pure and applied mathematicians ric series, series of Bessel func­ tions, etc.) and arranges sums of and control engineers, and as a textbook for graduate series according to the appearance students. of the general term in the series. Series of Bessel functions and February 1966, 324 pp., $13.50 Legendre functions are included. 1965, 135 pp., $6.95

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397 NEW ... TEACHABLE ... AUTHORIT AliVE • A FIRST COURSE IN ABSTRACT advanced subjects are discussed: lattice theory ALGEBRA aspects of point set topology; detailed accounts of both net theory and filter theory and the HIRAM PALEY and PAUL M. WEICHSEL, relationship between these theories; and prox­ both of the University of Illinois imity spaces. undergraduate This book is designed for the March 1966 256 pp. $9.00 tent. course in abstract algebra. It includes naive set theory, elementary number theory with modular • A SHORT COURSE IN arithmetic, the elementary theory of groups and AUTOMORPHIC FUNCTIONS rings, and a carefully selected list of special JOSEPH LEHNER, University of Maryland topics in groups and rings for special assignment Elaborates the theory of discontinuous groups and advanced reading. by the classical method of Poincare and develops March 1966 352 pp. $8.00 automorphic functions and forms via the Poin­ • A FIRST COURSE IN care series. The connections between the auto­ INTEGRATION morphic function theory and Riemann surface EDGAR ASPLUND, University of Stockholm, theory are also covered. (Athena Series) Sweden, LUTZ BUNGART, University of February 1966 160 pp. $5.00 California, Berkeley • APPROXIMATION OF FUNCTIONS undergraduate or A basic text for a one-semester G. G. LORENTZ, Syracuse University graduate course in the theory of Lebesgue inte­ Of great importance to the practical mathe­ gration. Since most students at this stage have matician, this work provides an introduction to more background in functions than in set theory, the approximation of real functions at full depth this book develops the technique of integrating up to very recent (1963-64) results. (Athena of approaching the Lebesgue functions as a means Series) theory earlier than usual in the same course. May 1966 192 pp. $4.00 tent. March 1966 512 pp. $10.50 tent. • TOPOLOGICAL STRUCTURES WOLFGANG J. THRON, University of Coloraoo A unique feature of this book is the description of the historical development of the various topics. Besides the usual theory, a number of

NEW MATHEMATICS BOOKS FROM P-H INTRODUCTION TO FINITE FRESHMAN MATHEMATICS FOR MATHEMATICS, Second Edition, 1966 UNIVERSITY STUDENTS John ·G. Kemeny and J. Laurie Snell, Dart­ Sheldon T. Rio, Southern Oregon College; mouth College; and Gerald L. Thompson, Frederick Lister, Western Washington State Carnegie Institute of Technology. The purpose College; and Walter J. Sanders, University of of this book is to provide a unified treatment lllinois. A classroom tested text designed to of topics such as: logic, set theory, game the­ prepare the pre-calculus student to read and ory, probability, linear algebra, etc. A revision understand the more rigorous calculus books. of one of the most unique undergraduate text­ January 1966, 448 pp., $8.50 books ever published. April 1966, approx. 500 pp., $8.95 CALCULUS AND ANALYTIC GEOMETRY, Second Edition, 1965 PRELUDE TO ANALYSIS Robert C. Fisher, The Ohio State University; and Allen D. Ziebur, Harpur College. An Paul C. Rosenbloom, Columbia University; and accurate understandable introduction to calcu­ Seymour Schuster, University of Minnesota. A lus and to analytic geometry rewritten to include new and novel pre-calculus mathematics text a discussion of line integral, up-dated problems, presenting a thorough grounding in analytic, as and differential equations. 1965, 768 pp., $10.95 well as algebraic, aspects of the real number system relative to the requirements of calculus. ELEMENTS OF PROBABILITY January 1966, 473 pp., $8.25 AND STATISTICS Elmer B. Mode, Emeritus, Boston University. (Prices shown for student use) Designed to develop the basic concepts and For approval copies, write: Box 903 ruies of mathematical probability. Examples PRENTICE-HALL from many fields. May 1966, approx. 336 pp., Englewood Cliffs, N.J. 07632 $8.00

