Mathematicians at Stockholm, the Council of the Society Will Meet Sweden

AMERICAN

MATHEMATICAL

SOCIETY

VOLUME 9, NUMBER 4 ISSUE NO. 62 AUGUST 1962

THE AMERICAN MATHEMATICAL SOCIETY

Edited by John W. Green and Gordon L. Walker

CONTENTS

MEETINGS Calendar of Meetings .•...... •.•.•...•.....•...... ••••.•. 248 Program of the Summer Meeting in Vancouver, B. C .•...... 249 Abstracts of the Meeting - pages 284-313 PRELIMINARY ANNOUNCEMENT OF MEETING •..•.••...... •..•. 263 ACTIVITIES OF OTHER ASSOCIATIONS .••.••••..•.....•...... ••• 265 NEWS ITEMS AND ANNOUNCEMENTS .••.•.....••.....•.•. 262, 267,276 PERSONAL ITEMS .•..•.••.....•.•.•.•...... •. 268 NEW AMS PUBLICATIONS •.....•..••••.....•••....•.....•.•...• 273 LETTERS TO THE EDITOR ....•....•....•...... •..••...... •.. 274 MEMORANDA TO MEMBERS The National Register of Scientific and Technical Personnel . . • • . • . . • . . 2 77 The Employment Register ....••.••.•.•...•.•.•.•.•..••....• 277 The Combined Membership List, 1962-1963 •...•.....•.....•...•. 277 Corporate Members •.••...... •.•....•.....•••..• 278 The Australian Mathematical Society ...... 278 Backlog of Mathematical Research Journals • ...... • . . • ...... • . 279 SUPPLEMENTARY PROGRAM - NO. 12 . . . . . • ...... 280 ABSTRACTS OF CONTRIBUTED PAPERS ...•...... ••.....•.• 284 INDEX TO ADVERTISERS ...•...•..•..•.....•.....•...... 355 RESERVATION FORM . • • ...... • . . • . • . • . • ...... • ...... 3 55 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 NOTICES 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*

593 October ?.7, 1962 Hanover, New Hampshire Sept. lZ 594 November 16-17, 196?. Tallahassee, Florida Oct. Z 595 November 17, 196?. Los Angeles, California Oct, Z 596 November ?.3, ?.4, 196?. Northwestern University Oct. Z 597 January ?.4-?.8, 1963 Berkeley, California Nov. ?.3 (69th Annual Meeting) April ?.6-Z 7, 1963 New Mexico State University August ?.6-30, 1963 Boulder, Colorado (68th Summer Meeting) January Z0-?.4, 1964 Miami, Florida (70th Annual Meeting) August, 1964 Ann Arbor, Michigan (69th Summer Meeting) January, 1965 Denver, Colorado (7lst Annual Meeting) August, 1965 Ithaca, New York August, 1966 New Brunswick, New Jersey * 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 lines also apply to news items. The next two deadline dates for by title abstracts are September 5 and September ?.5.

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The NOTICES of the American Mathematical Society is published by the Society in January, February, April, June, August, October and November. Price per annual volume is $7.00, Price per copy, $?..00, Special price for copies sold at registration desks of meetings of the Society, $1.00 per copy. Subscriptions, orders for back numbers (not available before 1958), and inquiries should be addressed to the American Mathematical Society, 190 Hope Street, Providence 6, Rhode Island. Second-class postage paid at Providence, Rhode Island, and additional mailing offices. Authorization is granted under the authority of the act of August ?.4, 191?., as amended by the act of August 4, 1947 (Sec. 34, Zl, P. L. and R.). Accepted for mailing at the special rate of postage provided for in section 34.40, paragraph (d).

PROGRAM

The sixty-seventh summer meeting August 30. The title of his talk is "Models of the American Mathematical Society will of complete theories". be held at the University of British Colum Sessions for contributed papers will bia, Vancouver, B. C., from Tuesday, be held in the Buchanan Building at 10:30 August ZB, to Friday, August 31, 196Z. A.M. on Wednesday, 10:30 A.M.and 3:3"0 During this week there will be meetings at P.M. on Thursday, and at Z:OO P.M. on the University of British Columbia of the Friday. Abstracts of the papers to be pre Mathematical Association of America, and sented at these sessions appear on pages the Society for Industrial and Applied 284-313 of these NOTICES. There are Mathematics. Additional information about cross references to the abstracts in the the programs of these other meetings will program. For example, the title of paper be found elsewhere in these NOTICES under (1) in the program is followed by (592-9) "Activities of Other Associations". indicating that the abstract can be found There will be no Colloquium Lec under the designation 59Z-9 among the ture at this meeting because of the occur published abstracts. There will be noses rence in August, 196Z of the International sion for late papers. Congress of Mathematicians at Stockholm, The Council of the Society will meet Sweden. at 4:00 P.M. on Thursday, August 30 in The Committee to Select Hour the Study Room of the International House. Speakers for Summer and Annual Meet There will be a Business Meeting of the ings has invited Professor C. C. Lin of Society at 10:00 A.M. on Friday, August31 the Massachusetts Institute of Technology, in the Auditorium Building. Professor Charles Loewner of Stanford The Mathematical Sciences Employ University, Professor Albert Nijenhuis of ment Register will be located in Room 1 ZZ 1 the University of Washington, Professor of the Buchanan Building. It will be main R. S. Pierce of the UniversityofWashing tained on Tuesday, Wednesday and Thurs ton, and Professor R. L. Vaught of the day from 9:00 A.M. to 5:00 P.M. During University of California, Berkeley to ad these same days, there will be book ex dress the Society. All of these lectures hibits in the main hall on the second floor will be given in the Auditorium Building of the Buchanan Building. about ZOO yards west of the Buchanan Building. Professor Lin's address, en titled "Hydrodynamic stability - a study REGISTRATION in applied mathematics", will be given at 9:00 A.M. on Wednesday, August Z9. Pro Registration headquarters will be in fessor Loewner will speak at Z:OO P.M. the main floor corridor of the Buchanan on Thursday, August 30. The title of his Building. The Registration Desk will be talk is "On semigroups in analysis and open from 2:00 P.M. to 8:00 P.M. on Sun geometry". Professor Nijenhuis' s talk on day, from 9:00A.M. to 5:00P.M. onMon "Derivations and structure" will be given day through Thursday, and from 9:00A.M. at Z:OO P.M. on Tuesday, August ZB.- Pro to 2:00 P.M. on Friday. All persons attend fessor Pierce will speak on "Representa ing the meeting are requested to register tions of lattices" at 9:00 A.M. on Friday, as soon as possible after their arrival. August 31. Professor Vaught's address _ The schedule of registration fees is as will be given at 9:00 A.M. on Thursday, follows:

249 Members of participating organi- matic laundry facilities are available. zations (except students) $2.00 Rooms in residences may be occupied from First nonmember in member's 2 :00 P .M.,Saturday, August 25 to 2:00P.M., family .50 Saturday, September 1. The nearest hotels or motels are Other nonmembers in mem- located in downtown Vancouver (about 6 ber's family free miles from the campus). The nearest pub Students free lic campsites are located at Alouette Lake, about 40 miles from the campus. Nonmembers not in any of the The main cafeteria located in the $5.00 above categories Auditorium building will be open all day A directory of all persons attending the from 8:30 A.M. Cafeteria service will also meetings will be located at registration be available in three dining rooms located headquarters. in the three main residence areas. Meal hours are: Breakfast 7:30-8:30 A.M.; ROOMS AND MEALS Lunch 11:45-1:00 P.M.; Dinner 5:00-6:30 Accommodations will be available P.M. Prices of regular cafeteria meals in the University residences to all attend are: Breakfast-50 cents; Lunch-65 cents; ing the meetings and to adult members of Dinner-85 cents. their families. Only a few units are avail RESERVATIONS able on campus which are suitable for families with children under twelve. Chil Reservations for University resi dren twelve and over can be accommodated dences and dormitories should be made by in the regular residences at the regular writing directly to the Conference Office, rates. Low cost dormitory space is avail Department of Extension, University of able in converted huts in Fort Camp and British Columbia, Vancouver 8, Canada at Acadia Camp. The cost of dormitory hous the earliest possible date and before ing will be $2,00 a night per person in a August 9. A reservation form for this pur single room. In a double room with separ pose can be found on the inside back cover ate beds the cost is $1,50 a nightper per of these NOTICES. The university also has a number of son. HOTELS AND MOTELS new, modern, three-story permanent resi dences containing mostly single rooms. Persons desiring hotel and motel The cost of permanent residence housing accommodations should make their reser will be $4.00 a night per person. A small vations directly with the appropriate number of double rooms with separate beds manager, and under no circumstances are available at $3.00 a night per person. write to the Conference Office which is in Towels and bedding will be provided. Auto- charge only of the University facilities.

Suggested Hotels and Motels Units or Minimum Rates Name Address Phone Rooms Single Double

Burrard Motel 1100 Burrard MU 1-2331 60 $ 8.00 $11.00 Devonshire Hotel 894 W. Georgia MU 1-5481 150 7,50 10,50 Doric Howe Motor Hotel 1061 Howe St. MU 2-3171 103 8.00 10,00 Georgia Hotel 801 W. Georgia MU3-1182 314 9.00 12.50 Georgian Towers Motor Hotel 1450 W. Georgia MU 1-4321 105 7.00 13,00 Kamlo Apt. Hotel 1120 Denman St. MU 4-7474 47 7.00 10.00 Travelodge Motel 1304 Howe St. MU 2-2767 74 7.00 9.00 Bayshore Inn 1601 Georgia St. MU 2-3377 308 12.00 15.00 Sands Motor Hotel 17 55 Davie St. MU2-1831 100 8.50 9.50 Sylvia Hotel 1154 Gilford St. MU 1-9321 125 5.00 8,00 Vancouver Hotel 900 W. Georgia MU4-3131 561 9.00 11.50

250 ENTERTAINMENT AND RECREATION CUSTOMS INFORMATION There will be an outdoor Salmon Crossing the United States - Can Barbecue on Wednesday, August Z9 at 6:00 adian border either way is made without P.M. in the vicinity of the University difficulty of delay by citizens or permanent Residence "Common Block". Tickets for residents of the United States. Passports the barbecue can be purchased at the Re are not needed. However, U. S. Citizens gistration Desk. The price of tickets is should carry some document establishing approximately $Z.50 for adults and $l.Z5 their citizenship. Alien permanent resi for children under twelve. dents of the United States are advised to A tea will be given between 4:00 have their Alien Resistration Receipt Card P.M. and 6:00 P.M. on Tuesday, August Z8 (U.S. Form 1-151). Visitors to the United in the courtyard of the Buchanan Building States who have only a single entry visa to if the weather permits, and otherwise in the the U.S.A. should present their documents International House. at an office of the United States Immigra Some of the athletic facilities of the tion and Naturalization Service before en University will be available to members tering Canada, to be sure that they have the and their guests. The Seattle World's Fair necessary documentation for their re-entry and the Vancouver International Festival to the United States, Further information will be in operation during the meeting, regarding entry to Canada and what mer chandise may be taken back into the United TRAVEL States may be obtained by writing to the Vancouver is approximately 140 Canadian Government Travel Bureau at miles north of Seattle, Washington on High one of the following addresses: Ottawa, way 99. The best route to the University Ontario, Canada; Canada House, 680 Fifth from the south is on Highway 99 via Deas Avenue, New York 19, New York; lOZ West Island Tunnel (toll), Oak Street bridge Monroe Street, Chicago 3, lllinois; 1 Sec (toll), West 41st Avenue and Marine Drive. ond Street, San Francisco 5, California. Vancouver is served by Canadian The Canadian currency system is Pacific, Qantas, Trans Canada, and United the same as that of the United States, but Airlines; the Canadian National, Canadian the rate of exchange between the American Pacific, and Great Northern Railroads; and Canadian dollar varies from time to and the Greyhound Busline. Automobile time. United States visitors are advised to ferry service is available from Vancouver exchange their currency for Canadian funds Island on British Columbia Government at a bank in Canada to be assured of re ferries from Nanaimo and Swartz Bay, and ceiving the prevailing rate of exchange. by Canadian Pacific from Nanaimo, Public transportation in Vancouver MAIL AND TELEGRAM is provided by buses from the city center to the Blanca loop ( 15 cent fare), transfer Communications with members of ring to the UBC bus running to the campus the Society and their guests should be ad (10 cents). dressed to them in care of the American Airport, Bus, and Limousine ser Mathematical Society, The University of vice. Bus and Limousine service costs British Columbia, Vancouver 8, Canada, $1.Z5, Cab from the airport to Vancouver about $4.50, Cab from the airport to U.B.C. about $6,00, and Cab from the city center COMMITTEE ON ARRANGEMENTS to U.B.C. about $3.00. For one or two per sons, the most economical way to travel H. L. Alder R. S. Pierce is to take the bus from the airport to N. J, Divinsky R. A. Restrepo Granville and Broadway streets, and con R. D, james W. H, Simons tinue by taxi to U.B.C. (total cost about B. N. Moyls G. L. Walker $4.00 for one person). (Chairman)

251 TIME TABLE (Pacific Daylight Time)

SUNDAY, August Z6 American Mathematical Mathematical Association Society for Industrial Society of America and Applied Mathematics

10:00 A.M. Board of Governors Meeting - Study Room, International House Z:OO P.M.- 8:00P.M REGISTRATION-- MAIN FLOOR -- BUCHANAN

MONDAY, August Z7 Society Association SIAM

9:00A.M. - 5:00P.M. REGISTRATION -- MAIN FLOOR -- BUCHANAN

The Auditorium 9:00A.M. Hedrick Lecture. I A. M. Gleason 10:10 A.M. Invited Address C. B. Lindquist 11:10 A.M. Invited Address A. W. Tucker Z:OO P.M. Hedrick Lecture. II A. M. Gleason Session on Teaching Machines 3:10P.M. L. M. Stolurow 4:00P.M. J. E. Forbes 4:55P.M. Film: The Kakeya Problem by A. S. Besicovitch 7:30P.M. Film: What is an Integral, by Edwin Hewitt

TUESDAY, August Z8 Society Association Pi Mu Epsilon-

8:00A.M. Dutch Treat Breakfast "Common Block" 9:00 A.M. - 5:00 P.M. REGISTRATION --MAIN FLOOR -- BUCHANAN EMPLOYMENT REGISTER --ROOM lZZl -- BUCHANAN EXHIBITS --MAIN FLOOR -- BUCHANAN

The Auditorium 9:00A.M. Hedrick Lecture. III A. M. Gleason 10:05 A.M. Business Meeting Session on Inequalities 10:30 A.M. George Polya 11:05 A.M. E. F. Beckenbach 11:40 A.M. M.D. Marcus Z:OO P.M. Invited Address - The Auditorium Albert Nijenhuis

252 TIME TABLE (Pacific Daylight Time)

American Mathematical Mathematical Association Society for Industrial Society of America and Applied Mathematics

TUESDAY, August Z8 3:15P.M. Film: The Theory of Limits, by E. J, McShane The Auditorium

4:00P.M. TEA-- BUCHANAN COURTYARD 7:00P.M. Film: Mathematical Induction, by Leon Henkin The Auditorium

7:30P.M. Meeting of Section Officers - Room 3Z05 Buchanan 8:10P.M. Film: The Kakeya Problem by A. S. Besicovitch (a repeat showing) The Auditorium

WEDNESDAY, August Z9 Society Association SIAM 9:00 A.M. - 5:00P.M. REGISTRATION-- MAIN FLOOR -- BUCHANAN EMPLOYMENT REGISTER -- ROOM 1ZZ1 --BUCHANAN EXHIBITS --MAIN FLOOR -- BUCHANAN

9:00A.M. Invited Address - The Auditorium C. C. Lin 10:15 A, M. Invited Address - The Auditorium S. P. Diliberto 10:30 A.M. Sessions for Contributed Papers - Buchanan Geometry: Room 100 Algebra: Room 10Z Th, of Nos:Roo·m 104 Analysis: Room 106 The Auditorium Z:OO P.M. A Report on the Interna tional Congress V. L. Klee, Jr. Session on Differential Equations Z:50 P.M. Hans Sagan 3:35P.M. R. W. McKelvey 4:ZO P.M. Fred Brauer 6:00P.M. SALMON BARBECUE-- COMMON BLOCK

253 TIME TABLE (Pacific Daylight Time)

American Mathematical Mathematical Association Society for Industrial Society of America and Applied Mathematics

THURSDAY, August 30 9:00A.M. -5:00P.M. REGISTRATION-- MAIN FLOOR-- BUCHANAN EMPLOYMENT REGISTER -- ROOM 1221 -- BUCHANAN EXHIBITS-- MAIN FLOOR -- BUCHANAN

9:00A.M. Invited Address - The Auditorium R. L. Vaught 10:15 A.M. Invited Address - The Auditorium H. B. Mann 10:30 A.M. Sessions for Contributed Papers- Buchanan - Topology: Room 100 Algebra: Room 102 Probability:Room 10~ 2:00P.M. Invited Address - The Auditorium Charles Loewner 3:15P.M. Invited Address - The Auditorium Frank Proschan 3:30P.M. Sessions for Contributed Papers - Buchanan - Algebra: Room 100 General: Room f02 Analysis: Room 104 4:00P.M. Council Meeting - Study Room, Interna tional House

FRIDAY, August 31 Society As so.ciation SIAM 9:00A.M.- 2:00P.M. REGISTRATION-- MAIN FLOOR-- BUCHANAN 9:00 A.M. Invited Address - The Auditorium R. S. Pierce 10:20 A.M. Sessions for Contributed Papers - Buchanan Rooms 100 and 102 10:30 A.M. Business Meeting The Auditorium 2:00P.M. Sessions for Contributed Papers - Buchanan Algebra and Logic: Room 100 Analysis and Applied Mathematics: Room 102 Analysis: Room 104

254 PROGRAM OF THE SESSIONS The time limit for each contributed paper is ten minutes. The contributed papers are scheduled at 15 minute intervals so that listeners can cir culate between the different sessions. To main tain this schedule, the time limit will be strictly enforced.

TUESDAY, 2:00P.M. Invited Address, The Auditorium Derivations and structure Professor Albert Nijenhuis, University of Washington

WEDNESDAY, 9:00 A.M. Invited Address, The Auditorium Hydrodynamic stability - a study in applied mathematics Professor C. C. Lin, Massachusetts Institute of Technology

WEDNESDAY, 10:30 A.M. Session on Geometry, Room 100, Buchanan 10:30 - 10:40 ( 1) A geometric proof of the Bruchat double coset lemma Professor Robert Hermann, University of California, Berkeley (592-9) 10:45 - 10:55 (2) The equality of Haantjes-Finsler and geodesic curvature of a curve in Rieman nian space Mr. D. C. Kay, Michigan State University (592- 63) 11:00- 11:10 (3) On Chow bunches for projective varieties. Preliminary report Dr. W. L. Hoyt, Brandeis University (592-65) 11:15- 11:25 (4) On quadratic transformations and ordered neighborhoods Professor J. W. Kenelly, Jr., University of Southwestern Louisiana, and Professor W. R. Hutcherson*, University of Florida (592-17) 11 : 30 - 11: 40 (5) Semi-translation planes Professor T. G. Ostrom, Washington State University (592-14) 11:45 - 11:55 (6) A limiting surface for the polar zonohedron Dr. B. L. Chilton, University of Buffalo, and Professor H.S.M. Coxeter*, University of Toronto (592-26)

WEDNESDAY, 10:30 A.M. Session on Algebra, Room 102, Buchanan 10:30 - 10:40 (7) On minimal sets of generators of pure inseparable field extensions Professor P. T. Rygg, University of South Dakota (592-75) 10:45 - 10:55 (8) Hereditary orders. Preliminary report Dr. Manabu Harada, Northwestern University (592-35)

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

255 11:00 - 11:10 (9) Uniqueness of invariant Wedderburn factors Professor E.]. Taft, Yale University (592-12) 11:15- 11:25 (10) Subrings of the maximal ring of quotients associated with closure operations Professor D. C. Murdoch, University of British Columbia (592-39) 11:30 - 11:40 (11) Classes of rings in Boolean algebras. Preliminary report Professor C. H. Cunkle, Utah State University (592-43) 11:45 - 11:55 (12) Non-zero idempotents in certain rings Professor B. R. Toskey, Seattle University (592-27) 12:00 - 12: 10 (13) Non-associative algebras satisfying identities of degree three Professor Frank Kosier* and Professor J. M. Osborn, University of Wisconsin (592-90)

WEDNESDAY, 10:30 A.M. Session on the Theory of Numbers, Room 104, Buchanan 10:30 - 10:40 (14) Generalized Dirichlet products of arithmetic functions. Preliminary report Mr. A. A. Gioia and Professor M. V. Subba Rao*, University of Missouri (592-67) 10:45 - 10:55 (15) Cauchy product estimates of arithmetical functions Professor Eckford Cohen, University of Tennessee (592-48) 11:00- 11:10' (16) On certain multiplicatively recurring sequences Dr. S. W. Golomb, Jet Propulsion Laboratory, California Institute of Technology (592-38) 11:15- 11:25 (17) Some integral matrix equations. Preliminary report Professor]. H. Hodges, University of Colorado (592-74) 11:30 - 11:40 (18) Concerning real numbers whose powers have nonintegral differences Professor Herman]. Cohen* and Professor Fred Supnick, The City Col lege of New York (592-57) 11:45 - 11:55 (19) Polynomials defined by generalized powers Professor Gloria Olive, Anderson College (592-55)

WEDNESDAY, 10:30 A.M. Session on Analysis, Room 106, Buchanan 10:30 - 10:40 (20) Integral representation of semi-norms on Euclidean n-space Professor E. K. McLachlan, Oklahoma State University (592- 53) 10:45 - 10:55 (21) An order-theoretic property of the weak topology in sequence spaces Dr. A. L. Peressini, University of Illinois (592 -40) 11:00- 11:10 (22) On completion and completeness of B(C )-spaces Professor Taqdir Husain, University of Ottawa (592-28) 11:15 - 11:25 (23) Interpolation spaces and interpolation methods Professor N. Aronszajn, University of Kansas, and Professor E. Gagliardo*, University of Genoa and University of Kansas (592-52)

256 11:30 - 11:40 (24) Homogeneous algebras on locally compact abelian groups Professor A. B. Willcox, Amherst College (592- 84)

THURSDAY, 9:00 A.M. Invited Address, The Auditorium Models of complete theories Professor R. L. Vaught, University of California, Berkeley

THURSDAY, 10:30 A.M. Session on Topology, Room 100, Buchanan 10:30 - 10:40 (25) Extended topology: Continuity II Professor P. C. Hammer, University of Wisconsin (592-29) 10:45 - 10:55 (26) A property of plane curves Professor Steve Armentrout, University of Iowa (592-61) 11:00 - 11:10 (27) A characterization of the composants of compact indecomposable continua Professor G. W. Henderson, University of Virginia (592-79) 11:15 - 11:25 (28) The inner and outer T set function in semigroups Professor P. M. Swingle, University of Miami (592-46) 11:30 - 11:40 (29) The max-chord structure of planar convex bodies Professor J. G. Cedar, University of California, Santa Barbara (592-2) 11:45 - 11:55 (30) Note on topological tensor products. Preliminary report Professor H. R. Fischer, Montana State College (592-69)

THURSDAY, 10:30 A.M. Session on Algebra, Room 102, Buchanan 10:30 - 10:40 (31) Adjoint geometry of linear transformations Professor Ali R. Amir-Moez, University of Florida (592-15) 10:45 - 10:55 (32) Inertia theorems for matrices: The semi-definite case Mr. D. H. Carlson and Professor Hans Schneider*, University of Wisconsin (592-25) 11:00- 11:10 (33) On some matrix equations Mr. P. Basavappa and Professor N. Kimura*, University of Saskatche wan (592-87) 11:15- 11:25 (34) Cyclic matrices Professor A. L. Dulmage* and Professor N. S. Mendelsohn, University of Manitoba (592-5) 11:30 - 11:40 (35) An algorithm related to the optimal assignment problem Professor A. L. Dulmage and Professor N. S. Mendelsohn*, University of Manitoba (592 -4) 11:45- 11:55 (36) Every planar nine point graph has a nonplanar complement Mr.J. Battle, Professor F. Harary, and Dr. Y. Kodama*, University of Michigan (592-45)

257 THURSDAY, 10:30 A.M. Session on Probability and Statistics, Room 104, Buchanan 10:30 - 10:40 (37) Boundaries for 17'"-continuous Markov chains and representations of excessive functions Professor S. C. Moy, Syracuse University (592-64) 10:45 - 10:55 (38) A sojourn density process of Brownian notion Professor F. B. Knight, University of Minnesota (592-56) 11:00 - 11:10 (39) Class D supermartingales Professor L. L. Helms* and Professor Guy Johnson, University of Illinois (592-24) 11:15- 11:25 (40) On adding stochastic processes Mr. T. S. Pitcher, Lincoln Laboratory, Massachusetts Institute of Tech nology (592-33) 11:30 - 11:40 (41) A condition for absolute continuity of infinitely divisible distribution functions Professor Marek Fisz* and Dr. V. S. Varadarajan, New York University (592-37) 11:45 - 11:55 (42) Sufficient conditions for fixed length confidence intervals in two stages Professor J. I. Rosenblatt, University of New Mexico and Sandia Corpo ration, Albuquerque, New Mexico (592-41) 12:00 - 12:10 (43) Recurrent events and fluctuation theory Mr. S.C. Port, The RAND Corporation, Santa Monica, California (592-80) 12:15 - 12:25 (44) Probabilities of independent events. Preliminary report Professor H. M. Lieberstein* and Professor D. E. Myers, University of Arizona (592-94)

THURSDAY, 2:00P.M. Invited Address, The Auditorium On semigroups in analysis and geometry Professor Charles Loewner, Stanford University

THURSDAY, 3:30P.M. Session on Algebra, Room 100, Buchanan 3:30 - 3:40 (45) The symmetrizer subgroup of the general linear group Dr. P. F. G. Stanek, Institute for Defense Analyses, Princeton, New Jersey (592-11) 3:45 - 3:55 (46) The product of the orders of certain centralizers in a finite group. Preliminary report Professor J. S. Frame*, Michigan State University, and Dr. Olaf Tamaschke, Universitat Tiibingen (592- 83) 4:00 - 4:10 (47) Inverse limits of solvable groups Mr. Ethan Balker, Harvard University (592-72)

258 4:15 - 4:25 (48) High extensions of Abelian groups Professor D. K. Harrison, New Mexico State University and University of Pennsylvania, and Professor J. M. Irwin, Miss C. L. Peercy, and Professor E. A. Walker*, New Mexico State University (592-66) 4:30 - 4:40 (49) A generalization of a theorem of F¢lner Dr. A. H. Frey, Jr., International Business Machines Corporation, Beth esda, Maryland (592-13) 4:45 - 4:55 (50) Fixes in finite semigroups and the Isbell property Professor Roy Leipnik and Professor Henryk Mine*, University of Florida (592-23) 5:00 - 5: 10 (51) Semigroups in which all subsemigroups are one-sided ideals Professor T. Tamura and Mr. R. B. Merkel*, University of California, Davis (592 -77) 5:15 - 5:25 (52) Semigroups whose proper subsemigroups are left ideals Professor T. Tamura, University of California, Davis (592-76)

THURSDAY, 3:30P.M. General Session, Room 102, Buchanan 3:30 - 3:40 (53) A doubly stochastic matrix equivalent to a given matrix Professor J. E. Maxfield* and Professor Henryk Mine, University of Florida (592-81) 3:45 - 3:55 (54) On best doubly stochastic estimates Mr. Richard Sinkhorn, Boeing Airplane Company, Wichita, Kansas (592-34) 4:00 - 4:10 (55) Further results on the Kantorovich inequality Professor Marvin Marcus, University of California, Santa Barbara, and Professor A. H. Cayford*, University of British Columbia (592-51) 4:15 - 4:25 (56) Equivalent classes of difference equations Professor W. A. Harris, Jr., University of Minnesota (592-86) 4:30 - 4:40 (57) Synthesis of n-port networks by a matrix Richards' theorem Mrs. E. K. Boyce, Rensselaer Polytechnic Institute, Professor R. J. Duffin, Carnegie Institute of Technology, Professor D. Hazony*, and Professor H. J. Nain, Case Institute of Technology (592-20) 4:45 - 4:55 (58) A general dual theorem for convex programs with convex constraints Professor Abraham Charnes*,Northwestern University, Professor W. W. Cooper, Carnegie Institute of Technology, and Mr. W. Kortanek, North western University (592-7) 5:00 - 5:10 (59) An extended simplex method Dr. Philip Wolfe, The RAND Corporation, Santa Monica, California (592-78) 5:15 - 5:25 (60) A note on the values of large games Dr. Irwin Mann* and Dr. L. S. Shapley, The RAND Corporation, Santa Monica, California (592- 92)

259 THURSDAY, 3:30P.M. Session on Analysis, Room 104, Buchanan 3:30 - 3:40 (61) Tangential limits of functions from the class Sa. Dr. J. R. Kinney, Lincoln Laboratory, Massachusetts Institute of Tech nology (592-31) 3:45 - 3:55 (62) Asymptotic values of holomorphic functions Professor G. R. MacLane, Rice University (592-60) 4:00 - 4:10 (63) The uniqueness of functions harmonic in the interior of the unit disc Professor V, L. Shapiro, University of Oregon (592-16) 4:15 - 4:25 (64) A convolution product for discrete analytic functions Professor R. J, Duffin, Carnegie Institute of Technology, and Professor C. S, Duris*, Michigan State University (592-22) 4:30 - 4:40 (65) On substitution theorems for the Lebesgue-Stieltjes integral and the Stieltjes mean sigma integral Dr. Marvin Mundt*, Valparaiso University, and Professor F. M. Wright, Iowa State University (592-89) 4:45 - 4:55 (66) Some problems in nonlinear Volterra integral equations Professor J. A. Nobel, University of Wisconsin (592-2 1) 5:00 - 5:10 (67) Higher and iterated Hausdorff derivatives Professor R, F. Rinehart*, Case Institute of Technology, and Professor J, C. Wilson, Florida Presbyterian College (592-10) 5:15 - 5:25 (68) Geodesics and Lebesgue area Professor Edward Silverman, Purdue University (592- 50)

FRIDAY, 9:00A.M. Invited Address, The Auditorium Representations of lattices R, S, Pierce, University of Washington

FRIDAY, 2:00P.M. Session on Algebra, Logic and Foundations, Room 100 Buchanan 2:00 - 2:10 (69) Relativization of a primitive recursive hierarchy Professor Paul Axt, Michigan State University (592-18) 2:15- 2:25 (70) The solvability of machine mappings of regular sets to regular sets Dr. Seymour Ginsburg and Mr. T. N. Hibbard*, System Development Corporation, Santa Monica, California (592-8) 2:30 - 2:40 (71) The star-height of regular events. Preliminary report Professor L. C. Eggan, University of Michigan (592-47) 2:45 - 2:55 (72) Self-dual structures, Preliminary report Dr, W. R. Hare, Jr., Duke University (592-88) 3:00 - 3:10 (73) Jordan-Holder theorems Professor C. E. Watts, University of Rochester (592-68)

260 3:15 - 3:25 (74) The existence of free unions in classes of abstract algebras Professor Gloria C. Hewitt, Montana State University (592-1) 3:30 - 3:40 (75) Finite free lattices Professor E. A. Nordhaus* and Mr. Paul Pennock, Michigan State Uni versity (592-54) 3:45 - 3:55 (76) Counting reflexive, symmetric and transitive relations on a finite set. Prelim inary report Captain J. R. Perkins, USAF Academy (592-82)

FRIDAY, 2:00P.M. Session on Analysis and Applied Mathematics, Room 102, Buchanan 2:00 - 2: 10 (77) The effect of radiation pressure and oblateness on the equatorial orbit of an earth satellite Dr. J. A. Morrison, Bell Telephone Laboratories, Murray Hill, New Jersey (592-3) 2:15 - 2:25 (78) Expansion coefficient of wave fronts in continuum mechanics Professor R. P. Kanwal, Pennsylvania State University (592-19) 2:30 - 2:40 (79) On the solution of axial-symmetric blast waves in fluid dynamics from differen tial-geometric aspects Dr. G. M. Schindler, General Electric Company, Santa Barbara, Califor nia (592-49) 2:45 - 2:55 (80) Approximate inversion of a class of Laplace transforms with application to supersonic flow problems Mr. Y. L. Luke, Midwest Research Institute, Kansas City, Missouri (592-30) 3:00 - 3:10 (81) The 'heat' equation in n-space and certain related functions Professor L. R. Bragg, Case Institute of Technology (592-71) 3:15 - 3:25 (82) A generalized boundary value problem for a hyperbolic partial differential equation Dr. James Conlan, U. S. Naval Ordnance Laboratory, Silver Spring, Maryland (592-59) 3:30 - 3:40 (83) Closed form solutions of a second order linear ordinary differential equation with n-regular singular points Professor H. L. Crowson, International Business Machines Corporation, Bethesda, Maryland (592-44) 3:45 - 3:55 (84) On the series solution of ordinary simultaneous equations Professor Stephen Kulik, Long Beach State College (592-36) 4:00 - 4: 10 (85) On an iterative technique for algebraic systems. Preliminary report Mr. Charles Bryan*, University of Arizona (592-93) (Introduced by Professor H. M. Lieberstein)

261 FRIDAY, 2:00P.M. Session on Analysis, Room 104, Buchanan 2:00 - 2: 10 (86) Pointwise divergence of approximating polynomials Professor P. C. Curtis, Jr., University of California, Los Angeles (592-73) 2:15 - 2:25 (87) Optimum norms for data smoothing Dr. j. R. Rice* and Dr. J. S, White, General Motors Research Labora tories, Warren, Michigan (592-70) 2:30 - 2:40 (88) On invariant subspaces for a class of one-parameter group of operators Dr. R. T. Harris, Duke University (592-32) 2:45 - 2:55 (89) An adjoint ergodic theorem Dr. S. P. Lloyd, Bell Telephone Laboratories, Murray Hill, New jersey (592- 6) 3:00 - 3:10 (90) A note on strongly positive operators Mr. W. S. Eberly, Washington State University (592-85) 3:15 - 3:25 (91) Quasi-positive operators Dr. D. W. Sasser, Sandia Corporation. Albuquerque, New Mexico (592-58) 3:30 - 3:40 (92) A spectral mapping theorem for functions ~f two commutative linear operators Dr. Anthony Trampus, General Electric Company, Santa Barbara, Cali fornia (592-91) R. S. Pierce Seattle, Washington Associate Secretary

NEWS ITEMS AND ANNOUNCEMENTS

THE BRITISH MATHEMATICAL through july 11, 1963. The symposium is COLLOQUIUM will hold its 15th Annual being organized by the Association for Meeting at the Royal Military College of Symbolic Logic, which is asking the Inter Science, Shrivenham on September 3rd- national Union for the History and Philo sophy of Science to serve as co-sponsor. 7th, 1963. Further information and appli Supporting funds are being requested to cation forms may be obtained from the enable The Organizing Committee to pro Secretary, Dr. L. W. Longdon, Royal Mili tary College of Science, Shrivenham, vide travel expenses for invited speakers. If possible, one or more sessions Swinden, Wiltshire. for contributed papers will also be sche duled. It is hoped that younger logicians, including students, will be encouraged to AN INTERNATIONAL SYMPOSIUM attend, and The Organizing Committee ON THE THEORY OF MODELS is planned hopes to find some funds to furnish partial at Berkeley, California, from june 25 support for a few of these.

