NATIONAL BUREAU OF STANDARDS REPORT

2688

PROJECTS and PUBLICATIONS

«

of the

NATIONAL APPLIED MATHEMATICS LABORATORIES

A Quarterly Report

April through June 1953

<NBp>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, 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 back

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 Chemistry. 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. Tropospneric 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 N8S REPORT

NAML 2688

PROJECTS and PUBLICATIONS

of flie

NATIONAL APPLIED MATHEMATICS LABORATORIES

April through June 1953

The pubi. Approved for public release by the „

unless permission li Director of the National Institute of andard8 , Washington

25, d.c. Such per Standards and Technology (NIST) port has been specifically prepared If tl on October 9, 2015 sport for Its own use.
NATIONAL APPLIED MATHEMATICS LABORATORIES

April 1 through June 30, 1953

ADMINISTRATIVE OFFICE

Franz L. Alt, Ph.D. , Acting Chief

fEdward W. Cannon, Ph.D., Assistant Chief

Olga Taussky-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. Rodriguez, M.A. , Chief Clerk

John B. Tallerico, B.C.S., Assistant Chief Clerk Margaret W. Dawson, M.A. , Secretary

Gloria S. Ford, B.S., 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

Magnus R. Hestenes, Ph.D... Assistant Director and UCLA Liaison Officer

fHarry D. Huskey, Ph.D Assistant Director for Mathematical Services

Gertrude Blanch, Ph.D.... Assistant to the Director (Numerical Analysis)

Albert S. Cahn, Jr., M.S Assistant to the Director (Administration) Research Staff

f Forman S. Acton, D.Sc Mathematician

Edward W. Barankin, Ph.D UCLA Research Associate (Mathematics)

George E. Forsythe, Ph.D Mathematician

Dick W. Hall, Ph.D Consultant

f Cornelius Lanczos, Ph.D Mathematician

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

Lowell J. Paige, Ph.D Mathematician

David S. Saxon, Ph.D Physicist

Dan Teichroew, M.A Statistician

Charles B. Tompkins, Ph.D Mathematician Joseph L. Walsh, Ph.D Consultant

Wolfgang R. Wasow, Ph.D Mathematician

Graduate Fellows

John W. Addison, Jr., M.S. John Selfridge, B.S. Walter I. Futterman, M.A. Marian I. Walter, B.A. (UCLA)

Stanley W. Mayer, M.A. Roger D. Woods, M.A. Mathematical Services Unit

Arnold D. Hestenes, Ph.D Mathematician

Frederick H. Hollander, M.A Mathematician

Marvin Howard, M.S Mathematician

t Roselyn S. Lipkis, B.A Mathematician

Robert R. Reynolds, M.S Mathematician

Thomas H. Southard, Ph.D Mathematician

Everett C. Yowell, Ph.D Mathematician

Comput ing Staff

Patricia B. Bremer, B.A. Helen V. Meek, B.S. Patricia L. Childress, B.A. Owen R. Mock, B.A. Leola Cutler, B.A. Elmer E. Osborne, Ph.D. Alexandra I. Forsythe, M.A. Mary M. Y. Quan Lillian Forthal, B.A. Nan N. Reynolds, M.S. Benjamin F. Handy, Jr., M.S. Laveme E. Rickard, M.A. Ruth B. Horgan, B.A. Albert H. Rosenthal, B.A. William Justice, Jr., M.A. Louise M. Straus, M.A. William P. Keating, M.A. Gladys P. Walker. B.S. Merry M. Kruse, B.S. Maria Weber, Ph.D. (UCLA)

Thomas D. Lakin, B.S. Marian E. Williams Shirley L. Marks, M.A. Lmdley S. Wilson, M.A. Frank B. Meek, M.A. Machine Engineering and Development Unit

Ragnar Thorensen, Ph.D Electronic Scientist

Biagio F. Ambrosio, B.S Electronic Engineer William Arsenault, M.S Electronic Engineer

Norman F. Loretz, B.S Electronic Engineer

Robert R. Schmidt Electronic Engineer

James W. Walsh Electronic Technician

Electronic Laboratory Staff

Blanche C. Eidem Charles H. Keller (UCLA)

John P. Francis, B.S. Howard K. Marks

Allen E. Garfein (UCLA) Edward D. Martinolich

Gaylord W. Jones Eugene M. Rector, B.S. Louis E. Justice, B.S. 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. Gordamer Lucy A. Moore

Dorothy M. Hibbard Anne B. Oates

Elsie L. Husman

|0b lei i during fiscal 1953.

COMPUTATION LABORATORY

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

Henry A. Antosiewicz, Ph.D. Philip Davis, Ph.D

William H. Durfee, Ph.D. . . . Ky Fan. Ph.D

Andre N. Gleyzal, Ph.D

Peter Henrici, Ph.D

Alan J. Hoffman, Ph.D

Murray Mannos, Ph.D

Morris Newman, Ph.D

Fritz Oberhettinger , Ph.D.. Philip Rabinowitz, Ph.D.... Daniel Sokolowsky, M.A

Irene A. Stegun, M.A

Charles J. Swift, Ph.D

Joseph H. Wegstein, M.S.... Norman A. Wiegmann, Ph.D... Helmut Wielandt, Ph.D

Computing Staff

Hans 0. Bremer, B.A. Rolla L. Brooks, B.S. 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. Seymour Domnitch

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

Myrtle C. Simerson

Lillian Sloane

Chief Assistant Chief

Mathematician

.... AU* Research Associate Mathematician

Mathematician

.... AU* Research Associate Mathematician

.... AU* Research Associate Mathematician Mathematician

Mathematician

.... AU* Research Associate Mathematician

Mathematician Mathematician

Physicist

Physicist Mathematician

.... AU* 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. Wallace R. Soderquist, M.S. Milton Stein Otto T. Steiner, B.A. Frank J. Stockmal, B.A. Gretchen Stolle, M.A. Ellen V. Stubblefield

Sally Tsingou, B.S. Bernard Urban, B.S. Bertha H. Walter

Ruth Zucker, B.A.

Marjorie E. Young

“'Aaericon Universi ty

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 Cocozzella

Laurel Pelzman Shirley C. Schleter, B.A

Chief

Assistant Chief

Technical Aid Mathematical Statistician Mathematician

Mathematician

Mathematician

Mathematician Mathematical Statistician Computer Student Assistant (Mathematics) Secretary

Secretary

Secretary

MACHINE DEVELOPMENT LABORATORY

f Edward W. Cannon, Ph.D Chief

Edith N. Reese, B.A Mathematician
APPLIED MATHEMATICS ADVISORY COUNCIL

The activities of the National Applied Mathematics Laboratories

are carried on with the advice of a pol ie y-formula t ing group known as

the Applied Mathematics Advisory Council. The membership of the council

for the fiscal year 1953 is as follows.

AGENCY REPRESENTATIVES AND THEIR ALTERNATES

National Science Foundation Raymond J. Seeger (Chairman, 1953)

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. Hansen James L. McPherson, Alternate

Weather Bureau Harry Wexler

R. A. Allen, Alternate

Atomic Energy Commission Captain John T. Hayward, USN Commander Charles M. MacDonald, USN, Alternate

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

of Technology

A. H. Taub

University of Illinois

J. Barkley Rosser

Cornell University

Edward Teller

Radiation Laboratory

University of California

Contents

Page

Index ....... vi

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

Computation Laboratory 29

(NBS Section 11.2)

Statistical Engineering Laboratory. • 59

(NBS Section 11.3)

Machine Development Laboratory. ... 65

(NBS Section 11.4)

Lectures and Symposia 68

Publication Activities ......... 7

2
Index of Aetiv® Research and Development Projects

Notes This index is not intended to cover the numerous special problem

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

agencies, which form an important part ©f the work of the National Applied Mathematics Laboratories * These services are, however, fully repre

sented in the body of the report,

A. Research? Pur® Mathematics

Miscellaneous studies in pure mathematic s .3, 30

B. Research; Numerical Analysis

Ba ker-Rausdorf f formula, .................... ,35

Classical numerical analysis, Research in, ........... ,29

Differential equations, Studies in numerical integration of, , , . 4

BiriefoXet problem for certain multiply connected domains, Investigation of Bergman's method for the solution of the . . , . ,-,29

Eigenvalues, eigenvectors, and eigenfunctions of linear operators,

Calculation of ....... , 2 Monte Carlo method, Solution of Laplace equation by ....... 31

Probability methods and sampling techniques ........... 5

Riemann-zeta-f unction, Computation of the complex zeros of. ... 4

Roots of algebraic and transcendental equations ......... 35

* SCAMP. .............................. 8

Solution of sets of simultaneous algebraic equations and

techniques for the inversion and iteration of matrices. .... 1 Variational methods, ...................... 6

€. Research; Applied Mathematics, Physics, Astronomy, and Automatic

Translation

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

*B.P.A» Studies ......................... .27

Compressible flow - method of orthogonal and kernel functions, . .37

Cooperative phenomena, An integral arising in the theory of. . . .35

Crystal structure, Analysis of................. .31

Cubic lattices, Distribution of normal modes of vibration of . . .3*+ Differential equation for nerve fiber reaction ......... .35

Digital computing machines, Studies in the theory of ...... .12

Discrete minimal spaces. .................... .23

Dynamic behavior of aircraft structures. ...... .® .... . .53

Finite matrix theory, Special problems in. ........... .32

Flow in supersonic nozzles ................... .5-7 Generalized Rsyleigh-Ritz method for eigenvalues of a clamped

plate. ............................. 8

Geomagnetic field, Analysis of ................ . .32

Hypergeometric functions .................... .36

Integral operators and interpolation series. .......... .38

Internal conversion coefficients for L-shell .......... .47

Language translation study 12

Linear programming, Research in. ................ .36

* Loran UNIVAC code. ........................56

Mathematical theory of program planning, Research in the ..... 9

Neutron diffusion. . 1+9 Neutron diffusion II ..................... .52

* Neutron diffusion III. ........... J .

'

* .’57 Painleve equation. ’ ***** * 1+5 Poisson-Boltzman equation. 1 !!!!!!'!!!!!! ! Pressure distribution on bodies of revolution. ......... .48 Pressure fields of potential flow past a body of revolution.” I III 5 Prolate spheroidal wave functions. ............... .20 Rayleigh scattering of light in the atmosphere ......... .24 Roots of polynomial equations. ^ . 0 . 31+ *New tasks
Index of Active Research and Development Projects

Russian mathematical progress. Study of. .......

Scattering functions. ...... » Single shot probabilities

Spherical blast. . . ................

Standard Loran tables - extension of rates 1L4, 1L 5, 1 Lo . . . . Survival probability in pattern bombing

Study of trajectories, Application of the theory of stochastic

processes to the

Subsonic compressible flow past oscillating airfoils; Reissner's

method.

Theoretical physics, Miscellaneous studies in

Three-body problem. .... ...

Transportation problem II .

. 8

.44

.23

.49

.55

.37

.63

18

7 -32

•57

D. Mathematical Statistics

Applications of mathematical statistics to problems of the

Chemical Corps, Research in ...... . ..... 64

Committee on Ship Steel, NRC, Statistical services for 64

Experiment design, Studies in the mathematics of. . «, 60

Latin square of order n, Search for magic sets in a 19

Miscellaneous studies in probability and statistics 62

NBS Administrative Operations, Statistical aspects of .63

NBS Research and testing, Collaboration on statistical aspects of.6l

Non-pa rametric statistics, Procedures of 60

Propagation of error, Law of 60

Spectral analysis of stationary time series. . . .54

E. Mathematical Tables

Antilogarithms, Table of

Arcsin for complex arguments, Table of.

Bivariate normal distribution function, Tables of the

Bivariate normal distribution function, Tables relating to the.

Collected short mathematical tables of the Computation Laboratory

Coulomb wave functions, Tables of . . . E^ (z ) , (z = x + iy), Tables of ....

Error function for complex arguments, Table of

Exponential function for negative arguments, Extension of tables

of the. ....

Gamma function for complex arguments, Table of the. .......

Gases, Table of thermodynamic properties of . Hyperbolic sines and cosines, Extension of the table of ... . I (x,c ), Table of (see task 1 1 02-50-51 26A9-1 3 ) ^ y • Integrals involving higher transcendental functions, Tables of.

Jacobi elliptic functions. . Lagrangian coefficients for sexagesimal interpolation, Table of.

Legendre functions

Mathematical tables, Revision of. ......

Mathieu functions II ....... ...

Modified Airy integral, Table of the. ....

Power points of analysis-of-va riance tests, Tables of

Probability tables for extreme values

Punched card library

Radial Mathieu functions

Random samples, Table to facilitate drawing

Rocket and comet orbits, Tables for

Secants and cosecants, Tables of

Sievert's integral. ..... . Spheroidal wave functions

Standard Loran tables - rates 2H2, 2H3, 2H4

39

4l

15

19

41

39

39

42

42

.39

58

42

46

38

44

40

18

41

13

42

45

4o

14

43

59

14

44

43

43

.55

*New tasks
Index of Active Research and Development Projects

Van der Pol Equation. ^3

Wave function for lithium . . . *+1

F. Manuals, Bibliographies, Indices, and Technical Information

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

Bibliography of. ........ ^0

Fitting straight lines, Manual on. . . ...... 59

Statistical literature, Bibliography and guide to. ....... 59

G. Computing Machine Development

Air Comptroller's computing machine 65

Army Map Service computing machine. 6 7

Bureau of the Census computing machine. ....... 65

SEAC: National Bureau of Standards Eastern Automatic Computer . . 66

SEAC: Number-theoretical test problems for. 30

Social Security Agency, Investigation of the applicability of

automatic digital electronic computing to problems of the ... 67

SWAC: National Bureau of Standards Western Automatic Computer . . 11 Wright Development Center computing machine . . 66
Status of Projects

June 30, 1953

I. Institute for Numerical Analysis

(Section 11.1)

1 . Fundamental Research

SOLUTION OF SETS OF SIMULTANEOUS ALGEBRAIC EQUATIONS AND

TECHNIQUES FOR TOE INVERSION AND ITERATION OF MATRICES

Task 1 101 -10-51 00 A9-AE2

(formerly 1 1 .1/1 -49-AE2)

Origin: NBS Authorized 2/1 5A9

Sponsor: Office of Naval Research, USN Revised 9/1 5A9

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

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

Status: CONTINUED. The SWAC experiments on the conjugate gradient

method for solving a linear system AX=k have been written up in publication

(5-). These were performed before SWAC ' s magnetic drum became available,

and hence were limited to matrices A of orders up to 12. The matrices A

were treated as nonsymmetrie . Let P=P(A*A) be the ratio of the largest

to the smallest of the eigenvalues of A*A. When P is sufficiently near 1,

the solution of Ax=k by th© conjugate gradient method offers no difficulty

and may not even require the use of floating-point operations. For P sufficiently large, it is 'probably impossible to solve the system Ax=k,

even with floating-point operations. In the SWAC experiments the values

of P ranged up to 10°, for matrices of orders 5 to 8, and solutions were

readily obtained with floating-point operations, using 28 significant bits.

The method failed completely on on® matrix of order 12 for which P is

unknown, but apparently exceedingly large.

A SWAC code ha§ been prepared by L. S. Wilson to solve Ax=k by

elimination for a general order n up to 45, using fixed binary-point operations and a system of scaling.

Item (10) under Publications immediately below gives a mathematically elegant treatment of a number of numerical methods of solving

linear systems and getting eigenvalues of finite matrices. There are many

numerical examples worked in detail.

Publications: (1 ) "Tentative classification of methods and bibliography on solving systems of linear equations," by G. E. Forsythe; to

appear in Simultaneous Linear Equations and the Determination of Eigenvalues,

Proceedings of an NBS Symposium held in Los Angeles, August 1951) NBS

Applied Mathematics Series 29, in press, U. S. Government Printing Office.