398 FOUR MAJOR MATH TEXTS

by John F. Randolph, University of Rochester. This best­ Calculus selling text, revised, incorporates treatments of set theory and notation, definition of a function, and vector analy­ and Analytic sis in addition to traditional topics .. New expositions and Geometry proofs have been added; new airbrush illustrations are included. Proofs to difficult problems appear in the ap­ 2nd edition pendix: their omission will not interrupt the continuity of a course. 1965. 640 pages. 6 x 9. Clothbound.

by Walter Leighton, Western Reserve University. An intro­ Ordinary duction to ideas fundamental for understanding of dif­ ferential equations, now revised to include many new Differential and challenging problems. Important elementary solution Equations methods are retained, but emphasis is mainly on theories of solutions of differential equations and systems. Includes 2nd edition material on the Laplace Transform., Liapunov theory, and oscillation theory. Just published. 240 pages. 6 x 9. Clothbound.

by Robert C. James, Harvey Mudd College. A clear and Advanced rigorous text featuring exercises of graded difficulty, a flexibility that makes it useful in courses ranging from the Calculus traditional to the rigorous and introductions to orthogonal functions and measure theory. Includes background on the historical development of theories and techniques. Just published. 560 pages. 6 x 9. Clothbound.

A Primer by Hugh J. Hamilton, Pomona College. Designed to give students a substantial course in complex variables without of Complex advanced calculus as a prerequisite. Contains outstanding Variables problems, worked examples, considerable emphasis on with an Introduction ordered pair notation, and exceptional organization. Just to Advanced Techniques published. 224 pages. 6 x 9. Clothbound. WADSWORTH Publishing Co., Inc. Belmont, California BoxN

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1. LECTURES ON BOOLEAN ALGEBRAS Paul R. Halmos, University of Michigan 147 pp. $2.95 2. LECTURES ON ELLIPTIC BOUNDARY VALUE PROBLEMS Shmuel Agmon, Hebrew University of Jerusalem 289 pp. $3.95 3. NOTES ON DIFFERENTIAL GEOMETRY Noel J. Hicks, University of Michigan 183 pp. $2.95 4. TOPOLOGY AND ORDER Leopolda Nachbin, University of Rochester 128 pp. $2.50 5. NOTES ON SPECTRAL THEORY Sterling K. Berberian, University of Iowa 121 pp. $2.50 6. NOTES ON LOGIC Roger C. Lyndon, University of Michigan 103 pp. $2.50 7. LECTURES ON CHOQUET'S THEOREM Robert R. Phelps, University of Washington 136 pp. $2.50 8. EXERCISES IN SET THEORY L. E. Sigler, Hofstra University 144 pp. $2.75 9. LECTURES ON THE THEORY OF FUNCTIONS OF A COMPLEX VARIABLE George W. Mackey, Harvard University 275 pp. $3.95 10. LECTURES ON QUASICONFORMAL MAPPINGS Lars Ahlfors, Harvard University 156 pp. $2.75 11. SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY J. Peter May, Yale University about 156 pp. prob. $2.75 12. TOPICS IN THE THEORY OF FUNCTIONS OF ONE COMPLEX VARIABLE Wolfgang Fuchs, Cornell University about 144 pp. prob. $2.75