262 PRELIMINARY ANNOUNCEMENT OF MEETING

Five Hundred Ninety-Third Meeting Dartmouth College Hanover, New Hampshire October 27, 1962

The five hundred ninety-third meet Green Lantern Inn, Main Street, Han ing of the American Mathematical Society over. Two blocks from center of town, will be held at Dartmouth College on Sat four blocks from Bradley Center. urday, October 27, 1962, It is probable that Phone 643-3410. all sessions will be held in the Albert Norwich Inn and Motel, Norwich, Ver Bradley Center for Mathematics. All times mont. One mile from Hanover. given below are Daylight Saving Time. Phone 649-1143. By invitation of the Committee to Select Hour Speakers for Eastern Sectional Chieftain Motel, Lyme Road, Hanover. Meetings, Dr. M. F. Atiyah will address Two miles north of Hanover on Route the Society at 2:00 P.M. in the Lincoln 10. Filene Auditorium, which is Room 101 Phone 643-2550, Bradley. The title of his address is Sunset Motel, Hanover Road, West "Topology and linear algebra". Lebanon, New Hampshire. Three miles There will be sessions for contri south of Hanover on Route 10. buted papers at 10:00 A.M. and at 3:15 Phone 298-8721. P.M. Abstracts of papers should be sent to the American Mathematical Society, 190 MEALS Hope Street, Providence 6, Rhode Island The noon meal will be available at so as to arrive prior to the deadline of the college dining hall. September 12, 1962. The registration desk will be on the TRAVEL first floor of Bradley. It will beopenfrom 9:00A.M. to 3:30P.M. Coatandconversa Hanover is on New Hampshire tion space will be nearby on the same Route 10. The natural highway approach floor. from the south is U. S. Route 5. From Boston, the suggested approach is via ROOMS Routes 3 and 4 through Concord, New Each traveler should make his own Hampshire to Lebanon, New Hampshire reservation of rooms. A list of inns and and then Route 120 to Hanover. Hanover is motels concludes this section. In seeking 264 miles from New York bycarandabout reservations at Hanover Inn, please make 135 miles from Boston. clear that you are attending the meeting of The Northeast Airlines give ser this Society. vice from Boston and from New York to the Lebanon Airport about five miles from Hanover Inn, Hanover, Center of town. Hanover. There is taxi and bus service Owned and operated byDartmouthCol from the airport to Hanover. lege, 25 rooms reserved, will be re There is limited rail service on the leased on October 1 if not reserved by Boston and Maine Railroad from Boston to individuals by that time. White River Junction, Vermont, which is Phone 643-4300, about 5 miles from Hanover. There is bus Hanover Inn Motor Lodge, Lebanon service from Boston to White River Junc Street, Hanover. One block from center tion. There is local bus service from White of town, three blocks from Bradley Cen River Junction to Hanover. ter. The Albert Bradley Center for Phone 643-4400. Mathematics is two blocks north of the

263 center of Hanover, directly behind Baker meeting will appear in the October issue of Library on the north side of the campus. the NOTICES. Baker Library is said to resemble a mag Everett Pitcher nified Independence Hall. Associate Secretary Further details of the program of the Bethlehem, Pennsylvania

NOTICE TO MEMBERS

At the next annual meeting in Berk planning to submit a ten minute paper eley, it is expected to hold in addition to which he believes to be of special interest the sessions for ten minute papers, five and suitable for expansion so as to fit into sessions for somewhat longer papers in a one of these sessions, is invited to submit selection of fields. These sessions will his usual abstract early enough so that it consist of about five papers of about can be considered by the committee in twenty minutes each. These fields, to charge. Therefore, anyone interested in gether with the person in charge of the having his paper so considered should sub session, are: General Topology: R. H. mit his abstract by November 9 instead of Bing; Analysis, especially Asymptotic the regular deadline date of November Z3. Methods and Singular Perturbations: A. Abstracts received by November 9 will be Erdelyi; Functions of One or Several Com considered by the selection committee for plex Variables: Halsey Royden; Algebraic inclusion in the special sessions. Topology: E. H. Spanier; Algebra: L. j. The restriction on regular ten-min Paige. These papers will be partly by ute papers to the figure of two hundred is special invitation and partly by selecting the same as in 196Z. from the contributed papers. A member

You can examine the following recent Ginn and Company titles at Booth No. 13 at the summer meeting in Vancouver: ORDINARY DIFFERENTIAL EQUATIONS by Garrett Birkhoff, Harvard University and Gian-Carlo Rota, Massachusetts Institute of Technology AN INTRODUCTION TO MATHEMATICAL STATISTICS by H. D. Brunk, University of California, Riverside ANALYTIC FUNCTION THEORY, VOLUMES ONE AND TWO by Einar Hille, Yale University MATHEMATICAL MODELS IN THE SOCIAL SCIENCES by John G. Kemeny and J. Laurie Snell, both of Dartmouth College INTRODUCTORY COLLEGE MATHEMATICS, THIRD EDITION by William E. Milne, Emeritus, Oregon State College and David R. Davis, East Carolina College ELEMENTS OF LINEAR ALGEBRA by Lowell J. Paige and J. Dean Swift, both of Princeton University

For further information write GINN AND COMPANY, College Department Back Bay P .0. Box 191, Boston I 7, Mass. Sales Offices: New York II Chicago 6 Atlanta 5 Dallas I Palo Alto Toronto 16

264 ACTIVITIES OF OTHER ASSOCIATIONS

VANCOUVER, B. C. CANADA- August 27-31, 1962

THE MATHEMATICAL Columbia, Vancouver, B. C. Canada, from ASSOCIATION OF AMERICA Monday, August 27 to Wednesday, August 31, 1962, in conjunction with summer meet The forty-third summer meeting of ings of the American Mathematical Society the Mathematical Association of America and the Society for Industrial and Applied will be held at the University of British Mathematics.

FIRST SESSION, MONDAY: 9:00 a.m. The Auditorium 9:00 - 10:00 The Earle Raymond Hedrick Lectures: The Coordinate Problem, Lecture I Professor A. M. Gleason, Harvard University 10:10 - 11:00 Survey of Mathematics Programs in Institutions Granting Bachelor's and Higher Degrees Dr. C. B. Lindquist, Office ofEducation,U. S. Department of Health, Education and Welfare, Washington, D. C. 11:10 - 12:10 The Problem of Staffing College Departments of Mathematics Professor A. W. Tucker, Princeton University (A brief talk followed by general discussion)

SECOND SESSION, MONDAY: 2:00 p.m. The Auditorium 2:00 - 3:00 The Earle Raymond Hedrick Lectures: Lecture II Professor A. M. Gleason

SESSION ON TEACHING MACHINES

3:10- 3:50 Research Problems and Findings in Programmed Instruction Dr. L. M. Stolurow, Training Research Laboratory, Urbana, illinois 4:00 - 4:40 The Role of Programmed Material in the Teaching of Mathematics Dr. J, E. Forbes, Director of Research in Mathematics, Britannica! Center for Studies in Learning and Motivation, Palo Alto, California

THIRD SESSION, TUESDAY: 9:00a.m. The Autidorium 9:00 - 10:00 The Earle Raymond Hedrick Lectures: Lecture III Professor A. M. Gleason 10:05 - 10:30 Business Meeting of the Association

SESSION ON INEQUALITIES

10:30 - 11:00 To Introduce Inequalities: Two Principles Professor George Polya, Stanford University 11:05 - 11:35 Some Inequalities in the Differential Geometry of Surfaces Professor E. F. Beckenbach, University of California, Los Angeles

11:40 - 12:10 Some Methods for Proving Matrix Inequalities Professor M. D. Marcus, University of California, Santa Barbara

FOURTH SESSION, WEDNESDAY: 2:00p.m. The Auditorium 2:00 - 2:40 A Report on the International Congress of Mathematicians at Stockholm Professor V. L. Klee, Jr., University of Washington

265 SESSION ON DIFFERENTIAL EQUATIONS 2:50 - 3:30 The Sturm-Liouville Problem Professor Hans Sagan, University of Idaho 3:35 - 4:15 Symmetric Differential Operators Professor R. W. McKelvey, University of Colorado 4:20 - 5:00 Stability and Asymptotic Behavior Professor Fred Brauer, University of Wisconsin

FILM PROGRAM- The Auditorium Monday, 4:55 p.m. - 6:00 p.m. "The Kakeya Problem" (in color and with animation) by Professor A. S. Besicovitch Monday, 7:30 p.m. - 8;30 p,m, "What Is an Integral?" (a kinescope) by Professor Edwin Hewitt Tuesday, 3:15 p.m. - 4;33 p,m, "Theory of Limits" by Professor E. J. McShane Tuesday, 7:00p.m. - 8:00p.m. "Mathematical Induction" (in color) by Professor Leon Henkin Tuesday, 8:00 p.m. - 9:15 p.m. "The Kakeya Problem" (in color and with animation) by Professor A. S. Besicovitch (a repeat showing: this film is shown twice since it is the only one not shown previously at an Association meeting)

PROGRAM COMMITTEE M. D. MARCUS (Chairman) R. A. BEAUMONT LEO MOSER IVAN NIVEN PATRICK SUPPES L. B. WILLIAMS

PI MU EPSILON The Auditorium 10:15 - Wednesday: Professor S. P. Di Pi Mu Epsilon will hold a Dutch liberto, University of California, Ber treat breakfast meeting on Tuesday at keley: "Application of Transformation 8:00 A.M. in the University Residence Theorems to Solution Expansion Prob "Common Block" .All members and friends lems" of Pi Mu Epsilon are invited to attend this meeting. 10:15 - Thursday: Professor H. B. Mann, Ohio State University: "The Addition Theorems of Group Theory and Their Applications" 3:15 - Thursday: Dr. Frank Proschan, Boeing Scientific Research Labora THE SOCIETY FOR INDUSTRIAL tories: "Probabilistic Models in Relia AND bility Theory" APPLIED MATHEMATICS Sessions for Contributed Papers The Society for Industrial and Ap will be held in the Buchanan Building on plied Mathematics will meetfrom Wednes Friday, August 31, at IO:ZO A.M. in Rooms day, August Z9 to Friday, August 31. 100 and IOZ.

266 NEWS ITEMS AND ANNOUNCEMENTS

A. Koranyi (Berkeley): Matrix-valued A CONFERENCE ON AUTOMOR factors of automorphy PHIC FUNCTIONS OF SEVERAL COM S. Bochner (Princeton): Almost periodic PLEX VARUBLES AND CONNECTED objects on manifolds TOPICS was held at Stanford University S. Kobayashi (British Columbia): On on june lZ-15, 196Z. The program included automorphisms of almost complex the following lectures: structures A. Selberg(Institute for Advanced Study): H. Royden (Stanford): Remark on enve On the evaluation of the trace formula lopes of holomorphy for analytic automorphic forms for a j. Sladkowska (Lodz, Poland): Bounds of non-compact fundamental domain. analytic functions in domains with the S. Bergman (Stanford): Value distribu distinguished boundary tion of meromorphic functions of sev H. Cohn (Arizona): Some pseudo-con eral variables formal mapping properties of Hilbert S. Chern(Berkeley):Holomorphic curves modular functions in the field of /3 in a Grassmann manifold R. Langlands (Princeton): The problem W. Stoll (Notre Dame): Construction of of calculating the dimension of spaces jacobian and Abelian functions with the of automorphic forms Kneser integral Sessions were under the chairman M. Schiffer (Stanford): Automorphic ship of Charles Loewner, George Polya, functions and conformal mapping Ralph S. Phillips, Hans Samelson and K. Hoffman (M.I.T.): The minimum Robert Finn (all from Stanford) and jan boundary for analytic polyhedron G. Vander Corput(Berkeley). The confer R. Gunning (Princeton): Differential op ence was arranged by Stefan Bergman and erators and modular forms S. Chern.

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267 PERSONAL ITEMS

Assistant Professor ALEXANDER BHARGAVA of the State Teachers College, ABIAN of the University of Pennsylvania Elizabeth City, North Carolina, has been has been appointed to an associate pro appointed to an assistant professorship at fessorship at The Ohio State University. Kent State University. Professor B. H. ARNOLD of Oregon Mr. H. J. BIESTERFELDT, JR. of State University will be on leave of ab Pennsylvania State University has been sence in 196Z-1963 and has received a appointed to an assistant professorship at Fulbright grant to lecture at the National Lebanon Valley College. University of Taiwan in Taipei, Taiwan, Dr. G. R. BLAKLEY of Harvard Uni Republic of China. versity has been appointed to an assistant Dr. E. F. ASSMUS, JR. of Columbia professorship at the University of lllinois. University has been appointed Lecturer at Mr. D. M. BLOOM of Harvard Univer Wesleyan University. sity has been appointed to an associate Dr. M. F. A TIY AH of the University professorship at the University of Massa of Oxford, Oxford, England has been elec chusetts. ted a Fellow of the Royal Society. Dr. E. K. BLUM of Space Technology Mr. LOUIS AUSLANDER of Yale Uni Laboratories has been appointed to a pro versity has been appointed to a professor fessorship and Director of the Comput&r ship at Purdue University. Laboratory at Wesleyan University. Mr. DOV A VIS HAL OM of Minnesota Mr. L. M. BOSTICK of CONVAIR has University has been appointed to an assist accepted a position as Senior Mathemati ant professorship at St. Thomas College. cian with General Electric Company, Syra Dean W. L. AYRES of Purdue Univer cuse, New York. sity has been appointed Vice President Associate Professor T. A. BOTTS on and Provost at Southern Methodist Univer leave from the University of Virginia has sity. been appointed to a visiting professorship Mr. D. G. BABBITT of the University at the University of Puerto Rico. of Michigan has been appointed to an as Professor J. W. BOWER on leave sistant professorship at the University of from Connecticut College will spend the California, Los Angeles. academic year 196Z-1963 atthe University Dr. N. W. BAZLEY of the National of Miami. Bureau of Standards has accepted a po Mr. A. S. BRAMSON of Sylvania sition as Research Mathematician at the Electric Products, Incorporated has ac Institute Battelle, Geneva, Switzerland. cepted a position as Staff Programmer Assistant Professor H. S. BEAR, JR. with International Business Machines Cor of the University of Washington has been poration, Cambridge, Massachusetts. appointed to an associate professorship at Professor EUGENIO CALABI on leave the University of California,Santa Barbara. from the University of Minnesota will Associate Professor JOHN BENDER spend the academic year 196Z-1963 at the of Lafayette College has been appointed University of Pisa, Italy. to an associate professorship at Rutgers, Mr. J. C. CANTRELL of the University The State University, Newark, New Jersey. of Tennessee has been appointed to an Mr. R. J. BENICE of Sylvania Elec assistant professorship at the University tronic Systems has accepted a position as of Georgia. Senior Associate Mathematician with In Mr. G. T. CROCKER of Auburn Uni ternational Business Machines Corpora versity has been appointed to an associate tion, Rockville, Maryland. profe~sorship at the University of Southern Dr. M. P. BERRI of the University of Mississippi. California has been appointed to an assist Mr. C. G. CULLEN of Case Institute ant professorship at Tulane University. of Technology has been appointed to an Associate Professor TRILOKI N. assistant professorship at the University

268 of Pittsburgh. Professor WALL ACE GIVENS of Dr. R. B. DARST of Massachusetts Northwestern University has accepted a Institute of Technology has been appointed position as Associate Director of the to an assistant professorship at Purdue Applied Mathematics Division at Argonne University. National Laboratory, Argonne, lllinois. Dr. W. F. DAVIS ON of Space General Mr. E. H. GREENE of the University Corporation has accepted a position as of Virginia has been appointed to an assist Manager, Quantum Electro-Optical Branch ant professorship at Beloit College. Research and Development Department Professor HANS RAJ GUPTA of Panjab with Texas Instruments, Incorporated. University, India, will spend the academic Professor JOHN DECICCO of DePaul year 196Z-1963 as LecturerattheUniver University has been appointed to a pro sity of Colorado. fessorship at lllinois Institute of Techno Professor P. C. HAMMER on leave logy. from the University of Wisconsin, will Assistant Professor A. E. DOLO of spend the year 196Z-1963 at the University Columbia University has been appointed of California, LaJolla. to a professorship at the Universitat Professor PETER HENRICI of the Zurich, Zurich, Switzerland. University of California, Los Angeles has Mr. F. E. DRISTY of Florida State been appointed to a professorship at the University has been appointed to an assist Eidgenossische Technische Hochschule, ant professorship at the Florida Presby Zurich, Switzerland. terian College. Dr. JOSEPH HERSHENOV of Massa Dr. S. I. DROBNIES of William Marsh chusetts Institute of Technology has been Rice University has accepted a position as appointed to a professorship at Queens Senior Operations Analyst with General College. Dynamics Corporation, Fort Worth, Texas. Mr. YUKITOSHI HINOHARA of the Dr. P. L. DUREN of Stanford Univer Tokyo Metropolitan University has been sity has been appointed to a professorship appointed to an assistant professorship at the University of Michigan. at the Kanto Gakuin University, Yokohama, Dr. LEON EHRENPREIS of the Insti Japan. tute for Advanced Study has been appointed Associate Professor L.A. HOSTINSKY to a professorship at Courant Institute, of Pennsylvania State University has been New York University. appointed to a professorship at Connecticut Associate Professor CARL FAITH of College. Pennsylvania State University has been ap Assistant Professor J. E. HOULE, JR. pointed to a professorship at Rutgers, The of Georgetown University has been ap State University. pointed to an associate professorship at Dr. R. S. FREEMAN of the University Seton Hall University. of California, Livermore has been appoint Associate Professor LEVI HOWARD ed to a professorship at the University of of Columbia University has been appointed Maryland. to a professorship at Hunter College. Assistant Professor W. F. FREIBER Dr. R. E. HUGHS of Lehigh University GER of Brown University has been award has accepted a position as Staff Mathe ed a Guggenheim fellowship and will matician with the Sandia Corporation, spend the academic year 196Z-1963 at the Albuquerque, New Mexico. University of Stockholm, Sweden. Professor F. B. JONES of the Univer Mr. E. R. GENTILE of the Institute sity of North Carolina has been appointed for Advanced Study has been appointed to a professorship at the University of Research Associate at the Instituto de California, Riverside. Matematica Universidad del Sur, Bahia Mr. T. F. KIMES of Westinghouse Blanca, Argentina. Electric Corporation has been appointed Assistant Professor R. c. GILBERT to an assistant professorship and chair of the University of California, Riverside man of the Mathematics Department at is on sabbatical leave at the U. S. Army Austin College, Research Center, University of Wisconsin. Dr. W. A. KIRK of the University of

269 Missouri has been appointed to an assist Assistant Professor S. S. MITRA of ant professorship at the University of the University of Idaho has been appointed California, Riverside. to an assistant professorship at the Uni Professor S. C. KLEENE of the Uni versity of Arizona. versity of Wisconsin has been appointed Assistant Professor KOTARO Chairman of the Numerical Analysis De OIKAWA of Hiroshima University has been partment of the University of Wisconsin appointed to an assistant professorship at for the year 1962-1963. the University of Tokyo, Tokyo, Japan. Professor ERWIN KLEINFELD of Mr. SA TIO OKADA of General Dynam Ohio State University has been appointed ics/Electronics has accepted apositionas to a professorship at Syracuse University. Staff Engineer with the Mitre Corporation, Mr. R. G. LAATSCH oftheUniversity Bedford, Massachusetts. of Tulsa has been appointed to an assist Dr. NARASIMHACHARI P ADMA has ant professorship at Miami University. returned to Annamalai University, India Mr. M. W. LODATO of Rutgers, The after a leave of absence at Connecticut State University has accepted a position as College. Scientist with the Laboratory for Electro Mrs. E. J. PITTS of the University of nics, Incorporated, Monterey, California. Alabama has been appointed to an assist Assistant Professor P. E. LONG of ant professorship at the Georgia Institute the Southern Illinois University has been of Technology. appointed to an assistant professorship at Dr. R. W. RANDALL has been assigned Oklahoma State University. as Operations Evaluation Group field re Mr. FUMI-YUKI MAEDA of the Uni presentative to the U. S. Navy Seventh versity of Illinois has been appointed to a Fleet. research assistant at Hiroshima Univer Professor R. F. RINEHART of Case sity, Hiroshima, Japan. Institute of Technology has accepted a po Mr. S. W. MALINOWSKI of General sition as Director of Research of the Dynamics Corporation has accepted a po Weapons Systems Evaluation Group of the sition as Research Engineer with Sylvania Department of Defense, Washington, D. C. Electronic Systems, Waltham, Massachu Mr. R. W. RITCHIE of Dartmouth setts. College has been appointed to an assistant Assistant Professor D. G. MALM of professorship at the University of Wash the State University of New York has been ington. appointed to a professorship at Michigan Professor D. C. RUSSELL of Mt. State University, Oakland. Allison University, Canada has been ap Associate Professor M. D. MARCUS pointed to a professorship and chairman of the University of British Columbia, of the Department of Mathematics at York Canada has been appointed to a professor University, Canada. ship at the University of California, Santa Professor CHARLES SALTZER of the Barbara. University of Cincinnati has been appointed Dr. R. E. MESSICK of the California to a professorship at The Ohio State Uni Institute of Technology will be on leave of versity. absence in 1962-1963 and has received a Professor A. T. SCHAFER of Connec Fulbright grant at the University of Sydney, ticut College has been appointed to a pro Sydney, Australia. fessorship at Wellesley College. Dr. D. V. MEYER has been appointed Assistant Professor E. C. SCHLE a Lecturer at the State University of Iowa. SINGER of Wesleyan University has been Professor K. S. MILLER on leave appointed to an assistant professorship at from New York University for the aca Connecticut College. demic year 1962-1963, has beenappointed Associate Professor L. L. SCOTT of to a visiting professorship of engineering Southwestern at Memphis has been ap mathematics at Columbia University. pointed to a professorship at the Univer Dr. W. L. MISER of Lincoln, Massa sity of Louisville. chusetts has has been appointed to a pro Assistant Professor G. F. SIMMONS fessorship at Rose Polytechnic Institute. of Williams College has been appointed to

270 an associate professorship at Colorado JIM DOUGLAS, JR., William Marsh College. Rice University, to a professor ship. Associate Professor PAUL SLEPIAN C. H. DOWKER, Birkbeck College, of the University of Arizona has been ap London, England, to a professorship. pointed to an associate professorship at Dr. L. L. GAVURIN, Brooklyn College, Rensselaer Polytechnic Institute. to an assistant professorship. Mr. M. H. SLUD of General Electric H. I. GROSS, Corning Community Col Company has accepted a position as Senior lege, to an associate professorship. Engineer with Radio Corporation of Amer MORISUKE HASUMI, Ibaraki Univer ica, Princeton, New jersey. sity, japan, to a Lecturer. Dr. j. R. SMARTofNewYorkUniver R. V. HOGG, State University of Iowa, sity has been appointed to an assistant to a professorship. professorship at the University of Wiscon S. P. HUGHART, Sacramento State sin. College, to a professorship. Dr. MITSUO SUGIURA of the Univer B. M. KIERNAN, JR., St. Peter's sity of Tokyo, Japan has been appointed College, to an assistant professorship. a Lecturer at Osaka University, japan. L. H. LANGE, San jose State College, Dr. j. D. THOMAS of Massachusetts to a professorship. Institute of Technology has been appointed CHING-HW A MENG,Sacramento State to an assistant professorship at New College, to an associate professorship. Mexico State University. j. P. NASH, Lockheed Missiles and Professor T. Y. THOMAS of Indiana Space Company, to Vice President of Re University has been appointed to a visiting search and Engineering. professorship at the University of Cali D. E. SOUTH, Florida Presbyterian fornia, La jolla. College, to a professorship. Professor W.j.THRONoftheUniver ROBERT STEINBERG, University of sity of Colorado will be on leave of absence California, to a professorship. in 1962-1963 and has received a Fulbright j. D. SWIFT, UniversityofCalifornia, grant to lecture at Punjab University, to a professorship. Chandigarh, India. H. G. TUCKER, University of Califor Professor MINORU TOMITA onleave nia, Riverside, to an associate professor from Okayama University, japan has been ship. appointed a visiting Lecturer at the State M. S. WEBSTER, Purdue University, University of Iowa. to a professorship. Associate Professor H. P. WIRTH of the University of Chicago has been ap The following appointments to instructor pointed to a professorship at the City Uni ships are announced: versity of New York. Dr. j. A. WOLF of the Institute for University of California, Riverside: Advanced Study has been appointed to an Dr. L. M. YOUNG; University of Chicago: assistant professorship at the University M.G. ROTHENBERG; Connecticut College: of California, at Berkeley. W. R. R. TRANSUE; Universityoflllinois: Dr. P. M. YOUNG of Kansas State D. R. SHERBERT; University of Massa University has been appointed Vice Presi chusetts: H. T. D' ALARCAO; Memphis dent for Academic Affairs at the Univer State University: L. E. DE NOYA; Pennsyl sity of Arkansas. vania State University: ANTON GLASER; Stanford University: Dr. W. F. POHL. The following promotions are announced: Deaths: A. G. AZPEITIA, University of Professor F. A. BEHREND of the Massachusetts, to a professorship. University of Melbourne, Victoria, Aus R. W. BAGLEY, University of Ala tralia died on May 27, 1962 at the age of 51. bama, to a professorship. Professor E. C. BLOM of Belmont GEORGE BURKE, State University of College died on February 16, 1962 at the Iowa, to an assistant professorship. age of 71.

271 Mr. F. L. LYNCH, JR. of Seton Hall University died on March 12, 1962 at the age of 42. Employment opportunities Professor R. L. SWAIN of Rutgers, The State University died on February 4, with PAN AM .1962 at the age of 49. at Cape Canaveral

Pan American, as prime contractor ERRATA to the U.S. Air Force, has fulfilled the The announcement on page 191 of the responsibility for planning, engineering, June issue of the NOTICES concerning Dr. operating and maintaining Cape Canav GERTRUDE BLANCH should read as fol eral and the Atlantic Missile Range since lows: 1953. Dr. GERTRUDE BLANCH has been Pan American staff members have appointed a Senior Scientist in the Depart participated intimately in all stages of ment of Defense. She remains at her posi the national programs since early V-2 tion at the Air Force Aeronautical Re tests through Project Mercury's series of search Laboratories, Wright-Patterson manned space flights. Air Force Base, Ohio. Significant opportunities for senior mathematicians and mathematical statisticians exist in the following areas: The announcement on page 191 of the June issue of the NOTICES concerning Time series and their application Mr. MAREK FISZ should read as follows: Numerical Analysis Dr. MAREK FISZ of the Warsaw Uni Design of Experiments versity, Warsaw, Poland has been appoint Regression Analysis ed to a professorship at New York Univer Stochastic Processes sity. Celestial Mechanics Trajectory Analysis Operations Analysis The announcement on page 191 of the June issue of the NOTICES concerning These positions will require senior Mr. S. S. HOLLAND should read as follows: mathematicians and mathematical statisti Dr. S. S. HOLLAND of the National cians possessing MS or PhD degrees. Academy of Sciences-National Research In addition to normal company Council has been appointed to an assistant benefits, Pan Am offers the unique professorship at Boston College. advantage of a 90% world-wide air travel discount. The announcement on page 193 of the You are invited to send your resume June issue of the NOTICES concerning in confidence to David P~ Bruner, Person Mr. N. R. STANLEY should read as follows: nel Manager, Guided Missiles Range Division, Pan American World Airways, Dr. N. R. STANLEY of Sperry Rand Inc., P.O. Box 4336, Patrick Air Force Corporation has accepted a position as Base, Florida. Senior Mathematical Physicist with The Perkin-Elmer Corporation, Norwalk, Con An Equal Opportunity Employer. necticut.

~- GUIDED MISSILES RANGE DIVISION IQ- i>'i· i} t . ,.~....:!!••... CAPE CANAVERAL, FLORIDA

272 NEW AMS PUBLICATIONS

Announcing a Translation of the Chinese SELECTED TRANSLATIONS Journal IN MATHEMATICAL STATISTICS AND PROBABILITY, VOLUME ll ACT A MA THEMATICA SINICA 19 Papers on Statistics and Probability Under the title CHINESE MATHE Z4Z pages; List Price $4.80; ZS% discount MA TICS - ACT A, the American Mathemat to members of IMS and AMS. ical Society, with the support of the Na The Institute of Mathematical Sta tional Science Foundation, will publish a tistics and the American Mathematical cover-to-cover translation into English Society jointly announce the publication of of Acta Mathematica Sinica, which is the Volume II of the series of books SELEC official journal, containing research on TED TRANSLATIONS IN MATHEMATI pure and applied mathematics, of the Insti CAL STATISTICS AND PROBABILITY. tute of Mathematics, the Academy of Sci This volume contains nineteen papers on ences, Peking. The translation is directed mathematical statistics and probability. by the Joint Committee on Translations of Volume I, containing ZS papers, the American Mathematical Society and was published in February, 1961. These the Institute of Mathematical Statistics. translations, made under a grant from the Under present conditions very few National Science Foundation, are published copies of the original are available except for the Institute of Mathematical Statistics in China, and since most Western mathe by the American Mathematical Society, maticians do not read the Chinese lan under the direction of a Joint AMS-IMS guage, the Society is publishing an English Committee on Translations. edition of the full journal. Our translation begins with Volume SELECTED TRANSLATIONS 10, for the year 1960, of the original, SERIES II, VOLUME ZO which in that year had only 3 issues, al though in all other years it has been a Six Papers on Partial Differential quarterly. Our issues will correspond Equations by B. M. Levitan, M.A. Naimark, exactly to those of the original, the trans 0. A. Ladyzenskaya, A. I. Koselev, E. M. lation for the 1960 volume being numbered Landis, and M. I. Visik. Volume 1. The number of pages in the 364 pages; List Price $5.30; ZS% discount translation will be about 500 a year. to members. Volume 1, Number 1 of CHINESE MATHEMATICS- ACTA was published TRANSLATIONS - SERIES I in July 196Z. The List Price is $ZO.OO per year; price to members $1Z.OO. Orders Selected Translations, Series I, may be placed with the American Mathe first published during 1949 -19 54 as 10 5 matical Society. pamphlets, has been arranged by subject and republished in hard covers averaging MATHEMATICAL REVIEWS 500 pages per volume. Series I was ori ginally translated and published under a All back issues of MATHEMATICAL grant from ONR; the new edition has been REVIEWS are again available in complete made possible through a grant from the volumes, beginning with Volume 1 (1940). NSF. Volumes 8, 9, and 17 have been reprinted Volumes 1, Z, 3, and 4 have been in bound volumes. issued in July, 196Z. Volumes 5 through List price per volume 1-16 (1940-1955) 11 are scheduled for publication in the $4Z.OO fall, 196Z. List price per volume 17-ZZ (1955-1961) $50.00 List Price per volume, $5.00; ZS% discount AMS members receive Z5o/o discount. to members.

273 LETTERS TO THE EDITOR

Editor, the NOTICES of the new degree could be tailored to fit the science backgrounds of those who The short article, An Alternative think that a doctorate in mathematics is Doctor of Arts, by Professor S. G. to the just what they need in their field. Professor Richard R. Gold Ghurye and The April 1962 Report of the Mathe provides the ingredient for a syn berg, matical Sciences Section of the National of two problems and their solutions, thesis Register of Scientific and Technical Per problems are the providing of col These sonnel on "Professional Characteristics mathematics teachers and the pro lege of Mathematical Scientists", states: viding of mathematical specialists in ap plications. "One possible explanation is the The fact that the proposal for a increasing application of mathemat Doctor of Arts degree in mathematics is ics to other branches of science. As receiving serious consideration is an in the other sciences--social as well dication of forces at work. Professor as natural- -become more highly de Melvin Henriksen's letter in the June issue veloped and precise, they tend to points out that any definite action taken by use more and more refined mathe the Society should be well defined and matical techniques. Thus some carefully thought through. With due respect mathematical sophistication is nec to his fears, I wish to make a case for the essary not only for the physicist and granting of a new degree, and one bearing engineer, but for the economist, the the title of "Doctor." I propose: biologist, and the experimental psy chologist as well. This need for a American Mathematical (1) that the mathematical background in the sci colleges and univer Society encourage ences has sent enrollments in math a program leading to the sities to offer ematics courses up faster than in of Mathematics; degree of Doctor any other area." (2) that the initiation, scope and heri degree be provided by the tage of the Although it might seem in prin and that it further encourage, as AMS, ciple," ... not at all necessary that a person requirement in common among those a have a 'doctorate' of some kind to become may be stipulated by the schools which a qualified college teacher of mathemat individually, AMS certification by ics," there is evidence of a different trend means of national (or international) in the recent book Compensation on the comprehensive examinations. Ca,!!l_E_Us, ed. by J. F. Wellemeyer, Jr. Whether or not anything is proposed (National Education Association; Wash other than a change from "Arts" to "Math ington, D. C., 1961), p. 340. In a salary ematics" depends perhaps on the point of scale chart effective September 19 59 at the view. To Leonard Gillman the Doctor of thirteen institutions forming the California Mathematics degree could be what is State College System, Professors, Asso awarded in place of Ph.D. candidacy; to ciate Professors, Assistant Professors, Melvin Henriksen the AMS certification and Instructors are divided into as a safeguard that the new re could act Class I - College De.gree to Doctorate not become a joke. Actually, quirements Class II - Doctorate or Equivalent the change of title is motivated from two directions. The first concerns an indirect In industry there are similar discrimina effect; although the new title raises its tions. Like it or not, the doctorate is a own questions, it avoids explaining in the durable status symbol. future why the Doctor of Arts is granted Certification by the AMS, in addition only in mathematics. The second concerns to having the advantages listed by Profes the distinction between the A.B. and the sor Ghurye and Professor Goldberg, would B.S. degrees; the general requirements make it possible for many colleges who do

274 not now offer doctoral programs in mathe "Many mathematicians have advised matics to do so, thereby training large undergraduates to drop out of ROTC pro• numbers of just those students most in grams rather than be faced with a serious clined to go into teaching, There would be interruption of their graduate training. I the satisfaction of meeting a national can also testify from personal experience standard for certification, carrying with that students who are commissioned at the it the impulse to continue to participate. end of their senior year find themselves A Doctor of Mathematics could pursue, with a serious handicap in competing for with an indeterminate handicap, mathe graduate fellowships, since it is known that matical research on the same basis as a the students would be called to active duty Ph, D. in no more than two years. "At the same time I should like to Gerald W. Kimble call to your attention that all three of the Armed Services continually beg academic institutions for more Ph.D.'s in mathemat Editor, the NOTICES ics for their research installations. If they could be persuaded to allow ROTC mathe The Committee on the Undergradu maticians to complete their work towards ate Program in Mathematics (CUPM) the Ph.D. before being called to active learned some time ago that U. S. Army duty, this might go a long waytowardsre policy did not allow ROTC students re lieving the shortage of competent mathe ceiving their bachelor's degrees to be de maticians in the Armed Services." ferred from undertaking their active duty commitments in order to continue their A reply from Carlisle P. Runge, studies, except for students in "medicine, Assistant Secretary of Defense (Manpower) dentistry, law (in certain cases), osteo on April Z6 indicated that there had been pathy, veterinary medicine, and theology." a change in policy, so on May 1 Professor A letter prepared by the Committee was Kemeny requested answers to two ques sent to appropriate authorities in arguing tions: the case that the acute shortage of mathe "1, Am I correct in understanding maticians should be realised, and that from your letter that a suitably qualified qualified ROTC students should be deferred candidate could be deferred until he has for graduate study in mathematics, How completed his Ph.D, in mathematics? ever, a separate exchange of letters be Naturally, I would understand that such tween Professor john G. Kemeny (of deferment would be subject to there not CUPM's Panel on Teacher Training) and being a critical shortage of military offi some officials in the Department of De cers, and that in any case he would be fense have been most illuminating, obligated to fulfill his military duties at On April 16 Professor Kemeny the completion of his graduate training. wrote the Secretary of Defense, Robert If I am correct in so interpreting you, this McNamara and presented the case for a would be very good news to many of us. change in policy as follows: "Z. Would you permit me to circu "As several Government surveys late your letter amongst mathematicians? have shown in recent years, the shortage I know that a vast majority of them have no of mathematicians is more acute than that idea that deferment to the Ph.D. is a possi of any other field of specialization. Since bility." during the last decade we have averaged The following answer was obtained: only ZSO Ph.D.'s in mathematics a year, even a small fraction of this force diverted "Dear Mr. Kemeny: due to ROTC commitments can have a seriously detrimental effect. A three-year This is in reply to your letter of interruption in the graduate training of May 1, to the Assistant Secretary of De such students will put them under a serious fense (Manpower) which has been referred disadvantage when they return to graduate to this office for a direct reply. school, and often it means that they are In reply to your first question the discouraged from completing their train answer is yes provided the individual who ing. is granted a delay fulfills his tour of active

275 military duty prior to the expiration of his requirements. · military obligation. Sincerely yours, There is no objection in circulating the Assistant Secretary of Defense (Man (Signed) LYNN D. SMITH power) letter of Apri126, addressed to you, Brigadier General, U. S. A. however, it should be kept in mind that such Acting Director, Reserve delay authority is contingent upon military Affairs and Readiness Plans"

NEWS ITEMS AND ANNOUNCEMENTS

THE MATHEMATISCHES FOR F. Norguet (Strasbourg): Application de SCHUNGSINSTITUT OBERWOLFACH re la theorie des residus ports that the March 26-30, 1962 Confer 0. Forster (Munchen): Funktionswerte ence on Complex Analysis, under the chair als Randintegrale in komplexen Rau manship of H. Grauert (Gottingen), R. men Remmert (Erlangen), and K. Stein (Mun N. Kuhlmann (Wurzburg): Uber die Auf chen), was attended by many foreign losung der Singularitaten 3-dimen mathematicians from England, France, sionaler komplexer Raume Italy, the Netherlands, Switzerland, and the W. Rothstein Munster): Analogon eines United States. Most of the lectures and Satzes von Hartogs bei analytischen discussions were in English. Mengen The following lectures were presented: E. Calabi (Minneapolis): Inclusion and vanishing theorems in compact mani R. Remmert (Erlangen): Uber homogene folds kompakte komplexe Mannigfaltig M. Atiyah (Oxford): Some remarks on keiten harmonic forms H. Holmann (Munster): Quotienten kom U. Hirzebruch (Munster): Halbraume plexer Raume nach komplexen Trans und Holomorphismengruppen formationsgruppen G. Scheja (Munster): Uberhomologische H. Kerner (Gottingen): Approximation Codimension in komplexen Raumen holomorpher Abbildungen in homo K. Spallek(Munster):Verallgemeinerung gene komplexe Mannigfaitigkeiten eines Satzes von Hartogs-Osgood fur H. G. Tillmann (Heidelberg): Distribu Funktionen auf Serreschen komplexen tionen als "Randwerte" holomorpher Raumen Funktionen A. Pfister ( Munchen): Uber das Koef T. van de Ven (Leiden): Drei Bemer fizientenproblem der beschrankten kungen uber homogene komplexe holomorphen Funktionen von zwei Mannigfaltigkeiten Veranderlichen P. Dolbeault (Malakoff): Classes d'hom H. Rohrl (Minneapolis): Uber das Rie ologie de certains cycles analytiques mann-Privalovsche Randwertproblem K. J. Ramspott (Munchen): Bemerkungen Automorphiefaktoren uber Rungesche Paare K. Konigsberger (Munchen) :Systeme von W. Thimm (Bonn): Luckengarben von Automorphiefaktoren koharenten analytischen Modulgarben

