THE AMERICAN MATHEMATICAL SOCIETY

Edited by John W. Green and Gordon L. Walker

CONTENTS

MEETINGS

Calendar of Meetings . • . • . • . • ...... • • • . . . • . . . . 168 PRELIMINARY ANNOUNCEMENT OF MEETING...... 169 DOCTORATES CONFERRED IN 1961 ••••••••••••••••••••••••••••.•• 173 NEWS ITEMS AND ANNOUNCEMENTS • • • • • • • • • • • • • • • • • • • 189, 194, 195, 202 PERSONAL ITEMS •••••••••••••••••••••••••••••••.••••••••••• 191 NEW AMS PUBLICATIONS •••••••••••••••••••••••••••••••••••.•• 195 LETTERS TO THE EDITOR •.•.•.•••••••••.•••••.••••••••••••••• 196 MEMORANDA TO MEMBERS

The Employment Register . . . • . . • . . . . • . . • • . . . • . . . • • . . • . • . . • 198 Dues of Reciprocity Members of the Mathematical Society of Japan • . • . • . 198 SUPPLEMENTARY PROGRAM -No. 11 ••••••••••••••••••••••••••••• 199 ABSTRACTS OF CONTRIBUTED PAPERS •.•••••••••••.••••••••••••• 203 ERRATA •••••••••••••••••••••••••••••••••••••••••••••••.• 231 INDEX TO ADVERTISERS • • • • • • • • • • • • • . • . • • • • • • • • • • • . • • • • • • • • • • 243 RESERVATION FORM •••.•.•••••••••••.•••••••••••.••••••••••• 243 MEETINGS

CALENDAR OF MEETINGS

NOTE: This Calendar lists all of the meetings which have been approved by the Council up to the date a~ch 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*

592 August 28-31, 1962 Vancouver, British Columbia July 6 (67th Summer Meeting) 593 October 27, 1962 Hanover, New Hampshire Sept. 12 594 November 16-17, 1962 Tallahassee, Florida Oct, 2 595 November 17, 1962 Los Angeles, California Oct. 2 596 November 23, 24, 1962 Northwestern University Oct, 2 January 24-28, 1963 Berkeley, California Nov. 23 (69th Annual Meeting) April 26-27, 1963 New Mexico State University August 26-30, 1963 Boulder, Colorado (68th Summer Meeting) January 20-24, 1964 Miami, Florida (70th Annual Meeting) August, 1964 Ann Arbor, Michigan (69th Summer Meeting) January, 1965 Denver, Colorado (71st 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- 9.\l.arters Offices of the S~ciety in· Providence, Rhode Island, on or before these deadlines. The dead­ lines also apply to news items. The next two deadline dates for t>y title abstracts are June 29 and September. 5.

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The NOTICES of the American Mathematical Society is published by the Society six times a year, in February, April, June, August, October and November. Price per annual volume is $7 ,00, Price per copy, $2.00, Special price for copies sold at registration desks of meetings of the Society, $1.00 per copy. Subscriptions, orders for back numbers (non 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 24, 1912, as amended by the act of August 4, 1947 (Sec. 34. 21, P. L. and R.). Accepted for mailing at the special rate of postage provided for in section 34.40, paragraph (d).

CoP:vright © 1962 by tbe America11 Matbematical Society Pri11ted in tbe United States of America -

168 PRELIMINARY ANNOUNCEMENT OF MEETING

SIXTY-SEVENTH SUMMER MEETING University of British Columbia Vancouver, B. C. August 28-31, 1962

The sixty- seventh summer meeting will be given at 9:00 A.M. on Thursday, of the American Mathematical Society will August 30. The title of his talk is "Models be held at the University of British Colum­ of complete theories". bia, Vancouver, B. C., from Tuesday, There will be sessions for contri­ August 28, to Friday, August 31, 1962. buted papers in Buchanan Building at During this week there will be meetings times to be announced in the Program of at the University of British Columbia of the Meeting which will appear in the the Mathematical Association of America, August NOTICES. No sessions will be held and the Society for Industrial and Applied for papers which fail to meet the july 6 Mathematics. The Hedrick Lectures of deadline. the Mathematical Association of America The Council of the Society will meet will be given by Professor Andrew Glea­ at 4:00 P.M. on Thursday, August 30 in son of Harvard University. the Study Room of the International House. There will be no Colloquium Lec­ There will be a Business Meeting of the tures at this meeting because of the occur­ Society at 10:30 A.M. on Friday, August rence in August 1962 of the International 31, in the Auditorium Building. Congress of Mathematicians at Stockholm, Sweden. REGISTRATION The Committee to Select Hour Speakers for Summer and Annual Meet­ Registration headquarters will be ings has invited Professor C. C. Lin of in the main floor corridor of the Buchanan the Massachusetts Institute of Technology, Building. All persons attending the meet­ Professor Charles Loewner of Stanford ing are requested to register immediately University, Professor Albert Nijenhuis of on arrival. The schedule of registration the University of Washington, Professor fees is as follows: R. S. Pierce of the UniversityofWashing­ of participating organi- ton, and Professor R. L. Vaught of the Members zations (except students) $2.00 University of California, Berkeley to ad­ dress the Society. All of these lectures First nonmember in member's will be given in the Auditorium Building family .SO (about 200 yards west of the Buchanan Other nonmembers in mem- Building). Professor Lin's address, en­ free titled "Hydrodynamic stability -- a study ber's family in applied mathematics", will be given at Students free 9:00 A.M. on Wednesday, August 29. Pro­ not in any of the fessor Loewner will speak at 2:00P.M. on Nonmembers Thursday, August 30. The title of his talk above categories $5.00 is "On semigroups in analysis and geome­ A directory of all persons attending the try". Professor N ijenhuis' s talk on" Deri­ meetings will be located at registration vations and structure" will be given at headquarters. The employment register 2:00 P.M. on Tuesday, August 28. Pro­ will be located in Room Bu-1221 of the fessor Pierce will speak on "Representa· Buchanan Building. There will be exhibits tions of lattices" at 9:00 A.M" on Friday, in the main hall on the second floor of the August 31. Professor Vaught's address Buchanan Building.

169 ROOMS AND MEALS The main cafeteria located in the Auditorium Building will be open all day Accommodations will be available from 8:30 A.M. Cafeteria service will also in the University residences to all attend­ be available in three dining rooms located ing the meetings and to adult members of in the three main residence areas. Meal their families. Only a few units are avail­ hours are: breakfast 7:30- 8:30 A.M.;· able on campus which are suitable for lunch 11:45-1:00 P.M.; dinner 5:00-6:30 families with children under twelve. Chil­ P.M. Prices of regular cafeteria meals dren twelve and over can be accommodated are: breakfast 50 cents; iunch 65 cents; in the regular residences at the regular dinner 85 cents. rates. Low cost dormitory space is avail­ able in converted huts in Fort Camp and RESERVATIONS Acadia Camp. The costofdormitoryhous­ Reservations for University resi­ ing will be $Z.OO a night per person in a dences and dormitories should be made by single room. In a double room with sep­ writing directly to the Conference Office, arate beds the cost is $1.50 a night per University of British Columbia, Vancouver person. The university also has a number 8, Canada at the earliest possible date and of new, modern, three-storey permanent before August 9. A reservation form for residences containing mostly single rooms. this purpose can be found on the inside The cost of permanent residence housing back cover of these NOTICES. will be $4.00 a night per person. A small number of double rooms with separate HOTELS AND MOTELS beds are available at $3.00 a night per person. Towels and bedding will be pro­ Persons desiring hotel and motel vided. Automatic laundry facilities are accommodations should make their reser­ available. Rooms in residences may be oc­ vations directly with the appropriate man­ cupied from Z:OO P.M., Saturday, August ager, and under no circumstances write to ZS to Z:OO P.M., Saturday, September I. the Conference Office which is in charge The nearest hotels or motels are only of the University facilities. located in downtown Vancouver (about 6 It is advisable to make reservations miles from the campus). Thenearestpub­ early because the Seattle World's Fair is lic campsites are located at Alouette Lake, expected to attract an unusually large num­ about 40 miles from the campus. her of tourists to the Pacific Northwest.

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 Hotel1061 Howe St. MU 2-3171 103 8.00 10.00 Georgia Hotel 80 1 W. Georgia MU 3-118Z 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 Z-2767 74 7.00 9.00 Bayshore Inn 1601 Georgia St. MU 2-3377 308 12.00 15.00 Sands Motor Hotel 1755 Davie St. MU 2-1831 100 8.50 9.50 Sylvia Hotel 1154 Gilford St. MU 1-9321 125 5.00 8.00 Vancouver Hotel 900 W. Georgia MU 4-3131 561 9.00 11.50

170 ENTERTAINMENT AND RECREATION feringto the U.B.C. bus running to the cam­ There will be a barbecue on Wed­ pus (10 cents). nesday, August 29 at 6:00 P.M. The cost Airport, bus, and limousine service. will not exceed $2.50 per adult, with half Bus and limousine service costs $1.25, price for children under twelve. A tea will cab from the airport to Vancouver about be given between 4:00 P.M. and 6:00 P.M. $4.50, cab from the airport to U.B.C. about on Tuesday, August 28. More complete $6.00, and cab from the city center to information concerning the tea and barbe­ U.B.C. about $3.00. For one or two per­ cue will be given in the Program of the sons, the most economical way to travel Meetings. is to take the bus from the airport to Gran­ Some of the athletic facilities of the ville and Broadway streets, and continue University will be available to members by taxi to U.B.C. (total cost about $4,00 for and their guests. The Seattle World's Fair one per son). and the Vancouver International Festival Steamship Lines. The only regularly will be in operation during the meeting. scheduled ships between the California ports and Vancouver are operated by the TRAVEL Pacific and Orient Lines, but their sailing Vancouver is approximately 140 and arrival dates make them completely miles north of Seattle, Washington on unsuitable for persons attending the sum­ Highway 99. The best route to the Univer­ mer meetings. There are other lines that sity from the south is on Highway 99-B operate freighters with excellent accom­ via De as Island Tunnel (toll), Oak Street modations for a small number of passen­ Bridge (toll), West 41st Avenue and Marine gers, but they do not keep very rigid Drive. schedules. The trip from Los Angeles to Vancouver is served by Canadian Vancouver takes between 5 and 8 days and Pacific, Qantas, Trans Canada, and United the fares vary from about $130 to $180 on Airlines; the Canadian National, Canadian the Holland American Line.Less expensive Pacific, and Great Northern Railroads; and accommodations are available on the the Greyhound Busline. Automobile ferry American Mail and the Canadian Blue Star service is available from Vancouver Island Lines. Reservations should be made in ad­ on British Columbia Government ferries vance. from Nanaimo and Swartz Bay, and by Persons who are planning to attend Canadian Pacific from Nanaimo. both the Vancouver meeting and the Inter­ Public transportation in Vancouver national Congress of Mathematicians in is provided by buses from the city center Stockholm may find the following flight to the Blanca loop (15 cent fare), trans- schedules helpful.

Saturday Flight Lv. Stockholm 9:50 A.M. (Sat.) SAS Flight 565 Ar. Amsterdam 11:05A.M.

Lv. Amsterdam 1:25 P.M.(Sat. or Sun.) CPA Flight 301 Ar. Vancouver 4:35P.M.

Sunday Flight Lv. Stockholm 6:05 P.M. (Sun.) SN Flight 174 Ar. Amsterdam 7:20P.M.

Lv. Amsterdam 8:30P.M. (Sun. only) CPA Flight 363 Ar. Vancouver 11:40 P.M.

171 Saturdax or Sundax Flight Lv. Stockholm 8:30A.M. (daily) SAS Flight 551 Ar. Copenhagen 9:45A.M. Lv. Copenhagen 10:30 A.M. SAS Flight 551 Ar. Amsterdam 11:55 A.M.

Lv. Amsterdam 1:25 P.M. (Sat. or Sun.) CPA Flight 301 Ar. Vancouver 4:35P.M.

Thursday, Friday, and Saturday Flights Lv. Stockholm 9:45 P .M.(Thurs., Fri., or Sat.)SAS Flight 935 (via Oslo, Copenhagen, and Greenland) Ar. Los Angeles 5:20 A.M.(Fri., Sat., or Sun.) Lv. Los Angeles 7:30 A.M.CFri., Sat., or Sun.) There would be an added cost of about $10 on this flight due to surcharge on jet travel in the U. S. Ar. Vancouver 12:00 Noon

With a Vancouver-Stockholm ticket, stopovers are permitted in European cities anywhere west and south of Stockholm at no added cost. IT IS ADVISABLE TO MAKE FLIGHT RESERVATIONS AS EARLY AS POSSIBLE.

CUSTOMS INFORMATION House, 680 Fifth Avenue, New York 19, Crossing the United States-Cana­ New York; 102 West Monroe Street, Chi­ dian border either way is made without cago 3, Illinois; 1 Second Street, San difficulty or delay by citizens or perma­ Francisco 5, California. nent residents of the United States. Pass­ The Canadian currency system is ports are not needed. However, U. s. citi­ the same as that of the United States, but zens should carry some document estab­ the rate of exchange between the American lishing their citizenship. Alien permanent and Canadian dollar varies from time to residents of the United States are advised time. U. S. currency is accepted by most to have their Alien Registration Receipt merchants in Canada. However, United Card (U.S. Form 1-151). Visitors to the States visitors are advised to exchange United States who have only a single entry their currency for Canadian funds at a visa to the U.S.A. should present their bank in Canada to be assured of receiving documents at an office of the United Im­ the prevailing rate of exchange. migration and Naturalization Service be­ MAIL AND TELEGRAM fore entering Canada, to be sure that they have the necessary documentation for their Communications with members of re-entry to the United States. Further in­ the Society and their guests should be ad­ formation regarding entry to Canada and dressed to them in care of the American what merchandise may be taken back into Mathematical Society, The University of British the United States may be obtained by Columbia, Vancouver 8, Canada. writing to the Canadian Government Tra­ R. S. Pierce vel Bureau at one of the following addres­ Associate Secretary ses: Ottawa, Ontario, Canada; Canada Berkeley, California

172 DOCTORATES CONFERRED IN 1961

The list of doctorates in the mathematical sciences and related subjects from universities in the United States and Canada has been published annually in the May issue of the BULLETIN. The Editors of the BULLETIN and NOTICES have decided to now pub­ lish the list in the june issue ofthe NOTICES. This list is based on information supplied by the National Research Council, National Academy of Sciences. In 1961, 370 individuals were granted a doctorate in the mathematical sciences. Following are the names of the individuals and in each case when available, the university, the month in which the degree was conferred, minor subjects (other than mathematics) and the title of the dissertation.

Acheson, Willard Phillips and Euclidean spaces University of Pittsburgh, january Minor - Physics Anderson, Stuart Lee Dynamics of an elastic-Bingham body University of Washington, December Lebesgue constants in a certain class Adler, Irving of groups Columbia University, November Composition rings Andrew, David Robert University of Pittsburgh, August Adler, Roy Lee Weak and strong continuities in topo­ Yale University, june logical spaces Some algebraic aspects of measure preserving transformation Andrews, Newton Steele Auburn University, june Albert, Eugene Orthogonal harmonic homogeneous University of Virginia, june polynomials in three variables Markov chains and eigenvectors Antliff, William Bruce Alberti, Furio Queen's University at Kingston, Ontario, illinois Institute of Technology, june Canada, May Ordinal invariants of quadratic forms The limit of (n)m(E(n)x) as n - oo over a quadratic field Armendariz, Armando A. Alexander, Forrest Doyle Rice University, june Peabody College, May Cluster sets and quasiconformal map­ Minor - Education Administration pings An experiment in teaching mathemat­ ics at the college level by closed­ Askey, Richard Allen circuit television Princeton University, April Mean convergence of orthogonal se­ Ali, Mir Maswood ries and conjugate series University of Toronto, june Minor - Geophysics Aswad, Alfred George Contributions to linear orderly esti­ University of California (Berkeley), june mates of location and scale para­ Some statistical problems arising in meters a certification procedure. Part 1: Esti­ mation of the density of a material Allaud, Guy Dante with irregular surfaces. Part II: Eco­ University of Wisconsin, january nomic sample size Fiber spaces and loop actions Avila, Geraldo Severo de Souza Alperin, jonathan Lazare New York University, june Princeton University, October The asymptotic field of A point sources On a special class of regular p-groups in an inhomogeneous medium Ax, james B. Andalafte, Edward Ziegler University of California (Berkeley), Sep­ University of Missouri, june tember A metric characterization of Banach The intersection of norm groups

173 Bailey, Duane W. Berman, Simeon M. University of Oregon, june Columbia University, june Semi-normed algebras with involution Limiting distribution of the maximum term in sequences of dependent ran­ Bailey, Paul Bernard dom variables University of Washington, june Removal of the log factor in the asymp­ Berri, Manuel Philip totic estimates of the membrane University of California (Los Angeles), eigenvalues August Minimal topological spaces Baldwin, George Lewis University of Oklahoma, june Bhat, Beliyar Ramdas Minor - Chemistry University of California(Berkeley), Sep­ Some questions on distributions, nets tember and duality Bayes sequential tests for Markov chains Banilower, Howard Purdue University, january Bishir, john William Certain ideals in algebras of extended North Carolina State College, May real-valued functions Two problems in the theory of sto­ chastic branching processes Bardwell, George Eldred University of Colorado, June Bland, Richard Park Certain discrete distributions University of North Carolina, june A minimum average risk solution for Bauserman, Thomas the problem of choosing the largest University of Pittsburgh, August mean Anti-involutions in the plane of two complex variables Bleicher, Michael N. Tulane University, August Bednarek, Alexander Robert Lattice coverings of n-dimensional University of Buffalo, june Euclidean space by equal radius On the Mickle-Rado theorems and spheres binary relations Bohachevsky, Ihor 0. Bell, Howard Edwin New York University, june University of Wisconsin, August Simple waves and shocks in a system Reduction of matrices to canonical of conservation laws form under generalized Lorentzian transformations Bohn, S. Elwood University of Nebraska, june Bell, Lloyd Franklin Minor - Physics Stanford University, April A sub-function study of the Dirichlet A mathematical theory of guarantee problem for a quasi-linear differential policies equation Bellar, Fred james, Jr. Bollinger, Richard Coleman University of Maryland, june University of Pittsburgh, August Minor-Physics Summability of power series Pointwise bounds in parabolic and elliptic partial differential equations Borelli, Mario Benton, Edward Rowell Indiana University, june Harvard University, March Residual varieties Minor - Aeronautical Engineering Brande, Edward W ., S. j. Aerodynamic origin of the Magnus St. Louis University, June effect on a finned missile Minor - Philosophy B ere iter, Mary Catharine, Sister, O.P. The representations of binary quad­ Wayne State University, june ratic forms by quinary quadratic Linear sequence spaces forms

174 Brayton, Robert King and prime summands. Massachusetts Institute of Technology, September Bush, George Clark Minor - Physics Queen's University at Kingston, Ontario, On the asymptotic behavior of the num­ Canada, May ber of trials necessary to complete a On embedding a semi-group in a group set with random selection Cantrell, james Cecil Brillinger, David Ross University of Tennessee, December Princeton University, june Separation of the n-sphere by an Asymptotic means and variances in the (n - I)-sphere k-dimensional case Cantrell, Robert Headden, jr. Briney, Robert Edward Harvard University, june Massachusetts Institute of Technology, Gas film effects in the linear pyro­ june lysis of solids Minor - Languages Carlyle, jack Webster Intersection theory on quotients of University of California (Berkeley), Sep­ algebraic varieties tember Browder, Andrew Equivalent stochastic sequential ma­ Massachusetts Institute of Technology, chines September Carter, Frank Sydney Minor - Music University of Southern California, june Cohomology of maximal ideal spaces Contributions to the theory of Hilbert and some Dirichlet algebras space operators Buley, Ernest Robert Cayford, Afton Herbert University of California (Berkeley), jan­ University of California (Los Angeles), uary june Differentiability of solutions of cer­ A class of integer valued entire func­ tain types of variational problems tions Burdick, Donald Smiley Chamberlain, Erling William Princeton University, October Columbia University, june Stage by stage modification of poly­ Families of principal solutions of or­ nomial estimators by the jackknife dinary differential equations method Chambers, john Carlton Bures, Donald john Charles Case Institute of Technology, june Princeton University, june On the theory of minimum-cost infor­ The type of certain factors construc­ mation-collection systems ted as infinite tensor products Channapragada, Rao S. Burke, George University of lllinois, june University of Missouri, August Singular integral equations with an Some generalizations of the Glivenko­ infinity of intersecting arcs Cantelli theorem Chao, Chong-Yun Burton, David MacGregor University of Michigan, june University of Rochester, june Nilpotent Lie algebras The solution of hyperbolic systems of partial differential equations by double Cheema, Mohindar Singh Laplace transforms University of California (Los Angeles), june Buser, Mary Paul, Sister, C. S. j. Vector partitions and permutation St. Louis University, june vectors Minor - Philosophy Chicks, Barbara jean Thomson On the coefficients of the approxima­ University of Oregon, june ting polynomial for partitions into unit k- sample rank order statistics

175 Chou, Tao-Hsiung Conner, Howard Emmett Iowa State University, Ames, July Massachusetts Institute of Technology, Minor - Economic Statistics June Mixed integer programming with ap­ Minor - Physics plication of dynamic and non-linear A limit theorem for a space-dependent models for agriculture branching process Chowla, Paromita Conway, Melvin Edward University of Colorado, August Case Institute of Technology, June The class-number of real quadratic A set-theoretic model for logic sys­ fields and some diophantine equations tems in cyclotomic fields Cooley, james William Chwe, Byoung-Song Columbia University, june University of California (Berkeley), Sep­ Computational methods for the study tember of diatomic molecules Relative homological algebra and Cooper, Dale Edward homological division of Lie algebra North Carolina State College, February Clark, Allan Hersh Minor - Soils Princeton University, October Available soil moisture as a stochas­ Some applications of Hopf algebras tic process and cohomology operations Cornelius sen, Arvid Helge, Jr. Clay, jesse Paul Brown University, June University of Pennsylvania, June Finite deformation of elastic mem­ Prox~mity and equicontinuity in trans­ branes with application to the stability formation groups of an inflated and extended circular tube Clay, Robert Edward Notre Dame University, August Crawley, Peter Linton Contributions to Meteorology California Institute of Technology, June A decomposition theory for lattices Clifton, Yeaton Hopley without chain conditions Columbia University, June Completely integrable systems of flat Cross, Myrle Vivian Pfaffian equations University of Michigan, june Minor - Philosophy Cohn, James Alan On spaces approximated by open sets Harvard University, June with compact closures Some results in the cohomology theory Cundiff, joyce Coleman of finite groups University of Florida, June Minor - Astronomy Cohn, Martin Confluent cases of second order linear Harvard University, june differential equations with four singu­ Switching function canonical forms lar points over integer fields Daniel, Klaus H. Coleman, Donald Brooks University of California (Berkeley), Sep­ Purdue University, June tember On group rings A delivery-lag inventory model with emergency Conley, Charles Cameron Massachusetts Institute of Technology, Das, Krishna Mohan September Carnegie Institute of Technology, june Minor - Electrical Engineering Singularity and zero-free solutions On periodic solutions of long period for a class of complex non-linear dif­ for the moon problem ferential equation

176 Davis, Edward D. Eagon, john Alonzo University of Chicago, August University of Chicago, August Over-rings of commutative rings Ideals generated by the subdetermin­ ants of a matrix Davis, Kenneth Samuel Stanford University, january Edwards, Harold Mortimer The study of certain hyperbolic dif­ Harvard University, june ferential equations and their corre­ Applications of intersection theory to sponding symmetric Hamburger mo­ differential equations ment problems Eigel, Edwin George, jr. Davis, Morton David St. Louis University, june University of California (Berkeley), june Minor - Philosophy On an infinite game of perfect infor­ Paraphrases and divisor functions mation Ellis, Homer Godsey DeMar, Richard Francis University of Texas, june University of Wisconsin, january Solutions of analytic equations and Existence of interpolating functions of solutions of first order, analytic dif­ exponential type ferential equations DeMarr, Ralph Algernon Engelhardt, jacob University of lllinois, October New York University, February Minor - Statistics Solitary waves in two layer systems Point-to-set mappings and semi-group where the upper layer has variable transformations, with applications to density and the lower layer constant fixed set theory density Descloux, Alfred Falb, Peter Lawrence University of North Carolina, August Harvard University, june On the covariance between the num­ On differentials in function fields ber of offered and the number of over­ Faust, Claude Marie, Sister, C.C.V.I. flow requests in systems withlimited University of Notre Dame, june capacity On the boundary behavior ofholomor­ Dinsmore, Robert Sunderland phic functions in the unit disc Stanford University, October Feigl, Polly C. Bartholomew Extremum problems for harmonic University of Minnesota, june polynomials in higher • Euclidean Design considerations for the post­ spaces operative evaluation of analgesic Dixon, john Douglas drugs McGill University, May Feinberg, Irwin On general group extension New York University, june Drobnies, Saul Isaac On the existence of two-dimensional University of Texas, june subsonic compressible jets Minor - Business Administratlon Feldman, Dorian Concerning the uniform polynomial University of California (Berkeley), june approximation of a bounded function Contributions to the two armed bandit problem Drufenbrock, Madeleine Sophie, Sister, o.s.F. deFigueiredo, Djairo Guedes University of lllinois, October New York University, june Minor - Physics The coerciveness problem for forms On some metabelian groups of order over vector-functions plO Fishman, Robert Sumner Boston University, june Duda, Edwin The integral as an average on a group University of Virginia, june of translations in a subinvariant meas­ Brouwer property spaces ure space