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400 From the HOLDEN-DAY Library of Pure and Applied Science Introductory Calculus: With Algebra and Trigonometry c. 352 pp, (Spring 1966). $8.50* (Answer Book, $1.00*) Modern University Calculus c. 700 pp, (Summer 1966). $10.75* (Answer Book, $1.50*) By Stoughton Bell, J. R. Blum, James V. Lewis, and Judah I. Rosenblatt, UNIVERSITY OF NEW MEXICO Introductory Calculus and the more rigorous and complete Modern University Calculus are intended as complementary texts, yet can be used independently of each other. Both books are notable for their careful motivation, their flexibility, direct approach, full statement of all theorems, carefully thought-out notation, wide-spread use of pictures, and plentiful supply of counter-examples. Both include teaching aids with clearly marked optional sections, a descriptive outline for course planning and numerous examples, drill exercises, and chal­ lenging problems. Supplementary teachers' manuals provide answers to all problems (many worked out in detail, and with graphs) and several day-by-day suggested course outlines. Introductory Calculus provides a rapid introduction to the basic ideas and techniques of calculus, with emphasis on the intuitive motivation behind the main results and concentra­ tion on the development of technical facility. It includes only those derivations that contribute rapidly to the student's development. Careful notation of the omissions, however, makes apparent the relationship between rigor and intuition. The book is designed to allow science and engineering students immediate application in their concurrent courses, or as a survey for social science, humanities, and business students. Modern University Calculus presents a careful, complete, self-contained exposition of calculus of one and several variables, including the usual coordinate ~eometry. It provides, as well, a unified treatment of the topics of matrix theory, differential equations, infinite series, and applications. It seeks to build theory around meaningful problems, to show the student where care is needed to avoid errors, and to provide him with powerful mathematical tools and results applicable for later work. It fulfills the need for a stimulating modern text that, assuming little background, will build a proper foundation for the advanced mathematics increasingly required in applications. It is suitable for courses ranging from 12 to 15 credit hours. OTHER TITLES OF INTEREST Elementary Partial Differential Equations. By Paul W. Berg and James L. McGregor, STANFORD UNIVERSITY. C. 383 pp, (Spring 1966) $10.50* Introduction to the Theory of Differential Equations with Deviating Arguments. By L. E. El'sgol'ts. Trans. by Robert J. Mclaughlin, HARVARD UNIVERSITY. 96 pp, (Spring 1966) $7.75* The Structure Of Lie Groups. By G. Hochschild, UNIVERSITY OF CALIFORNIA, BERKELEY. 240 pp, (1965). $11.75 Elements of General Topology. 224 pp, (1964). $8.95 Elements of Modern Algebra. 218 pp, (1965). $8.95 Homology Theory: A First Course in Algebraic Topology. 240 pp, (Summer 1966). $10.50* By Sze-Tsen Hu, UNIVERSITY OF CALIFORNIA, LOS ANGELES Measure and the Integral. By Henri Lebesgue. Trans. by Scripta Technica. Edited with a biographical essay by Kenneth 0. May, CARLETON COLLEGE. C. 200 pp, (Spring 1966). $6.95* Lectures on Calculus. Edited by Kenneth 0. May, CARLETON coLLEGE. c. 120 pp, (Summer 1966). $8.50* Set Theory for the Mathematician. By Jean E. Rubin, MICHIGAN STATE UNIVERSITY. c. 350 pp, (Summer 1966). $9.50* An Introduction to Sequences, Theories, and Improper Integrals. By 0. E. Stanaitis. ST. OLAF COLLEGE. C. 200 pp, (Spring 1966). $7.75* Challenging Mathematical Problems with Elementary Solutions, Volumes I and 11. By A. M. Yaglom and I. M. Yaglom. Trans. by James McCawley, Jr. Revised and ed. by Basil Gordon, UNIVERSITY OF CALIFORNIA, LOS ANGELES. Paperbound and cloth. Vol. I, 274 pp, (1964). Cloth, $5.95, Paper, $3.95. Vol. II, 350 pp, (Spring 1966). Cloth, $5.95, Paper, $3.95 *Estimated price 728 Montgomery Street, For the catalog, write to: HOLDEN • DAY , INC • San Francisco, California 94111

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403 TOPOLOGY FROM r------New THE DIFFERENTIABLE THRESHOLD LOGIC SZE-TSEN HU VIEWPOINT The first comprehensive treatment of the mathematics associated with threshold devices. Chapters discuss the conditions which switching functions must satisfy to be thresh­ old functions, as well as the algorithms for determining whether or not a given switch­ ing function is a threshold function and the ways for finding its most economic physical realization. A final chapter presents an effective and mathematically rigorous finite process for constructing a mini}Ilal threshold network that realizes an arbitrary switching function. In addition to contributing a num­ ber of original research results, Dr. Hu's book may be used as a reference work on the subject, constituting as it does the first com­ prehensive exposition of most of the contri­ John W. Milnor butions to date. $8.00 Department of Mathamatics, Princeton University. r V;! from California ix, 64 pp., 19 figs. 6x9. $3.00 ~ ~\ University of California Press "?:}) 1 Berkeley 94720