276 MEMORANDA TO MEMBERS

THE NATIONAL REGISTER OF the cost involved in handling anonymous SCIENTIFIC AND TECHNICAL listings. PERSONNEL job applicants and employers who wish to be listed will please write to the The Mathematical and Statistical Employment Register, 190 Hope Street, Sciences Section of the Register will main Providence 6, Rhode Island, for applica tain a desk during the Summer Meeting at tion forms or for position description the University of British Columbia at forms. These forms must be completed Vancouver, on August 28, Z9, and 30, The and returned to Providence not later than National Register Desk will be located in August 1, 1962, in order to be included Room 12Z1 of the Buchanan Building, The free of charge in the listings at the Sum attendants will be pleased to assist with mer Meeting in Vancouver. Forms which registrations and to supply information. arrive after this closing date, but before The National Register as a whole is a August 10, will be included in the register responsibility of the Nation·al Science at the meeting for a late registration fee Foundation. The Mathematical and Statisti of $3,00, and will also be included in the cal Sciences Section is operated by the printed listings, but not until ten days American Mathematical Society with the after the meeting. The printed listings cooperation of the Association for Com will be available for distribution both puting Machinery, the Association for during and after the meeting. Symbolic Logic, the Econometric Society, It is essential that applicants and the Industrial Mathematics Society, the employers register at the Employment Institute of Mathematical Statistics, the Register Desk promptly upon arrival at Mathematical Association of America, the the meeting to facilitate the arrangement Operations Research Society of America, of appointments. the Society for Industrial and Applied Mathematics, the Society of Actuaries, and the American Statistical Association, THE COMBINED MEMBERSHIP LIST 1962-1963 THE EMPLOYMENT REGISTER Members are advised thatthe dead The following item is repeated from line for changes of listings in the forth the june 1962 issue of the NOTICES and coming issue of the COMBINED MEMBER gives a more detailed time schedule, SHIP LIST is October 1, If you were listed Th,e Mathematical Sciences Employ incorrectly in the 1961-1962 Combined ment Register, established by the Ameri Membership List, or if you have changed can Mathematical Society, the Mathemati any part of your listing since October 1, cal Association of America, and the Society 1961, please send the information re for Industrial and Applied Mathematics, quested below to reach the Headquarters will be maintained at the Summer Meeting Offices by October 1, 1962. Please note at the University of British Columbia at the following explanations of listings: Vancouver, on August 28, 29, and 30, 1962, Students are listed at the college or uni The Register will be conducted from 9:00 versity without a job title; two concurrent A.M. to 5:00 P.M. on each of these three permanent positions are separated by a days in Room 1221 of the Buchanan Build semicolon: a position held while on leave ing. is shown as a separate listing; in all There is no charge for registering cases "Mathematics Department" is under either to job applicants or to employers, stood unless otherwise specified. except when the late registration fee for To insure accurate listing, please employers is applicable. Provision will be give us the following information: name in made for anonymity of applicants upon re full and highest earned degree; place of quest and upon payment of $3,00 to defray employment and job title (please give us

277 the complete business name and address); temporary job title, if you will be on leave, including the name and address of the employer and the duration of the tem porary job; an indication of secondary em TECH NEWS ployment, if you will hold two concurrent positions, including job title, business name and address; and finally, your mail for Scientists, Mathematicians ing address. Operations Evaluation Group

CORPORATE MEMBERS One of our analysts has returned As of June 20, 1962 the following from field assignment with the were supporting the Society through Cor fleet and told us a significant im· porate Memberships. provement resulted when one of his recommendations was put into Academic Press practice during fleet maneuvers. Aerospace Corporation OEG's field activities, assigned on a Bell Telephone Laboratories, Incorpo rotational basis, represent unique travel opportunities for rated scientists and mathematicians. There are E. I. duPont de Nemours andCompany, OEG men with the fleet in the Incorporated Mediterranean, the Far East, Ha· Eastman Kodak Company waii, Key West, Norfolk, and San Ford Motor Company Diego, and field representatives in Newport, R. I. and London, General Motors Corporation England. Hughes Aircraft Company OEG's technical management has been transferred to the International Business Machines Corpo- Frankl in Institute. We will operate as a part of the new ration Center of Naval Analyses, in a role that promises to be Lockheed Missiles and Space Company broader that our former one. Having just celebrated its Pergamon Press, Limited 20th anniversary of work for the U. S. Navy, OEG looks forward to an even more productive future. Procter and Gamble Company Radio Corporation of America "For 20 years, the Navy has consistently been the first of Remington-Rand UNIVAC the services to foresee the opportunities for operations RIAS research and the requirements on its part to assure its Shell Development Company success"-Dr. Jacinto Steinhardt, OEG director, at the OEG Socony Mobil Oil Company, Incorporated Vicennial conference. Space Technology Laboratories, Incor- OEG provides scientific analysis in diverse problem areas porated of Naval operations, including nuclear warfare, air, sub Standard Oil Company, Incorporated, marine, and anti-submarine warfare, logistics, and strategic New Jersey planning. OEG's present expansion has created a need for United Gas Corporation scientists, mathematicians. economists, and engineers with advanced degrees to fill career United States Steel Corporation positions whose potential is as outstanding as their challenge. Imaginative, enter prising scientists thrive on the complex problem-solving they do at OEG ... assignments that often involve impor· tant contributions to our national purpose. The positions THE AUSTRALIAN MATHEMATI are well paid and carry comprehensive peripheral benefits. CAL SOCIETY announced that annual mem Please send your resume to Dr. Frank Bothwell, Chief bership dues under the reciprocity agree Scientist, Center of Naval Analyses. ment with the American Mathematical So ciety are $6.00andnot $7.50 as previously announced. OES CENTER OF NAVAL ANALYSES Arlington Towers, Arlington 9, Virginia An equal opportunity employer.

278 BACKLOG OF MATHEMATICAL RESEARCH JOUKNAL::i

Information on this important matter is be but not yet accepted are being ignored.) ing published twice a year, in the February and the Column 4. Estimated by the editors (or the August issues of the NOTICES, with the kind co Editorial Department of the American Mathemati operation of the respective editorial boards. cal Society in the case of the Society's journals) It is important that the reader should inter and based on these factors; Manuscripts accepted, pret the data with full allowance for the wide and manuscripts received and under consideration, sometimes meaningless fluctuations which are manuscripts in galley, and rate of publication. characteristic of them. Waiting times in particular There is no fixed formula. are affected by many transient effects, which Column 5. The first quartile (Q 1) and the arise in part from the refereeing system. Ex third quartile ( Q3) are presented to give a measure treme waiting times as observed from the pub of the dispersion which will not be too much dis lished dates of receipt of manuscripts may be torted by meaningless extreme values. The median very misleauing, and for that reason, no data on (Med.) is used as the measure of location. The extremes are presented in the table at the bottom observations were made from the latest issue re of this page. ceived in the Headquarters Offices before the dead Some of the columns in the table are not line date for this issue oftheNOTICES. The wait quite self-explanatory, and here are some further ing times were measured by counting the months details on how the figures were computed. from receipt of manuscript in final revised form, Column 2.. These numbers are rounded off to month in which the issue was received at the to the nearest 50. Headquarters Offices. It should benotedthatwhen Column 3. For each journal, this is the a paper is revised, the waiting time between re estimate as of the indicated dates, of the total num ceipt by editors of the final revision and its publi ber of printed pages which will have been accepted cation may be much shorter than is the case for a by the next time that manuscripts are to be sent paper which is not revised, so these figures are to the printer, but which nevertheless will not be to that extent distorted on the low side. sent to the printer at that time. (Pages received

1 2 3 4 5 Estimated No. Approx. Backlog current Observed waiting time Issues no. pages 6/1/62. 11/30/61 waiting in latest issue per published time Q Med. Q JOURNAL year per year pages pages months months months months American ]. of Math. 4 NR* NR* NR* NR* 7 8 10 Annals of Math. 6 12.00 NR* 800 12 8 12 12 Annals of Math. Statistics 4 1500 100 0 15 6 8 9 Arch.Hist.Exact Sciences not fixed 200 0 NR* 8 5 7 8 Arch.Rational Mech.Anal. not fixed 1000 0 NR* 5 6 6 8 Canadian J. of Math. 4 700 275 350 12 12 15 19 Duke Math. J. 4 700 350 150 14 12 12 14 Illinois J. of Math. 4 700 425 300 16-18 14 15 16 ]. Math. Analyses Appl. 8 1000 500 NR* 6 ** ** ** ]. of Math. and Mechanics 6 1000 NR* NR* 10 ** ** ** ]. of Math. and Physics 4 350 NR* 120 9 7 8 12 J. of Mathematical Physics 6 1350 652 0 6 9 10 12. Michigan Math. ]. 4 400 70 40 6 7 9 9 Pacific J. of Math. 4 1400 600 1000 12. 16 17 19 Proceedings of the AMS 6 1000 900 900 14 13 14 14

Quarterly of Appl. Math. 4 NR* NR* NR* NR* 8 10 11 SIAM Journal 4 800-900 200 60 9-12 9 11 12. SIAM Review 4 400 0 0 9-12 5 6 8 Transactions of the AMS 12 2200 1500 850 13 11 11 12 • NR means that no response was recelVed to a request for information. * • Dates of receipt of manuscripts not indicated in this journal.

279 SUPPLEMENTARY PROGRAM-NO. 12

During the interval from April ZB, 196Z through june Z7, 196Z, the papers listed below were accepted by the American Mathematical Society for presentation by title. Readers may wish to refer to Page 713 of the November, 1960 issue {No. 49 of these NOTICES) where it is explained in detail that the presentation of papers by title is now dissociated from meetings of the Society. After each title on this program is an identifying number. The abstract of the paper will be found following the same number in the section on Abstracts of Contributed papers in this issue of these NOTICES. {I) C{S) spaces of P). type {10) Normal functions, Montel'sproperty, Professor Dan Amir, Hebrew Uni and interpolation in H versity {6ZT-194) Professor G. T. Cargo, Syracuse {Introduced by Professor Branko University {6ZT-Z35) Grunbaum) {11) Fractional order differences of the {Z) On a class of linear difference-inte- coefficients of polynomials gral equations Dr. G. T. Cargo and Dr. Oved Dr. P. M. Anse1one and Dr. Donald Shisha, National Bureau of Stand Greenspan, Mathematics Research ards, Washington, D.C. {6ZT-Z41) Center, U. S. Army, Madison, Wis {lZ) Problems in linked operators. I. consin {6ZT-ZS9) Professor Robert Carroll, Rutgers, {3) Some higher-order cohomology oper- The State University {6ZT-ZS3) ations {13) Entropies of several sets of real Professor W. D. Barcus, State Uni valued functions. Preliminary report versity of New York, Long Island Df. G. F. Clements, Syracuse Uni Center {6ZT-ZSZ) versity {6ZT-Z44) {4) On finite groups whose generators are {14) Linear and quadratic equations in a subgroup generators. II. Preliminary Galois field with applications to geo report metry Professor H. F. Bechtell, Lebanon Professor Eckford Cohen, Univer Valley College {6ZT-189) sity of Tennessee {6ZT-Zl0) {S) Meromorphic minimal surfaces {15) Arithmetical notes, XII. A sequel to Professor E. F. Beckenbach and note VI. Mr. G. A. Hutchison, University of Professor Eckford Cohen, Univer California, Los Angeles {6ZT-Z68) sity of Tennessee (6ZT-ZZ6) {6) Some results oninfinitevaluedpredi ( 16) Arithmetical notes, XIII. A sequel to cate logic. Preliminary report note IV. Mr. L. P. Beliuce, University of Professor Eckford Cohen, Univer California, Berkeley {6ZT-Z66) sity of Tennessee (6ZT-Z37) {7) Optimality in diophantine program- (17). Existence of stable payoff configura- ming tions for cooperative games Dr. Adi Ben-Israel and Professor Professor Morton Davis, Princeton Abraham Charnes, Northwestern University and Mr. Michael Masch University {6ZT-Z46) ler, Hebrew University (6ZT-ZOZ) (B) An arc is tame in 3-space if andonly (18) The degree of a rationalmatrixfunc- if it is strongly cellular tion Professor R. H. Bing, University of Professor R. J. Duffin, Carnegie Wisconsin and Dr. A. Kirkor, Uni Institute of Technology and Pro versity of Warsaw {6ZT-Zll) fessor Donald Hazony, Case Insti {9) The segmental variation of Blaschke tute of Technology (6ZT-ZlZ) products (19) The behavior of complex algebraic Professor G. T. Cargo, Syracuse varieties as integral currents University {6ZT-Z36) Professor Herbert Federer, Brown

280 University (62T-215) (32) A geometric remark concerning the (20) Provable well-orderings of and re bounded symmetric domains lations between predicative and rami Dr. Robert Hermann, University of fied analysis California, Berkeley (62T-191) Professor Solomon Feferman, Stan (33) A unified description of the symmet ford University (62T-206) ric bounded domains (21) A Skolem-type normal form for lan Dr. Robert Hermann, University of guages with a generalized quantifier California, Berkeley (62T-200) Mr. Gebhard Fuhrken, University of (34) A vector space of products of deter- California, Berkeley (62T-198) minants (22) Noncharacterizability of the ordering Professor A. F. Hillman, Mr. D. W. of the natural numbers Forslund, and Mr. G. j. Giaccai, Mr. Gebhard Fuhrken and Professor University of Santa Clara (62T-186) R. L. Vaught, University of Califor (35) Differential ideals of Wronskians nia, Berkeley (62T-199) Professor A. P. Hillman, Mr. D. W. (23) The cohomology structure of a ring Forslund, and Mr. G. j. Giaccai, Professor Murray Gerstenhaber, University of Santa Clara (62T-187) Institute for Defense Analyses and (36) Integral representations of the direct University of Pennsylvania (62T- product of groups 242) Mr. Alfredo jones, University of (24) Strongly divergent fourier series in lllinois (62T-196) FK-spaces (37) The number of indecomposable inte Professor G. W. Goes, University of gral representations Western Ontario (62T-245) Mr. Alfredo jones, University of (25) Integral inequalities for subadditive lllinois (b2T-197) functions (38) An extension of the Picard-Vessiot Professor R. P. Gosselin, Univer theory sity of Connecticut (62T-228) Professor H.F.Kreimer, Yale Uni (26) On amenable semigroups with a finite versity (62T-220) dimensional set of invariant means (39) The foundation for an extension of Professor E. E. Granirer, The differential algebra Hebrew University of jerusalem, Professor H. F. Kreimer, Yale Uni Israel (62T-240) versity (62T-221) (Introduced by Dr. Harry Kesten) (40) Mahlo' s operation and the existence (27) A note on entire functions and a con- of a.-complete prime ideals jecture of ErdCSs. II Dr. H. j. Keisler, Institute for Mr. Alfred Gray, University of Cali Defense Analyses, Princeton, New fornia, Los Angeles and Professor jersey (62T-257) S. M. Shah, University of Kansas (41) The equivalence of certain problems (62T-193) in set theory with problems in the (28) A classification of ordinal recursive theory of models functions Dr. H. j. Keisler, Institute for Mr. j. R. Guard, Princeton Univer Defense Analyses, Princeton, New sity (62T-255) jersey (62T-258) (29) Hierarchies of recursive arithmetics (42) On operator-roots of an analytic Mr. j. R. Guard, Princeton Univer function sity (62T-256) Professor Svetozar Kurepa, Institut (30) Some structure in the hierarchy of za primijenjenu matematiku, Za maximal principles in set theory greb, Yugoslavia (62T-224) Mr. j. D. Halpern, Univers-ity of (43) A theorem about similarity of opera- California, Berkeley (62T-178) tors (31) Homogeneous Riemannian manifolds Professor Svetozar Kurepa, Institut of nonpositive curvature za primijenjenu matematiku, Za Dr. Robert Hermann, University of greb, Yugoslavia (62T-225) California, Berkeley ( 62T-254) (44) The module type of a ring. Prelimin-

281 ary report Dr. Israel Navot, Israel Institute of Professor W. G. Leavitt and Mr. Technology, Haifa, Israel (6ZT-Z39) R. E. Peinado, University of Neb (Introduced by Professor R. M. raska (6ZT-Z16) Foster) (45) On the subgroup topology of Marshall (57) A new proof of de Rham's decomposi Hall algebra and the theory of num tion theorem for a reducible Rieman bers nian manifold Professor joseph Lehner, Michigan Professor Katsumi Nomizu, Brown State University (6ZT-Z64) University (6ZT-Z65) (46) An unconditionally stable, explicit, (58) On a class of weakly alternative rings. two-level method for solving para Preliminary report bolic partial differential equations Mr. D. L. Outcalt, The Ohio State Dr. M.S. Lynn, California Research University (6ZT-Z50) Corporation, La Habra, California (59) Recursive equivalence types whose 6ZT-Z34) predecessors are well-ordered (Introduced by Dr. D. P. Squier) Dr. W. H. Richter, Rutgers, The (47) Do infinite creative sequences exist? State University (6ZT-Z18) Mr. T. G. McLaughlin, University of (60) A generalized associator identity. California, Los Angeles (6ZT-Z63) Preliminary report (48) Stability theorems for systems of non- Mr. D. J. Rodabaugh, Illinois Insti linear ordinary differential equations tute of Technology (6ZT-Zl3) Dr. 0. L. Mangasarian, Shell De (61) Multiple valued functions on groups velopment Company, Emeryville, Professor R. H. Rosen, University California (6ZT-Z3Z) of Michigan (6ZT-Z31) (49) Exponential analogues of a general- (6Z) The Grothendieck group of torsion ized Lambert series free abelian groups of finite rank Sr. Gregory Marie Meyer and Pro Professor joseph Rotman, Univer fessor Francis Regan, St. Louis sity of Illinois (6ZT-Z01) University (6ZT-Z33) (63) Dual of a Moore space. Preliminary (50) Hamiltonian semigroups report Professor D. W. Miller, University Dr. Prabir Roy, Institute for Ad of Nebraska (6ZT-Z48) vanced Study (6ZT-Z19) (51) Raising the differentiability class of (64) Multiple quadrature with backward a manifold in euclidean space differences Professor D. A. Moran, University Dr. H. E. Salzer, General Dy of Illinois (6ZT-Z17) namics/ Astronautics Corporation, (5Z) On a cubic congruence in three vari San Diego, California (6ZT-188) ables. II (65) On subplanes of free planes. Prelim- Professor L. j. Mordell, University inary report of Arizona (6ZT-Z14) Mr. R. I. Sandler, Institute for De (53) On anhomomorphic mappings between fense Analyses, Princeton, New algebraic systems jersey (6ZT-Z09) Mr. A. A. Mullin, University of (66) Discrete groups acting on nilpotent Illinois (6ZT-183) Lie groups (54) Isolated components and the associ Mr. j. C. Sanwal, Cornell Univer ated primes of an ideal sity (6ZT-Z49) Professor D. C. Murdoch, Harvard (67) A general interpolation theorem. Pre University (6ZT-Z51) liminary report (55) Connected spaces without infinite Professor Martin Schechter, New a--connected sets York University (6ZT-19Z) Professor jan Mycielski,University (68) The circles of curvature of levelloci of California, Berkeley and Institute and orthogonal trajectories of har of Mathematics of the Polish Aca monic functions demy of Sciences (6ZT-179) Mrs, D. B. Shaffer, Harvard Uni (56) The Euler-Maclaurin functional for versity (6ZT-Z08) functions with a quasi- step discontin (69) Lemniscate surfaces in Rn uity Mrs. D. B. Shaffer, Harvard Univer-

282 sity (62T-229) ington, D. C. (62T-180) (70) On the factor spaces of the complex (82) Products of indecomposable, aperio doubly stochastic matrices dic, stochastic matrices Mr. Richard Sinkhorn, The Boeing Professor Jacob Wolfowitz, Cornell Company, Wichita, Kansas (62T- University (62T-227) 243) (83) A pseud

283 ABSTRACTS OF CONTRIBUTED PAPERS

The Summer Meeting in. Vancouver, B.C. August 28-31, 1962

592-l. G. C. HEWITT, Montana State University, Missoula, Montana. The existence of free unions in classes of abstract algebras.

Let ot be a class of abstract algebras of the same type. Let fA 11 I~ E A.} be a subset of Ot. A system

g~: B -+Ail, where B E m_, then there is a unique homomorphism g: B -A such that g 't = h'lo g for all A. This concept is dual to that of free products studied by D. J. Christensen and R. S. Pierce. Theorem. Let Ot. be a class of abstract algebras of the same type with the following properties: (l) any homomorphic image of an algebra of ~is in cJt.; (2) any free product of algebras of a is in Of. Let fA~I-1 E A} be a subset of Cit. Suppose there is a c in a and a family of homomorphisms h~: C -AI\ onto. Then the free union of fAi\IA E AJ- exists. If we add the hypothesis (3) every algebra of ot contains a one element subalgebra, or (3') otcontains free algebras of arbitrarily large genera ting sets, then every subset of ()t has a free union. In fact, we can conclude that, Theorem: If CJt sat;isfies: (l) any homomorphic image of an algebra of tJt is in t1l'; (2) any subalgebra of an algebra of tr is in LYt'; (3) every algebra of cJt contains a one element subalgebra; then every subset of~ has a free union if and only if it has a free product. (Received January 2, 1962.)

592-2. J. G. CEDER, University of California, University, California. The max-chord structure of planar convex bodies.

A max-chord of a planar convex body C is a chord having maximum length in a given direction. For x E C define N(x) to be the number of max-chords of C which pass through x. Various results are obtained on the topological structure of the sets N - 1(k) fork= l, 2, .•• ,oo and on the structure of the

range of the function N. For example: Suppose N has finite range where m = max~ange N - {oo}]. Then, if N- 1(m) nN- 1(oo)= ¢,then m is odd and N- 1(m) is open. (Received February 26, 1962.)

592-3. J. A. MORRISON, Room 3D-278 Bell Telephone Laboratories, Murray Hill, New Jersey. The effect of radiation pressure and oblateness on the equatorial orbit of an earth satellite.

The effect on the equatorial orbit of a satellite of an oblate earth, which is subject to a radiation force in a fixed direction parallel to the orbital plane, is considered. The radiation force is modified in such a way that the equations of motion possess a first integral and reduce, upon elimination of the time, to a second order differential equation for the reciprocal of the distance of the satellite from the center of the earth. By suitably modifying the oblateness force this equation becomes a

nonhomogeneous Lam~'s equation of order 2, the solution of which may be derived in terms of Jacobi's elliptic and theta functions. An exact expression is obtained for the change in the reciprocal radius after n revolutions of the satellite, and this expression is examined in detail in the case when the

284 radiation and oblateness forces are of the same order of magnitude. When the radiation force is not fixed but rotates in the orbital plane, this may be taken into account in an approximate manner that leads to a nonhomogeneous Lam6's equation of nonintegral order for the reciprocal radius. (Received March 19, 1962.)

592-4. A. L. DULMAGE and N. S. MENDELSOHN, University of Manitoba, Winnipeg 19, Manitoba. An algorithm related to the optimal assignment problem.

The authors have recently given an algorithm for finding the canonical decomposition of a bipartite graph. This algorithm can be used to determine the dimension of the dual solution space of an optimal assignment problem. It can be used also to reduce the problem of finding all dual solutions of an optimal assignment problem to the problem of finding all dual solutions when there is exactly one primal solution. (Received March 19, 1962.)

592-5. A. L. DULMAGE and N. S. MENDELSOHN, University of Manitoba, Winnipeg 19, Manitoba. Cyclic matrices.

A directed graph D with vertex set V = (1,2,. •• , n) is cyclically d-partite if V can be partitioned

into V 1' V 2, ••• , Vd such that if the ordered pair (i,j) is an edge of D and if i E V s• j E V t• then t - s :: 1 mod d. The directed graph D A of an n by n matrix A is defined by agreeing that (i,j) is an edge of D A if and only if aij 'I 0. The following results are noted. If the directed graph D A of an n by n matrix A is cyclically d-partite then there is a polynomial f(x) and an integer a. ;::; 0 such that (1) the characteristic polynomial of A is f(xd)xa.: (2) the characteristic polynomial of Ad is ~(x)}dxa. and,

(3) there is a permutation matrix P such that P- 1 AdP = Diag. (Ap A2, ••• ,Ad) and the characteristic a.· polynomial of Ai is f(x)x 1 fori= 1,2, •.• ,d with L_a.i = a.. Also, there exists a polynomial g and an integer such that the minimal polynomial of A is g(xd)x;..; The following application is immediate. Let A be an irreducible nonnegative matrix with index of imprimitivity d. There is a permutation

matrix P such that p-1Adp = Diag. (A 1,A2, ... ,Ad) where the Ai are primitive and hav·e the same characteristic equation except for a factor which is a power of x. This generalizes a well known result. (Received March 19, 1962.)

592-6. S. P. LLOYD, Bell Telephone Laboratories, Murray Hill, New Jersey. An adjoint ergodic theorem.

Let P(t), 0 < t

limt ..... ooiiP(t>ll < oo. It is shown that the closed convex hull of [P*(t), 0 < t < oo} in the weak*

operator topology contains projections of norm !iii M onto the subspace of vectors invariant under

all [P *(t), 0 < t < ooJ. In an application, a Doeblin-like decomposition of arbitrary Markov processes is obtained. (Received May 28, 1962.)

285 592-7, ABRAHAM CHARNES, W. W. COOPER and K. KORTANEK, Room 346-A, Technological Institute, Northwestern University, Evanston, Illinois. A general dual theorem for convex programs with convex constraints,

To date the dual theorem of E. Eisenberg (Duality in homogeneous programming, Proc, Amer. Math, Soc, 12 (1961) 783-787) for the problem of minimizing a (first-degree) homogeneous convex function subject to general linear inequality constraints is the only generalization of the linear pro gramming theorem in which the dual problem does not involve primal optimizing variables, Based on a little-known work of Haar, A, Charnes, W, W. Cooper and K, Kortanek (Duality, Haar programs, and generalized finite sequence spaces, Proc, Nat, Acad, Sci, May 1962) have generalized the major theorems of linear programming to pairing of a finite dimensional vector space and a "generalized finite sequence space" (g,f.s,s,). These results are applied to yield a dual theorem associating as dual problems minimization of an arbitrary convex function over an arbitrary convex set in n-space with maximization of a linear function in non-negative variables of a g,f,s,s, subject to a finite system of linear equations, In principle, this result includes Eisenberg's and all other special results, such as those of Dennis, Dorn, Wolfe, and the interesting special nonlinear theorem of Duffin (Dual pro grams and minimum cost, J, Soc, Indust, Appl. Math, 1962), The above type of maximization is essentially computable by standard methods, (Received April 2, 1962,)

592-8, SEYMOUR GINSBURG and T. N, HIBBARD, 2500 Colorado Avenue, Santa Monica,

California, The solvability of machine mappings of r~gular sets to regular sets.

The following three results are shown, (1) It is recursively solvable to determine of arbitrary regular sets U and V whether or not there exists a complete sequential machine which maps U into V. (2) It is recursively solvable to determine of arbitrary regular sets U and V whether or not there exists a generalized sequential machine which maps U into V so that the image of U is infinite if U is infinite, (3) It is recursively solvable to determine of arbitrary regular.sets U and V whether or not there exists a complete sequential machine whiclt maps U onto V. The generalized sequential machine onto problem is unresolved, Results (1) and (2) are demonstrated by showing that only a preassigned number of machines need be examined to see if any one of them maps U into V, Result (3) is proved by exhibiting a recursive procedure which ultimately yields a decision, Unfortunately it is not known how long this procedure must be applied. (Received by April 16, 1962,)

592-9, ROBERT HERMANN, University of California, Berkeley, California, A geometric proof ·of the Bruhat double coset lemma,

Let G be a complex simple Lie group, let S be a maximal connected solvable subgroup, and let M = G/S, M is known to be a homogeneous compact Kaehler manifold, The lemma in question states that the number of orbits of S acting on the usual way on G/S is equal to the Euler character istic of M, It is shown that there is a real valued function on M with nondegenerate critical points having precisely one critical point on each of the orbits and that the orbit of S at each critical point is the stable manifold of the gradient field of the function at that critical point, In general, this method gives sufficient conditions that certain foliations with singularities have only a finite number

286 of leaves and proves fixed point theorems for certain foliations that seem to be different from those obtainable by purely topological arguments. (Received April 16, 1962.)

592-10. R. F. RINEHART, Case Institute of Technology, Cleveland 6, Ohio, and J. C. WILSON, Florida Presbyterian College, St. Petersburg, Florida. Higher and iterated Hausdorff derivatives.

Let A be a finite dimensional, linear associative algebra with identity over the real or complex field, F, with basis e 1, e 2, ••.• en• e 1 being the identity of A. A function f(§) with domain (open) and range in A is called H(ausdorff)-differentiable of order m, if the mth differential dm(f($); S' •...• s

592-ll. P. F. G. STANEK, Von Neumann Hall, Princeton, New jersey. The symmetrizer sub group of the general linear group.

Let K be a field with more than three elements. Let G be the group of all n by n invertible matrices with entries from K; letS be the subgroup of matrices of determinant one. For X, Y E G, define the symmetrizer of X and Y by [X,Y] = X Ty T XY, where X T = X transpose. Also define G* to be the subgroup of G generated by all symmetrizers; and, by induction, set G(k) = (G(k- 1))*. Let H

= n~ 1G (k). Some results are: (i) G* always contains S; (ii) if K is a finite field or 0, the field of rational numbers, then H = S; and (iii) if K = R, the field of real numbers, H = fX E Gldet X> 0} = G*.

A discussion of the group H, for fields K, Q!;;;; K ~ R, will be given. (Received April 19, 1962.)

592-12. E. j. TAFT, Yale University, New Haven, Connecticut. Uniqueness of invariant Wedderburn factors.

Let A .be a finite-dimensional associative algebra over a field of characteristic not two. Let R be the radical of A, and assume that A/R is separable. Let G be a finite group of automorphisms and antiautomorphisms of A, whose order is not a multiple of the characteristic of the ground field.

It is known that Gleaves fixed a maximal separable subalgebra of A (Invariant Wedd~_EI:n1rn___ !_actors, Illinois J. Math. 1 (1957), 565- 573). Theorem: Let S be a G-fixed maximal separable subalgebra of A. Then T is a G-fixed maximal separable subalgebra of A if and only if there exists an element x in R such that 1 - x is G-orthogonal (i.e., left fixed by the automorphisms in G and sent into its inverse by the antiautomorphisms in G) and the inner automorphism of A determined by conjugation by 1 - x maps S onto T. If the characteristic is zero, this automorphism may be written in the form exp(Ad z), where z is a G-symmetric (i.e., left fixed by the automorphisms in G and sent into its negative by the antiautomorphisms in G) element of R. This result generalizes the theorem announced in Abstract 588-8, Notices Amer. Math. Soc. 9 {1962), 28. (Received May 3, 1962.)

287 592-13. A. H. FREY, Jr., 10301 Westlake Drive, Rockville, Maryland. A generalization of a theorem of F¢lner.

A semigroup S is said to be right amenable if there is a positive linear functional p of norm on the space M(S) of bounded real-valued functions on S with the property that p(f) = p(fy) for all f E M(S) and y € S (fy(x) = f(xy) for all x E S). Theorem. A semigroup S is right amenable if and only if, for each finite subset [w 1, w 2, ••• ,wnJ of Sand positive real number e, there exists a finite subset

F of S and a non-negative, real-valued function 'Yon S such that 'Y (x) = 0 for x E: F 1 (F' denotes the complement of F) and Ef=l ~xE'SI?'(x) - 'Eywi=X?'(Y) I< e Lxes.Y(x). This characterization leads to a generalization of a theorem of F!lilner [Math. Scand. 3 (1955), 243-254]. Theorem. For a semigroup

S to be right amenable it is {sufficient} [necessary} that for each finite subset {w 1, w 2, ••• , w nJ of S and positive real number e, there exists a finite subset F of S such that {card (FwinF) '5: (1 - e)

.card (F)} {card (Fwi nF') <. (s) card (F~ fori= 1, 2, ••• ,n. This gives a characterization of right amenable right cancellation semigroups. The necessary condition is seen not to be sufficient for S to be right amenable by considering any set S with two or more members and defining xy = y for all x, y € S. The question of the necessity of the sufficient condition is unanswered. (Received July 2, 1 962.)

592-14. T. G. OSTROM, University of Washington, Pullman, Washington. Semi-translation planes.

This paper is an investigation of some of the questions in connection with a class of planes constructed by the author. Investigate the nature of possible extensions of this class; collineations moving f ; dualities, and some aspects of the coordinate system. A semi-translation plane means 00 a finite projective plane of order q2 such that (1) For every line L, the group of elations with axis

J. is at most of order q2. (2) For some line J00 , there is a set of q + 1 points, each of which is the

center of a group of elations of order q with f 00 as axis. (Received May 7, 1962.)

592-15. A. R. AMIR-MOEZ, 208 Walker Hall, University of Florida, Gainesville, Florida. Adjoint geometry of linear transformations.

Let Al'···•Ak be a set of linear transformations on a unitary space. Imposing certain conditions on Ai we get interesting equalities and inequalities among the singular values of Ai' all i. For example,

let At~ = K, i < j, and A1Ai = E, where Eisa projection of rank p. Let B = (1/k) ~Ai and

~ 1 l: ••. ;:; ~n be the singular values of B. Suppose p 1 e; ••• l!; Pn are the real singular values of K. Then il.~+j-n ;::;- 1/k + ((k - 1)/k)pi' i lii p and .\f+j+l 1i ((k - 1)/k)Jlj• i ,.. p, and i + j !lin+ 1. The hypothesis also implies two more inequalities. There are many theorems of this nature which are proved in this paper. (Received May 11, 1962.)

592-16. V. L. SHAPIRO, University of Oregon, Eugene, Oregon. The uniqueness of functions harmonic in the interior of the unit disc.

With llf(r ,S) II designating the sup norm on concentric circles of radius r, a theorem is estab lished which has the following theorem as a corollary. Theorem 1. Let f(r,8) be harmonic for

0 o:i r <. 1. Suppose that (i) for every 8, f(r,S) -o ~ r -1. (ii) l!f(r,8) II= o [(l - r) - 2]~ r ~ 1.

288 Then f(r,8) is identically zero, Theorem l is best possible in three different senses, i.e. (i) cannot be weakened by replacing "for every 8" by "for every 8 but one mod 21T", (ii) cannot be weakened either by replacing 0 [(l - r) - 2] by 0 al - r) - 2] or by replacing the sup norm on concentric circle of radius r by the L 1 norm on concentric circles of radius r. The main theorem established is the following: Theorem 2, Let f(r,8) be a function harmonic in the interior of the unit disc and let E be a countable set contained in the interval [0,21T). Suppose that (i) lif(r,8)il = o[(l - r)-2] ~ r- l, (ii) f*(8) and f*(8) are finite for 8 in [9,211")- E, (iii) f .(8) and f*(8) are both in L 1 on [0,2"11"),

(iv)(l- r) f(r,8) -o ~ r ~1 for 8 in E. Then f*(8) = f*(8) almost everywhere in [9,211") and f(r,8) = .,--l j;f1TP(r,8 - ¢)f*(¢)d¢. In the above, f*(8) designates the lim infr__..l of f(r,8), f*(8) designates the lim sup, and P(r,8) is the usual Poisson kernel. (Received May 14, 1962.)

592-17. J. W. KENELL Y, JR., University of Southwestern Louisiana, Lafayette, Louisiana and W. R. HUTCHERSON, University of Florida, Gainesville, Florida, On quadratic transformations and ordered neighborhoods,

Repeated applications of some quadratic transformations on certain algebraic curves passing through a singular point on a projective plane produce number patterns which suggest a more general truth. [YV. R, Hutcherson and J, W, Kenelly, Jr,, Three branch points on a surface in a space of ten dimensions, Revista, Matematica y Fisica Teorica (Tucuman) l3 (1960), 36-40: J, C. Morelock, N. C. Perry, and W. R, Hutcherson, Fibonacci sequence applied to quadratic transformations, Revista (ibid) 12 (1959), 81-84]. We use the general equation of a polynomial, invariant under an involution of a period p (prime and positive), Assume (i + j + k) = [m 1 + m 2 + m 3]. The parentheses represent the degree of the term where i is the largest exponent of x 1• The brackets represent the degree of the general term, It is found that in this term the exponents of the variables x 1, x 2 , x 3 have a net increase in value of m 1 + m 2 , m 3 - k, and zero, respectively, with each application of the quadratic transformation R, x 1: x 2 : x 3 = zf: z 1z 2: z 2z 3. (Received May 14, 1962.)