(2) "On certain character matrices," by D. H. Lehmer; submitted to a technical journal. (3) "Punched-ca rd experiments with accelerated gradient
2 Status of Projects

methods for linear equations

,

11 by A. I. and G. E. Forsythe; to appear

in Contributions to the Solution of Systems of Linear Equations and

the Determination of Eigenvalues, MBS Applied Mathematics Series;

(4) "Some numerical examples on solving systems of linear equations

by the conjugate gradient method for non symmetric systems of equations,"

by M. R. Hestenes, U. Hochstrasser, and L, S, Wilson; IN MANUSCRIPT®

(5) "On the linear iteration procedures for symmetric matrices," by

A® M. 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; submitted to a technical journal. (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

us© on IBM equipment," by U. Hochstrasser; IN MANUSCRIPT. (9) "Solving

linear algebraic equations can be interesting," by G. E. Forsythes to

appear in the Bulletin of the American Mathematical Society. (10) "Computational methods of linear algebra," by V. W. Faddeeva, translated from the

Russian by C. D. Renster. (Chapter 1, edited by G. £. Forsythe, is an MRS report, now out of stock. Chapters 2 and 3 a re IN MANUSCRIPT without

displayed formulas.)

CALCULATION OF EIGENVALUES. EIGENVECTORS , AND EIGENFUNCTIONS

OF LINEAR OPERATORS

Task 1 101-10-5100/50-3

(formerly 11 , 1 /1 - 50 - 3 ) Origin; NRS Authorized 7/1 A9

Sponsor; Office of Naval Research, USN

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

Fell task description appears in July-Sept 1 9*+9 issue.

Status; CONTINUED, G. E» Forsythe has extended his results on the

relation of the fundamental eigenvalue of the Laplace operator A to that

of an approximating difference operator in two dimensions (see Jam-Mar

1953 issue, p. 2.) Let R be any convex domain in the plane, with boundary

C, Let X , u(x,y) be the fundamental eigenvalue and corresponding eigenfunction for R; i.e. , assume Au=u v+u = - Xu, in R, with u=0 on C. Let

a square net with mesh-width h be constructed in the plane of R. Let R^

denote the nodes of the met which fall inside R„ Over R^ one constructs

a certain self-adjoint difference operator A^ which approximates A. Let

X

h be the fundamental eigenvalue of A^„ Then it is proved that as h—i>0,

ik_ ^ 1 - Ah 2 + o(h 2

)

,

r> n where

A =

. < "KX

+ “yy )dsd5’ %u

2 sin 2

2 V dv

n

1 2 r

p 2

(u~ + updxdy

Here si is the normal derivative of u, while V is the angle between the

tangent to C smd the x-axis. As a consequence of (1), A is a lower bound

for A for all h^h^, for some sufficiently small h^; no estimate of is
Status of Projects 3

given. It is conjectured that V < <*• for all h. Since good lower bounds

are ordinarily very difficult to obtain, this result may have considerable

practical importance.

In solving one problem for an aircraft company (see task 1101-50-

5131/53-65 p. 19, the Institute for Numerical Analysis Mathematical Services Unit determined all 45 eigenvalues and eigenvectors of a symmetric

matrix of order 4-5 on the SWAC. For k=0 , 1

, 2, . .

. ,44, the method was as follows: After the largest k eigenvalues and vectors were found, the

gradient method of Hestenes and Karush (see "A method of gradients for the

calculation of the characteristic roots and vectors of a real symmetric

matrix," by M. R. Hestenes and W„ Karush; J. Res. MBS k2, ^+5— 6 1 (July 1951)) was used to find the largest eigenvalue in the subspace S^. orthogonal to

the k known eigenvectors. The maximizing vector was kept in by suc- cessively orthogonalizing it against eaeh known eigenvector. A combination

of "optimal -00 " and "fixed-it " procedures was followed. After the

magnetic drum of SWAC became available, the elapsed time to get one eigenvalue and vector averaged approximately one hour for each of 25 eigenvectors.

The method seems to have maintained great accuracy, as revealed by the

smallness of the Au . — A . u • , and by the fact that

X XX

trace (A)

traee(A)

„^_11 was approximately 10 . Publication (10) under task 1 1 01 -10-51 00/49-AE2 , p. 1 also

deals in part with eigenvalue problems.

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 1 101-10-5100/50-4

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

Origin: NBS Authorized 7/1/49

Sponsor: Office of Naval Research, USN

Managers: Various Staff Members

Full task description appears in July-Sept 1949 issue, see 11.1/1-50-4.

Status: CONTINUED. D. H. Lehmer and E. Lehmer have devised methods

for investigating whether a given prime is irregular in the sense of

Kuramer. Results obtained by H. S. Vandiver over the past 20 years were

rerun on the SWAC in two hours with nearly complete agreement. Three

cases of irregular primes have been overlooked by Vandiver, the smallest

being 389 which divides the 100th Bernoulli number. Thus the impossibility of

389 , 389 389 X + y

= z in nonzero integers remains unproved. An extended examination of

primes between 6l 7 and 1000, indicates that almost half of the primes

are irregular.
4

Status of Projects

A sieve process (see July-Sept 1952 issue, p. 16) for the solution of the diophantine equation

has been applied to the unsolved eases of D<100. A search up to x=10 , and much beyond in some cases, failed to reveal any solution not previously known.

Publications; (1) "On a recursion formula and on some Tauberian

theorems." by N. G. de Bruijn and P. Erdos; J. Res. NBS 50.1 6l -1 64 (Mar.

1953). (2) "Boolean Geometry I," by L. M. Blumenthal; to appear in Rendiconti del Circolo Matematico di Palermo, (3) "On Polya frequency functions III; The positivity of translation determinants with an application

to the interpolation problem by spline curves," by I. J. Schoenberg and

A, Whitney; Trans. Am. Math. Soe. 24, 246-259(Mar. 1953). (5-) "On smoothing operations and their generating functions," by I. J4 Schoenberg;

Bui. Am. Math, Soc. j?2, 199-230(1953). (5) "An isope rimetric inequality

for closed curves convex in even-dimensional Euclidean space," by I. J. Schoenberg; submitted to a technical journal. (6) "On the distribution

of totitives," by D. H. Lehrner; 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

Manager; D. H. Lehmer

Full task description appears in Apr-Jun 1950 issue, see 11.1/1-50-13.

Status; INACTIVE. For status to date see Jan-Mar 1953 issue.

STUDIES IN TOE NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS

Task 1101-10-5100/51-1

(formerly 1101-11 -51 00/51 -1 ) Origin; NBS Authorized 9/1/50

Sponsor; Office of Naval Research, USN

Managers: W. Wasow and G. Blanch

Full task description appears in July-Sept 1950 issue.

Status; CONTINUED. G. Blanch has completed the experiments with the

numerical method of Kaskind-Reis sne r for the computation of lift and moment

coefficients in subsonic (two-dimensional) compressible flow. Results obtained agree with Dietze's calculations considerably better than those

heretofore obtained by other investigators.

G. Blanch and I. Rhodes completed the table of characteristic

values of Mathieu's equations for large parameters for orders up to nine.

Some further subtabulations are required in functions of higher order,

but tables at intervals of 0.002 in t (where t=1 Afs ) are available, for

orders up to fifteen, [See National Bureau of Standards "Tables relating

to Mathieu functions," (Columbia University Press, New York, 1951) fo p definitions.] An examination of the entries shows that be (s) a o re

F
Status of Projects 5 bo^ (s) to eight decimals for t<. 025 , even when the order r is as large

®s 1*+. In this region it should b© possible to improve the known asymptotic

expansion. Meixner communicated an expression for the difference be^-bo^ . His formula was spot cheeked against the new table; and it appears that

for low orders r and small t, the formula gives very good results. However, for orders greater than five and t>.05, it is not possible to obtain

more than the order of magnitude of the functions in the region where the

formula is applicable.

A report on mathematical aspects of the theory of relaxation

oscillations with one or more degrees of freedom was prepared by W. Wasow

(see publication (7)) . It includes a simplified derivation of the first

perturbation term for the period of oscillation in Van der Pol's case.

In connection with this work the paper by A. A. Dorodnitsyn on "Asymptotic

solution of Van der Pol's equation" was translated from the Russian. It

contains a short account of a complete perturbation procedure.

The question as to when asymptotic series solutions in powers of

a parameter of a linear differential equation are convergent (see Jan-March

1953 issue, p. 4.) has been further studied by W. Wasow in collaboration

with R. M. Redheffer of the University of California at Los Angeles. Several sufficient conditions for convergence in special cases have been obtained. Other results shed some light on the nature of "inner friction

layers" occurring in the theory of hydrodynamic stability. A joint manuscript is in preparation.

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

differential equations," by G. Blanch; accepted by the Journal of Research

©f the NBS . (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 o.f hydrodynamic

stability in a domain containing a transition point," by W. Wasow; ac- cepted 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. (8) "Asymptotic solution of

Van der Pol's equation," by A. A. Dorodnitsyn, translated from Russian

by C. D. Benster, edited by W. Wasow; an NBS report.

PROBABILITY METHODS AND SAMPLING TECHNIQUES

Task 1101-10-5100/51-2

(formerly 1 101 -1 1 -5100/51 -2)

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Manager: W. Wasow

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. In connection with mathematical service work done

for the USAF School of Aviation Medicine under the guidance of D. Teichroew,

the transformation of an almost normal random variable t into a normal one

x and vice versa, was studied theoretically by W. Wasow. It was shown that

for a certain class of distributions the formal expansions in terms of a parameter obtained for these transformations are asymptotic. The coefficients of the expansions are polynomials in x or t, respectively. The class
6 Status of Projects

of distributions considered includes Student's distribution and the chi-square

distribution. A paper on this subject is being prepared.

P. Erdos and T. S. Motzkin solved a problem on frequencies left

open by Dvoretzky and Motzkin. If f(p,q, c*i

' ) is the probability that, in

a sequence of p zeros and q ones, every initial sequence should have at

least (or more than) <* times as many zeros as ones, it is shown that for

p —>oo , q-— oo , p/q the limit of f exists. The limit function has

jumps exactly at all rational «. . Recurrence relations for the probabilities are established which facilitate the computation of the limit. In

the simplest non trivial cases, the value of the limit is the quotient of

two hypergeometrie functions with immediately available arguments.

Publications; (1) "Additive functionals of a Markoff process,"

by R. Fortet; submitted to a technical journal. (2) "Metodi probabilistic

per la soluzione numeriea di alcuni problem! di analisi," by W. Wasow;

Rend. Mat. App. {Vj XI, 336-346 (Roma 1952); also issued separately as Pubblicazioni dell 1 Istituto per le Applicazioni del Calcolo N. 35*+ (Roma

1953). (3) "Statistical estimation of matrix quantities by means of a class of discrete Markov chains," by H. P. Edmundson; IN MANUSCRIPT.

(4) "Limits for permanent preponderance," by P. Erdos and T. S. Motzkin;

IN MANUSCRIPT.

VARIATIONAL METHODS

Task 1101-10-5100/51-3

Origin; NBS Authorized 9/1/50

Sponsor: Office ©f 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: (1) "On methods for obtaining solutions of fixed endpoint problems in the calculus of variations, " by M. L. Stein; J. Res.

NBS £0, 277-297 (May 1953). (2) "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.

STUDIES IN APPLIED MATHEMATICS

Task 1101-10-5100/51-4

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Managers: C. Lanczos, G. E. Forsythe, and D. Teichroew

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. C. B. Tompkins, T, S. Motzkin, and G. E. Forsythe

have organized a seminar on numerical analysis, a lecture and discussion

group meeting twice weekly, in which the public is invited to participate.

Eighteen meetings were held between April 20 and June 17. The central

topic of the seminar has been the use of automatic digital computers (especially SWAC) for the solution of analytical or combinatorial problems.

The computational problems discussed have dealt with: a military attrition

function, linear inequalities, the assignment problem, vibrational frequencies of membranes , eigenvalues of a 45th order matrix, matrix inversion,
n Status of Projects ( and the maximization of a function of many variables. (See Lectures and

Symposia for a detailed list of the sessions.) The seminar has attracted

participants from a variety of scientific organizations in the Los Angeles

a rea

.

Although divided differences are of basic importance in numerical

analysis, they are harder to interpret than ordinary differences, partly

because their properties differ radically from familiar ordinary differences. Thus, for sufficiently regular functions f(t) oyer a region R,

the divided difference or order n can be expressed by f^ n ^(y)/nl, while

the nth ordinary difference of f(t), at uniform intervals h, has the form

h n f(n)( z ), y and z in R. G. Blanch has shown that by introducing an ajiex.- aoe interval and modifying the definition of divided differences, the latter

acquire many of the characteristics of ordinary differences. They can be

used quite readily when functions are available only at unequal intervals

of the argument, and an examination of successive differences for error- patterns resembles the corresponding use of ordinary differences. A paper

on the subject is being prepared.

Of interest to numerical analysts is publication (7) described

below. It makes a large body of material in English available for the

first time to applied mathematicians. It gives lucid explanations, with

many references to the world literature, of numerical and analytical

methods for solving ordinary and partial differential equations and integral

equations. The chapter titles are: 1) "Methods based on representation

of the solution as an infinite series"; 2) "The approximate solution of

the integral equations of Fredholm"; 3) "The method of nets"; 4) "Variational methods"; 5) "The conformal transformation of regions"; 6) "The

principles of the application of conformal transformation to the solution

of the fundamental problems for canonical regions"; and 7) "Schwarz's

method"

.

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

statistics from a normal population," by J. B. Rosser; submitted to a technical journal. (2) "Sequential decision problems for processes with

continuous time parameter. Testing hypotheses," by A. Dvoretzky, J. Kiefer,

and J. Wolfowitz; Ann. Math. Stat. 24, No. 2, 254-264(1953). (3) "Changes

of sign of suras of random variables," by P. Erdos and G. Hunt; submitted

to a technical journal. (4) "a numerical analyst's 1 5-foot

shelf," by G. E. Forsythe; submitted to a technical journal. (5) "Seminar

on numerical analysis —• summary of presentations between April 20 and

May 13, 1953?" by C. B. Tompkins, multilithed typescript, 10 p. (6) "Seminar on numerical analysis — summary of presentations between May 18 and

June 8, 1953?" by G. E, Forsythe, multilithed typescript, 38 p. (7) "Approximate methods of higher analysis," by L. V. Kantorovich and V. I. Krylov, translated by C. D. Benster; IN MANUSCRIPT. (This draft translation is typed without displayed formulas.) (8) "Translations of Russian

articles on the Kolmogorov and Smirnov tests," by C. D. Benster, edited

by D. Teichroew; IN MANUSCRIPT.

MISCELLANEOUS STUDIES IN THEORETICAL PHYSICS

Task 1101 - 10 - 5100/51-5

Origin: Office of Naval Research, U5N Authorized 9/1/50

Sponsor: " " Manager: D. Saxon

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. A manuscript is in preparation on the application

of variational methods to quantum mechanical scattering problems (see OctDec 1952 issue, p. 7 ). Programs have essentially been completed, and
8 Status of Projects

numerical work is continuing on the calculation of the photo-disintegration

of the deuteron (see Oct-Dec 1952 issue, p. 7). R. Woods has begun a program of computation on the elastic scattering of protons from various

elements, under the guidance of D. Saxon. The main coding program has been

completed and checked; some preliminary calculations have been carried out.

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) "The torsion of anisotropic elastic cylinders by forces applied on the lateral surface," by H. Luxenberg; J. Res.

NBS £0, 2o3-276 (May 1953). (3) "A nonlinear model for the composite Pimeson," by S. G. Gasiorowicz; submitted to a technical journal. (4) "Acoustic radiation pressure on a circular disk," by H. Levine; to appear in the

Proceeedings of the Fifth Symposium on Applied Mathematics of the American

Mathematical Society. (5) "A numerical solution of Schroedinger ' s equation

in the continuum," by W. Futterman, E. Osborne, and D. S. Saxon; accepted

by the Journal of Research of the NBS.

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 RAYLEXGH-RXTZ 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. The programming of the computations for the

SWAC is in progress.

