NATIONAL BUREAU OF STANDARDS REPORT

29 IS

PROJECTS and PUBLICATIONS

of til

national applied mathematics laboratories

A Quarterly Report

July through September 1953

FOR OFFICIAL USE

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Aatin, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards is suggested in the following listing of

the divisions and sections engaged in technical work. In general, each section is engaged in specialized research, development, and engineering in the field indicated by its title. A brief description

of the activities, and of the resultant reports and publications, appears on the inside of the Rack

cover of this report.

Electricity. Resistance Measurements, Inductance and Capacitance. Electrical Instruments. Magnetic Measurements. Applied Electricity. Electrochemistry.

Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic

Technology. Length. Gage.

Heat and Power. Temperature Measurements. Thermodynamics. Cryogenics. Engines and

Lubrication. Engine Fuels. Cryogenic Engineering.

Atomic and Radiation Physics, Spectroscopy. Radiometry. Mass Spectrometry. Solid State

Physics. Electron Physics. Atomic Physics. Neutron Measurements. Infrared Spectroscopy.

Nuclear Physics. Radioactivity. X-Rays. Betatron. Nucleonic Instrumentation. Radiological Equipment. Atomic Energy Commission Instruments Branch.

Chemistry. Organic Coatings. Surface Cheipistry. Organic Chemistry. Analytical Chemistry.

Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry. Spectrochemistry. Pure Substances.

Mechanics. Sound. Mechanical Instruments. Aerodynamics. Engineering Mechanics. Hydraulics. Mass. Capacity, Density, and Fluid Meters. Organic and Fibrous Materials. Rubber. Textiles. Paper. Leather. Testing and Specifications. Polymer Structure. Organic Plastics. Dental Research.

Metallurgy. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy. Corrosion.

Mineral Products. Porcelain and Pottery. Glass. Refractories. Enameled Metals. Concreting Materials. Constitution and Microstructure. Chemistry of Mineral Products.

Building Technology. Structural Engineering. Fire Protection. Heating and Air Conditioning. Floor, Roof, and Wall Coverings. Codes and Specifications.

Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Machine

Development.

Electronics. Engineering Electronics. Electron Tubes. Electronic Computers. Electronic

Instrumentation.

Radio Propagation. Upper Atmosphere Research. Ionospheric Research. Regular Propagation Services. Frequency Utilization Research. Tropospheric Propagation Research. High

Frequency Standards. Microwave Standards.

Ordnance Development. These three divisions are engaged in a broad program of research

Electromechanical Ordnance, and development in advanced ordnance. Activities include

Ordnance Electronics. basic and applied research, engineering, pilot production, field testing, and evaluation of a wide variety of ordnance materiel. Special skills and facilities of other

NBS divisions also contribute to this program. The activity is sponsored by the Department of

Defense.

Missile Development. Missile research and development: engineering, dynamics, intelligence,

instrumentation, evaluation. Combustion in jet engines. These activities are sponsored by the

Department of Defense. • Office of Basic Instrumentation • Office of Weights and Measures.
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT

NAML NBS REPORT

2 ! 1 6

PROJECTS mi PUBLICATIONS

if the

NATIONAL APPLIED MATHEMATICS LABORATORIES

July through September 1953

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication,

unless permlssloi 25.D.C, Such p

cal ly prepared If

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

or In par t, Is prohibited

Standards, Washington

report has been spoclflreport for Its own use,
NATIONAL APPLIED MATHEMATICS LABORATORIES

July 1 through September 30, 1953 ADMINISTRATIVE OFFICE

Franz L. Alt, Ph.D., Acting Chief Edward W. Cannon, Ph.D., Assistant Chief

Olga Ta us sky-Todd , Ph.D., Mathematics Consultant

W. J. Youden, Ph.D., Mathematics Consultant

Eugene Lukacs, Ph.D., Mathematics Consultant

Ida Rhodes, M.A., Mathematician

Myrtle R. Kellington, M.A., Technical Aid

Luis 0. Rodriquez, M.A., Chief Clerk

John B. Tallerico, B.C.S., Assistant Chief Clerk

Margaret W. Dawson, M.A., Secretary

Louis J. Meyerle, Jr., Secretary

Esther L. Turner, Secretary

INSTITUTE FOR NUMERICAL ANALYSIS

(Los Angeles, California) Derrick H. Lehmer, Ph.D.. Director of Research Charles B. Tompkins, Ph.D Acting Director of Research

Magnus R. Hestenes, Ph .D. . .Assistant Director and UCLA Liaison Officer Harry D. Huskey, Ph.D Assistant Director for Mathematical Services

Gertrude Blanch, Ph .D . . . As sis ta nt to the Director (Numerical Analysis) Albert S. Cahn, Jr., M.S ... .Assistant to the Director (Administration)

Research Staff Forman S. Acton, D.Sc Mathematician

A. Adrian Albert, Ph.D UCLA Research Associate (Mathematics) Alfredo Banos, Jr., Ph.D UCLA Research Associate (Mathematics)

Richard H. Bruck, Ph.D Mathematician (Consultant) Kenneth A. Bush, Ph.D Ma thema tic ia n (Consultant) Randolph Church, Ph.D UCLA Research Associate (Mathematics)

Robert P. Dilworth, Ph.D Mathematician (Consultant) George E. Forsythe, Ph.D.. Mathematician

Richard A. Good, Ph.D Mathematician (Consultant)

John W. Green, Ph.D ..UCLA Research Associate (Mathematics) Dick Wick Hall, Ph.D Mathematician (Consultant)

Marshall Hall, Jr., Ph.D UCLA Research Associate (Mathematics) Erwin Kleinfeld, Ph.D Mathematician (Consultant) Harold W. Kuhn, Ph.D .Mathematician (Consultant) Cornelius Lanczos, Ph.D Mathematician

Jacob Marschak, Ph.D UCLA Research Associate (Mathematics) Theodore S. Motzkin, Ph.D UCLA Research Associate (Mathematics)

Lowell J. Paige, Ph.D Mathematician

Gordon Pall, Ph.D UCLA Research Associate (Mathematics) William A. Pierce, Ph.D Mathematician (Consultant) David A. Saxon, Ph.D Jonathan D. Swift, Ph.D Daniel Teichroew, Ph.D Joseph L. Walsh, Ph.D Wolfgang R. Wasow, Ph.D Graduate Fellows

John W. Addison, Jr., M.S. Walter I. Futterman, M.A. Eugene Levin, M.A. Genovevo C. Lopez, B.A. (UCLA) Stanley W. Mayer, M.A. Edwin Mookini, M.A. (UCLA) Mathematical Services Unit Arnold D. Hestenes, Ph.D Mathematician Frederick H. Hollander, M.A.; Mathematician

Marvin Howard, M.S. Mathematician Roselyn S. Lipkis, B.A , . Ma thema tic ian Robert R. Reynolds, M.S Mathematician

Thomas H. Southard, Ph.D Mathematician

Everett C. Yowell, Ph.D Mathematician Computing Staff

Physic is

t

UCLA Research Associate (Mathematics) Statistician

... o ...... Ma thema tic ia n (Consul ta nt)

« Ma thema tic ia n Mervin Muller, M.A. Lloyd Philipson, M.A. John Selfridge, B.S. Marian I. Walter, B.A. (UCLA) Roger D. Woods, M.A.

Patricia B. Bremer, B.A. Patricia L. Childress, B.A. Leola Cutler, B.A. Lillian Forthal, B.A. Benjamin F. Handy, Jr., M.S. Ruth B. Horgan, B.A. William Justice, Jr., M.A. William P. Keating, M.A. Merry M. Kruse, B.S. Thomas D. Lakin, B.S. Shirley L. Marks, M.A.

Frank B. Meek, M.A. Helen V. Meek, B.S. Owen R. Mock, B.A. Elmer E. Osborne, Ph.D. Nan N. Reynolds, M.S. Laverne E. Rickard, M.A. Albert H. Rosenthal, B.A. Louise M. Straus, M.A. Gladys P. Walker, B.S. Marian E. Williams

Lindley S. Wilson, M.A. Machine Engineering and Development Unit Ragnar Thorensen, Ph.D Electronic Scientist

Biagio F. Ambrosio, B.S Electronic Engineer William Arsena ult , M.S Electronic Engineer

Electronic Engineer

S Electronic Engineer

S.. Electronic Engineer

S Electronic Engineer

Electronic Engineer Electronic Technician

Howard K. Marks Edward D. Martinolich

John Francis, B.S

Louis F. Justice, Norman F. Loretz, Eugene M. Rector, Robert R. Schmidt

James W. Walsh . .

.

Electronic Laboratory Staff

Blanche C. Eidem

Gaylord W. Jones Charles H. Keller (UCLA) Library Section

F. Patricia K. Peed .Librarian Estelle H. Strauss Library Assistant

Administrative Unit

M. Evelyn Michaud ^ Administrative Officer

Mildred Vukmanic Administrative Assistant

Mildred B. Webb Administrative Clerk

Reve Vineyard Administrative Clerk

Miller A. Holt Property and Stock Control Clerk

Leo Moskowitz Procurement Clerk

Louis V. Adelizzi ..Miscellaneous Office Machine Operator Secretarial Staff

Mildred L. Dalton Mildred I. Martinolich

Vendla H. Gordanier Lucy A. Moore Dorothy M. Hibbard Anne B. Oates

Elsie L. Hus man On leave of absence.

COMPUTATION LABORATORY

John Todd, B.S Milton Abramowitz, Ph.D Ruth K. Anderson, M.A Henry A. Antosiewicz,

Philip Davis, Ph.D William H. Durfee,

Ky Fan, Ph.D Andre N. Gleyzal,

Peter Henrici, Alan J. Hoffman,

Murray Ma n n o s

,

Morris Newman, Fritz Oberhe ttinger

,

Philip Rabinowitz,

Daniel Sokolowsky,

Irene A. Stegun, Charles J. Swift,

Joseph H. Wegstein, Norman A. Wiegmann, Helmut Wielandt, Ph.D Computing Staff Hans 0. Bremer, B.A. Robert B. Bryce, B.A. William F. Cahill, M.S. Ruth E. Capuano

James G. Carroll

Kathryn Christoph, B.S. Anne R. Cock, B.A. Lionel A. Difford, M.A. Mary M. Dunlap, B.S. Anne M. Futterman, B.A. Leon Gainen, B.A. Billie R. Gill, M.S. Elizabeth F. Godefroy

Karl Goldberg, M.A. Allen A. Goldstein, M.A. Stanley D. Grant, Jr. William G. Hall, B.S. Genevie E. Hawkins, B.S. John G. Hershberger, M.S. Renee B. Jasper, B.A. Dorothea H. Jirauch, M.A. Lambert S. Joel, B.A. Secretarial Staff Gloria S. Ford, B.S. Lillian Sloane American University

Chief

.Assistant Chief Mathematician AU ++ Research Associate

Mathematic ian

Ma t h e ma t ic ia n AU** Research Associate

Ma thema t ic ian Research Associate

Ma thema tician

Ma thema t ic ia n Ma thema tician

AU* + Research Associate

Ma thema tician

Ma thema t ic ian Mathematic ian Physic is

t

Physic ist Mathematician

Research Associate

Marlee E. Johnson, B.S. Terry W. Ledley,M.A. Norman Levine, M.S. David S.Liepman

Ethel C.Marden, B.A. Theodore R. Mikell, M.S. Kermit C. Nelson

Peter J. O'Hara, B.S. Mary Orr Maxine L. Paulsen, B.S. Annabelle W. Phelps, M.A. Solomon L. Pollack, M.S.

B. Stanley Prusch Donald I. Rubin, M.S. Milton Stein Otto T. Steiner, B.A. Frank J. Stockmal, B.A. Gretchen Stolle, M.A. Sally Tsingou, B.S. Bernard Urban, B.S. Bertha H. Walter

Ruth Zucker, B.A. Marjorie E. Young

Ph.D

Phii) ::::::::::::::

Ph.D Ph.D AU**

Ph ,D Ph.D

Ph.D

Ph.D Ph.D

M.A

M.A. Ph.D

M.S Ph.D AU+*

STATISTICAL ENGINEERING LABORATORY Churchill Eisenhart, Ph.D... Joseph M. Cameron, M.S Lola S. Deming, M.A Willard H. Clatworthy, Ph.D. William S. Connor, Jr., Ph.D Edgar P. King, D.Sc

Julius Lieblein, Ph.D

I. Richard Savage, M.S Marvin Zelen, M.A Marion T. Carson

Thomas A. Gorman....

Yvette B. Cocozzella

Lela J. Hamilton Shirley C. Schleter, B.A....

Chief Assistant Chief

Technical Aid Mathematical Statistician

Ma thema tician

Ma thema tic ian

Ma thema tician

Ma thema tic ian Mathematical Statistician

.Computer Student Assistant (Mathematics) Secretary

Secretary

Secretary

MACH INF DEVELOPMENT LABORATORY Edward W. Cannon, Ph.D J Chief

Edith N. Reese, B.A Mathematician
APPLIED MATHEMATICS ADVISOR! COUNCIL

The activities of the National Applied Mathematics Laboratories

are carried on with the advice of a policy-formulating group known as

the Applied Mathematics Advisory Council. The membership of the council

is as follows.

AGENCY REPRESENTATIVES AND THEIR ALTERNATES

National Science Foundation Raymond J. Seeger (Chairman , 1963)

Department of the Army Major Ivan R. Hershner, Jr.

T. J- Killian, Alternate

Department of the Navy Mina Rees

C. V. L. Smith, Alternate

Department of the Air Force Colonel Oliver Haywood, Jr. Marshall K. Wood, Alternate

Department of Agriculture Earl E. Houseman

Glenn L. Burrows, Alternate

Bureau of the Census Morris H. Ransen

James L. McPherson, Alternate

Weather Bureau Harry Wesler

R. A. Allen, Alternate

Atomic Energy Commission Captain John T. Hayward, USN

Commander Charles H. MacDonald, USN, Alter not

National Advisory Committee

for Aeronautics

Carl Kaplan

I. E. Garrick, Alternate

Bureau of the Budget Ezra Glaser, Observer

James W. Clark, Alternate Observer

SCIENTIFIC ADVISERS

David Blackwell

Howard University

J. J. Stoker

Institute for Mathematics and Mechanics

New York University

Philip M. Morse Massachusetts Institute

A. H. Taub

University of Illinois

of Technology

Contents

Page

Index. vi

Status of Projects as of September 30, 1953. . 1 Institute for Numerical Analysis 1 (NBS Section 11.1)

Computation Laboratory ........... 26

(NBS Section 11.2)

Statistical Engineering Laboratory 5^+ (NBS Section 1 1 .3)

Machine Development Laboratory . . . .... 59

(NBS Section 11 .4)

Lectures and Symposia a 62

Publication Activities . 66
Index of Active Research and Development Projects

Note: This index is not intended to cover the numerous special problem

solutions, statistical analyses, and other ad hoc services to Government

agencies, which form an important part of the work of the National Applied

Mathematics Laboratories. These services are, however, fully represented

in the body of the report.

A. Research: Pure Mathematics

Miscellaneous studies in pure mathematics.. ............ 2,27

B. Research: Numerical Analysis

Baker-Hausdorf f formula. ..................... .31

Classical numerical analysis, Research in. ............ .26

Differential equations, Studies in numerical integration of. .... 3

Dirichlet problem for certain multiply connected domains, Investigation of Bergman’s method for the solution of the 26

Eigenvalues, eigenvectors, and eigenfunctions of linear operators, Calculation of ............... . . . 2

Monte Carlo method, Solution of Laplace equation by. ....... .28

Probability methods and sampling techniques 4

Riemann-zeta function, Computation of the complex zeros of ..... 3

SCAMP. 10

Solution of sets of simultaneous algebraic equations and

techniques for the inversion and iteration of matrices 1 Variational me t hod s ......................... 5 C. Research: Applied Mathematics, Physics, Astronomy, and Automatic

Translation

Applied mathematics, Studies in .................. 6

B.P.A. Studies. .............. ..... .22.

Compressible flow - method of orthogonal and kernel functions . . „3®+

Cooperative phenomena, An integral arising in the theory of . . . .30

Differential equation for nerve fiber reaction. . . .31 Digital computing machines, Studies in the theory of. .12

Discrete minimal spaces .............. . ...... .20

Dynamic behavior of aircraft structures, ........ 49

Finite matrix theory, Special problems in. .... ... 28

Flow in supersonic nozzles. ....... 44

Generalized Rayleigh-Ritz method for eigenvalues of a clamped

plate. ........ ..... 7 Geomagnetic field, Analysis of ...... . ........... .28

Hypergeometric functions.. ... ..... 32

Integral operators and interpolation series . .34

Language translation study .................... 1

1

Linear programming, Supporting research in . 35

Loran UN IVAC c ode ......................... 50

Mathematical theory of program planning, Research in the 7 Neutron diffusion II . . . . . . . . . . . . . .48

Neutron diffusion III 51 Painleve equation 43

Pois son-Boltzman equation 32

Pressure distribution on bodies of revolution. ..... 45

Prolate spheroidal wave functions. ................ 1 8

Russian mathematical progress, Study of. ... . 7

Scattering functions ............... 42

Single shot probabilities 20

Spherical blast. ......... ....... 46

^Standard Loran tables - Rates 1 L4, 1 L5 52

*New task
Index of Active Research and Development Projects

Study of trajectories, Application of the theory of stochastic

processes to the. 58

Theoretical physics, Miscellaneous studies in . . . 6

Three- body problem. 28

Transportation problem II 5

D. Mathematical Statistics

Applica tions of mathematical statistics to problems of the

Chemical Corps, Research in. . ..... .58

Committee on Ship Steel, NRC , Statistical services for 58

Experiment design, Studies in the mathematics of ........ 56

Miscellaneous studies in probability and statistics. 55. NBS Administrative Operations, Statistical aspects of. ..... 57

NBS Research and testing, Collaboration on statistical aspects

of. ........ 57

Non-parametr ic statistics, Procedures of. ........... .56

Propagation of error, Law of. ................. .55

Spectral analysis of stationary time series . . ... .50

E. Mathematical Tables

Antilogarithms, Table of ............. .37

Arcsin for complex arguments, Table of. ............ .5-0 Bivariate normal distribution function, Tables of the 15- Bivariate normal distribution function, Tables relating to the. .16

Collected short mathematical tables of the Computation Laboratory . ............ .39

Coulomb wave functions, Tables of . .37

E^ ( z) , (z = x + iy), Tables of. 37

Error function for complex arguments, Table of 5-1 Exponential function for negative arguments, Extension of tables

of the. .5-1 Gamma function for complex arguments, Table of the. 37

Gases, Table of thermodynamic properties of 52

Hyperbolic sines and cosines, Extension of the table of 5-0 Integrals involving higher transcendental functions, Tables of. .35- Lagrangian coefficients for sexagesimal interpolation, Table of .38

Legendre functions . . . . .16

Mathematical tables, Revision of 39

Mathieu functions II 13

Modified Airy integral, Table of the .5-1 Power points of analysis-of-variance tests, Tables of 5-3 Probability tables for extreme values 38

Punched card library 13

Radial Mathieu functions 5-2 Random samples, Table to facilitate drawing 5

5- Secants and cosecants, Tables of. 5-2 Sievert's integral 5-2 Spheroidal wave functions ....... 5-1 Standard Loran tables - rates 2H2, 2H3, 21i5- 50

Van der Pol Equation 5-2 Wave function for lithium 39

F. Manuals, Bibliographies, Indices, and Technical Information

A.d.c.m., Logical notation and block diagram symbolism for. . . .11 Coding procedures, Mathematical tables and numerical analysis,

Bibliography of • 39

Fitting straight lines, Manual on 55- Statistical literature, Bibliography and guide to 5

5-
Index to Active Research and Development Projects

G. Computing Machine Development

Air Comptroller's computing machine. 59

Army Map Service computing machine ............... .61

Bureau of the Census computing machine 59

SEAC : National Bureau of Standards Eastern Automatic Computer. „ .60

SEAC: Number-theoretical test problems for .28

Social Security Agency, Investigation of the applicability of

automatic digital electronic computing to problems of the. . . . 6l SWAC: National Bureau of Standards Western Automatic Computer. . .10

Wright Development Center computing machine. .......... . 60
S 1 a I ii s of Pro jecls

September 30. 1053

I. Institute for Numerical Analysis

(^Section I I . I)

1 o Fundamental Research

SOLUTION OF SETS OF SIMULTANEOUS ALGEBRAIC EQUATIONS AND

TECHNIQUES FOR THE INVERSION AND ITERATION OF MATRICES

Task 1 101-1 0-5100A9-AE2

(formerly 1 1 .1 /I -^9~AE2)

Origin: NBS Authorized 2/1 5/1+ 9

Sponsor: Office of Naval Research, USN Revised 9/15/+9

Managers: G.E. Forsythe, M. R. Hestenes, D. H. Lehmer

Full task description appears in July-Sept 1 9^9 issue.

Status: CONTINUED. M. R. Hestenes has continued his theoretical

investigation of the conjugate gradient method of solving linear systems

By extending the concept of gradient somewhat he shows that every n step

iterative procedure can be interpreted as a conjugate gradient method.

T. S. Motzkin has thought of an n-step process, believed new,

for solving a system of linear equations. Let the system be regarded as the intersection of n hyperplanes H. , . . . ,H . Select n+1 points P. ,

. . .

,

P

n+1

at random. Using one point, say Pn+ ^

, as a center of projection, project

the other points into H 1 , to obtain the points P (1) (D

( 1 )

‘1 as a center, project P O) 0 ) Using

( 2 ) __ _ 7 r

_ _ . . ,...,P onto H0 , obtaining P.

II / I n- 1 / \ C- I P

n

" . Continuing in this way, the point P^

n is at the intersection

of all hyperplanes, and thus a solution to the system is found.