177 Foland, Neal Eugene tions of the stable homotopy groups of University of Missouri, june spheres Continuous flows on arcs and Z-cells Gillilan, james Horace Forelli, Frank john, jr. University of lllinois, june University of California (Berkeley), june Minor - Astronomy Marcel Riesz's theorem on conjugate Generalized set functions functions Gilmer, Robert William, Jr. Fox, David Lawrence Louisiana State University, August New York University, june Rings in which every ideal has a reg­ Perturbation methods in a problem of ular representation waveguide theory Gindler, Herbert Aaron Frandsen, Henry University of California (Los Angeles), University of lllinois, june August Minor - Philosophy Some properties of operator spectra On metabelian groups of order plO Ginsburg, Michael with five generators Massachusetts Institute of Technology, Frederick, Karen Norma june New York University, October Minor - French Literature The Hopfian property for fundamental On the Lusternik-Schnirelman cate­ groups gory of space Freese, Raymond William Giri, Narayan Chandra University of Missouri, june Stanford University, October Ptolemaic metric spaces On tests with likelihood ratio criteria in some problems of multivariate Friedman, Manfred analysis New York University, February Two problems in transonic fluid dy­ Glasgow, Mark Othello namics University of Texas, june Some marginal distributions associ­ Furman, Walter Laurie ated with permutation cycles University of Florida, February Minor - Physics Gleiberman, Leon Mathematical research in Latin Amer­ University of Pittsburgh, August ican universities Representation of selected functions of a complex variable by surfaces in Galmarino, Alberto Raul Argund four space and resulting clas­ Massachusetts Institute of Technology, sification of Riemann surface theory September Minor - Economics Glick, Irving Isadore Isotropic diffusions University of Maryland, june On an analogue of the Euler-Cauchy Gangolli, Ramesh Anand polygon method for the partial differ­ Massachusetts Institute of Technology, ential equation llx x = f . june I··· n Minor - Economics Gluck, Herman Randolph Brownian motion and vector fields on Princeton University, june a Riemannian manifold The embedding of two-spheres in the four-sphere Geiger, David Scott University of lllinois, October Gold, Phillip john Metabelian groups of order pi 0 New York University, june The mapping class and symplectic modular groups Gershenson, Hillel Halkin University of Chicago, August Goldman, Aaron Simpson Some relationships between the Adams Oklahoma State University, May spectral sequence and Toda's calcula- On the determination of sample size

178 Goodman, Arnold Frank Harvey, John Grover, ll Stanford University, October Tulane University, June The selection of an optimum sampling Complete holomorphs and chains in partition partially ordered groups Govindarajulu, Zakkula Hayden, Thomas Lee University of Minnesota, March University of Texas, June Central limit theorems and asympto­ Minor - Physics tic efficiency for one sample non­ A convergence problem for continued parametric procedures. fractions Green, Paul Edgar Hellerstein, Simon University of Pennsylvania, December Syracuse University, September Minor - Economics Deficiencies of a meromorphic func­ Some intra-firm applications of Bay­ tion and distribution of values in angles esian decision theory to problems in and annuli business planning Hershenov, Joseph Greenleaf, Newcomb Massachusetts Institute of Technology, Princeton University, June September Local zeros of global forms Uniformly valid asymptotic solutions of the Orr-Sommerfeld equation and Guard, James Russell their application to plane Poiseville Princeton University, June flow The independence of transfinite induc­ tion up to ZO-ZO in recursive arithme­ Hetherington, Richard George tic University of Wisconsin, August Numerical integration over hyper­ Gurtin, Morton Edward shells Brown University, June Some theorems in the linear theory Hickman, James Charles of elasticity State University of Iowa, February On random sets, derived from sub­ Ha, Kwang Chul samples, for statistics based on the University of North Carolina, August entire sample The radical of topological abelian groups Hill, Bruce Marvin Stanford University, April Hachigian, Jack A test of linearity versus convexity Indiana University, June of a median regression curve 'Some further results on functions of Markov processes Hills, Norman Leslie New York University, October Haddock, Aubura Glen Semi-infinite diffraction gratings Oklahoma State University, August Some theorems concerning compact Hilt, Arthur Lincoln continua in the plane Lehigh University, October Vector fields and infinitesimal trans­ Hahn, Hwa Suk formations on Almast-Hermitian man­ University of illinois, October ifolds with boundary Minor - Astronomy On the relative growth of differences of partition functions Ho, Hung-Ta Brown University, June Hanna, Samuel Charles The compressible viscous layer in University of Pittsburgh, January rarefied hypersonic flow Metrization and uniform continuity Hare, William Ray, Jr. Ho, Yu-Chi University of Florida, June Harvard University, March Minor - Philosophy A study of the optimal control of dy­ Projective convexity namic systems

179 Holland, Samuel Shaheen, Jr. Johnsen, Eugene Carlyle Harvard University, March Ohio State University, December The Radon-Nikodym theorem in di­ Matrix rational completions satisfying mension lattices generalized incidence equations, and integral solutions to the .incidence Holland, Wilbur Charles, Jr. equation for finite projective plane Tulane University, June cases of orders n !i 2 (mod 4) Extensions of ordered algebraic struc­ tures Jones, Benjamin Franklin Rice University, June Hooker, William Weston The determination of a coefficient in University of California (Berkeley). Jan­ a parabolic differential equation uary Jones, Richard Hunn Lower bounds for the first eigenvalue Brown University, June of elliptic equations of orders two and Spectral e·stimates and their distribu­ four tions Hopper, Edgar Hugh Joseph, Roger David Auburn University, August Cornell University, September Orthogonal polynomials over discrete Contributions to perception theory sets Kahn, Donald W. Hudson, Anne Lester Yale University, June Tulane University, June On the real cohomology of the fibre On the structure of certain classes spaces of topological semigroups Kahn, Paul Markham Hunt, Robert Weldon University of Michigan, June University of Utah, June Some applications of collective risk The behavior of solutions of ordinary, theory to reinsurance and group ex­ self-adjoint differential equations of perienc~ rating arbitrary even order Kapadia, Chandrakant Harilal Iglehart, Donald Lee Oklahoma State University, August Stanford University, October Variance components in two way clas­ Dynamic programming and stationary sification models with intersection problems analyses of inventory Kaplan, Stanley James, Donald D. University of Pittsburgh, January Iowa State University, June Minor - Mechanical Engineering Minor - Philosophy Extensions and applications of the Periodic integral surfaces of periodic method of finite transform differential equations Karrass, Abraham Adelphi College, June Jarvis, Stephen, Jr. Parallel geometries and finite non­ Columbia University, June Desarguian planes Apparent wall shapes in viscous shear Keisler, H. Jerome flows University of California (Berkeley), June Ultra products and elementary classes John, Chelikuzhiel Thomas Kelly, Edgar Preston, Jr. Boston University, June Oklahoma State University, May Spherical bearing under axial and On abstract summability radial loading John, Floyd ldwal Kenelly, John Willis, Jr. Purdue University, August University of Florida, February On queueing, storage and related Minor - Philosophy topics An involution of period seventeen

180 Kennedy, Hubert Collings Lawson, Charles Lawrence St. Louis University, June University of California (Los Angeles), Minor - Philosophy June Group membership in rings Contributions to the theory of linear least maximum approximation Killam, Eleanor Yale University, June Leaf, Gary Kristian Locally M-convex algebras University of Illinois, October Minor - Physics Kishore, Nand A spectral theory for a class of linear University of California (Berkeley), Sep­ operators tember Arithmetical properties of Bessel Lee, Shing- Meng functions Cornell University, June On a two-parameter family of sum­ Koch, Charles Frederick mation methods University of Illinois, June Minor - Physics Leetch, James Frederick Density theorems in the product space Ohio State University, August of two abstract measure spaces Functions whose values are sets in a vector space Konsler, Thomas Rhinehart North Carolina State College, Raleigh, Lehman, Eugene H., Jr. May North Carolina State College, May Minor - Genetics Estimation of the scale parameter in A quantitative analysis of the growth the W eibull distribution and regrowth of a forage crop Leibowitz, Martin A. Kristensen, Leif Harvard University, March University of Chicago, August Rigorous derivation of Fermi age On the cohomology of two- stage Post­ theory nikov systems Levy, Lawrence Sherwin Krueger, Charles Gordon University of lllinois, February University of Illinois, February Minor - Music Minor - Philosophy Unique subdirect sums of rings Baire classes in a Denjoy-Tvjitzinsky Lick, Don Raymond measure space Michigan State University, June Kullback, Joseph Henry Sets of points of non-uniform conver­ Stanford University, January gence of Taylor series and trigono­ A quality control model for complex metric series items Lincoln, Andrew James Lange, John Edward Harvard University, June University of Wisconsin, June Magnetic devices in sampled analog Entire functions as limits of zero­ systems restricted polynomials Long, Paul Edward Larkin, William James, III Oklahoma State University, May University of Cincinnati, June Properties of certain non-continuous On the first-order homogeneous bi­ transformations linear algebraic differential equation Low, Leone Yarborough La Rue, James Arthur Oklahoma State University, May University of Pittsburgh, August Estimation of variance components, •comparison of relative strengths of Eisenhart's Model II, balanced on regular matrix methods and their re­ complete block and general two-way spective submethods classification

181 Lowengrub, Morton McCarty, George Schultz, jr. Duke University, june University of California (Los Angeles), Minor - Physics june Stress in the vicinity of a crack in a Homeotopy groups thick elastic plate McCoart, Richard Felix, Jr. Lubarsky, Bernard University of North Carolina, August Case Institute of Technology, june Irreducibility of certain classes of Generalized functions and the funda­ Legendre polynomials mental solutions for polyharmonic McCord, james Richard, III difference operations Massachusetts Institute of Technology, Lukawecki, Stanley Michael june Auburn University, August Minor - Physical Chemistry Diffraction of elastic waves by a semi­ Generalized multidimensional Pois­ infinite plane son distributions for finite Markov chains Lutts, john A., S. j. University of Pennsylvania, December McCoy, Thomas La Rue Some imbedding properties of locally University of Wisconsin, june compact Hausdorff spaces whichpos­ Entire functions with restraints on the sess unisolvent systems zeros of the partial sums MacGregor, Thomas Harold McCullough, Roger Stewart University of Pennsylvania, june Iowa State University, February Topics in the theory of Schlicht func­ Testing equality of means under vari­ tions ance heterogeneity MacRae, Robert Eugene McKinney, Earl Harry University of Chicago, june University of Pittsburgh, january Homological dimension of ideals in Vibration analysis of continuous beam­ Noetherian rings columns with uniformly distributed Maeda, Fumiyuki axial load Yale University, June McLean, Robert T. Spectral theory on locally convex University of Pittsburgh, August spaces Defining relations which characterize Majumdar, Kulendra Narayan groups having elements of the form Purdue University, january AxlAxz Axn 1 2 ••• n An investigation of the properties of incomplete block designs McQuillan, Donald Laurence Marcia, john Glen johns Hopkins University, june University of California (Berkeley), jan­ A generalization of a theorem of Heeke uary Mertz, Robert Theodore Almost periodic surfaces Columbia University, June Martin, Charles john On the continuation of solutions of a Rensselaer Polytechnic Institute, june linear ordinary differential equation Vibrations of an elastic plate under with polynomial coefficients uniform tension Metcalf, Frederic Thomas Martino, joseph Paul University of Maryland, August Ohio State University, December Bounds on apsidal angles Minimum-expected-cost filters for Meyer, Donald Lee arbitrary cost functions University of Minnesota, june Mayer, Alan Lionel Minor - Educational Psychology Princeton University, October Response surface designs with integer On the jacobi inversion problem valued factors

182 Mikhail, Wadie Faltas Moroney, Richard Morgan, jr. University of North Carolina, june Massachusetts Institute of Technology, On the monotonicity and admissibility june of some tests in multivariate analysis A class of characteristic value prob­ lems Mikulski, Piotr Witold University of California (Berkeley), Sep­ Morrison, David Dean tember University of California (Los Angeles), Some problems in the asymptotic june theory of testing statistical hypotheses Errors in the solution of eigenvalue - efficiency of nonparametric proce­ problems by finite difference methods dures Mundt, Marvin Glen Miller, Millage Clinton, III Iowa State University, November University of Oklahoma, August Minor- History of Science An investigation of the application of Sufficient conditions for a summability the method of steepest a"Scent in method for improper Lebesgue-Stielt­ medical research jes integrals involving an integral-to­ function transformation to be regular Miller, Robert Gerard Harvard University, june Murthy, Vrudhula Krishna Minor - Meteorology University of North Carolina, june An application of multiple discrimin­ Minor - Mathematical Economics ant analysis to the probabilistic pre­ On the general renewal process diction of meteorological conditions Muskat, joseph Baruch affecting operational decisions Massachusetts Institute of Technology, February Mitra, Shashanka Shekhar Minor - Electrical Engineering University of Washington, December Criteria for prime power residuacity Asymptotic rate of convergence for a process with stationary independent Myers, Earle Frederick increments University of Pittsburgh, january Behavior of curves and surfaces Moeller, james Walter through fundamental elements of hi­ New York University, February rational transformations On the spectra of some translation invariant spaces Nanda, jagdish Lal Indiana University, june Einstein's connections: degenerate Mohanty, Sri Gopal cases of the first class University of Alberta, Edmonton, Al­ berta, Canada, May Nassar, Hanna I. On some properties of compositions Lehigh University, October on an integer and their application to Parallel and central transformations probability theory and statistics of Riemannian manifolds Monahan, Irene Patricia Neumann, Peter Gabriel North Carolina State College, june Harvard University, june Incomplete variable designs in multi­ Efficient error-limiting variable­ variate experiments length codes Monk, james Donald Neuts, Marcel Fernand University of California( Berkeley), june Stanford University, April Studies in cylindric algebra Games on the unit- square with dis­ crete payoff Montzingo, Lloyd j., Jr. University of Buffalo, june Newman, Rogers joseph Minor - Physics University of Michigan, February Convergence of distribution functions Minor - Physics in a parameter space Capacity and Tchebycheff polynomials

183 Ney, Peter Ernest Peressini, Anthony Louis Columbia University, June Washington State University, February Some contributions to the theory of Topologies in ordered vector spaces cascades Perrin, Edward Burton Nieto-De-Pascual, jose Stanford University, january Iowa State University, Ames, February The estimation of parameters in sys­ Theory of minimum variance estima­ tems related to the observations by tion with applications an unknown monotone transformation Odeh, F arouk Mohamad Saleh Petrich, Mario University of California (Berkeley), jan­ University of Washington, june uary Semicharacters of direct products and Uniqueness theorems under the radia­ of certain other semigroups tion condition Petridis, Nicholas Constantine Ogawa, Hajimu University of Chicago, june University of California (Berkeley), jan­ Quasiconformal mappings andpseudo­ uary meromorphic curves On difference methods for the solution of partial differential equations of Petro, john William mixed type State University of Iowa, june Some results in the theory of pseudo­ Ogg, Andrew Pollard valuations Harvard University, june Cohomology of Abelian varieties over Petryshyn, Walter Volodymyr function fields Columbia University, june Direct and iterative methods for the Oglesby, Francis Clyde solution of linear operator equations Lehigh University, june in Hilbert space An examination of a decision proce­ dure Pixley, Alden French University of California (Berkeley), june Onishi, Hironori Clusters of algebras: identities and Massachusetts Institute of Technology, structure lattices june Minor - Philosophy Pohl, William Francis The general sieve method University of California (Berkeley), Sep­ tember Ono, Tamio Differential geometry of higher order University of California (Berkeley), Sep­ tember Posten, Harry 0. Local theory in function analysis Virginia Polytechnic Institute, june Power of the likelihood-ratio test of Ossesia, Michel Germain the general linear hypothesis in mul­ University of Pittsburgh, May tivariate analysis Some invariant hyper surfaces Poxon, Nancy jim Moody Paolucci, Dominic Anthony University of illinois, june Indiana University, june Minor - Physics The effects of higher order viscosity Representations of functions in terms terms on fluid flow of logarithmic potentials Patel, Mavibhai Sukhabhai Quesenberry, Charles Price University of North Carolina, june Virginia Polytechnic Institute, june Investigations on factorial designs Some tests for outliers Peach, Paul Rao, jonnagadda Nalini Kanth Case Institute of Technology, june Iowa State University, February Some results of queueing and vulner­ Minor - Mathematical Economics ability relevant to aircraft mainte­ Sampling procedures involving une­ nance qual probability selection

184 Ratliff, Louis jackson, jr. Rung, Donald Charles, Jr. State University of Iowa, june University of Notre Dame, August Affine rings over rank two regular Results on the order of functions de­ local rings fined in the unit disk, with applications to holomorphic functions Reinitz, Rudolf N. C. Stanford Unive;rsity, june Rushforth, Norman Burleigh An integrated job shop sequencing Cornell University, February problem Minor - Psychology A comparison of sample correlation matrices and a multivariate analysis Reinke, William Andrew of job concepts of selected industrial Western Reserve University, june executive groups Minor - Economics Dynamic production and inventory Rutenberg, Yechezkel Henryk control in the face of uncertainty Case Institute of Technology, june Sequential decision models Rhodes, Benjamin Thomas, Jr. Oklahoma State University, May Saber, Nicholas john The use of combined information in University of Pittsburgh, january interval estimation On the product of real valued uniform­ ly continuous functions in metric Rich, Robert Norman spaces University of Pennsylvania, December Simple and weakly almost periodic Sacks, Gerald Enoch transformation groups Cornell University, june On suborderings of degrees of recur­ Riehm, Carl Richard sive unsolvability Princeton University, October On the integral representations of Sandler, Reuben Irving quadratic forms over local fields University of Chicago, August Autotopism groups of some finite non­ associative algebra Ritchie, Robert Wells Princeton University, january Sanwal, jagdish C. Classes of recursive functions of pre­ Indiana University, june dictable complexity Minor - Physics On locally affine spaces Rodin, Burton University of California (Los Angeles). Sarason, Leonard june New York University, February Reproducing formulas on surfaces On boundary value problems for hy­ perbolic equations Romano, Albert Virginia Polytechnic Institute, june Sasieni, Maurice Wolf A trapezoidal approximation in spec­ Case Institute of Technology, june tral-density estimation Some problems in the analysis of queuing systems Rosenthal, Paul Lester Oregon State University, june Schrader, Walter Reginald On a generalization of Mahler's in­ University of Wisconsin, january version formula and some of its appli­ The epistola de proportione et pro­ cations portionalitate of Ametus Filius Iosephi Seshadri, Vanamamalai Roy, Prabir Oklahoma State University, May University of North Carolina,.August Estimation in the balanced incomplete Separability of metric spaces block design Rumsey, Howard Calvin, jr. Shepp, Lawrence Alan California Institute of Technology, june Princeton University, April Sets of visible points Recurrent sums of random variables

185 Sherman, Gordon Rae Smith, Thomas Jefferson Purdue University, January University of Wisconsin, August Minor - Economics Planar line families Combinatorial scheduling: on finding Sparer, Gerson H. a partition of a finite set which maxi­ New York University, June mizes a set function Units of algebraic number fields Sherry, Murray Elliot Sparkman, Mary Clarice, Sister Harvard University, March University of Texas, June Minor - Applied Physics Minor - Education Syntactic analysis in automatic trans­ Relationship and analogue between the lation finite calculus and the limit calculus Shinbrot, Marvin Sprague, Richard Howard Stanford University, April University of Kentucky, January Difference kernels Minor - Philosophy Shtern, Israel Hirsh Second variations of analytic functions McGill University, May Sreedharan, V. P. On some reducible diophantine equa­ Carnegie Institute of Technology, June tions Function theoretic solutions of cer­ Simpson, James Edward tain boundary value problems Yale University, June Stanek, Peter F. G. On spectral measures and spectral University of Chicago, August operators Two-element generation of the sym­ plectic group Sinclair, Roy George Starr, Selig Massachusetts Institute of Technology, George Washington University, June February Some algebraic aspects of the analysis Torsion and extension of helicoidal of variance strips Stasheff, James Dillon Singh, Shobh Nath Princeton University, June University of California (Berkeley), Homotopy associativity of H-spaces September On a hypothetical chance mechanism Stearns, Richard Edwin for variation in the number of births Princeton University, October per couple Three person cooperative games with­ out side payments Slagle, James Robert Paul Massachusetts Institute of Technology, Steele, William Frank June University of Pittsburgh, May For mal integration by a digital com­ Summability of infinite sequences by puter submatrix methods Steinberg, Leon Sledd, William Tazwell University of Pennsylvania, June University of Kentucky, May Partitions into bounded summands Minor - Physics from' arbitrary sets Some properties of Karamata summa­ bility matrices Stell, George Roger New York University, June Smart, John Roderick The solutions of the hierarchy equa­ Michigan State University, June tions of statistical mechanics for Modular forms of dimension - Z for mixtures subgroups of the modular group Stengle, Gilbert Allan Smith, Georgia Alberta Caldwell University of Wisconsin, June University of Pittsburgh, January A construction for solutions of an Some results on the anticenter of a Nth order differential equation in the group neighborhood of a turning point

186 Steward, Robert Franklin Tewarson, Reginald Prabhakar Auburn University, june Boston University, june Continuity and uncertainty of inter­ Solution of a generalized diffusion polating rational functions equation by difference methods Still, Harold Armstrong Thigpen, Charles Craig Virginia Polytechnic Institute, june Virginia Polytechnic Institute, june Analysis of multiple covariance when Distribution of the largest observation the regression coefficients depend on in normal samples under nonstandard the blocks conditions Stoddard, james Harwood Throckmorton, Thomas Neil University of Michigan, june Iowa State University, February Regular convergence in a paracompact Structures of classification data space Thurber, james Kent Stolzenberg, Louis Gabriel New York University, june Massachusetts Institute of Technology, An asymptotic method for determining june the lift distribution of a swept-back Minor - Linguistics wing of finite span A maximal ideal space with no analy­ Tilley, john Leonard tic structure University of Florida, February Stone, Charles joel Minor - Education Stanford University, October Stress distribution of a rotating lima­ Limit theorems for birth and death con processes and diffusion processes Trahan, Donald Herbert Stone, jeremy Judah University of Pittsburgh, August Stanford University, january Green's theorem in the complex do­ Exponential polynomials on commuta­ main and related theorems tive semigroups Trawinski, Benon john Struble, George Waring Virginia Polytechnic Institute, june University of Wisconsin, August Minor - Physics Orthogonal polynomials: variable­ Selection of the best treatment in a signed weight functions paired-comparison experiment Suh, Tae-il Troy, Allan Yale University, june University of lllinois, October On isomorphisms of Zittle projective Integral representations of cyclic groups of Cayley planes groups of order pz. Summers, john Willis Troy, Daniel joseph University of California (Berkeley), june St. Louis University, june On Kan fibre spaces Minor - Philosophy Hadamard operators on Lambert se­ Summers, George William ries Case Institute of Technology, june Financing and initial operations of Tucker, Patricia Anne new firms University of Wisconsin, August On the reduction of induced represen­ Suprunowicz, Konrad tations of finite groups University of Nebraska, February Minor - Physics Varley, Eric Diagram normal forms and their ap­ Brown University, june plications to the theory of models Flows of dilatant fluids Suryanarayan, E. Ramnath Von Wolff, Mary Robert, Sister University of Michigan, june University of Notre Dame, August The geometry of fluid flows in rela­ Densest admissible point sets for two tivity star domains