SCAMP

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

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

Sponsor: " " Manager: C. B. Tompkins

Objective: To develop numerical methods for discrete-variable problems arising in certain specialized quantitative aspects of military science,
Status of Projects 9

with special emphasis on the application of automatic computers.

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

sponsoring agency to bring together mathematicians for the purpose of con- tributing to the deep mathematical developments required.

Status: NEW. Preliminary work is now under way; full scale work

sta rts July 1

.

2. Applied Research

RESEARCH IN THE MATHEMATICAL THEORY OF PROGRAM PLANNING

Task 1101-10-5102/50-11

(formerly 11.1/1-50-11)

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

Sponsor: " 11 Managers: E. W. 8a ran kin and T. S. Motzkin

Full task description appears in Apr-Jun 1950 issue.

Status: CONTINUED. Investigation of the farthest projection method

for solving linear inequalities was continued. T. S. Motzkin studied the

problem of whether the solution necessarily becomes cyclic with respect

to the order in which the projections are made on the hyperplanes bounding

the regions defined by the inequalities. It is clear that the length of

such a limiting cycle may be greater than the number of inequalities,

and it is not clear that such a cycle need exist at all. One special re- sult established was that the set of systems of three incompatible inequalities whose bounding lines form an acute-angled triangle and for which

there are patterns of projections not cyclic after a sufficient number of

projections has measure at most zero in a natural metric of triangles.

The proof is through an argument facilitated by a choice of a special

function on the boundary of the triangle. In terms of this function the

projection operator is simple and an analysis (related to Borel's treatment of the measure of normal numbers) of the situations which may exist

is possible.

The code of L. S. Joel and B. Handy for Gleyzal's method of

solving the assignment problem was run on the SWAC with matrices up to or- der 8 . The method is one of remarkable computational efficiency in which

successive approximations to the desired permutation are made. Beginning

with an arbitrary permutation, modifications are made by seeking closed

circuits in the original matrix, each path in the matrix being either a vertical line across a row, and with the property that every second element

of the circuit is an element which will be in the trace of the matrix modified by the approximating permutation. Such a circuit is acceptable as a modifying circuit if every element not on the trace of the matrix modified

by the approximating permutation but lying on the circuit is larger than

a corresponding element above the circuit on the trace. Careful and ingenious modification of the matrix in a way which leaves a solution to the

problem invariant yields a code which takes the SWAC for problems of the

size mentioned only a few seconds.

D. H. Lehmer proposed a direct combinatorial method of attacking

the problem. It consists of an exhaustive search through permutations with

an added feature that many permutations may be rejected in a block. With

some simplications by T. S. Motzkin the method is being coded by B. Handy.
10 Status of Projects

The cyclic projection and acceleration method of Kaczmarz and

Tompkins was coded for the SWAC by R. B. Horgan and used in solving a large system of simple linear inequalities on the SWAC. The novel feature

of this attack was the use of the collator input to present the inequalities

systematically to the machine. Each inequality could be described with a small amount of data, and the machine could generate the coefficient of the

inequality. Two inequalities can be described on a single IBM input card,

and the machine can compute rapidly enough' to permit rapid input of data,

frequently with no interruption to the collator feed at all. The method

seems to be working successfully, and there can be no question of the

feasibility of this use of the collator as input.

E. W. Barankin has investigated convergence properties of the

technique described in July-Septembe r 1952 issue, p. 12. A simplifying

property of this iterative system has been discovered, and it is hoped that

this will lend insight toward a convergence theorem. The computational work

on this technique to date along with the intuitive motivation of the technique, gives reason to expect a very effective convergence theorem. Work

on stochastic linear inequalities is continuing.

The stochastic search for the maximum of a non convex function

has been studied by E. W. Barankin. Some results on this problem are already in hand, assuring that under moderate conditions a stochastic se- quence of points in the domain of a function does bring with it stochastic

convergence, to the maximum value of the function, of the largest values of

the function on the successive sets of points turned up in the stochastic

process.

As part of the program of computation for the purpose of a com- parative study of various methods of solution of linear systems, with and

without optimizers, E. E. Osborne has begun coding the traversal method

of Brown and Koopmans.

Related to the task are investigations by C. V. Tompkins into the

evaluation of a probabilistic game representing military attrition. A

method of successive approximation has been developed and described in a working paper. Some computational experience in the use of this method has

been obtained by T. H. Southard and P. L. Childress.

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

by T. S. Motzkin; IN MANUSCRIPT. (2) "The multi-index transporation problem," by T. S. Motzkin; IN MANUSCRIPT. (3) "Ray systems with maximum angle sum," by T. S. Motzkin; IN MANUSCRIPT. (*+ ) "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; IN MANUSCRIPT. (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 existence 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.
Status of Projects

3. Development

1

1

NATIONAL BUREAU OF STANDARDS WESTERN AUTOMATIC COMPUTER (SWAC)

(previously listed as Air Materiel Command Confuting Machine)

Task 1101-20-5103A9-1

(formerly 1 101 -34-5103A9-1 ) Origin: Aeronautical Research Laboratory Authorized 11/1A8

Wright Air Development Center, USAF

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

Full task description appears in Apr-Jun 1 9*+9 issue, see 11.1/22-49-1.

Status: CONTINUED. During the last quarter the SWAC performed 875

hours of computation, working on 32 different problems. Of particular

numerical interest has been the determination of the full set of eigenvalues and eigenvectors of a 45th order symmetric matrix.

A significant increase in the effectiveness of the computer was materialized during the latter part of the quarter when a magnetic drum

memory of 4096 words was put into operation as an integral part of the

machine. The drum memory has already been used for a variety of problems,

such as in the solution of the matrix problem mentioned above, the solution

of combinatorial and probability problems, and the computation of energy

flow in a large electric power system. During this time the magnetic drum

memory has performed with a very high degree of reliability. The only trouble

encountered was traceable to defective read-write heads which since have

been weeded out.

Other engineering changes of major importance include the installation in the SWAC of a new high level deflection system for the cathode ray tube memory and the start of construction of a new flexible breakpoint control. The high level deflection system has contributed significantly to the stability and reliability of the computer while the breakpoint control when completed will aid the operator both in code checking

and in problem solution.

Publications: (1) "An improved cathode ray tube storage system," by

R. Thorensen; to appear in the Proceedings of the Western Computer Conference of the AIEE-IRE-ACM, held in Los Angeles, Calif., February 4,5,6,

1953. (2) "The SWAC - design features and operating experience," by

H. D. Huskey, R. Thorensen, B. F. Ambrosia, and E. C. Yowell; submitted to

a technical journal.

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 5A9

Sponsor: Aeronautical Research Laboratory, Wright Air Development Center, USAF

Manager: H. D. Huskey

Full task description appears in Apr-Jun 1949 issue.

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

LANGUAGE TRANSLATION STUDY

Task 1101-20-5103/52-1

(formerly 1101-21-5104/52-1)

Origin: NBS Authorized 8/1 5/51

Sponsor: The Rockefeller Foundation Revised 10/15/51

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

Full task description appears In Oct-Dec 1951 issue*

Status: CONTINUED. In the publication below V. A. Oswald and R. H.

Lawson report on experiments in translating German texts on brain surgery

with a limited glossary of 4328 entries. It is found that about ninety

percent of the running text can be translated with this glossary. This

concludes Prof. Oswald's research, sponsored by The Rockefeller Foundation

and administered by the National Bureau of Standards, Los Angeles.

Other phases of the task are inactive.

Publication: "An idioglossary for mechanical translation," by V. A.

Oswald, Jr., and R. H. Lawson; hectographed typescript, University, of

California at Los Angeles, Department of Germanic Languages, 16 p. (June

1953).

STUDIES IN THE THEORY OF DIGITAL COMPUTING MACHINES

Task 1 101 -20-51 03/53-1

Origin: NBS Authorized 9/30/52

Sponsor: Aeronautical Research Laboratory, Wright Air Development Center, USAF

Manager: D. H. Lehmer

Full task description appears in July-Sept 1952 issue.

Status: CONTINUED. During the last quarter, a code was written to

adapt SWAC to a three address, CPC-type operation. This code reads ten

words from a card, obeys the ten instructions, and then calls for another

card. Since all the orders are read from cards, logical instructions

needed for controlling calculating loops are omitted. The arithmetic operations of addition, subtraction, multiplication, and division are included as well as square roots, sines, and cosines. Future plans call for

the incorporation of other functions by making use of the magnetic drum.

This mode of operation, known as SWACPEC, performs 1,200 operations per minute. It then represents roughly a geometric mean between CPC

speeds and SWAC speeds. It Is felt now that the extra ease in coding for

this mode of operation will counterbalance the increase in running time and

will yield more efficient operations for small problems.

Publications: (1) "A general-purpose control panel for a model XI CPC,”

by P. Bremer, D. Teichroew, and E. C. Yowell; to appear in the IBM Newsletter. (2) "SWAC coding guide," by R. Morgan; an NBS report. (3) "Distribution sampling with high-speed computers," by D. Teichroew; IN MANUSCRIPT. (4) "Systematic generation of permutations on an automatic com- puter and an application to a problem concerning finite groups," by L. J. Paige and C. B. Tompkins; IN MANUSCRIPT.
Status of Projects 13

COMPUTING SERVICES FOR RESEARCH STAFF OF

THE INSTITUTE FOR NUMERICAL ANALYSIS

Task 1101-40-11 11 A9-1 a (formerly 1 101 -53-1 101 A9-1 a and 1 101 -53-1 100A9-1 ) Origin: NBS Authorized 9/2A8

Sponsor: Office of Naval Research Revised 11/16A9

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

Full task description appears in July-Sept 1 9*+9 issue, see . task 11 .1/32-49-1.

Status: CONTINUED. The following problems were computed on IBM ma- chines: (1) For D. Saxon (see task 1 1 01 -1 0-51 00/51 -5? p. 7 ): The integration of matrix elements involved in the photoelectric disintegration

of the deuteron for a Yukawa potential. (2) For D. Teichroew: The key

punching of the inverse table,

x ( p ) , where p = £ [

1 + (x)]

and oc(x) .= f z(t)dt, z(x) = ~~1~~ exp (- ^x 2

), u s/2¥

for p = 0.5(. 001)0. 999", 1 0D.

Research staff problems involving SWAC were: (1) For D.- H.Lehmer

(see task 1 101 -1 0— 51 00/50-4, p. 3 ): All but nine numbers of the Mersenne

problem have been computed twice. Those remaining will be checked as soon

as time on the SWAC permits. (2) For S. Mayer (see task 1 101 -1 0-5100/51 -5?

p. 7 )s The calculations on electron density in crystal structure have

been completed. (3) M. Hestenes (see task 1 101 -10-51 00A9-AE2, p.1 ):

A small amount of calculation was done on the 12x12 matrix supplied by the

Douglas Aircraft Company. (4) For DeVogelaere: (Universite Laval, Quebec,

Canada). Integration of a differential equation of the form x=f(x).

(5) For M. Muller: Some checking has been carried out on a code which will

be used to compute, by sampling methods, the power function of sequential

tests of the general linear hypothesis.

4. Mathematical Services

MATHIE U FUNCTIONS II

Task 1 101 -40-5131 A5-1

(formerly 1 1 01 -53-1 1 01 A5-1 ) Origin: Applied Mathematics Panel NDRC Authorized 7/1 A

7

Sponsor: Aeronautical Research Laboratory, Wright Air Development Center, USAF

Manager: E. C. Yowell

Full task description appears in Apr-Jun 1949' issue, see 11.1/2-45-1.

Status: INACTIVE. For status to date see Oct-Dec 1950 issue.
14 Status of Projects

TABLES FOR ROCKET AND COMET ORBITS

Task 1101-40-5131 A8-3

(formerly 1101-53-1101/48-3)

Origin: MBS Authorized 5/25A8

Sponsor: Aeronautical Research Laboratory, Completed 6/30/53

Wright Air Development Center, USAF

Manager: G» Blanch

Objective: Confutation of (a) V=1 -eosx,S=s inx, and x as functions

of U=x-sinx, and (b) V^Ccosh x) -1 jS^sinh x, and x as functions of

U

h =(sinh x)-x for the range

u

\ = 10" 14

(10

_1

5

)io” l3 (io" l4 )io‘ 12

. . .io“ 2 (io~3 )io

_1 do”2

) 3.1 5.

Background: There are no similar tables now in existence. The

tables will be of value in rapidly solving Kepler's equation. This was

specifically requested by Dr. Samuel Herrick of the Astronomy Department

of the University of California at Los Angeles.

Status: COMPLETED. The tables have been issued as an AMS publication . Publication: "Tables for rocket and comet orbits," by S. Herrick,

NBS Applied Mathematics Series 20; Available from U. S. Government

Printing Office, Washington 25, D. C. , STl .75.

PUNCHED CARD LIBRARY

Task 1101-40-5131 A9-2

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

Origin: NBS Authorized 7/1 4A8

Sponsor: Aeronautical Research Laboratory, Wright Air Development Center,

Air Research and Development Command, USAF

Manager: P. Bremer

Full task description appears in Ap r- Jan 1949 issue, see 11.1/2-49-2.

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 inverse table has been keypunched;

x(p), where p= 1+ oc(x) ] , and z(x) = ~^/exp(- ^x2

)

,

sf2tt

for p=0

.

5( .001 )0

.

999; 10D. Here

-x

ct(x) =
Status, of Projects 15

STATISTICAL SMOOTHING

Task 1 101 -40-51 31/51 -19

(formerly 1101 -53-1 1 01 /5l -19)

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

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

Manager: M. Howard

Full task description appears in Jae-Mar 1951 issue.

Status: CONTINUED, Seven eases were completed this quarter.

TABLES OF THE BIVARIATE NORMAL DISTRIBUTION FUNCTION

Task 1 101-40-5131/51-32

(formerly 1 1 01 -53-1 1 01 /5l - 32 ) Origin: Division 13? NBS Authorized 5/31/51

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

Manager: G. Blanch

Full task description appears in Apr-Jun 1951 issue.

Status; INACTIVE. For status to date see Oct-Dee 1952 issue.

PRESSURE FIELDS OF POTENTIAL FLOW PAST A BODY OF REVOLUTION

Task 1 101-40-5131/51-33

(formerly 1 101 -53-1 101 /5l -33)

Origin: Naval Ordnance Test Station (Pasadena) Authorized 6/22/51

Sponsor: Bureau of Ordnance, USN Completed 6/30/53

Manager: R. R. Reynolds

Objective: To compute the potential, velocity, and pressure on the

surface of a body of revolution.

Background: The Naval Ordnance Test Station needs these results in

a research project to determine bodies of minimum resistance.

Comments: In task 1 1 01 -53-1 1 01 /50-1 3 (see July-Sept 1950 issue, p.

30) an approximate procedure was used to determine a body for which a given

velocity distribution was valid. Some of the results of that task are now being utilized to obtain the pressure distribution.

Status: COMPLETED. The results were sent to the originator.

SIMPLIFIED ROLLING PULLOUT EQUATIONS

Task 1101-40-5131/51 -34

(formerly 1 101 —53—1 101/51 — 34

)

Origin: Cornell Aeronautical Laboratory Authorized 6/22/51

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, USAF

Manager: E. C. Yowell

Full task description appears in Apr-Jun 1951 issue.
1 6 Status of Projects

Status; CONTINUED. Two integrations were completed this quarter.

The necessary coding was completed for transferring this problem from the

CPC to SWAC.

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: Office of Naval Research, USN

Manager: A, D. Hestenes

Full task description appears in Apr-Iun 1951 issue.

Status: CONTINUED. This task has been reactivated and now will be

computed with the aid of SWAC. Coding is well under way. It is expected

that several important auxiliary tables will be available as a bi-product

of these computations.

INHERENT ERROR ANALYSIS FIRE CONTROL EVALUATION PROGRAM

Task 1 101-40-5131/52-19

(formerly 1 101-53-1101/52-19)

Origin: Naval Ordnance Test Station (Inyokern) Authorized 4/1/52

Sponsor: Bureau of Ordnance, USN Completed 6/30/53

Manager: R. R. Reynolds

Objective: This study is concerned with the inherent error analysis

of the air to ground rocket fire control evaluation program of the client

and consists of four phases as follows:

(A) Review of evaluation program and determination of the mathematical expressions for the inherent errors.