Publications: (1) "Tentative classification of methods and bibliography on solving systems of linear equations," by G. E. Forsythe. Simultaneous linear equations and the determination of eigenvalues, Proceedings of an NBS Symposium held in Los Angeles, August 1951", NBS Applied

Mathematics Series 29, p . 1-28 (U. S. Government Printing Office, Washington, Do C.) (2) "On certain character matrices," by D. H. Lehmer;

submitted to a technical journal. (3) " Punched-ca rd experiments with

accelerated gradient methods for linear equations," by A. I. and G. E. Forsythe; to appear in Contributions to the solution of- systems of linear

equations and the determination of eigenvalues, NBS Applied Mathematics

Series; in press, U. S. Government Printing Office. ( + ) "Some numerical

examples on solving systems of linear equations by the conjugate gradient

method for nonsymmetric systems of equations," by M. R. Hestenes,

U. Hochstra sser , and L. S. Wilson; IN MANUSCRIPT. (5) "On the linear
2 Status of Projects

iteration procedures for symmetric matrices," by A. Pd. Ostrowski; submitted to a technical journal. (6) "On over and under relaxation in the

theory of the cyclic single step iteration," by A. M. Ostrowski; MTAC 43,

152-159 (July 1953). (7) "On the convergence of Gauss' alternating procedure in the method of the least squares, I," by A. M. Ostrowski; submitted to a technical journal. ( 8 ) "The solution of systems of linear

equations by the conjugate-gradient method for use on IBM equipment," by

U. Hochstrasser ; IN MANUSCRIPT. (9) "Solving linear algebraic equations

can be interesting," by G. E. Forsythe; Bui. Am. Math. Soc . 32? 299~329

(July 1953).

CALCULATION OF EIGENVALUES, EIGENVECTORS, AND EIGENFUNCTIONS

OF LINEAR OPERATORS

Task 1101-10-5100/50-3

(formerly 11.1/1-50-3)

Origins NBS Authorized 7/1/49

Sponsor: Office of Naval Research, USN

Managers: G. E. Forsythe and M. R. Hestenes

Full task description appears in July-Sept 1949 issue.

Status: INACTIVE. For status to date see Apr-June 1953 issue.

Publications: (1) "Completely continuous normal operators with

property L," by I. Kaplansky; submitted to a technical journal. (2) "Asymptotic lower bounds for the frequencies of polygonal membranes," by G. E.

Forsythe; submitted to a technical journal. (3) "On the spectrum of

a one parametric family of matrices," by A. M. Ostrowski; submitted to

a technical journal.

STUDIES IN PURE MATHEMATICS

Task 1101-10-5100/50-4

(formerly 1 101 -1 1 -5101/50-4)

Origin: NBS Authorized 7/1/49

Sponsor: Office of Naval Research, USN

Manager: C. B„ Tompkins

Full task description appears in July-Sept 1949 issue.

Status: CONTINUED. The calculations for the determination of

irregular primes were continued for all primes less than 2000. To do

this it was necessary to recode the problem to make use of the magnetic

drum memory. It was discovered that there are 1

1

8 irregular and 1 84

regular primes up to that limit. It has been proved by Kummer that

Fermat's Last Theorem is true for all regular primes. Further tests were deemed necessary in order to complete the proof for irregular primes.

Such a test - a modification of a Kummer criterion - was

programmed for the SWAC . This test was found to hold for all irregular primes less than 2000 thus proving Fermat's Last Theorem up to

that limit,

T. S. Motzkin, jointly with P. Erdos, E. G. Straus, and

J. D. Swift, studied the representations of a positive integer n as sura of three integers n=a

k+bk+c

k

> a k ” b

k “ c k” 1

’ k =1 , 2 , . . .

, r

,

such tha t a

1 c

1 = .

.

.-a

p br

c .
Status of Projects 3

Besides asymptotic properties it was established that sets

of distinct triples with arbitrarily large r exist, and that sets of

two distinct triples exist for every n>l 8 . Background studies on embedding of surfaces in Euclidean space

and in problems connected with minimizing Dirichlet ’ s Integral under

various conditions (the Problem of Plateau and problems of lesser difficulty) have been initiated by G. Blanch, ML R„ Hestenes, C. B. Tompkins,

and others. These studies will produce some partial differential equations to be attacked on SWAG

.

Discrete variable problems were attacked by various workers

including E. Kieinfeid, E Lehraer, D. H. Lehmer, and C. B. Tompkins.

(For discrete variable problems see also task 1 1 01 -1 0~5l 50/53_ 1 5 p. 10).

R. fi. Brock, M. R. Hestenes, H. W, Kuhn, T. S. Motzkin, C. B. Tompkins, J. L. Walsh, and W. Wasow participated in the American Mathematical Society's Sixth Symposium held in Santa Monica on August 26-28,

1953.

Publications: (1) "Boolean Geometry I," by L, M. Blumenthal;

Rend. Circ. Mat. Palermo, {II} , 1, 1 —1 8 (1952). (2) "An isoperimetric

inequality for closed curves convex in even-dimensional Euclidean space,"

by I. J. Schoenberg; submitted to a technical journal. (3) "On the distribution of totitives," by D. H. Lehmer; IN MANUSCRIPT.

COMPUTATION OF THE COMPLEX ZEROS OF THE RIEMANN-ZETA FUNCTION

Task 1101-10-5100/50-13

(formerly 1 101 - 1 1-5101 /50-1 3)

Origin: NBS Authorized 6/1/50

Sponsor: Office of Naval Research, USN

Managers: D. H. Lehmer and C. B. Tompkins

Full task description appears in Apr-June 1950 issue.

Status: CONTINUED. Careful analysis of the zeta function at the

5-00 spe cial Gram points is being carried out. (See Jan-Mar 1953 issue,

p. L.)

STUDIES IN THE NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS

Task 1101-10-5100/51-1

(formerly 1101-11-5100/51-1)

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Managers: G. Blanch and W„ Wasow

Full task description appears in Ju'ly-Sept 1950 issue.

Status: CONTINUED. Much of the activity of the past quarter centered around the symposium on numerical analysis (see p. 62 ) and on the

teaching of courses at UCLA during the summer session. In connection

with the latter, W. Wasow is preparing a set of lecture notes on the

asymptotic theory of linear differential equations.

The table of characteristic values of Mathieu's equation (see

Apr-June 1953 issue, p. 4) by G. Blanch and I. Rhodes is being put in

final form during spare hours. It is hoped that the table will shed more light on the asymptotic behavior of the characteristic values and

will stimulate research which will result in better asymptotic approximations .
4 Status of Projects

Publications: (1) "On the numerical solution of parabolic partial

differential equations," by G. Blanch; J. Res. NBS 3*+3~356 (June 1953).

(2) "On mildly nonlinear partial difference equations of elliptic type,"

by L. Bers; accepted by the Journal of Research of the NBS . (3) "Asymptotic solution of the differential equation of hydrodynamic stability

in a domain containing a transition point," by W. Wasow; accepted by the

Annals of Mathematics. (4-) "An expansion method for parabolic partial

differential equations." by J. W. Green; accepted by the Journal of Research of the NBS. (5) "On small disturbances of plane Couette flow,"

by W. Wasow; accepted by the Journal of Research of the NBS. ( 6 ) "Tables

of lift and moment coefficients for oscillating airfoils in subsonic

compressible flow," (formulation of Reissner and Haskind) by G. Blanch;

an NBS report. (7) "Singular perturbation methods for nonlinear oscillations," by W. Wasow; IN MANUSCRIPT.

PROBABILITY METHODS AND SAMPLING TECHNIQUES

Task 1 101-10-5100/51-2

(formerly 1 101 -1 1 -5100/51 - 2 ) Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Managers: D. Teichroew and W. Wasow

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. D. Teichroew has continued his investigation of

methods of generating random variates. A subroutine to compute values of

random variates on the SWAC by the polynomial transformation method has

been coded and checked. A routine to compute functions of sequences of

random variates computed by this method has also been coded and checked.

The functions are: (i) the first eight moments, (ii) the serial correlation, and (iii) the frequency distribution. These statistics are used

to test the hypothesis that the random variates have the desired distributions. Experience up to the present indicates that the hypothesis will

not have to be rejected.

An investigation of methods of computing probability distributions appropriate to high speed computers has been started by D . Teichroew.

A code for computing the incomplete gamma function (for 'integral values

of the parameter) has been checked and is being used to evaluate the

accuracy of other methods. (It is hoped that an opportunity will arise

to compute an extensive table of the incomplete gamma function). One

method which, to our knowledge, has not yet been investigated is that

based on the inversion of the characteristic" function. For example,

F (x) =

x

e -t /21 dt = 1-

J2jf 27F

, oo

-oo

- 1 2 /2 sin xt dt

From preliminary calculations it appears that, for many distributions,

numerical integration of an integral analogous to the integral on the

right-hand side may be a reasonable way of computing the value of the

distribution function for arbitrary values of the argument. Another

method being investigated is that based on asymptotic expansions in

terms of normal deviates. The expansion has been obtained for certain

distributions that tend to the normal as a parameter tends to a limit.

One advantage of this method is that it permits computations of variates

associated with a given probability and therefore eliminates the need

for inverse interpolation in tables of distribution functions. Routines

have been coded and checked for computing t and chi-square variates from
Status of Projects 5 normal deviates and vice versa, normal deviates from t and chi-square

variates. In connection with the incomplete gamma function the introduction to "Tables for the computation of the incomplete gamma function and

the probability function of X ," by E. E. Slutskii, has been translated

from the Russian by D. Benster.

A report is being prepared which contains the results of some sampling experiments (carried out on the SWAG by D. Teichroew) to determine the power of certain nonparametric tests, (see publica tion (3) below)

.

The following related articles have been translated from the

Russian by C. D. Benster: (1) "On the maximum discrepancy between two

empirical distributions," by B. V. Gnedenko and V. S. Koroliuk, (2) "On

a problem of the comparison of two empirical distributions," by B. V.

Gnedenko and E . L. Rvacheva, (3) "Some results on the maximum discrepancy

between two empirical distributions," by B. V. Gnedenko, (4) "On the

empirical distribution function in the case of grouping of the data,"

by I 0 I. Gikhman , (5) "On the distribution of the number of excesses of one empirical distribution function over another," by B. V. Gnedenko

and V. S. Mikhalevich, (6) "Two theorems on the behavior of empirical distribution functions," by B. V. Gnedenko and V. S. Mikhalevich, and (7)

"On the mutual disposition of two empirical distribution functions," by

V. S. Mikhalevich.

F. G. Foster, on leave from the London School of Economics and

Political Science, continued his research on a record test for trend in

a time series by using the SWAG to compute, by sampling, power points

of the test against normal alternatives, (see publication (4) below)

,

Publications: (1) "Statistical estimation of matrix quantities

by means of a class of discrete Markov chains," by H . P. Edmundson; IN

MANUSCRIPT. (2) "Limits for permanent preponderance," by P. Erdos and

T. S. Motzkin; IN MANUSCRIPT. (3) "Some sampling results on the power of

nonparametric tests against normal alternatives," by W. J. Dixon and

D. Teichroew; IN MANUSCRIPT. (4) "Power function of the record test for

trend in a time series," by F. G. Foster and D. Teichroew; IN MANUSCRIPT.

VARIATIONAL METHODS

Task 1101-10-5100/51-3

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Manager: M. R. Hestenes

Full task description appears in July-Sept 1950 issue.

Status: INACTIVE. For status to date see July-Sept 1952 issue.

Publications: "Iterative methods of solving linear problems on Hilbert space," by R. M. Hayes; to be included in Contributions to the

solution of systems of linear equations and the determination of eigenvalues, NBS Applied Mathematics Series.
6 Status of Projects

STUDIES IN APPLIED MATHEMATICS

Task 1101—1 0-51 00/51 -*+

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Managers: G. E. Forsythe and T. S. Motzkin

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. The seminar on numerical analysis continued

to meet through July and early August. Under the influence of the SCAMP

program in finite projective geometry, several talks were presented on computational aspects of discrete problems, and especially on the determination of perfect difference sets. There were also two talks on com- putational problems of theoretical physics. A detailed list of sessions

will be found under Lectures and Symposia in the back of this issue.

Summaries of the presentations have been prepared for the sessions from

April 20 through May 13 by C. B. Tompkins and for the sessions from

May 1 8 through June 8 by G. E. Forsythe.

T. S. Motzkin and J. L. Walsh continued their study of approximation by polynomials of a given degree on a real or complex finite

point set, in particular of the minimization of j f(z^)l P, where the

weight and the exponent p>0, as well as the set z^,...,z m

are given

(for p=GO replace the sum by max /u

^

| f(z^)|). The set of all polynomials

belonging, for given p>1 , to weights at least k of which are greater than

0, and the set of all inf ra polynomials (that is, such that no other

polynomial g(z) has the property g(z^)= f(z^), |-£^| <1 , (i=1 , . . . , m) were

determined and compared. Related investigations of real linear families

of complex polynomials and of generalizations to higher-dimensional

space were begun.

Publications: (1 ) "Numerical computation of low moments of order

statistics f.rom a normal population," by J. B. Rosser; submitted to a technical journal. (2) "Changes of sign of sums of random variables,"

by P. Erdos and G. Hunt; submitted to a technical journal. (3) "A num- erical analyst's 1 5~f oot shelf," by G. E, Forsythe; to appear in Mathematical Tables and Other Aids to Computation. (4) Seminar on numerical

analysis — summary of presentations between April 20 and' May 13, 1953,"

by C. B. Tompkins, multilithed typescript, 10 p. (5) "Seminar on numerical analysis — summary of presentations between May 1 8 and June 8,

1953," by G. E. Forsythe, multilithed typescript, 38 p.

MISCELLANEOUS STUDIES IN THEORETICAL PHYSICS

Task 1 101-10-5100/51-5

Origin: Office of Naval Research, USN Authorized 9/1/50

Sponsor: " " Managers: D. Saxon and C. B. Tompkins

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. Item (5) under publications below deals with

the application of variational methods to quantum mechanical scattering

problems. A variational formulation for general acoustic scattering

problems 'has also been developed (see publication (6) below).

Programming and preliminary numerical work is continuing on the calculation of the photo- disintegration of the deuteron (see Qct-Dec

1952 issue, p. 7). Woods is continuing a program of computation on
Status of Projects 7

the elastic scattering of protons from various elements. A. Banos developed an extension of the saddlepoint method of integration valid when

a pole exists in the vicinity of the saddlepoint.

Publications: (1) "Modes of vibrations of a suspended chain," by

D. S. Saxon and A. S. Cahn; accepted by Quarterly Journal of Mechanics

and Applied Mathematics (Oxford). (2) "A nonlinear model for the com- posite Pi-meson," by S. G. Gasiorowicz; submitted to a technical journal.

(3) "Acoustic radiation pressure on a circular disk," by H. Levine; to

appear in the Proceedings of the Fifth Symposium on Applied Mathematics

of the American Mathematical Society. (4) "A numerical solution of

Schroedinger 1 s equation in the continuum," by W. Futterman, E. Osborne,

and D. S. Saxon; accepted by the Journal of Research of the NBS. (5) "Application of variational methods to intermediate and high energy scattering," by E. Gerjuoy and D. S. Saxon; IN MANUSCRIPT. (6) "Variational

principles for the acoustic field," by E. Gerjuoy and D. S. Saxon; IN

MANUSCRIPT'.

STUDY OF RUSSIAN MATHEMATICAL PROGRESS

Task 1 101-10-5100/52-1

Origin: NBS Authorized 3/1 5/52

Sponsor Office of Naval Research, USN

Manager: G. E. Forsythe

Full task description appears in Jan-Mar 1952 issue.

Status: CONTINUED. The task manager is continuing to accumulate

bibliographical cards on Russian mathematical monographs (see Jan-Mar

1952 issue, p. 11 and Oct-Dec 1952 issue, p. 8). The translations formerly reported under this task are now reported in connection with the

pertinent tasks of the Institute for Numerical Analysis.

GENERALIZED RA YLEIGH-RITZ METHOD FOR EIGENVALUES OF A CLAMPED PLATE

Task 1 101-10-5100/53-1

Origin: NBS Authorized 12/29/52

Sponsor: Office of Naval Research, USN

Manager: G. Blanch

Full task description appears in Oct-Dec 1952 issue.

Status: CONTINUED. Some progress has been made in programming

the computations for SWAC

.

RESEARCH IN THE MATHEMATICAL THEORY OF PROGRAM PLANNING

Task 1 101-10-5102/50-1 1 (formerly 11.1/1-50-11)

Origin: Office of Air Comptroller, USAF Authorized 6/1 5/50

Sponsor: " " Manager: T. S. Motzkin

Full task description appears in Apr-June 1950 issue.

Status: CONTINUED. T. S. Motzkin investigated the assignment
Status of Projects

problem from two different points-of-view. If the entries are sufficiently rough and the matrices sufficiently large, it turns out to be

always feasible, even by slightly modified direct combinatorial trials,

to obtain a permutation consisting of maxima in all rows and columns.

The corresponding computational method, proposed by D. H. Lehmer and

coded by B. Handy, was run for a 12x12 matrix and found to* be too long,

especially due to the large percentage of time consumed for access to the

drum. On the other hand, in view of evaluation of application of the

simplex method, the polyhedra corresponding to the assignment problem

and some of its generalizations were studied and the faces of all dimensions determined.

Preparation of an expository account of the theory of inequalities was begun by T. S. Motzkin and H. W. Kuhn. This presentation has

been tentatively divided into four sections? (1) duality theorems,

(2) rank theorems, (3) criteria for solvability, and (4) methods of solution; the latter will be worked out by A. J. Hoffman. Section (1) also

contains systematic and elementary proofs of the incompatibility and implication theorems.

H. W. Kuhn investigated the feasibility of the face approach for

the "traveling salesman" problem. It was found that, although there are

but ( n—1 ) i permutations to be considered, the convex hull has many more

than ni faces in general. The exact number is known only for n=2,3,4;

there are 450 distinct faces known for n=5. All of the conditions found

thus far can be characterized in terms of a "blocking set", i.e., a minimal set of pairs (i,j) such that each admissible permutation takes some

i into j. With the assistance of T. S. Motzkin, a finite zero-sum two-person

game equivalent to the "traveling salesman" problem was constructed. The

experience gained from this line of investigation seems to indicate that

the problem is one of direct combinatorial enumeration.

A connection between convexity and contractibility was established by H. W. Kuhn who found that a contractible polyhedron in n-space

with contractible faces must be convex. This result is related to recent

econometric studies by Debreu and Arrow.

J. Marschak, in collaboration with R. Radner, studied optimal

networks and rules for teams. The goal of a team is defined by the payoff

function u(a,x) where a= (i=1,...,n) and x=

£ x (j=1,...,m) are,

respectively, the vectors of

controlled (or external) var

§ (x), the problem is to cho

best communication network ft network N corresponds a cost,

ture," S=S(N), whose element

does not depend on Xj. Let

s. .=1 . Then the rule of act

1 J (':') that a.- (x<-4 '). The bes

1 X o

controlled (or action) variables and of non- iables. Given the probabilityA distribution

ose the best rule of actions, , and the

, defined as follows? To eacfi communication

C=C(N), and a matrix of "information strucs. .= 1 or 0 according to whether a. does or

! 1 J

( j ) 1 the vector Xg consist of all Xj such that

ions is the vector of functions such

t expected gross payoff is

U(S) = U(S (N) , °0 = max u [^

1 (Xg (1

the best expected net payoff is

^

n (x

s

( ^

)) ,x]d $(x);

V(N,£) = max [U(S(N),£) - C(N)].

N

The problem was studied concave function of a non-bounded

of a bounded vector a. Case (II)

uncertainty.

for two cases? (I) u, a strictly

vector a; (II) u, a linear function

represents linear programming under
Status of Projects 9

The economic example for case (I) was a firm in a competitive

market, earning a profit u=aQa 1 -xa ' , where a is the vector of inputs,

x is the vector of their respective prices (the price of a unit of output

being =1), and Q is a matrix characterizing the production technology.

The above profit function is a special case of a general quadratic function of a and x. Such a quadratic function can, without loss

of generality, be assumed to be of the form

u(a,x) = aQa' - 2xRa 1 ,

where the expected value of x is zero. It is also assumed that for every

x, u has a unique maximum in a, i.e., Q is negative definite. A For a given information structure S, the best rule of action oc is the solution of the system of integral equations

E(oc(x)Q-xR|x ('

;'')) = 0, i = 1 , .

(i)

, n

.

If x has a normal distribution, then each ot(x v '

)

is a homogeneous

linear function of x(i). Formulas for the feest rule of action and best

expected gross payoff have been obtained under various special conditions

on the distribution of x and on the information structures.

The economic example for case (II) was a firm engaged in speculative shipping of a commodity. If t^j is the cost of transporting the

commodity from city i to city j, if x is the vector of its prices in n cities, and if not more than one unit can be shipped from each city, the

best expected gross payoff is

U ( S ) = \ E ( ma x E ( x - fe J

J - x • t . .) ij

(i)

U)

the information structure is S. For n=2 , with t. .= (1-6. . ) t

,

with c=cost

J J 2

of two-way communication between any two cities, and with Ex^=0, Ex/=1 , and E Xi x

2=/>, the best networks were »c omputed manually for given c, t,

and/3

. A similar computation was made for n=3 based on the assumption

of two alternative simple discrete distributions of x, one involving

statistical dependence and the other free of it. Moreover, assume the

latter distribution (with equal probabilities for each z^ to take the

values 1 and -1) and permit n to change. Then, the optimal size of "subteams" (consisting of members who communicate with each other but not

with members outside of the sub-team) turns out to be independent of n (it is equal to the largest integer that is smaller than or equal to

-log^c), provided the total number n of members is sufficiently large

and t is sufficiently small.