187 Wachman, Murray Weinberg, Israel Jacob New York University, February Massachusetts Institute of Technology, Essential isolated singularities in February solution of elliptic partial differential Minor - Physics equations with analytic coefficients Symmetric finite deflections of cir­ cular plates subjected to edge forces Wagner, Norman Rayfield, Jr. and edge moments of Technology, Massachusetts Institute Weitkamp, Harvey Meredith June University of Connecticut, June Engineering Minor - Electrical Minor - Education Existence theorem for a nonlinear The equation xn + yn + zn 0 in problem in ordinary boundary value rational binary matrices differential equations Wells, Mark Brimhall University of California (Berkeley), June W ainger, Stephen Simplification ofnormalform expres­ University of Chicago, August sions for Boolean functions of many Special trigonometric series in higher variables dimensions White, Alvin Murray Wall, Francis Joseph Stanford University, January University of Minnesota, June Singularities of harmonic functions of Minor - Industrial Engineering three variables generated by a Whit­ Biostatistical linear models in longi­ taker-Bergman operator tudinal medical research Whiteman, Russell A. January Walston, Dale Edouard illinois Institute of Technology, of analytic University of Texas, August Concerning certain classes Hadamard product An existence and uniqueness theorem functions and the for integro-partial differential equa­ Whitman, Andrew Peter tions Catholic University of America, Febru­ Walter, Everett Leon ary New Mexico State University, August Minor - Philosophy principal fi­ Generalized conjugate spaces Invariant connections in ber bundles over locally homogeneous W arten, Ralph Martin spaces Purdue University, August On the approximate solution of axially Whitten, Wilbur Carrington, Jr. symmetric problems by means of University of Pittsburgh, August rigid mo­ finite differences Projective coordinates and tions in the plane of two complex vari­ Webster, Porter Grigsby ables interpreted in argand four- space Auburn University, June Determination of the convergence of Williams, James Stanley Lagrange interpolation by Runge's Washington State College, May the worth of some method An evaluation of selection indices Weidlich, John Edward, Jr. University of California (Berkeley), Jan­ Williams, Leland Hendry uary Duke University, June The asymptotic behavior of the real Minor - Physics in solutions of certain first order non­ Extended algebra of polynomials for a digital com­ linear differential systems several variables puter

Weinberg, Elliot Carl Williams, Vincent Cassell Purdue University, January Harvard University, March Higher degrees of distributivity in On conformal maps of regions of in­ lattices on continuous functions finite connectivity

188 Wilson, Atholl Livingstone field of rational numbers University of California (Berkeley), june Youngdahl, Carl Kerr An approach to n-person games Brown University, june Wiser, Horace Clare Transient thermal stresses in a cir­ University of Utah, june cular cylinder Decomposition and homogeneity of Yozwiak, Bernard james continua on a 2-manifold University of Pittsburgh, May Woods, Walter Max Relations between the Lototsky and Stanford University, january certain other methods of summability Variables sampling inspection pro­ Zaki, Ramzi Mohamed cedures which guarantee acceptance North Carolina State College, May of perfectly screened lots Applications of linear programming Worrell, John Mays, Jr. techniques to some problems of pro­ University of Texas, August duction planning over time Minor - Zoology Zechmann, Albert William Concerning scattered point sets Iowa State University, November Yaspan, Arthur joel Minor - Physics Case Institute of Technology, June Problems in viscoelasticity The inventory problem with periodic Ziemer, William Paul reorder Brown University, june Yearout, Paul Harmon Integral currents mod 2 University of Washington, December Zimmerman, jack Miles A study of (m,n) distributive rings University of Southern California, june Young, Lael Marcus Band-limited functions and improper Stanford University, june problems in partial differential equa­ On certain cyclic extensions of the tions

NEWS ITEMS AND ANNOUNCEMENTS

THE FOURTH SUMMER INSTI­ The Summer Institute in Dynamical TUTE IN DYNAMICAL ASTRONOMY, Astronomy is considered to be of special jointly sponsored by the Yale University interest to mathematicians at this time, Observatory and the National Science in view of the plan to combine it with the Foundation, will be held at the Yale Uni­ American Mathematical Society's Summer versity Observatory, New Haven, Connec­ Seminar in Applied Mathematics in 1963, ticut, from june 25 to August 3, 1962, to be held at Cornell University. The Ap­ The Institute is open to astronomers, phy­ plied Mathematics Committee of the AMS sicists, and mathematicians interested in selected "Space Mathematics" as the topic the field of celestial mechanics. The Insti­ for the 1963 Summer Seminar, and its rec­ tute wishes particularly to call the atten­ ommendation for a joint program with the tion of mathematicians to the last two Yale Observatory was approved by the weeks of the program, which will empha­ Council at the Atlantic City meeting of the size the mathematical aspects of celestial Society. The Organizing Committee will mechanics. consist of J, Barkley Rosser and Dirk Academic applicants who are ac­ Brouwer, and several others yet to be se­ cepted will receive stipends to defray ex­ lected. penses for themselves and their families, Plans now call for a program of six plus a travel allowance. For further infor­ weeks duration, subdivided into sections mation write Professor Dirk Brouwer, devoted to narrower topics. Although the Yale University Observatory, Box 2023, sections will be related in the same broad Yale Station, New Haven, Connecticut. field, each may also be considered as an

189 independent unit, so that individuals may Glass Works. For further information write conveniently attend for shorter periods on to Little, Brown and Company, 34 Beacon just those topics of special interest to them. Street, Boston 6, Massachusetts. Further announcements on the Sum­ mer Seminar will be made as plans pro­ gress. AUTOMORPHIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES is the A $10,000 CORNING SCIENCE subject of a conference to be held atStan­ PRIZE to be awarded to the best book ford University on June 12-15, 1962. For manuscript in the field of the natural or further information write to Professor physical sciences for the general reader Stefan Bergmann at the Department of has been announced by Little, Brown and Mathematics at Stanford University, Stan­ Company, in co-operation with Corning ford, California.

NSF POSTDOCTORAL FELLOW­ for advanced study and research in the SHIP AWARDS for March were announced sciences, thirteen were in mathematics. March 22. Of the 185 fellowships granted NSF Postdoctoral Fellowship Awards Name Present Institution Fellowship Institution Bass, Hyman University of Chicago College de France Dudley, Richard M. Princeton University University of California (Berkeley) Edwards, Harold M. Harvard University Institute for Advanced Study Guillemin, Victor W. Harvard University University of Paris Hales, Alfred W. California Institute of Tech- nology Cambridge University Knill, Ronald J. University of Notre Dame Northwestern University Martin, Joseph M. State University of Iowa Institute for Advanced Study McMillan, Daniel R., Jr. University of Wisconsin Institute for Advanced Study Resnikoff, Howard L. University of California (Berkeley) University of Miinster Rhodes, John L. Massachusetts Institute of Institut des Hautes Etudes, Technology Paris Topping, David M, Tulane University Massachusetts Institute of Technology Vandeneynden, Charles L. University of Oregon University of Michigan Vasquez, Alphonse T. Uniyersity of California (Berkeley) Princeton University

FORTY-NINE NATO POSTDOC­ tists from NATO countries will come to the TORAL FELLOWSHIPS IN SCIENCE have United States for further training. been announced by the State Department Of the awards just announced, 15 are and the National Science Foundation. De­ in the life sciences, one in the social sci­ signed to encourage further study abroad, ences, and 33 in the physical sciences and these awards will enable U.S. Fellows to mathematics. The four awards in mathe­ study in ten foreign nations, while scien- matics were as follows: Name Present Institution Fellowship Institution Foulser, David A. University of Michigan Oxford University Hudson, Anne L. Tulane University University of Tubingen Mayer, Alan L. Princeton University University of Pisa Topping, David M. Tulane University University of Paris

190 PERSONAL ITEMS

Mr. D. R. BAKERofSystem Devel-· ciate professorship at the University of opment Corporation has accepted a posi­ Toronto. tion as Computer Programmer with Inter­ Dr. R. M. DAVIS of David Taylor national Telephone and Telegraph, Para­ Model Basin has accepted a position as mus, New jersey. Staff Assistant with the Department of De­ Mrs. R. A.BATTERTONofThomp­ fense, Washington, D. C. son Ramo-Wooldridge, Incorporated has Mr. D. G. de FIGUEIREDO of accepted a position as an Associate with Universidade do Ceara has been appointed Planning Research Corporation, Los An­ to a research professorship at the Instituto geles, California. de Matematica Pura e Aplicada, Rio de Dr .GERTRUDE BLANCH of Wright­ janeiro, Brazil. Patterson Air Force Base has accepted a Captain R. L. EISENMAN oftheAir position as Senior Scientist in the Depart­ Force Academy will be assigned for the ment of Defense, Washington, D. C. academic year 196Z-1963 to the new Re­ Dr. JOSEPH BRAM of David Taylor search Department of the Air Force Aca­ Model Basin has accepted a position as a demy, Colorado. Mathematical Analyst with Operations Mr. MAREK FISZ of Columbia Uni­ Evaluation Group, Massachusetts Institute versity has been appointed to a professor­ of Technology, Arlington, Virginia. ship at New York University. Professor j. R. BRITTON hasbeen Dr. LEONARD GELLER of Com­ appointed Chairman of the Department of bustion Engineering Company has accepted Applied Mathematics at the University of a position as Senior Associate with S. M. Colorado. Stoller Associates, New York. Mr. H. H. BROWN of North Ameri­ Mr. j. J. GILVARRYofAllis-Chal­ can Aviation Incorporated has accepted a mers Manufacturing Company has accepted position as Mathematician with National a position as Senior Staff Scientist at Gen­ Aeronautics and Space Administration, eral Dynamics Astronautics, San Diego, Greenbelt, Maryland. California. Mr. j.R.BRUGGEMANofTheMar­ Associate Professor DONALD tin Company has accepted a position as GREENSPAN of Purdue University has ac­ Senior Engineer at Thiokol Chemical Cor­ cepted a position as Staff Member at the poration, Brigham City, Utah. Mathematics Research Center, United Dr. E. W. CANNON of the National States Army, University of Wisconsin, Bureau of Standards, has been awarded the Mr. SIN HITOTUMATU of the Unit;> Department of Commerce Gold Medal for versity of Tokyo has been appointed to a Exceptional Service. professorship at St. Paul's University, Mr. E. D. CAREY of Federal Tele­ Tokyo, japan. communications Laboratories has accept­ Mr. S. S. HOLLAND of the National ed a position as Mathematician with the Academy of Science has been appointed to General Electric Company, MSVD, Phila­ an assistant professorship at Boston Col­ delphia, Pennsylvania. lege. Dr. J. F. CARPENTER of Dalmo Mr. R. E. HOLZMAN of Sylvania Victor Company has accepted a position as Electric Products Incorporated has ac­ Section Head with Aerospace Corporation, cepted a position as Research Engineer El Segundo, California. with jet Propulsion Laboratoryofthe Uni­ Mr. C. H. COOK of The Martin versity of California,P as adena, California. Company has been appointed to an assist­ Assistant Professor W. B. HOUS­ ant professorship at the University of TON, JR. of Carleton Collegehasbeenap­ Oklahoma. pointed to an assistant professorship at Dr. CHANDLER DAVISofNewYork Antioch College. University has been appointed to an asso- Mrs. A. IONESCU TULCEA of Yale

191 University has been appointed Research Company Incorporated has accepted a po­ Associate at the University of Pennsyl­ sition as Senior Engineer with Edgerton, vania. Gemeshausen and Grier Incorporated, Mr. C. IONESCU TULCEA of Yale Boston, Massachusetts. University has been appointed to an asso­ Mr. G. T. McALLISTER of the Uni­ ciate professorship at the University of versity of California has accepted a posi­ Pennsylvania. tion as Research Mathematician with Dr. R. C. KAO of System Develop­ Sperry Rand Corporation, Sudbury, Massa­ ment Corporation has accepted a position chusetts. as Research Economist with The Rand Dr. j. GAYLORD MAY of the U.S. Corporation, Santa Monica, California. Army Ordnance, Aberdeen Proving Ground Mr. MASUDA KATSUHIKO ofYam­ has been appointed to an assistantprofes­ agata University, japan has been appointed sorship at Wake Forest College. to a professorship at Okayama University, Mr. R. A. MICHELSON of the Uni­ japan. versity of Washington has accepted a Dr. R. B. KELMAN of Remington position as Associate Research Engineer Rand UNIVAC Division of the Sperry Rand with The Boeing Company, Renton, Wash­ Corporation has been appointed a Lecturer ington. at Howard University. Mr. DIETRICH MORGENSTERN of Dr. H. P. KERFOOT, JR. of the the University of Munster, Germany has University of Southern California, has been appointed to a professorship at the accepted a position as Director, Mechani­ Universitat Freiburg, Freiburg, Germany. cal and Mathematical Sciences Laboratory Mr. D. E. MORRILL of Thiokol of Lockheed Missile and Space Company, Chemical Corporation has accepted a posi­ Palo Alto, California. tion with International Business Machines Mr. G. W. KIMBLE of the University Corporation, Poughkeepsie, New York. of California has been appointed to an Mr. P. D. OYER of Westinghouse assistant professorship at Long Beach Electric Company has accepted a position State College. as Research Mathematician with Armour Mr. W. E. KOPKA of Service Bu­ Research Foundation, Annapolis, Mary­ reau Corporation has accepted a position land. as Advisory Programmer, International Dr. L. G. PECK has beenappointed Business Machines Corporation, Pough­ Chairman of the Mathematics Department keepsie, New York. at the State University of New York. Professor TADAO KUBO of Kyoto Dr. A. J. PENICO of General Tele­ Prefectural University, japan has been ap­ phone and Electronics Laboratories Incor­ pointed to a professorship at Kobe Univer­ porated has been appointed Research Math­ sity, Japan. ematician at Stanford University. Dr. H. E. KYBURGofWesleyan Uni­ Dr. W. B. PENNINGTON of West­ versity has been appointed to an associate field College, University of London has professorship at the University of Denver. been appointed to a professorship at the Mr. L. L. LAYTON ofDavidTaylor University College of Wales. Model Basin, Washington, D. C. has ac­ Mr. H. K. PHILIPS of General Pre­ cepted a position as Aerospace Technolo­ cision Incorporated has accepted a position gist at the Nationai Aeronautics and Space as Programmer Analyst with National Administration, Greenbelt, Maryland. Computer Analysts, Incorporated, Prince­ Dr. j. H. LEVIN of Datamatic Cor­ ton, New jersey. poration has accepted a position as Direc­ Dr. R. M. RAU of the University of tor, Programming Systems Division with lllinois has been appointed to an assistant Minneapolis Honeywell Regulator Com­ professorship at Ohio State University. pany, Wellesley Hills, Massachusetts. Dr. R. W. RECTOR of Space Tech­ Dr. G. E. LEWIS of New York Uni­ nology Laboratories Incorporated has ac­ versity has been appointed to an assistant cepted a position as Administrative Mana­ professorship at the University of Colo­ ger of the Aerospace Corporation, Los rado. Angeles, California. Mr. W. G. MADISON of National Mr. G. S. RINEHART of the Univer-

192 sity of California, Berkeley, California has Mr. R. J. WERNICK of General been awarded a National Academy of Electric Company has accepted a position Sciences-National Research Council Post­ as Applied Mathematician with Mechanical doctoral Research Fellowship. Technology Incorporated, Schenectady, Professor HANNO RUND of the New York. University of Natal, South Africa has been Dr. NEAL ZIERLER after a leave appointed to a professorship at the Univer­ of absence has returned to Lincoln Labo­ sity of South Africa. ratory, Massachusetts Institute of Tech­ Mr. D. A. RUSSO of GrummanAir­ nology. craft Engineering Corporation has ac­ cepted a position as Project Engineer with The following promotions are announced: the Defense Electronics Products, Moores­ town, New Jersey. C. B. BELL, San Diego State Col­ Dr. W. C. SANGREN of General lege, to an associate professorship. Dynamics Corporation has accepted a J. L. BOAL, University of South position as Vice President and Director Carolina, to an associate professorship. of West Coast Operations, Computer Appli­ 0. S. BOWMAN, Bridgewater Col­ cations Incorporated, San Diego, Califor­ lege, to an assistantprofessorship. nia. J. CHOVER, University of Wiscon­ Mr. H. E. SCHLOSS of the Hughes sin, to an associate professorship. Aircraft Company has accepted a position L. GARFIN Pacific Mutual Life In­ as a Mathematician with the Burroughs surance Company, Los Angeles, Califor­ Corporation, Pas adena, California. nia, to an Actuary. Mr. J. D. SIDLEY of Georgetown C. A. HUTCHINSON, University of University has accepted a position as Colorado, to Associate Dean ofthe College Mathematician with the Bureau of Naval of Engineering. Weapons, Washington, D. C. R. A. JORGENSEN, JR. Andrews Mr. M. S. SKAFF of The University University, to an assistant professorship. of Illinois has accepted a position as Com­ F, B. KNIGHT, University of Min­ puting Analyst with the Douglas Aircraft nesota, to a professorship. Company, Santa Monica, California. P. J. LAUFER, College Militaire Dr. C. V. L. SMITH of National Royal de St. Jean, to a professorship. Aeronautics and Space Administrationhas J. R. McCORD, Massachusetts In­ accepted a position as Head of the Mathe­ stitute of Technology, to a research asso­ matics and Computer Programs, Research ciate. Division, Atomic Energy Commission, E. E. POSEY, Virginia Polytechnic Germantown, Maryland. Institute, to a professorship, Mr. N. R. STANLEYofSperryRand Y. SIBUYA, Ochanomizu University, Corporation has accepted a position as Tokyo, Japan, to an assistant professor­ Senior Mathematical Physicist with The ship. Perkin-Elmer Corporation, Norwalk, T. H. SLOOK, Temple University, Connecticut. to an assistant professorship. Mr. H. R. STEVENS of Duke Uni­ J. SONNER, University of South versity has been appointed to an assistant Carolina, to a professorship. professorship at Pennsylvania State Uni­ J. L. ZEMMER, JR., University of versity. Missouri, to a professorship. Dr. HARUO SUZUKI ofTohoku Uni­ The following appointments to instructor­ versity, Sendai, Japan has been appointed ships are announced: to an assistant professorship at Kyushu University, Fukuoka, Japan. Cornell Univer~ity: D. S. GILLMAN; Mr. J. T. WARFIELD of Johns lvanovo State Furmanov Teachers' Col­ Hopkins University has accepted a position lege, U.S.S.R.: M. GREENDLINGER; Keu­ as Senior Engineer with the Martin­ Kuk University, Seoul, Korea: S. K. KIM; Marietta Corporation, Baltimore, Mary­ Montana State University: M. E. MANIS; land. Xavier University: W. A. HORN.

193 Deaths: NEWS ITEMS

Professor ODDV AR BJORGUM of THE MATHE MA TISCHE FOR- the University of Bergen, Norway died 01;1. SCHUNGSINSTITUT OBERWOLFACH an­ December ZZ, 1961 at the age of 45. nounces its schedule of meetings for 196Z Dr. H. T. ENGSTROM of Remington as follows: Rand Division of Sperry Rand Corporation june 1Z-16. Foundation of Geometry died on March 8, 196Z at the age of 60. Chairmen: F. Bachmann, Kiel; E. Sper­ Professor Emeritus M. G. GABA ner, Hamburg; R. Baer, Frankfurt. of The University of Nebraska died on February 11, 196Z at the age of 78. june Z1-Z5. Workshop Mrs. L. F. HUTCHINSON of Bryn Chairman: R. Baer, Frankfurt. Mawr College, died on january Z9, 196Z july Z3-Z8. Discrete Geometry at the age of 48. Chairman: L. Fejes Toth, Budapest Professor Emeritus F. S. NOWLAN of the University of British Columbia, Van­ August 6-10. Group Theory couver, Canada died on September 8, 1961 Chairmen: R. Baer, Frankfurt; H. Wie­ at the age of 75. He had been a member of landt, Tiibingen the Society for 49 years. August Z7-31. Workshop Professor L. M. REAGAN of the Chairman: H. Konig, Aachen. University of WichitadiedonDecemberZ9, September 3-7. Partial Differential Equa­ 1961 at the age of 57. He had been a mem­ tions ber of the Society for ZZ years. Chairmen: W. Haack and G. Hellwig, ~ Berlin Septembe.r 17-ZO. History of Mathematics ERRATA Chairman: j. E. Hofmann, Ichenhausen

The announcement on page 91 of the September Z4-Z9. Geometry April issue of the NOTICES concerning Chairman: K. H. Weise, Kiel Dr. B. Bernholtz should read as follows: October 1-6. Functional Analysis Dr. B. BERNHOLTZ of the Hydro­ Chairman: G. Kothe, Heidelberg Electric Power Commission of Ontario, Any additional meetings or changes will Canada has been appointed to an Associate be announced. Professorship in the Department of Indus­ trial Engineering at the University of Toronto, Toronto, Canada. THIRTY-SIX FELLOWSHIPS FOR AMERICAN WOMEN will be awarded by the American Association of University Wom­ The announcement on page 549 of an Educational Foundation for the period the December issue of the NOTICES con­ july 1, 1963, to june 30, 1964. These cerning Dr. H. E. CONNER shouldreadas awards, in varying amounts, are unre­ follows: stricted as to age or field and may be used Dr. H. E. CONNER on leave of ab­ in the United States or abroad. sense from Lincoln Laboratory, Massa­ Application forms are available chusetts Institute of Technology has been from Fellowships Office, AAUW Educa­ appointed a member of the Mathematics tional Foundation, Z401 Virginia Avenue, Research Center, United States Army, Uni­ N. W., Washington 7, D. C. Requests for versity of Wisconsin. applications must indicate present aca­ demic status. The deadline for filing appli­ cations is December 1, 196Z, and notifica­ ';rhe announcement on page 9Z of the tion of awards will be made on March 1, April issue of the NOTICES concerning 1963. Mr. D. j. OSTROFF shouldreadasfollows: For information about fellowships Instructorship: San Fernando Valley State offered for women of other countries, College: D. J. OSTROFF. write to the above address.

194 NEW AMS PUBLICATIONS

PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS

Volume 13 City, April 1960. Papers by: C. C. Lin HYDRODYNAMIC INSTABILITY and D. j. Benney; K. M. Case; Marshall Edited by G. Birkhoff, R. Bellman, N. Rosenbluth; R. J. Donnelly; Garrett and C. C. Lin Birkhoff; Russell E. Duff; John W. Miles; 0. M. Phillips; M. S. Longuet-Higgins; 319 pages; $7.90 List Price; 25o/o discount M. St. Denis; Eberhard Hopf; J. Kampe de to members. Feriet; Robert H. Kraichnan; joseph B. Proceedings of the Symposium in Keller; S. Ulam; Brockway McMillan; Applied Mathematics of the American Richard Bellman; R. E. Kalman; j. P. Mathematical Society, held in New York LaSalle.