(B) Recommendation of statistical experiments by which NOTS can determine measurement errors, reading errors, etc., required for a numerical evaluation of the expressions obtained in (A).

(C) Study of the permissible measurement errors for a desired

accuracy in the resulting data.

(D) Numerical evaluation of the mathematical expressions obtained

in (A) using numerical values of errors obtained as a result of (B)

.

Background: This study is needed to determine the validity of the

results of fire control evaluation programs.

Status: COMPLETED. Data supplied by the Test Station have been

evaluated

.

SYSTEMATIC AND RANDOM ERRORS

Task 1 101 -40-5131/52-29

(formerly 1 101 -53~1 101/52-29)

Origin: North American Aviation Co. Authorized 4/1/52

Sponsor: Aeronautical Research Laboratory, Completed 6/30/53

Wright Air Development Center, USAF
Status of Project 17

Managers A, D. Hesteoes

Objective: To compute the salvo kill probability of a square target

of side 2a as a function of the parameters: aiming (systematic) error,

ammunition dispersion (random error), and salvo size,. The expression to

be evaluated is

oo oo

PSK = 2 2Z 2Z Q(i»jjN) pA

(i>j) oK i=0 j=0 A

where

Q(i,j,N) = 1 - [1 - P

k Ph (i,j)]N

Ph (i,j) = (1 i+

n

)) + cf> (-] + ))

(i - M)) + cf (^( i + Jii))

PA (i»j) = '<tdl it!) - <?(

o' n A

n

a«/2”

cr« n J

<f(— (hi - k» - i- k »

f

(x)

u

c 2~

du

.

sfnt

The distributions are assumed to be circular normal distributions with standard deviations cr^for the random error and o' for the aiming

error. N is the salvo size and y^ka, Xq=0 represents the center of the

aiming point distribution. P, is a probability that a target will be

destroyed if hit by a single shot independent of the effect of other shots,

a/n represents the mesh size. n is to be chosen such as is correct to

three decimals. Pg^ is to be evaluated for the following values of the

parameters P^O.2, 0.4, 0 .7? and 1.0; N=1 , 5>1 0 , 25, 50 , 1 00 , 1 50 , 200;

yQ=0,a ,2a ,3a , 5a ?10a 7 1 5a ,20a

; ^=a ,2a ,3a , 5a ,1 Oa , 1 5a ,20a ; and a^=a,2a,3a,

5a , 1 0a ,1 5a ,20a . Background; These computations are needed to predict performance

of new (as well as old) weapons.

Comments: Although much work of this nature has been done piecemeal

in the past, the program outlined above goes beyond the previous work

and represents a comprehensive program which will be useful to many or- ganizations interested in defense programs.

Status: COMPLETED. The results have been submitted to the originator. A manuscript has been prepared for possible publication.
18 Status of Projects

SIERRA WAVE PROJECT

Task 1101-40-5131/52-36

(formerly 1 1 01 -53-1 1 01 /52-36)

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

Sponsors Aeronautical Research Laboratory, Wright Air

Development Center, USAF

Manager: T. H. Southard

Full task description appears in Jan-Mar 1952 issue.

Status: CONTINUED. Theodolite data for several flights were reduced,

and the results were submitted to the originator.

SUBSONIC COMPRESSIBLE FLOW PAST OSCILLATING AIRFOILS;

REISSNER'S METHOD

Task 1101-40-5131/52-41

(formerly 1 101 - 53-1 101/52-41

)

Origin: Air Materiel Command, Wright Field Authorized 6/23/52

Sponsor: Aeronautical Research Laboratory, Wright Completed 6/30/53

Air Development Center, USAF

Manager: G. Blanch

Objective: To test Reissner's method, outlined in NACA Technical

Note 2363, by computing lift and moment coefficients for certain specified

parameters

.

Background: The problem of two-dimensional subsonic compressible

flow was treated by Possio in 1938. In 1944 DIetze gave tables and curves

for lift and moment coefficients. His computations are based on an iterative method, using the known results for the incompressible case as a starting point. H. E. Fettis, using a somewhat different technique, made

some further computations, and his results agreed well with those of Dietze.

In December, 1951, there appeared in the Journal of Aeronautical Sciences

a paper by Tinman, Van d© Vooren, and Greidanus, with a treatment of the same problem, based on the use of Mathieu functions. Tables are given which

agree with those of Dietze over a fairly wide range, but which disagree

sharply with Dietze's in other regions. The authors could not account for

the disagreement. In 1950, this laboratory mad© certain computations for

Wright Field, using a method developed by Has kind , involving Mathieu functions. The results differ considerably (though not in order of magnitude)

from those of Dietze. Reissner, in th© note already cited, also gave a treatment of the same problem. This treatment is based on his earlier

studies, and modified somewhat by taking account of Haskind's method. It

is now the purpose to make some computations using Reissner's method, to see to what extent results agree with those of Dietze and Haskind. If possible, an attempt will be made to discover the reasons for differences in

results based on the various methods. The work will be done in collaboration with H. E. Fettis of Wright Field, who has made extensive studies of

the problem.

Status: COMPLETED. The results have been transmitted to the sponsor

in the form of an NBS report (se© publication).

Publication: "Tables of lift and moment coefficients for oscillating

airfoils in subsonic compressible flow," by G. Blanch; an NBS report.
Status of Projects 19

LEGENDRE FUNCTIONS

Task 11 01 ~4-0-5l 3.1/52-48

Origin; Hughes Aircraft Company Authorized 6/30/52

Sponsor; Aeronautical Research Laboratory, Wright Air

Development Center, USAF

Manager; E, C. Yowell

Full task description appears in July-Sept 1952 issueStatus; INACTIVE. For status to date see Jan-Mar 1953 issue.

TABLES RELATING TO THE BIVARIATE NORMAL DISTRIBUTION FUNCTION

Task 1 101 -*+0-51 31/52-50

(formerly 1 101-53-1 101/52-50)

Origin; Division 13? NBS Authorized 6/1/52

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, USAF

Manager: G. Blanch

Full task description appears in Ap r- Jim 1952 issue.

Status: INACTIVE. For status to date see Oct-Dec 1952 issue.

SEARCH FOR MAGIC SETS IN A LATIN SQUARE OF ORDER n Task 1101-40-5131/53-3

Origin: Summer Symposium Group

Sponsor: Project SCAMP, 0NR

Manager: A. D. Hestenes

Objective: To determine the magic sets of a 10x10 latin square.

Background: A given n-th order latin square has a latin square or- thogonal to it if, and only if, the given latin square has n nonoverlapping

magic sets. A complete set of n-1 orthogonal latin squares determines a projective geometry of order n.

Status: COMPLETED. This task was initiated to study machine methods.

As present work has grown beyond the scope of the task as written, the

program will be continued under a future task which will represent general

computing for the SCAMP Project.

Authorized 7/28/52

Completed 6/30/53

EIGENVALUES

Task 1101-40-5131/53-6

Origin: Convair Authorized 9/26/52

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, USAF

Manager: E. C. Yowell

Full task description appears in July-Sept 1952 issue.

Status: CONTINUED. The eigenvalues and eigenvectors of this matrix
20 Status of Projects

have been transmitted to the contractor. The determination of the influence coefficients has not been started.

METEOROLOGICAL MEANS

Task 1101-40-5131/53-10

Origins Meteorology Department, UCLA Authorized 9/30/52

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, U5AF

Manager: F. H. Hollander

Full task description appears in JuXy-S@pt 1952 issue.

Status: CONTINUED, The computation of u. . i, and v. . for all

? J 2” 9 J levels has been completed on the SWAC. Listings of this information have

been made on the IBM tabulator and forwarded to the UCLA Meteorology Department® Coding has been completed on routines for time averages (based

on daily readings for one month) and space averages (based on readings,

for a given day, around an entire latitude circle) of the wind velocity,

temperature, and height. For the sea level, pressure is computed instead

of height.

Computing has been completed on the time averages, and has been

begun, but not completed, on the space averages. Checking for these

computations has been done by repeating all calculations. The results of

the two runs, on punched cards, are compared in an IBM reproducer. Averages

in space, of the time averages, and averages in time, of the space averages, have also been computed. This has been done on an IBM calculating

punch, type 604, rather than on SWAC.. It is estimated that this problem

is more than eighty percent completed, (See Jan-Mar 1950 issue, project

1 1 .1/31-50-17, p. 15.)

PROLATE SPHEROIDAL WAVE FUNCTIONS

Task 1101-40-5131/53-11

Origin: Stanford Research Institute Authorized 9/30/52

Sponsors Aeronautical Research Laboratory, Wright Air

Development Center, USAF

Manager: R. R. Reynolds

Full task description appears in July-Sept 1952 issue.

Status:

and n=9(2)19,

been computed

These values

dance

.

CONTINUED . For c=1.2, tt/2

,

the coefficients and b^n

for k= ( 1

) ^ + 5 , wh e re 2a

are now being substituted in

2, 3V4, 2.5. 2.8, 3, v , 3.2,

as well as K; 1 > and O. have in vln . is the lower index of max c^.

the general formulas for impeNONLXNEAR SERVO PROBLEM

Task 1101-40-5131/53-14

Origin: Douglas Aircraft, Santa Monica Authorized 12/15/52 Sponsor: Aeronautical Research Laboratory, Terminated 6/30/53 Wright Air Development Center, Air Research

and Development Command, USAF
Status of Projects 21

Managers E. C. Yowell

Objectives T® solve the set of differential equations

®s

= K

1®s

+ E

9

s K

2 + E9/3 + K

4 ®s

+ ' K

6 5

q

+ k

7

9

Q

+ K

8

F

s

®q = k

28fz

+ k

29 f + k

30

+ k

31 es + k

33

+ K

34 + K35^ + K

36

6

6 = K

37

Fz

+ K

3 q

F + K

39 0 s + K1+09

g + K^29

Q

+ K44 6 +

*4+5 5

where are all constants, are + 1 or -1 , F is given as I5sin87rt , and F are analytic functions of 6 and 9^.

Background; This problem arises in the study of a servo control

system. This solution is to provide a calibration check on a large set

of differential analyser solutions of this problem.

Status; TERMINATED. At the contractor's request, this problem

has been terminated. The work to date and the results obtained are sum- marized for possible future reference by the originator.

NONSYMMETRIC MATRIX

Task 1101-40-5131/53-1 5 Origins Aeronautical Research Laboratory, Wright Air Authorized 1 2/1 5/52

Development Center, Air Research and Terminated 6/30/53

Development Command, USAF

Sponsor; " " Managers M. W. Steinberg

Objective: Find the eigenvalues of an 1 8x1 8 nonsymmetrix matrix.

Background: This matrix originated at the Aeronautical Research

Laboratory and was sent to us for the purpose of developing methods of

handling problems of this nature.

Status; TERMINATED.

FLUTTER MATRICES

Task 1 101-40-5131/53-17

Origin: Boeing Airplane Company Authorized 1 2/1 5/52

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, Air Research and

Development Command, USAF

Manager: E. E. Osborne

Full task description appears in Oct-Dec 1952 issue.

Status: CONTINUED. The eigenvalues and eigenvectors of the fifteen

6x6 matrices of complex elements have been found. The results have been

sent to the requesting contractor. It is planned that a report describing

the method used will be prepared and forwarded to the originator.
22 Status of Projects

DISCRIMINANT FUNCTIONS

Task 11 01 -1+0-5131/53-1 8

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

Sponsor; " Manager; Do Teichroew

Full task description appears in Oct-Dee 1952 issue.

Status; CONTINUED* Coding of the problem for the SWAC is continuing

POWERS OF MATRICES

Task 1101-40-5131/53-19

Origin; Consolidated Vuitee Aircraft Corporation Authorized 1 2/1 5/52

(Convair) Terminated 6/30/53

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, Air Research and

Development Command, USAF

Manager: M„ W. Steinberg

Objective: Find the last rows of the 1st, 2nd, 3rd, and 4th powers

of the matrix

(q) 5 0 , Oj , »

,

(J), P(1), (]) P(1), 0,...

(q) P

2

(2), ( 2

) p(2) P(2), (|) P

2

(2),

\

4. (q) PN

(N), (![

, )PN" 1 (N) P(N), (N)PN“2 (N)P2

(N,...(JJ)P

N

(N)

with

J

P(j) = (1 --|) n

, P(j) = 1 - P(j)

and for the following values of the parameters N=100; n=1 00 , 200 , 500, 1 000,

1200s and p=0.3,0o5,0.7,1 .0.

Backgrounds This problem is associated with the contractor's Air

Force projects.

Status: TERMINATED.

LIGHT SCATTERING FUNCTIONS

Task 1101-50-5131/53-22

Origin: Naval Radiological Defense Laboratory, Authorized 1 2/1 5/52

San Francisco Terminated 6/30/53

Sponsor: " " " Manager: T. H. Southard

Objective: A light beam is scattered by a spherical droplet. It is
States of Projects 23

desi .”ed to compute the intensities of light being transmitted in various

directions, for various kinds and sizes of droplets.

Background: The problem arises in connection with the work of the

originator. Intermediate results have been tabulated by Gumprecht and

Sliepcevich in "Light scattering functions for spherical particles". In

addition, the intensity functions have been tabulated for.® different parameter field in "Tables of scattering functions for spherical particles,"

National Bureau of Standards Applied Mathematics Series 4(1948, U. S. Government Printing Office, Washington, D. G.)

Status: TERMINATED.

BIO-ASSAY PROBLEM

Task 11 01 -40-51 31/53-2^

Origins 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. A table for *>=1(1)17; *>=-5(.5)5; /S=.5(.5)10

has been confuted and is being cheeked.

SINGLE SHOT PROBABILITIES

Task 1101-40-5131/53-25

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. A report on the theoretical investigation is

being prepared.

DISCRETE MINIMAL SPACES

Task 1101-40-5131/53-26

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: CONTINUED. This task was coded for SWAC by the originator.

Several solutions have been made. These are being studied before the

next set of solutions will be computed.
24 Status of Projects

RAYLEIGH SCATTERING OF LIGHT IN THE ATMOSPHERE

Task 1101 -5+0-51 31/53-28

Origin: Naval Ordnance Test Station (Inyokern) Authorized 3/31/53

Sponsors Bureau of Ordnance, USN Completed 6/30/53

Manager: G. Blanch

Objective: To produce tables similar to those computed in connection

with task 1 101 -50-51 31 /5l -25, "Confutation in connection with a study of

polarization of light," for five additional values of , namely V= .0001

,

.001 , .01 , .02, .OR. (These values may be replaced by slightly different

ones after further consultation with Z. Sekera and sponsor.)

Background: The following report by Prof. Zdenek Sekera will be

basic in this task: "Tables relating to Rayleigh scattering of light in

the atmosphere," Scientific Report Number 3? UCLA Department of Meteorology, Contract AF1 9(1 22)-239?November 1952. The functions tabulated in

this report depend on Chandrasekhar's "X" and "Y" functions, the computation of which involves the solution of nonlinear integral equations by

lengthy’ iteration processes. It turns out that for small values of the

parameter V , it is possible to use a much simpler approximation, for the

accuracy aimed at in the present task. Once these values of X and Y are available, there remain to be computed the various scattering functions

and "intensities" (about 15 functions).

Status: COMPLETED. Th© contractor expects to issue a report.

LIGHT SCATTERING COMPUTATIONS

Task 1101-40-5131/53-30

Origin: University of California at Los Angeles Authorized 3/31/53

Sponsor: Atomic Energy Commission Completed 6/30/53

Manager: P. Bremer

Objective: To reduce data in accordance with the contractor's

formulas. Approximately two sets of observations are processed on IBM

machines ©very week.

Background: These computations arise from th© work of the AEC Medical

Group at the University of California at Los Angeles.