For general normal distribution of x, with n=4 or larger, a SWAC procedure has been set up to compute optimal gross payoffs U(S) for

all information matrices S. The procedure will use random samples of x, with population covariances <r^ j= k Jbo , and alternatively -1< j=1 -k | i- jl <1 With covariance matrices just mentioned and with n increasing

indefinitely, it is also hoped to gain insight into the properties of a continuous version of the network problem.

Publications: (1) "Basic solutions of the transportation problem,"

by T. S. Motzkin; IN MANUSCRIPT. (2) "The multi-index transportation
10 Status of Projects

problem," by T. S. Motzkin; IN MANUSCRIPT. (3) "Ray systems with maximum

angle sum," by T. S. Motzkin; IN MANUSCRIPT. (5-) "On Fejer sets in linear

and spherical spaces," by T. S. Motzkin and I. J. Schoenberg; accepted by

the Annals of Mathematics. (5) "On the relaxation method for linear inequalities," by T. S. Motzkin and I. J. Schoenberg; submitted to a technical

journal. (6) "On the optimal character of the (s,S) policy in inventory

theory," by A. Dvoretzky, J. Kiefer, and J. Wolfowitz; submitted to a technical journal. (7) "Least p-th power polynomials on a real finite

point set," by T. S. Motzkin and J. L. Walsh; submitted to a technical

journal. (8) "Least p-th power polynomials on a complex finite point set,"

by T. S. Motzkin and J. L. Walsh; IN MANUSCRIPT. (9) "A new type of ex- istence theorem for systems of linear inequalities," by J. W. Gaddum; IN

MANUSCRIPT. (10) "Boundedness of sequential projections," by T. S. Motzkin

and C. B. Tompkins; IN MANUSCRIPT. (11) "Towards an economic theory of

organization and information," by J. Marschak; to appear in a volume on Decision Processes edited by C. Coombs and R. Thrall. (The original

manuscript written in December 1952, was substantially revised in August

during the author's stay at the Institute.) „

SCAMP

Task 1 101-10-51 50/53-1

Origin: Office of Naval Research Authorized 6/10/53

Sponsor: " " Manager: C. B. Tompkins

Full task description appears in Apr-June 1953 issue.

Status: CONTINUED. A group of about 20 mathematicians participated

in the most intensive part of the program, which lasted from 1 July through

28 August. A formal report has been submitted by the visiting chairman,

Prof. S . S. Cairns, to the Department of Defense. Computations have

been turned over to be carried out under task 1 1 01 —^t-O— 5 1 31 /55— 1 (see p, 2V )

.

2 . Development

NATIONAL BUREAU OF STANDARDS WESTERN AUTOMATIC COMPUTER (SWAC)

(previously listed as Air Materiel Command Computing Machine)

Task 1 101-20-5103A9-1

(formerly 1 1 01 -35~5l 03A9-1 ) Origin: Aeronautical Research Laboratory Authorized 11/1/5-8

Wright Air Development Center, ARDC,USAF

Sponsor: " " " Managers: H. D. Huskey and R. Thorensen

Full task description appears in Apr-June 195-9 issue.

Status: CONTINUED. During the last quarter the SWAC was used for

29 different problems for a computing time of 623 hours out of a scheduled

time of 85-5 hours. A full set of eigenvalues and eigenvectors for a 32d

order matrix was computed to 10 significant decimal digits. The complete

solution was obtained in 1

8

hours of computing time, with the aid of the
Status of Projects 1 1 magnetic drum memory.

Only minor engineer ing changes have be en made on the c omp uter

dur ing this past qua rter. Th ese changes have be en limited to th e c ompie tion of projects in progre s s and include such it ems as c onstr uc t ion

of a new breakpoint f ac il ity a nd installation of new power suppl ies and

vol tage regulators f or an imp roved cathode ray t ube deflect ion s yst em. No new d evelopment projects h a ve been initiated. and it is e stima ted tha

the SWAC may be kept in effec tive operation on a tw 0 shift basis wi th

the services of only two main tenance engineers.

LOGICAL NOTATION AND BLOCK DIAGRAM SYMBOLISM FOR A.D.C.M.

Task 1 101-20-5103A9-2

(formerly 11.1/22-49-2)

Origin: NBS Authorized 2/1 5/49

Sponsor: Aeronautical Research Laboratory, Wright Air Development

Center, ARDC , USAF

Manager: H. D. Huskey

Full task description appears in Apr-June 1949 issue.

Status: INACTIVE. For status to date see Apr-June 1952 issue.

LANGUAGE

Task 11 (formerly

TRANSLATION STUDY

01-20-5103/52-1

1 101-21-5104/52-1 ) Origin: NBS

Sponsor: The Rockefeller Foundation

Managers: H. D . Huskey and G. E. Forsythe

Authorized 8/1 5/51

Revised 1 0/1 5/51

Terminated 9/30/53

Objective: This is a study of certain statistical and logical

aspects of language translation, with a view to obtaining at least some partial answers to questions concerning the feasibility and practicality

of performing language translations by automatic computing machines.

Statistical aspects will include investigations of word frequencies and

key-word positions in various types of vocabularies; logical aspects

will include the study of the possibility of coding the elements of

sentence structure in mathematical symbols understood by a computing

machine so that the machine can make a beginning at constructing a real

translation rather than doing just a word-substitution.

Background: This task originated from the interest of the Library

of Congress and State Department in automatic language translation

machines, as stated in a conference between the Librarian, Dr. Luther

Evans, the Assistant Librarian, Dr. V. W. Clapp, and representatives of

the State Department and of the National Bureau of Standards in January

1950.

Comments: This is

Apr-June 1951 issue, p. 1 being done by members of

a continuation of task 1101-21-1102/50-10 (see

1). The major portion of the work on this is

the language departments at UCLA.

Status: TERMINATED
12 Status of Projects

STUDIES IN THE THEORY OF DIGITAL COMPUTING MACHINES

Task 1101-20-5103/53-1

Origin: NBS Authorized 9/30/52

Sponsor: Aeronautical Research Laboratory, Wright Air Development

Center, ARDC , USAF

Manager: H. D. Huskey

Full task description appears in July-Sept 1952 issue.

Status: CONTINUED. Methods of facilitating coding and new systems

for using the drum in conjunction with the high-speed memory are being

considered

.

3 . Mathematical Services

COMPUTING SERVICES FOR RESEARCH STAFF OF

THE INSTITUTE FOR NUMERICAL ANALYSIS

Task 1 101-40-5130/54-1

(formerly 1101 —40— 1 111 /49— 1 a

)

Origin: NBS Authorized 9/2/48

Sponsor: Aeronautical Research Laboratory, Wright Revised 11/16/49

Air Development Center, ARDC, USAF

Managers: M. Howard, F . Hollander and P. Bremer

Full task description appears in July-Sept 1949 issue.

Status: CONTINUED. Research staff problems involving SWAC were:

(1) For D. H. Lehmer: (a) The Mersenne problem has been completed. All

numbers have now been computed twice as a check. (b) Turan's conjecture

has been tested at 125,000 and checked by repeating the computation.

(c) Certain prime numbers were tested by Alway's method. (d ) Irregular

primes have been tested in an investigation of Fermat's last theorem.

(2) For M. Hestenes: Code checking was done on a conjugate gradient

method of solving linear systems of equations. (3) For R. Woods: Code

checking and some calculation was done on an optical model for calculating the scattering of protons by heavy nuclei. (4) For G. Blanch:

An extension of the table of the bivariate normal distribution function

was computed. (5) For E. C. Yowell: Several additional routines were

checked out for the problem of the determination of periods of light

variation of variable stars. ( 6 ) For D. Teichroew: SWAC has been used

to compute a relatively large number of random variates by the polynomial

transformation method, to check the efficiency of the method. Variates

computed in this way have been used to compute power functions of certain

nonparametric tests. (7) For the Geophysics Dept, of UCLA: Code checking

was completed and calculation started on the solution, by the method of

least squares, of the problem of high order tidal constituents for

earth and ocean tides. ( 8 ) For the Chemistry Dept, of UCLA: Several

routines concerned with the problem of crystal structure have been

checked. Computations are continuing. (9) For the Physics Dept, of UCLA:

Code checking is continuing on the routine for the solution of the molecular wave function for the Li 9 molecule. (10) For the Education Dept,

of UCLA: Computations are being performed on item analysis of certain

psychological instruments. (11) For the Business Administration Dept, of

UCLA: A set of five linear equations in five unknowns was solved on SWAC.
Status of Projects 13

math ie u functions ii

Task 1 101-40-5131 A5-1

(formerly 1 1 01 -53_ 1 1 01 /45-1

)

Origin: Applied Mathematics Panel, NDRC Authorized 7/1/47

Sponsor: Aeronautical Research Laboratory, Terminated 9/30/53

Wright Air Development Center, ARDC, USAF

Manager: E. C. Yowell

Objective: To prepare a table of the periodic solutions:

oo Se

r (s,t) = \, De

2 n+

pC os ( 2 n+p) t

,

(P = 0,1 ) n=o

00 So

r (s,t) = 'y Do 9 n _pSin( 2 n-p) t, (P = 0,1 ) n=1

for r=0 (1 ) 1

5

9 t=0(1 °)90° over the range s=0 to s=100, of the Mathieu

differential equation:

2 y" + (b - s cos t)y = 0 .

Backgrounds Mathieu functions arise in the solution of the wave

equation for elliptical domains. Numerous physical applications involving

Mathieu functions are described in "Theory and Applications of Mathieu

Functions" by N. W. McLachlan (Oxford Press 1947). The existing tables

by Ince and Goldstein are inadequate. The task was originally proposed

by Dr. Philip Morse and Dr. James Wakelin.

Status: TERMINATED.

PUNCHED CARD LIBRARY

Task 1 101-40-5131/49-2

(formerly 1101-53-1101/49-2)

Origin: NBS Authorized 7/14/48

Sponsor: Aeronautical Research Laboratory, Wright Air Development

Center, ARDC, USAF

Manager: P. Bremer

Full task description appears in Apr-June 1949 issue.

Comments: A catalog of tables on punched cards which are on file

at the Institute may be obtained by addressing the Institute for Numerical

Analysis, 405 Hilgard Avenue, Los Angeles, 24, California. Within the

limits of the program of the computation unit of the Institute, tables

will be duplicated upon request, provided the requester furnishes the

blank cards. Requests should be addressed directly to the Institute.

Status: CONTINUED. The following tables were added:

(1) Table 5020,

G(b;*,/s) ((3+ 1) [F(x+b)]°

C [1-F(x )]

/9 f (x)dx,

-oo
14

where

Status of Projects

,-*2/2

f(x) = — an<3 = u

-oo

«: = 0(1 )4; p= 1 (1 )9; 6 = -3.20(.05)0(.01 )6.4( .1)7.1 ;9d,

(2) Table 5021,

(3) Table 5022

where

0(x,y,r

)

,00 00 L(h,k,r) = dx

J

dy 0(x,y,r)

,

k

n

h Ax

V(h, Ah) = dx P dy[z (x)z (y) ]

2vJ 1 - r —— exp(--Jt^)

.

sf2jr

[See tasks 1101 —53—1 1 01/51 —32 7 Apr-June 1 951 , p. 25. and 1 101 — 53— 1 101 /52-50,

Apr- June 1952, p. 28, for background.] For L(h,k,r): + r=0 ( .05)0 . 9 5 ( .01 ) 1

6d for positive values of r; 7D for negative values of r. h,k=0(.1)4; or up to point where L=0 to 7 decimals. For V(h, Ah): h=0 ( .01 )4( .02)4. 6( . 1 ) 5.6

and oo. For V(Ah,h): h=0( .01 )4( .02)5.6 and oo . Both V(h,Ah) and

V(Ah,h) are given forA = .1(.1)1, 7D,with last place uncertain by two units.

STATISTICAL SMOOTHING

Task 1101-40-5131/51-19

(formerly 1 101-53-1 101/51-19)

Origin: Stanford Research Institute, Stanford University Authorized 1/15/51

Sponsor: Office of Research Operations, U.S. Array

Manager: M. Howard

Full task description appears in Jan-Mar 1951 issue.

Status: INACTIVE. For status to date see Apr-June 1953 issue.

TABLES OF THE BIVARIATE NORMAL DISTRIBUTION FUNCTION

Task 1 101-40-5131/51-32

(formerly 1 101-53-1

1

01 /5l -32)

Origin: Division 13, NBS Authorized 5/31/51

Sponsor: Office of Chief of Ordnance, U. S. Array

Manager: G. Blanch

full task description appears in Apr-June 1951 issue.
Status of Projects 15

Status: CONTINUED. A preliminary manuscript has been prepared

with an introduction by the task manager concerning methods of interpolation and the method used to collate the tables. The collated volume

also includes the function V(h,k), described in task 1 101 -4-0-51 31 /52-50,

p. 1 6. The final manuscript will be prepared from the preliminary one

on special forms. All values are now on punched cards.

SIMPLIFIED ROLLING PULLOUT EQUATIONS

Task 1101 -1+0-51 31/51 -3^

(f ormerly 1 101 -53-1 101 /5l -34)

Origin: Cornell Aeronautical Laboratory Authorized 6/22/51

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, ARDC , USA!' Manager: E. C. Yowell

Full task description appears in Apr-June 1951 issue.

Status: CONTINUED. Integrations involving three sets of observed

data were completed during this quarter.

LOW MOMENTS OF ORDER STATISTICS

Task 1 101-40-5131/51-36

(formerly 1101 -53-1 101/51 -36)

Origin: University of Oregon Authorized 6/22/51

Sponsor: Offi.ce of Nlaval Research, USN

Manager: A. D. Hestenes

Full task description appears in Apr-June 1951 issue.

Status: CONTINUED. The Rosser method is being coded for SWAC

solution. In order to achieve the desired results triple precision multiplication, addition, etc., is used. At this point values of

-_t 2 f(t) = e 2//2tt

have been computed to 30 decimals at intervals of .02 units. The code

for computing the integral of f(t) is well under way. Some work has been

done on a few of the remaining routines.

SIERRA WAVE PROJECT

Task 1 101-40-5131/52-36

(formerly 1 101-53-1 101/52-36)

Origin: Department of Meteorology, U.C.L.A. Authorized 4/1/52

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, ARDC, USAF

Manager: T. H. Southard

Full task description appears in Jan-Mar 1952 issue.

Status: INACTIVE. For status to date see Apr-June 1953 issue.
Status of Projects

l4

whe

(2

(3

wl

LEGENDRE FUNCTIONS

Task 1101-50-5131/52-^8

Origin: Hughes Aircraft Company Authorized 6/30/52

Sponsor: Aeronautical Research Laboratory, Terminated 9/30/53

Wright Air Development Center, ARDC, USAF

Manager: E. C. Yowell

Objective: To determine the values of Y for which Py(x) = 0,

x=x^

where PTM(x) is the associated Legendre function. The first six values

of r for m=0(1)10 and for two values of Xq are required. Also to determine

*

[ Py(x) ] 2 dX,

where Y is determined from the first problem.

Background: These eigenvalues are necessary for the expansion

in orthogonal functions of the radiation fields from certain antennae.

Status: TERMINATED.

[A

6

uaV

TABLES RELATING TO THE BIVARIATE NORMAL DISTRIBUTION FUNCTION

Task 1 101-40-5131/52-50

(formerly 1 1 01.-53 - 1 1 01 /52-50)

Origin: Division 13? NBS

Sponsor: Aeronautical Research Laboratory,

Wright Air Development Center, ARDC, USAF

Manager: G. Blanch

Objective: To provide tables of V(,h,k) and V(k,h), relating to

the bivariate normal distribution function, where

Authorized 6/1/52

Completed 9/30/53

V(h , k) =

1_

27T

xk/h

dx exp( y )dy.

Parameters: h=0(.01)4, and h=oo ; k/h=0 . 1 ( . 1 ) 1

Accuracy 6 decimals.

Background: A seven-place table of V(h,k), at intervals of 0.05

or 0.1 in h, was computed by the Mathematical Tables Project, under the

auspices of NDRC , at the request of H. H. Germond. The latter subtabulated this table, and published an extensive table of V(h,k),but not of

V(k,h) in a restricted NDRC report, which has since been declassified.

There is an intimate relationship between the function V(h , k) and the

bivariate normal distribution function L(h , k, r ) , where

L(h,k,r)

2irJ] -r" S

dx 0(x, y, r )dy
Status of Projects 17

and

0(x,y,r) = exp [ - (x

2+y

2-2rxy)/2 (1 -r

2

) ]

It can be shown that

L(h,k,r) = V h, •fh

p )

+ V (k, *y-r -k-p~ ' )

+ A [ 1 -oc(h)-oc( k) ] v ( 1 -r ) ) \ J(1 -r ) /

4

‘

where

arc sin r 2 TT

oc{\) -

•J2 TT

Af-x

exp (- i t 2 )dt.

For some purposes, the function V(h,k) is more convenient than L(h,k,r),

and it is deemed desirable to include a table of both V(h,k) and V(k,h)

in the volume of collated tables L(h , k , r ) , described under task 1101 - 53- 1101 /

51 - 32 . Status: COMPLETED. The table is included in the collated tables

of the Bivariate Normal Distribution Function. See task 1101 -40- 5131 /

51 - 32 , P . 14 .

EIGENVALUES

Task 1 101 -40- 5131 /53-6

Origin: Convair Authorized 9/26/52

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, ARDC , USAF

Manager: E. C. Yowell

Full task description appears in July-Sept 1952 issue.

Status: CONTINUED. The first portion of the problem was completed

by the determination of all eigenvalues and eigenvectors of the 45th

order matrix, and these results have been transmitted to the proposer

of the problem. The second portion, which involves the inversion of

this matrix, has not yet been started.

METEOROLOGICAL MEANS

Task 1101 -40 - 5131 /53-10

Origin: Meteorology Department, UCLA Authorized 9/30/52

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, ARDC, USAF

Manager: F. H. Hollander

Full task description appears in July-Sept 1952 issue.

Status: CONTINUED. Coding has been completed for the 850 mb level

calculations. The programming for the "Mountain Term" is complete, and
13 Status of Projects

coding has begun. Time averages for two levels remain to be coded.

The problem has been extended by the addition of data for the 100 mb

level for January-Febr uary 19^9 (see project 1 1 . 1 /31 — 50—1 7, Jan-Mar

1950 issue, p. 1 5) . The calculation of results for these data will use

the codes already formed.

PROLATE SPHEROIDAL WAVE FUNCTIONS

Task 1101-40-5131/53-11

Origin: Stanford Research Institute Authorized 9/30/52

Sponsor: Aeronautical Research Laboratory, Terminated 9/30/53

Wright Air Development Center, ARDC, USAF

Manager: R . R. Reynolds

Objective: To compute

2

Y = 30 (log \)

2 -TV A [ I 0 - E (I, ) J n=1 ,3, 5} 7 n 2n n In

a nd

Z = Y

»

for the 108 sets of values V. 1= 1+10 ra (m= 2(2)8)

3^/4 ,2.5,2. ro fc rn OO .2. A

n and Bn are tabulated

and

%

1)"^

*1n

= J yi2 - R

1 (3)n (c, V d * %

*2n

= 2 f

y^2 - 1 R

1 (3)

n (0,

4o

W (3)

( n c,y) = R

1 (1 ^ n (c, y

)

dR 1

' 1 n (c ’\) )/d j

^0

(3)

V(v

e-1) y >

^jd(1 £, Wj t

JL b

= [1 + (b/a)^ (50

" - I)] 2

,

(3) - u(D

^ J> (c,4) - R)^(c,>) + i R-, ^ 2)

(c ,'5>) , i" = -1 ,

( 1 ) (2) and R^ n and R

j

n are the first and second solutions of

.2

[d- L <y -'’Ik!"

with the expansions

R li

n = [K li^

1 - 1 ^

00

k=0

1 n

: 2k (-) K 0T ~ I)

1

00 "i/ 1 = 5 Qln // log + D/(> - 1 ) ] + MV2 -!)

-*

i 1^

b“'C‘~n 1n,% 2

*
Status of Projects 19

( 1 )

[ Q1 n and are tabulated real functions of c, and 1^^<oo.]

Background: Y and Z are the input admittance and impedance of a monopole antenna in the form of half a prolate spheroid which is fed by

a coaxial line whose outer conductor spreads out into a conducting plane.

Comments: The coefficients c-J^ and b^ n are tabulated as real

functions of c for n=1(2)7, k=0’(1)5 in "Prolate spheroidal wave functions,"

by C. Flammer, Technical Report 1-6, Stanford Research Institute, 1951.

Status: TERMINATED.

FLUTTER MATRICES

Task 1101-40-5131/53-17

Origin: Boeing Airplane Company Authorized 12/15/52

Sponsor: Aeronautical Research Laboratory, Completed 9/30/53

Wright Air Development Center, ARDC , USAF

Manager: E. E. Osborne

Objective: To find the eigenvalues of each of fifteen 6x6 non- symmetric complex flutter matrices.

Background: The problem is associated with the contractor's work

on flutter.

Status: COMPLETED. The final report is being reproduced.

DISCRIMINANT FUNCTIONS

Task 1 101-40-5131/53-1

8

Origin: Randolph Field, USAF Authorized 12/15/52

Sponsor: " " Manager: D. Teichroew

Full t»sk description appears in Oct-Dec 1952 issue.

Status: CONTINUED. SWAC codes have been prepared and checked.

Preliminary results have been submitted to the contractor.

BIO-ASSAY PROBLEM

Task 1 101-40-5131/53-24

Origin: Stanford University Authorized 3/31/53

Sponsor: Office of Naval Research

Manager: D. Teichroew

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. The table which was computed in the previous

quarter has been submitted to the contractor.
20 Status of Projects

SINGLE SHOT PROBABILITIES

Task 1 101-40-5131/53-25

Origin: Naval Air Missile Test Center, Point Mugu Authorized 3/31/53

Sponsor: Bureau of Aeronautics, USN

Manager: D. Teichroew

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. Several additional problems connected with the

estimation of probabilities were submitted and have been more rigorously

formulated

.