592-18, PAUL AXT, Michigan State University. East Lansing, Michigan. Relativization ·of a primitive recursive hierarchy.

In a previous paper (Abstract 62T-124) a hierarchy of classes of primitive recursive functions, En, n = 0, 1, 2, ... , established by an enumeration procedure of Kleene, as applied at successor ordinals, is shown to be identical to G: n+4 , n = 0, l, 2, ... , of the Grzegorczyk hierarchy of primitive recursive functions. A relativized form of this hierarchy, E;, n = 0, l, 2, ... , is investigated here which as shown gives an analogous classification of the functions primitive recursive in an assumed

function X. This relativized hierarchy is used to give a hierarchical classification of length t>J2 of the 2-recursive functions which has as its cv segment En, n = 0, l, 2, .. ,, . (Received July 6, 1962,)

592-19. R. P. KANW AL, The Pennsylvania State University, University Park, Pennsylvania, Expansion coefficient of wave fronts in continuum mechanics,

In continuum mechanics the intensity of a wave front is measured by the absolute value of the jump of a representative physical parameter across that front, The purpose of this paper is to point

289 out that expansion and contraction of these waves can be evaluated by studying the variation of the wave intensity along the orthogonal trajectories to the wave fronts. (Received May 17, 1962.)

592-20. E. K. BOYCE, Rensselaer Polytechnic Institute, Troy, New York, R. J. DUFFIN, Carnegie Institute of Technology, Pittsburgh, Pennsylvania, D. HAZONY and H. J. NAIN, Case Institute of Technology, Cleveland, Ohio. Synthesis of n-port networks by a matrix Richards' theorem.

Let Z(s) be a positive real n by n rational matrix function of the complex frequency variable s. A synthesis method is developed to obtain an n-port network whose impedance matrix is Z(s). The method is based on the Bott-Duffin synthesis procedure. Each cycle of the synthesis represents Z(s) in terms of a prescribed realizable loss-less 2n-port buffer network terminated in a positive real n-port matrix function of reduced degree. To carry this out, an extension of Richards' theorem for matrix functions is employed which is due to Boyce and Duffin (Bull. Amer. Math. Soc. 61 (1955), 161). Unlike the usual one port method the present n-port method employs two Richards' transformations per cycle as indicated in the method of zero cancellation (D. Hazony, IRE Trans. on Circuit Theory, Vol. CT-8, pp. 114-120, June 1961). Each cycle reduces the degree of the matrix by at least two. (Received May 28, 1962.)

592-21. J. A. NOHEL, University of Wisconsin, Madison 6, Wisconsin. Some problems in nonlinear Volterra integral equations.

Upper (lower) bounds for the norm of solutions of the system of integral equations (1) x(t) = h(t) + .foq(t- r)f(r,x(T))dr, t: 0, where h,f are given vectors with n components and q is a given n by n matrix defined on 0 :it < t 0, lxl < oo for some t 0 > 0 (to= + oo not excluded), are obtained in terms of maximum (minimum) solutions of a related scalar integral equation. (2) r(t)

= H(t) + J'J Q(t - r)w( -r,r(T))dT, 0 ~ t < t0• In (2) H, Q, w are nonnegative functions such that lh(t) I ~ H(t), lq(t) I ;!! Q(t), lf(t,x) I ~ w(t lx 1), for 0 l!i t < t 0, lx I < oo; it is assumed that h, f, H, w are continuous, q,Q are integrable on every finite subinterval of I!J,to) and w(t,r) is nondecreasing (non increasing) in r for each fixed t. In particular, global existence and boundedness theorems for solu tions of (1) are obtained; the method of comparing solutions of (1) with those of (2) is also adapted to yield general uniqueness and convergence of successive approximation results for solutions of (1). Known results of this type for systems of first order differential equations occur as special cases. (Received June 4, 1962.)

592-22. R. J. DUFFIN, Carnegie Institute of Technology, Pittsburgh, Pennsylvania, and C. S. DURIS, 1714 Grandview Avenue, North Braddock, Pennsylvania. A convolution product for discrete analytic functions.

This paper concerns complex valued functions defined at the points of the complex plane with integer coordinates. The functions of particular interest, called discrete analytic functions, satisfy difference equations analogous to the Cauchy-Riemann equations. Previous work has shown that it is difficult to define a suitable analog for the product of discrete analytic functions which preserves this analyticity. To obtain such an analog this paper considers a "convolution product", which is a

290 summation resembling the convolution integral appearing in the theory of the Laplace transform, The path over which the discrete convolution is taken is an arbitrary chain of lattice points in the complex plane, This product is commutative, associative and distributive; and if the factors involved are discrete analytic so is the product, Moreover the convolution product may be related to a discrete analog of the Laplace transform, With this machinery it is possible to develop a discrete theory similar to that of Volterra integral equations, The solutions of these "integral equations" are dis crete analytic, In turn, this gives a procedure for continuing the solutions of difference equations on the real axis to discrete analytic functions in the complex plane, (Received June 7, 196Z.)

59Z-Z3, ROY LEIPNIK and HENRYK MINC, University of Florida, Gainesville, Florida,

Fixes in finite semi-groups and the Isbell property,

The fix Fa of an element a in a semi-group S is the set of all x E S such that xa = ax = x, Some basic properties of fixes are obtained, A pair of commuting elements a, b of S is said to possess the Isbell property in case Fa"# {1, Fb "#{I imply Fa r1 FbI {1. Sis said to possess the Isbell property if each pair of commuting elements does, The Isbell property is shown to hold in all finite semi groups of order less than 8, A counter-example is exhibited for order 8, (Received June 7, 196Z.)

59Z-Z4, L. L. HELMS, University of Illinois, Urbana, Illinois, and GUY JOHNSON, William Marsh Rice University, Houston 1, Texas, Class D supermartingales,

Meyer has shown that a potential (i.e,, a non-negative uniformly integrable supermartingale

{Yt• F t: 0 li t IIi oo} with right-continuous sample functions and limt-ooYt = 0 with probability 1} can be decomposed into a difference of a martingale and a process with increasing sample functions if and only if the family fyT: T E 1}, where I is the class of stopping times for the process, is uniformly integrable (Meyer, A decomposition theorem for supermartingales, Illinois J, Math. 6 (196Z)), An example of a potential which cannot be decomposed in the above manner is constructed by composing the superharmonic function u(p) = IIPII- 1, where p E E3 , with a Brownian motion process starting from (1,0,0) in E 3• Theorem, Let {Yt• F t: 0 lit ~ oo} be a non-negative supermartingale with right-continuous sample functions, If the family {YT: T E I} is uniformly integrable, then

limn-00nP (.sup 0 ~ t;;ooYt > n] = 0, The converse holds if the supermartingale has sample functions which are. continuous with probability 1, (Received June 7, 196Z,)

59Z-Z5, D. H. CARLSON and HANS SCHNEIDER, University of Wisconsin, Madison 6, Wisconsin, Inertia theorems for matrices: The semi-definite case,

Generalizations and applications of the Main Inertia Theorem for Matrices (Ostrowski-

Schneider, J, Math, Anal, Appl, 4 (196Z), n-841 0, Taussky J, Soc. lndust, Appl. Math 9 (1961), 640-643) are discussed in the case j('(AH) l: 0, In two special cases the relation between In H and In A is determined, It is shown that for all A there exists a non-singular Hermitian H such that

)f (AH) ii: 0 and a necessary and sufficient condition is given for the existence of H > 0 for which

'f(AH) iii: 0, Conditions are found for the existence of a Hermitian K such that if(AK) ill 0 and for a given subspace 'J?, and ?{ = ?((K) = ?{(~(AK)). (Here ~(K) is the null-space of K,) If such a K exists,

291 further conditions are found under which ?((H)= ?( for all Hermitian H with J!'(AH) e;. 0 and ??

592-26. B. L. CHILTON, University of Buffalo, Buffalo 14, New York and H. S. M. COXETER, University of Toronto, Toronto 5, Ontario, Canada. A limiting surface for the polar zonohedron.

A polar zonohedron [Coxeter, Regular polytopes, Macmillan, New York, 1962, p. 29] is a convex polyhedron having n(n - 1) rhombic faces and 2n(n - 1) edges: n - 1 pairs of opposite edges in each of n directions, namely the directions of n evenly spaced generators of a cone of revolution. The 2(n - 1) edges in each direction belong to the same number of faces, forming a zone. If n > 2, the remaining edges of these faces form two skew 2(n - 1)-gons. The vertices of each skew 2(n - 1)-gon lie on a twisted curve consisting of arcs of two circular helices on cylinders with parallel axes. The n(n - 1) + 2 vertices of the whole polyhedron lie on the surface obtained by revolving either of the helical arcs about the common generator of the two cylinders. The meridian of this sur face of revolution is an arc of a sine curve. When n increases, the polyhedron approximates the surface more and more closely. (Received June 15, 1962.)

592-27. B. R. TOSKEY, Seattle University, Seattle 22, Washington. Nonzero idempotents in certain rings.

Multiplication in an associative ring whose additive group is the direct sum of two infinite cyclic groups generated by u 1 and u2 is uniquely determined by a set of numbers gijk defined by uiuj = gijlul + gij2u 2 and subject to the associativity conditions gijlglkp + gij2g 2kp = gjklgilp + gjk2g 12p (see Beaumont, Duke Math. J. 15 (1948), 367-369). It is shown that if g 111, g 112, g 221, and g 222 are given integers, then integers g 121, g 122, g 211 , and g 212 can be chosen such that the numbers g ijk are the multiplication coefficients of a ring containing nonzero idempotent elements if and only if jg111 1+ jg222 1+ jg112g 221 1 'f-0. In the proof of this result, all such rings are found, up to isomorphism, which contain nonzero idempotent elements. (Received June 15, 1962.)

592-28. T AQDIR HUSAIN, University of Ottawa, Ottawa 2, Ontario, Canada. On completion and completeness of B(C)-spaces. Preliminary report.

Let E be a B((?)-space (see Notices Amer. Math. Soc. 7 (1960), 943, Abstract No. 576-135). It is known that E is B-complete (Bull. Soc. Math. France, 86 (1958), 41-74) if and only if it is a B( C)-space where C = .fl., the class of all locally convex (I.e.) spaces. Here it is shown that a com plete l.c. space E cannot be characterized in terms of B(e)-spaces for any class C of l.c. spaces. It has been shown by exhibiting that a quotient space of a B(C)-space is always a B(C)-space. How ever, it is well-known by an example due to G. Kothe that a complete space may fail to have this permanence property. Further, it is shown that the completion E of a B(C)-space E is also a B(C )-space provided every subspace F of any I.e. space Gin C is also in C. From this it follows that the completion of a B(?J{)-space is always a B(~)-space, where:?( is the class of all metrizable l.c. or normed spaces. (Received June 15, 1962.)

292 592-29. p. C. HAMMER, Numerical Analysis Department, 5534 Sterling Hall, University of

Wisconsin, Madison 6, Wisconsin. Extended topology: Continuity, II.

A separation S is a hereditary binary relation in the class of all subsets of a space M. A Wallace separation is a symmetric separation which is disjunctive, i.e. (X,Y) E S implies X and Y are disjoint. Let S and S 1 respectively be separations for spaces M and M 1. Then t:M ~ M 1 is

(S,S 1)-continuous provided X, Y (;;; M, (X, Y) j{ S implies (tX,tY) i S 1• Properties of such continuitie are derived, Theorem. 1. To each Wallace separation S for M there corresponds a unique maxim•

Wallace separation, S*, defining the same class of connected sets asS, 2, To each class(! of sut of M there exists a unique minimum superclass C* of sets each of which is connected with respec1 a Wallace separation. 3, Let S and s 1 be Wallace separations for spaces M and M 1 respectively. necessary and sufficient condition that t:M- M 1 map S-connected sets onto S 1 -connected sets alw

is that t be (S*,S 1)-continuous. Remark. As applied to real-valued functions of a real variable, S*

cannot be replaced by the separation of any topology. The separation S 1 in the conclusion of (3) m8 be replaced by Si. Part (2) permits application of (3) to preserve arcwise-connectedness of mappi of the plane onto itself. That is, there exists a Wallace separationS with precisely the arcwise connected sets as connected sets. (Received June 18, 1962.)

592-30. Y. L. LUKE, Midwest Research Institute, 425 Volker Boulevard, Kansas City 10, Missouri. Approximate inversion of a class of Laplace transforms with application to supersonic flow problems.

Solution of numerous applied problems requires the inversion of a Laplace transform which involves functions of hypergeometric type. An approximate procedure for inverting transforms of

this kind is presented, The method inverts rational function approximations of these transcendent~

(see Y. L. Luke. On economic representations of transcendental functions, J. Mathematical Phys.

38 (1960), 279-294) and yields simple and useful expressions for the desired inverse functions. To illustrate the technique, consider the solution of the wave equation with a particular set of boundar:

conditions. The functions which arise have numerous applications in supersonic flow, acoustics an heat conduction. In the field of aerodynamics, these include the velocity field of a circular cylinde impulsively moved in the axial direction with constant axial force, the flow past quasi-cylindrical

bodies, shock-body interference, wing-body interference and the transient motion of a body of revo tion in supersonic flow, (Received June 19, 1962.)

592-31. J, R. KINNEY, Massachusetts Jnstitute of Technology, Lincoln Laboratory, Lexingt 73, Massachusetts. Tangenitiallimits of functions from the class Sa..

2 Suppose f(z) = Ln ,.0 cnzn, L na.lcn 1 < oo for some a., 0 < a. < 1, Then f(z) approaches limr_,.1f(rei6 ) as z approaches ei6 with 1- lzl;:;. lei6 - zl7'", for all -r < (1- r)/(1- a.), except possibly for a set E whose capacity of order 1 - r is zero. Similar results are proved for the fractional integral and fractional derivative of f(z). (Received June 19, 1962.)

293 5\lZ-32. R. T. HAKlU:S, U3 Physics lJuilding, Uul

Let D and E be banach spaces with conjugate duals E 1 and D1 where D !;;; E ~ E 1 ~ D 1 with all maps induced by inclusion required to be continuous with dense range; in addition, assume that the duality between D and D 1 coincides with that between D and E 1 when both are restricted to D X E 1 •

If {R(t): - oo < t < + oo} is a strongly continuous one-parameter group of operators in L(E) such that for all x E E,

592-33. T. S. PITCHER, Massachusetts Institute of Technology, Lincoln Laboratory, Lexington 73, Massachusetts. On adding stochastic processes.

Let x(t) be a Gaussian stochastic process on a finite interval and y(t) another process independ ent of it. If P x and P x+y are the measures induced on sample space by the processes x(t) and x(t) + y(t), then Px+y is absolutely continuous with respect to Px if and only if Py(~IPx+f <. Px]> = 1. (Received June 21, 1962.)

592-34. RICHARD SINKHORN, 1144 North Market, Nichita 14, Kansas. On best doubly stochastic estimates.

To any given strictly positive square matrix A there corresponds a unique doubly stochastic

matrix T A which can be expressed in the form TA = D 1AD2 where 0 1 and D2 are diagonal matrices with strictly positive diagonals. The matrix T A can be obtained from A as a limit to the iteration defined by alternately normalizing rows and columns of A. The rate of convergence is geometric;

there exists a number ?", 0 <= r..::. 1, such that liT A - A (n) 111i C-rn for some constant C where A (n) is the nth stage of the iteration on A. The convergence is uniform with respect to A on compact subsets of the strictly positive N X N matrices and consequently T A is a continuous f\lnction of A when A is strictly positive. When A is non-negative but contains zero entries certain peculiarities develop and none of the above results need hold. (Received June 22, 1962.)

592-35. MANABU HARADA, 1725 Orrington Avenue, Apartment 520, Evanston, Illinois. Hereditary orders. Preliminary report.

Let R be an integral domain with the quotient field K, and 2:: a semi-simple K-algebra with finite dimension over K. If an order A over R in l:: is hereditary as a ring, call it an h-order (hereditary order). Theorem 1. Be restricted to the case where L: is simple and R is a Dedekind domain. Assume that R is a discrete rank one valuation ring and :E is simple. Theorem 2. h-orders n containing A can be completely determined by the form n =A r;_, f'i maximal orders containing A, and the number of those orders is equal to 2n - 1, where n is the number of maximal ideals in A.

294 Theorem 3. The inversible two-sided fractional ideals of A is a cyclic jSroup generated by the radical of A. Finally, h-orders in quaternions are considered as an example. (Received June 28, 1962.)

592-36. STEPHEN KULIK, 240 Prospect Avenue, Long Beach 3, California. On the series solution of ordinary simultaneous equations.

Let (1), fi(X) = 0, i = 1,2, •.. , n; X= (xl'x2, ••• ,xn), be a system of n simultaneous equations inn unkno-vns. Under conditions given in the paper, the following solution of (1) is constructed. A system of equations (2), F i(X;u) = 0, can be formed in various ways such that F i(X;u0) =fi (X), where u0 is a constant. System (2) has a unique solution (3), xi= ki(u), in a neighborhood of the solution A=

(a 1,a2, •.• ,an) of (1), and ki(u) have the series expansions there, ki(u) = ki(u 1) + (u- u 1)ki(u 1) + ..• , where kfm>(u 1) depend on the values of Fi(X 1; u 1) and their partial derivatives with respect to to ai, then the series x 1, x 2, ..• ,xn; x 1,u1 being the solution of (2). Now, if xil = ki(u 1) is close enough solution of (1) is ai = ki(u1) + (u0 - u 1)k'(u1) + .•.. (Received June 28, 1962.)

592-37. MAREK FISZ and V. S. VARADARAJAN, 220 West 104th Street, Apartment 12, New York, New York. A condition for absolute continuity of infinitely divisible distribution functions.

Let F (x) be an infinitely divisible distribution function, and let H(u) be the function assigned to F(x) by Khintchine's formula. If, for some uo > 0, in one at least of the intervals (- uo,O), (0, uo). H(u) is both unbounded and absolutely continuous, F(x) is absolutely continuous. If follows from this and from a result of the first author (On the orthogonality of measures induced by L-processes, Trans. Amer. Math. Soc. in print) that any nonsingular L-distribution function is abso lutely continuous. (Received June 28, 1962.)

592-38. S. W. GOLOMB, Building 185, 4800 Oak Grove Drive, Pasadena, California. On certain multiplicatively recurring sequences.

Let tb~)J be the sequence defined, for each non-negative integer r, by the recursion (r) _ (r) _ n (r) (0) _ (1) _ (1) bQ - 1, bn+ 1 - TTi=Obi + r for n E; O. Thus {bn f- {1,1,1,1,1,1, ..• J. fbn }- fl,2,3,7,43,1807, ••• J. {b~2 ) = {1,3,5,17,257,65537, .•• J. etc. The recursion (1) is equivalent to (2) bJr)= 1, bir) = 1 + r, (r) (r) 2 2 2 bn+1 = (bn - r/2) + (r- r /4) for n ~ 1. For the roots of r/2 = r- r /4, viz. r = 0 and r = 2, the 2 1 corresponding sequences {b~r)J have rational representations, viz. b~O) = 1 and b~2 ) = 2 n- + 1 (for n > 0). For all other values of r, .there is a constant Sr such that b~r) - r/2 = S~n- & ~r) , with 0 <. eir) <. 1/bi~\ for large enough n. For each value of r, the terms of fb~r)J are pairwise relatively prime. The sequence {b~2 )} was the subject of a famous false conjecture of Fermat. The 1 1 sequence fb~l)J has many remarkable properties, including 2:::: 11/bi ) = 1 with L~= 1 1/b~ ) being the closest strict underestimation of 1 which is the sum of k reciprocal integers. For any modulus m, fb~r)J is ultimately periodic mod n. (Received June 27, 1962.)

295 592-39. D. C. MURDOCH, Harvard University, 2 Divinity Avenue, Cambridge 38, Massachu setts. Subrings of the maximal ring of quotients associated with closure operations.

Let R be a ring with maximal right quotient ring Q in the sense. of Utumi [Osaka Math. J. 8 (1956) 1-18]. Let .;:rx, and£ be the lattices of right and two sided ideals of R. A Closure operation

-<:. in ;;er is bilateral if ~; G..<::': To each bilateral closure ;e. corresponds a ring R-c• between R and Q which is constructed from the semi-endomorphisms a of R (as a right R-module) such that J E,rr•

J !;. dom a ~ aJ c.;;, J<. The bilateral closures are partially ordered and for each..::. there is a unique

maximal iC. such that R~ = R-t:. These maximal closures form a complete lattice which is dual isomorphic to the lattice of closure subrings R.e of Q. A bilateral closure .c.(P) is associated with each prime ideal P of R. In a commutative Noetherian ring, if Q is th.e full ring of quotients and IC(P) is maximal, R.c. (P) = Rp• the local extension of R with respect to P. Sufficient conditions for a similar theorem in the nonc.ommutative case are discussed. (Received June 27, 1962.)

592-40. A. L. PERESSINI, Room 213, Altgeld Hall, University of Illinois, Urbana, Illinois. An order-theoretic property of the weak topology in sequence spaces.

Suppose that A is a vector space of real sequence such that ~ contains the vector space vi of sequences with only finitely many nonzero coordinates. G. Kothe defines the a-dual A. x of A as )..x = [u = (ui) E c.>: E~ 1 1xiud '""- + oo for all x =(xi) in A.J where c.> denotes the vector space of all real sequences. The spaces A. and Xx are in duality with respect to the bilinear form (x,u)--+ (x,u) = 2:~ 1 xiui and both spaces are ordered vector spaces for the partial order defined by x i: y if xi !; y i for i = 1,2,... • Theorem 1. If A is a vector lattice (resp. if A is perfect, i.e., )\ = (.),x)x) then the lattice operations in A are o-(~.Ax)-continuous if and only if Ax= vi (resp • .\ = w). Theorem 2. If ).. is a lattice, the lattice operations in A are always o-(.\,/lx)-sequentially continuous. The proof of the second result makes use of Kothe's projective limit characterization of the weak topology in perfect spaces. Generalizations of these results to Kothe (function) spaces are discussed. (Received June 25, 1962.)

592-41. J. I. ROSENBLATT, The University of New Mexico, Albuquerque, New Mexico. Sufficient c"mditions for fixed length confidence intervals in two stages.

Let X 1, x2 •••. be independent random variables with common distribution function F € Jf, and let 9p be a scalar associated with F. Useful simply verified sufficient conditions are given in

order to insure that for fixed S > 0 and fixed a E (0,1), Pp flgn(Xk+ 1 .... ,Xk+n> - 9p lli 8} 5o 1 - a

where n is a random variable measurable with respect to the a--algebra generated by x1, ••. ,Xk• {gj}is a sequence of real valued Baire functions, gj having domain Rj. (Received June 25, 1962_)

592-42. WITHDRAWN

296 592-43, C. H. CUNKLE, Utah State University, Logan, Utah. Classes of rings in Boolean algebras. Preliminary report.

The rings which can be formed from Boolean algebras are those with operations x E!l y

= e • x • y, x 0 y = b X y + e (x + y +b), where x • y denotes the symmetric difference, x y' + x'y, and (e,b) is a pair of arbitrary constants uniquely determining a ring, The relation of being isomor phic separates these rings into equivalence classes. A class can be represented by (e, e • c), where cis a fixed element and e ranges over all elements of the Boolean algebra. It contains the ring (O,c) and no other of this form. Each class contains the same number of rings as there are elements of the Boolean algebra. All rings are commutative, and the class containing (0,1) consists of the idempotent rings, which are identical to those with unity element and thus the Boolean rings. (Received July 2,

1962.)

592-44, H. L. CROWSON, P. 0. Box 5983, Bethesda 14, Maryland. Closed form solutions of a second order linear ordinary differential equation with n-regular singular points.

The following equation z(z- 1) TTf;{

592-45. J. BATTLE, F. HARARY, andY. KODAMA, University of Michigan, Ann Arbor, Michigan. Every planar nine point graph has a nonplanar complement.

Let G be a graph with p points and q lines, and let G be its complement with (~)-q lines. It is easy to show that G or G is nonplanar when p ~ 11, using the Euler polyhedron formula. Simple examples verify that there exist graphs G with p lii 8 such that both G and G are planar. It is shown here that for p = 9 (and p = 10), G or G is nonplanar, proving a conjecture of John L. Selfridge. Let G

297 be a planar graph with 9 points and let T be the !-skeleton of a triangulation of the sphere with 9 vertices, which contains G as a subgraph. Let di be the degree of the point vi of T, and call the partition ofT the vector 1J'"(T) = (dp d2, ... ,d9). Of course :L:di = 42 since Tis a triangulation. The following sequence of propositions proves the conjecture. (1) The complement T of T has at most 3 components. (2) If 'i' has 2 or 3 components, it is nonplanar. (3) Thus if 'f is disconnected, it is nonplanar. (4) If 'f is connected and planar, then 7r(T) does not contain two 4's and also does not contain two 3's. (5) If Tis connected, then 3 l!i di i!i 7 for each di in 11'"(T). (6) There exists a unique partition 17"o of 42 into 9 summands satisfying the conclusions of Propositions 4 and 5, namely

,.-0 = (5,5,5,5,5,5,5,4,3). (7) There is no triangulation T0 of the sphere with p = 9 such that 1t"(T0) = "17"0• (8) Thus if 'F is connected, it is nonplanar. Combining Propositions 3 and 8 proves the theorem. (Received July 2, 1962.)

592-46. P. M. SWINGLE, 4970 N. Kendall Drive, Miami 56, Florida. The inner and outer T set function in semigroups.

Let S be a compact Hausdorff continuum and semigroup, S = ESE. Below Q is open set, W is a continuum, A C S, p E S, K is minimal ideal, iT is outer T function and Tithe inner. Define: i T(A) = {S - y: there exist Q, ideal W such that y E Q C W C S - A} and Ti(A) = S - {y: there exist Q, W, but W not an ideal, such that y E Q C W C S - A}. Prove: Ti(A) U iT(A) = S and Ti(A) n iT(A) = T(p); if S 'I iT(A), then S - iT(A) is a connected open ideal; S - iT(p) U T(p) is connected; iT(A) = S if and only if An K '/ !1; iT2(p) = iT(p); T/(A) = Ti(A) if and only if TTi(A) = Ti(A); S =Kif and only if, for all p, iT(p) = S; or S =Kif and only if, for all p, Ti(p) = T(p). Similarly functions liT' Tli can be defined using left ideal in place of ideal and related theorems obtained; similarly for right ideal; iT' liT' riT are all expansive closed set functions, but Ti' T 1i' Tri are antitonic although enlarging. (Received July 2, 1962.)

592-47. L. C. EGGAN, Angell Hall, University of Michigan, Ann Arl?or, Michigan. The star height of regular events. Preliminary report.

Let CJt be the free multiplicative semigroup with identity e generated by Cl = fA 1, .•• ,Anf· Let G·= v/.v.·,*,) be the free algebra generated by ;;I =a u {.A..B}, where A.a ¢.a. with associa tive binary operations V and •, and the unary operation *. For rr E G. define the set denoted by rr, written lrrl, inductively as follows: lAd= fAif• IAI = ¢, 1e1 = {eJ, lrrvwl = lrrl U lwl, lrr·wl = f.xy; xclo-1, yelcul}• lrr*l = 1e1 Ulrrl Ulrr·a-1 U ••.. Thus I I maps G into 2~. Let R, called the class of regular events, be the range of this function. Finally, define h: ~ -z by h(s) = 0 if s E p/. h(cr V(A)) = h(rr"') = max{h(rr), h(cu)J and h(o-*) = h(rr) + 1, for rr, c., E ~; and if 2 E R, H(:E) = min(h(o-); lrr I = l:}. Call H(l:) the star-height of .t:. As a corollary to the main result, it is shown that for each positive integer n, there exists a regular event l: of star-height n. The main result gives certain sufficient conditions for a regular event to have star-height greater than or equal to n. It is also shown that for each positive integer k, there exists a regular event l: of star-height k for which the application of the star is nontrivial (this means that (Vn) (I:n f.~*)), but such that I:* also has height k. Finally, this is related to the cycle structure of logical nets and state graphs in the theory of finite automata. (Received July 2, 1962.)

298 592-48. ECKFORD COHEN, University of Tennessee, Knoxville, Tennessee. Cauchy product estimates of arithemetical functions.

This paper is concerned with asymptotic estimates for the Cauchy product, h(n) = Lacnf(a)g(n - a) of arithmetical functions, f(n), g(n), of divisor and totient type. The method is based upon trigonometric series representations of such functions, and, unlike other elementary methods applicable to similar problems, does not require estimates for sums, either finite or infinite. The. method is comparatively effective for a wide class of functions. For example, it gives a satisfactory

estimate for the number of representations of n as a sum of two k-free integers for all k > 2. In cer tain cases, the results are as good as those obtainable by methods based on summation processes. In particular, let sk(n) denote the sum of the reciprocals of the kth power divisors of n; it is shown that ::La<.nsk(a)sk(n - a) = ((:" 2 (2k)/~(4k)) T k(n)n + 0(1), where Tk (n) is the sum of the cubes of the reciprocals of the kth power divisors of n, k assumed > 1. (Received July 2, 1962.)

592-49. G. M. SCHINDLER, 753 State Street, Santa Barbara, California. On the solution of axial-symmetric blast waves in fluid dynamics from differential-geometric aspects.

An approach is presented which uses differential-geometric tools for determining an axial symmetric blast wave resulting from a point source explosion in a specified medium. The basic concept is to make use of the geometric shape of the shock wave itself by writing the fluid dynamic equations in the invariant form referred to a still undetermined curvilinear system of coordinates

($, 17.~) and assuming that the shock front coincides at any instant t with one of the surfaces t = const.

The fluid dynamic equations along with the orthogonality condition of

formation formulae relating the Cartesian system (x,y,z) and the curvilinear system (.J•?7•~). The developments carried out here exhibit very clearly the close relations between the geometric shape of the wave and the physical properties of the medium. In a particular case the geometric shape of the blast wave is determined explicitly. (Received July 2, 1962,)

592-50. EDWARD SIL YERMAN, Division of Mathematical Sciences, Purdue University, Lafayette, Indiana. Geodesics and Lebesgue area.

Let C be the space of continuous functions on a k-cell Q into m, the space of bounded sequences.

If X E c let r6x(p,q) = llx(p) - x(q) n. There exists a real-valued function L on c having the following

properties: (1) Lx If Ly if ~x· ~ r6y, (2) Lx = EL(xiAi) if xis Lipschitzian and Q is subdivided into a finite number of k-simplexes 6i, (3) L is lower semi-continuous on C, (4) there exist quasilinear functions Yp E C such that Yp- x and (geometric area Yp)-+ Lx, and (5) if x is continuous on Q into

En, which can be assumed to be isometrically embedded in m, then (Peano area of x) li Lx l!i (Lebesgue area of x). Let x = i\p be a monotone-light factorization of x, and Gx(p,q) = inf length .H where f is continuous on [0, 1] into range p with f(O) = p(p) and f(1) = p(q). Thus Gx(p,q) is the geodetic distance

for x between p and q if xis light. Theorem. If Gx = Gy then Lx = Ly. (Received July 2, 1962.)

299 592-51. MARVIN MARCUS, University of California, Santa Barbara, California, and A. H. CAYFORD, University of British Columbia, Vancouver, B. C. Further results on the Kantorovich inequality.

Let CT = (cr1, ••• ,o-n) be on the unit (n- I)-simplex sn-l and define the real valued function F (o-) = Lf= 1).iO" iLf= 1f(A.i )CTi where f is a non-negative monotone decreasing strictly convex function defined on the closed interval [>. 1 .~n] and 0 < A. 1 :;; .;1. 2 ;:; •.• ~ ~n· The structure of the points CT E sn-l for which F(CT) is a maximum is determined. This is then applied to the various known generalizations of the Kantorovich inequality (M. Newman, J. Res. Nat. Bur. Standards 64B 1 (1959), 33-34; A. H. Schopf, Numer. Math. 2 (1960), 344-346; M. Marcus and N. Khan, Portugal. Math. 20, 1 (1961), 33-38) to determine the cases of equality. Also, some new generalizations of the Kantorovich inequality are obtained by applying the results to various associated compound matrices. (Received July 2, 1962.)

592-52. N. ARONSZAJN and E. GAGLIARDO, University of Kansas, Lawrence, Kansas. Interpolation spaces and interpolation methods.

Two Banach spaces A,B are compatible if they are continuously imbedded in the same Hausdorff vector space; An B and A+ B are then Banach spaces with well determined norms. An intermediate space Cis a Banach space, with An B C C, continuously imbedded in A+ B. Let f(A,B,A',B') be the class of linear transformations T of A + B into A' + B 1 , such that T(A) C A', T(B) C B ',

ITIA,A' l:i! 1, ITIB,B' ;a 1. An interpolation space between A and B is an intermediate space such that for every (compatible) A',B' there is at least one intermediate space C' such that a bilateral inter polation theorem holds between [A,B,C) and fA',B',C'}, i.e. that for every T E" ,7(A',B',A,B) one has T(C) C C' and for every T' C .:T(A',B',A,B) one has T'(C') C C. Now let C be an intermediate space between A and B such that for every S E .J(A,B,A,B) one has S(C) C C; it can be proved that C is an interpolation space and gives rise to two extreme interpolation methods, i.e. for every A',B' one can determine two interpolation spaces Ci = F~~c(A',B'), i = 1,2, such that: (I) F~~c(A,B) = C, (i = 1,2); (2) for any two couples {A',B'}, {A",B''} a bilateral interpolation theorem holds between fA',B',Ci} and fA",B",Ci'J• (i = 1,2); (3) for any interpolation method F(A',B') with F(A,B) = C

~ne has C]_ :::::> F(A',B') :J C~. (Received July 2, 1962.)

592-53. E. K. McLACHLAN, Oklahoma State University, Stillwater, Oklahoma. Integral representation of semi-norms on Euclidean n-space.

Let Po be a nonzero semi-norm on En and let c 0 be the sub-cone of semi-norms p of C (Notices, Amer. Math. Soc. 8 (1961), 255) such that p(x) > 0 where xis such that Ib (x) > 0. A topology is selected such that c 0 - c 0 is locally convex. In c 0 a convex compact set B is found such that B meets every ray of c 0 once and only once and B does not contain the origin. By a theorem of Choquet there exists a positive Radon measure u on the closure of the extreme points of B, i.e. extremal elements of c 0 , such that Po= jp du. The cone c 0 contains extremal elements of the form If I where f is a linear functional on En. For n = 2 the extremal elements of c 0 are the extremal elements of C in c 0 and, furthermore, the extremal elements of c 0 are not dense in c 0• (Received July 2, 1962.)

300 592-54. E. A. NORDHAUS and PAUL PENNOCK, Michigan State University, East Lansing, Michigan. Finite free lattices.

The free lattice L generated by a partly ordered set P is by definition the free lattice generated by the elements of P and preserving bounds, whenever they exist, of pairs of elements of P. Thus (a) L is a lattice containing P as a subset and in which the order of P is preserved, (b) every element of L is a lattice polynomial in the elements of P, and (c) every lattice which satisfies (a) and (b) is a homomorphic image of L. It is shown that the free lattice 4 + l is infinite, and the method used also shows the (known) result that the free lattice 2 + 2 is infinite. All finite free lattices generated by chains, and consequently those generated by partly ordered sets are then readily deter mines. (Received July 2, 1962.)

592-55. GLORIA OLIVE, 1262 Adams, Corvallis, Oregon. Polynomials defined by generalized powers.

By successive use of the definition in Abstract (589-6) of these Notices, Amer. Math. Soc. 9

(1962), 105, it is found that x(b) = L~=OP~(b)xi where P~ (b) is a polynomial in b of deg:r.ee ct,2a_r• with rational coefficients when 0 ...,. j l:! t, and Pt(b) = 8 Ot (Kronecker delta). Other properties of P!(b) t t t j+t t'' t t are (l) Lj=OPj(b) = l; {2) Pj(b) = (- l) b PjCl,.£ if b 10 (thus the coefficients of Pi (b) are symmetric or anti-symmetric according as j +tis even or odd); and (3) 2::1=j ci,j~ (b)= Ll=j N~(b)P}(b). II t . t When x(b) is expressed in the form L~=OQi(x)b 1 it is found that Qi(x) is a polynomial in x of degree t with rational coefficients. Other properties of Q~(x) are {l) L~:'0 Qf(x) = xt and (2) Q~"-i(x) = (- l)tQ~(- x) if 0 t5 i ;at". (Received July 3, 1962.)

592-56. E. B. KNIGHT, University of Minnesota, Minneapolis 14, Minnesota. A sojourn density process of Brownian motion.

Let X(t), t i: 0, X(O) = 0, be a separable Brownian motion process in l dimension, and let I(a,b) (z) = l or 0 according to whether z E' (a,b) or z ¢ (a,b). For a. ~ 0, set T = inf.t: (d/dx),/oi(-oo,x) (X("t"))d't"]x=O 5; a.. Define Y(x) for x!!;. 0 by Y(x) = (d/dx)./6\-oo,x)(X('t"))d't. Theorem. Y(x) is the diffusion process with initial value a., infinitesimal generator y d 2/dy2, and absorbing barrier at 0. (Received July 3, 1962.)