Status: COMPLETED. Additional data reduction was performed, and

the results were fitted with straight lines by the method of least

squares. (See Jan-Mar 1953 issue, p. 25.)

EXPERIMENT DESIGN

Task 1101-40-5131/53-31

Origin: Radioplane Corporation Authorized 3/31/53

Sponsor: Aeronautical Research Laboratory, Wright Completed 6/30/53

Air Development Center, Air Research and

Development Command, USAF

Manager: D. Teichroew

Objective: To study data submitted by the originator and recommend

a suitable design to determine a combination of parameters which will
Statius of Projects 25

maximize a desired function.

Backgrounds The Parachute Testing Panel of the Committee of Aeronautic Equipment of the Research Development Board has suggested the parachute testing activities should make use of statistical techniques in its * programs. This is an attempt to give assistance in that direction.

Status: COMPLETED*, A report has been submitted to the contractor.

ROOTS OF DETERMINANTS

Task 11 01-40-51 31/53-32

Origin: North American Aviation Authorized 3/31/53

Sponsor: Atomic Energy Commission

Manager: E. C. Yoweli

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. Eight cases were solved this quarter. Recoding

for use with the magnetic drum memory of SWAC is now in progress.

PLANE SHOCK WAVES

Task 1101-40-5131/53-33

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

Development Center, Air Research and Development Command, USAF

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

Full task description appears in Jan-Mar 1953 issue.

Status: INACTIVE. For status to date see Jan-Mar 1953 issue.

LINEAR EQUATIONS (CONVAIR)

Task 1101-40-5131/53-35

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. This task is held up temporarily pending the

receipt of data.

MISCELLANEOUS CORONA COMPUTATIONS

Task 1101-40-5131/53-36

Origin: NBS, Div. 15

Sponsor: " Authorized 3/31/53
26 Statras of Projects

Managers P. Bremer

Full task description appears in Jan~Msr 1953 issue.

Status; INACTIVE. For status to date se© Jan-Mar 1953 issue.

ATTRITION FUNCTIONS

Task 1 101-40-5131/53-37

Origin; Logistic Research Authorized 3/31/53

Sponsor; Office of Naval Research, Logistics Completed 6/30/53

Branch

Manager: C. B. Tompkins

Objective; To solve some simple arithmetically described battlelike

games with probabilistic attrition in order to create models of evaluation

functions applicable to military or to other competitive situations; to

present these to experienced students of military science (quantitatively

inclined officers of the Armed Forces) in a small working group discussion

in order to guide the sponsor and the workers in future studies of war games.

Backgrounds Studies of this kind have been carried on with low

priority for several years by several groups. This task is a natural application of the large computing instrument to these studies which have

heretofore been hampered by inadequate computational facilities.

Status: COMPLETED.

FLUTTER PROBLEM

Task 11 01 -4C-51 31/53-38

Origin: Douglas Aircraft Company Authorized 3/31/53

Sponsor: Aeronautical Research Laboratory, Wright Terminated 6/30/53

Air Development Center, Air Research and

Development Command, USAF

Manager: F. Hollander

Objective: To obtain the powers of a 9x9 matrix with complex elements, using doable precision programming.

Background: The originator coded the problem for SWAC and supplied

three matrices, one of which was chosen as a test problem. The results

are to be compared with answers obtained by the originator on his IBM

equipment

.

Status: TERMINATED.

MONTE CARLO STUDIES

Task 1101-40-5131/53-39

Origin: RAND Corporation Authorized 3/31/53

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, Air Research and Development

Command, USAF
Status of Projects 27

Manager; A. D. Hcstenes

Full task description appears in Jan-Ma r 1953 issue.

Status; CONTINUED. This problem was coded for SWAC by RAND personnel. The XNA has provided machine and operator time. Several cases have been computed, and an equal amount of computing is anticipated in

the future.

RANKING PROBABILITIES

Task 11 01-40-51 31/53-40

Origin; Cornell University Authorized 3/31/53

Sponsors Office of Naval Research

Manager; D. Teichroew

Full task description appears in Jan-Mar 1953 issue.

Status; CONTINUED. The problem has been coded for the SWAC, and

the routine is being checked.

ACCEPTANCE TESTS

Task 1101-40-5131/53-41

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

Institute of Technology

Sponsor: Army Ordnance Corps, U. S» Army

Manager: E. C. Yowell

Objective: To design and administer the acceptance tests for the

high speed digital counter being purchased by Jet Propulsion Laboratories,

Background: The amount of computation being performed at JPL has

made it advisable to purchase a large scale automatic digital computer.

Since JPL ! s past computing experience has been primarily analogue, it was

felt that INA could be ©f assistance to them, particularly in evaluating

a new machine.

Status; NEW.

B. P. A. Studies

Task H01-40-5131/53-42

Origin: Bonneville Power Administration Authorized 6/29/53

Sponsor: " " Manager: E. C. Yowell

Objective: To evaluate the utility of digital computing machinery

in solving power transmission problems.

Background: Many problems arising in the work of the BPA are being

solved on analogue equipment. This investigation seeks to determine which

of their problems can be more effectively solved on digital equipment.

All coding is being done by BPA personnel.

Status: NEW
28 Status of Projects

ROOT’S OF A TRANSCENDENTAL EQUATION

Task 1101-40-5131/53-43

Origin; U. S. Naval Ordnance Test Station, Pasadena Authorized 5/20/53

Sponsor: " 11 " Completed 6/30/53

Manager: A. D» He s tones

Objective: To obtain «* (if suefa exists) satisfying

sin(«:- 0 ) =/° 08iii(<*-0

- 0)

<x.Q

<

oc 4 & + io°

where /’(«,/') is given empirically, and

0 = a re tan 0 -< n

x

1 - cosy' 2 2

, _ Asia A

A - • •

f is to be taken at. 10° intervals in the range -v 4 f 4 t. Th© parameter!

have the following values: x= 1 *5,3? A=^/6,Tr/4,Tr/3,y= =1 500 , 1 250 , 1 000

,

750,500, and ^

0=-l5 0

S “12^0

, oe o,12^0 ,l5°»

Status: COMPLETED (NEW), Th© roots were obtained using a suitable

SWAC code and have been submitted to th© originator.

HINGE MOMENTS

Task 1101-40-5131/53-45

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

Air Development Center, Air Research and

Development Command, USAF

Sponsor: 88 81 18 Manager: G. Blanch

Objective: To compute the integrals RP, (c), RT. (c), RP„(c), RT.(c),

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

n n

Background: In th® process of computing lift and moment coefficients for task 1 1 01 -53-1 1 01 /52-4l , there were computed certain constants

(21 A and B, in terms of which Meissner's p for the circulatory flow can be

determined. Once this is done, th© following integrals are required:

RP

h (c)

RP*(e)

[p a

] h dx 5

2

|

[pj^dx;

RTh (c)

RToc( c

)

c) [P a

] h

dx

c) [pj^dx

where R is some constant.

Status: NEW.
II. Computation Laboratory

(Section 11.2)

1 » Resea reh

RESEARCH IN CLASSICAL NUMERICAL ANALYSIS

Task 1102-10-1104/50-1

(formerly 11.2/11-50-1)

Origins NBS Authorized 1/1/50

Managerss J. Todd, M» Abramowitz, and H. A, Antosiewicz

Full task description appears in Jan-Mar 1950 issue.

Status; CONTINUED. H. Antosiewicz investigated the differential

equation x+k[f (x)+g(x)x]x+h(x)=ke( t) . A paper including his recent

as well as all his previous results on this equation has been prepared

(see publication (1) below).

M. Abramowitz has been engaged in a further study of the problem

of heat convection in laminar flow of fluids inside a tube, an extension of

his previous work in this field. A paper on this has been prepared (see

publication (3) below).

He has also been investigating the numerical inversion of the

Laplace transform.

Publications; (-1) "On the differential equation x+k[ f (x)+g(x)x]x

+h(x)=ke(t) , " by H. A. Antosiewicz; submitted to a technical journal.

(2) "Some implications of Liapunov's conditions for stability," by H. A. Antosiewicz and P. Davis; submitted to a technical journal. (3) "On

forced convection in laminar flow through a tube," by M. Abramowitz; 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 1102-10-1104/50-2

(formerly 11.2/11-50-2)

Origin: NBS Authorized 3/1/50

Manager: P. Davis

Full task description appears in Jan-Mar 1950 issue.

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

29
30 Status of Projects

MISCELLANEOUS STUDIES IN PURE MATHEMATICS

Task 1102-10-110^/50-*+

(formerly 1102/11 -50-*+)

Origin: MBS Authorized 1/1/50

Managers? 0. Tans sky Todd, J. Todd, M. Abramowitz, and A. Hoffman

Full task description appears is Jam-Mar 1950 issue*

Status; CONTINUED. The manuscript for a proposed volume in the

Applied Mathematics Series whieh deals with numerical experiments in

conformal mapping is in preparation. The volume will contain articles

by A. M« Ostrowski, S. E. Warsehawski, J. Todd, L. Ahlfors, and G. Blanch

and L. K. Jackson.

In connection with this work elaborate experiments have been

carried out on SEAC by I. Rhodes on the mapping of an ellipse with axis

ratio 5:1 on a circle. Many interesting phenomena have turned up, one

of which lead to the research on the eigenvalues of integral equations

(mentioned by H. Wielandt in task 1 102-10-1 10*+/ 52-3*+, p .32 ) . It has

S>een found, .for example, that although some fifty iterations would be

necessary to obtain results correct to 9 decimal places directly, some 12 are sufficient when these are subjected to multiple accelerations of the

Aitken form. In order to obtain 9 decimal places it is necessary to use

a 1° interval in the quadrature, and some 25 minutes were required for

each iteration. This indicates the economy effected by use' of this process. Standard subroutines for the Aitken process have been prepared by

P. Henrici.

Publication: "Linear functional equations and interpolation series,"

by P. Davis 5 submitted to a technical journal.

NUMBER' THEORETICAL TEST PROBLEMS FOR SEAC

Task 11 02-1 0-1 10*+/ 50-5®

(formerly 1 1 .2/1 1-50-5)

Origin: NBS Authorized 6/1/50

Managers s 0. Taussky Todd and K. Goldberg

Full task description appears in Apr-Jun 1950 issue.

Status? CONTINUED. The table of least positive primitive roots

is being extended when time is available on SEAC. The preparation of an index table is under consideration.

Publications "A table of Wilson quotients and the third Wilson

prim©," by K. Goldberg; J. London Math. Soe», 28, Pt. 2, 252-256 (Apr.

1953) . Reprints available.
Status of Projects 31

ANALYSIS OF CRYSTAL STRUCTURE

Task 1102-10-1104/51-3

Authorized 9/28/50

Terminated 6/30/53

Origin: NBS

Manager: Ethel C. Marden

Objective: Calculation of electron density /> (x,z) corresponding

to structure factors F(h,0,-£) satisfying the reciprocal relationships

CD oo /o(x,z) = y

’ y

' F (h ,£) cos 27T(hx- £z)

h=-oo h=-oo

F(h,i) = ^o(x,z) cos 27r(hx + /z)dx dz.

Background: This problem arises in the study of crystal structures.

It is the two-dimensional special case of a more general three-dimensional

problem in which /o is a function of x,y,z and F depends on three parameters

(Miller indices) h, k,j&. The function measures the distribution of

electrons in the unit cell of a crystal, treated as a continuous function,

and F measures the intensity of light reflected from a crystal surface with

Miller indices h,k,l5.

The approach is to start with a preliminary experimentally

determined set of structure factors F (h ,0 ,£) , and calculate the corres- ponding /) ' s . Then the /o 's are modified so as to make the distribution

more "acceptable," in particular so that /O is non-negative and conforms

to certain preconceived ideas about the shape of such distribution functions. Changes in /O are to be accomplished in such a way as to keep

F as close as possible to the observed values.

Comments: This work is part of an investigation of crystal structures being carried on by the Portland Cement Association Fellowship.

The problem was proposed by Dr. F. Ordway, who suggested a method to be

followed

.

Status: TERMINATED.

SOLUTION OF LAPLACE EQUATION BY MONTE CARLO METHOD

Task 1102-10-1104/51-6

Origin: NBS Authorized 9/28/50

Manager: M. Abramowltz

Full task description appears in July-sept 1950 issue.

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

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-Dee 1951 issue.

SPECIAL PROBLEMS IN FINME MATRIX THEORY

Task 1102-10-1104/52-34

Origin: NBS Authorized 11/6/51

Manager: 0. Taussky Todd

Full task description appears in Qet-Dee 1951 issue.

Status: CONTINUED. A. J. Hoffman and 0. Taussky Todd continued

their investigations concerning characterizations of normal matrices (see

publication (1) below). T. S. Motzkin and Mrs. Todd continued their work

on matrices with property L (see publication (2) below). They were also

trying to find an alternative treatment of their problem by a function

theoretic method. This has now been taken up successfully by H. Wielandt.

0. Taussky Todd performed mar© work on commutators of matrices and

permutations of products of matrices.

A new volume in the Applied Mathematics Series was edited by

0. Taussky Todd (see publication (3) below), a sequel to AiV629, "Simultaneous linear equations and the determination of eigenvalues," and includes: (1) "Practical solution of linear equations and inversion of

matrices," by L. Fox; (2) "Punched card experiments with accelerated

gradient methods for linear equations," by A. I. Forsythe and G. E. Forsythe;

(3) "Iterated methods ®f solving linear problems on Hilbert space," by

R„ M. Hayes; (4) "Tables of inverses ©f finite segments of the Hilbert

matrix," by I. R. Savage and E. Lukacs; (5) "The condition of the finite

segments of the Hilbert matrix," by J. Todd; and (6) "Lower bounds for the

rank and location of the eigenvalues of a matrix," by Ky Fan and A. J. Hoffman. Descriptions of all these papers have appeared earlier except

that of J. Todd. This paper is an examination of the condition of the

Hilbert matrix H and a study of its inversion on SEAC. The condition

number of H

r is exponentially large; the average condition number of a matrix is 0(n). While it has been possible to invert matrices with

condition number O(n^) of order 50 or more on SEAC, obtaining about seven significant figures, it is not possible to invert a 6x6 Hilbert matrix.

The process used is a modification of the Gauss elimination process as presented by von Neumann and Goldstine which was coded for SEAC by

M. Newman and S. L. Pollack.

H. Wielandt extended his earlier results concerning matrices
Status of Projects 33

with property L (see publication (^f) below). He found alternative proofs

and generalizations of the the theorems in publication ( 2 ) below. These

proofs are based on function-theoretical arguments, and the generalizations

refer to the case of matrices whose elements are polynomials of arbitrary

degree (not only linear) in a parameter z. Dr. Wielandt found new inequalities for the eigenvalues Yi , of

the sums C=A+B of two Hermitian matrices with given eigenvalues <k. . These inequalities have the form 1 x

+ **’ /VL +, ** + ^/’s and include inequalities of H. Weyl and Lidskii. The proofs are based

on a maximum-minimum characterization of oc„ , which includes the

"l "s’

maximum characterization of «-. + ...+ oc given by Ky Fan. H. Wielandt

determined those values y which may occur as eigenvalues of A+B , where

A and B run over all normal (not necessarily Hermitian) matrices with

given eigenvalues. He found the following comparison theorem for the

eigenvalues <*

, of normal matrices A,B which are "near" in the sense

that their difference A-B has a given rank r < n: Every circle containing

b eigenvalues of B contains at least b-r eigenvalues of A. He proved a similar theorem for the eigenvalues of normal segments of normal matrices.

He further determined the possible spectra of the segments of all Hermitian matrices with a given set of eigenvalues. Dr. Wielandt developed

a method for estimating the error of the eigenvalues which occur if an Hermitian integral equation J'K(x^ )yOpd %

= ky(x) is solved approximately by use of any given formula for numerical quadrature. The method

can be applied without any previous knowledge concerning the eigenfunctions or eigenvalues. It seems the first known method to determine in

advance a number of interpolation points sufficient to guarantee a prescribed accuracy.