DISCRETE MINIMAL SPACES

Task 1 101-40-5131/53-26

Origin: Gilfillan Brothers, Inc. Authorized 3/31/53

Sponsor: Evans Signal Laboratories, Army Signal Corps

Manager: A. Hestenes

Full task description appears In Jan-Mar 1953 issue.

Status: INACTIVE. For status to date see Apr- June 1 953 issue.

ROOTS OF DETERMINANTS

Task 1 101-40-5131/53-32

Origin: North American Aviation Authorized 3/31/53

Sponsor: Atomic Energy Commission

Manager: E. C. Yowell

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. Eight cases were solved during jthis quarter.

PLANE SHOCK WAVES

Task 1101-40-5131/53-33

Origin: Aeronautical Research Laboratory, Wright Authorized 3/31/53

Air Development Center, ARDC , USAF Terminated 9/30/53

Sponsor: " 11 " Manager: M. W. Steinberg

Objective: To solve numerically two nonlinear second-order

differential equations arising in the problem of motion of a compressible

fluid. These equations are

e = (8©)/(3w0) [w - of 1 (w + e w

-1 - 1)]

-2/3(3e+^)/(9+^)w ,2-3/2(9+3^)/(9+^)9' 20"" 1

-2[ (w'9)w

_1

]

'
Status of Projects 21

w=4/

(

230 ) ^28oe/(45w0) [9 '- 5/70 1 (0 - 0(1 -w) 2 +«)]

-(115© + 96/S)A(0+^)w'9 , -2ew' 2 w

_1

j

7/2 where and P are given constants and 0 = 9

J /(©+ z3

) . The prime means differentiation with respect to x. There are four different cases with

two-point boundary conditions; w and 9 are given for x=-o© and x=+oo . Background: The theoretical discussion of these equations is

found in L. H. Thomas' "Note on Becker's Theory of the shock front,"

J. Chem. Phys. 1_2, No. 11 (Nov. 1944). Numerical work and additional

discussion is given in K. Zoller's "Zur Struktur des Verdichtungsstosses,"

Zeit. Phys. 130 (1951).

Status: TERMINATED.

LINEAR EQUATIONS (CONVAIR)

Task 1101 —40 —

51 31 /53—35

Origin: Consolidated Vultee Aircraft Corporation Authorized 3/31/53

Sponsor: Bureau of Aeronautics, USN

Manager: R. R. Reynolds

Full task description appears in Jan-Mar 1953 issue.

• Status: CONTINUED. As a single precision SWAC code was ineffectual, a double precision code was prepared. It is hoped that the system

will be solved early next quarter.

MISCELLANEOUS CORONA COMPUTATIONS

Task 1 101-40-5131/53-36

Origin: NBS , Div. 15

Sponsor: " Manager: P. Bremer

Objective: This task was established to perform small miscellaneous

computations for the NBS Corona Laboratories.

Comments: From time to time other small tasks will be reported.

The objective of these tasks will be included in the current issues of

Projects and Publications.

Authorized 3/31/53

Terminated 9/30/53

Status: TERMINATED.
22 Status of Projects

MONTE CARLO STUDIES

Task 1101-40-5131/53-39

Origin: RAND Corporation Authorized 3/31/53

Sponsor: Aeronautical Research Laboratory, Wright

Air Development Center, ARDC, USAF

Manager : A. D . Hestenes

Full task description appears in Jan-Mar 1953 issue.

Status: INACTIVE. For status to date see Apr- June 1953 issue.

RANKING PROBABILITIES

Task 1 101-40-5131/53-40

Origin: Cornell University Authorized 3/31/53

Sponsor: Office of Naval Research

Manager: D. Teichroew

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. A table of P(h/r,k) f or V2h=0( .01 )6.4 end the

required values of r and k have been computed, checked, and submitted

to the contractor.

ACCEPTANCE TESTS

Task 1 101-40-5131/53-41

Origin: Jet Propulsion Laboratory, California Authorized 6/29/53

Institute of Technology

Sponsor: Army Ordnance Corps, U.S. Army

Manager: E. C. Yowell

Full task description appears in Apr-June 1953 issue.

Status: INACTIVE. For status to date see Apr-June 1953 issue.

B. P. A. Studies

Task 1101-40-5131/53-42

Origin: Bonneville Power Administration Authorized 6/29/53

Sponsor: " " Manager: E. C. Yowell

Full task description appears in Apr-June 1953 issue.

Status: INACTIVE. For status to date see Apr-June 1953 issue.
Status of Projects 23

HELICOPTER STABILITY STUDIES

Task 1 101-40-5131/53-5-5-

Origin: J. B. Rea Company Authorized 6/30/53

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, ARDC , USAF

Manager: F. H. Hollander

Objective: To determine the coefficients of the characteristic

equations, the characteristic roots, and some of the eigenvectors of

certain loth-order matrices, and to obtain various transient responses

by Fourier synthesis techniques, using these frequency solutions.

Background: Dynamic stability characteristics for various flight

velocities of aircraft are needed. This can best be accomplished on high speed digital computing equipment such as SWAG . Forty-four sets of

sixteen simultaneous linear equations in sixteen unknowns have been

supplied, on which the above operations are to be performed.

Status: NEW. Programming for the calculation of the characteristic

polynomial has been completed. Hand-punching of data has been done.

The remaining parts of the problem have been partially programmed.

Origin: Aeronautical Research Laboratory, Wright Authorized 6/30/53

Air Development Center, ARDC, USAF Terminated 9/30/53

Sponsor: " " Manager: G. Blanch

Objective: To compute the integrals RP^Cc), RT^(c), RP^Cc), RT^c),

for c = -.9( .1 )9.

Background: In the process of computing lift and moment coefficients for task 1101-53-1101/52-5-1, there were computed certain constants

( 2 ) A and B, in terms of which Reissner ’ s p^ for the circulatory flow can

be determined. Once this is done, the following integrals are required:

HINGE MOMENTS

Task 11 01 -Mo-51 31/53-^5

c c

c . c where R is some constant

Status: TERMINATED.
24 Status of Projects

COMPUTATIONS FOR SCAMP

Task 1101-40-5131/54-1

Origin: Office of Naval Research Authorised 9/29/53

Sponsor: " . " Manager: B. Tompkins

Objective: Calculations to support SCAMP (see task 1 1 01 -1 0—51 50/

53-1 , p. 10).

Background: This task is the result of a desire on the part of the

sponsoring agency to bring together mathematicians for the purpose of

contributing to the deep mathematical developments required.

Status: NEW. Some calculations were completed during the quarter.

Calculations involving exhaustive search were started and are continuing.

About 375 sets of Steiner Triples of order 19 were calculated? an inconsiderable fraction of the ones which probably can be found. New

calculations will be started.

EIGENVALUES AND EIGENVECTORS

Task 1 101-40-5131/54-2

Origin: Consolidated Vultee Aircraft Corporation Authorized 9/29/53

Sponsor: Aeronautical Research Laboratory, Completed 9/30/53

Wright Air Development Center, ARDC , USAF

Manager: A, D. Hestenes

Objective: Determine the eigenvalues and eigenvectors of a 30x30

symmetric matrix.

Status: COMPLETED (NEW). Routines coded for task 1 \ 01 -40~5l 31 /53~6,

p. 17 , were used. Results have been forwarded to the originator.

CRUISE CONTROLLER

Task 1 101 -40-5131/5^-3

Origin: J. B. Rea Company Authorized 9/29/53

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, ARDC, USAF

Manager: F. H. Hollander

Objective: A cruise controller is an instrument which is designed

to determine the appropriate throttle setting to obtain the maximum

miles per pound. The associated SWAC problem is to determine the type

of throttle input which results in the smallest hunting period.

Comments: The originator is analyzing this problem to determine

its appropriate mathematical form.

Status: NEW
Status of Projects 25

RAM ROCKET TRAJECTORIES - PARAMETER STUDY

Task 1 101-^0-5131/5^

Origin: Marquardt Aircraft Company Authorized 9/29/53

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, ARDC , USAF

Manager: F. H. Hollander

Objective: To determine engine characteristics for optimum performance over a wide range of trajectory variables. The analysis and

coding is to be performed by the originator. INA will supply SWAC time

and coding consultation as is necessary.

Status: NEW. Some routines have been prepared and are presently

being checked on SWAC.
II. Computation Laboratory

(Section 11.2)

1 , Research

RESEARCH IN CLASSICAL NUMERICAL ANALYSIS

Task 1102-10-1104/50-1

(formerly 11 . 2/11 - 50 - 1 ) Origin: NBS Authorized 1/1/50

Managers: J. Todd, M. Abramowitz, and H . A. Antosiewicz

Full task description appears in Jan-Mar 1950 issue.

Status: CONTINUED. H. A. Antosiewicz investigated the integral

1 -kx2

t J0 (rx)Y0 (ax) " J0

( ax ^ Y

0 (rx) dx . — e —

2

2 x

* or 1 ^°°

u 0

Jq (ax) + Yq (ax)

which arises in the solution of the equation of heat flow in a region

Internally bounded by a circular cylinder. This problem was suggested

by B. A. Peavey of the NBS Building Technology Division (Div. 10).

M. Abramowitz has completed an investigation of forced heat

convection in laminar flow through a tube. M, Abramowitz and W. Cahill

are investigating the problem of vibrations of plates by the method of

finite differences.

In continuation of earlier work of Gohner (lag. Arch. 1, 619

(1930) and Freiberger (Australian J. Sc. Research {A] 2, 354 (19^+9),

P. Henrici investigated the stress distribution in a hejLical spring of

finite circular cross-section and developed the stress-concentration

factor into a power series in terms of the spring index.

W. Cahill together with S. Levy of the NBS Engineering Mechanics Section are investigating the problem of modes of vibration of an airplane wing.

Publications: (1) "On a certain integral involving Bessel functions"

by H. A. Antosiewicz; IN MANUSCRIPT. (2) "Forced heat convection in

laminar flow through a tube" by M. Abramowitz; IN MANUSCRIPT. (3) "On

helical springs of finite thickness by P. Henrici; IN MANUSCRIPT. (4) "Computation of vibration modes and frequencies on SEAC" by W. Cahill and

S. Levy; IN MANUSCRIPT. (5) "On a problem in the theory of mechanical

quadratures" by P. Davis; IN MANUSCRIPT. (6) "On the estimation of quadrature errors for analytic functions" by P. Davis and P. Rabinowitz;

IN MANUSCRIPT.

RESEARCH IN MODERN NUMERICAL ANALYSIS: INVESTIGATION

OF BERGMAN'S METHOD FOR THE SOLUTION OF THE DIRICHLET

PROBLEM FOR CERTAIN MULTIPLY CONNECTED DOMAINS

Task 1102-10-1 104/50-2

(formerly 11.2/11-50-2)

Origin: NBS

Manager: P. Davis

26

Authorized 3/1/50
Status of Projects

Full task description appears in Jan-Mar 1950 issue.

27

Status: INACTIVE. For status to date see Jan-Mar 1952 issue.

MISCELLANEOUS STUDIES IN PURE MATHEMATICS

Task 1102-10-1104/50-4

(formerly 11.2/11-50-4)

Origin: NBS Authorized 1/1/50

Managers: 0. Taussky Todd, J. Todd, M. Abramowitz, and A. Hoffman

Full task description appears in Jan-Mar 1950 issue.

Status: CONTINUED. 0. Taussky Todd made a survey of computational

problems in algebraic number theory and gave a report on this subject

to the Applied Mathematics Symposium of the American Mathematical Society

which was held at Santa Monica, California, in September. Computational

problems connected with the following topics were mentioned: (1) integral

bases, (2) factorization of rational primes in number fields, (3) algebraic units (with special attention given to a conjecture of Ankeny, Artin,

and Chowla concerning the (quantity u in the fundamental unit Jr(t + i*/p)

in the field generated by vpj at present under examination on SEAC ,

)

(4) ideal classes and class numbers, and (5) the determination of the

ideal class of a field F with non-cyclic ^-class group which becomes

the principal class in an unramified relatively cyclic extention field

of F of relative degree X . A. Hoffman, M. Newman, and 0. Taussky Todd, in collaboration with

E. Straus of U.C.L.A., present in their paper (see item (1) under publications below) a new approach to a problem treated earlier by R. Baer

and R. W. Ball. By a judicious numbering of the points and lines of a finite projective plane, the number of absolute points becomes trace A,

where A is the incidence matrix of the finite projective plane, which

is studied in terms of the relation between the characteristic polynomial

of A and the transitivity properties of the collineation which is the

square of the given correlation. This approach is capable of easy generalization to combinatorial configurations more complicated than projective planes and even in the case of planes obtains information apparently stronger than previously known.

K. Goldberg and M. Newman, in collaboration with E. Straus

and D. Swift of U.C.L.A., have completed a paper (see item (2) under

publications below) in which the integers represented by quotients of

certain binary quadratic forms are studied.

The preparation of a code to determine a certain number-theoretic

invariant associated with finite projective planes was begun by M. Newman

in collaboration with G. Pall. The problem will be run on SEAC for the

incidence matrices of finite projective planes of order nine (for which

the incidence matrices are of order 91), both for Desarguian and nonDesarguian finite projective planes.

Publicat

by A. Hoffman,

(2) "The repre

by K. Goldberg

nical journal.

IN MANUSCRIPT,

functions and ear functional

to a technical

ions: (1). "The number of absolute points of a correlation,"

M. Newman, 0. Taussky, and E. Straus; IN MANUSCRIPT,

sentation of integers by binary quadratic rational forms,"

, M. Newman, E. Straus, and D. Swift; submitted to a tech-

(3) "Ueber die Funktionen von Gegenbauer," by P. Henrici;

(4) "On certain series expansions involving Whittaker

Jacobi polynomials," by P. Henrici; IN MANUSCRIPT. (5) "Linequations and interpolation series," by P. Davis; submitted

journal

.
28 Status of Projects

NUMBER THEORETICAL TEST PROBLEMS FOR SEAC

Task 1102-10-1104/50-53

(formerly 11.2/11-50-5)

Origin: NBS Authorized 6/1/50

Managers: 0. Taussky Todd and K. Goldberg

Full task description appears in Apr- Jun 1950 issue.

Status: CONTINUED. A routine to determine the primes p / 100,000

for which 2P~1-1 is divisible by p2 has been prepared and checked.

A routine to determine the fundamental unit of a real quadratic

field has been prepared and checked.

The table of least positive primitive roots is being extended

as time is available.

SOLUTION OF LAPLACE EQUATION BY MONTE CARLO METHOD

Task 1102-10-1104/51-6

Origin: NBS Authorized 9/28/50

Manager: M. Abramowitz

Full task description appears in July-Sept 1950 issue.

Status: INACTIVE. For status to date see Apr-Jun 1952 issue.

THREE-BODY PROBLEM

Task 1102-10-1104/52-4

Origin: NBS Authorized 10/5/51

Manager: A. Goldstein

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see July-Sept 1952 issue.

ANALYSIS OF GEOMAGNETIC FIELD

Task 1102-10-1104/52-8

Origin: NBS Authorized 8/10/51

Manager: C. J. Swift

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see Oct-Dec 1951 issue.

SPECIAL PROBLEMS IN FINITE MATRIX THEORY

Task 1102-10-1104/52-34

Origin: NBS Authorized 11/6/51

Manager: 0. Taussky Todd

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. 0. Taussky Todd continued work with T.S . Motzkin
Status of Projects 29

on matrices with property L. In particular they extended theorem II in

their report "Matrices with property L, II" to more general matrices.

Ky Fan prepared a manuscript (see item (1) under publications) concerning

his results on commutators of matrices. He generalized a result of 0 . Taussky Todd concerning a pair of matrices with equal determinant to two

elements x,y in a group for which xy~^ is a commutator, and proved a cor- responding and generalized result. It was also shown that any unimodular

unitary matrix is a commutator in the unitary group, and that any unimodular normal matrix is a commutator of normal matrices.

Ky Fan developed in publication ( 2 ) some new inequalities for

eigenvalues of Hermitian matrices. The first section presents inequalities for the successive eigenvalues of a Hermitian matrix A=(a^j) in terms

of the quantities a^ and

/>, = < X Kji

2 )* . j>i J

A typical result is: if

numbers such that

c i > d

i l

c ^ • 5 c n ^ ^ nd (J n 1 'n-1

c i

d.

c . -1 u i+1 5 l

a u! + c

i A i d i

nonnegative

then

d n-i+ 1

where A. =,..-h are the eigenvalues of A. The second section compares

the eigenvalues of A with those of B=(b^j), where B is Hermitian and consists of blocks on the diagonal agreeing with A and of zeros elsewhere.

Let the eigenvalues of B be K^... 2 K

n

. One of the results is that for

any h, where 1<Ih^n, we have

hFurther, if A is positive definite, then

TT X n-i+1 = TT Kn-i+1 ' i=1 i=1

Ky Fan and A. Hoffman presented in publication ( 3 ) matricial

analogues of some well-known inequalities for complex numbers. For example, let us consider the inequalities (i) x real, z complex which implies

that

1^1 z + z I

I z - x | > | z — — ^ [ iQ and (ii) z = p e , ^ 0, 'wj - 1 which imply that

I i0 | v jz+e | | . | i© | > z-w > z-e

These inequalities are generalized by replacing a complex number by a matrix, a real number by a Hermitian matrix, the conjugate of a complex
30 Status of Projects

number by the adjoint of a matrix, a nonnegative number by a positivesemi-definite matrix, a number of absolute value unity by a unitary

matrix, and absolute value by any unitarily invariant norm,

A. Hoffman together with R. Bellman of the RAND Corporation,

derived in publication (*+) , alternative methods of proving the theorem of

A. M. Ostrowski and 0. Taussky that A=A*, B-B*, A>0, C=A+iB imply

[det C | 2. det A. This theorem and others are shown to be consequences

of (1 ) the formula

= P e-< c x,*> dx ; (det C ) d

j,n

or (2) convexity arguments based on a geometric formulation of results

of K. Fan,

J, Todd completed a study of the condition of the system of

equations which approximates to the biharmonic equation. It was found

that the P-condition number of this system is the square of that of the

system corresponding to the harmonic equation. The order of the condition number, as a function of the reciprocal of the mesh length, is

independent of the number of independent variables. Use was made of a theorem of Ledermann later generalized by H. Wielandt and K, Fan about

the effect on the characteristic roots of finite changes in a small

number of elements of a matrix. These theoretical results have been confirmed by extensive experiments on SEAC

.

H. Wielandt continued his investigations concerning the numerical

calculation of the eigenvalues of integral equations with real symmetric

kernels; he finished his work concerning the method based on numerical

quadrature (see publication (7)) and developed error estimates for the

variational method which corresponds to the Rayleigh-Ritz method in differential equations. The latter estimates are consequences of new inequalities which connect the eigenvalues of a Hermitian (finite or infinite completely continuous) matrix with the eigenvalues of a segment

of the matrix (see publication (8)).

Publications: (1) "Some remarks on commutators of Matrices, " by

K. Fan; submitted to a technical journal. (2) "Inequalities for eigenvalues of Hermitian matrices," by K. Fan; to be included in Contributions

to the solution of systems of linear equations and the determination of

eigenvalues, NBS Applied Mathematics Series. (3) "Some metric inequalities in the space of matrices," by K. Fan and A. Hoffman; IN MANUSCRIPT.

(4) "On the theorem of Ostrowski and Taussky," by R. Bellman and A.

Hoffman; IN MANUSCRIPT. (5) "Pairs of matrices with property L, II" by

T. S. Motzkim and 0. Taussky; accepted for publication in the Proceedings

of the National Academy of Science. (6) "The condition of certain ma- trices, II" by J. Todd; IN MANUSCRIPT. (7) "Error bounds for eigenvalues

of symmetric integral equations," by H. Wielandt; IN MANUSCRIPT. (8) "Einschliessung von Eigenwerten hermite scher Matrizen nach dem Abschnitts ver- fahren," by H. Wielandt; IN MANUSCRIPT, (9) "Characteristic roots of

quaternion matrices," by 0. Taussky; to appear in Archiv der Mathematik.

AN INTEGRAL ARISING IN

Task

THE THEORY OF COOPERATIVE PHENOMENA

1102-10-1 10^-/52-69

Origin: NBS

Manager: J. Todd

Authorized 5/9/52

Terminated 9/30/53
Status of Projects 31 Objective: To tabulate

TT TT tt 0 r> o F(b) = -4

dw J| dw2dw-^

for %=-/*"= .01 ( .01 )1 .

u O ooo

3b - (cos w^+cos W

2+C 0 S w^)

Background: Integrals of the form

f(3b - cos w^-cos w

ir ir ir n 0 0

u,0 4A

2 cos w^)dw

1 dw2dw^

occur in problems involving averages of the characteristic values, , of the difference equation Y# - X Y (where is the Laplacian -t, ra, n -t, m, n

r second difference operator) with the periodic boundary condition

Y - Y — y y

/,m,n /+N,m,n /,nH-N,n = ^,m,N+n*

Such averages arise in the theory of ferromagnetism, cooperative phenomena in solids, and critical fluctuations in liquids.

Status: TERMINATED. It is hoped to resume the study of this integral by Monte Carlo methods later.

DIFFERENTIAL EQUATION FOR NERVE FIBER REACTION

Task 1102-10-1104/53-15

Origin: National Naval Medical Institute Authorized 12/8/52

Sponsor: " " Managers: H. A. Antosiewicz and P. Rabinowitz

Full task description appears in Oct-Dec 1952 issue.

Status: CONTINUED. Computations are performed as requested by

the sponsor.

BAKER -HAUSDORFF FORMULA

Task 1102-10-1104/53-16

Origin: NBS Authorized 12/8/52

Manager: 0. Taussky Todd and K. Goldberg

Full task description appears in Oct-Dec 1952 issue.

Status: CONTINUED. The publication below describes a method for

the calculation of the coefficients in the Baker-Hausdorf f equation.