As might be expected from a recipient of the Field medal in mathematics in 1962, Professor From our publishing programme: Milnor's book is a simple and elegant exposition Singulare lntegralgleichungen of complicated mathematical ideas. An excellent (Singular Integral Calculus) introduction to topology and global analysis, it Boundary value problems on the functional theory and will be extremely useful to srudents. the applications in mathematical physics By N: I. Muschelischwili CONTENTS. Preface. 1. SMOOTH MANIFOLDS (Translation from the Russian into German); Published by lothar Berg and Hans Schubert AND SMOOTH MAPS. Tangent Spaces and Deriva­ 1965, XIV, 564 pages, 25 illustrations, cloth-bound, tives. Regular Values. The Fundamental Theo­ $14.32 rem of Algebra. 2. THE THEOREM OF SARD Allgemeine Mengenlehre-Ein Fundament der Mathematik AND BROWN. Manifolds with Boundary. The (General Theory of Sets) Brouwer Fixed Point Theorem. 3. PROOF OF By Dieter Klaua SARD'S THEOREM. 4. THE DEGREE MODULO 2 1964. VIII, 581 pages, cloth-bound, $13.13 OF A MAPPING. Smooth Homotopy and Smooth Fourth edition-just published: Isotopy. 5. ORIENTED MANIFOLDS. The Brou­ Lehrbuch der Wahrscheinlichkeitsrechnung wer Degree. 6. VECTOR FIELDS AND THE EULER (Textbook of Probability Calculus) NUMBER. 7. FRAMED COBORDISM: THE PoN­ By B. W. Gnedenko (Translation from the Russian into German); Published by TRYAGIN CONSTRUCTION. The Hopf Theorem. H. J. Rossberg 8. EXERCISES. APPENDIX: Classifying 1-Mani­ 1965. XIII, 393 pages, 20 illustrations, 21 plates, cloth-bound, $7.04 folds. BIBLIOGRAPHY. INDEX. Please order through your local bookdealers If you will send us your address indicating your special interests, we shall be happy to supply you with regular University Press of Virginia information AKADEMIE-VERLAG · BERUN Charlottesville 101 Berlin, leipziger Strasse 3-4, German Demauatic Repulllic

404 INTRODUCTION TO ALGEBRA GENERAL THEORY OF FUNCTIONS AND by Sam Perlis, Purdue University INTEGRATION An undergraduate text intended for a one­ year introductory course in modern alge­ by Angus E. Taylor, University of bra, this text is arranged so that it can be California, Los Angeles used for separate courses in linear algebra Intended for use in a course for fist-year and in abstract algebra. Designed to pre­ graduate students, this book may be used sent a well-integrated, solid basis of infor­ as a guide in independent study. The book mation and understanding of the subject, is arranged so as to provide a natural this text sets upper bounds against an transition from the classical theory of excess of difficult abstractions and deep point sets in Euclidean space, and the theorems. 1966. In press. theory of functions of one or more real variables, to the more abstract settings in which the ideas of topology, continuous functions, and integration find their most natural development. 1965. 437 pp. $12.50 TOPICS IN ALGEBRA A FIRST COURSE IN PARTIAL DIFFERENTIAL EQUATIONS by I. N. Herstein, University of Chicago by H. F. Weinberger, Basic algebraic systems are presented for University of Minnesota intermediate and advanced students. Pro­ ceeding from the abstract to the particular, Designed as a text for a one-year intro­ the topics covered are treated in depth, re­ ductory course in the solution of partial ducing the danger involved in introducing differential equations, this book includes abstract ideas before a sufficient base of the separation of variables and the asso­ examples has been established to render ciated expansion theory, the elementary them credible and natural. Exercises are theory of complex variables, Fournier included. 1964. 342 pp. $9.00 and Laplace Transforms, and numerical methods of solution. The general proper­ ties of partial differential equations such as characteristics, domains of dependence, and maximum principles are clearly shown; solution· ·methods are then tied to this framework. 1965. 446 pp. $12.50 FUNCTIONAL ANALYSIS METHODS OF REAL ANALYSIS by Albert Wilansky, Lehigh University by Richard R. Goldberg, Appropriate for a two-semester .course o.n Northwestern University the senior or graduate level, this book IS designed for an introductory functional For students who have completed a course analysis course. The first half of the text in calculus and differential equations, this carries the subject through linear, metric, text on elementary theory of functions of Frechet, Banach, and Hilbert space, and a real variable contains material on se­ includes the Hahn-Banach theorem and the quences, series, and the foundations of uniform boundedness principle. The second calculus. The text also includes topics ordin­ half treats topology and nets, linear topolo­ arily not taught at this level, such as metric gical space, the closed graph theorem, local spaces and Lebesgue integration. Methods convexity, duality, and Banach algebra. of Real Analysis prepares students for ad­ Applications are stressed throughout. A vanced courses in mathematics, including feature of this book is the abundance of topology, measure theory, and functional problems, most of which are elementary, analysis. 1964. 359 pp. $9.00 designed to test the reader's understanding. 1964. 291 pp. $10.50