MEMOIR 42 THE FACTORIZATION OF CYCLIC REDUCED POWERS BY SECONDARY COHOMOLOGY OPERATIONS by Arunas Liulevicius 113 pages; $1.90 List Price; 25o/o discount tions. The analogous theorem for cyclic to members. reduced powers pPs s i!: 1 (pan odd prime) is proved here. One consequence of the j. F. Adams has proved that the result is the triviality of the mod p Hopf Steenrod squares Sq 2rr i!:' 4 can be factored invariant in most dimensions. into stable secondary cohomology opera-

NEWS ITEMS AND ANNOUNCEMENTS

THE COMMITTEE ON INTERNA­ States if arrangements are completed for TIONAL EXCHANGE OF PERSONS of the a lecturing or research appointment at an Conference Board of Associated Research American university or college. Councils has for distribution a list of Mathematicians on this year's list foreign scholars available for remunera­ are Godrey L. Isaacs, Head of the Depart­ tive positions in American universities ment of Mathematics at the University of and colleges for the Summer Term 1962 Witwatersrand, South Africa, and Vasanti and Academic Year 1962-1963. This list S. Huzurbazar, Head of the Department of is compiled annually by the Committee and Mathematics and Statistics at the Univer­ includes information about scholars rec­ sity of Poona, India. For details write ommended by the United States Educa­ Mrs. john D. Leary, Program Officer, tional Commissions and Foundations Conference Board of Associated Research abroad. Each scholar is eligible for a Councils, 2101 Constitution Avenue, N. W., Government travel grant covering costs of Washington 25, D. C. round-trip transportation to the United

195 LETTERS TO THE EDITOR

Editor, the NOTICES fields closely related to mathematics, Several years ago J. P. Phillips, such as astrophysics, mechanics, etc. in an article "The individual in chemical There would be common agreement to ex­ research" (Science, vol. 1Z1 (1955), pp. clude certain of these fields as not being 311· 31 Z), gave a rough measure of the de­ mathematics but there are others about crease in individual research in contrast which there would be some debate. I've to team research in chemistry as meas­ arbitrarily settled this by excluding those ured by the number of papers published fields now reviewed in Part B (the gr~en­ each year by a single author compared stripe part) of Mathematical Reviews even with the number of papers published and though this excludes probability and statis­ having two, or more, authors. This study tics, both of which are usually considered was based on a single journal, the journal as genuine mathematics. of the American Chemical Society, with TABLE 1. MATHEMATICAL REVIEWS partial results for the journal of Chemical Physics. This was not a deep and ex­ Number Number Percent haustive study of the increase in team re­ of jointly jointly search but it was, as the author claimed, Year titles authored authored interesting to learn that between 1918 and 1940 1579 9Z 5.8 1950 the number of papers published by a 1950 3Z98 Zl4 6.5 single author in the journal of the Ameri­ 1960 4393 473 10.8 can Chemical Society decreased monotoni­ cally from 45o/o to 1411/o of the total papers Mathematical Reviews is an inter­ published in that journal. national publication in that it reviews es­ The last paragraph of the article sentially all of the mathematics published of Phillips follows. "The real frontier throughout the world. Since it only began for the individual in science, however, publishing in 1940, it is necessary togo to would appear to be in mathematics. In other sources for evidence prior to 1940. 19 50, 9 8 percent of the papers in the Rather than examine an older review jour­ American Mathematical Monthly were nal, I've selected, arbitrarily, three of the written by individuals." older research journals of the United Evidence from a large sample of States in forming Table Z. The journals are papers is provided in the following para­ the American journal of Mathematics, the graphs. This census, like that of Phillips, Annals of Mathematics and the Transac­ is informal and is not intended to have tions of the American Mathematical Soci­ breadth or depth. Also, I join Phillips in ~ In Table Z the numbers following the not drawing conclusions, moral or other­ year are the totals for all three of these wise. journals. Since Vol. Z 1 of the Annals of Table 1 is based on titles in Mathe­ Mathematics is dated 1919-19ZO and Vol. matical Reviews Vol. 1 (1940), Vol. 11 ZZ is dated 19Z0-19Zl, I've included both (1950) and Vol. Z1 (1960). These are volumes in the tabulation for the year 19ZO. "titles" in contrast to "research papers" to emphasize that I have counted the titles TABLE Z. THREE U.S. JOURNALS without making judgments as to which papers represent research and which do Number Number Percent not represent research except in the opin­ of papers jointly jointly ion of their authors. Books containingnew Year Eublish~d authored authored results as well as some advanced text­ 19ZO 89 z z.z books are reviewed in this journal and 1930 170 7 4.1 their titles were counted. 1940 170 31 18.Z Mathematical Reviews has always 1950 Z14 39 18.Z reviewed some of the papers· in some 1960 ZZ8 Z9 1Z,7

196 The percentages in Table 2 are this proposal, in the present instance to conspicuously not monotone. In each of the Professor Bing's committee, but with what three years 1940, 1950, 1960 the American results I do not know. Journal of Mathematics had a greater per­ Relative to this proposal let me cent of jointly authored papers than either make clear that it is in no sense whatso­ of the other two journals. Even if this ever an accrediting device. journal is excluded and one makes the table It would be a sort of "consultant's for the other two, the percentages are still bureau" comprised of mathematicians of not monotone, In fact, the percentage col­ repute, not necessarily always those of umn will not be monotone if formed for great distinction, but mathematicians who any combination of two of these journals (i) are held in some esteem for their re­ for the years considered in Table 2. search, (ii) have had experience in direct­ ing dissertations, (iii) are now and have W, R. Utz been concerned with formulating and main­ taining a reputable graduate program, (iv) Editor, the NOTICES have some sense of the financial difficul­ ties involved and (v) are cognizant of the The number of those institutions present national state of our art in most whose Departments of Mathematics have of its ramifications - teaching loads, recently begun, or are in the process of availability of mathematicians with certain starting doctoral programs is, to say the training and talent, and so on. Of course least, surprisingly large. Whether the they should be wise beyond measure and magnitude is to be regarded with alarm or tactful beyond reproach. (Needless to say, satisfaction is not the present con­ with in a mathematical sense.) cern, which is rather that they should be The first duty of such a group doctoral programs. good would be to prepare a prospectus for a Institutions inaugurate doctoral pro­ sound doctoral program - necessary li­ for, no doubt, a variety of reasons. grams brary facilities, suitable courses (and these are their desires to meet the Among texts:), suggestions as to seminars, teach­ repeated pleas of the smaller colleges for ing loads, salaries, and other pertinent doctors, their desire to mathematical items. They should then - in the most dis­ meet their own staffing problems (by way creet fashion possible - advertise their teaching assistants), their normal as­ of existence and their availability. for bettering themselves and, pirations Should it be so that a Department of their hope offinancial assistance. possibly, Mathematics and the administration of an In some instances the pressure for a doc­ institution made clear the desire and ne­ program comes from within the De­ toral cessity for a consultant, one would be of Mathematics and in other in­ partment assigned by the committee. Upon visiting it is from the administration. stances the institution he would consult with all The real question here is not wheth­ parties, give whatever advice tact and the er there should be more doctoral programs situation called for and report to the insti­ are more and there are more coming (there tution and to the committee upon his visit, into existence) but whether that body of Such reports would, of course, be confi­ who have, in recent mathematicians dential. been so seriously concerned with months, It is not my thought, even in this maintaining standards in this context, can best possible of worlds, that such a bu­ will play the role of mathematical and reau would solve all problems inherent in That this sort of obstetrical midwives. our present position. Nothing will. It assistance is needed, seems to me beyond could, I think, do much good over the next question. ten years. Reputable doctoral programs A few years ago, when this matter are not brought into existence with pen had not reached its present proportions, I and paper, even if some of the latter is of suggested to the Council that it should ap­ the appropriate green. It is not a short­ point a visiting committee to give aid and term program and !contemplate the notion assistance to those Departments of Mathe­ that a given institution might be visited matics who desired it and who might profit several times over a period of years. from it. Recently, I have again transmitted Alexander Doniphan Wallace

197 MEMORANDA TO MEMBERS

THEEMPLOYMENTREGffiTER DUES OF RECIPROCITY MEMBERS OF THE The Mathematical Sciences Employ­ MATHEMATICAL SOCIETY OF JAPAN ment Register, established by the Ameri­ can Mathematical Society, the Mathemati­ Owing to increasing printing and cal Association of America, and the Soci­ other costs, the Mathematical Society etyfor Industrial and Applied Mathematics, of Japan has found it necessary to raise will be maintained at the Summer Meet­ members' dues for 1962. Atthesametime ing at the University of British Columbia it has been decided to give reciprocity at Vancouver, British Columbia, Canada, members a choice of journal privileges, on August 28, 29, and 30,1962. The Regis­ with the dues scaled accordingly. Reci­ ter will be conducted from 9:00 A.M. to procity members in the United States will 5:00 P.M. on each of these three days in belong to either of two categories, accord­ Room 1221 of the Buchanan Building, ing to whether they are (A) interested only There is no charge for registration, in the journal of the Mathematical Society either to job applicants or to employers, of Japan (in European languages), or (B) in­ except when the late registration fee for terested also in receiving Sugaku (in Japan­ employers is applicable. Provision will ese). The dues for these two groups are be made for anonymity of applicants upon now as follows: request and upon payment of $3.00 to de­ fray the cost involved in handling anony­ 1961 1962 mous listings. Group A $Z.OO $Z.50 Job applicants and employers who Group B $3.50 $4.00 wish to be listed will please write to the Employment Register, 190 Hope Street, Providence 6, Rhode Island, for applica­ The NOTICES of the Society (in tion forms or for position description Japanese) are sent to all members in both forms. These forms must be completed groups. and returned to Providence not later than August 3, 1962, in order to be included Most reciprocity members paid free of charge in the listings at the Sum­ $Z.50 in dues for 1961, and have been sent mer Meeting in Vancouver. Forms which issues No. 1 and No. 2 of Volume 13 of arrive after this closing date, but before Sugaku, issued in fiscal year 1961, which August 10, will be included in the register ended March 31, 1962. The new policy at the meeting for a late registration fee change was only recently agreed upon, and of $3.00, and will also be included in the it was decided to put it into effect imm edi­ printed listings, but not until ten days after ately. Therefore the remaining issues of the meeting. The printed listings will be Volume 13 of Sugaku will be sent only to available for distribution both during and those members who specifically request after the meeting. it, and make an additional payment of $1.00 It is essential that applicants and to dues for 1961, employers register at the Employment Register Desk promptly upon arrival at New applicants for membership are the meeting to facilitate the arrangement requested to indicate whether they wish to of appointments. belong in Group A or Group B.

198 SUPPLEMENTARY PROGRAM- No. 11

During the interval from February Zl, 196Z through April Z7, 196Z, the papers listed below were accepted by the American Mathematical Society for presentation by title. Readers maywishtorefertoPage713 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. ( 1) Concerning Abelian differentials on ( 11) An elementary estimate for the k-free parabolic Riemann surfaces. Prelim­ integers inary report Professor Eckford Cohen, Univer­ Professor R. D. M. Accola, Brown sity of Tennessee (6ZT-1Z5) University (6ZT-151) (1Z) On cartesian products which are (Z) On generalized harmonic measures homotopy manifolds of Riemann surfaces. I. Preliminary Professor M. L. Curtis and Pro­ report fessor D. R. McMillan, Florida Professor R. D. M. Accola, Brown State University (6ZT-1Z7) University (6ZT-15Z) (13) On certain sequence to sequence (3) On generalized harmonic measures transformations which preserve con­ on Riemann surfaces. II. Preliminary vergence report Professor D. F. Dawson, North Professor R. D. M. Accola, Brown Texas State University (6ZT-157) University (6ZT-153) ( 14) Concerning a theorem of Hadamard (4) On Ta.·functions Professor D. F. Dawson, North Professor M. A. Al-Bassam, Texas Texas State University ( 6ZT-158) Technological College (6ZT-176) (15) Convergence and divergence of con- (5) Some existence theorems of differen­ tinued fractions tial equations of noninteger order Professor D. F. Dawson, North Professor M. A. Al-Bassam, Texas Texas State University (6ZT-159) Technological College (6ZT-177) (16) A solvable infinite class of AEA for- (6) Solution of loop equations by adjunc- mulas with axioms of infinity tion Professor Burton Dreben, Mr. A. S. Professor Rafael Artzy, Rutgers, Kahr and Professor Hao Wang, The State University (6ZT-163) Harvard University (6ZT-106) (7) Enumeration and the Grzegorczyk ( 17) Invariant subspaces of tridiagonal hierarchy operators Professor Paul Axt, MichiganState Dr. P. L. Duren, Stanford Univer­ University (6ZT-1Z4) sity (6ZT-114) (8) On the construction of models of (18) Open 3-manifolds which are simply analytic polyhedra connected at infinity Professor Stefan Bergman, Stanford Professor C. H. Edwards, Jr., Uni­ University ( 6ZT-1ZO) versity of Wisconsin (6ZT-1Z8) (9) Some Abelian results for Dirichlet (19) On the rate of convergence of a sym­ series metric successive overrelaxation Professor W. E. Briggs, University iterative method. Preliminary report College, London, England(6ZT-130) Mr. L. W. Ehrlich, The University (10) Formal power series of Texas (6ZT-16Z) Dr. E. D. Cashwell and Dr. C. J. (ZO) Hierarchies in recursive function Everett, Los Alamos Scientific Lab­ theory oratory, Los Alamos, New Mexico Mr. H. B. Enderton, Harvard Uni­ (6ZT-131) versity (6ZT-148)

199 (Z 1) Directed distance geometry structure on strictly convex surfaces Professor C. M. Fulton, University Professor T. S. Klotz, University of California, Davis (6ZT-171) of California, Los Angeles (6ZT- (ZZ) Partial recursive functionals of finite 146) type (34) Non-Desarguesian planes of order Dr. R. 0. Gandy, The University, zZm+l. Preliminary report Manchester, England (6ZT-149) Mr. D. E. Knuth, California Institute (Z3) Complete predicates for the analytic of Technology (6ZT-137) hierarchy (35) New characterizations of the class of Dr. R. 0. Gandy, The University, hyper arithmetic sets of natural num­ Manchester, England (6ZT-150) bers. (Z4) Brownian motion and harmonic forms Professor Georg Kreisel, Institut on a compact manifold. Preliminary Henri Poincare, Paris, France report (6ZT-174) Dr. R. A. Gangolli, Massachusetts (Introduced by Dr. S. H. Gould) Institute of Technology ( 6ZT-1Z3) (36) Formalizing predicative analysis (Z5) A deformation theory for algebras Professor Georg Kreisel, lnstitut Professor Murray Gerstenhaber, Henri Poincare, Paris, France University of Pennsylvania and In­ (6ZT-175) stitute for defense Analyses, (Introduced by Dr. S. H. Gould) Princeton, New jersey (6ZT-1ZZ) (37) Traffic in a polar coordinate net of (Z6) Poisson's equation and generalized streets axially symmetric potential theory Professor E. 0. A. Kreyszig, Ohio Professor R. P. Gilbert, Institute State University (6ZT-116) for Fluid Dynamics and Applied (38) Traffic in a city having radial and Mathematics, University of Mary­ circular streets land (6ZT-1Z9) Professor E. 0. A. Kreyszig, Ohio (Z7) Note on difference approximations State University (6ZT-117) with non-negative coefficients for (39) On a class of integral operators partial differential equations Professor E. 0. A. Kreyszig, Ohio Dr. Donald Greenspan, Mathematics State University (6ZT-118) Research Center, University of (40) Integral operators for generating har­ Wisconsin (6Z.T-165) monic functions of four variables (ZS) On a problem of Erdos and Tarski Professor E. 0. A. Kreyszig, Ohio Mr. W. P. Hanf, IBM Corporation State University (6ZT-119) and University of California, Berke­ (41) Imbedding and immersion of projec­ ley ( 6ZT-17Z) tive spaces (Z9) A two dimensional analogue to boolean Mr. J. P. Levine, Massachusetts algebras Institute of Technology ( 6ZT-14Z) Mr. C. M. Howard, University of (4Z) Imbedding and immersion of mani­ California, Berkeley (6ZT-140) folds (Introduced by William Craig) Mr. J. P. Levine, Massachusetts (30) Indeterminate forms on En Institute of Technology (6ZT-143) Mr. Seymour Kass and Mr. Arthur (43) Convergence of series of iterates Fine, Illinois Institute of Techno­ Professor M. A. McKiernan, Uni­ logy (6ZT-156) versity of Waterloo, Ontario, Can­ (31) A new potential operator for recur- ada (6ZT-173) rent Markov chains (Introduced by Professor Gerald Professor J. G. Kemeny and Pro­ Berman) fessor J. L. Snell, Dartmouth Col­ (44) Two theorems on contraproductive lege (6ZT-170) centers (3Z) On metric compactifications. Prelim­ Mr. T. G. McLaughlin, University inary report ofCalifornia,Los Angeles (6ZT-1Z6) Mr. L. R. King, Duke University (45) A note on pseudo D. C. pairs (6ZT-154) Mr. T. G. McLaughlin, University (33) Some uses of the second conformal of California, Los Angeles (6ZT-144)

200 (46) Stiefel-Whitney classes of a nonori­ bay University, India (62T-109) entable manifold (59) Existence and uniqueness of solutions Professor W. S. Massey, Yale Uni­ of the Poisson interface problem versity (62T-141) Dr. D. P. Squier, California Re­ (47) Categoricity in power search Corporation, La Habra, Cali­ Mr. M.D. Morley, Berkeley, Cali­ fornia (62T-160) fornia ( 62T-136) (60) On the existence and analyticity of (48) On some algebraico-topological ana­ solutions of regular variational prob­ logs lems Mr. A. A. Mullin, University of Professor Guido Stampacchia, lllinois (62T-103) Courant Institute of Mathematical (49) On the mutant double-cosets of any Sciences, New York University and group University of Pisa, Italy (62T-104) Mr. A. A. Mullin, University of (61) Kernel constructions and Borel sets lllinois (62T-138) Professor A. H. Stone, University (50) On differences of algebraic structures of Rochester (62T-134) Mr. A. A. Mullin, University of (62) Borel isomorphism and o -discrete­ lllinois (62T-168) ness (51) On complete sets of mutually ortho­ Professor A. H. Stone, University gonal latin squares of Euler conjec­ of Rochester (6

201 of Technology (62T-132) Advanced Study (62T-167) (71) Elliptic spaces in Grassmann mani­ (74) Geometry of n-planes in a real or folds complex Euclidean space. Prelimin­ Dr. J. A. Wolf, The Institute for ary report Advanced Study (62T-101) Professor Y.-c. Wong, University (72) On locally symmetric spaces of non­ of Hong Kong (62T-161) negative curvature and certain other (75) Timelike initial value problems for locally homogeneous spaces hyperbolic equations Dr. J. A. Wolf, The Institute for Mr. J. M. Zimmerman, Courant Advanced Study (62T-166) Institute of Mathematical Sciences, (73) Riemannian nilmanifolds New York University (62T-145) Dr. j. A. Wolf, The Institute for

NEWS ITEMS AND ANNOUNCEMENTS

AN INSTRUCTIONAL CONFER- followed by a course of basic lectures on ENCE ON MATHEMATICAL PROBABIL­ the fundamentals of the subject; these will ITY is to be held during the fortnight be supplemented by classes for informal March 28 to April 11, 1963, intheDepart­ discussion. ment of Mathematics, Durham Colleges, The second part of the Conference University of Durham, under the auspices will be devoted to three series of advanced of The London Mathematical Society. lectures. Each of these, starting where the This will be the second in the Lon­ basic course leaves off, will attempt to don Mathematical Society's series of In­ give a thorough review of some branch of structional Conferences;the first, on Func­ the subject in which recent research has tional Analysis and some of its applica­ been most active. tions, was held in April, 1961. On this Arrangements are being made for occasion the subject will be Mathematical accommodations and full board to be pro­ Probability, and while it is hoped that there vided at the Durham Colleges at a charge will be much in the program to interest of thirty shillings per day per person, in­ specialists the main purpose of the Con­ clusive of gratuities. Full details of the ference is to meet the needs of mathemati­ organisation and program of the Confer­ cians and statisticians who would like a ence will be advertised in due course, but thorough introduction to this field. those wishing to be kept informed of the The Conference will begin with two arrangements are encouraged to register series of introductory lectures delivered now. The registration fee of two guineas in parallel on the first day; the first of (which covers participation in all the sci­ these, on the applications of probability entific activities of the Conference) should theory, will consist of informal accounts of be sent to the Secretary, Science Labora­ some practical problems whi&h have moti­ tories, South Road, Durham, England, and vate~ much recent theoretical work, while the envelope should be marked 'Probabil­ the second will survey some of the mathe­ ity Conference;' remittances should be matical prerequisites for the subsequent made payable to 'The London Mathematical courses. Society.' The introductory lectures will be

202 ABSTRACTS OF CONTRIBUTED PAPERS

The April Meeting in Atlantic City April 16 -19, 1962

590-40, MELVIN DRESHER, RAND Corporation, 1700 Main Street, Santa Monica, California. On an infinite game with a discontinuous game value,

Suppose the payoff M(x,y) in a game over the unit square takes on the three values b, a, or 1 depending upon whether y ;;_ x, x < y ;; x + t, or x + t <:: y, respectively (a< b ~ 1, t < 1). It is shown that this game has optimal mixed strategies and a game value which is discontinuous. (Received February 28, 1962.)

590-41. S, P. LLOYD, Bell Telephone Laboratories, Murray Hill, New Jersey. On positive projections of unit norm.

With X compact Hausdorff, let T be a positive projection in C(X) of unit norm. There is an upper semicontinuous decomposition of X into closed equivalence classes, such that all functions Tf are constant on equivalence classes, It is shown that an equivalence class is either ergodic or dis- sipative, as follows. The value of Tf on an ergodic class is the average of f with respect to a regular Borel probability measure living on the ergodic class. The value of Tf on a dissipative class is a subaverage off over ergodic classes, An application to Markov chains is given. (Received February 28, 1962,)

590-42, J, D. STASHEFF, 2-178 Massachusetts Institute of Technology, Cambridge, Massachusetts. A classification theorem for fibre spaces, Preliminary report.

Let F be a finite CW -complex. Let H be the topological monoid of all homotopy equivalence ofF onto itself. Dold and Lashof have constructed a space BH such that H;; nBH. Theorem. Fibre homotopy equivalence classes of fibrings p: E ->X with fibres of the homotopy type of F are in 1-1 correspondence with homotopy classes of maps of X into BH (at least for reasonable X). The crucial step is the construction of fibrings by a generalization of the Hopf construction. (Received February 28, 1962,)

590-43. R. A. GANGOLLI, Massachusetts Institute of Technology, Cambridge 39, Massachusetts. Diffusions and Brownian motions on a Riemannian manifold.

Let M be a second countable orientable Riemannian manifold. By dissecting the sample paths of skew Brownian motions, one in each tangent space, and injecting these into M via the exponential map, an almost Markovian path is constructed on M. It is shown that as the mesh of the dissection approaches zero, these paths converge uniformly on compacts of [O,oo) to the paths of a diffusion on M whose generator is identified, This leads to geometrical descriptions of the Laplace-Beltrami Brownian motion on M. When M is homogeneous = G/H, G,H being the Isometry and Isotropy groups of M, a Brownian motion on M is defined as a diffusion whose law is invariant under G. Under various

203 hypotheses on M, such motions are described. Sample results: Let M be symmetric. Let g,h be the Lie algebras of G,H, g = h + m, Ad(H)m C m; (i) Every probability measure dp with global support on the space of orbits of points of the unit sphere of m under the action of Ad(H) gives rise to a Brownian motion on M in a natural way. (ii) Every decent Brownian motion on M arises in this way from such a measure dp. (Received March 1, 1962.)

590-44. J. M. WORRELL, JR., 1509 Windsor Road, Austin 3, Texas. Concerning upper semi­ continuous collections of mutually exclusive closed and compact point sets.

Axiom I..,. There exists a sequence G 1, G 2, G 3, .•.• satisfying the conditions of R. L. Moore's Axiom 13 (Foundations of point set theory, Amer. Math. Soc. Colloquium Publications, Vol. 13) such that if R1 , R2 , R 3 , •.• , is a sequence such that for each n, Rn belongs to Gn and Rn contains Rnt 1, then there is a point P which belongs to Rn for each n. Suppose that :E is a space satisfying Axioms 0 and !3 of R. L. Moore and G is an upper semi-continuous collection of mutually exclusive closed and compact point sets covering the set S of all points. Let :E' denote the space obtained by interpreting "point" to mean "element of G" and "region" to mean "subcollection G' of G such that G* is a domain in :E." The author shows that Axiom 13 holds for ~',thus settling a question raised by R. L. Moore, and that if Axiom lc.> holds for ~. then Axiom IGJ holds for 1:: '. R. L. Moore has settled an analagous question with respect to a considerably stronger axiom called Axiom 1' in Foundations of point set theory. (Received February 28, 1962.)

The April Meeting in Monterey April 28, 1962

591-26. W. E. MILNE, 3850 West Hills Road, Corvallis, Oregon and R. R. REYNOLDS, 600, 9th Avenue, No. 208, Seattle, Washington. Note on the numerical solution of ordinary differential equations.

The authors propose an extension of the usual "predict-correct" procedure for the numerical solution of ordinary differential equations. The purpose of the extension is two fold: (l) to increase the accuracy without increase in the number of values of the dependent variable and its derivatives that must be carried along; and (2) to insure a high degree of stability in the computed values. The

process consists in predicting the value of Ynt1 by an open type quadrature formula, computing

dy/dx = Y~tl' and correcting Yn+I by a closed type formula, recomputing y~+ 1 ' and finally (the new feature) recomputing Yn by a stable formula of higher accuracy than is available for Yntl. (Received February 28, 1962.)

591-27. T. K. BOEHME, University of Washington, Seattle 5, Washington. Convolution algebras on the half-line.

Every f whose pth power is locally integrable on the half-line t ;;:; 0 can be expressed as f(t) = JJ g(t - u)da.(u) (a. e. t ~ 0) where g is a function whose pth power is locally integrable, and a. is locally of bounded variation and a(O+) = 0. A necessary and sufficient condition that a continuous

204 function c can be expressed as j'0tc2(t - u) da.(u) with c 2 continuous and a. as above is that c 1 (0) = 0. There are continuous c, which cannot be so expressed if it is required that a. be absolutely continuous and zero at the origin. (Received February 28, 1962.)