Publication: "The equation e

Z = e XV in a free associative ring,"

by K. Goldberg; IN MANUSCRIPT.
32 Status of Projects

POISSON-BOLTZ MAN EQUATION

Task 1102-10-1104/53-31

Origin: NBS

Manager: M. Abramowitz

Objective: To determine the

non-linear differential equation

Authorized 1/23/53

Terminated 9/30/53

maximum values of 6 for which the

i = -6e9

Z dz

has a solution satisfying the conditions 9 '=0 for z=0, 9=0 for z=1 . The

parameter k is a positive integer.

Background? The differential equation occurs in the theory of

thermal explosion under the assumption that the critical condition for

inflammability is reached when the amount of heat developed is equal to

the heat lost to surroundings.

Status: TERMINATED.

HYPER GEOMETRIC FUNCTIONS

Task 1 102-10-1 104/53-35

Origin: NBS Authorized 3/25/53

Managers: P. Rabinowitz, W. Cahill

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. The code for the hypergeome trie function has

been used to compute the complex error function Erf(z), z-x+iy for x,y

= 0 ( . 2) 1 ,2(.4)3.2.

BASIC RESEARCH IN LINEAR PROGRAMMING

Task 1 102-10-51 1 6/50-2

Origin: Office of Scientific Research, ARDC , USAF and Authorized 3/31/50

Office of the Air Comptroller, USAF

Sponsor: Office of Scientific Research, ARDC, USAF

Managers: J. Todd and A. Hoffman

Full task description appears in Jan-Mar 1950 issue, see 11.2/12-50-1.

Status: CONTINUED. Applications in support of requirements of the

Air Comptroller's Office, formerly reported under this task, will be re- ported under task 1 1 02-1 0-51 1 6/54-9 , p. 35. The basic aspects of the

program will continue to be reported under this heading.

Most of the machine work during the quarter was confined to the

development and testing of a matrix inversion code using tapes for temporary storage. This code will be used in a revision of the simplex

method to obtain greater efficiency for sparse matrices. In addition,

some experimentation was undertaken, using the simplex method, in support

of research of H. W. Kuhn on the faces of the convex hull of the cyclic

permutation matrices.

A fully automatic machine procedure for solving the so-called

two-parameter problem (i.e. for each X , ft , minimize (a+ Ab+ ^c, x) ,
Status of Projects 33

where a , b, c are given vectors and x ranges over a convex polyhedron) has

been devised and awaits trial. Briefly, it consists of (i) noting, with

each solution corresponding to a given X, , the sides of the convex

polygon in the (A,y«0 plane in which that solution minimizes, (ii) introducing into the basis all vectors which contribute sides of the polygon,

and (iii) keeping track of past introductions so as to minimize superfluous changes of basis.

The following variation of the smoothing problem has been considered: r^,r 9 ,...,r are given nonnegative constants, representing re- quirements in each month; and x^,...,x

n

, s -p"*«? s n are given nonnegative

variables denoting production and storage, respectively, for all values

of the variables such that

x

1 + . . . + x^ - s^ = r

1 + ... + (where h = 1,...,n)

minimize, for each 3-5 o, > js^ - s^-jj + A^r~(x^-x^ _.j )+ . In the case

r h < r^

+ ^ (where h=1 ,

. . .

, n-1 ) , the answer may be described as follows: for

A > + . . , + r ,+ r set x

h =

for h=1 ,2, . . . , n. For smaller A, if 0<( A ^2m (where m=0, 1,2,...), set

x r , x , r , ... , x t-i, x, r , , X/, r^,, n n-ro 7 n-1 n-m 7 7 n-k+1 n-m 7 1 1

7 2 27 7

n-k

n-k-1

s;

i=1

si x

i - r

- x

i + A

r ' i= n-k+1 i=1

where k, is determined by the property

n

k r < Y r n-m - ( k+ 1 ) r m > Y r.

l ’ n-m ^2 i

i-n-k+1 i=n-k

Publications: (1) "On 'overshoot 1 in the furthest hyperplane

method," by R. Bryce; IN MANUSCRIPT. (2) "Lower bounds for the rank and

location of the eigenvalues of a matrix," by K. Fan and A. J. Hoffman; to

be included in Contributions to the solution of systems of linear equations and the determination of eigenvalues, NBS Applied Mathematics Series.

(3) "On the caterer problem," by J. VV. Gaddum, A. J. Hoffman, and D.

Sokolowsky; INH1ANUSCRIPT. (4) "Remarks on the smoothing problem," by

A. J. Hoffman; IN MANUSCRIPT. (5) "Computational experience in solving

linear programs," by A. J. Hoffman, M. Mannos, D. Sokolowsky, and N. A.

Wiegmann; J. Soc. Ind . App. Math. I, 1

7

_

3

l+ (Sept. 1953). (o) "A characterization of normal matrices," by A. J. Hoffman and 0. Taussky; to

appear in the Journal of Research of the NBS. (7) "On block relaxation,"

by L. S. Joel; IN MANUSCRIPT. (8) "Eigenvectors of matric polynomials,"

by M. Mannos; J. Res. NBS, 51, 33-36 (July 1953). (9) "Experimental re- sults with the double description method, by S. Pollack; IN MANUSCRIPT.

(10)"A method of solving the transportation problem," by A. Gleyzal; IN

MANUSCRIPT.
35- Status of Projects

COMPRESSIBLE FLOW - METHOD OF ORTHOGONAL AND KERNEL FUNCTIONS

Task 1 102- 10-51 16 /52-1 6

Origin: Aeronautical Research Laboratory, Wright Air Authorized 9/29/51

Development Center, USAF , and Harvard University

Sponsor: Aeronautical Research Laboratory, USAF

Managers: P, Davis and F. L. Alt

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. s The computation of the 12 particular solutions

of the stream equation +4-F (A) 'f =0 has been completed. These solutions are given by

Im

f(Z±£) dzdz + . .

.

( n=0 ,

1

5).

The local accuracy of these solutions is now being checked.

TABLES OF INTEGRALS INVOLVING THE HIGHER TRANSCENDENTAL FUNCTIONS

Task 1 102-10-51 16/52-33

(formerly 1 102-21 -51 1 7/52-33)

Origin: NBS Authorized 10/11/51

Manager: F. Oberhettinger

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The table of integral transforms has been varityped. The rest of the table is being prepared for varityping. Dr.

Oberhettinger has added a section on the Lebedev transform to the table

of integral transforms.

INTEGRAL OPERATORS AND INTERPOLATION SERIES;

Task 1 102-20-51 16/53-2

Origin: Flight Research Laboratory, Wright Air

Development Center, USAF

Sponsor: " " " Managers: F. L. Alt, P. Davis

Objective: To investigate the relationship between the method of

integral operators of the Bergman type, the symbolic operator method of

Cauchy-Hea visid e , and the method of analytic functionals as developed

in Italy by L. Fantappie. Also to investigate the generation of complete

sets of solutions of differential equations by the method of orthogonal

complements

.

Background: The first of these investigations is to extend to the

theory of ordinary linear differential equations with nonconstant co- efficients as well as to the case of linear partial differential equations.

In the case of ordinary equations, Dr. Davis has recently obtained a number

of results, showing how the Bergman integral operator provides new re- presentations for the Duhamel kernel of the equation. This work is re- lated to some results of I. M. Schiffer (Tokyo Journal, 1932), and has

potential application to the theory of variable networks and to the

Authorized 9/26/52

Revised 9/30/52

Terminated 9/30/53
Status of Projects

35

statistical theory of noise in such networks. It is desired to press

this investigation to the field of partial differential equations, and new representations for the solution of initial value problems are anticipated . The method of orthogonal complements would be intimately re- lated to the theory of kernel functions of Bergman type and to the theory

of interpolation series for analytic functions, inasmuch as the kernel

of the orthogonal complement of the subspace of solutions is to be obtained by the method of interpolation series. This investigation may be

regarded as a continuation of some recent work in the theory of interpolation series by Walsh and Davis ("Interpolation and orthonormal

systems," by J. L. Walsh and P. Davis; to appear in Journal D'Analyse

Ma theraa tique ) . It is to be carried out for differential equations of

arbitrary order and for an arbitrary number of variables, but for analytic

coefficients. By the introduction of kernel tensor fields, it will also

be possible to treat systems of linear partial differential equations.

Some theoretical work on this investigation has been done; in particular,

it has been shown how, by the use of interpolation series, we may generate a kernel K (Z,w) such that the transform g(z) = P K (z,w)f (w)dV maps

the space of analytic functions of n complex variables onto the subspace

of solutions.

Status: TERMINATED. The results obtained have been incorporated

by P. Davis into a paper (see publication below).

Publication: "Linear functional equations and interpolation series,"

by P. Davis; submitted to a technical journal.

WATER WAVES

Task 1 102- 10- 51 16 /53-55

Origin: NBS, Division 6 Authorized 6 /1/53

Sponsor: Office of Naval Research

Manager: P. Davis

Full task description appears in Apr-June 1953 issue.

Status: CONTINUED. A joint paper by P. Davis and R. F. Dressier

entitled "A computational scheme for a free boundary problem" is now being prepared. The possibility of using the formulation of the boundary

value problem in Lagrangian coordinates as a basis for computation is

also being studied.

SUPPORTING RESEARCH IN LINEAR PROGRAMMING

Task 1 102- 10 - 5116/55-9

Origin: Office of the Air Comptroller, USAF Authorized 9/25/53

Sponsor: Office of Scientific Research, ARDC , USAF

Managers: J. Todd and A. J. Hoffman

Objective: (a) The. computation of solutions to specific programming

problems originating in the Planning and Research Division, AFACO. (b)

The development of improved computation schemes for solving general dynamic linear programs. (c) The development of computation schemes for

solving special dynamic programs.
36 Status of Projects

Background: (a) Based on past experience, the specific programming

problems of the Planning and Research Division, AFACO, will frequently be

in the nature of experimental computations to verify or deny intuition

and preliminary heuristic analysis, or to test the validity of the mathematical formulation of a practical situation.

(b) The improvements in the computation schemes for solving

general dynamic linear programs are expected to be modifications of the

basic principles of the simplex algorithm, in order to facilitate

processing of problems by high-speed computers. Where the problem

deals with a single objective, we are now in possession of what appears

to be sound ideas for taking advantage of the qua si-canonical format of

dynamic programs, and these await the test of machine trial. Where the

problem is concerned with a class of objectives, (e.g. , where the objective cannot be precisely formulated), further study is necessary. The

success already attained, however, for the case that the class of objectives

is linear and one-dimensional, suggests the direction of research for the more general multi-parameter problem. One line of attack, recently devised for the two-parameter problem, deserves experimentation. The development of these new tools will substantially enlarge the domain of

applicability of linear programming.

(c) Many important special dynamic problems occur so frequently

in military and logistic planning that it is worthwhile, if possible, to

find either a formula or an ingenious method for the solution.

Over the past few years, and especially during the last year,

the Air Force and National Bureau of Standards personnel have, as a team,

acquired "technique" in the formulation and solution of these special

problems, and further co-operation along these lines should produce additional contributions to saving and efficiency in the military program.

Comments: This is a continuation of certain phases of former task

1 102—10—51 1 6/50—2 "Research in Linear Programming."

Status: NEW. One problem was attacked during this quarter, and its

solution was completed shortly after the end of the quarter. This is a generalization of the so-called "Transportation Problem", which arises in

the awarding of purchase contracts. The problem actually occurs in the

operation of such agencies as the Textiles and Apparel Procurement Agency,

which purchases uniforms and similar articles for the Array and Air Force.

The problem is to select, from all the potential suppliers who submit bids

on a given requirement, a number of suppliers in such a way as to minimize the total cost, including cost of shipment. The problem differs

from the classical transportation problem in a' number of special conditions such as upper and lower limits on bids, volume discounts, etc.

Two amplifications of this problem have been considered but not satisfactorily solved. These refer to the conditions that the quota of each suc- cessful bidder be distributed over several different sizes (in the case

of apparel) in fixed proportions, or be scheduled uniformly over a period of time. These problems lead to very difficult questions in the

theory of inequalities. We are sure of one method of solution, namely,

the general simplex method; this, however, is prohibitively long. We sus- pect at present that no shortcuts to the simplex method exist for these

two problems. If this conjecture is confirmed, attention will be con- centrated on methods for approximate solution.

These problems are closely related to earlier work done here

for the Air Comptroller's Office and for the Logistics Research Project

of George Washington University (see task 1 1 02-^0-51 26/53~55, p. 5 1).
Status of Projects 37

2 . Mathematical Tables and Experimental Computations

TABLES OF E^(z), (z = x + iy)

Task 1 102-10-1 104- A3-3

(formerly 1102-10-1110/4-3-3)

Origin: Canadian National Research Council Authorized 7/1 A7

Manager: I. A. Stegun

Full task description appears in Apr-June 1949 issue, see task 11.2/2-43-3

Status: CONTINUED. Revisions of the manuscript are in progress.

TABLE OF. THE GAMMA FUNCTIONS FOR COMPLEX ARGUMENTS

Task 1 102-10-1 104A6-1

(formerly 1102-10-1110/46-1)

Origin: NBS Authorized 7/1/47

Manager: J. Todd

Full task description appears in Apr-June 1949 issue, see task 11.2/2-46-1

Status: CONTINUED. The page proof has been received for this

volume

.

Publication: "Table of the Gamma function for complex arguments,"

NBS Applied Mathematics Series 34; in press, U. S. Government Printing

Office

.

TABLES OF COULOMB WAVE FUNCTIONS

Task 1 102-1 0-1 104A7-2

(formerly 1102-10-1110/47-2)

Origin: NBS Authorized 7/1 A7

Managers: M. Abramowitz and P. Rabinowitz

Full task description appears in Apr-June 1949 issue, see task 11.2/2-47-2

Status: CONTINUED. The tables of the regular and irregular functions together with their derivatives for y°=1(1)10, >1=1(1)10, L=0,5>

are being proofread, and an introduction is being prepared.

TABLE OF ANT I LOGARITHMS

Task 1 102-10-1 104A7-3

(formerly 1 1 02-1 0-1 1 1 0A7~3)

Origin: NBS Authorized 7/1 A

7

Manager: J. Todd

Full task description appears in Apr-June 194-9 issue, see task 11.2/2-47-3

Status: CONTINUED. This volume is now in press.

Publication: "Tables of 10 x

," NBS Applied Mathematics Series 27;

in press, U. S. Government Printing Office.
38 Status of Projects

TABLE OF LAGRANGIAN COEFFICIENTS

FOR SEXAGESIMAL INTERPOLATION

Task 1 102-10-1 104A8-2

(formerly 1 1 02-1 0-1 1 1 0A8-2)

Origin; NBS Authorized 5/25A8

Manager; J. Todd

Full task description appears in Apr-June 19.49 issue, see task 11.2/2-48-2

Status; CONTINUED. The page proof has been received for this

volume

.

Publications "Tables of Lagrangian coefficients for sexagesimal

interpolation," NBS Applied Mathematics Series 35; in press, U. S.

Government Printing Office.

PROBABILITY TABLES FOR EXTREME VALUES

Task 11 02-1 0-1 1 04/50 Aa

(formerly 1 1 02-1 0-11 1 0/50 Aa ) Origin; NBS, Section 11.3

Sponsor; Aeronautical Research Laboratory,

Wright Air Development Center, ARDC , USAF

Manager: J. Todd

Objective: To prepare a set of short tables relating to the

statistical theory of extreme values, including;

(1 ) The inverse of the (cumulative) distribution function of

extremes, y=Xog(-Xog x), x= .0001 ( .0001 ) .005( .001 ) ,988( .0001 ) .9999, 5d.

(2) The direct (cumulative) distribution of extremes,

x=exp(-exp(-v) ) , y=-3( . 1

) -2 A( ,05)0( . 1 )4( . 2) 8( . 5)1 7_> 7Do_

v (3) Probability density of extremes, x'=e Ye _e ", range of y same as (2), 7B.

(4) Probability density of extremes as a function of the

probability, x'=-xlogx, x= .0001 ( .0001 ) .01 ( .001 ) .999, 5d.

(5) Probability points for n'th extremes.

(6) Distribution and density function for the range T'' (R) and

y*(R), as defined by y^"+ -y>' -e

_R

'f' =0; R between -4.6 and 20; 6 or 7D.

Background: The theory of extremes has been developed in recent

years, primarily by Dr. E. J. Gumbel. At his suggestion this compilation

of pertinent tables has been undertaken. The table listed as (1) above

is an extension of the one computed by this Laboratory under project

11.2/33A9-7 (see Qct-Dec 1948 issue, p. 50). Tables (2) to (5) are taken

from unpublished manuscripts of A. J. Greenwood, table (6) from an un- published table computed at Dahlgren Proving Ground, with additions by

A. J. Greenwood. All tables are to be checked and edited by this Laboratory. Dr. Gumbel participated in the detailed planning.

Status: COMPLETED.

Publication: "Probability tables for analysis of extreme-value data,

NBS Applied Mathematics Series 22. Available from U. S. Government

Printing Office, Washington, 25, D. C.

Authorized 12/31A9

Completed 9/30/53
Status of Projects 39

BIBLIOGRAPHY OF CODING PROCEDURES, MATHEMATICAL TABLES

AND NUMERICAL ANALYSIS

Task 1102-10-1 104/50-5

(formerly 1102-10-1110/50-5)

Origin: NBS Authorized 3/1/50

Managers: J. Todd, J. H. Wegstein, and Revised 1/9/53

P. Rabinowitz

Full task description appears in the Oct-Dec 1952 issue.

Status: CONTINUED. Preparation of the bibliographies continues,

and reports prepared by J. H. Wegstein on the bibliographies are issued

periodically in Mathematical Tables and Other Aids to Computation.

WAVE FUNCTION FOR LITHIUM

Task 1 102-10-1 104/50-7

(formerly 1 1 02-1 0-1 1 1 0/50-7)

Origin: NBS Authorized 6/1/50

Sponsor: Bureau of Ordnance, USN

Managers: D. I » Rubin and W'. H. Durfee

Full task description appears in Apr- June 1950 issue.

Status: CONTINUED. Computation on SEAC continues as time is

available

.

COLLECTED SHORT MATHEMATICAL TABLES OF THE COMPUTATION LABORATORY

Task 1102-10-1104/51-4

(formerly 1102-10-1110/51-4)

Origin: NBS Authorized 9/28/50

Manager: J. Todd

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. The first volume of such tables is in press.

Publication: "Tables of functions and of zeros of functions,"

Volume I of Collected Short Tables of the Computation Laboratory; NBS

Applied Mathematics Series 37? in press.

REVISION OF MATHEMATICAL TABLES

Task 1 102-10-1 104/52-7

(formerly 1

1

02-1 0-1 1

1

0/52-7)

Origin: NBS Authorized 8/10/51

Managers: J. Todd? W. F. Cahill and I. Stegun

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The following action is being taken in connec-
q

Status of Projects

tion with mathematical tables the sales stock of which has been ex- hausted :

Tables of normal probability functions, originally

MT1 4 (Tables of probability functions, vol. II),

reissued as AM523: available from U. S. Government

Printing Office, Washington, D. C. Table of natural logarithms for arguments between zero

and five to sixteen decimal places, originally

MT10, to be reissued as AMS31 s in press.

Table of sine and cosine integrals for arguments from

10 to 100, originally MT13? to be reissued as AMS 32 : in press.

Tables of circular and hyperbolic sines and cosines

for radian arguments, originally MT5, to be re- issued as AMB 36 : in press.

Table of probability functions, vol. I, originally

MI'S: revision completed for reissue in the AMS.

Tables of sines and cosines for radian arguments,

IVTT4: revision in progress.

Table of natural logarithms, MT12: revision in progress.

The following two volumes are now out of print:

Tables of sine, cosine, and exponential integrals

vol. I, MT5 (1940;: reissue under consideration.

Tables of scattering functions for spherical particles,

AMS4(1 9^9) s not to be reissued.

TABLE OF ARC SIN FOR COMPLEX ARGUMENTS

Task 1102-10-1 104/52-1

4

(formerly 1102-10-1110/52-14)

Origin: NBS Authorized 10/1/51

Manager: A. A. Goldstein

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The information has been put on punched

cards for checking and editing.

EXTENSION OF THE TABLE OF HYPERBOLIC SINES AND COSINES

Task 1102-10-1104/52-18

(formerly 1102-10-1110/52-18)

Origin: NBS Authorized 9/1 7/51

Manager: W. F. Cahill

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The punched cards are being processed for

typing.
Status of Projects 4i

TABLE OF THE MODIFIED AIRY INTEGRAL

Task 1102-10-1104/52-23

(formerly 1102-10-1110/52-23)

Origin: NBS Authorized 10/4/51

Manager; P. Rabinowitz

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The tabular manuscript is in process of preparation for publication. An introduction to accompany the table has

been completed? it consists of two parts by E. E. Osborne and P. Rabinowitz

.

TABLE OF ERROR FUNCTION FOR COMPLEX ARGUMENTS

Task 1 102-10-1 104/52-25

(formerly 1 102- 10- 1

1

10/52-25)

Origin: NBS Authorized 10/5/51

Managers: M. Abramowitz and F. J, Stockmal

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The table for x,y = 0(.1)2 has been computed.

Additional values were also obtained for x,y=2( .4)3.2.

EXTENSION OF TABLES OF THE EXPONENTIAL FUNCTION FOR

NEGATIVE ARGUMENTS

Task 1 102-10-1 104/52-7

(formerly 1 1 02-1 0-1 1 1 0/52-7)

Origin: NBS Authorized 10/9/51

Manager: E. Marden

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see Apr-June 1953 issue.