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405 Available Reprints tn• Mathematics Duke Mathematical Journal Mathematical Society Vols.l-27. Durham 1935-1960. of Japan: Journal (Portly in the original edition) Paper bound set ...... $675.00 Vols.l-13. Tokyo 1948-1961. Vols. 2-13, 1936-1946 Cloth bound set ...... $230.00 Paper bound set ...... 200.00 Per volume, paper bound ...... 25.00 Per volume, paper bound ...... 16.00 Vols. 14, 15, 1947 Per volume, paper bound ...... 35.00 Vols.16·19,21,23, 1949-1956 Per volume, paper bound ...... 25.00 Mathematische Annalen Vols. 20, 24, 25, 1953, 1957, 1958 Vols. 1-80. Leipzig; Berlin 1869-1920. (Partly in the original edition) Cloth bound set ...... $2,300.00 Per volume, paper bound ...... 25.00 Paper bound set ...... 2,120.00 Voi.1,Nos.1,3,4, 1935 Voi.24,Nos.1,3, 1957 Vols. 1-25, 1869-1885 Vol. 16, Nos. 1-4, 1949 Vol. 25, No. 2, 1958 Cloth bound set ..•...... 675.00 Vol. 20, Nos. 1, 2, 4, 1953 Paper bound set ...... 620.00 Per issue, paperbound ...... 6.50 Per volume, paper bound ...... 25.00 Vols. 26-80,1886-1920 Journal of Mathematics Cloth bound set ...... 1,625.00 and Physics Paper bo,und set ...... 1,500.00 Per volume, paper bound ...... ,...... 27.50 Vols. 1-33. Cambridge, Mass. 1921-1945. (Partly in the original edition) Paper bound set ...... $600.00 Vol. 1-20 Mathematische Nachrichten Cloth bound set ...... , ...... 365.00 Paper bound set ...... 320.00 Vols. 1-20. Berlin 1948-1959. Cloth bound set...... $450.00 Per volume, paper bound ...... 16.00 Paper bound set ...... 400.00 Vols. 21-33 Per volume, paper bound ...... 20.00 Per volume, paper bound ...... 22.00 In the original edition Vols. 35, 36, 1956, 1957. Vols. 21, 22, 1960 (Partly in the original edition) Per volume, unbound ...... 25.00 Per volume, paper bound ...... 22.00 Vol. 35, No.1, 1956 Vol. 36, No.I, 1957 Per issue, paper bound ...... 5.50 Mathesis: Recueil Mathematique Mathematical Gazette Vols. 1-7. Ghent 1881-1887. (Mathematical Association) Cloth bound set ...... $130.00 Nos. 1-6. London 1894-1895 Paper bound set ...... 110.00 New Series. Vols. 1-15, 1896-1931. Per volume, paper bound ...... 16.00 (Including lndex1894-1931) Cloth bound set ...... $345.00 Paper bound set ...... 300.00 Nos. 1-6 Michigan Mathematical Journal Paper bound in one volume ...... 10.00 Vols.l-7. Ann Arbor 1952-1960. New Series. Vols. 1-15 Cloth bound set...... $125.00 Per volume, paper bound ...... 20.00 Paper bound set ...... 105.00 Index 1894-1931, paper bound ...... 10.00 Per volume, paper bound ...... •...... 15.00 JOHNSON REPRINT CORPORATION ®JOHNSON REPR~~~h~~;;~~~N-~~; Berkeley Square House, London W.l, England

406 NEW MATHEMATICAL PUBLICATIONS FROM POLAND

Stanislaw Saks and Antoni Zygmund

ANALYTIC FUNCTIONS 2nd Edition Published in 1965, this new edition in English contains an impor­ tant new chapter on subharmonic functions. The first edition has been used as a textbook in many American universities.