In connection with 0. Taussky Todd's work on commutators of

matrices, Ky Fan determined the commutators of the unitary group and

the orthogonal group. Ky Fan continued his investigation on inequalities

concerning the eigenvalues of Hermitian matrices and on unitary-invariant

metric properties of the space of matrices.

Publications: (1) "A characterization of normal matrices," by

A. J. Hoffman and 0. Taussky; IN MANUSCRIPT. ( 2 ) "Pairs of matrices

with property L, II," (summary) by T. S. Motzkin and 0. Taussky; to

appear in the Proceedings of the National Academy of Science. ( 3 ) "Contributions to the solution of systems of linear equations and the determination of eigenvalues," edited by 0. Taussky; to appear in the NBS

Applied Mathematics Series. (4) "Pairs of matrices with property L,"

by H. Wielandt; accepted for publication by the Journal of Research of

the NBS

.
34

Status of Projects

ROOTS OF POLYNOMIAL EQUATIONS

Task 1102-10-1104/52-51

Origins NBS Authorised 2/1/52

Managers D. I. Rubin Terminated 6/30/53

Objectives To produce a SEAC routine which will determine the

roots, real and complex, of any given polynomial equation,, provided its

degree is not so large that computation time becomes excessive*

Background; Let C be a closed

f(z) a function which is meromorphic

Co Then

curve in the complex plane, and

within, and has no singularities on

1 2x1

where , r^ » • • * ar® the orders of the zeros and s

.j , S£

;

s

^

* the orders

of the poles of f(z) within C» Since a polynomial has no poles, it can

be determined whether the polynomial has zeros in a given region, and, if so, the region can be subdivided and the process repeated until the roots are known to any desired accuracy.

Status: TERMINATED. This task was terminated in favor of tasks

of higher priority.

DISTRIBUTION OF NORMAL MODES OF VIBRATION OF CUBIC LATTICES

Task 1 102-1 Ot-1 104/52-62

Origins NBS

Sponsor: " Manager: T. W. Ledley and F. J. Stockmal

Authorized 2/25/52

Completed 6/30/53

Objective: To compute the distribution of normal modes of vibration

xn cubic lattices, on which depend the vibrational contribution to the thermodynamic properties of polyatomic molecules and crystals.

Exact distribution functions have been found for two-dimensional lattices, but the amount of computation necessary in the three-dimensional

ease has hitherto been prohibitive. This program will investigate ap- plication of high-speed computing devices to the problem.

Background:^ The characteristic frequencies of normal modes of vibration of a cubic . la ttice are roots of cubic equations. Since the number of equations is of the order of the number of particles in the lattice (0(10^)), the time required for the calculation of the frequen- cies

_ is tremendous. It is expected that by taking small crystals (0(1 q4) lattice points) one would obtain a considerable amount of information concerning the distribution.

Status: COMPLETED. The results were transmitted to the originator.
Status of Projects

35

AN INTEGRAL ARISING IN THE THEORY OF COOPERATIVE PHENOMENA

Task 1102-10-1 10m-/ 52-69

Origin: NBS Authorized 5/9/52

Manager: J. Todd

Full task description appears in Apr-Jun 1952 issue.

Status: CONTINUED., It has not yet been possible to carry out ex- periments on handling this work by Monte Carlo methods. The work done

on this task is described in the publication below.

Publication: "The tabulation of an integral arising in the theory

of cooperative phenomena," by M. Tiksoni; J. Res. NBS 50, 177-178 (Mar.

1953).

ROOTS OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS

Task 11 02-1 0-110^/52-76

Origin: NBS

Manager: D. I. Rubin

Full task description appears in Apr-Jun 1952 issue.

Authorized 5/21/52

Terminated 6/30/53

Status: TERMINATED. This task was terminated in favor of tasks

of higher priority.

DIFFERENTIAL EQUATION FOR NERVE FIBER REACTION

Task 1102-10-110V53-15

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

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

Full task description appears in Oet-Dec 1952 issue.

Status: CONTINUED. Further results were obtained for various

parameter values 1^,1^. Also a periodic solution V(t) was found by

numerical integration of the system. These results were transmitted to

the sponsor.

BAKER-HAUS DORFF FORMULA

Task 11 02-10-1104/53-1 6

Origin: NBS Authorized 12/8/52

Manager: 0. Taussky Todd and K. Goldberg

Full task description appears in Oet-Dee 1952 issue.

Status: CONTINUED. K. Goldberg has derived a formula for evaluating the coefficients of the powers of x and y in z, where ez=ex .ey

and xy/yx.
36 Status of Projects

POISS GN-B OLTZMAN EQUATION

Task 1 102-10-1 1 04/53-31

Origins NBS Authorised 1/23/53

Managers M. Abramowitz

Full task description appears in Jan-Mar 1953 issue.

Status: INACTIVE. For status to date see Jan-Mar 1953 issue.

HYPERGEOMETRIC FUNCTIONS

Task 1102-10-1104/53-35

Origins NBS _ Authorized 3/25/53

Manager; P. ftabinowitz, W. Cahill

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. The eod© for the hyper geometric function with

complex parameters and arguments has been modified to compute the confluent hypergeometric function. A report on the code for the general

hype rgeome trie function is in preparation.

RESEARCH IN LINEAR PROGRAMMING

Task 1 102-10-51 1 5/50-2

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

Office of the Air Comptroller, ISAF

Sponsor: " " " Managers: J. Todd and A. Hoffman

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

Status: CONTINUED. The series of lectures on the "Theory of Games

and Its Applications," jointly sponsored by the National Bureau of Standards and American University, which was reported in Jan-Mar 1953 issue, p. 37?

continued into the current quarter. A list of the titles and speakers

of the talks presented this quarter is given in Lectures and Symposia,

p.68 . The principal work of the quarter was the experimental investigation of various methods of solving linear programs. The "block

relaxation" method is a variation of the furthest hyperplane scheme. Instead of moving to the furthest hyperplane, one moves to the intersection

of the two furthest hyperplanes. The results of tests with this method

are reported in publication (7) below. Further experiments with the

furthest hyperplane procedure, in order to determine the optimal "overshoot", are reported in publication (1) below. The use of double description is described in publication (9) below.

A code for solving linear programs by a method proposed by

Brown and Koopmans has been prepared by K. Christoph. New procedures

for solving linear programs in which the objective function depends linearly on several parameters are under study.

Publications: (1) ' jn 'overshoot' 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 Ky 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.
Status of Projects 37

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

D. Sokolowsky; IN MANUSCRIPT. (4) "Remarks on the smoothing problem,"

by A . J. Hoffman; IN MANISCRXPT. (5) "Computational experience in

solving linear programs," by A. J. Hoffman, M„ Msnnos, D. Sokolowsky,

and N. A. Wiegmann; accepted for publication in the Journal of the

Society for Industrial and Applied Mathematics. (6) "A characterization

of normal matrices," by A. J. Hoffman and 0. Taussky; IN MANUSCRIPT.

(7) "On block relaxation," by L. S. Joel; IN MANUSCRIPT. (8) "Eigenvectors

of matrix polynomials" by M. Mannos; to appear in the Journal of Research

of the NBS . (9) "Experimental results with the double description method,"

by S. Pollack; IN MANUSCRIPT.

SURVIVAL PROBABILITY IN PATTERN BOMBING

Task 1 102-10-51 1 5/53-13

Origin: Deputy Chief of Staff Operations, Authorized 12/8/52

Directorate of Operations, Operations Completed 6/30/53

Analysis, LSAF

Sponsor: Office of the Air Comptroller, U5AF

Managers: E. Marden and R. K. Anderson

Full task description appears in Oet-Dee 1952 issue.

Status: COMPLETED. Results were transmitted to the originator.

COMPRESSIBLE FLOW - METHOD OF ORTHOGONAL AND KERNEL FUNCTIONS

Task 1102-10-5116/52-16

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

Development Center, U5AF 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. Computation of 1 2 particular solutions of

stream equation A Y +

tfF(A)'f- =0 is proceeding. These solutions are given by

z

n f(^)dzdij (n = 0,1

0 zQ J zq

5).

the

When reduced to real form, each ©f the above complex integrals is equivalent, for fixed n, to six real double integrals. Sixty of the seventy-two

integrals have been computed. A code is being written to check these

particular solutions in the differential equation itself.
38 Status of Projects

TABLES OF INTEGRALS INVOLVING THE HIGHER TRANSCENDENTAL FUNCTIONS

Task 1102-10-5116/52-33

(formerly 1 1 02-21 -51 1 7/52-33

)

Origin? MBS Authorised 10/11/51

Managers F. Qberhettinger

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED, The tables have been compiled and are now being edited.

INTEGRAL OPERATORS AND INTERPOLATION SERIES

Task 1 102-10-51 16/53-2

Origins Aeronautical Research Laboratory, Wright Air Authorised 9/26/52

Development Center, LEAF Revised 9/30/52

Sponsor? " " Manager? P. Davis

Full task description appear*? in July-S ept 1952 issue.

Status? CONTINUED, Theoretical investigation of Part II (see

Jan-Mar 1953 issue, p. 38), continues.

WATER WAVES

Task 1 102-1 0-511 6/53-5^

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

Sponsor? Office of Naval Research

Manager; P. Davis

Objective; To study the partial differential equations governing

the motion of water waves on a shallow beach and to formulate numerical

methods for the solution of these equations on SEAC.

Background: The problem of th© breaking of water waves requires

the solution ©f the Laplace equation over a region bounded by a bottom

of variable depth and a free surface whose location is governed by

nonlinear boundary conditions. In their full generality, these equations

cannot b© solved in closed form, and it is therefore of great interest

to obtain numerical solutions.

Status? NEW
Status of Projects 39

2. Applied Research: Tables and Experimental Computations

TABLES OF E , ( z ) , (z = x + iy)

Task I162-IO-IIIOA3-3

(formerly 1102-21-1110/43-3)

Origin: Canadian National Research Council Authorized 7/1/47

Manager: I. A. Stegun

Full task description appears in Apr-Jun 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 1102-10-1110/46-1

(formerly 1102-21-1110/46-1)

Origin: NBS Authorized 7/1/47

Manager: J. Todd

Full task description appears in Apr-Jun 1949 issue, see task 11.2/2-46-1

Status: CONTINUED. This volume is now in press.

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 1102-10-1110/47-2

(formerly 1102-21-1110/47-2)

Origin: NBS Authorized 7/1/47

Managers: M. Abramowitz and P. Rabinowitz

Full task description appears in Apr-Jun 1949 issue, see task 11.2/2-47-2

Status: CONTINUED. The regular and irregular functions together

with their derivatives have been computed on SEAC f or /o =1 (1 )10,

=1(1)10, L=0,5. These data are being processed.

TABLE OF ANTILOGARITHMS

Task 1102-10-1110/47-3

(formerly 1102-21-1110/47-3)

Origin: NBS Authorized 7/1/47

Manager: J. Todd

Full task description appears in Apr-Jun 1949 issue, see 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.
Status of Projects

4o

TABLE OF LAGRANGIAN COEFFICIENTS

FOR SEXAGESIMAL INTERPOLATION

Task 1102-10-1110A8-2

(formerly 1102/21/1110/48-2)

Origin; NBS Authorised 5/25/48

Manager; J. Todd . _ Fall task description appears in Apr-Jan 1949 issue, see 1 1 „ 2/ 2-M-0-2

,

Status; CONTINUED. This volume is in press.

Publication; "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 1102-10-1110/50-4®

(formerly 1 102-21-1 110/50-4®)

Origin; NBS, Section 11.3 Authorized 12/31/43

Manager; J. Todd

Full task description appears in Oct-Dee 1949 issue, see 11.2/2-50-4.

Status; CONTINUED.

Publication; "Probability tables for analysis of extreme-value

data," NBS Applied Mathematics Series 22; in press, U. S. Government

Printing Office.

BIBLIOGRAPHY OF CODING PROCEDURES, MATHEMATICAL TABLES

AND NUMERICAL ANALYSIS

Task 1 102-10-1 1 10/50-5

Origins NBS Authorized 3/1/50

Managers; J. Todd, J. H. Wegstein, and Revised 1/9/53

P. Rsbinowitz

Full task description appears in the Oct-Dec 1952 issue.

Status: CONTINUED, Items for the coding library are being reeeiv<

from laboratories in this country and abroad. A comprehensive card cata.

is being maintained, and the more notable acquisitions are described in

Mathematical Tables and Other Aids to Computation. The file of reviews

of mathematical tables and papers in numerical analysis is being kept up

to date and is being subdivided according to topic.
Status of Projects 41 WAVE FUNCTION FOR LITHIUM

Task 1102-10-1110/50-7

(formerly 1 1 02-21 -1 1 04/50-7)

Origin: NBS Authorized 6/1/50

Sponsor: Bureau of Ordnance, USN

Managers: D. I. Rubin and W. Ho Durfee

Full task description appears in Apr-Jun 1950 issue.

Status: CONTINUED. Computation on SEAC continues as time is

available

.

COLLECTED SHORT MATHEMATICAL TABLES OF THE COMPUTATION LABORATORY

Task 1102-10-1110/51-4

(formerly 1102-21-1104/51-4)

Origin: NBS Authorized 9/28/50

Manager: J. Todd

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED, This volume 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 1102-10-1110/52-7

Origin: NBS Authorized 8/10/51

Managers: J. Todd, W. F. Cahill, and I. Stegun

Full task description appears in Jwly-Sept 1951 issue.

Status: CONTINUED. The following action is being taken in connection with a mathematical table the sales stock of which has been ex- hausted:

Tables of circular and hyperbolic sines and cosines

for radian arguments (a reissue of MT3) AMS36:

revision completed, in press.

The following volume is now out of print:

Table of natural logarithms, log x: x=5( .0001 )9.9999>

( 1 941 ) . MT12; reissue is planned.

TABLE OF ARCS IN FOR COMPLEX ARGUMENTS

Task 1102-10-1110/52-14

(formerly 1102-21-1110/52-14)

Origin: NBS Authorized 10/1/51

Manager: A. A. Goldstein

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The processing of punched cards continues
42 Status ©i Projects

EXTENSION OF THE TABLE OF HYPERBOLIC SINES AND COSINES

Task 1102-10-1110/52-18

(formerly 1 1 02-21 -1 1 1

0

/ 52-1 8)

Origins NBS Authorized 9/17/51

Manager s W. F» Cahill

Fall task description appears in Jnly-Sept 1951 issue.

Status; CONTINUED. Comp a tat ion on the SEAC is completed, and

th© results have been transferred t© IBM cards.

TABLE OF THE MODIFIED AIRY INTEGRAL

Task 1102-10-1110/52-23

(formerly 1102-21-1110/52-23)

Origins NBS Authorized 10/4/51

Managers P. Rabinowitz

Fm.11 task description appears in July-Sept 1951 issue.

Status s CONTINUED. The introduction to the table has been revised

and will include a derivation of the asymptotic formula for

-> xA

0 (x)dx„

VJ 0

TABLE OF ERROR FUNCTION FOR COMPLEX ARGUMENTS

Task 1102-10-1110/52-25

(formerly 1 1 02-21 -1 1 1 0/52-25)

Origins NBS Authorized 10/5/51

Managers s M. Abramowitz and F. J„ Stoekmal

Full task description appears in July-Sept 1951 issue.

Status s INACTIVE. For status to date see Jan-Mar 1953 issue.

EXTENSION OF TABLES OF THE EXPONENTIAL FUNCTION FOR

NEGATIVE ARGUMENTS

Task 1102-10-1110/52-31

(formerly 1102-21-1110/52-31 ) Origins NBS Authorized 10/9/51

Managers E„ Harden

Full task description appears in July-Sept 1951 issue.

Status s INACTIVE . For status to date see Jam-Mar 1953 issue.
Status of Projects 43

SPHEROIDAL WAVE FUNCTIONS

Task 1102-10-1110/52-37

(formerly 1 1 02-21 -1 1 1

0

/ 52-37 ')

Origin: NBS Authorized 11/28/51

Manager: T. Ledley

Full task description appears in Oct-Dee 1951 issue.