SPHEROIDAL WAVE FUNCTIONS

Task 1 102-10-1 104/52-37

(formerly 1 102- 10 - 1

1

10/52-3 7)

Origin: NBS Authorized 11/28/51

Manager: T. Ledley

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. The computation of the spheroidal wave functions for the prolate case for m=0 , 1 ,2 , JL =0 ( 1 ) 1 0 ; m=3, ^=0(1)6; and

m=4, 5? & -0 ( 1 ) 5 has been completed. Also the computation for the oblate

case for m=0 ,

1 , ^=0(1 )1 0; iip 2,3, /=0(1 )6; and m=4, 5? -^=0(1 ) 5 has been

completed

.
42 Status of Projects

VAN DER POL EQUATION

Task 1 1 02-10-1 1

0

Lt-/52-1+3

(formerly 1102- 10- 1110/ 52-43 )

Origin: NBS Authorized 11/28/51

Manager: W. F. Cahill

Full task description appears in Oct-Dec 1951 issue.

Status: INACTIVE. For status to date see Apr- June 1952 issue.

RADIAL MATH IE U FUNCTIONS

Task 1102-10-1104/52-49

(formerly 1102-10-1110/52-49)

Origin: NBS Authorized 2/1/52

Managers: J. Todd, I. Rhodes, and G. Blanch

Status: CONTINUED. All the subtabulation has been completed.

The results are being processed for punched cards. This work is being

carried out under the supervision of I. Rhodes.

SIEVERT'S INTEGRAL

Task 1 102-10-1 104/52-57

(formerly 1 102-10-1 1 10/52-57)

Origin: NBS Authorized 2/12/52

Managers: 0. Steiner and R. B. Jasper

Full task description appears in Jan-Mar 1952 issue.

Status: CONTINUED. Computation of the tables on SEAC has been

completed, and the results have been transferred to punched cards for

differencing and future printing.

SCATTERING FUNCTIONS

Task 1102-10-1104/52-63

(formerly 1102-10-1110/52-63)

Origin: NBS Authorized 3/10/52

Manager: A. Gleyzal

Full task description appears in Jan-Mair 1952 issue.

Status: INACTIVE. For status to date see Apr-June 1953 issue.

TABLE OF SECANTS AND COSECANTS

Task 1102-10-1104/52-81

(formerly 1102-10-1110/52-81)

Origin: NBS Authorized 7/1 /52

Managers: K. C. Nelson and I. A. Stegun

Full task description appears in July-Sept 1952 issue.

Status: CONTINUED.
Status of Projects 43

Publication: "Table of secants and cosecants to nine significant

figures at hundredths of a degree"; to appear in the National Applied

Mathematics Series.

PAINLEVE EQUATION

Task 1102-10-1104/53-3

(formerly 1102-10-1110/53-3)

Origin: NBS Authorized 8/11/52

Managers: J. Todd and H. A. Antosiewicz

Full task description appears in July- Sept 1952 issue.

Status: INACTIVE. For status to date see Jan-Mar 1953 issue.

L-SHELL CONVERSION COEFFICIENTS

Task 1102-10-1104/53-52

(formerly 1 102- 1

0

- 1

1

10/53-52)

Origin: Oak Ridge National Laboratory Authorized 5/20/53

Manager: C. J. Swift

Full task description appears in Apr-June 1953 issue.

Status: CONTINUED. The code has been completed and is being

checked. More sample computations have been performed.

TABLES OF POWER POINTS OF ANALYSIS OF VARIANCE TESTS

Task 1304-34-6351/51-8

Origin: Section 11.3: NBS Authorized 3/26/51

Managers: A. Hoffman and L. Joel

Full task description appears in Apr-June 1951 issue.

Status: INACTIVE. For status to date see Jan-Mar 1953 issue.

3 . Mathematical Services

MOLECULAR STRUCTURE CALCULATIONS, II

Task 1102-40-5126/50-16

(formerly 1 102-53“1 106/50-1 6)

Origin: Naval Research Laboratory, USN Authorized 3/31/50

Sponsor: " " Manager: P. J. O'Hara

Full task description appears in Jan-Mar 1950 issue, see task 11. 2/33 _ 50— 1 6

.

Status: CONTINUED. Computations were performed as requested.
Status of Projects

PROGRAM COMPUTATION ON THE SEAC

Task 1 102-40-5126/51-7

(formerly 1 1 02—53—1 1 06/51 -7)

Origin: Office of the Air Comptroller, USAF Authorized 9/1/50

Sponsor: " " Terminated 9/30/53

Manager: A. Hoffman

Objective:. To compute Air Force programs on the SEAC by the

"triangular" method.

Status: TERMINATED. It is expected that the computing equipment

of the Air Comptroller's Office will be adequate to handle this task

in the future.

FLOW IN SUPERSONIC NOZZLES

Task 1 102-40- 5126/51-13

(formerly 1 1 02- 53 - 1 1 06/51 - 1

3

) Origin: Naval Ordnance Laboratory Authorized 12 /15/50

Sponsor: " " Terminated 9/30/53

Manager: N. Levine

Objective: To calculate the variation of flow in a test section

of a supersonic nozzle for a given range of Mach numbers.

Background: The contour of a supersonic nozzle can be determined,

neglecting boundary layer effects, such that the flow in a test section

of nozzle will be uniform for one assumed Mach number. If the ratio of

the nozzle diameter at the throat to the nozzle diameter at the test

section is varied, the Mach number of the flow will change and the flow

will become nonuniform. The problem is to calculate the amount of this

nonuniformity in the flow. Analytically, the problem is one in potential theory where it is necessary to integrate the following system

the area of a meridional nozzle section:

1 8M

m( i + a y

m je

( M

^-1 ) (1 + 3 f

where 0 are the potential lines of the flow

are the stream lines of the flow

M is the Mach number

9 is the angle of the stream line with respect to the axis of

symmetry of the nozzle.

of hyperbolic equations over

99

90

9M

90

Status: TERMINATED.
Status of Projects 45

LIQUID-VAPOR TRANSITION

Task 1102-40-5126/51-22

(formerly 1102— 53— 1 106/51 — 22 ) Origin: Naval Medical Research Institute Authorized 2/1/51

Sponsor: " " Manager: I. Stegun

Full task description appears in Jan-Mar 1951 issue.

Status: INACTIVE. For status to date see July-Sept 1952 issue.

MOLECULAR STRUCTURE, III

Task 1102-40-5126/51-37

(formerly 1102-53-1106/51-37)

Origin: Naval Research Laboratory, USN Authorized 8/10/51

Managers: P. J. O’Hara and I. A. Stegun

Full task description appears in July-sept 1951 issue.

Status: CONTINUED. Computations were performed as requested.

SHOCK WAVE PARAMETERS, II

Task 1 102-40-5126/51-38

(formerly 1102-53-1106/51-38)

Origin: Bureau of Ordnance, USN Authorized 6/28/51

Sponsor: " " Manager: I. A. Stegun

Full task description appears in Apr-June 1951 issue.

Status: INACTIVE. For status to date see Apr-June 1953 issue.

PRESSURE DISTRIBUTION ON BODIES OF REVOLUTION

Task 1 102-40-5126/52-3

(formerly 1

1

02-53-1 1 06/52-3

)

Origin: David Taylor Model Basin, USN

Sponsor: U. S. Navy

Managers: A. Gleyzal and 0. Steiner

Objective: To calculate the axially symmetric potential flow about

elongated bodies of revolution for 30 different shapes. An iteration

method is employed to solve a Fredholm integral equation of the first kind.

Authorized 8/10/51

Terminated 9/30/53

utai/lri ds

2r J

1

where U(x) is the unknown velocity function and the kernel is

The functions y^(x) are given polynomials of 6th degree which

the shape of the body, r^= (x-t ) 2+y2 , and ds is arc length.

2 /o 3

y /2r . describe
46 Status of Projects

The iteration formula used is

U

n+1

(t)

- U

n (t) + cos Y (t)

P 2, x X ( x ) 0 2r ^ U

n (x)ds

(J|(t) = 1 + k cos>- , r- 4^

where k is the longitudinal mass coefficient of an "equivalent" prolate

spheroid

.

Background: The origin of the problem and the method of solution

are described in TMB Report No. 7&1 , "Axially symmetric potential flow

about elongated bodies of revolution," by L. Landweber.

Status: TERMINATED.

powder diffraction

Task 1 102-40-5126/52-6

(formerly 1 102— 53 — 1 106/52 — 6 ) Origin: NBS , Section 9.7 Authorized 9/17/51

Manager: E. Marden

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see July-Sept 1951 issue.

PRECISE DETERMINATION OF THE PARAMETER OF DISPERSION

EQUATION FOR SEVERAL TYPES OF OPTICAL GL^SS

Task 1102-40-5126/52-17

(formerly 1 1 02-53-1 106/52-1 7)

Origin: NBS, Division 2 Authorized 9/29/51

Sponsor: " Manager: I. A. Stegun

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see Apr-June 1953 issue.

SPHERICAL BLAST

Task 1102-40-5126/52-20

(formerly 1 1 02-53-1 1 06 /52 - 20 ) Origin: Naval Ordnance Laboratory Authorized 9/27/51

Sponsor: " " Manager: D. H. Jirauch

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see Oct-Dec 1952 issue.
Status of Projects *+7

MAGNETIC FIELD EXTRAPOLATION

Task 1102-40-5126/52-22

(formerly 1 1 02-53-1

1

06 /52 - 22 ) Authorized 10/1/51

Terminated 9/30/53

Origin: Naval Ordnance Laboratory

Sponsor: " " " Manager: N. Levine

Objective: To evaluate the integral

x +oo H (^ , (z-/0d ^

d r\

1 + oo + 00 R

2 TV J J f -co - oo ^

i 2+ (y-\) 2+ ( t -/>)

j 3/2

for several specified values of z and a sufficiently fine grid of pairs

x,y. H^(x,y,z) is the unknown vertical component of the magnetic intensity at any point in the x,y plane below a given % , ^ plane in which

the magnetic intensity at every point is known.

Background: Formerly empirical means were used to measure the

field generated by a given electromagnetic apparatus. This calculation,

if feasible and sufficiently accurate, should replace costly measurement

procedures

.

Status: TERMINATED.

CALCULATIONS FOR d S PACINGS

Task 1 102-40-5126/52-44

(formerly 1102-53-1106/52-44)

Origin: NBS , Div. 9 Authorized 12/7/51

Sponsor: " Manager: I. Stegun

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. Computations were performed as requested.

GAS ADSORPTION BY HIGH POLYMERS

Task 1102-40-5126/52-70

(formerly 1 1 02-53-1 1 06/52-70)

Origin: Bethesda Naval Medical Center Authorized 4/1/52

Sponsor: " " Manager: I. Stegun

Full task description appears in Jan-Mar 1952 issue.

Status: INACTIVE. For status to date see Apr-June 1952 issue.
4-8 Status of Projects

CHEMICAL TRANSITION PROBABILITIES

Task 1 102-40-5126/52-82

(formerly 1102- 53 - 1106/52- 82 ) Origin: NBS , Section 3.2 and Hydrocarbon Research Authorized 6/1/52

Corp.

Sponsor: Office of Naval Research

Managers: A. Gleyzal and A. A. Goldstein

Full task description appears in Apr-June 1952 issue.

Status: CONTINUED. A code for the second part of the problem has

been prepared and checked.

NEUTRON DIFFUSION, II

Task 1 102-40-51 26/53-4

Origin: Atomic Energy Commission, New York Authorized 9/30/52

Office, (NDA)

Sponsor: " " " Managers: 0. Steiner and N. Levine

Full task description appears in July-Sept 1952 issue.

Status: CONTINUED. Computations are being performed as requested,

and results are transmitted to the sponsor.

STRENGTH OF WING COMPONENTS

Task 1 102-40-5126/53-1

1

Origin: National Advisory Committee for Aeronautics Authorized 12/8/52

Langley Field, Va.

Sponsor: " " " Manager: W. H. Durfee

Full task description appears in Oct-Dec 1952 issue.

Status: CONTINUED.

RADIANT HEATING OF SOLIDS

Task 1102-40-5126/53-20

Origin: NBS, Section 10.2 Authorized 12/15/52

Sponsor: " Manager: W. F. Cahill

Full task description appears in Oct-Dec 1952 issue.

Status: CONTINUED. Solutions are being computed for various

values of the parameters and transmitted to the sponsor as completed.
Status of Projects 49

COMPUTATION OF THERMODYNAMIC FUNCTIONS

Task 1.102-40-5126/53-27

Origin: NBS, Division 5 Authorized 3/18/53

Sponsor: " Manager: E. Marden

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. Computation of the tables continues.

STUDY OF TRICADCIUM ACUMINATE

Task 1102-40-5126/53-28

Origin: NBS, Division 9 Authorized 3/30/53

Sponsor: 11 Manager: R. Anderson

Full task description appears in Jan-Mar 1953 issue.

Status: INACTIVE. For status to date see Jan-Mar 1953 issue.

DYNAMIC BEHAVIOR OF AIRCRAFT STRUCTURES

Task 1102-40-5126/53-29

Origin: NBS, Section 6.4 Authorized 1/23/53

Sponsor: " Manager: I. Rhodes

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. Computations are being performed as requested,

and results are being transmitted to the sponsor.

REDUCTION OF ECLIPSE DATA

Task 1 102-40-5126/53-34

Origin: Air Photographing and Charting Services, Authorized 3/30/53

USAF

Sponsor: " " Manager: A. Goldstein

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. The following results have been obtained to

diminish the required calculations: Defining the light function L as

L = 5 c L , where

n n

7 n -0

an

L =

n b-A

(a‘

n 2x2 -/> ) arccos

b^ - t\2A/> dr

it is found that

oo ALa + aL + bL, = 2L + n C L , A a b z. n n

n=0
50 Status of Projects

and an expansion for L as an infinite series is given involving P -functions . A least squares solution has been carried out for Bangui with

differential corrections for the solar and lunar radii and the time of

mid-totality. The L 's are now being computed for Khartoum.

SPECTRAL ANALYSIS OF STATIONARY TIME SERIES

Task 1 102-4-0-5126/53-3 7 Origin: Statistical Research Center, University Authorized 4/20/53

of Chicago

Sponsor: Office of Naval Research, USN

Manager: I. Stegun

Full task description appears in Apr- June 1953 issue.

Status: CONTINUED. A report is being prepared.

ACOUSTICAL IMPEDANCES

Task 1102-40-5126/53-39

Origin: NBS , Section 6.1 Authorized 6/30/53

Sponsor: " Managers: S. Prusch and K. Nelson

Full task description appears in Apr-June 1953 issue.

Status: INACTIVE. For status to date see Apr-June 1953 issue.

STANDARD LORAN TABLES - Rates 2H2,2H3,2HH

Task 1 102-40-5126/53-40

Origin: Hydrographic Office, U. S. Navy Authorized 4/20/53

Sponsor: " " Terminated 9/30/53

Managers: W. H. Durfee, D. Jirauch, and K. Nelson

Objective: To prepare tables giving coordinates of hyperbolic

lines of position.

Background: Standard Loran navigation tables are necessary for

preparation of charts used by navigators in determining their positions

with the aid of certain electronic equipment.

Status: TERMINATED. The results were transmitted to the sponsor.

LORAN UNIVAC CODE

Task 1 102-40-5126/53-41

Origin: Hydrographic Office, U. S. Navy Authorized 4/20/53

Sponsor: " " Managers: W. H. Durfee, I. Rhodes, and D. Rubin

Full task description appears in Apr-June 1953 issue.

Status: CONTINUED. The code has been completed and is almost

completely checked.
51 Status of Projects

NOMOGRAMS FOR RF PERMEAMETER

Task 1 102-40 - 51 26/53-42

Origin: NBS, Division 14.8

Sponsor: " Manager: I. Stegun

Objective: To perform analysis as necessary

Background: The problem arises in connection

R F pe r me a me t e r

.

Status: TERMINATED.

Authorized 6 /29/53

Terminated 9/30/53

to prepare nomograms,

with the use of the

OPTIMUM SECTIONS FOR DELTA WINGS

Task 1 102-40 - 5126 /53-49

Origin: National Advisory Committee for Authorized 4/29/53

Aeronautics, Langley Field, Va

.

Sponsor: " " Manager: 0. Steiner

Full task description appears in Apr-June 1953 issue.

Status: CONTINUED. The computations desired were completed, and

the results were transmitted to the sponsor.

NEUTRON DIFFUSION III

Task 1 102-40- 5126/53-51

Origin: NBS, Section 4.8 Authorized 6 /9/53

Sponsor: Armed Forces Special Weapons Project

Manager: F. Stockmal

Full task description appears in Apr-June 1953 issue.

Status: CONTINUED. The code for the first part of the problem has

been completed and checked and a sample run of 50 histories has been

made. Coding for the second part of the problem is underway.

TRANSPORTATION PROBLEM II

Task 1 102-40- 5126/53-55

Origin: Logistics Research Project, Authorized 6 /9/53

George Washington University

Sponsor: Office of Naval Research

Managers: A. J. Hoffman and L. Gainen

Full task description appears in Apr-June 1953 issue.

Status: CONTINUED. All necessary codes for SEAC for the first part

of the problem have been completed and checked.
52 Status of Projects

STANDARD LORAN TABLES - Rates 1L4,1L5

Task 1 102-40-5126/5^-2

Origin: Hydrographic Office, U. S. Navy Authorized 9/16/53

Sponsor: " " Manager: W. H. Durfee

Objective: T© prepare tables giving coordinates of hyperbolic

lines of position.

Background: Standard Loran navigation tables are necessary for

preparation of charts used by navigators in determining their positions

with the aid of certain electronic equipment.

Status: NEW. Computations are in progress.

TABLES OF THERMODYNAMIC PROPERTIES OF GASES

Task 0302-40-2606A9-5

(formerly 11.2/33-49-5)

Origin: NBS , Section 3.2 Authorized 11/29/48

Sponsor: National Advisory Committee for Aeronautics

Manager: F. L. Alt

Full task description appears in Apr-June 1949 issue.

Status: INACTIVE. For status to date see July-Sept 1952 issue.

BASIC IONOSPHERIC DATA

Task 1 401 -34-1 473A9-1

4

(formerly 1 1 .2/33A9-1 4)

Origin: NBS, Section 1 4 .

3

Sponsor: " Manager: M. Stein

Authorized 3/31 A9

Terminated 9/30/53

Objective: To analyze multifrequency, vertical-incidence observations of the ionosphere. The observations made at a number of observing

stations are to be transcribed on punch cards. Various statistical

analyses are to be performed and, in the case of current data, tables

are to be prepared for publication.

Background: The observations consist of virtual heights and critical frequencies of the regularly ionized layers of the earth's upper

atmosphere. These are measured hourly at 1 5 stations operated by or under agreement with the National Bureau of Standards. About 60 stationyears of past data exist. Certain simple manipulations with this kind

of data have been standard for many years, although the huge number of

observations - some 6000 per station per month - has precluded much

detailed investigation. Some analyses serve to extract typical ionospheric conditions to be applied in predictions of usable frequencies

for long distance radio transmissions. Others will investigate statistically, changes in the characteristics of the ionospheric layers

accompanying other geomagnetic phenomena, solar activity, etc., to in- crease knowledge on the origin and structure of the ionosphere and its

effect on radio wave propagation.
Comment: Dr. A. H. Shapley of Section 14.1 is

Status: TERMINATED.

RAY TRACING

Task 0202-10-2308/50-13

(formerly 11.2/33-50-13)

Origin: NBS , Section 2 0 2

Sponsor: " Managers: R. K. Anderson and D. Rubin

Full task description appears in Jan-Mar 1950 issue.

directing this work.

Authorized 3/1/50

Status: CONTINUED. Computations are being performed as requested.
III. Statistical Engineering Laboratory

(Section 11.3)

1 . Fundamental Research in Mathematical Statistics

BIBLIOGRAPHY AND GUIDE TO STATISTICAL LITERATURE

Task 1103-10-1 107A9-1 a (formerly 11.3/2-49-1)

Origin: NBS Authorized 1/9/49

Manager: L. S. Doming

Full task description appears in Jan-Mar 1949 issue.

Status: CONTINUED. Abstracts taken from Mathematical Reviews and

Zentralblatt fur Mathematik are put onto file cards to form a bibliography

of literature dealing with probability, statistics, and their applications. Each card is classified by a code number according to its subject

and arranged alphabetically by author for easy reference.

Abstracts from Mathematical Reviews cover the literature from

1939 to the present. Abstracts from Zentralblatt start with 1940 and work

back through the years as far as there is activity in the statistical

field. The file at present includes Zentralblatt abstracts back through

1935.

MANUAL ON FITTING STRAIGHT LINES

Task 1103-10-1107/50-2

(formerly 11 . 3 /2- 50 - 2 ) Origin: NBS Authorized 3/1/50

Manager: F. S. Acton

Full task description appears in Jan-Mar 1950 issue.

Status: INACTIVE. For status to date see Oct-Dec 1952 issue.

TABLE TO FACILITATE DRAWING RANDOM SAMPLES

Task 1103-10-1107/51-1

Origin: NBS Authorized 7/1 /50

Managers: C. Eisenhart and L. S. Deming

Full task description appears in July-Sept 1950 issue.

Status: INACTIVE. For status to date see July-Sept 1952 issue.
Status of Projects 55

MISCELLANEOUS STUDIES IN PROBABILITY AND STATISTICS

Task 1103-10-1 107/51-2

Origin: NBS Authorized 7/1/50

Manager: E. Lukacs

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. (1) J. Lieblein found that a method which he had

used in an earlier paper (see "On the exact evaluation of the variances

and covariances of order statistics in samples from the extreme value

distribution," Ann. Math. Stat. 24, 282-287 (June 1953) for obtaining the

exact covariances of order statistics for one type of extreme-value distribution can be applied to two other types as well. (2) J. Lieblein

prepared two working papers, one dealing with a conjecture of E. J. Gumbel

concerning the asymptotic relationship between the expected mean, expected

variance, and expected largest value of an extreme variate ; and the other

treating a problem of rejection in connection with duplicate measurements.