CHAPTERS: Theory of Sets Analytical Functions Functions of a Complex Entire Functions Variable Elliptic Functions Holomorphic Functions Meromorphic Functions Functions 7 and r Dirichlet Series Elementary Geometrical Functions Harmonic and Subharmonic Conformal Transformations Functions

Published by Polish Scientific Publishers. 508 pp. Cloth. $10.00

Recommended Volumes in English from the MONOGRAFIE MATEMATYCZNE Series

Vol. 41 H. Rasiowa and H. Sikorski, THE MATHEMATICS OF METAMATHEMATICS, 1964. $12.00 Vol.43 J. Szarski, DIFFERENTIAL INEQUALITIES, 3rd Ed., 1965. $8.00 Vol.34 W. Sierpinski, CARDINAL AND ORDINAL NUMBERS, 3rd Revised Ed., 1965. $10.00

ORDER FROM: ARS POLONA FOREIGN TRADE ENTERPRISE 7 Krakowskie Przedmiescie, WARSAW, Poland

407 THE MATHEMATICS AND STATISTICS SECTION of the NATIONAL REGISTER OF SCIENTIFIC AND TECHNICAL PERSONNEL

Cooperating Societies:

American Mathematical Society The Institute of Mathematical Statistics American Statistical Association Mathematical Association of America Association for Computing Machinery Operations Research Society of America Association for Symbolic Logic Society for Industrial and Applied Biometric Society Mathematics Econometric Society Society of Actuaries Industrial Mathematics Society

The American Mathematical Society, at the request of the National Science Foundation, will mail the 1966 reporting forms to the Mathe­ matical and Statistics Section of the National Register. The main objective of the National Register is to provide up-to-date information on the scientific manpower resources of the United States. It is also increasingly valuable to our profession as a source of statistical information.

When you receive a National Register questionnaire, please fill it out and return it promptly to the Headquarters offices of the Society at P.O. Box 6248, Providence, Rhode Island 02904.

If you have never received a questionnaire and feel that you are qualified for inclusion in the National Register, please write to us at the above address.

Gordon L. Walker Executive Director

408 Are you a PhD in math, strong on statistics and probability, with a Renaissance man's capability to absorb other disciplines?

Astropower, a research laboratory of the Douglas Aircraft Company, is conducting basic research in the field of artificial intelligence. Over eight different technologies are involved and you will assimilate something of each. Experience or training in these areas is an asset. As for your normal profession, you must be a PhD in math, strong on statistics and probability. And you should have a working knowledge of adaptive or learning systems. If you did your dissertation on that subject, it is sufficient. Your primary responsibility will be basic research in the field of artificial intelligence systems. Currently, we are seeking to develop learning systems capable of recognizing specific patterns or objects of interest in complex photographic imagery such as that received from the Tiros or Ranger scientific satellites. If your professional qualifications meet our requirements, and if you have the capability and interest to absorb other disciplines, please send your professional biography to Mr. E. F. MacDonald, 2750 Ocean Park Blvd., Santa Monica, Calif. DOUGLAS An equal opportunity employer, males & females may apply. MISSILE & SPACE SYSTEMS DIVISION

409 Applied Mathematicians Mathematicians Operations Analysts Appl1ed Physicists and Statisticians CAREER APPOINTMENTS looking for an opportunity

to apply your special skills and your analytical abilities? The Center for Naval Analyses of The Franklin Institute is inter­ ested in men who can derive mathematical bases for new tactical and operational proce­ dures for the U.S. Navy.

Your ability to develop and test mathematical models will be valuable in formulating opera­ tional requirements for equip­ ment under evaluation.