Status: CONTINUED,, Spheroidal wave functions have been computed

for the following cases: (1) Prolate : 1/c=t= .01 ( .01 ) 1 , c 2=1(.01)0, where

m=0 and A =0, 1 ,2,3 ,4, 5j m=1 and -^=0,1 ,2,3; m=2 and /=0, 1 , 2, 3 ; and m=3

and ^=0,1. 1/e=t=.005(. 005)1 , c -1(.01)0, where m=1 and -£=4,5; m=2 and

-^=4,5; m=3 and -£=2.3; m=4 and i! =0,1 ,2,3,4; and m=5 and>£=0. (2) Oblate : 1/e2=t=.005( .005)1 j c2=i(.01)0, where m=0 and /=0, 1 ,2,3. More cases will be computed whenever machine time is available.

VAN DER POL EQUATION

Task 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-Jun 1952 issue.

RADIAL MATHIE U FUNCTIONS

Task 1102-10-1110/52-49

Origin: NBS Authorized 2/1/52

Managers: J. Todd, I. Rhodes, and G. Blanch

Status: CONTINUED. New tables which are suitable for differencing

were computed, but some further subtabulation will be necessary.

SIEVERT'S INTEGRAL

Task 1102-10-1110/52-57

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 continues

as time is available. Values in the range 0( .01 )2( .02) 5 have been

computed. Results are being transferred to punched cards for future

printing. Seventy-five percent of the anticipated computation is finished .
44 States of Projects

SCATTERING FUNCTIONS

Task 1102-10-1110/52-63

Origins MBS Authorized 3/10/52

Managers A. GleyzaX

Fall task description appears in Jan-llar 1952 issue.

Status: CONTINUED, Coding for the computation of the' general

solution of the wav® equation is underway.

JACOBI ELLIPTIC FUNCTIONS

Task 1102-20-1110/52-7'+

Origins MBS

Managers T. Lediey

Authorized 7/1/52

Terminated 6/30/53

Objectives To tabulate the Jacobi elliptic

sm(u,k)=sin 0, en(u,k)=eos 0, dn(u,k)= n/1 -k 2

s ira 20

,

functions

whe re

n

*(i - k

2

S i„2 6)-i

n = pK,

K =

0 is defined by u =

and k 2=0(.01 )1 , p=0(.02)1

.

dO,

(1 - t 2 )"£(1 - k

2

t 2 )"^dt,

Background: A coarse tabulation of these functions (for p=0(.1)1)

was undertaken under project 43D2-4 (see July-Dee 1 9*+7 issue, p. 4). The

results have never been published. The purpose of this task is to produce

a table, to a moderate number of decimals, probably 6 or 7? which is

interpolable conveniently in the u direction. Every fifth value so obtained will be checked against the original hand-computations which were

to about 16 decimals. Computations are to be done on SEAC

.

Status: TCRMINATED. sn(pK,k), cn(pK,k), and dn(pK,k) have been

computed for k2=0(.01)1, p=0(.02;1. The results are stored on punched

paper tape from which Flexowriter copies can be made.

TABLE OF SECANTS AND COSECANTS

Task 1102-10-1 1 10/52-81

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. The editing of the introductory material was completed

.
45

Status of Projects

PAINLEVE EQUATION

Task 1102-10-1110/53-3

Origin; MBS Authorized 8/11/52

Managers; J. Todd and H. A. Antosiewiez

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 1102-10-1110/53-52

Origin: Oak Ridge National Laboratory Authorized 5/20/53

Manager: C. J. Swift

Objective: To compute on the SEAC tables of internal conversion

coefficients for the K and L shells of the atoms of atomic number 5(10)95.

Background; This is a continuation of task 1 1 02-40-51 26/51 -1 9

»

p.47. For background, refer to that task.

Status; NEW. The code is in the final stage of checking, and a few

preliminary answers have been obtained.

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-Jun 1951 issue.

Status: INACTIVE. For status to date see Jan-Mar 1953 issue.

3. Mathematical Services

LINEAR PROGRAMMING ON STANDARD PUNCHED CARD MACHINES

Task 1102-40-5126/49-3

(formerly 1102-53-1106/49-3)

Origin: Office of the Air Comptroller, USAF Authorized 3/21/49

Sponsor: " " Terminated 6/30/53

Manager: A. Hoffman

Full task description appears in Apr-Jun 1949 issue, see task 11.2/36-49-3.

Status: TERMINATED. The sponsoring agency, the Office of the Air

Comptroller, will perform all future computations required in their program

planning with their own computing facilities.
46 Status of Projects

SHOCK WAVE PARAMETERS , I Task 1102-40-5126/49-13

(formerly 1102-53-1106/49-13)

Origin; Bureau of Ordnance, Department Authorized 3/31/49

of the Navy Terminated 6/30/53

Sponsor; " 11 Manager; I, A. Stegun

Objective; To provide graphs and tables for the rapid determination

of the parameters of spherical shock waves emitted by explosions. In

particular, (a) to prepare basic tables of certain functions I

n (x,c

y

) [n=0,1 ,2, . . .

,

10 , c y=specific heat at constant volume] needed in the

following parts ©I the problem, (b) to determine (the initial pressure

at the shock front) in its dependence on p (the static pressure of the

explosion gases) and a parameter K which measures the effect of density,

temperature, etc.; (c) to compute tables of certain functions which facilitate the determination of K for a given explosive.

Background; OSRD Reports 1030, 2022, 3550, and 5649 contain tables

describing the entire course of shock waves in air, fresh water, and salt

water as a function of the initial conditions at th© instant of explosion.

The present task aims at facilitating the determination of these initial

conditions from a knowledge of the chemical composition of the explosive,

and thus t© aid in studies of the effectiveness of explosives. The initial

conditions, particularly pressure and velocity, are determined by the

fact that they satisfy the equation of state of the explosion gas and

th© Hugoniot condition for th© surrounding medium (water). The latter is

tabulated in OSRD Reports 670 and 8l 3 5 for the former the Wilson-Kis tiakowsky

equation may be used. Kirkwood and Montroll (OSRD Report 67O) developed

the theory which forms the basis of th® present computations. They also

introduced and tabulated the auxiliary functions I (x,e v

). The range of

their tables is, however, not quite sufficient, and subtabulation is

needed for convenient use. Comments; The problem was propsed by Dr. T. L. Brownyard of the

Bureau of Ordnance and Dr. H. G. Snay of the Naval Research Laboratory.

Th® latter developed th© method used in this task.

Status; TERMINATED.

MOLECULAR STRUCTURE CALCULATIONS, II

Task 1102-40-5126/50-16

(formerly 1

1

02-53-1 1 06/50—1 6 ) Origin: Naval Research Laboratory, U5N Authorized 3/31/50

Sponsor: " " Manager: P. J. O'Hara

Full task description appears in Jan-Mar 1950 issue, see task 1 1 „ 2/33-^50—1 6 . Status: CONTINUED. Computations were performed as requested.
Status of Projects 47

PROGRAM COMPUTATION ON THE SEAC

Task 1 102-40-51 26/51-7

(formerly 1 1 02-53-1 1 06/51 -7)

Origin: Office of the Air Comptroller, U5AF Authorized 9/1/5C

Sponsor: " " Manager: A. Hoffman

Full task description appears in July-Sept 1950 issue.

Status: INACTIVE. For status to date see Oct-Dec 1952 issue.

FLOW IN SUPERSONIC NOZZLES

Task 1 102-40-51 26/51-13

(formerly 1 102-53-1 106/51 -1 3)

Origin: Naval Ordnance Laboratory * Authorized 1 2/1 5/50

Sponsor: " Manager: N. Levine

Full task description appears in Oct-Dec 1950 issue.

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

INTERNAL CONVERSION COEFFICIENTS FOR L-SHELL

Task 1 102-40-51 10/51-19

(formerly 1 1 02-50-51 26/51 -1 9 and 1 102-53-1 106/51 -1 9)

Origin: Atomic Energy Commission, Oak Ridge Authorized 3/30/51

National Laboratories Terminated 6/30/53

Sponsor: " " 11 Manager: C. J. Swift

Objective: To compute on the SEAC tables of internal conversion

coefficients for the K and L shells of atoms of atomic number 5(10)95.

Background: A radioactive nucleus may make a transition in which

the energy is carried off either as a photon (gamma ray) or by one of

the extra-nuclear electrons. The ratio of the number of electrons to

photons observed is known as the internal conversion coefficient. A

comparison of calculated and observed coefficients is a tool used in

classifying the radioactive transition and in understanding the decay

scheme

.

Since extra-nuclear (atomic) electrons are present in K,L,M...

shells one may consider conversion coefficients for these various shells.

Due to the difference in energy of converted K,L,M electrons, they are experimentally distinguishable. In fact, the ratio of K to L conversion

is easily measurable, and corresponding calculations are desirable.

At present, some accurate K-shell calculations exist. However only a few scattered calculations on the L-shell exist in addition

to some rough approximate formulae. The present program is intended

to obtain a wide range of L-shell internal conversion values accurately

calculated with due attention to all important physical effects. In

addition, the gaps in the present K-shell tables should be filled in.

Thus, at the end of the present calculations there will exist

tables of K-shell and L-shell internal conversion coefficients in sufficient completeness that the experimenter will have a powerful tool at
48 Status of Projects

his disposal to aid in classification of nuclear states and decay schemes.

Status s TERMINATED . See task 1102-10-1110/53-52. p. 45 .

LIQUID-VAPOR TRANSITION

Task 1102-40-5126/51-22

(formerly 1 102—53—1 106/51 -22)

Origin? Naval Medical Research Institute Authorised 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 1 102-53-1 106/51-37)

Origin: Naval Research Laboratory, ISN 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 1102-40-5126/51-38

(formerly 1 102-53-1 106/51 -38)

Origin: Bureau of Ordnance, U5N Authorized 6/28/51

Sponsor: ” " Manager: I. A. Stegun

Full task description appears in Apr-Jun 1951 issue.

Status: CONTINUED. Exploratory and test cases have been completed for several explosives.

PRESSURE DISTRIBUTION ON BODIES OF REVOLUTION

Task 1 102-40-51 26/52-3

(formerly 1 1 02-53-1 1 06/52—3

)

Origin: David Taylor Model Basin, USN Authorized 8/10/51

Managers: A. Gieyzal and 0. Steiner

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see Jars-Mar 1953 issue
Status of Projects 5-9

POWDER DIFFRACTION

Task 1102-40-5'l 26/52-6

(formerly 1 1 02-53-1 1 06/52-6)

Origin: NBS , Section 9.7 Authorized 9/1 7/51

Manager: E. Warden

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see July-Sept 1951 issue.

NEUTRON DIFFUSION

Task 1102-5-0-5111/52-15

(formerly 1 1 02-50-51 26/52-1 5 and 1 102-53-1 106/52-1 5)

Origin: Rand Corporation Authorized 9/1 7/51

Sponsor: Air Materiel Command, U5AF Terminated 6/30/53

Manager: C. J. Swift

Objective: To determine depth of penetration and other theoretical

data for gamma radiation entering a semi-infinite medium, using the Monte

Carlo technique.

Background: Some information has been obtained from ENXAC , but

SEAC is a better machine to give the extensive information wanted.

Status: TERMINATED,

PRECISE DETERMINATION OF THE PARAMETER OF DISPERSION

EQUATION FOR SEVERAL TYPES OF OPTICAL GLASS

Task 1102-5-0-5126/52-17

(formerly 1 102-53-1 1 06/52-1 7)

Origin: NBS, Division 2 Authorized 9/29/51

Sponsor: ” Manager: I. A. Stegun

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. Computations were performed as requested.

SPHERICAL BLAST

Task M 02-5-0-51 26/52-20

(formerly 1102-53-1106/52-20)

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
50 Status of Projects

MAGNETIC FIELD EXTRAPOLATION

Task 1102-40-5126/52-22

(forme rly 1 1 02-53-1 1 06/52-22)

Origins Naval Ordnance Laboratory, UBN Authorized 10/1/51

Sponsors ” Managers N. Levine

Full task description appears in July-Sept 1951 issue,.

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

CALCULATIONS FOR d S PACINGS

Task 1102-40-5126/52-44

(formerly 1 102-53-1106/52-44)

Origins MBS, Div. 9 Authorized 1 2/7/51

Sponsors

Manager; I* Stegun

Full task description appears in Oet-Dec 1951 issue*

Status; CONTINUED. Computations were performed as requested.

INTENSITIES OF SPECTRAL LINES II

Task 1102-40-5126/52-46

(formerly 1102-53-1106/52-46)

Origin: NBS , Div. 14 Authorized 12/10/51

Sponsor: " Terminated 6/30/53

Manager: R, Zueker

Objective: To perform additional calculations on the theoretical

intensities of the rotational spectral lines of water vapor.

Background: This task is an outgrowth of task 1 409-31 -1 457/51 -5?

Jan-Mar 1951 issue, p. 48.

Status: TERMINATED.

MULTIPLE COMPTON SCATTERING OF LOW ENERGY GAMMA RADIATION

Task 1 102-40-5126/52-65

(formerly 1 1 02-53-1 106/52-65)

Origin: Naval Research Laboratory Authorized 4/1/52

Sponsor: " 11 Terminated 6/30/53

Manager: I. Stegun

Objective: To evaluate Fourier integrals for a selected set of

pa rameters

.

Background: The integrals arise in the theoretical studies being

carried out at the Naval Research Laboratory. These computations were

specifically requested by Mr. O'Rourke.

Status: TERMINATED
Status of Projects

LONG PATH USABLE FREQUENCY PREDICTIONS

Task 1 1 02-40-51 26/52-66

(formerly 1102-53-5126/52-66)

Origin: NBS , Section 1 . M- Sponsor:

Manager: M. Stein

Objective: To analyze available radio traffic and field strength

data to determine observed maximum usable frequencies over long paths.

To develop empirical methods of calculating long path maximum usable

frequencies to bring them into agreement with observations.

Background: Experience has shown that the maximum usable frequencies predicated by methods currently in use at the Central Radio

Propagation Laboratory are discrepant with and in general lower than

those actually observed. It is believed that empirical methods can be

devised to improve this situation.

Status: COMPLETED. The results have been transmitted to the

spons or.

GAS ADSORPTION BY HIGH POLYMERS

Task 1102-40-5126/52-70

(formerly 1 102-53-1 106/52-70)

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.

Authorized 4/1/52

Completed 6/30/53

STANDARD LORAN TABLES

Task 1 102-40-5126/52-77

(formerly 1 1 02-53-1 1 06/52-77)

Origin: U. S. Navy Hydrographic Office Authorized 5/15/52

Sponsor: " " Completed 6/30/53

Manager: M. Abramowitz

Objective: Preparation of tables giving coordinates of hyperbolic

lines of positions.

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: COMPLETED. The results were transmitted to the sponsor.
52 Status of Projects

CHEMICAL TRANSITION PROBABILITIES

Task 1 102-40-5126/52-82

(formerly 1 1 02-53-1 1 06/52-82)

Origin: NBS , Section 3.2 and Hydrocarbon Research Authorised 6/1/52

Corp

.

Sponsor: Office of Naval Research

Managers: A. Gleyzal and A. A. Goldstein

Full task description appears in Apr-Jun 1952 issue.

Status: CONTINUED. Code checking of the first portion of the

problem has been completed. Code cheeking of the second portion, using

dummy routines for certain spherical Bessel functions, has also been

completed

,

NEUTRON DIFFUSION, II

Task 1 102-40-51 26/53-4

Origin: Atomic Energy Commission, New York Authorized 9/30/52

Office, (NDA)

Sponsor: " 11 " Managers: 0. Steiner and N, Levine

Full task description appears in July-Sept 1952 issue.

Status: CONTINUED. Ninety percent of the original program has

been completed, and the results have been transmitted to the sponsor.

STRENGTH OF WING COMPONENTS

Task 1102-40-5126/53-11

Origin: National Advisory Committee for Aeronautics Authorized 12/8/52

Langley Field, Va

.