(3) M. Zelen found bounds on a cumulative distribution function, which are functions of the moments up to order four. Explicit expressions for the

Tchebycheff Markoff inequalities are given.

Publications: (1) "On absolute measurements," by E. N. Dorsey and

C. Eisenhart; Sci. Monthly ZZ? 103-109 (August 1953). (2) "On some procedures for rejection of suspected data," by E. P, King; J. Am. Stat.

Assoc. 48, 531-533 (September 1953). (3) "A property of the normal distribution," by E. P. King and E . Lukacs; submitted to a technical journal,

(b-) "A new method of analyzing extreme value data," by J. Lieblein; ac- cepted for publication as an NACA Technical Note. (5) "On strongly con- tinuous stochastic processes," by E. Lukacs; submitted to a technical

journal. (6) "Certain Fourier transforms of distributions (II ),” by

E. Lukacs and 0. Szasz; accepted for publication in the Canadian Journal

of Mathematics. (7) "Nonnegative trigonometric polynomials and certain

rational characteristic functions," by E. Lukacs and 0. Szasz; accepted

for publication in the Journal of Research of the NBS. (8) "A historical

note on the application of the "weakest-link" idea to tensile strengths,"

by J. Lieblein; submitted to a technical journal. (9) "Tables of expected

values of 1 /X for positive Bernoulli and Poisson variables," by E. Grab

and I. R. Savage; submitted to a technical journal. (10) "Tables of the

inverses of finite segments of the Hilbert matrix," by I. R. Savage and

E. Lukacs; to be. included in Contributions to the solution of systems

of linear equations and the determination of eigenvalues, NBS Applied

Mathematics Series. (11) "On optimum grouping in one-criterion variance

components analysis," by E u P. King; IN MANUSCRIPT. (12) "On the variances

and covariances of order statistics from the VVeibull distribution," by

J. Lieblein; IN MANUSCRIPT.

LAW OF PROPAGATION OF ERROR

Task 1 103-10-1 107/52-1

Origin: NBS Authorized 6/23/51

Managers: C. Eisenhart and I. R. Savage

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see July-Sept 1951 issue.
56 Status of Projects

procedures of non- parametric statistics

Task 1 103- 10 - 1 107/52-2

Origin: NBS Authorized 9/1 7/51

Manager: I. R. Savage

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED.

Publication: "Bibliography of non-parametric statistics and related

topics," by I. R. Savage; accepted for publication by the Journal of the

American Statistical Association.

STUDIES IN THE MATHEMATICS OF EXPERIMENT DESIGN

Task 1 103-10-1 107/53-1

Origin: NBS Authorized 10/15/52

Manager: W. S. Connor

Full task description appears in the Oct-Dec 1952 issue.

Status: CONTINUED. (1) M. Zelen proved that a partially balanced

incomplete block (PBIB) design with (nn-1 ) associate classes can be obtained

from a PBIB design with m associate classes by replacing each variety of

the design by n new varieties. (See item (5) under publications below.)

( 2 ) M. Zelen derived the intra- and inter-block analysis of covariance

for PBIB designs with two associate classes. (3) W. H. Clatworthy studied

the class of PBIB designs with two associate classes, two plots per block,

and ten or fewer replications of each variety. He either proved impossible or constructed all but four of the designs. (See item (2) under

publications below.) (5-) W. J. Youden discovered a new class of incomplete block designs in which each variety is replicated twice. (5) W. S.

Connor and W. J. Youden prepared a paper which describes by example the use and methods of analysis of new experimental arrangements with two

plots per block. The designs treated are the " two-group" and "rectangular"

arrangements. (See item (7) under publications below.)

Publications: (1) "The chain block design," by W. J. Youden and

W. S. Connor; Biometrics 2, 127-15-0 (June 1953). (2) "Partially balanced

incomplete block designs with two associate classes and two treatments

per block," by W. H. Clatworthy; submitted to a technical journal.

(3) "Some theorems for partially balanced designs," by W. S. Connor and

W. H. Clatworthy; accepted for publication in the Annals of Mathematical

Statistics. (4) "An embedding theorem for balanced incomplete block

designs," by M. Hall, Jr., (Ohio State University) and W. S. Connor;

accepted for publication in the Canadian Journal of Mathematics. (5) "A

note on partially balanced designs," by M. Zelen; submitted to a technical

journal. ( 6 ) "Analysis for some incomplete block designs having a missing block," by M. Zelen; submitted to a technical journal. ( 7 ) "Designs for paired observations," by IV. S. Connor and W. J. Youden; IN

MANUSCRIPT.
Status of Projects

2 . Applied Research in Mathematical Statistics

57

COLLABORATION ON STATISTICAL ASPECTS OF NBS

RESEARCH AND TESTING

Task 3737-60-0002/51-1

(formerly 301

1

- 60 -0002/51 - 1 ) Origin: NBS Authorized 7/1/50

Manager: W. J. Youden

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. Activity under this task fell into two main

categories:

A. Design of Experiments: A partially balanced incomplete block

design involving two items per block was constructed for use in an ex- periment on spark plug performance. Two-cylinder engines were involved

in the test, and a two way elimination (of cylinder position differences

and of engine differences) was effected in a case where this elimination

could not have been achieved in the corresponding completely balanced

design

.

B. Development or Selection of the Appropriate Method for Analysis

and Interpretation of Data: (a) A nonor thogonal analysis of covariance was worked out for testing the statistical significance of the effects of

heaters and the effect of temperature rate in the measurement of the

thermal conductivity of aluminum, and (b) some investigations were made

concerning the adequacy of certain statistical distributions such as the

Weibull and log-normal in representing fatigue data.

Publications: (1) "Confidence and tolerance intervals for the

normal distribution," by F. Proschan; J. Am. Stat. Assoc. 48, 550-564

(Sept . 1953). (2) "Estimating the standard deviation of a normal population," by E. P. King; Ind. Quality Control IQ, 30-33 (Sept. 1953).

(3) "Sets of three measurements," by W. J. Youden; Sci. Monthly QZ, 1*+3“

1 47 (Sept. 1953). (5-) "Units for measuring variations in measurements,"

by W. J. Youden; Metals Progress 64, 91-96 (Sept. 1953). (5) "Acceptance

sampling of electroplated articles," by J. M. Cameron and F. Ogburn (5.6);

accepted for publication in Plating. ( 6 ) "Probability limits for the

average chart when process standards are unspecified," by E. P. King;

submitted to a technical journal. ( 7 ) "Rejection of outlying observations," by F. Proschan; accepted for publication in the American Journal

of Physics. ( 8 ) "The principles of experimental design," by IV. J. Youden;

accepted for publication in Selection, Training and Use of Personnel in

Industrial Research. Proceedings of the Third Annual Conference on Industrial Research. (9) "Making one measurement do the work of two," by

W. J. Youden and W . S. Connor; accepted for publication in Chemical

Engineering Progress. (10) "Performance of inspectors and gasoline pumps,

by W. J. Youden and M. W. Jensen; submitted to a technical journal.

STATISTICAL ASPECTS OF NBS ADMINISTRATIVE OPERATIONS

Task 3737-60-0002/52-1

(formerly 3011 - 60-0002/52 - 1 )

Origin: NBS Authorized 10/1/51

Manager: I. R. Savage

Full task description appears in Oct-Dec 1951 issue.

Status: INACTIVE. For status to date see July-Sept 1952 issue.
58 Status of Projects

APPLICATION OF THE THEORY OF STOCHASTIC PROCESSES TO

THE STUDY OF TRAJECTORIES

Task 1 103-21-51 19/52-1

Origin: U. S. Naval Ordnance Test Station, Inyokern Authorized 1/1/52

Sponsor: " " Terminated 9/30/53

Manager: E . Lukacs

Objective: To determine whether mathematical-statistical tools

associated with the theory of stochastic processes can be profitably

applied to the analysis of trajectory data of the type gathered at the

Naval Ordnance Test Station, Inyokern.

Background: The theory of stochastic processes deals with timedependent phenomena in which there is a probability relationship between

a state at a given instant and one or more states at preceding instants.

Some of the difficulties encountered in analyzing ordnance data can be overcome by considering a trajectory to be a stochastic process, thus

eliminating the difficulties created by the fact that only one observation is available for the position of the missile at each instant.

Status: TERMINATED.

STATISTICAL SERVICES FOR COMMITTEE ON SHIP STEEL, NRC

Task 1 1 03-50-51 05/52—

1

Origin: Ship Structure Committee, NRC Authorized 12/1/51

Sponsor: " " Manager: W 0 J. Youden

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. Data from an interlaboratory study in which

thirteen laboratories participated are being analyzed. The data provide

comparisons among the laboratories of their notching and testing techniques, and estimates of experimental error.

RESEARCH IN APPLICATIONS OF MATHEMATICAL STATISTICS TO

PROBLEMS OF THE CHEMICAL CORPS

Task 1 103-50-51 1 8/52-1

Origin: Biological Laboratories, Chemical Corps Authorized 10/1/51

Dept, of the Army

Sponsor: " " " Manager: C. Eisenhart

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. Reports on two special problems were transmitted to the sponsor.
IV. Machine Development Laboratory

(Section 1 1 . t)

in ronprral ion will) Electronic Computer Section

(Section 1 2.3)

1 . Development: Design and Construction of Automatic Digital

Computing Machines

THE BUREAU OF THE CENSUS COMPUTING MACHINE

Task 11 04-3^-51 07A7-1

(formerly 11.4/21-47-1)

Origin: The Bureau of the Census Authorized 7/1/47

Sponsor: " " Full task description appears in Apr-June 1949 issue.

Status: CONTINUED. The first UNTVAC installation has continued

to be operated on an around-the-clock basis by the Bureau of the Census.

The machine is maintained by a crew of one contract engineer plus the

Census personnel. The National Bureau of Standards has continued its

obligation of obtaining a stock pile of replacement parts under con- tract with Remington Rand Inc., Eckert-Mauchly Division. A complete

set of acceptable maintenance manuals has not yet been furnished by the

company for any of the UNIVAC installations procured under NBS contract.

THE AIR COMPTROLLER'S COMPUTING MACHINE

Task 1 104-34-5107/47-3

(formerly 11.4/24-47-3)

Origin: Office of the Air Comptroller, USAF Authorized 7/1/47

Sponsor: " " Full task description appears in Apr-June 1949 issue.

Status: CONTINUED. The second UNIVAC System has continued to

be operated and maintained by personnel of the Office of the Air Comptroller. However, a reduction in maintenance personnel as well as in

workload has necessitated a reduction from an around-the-clock seven-day

a-week schedule to a schedule consisting of three 40-hour shifts per week.

A minor amount of specialized procurement of parts has been handled by

the National Bureau of Standards during this quarter, now that channels

for the procurement of UNIVAC parts have been firmly established at the

Pentagon 0

59
60 Status of Projects

NATIONAL BUREAU OF STANDARDS EASTERN AUTOMATIC COMPUTER (SEAC)

Task 1 104-3>+-5l07A9-1

(formerly 1 1 .4/24-49-1 ) Origin: NBS Authorized 12/15/48

Sponsor; Office of the Air Comptroller, USAF

Full task description appears in Apr-June 1 9*+9 issue.

Status: CONTINUED. Operating efficiency for the quarter July

through September 1953 continued at the relatively high level of 81 °/0 (which represents total productive time over total time scheduled).

The electrostatic memory, now available for problems which require more

than the regular 512 words of acoustic memory, was used on at least

45 different occasions, with error-free runs ranging from 10 minutes

to nine hours in duration. The acoustic memory has been provided with

an odd-even (parity) check for all words read from the memory, which

has added to the efficiency of machine operation. An additional magnetic wire input-output dumper has been added to the installation. Using this dumper, a code has been devised which will check-sum all but

eight words of the 1 029—word memory, transfer the entire contents of the

memory to a removable wire cartridge, check the complete recording for

accuracy, and enable the machine automatically to resume computation,

all within 90 seconds.

WRIGHT DEVELOPMENT CENTER COMPUTING MACHINE

(formerly AIR MATERIEL COMMAND COMPUTING MACHINE)

Task 1 104-34-51 07A9-1 a (formerly 1 1 .4/23-49-1

)

Origin: Aeronautical Research Laboratory, Authorized 3/21/49

Wright Air Development Center, USAF

Sponsor: " " Manager: E. W. Cannon

Objective: To design and construct an automatic-sequenced electronic digital computing machine suitable for use by the Wright Air

Development Center in connection with air research problems.

Background: The Wright Air Development Center requires a highspeed and flexible computing machine to perform the involved calculations arising in the multifold problems associated with the design and

development of aeronautical equipment and accessories. The problems

involved range from the computation and evaluation of voluminous test

data to the solution of intricate problems related to the design and

performance prediction of airfoils, propellers, propulsive equipment

and both subsonic and supersonic missiles. The computer should have a large memory capacity and a precision sufficiently high to furnish

significant answers to problems like those in flutter analysis, which

involve the treatment of large linear systems.

Comments: This task is related to projects 11.4/21-47-1, 11.4/22-

47-2 and 1 1 .4/24-47~3a (see Apr-June 1949 issue, p. 8 7, 88, and 90,

respectively)

.

Status: TERMINATED. With the delivery of manuals and diagrams

to Wright Air Development Center during this quarter, the contractual

obligations of the General Electric Company to the National Bureau of

Standards were completed and the contract terminated.
Status of Projects 6l ARMY MAP SERVICE COMPUTING MACHINE

Task 1 104-34-51 07A9-1 b (formerly 11 .4/25-49-1)

Origin: Army Map Service, USA

Sponsor: " " Manager: E. W. Cannon

Objective: To develop specifications for and to construct an automatic-sequenced electronic digital computing machine suitable for

use by the Army Map Service in cartography.

Background: In the development of maps, particularly where various

maps based on different projection systems are combined into one large

map based upon a single projection system suitable for military application, there is required mass production of detailed calculations on point coordinates. A suitable computer must be able to handle huge masses

of raw data, perform a calculation on each item of data and print output

results in usable tabular form with speed comparable to that at which

data enters the machine.

Comments: This task and projects 11.4/21-47-1, 11.4/22-47-2 and

11.4/3-47-4 (see Apr- June 1949 issue, p. 87, 88 and 93, respectively),

are interrelated. Project 11.4/3-47-4 serves to coordinate the mathematical direction of the three computing machine projects.

Status: TERMINATED.

Authorized 12/15/48

Terminated 9/30/53

INVESTIGATION OF THE APPLICABILITY OF AUTOMATIC DIGITAL

ELECTRONIC COMPUTING TO PROBLEMS OF THE SOCIAL SECURITY

AGENCY

Task 1 104-53-5108/51-1

Origin: Social Security Agency

Sponsor: " " Manager: I. Rhodes

Objective: The purpose of the survey will be to investigate the

applicability of automatic digital electronic computing to the record

keeping and statistical operation of the Division of Accounting Operations of the Social Security Agency.

Background: The Social Security Agency has requested this survey

to be performed by staff members of the Machine Development Laboratory

of the National Bureau of Standards, with the cooperation of procedural

consultants of the Division of Accounting Operations. Equipment re- quirements are to be evaluated by the Electronics Division of the

Bureau. In connection with the survey, representative problems of the

Division are to be tried out on existing computing machines. At the

conclusion of the survey, a report of findings and recommendations are

to be submitted to the Division.

Status: TERMINATED. This task is being terminated with the

submission of the final report which is in preparation.

Authorized 12/31/50

Terminated 9/30/53
Lectures and Symposia

Note: In general, copies of papers or talks listed in this section are

not available from the National Bureau of Standards. If and when a paper is to be published, it will be listed in the section of this report

on Publication Activities.

American Mathematical Society

Sixth Symposium in Applied Mathematics : Numerical Analysis

The National Bureau of Standards was a co-sponsor with the

American Mathematical Society of a Symposium on Numerical Analysis held

at Santa Monica City College, Santa Monica, California, August 26-28.

The program was as follows:

August 26:

RIDDICK, Dr. Morford (Director of the General Division, Santa Monica

City College). Greetings.

TOMPKINS, C. B. (National Bureau of Standards). Application of automatic

digital computers to problems with discrete variables.

BRUCK, Ft. H. (University of Wisconsin). Computational aspects of certain

combinatorial problems.

TAUSSKY, Olga (National Bureau of Standards). Some computational problems In algebraic number theory.

LEHMER, Emma (Berkeley, California). Number theory on the SWAC

.

WIELANDT, H. (University of Tubingen). Error bounds for eigenvalues of

symmetric integral equations.

GIVENS, W. (Oak Ridge National Laboratory). A method of error analysis.

YOUNG, D. M. , Jr. (University of Maryland). On the solution of linear

systems by iteration.

HESTENES, M. R. (University of California at Los Angeles and National

Bureau of Standards). The conjugate gradient method for linear

sys terns . August 27:

BERGMAN , S . (Stanford University). Some methods for solutions of

boundary value problems of linear partial differential equations.

MOTZKIN, T. S. (University of California at Los Angeles). The assignment problem.

BELLMAN, R. (The RAND Corporation). On some computational problems in

the theory of dynamic programming.

KUHN, H. W. (Bryn Mawr College). The traveling salesman problem.

August 28:

WARSCHAWSKX, S. E. (University of Minnesota). Recent results in numerical methods of conformal mapping.

TAUB, A. H. (University of Illinois). Numerical solution of the equations of pseudo-stationary flow.

FISCHBACH, J. W. (Ballistic Research Laboratories). Some applications

of gradient methods.

FRANKEL, S. P. (California Institute of Technology). Stability problems

in partial differential equations.
Lectures and Symposia

WALSH, J. L. (Harvard University). Best-approximation polynomials of

given degree.

HASTINGS, C., Jr. (The RAND Corporation). Approximations in numerical

analysis

.

WASCW , W. R. (National Bureau of Standards). On the asymptotic transformation of certain probability distributions.

ROSENBLOOM, P. C. (University of Minnesota). The method of steepest

descent.

SARD, A. (Queens College). Function spaces and approximation.

Numerical Analysis Colloquium Series

(Los Angeles, California)

PALL, G. (Illinois Institute of Technology). Some properties of finite

projective planes. July 13.

LEFSCHETZ, S. (Princeton University). Some recent contributions to

non-linear differential equations. July 16.

MARSCHAK, J. and RADNER, R. (Cowles Commission for Research in Economics,

University of Chicago). Optimal communication in teams.

August 10.

BANOS, A., Jr. (University of California, Los Angeles). On Sommerfeld’s

electromagnetic surface waves. August 17.

FOSTER, F. G. (London School of Economics, University of London). A

review of some recent work in the theory of queues. September 21 .

Applied Mathematics Division Technical Meetings

At Los Angeles, California:

YOWELL, E. C. Coding for SWAC . July 7 and July 9.

BRUCK, R. H. (University of Wisconsin). Abelian difference sets. July 10.

ALBERT, A. A. (University of Chicago and University of California). On

proper solutions of the incidence equation for finite projective planes. August 11.

ROSENBLOOM, P. (University of Minnesota). Methods of steepest descent

applied to calculus of variations. September 9, 11, 14.

At Washington, D. C .

FAN, K. A combinatorial lemma and antipodal point theorem. July 9.

CONNOR, W. A problem in the design of experiments. July 23.

DAVIS, P. Errors of numerical approximation for analytic functions.

August 14.

WIELANDT, H. The converse of Abel's convergence theorem. August 19.
64 Lectures and Symposia

Seminar on Numerical and Computational Aspects of Linear Problems : Games, Linear Equalities, Linear Inequalities. Programming. . .. (cont ' d .

)

(Los Angeles, California)

SOUTHARD, T. H., Numerical aspects of projecting and war games. July 6.

LEHMER, D. H., and LEHMER, EMMA. Programming for discrete variable probmens. July 8, 13.

HOFFMAN, A. J. , NEWMAN, M., TAUSSKY, 0., TODD , J. Recent work on SEAC

.

July 1 5. PAIGE, L. J. A continuous method in a discrete problem. July 20.

SAXON, D. S. Problems for theoretical physics suitable for SWAC computation. July 22, 30.

ROBERTS, A. E. Projection methods in discrete problems. July 28.

TOMPKINS, C. B. A continuous manifold of quasi-solutions to a discrete

problem. August 4.

BANOS, A., Jr. On the saddle-point method of integration. August 6.

MARSCHAK, J. , and RADNER^R. Some problems in the mathematical theory

of organization. August 13.

TOMPKINS, C. B. Numerical local embedding of a surface with Riemannian

metric into Euclidean 3~§Pa ce, I. Geometrical introduction and classical formulas. September 29.

£a_pe r and,,13yit ejjL Ta 1 ks

Presented by Members of, the Staff

at_ JfefitiB.g.s. ff.L .Ou t p.i.de._Orqani^a tions

Papers presented at the joint meeting of the American Mathematical Society and the Institute of Mathematical Statistics, Kingston, Ontario,

August 31 - September 5* ANTOSIEWICZ, H., and DAVIS, P. Some implications of Liapunov's conditions

for stability.

FORSYTHE, G. E„ Arbitrarily close lower bounds for the fundamental frequency of a convex membrane.

LIEBLEIN, J. Estimation of extremal parameters by use of order statistics

Papers presented at the Summer Statistical Seminar, University of Connecticut, Stores, Conn., August 12, 1953s

EISENHART, C. Statistical method in experimentation: Individual experiments.

KING, E. P. Statistical resources for physicists.

Papers presented at the meeting of the Association for Computing Machinery

held at the Massachusetts Institute of Technology, Cambridge, Mass.,
Lectures and Symposia 65

September 9~1 1 : ALT, F. L. Numerical solution of some nonlinear heat transfer problems.

FORSYTHE, G. E., and HANDY 5 B. F. Finite-difference approximations to the

fundamental frequency of a vibrating membrane.

Papers presented elsewhere:

CAMERON, J. M. , and CONNOR, W„ S., Statistical engineering at the National

Bureau of Standards. Presented at a Seminar on Mathematical

Statistics held at the Bureau of Ordnance, Department of the

Navy, Washington, D. C. , July 16.