591-28. BERTHOLD SCHWEIZER, University of Arizona, Tucson, Arizona, and ABE SKLAR, Illinois Institute of Technology, Chicago 16, Illinois. Best-possible triangle inequalities for a class of statistical metric spaces.

{For terminology and notation see our papers in Pacific J, Math. 10 (1960), 313-334; Publ. Math. Debrecen 8 (1961), 169-186]. Let (S,d) be a metric space, a. ~ 0, and G a distribution function which is strictly increasing on (O,oo), continuous, and such that G(O) = 0. For every pair of points p, q inS define a distribution functionJf(p,q) = Fpq via Fpq(x) = G(x/[!l(p,q)f). Then (S,;:f) is a statistical metric space -which we call the a.-simple SM space generated by G and d. For 0;:; a.~ 1, this SM space is a Menger space under the t-norm Min, independently of G; for a. > 1, it is a Menger space under the t-norm generated by (G*) l/l-a., where G* is the inverse of G. This result is best­ possible. (For related results cf. E. 0. Thorp, Proc. Amer. Math. Soc. 11 (1960), 734-740.) (Received February 28, 1962,)

591-29. ALFRED TARSKI, 462 Michigan Avenue, Berkeley 7, California. Solution of the decision problem for the elementary theory of commutative semigroups.

A commutative semigroup 2J<= (§. •) is an algebraic structure with a binary operation • satisfying the commutative, associative, and cancellation laws, and having a unit 1; an element

~ E ~ ~f 1, is indecomposable if x = y•z, y f 1, always implies z = 1. If, in particular,§_ is a set of positive integers closed under the ordinary multiplication •, ?- = (§_. •) is called an M-semigroup.

Let l?O• _£1 ..... be the increasing sequence of all primes. For any natural !!h!! let ill"!!= 2ID+l · (2!! + 1). Let mRn mean that, for some odd number.!_, 2ID·.f :!i g <2m •

all the numbers P~m+ 1• Pm•n and E~n+ 1 • .Pm• n for which !!!BJ!. B' is elementaril;definable in 3'•; in fact, xR'y iff b;are i~:Compos~ble (:U ~')and .!3 •.!! = y_2 •J. for some indecomposable~·~· Hence, using terminology and results of Tarski-Mostowski-Robinson's Undecidable theories, one proves: Theorem l. The elementary theory of the M-semigroup ~· is hereditarily undecidable. Corollary 2. The elementary theories of the classes ;r&1 of all M-semigroups and,Kz of all commutative semigroups are undecidable. (Received February 28, 1962.)

591-30. E.]. TULLY, JR., 534 Midvale Avenue, Los Angeles, California. A class of commuta­ tive semigroups having a group-like· property.

Let S be a commutative semigroup with maximum condition on principal ideals. Let b t S be such that [x: xfbJ =I {x: x

205 a (possibly uncountable or empty) set I= {o •••• }. S = G U{(i,n): i € I, nan integer, 0!! n < ni, (i,n) f. (0,0)} with an optional zero element adjoined. Define multiplication by (i,n)g = g (g E G),

(i,n)(j,m) = (i,n + m) if i = j and n + m <. ni' (i,n)(j,m) = b otherwise. II. Let b be any element of a group G, S = G Ufc}, with an optional zero element. Define cg = bg (g E G), c 2 = b 2• III. S =either {0,1, ••• ,b + 1} or {1 ••••• b + 1J, mn =min (m + n, b + 1). IV. S = {0,1, •••• b + 1} U {c}, Oc = b, xc = b + l if x f. 0, mn = min (m + n, b + 1) if m 1- c 1- n. Corollary. If S is as in the first sentence, and

all the congruence relations on S commute, then S is of one of the following types. V. Let G0,Gl' ... 'Gn be groups, 11j_ (for 11i i ~ n) a homomorphism of Gi-l onto Gi' S = UGp with a zero element adjoined.

If x €. Gi andy E Gj' let xoy = (x"i+1ri+2 •••1Ti+j) (y'll'"j+lrj+2 .•• rj+i) if i + j ;in, x o y = 0 otherwise. VI. S = fl,2, ••• ,n}, ij = min(i + j, n). VII. LetS be any group. (Received February 28, 1962.)

591-3·1. STEFAN BERGMAN, Stanford University, Stanford, California, On singularities of harmonic functions of three variables.

Let!! be a linear space of harmonic functions regular at the origin with a basis {T'nJ<(X), n = 0,1, •••• f = 1,. •• ,2n}. Here: X= (R,8,fll); 1' nii(X) = Cn11RnP n,I·K-nl(cos 8) exp(ir6( o\' - n)); R, 1!,!11 are polar coordinates; P n,p-the associated Legendre functions; Cntt- numerical constants. H is an algebra with respect to the addition and the composition defined in (1) Math. Zeit. 24, p. 647 (see also (2) Arch. Rational Mech. Anal. 8, p. 216 and (3) Ergebnisse der Math. NF 23). Let .J llp be the sub-

space of,!! with the basis {Cni(R npn,lp.l (cos ~) exp(ipr6), n = Ipl. IPI + l, ••. J. For R > 0, sufficiently small, J 11}l is assumed to be isomorphic to the algebra of meromorphic functions (regular at the origin) of a complex variable. Let S}l(X) = E: IPian,n+p T'n,n+p(X) E r llp' and let 11n' n = l, •••• N. be arbitrary complex constants. The function G}l (X) reciprocal to ~p (X) - ~~= jp 111n T' n+p(X)] has branchlines b!(11) = [R cos !9 = Re A>'(11), R sin i = Im A,(11)], Y = 1,2, ••• , see (3), pp. 45-47. Further, if limn-.oo [n log (- loglan,n+pl> -l] ~ f• then L~ 1 1A>'(11) ~-~ <. oo for .:\ ':>'f. (Received April 2, 1962.)

• 591-32. MARTIN BLUMBERG, Department 58-22, Building 204, Lockheed Missiles and Space Company, 3251 Hanover Street, Palo Alto, California. Some formulas involving the products of trigonometric functions spaced at regular intervals.

In the study of conformal maps of infinite arrays of line charges there arise functions in­ volving the products of sines and of cosines spaced at regular intervals. A typical formula is (- l/2)n- 1sin nz = TT!_:,bsin(z - m1T/n). The value of the constant, (- 1/2)n- 1, is obtained by several methods, one utilizing a formula due to Euler. Many other relations are derived, some involving sums of tangents and cotangents spaced at regular intervals. The manipulations leading to these are also of interest. (Received April 12, 1962.)

206 Abstracts Presented by Title

62T-101. J. A. WOLF, The Institute for Advanced Study, Princeton, New Jersey. Elliptic spaces in Grassmann manifolds.

Let Fk denote a positive definite left hermitian vectorspace over F = reals, complex numbers, or quaternions; if 0 < n < k, let Gn,k(F) be the Grassmann manifold of n-dimensional subspaces of Fk, with its usual structures as a Riemannian symmetric space. lf B is a collection of pairwise isoclinic sub spaces of Fk, then it is shown that there is an orthogonal decomposition Fk = V 1 fB ••• e Vm' and that there are connected totally geodesic submanifolds Bi of Gn,k(F), such that (1) every element of Bi lies in vi. (2) B c UBi, and (3) UBi is a collection of pairwise isoclinic subspaces of Fk. Results of an earlier paper (Geodesic spheres in Grassmann manifolds. See Abstract 62T-65, Notices Amer. Soc. 9 (1962), 143) show that each Bi is a Riemannian symmetric space of rank one, and give the classification of those Bi which are spheres. Here, it is shown that Bi is a real, complex or quater­ nion projective space if it is not a sphere, and those Bi are classified. (Received February 21, 1962.)

62T-102. L. B. TREYBIG, 608 Jefferson Park Avenue, New Orleans 21, Louisiana. Concerning homogeniety in totally ordered, connected topological spaces.

Suppose L is a totally ordered, connected topological space such that (1) the topology of L is the order topology, and (2) L has no first or last point. Theorem 1. lf L is homogeneous, then L satisfies the first axiom of countability. Theorem 2. L is homogeneous if and only if each pair of closed intervals lying in L are homeomorphic. Theorem 3. Suppose L is homogeneous and I is a closed interval lying in L. Then, if for each positive integer p, Ip denotes a copy of I, then the space X = L X 11 X 12 X 13 X ••• with the topology induced by the lexicographic order is also homo­ geneous. (Received February 23, 1962.)

62T-103. A. A. MULLIN, University of Illinois, Urbana, Illinois. On some algebraico­ topological analogs.

Theorem 1: LetT be a transformation from a topological space T 1 into a topological space T2• Let S be a homomorphism from an algebraic system S 1 into an algebraic system s2• If there exists a mapping a from the mutant sets of S 1 into the open (closed) sets of T 1 and if there exists a mapping (3 from the open (closed) sets of T 2 into the mutant sets of s2 such that T- 1(P) = a(S- 1(f3(P))), where P is any open (closed) set of T 2, then T is continuous. Definition. A homomorphism 91 from an algebraic system A 1 into an algebraic system A2 is said to be stationary provided 9l{A) ~ {6(A) for every mutant set A of A 1• Theorem 2. A stationary homomorphism is mutant preserving. Meta­ theorem 1. If one sets up the appropriate correspondences between the symbols of the two syntax languages of Theorem 1 then "continuous transformations of the topological spaces" corresponds to "stationary homomorphisms of the algebraic systems," and vice versa. The theory of abelian U -dia­ grams can be used to treat the more special cases of isomorphic algebraic systems and homeomor­ phic topological spaces. (Received February 26, 1962.)

207 62T-104. GUIDO STAMPACCHIA, Courant Institute of Mathematical Sciences, 4 Washington Place, New York 3, New York. On the existence and analyticity of solutions of regular variational problems.

Let f(p) be a convex function (p = (p l•P2• .... Pn»• with the quadratic form: ~iPj (p)Ji ~j, positive definite at every point p (not necessarily uniformly). Consider the regular variational problem:

(1) I(u) = j'0 f(grad u)dx = min. It is supposed that n is a bounded domain of En and that u equals a given function sil(x) on the boundary 'aSJ of Sl. Let r denote the (n - I)-dimensional manifold of

En+l (xl'x2, ... ,xn,u) {x E' ~n. u = sil(x)J. Let the boundary values sil(x) satisfy "the bounded slope condition K II if for every point Po of r there exist two planes .,._ and .,.+ of slope ;i K passing through PO such that r is between fr_ and 71"+. Consider the variational problem (1) for all Lipschitz functions in n having boundary values rJ satisfying "the bounded slope condition K." The existence and smooth­ ness of the solution is shown, and in particular, if the data of the problem are analytic also the solu­ tion is analytic in the closure of n. The result, which is related, for n = 2, to a classic theorem by Haar, requires no conditions on the growth of the function f at infinity. The proof is based on De Giorgi's result and its extension at the boundary and uses an a priori bound for the maximum of the first derivatives of the solution. (Received February 26, 1962.)

62T-105. PETER WERNER, AACHEN Techniche Hochschule, institut fur Reine und Angewandte Mathematik, Templergraben. An interface problem in the theory of acoustic wave fields.

Stationary acoustic wave fields in nonhomogeneous media are described by potentials U(x) satisfying (1) pV • (p -l V U) + K 2u = f with given variable coefficients p, 1(, f. Study the case in whic.h p. I(, f are discontinuous on a closed surface S. The physical experience that pressure and normal velocity are continuous across S leads to two interface conditions (2), (3) linking the interior and the exterior field. Under certain assumptions on P• c, fat infinity, and additional regularity requirements for these coefficients and S, it can be shown that a uniquely determined U exists satisfying (1), (2), (3) and Sommerfeld's radiation condition. This continues investigations of Sternberg, Freudenthal, Kupradse, Muller and Barrar-Dolph concerning interface problems for acoustic, electro-magnetic and elastic fields in homogeneous media. (Received February 26, 1962.)

62T-106. BURTON DREBEN, A. S. KAHR and HAO WANG, Computation Laboratory, 33 Oxford Street, Cambridge 38, Massachusetts. A solvable infinite class of AEA formulas with axioms of infinity.

Let J be the class of all AEA formulas AxEuAyMxuy with only dyadic predicate letters G1,G2, etc. and monadic ones such that, for every i, Giyx, Giyu never occur in the matrix Mxuy. It is proved by a new type of argument that J is solvable. A similar procedure also solves the class J* in which Gixy,Giuy never occur. In both cases, it is further shown that (1) the formulas containing atomic formulas only of the forms Gixy, Giuy or only of the forms Giyx, Giyu do not include any axiom of infinity; (2) there are axioms of infinity in any class of all formulas containing three (or more) forms of atomic formulas which include either the two forms Gixy or Giuy, or the two forms Giyx, Giyu.

For example, -G1xx & (G1xy-G1uy) & (G2xy = G1uy) &Gzxx is an axiom of infinity. It is

208 worth noting that the class in which Giuy, Giyu never occur and the class in which Gixy, Giyx never occur are solvable and indeed contain no axioms of infinity. (Received February 27, 1962.)

62T-107. LOUIS SUCHESTON, 3702 N. Murray, Shorewood 11, Wisconsin. An ergodic applica­ tion of Banach limits. II.

Let T be a measurable one-to-one transformation on a probability space (0, t:(, P) such that if P(A) = 0 then P(TA) = P(T-1A) = 0. A necessary and sufficient condition that there exist a finite invariant measure equivalent with P is that for each measurable set A all Banach limits on the sequence {P (TnA)} coincide (cf. Abstract 584-2, Notices Amer. Math. Soc. 8 (1961), 496). (Received February 26, 1962.)

62T-108. G. j. RIEGER, Dahlienstrasse 24, Ottobrunn near Munich, West Germany. On the representation of algebraic integers as the sum of four squares.

Let K be an arbitrary algebraic number field. Theorem. The 4-fold Schnirelmann sum of the sequence of all squares of a sequence of totally positive integers of K with positive Schnirelmann density has positive Schnirelmann density. The proof is elementary and uses ideas of Linnik. (Received February 26, 1962.)

62T-109. S. S. SHRIKHANDE, D. RAGHAV RAO and S. K. THARTHARE, Banaras Hindu University, Varanasi 5, India. Non-existence of some unsymmetrical partially balanced incomplete block designs.

In the case of a partially balanced incomplete block (PBIB) design with two associate classes, let v,b,r,k have their usual meanings and let N be the incidence matrix of the design. Then the characteristic roots of NN' can be found in terms of the parameters of the design by the methods of W. S. Connor and W. H. Clatworthy (Ann. Math. Statist. 25 (1954), 100-112). One considers the case when b = v - a. and zero is a characteristic root of NN' of multiplicity a., while fo = rk is a root of

multiplicity 1 and the remaining root is a rational number p 1 with multiplicity a. 1 = b - 1. Let Q1 and Q be the gramians of the rational characteristic vectors corresponding to the roots ('1 and zero respectively. Then it can be proved that necessary conditions for the existence of the design are 1 (i) p0 p1 a.1 IQI is a perfect square and if (i) satisfied, then (ii) (fb.-v h a.1+ )p

62T-110. j. L. WALSH, Harvard University, 474 Widener Library, Cambridge 38, Massachu­ setts. A sequence of rational functions with application to approximation by bounded analytic functions.

Let region D be bounded by mutually disjoint jordan curves BpB2, ••• ,Bp,Cl'C2, •• .,C')I, the Bj having smoothness properties. Let u(z) be harmonic in D, continuous in its closure, zero on B = ~B j

209 and unity on C = ~Cj• By a conformal map [H. J. Landau] we choose D interior to c 1, and interior to each Cj (j .,. 1) choose u(z) harmonic with a single singularity, of form -njloglz - ajl; u(z) has a corresponding singularity at a 1 = oo. Points a 1, a2 , •••• each some '1• exist [Ann. Mat. Pura. Appl. 39 (1955), 267-777, Lemma 1] distributed among the aj asymptotically in the proportions nj• Points Pnk(k = 1,2, ••. ,n + 1; n = 1,2, ••• ) exist [same Lemma] equally spaced on each Bj with respect to the conjugate v(z) of u(z), distributed among the Bj asymptotically in the proportions /'Bjdv. Rational functions with poles in al' ••• ,an interpolating in the points flnk to an arbitrary f(z) analytic on and within the B j can be used to generalize previous results on approximation; the curves B j need not be analytic. In particular ::V = 1 is a new treatment of Walsh and Sewell, Duke Math. j. 6 (1940), 658-705. (Received February 28, 1962.)

62T-111. IRVING REINER, University of Illinois, Urbana, Illinois. Failure of the Krull­ Schmidt theorem for integral representations.

Let RG denote the group ring of a finite group G over the ring R of all algebraic integers in an algebraic number field K. Consider RG-modules which are finitely-generated and R-torsion-free. It is shown that if G is not a p-group, and if G contains a normal subgroup of prime index, then the Krull-Schmidt theorem cannot be valid for RG-modules. Indeed, in a direct sum of indecomposable RG-modules, the R-ranks of the summands need not be uniquely determined by the direct sum. The proof depends on showing that if G satisfies the above conditions, then there exist RG-modules A, B and C such that KA, KB and KC are irreducible KG-modules no two of which are isomorphic, and such that there exist nonzero elements F E ExtkG(B,A) and F' € ExtiG(C,A) whose orders are relatively prime. (Received March 5, 1962.)

62T-112. B. W. ROMBERG, 1931 Sherman Avenue, Evanston, Illinois. The dual space of Hp. Preliminary report.

The space Hp is defined to be the class of all functions f(z) =}:anzn, analytic for lz I < 1 and such that sup f.l..,._,. lf(reix) 1Pdx: 0 1iii r < 1] < oo. In the following, p is restricted to 0 < p < 1. It was previously known that T E Hp if and only if there is a function g(z) = :5:_bnzn, analytic for lzl < 1 and such that Tf = limr-..1 ~anbnrn. Such a g is said to be a kernel of T. The following theorems give a complete characterization of Hp for the case 1/p not an integer, and a partial characterization when 1/p is an integer. Define k as the greatest integer less than 1/p - 1 and lip(l/p - 1) as the class of functions g(z), analytic for lz I < 1 and such that limr-..,l g(k)(reit) exists for almost all t and satisfies a Lipschitz condition of order 1/p - (k 1). Then we have: Theorem I. 1!_1/p is not an integer, then g is the kernel of some T E H; if and only if g E lip(l/p - 1). Theorem II. Suppose 1/p - 1 = j for some integer j. Jf_g(z) = }:bnzn is analytic for lzl < 1 and there exists a·function F(x)"'2:cneinx, bounded in the unit Circle and SUCh that CD ( r (D + l - j)/ r (n + l))bD' then g is the kernel of some T E Hp. Conversely,~ T E Hp, then T has a kernel g(z) such that limr-l g(k)(reit) exists and has the modulus of continuity w(t) = 0 ~(ln 1/t)lfP]. (Received March 5, 1962.)

210 62T-ll3. D. M. TOPPING, Tulane University, New Orleans 18, Louisiana. Extreme points in order intervals of self-adjoint operators.

Let e l: 0 be a s.a. operator and let (O,e] = {a: a is s.a. and 0 ~ a 1i eJ. Clearly (O,e] is convex and is easily seen to be compact in the weak operator topology; extreme points abound. Theorem. For v E' [O,e] the following are equivalent: (l) vis an extreme point of [O,e] with ve = ev; (2) ve = v 2 (= ev); (3) ve = ev and for each !II E N(v) ... , vfll = efll (here N(v) denotes the null space of v). Moreover, if vfll = efll, for each !II t N(v).L, then v is extreme. If e is a projection, the extreme points of [O,e) are just the sub-projections of e and, as is well-known, e commutes with each a E [O,e]. The extreme points need not commute among themselves (as the case e = l clearly shows), nor need they even commute with e. The proof employs simple lattice techniques locally, where commutativity and lattice structure are available. (Received March 8, 1962.)

62T- 114. P. L. DUREN, Stanford University, Stanford, California. Invariant subspaces of tridiagonal operators.

Let Tx = y be a bounded operator defined on the complex Hilbert sequence space 12 by Yn = ?lnXn-1 + anXn + ~ nXn+l• n 0, x_l = 0, where .Y n• an, Pn are complex numbers and /Jn I 0.

For complex A the vector equation (T - A.I)x = 0, xo = l, recursively generates a sequence Xn = PnG~.). Suppose the polynomials Pn().) are orthogonal, with respect to a positive continuous weight function, over some rectifiable Jordan curve bounding a Smirnov domain, and suppose the norms I:Pnll are bounded away from 0 and oo. Then the invariant subspaces ofT have the same lattice structure as those of the shift operator (?'n;;; 0, an= 0, fln = 1). Several specific applications can be made. (Received March 8, 1962.)

62T-ll5. MAR TIN SCHECHTER, 25 Waverly Place, New York 3, New York. Coerciveness in LP for bilinear forms. Preliminary report.

Let G be a bounded domain in En and [u,v] =j'Gipl.lt"l~m~TD,UuD,_v dx a bilinear integra­ differential form of order m. Let B l'···•Br be boundary operators which are nowhere characteristic to the boundary C}G of G and have distinct orders. Let V be the set of those u E C 00(G) which satisfy Bju = 0 on C3G, l a j-& r. The form [u,v) is said to be coercive over V if Uull~, 2 11 const. 2 (Re [u, u] + Uu llo , 2) for all u € V, where for any real p > l and s the norm 11·11 s,p is defined as in Abstract 62T-20, Notices Amer. Math. Soc. 9 (1962), 40. Necessary and sufficient conditions for [u,v] to be coercive over V have been given by Agmon (J. Analyse Math. 6 (1958), 216). Set [u) s,p = supvEVI [u,v] I llvll;;_s,q' wher~ q = p/(p - 1). Theorem. Under suitable regularity conditions, if

[u,v] is coercive over V, then for each real s;;, m, llull ~ const. ([u] + Uull ) for all s,p s-m,p s-m,p --- u E V. For each integer s, llull ;&_ const. ([u]s + 2:~- (B .)t. + llull ) for all u C 00 s,p -m,p J- 1 J J s-m,p --- E (G), where the<·) t· are appropriate boundary norms. (Received March 1, 1962.) J

211 62T-116. E. 0, A. KREYSZIG, Ohio State University, Columbus, Ohio. Traffic in a polar coordinate net of streets.

It is shown that a previously developed theory for estimating the density of traffic in a net of radial and corresponding orthogonal circular streets can be generalized to include the case where the distribution of the population is completely arbitrary. The application to freeway systems is illus­ trated by an example. The situation is treated as a discrete problem using a suitable subdivision of the city area into small portions. (Received March 9, 1962.)

62T-117. E. 0. A. KREYSZIG, Ohio State University, Columbus, Ohio. Traffic in a city having radial and circular streets.

Consider a city whose main traffic lines are concentric circles and straight streets originating from the center. Subdividing the city into "cells" (small portions around the points of intersection of the streets) and assuming that the traffic flows along the lines of minimum distance in the net of streets, simple formulas for computing the number of vehicles passing through the various portions of the net of streets are derived. Here it is assumed that the density of the population depends only on the distance from the center. By considering particular densities the influence of the geometrical shape of the shortest paths in the net can be estimated. (Received March 9, 1962,)

62T-ll8. E. 0. A. KREYSZIG, Ohio State University, Columbus, Ohio. On a class of integral operators.

The present paper is concerned with a class of integral operators for generating harmonic functions of three variables from analytic functions. The class includes the well-known Bergman­ Whittaker operator and a similar operator used by A. Erdelyi. Each operator of the class generates infinitely many sets of homogeneous harmonic polynomials of degree m, where m = 0, 1,... . Each set is orthogonal on the unit sphere. The totality of these polynomials constitutes a base for the harmonic functions H of three variables in the sense that any such function H can be represented by a series of the polynomials. This representation converges absolutely and uniformly in a sufficiently small neighborhood of a point at which H is regular. The polynomials can be represented in terms of spherical harmonics. Each operator has an inverse in a restricted sense, This inverse can be written in the form of a single definite integral. The operators can be used for investigating the coefficient problem, in particular with respect to the position and nature of curves of singularities. Some of the results can be translated into results about solutions of more general partial differential equations by the use of Bergman's theory. (Received March 9, 1962.)

62T-119. E. 0. A. KREYSZIG, Ohio State University, Columbus, Ohio. Integral operators for generating harmonic functions of four variables.