592-57. H. J. COHEN and FRED SUPNICK, The City College, New York 31, New York. Concerning real numbers whose powers have nonintegral differences.

Various classes of real numbers a. have been considered for which a.s - a.r (s,r positive integers, s ~ r) is never integral (see pp. 244-246 ofF. Supnick, H. J. Cohen, J. F. Keaton, On the powers of a real number reduced modulo one, Trans. Amer. Math. Soc. 94 {1960), 244-257). This note adds another class to this category. If one considers the numbers a.= a+ b(2)112, one finds that if a and b are positive and rational, a ~ l, then a. 8 - a.r is nonintegral. This result is now generalized:

Theorem. Let ;j be any algebraic number having a minimal polynomial of the form xn - b 1xn-l - b 2xn-2 - .•• - bn_ 1x- bn• where each bi is rational, bi ~ 0, n > l. Now, let the number a. be defined dl d2 dm by a.= c0 + c 1f' + c 2p + ••. + cmlf , where each ci > 0, c0 ;;:; l, m ;;. l, l ~ d 1 """ d 2 < ••• <. dm

301 < n; ci rational, di integral. Then as - ar is nonintegral for all positive integers s,r (s ;o!:-r). (Received July 3, 1962.)

592-58. D. W. SASSER, Applied Mathematics Division, 5421 Sandia Base, Albuquerque, New Mexico. Quasi-positive operators.

Let B be a real Banach space, K a closed proper cone (closed convex set containing all posi tive multiples and not containing both x and -x unless x = 0) in B and K+ the dual cone in B*(K+ =

{x* lx*(x) i:; 0 for all x E K} ). A linear operator Tis called quasi-positive (with respect to K) if for

each x E- K, x* E K+, x*(Tnx) !; 0 for n;;; n(x,x*). In particular, if TK r;;; K, T is quasi-positive. T is called strictly quasi-positive if x E K, x* E K+, x -j; 0, x* f. 0 imply x*(Tnx) > 0 for n;:. n(x,x*) and T

is called strongly quasi-positive if x E K, x* E K+, x t 0, x* f. 0 imply lim infn_ 00x*(Tnx)/11Tnll > 0. Under the assumptions that T is compact (a slightly weaker condition will suffice) with spectral radius 1 and the linear span of K is dense in B, the following theorems are proved. Theorem 1. If T is quasi-positive, then 1 is in the spectrum ofT and there exist nontrivial elements u E K, u* E K+ such that Tu = u, T*u* = u*. Theorem 2. If Tis strictly quasi-positive, then 1 is a simple eigenvalue and the corresponding eigenspace is one-dimensional. Theorem 3. Tis strongly quasi-positive if and only if (1) there exists u E K such that Tu = u and x*(u) > 0 for all x* E K+, x* i 0, (2) there exists

u* E K+ such that T*u* = u* and u*(x) > 0 for all x E K, xi 0, (3) Tx = x implies x = au, and (4) for~ in the spectrum of T, either A= 1 or lA. I < 1. (Received June 29, 1962.)

592-59. JAMES CONLAN, U. S. Naval Ordnance Laboratory, White Oak, Silver Spring, Maryland. A generalized boundary value problem for a hyperbolic partial differential equation.

By means of a finite difference scheme, an existence theorem is proved for the following

boundary value problem: uxy = f(x,y,u,ux,uy)' ux =

592-60. G. R. MAC LANE, Rice University, Houston 1, Texas. Asymptotic values of holomor phic functions.

The function f(z), holomorphic and nonconstant in lz I < 1, belongs to class A iff has asymptotic values at a dense subset of lz I = 1. The curves associated with the asymptotic values are unrestricted: they may be any curves in lz I < 1, each of which ends at a definite point of lz I= 1. Similarly, f belongs to class B if there is a set of curves, with end points dense on lz I = 1, such that on each curve either

f is bounded or f--> oo. f belongs to class L if for each pair ~ > 0, $ > 0, there exists "l > 0 such

that every component of £zf lf(z) I = ,\, 1 - '? <. lz I < 1} has diameter < $. Theorem. A = B = L. Various sufficient conditions for f E A are derived; one is ./o 1(1 - r) log+ lf(rei8) I dr < oo for 8 E. ®,

where fl) is dense on [0,2'77"]. The points on lz I= 1 at which f E A has asymptotic values may be only countable; e.g., the modular function. However, if J is an arc of lz I= 1 such that f does not have the

302 asymptotic value oo at any point of J then f has asymptotic values on a subset E C J with meas(E) :> 0.

In general however meas(E) < meas(J). (Received July 5, 1962.)

592-61. STEVE ARMENTROUT, University of Iowa, Iowa City, Iowa. A property of plane curves.

Suppose that K is a nondegenerate compact continuum in a plane S, p E K, and J is a simple closed curve inS enclosing p and intersecting K. A is an independent set of arcs for J, p, and K if and only if A is a finite set such that (1) if a € A, a is an arc spanning J (interiorly) and such that any point common to a and K is an endpoint of a., and (2) if, for each arc a of A, Ua. is the component of (Int J) - a. not containing p, then if a. and ,S are distinct arcs of A, there is neither an arc nor a simple closed curve 'Y spanning J (interiorly) such that r- J is disjoint from K and r separates

U a. U U~ from p in (Int J). If there is a finite maximal independent set of arcs for J, p, and K, any two such sets have the same number of elements. Suppose K is nowhere-dense inS. If there is a positive integer n such that for each positive number s, there is a simple closed curve J 8 enclosing p, intersecting K in a totally disconnected set, and such that there is a maximal independent set of arcs for J 8 , p, and K having exactly n distinct elements, define the index of K E p to be the least such positive integer n. Certain properties of this index are established. (Received July 5, 1962.)

592-62. WITHDRAWN

592-63. D. C. KAY, 1905 New York Avenue, Lansing 6, Michigan. The equality of Haantjes- Finsler and geodesic curvature of a curve in Riemannian space.

Let ?' be a curve of class c3 in a Riemannian space R coordinatized by real number n-tuples (x1,x2, ••• ,xn) with metric ds2 = gijdxidxj (using the summation convention). Let 0 be any point on rand P a neighboring point, and suppose s and cr are respectively arc length (along?"') and chord length (along the geodesic) from 0 to P. It is proved that 4!(s - o-)/s3 tends to (kg)2 as p approaches 0, where kg is the geodesic curvature of r at 0 while the former quantity is the square of the Haantjes-Finsler curvature of?" at 0 applied to R as a metric space. The method of proof is by Taylor expansions and the deducing of coefficients of certain variables up to the third order of small magnitude. (Received July 5, 1962.)

303 592-64. S. c. MOY, Syracuse University, Syracuse 10, New York. Boundaries for 11-continuous Markov chains and representations of excessive functions.

In 1959 J. L. Doob introduced the Martin boundary for Markov chains with a countable state space (J. Math. Mech. 8 (1959); Math. Reviews 21 No. 5825). The boundary has the property that almost all paths either terminate at a finite time or converge to boundary points. G. A. Hunt discussed the boundary in further depth [j:Ilinois J. Math. 4 (1960); Math Reviews 23 No~ A691]. The present paper is concerned with extending the Martin boundary theory to Markov chains with a general state space. A transition function P is said to be 1r-continuous if the transition probabilities admit density functions with respect to a probability measure -rr. A tr-continuous transition function is said to be transient if 2:7rPn is IT-finite. For a '17'-continuous transient transition function, a kernel K(x,y) and a boundary based on the kernel are introduced. Similar convergence and representation of excessive functions theorems are proved. (Received July 5, 1962.)

592-65. W. L. HOYT, Brandeis University, Waltham 54, Massachusetts. On Chow bunches for projective varieties. Preliminary report.

For each biholmorphic map f: V _____.. P of a complete irreducible variety V into a projective space P, let C(V, f, P) denote the union of all Chow bunches for positive homogeneous cycles X in P such that Supp XC f(V). Theorem. There exist a union C(V) of closed subsets of projective spaces and holomorphic maps g(f,P): C(V)-C(V, f, P) such that (1) the g(f,P) induce homeomorphisms of the underlying spaces; (2) for each finite union E of subvarieties of C(V), there exists fE: V -PE such that g(fE, PE) restricted to E is biholomorphic; and (3) if Vis normal and X is an ample positive divisor on V, there exists mE such that one may choose fE to be an embedding determined by lmXI for any m ;: mE. C(V) is the inverse limit of the system consisting of the C(V, f, P) together with holomorphic maps from C(V, h o (f 1 X f2 ), P 3) onto C(V, fi• Pi) (i = 1, 2; h: P 1 X P 2- P 3 a Segre embedding) which are obtained from a relation between the Chow polynomial for h(W) C P 3 (W a sub variety on P 1 X P 2) and the Chow polynomials for the components of the sets W)l = fa E P1 ldim W n (a X P 2) ~ p} (p = 0, 1, 2, .•• ). (Received July 5, 1962.)

592-66. D. K. HARRISON, J. M. IRWIN, C. L. PEERCY and E. A. WALKER, New Mexico State University, University Park, New Mexico. High extensions of Abelian groups.

A is a high subgroup of G if A is max disjoint from G1 = nnG. It is well known that if A is high in G then A is pure in G and G/ A is divisible. An exact sequence 0-+ A--> G--. B -o is a high sequence if A is high in G. When A 1 = 0 and D is divisible, let Hext(D,A) be the high sequences in Ext(D,A). Let Shom(X,Y) be the subgroup of elements of Hom(X,Y) whose kernels are essential sub groups of X. If 0 -+A~ C - C/ A -> 0 is any pure injective resolution of A, then the sequence Shom(D,C/A)-+Hext(D,A)-. 0 is exact. In particular, Hext(D,A) is a subgroup of Pext(D,A). The definition of high is extended by calling a sequence 0 -H -G-+ K--+ 0 pure-high if it is in the image of the map Shom(K,C/H)- Pext(K,H), where 0-H- C--+ C/H _,. 0 is a pure injective resolution of H. This image is denoted Hextp(K,H) and is independent of the resolution. It follows that a sub group of G is pure-high if and only if there exists a subgroup K of G such that H is max disjoint from

304 K and (H + K)/K is pure in G/K. If K is torsion, then Hextp(K,H) = npPext(K,H), p prime. Derived functors of Shorn are related to Hextp(X,Y) and to the subgroup Next(X,Y) of all neat sequences in Ext(X, Y). Analogous statements hold for neat-high sequences. (Received July 5, 1962.)

592-67. A. A. GIOIA and M. V. SUBBA RAO, University of Missouri, Columbia, Missouri. Generalized Dirichlet products of arithmetic functions. Preliminary report.

This paper defines the generalized Dirichlet product, (f• g)(n), of two complex-valued arithmetic functions f(n) and g(n) by (f•g)(n) = 2:f(a)g(b)-\(c), where c = (a,b), ab = n, and .\ is a fixed multiplicati'

function satisfying A.r1a,b)] •X[(ab,c)] = ~ [(a,c)J •A.[{ac,b)] for all positive integers a,b, and c. The

..\ -g.c.d., (a,b);~., of two positive integers is defined to be max[d: d l(a,b),~ [(d,b/d)] = lJ and a ~-reduce residue· system (mod n) as the set of incongruent integers (mod n) for which (a,n)A = 1. Defining p*(n: to be the multiplicative function given by ,u*(n) = - [1 - .\ (p)]k-l, p being a prime, the authors obtain

the generalized Mobius inversion formula F(n) =~f(a)~(c)~f(n) = E.F (a),u*(b).( (c), where in each summation, c = (a,b) and ab = n. They also define the generalized Ramanujan sum Cll. (m,n) to be

L exp(27rimx/n), (x,n)~ = 1, and investigate its properties. The usual Mobius inversion formula and Ramanujan sum arise on taking )..(n) = 1, for all n; and E. Cohen's results on unitary products [Math.

z. 74 (1960), 66-80; Duke Math. J. 28 (1961), 475-485] arise on taking~ (1) = 1, ~ (n) = 0, n > 1. (Received July 5, 1962.) •

592-68. C. E. WATTS, University of Rochester, Rochester 20, New York. Jordan-Holder theorems.

Let L: be the set of isomorphism classes of finite groups, F the free abelian group generated

by .:!:: , and let~ be a class of finite groups closed under taking normal subgroups and factor groups. A group is called .$-simple iff none of its proper factor groups belong to J!J. Let N be the subgroup of F generated by all elements G - H - G/H, where G E J!J and H is normal in G. Theorem. F /N is a free abelian group on the .ll-simple groups. If /1) is the class of all finite groups, this is equivalent tc the classical Jordan-Holder theorem. For other choices of i), generalizations of the classical theore are obtained. (Received July 5, 1962.)

592-69. H. R. FISCHER, Montana State College, Bozeman, Montana. Note on topological tensor products. Preliminart report.

The tensor product E 181 F, E and F locally convex real vector spaces, is given a topology such

that for every I.e. G, the space L(E 0 F ,G) of continuous linear maps E 0 F ~ G is isomorphic to the separately continuous, (Y, .7)-hypocontinuous bilinear maps E X F - G, ?"and :T being sets of boundc subsets of E,F resp. These topologies are Hausdorff provided E,F be Hausdorff. If E and F are barreled, the tensor product topologies considered all coincide with Grothendieck's "inductive topo logical tensor product." In particular, in this case, E 0 F is barreled and thus the Banach-Steinhaus theorem and the principle of uniform boundedness hold in L(E 181 F ,G). The construction generalizes immediately to arbitrary tensor powers 0PE. By taking quotients of these, one obtains "reasonable" topologies on the exterior powers .<1.PE and on the symmetric powers sPE, whence direct sum

305 topologies on A(E) = ~pil:O..t\.PE and S(E) = ~p~c;;%. These two algebras become locally convex algebras with hypocontinuous product. The exterior algebra A(E) leads to a theory of differential forms on E, while S(E) has applications to the theory of (locally) holomorphic maps E -F. The choice of the above topologies is motivated by the definition of higher derivatives of maps E -.. F. (Received July 5, 1962.)

592-70. J. R. RICE and J. S. WHITE, General Motors Research Laboratories, 12 Mile and Mpund Roads, Warren, Michigan. Optimum norms for data smoothing.

Assume given Yi = !'(A*,x) + ei•~(A,x) = 2:aiflli(x), ei =random errors with probability density function f(z)]. To estimate A* from the data Yi one may approximate the Yi by P(A,x) so as to minimize some measure of approximation, i.e. smooth the data. In particular let .f(Ap• x) be the best Lp approximation to the Yi· The parameters Ap are random variables and we set o-(Ap) = standai deviation of (A* - Ap). A definition of optimum and best Lp norms for smoothing is given in terms of comparisons of o-(Ap). Analytical and Monte Carlo results are given which show that optimum and best Lp norms for smoothing depend heavily on f(z). For example the L 00 norm is an order of magni tude better than the Lz norm for smoothing in the presence of uniformly distributed errors. (Receivec July 5, 1962.)

592-71. L. R. BRAGG, Case Institute of Technology, Cleveland 6, Ohio. The 'Heat' equation in n-space and certain related functions.

The solution of the 'heat' problem (*) ut(x,t) = ~nu(x,t), u(x,O) = j1(x) has the representation (**) u(x,t) = e tAn. !1(x) for suitable j1(x) where An is the Laplacian operator inn-space. The paper considers reduction properties of the operator etAn suitable for the study of special functions related to (*) that are useful in the representation of solutions of (*). Particular attention is given to the problem (*) in which 1iJ is dependent only on r, the radial distance from a fixed point. One of the main results is: Let k be a non-negative integer and let !l(r) = rk. Then (1), if k = 2m, the right member of (**) is m! 4 ~mL~/ 2 - 1 >(- r 2 /4t), L~/ 2 - 1 ) being the generalized Laguerre polynomial and (2), if k = 2m + 1, then the right-member of ("*) reduces to a terminating series in negative powers of r onl) if n is odd with r -n+2 being the smallest occuring power of r. Expressions for these functions through the use of a repeated operator are also noted. (Received July 5, 1962.)

592-72. ETHAN BOLKER, Harvard University, 2 Divinity Avenue, Cambridge 38, Massachu setts. Inverse limits of solvable groups.

A supernatural number (Tate) is a formal product of powers of primes. Infinitely many "factors" with possibly infinite exponents are allowed. If G is the inverse limit of the finite groups {Ga.} the order of G may be defined as the (supernatural) l.c.m. of the orders of the {Ga.j. Tate has defined Sylow subgroups of G and proved their existence and conjugacy. When each Ga. is solvable most of P. Hall's results on the Sylow systems of solvable groups g. London Math. Soc. 3 (1928), 99; J. London Math. Soc 1Z (1937), 198-200; Proc. London Math. Soc. 43 (1937), 316-323) can be proved for the limit G. The proofs use Hall's theorems, some of his methods, and compactness arguments in G. (Received July 5, 1962.) 306 592-73, P. C. CURTIS, JR. University of California, Los Angeles 24, California, Pointwise divergence of approximating polynomials.

Let Pn be a projection of the continuous functions on the circle D onto the trigonometric poly nomials of degree :3 n. Then by a theorem of Nikolaev liP n II ~ (l/4v"ti') log n. Call a point x E-D a point of divergence if supn(P nf)(x) = oo for some f. A natural conjecture is that each sequence of such projections has points of divergence. Under the following additional hypotheses this is indeed the case. LetT a be the translation operator, i.e. Taf(x) = f(a + x), and let m be an integer s:; 2n + l, Theorem, Let P n be a sequence of the above mentioned projections such that for infinitely many integers n,PnjT217'"/mj = T 27r/mlnj' If supjmj = oo, then the points of divergence for the sequence are of the second category, If lim supjm/nj > 'T'f/'{2, then every point is a point of divergence, Projec tions satisfying the above conditions include interpolating and least-squares approximating projections on sets of equally spaced points, (Received July 5, 1962.)

592-74, J, H. HODGES, ll4E Hellems, University of Colorado, Boulder, Colorado, Some integral matrix equations. Preliminary report,

Let m denote an arbitrary integer > 0 and M denote the 2 X 2 matrix diag (m, l), For arbitrary

integer s ;?; 2, let N(m,s) denote the number of pairs U, V of 2 X s and s X 2 matrices, respectively, having nonnegative integral elements ;a. m and such that UV = M. In general, N(m,s) is the number of solutions of a system of bilinear diophantine equations and it is shown that this number can be ex pressed in terms of multiple sums of products of the function d(n), the number of positive divisors of the positive integer n. For small s, some results of S. Ramanujan (On certain arithmetical functions, Trans, Camb. Phil. Soc, 22 (1916), 159-184) are applied to yield explicit formulae for N(m,s). Similar but somewhat simpler, results hold if elements of U and V are all required to be odd, The analogous quadratic matrix equation, whose solution involves representations as sums of squares, has been treated by L. Garlitz (A note on representations of quadratic forms, Portugal, Math, 15 (1956), 79-81), (Received July 5, 1962,)

592-75, P. T, RYGG, University of South Dakota, Vermillion, South Dakota, On minimal sets of generators of pure inseparable field extensions.

Let F be an extension field of K. M is a minimal set of generators of F over K if F = K(M) and K(M') =IF for every proper subset M' of M. Theorem, IfF is a pure inseparable extension of K with finite exponent, then there exist minimal sets of generators of F over K and any two such sets have the.same cardinal number. The proof follows from the fact that M is a minimal set of generators of F over FP(K) if and only if M is a minimal set of generators of F over K. (Received July 5, 1962,)

592-76. TAKAYUKI TAMURA, University of California, Davis, California. Semigroups whose proper subsemigroups are left ideals.

The structure of a ). -semigroup, a semigroup whose subsemigroups are all left ideals, is obtained by Kimura in his paper to be published, In the present paper a semigroup S is called a A-semigroup if every proper subsemigroup of S is a left ideal of S. Every element of S is of finite

307 order and there are three kinds of idempotents, accumulated idempotents, inner isolated idempotents and outer isolated idempotents, which play an important part. Theorem. LetS= L~£Q'l.f be the greatest semilattice decomposition of S. Then the semilattice Q is of length I; and if IQI > I, So is a 4-semigroup and each Sg

592-77. TAKAYUKI TAMURA and R. B. MERKEL, University of California, Davis, California. Semigroups in which all subsemigroups are one-sided ideals.

This is a generalization of ideal semigroups treated in Abstract 59 I- 10 fuhese Notices 9, (I962), 134] and N. Kimura's A- (f-) semigroups in which every subsemigroup is a left (right) ideal. The authors treat semigroups in which every subsemigroup is a one-sided ideal and in which at least one subsemigroup is a left ideal but not a right ideal and at least one other subsemigroup is a right ideal but not a left ideal. Such semigroups are called osi-semigroups. Let S be an osi-semigroup. S is unipotent with zero; abc = 0 for every a, b, c E s; S = A U B, A is a unipotent A. -semigroup, B is a unipotent f-semigroup, A and B satisfy certain restrictions; conversely, from such A and B, an osi-semigroup can be constructed. (Received July 5, I962.)

592-78. PHILIP WOLFE, The RAND Corporation, I700 Main Street, Santa Monica, California. An extended simplex method.

A mild extension of Dantzig's simplex method for linear programming permits the handling of nonvertex solutions of a set of linear constraints, allowing use of the simplex tableau in interior methods for linear or nonlinear programming. For one step of the procedure: Let a feasible point x = (x 1•••. ,xn) l!: 0 and any simplex method basis be given for the problem of maximizing f(x) under linear constraints Ax = b. Let c I•···•cn be reduced costs with respect to the current basis of the gradient of f (those corresponding to basic variables vanish). Analogously to a gradient method, set L'!.xi = ci if xj > 0 or ci > 0 and Llxi = 0 otherwise, for xi nonbasic; for xi basic, define Axi so that A l:lx = o. Determine the length of a step in the direction X by & = Max {G: X + a Llx ~ 0}; x = x + 8 Ax is the new point. If a nonbasic variable in x has vanished, nothing additional is done; if a basic variable has vanished, pivot in the tableau so that it becomes nonbasic and the nonbasic variable achieving Max xi becomes basic. If f is linear and the problem is primal nondegenerate and has a solution, the procedure terminates. (Received July 5, 1962.)

592-79. G. W. HENDERSON, University of Virginia, Charlottesville, Virginia. A characteriza· tion of the composants of compact indecomposable ·continua.

A. Lelek [fund. Math. vol. XLV I, I957] asks "is it true that each composant (of dimension n 5: I) of an indecomposable continuum is homeomorphic to a dense subset of Euclidean space (of dimension n + I)." The question is answered in the negative through the use of a characterization of the composants of compact indecomposable continua. (Received July 5, I962.)

308 592-80. S.C. PORT, The RAND Corporation, 1700 Main Street, Santa Monica, California. Recurrent events and fluctuation theory.

Let {Sn} be the sequence of partial sums of independent random variables with a common characteristic function¢(),.). If ~Wkt are the waiting times for the ladder points; z 1 = Sw 1, z 1 + z 2 = Sw +W Z + •.. + Zk = Sw + +W . and if Ln is the index of the first maximum of l 2•···· 1 1 ... k•···· {o.s 1, s 2 , •..• Snr then Z~otnE ~tASnxLn) = u(-\; tx)/u(.i\; t) [l - t¢(~ )), where It I ""- l, lx I ~ l, and u(.:\; t) = [l - E [ei).Z l t W l JJ- l. Differentiation with respect to x at 1, and use of the equivalence principle shows that u(A, ;t) = exp fn- 1tE[ei.\Sn; Sn :> 0) . From this Spitzer's identity (Trans. Amer. Math. Soc. 82 (1956), 323-339) and all related such identities can be derived by completely elementary and simple arguments. (Received July 5, 1962.)

592-81. J. E. MAXFIELD and HENRYK MINC, 205 Walker Hall, University of Florida, Gainesville, Florida. A doubly stochastic matrix equivalent to a given matrix.

Marvin Marcus and Morris Newman have shown in a recent paper that if A is ann-square positive definite symmetric matrix having non-negative entries there exists ann-square diagonal matrix D such that DAD is doubly stochastic. In this paper given the matrix A, there is displayed an iterative procedure for constructing the matrix D. (Received July 5, 1962.)

592-82. J. R. PERKINS, USAF Academy, Colorado. Counting reflexive, symmetric and transitive relations on a finite set. Preliminary report.

This report displays the number of relations with the properties of reflexiveness, symmetry and transitiveness that can be defined on a finite set. A table representation of the Cartesian product of a set With itself is introduced which facilitates counting the relations. The number of relations that can be defined on a finite set are counted. The properties that a relation can have are considered one at a time and the number of relations is derived for each of these properties. No method was dis- covered for counting the transitive relations. Relations that satisfy two properties are considered and the number of relations that satisfy each of the compatible combinations of reflexiveness and sym metry is found. Finally, the number of equivalence relations that can be defined on a finite set is derived. (Received July 5, 1962.)

592-83. J. S. FRAME, 704 17th Street N. W., Washington 6, D. C. and OLAF TAMASCHKE, Universitat Tiibingen, Tubingen, Germany. The product of the orders of certain centralizers in a finite group .. Preliminary report.

Let Ci (i = l, 2 •..•• m) denote the classes of conjugate cyclic subgroups in a finite group 7 of order h, let c i be the order of the centralizer of any one of the subgroups in Ci, and let c = TTc i denote the product of the m integers ci• It is conjectured that the integer c is a rational square for all finite groups. For a certain rational number v defined below, it is proved that cv is a rational square, and it is conjectured that v = l. Consider the field Qh of all the rational combinations of the complex hth roots of unity, which contains all the values of the complex-irreducible characters of ~ . The group

(){.of the automorphisms of Qh over Q1 induces a group of permutations P on the classes off with

309 transitive constituents of lengths vp ••• ,vm• and also a group of permutations P# on the irreducible characters of 7 ,with transitive constituents of lengths vt Corresp"onding permutation matrices. of P and P"" are intertwined by the character matrix, The rational number v above is v = lTvt/vi. The conjecture v = 1 (and c a square) is proved whenever the group P is cyclic, This includes the case that the complex-irreducible characters of J are all rational, (Received July 5, 1962,)

592-84, A. B. WILLCOX, Amherst College, Amherst, Massachusetts. Homogeneous algebras on locally compact abelian groups,

This paper is a continuation of a paper by the author (Pacific J, Math, 9 (1959), 1279-1294) which will be referred to as (I). Let G be a l,c, abelian group, p a measurable real function on G for which p(t) ~ 1, p(t1t 2) ~ p(t1)p(t2) on G. Let R(p) be the B-algebra (convolution) of all complex f for which llf II = j G lf(t) lp(t)dt < oo. Such algebras have been studied by Beurling, Wermer, Do mar and others. Under various additional conditions on p, R(p) is a homogeneous algebra on the character group G in the sense of (I), and general structure theorems in (I) yield new proofs of interesting properties of R(p). If K is a generating subsemigroup of G containing the identity then the B -algebra R(p,K) of all functions in R(p) which vanish a.e. outside of K-l is almost, but not quite, a homogeneous algebra on G. Its set of maximal regular ideals is essentially the set of homomorphisms r6 of K into the complexes for which lr6(t) I ;a p(t), t E K. This example suggests general theorems for algebras which are similarly almost homogeneous on connected I.e. abelian groups. Several such theorems are proved which generalize results of de Leeuw (Trans, Amer. Math. Soc. 87 (1958), 372-386). (Received July 6, 1962,)

592-85, W. S. EBERLY, 620 8th Avenue, Seattle 4, Washington •. A note on strongly positive operators.

Let E be a Banach space partially ordered by the cone K, and .t'(E) the Banach algebra of endomorphisms on E. We assume that the spectral radius r of the positive operator T E ,;ctE) is a pole of the resolvent, and that x 1 and f are corresponding characteristic vectors for T and T' respect ively. Krein and Rutman define an operator T to be strongly positive if given nonzero x in K, there exists a positive integer n such that Tnx is in the interior of K. Helmut Schaefer defines an operator

T to be quasi-interior if there exists a~ > r such that the linear hull of the order interval [9, TR).x] is dense in E for every nonzero x inK. Theorem 1, If x 1 is in the interior of K, f is strictly positive and the dimension of the null space of r - T is unity, then T (resp. I + T) is strongly positive if r is (resp. is not) the only point of

592-86. W. A, HARRIS, JR, University of Minnesota, Minneapolis 14, Minnesota, Equivalent classes of difference equations,

Consider the difference equation w(z + 1) = A(z)w(z) where w is an n-vector and A(z) is an n by n matrix such that z -p A(z) and z -qA- \z) (p and q integers) have convergent factorial series

310 representations. A necessary and sufficient condition that this difference equation have a fundamental matrix of the form W(z) = S(z)zR such that S(z) and z-rs- 1(z) have convergent factorial series repre sentations (r an integer) is that A(z) be equivalent to B(z) with a matrix T(z) (B(z) = T- 1(z + 1)A(z)T(z) such that T(z) and z -sT- 1(z) have convergent series representations (s an integer) where B(z) = I +E_~ 1 (B p+J!z(z + 1) ••• (z + p)). This paper presents a method of deciding when a given matrix A(z) has this property and how a suitable matrix T(z) may be constructed. (Received July 6, 1962.)

592-87. P. BASAVAPPA and NAOKI KIMURA, University of Saskatchewan, Saskatoon, Saskatchewan, Canada. On some matrix equations.

Let f(x,y) and g(x,y) be monomials in x andy of the form xy ••• xy, xy ••• yx, yx •.. xy or yx .•• yx. Matrix equations of the form f(X,X*) = g(X,X*), where X* is the complex transpose of the matrix X, will be completely solved. These equations are classified into six distinct cases except a trivial one. Also a similar result will be given when X is a bound linear operator in a Hilbert space. (Received July 6, 1962.)

592-88. W. R. HARE, JR., Duke University, Durham, North Carolina. Self-dual structures. Preliminary report.

A set S with a symmetric and reflexive relation d is called a structure. For a given symmetri relation d the conjugate relation d* is defined by xd*y iff xdy or x = y. (Thus, D U D* = S X S.) A structure (S,d) is called self-dual iff there exists a 1-1 mapping f of S onto S such that xdy iff f(x)d *f(y). In this paper a conjecture of L. Sathre (Thesis, University of Florida, 1960) is proved. Theorem. Let Sn be a set of n elements, nan integer. Then there exists a symmetric and reflexive relation d defined on Snsuch that (Swd) is a self-dual structure iff n = 4k or n = 4k + 1, k an integer. (Received July 6, 1962.)

592-89. MARVIN MUNDT, Valparaiso University, Valparaiso, Indiana, and F. M. WRIGHT, Iowa State University, Ames, Iowa. On substitution theorems for the Lebesgue-Stieltjes integral and the Stieltjes mean sigma integral.

The authors first establish in a relatively simple manner a known substitution theorem for the

Lebesgue-Stieltjes integral involving an integrator function 11 which is a monotone nondecreasing real-valued function on the entire real axis, and involving in the integrand real-valued functions on an infinite interval of the real axis. The authors have found this theorem to be useful in studying summability methods for improper Lebesgue-Stieltjes integrals. The proof developed here for the theorem in question is based essentially on the analogous result for the Riemann-Stieltjes integral, th.e relationship between the Riemann-Stieltjes integral and the Lebesgue-Stieltjes integral over a nondegenerate finite closed interval of the real axis, and the relationship between bounded a-measurab"le real-valued functions and continuous real-valued functions on a nondegenerate finite closed interval of the real axis. The analogous result for the Stieltjes mean sigma integral is then considered (Theorem 4.2 given by R. E. Lane, Proc. Amer. Math. Soc. 5 (1954), 59-66). (Received July 6, 1962.)

311 592-90. FRANK KOSIER and J. M. OSBORN, University of Wisconsin, Madison 6, Wisconsin. Nonassociative algebras satisfying identities of degree three.

Let A be an algebra over a field F and suppose that A satisfies a nontrivial homogeneous identity of degree 3 and 1 E A. Set (x,y,z) = (xy)z - x(yz) and ~.y] = xy - yx. Theorem. A is quasi equivalent to an algebra satisfying one of six specific identities or (1) p(y,x,x) - (p. + 1)(x,y,x) + (x,x,y) = 0 for some scalar p. Theorem. Let A be a ring of characteristic p, p prime to 6, satisfying (1) and (2) (x,x,x) = 0. (a) If p 1:- 5 and p of 1 then A is power-associative; (b) If e is an idempotent of A and p. f 1, -2, -1/2, then A = A11 + A 10 + A0 1 + A00; and if A is simple and A10 + A0 1 'f. 0, then A is alternative; (d) If p 'f. -1 and A is a semi-simple finite-dimensional power-associative algebra, then A is the direct sum of simple algebras and has a unity element. Theorem. Let A be a simple algebra over F of char. p.;:. 2,3 with 1 E A and e E A, e an idempotent not 1. If A satisfies (2) and an identity of degree 3 not implied by (2), then one of the following holds: (i) A+ is associative; (ii) (x,y,x) = 0 in A;

(iii) ~. [x,y]] = 0 in A; (iv) (x,y,z) + (y,z,x) + (z,x,y) = 0 in A; (v) A is quasi-equivalent to an algebra satisfying x(xy) + (yx)x = 2(xy)x or (x,x,y) = (y,x,x). (Received July 6, 1962.)

592-91. ANTHONY TRAMPUS, P. 0. Drawer QQ, Santa Barbara, California. A spectral mapping theorem for functions of two commutative linear operators.

Let z be a complex Banach space and Z* the dual space of z. Let tl and a· be Banach algebras of endomorphisms of :Z and:£*, respectively, and let/9 be the Banach algebra of endomor phisms of a. considered as a Banach space. With any U, V E a associate operators u+, v- E ;(j defined by u+(X) = UX and v-(X) = XV. Note that u+v- = v-u+. Let f( t;'l' ~2 ) be a function of two complex variables that is analytic in both variables in a domain containing CT(U) X CT(V). Then f(U+,v-) is defined as in 1). Schwartz, Two perturbation formulae, Comm. Pure Appl. Math. 8 (1955),

371-376]. Theorem 1. CT(f(U+,v-)) = f( if ~ 1 t- ~2 and g(~l' «:2) = g'(~ 1 ) if ~1 = <;'2• Corollary. o-(g 1(U)) = g(CT(U), CT(U)). Theorem 2. If p and ")) are characteristic values of U and V* (the adjoint of V) with corresponding characteristic vectors u € .:C and v* E :;(•, respectively, then f(p., J)) is a characteristic value of f(U+, v-) with corresponding characteristic vector u 8 v* E Cl defined by• (u 8 v*) z = v*(z)x. (Received July 6, 1962.)

592-92. IRWIN MANN and L. S. SHAPLEY, The RAND Corporation, 1700 Main Street, Santa Monica, California. A note on the values of large games.

The winning coalitions in a weighted majority game [q; w 1,. .. ,wn] are those sets Shaving weight

::EiE:Swi Or; q. The value 91i for a player i may be defined as 91i = E

312 592-93. CHARLES BRYAN, University of Arizona, Tucson, Arizona. On an iterative technique for algebraic systems. Preliminary report.

Consider the iteration xn+l = xn - VF - 1(xn;F (xn)) where F:Sl _..En with ith component fi (x), VF (x;h) is a linear operator on En which is represented by the diagonal portion of the Jacobian· matrix of F(x), and v- 1F(x;h)·denotes the inverse with respect to h. This iteration will converge to x* such that F (x*) = 0 if (1) there exists r > 0 and xo E Sl such that F is

twice Frc!chet differentiable and IIF "(x) II ;:;; N when x E S(x0,r), (2) VF (x0;h) exists and has a -1 -1 -1 -1 bounded inverse F 0 such that liFo II;; B, liFo (F(x0))11 ~ Q, liFo (F'(x0)- F 0)illi H < 1 and (3) h = NBQ/(1- H)2 ;:>!1/4, Q(1- 1- 4h)/ 2h(1 -H)< r. Furthermore llx*- xnll ~ anQ/(1 -a) where a = (H + 1 -1<1 - H)2 - 4QBI•n/(H + 1 + ./(1 - H) 2 - 4QBN). The proof is a modification of a method due to J. Schroder (Arch. Rational Mech. Anal. 1 (1957), 154-180). This iteration was proposed in 1959 by H. M. Lieberstein (Cf. MRC Rept. 80). The above criteria was used to establish existence

for the test problem of MRC Rept. 122 by using for x0 the final iterate obtained by the Nonlinear overrelaxation scheme which was also proposed in MRC Rept. 80. A maximum error bound of 10-5 was obtained. (Received July 9, 1962.)

592-94. H. M. LIEBERSTEIN and D. E. MYERS, University of Arizona, Tucson, Arizona. Probabilities of independent events. Preliminary report.