CNA is a private scientific organization engaged in opera­ Vision tions research and systems analysis for the U.S. Navy and Marine Corps. It offers a prom­ ising career with good salary and the accepted concomi­

tants of a position that de­ Vision-at Booz•AIIen Applied Research­ mands responsibility and is the union of insight and farsightedness in the service of military, governmental and returns satisfaction. industrial clients. The clarity of this vision has been evident in our organization's repu­ tation for competence, our increasing re­ sponsibilities ... and our continuing need For an interview, write, for additional new talent. enclosing resume, to: Your career growth at Booz•AIIen Ap­ plied Research will be as rapid as your James M. Hibarger talents permit. Because our breadth of interests el)compasses astronautics, CENTER FOR NAVAL ANALYSES communications, computer technology, mathematics and statistics, meteorology, 1401 Wilson Boulevard operations research, reliability, and a dozen Arlington, Virginia 22209 more, each professional staff member is able to participate in a wide range of inter­ disciplinary assignments. If you wish to share an outstanding record of diversified achievement and can bring us appropriate abilities and experience, we would like to hear from you. Please write Mr. Robert H. Flint, Director of Professional Appoint­ ments. CENTER FOR NAVAL ANALYSES OF THE FRANKLIN INSTITUTE BdOZ• ALLEN APPLIED RESEARCH Inc. INS· Institute of Naval Studies 135 South LaSalle Street, Chicago, Illinois 60603 SEG • Systems Evaluation Group New York • Washington OEG • Operations Evaluation Group Cleveland • Chicago NAVWAG • Naval Warfare Analysis Group Los Angeles MCOAG • Marine Corps Operations Analysis An equal opportunity employer Group An equal opportunity employer

410 ENGINEERS o PHYSICISTS o MATHEMATICIANS

NAVAL PROBLEM SOLVING

The Applied Physics Laboratory, The Johns Hopkins University now offers both senior and associate staff appointments in naval operations analysis covering broadly defined areas of submarine, surface, and air operations ... present and future.

Primary requirements include (I) interest in solving complex problems of conflict that go far beyond the limitations of Game Theory, and (2) ability to abstract operational models from the complexity of real or proposed systems. Background in physics, engineering or mathematics is most applicable; knowledge of operations research techniques and modern computer usage in combat simulation is useful but not mandatory.

The APL location in suburban Washington affords a choice of city, suburban or country living and offers many benefits including complete service and recreational facilities and superior public and private schools. Programs leading to advanced degrees from The Johns Hopkins University may be pursued at APL. In addition, staff members may continue their education at six other area universities with APL.financial support.

Direct your inquiry to: Mr. W. S. Kirby, Professional Staffing

The Applied Physics Laboratory · The Johns Hopkins University

8638 Georgia Avenue, Silver Spring, Maryland (Suburb of Washington, D. C.) An equal opportunity employer

411 A MESSAGE OF CAREER INTEREST TO THE ADVANCED, CREATIVE MATHEMATICIAN FROM National Security Agency

A unique organization operating within the structure of the Federal Government, the National Security Agency is responsible for developing "secure" (i.e., invulnerable) communications systems to transmit and receive vital information.

Mathematicians are key members of NSA's scientific fraternity. And the Agency's advanced career opportunities for senior mathematicians have never been more exciting and attractive than they are today ... in their challenge, their scope and their potential.

Theoretical Research Opportunities There are presently a number of NSA groups engaged in theoretical mathematical research programs. Many other branches of mathematics which might fruitfully advance the Agency's work simply await the capabilities and interest of new mathematicians. In addition, there is considerable opportunity to make substantive scientific contributions in bridging gaps between the theoretical and the practical. Computer software and applications comprise still another vital area for potential contributions.

NSA mathematicians are frequently called in to help define new problems. Observing their origins, characteristics and the trends of any data associated with them, the mathematician must determine whether the problem and ~ata are susceptible to mathematical solution. If so, he may then help define and determine the method of that solution.

Advanced Facilities, Educational Programs NSA will encourage you to pursue advanced mathematical study at any of several area universities, including the highly regarded University of Maryland. Educational assistance ranges from tuition payment for part time study-to full time, full salary fellowships.

As a professional staff member of NSA, you enjoy all the benefits of Federal employment without the necessity of Civil Service certi­ fication. The Agency is located between Washington and Baltimore, its site permitting your choice of city, suburban or country living and offering a wide range of nearby nsa recreational/cultural/historical attractions. NATIONAL SECURITY AGENCY Suite 10,4435 Wisconsin Ave., N.W. Washington, D.C. 20016 For further career information, please An equal opportunity employer (M & F) send us your resume now. . .. where imagination is the essential qualification