If~ is an isotropic vector, i = (x1•·· .. Xu), u = 1 :t and f(z) is analytic then f(u) is a harmonic function of n variables. The operators under consideration involve two integrations in the complex plane, the integrands being analytic functions of u, depending on two parameters, where n = 4. These

212 analytic functions are called associated functions of the corresponding harmonic functions with respect to the operator. The operators generate harmonic polynomials which are homogeneous and can be rep­ resented in terms of Jacobian polynomials, multiplied by some simple functions, lt follows that these polynomials are orthogonal on the surface of the hypersphere of radius 1 in four-space. Choosing a fixed operator and introducing a certain class of associated functions the correspondence between these functions and the harmonic functions generated by that operator becomes one-to-one. Then the inverse of that operator can be defined, lt is represented by a relatively simple double integral, The class of operators under consideration includes an operator for generating solutions of the three­ dimensional wave equation introduced by R. P. Gilbert. Any harmonic function of four variables which is regular at the origin can be represented by a series of those polynomials. The proof is similar to that in the three -dimensional case. (Received March 9, 1962.)

62T-120. STEFAN BERGMAN, Stanford University, Stanford, California. On the construction of models of analytic polyhedra.

The author describes a combined camera- and computing machine for the construction of models of analytic polyhedra (domains with a distinguished boundary surface), as described in (1) Proc. of Intern. Congress of Math. I (1950), 363 ff. An analytic polyhedron A 4 can be described analytically by several equations (see (2.1a)- (2.1d) of (1)). Let xpy1,x2,y2 be the coordinates of the four-dimensional space. For a number of discrete values x~"'), y~l), :V = 1,2,.,,,n, the machine com­ putes the boundary curve of the intersection of A 4 with x 2 = x~"), y2 = /J:>, )J = 1,2,,,,,n, Then it cuts . . 2 4 (:») (1) 2 . out from cardboard the mtersectwns Sy = A n (x2 = x2 ' Y2 = Y2 ) and places s,.. on the helght x 2 = x~"">. The still obtained in this way is photographed by the camera. This is repeated for anum­ ber of values of y 2 to obtain stills indicated in Fig. 1.1 - 1.4 of (1). These stills are used to prepare a film. Projecting this film, we will obtain the movie model. 1t is useful to make pictures which correspond to the image of the body using different coordinates x~n), y~n), x<£->, y<£"> which one obtains by orthogonal linear transformations. In this way is "shown" the four-dimensional body from different angles. Compare p. 365 and p. 367 of (1) where the same body is shown from different angles, (Received March 12, 1962,)

62T-121. D. j. RODABAUGH, Illinois Institute of Technology, Chicago, Illinois. A generalization of the flexible law.

Algebras satisfying (x,x,x) = 0 and (xpx2,x3) = e(x7r(1)' x77( 2), x~ 3 }) with e in F, the base field and 7r in s3 satisfy (for char. i 2,3): (1) (x,y,z) = (z,y,x) and (x,x,x) = 0, (2) (x,y,z) = (y,z,x) or are well known. Those satisfying (2) are power associative for char. f. 2,3,5 and have an associative type decomposition. Such a semisimple algebra is the direct sum of associative or nodal algebras if F is algebraically closed. Algebras satisfying (1) (studied by F. Kosier, Trans. Amer. Math. Soc. 102, No. 2, 299-319) are power associative for char. f. 2, For such a semisimple algebra A, a necessary and sufficient condition is given so that A has an identity and is the direct sum of simple algebras, which are nodal or associative. In a semisimple algebra with identity satisfying (1) either A 10 (e) t Ao1(e) = 0 or Aw(e)Ao1(e) = Au(e) where e is primitive and always A 11(e)2 n Au(e)/t;; Z = {x: (x,y) = 0}. If F is algebraically closed, altering products by elements of Z changes a semi-

213 simple algebra into a direct sum of nil, matrix and degree 1 algebras and a simple algebra is changed into a direct sum of degree 1 algebras. Partial converses are proved. A simple non-nil algebra has an identity. For a nodal algebra A with A+ associative, alteration of multiplication produces A from A+ and conversely. (Received March 12, 1962.)

62T-122. MURRAY GERSTENHABER, Institute for Defense Analyses, Princeton, New Jersey. A deformation theory for algebras.

Defining a deformation of a Lie or associative algebra A to be a multiplication of the same type in the tensor product of A with a power series ring in t specializing at t = 0 to the original multiplication, H2(A,A) becomes naturally the space of infinitesimal deformations. These have successive obstructions in H3• If H3 = 0, all elements of H2 are "integrable" to deformations. A deformation has an algebraic variety of moduli (analogous statements hold for module structures over A). Deformation of an algebra of characteristic p to characteristic zero may be defined, follow­ ing Goldman, using rings complete in a valuation; here there are further obstructions to integrating an infinitesimal deformation. (Received March 12, 196Z.)

62T-123. R. A. GANGOLLI, Massachusetts Institute of Technology, Cambridge 39, Massachu­ setts. Brownian motion and harmonic forms on a compact manifold. Preliminary report.

Let M be a compact Riemannian manifold, a. be a 1-form, 6. the Laplace-Beltrami operator. The solution ,e(t) = Tta. of 1::.{3 = l)p/0 t, the heat equation for !-forms can be written explicitly in terms of the Brownian motion on Mas follows. (Tta.)(x) = Exi.B(t)a.(xt)], where xt is the Brownian motion on M, Ex is the expectation operator for this motion starting at x and B(t) is a multiplicative matrix-valued functional of the Brownian path in the sense of Dynkin. In fact if R = Rij is the mixed

Ricci tensor of M, B(t) = limk_ooTI~=-Jexp [- R(xm2-kt)2-k), where exp is matrix exponentiation and lT is matrix multiplication, the limit existing uniformly with Probability 1. From this and similar formulas, it is possible to present simple proof of various known theorems of Hodge, Bochner and Yano as ergodic theorems. (Received March 13, 1962.)

6ZT-1Z4. PAUL AXT, Michi-gan State University, East Lansing, Michigan. Enumeration and the Grzegorczyk hierarchy.

The method of extending classes of functions used by Kleene (Extension of an effectively generated class of functions by enumeration, Colloq. Math. 6 (1958), 67-78) is applied to give a hierarchy having as its initial class the class e 4 of the Grzegorczyk hierarchy (Some classes of recursive functions, Rozprawy Mat. Warsaw (1953)), the latter having been established by the use of a scheme of bounded primitive recursion and a version of the Ackermann majorizing function. The segment of the resulting hierarchy corresponding to the finite ordinals is shown to be identical to the Grzegorczyk hierarchy beginning with class e4• (Received March 5, 1962.)

214 62T-125. ECKFORD COHEN, University of Tennessee, Knoxville, Tennessee. An elementary estimate for the k-free integers.

Let k denote an integer > 1 and let Qk(x) represent the number of k-free integers ~ x. It is

proved in this note, using no analytical fact beyond the mere convergence of L.l/ns (s > 1), that

Qk (x) = ~-l (k)x + O(xl/k+c) for all e > 0, where ~ (k) = ::L~ 1 1/nk. The proof is based on an expression for the characteristic function of the k -free integers, which involves the trigonometric sum ck(n,r) of the author. (Received March 14, 1962.)

62T-126. T. G. McLAUGHLIN, University of California, 405 Hilgard Avenue, Los Angeles 24, California. Two theorems on contraproductive centers.

It has been shown, by Carol Karp, that any productive set a has ~O mutually disjoint productivf centers. A relatively simple proof is given (using the result of Myhill that any productive set is

P -productive) that every productive set admits .1{0 productive functions with mutually disjoint ranges (which of course implies the above result). Further, in the presence of a simple lemma on existence of 1-1 contraproductive functions, the same argument yields Theorem 1. Every P -contraproductive

set admits ~O contraproductive functions with mutually disjoint ranges. Theorem 2. If a is a set productive w.r.t. the partial function p(x), and if range (p) n a: is r.e., then a has a contraproductive center lying inside range (p) n a. The hypotheses of Theorem 2 can be satisfied in such a way that a- (range(p) n 0:) is r.e., but ~ ( 3 ?')( ris recursive & 1i-(range(p) n 'ii) c;; ,,~ 0:). (Received March 29, 1962.)

62T-127. M. L. CURTIS and D. R. McMILLAN, Florida State University, Tallahassee, Florida. On cartesian products which are homotopy manifolds.

Let rr be an arc (preferably wild) in R 3, and let X denote R 3 modulo n-(i.e. the quotient space

obtained by collapsing 7r to a point). Then X X X is a homotopy manifold. It is known that X may fail to be a homotopy manifold. It is not known if X X X is always a manifold, and, indeed, there is no known example of a nonmanifold X such that X X X is a manifold. The theorem is proved by using a lemma of Penrose-Whitehead-Zeeman (Ann. of Math. (1960), 620), M. Brown's theorem on cellular sets (Bull. Amer. Math. Soc. 1960), and Kwun's theorem on decompositions of Sn which give homotopy manifolds (Proc. Amer. Math. Soc. 1961). More general results are also obtained by using these same techniques. (Received March 19, 1962.)

62T-128. C. H. EDWARDS, JR., University of Wisconsin, Madison, Wisconsin. Open 3-manifolds which are simply connected at infinity.

Let a triangulated open manifold M be called simply connected at infinity if each compact subset C of M is contained in a compact polyhedron P in M such that each component of M - p is simply connected. Now suppose that U is an open 3-manifold which is simply connected at infinity, and such that each compact subset of U can be imbedded in E 3. It is then shown that U is the union of a sequence {Ci}f of punctured cubes, with Ci C Int Ci+l for each i. This implies that there is a totally disconnected subset Y of E 3 such that U and E 3 - Y are homeomorphic. If in addition U is

215 contractible, it follows that U is homeomorphic to E3 • In order to prove the 3-dimensional Poincare conjecture, it would suffice to obtain this last result without the hypothesis that each compact subset of U can be imbedded in E3, (Received March 19, 1962.)

62T-l29, R. P. GILBERT, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland, Poisson's equation and generalized axially symmetric potential theory,

In this paper a method is developed by which one may obtain solutions to the nonhomogeneous equation of generalized axially symmetric potential theory (GASPT). Representation formulae are obtained and the method is shown to be extendable under certain symmetry conditions to the Poisson equation in three variables. (Received March 19, 1962,)

62T-l30, W. E. BRIGGS, Gower Street, London WC 1, England, Some Abelian results for Dirichlet series.

If f(s) is the analytic function defined by~ ann -s, certain properties of f(s) can be deduced from information about A(x) = Ln. 2 xan, (an Abelian theorem), In this paper it is shown that if A(x) = Cx logkx + O(xb), where k is a positive integer and 0 < b < 1, then the coefficients in the Laurent series for f(s) at s = l are related to constants which appear in the estimates for E.n;i:lCn- 1anlogrn. If A(x) = Cx + O(x/logtx) for arbitrarily large t, a similar result will follow under the assumption that f(s) has a pole at s = 1. This theorem includes the important special case of f(s) = -£''(s) /~(s) which arises when an= An. (Received March 22, 1962,)

62T-131. E. D. CASHWELL and C. J, EVERETT, University of California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, Formal power series,

Generalizing our previous result (Pacific J, Math, 9 (1959), 975-985) we prove that the ring P of formal power series, over an arbitrary domain of integrity C, in an arbitrary (cardinal) number of variables, is factorial (unique factorization domain) if and only if the rings C[[x1,. .. ,xnJ1 of such series in every finite number n of variables are factorial. From a recent theorem of D. Buchsbaum (J. Math, Mech, 10 (1961), 749-753) if follows that P is factorial whenever the coefficient domain C is a principal ideal ring, In the countable case it appears that the ring of all integer-valued arithmetic functions is a unique factorization domain. These statements appear as corollaries of a general theo­ rem on semigroups, (Received March 26, 1962,)

62T-132, W. G. WITTHOFT, Illinois Institute of Technology, Chicago 16, Illinois, A class of nilstable algebras,

Let A be a simple algebra over an algebraically closed field of characteristic 'I 2,3,5 satisfying x(xa) + (ax)x = 2(xa)x and x 2x 2 = x 3x. Then A is power associative, has a unit, l, and has been shown by Kosier (Trans, Amer. Math. Soc. 102 (1962), 299-318) to be Jordan or quasiassociative whenever degree A ;:; 3, Suppose A is of degree 2; then 1 = e + f with e and f orthogonal primitive idempotents,

216 A is the direct sum of subspaces Ae(A) where (ex+ xe)/2 = ~ x for all x E" Ae(-\) and ~ = 1, 1/2, or 0, Ae(l) = eF + N 1, and Ae(O) = fF +No with N 1 and No nilsubalgebras of Ae(1) and Ae(O), respectively. Suppose A is nil stable, i.e., that Au(..\)Au(l/2) and Au(l/2)Au(~)!; Au(1/2) + N 1_).. for J. = 1,0 and for every idempotent, u. Then, following Oehmke (Proc. Amer. Math. Soc. 12 (1961), 151-158) and Kokoris in an article on nilstable flexible algebras to appear in the Proceedings of the Amer. Math. Soc., A is shown to be a noncommutative Jordan algebra. In case ex= x/2 for all x E Ae(l/2), A is commutative Jordan. Should A fail to be nilstable, A might be of a class of commutative power asso­ ciative algebras of degree two that was constructed by Kokoris (Proc. Nat. Acad. Sci. 38 (1952), 534-537) and shown by him not to satisfy the Jordan identity. (Received March 26, 1962.)

62T-l33. E. T. PARKER, Remington Rand UNIVAC, Univac Park, St. Paul 16, Minnesota. On complete sets of mutually orthogonal latin squares of Euler conjecture orders. Preliminary report.

Let L be a latin square of order 4t + 2, t a positive integer. Let B be a (2t + 1) x (2t + 1) block of L. Designate as a special cell any cell of B containing a digit in a fixed subset of 2t + 1 digits. In Bull. Amer. Math. Soc. 50 (1944), 249-257, Mann proves: If L has an orthogonal mate, then B contains at least t + 1 special cells. Under a more stringent hypothesis, the author has sharp­ ened this inequality slightly. Theorem. If L is contained in a complete set of mutually orthogonal latin squares (equivalent to an affine plane of an Euler conjecture order), then B contains at least t + 1 + f(t) special cells; f(t) grows as ct1/ 2, and is positive if t l!; 3. For the most interesting order 4t + 2 = 10, the author's techniques give some information, but show no prospect of improving the inequality. (Received Aprilll, 1962.)

62T-134. A. H. STONE, University of Rochester, Rochester 20, New York. Kernel construc­ tions and Borel sets.

Let P be a class of spaces such that, whenever B is a closed subspace of A E P, then B E P. The "non-locally-P kernel" K = K(P,X) of a space X is the largest closed subset of X in which no neighborhood is in P. Suppose for simplicity that all the spaces mentioned are metrizable. General­ izing the Cantor-Bendixson theorem, it is shown that X - K is the union of a sequence of closed locally P sets. Generalizations of a theorem of Banach are deduced, including: if XC Y and every non­ empty closed subset of X is somewhere of the first category in Y, then X is of the first category in y. Similar generalizations are given of a theorem of Montgomery (roughly that locally Borel sets are

Borel). The spaces which are absolutely both Fa- and G& are shown to be those for which the non­ locally-compact kernel is empty, and the spaces all of whose subsets are absolutely both FCT and Gs are shown to coincide with the scattered spaces, as do those all of whose nonempty subsets are of the second category (in themselves). (Received March 27, 1962.)

62T-l35. A. H. STONE, University of Rochester, Rochester 20, New York. Borel isomorphism and a--discreteness.

All spaces considered are metrizable and absolutely Borel. A Borel isomorphism is a 1-1

217 correspondence f such that both f and f-• take Borel sets into Borel sets. Theorem. The property of being O"-discrete is invariant under Borel isomorphism. The proof uses the methods and results of earlier work of the author (Rozprawy Mat. 28). It follows that every (Borel) space X is either

62T-136. M.D. MORLEY, 2140 Roosevelt, Berkeley 3, California. Categoricity in power.

Theorem. If a first order theory is categorical in one uncountable power, then it is categorical in every uncountable power. This completely solves the problem raised by JjJ§ in On the categoricity in power of elementary deductive systems, Colloq. Math. 3 (1954), 58-62. (Received March 28, 1962.)

62T-137. D. E. KNUTH, California Institute of Technology, Pasadena, California. Non-Desarguesian planes of order 22m+l. Preliminary report.

Let x be a primitive element of GF(22m+1), m > 1. A new multiplication is defined on the elements of this field by letting xa o xb = xaxb, x 2m o xa = xa o x 2m = xa(x2m + xa + 1), and x2m o x2m = (x2m + 1)2, where 0 li a,b < 2m. The resulting system is a commutative non-associative division algebra. For if s and t are arbitrary nonzero polynomials in x of degree less than 2m, s o t

= st 1' 0, and (x2m + s) o t = (~m + s + t + l)t i 0, and (x2m + s) o (x2m + t) = (s + t)2 + (s + t)y + y2 f. C where y = x2m + s + 1. Furthermore, (xm o xm) o xI xm o (xm o x), so this system coordinatizes a non-Desarguesian projective plane. (Received March 29, 1962.)

62T-l38. A. A. MULLIN, 108d EERL, University of Illinois, Urbana, Illinois. On the mutant double-cosets of any group.

Theorem: Let (G, o) be any group. Then every double-coset (modulo the normal subgroups

(H,o) and (K,o)), with the exceptions of H ° K (clearly (H o K, o) is a subgroup) and(}>" H o K)-mutant double-cosets, is a (~ t 1, G)-mutant set. No systematic method has appeared to the author for de­ termining which, if any, of the abundance of mutant double-cosets are nontrivial maximal ones over some set N ~G. (Received March 30, 196-2.)

62T-139. L. C. ROBERTSON and S. P. FRANKLIN, University of California, Los Angeles 24, California. a-sequences and 0-nets.

An 0-net (0-sequence) is a net (sequence) which is either eventually inside ·or eventually out­ side any given open set. (See Levine, Amer. Math. Monthly 67 {1959), 667-668). The relationship between 0-nets and universal nets shows that convergence of every 0-net is equivalent to compact­ ness (not necessarily T 2). Tychonoff's theorem is an easy corollary, using familiar techniques. Con­ vergence of every 0-sequence is equivalent to nonvoid intersection of every countable nest of point

218 closures. This result can be formulated in terms of Davis' characterization of topological spaces (Amer. Math. Monthly 68 (1961), 886-894). Davis' condition Ro (point closures form a partition) is related in that an a-sequence converges to almost all points of its range in Ro spaces. Spaces in which 0-sequences have finite range or are eventually constant are characterized in terms of the possibility of separating countably infinite sets into two infinite subsets by intersection with an open set and its complement. It sometimes suffices to require separation of compact countable subsets.

Relationship to separation axioms To, T 1, and T2 is displayed. (Received April 2, 1962).

62 T-140. C. M. HOWARD, University of California, Berkeley, California. A two dimensional analogue to boolean algebras.

A concrete E 2-algebra is an algebra {A,+, •, -, S, D, d) such that for some set B, +, •,- are union, intersection, and complement with respect to B X B, 0 I A~ P(B X B), and for each a E A, S(a)={ (u,v)l(v,u) E aj,D(a)= f {u,v)l(u,u) E a, v E BJ, d = t(u,u)lu E B}. (1) The concrete E 2-algebras are those classes A of binary relations on a set B, such that A is closed under all the operations expressible without quantifiers in the first order binary predicate calculus with identity. (2) A natural, finite, set of identities I together with two universal sentences can be shown to characterize all concrete E 2-algebras. (3) The finitely generated algebras satisfying I are finite. (4) For the algebras satisfying I, the notions of being (a) simple, (b) subdirectly irreducible, (c) weak­ ly subdirectly irreducible, (d) concrete, (e) directly irreducible can be expressed by a universal sentence. (5) Each property (a) - (e) implies the next with only (b) and (c) coinciding. (Received April 2, 1962).

62T-141. W. S. MASSEY, Box 2155, Yale Station, New Haven, Connecticut. Stiefel-Whitney classes of a nonorientable manifold.

The following theorem is proved: Let yn be a compact connected non-orientable n-dimensional manifold with n even and not a power of 2. Then the dual Stiefel-Whitney class of yn in dimension n­ (with twisted integer coefficients) vanishes. The restriction that n is not a power of 2 is essential here, as the example of real projective space of dimension 2k shows. In view of recent results of M. Hirsch and A. Haefliger (as yet unpublished), this theorem has implications regarding the possible imbeddings of such a yn in Euclidean space. (Received April 2, 1962.)

62T-142. J.P. LEVINE, Room 2-174, Massachusetts Institute of Technology, Cambridge, Massachusetts. Imbedding and immersion of projective spaces.

The following results on imbedding and immersing projective spaces in Euclidean space are obtained: (A) If n - 1 is a power of two, pn is not imbeddable in E2n-2, cpn not in ~n-5, Qpll not in E 8n-ll; (B) If n > 7, Pn is not immersible in En+2. (A) is proved by examining the cohomology of the normal bundle, !lla Massey (Pacific.]. Math. 9 (1959)); (B) is proved by a study of 2-plane bundles over Pn. (Received April 4, 1962.)

219 62T-143. J. P. LEVINE, Room Z-174, Massachusetts Institute of Technology, Cambridge, Massachusetts. Imbedding and immersion of manifolds.

It has been proved by Haefliger (see Comment. Math. Helv. 36 (1961),47) that any closed q-connected n-manifold can be differentiably imbedded in E 2n-q if n ;: Zq + 3. This result is improved as follows. Let M be an orientable (q - I)-connected n-manifold with normal Stiefel-Whitney class wn-q equal to zero. Then, for n e; 2q + 3, M can be imbedded in EZn-q. Furthermore, if the homo- morphism 7r (sn-q-l) -1r (v +l) induced by the natural inclusion is zero, M can be iro- n- 1 n- 1 2 n-q,n mersed in EZn-q-1 (e.g. if (n,q) = (4k + 1, 1), (4k + 1,2), (4k + 2,2), (8k + 3,3), (n,4) or (n,5)). Some corollaries are: (1) Any orientable closed n-manifold can be imbedded in EZn-l. (Z) If n is not a power of two, CPn is imbeddable in E4n-Z, QPn in E 8n-4 and QPn is immersible in E 8n- 5• (Received April 4, 1962.)

62T-144, T. G. McLAUGHLIN, University of California, 405 Hilgard Avenue, Los Angeles 24, California, A note on pseudo D. C.+ pairs,

R. M. Smullyan has suggested the name "pseudo doubly productive" for the pairs defined as doubly productive in his Theory of formal systems (Annals of Mathematics Studies 47, Princeton, Uni- versity Press, Princeton, 1960). Define accordingly the notions "pseudo doubly creative" and "pseudo D. C. +u (cf. Smullyan, Theory of formal systems, pp. 109, 120), and verify the existence of pseudo D. C.+ pairs (a.,(J) such that both of a. - f3, jj- a. (and hence also the symmetric difference of a.,(8) are: (i) recursive; (ii) r .e. but not recursive; (iii) not r.e. (for the reason for the introduction of the adjective "pseudo," see Abstract 588-28, Notices Amer. Math. Soc, 9 (1962), 34). (Received April 9, 1962.)

62T-145. j. M. Zimmerman, 100 West 12th Street, New York 11,-New York. Timelike initial value problems for hyperbolic equations.

We consider equations of the form (1) a(y)utt = b(y)uxx + uyy• a f: c2 , a > 0, bE' cO, and seek c 2 solutions in a region R = { (t,x,y), - oo < t, x < oo, 0 ':i y ~ h < ooJ, given initial data (2) u(t,x,O) = f(t,x), uy(t,x,O) = g(t,x) on the timelike manifold y = 0. When b(y) ,.. 0 for at least some subinterval of [!l,h], such problems are improper in the sense of Hadamard, We have previously shown 1). Math. Mech, 11, no. 2 (1962)] that if f(t,x), g(t,x) E Lz with respect tot and x, and their Fourier transforms

F('t',c.>), G('t",c.J) have compact support in both variables, then there exists a unique L 2 solution which depends Lipschitz continuously (in the Lz norm) on the initial data f,g, In this paper we show that for the problem (1), (2) it is sufficient to assume only that F (I,Ci)), G(t',c.J) have compact supports in c.> in order to obtain the same results. Our methods generalize to the initial value problem for the ultrahyperbolic equation L: 1 ai(y)utiti = 2:,j= lbj(y)uXjXj + uyy· In the case m = n = 1 a more refined result is obtained relaxing the assumption that f,g E: Lz with respect to t, and using the norm lllu(y)lll = suptf.f-~ lu(t,x,y)i2dx} 1/ 2• In this case the coefficients a,b may depend on t andy. (Received April 10, 1962.)