CONNOR, W. S. Some new experimental designs. (1) Presented at the Gordon

Research Conference, New Hampton, N. H., July 27- 31 . ( 2 ) Also

presented at a meeting of the American Statistical Association, Chicago, 111., September 7 . FORSYTHE, G. E. Solving linear algebraic equations can be interesting.

Presented to the Department of Mathematics, University of

Michigan, Ann Arbor, Mich., September 15.

LUKACS, E. Remarks on characteristic functions. Presented at a meeting

of the Mathematical Society, University of Basel, Basel,

Switzerland, July 6.

TEICHROEW, D. Use of continued fractions in high speed computing. Presented at a meeting of the Digital Computers Association,

Los Angeles, Cal., July 24.

THORENSEN, R. (1) Memory systems: Introduction and survey; (2) The

magnetic drum memory; ( 3 ) Selection and switching; ( 4 ) Magnetic drum memory system design; ( 5 ) Magnetic tape memories;

( 6 ) Megabit magnetic memories; ( 7 ) Electrostatic memories;

( 8 ) The Williams system; ( 9 ) Limitation of the Williams tube

memory system and discussion of current research; (10) Survey

of various types of memories. Lectures given at Summer Course

in Computer Applications and Components, held at Wayne University Computation Laboratory, Detroit, Michigan, August 10-21.

WIELANDT, H. Error bounds for the eigenvalues of integral equations. Presented at the Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md . , September 25 . YOUDEN, W. J. Statistical units of measurement. Presented at a Symposium

on Modern Statistical Methods held by the Charleston, West

Virginia Section of the American Institute of Chemical Engineers, at Charleston, W. Va . , September 24 . YOWELL, E. C. (1) Problem formulation; ( 2 ) Linear problems: Matrix inversion

( 3 ) Linear problems: Characteristic values; ( 4 ) Approximation

problems; ( 5 ) Monte Carlo techniques; ( 6 ) Subroutines;

( 7 ) Relative and abstract coding; ( 8 ) Checking procedures;

(9) Machine design; (10) Operating efficiency. Lectures

given at Summer Course in Computer Applications and Components, held at Wayne University Computation Laboratory, Detroit, Michigan, August 10-21

.
Publication Activities

1 . PUBLICATIONS WHICH APPEARED DURING THE QUARTER

1.1 Mathematical Tables

(1) Probability tables for analysis of extreme-value data. NBS

Applied Mathematics Series 22. Available from U. S. Government

Printing Office, Washington 25, D. C., 25 cents.

(2) Tables of normal probability functions. NBS Applied Mathematics

Series 23. (Supersedes NBS Mathematical Table MTl4, Tables of

probability functions, Volume II.) Available from U. S. Government Printing Office, Washington 25, D. C., $'2.75 (blue

buckram)

.

(3) Tables of coefficients for the numerical calculation of Laplace

transforms, H. E. Salzer. NBS Applied Mathematics Series 30.

Available from U. S. Government Printing Office, Washington 25,

D. C., 25 cents.

1 .3 Technical Papers

(1 ) On the numerical solution of parabolic partial differential equations. G. Blanch. J. Res. NBS 50, 3^3-356 (June } 953) 5 RP2424.

(2) Boolean geometry I. L. M. Blumenthal. Rend. Circ* Mat. Palermo

II , 1_, 1-1 8 (1952). Reprints available.

(3) On absolute measurement. N. E. Dorsey and C. Eisenhart. Scientific Monthly ZZ» 103-109 (Aug. 1953). Reprints available.

(5-) Solving linear algebraic equations can be interesting. G. E. Forsythe. Bui. Am. Math. Soc. 52? 299~329 (July 1953). Reprints

available

.

(5) Tentative classification of methods and bibliography on solving

systems of linear equations. G. E. Forsythe. Simultaneous

linear equations and the determination of eigenvalues, Proceedings of an NBS Symposium held in Los Angeles, August 1951?

NBS Applied Mathematics Series 29, pp. 1 -28. (Available from

U. S. Government Printing Office, Washington, D. c. ? $5.50).

(6) A non-harmonic Fourier series. J. M. Hammer sley. Acta Math. 89 . 25-3-260 (1953). Reprints available.

(7) Computational experience in solving linear programs. A. J. Hoffman, M. Mannos, D. Sokolowsky, and N. Weigmann. J. Soc. Ind.

App. Math. I, 17-34 (Sept. 1953).

(8) Estimating the standard deviation of a normal population. E. P. King. Ind. Quality Control 10, 30-33 (Sept. 1953). Reprints

available

.

66
Publication Activities 67 (9)

On some procedures for the rejection of suspected data. E. P. King. J. Am, Stat. Assoc. 48, 531-533 (Sept. 1953) » Reprints

available

.

(10) Eigenvectors of matric polynomials. M. Mannos. J. Res. NBS H,

33-36 (July 1953); RP2429.

(11) The coefficients of certain infinite products. M. Newman. Proc.

Am. Math. Soc. 4, 435-439 (June 1953). Reprints available.

(12) On over and under relaxation in the theory of the cyclic single

step iteration. A. ©strowski. MAC 43., 1 52-1 59 (July 1953).

(13) Confidence and tolerance intervals for the normal distribution.

F. Proschan. J. Am. Stat. Assoc. 48, 550-564 (Sept. 1953).

(1 4) Sur les problemes de perturbation singuliers dans la theorie des

vibrations non lineaires (On singular perturbation problems in

the theory of non-linear vibrations). W. Wasow. Appeared in

Actes du Collogue International des vibrations non lineaires,

lie de Porquerolles , France, 1951 (Publications Sc ientif iques

et Techniques du Ministere de 1

! Air , No. 2 8l )

.

(1 5) Pairs of normal matrices with property L. H. Wielandt. J. Res.

NBS 11, 89-90 (Aug. 1953).

(16) Sets of three measurements. W. J. Youden. Sci. Monthly 77 . l43-l47 (Sept. 1953). Reprints available.

(17) Units for measuring variations in measurements. W. J. Youden.

Metals Progress 64, 91-96 (Sept. 1953).

(18) The chain block design. W. J. Youden and W. S. Connor. Biometrics 1, 127-l4o (June 1953). Reprints available.

1.5 Miscellaneous Publications

(1)

Simultaneous linear equations and the determination of eigenvalues. Proceedings of an NBS Symposium held in Los Angeles,

August 1951. NBS Applied Mathematics Series 29. Available from

U. S. Government Printing Office, Washington 25, D. C., S’1.50.

2. MANUSCRIPTS IN THE PROCESS OF PUBLICATION September 30, 1953.

2.1 Mathematical Tables

(1) Tables of 10K

. NBS Applied Mathematics Series 27. In press,

Government Printing Office.

(2) Table of natural logarithms for arguments between zero and five

to sixteen decimal places. (A reissue of NBS Mathematical Table

IWT10, Table of natural logarithms, vol. III.) NBS Applied

Mathematics Series 31. In press, Government Printing Office.

(3) Table of sine and cosine integrals for arguments from 10 to 100.

(A reissue of NBS Mathematical Table MT13.) NBS Applied Mathematics Series 32. In press, Government Printing Office.

(4) Table of the Gamma function for complex arguments. NBS Applied

Mathematics Series 34. In press, Government Printing Office.
68 Publication Activities

(5) Tables of Lagrangian coefficients for sexigesimal interpolation.

NBS Applied Mathematics Series 35. In press, Government Printing

Office

.

(6) Tables of circular and hyperbolic sines and cosines for radian

arguments. (A reissue of NBS Mathematical Table MT3.) NBS

Applied Mathematics Series 36. In press, Government Printing Office

(7) Tables of functions and of zeros of functions. Volume I of Collected short tables of the Computation Laboratory. NBS Applied

Mathematics Series 37. In press, Government Printing Office.

(8) Table of secants and cosecants to nine significant figures at

hundredths of a degree. To be issued in the NBS Applied Mathematics Series.

(9) Tables of expected values of 1 /X for positive Bernoulli and

Poisson variables. E. Grab and I. R. Savage. Submitted to a technical journal.

(10)

Tables of the inverses of finite segments of the Hilbert matrix.

I. R. Savage and E. Lukacs. To be included in Contributions to

the solution of systems of linear equations and the determination of eigenvalues, NBS Applied Mathematics Series.

2.2 Manuals, Bibliographies, Indices

(1)

The statistical theory of extreme values and some practical applications. A series of lectures by E. J. Gumbel. To appear

in the NBS Applied Mathematics Series.

(2)

Bibliography of nonparametric statistics and related topics.

I. R. Savage. Accepted for publication in the' Journal of the

American Statistical Association.

2.3 Technical Papers

( 1 ) Evaluation of

Accepted for

Physics.

oo the integral ' J 0

publication in the

e u (x/u )du. M. Abramowitz.

Journal of Mathematics and

(2) On the solution of the differential equation occurring in the

problem of heat convection in laminar flow through a tube.

M. Abramowitz. Accepted for publication in the Journal of

Mathematics and Physics.

(3) Regular and irregular Coulomb wave functions expressed in terms

of Bessel-Clif f ord functions. M. Abramowitz. Accepted for

publication in Journal of Mathematics and Physics.

(4) Approximate method for rapid Loran computation. M. Abramowitz,

D. H. Call, and J. C. Mathews. Submitted to a technical journal.

(5) The relaxation method for linear inequalities. S. Agmon. Submitted to a technical journal.

(6) On the differential equation x + k(f(x) + g(x)x)x = ke(t).

H. A. Antosiewicz. Submitted to a technical journal.
Publication Activities 69

(7) Some implications of Liapunov's conditions for stability. H. A.

Antosiewicz and P. Davis. Submitted to a technical journal.

( 8 ) The convergence of numerical iteration. H. A. Antosiewicz and

J. M. Hammersley. Accepted for publication in the American

Mathematical Monthly.

(9) On mildly nonlinear partial difference equations of elliptic

type. L. Bers. Accepted for publication in the Journal of

Research of the NBS

.

(10) A general-purpose control panel for a model II CPC. P. B. Bremer,

D. Teichroew, and E. C. Yowell. To be published in the IBM

Newsletter.

(11) Programs for computing the hypergeometric series. W. F. Cahill.

Submitted to a technical journal.

(12) Mathematical services useful in industry. A. S. Cahn, Jr. To

appear in the Proceedings of a Symposium on Industrial Applications of Automatic Computing Equipment held by the Midwest

Research Institute, Kansas City, Mo., January 8 , 1953.

(13) Acceptance sampling of electroplated articles. J. M. Cameron

and F. Ogburn. Accepted for publication in Plating.

(14) Partially balanced incomplete block designs with two associate

classes and two treatments per block. W. H. Clatworthy. Submitted to a technical journal.

(15) Some theorems for partially balanced designs. W. S. Connor and

W. H. Clatworthy. Accepted for publication in the Annals of

Mathematical Statistics.

(16) Time-discrete stochastic processes in arbitrary sets, with

applications to processes with absorbing regions and to the

problem of loops in Markoff chains. D. van Dantzig. Submitted

to a technical journal.

(17) Linear functional equations and interpolation series. P. Davis.

Submitted to a technical journal.

p (18) Some J. aspects of Faber polynomials. P. Davis and H. Pollack.

Submitted to a technical journal.

( 19 ) On representations and extensions of bounded linear functionals

defined on classes of analytic functions. P. Davis and J. L. Walsh. Submitted to a technical journal.

(20) On the optimal character of the (s,S) policy in inventory theory.

A. Dvoretzky, J. Kiefer, and J. Wolfowitz. Submitted to a technical journal.

(21 ) A test for statistical control applicable to a short series of

observations. C. Eisenhart and E. P. King. Submitted to a technical journal.

(22) Changes of sign of sums of random variables. P. Erdos and G. A.

Hunt. Submitted to a technical journal.
70 Publication Activities

(23) Inequalities for eigenvalues of Hermitian matrices. K. Fan.

To be included in Contributions to the solution of systems of

linear equations and the determination of eigenvalues, NBS Applied Mathematics Series.

(24) Some remarks on commutators of matrices. K. Fan. Submitted to

a technical journal.

(25) Lower bounds for the rank and location of the eigenvalues of a matrix. K. Fan and A. J. Hoffman. To be included in Contributions to the solution of systems of linear equations and the

determination of eigenvalues, NBS Applied Mathematics Series.

(26) A numerical analyst's fifteen-foot shelf. G. E. Forsythe.

Accepted for publication in Mathematical Tables and Other Aids

to Computation.

(27) Asymptotic lower bounds for the frequencies of polygonal raenr- branes. G. E. Forsythe. Submitted to a technical journal.

(28) Punched-card experiments with accelerated gradient methods for

linear equations. A. I. and G. E. Forsythe. To be included in

Contributions to the solution of systems of linear equations and

the determination of eigenvalues, NBS Applied Mathematics Series. In press. Government Printing Office.

(29) Practical solution of linear equations and inversion of matrices.

L. Fox. To be included in Contributions to the solution of systems of linear equations and the determination of eigenvalues.

NBS Applied Mathematics Series.

(30) A numerical solution of Schroedinger ' s equation in the continuum.

W. Futterman, E, Osborne, and D. S. Saxon. Accepted for publication in the Journal of Research of the NBS.

(31) A nonlinear model for the composite pi-meson. S, G. Gasiorowicz.

Submitted to a technical journal.

(32) Equations of physics in general Newtonian space-time. A. N.

Gleyzal. Submitted to a technical journal.

(33) A 2-basic set of density zero. K. Goldberg. Submitted to a technical journal,

(34) The representation of integers by binary quadratic rational

forms. K. Goldberg, M. Newman, E. G. Strauss, and J. D. Swift.

Submitted to a technical journal.

(35) An expansion method for parabolic partial differential equations.

J. W. Green. Accepted for publication in the Journal of Research

of the NBS.

(36) The embedding theorem for balanced incomplete block designs.

M. Hall, Jr., and W. S. Connor. Accepted for publication in the

Canadian Journal of Mathematics.

(37) Iterative methods of solving linear problems on Hilbert space.

R. M. Hayes. To be included in Contributions to the solution of

systems of linear equations and the determination of eigenvalues,

NBS Applied Mathematics Series.
Publication Activities

71 (38) On a combinatorial theorem. A. J. Hoffman. Submitted to a technical journal.

(39) A characterization of normal matrices. A. J. Hoffman and 0.

Taussky. Accepted for publication in the Journal of Research

of the NBS.

(40) The SWAC —design features and operating experience. H. Huskey,

R. Thorensen, B. F. Ambrosio, and E. C. Yowell. To appear in

the Proceedings of the Institute of Radio Engineers.

(4l ) Completely continuous normal operators with property L. I. Kaplansky. Submitted to a technical journal.

(42) Probability limits for the average chart when process standards

are unspecified. E. P. King. Submitted to a technical journal.

(43) A property of the normal distribution. E. P. King and E. Lukacs.

Submitted to a technical journal.

(44) On certain character matrices. D. H. Lehmer. Submitted to a technical journal.

(45) Acoustic radiation pressure on a circular disk. H. Levine. To

appear in the Proceedings of the Fifth Symposium on Applied

Mathematics of the American Mathematical Society.

(46) On strongly continuous stochastic processes. E. Lukacs. Submitted to a technical journal.

(47) Certain Fourier transforms of distributions (II). E. Lukacs and

0. Szasz. Accepted for publication in the Canadian Journal of

Mathematics

.

(48) Noenegative trigonometric polynomials and certain rational

characteristic functions. E. Lukacs and 0. Sz«Tsz. Accepted for

publication in the Journal of Research of the NBS.

(49) On Fejer sets in linear and spherical spaces. T. S. Motzkin

and I. J. Schoenberg. Accepted for publication in Annals of

Mathematics.

(50) On the relaxation method for linear inequalities. T. S. Motzkin

and I. J. Schoenberg. Submitted to a technical journal.

(51 ) Pairs of matrices with property L, II (summary). T. S. Motzkin

and 0. Taussky. Accepted for publication in the Proceedings

of the National Academy of Science.

(52) Least p-th power polynomials on a real finite point set. T. S. Motzkin and J. L. Walsh. Submitted to a technical journal.

(53) The arithmetic structure of certain modular subgroups. M. Newman.

Submitted to a technical journal.

(54) On the Lerch zeta function. F. Oberhettinger . Submitted to a technical journal.

(55) On two problems in abstract algebra connected with Horner's

rule. A. M. Ostrowski. Submitted to a technical journal.
72 Publication Activities

(56) On nearly triangular matrices. A. Ostrowski. Submitted to a technical journal.

(57) Determinanten mit ueberwiegender Hauptdiagonale und die absolute

Konvergenz von linearen Iterationsprozessen. A. M. Ostrowski.

Submitted to a technical journal.

(58) On the convergence of Gauss' alternating procedure in the method

of the least squares, I. A. M. Ostrowski. Submitted to a technical journal.

(59) On the linear iteration procedures for symmetric matrices. A.

M. Ostrowski. Submitted to a technical journal.

(60) On the spectrum of a one parametric family of matrices. A. M. Ostrowski. Submitted to a technical journal.

(61 ) On Gauss' speeding up device in the theory of single step iteration. A. M. Ostrowski. Submitted to a technical journal.

(62) On. absolute convergence of linear iteration processes. A. M.

Ostrowski. Submitted to a technical journal.

(63) Rejection of outlying observations. F. Proschan. Accepted for

publication in the American Journal of Physics.

(65-) On spectra of second-order differential operators. D. Ray.

Submitted to a technical journal.

(65) Numerical computation of low moments of order statistics from a normal population. J. B. Rosser. Submitted to a technical journal.

(66) Modes of vibration of a suspended chain. D. S. Saxon and A. S.

Cahn. Accepted for publication in the Quarterly (Journal of

Mechanics and Applied Mathematics.

( 67 ) An isoperimetric inequality for closed curves convex in even- dimensional Euclidean space. I. J. Schoenberg. Submitted to a technical journal.

(68) Characteristic roots of quaternion matrices. 0, Taussky. Accepted for publication in Archiv der Mathematik.

( 69 ) Generalized commutators of matrices and permutations of factors

in a product of three matrices. 0. Taussky. For publication

in the von Mises Anniversary volume.

(70) An improved cathode ray tube storage system. R. Thorensen. To

appear in the Proceedings of the Western Computer Conference of’ the AXEE-IRE-ACM held in Los Angeles, Calif., February *+,5,6,

1953.

(71) The condition of the finite segments of the Hilbert matrix.

J. Todd. To be included in Contributions to the solution of

systems of linear equations and the determination of eigenvalues,

NBS Applied Mathematics Series.

(72) On the accuracy of the numerical solution of the Dirichlet problem by finite differences. J. L. Walsh and D. Young. Accepted

for publication in the Journal of Research of the NBS.
Publication Activities 73

(73) Asymptotic solution of the differential equation of hydrodynamic

stability in a domain containing a transition point. W. Wasow.

Accepted for publication in the Annals of Mathematics.

(7*+) On small disturbances of plane Couette flow. W. Wasow. Accepted

for publication in the NBS Journal of Research.

(75) Singular perturbation methods for honlinear oscillations. W.

Wasow. To appear in the Proceedings of a Symposium on Nonlinear

Circuit Analysis, held by the Polytechnic Institute of Brooklyn, N. Y.

(76) On the eigenvalues of A+B and AB. H. Wielandt. Submitted to a technical journal.

(77) The principles of experimental design. W. J. Youden. Accepted

for publication in Selection, Training, and Use of Personnel

in Industrial Research. Proceedings of the Third Annual Conference on Industrial Research.

( 78 ) Making one measurement do the work of two. W. J. Youden and

W. S. Connor. Accepted for publication in Chemical Engineering

Progress

.

(79) Performance of inspectors and gasoline pumps. W. J. Youden and

M. W. Jensen. Submitted to a technical journal.

(80) A note on partially balanced designs. M. Zelen. Submitted to a technical journal.

( 8 1) Analysis for some incomplete block designs having a missing

block. M. Zelen. Submitted to a technical journal.

Miscellaneous Publications

(1) Contributions to the solution of systems of linear equations and

the determination of eigenvalues. To be issued in the NBS Applied Mathematics Series.

(2) A historical note on the application of the "wea kest-link" idea

to tensile strengths. J. Lieblein. Submitted to a technical

journal

.
'
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and

maintenance of the national standards of measurement and the provision of means and methods

for making measurements consistent with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials,

devices, and structures; advisory services to Government Agencies on scientific and technical

problems; invention and development of devices to serve special needs of the Government; and the

development of standard practices, codes, and specifications. The work includes basic and applied

research, development, engineering, instrumentation, testing, evaluation, calibration services, and

various consultation and information services. A major portion of the Bureau’s work is performed

for other Government Agencies, particularly the Department of Defense and the Atomic Energy

Commission. The scope of activities is suggested by the listing of divisions and sections on the

inside of the front cover.

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The results of the Bureau’s work take the form of either actual equipment and devices or published papers and reports. Reports are issued to the sponsoring agency of a particular project or program. Published papers appear either in the Bureau’s own series of publications or in the

journals of professional and scientific societies. The Bureau itself publishes three monthly periodicals, available from the Government Printing Office: The Journal of Research, which presents

complete papers reporting technical investigations; the Technical News Bulletin, which presents

summary and preliminary reports on work in progress; and Basic Radio Propagation Predictions,

which provides data for determining the best frequencies to use for radio communications throughout

the world. There are also five series of nonperiodical publications: The Applied Mathematics

Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous

Publications.

Information on the Bureau’s publications can be found in NBS Circular 460, Publications of

the National Bureau of Standards ($1.00). Information on calibration services and fees can be

found in NBS Circular 483, Testing by the National Bureau of Standards (25 cents). Both are

available from the Government Printing Office. Inquiries regarding the Bureau’s reports and

publications should be addressed to the Office of Scientific Publications, National Bureau of Standards, Washington 25, D. C.