412 CUSHING-MALLOY, INC. FIVE-YEAR MATHEMATICAL REVIEWS INDEX 1350 North Main St., P. 0. Box 632

This index, in two volumes, covers the years Ann Arbor, Michigan 1960-1964, volumes 21-28 of MATHE­ 48107 MATICAL REVIEWS. It follows the same LITHOPRINTERS format, includes over 6,000 reviews, and is nearly as large as the Twenty-Year Index. • In addition, the new index presents a list of common Chinese characters, a table Known for including all current systems of translitera­ QUALITY- ECONOMY tions of Chinese names, and an errata applicable to the Twenty-Year Index. SERVICE List Price $39.30 Member Price $29.48 •

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Name P~ge Academic Press Inc...... 397 Addison-WesleyPublishingCompany, Inc . 396 American Mathematical Society . . . 413 Ars Polona ...... 407 W.A. Benjamin, Inc ...... 403 Blaisdell Publishing Company . . . . 405 Booz · Allen Applied Research Inc . . 410 University of California Press. . . . . 404 Centerfor Naval Analyses . . . . . 410 Cushing-Malloy, Inc ...... 413 Deutscher Buch-Export und lmportgesellschaft mbH. 404 Douglas Aircraft Company, Inc. . 409 Dover Publications, Inc ...... 395 D. C. Heath and Company ...... 394 Holden-Day, Inc...... 401 Holt, Rinehart and Winston, Inc ...... 398 The Johns Hopkins University, The Applied Physics Laboratory . 411 Johnson Reprint Corporation...... 406 Monroe International, Inc...... 393 National Register of Scientific and Technical Per8onnel...... 408 National Security Agency ...... 412 Pergamon Press, Inc...... outside back cover Prentice-Hall, Inc ...... 398 Ruch, Export and Import Enterprise . 402 D. VanNostrandCompany,lnc ...... 400 The University Press of Virginia ...... 404 Wadsworth Publishing Company ...... 399 John Wiley & Sons, Inc ...... inside back cover

413 ADVANCE REGISTRATION FORM Rutgers-The State University New Brunswick, New Jersey August 29-September 2, 1966

Please fill out the form below and return with your payment no later than August 13, 1966 to:

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414 Wiley and lnterscience-specialists in "how to" publications ~ HOW to...

~employ: Differential Equations of Applied Mathematics, by G. F. D. DUFF, University of Toronto; and D. NAYLOR, University of Western Ontario. 1966. 423 pages. $11.95 .

.... use statistical and probability techniques: An Introduction to Probability Theory and Its Measurement and Analysis of Random Data, Applications, Volume II, by WILLIAM FELLER. by JULIUS S. BENDAT, and ALLAN G. PIERSOL. 1966. 626 pages. $12.00. 1966. 390 pages. $17.75. Linear Statistical Inference and Its Applica­ The Theory of Stochastic Processes, by D. R. tions, by C. RADHAKRISHNA RAO. COX and H. D. MILLER. 1965. 522 pages. $14.95 . 1965. 398 pages. $11.50.

.... become proficient with programming and computers: Programming the IBM System/360, by the Staff of Computer Usage Company; ASCHER OPLER, Editor. 1966. 316 pages. $7.50. A Guide to Fortran IV Programming, by DANIEL D. McCRACKEN. 1965. 151 pages. Paperbound: $3.95. Error in Digital Computation, Volume II, edited by L. B. RALL. 1965. 288 pages. $6.75 . .... apply mathematics to specialized fields: Matrix Algebra for the Biological Sciences (In­ Asymptotic Expansions for Ordin~try Differen­ cluding Applications in Statistics), by S. R. tial Equations, by WOLFGANG WASOW. SEARLE. 1966. 296 pages. $9.95. Vol. 14 of the lnterscience Pure and Applied Mathematics Series. $14.00. An Introduction to Mathematical Learning The­ 1966. 362 pages. ory, by RICHARD C. ATKINSON, GORDON H. Programming, Games and Transportation Net­ BOWER and EDWARD j. CROTHERS. works, by C. BERGE and A. GHOUILA-HOURI. 1965. 429 pages. $9.95. 1966. 272 pages. $8.75.

~ keep up with new methods and applications: Communications on Pure and Applied Mathematics A journal Issued Quarterly by The Courant Institute of Mathematical Sciences, New York University. An lnterscience Publication. Subscription for the calendar year 1966 (Vol. XIX): $12.00. Foreign postage $1.00/vo/ume.

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