220 62T-146, T. S, KLOTZ, University of California, Los Angeles 24, California, Some uses of the second conformal structure on strictly convex surfaces,

Sis immersed c 3 in E3 with K > 0, oriented so that H > 0, Form a Riemann surfaceR by using conformal parameters z = x t iy on S, where x,y are bisothermal coordinates making n = p(x,y) •(dx2 + dy2), Then S1 = flidz2 with fli = (E - G- iF)/2 is a quadratic differential on R. Lemma, S1 is holomorphic on R iff K = constant on S. Corollary. Liebmann's Theorem. Corollary. If K =constant, then. S is bisothermal, i,e,, bisothermal lines-of-curvature coordinates exist in the neighborhood of all non-umbilics on S. Lemma, The index of an isolated umbilic U is j = (- 1/2)((1/2-.r)A.arg Xz•Xz) where X describes the immersion of S in E 3. (Note that only first derivatives of X appear above,)

Corollary. Caratheodory's conjecture (that j !!! 1) holds if K E constant near U. Analagous facts have long been known involving ordinary conformal structure on S, isothermal for bisothermal co­ ordinates, H forK, and (L - N - iM)/2 for fli, (See Heinz Hopf, Lectures on differential geometry in the large, mimeographed notes, Stanford University, 1956,) (Received April 13, 1962.)

62T-l47. P.M. TREUENFELS, 2600 Ridgway Road, Minneapolis 40, Minnesota, On the truncation error associated with a class of elliptic equations.

For rectangular domains it is shown that the truncation error associated with elliptic partial differential equations of the type a(x,y)uxx + b(x,y)uyy + c(x,y)ux + d(x,y)uy - e(x,y)u = 0, u prescribed on the boundary, a >0, b > 0, e ;:; 0, tends to zero with the mesh size, to the order (h2 + k2)1-e for any e >- 0, where h and k are the sides of the mesh. (Received April 13, 1962.)

62T-148. H. B. ENDERTON, 18 Prentiss Street, Watertown, Massachusetts. Hierarchies in recursive function theory.

Let <: 1 be a partial well-ordering, For each b in the field D of <: 1 the ordinal lb I can be defined inductively to be the least ordinal r:r such that (\>'y < 1 b) IYI < r:r, Consider only orderings which are neat and reasonable at successors, but allow them to be completely wild at limit points, Given either the jump or the hyperjump operation, a hierarchy of sets Hb indexed by the members of D can be constructed, This is done by following the example of the hyperarithmetic hierarchy, except

that, when lb I is a limit ordinal, let Hb = [2x3y: y < 1 b & x E Hy}· Theorem. If the set of predeces­ sors of each member of D is. uniformly recursive in an earlier H-set, then the Turing degree of Hb depends only on the ordinal lb 1. and not on the choice of b or even on the choice of the particular ordering < 1• This theorem allows construction of a hierarchy of Turing degrees indexed by a seg­ ment of the ordinals, without using a particular ordering of integers. In the case of the jump operation just the hyperarithmetic hierarchy is attained. With the hyper jump a hierarchy of 1:.~ nIT~ degrees is obtained, (Received April 16, 1962,)

62T-149, R. 0. GANDY, The University, Manchester, England, Partial recursive functionals of finite type.

For notations see Kleene, Trans. Amer. Math, Soc, 91 (1959), 1-52, Theorem, There are '))-operators, recursive in E 2 (the representing functional for A.a. 1(EI()(a.1(1C) = 0)) for partial recursive

221 functionals ('p.r.f.') whose arguments are of type not greater than 2, E.g. there is a p.r.f. J7(z,al,a2,p2) such that[z}()}(z,al,a2,E2),al,a2) = 0 whenever (EK)([z} (li',al,a2) = 0). The theorem has a number of obvious but interesting corollaries, E.g. (1) Let F 2 be any type 2 functional in which E 2 is recursive, then (K)(Ea1 E [F 2J) R (l(,a1,F 2) ::> (Ea1 E [F 2])(K) R (I(,~ t a 1(21f3\F l where [f' 2] is the class of functions recursive in F 2. (2) Let Ef be the representing functional for A.a 1(p1)(Etr) · (a1(,;3 1(1f)) = 0); if ?/l(a) = o =(a 1 E (EI]HEp-1 E f.EIJ)A(a 1,/,a):: (Ea1 E [EIJH/ E [E~)B(a\,o 1 ,a) with arithmetic A, B, then jtr f [E iJ. The proof of the theorem depends on the fact that the significant positions on the Kleene tree for a (defined or undefined) computation are well-ordered lexicographically

The theorem can be extended to p.r.f.'s with arguments of type 'liO j, but the ))-operators then require a well-ordering of the type j - 2 as an argument as well as the representing functional for ilaj-1(E,.&j-2)(aj- 1(pj-2) = 0). (Received April 16, 1962.)

62T-150. R. 0. GANDY, The University, Manchester, England, Complete predicates for the analytic hierarchy.

Let t::.,3 be the representing functional for /l.a2(Ea1 E [a2J)(a 2(a1) = 0), and note that E 2, Ey, are recursive in t~. 3 ; (notation as in previous abstract). Theorem l. There is a p.r. function 0 (z) such that if ~z) (a, t:,3 ) is defined for all a, then -r;(z) is an index number for the expression of (z}(a, .6. 3) = 0 in both 2-function-quantifier forms. (But there are predicates of ~ i 1\ rri which are not recursive in l:} .) Let functionals n! and predicates F n+2 be defined as follows: D~ is 'Ia 10; D~+ 1 is the 1 1 1 3 representing functional for A, a ( fa (0)} (.~ ta (t + 1), n;, A. ) is defined) and F n+2 is A.z( {z} (D;, i:::>h is defined). Theorem 2, Fn+2 is a complete predicate of the form TI~+ 2 (n = 0,1, ... ). Thus every analytic predicate is recursive in n; for some n. The proof of Theorem 1 starts from the observation that given both 2-function-quantifier forms for the graph of a 2 we can, usingKleene's XXVI and XXVII, write down the 2-function-quantifier forms for L'l.. 3(a2) = 0, and then proceeds by induction on tzJ(-). Theorem 2 follows fairly directly, using Kleene's definition of 'is defined'. (Received April 16, 1962,)

62T-l51. R. D. ACCOLA, Brown University, Providence 12, Rhode Island. Concerning abelian differentials on parabolic Riemann surfaces. Preliminary report.

Let ra stand for the Hilbert space of square integrable abelian differentials on a parabolic

Riemann surface w of infinite genus. Let ra I be the subset of r a consisting of differentials which are exact in a neighborhood of the ideal boundary. Lemma: If a and f3 E ra' then the function a/;s is a meromorphic function of finite valence, Theorem: If dim ra' ;;:- 2 then there exists a closed surface W' and an analytic map f of finite valence from W into W' such that every differential in r a' is an abelian differential on W' raised via f. Corollary: ra' is a finite dimensional subspace. Proofs: The lemma follows in the case where the ideal boundary of W has a finite number of points from a theorem of M. Heins (Riemann surfaces of infinite genus, Ann, of Math. 55 (1952), Theorem 5.1.) In the general situation it follows from a theorem of H. Royden (unpublished). The theorem follows from another theorem of M. Heins since W admits a nonconstant meromorphic function of finite valence. (On certain meromorphic function of bounded valence, Rev. Math. Pures Appl, II (1957). Theorem 1,) (Received April 16, 1962.)

222 62T-l52, R. D. ACCOLA, Brown University, Providence 12, Rhode Island. On generalized harmonic measures of Riemann surfaces, I, Preliminary report.

Definitions. Let rh(W) be the Hilbert space of square integrable harmonic differentials on a Riemann surface W. Let rhe be the exact forms in rh and let rho = rhe *J.. A harmonic function, u, will be called a generalized harmonic measure (g.h.m,) if the greatest harmonic minorant of u and 1 - u is zero. Theorem. Let u be a g.h.m, on W such that du E r he" Then: (l) The double of the surface[e '!i u ~ 1- sJ is parabolic for all s >0. (2) The set oft, such that the level curves fu = t} contain a noncompact component, has measure zero. (3) For CJ" E: rh' /u=trr converges absolutely to (

lldu n- du II - 0 as n - oo, (7) Let A be a nonbranched component of the set of level curves { u = t 0J. . J'f(s) ./ . Parametnze A by f:[O,a] -A so that s =<.lf(O)du* and a= A du*. Let l(s) be the traJectory through f(s) orthogonal to the level curves of u. Then the set of s, such that the variation of u on l(s) is less than one, has measure zero. Proofs. Part (l) follows by an extention of the methods of Kuramochi (Capacity of subsets of the ideal boundary, Proc. Japan Acad, 32 (1956), lll-116). The remaining parts follow from part (1) via theorems of the author as reported in the abstract Semi-parabolic Riemann surfaces, Notices Amer. Math. Soc. 7 (1960), 650. (Received April 16, 1962,)

62T-153, R. D. ACCOLA, Brown University, Providence 12, Rhode Island, On generalized harmonic measures on Riemann surfaces. II. Preliminary report.

Continue the notation of the preceding abstract, Theorem 2: Suppose r he (W) is finite dimen­ sional. If du E rhe and a ;:; u !!iii b, then there exists a partition of [a,b], a= r0 < t 1 < •.. <. tn = b such that (u - tk_ 1)/(tk - tk_ 1) is a g.h.m. on the surface ftk-l < u < tk}• Corollary. Suppose W can be embedded in W' such that lJW is piecewise analytic and rhe(W') is finite dimensional. If du E: rhe(W) then the set oft, such that the level curves {u = tJ contain a noncompact component, has measure zero. Remark. There exists a surface W with du E r hO(W) such that the level curves of u do not satisfy the condition of the corollary. Theorem 3, Let u be a singular harmonic function on W

such that for some a :> 0, du has finite norm on fO < u < aJ. Then (u - a)/(b - a) is a g.h.m. with finite norm on the surface (a< u < b} for any pair of nonnegative numbers, a,b, (Received April 16, 1962.)

62T-l54. L. R, KING, 311 West Trinity Avenue, Durham, North Carolina. On metric compact­ ifications. Preliminary report,

]. deGroot and R. H. McDowell have raised the following question (Fund. Math 48 (1959/60), 256): Let X be a metric space, dim X = 0, (i.e., X* is a

223 numbers in the unit interval I, = all continuous functions !11: X -X with an extension to I is a counterexample. The second question is answered by the following theorem: Let X be a separable

metric space, dim X g; n, = {llik E C(X,X): k ;: lJ. Then the set of homeomorphisms h E: C(X,I2n+ 1) satisfying dim h(X) :;;: n and h(X) is a <1>-compactification of X contains a dense G-delta set in C(X,I2n+1), for any positive integer n. (Received April 19, 1962.)

62T-155. WITHDRAWN.

62T-156. SEYMOUR KASS and ARTHUR FINE, Illinois Institute of Technology, Technology Center, Chicago 16, Illinois. Indeterminate forms on En.

The following is a generalization of the classical theorems of L 'Hospital and Stolz for indeter­

minate forms of 1-place functions. For simplicity, the theorem is stated for 2-place functions and a straightforward extension to n-place functions is indicated in the proof. Theorem. Let A: (a, b) E' R, where R C E 2 is star-shaped with respect to A. Let f and g be functions defined on R which, for P E: R, are both differentiable on the segment PA, except possibly at A. Moreover, let

(x- x')g1(P) + (y- y')g2(P) ol 0 on every segment of the form QPQ'A, where Q:(x,y}, Q':(x',y') and P are in R (perhaps Q' = A). With the understanding that all limits are taken from within R, there are

two cases. (1) f,g- 0 at (a,b}, (2) JgJ---.. oo at (a,b}. In either case, we have that if lim(a,bPf/Dg = L then lim(a,b//g = L, where D is a derivation on the ring of all (infinitely) differentiable functions,

defined by (Df}(x,y) = (x - a}f 1 (x,y) + (y - b)f2 (x,y). It should be noted that the non vanishing of (x - x')g 1(P) + (y - y')g2 (P) in the hypothesis of the theorem is a simple matter to investigate if one takes into account that the points involved lie on radial sevments. Moreover, Euler's theorem on homogeneous functions makes it possible to develop a calculus for the derivation D which makes the theorem easy to apply. (Received April 19, 1962.)

62T-157. D. F. DAWSON, North Texas State University, Denton, Texas. On certain sequence to sequence transformations which preserve convergence.

The following question is considered. For what subsets U of the set S of all complex sequences is it true that any triangular matrix A (Apq = 0 if q > p) which is convergence preserving over U is convergence preserving? Several subsets of S having this property are given here. The methods of proof generally do not make use of the three well-known conditions of Silverman and Toeplitz which are necessary and sufficient for such a matrix to be convergence preserving. Typical results are as follows. (1) Suppose X is a real sequence which converges to k o1 0. If A is convergence pre­ serving over the set of all real sequences which converge to k slower than X, then A is convergence

224 preserving. (Here "{ap} converges slower than fbpJ" means Lim ap =a, Lim bp = b, and Lim (bp - b)/(ap - a) = 0.) (Z) If A is convergence preserving over the set of all sequences whose terms are elements of the Cantor set, then A is convergence preserving. (Received April Z3, 1962.)

6ZT-158. D. F. DAWSON, North Texas State University, Denton, Texas. Concerning a theorem of Hadamard.

The following theorem essentially combines two results of Hadamard (Acta Math. Z7 (1903), 177-183), and includes theorems of Abel, Dedekind, and others. Theorem. In order that the complex number sequence f piJ be such that the series 2:::: ai and LPi 1\ converge or diverge together for every choice of zai' it is necessary and sufficient that2:_~11 - pi+ 1 /pil < oo for some N. The proof is based on the following property of complex number sequences. If zi---+ z as i---+ oo, then

I:~l(zi- zi+1)/(zi- z)l converges for noN. This result, which includes several theorems of Dini, Stipanic, and others on positive term series, is equivalent to the fact that a product TT(l + ai) con­ verges absolutely if and only if L ai converges absolutely. The theorem above is equivalent to a result of Agnew (Pacific j. Math. 1 (1951), 1-3): Suppose Ci =f -1, i = 1,Z,3,. ..• In order that {cp] be

such that Lap and Lbp converge or diverge together whenever an+/an = (bn+1/bn)(l +en), n = 1,Z,3, •.. , it is necessary and sufficient that2:::Jcpl <. oo. (Received April Z3, 196Z.)

6ZT-159. D. F. DAWSON, North Texas State University, Denton, Texas. Convergence and divergence of continued fractions.

Let g(b) denote the continued fraction 1/fTl/IJZIT~ with pth approximant gp = Cp/Dp. Lane and Wall (Trans. Amer. Math. Soc. 67 (1949), 368-380) studied the convergence and divergence

of g(b) in terms of properties of £hp~ ~ 1 defined as follows: h1 = -b2, hp+ 1 = (1 - hp)bp+ 1bp+2/hp, p = 1,Z,3, ..• , this sequence having interesting properties with respect to the linear fractional trans­ formations which generate the continued fraction equivalent to g(b) which has partial denominators 1. In this paper the following theorem is proved and is used to obtain several theorems on the conver­

gence and divergence of g(b). Theorem. Suppose fzpi ~ 1 is a complex number sequence whose terms are distinct from 0 and 1. Let B denote the set of all continued fractions g(b) such that hzp = zp,

p = 1,Z,3, •••• Then the following two statements are equivalent: (1) If g(b) ~ B, then fDzpS and

L:b2p both converge or both diverge, and (Z) L_ lzp I <. oo. A similar result holds if the roles of even and odd indices are interchanged. (Received April 24, 1962.)

62T-160. D. P. SQUIER, 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 Kau+ /(}n = ou- /(}non an arc F (the interface) in D which terminates on C. Here K is a positive constant. Existence and uniqueness of the solution are proved under suitable regularity conditions on the given data when F is an analytic arc. The method may also be used for the condition Ku + = u- on F. Existence and uniqueness are shown also when there are seyeral nonintersecting analytic

225 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-164. HAO WANG, 33 Oxford Street, Cambridge 38, Massachusetts, A direct characteri- zation of recursive predicates.

There are infinitely many individual variables and predicate letters, In addition, there are the individual constant 0, the equality predicate, the successor function, the truth functions, all with the intended interpretation, Quantifiers are not permitted, An individual variable or 0 is a term, applying the successor operation on a term gives again a term, A predicate followed by a suitable number of terms is a proposition, applying truth functions to propositions again give a proposition. A proposition is a sequential definition if it has a unique model for the occurring predicates, relative to zero, equality, and successor, Theorem 1, A predicate is recursive in the usual sense if and only if it is definable by a sequential definition, By using predicates instead of functions, one can apply here the infinity lemma, so that the Herbrand type of definition in terms of unique models determines the same set of predicates as the Godel type of definition in terms of derivability. Theorem 2, A propo­ sition p is a sequential definition if and only if, for every occurring predicate R and every set of constant arguments a, ... ,b, either Ra ... b or 1 Ra ... b is derivable from p but not both, (Received April 24, 1962,)

62T-165. DONALD GREENSPAN, Mathematics Research Center, United States Army, University of Wisconsin, Madison, Wisconsin. Note on difference approximations with non-negative coefficients for partial differential equations,

On a region R, the linear operator L (u(x,y)] = uxx + 2b(x,y)uxy + c(x,y)uyy• c(x,y) :> 0, is considered, Conditions are sought under which at (X,y) € R, L has a difference approximation Lh [u]:: L:P=o aiui with non-negative coefficients. If b and c are continuous and u E: c2(R); or if b and c are analytic, u E: c 2 (R), and u is a solution of L [u] = 0, then it is proved that a necessary and sufficient condition that Lh exist is that lb(x,y) llli min[1,c(x,y)]. When the latter inequality is valid, Lh is given constructively. Application is made to an extension of a numerical method of Bers for mildly nonlinear elliptic differential equations, (Received April 24, 1962.)

62T-166, J. A, WOLF, The Institute for Advanced Study, Princeton, New Jersey, On locally symmetric Sflaces 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' = EXT X M' where E is a Euclidean space, Tis a torus, M' is a compact simply connected Riemannian symmetric space, and the retraction of N lifts

226 interfaces. The method of proof uses the Schwarz iteration process and conformal mapping to extend the domain of existence from symmetric domains of existence. (Received by April 23, 1962.)

62T-161. Y. C. WONG, University of Hong Kong, Hong Kong. Geometry of n-planes in a real or complex Euclidean space. Preliminary report.

The objects of study are the n-planes (i.e. n-dimensional subspaces) in a real or complex inner-product vector space of dimension > n. Given any two n-planes A and B, we define in a natural way the (real) angle 6(u) between a direction u in A and its orthogonal projection in B. The n extreme values of 6(u) as u varies in A are called the angles between A and B, and the directions in A giving these extreme values the angle directions of A with respect to B. Theorem 1. Semi-perpendicular to both A and B, there are n mutually orthogonal 2-planes (some of which may degenerate into !-planes) having the property that each of them cuts A in an angle direction of A with respect to B and cuts B in an angle direction of B with respect to A. Theorem 2. The n angles between A and B completely determine the relative position of A and B. These results and others will form the basis of a study of the differential geometry of the Grassmann manifolds. As in Memoirs Amer. Math. Soc. 41: Isoclinic n-planes in Euclidean 2n-space, etc., the method of matrices is used. (Received April 23, 1962.)

62T-l62. L. W. EHRLICH, Computation Center, The University of Texas, Austin 12, Texas. On the rate of convergence of a symmetric successive overrelaxation iterative method. Preliminary report.

Sheldon [Math. Tables Aids Com put. 9 (1955), 10 l-112) suggested the use of a (point) symmetric successive overrelaxation (SSOR) iterative method with a semi-iteration acceleration procedure [Varga, J. Soc. Indust, Appl. Math. 5 (1957), 39-46) for the solution of a linear system of algebraic equations. Let R I(«>) denote the average rate of convergence [Young, J. Math. Physics 32 (1954),

243-255] of the SSOR method with semi-iteration and R 2(cv) denote the rate of convergence [Young, Trans. A mer. Math. Soc. 76 (l \154), 92 -lll] of the successive overrelaxation (SOR) method [Young, Trans. A mer. Math. Soc., op. cit.]. It is shown that for a symmetric and positive definite matrix with property A'rr' [Young, Trans. Amer. Math. Soc., op.cit.; Arms, Gates and Zondek, ]. Soc. Indust. Appl. Math. 4 (1956), 220-229] using an C

356-362) one has R 1(cv0);; kR2(wb) for some constant 3-l/2 < k < l, asymptotically as fi ~ 1. Here "1:, is the "best" c.> [Young, Trans. Amer. Math. Soc., op. cit.]. In particular, for the Dirichlet

problem in a square or a subregion of a square, for point SSOR, R 1 (cu0) ~ 2(n-h) 112 for sufficiently small mesh size h. For block SSOR in a square, if cu1 = ((J/2)(j3- (/3 2 - 4) 112), where tS = 4 - 2 cos 'rr'h, then the largest eigenvalue of the method is w 1 - l, and R 1 {4l1) ~ 2(27rh) 112 for small h. (Received April 24, 1962.)

62T-163. RAFAEL ARTZY, 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

227 to a deformation retraction of N 1 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, X.(N) t: 0. Finally, if "X(N) f= 0 then the fundamental group ;r-1 (N) is a finite 2-group. The rest of the paper is devoted to classification of the manifolds N with "X(N) i- 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.)

62T-167. j. A. WOLF, The Institute for Advanced Study, Princeton, New jersey. Riemannian nil manifolds.

A Riemannian nilmanifold is a Riemannian manifold which admits a transitive nilpotent group of isometries. Let N be a connected Riemannian nilmanifold. It is shown that N is isometric to a connected nilpotent Lie group in a left invariant Riemannian metric, that the nilradical of the full group of isometries of N is the only nilpotent connected transitive group of isometries of N, and that the full group of isometries of N is the semidirect product of this nilradical with any isotropy sub­ group. If N is simply connected and D is the group of deck transformations of a Riemannian covering N- M, then these are equivalent: _(1) M is a Riemannian nilmanifold, (2) M is Riemannian homo­ geneous, (3) D consists of Clifford translations (isometries of constant displacement), and (4) D con­ sists of isometries of bounded displacement. Other results are obtained on Riemannian coverings involving Riemannian nilmanifolds. (Received April 25, 1962.)

62T-168. A. A. MULLIN, 108d EERL, University of Illinois, Urbana, Illinois. On differences of algebraic structures.

The theory of mutant sets investigates relations between similarities of algebraic structures and differences of algebraic structures, permitting one to focus on the differences, if one wishes to do so. Although the property of being a mutant set is an algebraic property one can show that, e.g., under a very weak hypothesis an anti-homomorphism between two algebraic systems is bidirectionally mutant preserving. Under the same hypothesis the image of a maximal mutant under an anti-homo­ morphism is a maximal mutant. Those two lemmas provide the tools for establishing nontrivial con­ ditions under which the image of a maximal multiply mutant set under an anti-homomorphism is a maximal multiply mutant set. The case of anti-isomorphism is considered implicitly, but not explicitly. (Received April 25, 1962.)-

62T-169. WITHDRAWN 62T-170. j. G. KEMENY and j. L. SNELL, Dartmouth College, Hanover, New Hampshire. A new potential operator for recurrent Markov chains.

An analogue of the theory of logarithmic potentials for recurrent Markov chains was worked out by the authors IJ. Math. Anal. Appl. October, 1961]. This theory was based on a pair of dual poten­ tial operators C and G. Potential measures of the form :J) = -pC and functions of the form g = - Gf where the charges p and f were of total charge 0 were considered. A new self-dual operator K defined by Kij = Cij + (Goj - Coj) = Gij + (Cw - Gi0)"/"0• where 0 is any fixed state, was introduced.

228 K exists if and only if the chain is normal (C and G both exist), and it serves as a potential operator in place of C and G for both measures and functions, Extend the class of potentials considered to include functions g = - Kf and measures )I= - pK where f and p have finite support. It is shown that an equilibrium potential exists for every finite set and the probabilistic interpretation for its charge is given. The capacity of a set is the value of its equilibrium charge on the set, This capacity satisfies the capacity inequalities. Versions of classical principles, e.g., balayage and domination, are established, (Received April 25, 1962.)

62T-171, C. M. FULTON, University of California, Davis, California. Directed distance geometry.

Letting AB denote the distance from point A to point B, the geometry in question is based on the following two axioms, Axiom (I). AB•CD + BC•AD + CA•BD = 0. Axiom (II). If AB '! 0, AC =AD, BC = BD, then c and D are the same point. Using these axioms, definitions of straight lines and angles are given and the basic trigonometry is developed. Coordinates and their transformations are also dealt with, (Received February 9, 1962.)

62T-172. W. P. HANF, University of California, Berkeley 4, California. On a problem of Erdiis and Tarski.