Let elements X 1,x2, ••• ,Xn of an abstract set A of order n 6:; 2 be called n-ary events and real numbers x 1,x2, ••• ,xn E [9,1], such Lxi = 1, assigned to them, be called probabilities of n-ary events. Where ,£.1 = fx,y E (9,1] X I!J,1] lx - y = 0} and ) 2 = {- lx + y ~ 1} a probability product is defined to

be a function f: )p J,2 -'-+ I!J,1] such that (i) on .11u) 2, f(x,y) ~ x, (ii) on ./1u)- 2, f(x,y) = f(y,x), and (iii) on ,L.2, Zf(x,y) + 2f(x,1 - x - y) + Zf(y, 1 - x - y)+ f(x,x) + f(x,x) + (y,y) + f(1 - x - y, 1 - x - y) = 1. Elements (a,a) of A X A are said to be independent events for n-ary events if for x the probability of 'a' andy the probability of a, a number called the probability of event (a, a) is assigned to the element

(a,a) by a probability product f. Forb E (!J,1], f(x,y) = 1/2(1 + b)xy for x > 1/2 or y ~ 1/2 and

x + y ~ 1, f(x,y) = xy for x ~ 1/2 and y l§ 1/2, f(x,y) = bx2 + (1 - b)x for x = y > 1/2, is a probability

product. For n = 2, let the triangle (0,0) {1,1) (p,2p- 1), 1/2 ~ p ~ 3/4, be denoted by Tp. Let any function represented by a curve, passing through {1/2, 1/4), all points of which are elements of Tp, be denoted by F. Then functions f defined by f(x,x) = F (x), f(x, 1 - x) = 1/2 - 1/2lf (x) + F (1 - x)] are probability products. (Received July 9, 1962.)

313 ABSTRACTS PRESENTED BY TITLE

The first three abstracts listed (Numbers 62T-l60, 62T-l63 and 62T-l66) were originafly print ed in the June 1962 issue of these NOTICES. They have been entirely reprinted below because of an error in the first printing.

62T-l60. D. P. SQOIER, Box 446, La Habra, California. Existence and uniqueness of solutions of the Poisson interface problem.

The Poisson interface problem asks for a function u satisfying Poisson's equation in a simply connected bounded domain D, taking prescribed values on the boundary C, and satisfying K

62T-l63. RAFAEL AR TZY, Rutgers, The State University, New Brunswick, New Jersey. Solution of loop equations by adjunction.

Let w be a word whose letters are a symbol x and elements of a loop L. Let n be the number of times that x appears in w. Form f(x) from w by inserting parentheses between its letters so as to make it into a well-defined product if juxtaposition means loop-multiplication, If r E L, f(x) = r is said to be an integral equation over L, n its degree. Every integral equation has a solution in a suitably constructed extension loop (Cn,L), Cn being the cyclic group of order n. The number of solutions is studied, and additional results pertaining to loops with identities are obtained for the special case n = 2. (Received April 24, 1962.)

62T-l66. J. A. WOLF, The Institute for Advanced Study, Princeton, New Jersey. On locally symmetric spaces of non-negative curvature and certain other locally homogeneous spaces.

Let N be a complete connected locally symmetric Riemannian manifold with every sectional curvature ~ 0. Then there is a finite covering N ·-N and a deformation retraction of N onto a compact totally geodesic submanifold, such that N' = E X T X M' whe·re E is a Euclidean space, T is a torus, M' is a compact simply connected Riemannian symmetric space, and the retraction of N lifts to a deformation retraction of N' onto T X M '. In particular, the betti numbers (singular theory) of N are finite and the Euler-Poincare characteristic X,.(N) is defined. Furthermore, /(,(N) !; 0. Finally, if -'\.(N) ;t-O then the fundamental group 7r1(N) is a finite 2-group. The rest of the paper is devoted to classification of the manifolds N with ;;>((N) #0 in a special case which includes the case where the non-Euclidean part of the universal Riemannian covering manifold of N is irreducible, and to proving some of the results on the structure of N under weakened hypotheses. (Received April 25, 1962.)

314 62T-178. J. D. HALPERN, 1736 Buena Avenue, Berkeley 3, California. Some structure in the hierarchy of maximal principles in set theory.

Let S be any one of the usual formalizations of set theory (e.g., Zermelo-Fraenkel or Godel Bernays set theory) with the axioms of choice and regularity deleted. Denote by I, II, III, respectively, the statements: Every Boolean algebra has a prime ideal, every infinite set algebra has a non principal prime ideal, every family of sets includes a maximal subfamily of pairwise incomparable sets. (Two sets are comparable if one is a subset of the other.) (1) If Sis consistent, I is not deduc ible in S from II. (2) If S is consistent, III is not deducible in S from I, (3) If S is consistent, I is not deducible inS from III. (1) answers a problem posed by Tarski (Bull. Amer. Math. Soc. 60 (1954), 391). From a result of D. Kurepa it follows that the conjunction of I and III is equivalent inS to AC (axiom of choice). (2) and (3) thus show that I and III constitute a decomposition of AC into the con junction of two mutually independent maximal principles. Tarski has asked what must be added to I to obtain AC. III seems to be a rather good answer to this problem since the proof of (3) shows that ACn• the axiom of choice for families consisting of sets with cardinality n, is not deducible from Ill where n is any natural number greater than or equal to 2. The ACn's appear to be the weakest forms of AC deducible from I. (Received April 30, 1962.)

62T-179. JAN MYCIELSKI, University of California, Berkeley 4, California. Connected spaces without infinite a--connected sets.

A space is o--connected if it is not a union of a~i 0 disjoint nonempty closed subsets. Two infinite sets M 1 and M2 are constructed in the Euclidean plane. They are connected, neither of them contains an infinite a--connected set, M 1 is locally connected and (I) M 1 = A U B, where A E Ga and 'n = ~ and (II) M2 E o8• It is known (A. Lelek, Fund. Math. 47 (1959), 265-276) that neither (I) can be re placed by M 1 € Gs nor (II) by M2 E F a-• Both examples ~ are obtained as intersections of descend ing sequences ufl>::> uf2> ::> ••• where uij) are modifications of the set constructed in Ex. 3 (Knaster, Lelek, Mycielski, Colloquium Math. 6 (1958), 227-246) and U~j) are modifications of the set constructed in _ti2, III, 6 (Kuratowski, Topologie, Vol. II 1961, 115). The modifications uij) [tJ~j)J contain closed [!lalf-closed] domains instead of the one-dimensional elements in the original examples. (Received April 30, 1962.)

62T-180. J. L. WALSH, 474 Widener Library, Harvard University, Cambridge 38, Massachusetts and OVED SHISHA, Numerical Analysis Section, National Bureau of Standards, Washington 25, D. C. Infrapolynomials with prescribed derivatives at given points.

Hitherto there have been considered· infrapolynomials A(z) =L~o a,y.")} with some a,= A (>')(0)/')1! prescribed. We generalize now as follows. Let z1 , z2 •····~ be distinct, and for j = 1,2, .• ;k let there be given values w~ 0 ), w1 1>•••• ,w~mj). Let A be the set of all polynomials A(z) satisfying A(')))(zj) = wf">, .)) = 0, l, ••• ,mj' j = 1,2, ••• ,k. Let an integer n ( ~ 1) and a pointset S be given. bJ!_ (n,A.,S) infra-

315 polynomial is an A(z) E 11 of degree -:;; n, having the property: there does not exist a B(z) E A of degree< n such that B(z) f A(z), jB(z)j < jA(z)j whenever z E Sand A(z) f. 0, B(z) = 0 whenever z E S and A(z) = 0. SupposeS is disjoint from T = [z 1,z2, ••• ,zkJ' finite (nonempty), its number of points being N = n + 2 - 2:~ 1(m. + l) (the smallest interesting). Let P (z) be the (unique) element of A of J- l k m+l degree Jlii n + 1- N, and L(z) the Lagrange interpolation polynomial to P(z)/TTj=l(z- zj) J on S. k m.+l Set Ao(z) 5 P(z)- L(z)Tij=l(z- zj) J • (Thus, A0(z) vanishes throughout S.) Suppose that two disjoint disks c 1:iz- c 1 1 ~ rl'C2:iz- c 2 i ;§ r 2 contain, respectively, T and the zeros of A0(z). Then c2 together with the disks iz- (c 1 - ec2)/(l- e) I~ (r1 + r 2)/ll- ej, en-N+2 = l, sf. l, contains all zeros of every (n, .A.,S) infrapolynomial. Analogues of the theorems of B8cher and Jensen hold. (Received April 30, 1962.)

62T-l8l. R. ]. TROYER and W. P. ZIEMER, Indiana University, Bloomington, Indiana. Topologies generated by outer measures.

In the papers [l) Density topology and approximate continuity, (Duke Math. J., vol. 28), and [2] Approximately continuous transformations, (Proc. Amer. Math. Soc., vol. 12), it was shown that a dimensional Lebesgue measure induces a topology on En (Euclidean n-space). In this paper, it is shown that a regular outer Caratheodory measure, Jl$, which is finite on compact sets also induces a topology on En in a similar manner. As in [l], it is found that this topology is not normal, but that it is completely regular. Complete regularity follows from an analogue of the Lusin-Menchoff Theorem. A transformation, f, from En into a metric space S is said to be fl$-approximately continuous if it is continuous with respect to the topology generated by fl$. As in [ZJ, f(En) is a separable subspace of S and f is of Baire class l. It is no longer true that En is connected in this topology, but with an additional regularity condition on fl$, the result is valid. (Received May 3, 1962.)

62T-l82. L. B. TREYBIG, Tulane University, New Orleans, Louisiana. Concerning continuous images of compact ordered spaces.

Mardelfi~ and Papi~ have proved that if each of A and B is a nondegenerate compact Hausdorff continuum and A X B is the continuous image of a compact ordered continuum, then both A and B are metrizable. The author·, after trying in vain to apply the above result to a particular nonconnected space by use of certain constructions, found a generalization of the aforementioned result. Theorem. If each of A and B is a compact Hausdorff space which contains infinitely many points and A X B is the continuous image of a compact ordered space, then both A and B are metrizable. (Received May 4, 1962.)

62T-l83. A. A, MULLIN, l08d EERL, University of Illinois, Urbana, Illinois. On anhomomor phic mappings between algebraic systems.

Definitions: If~ mapping from one structured set into (i.e., onto a subset of) another struc tured set is a homomorphism then the second set is said to be anhomomorphic to the first set; similarly, one can define an anisomorphic mapping. Lemma; If a structured set is anhomomorphic to a given structured set then it is anisomorphic to the given set. E. g., Theorem: Let (A,*) and

316 (B, o) be two algebraic systems that satisfy left-associative (right-associative or both) binary com position laws when three or more compositions are performed. Let M be a (~,S)-mutant set of (B, o). Let (T,*) be a subsystem of (A,*). Then M is anhomomorphic toT and, a fortiori, M is anisomorphic to T. The proofs avoid the use of the law of excluded middle. As a trivial corollary to the given theorem: in a group, a coset (modulo a subgroup) is anhomomorphic to the group iff the coset is different from the subgroup. (Received May 7, 1962.)

62T-l84. S. K. STEIN, University of California, Davis, California. Chains in partially ordered sets.

R. P. Dilworth [Ann. of Math. 51 (1950), 161-166] proved that if Pis a finite partially ordered set then the number of chains in a minimal decomposition of P is equal to the maximal number of mutually unrelated elements of P. From this follows immediately Theorem; If P is a partially ordered set with p elements and mn < p then either P has a chain of length m or P has n mu~ually unrelated elements. As a special case of the Theorem we have a result of Erdos and Szekeres [Compositio Math. 2 (1935}, 463-470]: In a sequence of more than mn distinct real numbers there is a decreasing subsequence of m terms or an increasing sequence of n terms. (Received May 7, 1962.)

62T-185. WITHDRAWN

62T-l86. A. P. HILLMAN, D. W. FORSLUND and G. J. GIACCAI, University of Santa Clara, Santa Clara, California. A vector space of products of determinants.

Let Yij (i = l, .•• ,n; j = 1, ... ,2n) be 2n2 indeterminates over R; the field of rational numbers.

Let (a1 •••• ,an) with 1 a ah ;:::; ah+l li- 2n be the determinant whose hth row is ylj , ••• , ynj with j = ah. Let P = F 1 ..• F d where Fk = (alk•· .. •llruc ). Call P an a-product if for each i and j either ahi ~ ahj for all h or ahi <: ahj for all h. Let G = [a 1, ••. ,a2n; b 1, .•• ,b2nJ be the determinant whose hth row is y li"··•Yni, y lj•···•Ynj with i = ah and j = bh. If ah = bh for n + 1 values of h, G vanishes and a Laplace development provides an identity in products (c 1, •.• ,cn}(d 1, .•• ,dn}. Using such identities, it is shown that the a-products are a basis for the vector space over R generated by all products P. This is applied in the exponent problem for the differential ideal of the Wronskian of n differential indetermin ates. (Received May 4, 1962.)

62T-186. WITHDRAWN

62T-l87. A. P. HILLMAN, D. W. FORSLUND and G.]. GIACCAI, University of Santa Clara, Santa Clara, California. Differential ideals of Wronskians.

Let Yl'···•Yn be differential indeterminates over the rational numbers. Let /::;. = n(n - 1)/2.

Let F = (a 1, ..• ,an) be the determinant whose ith row consists of the aith derivatives of y 1, .•. ,yn. A product P = F 1 .•• Fd of such F's of total weight w is in the differential ideal !}¥]of the Wronskian W = (0, l, ••• ,n - 1) if w < f(d} where for n > 2, f(d} = [(d + 2!:;, + 5}d - 2]/2. Also Fq E !}¥]for q;;; g(v)

where vis the weight ofF and for n ;> 2, g(v) =max (l,v- A) for v ~A + 3 and g(v) = 2(v - !:;, - 2)

for v ;> /::;. + 3. For n = 2, f(d} = d 2 + 2d and g(v) = max(l,v - 1). This furnishes partial answers to Ritt'.s Questions for Investigation, Nos. 3 and 6 (Differential algebra, Amer. Math. Soc. Colloq. Publ.

317 val. 33, p 117) in a form similar to H. Levi's work on l}rPJ and ~v] (Trans. Amer. Math. Soc. 51 (1942.), 532.-568). (Received May 4, 1962..)

62.T-188, H. E. SALZER, General Dynamics/Astronautics, Mail Zone 591-0, Building 4, San Diego 12., California. Multiple quadrature with backward differences.

The differential equation (1) y(n) = f(x,y ,y' •••• ,y 1, may be integrated stepwise at

intervals of h, using the following n-fold quadrature formulas: open or predictor type, (2.) 'V~ 1 ;= nr ~M (n) m n _ n{ "'M (n) m h lfo + ~m=.lPm V fo+ ••• J. and closed or corrector type, (3) V y1 - h f 1 + L-m=lcm V f 1 + ••• J. 1n terms of rth Bernoulli polynomials of the rth order, BJr>(x), where B;.r):; B~r)(O), (n) m (m-n+l) (n) m (m-n+l) (r) (4) P m = (- 1) Bm /m! and (5) Cm = (- 1) Bm (1)/m! To calculate y , replace y by y(r) and n by n- r in (2) and (3). This scheme allows independent parallel computation of y, y', •• ,,y

from the same set of differences off 5 f(x,y,y' .... ,y~) and(- l)IDc~>. (Received May 9, i962..)

62.T-189. H. F. BECHTELL, Lebanon Valley College, Annville, Pennsylvania. On finite groups whose generators are subgroup generators. II. Preliminary report.

A Ll.-group is defined as a group, G, in which the Frattini subgroup, 4'(H), for each subgroup H s; G, is the identity element (cf. Abstract No. 62.T-44, Notices Amer. Math. Soc. 9 (1962.), 46). A unique, characteristic subgroup, called the .A-commutator subgroup, [G, A], is defined as the intersection of the normal subgroups N of G having the property that G/N is a A-group. It is

characterized by 4'(H) !;;;;; (G, A), for all subgroups H ~ G and furthermore is generated by the if'(H) for all H s; G. An immediate corollary is that the alternating group An= (Sn' A], for the symmetric group Sn. There exist no simple Ll-groups of odd order. Finally, if G = A ® B, then [G,.O.] = [A,6) ® (!:I,Ll.]. (Received May 9, 1962..)

62.T-190. C. D. WYMAN, Dartmouth College, Hanover, New Hampshire. A pseudo-simple set with no recursive center.

Mesoic sets were introduced by Dekker and Myhill (Proc. Ainer. Math. Soc. 4 (1953), 495-501, and Zeit. F. Math. Logik und Grundlagen der Math. 1 (1955), 97-108, respectively) as a means of separating the one-one and many-one reducibilities. Certain mesoic sets, among them those dis played by Dekker and Myhill, have come to be known as pseudo-simple. A pseudo-simple set is a recursively enumerable, but not recursive, set whose complement is the union of an infinite recur sively enumerable set (called the center) and an immune set. The question has been raised as to whether every such set has a recursive center. If one observes that a pseudo-simple set A has no

recursive center iff it has a center B for which there is no recursive set R, B ~ R ~A. then a

318 negative reply may be given by exhibiting two disjoint recursively enumerable recursively inseparable sets whose union is simple. Two such sets may be generated by enumerating simultaneously all recursively enumerable sets and all recursive sets, causing each of the former to intersect A or B and each of the latter not contained in A or B to intersect both while taking care to leave A U B infinite, (Received May ·10, 1962.)

62T-191, ROBERT HERMANN, University of California, Berkeley, California, A geometric remark concerning the bounded symmetric domains.

Let G be a compact connected simple Lie group, K a symmetric subgroup with nondiscrete center, M = G/K has an invariant Kahler metric, Gc• the complexification of G, also acts on G/K. Let G' be the subgroup of Gc defining the real form of Gc corresponding to K. D = G'• x 0, the orbit of G' at x0, the identity coset of M, is then a noncompact, irreducible symmetric Kahler manifold of nonpositive curvature, isomorphic to G' /K, and an open subset of M. These facts are well known

(A. Borel, Proc, Nat, Acad, Sci, 40 (1954), 1147-1151). For each unit tangent vector v to x0, let d0(v) (resp, d 1 (v)) be the length of the largest one-sided open geodesic of M starting at x0, tangent there to v, that lies completely in D (resp. that contains no conjugate points of x0). Theorem 1. d 1 (v) = 2d0 (v). Theorem 2, Dis covered by the set of all geodesics of M of length less than d0 (v) starting at x0 and tangent there to v, where v runs over the set of all unit tangent vectors at x0 • Roughly, Dis contained within half the con]ugate (hence also the minimum) locus of M at x 0• As corollary, every such G' /K is imbedded as a bounded domain in the tangent space to M at x0, a complex Euclidean space, This proof is considerably simpler than Harish-Chandra' s proof that all such G'/K are bounded do.mains (Amer. J, Math, 77 (1955), 743-777). (Received May 11, 1962,)

62T-192. MARTIN SCHECHTER, New York University, 25 Wave.rly Place, New York 3, New York. A general interpolation theorem. Preliminary report,

Let B0 , B, C be Banach spaces such that B0 C B with continuous injection. Let H be the set of all bounded linear mappings of B into C with norms ;a! 1, and let J be any subset of H. A Banach

space Co has property (Bo,J) if every mapping 7r E J maps Bo into Co anli there is a constant K such

that 11»-x II Co \li K llx IIBo for all 1T E. J and x E Bo. Theorem. There is a unique Banach space c0 such that (a) Co has property (Bo,J) and (b) a Banach space C' has property (Bo,J) if and only if c0 C C' with continuous injection, In particular, if B 1 and C 1 are Banach spaces such that B 1 C B0, C 1 C C with continuous injections and J consists of those mappings in H which take B 1 boundedly into C 1, then B0 , c 0 form an interpolation couple. (Received May 11, 1962,)

62T-193. ALFRED GRAY, University of California, Los Angeles and S.M. SHAH, University of Kansas, Lawrence, A note on entire functions and a conjecture of Erdos. II.

Let p(r) and M(r) denote as usual the maximum term and the maximum modulus .of an entire f(z) and let l'n = f'(n) denote the jump points of the rank :Y(r). (See Alfred Gray and S, M. Shah, !!.... note on entire functions and a conjecture of Erdos, Notices Amer. Math, Soc. 8 (1961), 572). The

119 following results have been proved. (i) If L = lim supn-+ 00p(n + 1)/f'(n) :>1 or 2 (ii) lim infr....,.00logp(r)/(log r) <. oo, then lim supr-ooJl(r)/M(r) >lim infr ..... ooP(r)/M(r). (iii) If L = 1,f

62T-194. DAN AMIR, Hebrew University, Jerusalem, Israel. C(S) spaces of Ph, type.

A separable Banach space B is called a P;{ space if from every separable Banach space Z :::J B

there is a projection on B with norm ~ ).. Let C(S) be the Banach space of all the continuous real valued functions on a compact space S with the maximum norm, then the following are equivalent: (1) C(S) is a P,k space for some finite).. (2) S is homeomorphic to the set of ordinals r(,S") = {'[; 'l;::; > < w'j with the order topology. (3) C(S) is isomorphic to co. (Received May 14, 1962.)

62T-195. S. ·W. YOUNG, 2701 Swisher, Austin, Texas. A characterization of closed convex sets in a complete inner-product space. Preliminary report.

Theorem. If M is a closed point set in a complete inner-product space, then M is convex if and only if there exist a point set K and a point P of K such that M is the set to which X belongs if and only if P. is the nearest point of K to X. The theorem is proved by the application of some basic theorems on convex sets and bounding planes. An example is given of a convex set M which is not closed for which there does not exist such a point set K. (Received May 14, 1962.)

62T-196. ALFREDO JONES, University of Illinois, Urbana, Illinois. Integral representations of the direct product of groups.

Let G 1 and G2 be finite groups, K an algebraic number field, R the algebraic integers of K, and if P is a prime ideal of R let Rp be the ring of the P-adic valuation of K. It is shown that if K is a splitting field for G 1• and if P is prime to the order of G1, then every indecomposable Rp (G1 x G 2) module is the tensor product of indecomposable RpG 1 and RpG2-modules. If K is a splitting field for G 1 X G 2• and if G l•G 2 have relatively prime orders, then every irreducible R(G 1 x G2)-module is the tensor product of irreducible RG 1 and RG2-modules. In general the hypothesis about the orders of G 1 and G2 can not be dropped. (Received May 21, 1962.)

62T-197. ALFREDO JONES, University of Illinois, Urbana, Illinois. The number of indecom posable integral representations.

It is shown that a finite group G has a finite number of inequivalent indecomposable represen tations over the rational integers if and only if for each p, each p-Sylow subgroup of G is cyclic of order at most p2• The results of Heller and Reiner (to appear in Ann. of Math.) are used in the proof. (Received May 21, 1962.)

62T-198. GEBHARD FUHRKEN, University of California, Berkeley 4, California. A Skolem-type normal form for languages with a generalized quantifier.

Let L be a language obtained from a first-order language by adding a new quantifier Q.

320 Theorem 1. For every countable set r of sentences of L, there is a countable set r• of first-order sentences (among whose symbols are new predicates U and V) having the property: If Q is read

"there are at least ~a+ 1 ••• ", then a relational system tJt of power at least Jl\a+1 is a model of riff CJt can be expanded to a model of f'* U{Qv Uv, -,Qv Vvj by adjoining additional relations. Using one of a number of variants of Theorem 1 and Vaught's Lowenheim-Skolem theorem for two cardinals in Abstract 578-58, one obtains: Theorem 2. If a sentence rJ of L is logically valid when Q is read

"there are at least ~ 1 .•• ", then rJ is logically valid when Q is read "there are at least .li" a+ 1 ••. ". For another application of Theorem 1, see the next abstract. (Received May 21, 1962,)

62T-199. GEBHARD FUHRKEN and R. L. VAUGHT, University of California, Berkeley 4, California. Noncharacterizability of the ordering of the natural numbers.

Theorem. The relational system (cu.<) cannot be relatively characterized by a countable set of sentences of the language L of the previous abstract, when Q is read "there are at least *q ... ", "Relatively characterized" means characterized up to isomorphism, using additional relations (for a precise definition cf. Abstract 62T-95 by W. Craig and W. Hanf). The proof uses a variant of Theorem 1 of the preceding abstract together with a strengthened version of the theorem in Abstract 578-58. The Theorem should be compared with the result of Abstract 591-4 that((<),<) £!!!_be so characterized using -~ 1 sentences. (Received May 21, 1962.)

62T-200. ROBERT HERMANN, University of California, Berkeley, California, A unified description of the symmetric bounded domains.

Let G' be a noncompact, simple Lie group, K a maximal compact connected subgroup whose center is nondiscrete. It is known that D = G' /K is a symmetric bounded domain in the sense of E. Cartan. We present here a unified way of describing the imbedding. Let Q' (res{'. !_) be the Lie algebra of G' (resp. K) and let M. be the orthogonal complement of! in Q' with respect to the Cartan Killing form. It is known that M can be made into a complex vector space so that AdK preserves the complex structure. For X E M• let u(X) be the largest of the absolute values of the eigenvalues of Ad X. Theorem. Dis isomorphic to fX E M: p(X) < 1r]. In case Dis a symmetric space of rank one, u is a constant multiple of the distance from the origin, hence it may be said that this theorem gives a representation of the bounded symmetric domains as the interiors of "generalized spheres." (Received May 21, 1962.)

62T-201. JOSEPH ROTMAN, University of Illinois, Urbana, Illinois. The Grothendieck group of torsion-free abelian groups of finite rank.

Let e denote the category of torsion-free abelian groups of finite rank; let Z denote the rational integers and N the positive integers. Theorem. The Grothendieck group K(e) is a ring which is isomorphic to the ring of all functions from N to Z which take on only finitely many values. If A E: (?,

a composition series of A is a chain A = A, 0 ::::;:1 A 1 ::::;:1 ... :J An = 0 whose factor groups A/Ai+ 1 are torsion-free of rank 1. Corollary. Even though the Jordan-Holder theorem in(! is false, the number of p-reduced factor groups of a composition series of A is an invariant of A, and, indeed, is equal to the p-rank of A. (Received May 21, 1962.)

321 62T-202. MORTON DAVIS and MICHAEL MASCHLER, Econometric Research Program, Princeton University, 92-A Nassau Street, Princeton, New Jersey. Existence of stable payoff con figurations for cooperative games.

A payoff configuration (x;:&) =(x 1, x 2, ••• ,xn; B 1, ••• ,Bk) for a cooperative game given in charac teristic function form is defined to be stable as in these Notices' 8 (1961), p. 261, except that (ii) is re placed by individual rationality requirement (ii)' xi ~ v(i), i = 1, 2,. .. ,n; and that only~ player can object against another~ player, both in a B j E fi9. A player a is called stronger than a player f3 w.r.t. (x;~, if a has an objection against IJ• which cannot be countered. This binary partial relation is asymmetric and not intransitive; (however, as examples show, it is not necessarily transitive). From this, using Knaster-Kuratowski-Mazurkiewicz application of Sperner's Lemma (Fund. Math. 14

(1929), 132-138), one proves that for a given (x;~). one can modify the payoffs to members of one coalition so that no player in this coalition is then stronger than another. In particular, there always exist stable imputations. By Eilenberg-Montgomery fixed point theorem (Amer. J. Math. 68 (1946),

210-222), if each coalition in~ contains 3 players at most, there always exists a payoff x such that (x;~) is stable. Conjecture: the same holds without any limitation on the coaliti.ons of 27. (Received May 22, 1962.)

62T-203. JANINA SLADKOWSKA, Applied Mathematics and Statistics Laboratory, Stanford University, Stanford, California. A class of domains with Bergman-Shilov boundary.

Let 7'3 be a domain in the space of 2 complex variables which possesses the following properties. (1) Its three-dimensional boundary J) consists of n segments Jl~, k = 1, ••• ,n, of analytic hypersurt'aces. Each of the segments is a one-parameter family of analytic surfaces, and it is possible to express it in the form: (*) zl( = hlik(Zk,~k), l(= 1,2. IZkl ;a 1, 0 !Iii Ak ;a 2'11'", where hlfk are continuous by differentiable functions. The se~ (*) for fixed A.k is called a lamina and designated by p~( A.J2. (2) h.-k(Zk,;l.k) ih(Zk.~k) if (Zk,),k) i(Zk,~k>· (3) For every IZ~O)I <-1, A.~0 >and for sufficiently small tr > 0, the set of points (*) which corresponds to the values IZk - z~O) I< ct, l~k - ~~O) I~ ct of the parameters, contains all the points of J.3 lying sufficiently close to z~) = hl{k(z~0 >,.-..~) ). ·.1fl, which was introduced by Bergman (Math: z. 39 (1934), 76), is called an an analytic polyhedron. On J.3 lies a two-dimensional manifold 5 2, the Bergman-Shilov boundary surface of lJ. Let#~ designate an analytic surface which intersects J} along a line f 1• Jl}'2 = #~ n tJ and 11 = ;92 n ,63 possess the properties 1-3 (Bergman, Math. z. 63 (1955), 188); 5 is replaced by a weaker condition. The intro duction of domains, possessing an intersection surface of the structure described above, is of interest. Many theorems of one complex variable can be generalized in the domains 13. (Received April 23, 1962.)

62T-204. JANINA SLADKOWSKA, Applied Mathematics and Statistics Laboratory, Stanford University, Stanford, California. Bounds for analytic functions in domains with Bergman-Shilov boundary.

In the following is considered analytic polyhedra (as introduced by Bergman) and possessing

322 an intersection surface with properties described in a previous abstract. Let f be a function of two 2 2 complex variables z 1,z 2, regular in ffi- :1 , continuous in fi- .:; + /l n J 2. f E ,Ety (JI~,P), 2 p :;;> 0, if (l) f(z 1,z2) # 0 in 73 t ~l n :fi ; (2) in every lamina ~~(~k) Which is intersected by~~' f is mean multivalent of the order pk(~k) in the sense of Spencer or Biernacki; (3) pk()tk) is square integrable in 4• where -"k is the set of Ak' which corresponds to the laminas mentioned above; (4) """""·- ((l /211") I' pk2 . (.ilk )d .:tk) 1/ 2 ;:f; JP where J designates the number of segments~~ intersected ~J-l C/.<:Jkj J j J by .0'5; kj' j = l,. .. ,J, are the indexes of these segments. The author gives the lower and upper bounds for jf J on ;!12 for every f E fxr<.f'5,P) in terms of (a) constants which depend only on the minimum and the maximum of jfJ on an arbitrary cine-dimensional manifold of J.3; (b) constants depending only on "t3 and_#~. (Received April 23, 1962.)

62T-205. WITHDRAWN

62T-206. SOLOMON FEFERMAN, Stanford University, Stanford, California. Provable well- orderings of and relations between predicative and ramified analysis.

Reference is made to Abstract 583-47 and the abstract Formalizing predicative analysis (FPA) by G. Kreis~! in these Notices. In addition to the schemes of (FPA) we consider also EPA(-), where no free function variables appear. Let TO", T~>, T~) be progressions of theories (associated with notations a-) based on the formalizedw-rule and EPA, (-), (+) resp. Let Ra- be a progression of ramified theories, with the following main rules: (i) functions a. (at l) can be introduced in Ra-t 1 by definitions using only quantification over ;J( r), various .,.. ~ a-; (ii) formalization of the rule, for limit a-: if for all r< CT, J\a.(T)Q(a.(r)) is provable at an earlier stage, then 1\ a.(cr~(a.(a'~. Define ordinals ::\: ("))) by: X (O) = e ; for )) 1 0, X(>') = a.th critical X.()-!) number for all p. < :V. Let a a. a. a. -Y, = x.b-.>) 'T =least critical r-number. Note ?'2 =the .:to of Abstract (583-47). We use ;:::d in the following to indicate that a certain intertranslatability condition holds, preserving TT ~ statements. Theorems: l. T (-) ,. U R (n < ). 2. T = U R (a-< G)GJ). 3. U T .., R , a(T') ranging over 0 n 1J o o U ,... T-(R-) autonomous notions. 4. The provable well-orderings of To are exactly those <.c.>"':

5. The provable well-orderings (autonomous notations) of the T CT (hence by 3 of the R..J are exactly those <:. T. (The corresponding result for the T~) with bound ;!.0 is the precise content of Theorem 2 of 583-47.) 6. No new well-orderings are provable using the T~+) or an autonomous infinitary .v-rule. Theorem 5 for the ~has been independently obtained by K. Schutte, at about the same time as our work. (Received January 2, 1962.)

62T-207. MITSURU Y ASUHARA, University of California, Berkeley 4, California. On categori cal PC classes of an extended first order language.

The following notation is used; L for a first order language with equality; M for a language obtained from L by adding a new quantifier expressing "There are infinitely many---"; WS for a week second order language; PC(M) and PC(WS) for PC classes of M and of WS; I(ot) for the class of all relational system isomorphic to rJt. Theorem 1. For any ordinal number ?( and its well-ordering <,

I( ( "?, <. {) E PC(M) iff "'{is constructive. Theorem 2. (a) For any hyper arithmetical predicates,

Rl•···• Rn, I( ( c..>o, Rl•···• Rn l) EPC(M). (b) If I( ( a.~0 , <,R 1, ... , Rn)) EPC(M), then R 1, ... , Rn

323 are hyperarithmetical, where < is the well-ordeiing of aJ0 [cf, 3,3A, p. 199, Journal Symb, Logic, vol. Z3 ],In Theorem 2b, whether ( aJ-0, <,Rr'"'' Rn) can be replaced by·(4J0,Rr'"'' Rn} is an open problem. For the proof of Theorem Za, note that I( ( C

6ZT-ZOB. D. B. SHAFFER, r56 Intervale Road, Stamford, Connecticut. The circles of curvature of level loci and orthogonal trajectories of harmonic functions,

Bounds are derived for the curvature K of the following loci; the level loci and orthogonal trajectories of a polynomial p(z) all of whose zeros are contained in a disc D; and the level loci of a rational function R(z) all of whose zeros and poles are contained in discs Dr and Dz respectively, Dr n Dz = 0, The bounds for L}l: lp(z)l = p hold for the level curves of Green's function with pole at infinity of a region whose boundary C D. The bounds for IR(z) I= p are valid for a harmonic function

u = r on a set of curves C li C D1 and u = 0 on Czi C Dz. For the former loci the bounds for K at P, p ¢ D,-are expressed in terms of the reciprocals a,b (a !lii b) of the intercepts of the nofmal to L }1 at P with the boundary C of D: (Walsh, Amer, Math. Monthly, 1935, proved that the normal to L}l

must intersect D) K ;; b; a:; K for Za e: b and - a - b + Z f?ab]1/Z a K, for Za ;i b, For the orthogonal trajectories IK I ;Iii ([b]1/Z - [a]r;z)z. It follows that the set L,.u is convex exterior to a concentric disc of radius lf}1/Z radius of D. (Received May 29, 1962,)

62T-209. R. I. SANDLER, Institute for Defense Analyses, Princeton, New Jersey. On sub planes of free planes, Preliminary report.

Following the lead of L. I. Kopejkina, the author has continued the investigation of the properties of subplanes of free planes, A theorem concerning the existence of "minimal sets of free generators" has been proven, and from it, one can deduce various theorems, e.g., Theorem 1: The set Sm (ordered

by inclusion) of all subplanes of rank ;l!i m of a free plane ,.k of finite rank, satisfies the maximal condition, Theorem 2: A free plane of finite rank always has subplanes of infinite rank. Work is con tinuing, and it appears as though the main tool theorem (analogous to Theorem 5* p. 160 of Specht's Gruppentheorie) will have fairly wide application in this type of study, (Received May 28, 1962.)

62T-2rO. ECKFORD COHEN, University of Tennessee, Knoxville, Tennessee. Linear and quadratic equations in a Galois field with applications to geometry.

Let F denote a Galois field of odd order and let Ns,t(a,b) denote, for arbitrary a,b of F, the 2 2 number of solutions in xr, ... ,xt of a= a 1xr + ... + atxt, b = b 1x 1 + ... + btxt, where a 1, ... ,at are nonzero elements ofF and exactly s of the elements b 1, ... ,bt are nonzero (1 ~ s l!; t). Explicit formulas for N 8 ,t(a,b) are determined, The case s = twas considered previously on the basis of exponential sums. The method of the present paper involves the reduction of the problem, for arbi-

324 62T-211, R. H. BING, University of Wisconsin, Madison 6, Wisconsin and A, KIRKOR, Warsaw University, Warsaw, Poland, An arc is tame in 3-space if and only if it is strongly cellular.

An arc A in E 3 is strongly cellular if A is a strong deformation retract of a 3-cell Q in E3; i,e, there exists a mapping h: Q X I -Q such that if ht(x) = h(x,t) and S = boundary of Q, then:

(1) ho = id and htiA = id for all t, (2) h 1 (Q) = A, (3) htls is a homeomorphism for t < 1,

(4) ht(S) o htt(S) = 0 if t =It'. By M. Brown's generalization of the Schoenflies theorem there will be no loss of generality if the cell Q is assumed to be a geometric ball. Hence the theorem in the title is the corollary of the following one: Let Q* be the ball x2 + l + z 2 ~ 1 and A* be the segment

- 1/2 § x ;iii 1/2, y = z = 0, Given a homeomorphism H of A* onto A there exists a homeomorphism H* of Q* onto Q which is an extension of H. The construction of H* repeatedly uses the following lemma: For any point p of A there exists such a disk Dp that: (1) Dp • ht(S) is a simple closed curve fort < 1 or the point p for t = 1, (2) given any other point q of A there exists such a number tq that 0 ~ tq < 1 and Dp • ht (Q) • h~ 1 (q) = 0. (Received May 28, 1962,) q

62T-212. R. J, DUFFIN, Carnegie Institute of Technology, Pittsburgh, Pennsylvania, and D. HAZONY, Case Institute of Technology, Cleveland, Ohio, The degree of a rational matrix function,

Let Z(s) be an n by n matrix whose matrix elements are rational functions of the complex variable s, The degree of Z(s) is here defined to be the maximum of the degree of the numerator polynomial of det(Z + A) for any constant matrix A, Brockway McMillan proved theorems on the degree of Z which are analogous to properties of the degree of a scalar rational function (Bell System Tech, J, 21 (1952), 217-299, 545-600), A different and simplified proof of McMillan's theorems is given here, Several additional theorems are also proved, Thus it is shown that the degree of a minor matrix cannot exceed the degree of the whole. Other new results concern the degree of the product of two matrices. The properties .of the degree have application in the synthesis of an impedance matrix of an electrical network. (Received May 28, 1962.)