An a-complete Boolean algebra(){..= (A, =') is said to have an ordered a-basis if there exists

a set B !;;;_ A such that B is simply ordered by :0 and every element of A is a sum of fewer than a

differences of elements of B. Theorem 1, If a is a strong limit number, then every a-complete and a-distributive Boolean algebra of power at most a has an ordered a-basis, Theorem 2, Suppose there

exists an a-complete and a-distributive Boolean algebra tJt of power a such that every a-complete prime ideal of CJt is principal, Then there exists a simply ordered set of power a which has no well ordered or inversely well ordered subset of power a. Combining Theorem 2 with known results (see P. Erdos and A. Tarski, Essays on the foundations of mathematics, Jerusalem, 1961, 50-82), one sees

that conditions P 1' P 2, R, and Q of Erdos-Tarski are all equivalent for inaccessible cardinals, Thus P 1 (i.e, the conclusion of Theorem 2 above) and P 2 apply to a comprehensive class of inaccessible cardinals as well as to all accessible cardinals. Abstract 61 T-240 shows that p 3 does not imply p2 (unless all cardinals a > cu satisfy P 2); the problem remains, does P4 imply p 3 ? (Received April 27, 1962,)

62T-173. M. A. McKIERNAN, University of Waterloo, Waterloo, Ontario. Convergence of

series of iterates.

Let f(x) be analytic about x = 0, f(O) = 0 and f'(O) =a for 0 'JaJ '1, Define f[n](x) recursively by f [OJ(x) = x and f[n+ 1l(x) = f{f[n] (x)J. Form the generating function !ll(x;z) = :L:of [nJ(x) z- (n+1>. The singu.larities in z of j1.1(x;z) are among the points z = an for n = 1,2,... . Let z = 0 be a limit point

of these singularities, and let h(z) = L~Oan(z - (J)n converge for Jz - /31 < I(JJ. Then Theorem. A sufficient condition for these series L~OanL~=O(~)(-p>)n-rf(r](x) to converge uniformly for all -» N nl x in some e neighbourhood of x = 0 is that there exist a Y > 0 such that N 12: n= 1a n(S is bounded

inN, and that the singularities in z ~f j1.1(x;z), with the exception of z = 0, lie within the intersection of

229 the regions lz - tJI < I;SI and the angular sector arg j8 - tr/2 + e ;€! arg z l! arg j8 + 1rj2 - e for some e ~ 0. Corollary. If a is real, 0 < a < l, then for any complex s, the Cayley-Schroder series "'Voo (s)-.:::-n (n)(- l)n-rf [r](x) converges uniformly in some s neighbourhood of x = 0 to a generalized L..n=O n L..r=O r iterate f (s)(x). (Received April 26, 1962.)

62T-l74. GEORG KREISEL, Institut Henri Poincare, Paris, France. New characterizations of the class of hyperarithmetic sets of natural numbers.

Rules (i) and (ii) of F P A(I) are suggested by the following theorems, dealin6 with hyperarithme­ tic sets, functions, resp.: (l) HYP is the smallest class C of sets closed under arithmetic operations and such that, with RandS arithmetic, ;\n[(VX E C) 1\mR(m,n,X)...... ,. (/IX E: C)VmS(m,n,X)]

- {n: (VX E C)J\mR(m,n,X)} E. C. (2) HYP is the smallest class C of functions closed under recursive operations and such that /\m(Va E C) AnR(m,n,a) ~ (Va1 E. C)/\m AnR'(m,n,a1). Direct predicative justification of the rules (i), (ii) is also given. Much of classical analysis, including the theory of Lebesgue measure, is readily developed in~ and ~;(Received January 2, 1962.)

62T-175. GEORG KREISEL, Institut Henri Poincare, Paris, France. Formalizing predicative analysis.

The traditional way of making the notion of predicative definition and proof explicit is by use of ramified hierarchies and theories. These have the defect that the relation between the contents of statements in predicative and impredicative (ordinary) analysis is not at all clear; for a universal (existential) quantifier over sets of level g: is weaker (stronger) than the use of the corresponding

quantifier over arbitrary sets. The following alternative formalization of predicative analysis~

the same language as impredicative analysis 1:;. is proposed. For simplicity, consider only variables n, m •.•. ranging over the set N of natural numbers and a, f3 , •.• ranging over functions in NN, a 1, ;9 1 , ... over NN X N, etc. Besides the usual logical and number-theoretical axioms (with induction applied to arbitrary formulas) consider the following schemes of inference (R,S, any primitive recursive relations): (i) ~ (Elementary predicative analysis). From 1\m {V,A9 ;\ n R(m,n,j3,'>") ~ A;3Vn S(m,n,ft,'>")} infer Val\ m fa(m) = 0 +--> VjJ An R(m,n,fS,r)J. (ii) ~+ (Constructive axiom

of choice). From 1\mVa/\nR(m,n,a,?') infer Va 1/\m/\nR'(m,n,ap?"), where R'(m,n,a 1,?") is obtained from R(m,n,a,?') by replacing ,\ta(t) by Jl.ta 1(m,t). It is shown that .!IE.A f; EPA+~ A,. (Received January 2, 1962.)

62T-l76. M. A. AL -BASSAM, Texas Technological College, P .O.Box 4346 Tech. Station, Lubbock, Texas. On Ta-functions.

Definition. f(x) E. Ta <=> (S): f(x) = L:oan(x - a)na, a > 0, on L: 0 ~ a ~ x ;€! b """I' where p is the radius of convergence of (S). It is shown that: (l) If g(x) = '2:~obn(x - a)na such that lim supn-+oolbnll/na = 1/r, or limD-+ 00 1bn+rfbnl = 1/ra, r > 0, then g(x) ETa. (2) If f1 (x), f 2 (x) E Ta, then (f l + f z) e Ta, and (f t f 2) E Ta. (3) If f E Ta, then (i) f is continuous at X = a, and at every interior point of L. (ii) f~(x) -the nth derivative off with respect to (x - a)a-, and

230 ~'(x)](na.-l) = I:l-a.nf', (for details see the author's work: Ann. Sci. Norm. Sup. Pisa, XV, Serie III (1961), 1-2-!Q, are continuous on L for every positive integer n. (iii) an= f~a)l r

62T-177. M.A. AL-BASSAM, Texas Technological College, P. 0. Box 4346 Tech. Station, Lubbock, Texas. Some existence theorems of differential equations of noninteger order.

Theorem A. If p(x) € ro. on L, and a is an ordinary point, then there exists a solution y = Y(x) E' To. which satisfies the differential equation of order a. 1: y(a.) + p(x)y = 0, and is such that Y = Y(x;ao). Example. By assuming y = ~Oan(x - a)na.+c, c > 0, n < a.< n + 1, it is found that the equation y(a.) +my= 0, is satisfied by y = ::;E:~=lCkyk' where Ck are arbitrary constants and yk = E~k-l)(- m; x). A similar result was obtained by J. H. Barrett (Canad. J. Math. (1954) 535-538)) by solving the equivalent integral equation by the method of successive substitution. Theorem B. If P(x) and R(x) E C on the interval (A_, JI), then the differential equation y(a.) + P(x)y = R(x), n < a.< n + 1, admits of a unique solution which together with y(a.-k)(x), (k = 1,2, ••• ,n), is continuous on (:>t.,p) and satisfies the conditions y(a.-k)(a) = 'lk• where a is a point of (.\,JI). Clearly for a.= n, this represents the well-known existence theorem of the nth order equation of this type. The unique solution satisfying the equation of the above mentioned example and these initial conditions is found to be the one in which Ck = >[k. (Received April 18, 1962.)

ERRATA- Volume 9

D. C. RUNG. Results on cluster sets of normalfunctions defined in the unit disk. Page 43, Abstract 62T-33.

Line 6. For "~ 1 and /J. 2" read " A1 and .6. 2 with .6.2 :J .A1". i9 i9 Line 6. For "R AI (f,e )" read "R t::, 2(f,e )". 9 9 9 Line 7. For "Rt::,(f,ei ) -TI t::, 2(f,ei ) is an open set" read "R ll.1(f,ei ) 9 9 -ITl.. 2(f,ei ) is contained in R~ 2 (f,ei )".

T. G. McLAUGHLIN. Some remarks about productive and contraproductive cen­ ters. Page 148, Abstract 62T-80. This abstract was published in error.

WILLIAM CRAIG and WILLIAM HANF. On relative characterizability in a Ian­

~· Page 152, Abstract 62T-95. Page 153, Line 3. For "hasno nontrivial automorphism" read "is not isomor­ phic to any of its proper subsystems".

231 P. S. Alexandroff Stability Die topologischen Dualitats~atze, Band II Mathematische Forschungsberichte, Bd. XIV of Motion Obersetzung aus dem Russischen 148 Seiten, 3 Abbildungen, Gr.8°, broschiert, APPLICATIONS OF LYAPUNOV'S etwa OM 20,- SECOND METHOD TO L. Berg DIFFERENTIAL SYSTEMS AND Einfiihrung in die Operatorenrechnung EQUATIONS WITH DELAY Mathematik ffir Naturwissenschaft und Technik, Bd. 6 N. N. Krasovskii Etwa 250 Seiten, 35 Abbildungen, Gr.so, Translated and Edited by Kunstleder, etwa OM 19,- ]. L. Brenner I. M. Gelfand - G. E. Schilow This rigorous analysis extends the fun­ Verallgemeinerte Funktionen !Distributionenl damental work of Lyapunov and applies his direct method to a variety of prob­ Band II: Lineare topologische Raume, Raume von lems. Lyapunov's theorems on asymp­ Grundfunktionen und verallgemeinerte totic stability and instability are gen­ Funktionen eralized; the problems of mth-order Obersetzung aus dem Russischen stability and persistent disturbances Etwa 300 Seiten, 5 Abbildungen, Gr.so, and related questions are explored. Leinen, etwa DM 30, - July. About $6.00 VEB DEUTSCHER VERLAG Order from your bookstore, please DER WISSENSCHAFTEN • BERLIN W 8 STANFORD UNIVERSITY PRESS

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232 Volume 11 CURVATURE AND HOMOLOGY By SAMUEL I. GOLDBERG University of Illinois, Urbana NEW VOLUMES June 1962, 316 pp., $8.50 OF In recent years, considerable attention has been given to the theory of harmonic integrals and its ap­ PURE and plications to global differential geometry. During the past two decades there have been many fruitful in­ vestigations on the relationship between the curva­ APPLIED ture properties of a Riemannian manifold and its homology structure. MATHEMATICS This book is the first comprehensive treatment of this important branch of modern mathematics. The A SERIES work is self-contained, requiring no specialized OF MONOGRAPHS knowledge of either topology or differential geometry. Special features AND TEXTBOOKS include the Kodaira vanishing theorems, 8-pinched manifolds, and a proof along modern lines of the existence theorems of de Rham. Included in the coverage are several results of recent Edited by studies by the author in cooperation with S. Koba­ PAUL A. SMITH yashi on the conformal transformation group of a and homogeneous Riemannian manifold. SAMUEL ElLENBERG Contents: Columbia University Riemannian Manifolds. Topology of Differentiable Mani­ folds. Curvature and Homology of Riemannian Manifolds. Compact lie Groups. Complex Manifolds. Curvature and Homology of Kaehler Manifolds. Groups of Transforma­ tions of Kaehler and Almost Kaehler Manifolds. Appen­ dix A: de Rham's Theorems. Appendix B: The Cup Prod­ uct. Appendix C: The Hodge Existence Theorem. Appendix D: Partition of Unity. References. Author Index-Subject Index.

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AN EQUAL OPPORTUNITY EMPLOYER

237 MATHEMATICIAN TECH NEWS for joint appointment as a faculty member of the Mathematics Depart­ for Scientists, Mathematicians ment and as a member of the staff of Operations Evaluation Group the University's Computer Labora­ tory. Recent Doctorate Degree in Mathematics, with experience in the May 1962 certainly stood out as "Operations Research formulation and development of Month"-in fact, if not by official proclamation. May saw the lOth anniversary meeting in Washington of the mathematical models of scientific and Operations Research Society of America (ORSA} as well engineering problems. as OEG's 20th Anniver- sary Conference on Op- 20 ~rations. Research, also m Washmgton. T H OEG, created in May For further information write to: 1942, is the oldest con- tinuing military opera- tions research organiza- y E A R Lee Brogan, Computer Laboratory tion in the country. The University of Denver 20th Anniversary Con- ference-of interna- 0 E ~ Denver 10, Colorado tional scope-reviewed ~ applications of OR in

NAT~, in industry a~d tC4'4'tC4'tCtC 1 public health, and m fiscal planning for defense, as well as considering OR education. OEG acts as civilian scientific advisor to the Chief of Naval Operations and the Commandant, U. S. Marine Corps, functioning in diverse problem areas. Typical MATHEMATICS problems OEG has been called upon to solve include: Lockheed Missiles and Space • What Navy vehicles would profit most from nuclear Company is seeking a talented propulsion? professional mathematician to • Can a workable radar theory be formulated at the complement an outstanding present state of the art? small group studying satellite orbits and interplanetary trajec­ • Set up exercise conditions to test the effectiveness tories. The position will be in of fleet air defense. Lockheed's Research Labora­ • What are the requirements for fire support of future tory in Palo Alto, California. amphibious operations? FOR INFORMATION, please This is complex and essential work. It requires scien­ write to Mr. R. C. Birdsall, tists, preferably with advanced degrees, who can com­ Employment Manager, Dept. bine analytical talent with a certain amount of enterprise. M-34B, 599 N. Mathilda Ave., If you are a mathematician, physicist, engineer or econ­ Sunnyvale, California. omist, and the potential inherent in OEG's program sounds rewarding, please send your resume to the LOCKHEED Director, Dr. Jacinto Steinhardt. MISSILES & SPACE COMPANY A GROUP DIVISION OF LOCKHEED AIRCRAFT CORPORATION OECS SUNNYVALE, PALO ALTO, VAN NUYS, SANTA CRUZ, SANTA MARIA, CALIFORNIA OPERATIONS EVALUATION GROUP CAPE CANAVERAL, FLORIDA • HAWAII Arlington Towers, Arlington 9, Virginia AN EQUAL OPPORTUNITY EMPLOYER An equal opportunity employer

238 AN IMPORTANT ANNOUNCEMENT TO Operations Research Scientists

The world's largest and most comprehensive Land Combat War Gaming Model is being developed by STAG ... The U. S. Army's Strategy and Tactioa Analysis Group

The Department of the Army recently ing a War Gaming Symposium at its established STAG to support Army Bethesda, Maryland headquarters in operational planning and evaluation September, 1962. activities by war gaming and other STAG headquarters is an eight-story scientific techniques, with the aid of building newly built and especially large scale computer systems. equipped to provide the environment for STAG has been assigned War Gaming. It is ideally located in the heart of Bethesda, a select residential these specific tasks: area just outside Washington, D. C. 1. Develop a Land Combat War Already functioning are IBM 1401 and Gaming Model with which to test Army 7090 systems and nearing completion is Plans and potential. This model will be a two-story Command Control Pent­ of sufficient size, scope, and flexibility to house featuring computer-generated vis­ permit war gaming of any or all phases ual display equipment, being developed of combat and support operations. It will especially for STAG to provide imme­ be capable of handling strategic or tac­ diate information access to any part of a tical operations ranging from a single problem. The Red and Blue forces have battle group or battalion up to and in­ monitoring equipment that allows in­ cluding the complete Theatre Army­ stant two-way visual communication. and supporting Sea, Air and Space STAG is accepting applications for forces. Programmed to be run on the civilian members of numerous military­ IBM 7090, the Land Combat War Gam­ civilian-computer teams being set up ing Model will be the most ambitious within STAG to provide the military, and far-reaching undertaking in the scientific and computer integration science of War Gaming. necessary for solving those problems 2. Conduct studies of other Army which lend themselves to the gaming or problems that will respond to War simulation type technique. Immediate Gaming techniques, and maintain liaison openings exist for Operations Research with other Army and Army-supported Analysts, Mathematicians and Mathe­ groups. One such program, nearing com­ matical Programmers. pletion at STAG is a computer-pro­ grammed mathematical model of a Field If you are interested in participating in the Army Ballistic Missile Defense System most complex and comprehensive War Gam­ (FABMDS). ing program ever undertaken, ple!'lse contact: 3. Provide Army participation in Chief, Army General Staff Civilian Personnel joint exercises and technological advice and assistance to other war games Office-STAG (M), Room 3-C-674, The groups. For example, STAG is sponsor- Pentagon, Washington 25, D. C.

Strategy and Tactics Analysis Group

239 ...J,.,.<••~ru. Sometime in 1963,000 distribution and direction of interplanetary dust in the (NASA's Orbiting Observatory) will be vicinity of earth. Magnetic fields, their intensity, direction launched into an elliptical the earth. It will and variation near earth and in space. Atmospheric gather, process and the physics of near- measurements, to study the pressure, temperature and earth and cislunar space. some of the studies composition of earth and cislunar space. Ultraviolet OGO may undertake in this flight: Energetic scattering, from hydrogen in space. Gegenschein photom­ particles, with nine separate on the flux and etry, to study sunlight scattered by interplanetary matter. characteristics of these OGO will be launched into a wide range of orbits and may and plasma studies). carry as many as 50 different experiments on each of its through measurements of ambient ~"'"A"n~"'"' missions. This Orbiting Geophysical Observatory will be sible from earth. Micrometeoroids, to deta);:mi.ne one of the most versatile earth satellites man has ever built.

Energetic Particles M icrometeoroids Ultraviolet Scattering

OGO: its challenge. Today OGO demands such as Aerodynamics; Sna\\f,craftHeat Transfer; Analog advanced techniques in spacecraft design and develop­ and Digital Computers; Mathematics; Electronic ment to meet its need for flexibility. It is a challenging Ground Systems; Power Instrumentation Sys- responsibility to STL engineers, scientists and supporting tems; Propellant Controls; System personnel, who design it, fabricate it, integrate it, and test Analysis; Thermal Analysis. For it. This versatile spacecraft will be manufactured at STL's Southern California or Cape positions, write vast Space Technology Center where expanding space Dr. R. C. Potter, One Space Pa Department -0, projects (000, Vela Hotel and other programs) create Redondo Beach, California, or P. Box 4277, Patrick immediate openings for engineers and scientists in fields AFB, Florida. STL is an equal' employer.

VLF Radio Propagation'-...._.. · Magnetic Fields~-

A. SPACE TECHNOLOGY LABORATORIES, INC. v,; a subsidiary of Thompson Ramo Wooldridge Inc. Los Angeles • Vandenberg AFB • Norton AFB, San Bernardino • Cape Canaveral • Washington, D.C. • Boston • Huntsville • Dayton.

Space Technology Laboratories, Inc. Job Order STL-51, Ad No. 433 Trade Magazines I Page, B&W Fuller & Smith & Ross, Inc. 3325 Wilshire Blvd., Los Angeles 5 240 DISCOVERING SOLIDS A new solid ~eometry film strip package

CI!NCO I!DUCATIONAL PILM8 Cenco announces a complete solid geometry unit divided into four 35 mm film strips in full color. Perfect for introduction as well as review, two hours and twenty minutes of informative and purposeful narration are included on two 12 inch, 33~ rpm, long-playing records, plus a specially designed Teacher's Guide for each strip. Produced in collaboration with Dr. E. H. C. Hildbrandt of Northwestern University. The four film strips in the "Discovering Solids" series include: • Surface Areas of Solids I: Cubes, Prisms and Pyramids (66 frames) • Surface Areas of Solids II: Cylinders, Cones and Spheres (74 frames) • Volumes of Cubes, Prisms and Cylinders (102 frames) • Volumes of Pyramids, Cones and Spheres (96 frames) The complete package; two 12 inch long-play­ ing records, the four film strips and four Teach­ er's Guides comes in a handsome storage carton, only $27.00. Use the handy order form below.

CENCO EDUCATIONAL FILMS a division of Cenco Instruments Corp, 1700 Irving Park Road • Chicago 13, Illinois

Mountainside, N.J.. Toronlu 4, Ontario, los Angeles 22, Calif., 237 Sheffield St. 146 Kendal Ave. 6446 Telegraph Rd. [E ~~~e~~~;~i~:t~r\ St. n3~~~~~3~r8Sf:· ~~~~ ~~~~~~~~~- •• Birmingham 4, Ala., Edmonton, Alberta, Tulsa 12, Okla., 3232 Eleventh Ave., N. 14920 Stony Plain Rd. 6901 East 12th St. ~ Montreal14, Quebec, onawa, Ontario, Houston 11, Texas, 7275 St. Urbain St. !30 Sparks St. 6610 Stillwell

GENTLEMEN: PLEASE SEND: 0 No. 59506, ~or.rr!'::~;~~::i~::r~n:d ~~~~::~:;ckaoe, $27.00 0 No. 59505·2(2~~~~s 5~~P~; ~~~~~=~= :~~da:5 o:n~~~i~es.;~,:~5.00 0 No. 59505-1(2 ~~~·.~~\!.•~~~~~::~";.~~dO:ssa~l~d1s,;~:~~d~O D Information on Cenco's 16 mm films

NAME TITLE

INSTITUTION

ADDRESS

CITY STATE

241 CUSHING-MALLOY, INC. Are you interested in University teach­ 1350 N. Main St. ing? If you have a doctor's degree, and P. 0. Box ll87 experience in numerical analysis and Ann Arbor, Michigan machine computation, we need you. LITHOPRINTERS Write: Printers of the NOTICES Professor Roy F. Reeves, Director

The Ohio State University Known for Numerical Computation Laboratory QUALITY- ECONOMY 1314 Kinnear Road SERVICE Columbus 12, Ohio Let us quote on your next printing

I I__.. I 500 I Over 500 different characters on one typewriter. Fits any standard typewriter-insert symbols as you type. Call your local oHice equipment dealer for additional information I or write to address below.

3 ~ 1@:1 ~ • 10 £ 2Ti + Br2 + ZH20 ..... 2Br- + TiO* + 4H+ Sr90 , 0 /\ A oc c:se M = zoo • h ... f V x = 'rllJ ~ Cos crt I I 0 C-CH2 -CH3 TYPJT® of course a product of ... mechanical enterprises, Inc. 3127 Colvin Street, Alexandria 11, VIrginia

242 INDEX TO ADVERTISERS

Name Page Name Page

Academic Press, Inc. 233 Mechanical Enterprises, Inc ...... •...... 242

American Mathematical Society 244 National Register of Scientific and Technical Personnel ...... 234 Cenco Instruments Corporation 241 The Ohio State University ...... • • . . . 242 Cushing-Malloy, Inc ...... 242 Operations Evaluation Group . . . • ...... 238 University of Denver ...... • ...... 238 Pergamon Press, Inc ...... •.•..•..... 235 Deutscher Verlag der Wissenschaften ...... 232 Space Technology Laboratories, Inc .....•. 240 Gordon and Breach, Inc...... •...... 236 Sperry Rand Research Center • . . . . • . . . • . . . 235 Holden-Day, Inc. 232 Stanford University Press ...... •...... 232 The Johns Hopkins University- The Applied Physics Laboratory • • . . • ...... 237 U. S. Army- STAG 239

Lockheed Missiles & Space Company • • ...... 238 University of Wisconsin Press • ...... 236

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. ------(Jetach-on1h1s11nefThose who wish to stay at a hotel or motel should make their reservations ______directly with the hotel or motel. _

American Mathematical Society Housing Reservation Form - Vancouver Me~ting

Name ______P~l'e-a-se--p-r 7in~t-(~la-s~t7)------~~~ir-s~t7) ______Address ______Institution ------Arrival Date ______Time ____ Departure Date ______Time _____

Housing available from 2:00p.m. Saturday, August 25th, to 2:00p.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 -----

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

Return Requested

ANNOUNCING A NEW PUBLICATION Proceedings of Symposia in Applied Mathematics, Volume 13 HYDRODYNAMIC INSTABILITY Edited by G. Birkhoff, R. Bellman and C. C. Lin Proceedings of the Symposium in Applied Mathematics of the American Mathematical Society, held in New York City, April 14·15, 1960. This volume deals with the following loosely related subjects: hydro­ dynamic instability, the statistical theory of turbulence, water waves, waves in a medium with random properties, and the theory of guidance and control. These subjects all use similar mathematical methods and ideas, and it seemed impossible to give an adequate picture of contemporary views about hydro· dynamic instability without touching on them all. The fact that such a diver· sHied group of problems are exhibited in hydrodynamics alone can only make one admire the inexhaustibility of fascinating problems that occur in nature. Yet we have hardly tapped on the range of problems that arise in astrophysics. 319 pages $7.90 25% discount to members

AMERICAN MATHEMATICAL SOCIETY 190 Hope Street, Providence 6, Rhode Island