62T-213. D. J, RODABAUGH, Apartment 310, 2851 South Parkway, Chicago, Illinois, A generalized associator identity. Preliminary report,

5 Algebras are studied satisfying an identity of the form (x,y,z) = ~i= 1ei(7ri(x), 17"i(y), 1T'i(z)) where ?ri is not the identity but is in s 3 and ei is in F the base field which for some results is assumed to be algebraically closed. The identity reduces to (x,y,x) = a(x,x,y) where a f. 0, - 1 or is a well-known identity, Such an algebra has the idempotent decomtJosition of associative algebras but not the usual multiplication of subspaces. If A is semisimple and a I' 1, - 1/2, - 2 then A is a direct sum of simple algebras, If A is simple and a i= - 2 then A is associative or nodal, If A is simple and a = - 2 then A 11 + Aoo = A for every idempotent or A is associative, If A is semisimple, a = - 2 and

e primitive then A10A01 = Au or A10 + Ao 1 = 0 and A can be written as the vector sum of an assoc-

325 iative ideal and subspaces of the fol'rn A 11 (e) for e primitive. Examples exist showing the limits of the theory. (Received May 31, 1962.)

62T-214_. L. J. MORDELL, University of Arizona, Tucson, Arizona. On a cubic congruence in three variables. II.

Let p be a prime and f(x,y,z) a cubic polynomial with integer coefficients which is not a function of only two independent variables and which is irreducible algebraically. Let N be a number of solutions of f(x,y,z):: 0 (mod p). Then possibly IN - li < kp, k an absolute constant. Part I, proves this when f(x,y,z) = ax3 + bx2y + cxy2 + dy3 + k. It is now proved for f(x,y,z) = ax3 + bx2y + cxy2 + dy3 + lx +my. In Part III, it is proved for f(x,y,z) = ax3 + by3 + cxy +d. There is now a simple closed expression for N. All three are cases of congruences, say, g(x,y,z) k in a parameter k with N(k) solutions. In general, no simple relation connects the N(k). When N(k) has the same value for many k e.g., if k is a quadratic residue and also when k is a nonquadratic residue, then results can sometimes be found by considering g(x,y,z) = k as a congruence in four variables. By using the result of Part III, progress can be made with estimating the exponential sumS= ~e ((211i/p) f(x,y)), where f is a cubic polynomial. The conjecture S = O(p) can be proved when f(x, y) = ax3 + by3 + cxy. (Received May 31, 1962.)

62T-215. HERBERT FEDERER, Brown University, Providence 6, Rhode Island. The behavior of complex algebraic varieties as integral currents.

Theorem 1. The operation of integrating forms over (the nonsingular part of) a complex projec tive variety is an integral current, with boundary zero; hence virtual varieties can be identified with certain integral cycles. Theorem 2. The intersection V·W, where Vis an rn-dirnensional variety and W is an n-dirnensional projective subspace, depends continuously (as a current) on W, whenever it is defined (algebraically). Theorem 3. The algebraic tangent cone of a variety, at any point, coincides with the tangent cone defined through geometric measure theory (the current P described in Ann. of Math. vol. 72, p. 519 ). (Received June 1, 1962.)

62T-216. W. G. LEAVITT and R. E. PEINADO, University of Nebraska, Lincoln, Nebraska. The module type of a ring. Preliminary report.

If there exists a left R-rnodule M containing a finite set of generators in terms of which each a. E M is uniquely representable, then R must have a right unit. The module type of such a ring may be defined, generalizing thin introduced by· one of the authors [W. G. Leavitt, The module type Of a ring, Trans. A mer. Math. Soc. 103 (1962), 113 -130]. Most of the theorems of this paper continue to apply. Among the differences: (1) the "matrix criterion" jp. ll5] becomes: There exist n + k by n and n by n + k matrices A, B such that XAB = X and YBA = Y for arbitrary vectors X, Y. (2) In the mapping theorem [Theorem 2, p. 115] replace "unit preserving" by "right-unit-preserving" and add the condition that the image of a right annihilator is a right annihilator. Additional results include: 1. There exist simple rings of arbitrary type. 2. There exists a ring of arbitrary type a, having homomorphic images of all types ~ a. 3. Non-semi-simple rings of arbitrary type exist. Further

326 generalizations are possible, such as a definition of type of a ring by means of the module type of its factor ring modulo a properly selected ideal. (Received June I, 1962.)

62T-217. D. A. MORAN, University of Illinois, Urbana, Illinois. Raising the differentiability class of a manifold in euclidean space.

Using methods similar to those expounded in (J. Indian Math. Soc. 25 (1960), 379-400), Morse's approximation theorem contained therein is broadened to include (possibly noncom pact) manifolds imbedded in euclidean space of arbitrary codimension, Thus if M is an n-manifold of class Cr, r ,. 0, imbedded in E = Rn+k, there exists a Cr -diffeomorphism T of E onto E such that T(M) is of class c 00 and approximates M arbitrarily closely. Moreover, it is shown that T can be chosen so as to be isotopic to the identity, and that if M is compact or k = 1, this isotopy can be chosen so as to be differentiable. The extension to submanifolds of C 00 -manifolds is obvious. Further, if L ___!.__. M ~N, where L, M, N are differentiable manifolds of dimensions n, n + k, n + k + m, re spectively, N is of class C 00 , and f and g are of class C r, 0 ;:;;; r <: oo, there is in general no homeo morphism T of N into N such that Tg(M) and Tgf(L) are both C 00 -submanifolds of T(N). (The above are the principal results of the author's thesis, University of Illinois, 1962, written under the direction of ProfessorS. S. Cairns. Research partly suppo'rted by NSF Grant 14431.) (Received June I, 1962,)

62T-218, W. H. RICHTER, Rutgers, The State University, New Brunswick, New Jersey, Recursive equivalence types whose predecessors are well-ordered,

Theorem. Let a be a nonrecursively enumerable superset of a maximal simple set, and let

A= Reqa, Then, (i) B < A~B ~ R, (ii) B + C = A ---+B = A or C = A. (iii) A <. A + A. Comment, It follows that there exist c such recursive equivalence types A. (i) and (ii) answer questions raised in J. C. E. Dekker and J, Myhill, Recursive equivalence types, University of California Press, 1960, pp. 93 and 10 I respectively. (Received June 1, 1962.)

62T-219, PRABIR ROY, Institute for Advanced Study, Princeton, New Jersey, Dual of a Moore space, Preliminary report.

Urysohn raised the question: is there a regular space on which every real valued continuous function is constant? This has been answered in the affirmative by several examples of various degrees of well-behavior. Here is another example--perhaps the best behaved of the lot. Theorem, There is a connected, locally connected, complete, separable, Moore space on which every real valued continuous function is constant (Moore space= regular, developable space [):ling, Canad, J, Math. 3 (1951), 176] ). For comparison note these known results: a connected, locally connected, .11'1 complete, Moore space is arc-wise connected; and-in presence of 2 > 2 z.to -a separable, normal, Moore space is metrizable, The example is constructed as follows: in a two-sided version-with two critical points-of a space due to Jones [Proc, Amer. Math. Soc, 9 (1958), 483], copies of the space itself are used to join certain pairs of points from a countable dense subset-identifying the two critical points in each copy with the two points to be joined by it; the process is repeated on each copy just introduced and continued to be repeated-for countably many stages-on each successive

327 new copy; and finally, the resulting space-which is so constructed as to be amenable to completing is completed. (Received April 27, 1962.)

62T-220. H. F. KREIMER, 3553 Raymar Boulevard, Cincinnati 8, Ohio. An extension of the Picard-Vessiot theory.

An element c of an M-ring R (defined in a previous abstract by the author) is a constant if multiplication by c commutes with operations on R by elements of M. A necessary condition for elements of an M-integral domain R to be linearly dependent over constants is the vanishing of determinants of Wronskian type. The condition is sufficient if R is an M-field. A Picard-Vessiot extension of an M-field is defined analogously to Kolchin's definition of a P-V extension of a partial differential field (Proc. Amer. Math. Soc. 3 (1952), 596-603). Let K be a P-V extension of an M-field L, and let G be the group of all M-automorphisms of K over L. G is an algebraic matrix group over the constants of L; the Galois closed subgroups of G coincide with the closed subgroups in the Zariski topology; and, if the constants of L are an algebraically closed field, there is a one-to-one correspond ence between the closed, connected subgroups of G and the intermediate M-fields over which K is regu lar. Furthermore, if K and L are differential fields or fields with higher derivations, there is a one to-one correspondence between closed subgroups of G and intermediate M-fields over which K is separable. Solvability of G is interpreted an!! examples of P-V extensions are given. (Received june 4, 1962.)

62T-221. H. F. KREIMER, 3553 Raymar Boulevard, Cincinnati 8, Ohio. The foundation for an extension of differential algebra.

An M-ring consists of a commutative ring R with identity, a set M and an associative, commu tative coalgebra C over R with identity for which M is a basis, and a homomorphism !II of the ring R into the conjugate algebra C *. If a E R and m EM, the formula am = (a!ll)(m) defines operation on R by elements of M. Examples of M-rings are differential rings, difference rings, partial difference differential rings with operators which need not commute, and rings with higher derivations. The concepts of M-homomorphism and M-extension ring are defined. Necessary and sufficient conditions that the structure of an M-ring be extendable from an integral domain to its field of fractions are found. If I is an M-integral domain which is an M-extension of an M-field F and I is regular over F; then I is compatible with any M-integral domain which is an M-extension of F, an M-isomorphism of F into I is extendable to an admissible M-isomorphism on I, and there exists an admissible M-isomorphism of I over F which moves any element of I not in F. If I and F are differential rings or rings with higher derivations, it is sufficient that I be separable over F; and, if I and F are difference rings, it is sufficient that F be algebraically closed in I. (Received june 4, 1962.)

62T-222. R. E. STONG, 5442 South Harper, Chicago, Illinois. Determination of H*(BO(k, ••• ),z2 ) and H*(BU(k, •.• ),Z2).

By using spectral sequences, secondary cohomology operations and exact sequences in the

Steenrod algebra, one can show that H*(BO(k, ••. ,oo),Zz) is {.H*(K('"k,k),Z2)/I(Qkik)J 8 Zz [ai IL(i) > !li(O,k)], where 7rk = ?rk(BO); Qk Is scf, scf, Sq3, or Sq5 as k =0, 1, 2, or 4 (modulo 8);

328 L (i) is one plus the number of one's in the dyadic expansion of i - 1; fli(O,k) is the number of integers s such that 0 -= s ;:; k with s =0, 1,2,4 (modulo 8); and Si are classes in H*(BO, Z2) congruent to 3 wi mod decomposable elements. Similarly, H*(BU(2p, ... ,oo), Z 2) is {H*(K(Z,2p),Z 2)/I(Sq i 2p>} @ z 2 [82i JL(2i) > p + 1]. Further, fairly complete determination is made of all the groups H*(K(mn),Z2)/I involved, all being z 2 polynomial algebras. (Received June 5, 1962.)

62T-223. R. E. STONG, 5442 South Harper, Chicago, Illinois. Cobordism classes of k-trivial manifolds.

By using the author's determination of H*(BO(k, ••. ,oo), Z 2) and H*(BU(k, •• ,oo), Z 2 ) one can show: Theorem. A k-trivial manifold of dimension less than 2fli(O,k)+l is cobordic to zero (in the unoriented

sense), where fli(O,k) is the number of integers s such that 0 "" s 'lS k with s = 0, 1,2, or 4 (modulo 8). Theorem. A k-trivial weakly complex manifold of dimension less than 2[)<:/ 2]+2 is cobordic to zero (in the unoriented sense), where \!c/2] is the integer part of k/2. (Received June 5, 1962.)

62T-224. SVETOZAR KUREPA, Institut za primijenjenu matematiku, Zagreb, Marulicev trg 19, Yugoslavia. On operator-roots of an analytic function.

Theorem. Let X be a Hilbert space, T:X -x a bounded linear operator, o-(T) the spectrum of T and cP(T) the set of all analytic functions f which are analytic in some neighbourhood of o-(T). Suppose that there exists a function f E .1"(T), f i 0, such that: (1) f has simple zeros in the circle, {.>..I lAili IITII}. (2) f(T) = 0. Then o-(T) is a finite set and Tis similar to a normal operator. (Received June 6, 1962.)

62T-225. SVETOZAR KUREPA, Institut za primijenjenu matematiku, Zagreb, Marulicev trg 19, Yugoslavia. A theorem about similarity of operators.

Theorem. Let X be a Hilbert space, T: X___,. X a bounded linear operator. If H = sin T = L~(- l)nT2n+l /(2n + 1)~ is selfadjoint and IIH II < 1, then, a bounded regular positive definite self adjoint operator Q exists such that: (1) QH = HQ, (2) Ho = QTQ -l is a bounded selfadjoint operator and, (3) sin Ho = H. (Received June 6, 1962.)

62T-226. ECKFORD COHEN, University of Tennessee, Knoxville, Tennessee. Arithmetical notes, XII. A sequel to note VI.

Let Q(m,n) denote the number of positive integers x-<: m, y < n, such that (x,y) = (m - x, n - y) = 1. In an earlier note (to appear in the Michigan Journal of Mathematics) an asymptotic formula for Q(m,n) was proved; another proof with a greatly improved remainder term has also been given (Notices, 8 (1961), 54, Abstract 61T-6). In the present note a different and much simpler method is used in treating this problem. In particular cases the estimate obtained is superior to that proved previously. Example: Q(m,m2) = u(m)m 3 + O(m2log2m) where u(m) is a certain bounded func tion of m. (Received June 6, 1962.)

329 62T-227. JACOB WOLFOWITZ, Cornell University, Ithaca, New York. Products of indecomposable, aperiodic, stochastic matrices,

A finite square stochastic matrix P is indecomposable and aperiodic (SIA) if lim n-.ooPn exists and has identical rows. Define S(P) = max -maxi i IPi - -Pi -1; Pi]. is the element in the ith row 1 1• 2 1 1 2 1 and jth column. Let A 1, ... ,Ak be square matrices of the same order such that any product of a finite number of A's is SIA. Let e > 0 be arbitrary. The author proves that there exists N(e) such that

any product Q of n !; N (e) matrices A satisfies 8(Q) < e. This result has applications in the study of inhomogeneous Markov chains and of indecomposable channels in information theory. (Received June 6, 1962.)

62T-228. R. P. GOSSELIN, University of Connecticut, Storrs, Connecticut. Integral inequalities for subadditive functions.

Let f be a positive, measurable, and subadditive function on En• Let w 1, w2, ••• ,wn be linearly

independent unit vectors. Let p $'; 1 and a.p > -1. There exist A~ B depending only on a., p, and the

~ w1 such that A jfP(x)/lxln+pa.dx ~ L:_~= 1jfP(wix)/lxl 1 +pa.dx ~ B,/fP(x)/lxln+pa.dx. In the extreme integrals, integration is over En, and in the center integrals, integration is over E 1• The result makes it possible to extend to n dimensions an inequality known for subadditive functions of a single variable (cf. Soine integral inequalities, Proc. Amer. Math. Soc., to appear). Applications are given pertaining to functions satisfying integral Lipschitz conditions and to Bessel potentials. (Received June 7, 1962.)

62T-229. D. B. SHAFFER, 156 Intervale Road, Stamford, Connecticut. Lemniscate surfaces

The concept of a lemniscate in the complex plane is generalized by defining a lemniscate sur face L in R n by F (x) = lT~ 1 11x - ai II = Jl• «> = grad F dx = dF. A critical point of GV is a point where BF /axi = 0, i = l, ••• ,n. This definition differs from Schurrer (Trans. Amer, Math, Soc. 89 (1958), 100-112) and Nagy (Bull. Amer. Math. Soc, 55 (1949), 329-342) who define lemniscate surfaces as polynomials in n variables. By a generalized Gauss theorem all critical points of cv are contained in the convex hull of the xi. Let A and B be two points on L. Then the hyperplane orthogonal to AB through the midpoint of the segment AB must intersect the convex hull of the ai· Let P E Lin E 3• Let c denote the shortest distance from P to the convex hull of the ai along the normal to L of p. Then the absolute value of the mean normal curvature of L at P is less than 1/c. The results can be extended to rational functions ·and harmonic functions u = 0 on a set of surfaces S li and u = 1 on s2i• (Received June 7, 1962.)

62T-230. WITHDRAWN

62T-231. R. H. ROSEN, University of Michigan, Ann Arbor, Michigan. Multiple valued functions on groups.

In the hope of eventually being able to apply algebraic techniques in the study of upper semi continuous point to set functions defined on topological spaces, an attempt is made to define similar algebraic operations. A multivalued function of a set A into a set B sends elements of A into nonempty

330 subsets of B. If C and D are subsets of a group G, CD is the set of all multiples cd for c in c and d in D. A multivalued function f from a group G into a group H will be called a multimorphism (polymor phism might be suitable as well) provided (a) g 1, g2 in G implies f(g1)f(g2) <;;;_ f(g 1g 2) and (b) if e is the identity of G then f(e) is a subgroup of H. Theorem. Let f be a multiva1ued function of a group G into a group H. Then f is a multimorphism if and only if there exist groups K <;. H and L, and homomor phisms ¢: G ~L, 6:K------. L such that f = B - 1 " ¢. By making obvious extensions an analogous result is proved for rings. The theorem suggests that the conditions for a multimorphism may be too stringent to apply to the topological case. (Received June 11, 1962.)

62T-232. 0. L. MANGASARIAN, Shell Development Company, Emeryville, California. Stability theorems for systems of nonlinear ordinary differential equations.

Consider the system of ordinary differential equations i = f(t,x) where x and f are n-dimensional

vectors, and 0 ~ t < oo. Assume that f(t,x) is continuous for 0 ;:; t < oo and x'x < oo, where the prime

denotes the transpose. Assume also that f(t,O) = 0 for 0 lii t < oo. Theorem I. If for 0 ~ t < oo the function x'f(t,x) is a concave function of x for all x, then the point x =0 is a stable equilibrium point. Theorem II. If for 0 ::; t ;;; oo the function x'f(t,x) is a strictly concave function of x for all x, then the point x = 0 is a uniformly asymptotically stable point in the large. Theorem III. If for 0 lii t 1S oo the function x'f(t,x) is a strictly convex function of x for all x, then the point x = 0 is unstable.

(Received May 7, 1962.)

62T-233. G. M. MEYER and FRANCIS REGAN, St. Louis University, St. Louis 3, Missouri. Exponential analogues of a generalized Lambert series.

Properties of S(z) = :;[:anexp(- A.nz)/(1 - exp(- JlnZ)) where .\nand un are real, positive, monotonely increasing, unbounded sequences, are discussed. The region of ordinary convergence of this series is established, together with the regions of uniform and absolute convergence. Since the behavior of this series is closely related to that of 2:. an exp(Jln - 4.n)z in the half plane R(z) <:. 0, the properties of this latter series are also discussed. Further, in regions of absolute convergence it is shown that the series S(z) may be expressed as a general Dirichlet series, which represents the same analytic function as the original series. Conversely, a general Dirichlet series can be expressed

as a series of this type. The P -Q series, resulting when ~n = p(ln il) and un = q(1n n}, is also con sidered. A P -Q series can always be expressed as an ordinary Dirichlet series. On the other hand, an ordinary Dirichlet series can be expanded in a P -Q series if and only if q is a multiple of p. Explicit formulas relating the coefficients of ordinary Dirichlet series and P-Q series are given. Finally, conditions under which they-axis is a natural boundary for the P-Q series are determined. (Received May 15, 1962.)

62T-234. M. S. LYNN, California Research Corporatioh, Box 446, La Habra, California. An unconditionally stable, explicit, two-level method for solving parabolic partial differential equations.

Consider the equation au(t,x)/8t = a2u(t,x)/8x2 together with the initial condition u(O,x) = f(x) and boundary conditions u(t,a) = a, u(t,b) = li'· The usual two-level discretization can be expressed in

the form ui+1,j - ui,j = A[ll(ui+1,j-1 - 2ui+1,j + ui+ 1,j-1l + (1 - 8) (ui,j-l- 2ui,j + ui,j+l)] where ui,j = u(i Ll t, i Ll.x), i\ = /1t/(/:J.x)2 and 0 ;!; 8 ;:; l. For B = 0 this method is explicit but only conditionally

331 stable, A ~ 1/2 being sufficient to ensure stability. For 1/2 ~ S 0. Hence, by Lax and Richt- myer's theorem, it is also convergent. It can be seen that it is thus a two-level equivalent of the method of Dufort and Frankel The method is extended to parabolic equations of higher complexity in several space dimensions. (Received June 12, 1962.)

62T-235. G. T. CARGO, Syracuse University, Syracuse 10, New York. Normal functions, Monte!' s property, and interpolation in H00

This paper exhibits an interrelationship among the subjects mentioned in the title. The follow ing theorem serves as a unifying agent: Theorem. Let D denote the open unit disk, let S be a subset of D having at least one accumulation point on the unit circumference, and let p(z l'z2) denote the hyperbolic non-Euclidean distance between two points z 1 and z 2 in D. Then, corresponding to each positive number?', there is a denumerable subset [z pz2, ••• j of S such that the non-Euclidean disks {z: p(z,zn) < r} (n = 1,2, .•• ) are disjoint and the Blaschke product 1Tflznl(zn- z)/zn(l- znz)} is bounded away from zero on the complement of their union. That this theorem is sharp follows from some research ofF. Bagemihl and W. Seidel in connection with normal functions (Sequential and continuous limits of merom orphic functions, Ann. Acad. Sci. F enn. Ser. A I 280 (1960), 17 pp.). The proof of the theorem is implicitly contained in D. J. Newman's paper Interpolation in H 00 , Trans. Amer. Math. Soc. 92 (1959), 501-507. The theorem yields a new proof of the main result in P. Lappan's paper Non-normal sums and products of unbounded normal functions, Michigan Math. J. 8 (1961), 187-192; in addition, it is used to shorten certain Work of the author dealing with radial and angular limits of meromorphic functions in D. (Received June 13, ·1962.)

62T-236. G. T, CARGO, Syracuse University, Syracuse 10, New York. The segmental variation of Blaschke products.

A function which is regular in the open unit disk D has finite segmental variation at a point iS ·s e (S real) provided every line segment connecting e 1 to a point of Dis mapped onto a curve of finite length by the function. Let B be a Blaschke product, and let {zn} be its sequence of zeros. Theorem 1. All the subproducts of B have finite segmental variation at the point eiS if and only if "'""(1 -lz I)/ leiS - z I < oo. Theorem 2. If""' (l -lz I> a < oo for some a (0 < a < 1), then B has finite L- n n L... n segmentai variation at each point of the unit circumference except on a set whose a-dimensional Haus- dorff outer measure is zero. Theorem 3. lf2:(1- lznl)logfl/(1- lznl>} < oo, then B has finite seg mental variation almost everywhere. (In contradistinction to this, W. Rudin has established the existence of a Blaschke product which maps almost every radius of D onto a curve of infinite length.) These results will appear in the Duke Mathematical Journal. (Received June 13, 1962.)

62T-237. ECKFORD COHEN, University of Tennessee, Knoxville, Tennessee. Arithmetical

notes, XIII. A sequel to note IV.

Let /!::. = "\: (n) denote the complementary divisor of the maximal kth power divisor of the 332 integer n > 0. This note contains elementary estimates for the number Vk,t (x) of integers. n l:i x such that the maximal exponent to which a prime divides A has a prescribed value t. In particular, it is shown that for 1 < t < k - 1, vk,t(x) = t(k) u;-- 1(t + 1) - ~- \t))x + 0(~. where ~(s) has its usual meaning. (Received June 14, 1962.)

62T-238. B. R. TOSKEY, Seattle University, Seattle 22, Washington. Rings on a direct sum of cyclic groups.

Let G be a ring whose additive group is a direct sum 2:o.c:S$fuo.\, of cyclic groups {uo.}• where S is an ordinal number which is partitioned appropriately to identify the orders of the cyclic summands. By considering certain types of r by S row-finite matrices and choosing a matrix notation for the set of multiplication coefficients in G, necessary and sufficient conditions are found for another ring H with the same additive group to be isomorphic to G. The result is an extension of Beaumont's result (Publ. Math. Debrecen 4 (1956}, 469-480) on the isomorphism of algebras over a ring. (Received June 15, 1962.}

62T-239. ISRAEL NAVOT, Israel Institute of Technology, Haifa, Israel. The Euler-Maclaurin functional for functions with a quasi-step discontinuity.

Following previous extensions (Notices, Amer. Math. Soc. 7 (1960}, 515 and 8 (1961}, 347 (abstracts)) by the author of the form assumed by the Euler-Maclaurin functional (1): E {f }a=

2::"_~~6f((v + a)/n) - n.fo1f(x)dx; 0 < a lii 1, for functions f(x) with integrable branch, logarithmic and both branch and logarithmic singularities at x = 0, consideration is now given to this functional for functions f(x,o.) which depend on o. in such a manner that foro.= 0 there is limO<.x-of(x,O) = f(O,O) + C where C f. 0 is a numerical constant. For f(x,o.} of this type it is generally true that for small values of o. their derivatives with respect to x oscillate strongly in the vicinity of x = 0, the peaks of the successive derivatives being proportional to successive negative powers of o.. Conse quently, when no. is of the order of unity, or less, the negative powers of no. in the ordinary Euler Maclaurin asymptotic series will render it useless for the numerical evaluation of the integral or sum in (1) since it may diverge from the beginning of the series. A method which eliminates this difficulty in the case where the quasi-step discontinuity is introduced by arctanx/o. and other closely related functions is developed in the present paper. (Received June 18, 1962.)

62T-240. E. E. GRANIRER, The Hebrew University, Jerusalem, Israel. On amenable semigroups with a finite dimensional set of invariant means.

Let G be a semigroup and m(G) be the real valued bounded functions on G with the sup. norm. Let Ml(G) C m(G)* be the set of left invariant means and dim M1(G) be the dimension of the linear manifold spanned by Ml(G). The following result is obtained: Let G be a semigroup with left cancel

lation. Then dim M 1 (G) = n, 0 < n <. co if and only if G is finite and is the union of n finite disjoint groups each isomorphic to the other and each of which is a left ideal in G. This result implies that a semigroup with left cancellation has a unique left invariant mean if and only if it is a finite group. P. Civin and B. Yood prove in Pacific J. Math. 11 (1961), 853 that the radical of the second conjugate

333 algebra m(Z)* (Z is the additive group of integers) is infinite dimensional. This is conjectured there for any commutative infinite group. By applying the above obtained result to infinite amenable groups one gets easily that dim J 1 = oo where J 1 C m(G)* is the set of left invariant elements which annihilate the constant one function of m(G). Since J 1 is contained in the radical of m(G)* one has: The radicaJ. of m(G)* for any infinite amenable group is infinite dimensional, which proves more than the above conjecture. (Received June 18, 1962.)

62T-241. G. T. CARGO and OVED SHISHA, National Bureau of Standards, Washington 25, D. C. Fractional order differences of the coefficients of polynomials.

A classical result of Enestrom (Ofversigt af Kongl. Vetenskaps Akademiens Forhandlingar, 50 (1893), 405-415) states that if the coefficients of a polynomial E(z) =21=0ckzk( +0) satisfy Vck :ii 0 (k = 1,2, ••• ,n + 1), then E(z) fO throughout lzl < 1. Here Vck =. ck- ck_ 1 (and cn+1 is taken as zero). This result is generalized as follows. Theorem 1. Let 0 < a li 1 and let a nonconstant polynomial E(z):: L~=O ckzk satisfy ck!: 0 (k = 0,1,. •• ,n), Va'l; :i 0 (k = 1,2, ... ,n), where 'i7 a.ck E :L:,~= 0 (- l)m(~)ck-m• Then E(z) f. 0 throughout lz I <. 1. When a= 1, Theorem 1 becomes Enestrom's result. Examples show that Theorem 1 may be applicable to polynomials E(z) to which

Enestrom's result does not apply. Theorem 2. Let 0

angular region with vertex at the origin whose angular measure 29 satisfies 0 li 29 < 1r. Then each of the following three hypotheses implies that E(z) has no zero in lz I< cos 9: (I) -ck (k = 0,1, •• .,n) and Vack (k = 1,2, ••• ,n) belong to S; (II) Vack E S (k = 1,2, ••• ,n,n + 1, ••• ); (III) 'Vck E S (k = 1,2, ••• ,n + 1) (Received June 11, 1962.)

62T-242. MURRAY GERSTENHABER, University of Pennsylvania, Philadelphia 4, Pennsylvania. The cohomology structure of a ring.

Let A be an associative ring, Hn denote the nth cohomology group of A with coefficients in A, and H* be the direct sum of the groups Hn. Then: (1) The cup product in H* is commutative, i.e., if am 'C. Hm, jJn E Hn then am UjSn = (- 1)mnP'n Uam. (2) There is defined in H* a second multiplica tion, the bracket product, [, ], in which H* becomes a graded Lie algebra. One has I}Im, Hn] C Hm+n-1, [am, ,Bn] = _ (- 1)(m-1)(n-l) r,en, am], and (- 1)(m-1)(p-1) [[am,,Bn], rPJ + (- 1)(n-1)(m-1) x_[(/, .,..P] amJ + (- 1)(P- 1Hn-l)[[?'P, am], pP] = 0. If m = n = 1, the bracket product is the ordinary Poisson bracket of derivations. (3) One has [am Upn. ?'P] = [am, ?'P] u~ + (- 1)m(p-1)am u [~,,.P],

i.e., the additive endomorphism of H* defined by a- (a, ?" P] is a derivation of degree p - 1 of H* considered as a ring under the cup product. (Received June 21, 1962.)

62T-243. RICHARD SINKHORN, 1144 North Market, Wichita 14, Kansas. On the factor spaces of the complex doubly stochastic matrices.

Let .?fN denote the complex N X N matrices and let 2J denote those members of ??(N which are complex doubly stochastic. If a. and p denote respectively the N dimensional vectors (1,1,1, ••• ,1) and (1,0,0, ••• ,0), and if r>= {P E ~la.P = Nft. Pf9T = a'Ij and~= iQ E '?J;NI/Q = a/N, qaT= ,.o'TJ, then

334 .fJ = ?:Z., and in fact, if P and Q are arbitrary nonsingular members of JO and :l, respectively, 2J = P.2. = )CQ. At the same time :Z. 'P = .2. P = QJO = 1 0 ~N _1• This result yields a simple isomor phism between 2J and ~N- 1 with respect to multiplication and convex addition. The factorization is unique in the following sense: Theorem. If x s;; ~N and y s;.?fN are such that :i:J = xy and yx = 1 1 E8 ?fN _1 , then x !;; /f> and y ~ f- .2.. for some complex number f f. 0, Both inclusions may be proper, (Received June 22, 1962,)

62T-244, G. F. CLEMENTS, Syracuse University, Syracuse 10, New York, Entropies of several sets of real valued functions, Preliminary report.

Let Fn (C,M) denote the class of real valued functions f defined on the unit cube in the p+a Euclidean n space which satisfy lf(x)l li C and have all partial derivatives of order k l!ii p, with the pth order derivatives satisfying a Lipschitz condition of order a with Lipschitz constant M, Under the n 1/(p+a) . uniform metric Kolmogorov has shown: Theorem. HE(F p+a(C,M))I((1/e) , (For notat10n and definitions see Kolmogorov, Tihomirov, e-entropy and e-capacity of sets in function spaces, Uspehi Mat, Nauk 14 (1959), 3-86 (Russian), or the translation, Arbeiten zur Informationstheorie, III, VEB

Deutscher Verlag der W"issenschaften, Berlin, 1960,) In this note it is shown using a simple counting

technique that this theorem also holds under the L 1 metric, Also, with Aa• 0 < a ;i 1, the set of continuous functions f defined on I!J,1], bounded by C, with modulus of smoothnesst"

62T-245, G. W, GOES, University of IV estern Ontario, London, Ontario, Canada, Strongly divergent Fourier series in FK- spaces,

Let F be a FK-space and E a BK-space of sequences c = {ckf of Fourier-coefficients (f -L,ckeikt). Let sn(c) be the truncated sequence of c corresponding to Lk=-nckeikt, Further

define: c E EN ~ llsm(c)- sn(c)IIE ~o. c E Es ~ supnllsn(c)IIE < oo. Then we have the Theorem. If llsn(c) - c liE= gn(c) is a continuous functional for every c E F and n = 1,2, ... and F n EN is not closed in the topology of F, then F f'I(EB - EN) and F - EB are not empty, This is an extension of a theorem of Meyer-Konig and Zeller (Math, Z. 78 (1962), 143-148), Applications will be given, (Received June 25, 1962,)

62T-246, ADI BEN-ISRAEL and ABRAHAM CHARNES, The Engineering Science Department, The Technological Institute, Northwestern University, Evanston, Illinois. Optimality in diophantine programming.

Let P = (P1 , ... , Pj ..... Pn) be an m X n matrix, B an m X p submatrix of P, L +lBJ = {L_njPj: Pj rl B,nj integer OJ. ForcE En, cT = (c1, ... , cj'"'' en)• let cB be the subvector of c corresponding (same ordering) to B. For Pk E L+{BJ define ck =2:njcj (corresponding to Pk =£njcj). Assumption 1: 3 an m X p B 3 P j = ByBj for j = 1, ... , n; YBj all integer E. EP (not necessarily unique). For F'!c E L+ fB} define YBk =~njYBj· Let Po E Em be given, Assumption 2: 3 B satisfying assumption 1 3 Po= ByBO' yBO unique, ;::; 0, all integer E EP. Theorem. Let B satisfy assumptions

335 1 and 2.. Then the following are equivalent: (a) c~y80 = max{cTx: Px =PO, x all integer, ;:. OJ. (b) For every Pk € L+{B}, c~YBk < ck =9 gBO - nyBk 3!: 0 for no positive integer n.l (In (b) compare ck' yBk corresponding to same nj in Pk =L:njP j") A primal algorithm for integer programming based on the above is described.. (Received june 2.5, 1962.)

6ZT-Z47. ]. A. SMOLLER, Division of Mathematical Sciences, Purdue University, Lafayette, Indiana. Translation-invariant linear functionals on spaces of functions of two variables. Preliminary report.

Let C c (R 2.) be the space of continuous functions with compact support in the plane and let X be a translation-.invariant linear subspace of Cc(R 2.); i.e., f(x,y) E X implies fst (x,y) = f(x + s, y + t) E X for all (s,t) E R z. A linear functional F on X is called translation-invariant on X if F (fst> = F (f) for all f E X and (s,t) 1: R 2 • For every pair of non-negative integers (i,j), let the linear functional Fi,j be defined on X by Fi,j(f) = [ai+jf(u,v)/auiavjJu=O,v=O' where f is the Fourier transform of f. LetT be the space of functionals on X that are finite linear combinations of the F i,/s and are also transla tion-invariant on X. Let k =min {i + j: 3Fi,j # 0 on XJ. Theorem I. dim T ~ Zk + I. Theorem 2.

If K = {Fi,k-i: F i,k-i 1 0 on X} is a linearly independent set on X, then dim T ~ k + 1. Theorem 3. Let (0,0) be an isolated point of nfExf(a,(.3) E RZ: f(n,fJ) =OJ. IfF is any translation-invariant linear functional on X which has a Fourier transform in the sense of L. Schwartz (Th6orie des distributions, Tome II), then F E T. These results are analogues of Theorem 4 of jerison and Rudin (Translation-invariant functionals, Proc. Amer. Math. Soc. 13 (1962.)). (Received June 2.5, 1962..)

6ZT-2.48. D. W. MILLER, 1836 Morningside Drive, Lincoln 6, Nebraska. Hamiltonian semigroups.

A semigroup S is said to be Hamiltonian if every nontrivial subsemigroup of S is the 0-kernel of a homomorphism of S onto a semigroup with zero; Kronecker if ab = a or ab = 0 according as b = a or b ;. a; the orthogonal product of semigroups A and B if S = A U B U 0 and if ab = ba = 0, all a E A, bE B; diagonal nilpotent of (finite) index n if Sn-l;. Sn = 0 and, for all a, bE S, (a) az = bz implies ab = az or ab = 0, and (b) az # bz implies ab = 0. If T is a semigroup, if a E T, and if we define, for some x / T, xt = at, tx = ta, xz = az (all t E T), then T U a is a semigroup in which xis called a replica of a. Theorem. A semigroup is Hamiltonian iff it is one of (1) a semigroup of order 2.; (2.) a Kronecker semigroup with n (!:; 0) replicas of the zero; (3) the orthogonal product of a Kronecker semigroup and a diagonal nilpotent semigroup of index 3; (4) a group G of prime order with n ( .0:. 0) replicas of elements of G '\I; (5) a certain extension of a group of odd prime order by a diagonal nilpotent semigroup of index 3. Theorem. Let S be a semigroup each of whose nontrivial sub semigroups is the !-kernel of a homomorphism of S onto a semigroup with identity. Then S is one of (1) a semigroup of order 2.; (2.) an abelian group; (3) a Hamiltonian group. (Received june 2.5, 1962.)

6ZT-Z49. ]. C. SANWAL, White Hall, Cornell University, Ithaca, New York. Discrete groups acting on nilpotent Lie groups.

Using a result of L. Auslander, Bieberbach theorems on space groups and discrete uniform

336 subgroups of Lie groups, Amer, J, Math, (1961), 276-280; it is shown that: (l) A complete locally nilpotent homogeneous space (in the sense of Ehresmann) is finitely covered by a nilmanifold, (2) The fundamental group of a complete locally Euclidean space can be realised as the fundamental group of a compact locally Euclidean space. (3) There do not exist complete locally Euclidean spaces with nilpotent fundamental groups of class strictly greater than 1. An example due to Mal'cev, On a class of homogeneous spaces, Amer, Math. Soc. Trans!, Ser, 1, no, 39, shows that there are infinite non homeomorphic 3-dimensional complete locally nilpotent homogeneous spaces, (Received June 25, 1962,)

62T-250, D. L. OUTCALT, Ohio State University, 216 North Oval Drive, Columbus 10, Ohio. On a class of weakly alternative rings, Preliminary report.

Let R be a ring of characteristic not 2 or 3 in which (1) (x,y,z) = (y,z,x) is an identity, If in addition (2) (x,x,x) = 0 is an identity in R, then R is trivially alternative, Conversely, (1) and (2) are identities in an alternative ring. It is shown that (3) (x,x,x)2 = 0 is an identity in R; hence R is alternative if a 2 = 0 implies a= 0 in R, That (3) holds in R was indicated some time ago by P, Jordan (M. Zorn, Alternativekorper und quadratische Systeme, Abh. Math, Sem, Univ, Hamburg 9 (1933), 395-402), but apparently no proof has appeared in the literature, Theorem 1, If R is simple with an idempotent e '1- 1, then R is alternative hence a Cayley-matrix algebra of dimension 8 over its center, Since R is not power-associative if R is not alternative, the following definition is tnade: An element x of R is nilpotent provided there exists a positive integer k such that xk = 0 no matter how associated. _Theorem 2, If R is a finite-dimensional algebra over a field F, then the following are equivalent: R is nil, R is nilpotent, R is solvable, Corollary. If R is a simple nil, nilpotent, or solvable finite-dimensional algebra over a field F, then R is trivial, (Received June 25, 1962,)

62T-251, D. C, MURDOCH, Harvard University, 2 Divinity Avenue, Cambridge 38, Massachu setts, Isolated components and the associated primes of an ideal,

Let R be a ring with ACC for two-sided ideals and let P be a prime ideal in R, For any ideal A, the upper and lower P-components u(A,P) and )(A,P) [D. C. Murdoch, Canad, J, Math, 4 (1952), 43-57] are then equal, Define the associated primes of A to be the tertiary radicals that occur in any tertiary decomposition of A [L. Lesieur and R, Croisot, Math. Ann, 134 (1958), 458-476]. It is proved that, (1) u(A,P) is the intersection of all ideals X containing A such that all the associated primes of X are contained in P. (2) If P is an associated prime of A then u(A,P) is P-primal. The converse is false in general but true if A has a right primary decomposition, (3) If P is a minimal prime of A then u(A,P) is P-tertiary. (4) If B 2 A and u(A,B) is the upper B-component of A [W. E. Barnes, Trans, Amer, Math, Soc, 82 (1956), 1-16] then u(A,B) = niu(A,Pi) where the Pi are the associated primes of B. (Received June 27, 1962,)

62T-252, W, D. BARCUS, State University of New York, Oyster Bay, Long Island, New York. Some higher-order cohomology operations.

A system of generalized Massey products is defined algebraically, and in the simpler cases by relations and cochain formulas, Example: A secondary operation on five variables, defined if

337 b'a' = b'c' = b'e' = 0 and d'a' = d'c' = d'e' = 0, arising from the relation (a',b',c')d' ;t (c',d',a')b' = 0,

and having cochain formula a ubcude ± uabc ude ± uab ucd e ± ucd a ube ± c uda ube ± ubc uda e ±. ((b,c,d) ~ 1 a± (c,d,a) ~ 1 b)e. Here a is a cocycle with cohomology class a', Suab = ab, etc., ' the Massey products are formed with these particular cochains u, and signs have been omitted. These operations arise in computing the Postnikov system of a space which is a suspension, and whose mod p cohomology is free over the Steenrod algebra up to a certain dimension. (Received June 28, 1962.)

62T-253. ROBERT CARROLL, Rutgers, The State University, New Brunswick, New jersey. Problems in linked operators. I.

Let E and H be separable Hilbert spaces with L and A closed, densely-defined, linear operators in E and H respectively with domains D(L) and D(A). Assume A is self-adjoint positive, (Au,u)

5; c llu 11 2, and that (ei) (resp. (hi)) is a defining orthonormal base for L (resp. A) chosen from D(L) n D(L*) (resp. D(A)). Let G be the set of finite linear sums u = Laij(ei 18 h.i) with scalar product (u,v) = L aijft ij' v = L/ij(ei ® ~ ). This defines a cross -norm cT on E ® H and a completion E ®cT H which is a Hilbert space. Let L 0, A0, s0 , s0 be the closures of L 18 I, I ® A, L ® I + I ® A, L* ®I+ I® A on G. LetS= Lo + A0 and S = L('j + A0 = L(j + A0 and assume -Lis maximal dissipative (Re(Lu,u) ~ 0; see R. S. Phillips, Trans. Amer.Math. Soc. (1959), 193-254). Then (S 0 , S0) is a formally adjoint pair, as is (S, S), and both possess solvable realization operators (see F. Browder, Math. Ann. (1959), 55-79). In particularS* is a solvable realization operator for both pairs. These results are related to certain Cauchy problems where L = d/dt with suitable boundary conditions. (Received June 28, 1962.)

62T-254. ROBERT HERMANN, University of California, Berkeley, California. Homogeneous Riemannian manifolds of nonpositive curvature.

Theorem. Let M be a homogeneous Riemannian manifold of nonpositive sectional curvature. There is a covering M' of M by a finite abelian group all of whose elements are of order two such that M' is a product of a connected solvable Lie group and a Euclidean space. (Received June 28, 1962.)

62T-255. ]. R. GUARD, Fine Hall, Princeton University, Princeton, New jersey. A classifica tion of ordinal recursive functions.

Let a be a "standard" primitive recursive well-ordering (sprwo) of the natural numbers of order type Jl"(cf., W. W. Tait, Math. Ann. 143 (1961), 236-250). This fact is abbreviated as lal = ?'. Let a·w, (w· a, wU, an, respectively, aw) be the sprwo of order type?'·w (w·?", w-r, ?'n, resp., -yw) obtained from a by the pairing functions and prime factorization in the natural fashion. Let [a-rec.] (resp., [nested a-rec.]) denote extensionally the class of functions definable from the primitive recur sive functions by composition and unnested (resp., nested) ordinal recursion on the sprwo a. (Cf., Tait, ibid., for definitions.) Tait has sho.vn [!tested a-rec.J C [wa-rec.] and [wa-rec.J c [nested w• a-rec.J. The following new results are proved: Theorem. If lal?; ww, then [wa-rec.] = ~ested a-rec.]. Theorem. __!!_ wn ~ Ia I < wn+ 1, then ~a -rec.] = [nested w· a-rec.] = {Rozsa Peter's n + 1-fold re cursive functions J. Theorem. [nested a-rec.] = [nested a· n-rec.], while [nested a-rec.J c;

338 [nested a.·w-rec.]. Theorem. [a.-rec.] = [a.n-rec.], while [a.-rec.J ~ [a.w -rec.]. (Received June 28, 1962.)

62T-256. J. R. GUARD, Fine Hall, Princeton University, Princeton, New Jersey. Hierarchies of recursive arithmetics.

The notation of the preceding abstract is assumed. For a sprwo a., a-induction is the rule- from P(O), xi 0 ---..f(x)ax, and P(f(x))---+P(x) to infer P(x). Let Ra (respectively, Sa) be primitive recursive arithmetic augmented by a-induction (resp., definition by unnested a-rec.). It is easy to verify that a-induction is a derived rule of Sa. The following theorems are an extension of an earlier result of this writer (cf. Abstract 578-5, Notices Amer. Math. Soc. 8 (1961), 140). Theorem. If

Ia 1 s;- w w, then the consistency of Ra can be proved in R a•W. Theorem. The consistency of Sa can be proved in Saw. The results of this and the preceding abstract answer questions raised in Tait, ibid., (cf. p. 250), and in R. L. Goodstein, Recursive analysis, North-Holland, Amsterdam, 1961 (cf. pp; 2, 108, 112). (Received June 28, 10962.)

62T-257. H. J. KEISLER, Institute for Defense Analyses, Von Neumann Hall, Princeton, New

Jersey. Mahlo 1s operation and the existence of a.-complete prime ideals.

Denote by OR, C, and AC, the classes of all ordinals, all infinite cardinals, and all weakly accessible cardinals, resp. If X~ C, let M(X) = fU y: 0 i y ~X, and if 0 f z <;;; y then Uz E y or

Uz = lJyj (Mahlo 1 s operation). Suppose K is a class of (binary) relational systems and R <;;;,.OR X OR; let KR = K nr< a, Rna X a): a E cJ and let IK I= {a E C: K has a member of power aJ. Let c 1 = {a E C: every a-complete prime ideal in the field of all subsets of a is principal}. Following Hanf Scott 61 T-240, let 2:: i be the family of classes of relational systems characterized by single sentences of third order logic all of whose third order quantifiers are existential and occur at the beginning.

There exist K 1 , K" E" Ei such that IK 1 - IKR I r;;, C 1• This implies the following result of 61T-240. If K E Ei. R ~OR X OR, and IKRI i c, then n(c -IKRI> E c1. Theorem 2. Let Fa be the least family F of subclasses of G such that: (i) if K E :Ef and R ~OR X OR, then IKR I E F; (ii) if G !,;F, then nG EF; (iii) if X$ E.: F for all S E OR, then {a E C: a E U}<

62T-258. H. J. KEISLER, Institute for Defense Analyses, Von Neumann Hall, Princeton, New Jersey. The equivalence of certain problems in set theory with problems in the theory of models.

For terminology see Erdos-Tarski (ET), pp. 50-82 in Essays on the foundations of mathematics,

Jerusalem 1961, and Tarski-Vaught, Comp. Math. 13, pp. 81-102. Let a, r be infinite cardinals with a lii ?'. Denote by L the natural well ordering on ?'. A (finitary) relational system c::t =

system ot = {r, L, Rt) tET which is fixed by a. Theorem 2. If ,.-a= r, then the following are

equivalent: (i 1 ) there is an a-complete field of subsets of 'Y which contains all finite subsets of r,

339 A has power ?', and in which every a-complete prime ideal is principal; (ii') there is a system that = ('r, L, R$)s

62T-259. P.M. ANSELONE and DONALD GREENSPAN, Mathematics Research Center, U.S. Army, Madison, Wisconsin. On a class of linear difference-integral equations.

Consider (*) 2::::/=0aj!li(t - j) = ./J K(s)!li(t- s)ds, t ;;: J, for a continuous solution !li(t), t <= 0, with to avoid assumptions required by K E L 1 (O,J) and a 0 t 0. Functional analysis methods are used a unique transform methods. Results: Each continuous l(t), 0 ;:; t ;;;; J, which satisfies (*)at t = J has

111 of (*) continuous extension to [O,oo) which satisfies (*) for t s: J. Let X = C l]:l, 1]. For each solution

!11 (t) = !li(t + n), 0 t !S 1. Then "' = ""J 1u ·"' ., where the U. are and n ~ 0, define !lin E X by n ~ "'n L...J= J"'n- J J operator operators defined in terms of (*). Define~ E xJ, n ~ 0, by "tn = (!lin•···•!lin+ J _ 1). Define the = Then Ton xJ by T(f0, .•• ,fJ-l) = (fl' ... ,fJ), fJ = 2::::/=lU/J•j" Then~= Tnf0• Let P(A.) 2:/=0aj~J-i. with P(T) is compact, so that limit points of cr are zero·s of P. The eigenvectors ofT associated = The ~ E cr, P (i\) t 0, correspond to solutions !li(t) = (c0 + c 1t + .•• + em tm)ezt of (*) with ez A_. solution manifold of (*) has decompositions corresponding to spectral decompositions of T and "inner" and "outer" spectral radii of T. Asymp r ~ limt-+00sup l!li(t) ll/t !!iF R, where r and R are the at totic expansions are obtained. Results are generalized to solutions 9i with simple discontinuities t = 1,2, ••.• (Received May 7, 1962.)

62T-260, 62T-26l, 62T-262. WITHDRAWN

62T-263. T. G. McLAUGHLIN, University of California, Los Angeles 24, California.

Do infinite creative sequences exist?

In his recent article Creative functions (Z. Math. Logik Grundlagen Math. 3 {1961), 205-212), interesting J. p. Cleave defines the notion of a creative sequence of infinite length, and proves several and theorems about such sequences, but neglects to verify that (1) infinite creative sequences exist, with the that (2) the informal suggestion, on p. 205, for a definition of creative function, matches up l it formal definition on p. 206. (2) turns out, indeed, to be the case; and from Cleave's Theorem proof of would then follow that infinite creative sequences exist. However, a key portion of Cleave's the this theorem contains a gap. Viz. (see p. 207 of the cited paper), the argument proceeds from fs is assumption that fh*(i) is defined at h(s) (which it is desired to show impossible) to a claim that support also defined at h(s), fg being complementary to fh*(i)· But the validity of this passage, without to fh*(i)• ing lemmas, is quite nonobvious, since the nowhere-defined function (e.g.) is complementary fs being and it is not a priori evident that (fh(s)• h(s)) cannot have a cycle of length fh*(i)(h(s)) without

defined at h(s). (Received June 22, 1962.)

62T-264. JOSEPH LEHNER, Michigan State University, East Lansing, Michigan. On the

subgroup topology of Marshall Hall algebra and the theory of numbers.

Marshall Hall (Ann.of Math. 52 (1950), 127-129) has defined a "subgroup topology" for in G groups G satisfying the following Property A: the intersection of all subgroups of finite index 340 is the identity, He proved that all free groups have Property A. In the present paper it is shown that a larger class of groups possesses this property. Denote by F (g; jl',l2, ... , .ls;t) the group with generators Al'Bl'"''Ag,Bg; E 1.... ,Es; P 1, ... ,Pt; and relations E{1 = •.• = E~s = 1, -1 -1 -1 -1 . . . P 1 ... PtE 1 ... Es A 1B 1A 1 B 1 AgBgAg Bg = 1, These groups are unportant m theory of discon- tinuous groups. Then: for s > 0, t > 0 or for g > 0, s = t = 0, F has Property A. The proof is made by constructing a representation of F as a group of 2 X 2 matrices with entries from the domain of integers of a fixed algebraic number field (depending on F) and utilizing a simple theorem on the con gruence subgroups of the matrix group. In the case s > 0, t > 0, the representation was exhibited in a previous paper (Mich. Math. J, 7 (1960), 233-236). Hall's result on free groups is accessible by this method only for countably generated groups. The representation is constructed by means of isometric circles. (Received June 29,. 1962,)

62T-265, KATSUMI NOMIZU, Brown University, Providence 12, Rhode Island. A new proof of de Rham's decomposition theorem for a reducible Riemannian manifold.

Let M be a simply connected complete Riemannian manifold and let T x(M) = TJr + T~ be a decomposition of the tangent space T x(M) into two mutually orthogonal subs paces invariant by the homogeneous holonomy group, Let M' and M" be the maximal integral manifolds through x of the parallel distributions obtained from TJr and T~ respectively, G. de Rham proved that M is isometric to the direct product M' X M" ti'ur la r6ductibilit6 d'un espace de Riemann, Comment, Math, Helv, 26 (1952), 328-344]. A rather simple and more geometric proof is given by making use of the notion of development of curves in M into T x

T x(M). Since the Euclidean space T x(M) is the direct product of two Euclidean subspaces T::C: and T~'· Ct is represe·nted as a pair (At,Bt), where At and Bt are certain curves in T::C: and T::c:" respectively, Let xt be the unique curve in M' which is developed upon At. It is proved that the end point of xt depends only on the homotopy class of zt, thus giving rise to a projection '11'1 of M onto M 1• Similarly, a projection 'Tr"" of M onto M" is defined, It follows that M 1 and M" are simply connected, It is easy to show that Tr1 X7r" is a global isometry of M onto M 1 X M". (Received June 29, 1962,)

62T-266, L. P. BELLUCE, 2114 1/2 Sacramento Street, Berkeley 2, California. Some results on infinite valued predicate logic. Preliminary report.

For reference see Abstracts 571-153, 571-154, Notices Amer, Math, Soc. October, 1960. Let

L be a first order language interpreted in R = (9,1]. Let 0 l!i r ~ 1, Definition. Cr (Dr) is the set of wffs P of L such that for all R-assignments f, fP 1:::. r(fP > r), Definition, r is a weak recursive real number (wrr) if there are recursive functions p,q such that q(n) > 0 and r = irun p(n)/q(n). Theorem. Dr is recursively enumerable iff r is a wrr. Moreover if r is rational an explicit axiomatization is given, This theorem is compared with the following unpublished result of C. C. Chang: if r "!- 0 is rational then Cr is not recursively enumerable, For each wff P of L let P* = sup {fP: f is an R-assignmentj. Theorem, For each wff P, P* is a wrr. Theorem, C * is not recursively enumer p able for any wff P of L. Corollary, If Cr "f Dr then Cr is not recursively enumerable, (Received June 29, 1962,)

341 62T-267. M. V. SUBBARAO and R. A. MELTER, University of Missouri, Columbia, Missouri. An isomorphism between two rings of arithmetic functions.

Let (a,b), [a,b] denote respectively the g.c.d. and l.c.m. of the positive integers a and band let A. (n) be a non vanishing, complex valued, completely multiplicative arithmetic function. The authors define the star ·product f * g of any two complex valued arithmetic functions f(n) and g(n) to be given by (f * g)(n) = L f(a)g(b)). (c), c = (a,b), summed over all positive integers a,b such that [a,b} = n. It is shown that the set S of all complex valued arithmetic functions forms a ring with respect to the natural sum and star product as ring addition and ring multiplication respectively; and that this ring is isomorphic to the ring formed on S with natural sum and natural product as the ring operations. Various consequences of this result are obtained, including inversion formulas for arithmetic func tions involving the star product. (Received June 29, 1962.)

62T-268. E. F. BECKENBACH and G. A. HUTCHISON, University of California, Los Angeles 24, California. Meromorphic minimal surfaces.

The Nevanlinna theory of meromorphic functions of a complex variable is based primarily on Jensen's formula. The authors present a generalization of this formula for minimal surfaces and show how the formula thus generalized can be applied to yield an extension of the theory to include these surfaces. The affinity of a minimal surface for a point in space is defined as the sum of an enumerative function, a proximity function, and a skewness function. Since the results apply in particular to plane conformal maps, they yield a measure of the affinity of a meromorphic function of a complex variable to any point of space, not merely to points of the complex plane. (Received May 16, 1962.)

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342 VOLUME 12 DIFFERENTIAL GEOMETRY Symmetric Spaces and Affine locally symmetric spaces. Groups of isometries. Rie· mannian globally symmetric spaces. The exponential mapping SYMMETRIC and the curvature. Locally and globally symmetric spaces. Com· pact Lie groups. Totally geodesic submanifolds; Lie triple sys SPACES tems. Exercises. Notes. By SIGURDUR HELGASON Decomposition of Symmetric Spaces algebras. The duality. Sectional Massachusetts Institute of Technology Orthogonal symmetric Lie curvature of symmetric spaces. Symmetric spaces with semi· August 1962, about 500 pp., $12.50 simple groups of isometries. Notational conventions. Rank of symmetric spaces. Exercises. Notes. This book, the first in an important area of con· Symmetric Spaces of the Noncom pact Type temporary mathematics, presents an introduction to Decomposition of a semi-simple Lie group. Maximal compact mod.er':l differential geometry, and to the theory of subgroups and their conjugacy. The lwasawa decomposition. sem1-s1mple L1e groups. These two disciplines are Nilpotent Lie groups. Global decompositions. The complex case. combined in an account of E. Cartan's theory of Exercises. Notes. subsequent contributions symmetric spaces. Certain of the Compact Type to the theory have been taken into account and a Symmetric Spaces type and the noncompact chapter on the theory of functions on symmetric The contrast between the compact points; Singular points· The dia· spaces has been included. Some familiarity with type. The Weyl group. Conjugate groups. Control over singular elementary point-set topology is assumed. gram. Applications to compact th~ set. The fundamental group and the center. Application to the Each chapter is followed by exercises and a bibli· symmetric space U/K. Classification of locally isometric spaces. which will serve as a guide to the original ography Appendix; Results from dimension theory. Exercises. Notes. papers. The book is suitable for text book in gradu· ate courses, and will be an important monograph Hermitian Symmetric Spaces for all mathematicians interested in the subject. Almost complex manifolds. Complex tensor fields; The Ricci Bounded domains; The Kernel function. Hermitian sym· Contents: curvature. and the noncompact type. Geometry metric spaces of the compact type Elementary Differential Irreducible orthogonal symmetric Lie algebras. Irreducible her· fields. Mappings. Affine connections. Manifolds. Tensor mitian symmetric spaces. Bounded symmetric domains. Exercises. Parallelism. The exponential mapping. Covariant differentiation. Notes. 1he structural equatioos. The Riemannian connection. Complete Riemannian manifolas. lsometries. Sectional curvature. Rieman· On the Classification of Symmetric Spaces nian manifolds of negative curvature. Totally geodesic submani· Reduction of the problem. Automorphisms. lnvolutive auto· folds. Exercises. Notes. morphisms. E. Cartan's list of irreducible Riemannian globally symmetric spaces. Two-point homogeneous spaces; Symmetric Lie Groups and Lie Algebras spaces of rank one; Closed geodesics. Exercises. Notes. The exponential mapping. Lie subgroups and subalgebras. Lie transformation groups. Coset spaces and homogeneous Functions on Symmetric Spaces spaces. The adjoint group. Semi-simple Lie groups. Exercises. Integral form~las. Invariant differential operators. Spherical Notes. functiOns; Def1mt10n and examples. Elementary properties of The formula for the Structure of Semi-simple Lie Algebras sphencal functiOns. Some algebraic tools. spherical function. Mean value theorems. Exercises. Notes. Preliminaries. Theorems of Lie and Engel. Cartan subalgebras. Root space decomposition. Significance of the root pattern. Real Bibliography-List of Notational Conventions forms. Cartan decompositions. Exercises. Notes. Subject Index.

Just published: Volume 11 CURVATURE and HOMOLOGY By SAMUEL I. GOLDBERG, University of Illinois June 1962, 315 pp., $8.50 Forthcoming: Volume 13 Introduction to the THEORY of INTEGRATION By T. H. HILDEBRAND"!; University of Michigan Early fall 1962, about 300 pp.

343 SCHOOL l\1ATHEMATICS STUDY GROUP The Yale University Press will continue to print and distribute the School Mathematics Study Group texts and commentaries for Grades 7-12 during 1962-63. These materials are designed to improve substantially the curriculum of school mathematics by offering the student not only the basic mathematical skills but also a deeper understanding of the basic concepts and structure of mathematics. The 1962 edition of the SMSG texts and commentaries has a "new look," but the subject material is essentially the same. Minor mathematical or grammat ical corrections have been made. Orders will be filled on a first come, first served basis. Your inquiries about these volumes are cordially invited. Please address all correspondence to: Yale University Press, School Mathematics Study Group, 92A Yale Station, New Haven, Connecticut.

Math for Junior High School, Volume 1, Student's Text, Parts I and II Math for Junior High School, Volume 1, Teacher's Commentary, Parts I and II Math for Junior High School, Volume 2, Student's Text, Parts I and II Math for Junior High School, Volume 2, Teacher's Commentary, Parts I and II First Course in Algebra, Student's Text, Parts I and II First Course in Algebra, Teacher's Commentary, Parts I and II Geometry, Student's Text, Parts I and II Geometry, Teacher's Commentary, Parts I and II Intermediate Mathematics, Student's Text, Parts I and II Intermediate Mathematics, Teacher's Commentary, Parts I and II Elementary Functions, Student's Text Elementary Functions, Teacher's Commentary Introduction to Matrix Algebra, Student's Text Introduction to Matrix Algebra, Teacher's Commentary

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ELLIOTT MoNTROLL, Editor Associate Editors Term ending Term ending Term ending Decemb_er 31, 1962 December 31, 1963 December 31, 1964 G. F. Chew Arthur Erdelyi Leslie L. Foldy N. Marcuvitz Stanley Desser Bernard Friedman L. I. Schiff Roy Glauber George Uhlenbeck A. N. Taub Walter Kohn John C. Ward J. A. Wheeler T. D. Lee Arthur S. Wightman C. N. Yang Arnold Siegert Eugene P. Wigner JOURNAL OF MATHEMATICAL PHYSICS ... a medium for highly mathematical physics articles and papers on branches of mathematics which are cur rently or potentially useful for the development of theoretical physics. Publication commenced in 1960 on a bimonthly basis; the journal will appear monthly, starting with the Janu ary, 1963 issue (Vol. 4, No. 1). Physicists and mathematicians concerned with mathe matical methods for the solution of physical problems, as well as original research furthered by such methods, are invited to subscribe.

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347 BANACH SPACES MAA Studies OF ANALYTIC FUNCTIONS by Kenneth Hoffman, Massachusetts Institute of in Mathematics Technology. Beginning with the publication of A study of some important Banach spaces of Ana STUDIES IN MODERN ANALY lytic functions in the unit disk. The book offers a SIS, the Mathematical Association of unified treatment of fundamental theorems on ana America launched a new series of books lytic functions in the Hardy class HP. Where possi in mathematics under the guidance of Robert P. Dilworth of the California ble, results are derived by the methods of functional Institute of Technology and Chairman analysis. The adoption of this point of view provides of the Committee on Publications for elegant proof of some important theorems in classical the MAA. analytic function theory. In The Modern Analysis This series of books will bring to the Series, edited by R. Creighton Buck. members of the Association and to the 1962 217 pages Text price: $8.25 general mathematical community, ex pository articles at the college and grad uate level on recent developments in TOPOLOGY OF 3-MANIFOLDS mathematics and the teaching of math ematics. One major aim of the series AND RELATED TOPICS is to help overcome the communication barrier which has arisen as a natural edited by M. K. Fort, Jr., University of Georgia consequence of the tremendous accel Contains the proceedings of the topology institute eration in mathematical development within the past 25 years. which was held at the University of Georgia in 1961. The articles which appear in the book have been con These short-paper volumes may be tributed by prominent topologists who attended the used effectively as a basis for seminars, reports, informal talks, and as supple institute. The majority of the papers deal with mani mentary material to provide background fold topology and are of a geometric nature, thus knowledge for both students and faculty. offering an excellent introduction to and summary of Each volume offers wide coverage - current research in this branch of mathematics. containing articles at various levels of i962 256 pages Text price: $7.50 difficulty. One or two volumes will be published each year. Volume I: STUDIES IN MATHEMATICAL STATISTICS MODERN ANALYSIS is edited by R. by John E. Freund, Arizona State University Creighton Buck, University of Wiscon sin. It includes papers by E. J. McShane, A modem treatment of mathematical statistics pre University of Virginia, M. H. Stone, senting a carefully designed balance between theory University of Chicago, E. R. Lorch, Columbia University, and Casper Goff- and application. The text features a sound introduc man, Purdue University. · tion to probability based on the theory of sets and develops an appreciation for statistical applications 1962 182 pages through its careful use of problems. Price: $4.00 (Prepaid -to non-MAA members) Special MAA Member price for one copy 1962 390 pages Text price: $7.50 only: $2.00 (When purchased through MAA)

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BAY AREA AssiGN grams. o Urgent requirements exist for highly qualified MENTS lln addition to the El Segundo activity, Aerospace engineers, physicists and mathematicians in the following Corporation has assignments for qualified personnel to areas: CoMPUTER PROGRAMMERS I Emphasis will be perform computing functions in the San Francisco area. placed on scientific programming in one or more of these Opportunities here involve systems engineering for pro areas : orbital mechanics and trajectory analysis, aero gramming real time satellite control, and various phases of dynamics, thermodynamics, missile dynamics, control sys high speed computing. For these positions a degree in tems and structures, simulation of guidance systems and Mathematics, Physics or Electrical Engineering is required computers. Degrees in Engineering, Physics, or Mathe together with at least four years experience in technical matics with programming experience for the IBM 7090 are computer programming. D Qualified applicants are in desirable for these applications. Other areas include man vited to contact Aerospace Corporation, an equal oppor agement data processing, compiler and monitor design and tunity employer. o To arrange convenient interviews appli development. NUMERICAL ANALYSTS I Individuals with cants should write to Mr. Charles Lodwick, Room 214, advanced degrees in Mathematics are desired for both Aerospace Corporation, Los Angeles 45, California. J ------fA\ AEROSPACE ~CORPORATION Organized in the public interest and dedicated to providing objective leadership in the advancement and application of space scieizce and technology for the United States Government. Research Scientists Boeing Openings for to work with Computer Programmers, Advanced Computer Methods Analysts, Systems Research Mathematicians Experienced research scientists with PhD or equivalent background in pure These positions, with Boeing's Aero-Space and applied math, statistics, theoreti Division offer challenging assignments and cal physics and celestial mechanics exceptional opportunities to advance to will have a unique opportunity to higher levels of responsibility and income. perform original work on a variety of Openings are available, now, in the follow classified research programs in the ing areas: following areas: COMPUTER PROGRAMMERS ... to perform digital com puter programming for solution of scientific and engi • Mathematical • War Gaming neering problems in connection with missile and space Physics • Research in vehicles, as well as performance of physical, mathemati • Ballistic Missile Mathematics cal and numerical analyses. Systems eComputer to develop or con Ballistics Sciences COMPUTER METHODS ANALYSTS ... • Exterior duct research in computer system methods, including • Astronautics • Operations implementation and programming for high Research specification, • Management Data speed, large-scale digital computer systems. Processing and • Mathematical Analysis Statistics RESEARCH MATHEMATICIANS ... to conduct research into, and explore applications of, advanced mathematical techniques applied to the development or improvement Latest Computers Available of specific scientific and engineering problem solutions. We can assure you the use of the most Requirements include a B.S. degree in Engi modern computers in your work includ ing the Navy's famous NORC (Naval neering, Mathematics or Physics, plus a Ordnance Research Calculator), and the minimum of four years of applicable experi IBM 7090. This month, the IBM ence. Salaries are competitively commensu STRETCH will be installed, and will DOUBLE the present computing capac rate with experience. ity. In addition to the computers themselves, capable Junior Scientists Send your resume today, to Mr. Lawrence will help expedite your work. W. Blakeley, the Boeing Company, P. 0. Box 3822- !ME, Seattle 24, Washington. The Boeing Company is an equal oppor The Naval Weapons Lab is located on the tunity employer. Potomac, about 55 miles from downtown Washington (fast becoming the Science Capital of the nation), and right in the middle of year-round resort living. Housing, World's Fair Invitation schools and shopping are pleasant and Many American Mathematical Society inexpensive throughout the area, and you members attending the Summer Meeting in will have the use of a variety of Naval Vancouver will want to arrange their trip to recreational facilities. include a visit to the science-oriented Seattle World's Fair. This is your invitation, while Starting salaries range up to $13,730, plus in Seattle, to call at the Boeing Professional the very real benefits of Career Civil Service. Employment Center, located in the MEBA For further information, write the Director, Building, Second Avenue at Broad Street, Computation and Analysis Laboratory near the Science Pavilion entrance to the Fair. Hours are 8:30 a.m. to 5:00 p.m., Monday through Friday. Free parking for U.S. Naval Boeing visitors. For further information phone Mr. Carl A. Anderson at JU 3-0858. Weapons Laboratory Department of the Navy Dahlgren, Virginia BOEING

354 INDEX TO ADVERTISERS Name Page Name Page

Academic Press, Inc. • • • • • • • • • • • • • • • • . 343 Houghton Mifflin Company •••••.• , • • • • • • 350 Aerospace Corporation . • . • . • . • • • . • • . • • 353 Hughes Aircraft Company - American Institute of Physics. • • • • • • • • • • 346 Aerospace Divisions • • • • • . . • • • • • • • • • • • • 351 American Mathematical Society. • • . • • . • • 356 The Johns Hopkins University - Atomics International The 4Jplied Physics Laboratory. • • • • • . • • 352 A Division of North American Aviation, Inc. 351 The Marathon Oil Company • . . • • . • • • . • • • 267 Boeing Company. • • • • • • • • • • • . • . • • • . • • • . 354 Operations Evaluation Group...... 278 Chrysler Corporation Missile Division •..•••.....••••••..••• 342 Pan American World Airways, Inc. Guided Missiles Range Division • • • • . . . • . 272 Cushing-Malloy, Inc. . • • . . • • . • • • • • • • • • • 350 Philco Corporation . • • • • . • . . . • • • . . . • • • • 283 Encyclopaedia Britannica Press ••••.•• , • 349 Prentice-Hall, Inc...... 348 General Electric Knolls Atomic Power Laboratory. • . • • • . • • . . • • • • . • • • • • 350 Space Technology Laboratories, Inc.. . . • . 345

Ginn and Company. • • • • • • • • • • • • • • • • • • • • 264 u.S. Naval Weapons Laboratory...... 354

Harper & Row, Publishers...... 347 Yale University Press. • • . • • • • • . • • • • • . • . 344

RESERVATION FORM

UNIVERSITY OF BRITISH COLUMBIA MEETING, Vancouver, Canada August 26 - 31, 1962

UNIVERSITY HOUSING RESERVATION FORM

Please make reservations by using the form below. It should be mailed to Conference Office, University of British Columbia, Vancouver 8, Canada, at the earliest possible date. Those who wish to stay at a hotel or motel should make their reservations ------(cteiaeh-onth1s11ne) ______directly with the hotel or motel. _

American Mathematical Society Housing Reservation Form -Vancouver Meeting

Name _____,~~--~~~~------7.'-~~------Please print (last) (first) Institution------Address ------Arrival Date ______Time ____: Departure Date ______Time _____

Housing available from 2:00 p.m. Saturday, August 25th, to 2:00 p.m. Saturday, September 1st.

Number of persons requiring accommodation (including children 12 years and over):

Male Female Male Female Husband + Wife Permanent residences - single rooms at $4 in double rooms at $3 per person

Dormitories - in single rooms at $2 in double rooms at $1. 50 per person

Only a few housing units suitable for families with young children are available on campus. Please apply as early as possible. You will receive an immediate reply. Total number in family: ; number of adults ; ages of children Can you provide your own crib/camp bed and sleeping bag/bedding for childre-n-.-."'Y"'e_s ____ -:N-=o-_:_-~~~- Number expecting to attend barbecue _____

355 AMERICAN MATHEMATICAL SOCIETY SECOND-CLASS POSTAGE 190 PAID AT Hope Street PROVIDENCE, RHODE ISLAND Providence 6, Rhode Island AND ANN ARBOR, MICHIGAN

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MATHEMATICS OF COMPUTATION Beginning with Volume 16, Number 7, January 1962 published by The American Mathematical Society for the National Academy of Sciences- National Research Council TABLE OF CONTENTS July 1962 Some Relations and Values for the Generalized Riemann Zeta Function ELDON R. HANSEN & MERRELL L. PATRICK Euler's Constant to 1271 Places ...... DONALD E. KNUTH A Calculation of the Number of Lattice Points in the Circle and Sphere W. FRASER & C. C. GOTLIEB Sign Wave Analysis in Matrix Eigenvalue Problems K. M. BROWN & P. HENRICI Acceleration Techniques for Iterated Vector and Matrix Problems P. WYNN Computing Error Bounds in Solving Linear Systems...... J. SCHRODER A Note on Finite Difference Methods for Solving the Eigenvalue Problems of Second-Order Differential Equations. . .. M. R. OSBORNE

A Method for Computing the Circular Coverage Function A. R. DIDONATO & M. P. JARNAGIN Optimum-Point Formulas foJ/ Osculatory and Hyperosculatory Interpolation HERBER E. SALZER The journal is published quarterly iu one volume per year. Subscription per year . $8.00 Single copies $2.50 Send Orders to AMERICAN MATHEMATICAL SOCIETY 190 Hope Street, Providence 6, Rhode Island