Sponsor: " 11 " Manager: W. H. Durfee

Full task description appears in Oet-Dec 1952 issue.

Status: CONTINUED.

RADIANT HEATING OF SOLIDS

Task 1 102-40-51 26/53-20

Origin: NBS, Section 10.2 Authorized 12/15/52

Sponsor: " Manager: W. F. Cahill

Full task description appears in Oet-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 53

STANDARD LQRAN TABLES - EXTENSION OF RATES 1L4, 1L5, 1 L6

Task 1102-40-5126/53-26

Origin: Hydrographic Office, U. S. Navy Authorized 1/23/53

Sponsor: " *' Terminated 6/30/53

Manager: W. H. Durfee

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.

COMPUTATION OF THERMODYNAMIC FUNCTIONS

Task 1102-40-5126/53-27

Origin: NBS , Division 5 Authorized 3/18/53

Sponsor: " Manager: E. Marden

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. Tables of thermodynamic functions and their

differences were computed for additional molecules. Modification of the

code to take into account additional equations is in process.

STUDY OF TRICALCIUM ACUMINATE

Task 1102-40-5126/53-28

Origin: NBS, Division 9 Authorized 3/30/53

Sponsor: " 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 were performed as requested and

transmitted to the sponsor.
54 Status of Projects

SKYWAVE TRAINER DELAY CURVES

Task 1 102-40-51 26/53-32

Origin: U. S. Navy Hydrographic Office Authorized 3/30/53

Sponsor: '• " Terminated 6/30/53

Manager: W. H. Durfee

Objective: To prepare tables of skywave corrections for certain

existing standard Loran tables giving coordinates of hyperbolic lines

of positions.

Background: These tables will be used in the training of military

personnel in the use of Loran equipment.

Status: TERMINATED. The results were transmitted to the sponsor.

REDUCTION OF ECLIPSE DATA

Task 1102-40-5126/53-34

Origin: Air Photographing and Charting Services, LEAF Authorized 3/30/53

Sponsor: " 11 " Manager: A. Goldstein

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. To start the program it was necessary to solve

an integral equation of the type of Volterra's integral equation of the

first kind. A method was developed and, using the SEAC, results were

obtained. In addition, investigations were carried out leading to re- commendations for the observations of the next eclipse. An interim re- port has been issued giving details of the accomplishments to date.

SPECTRAL ANALYSIS OF STATIONARY TIME SERIES

Task 1 102-40-51 26/53-37

Origin: Statistical Research Center, University Authorized 4/20/53

of Chicago

Sponsor: Office of Naval Research, IBN

Manager: I. Stegun

Objective: To compute the serial products

N—| vj N- | s %= Hvw

n-1

and = n

n=1

which are used in estimating the spectral distribution function and its

kurtosis, respectively.

Background: The present statistical theory of analysis of stationary time series has assumed complete knowledge of the covariance

sequence or, equivalently, of the spectrum of the process. It is, therefore, important to be able to estimate one of these. Knowledge of the

spectrum seems to yield greater immediate insight into the structure of

the process.

2 2

x x , n n+
Status of Projects 55

Comments: This problem was proposed by M. Rosenblatt and U. Grenander, SRC, University of Chicago.

Status: NEW. Computations have been completed, and checking is

in progress.

ACOUSTICAL IMPEDANCES

Task 1 102-40-51 26/53-39

Origin: NBS , Section 6.1 Authorized 6/30/53

Sponsor: " Managers: S. Prusch and K. Nelson

Objective: To solve the following simultaneous equations for Z

§ (internal impedance of the source) and ZY (the internal impedance of the

ea r) , _

E e Z

1 E

B Z

2 f _

E

B ZX

1 " Z

i +Z

s

’ 2 “ Z

2

+Z

s

’ X “ VZ

s

’

where and E

2 aw the probe output voltages observed for lengths J.. and £r^ a tube, respectively, E^ is the complex voltage put out by the

microphone, and and 2^ are the corresponding impedances calculated for

the tube. Tne impedance of the tube for a length ^ is given by

z=-i S /> c cot k(£ d)» where /> is the density of air, S is the cross- section area of the tube, c is the velocity of sound and k = (27r/wavelength)

.

Background: The problem arises in the present work of the Sound

Laboratory which is directed at obtaining an artificial ear that is more nearly representative of the human ear, at devising measuring equipment

that will minimize the effects of human variations, and, ultimately, to

set up instrumentation for a more precise determination of the physical

aspects of normal hearing.

Status: NEW. Solutions of equations for various parameters have

been sent to the Sound Laboratory for further study and investigation.

STANDARD LORAN TABLES - Rates 2H2,2H3,2H4

Task 1 102-40-5126/53-40

Origin: Hydrographic Office, U. S. Navy Authorized 4/20/53

Sponsor: " " 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: NEW. The computations were performed as requested, and

charting coordinates were given to the sponsor. The tables are now being

edited

.
56 S tat us of Projects

LORAN UNXVAC CODE

Task 1 102-40-51 26/53-41

Origin: Hydrographic Office, U. S. Navy Authorized 4/20/53

Sponsors " " Managers: W. H. Durfee, I. Rhodes, and D. Rubin

Objective: To prepare a code for computing on the UNXVAC standard

Loran tables with sky wave corrections.

Background; A code for performing these computations on the SEAC

already exists. In view of the importance of this work it was deemed

advisable to be able to prepare Loran tables, if necessary, on another

computer.

Status; NEW. Th© code is being prepared, and parts of it have

been checked in.

NOMOGRAMS FOR RF PERMEAMETER

Task 1 102-40-51 26/53-42

Origin: MBS , Division 1

4

. 8 Authorized 6/29/53

Sponsor: " Manager: I. Stegun

Objective: To perform analysis as necessary to prepare nomograms.

Background: The problem arises in connection with the use of

the RF permeameter.

Status: NEW. Several nomograms have been set up, and analysis is

being performed for other combinations of nomograms.

OPTIMUM SECTIONS FOR DELTA WINGS

Task 1102-40-5126/53-49

Origin: National Advisory Committee for Authorized 4/29/53

Aeronautics, Langley Field, Va

.

Sponsor: " " Manager: 0. Steiner

Objective: To perform computations to determine optimum sections

for delta wings at supersonic speeds.

Status: NEW. All the codes necessary to run this problem were

completed and checked. The pilot computations were performed, and the

results were turned over to the sponsor, who will determine what further

computations are needed.
Status of Projects 5?

WEATHER INFORMATION

Task 1102-40-5126/53-50

Origin: Weather Bureau Authorized 5/20/53

Sponsor; Office of Naval Research Completed 6/30/53

Manager: P. J. O'Hara

Objective: To compute a number of sea level pressure indices for

each day of the winter seasons from 1920 - 195-0. These indices are based

on weather data collected from the entire northern hemisphere.

Background: Pressure data obtained from the northern hemisphere

daily weather map can be summarized in a way to provide various indices

of the general circulation of the atmosphere. These indices can then be

used to study the reason for weather changes and should lead to inf>roved

forecasts. Original data obtained from the Weather Bureau list daily

pressure observations from approximately one thousand stations.

Status: COMPLETED. (NEW). The results obtained have been submitted

to the sponsor.

NEUTRON DIFFUSION III

Task 1 102-40-51 26/53-51

Origin: NBS , Section 4.8 Authorized 6/9/53

Sponsor: Armed Forces Special Weapons Project

Manager: F. Stockmal

Objective: To solve boundary-value problems in gamma-ray diffusion

by means of the Monte Carlo method.

Background: This problem arises in connection with theoretical work

on gamma-ray propagation performed by the Nuclear Physics Section of the

Radiation Laboratory. The nature of this problem is such that while a purely analytical treatment is extremely difficult, accurate results can be obtained with relative ease by means of random sampling techniques.

Status: NEW. A code has been completed for the first phase of this

problem, the computation of energy and angular histories. Results have

been verified by hand computation. Phase two will be concerned with the

spatial histories corresponding to the results of phase one, with imposition of various boundary conditions.

TRANSPORTATION PROBLEM II

Task 1 102-40-51 26/53-55

Origin: Logistics Research Project, Authorized 6/9/53

George Washington University

Sponsor: Office of Naval Research

Managers: A. J. Hoffman and L. Gainen

Objective: To solve systems of equations arising in the problem

of allocating shipments from bidders to depots minimizing the total cost

of the operation. To investigate problems arising from conditions imposed by bidders by time phasing of requirements, etc., and to formulate

these conditions for computation.
58 Status of Projects

Background; The technique of G. B . Dantzig in applying the simplex

method in finding solutions of the transportation problem of HitchcockKoopmans is used in solving standard allocation problems.

Status; NEW.

TABLES OF THERMODYNAMIC PROPERTIES OF GASES

Task 0302-5-0-2606/5-9-5

(formerly 11.2/33-5-9-5)

Origin; NBS , Section 3.2 Authorized 11/29/5-8

Sponsor: National Advisory Committee for Aeronautics

Manager: F. L. Alt

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

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

BASIC IONOSPHERIC DATA

Task 1 5-01 -35—1 5-73/5-9-1 5-

( formerly 11.2/33-5-9-14)

Origin: NBS, Section 1 5- . 3 Authorized 3/31/5-9

Sponsor: " Manager: M. Stein

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

Status: CONTINUED. The results are being transmitted to the

sponsor as completed.

RAY TRACING

Task 0202-10-2308/50-13

(formerly 11.2/33-50-13)

Origin: NBS, Section 2.2 Authorized 3/1/50

Sponsor: " Managers: R, K. Anderson and D. Rubin

Full task description appears in Jan-Mar 1950 issue.

Status: CONTINUED, A code for SEAC has been completed and several

runs have been made. Additional runs will be made when new data are obtained

.
III. Statistical Engineering Laboratory

(Section li.3)

1. Fundamental Research in. Mathematical Statistics

BIBLIOGRAPHY AND GUIDE TO STATISTICAL LITERATURE

Task 1 103-10-1 107A9-1 a (formerly 11.3/2-49-1)

Origin; MBS Authorized 1/9/49

Manager; L. S. Doming

Full task description appears in Jan-Mar 1949 issue.

Status: CONTINUED. The card file consisting of abstracts of

statistical literature is steadily growing as each issue of Mathematical

Reviews appears and as additional pertinent selections from back issues

of Zentralblatt are being photostated and prepared for inclusion in it.

MANUAL ON FITTING STRAIGHT LINES

Task 1103-10-1107/50-2

(formerly 11.3/2-50-2)

Origin: NBS Authorized 3/1/50

Managers F. S. Acton

Full task description appears in Jan-Mar 1950 issue.

Status; INACTIVE. For status to date see Oct-Dee 1952 issue.

Publication: "Analyzing straight line date," by F. S. Acton;

J. Chens. Ed. £0, 128-133 (Mar. 1953).

TABLE TO FACILITATE DRAWING RANDOM SAMPLES

Task 1103-10-1107/51-1

Origin: NBS

Managers: C. Eisenhart and L. Full task description appears

Authorized 7/1/50

S. Deming

in July-Sept 1950 issue.

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

59
Status of Projects

MISCELLANEOUS STUDIES IN PROBABILITY AND STATISTICS

Task 1 103-10-1107/51-2

Origin: MBS Authorized 7/1/50

Manager; E. Lukacs

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. (1) E. Lukacs and E, P. King continued their

studies of the independence of linear statistics. They proved the following theorem;

Let X, 5 X2

5

» . * ,X

n be n independently but not necessarily identically distributed variables and assume that the nth moment of each

X

s (s=1,2,., e ,!s) exists. The necessary end sufficient conditions

for the existence of two statistically independent linear forms

Y

i = y w y2 = y b

sx

s

are: (A) Each random variable which has a nonzero coefficient in both

forms is normally distributed 5 arad

(B) y ys < = °-

S = 1 2

Her© er" s is the variance of X

g (s=1 For n=2 this reduces to a theorem of Serge Bernstein [see "Generalisations de la loi de probability

do Laplace," by M, Frsehet; Annales de I'Institut Henri Poincare, XII

(1951)]. (2) E. P. King continued his study of grouping control chart

data. Mont© Carlo experiments were initiated to check some of the re- sults obtained by an approximate analytic method, (3) M. Zelen considered

various extensions of the Tehebycheff inequalities which ere based on , second, order moments. Bounds have been obtained for a univariate distribution function when a point of the distribution function is known,

and for a bivariate distribution.

Publications: (1) "Contribution to the theory of Markov chains,"

by Kai Lai Chung; J. Has, MBS 50, 203-208 (Apr. 1953). (2) "On the

exact evaluation of the variances and covariances of order statistics

in samples from the extreme-value distribution," by J. Lieblein; Ann.

Math. Slat. 24, 282-287 (June 1953). (3) "On some procedures for *he

rejection of suspected data," by E . P. King; accepted by the Journal of

the American Statistical Association. (

!+) "A property of the normal

distribution related to a theorem of S. Bernstein," by E. P. King and

E. Lukacs; submitted to a technical journal. (5) "Certain Fourier transforms of distribution, II," by E. Lukacs and 0, Szasz; submitted to a technical journal, ( 6 ) "Monnegative trigonometric polynomials and

certain rational characteristic functions," by E. Lukacs end O. bzasz;

accepted by the Journal of Research of the NBS. (7) "On absolute measurement,'

by N. E. Dorsey and C, Eisenhart; accepted by the Scientific Monthly.

( 8 ) "A historical note on the application of the 'weakest link' idea to

tensile strengths," by J, Lieblein; submitted to a technical journal.
Status of Projects 61 LAW OF PROPAGATION OF ERROR

Task 1103-10-1107/52-1

Origin: NBS Authorized 6/23/51

Managers: C. Eisenha rt and I. R. Savage

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE. For status to date see July-Sept 1951 issue.

PROCEDURES OF NON-PARAMETRIC STATISTICS

Task 1 103-10-1 107/52-2

Origin: NBS Authorized 9/17/51

Manager: I. R. Savage

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED.

Publication: "Bibliography of non-pa ramet rie statistics and related

topics," by I. R . Savage; IN MANUSCRIPT.

STUDIES IN THE MATHEMATICS OF EXPERIMENT DESIGN

Task 1103-10-1107/53-1

Origin: NBS Authorized 10/15/52

Manager: W. S. Connor

Full task description appears in the Oct-Dec 1952 issue.

Status: CONTINUED. (1) W. S. Connor and W. M. Clatworthy continued the development of necessary conditions for the existence of

partially balanced incomplete block (P.B. I. B.) designs, a study first

referred to in the Oet-Dec 1952 issue, p. 60. The conditions arise from

consideration of | NN'| , where N is the incidence matrix of a P.B.I.B.

design. This determinant has been evaluated for two, three, and

four associate classes, and detailed necessary conditions have been

obtained for two associate classes. Bounds on b, the number of blocks

in the design, have been obtained for all P.B.I.B. designs. The manuscript, "Necessary conditions for the existence of partially balanced

incomplete block designs with two associate classes," by W. S. Connor

and W. H. Clatworthy, which was submitted to a technical journal (see

Jan-Mar 1953 issue, p. 76 ), has been withdrawn in* order to incorporate

the new results. (2) M. Zelen has prepared a manuscript (see publication

(3) below) that describes the analysis which is appropriate when a block

is lost from a design or from certain P.B.I.B. designs.

Publications: (1) "An embedding theorem for balanced incomplete

block designs," by W. S. Connor and M. Hall, Jr.; accepted by the Canadian Journal of Mathematics. (2) "The chain block design," by W. J. Youden and W. S. Connor; accepted by Biometrics. (3) "Analysis for some incomplete block designs having a missing block," by M. Zelen; IN MANUSCRIPT.
62 Status of Projects

2. Applied Research in Mathematical Statistics

COLLABORATION ON STATISTICAL ASPECTS OF NB5

RESEARCH AND 'TESTING

Task 3737-60-0002/51-1

(formerly 3011-60-0002/51-1)

Origin: MBS 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: For example a 1/9 fractional replication design was constructed using the general theory of a 2-way elimination for a 3x3x4x27 factorial design to be used in tests of metal

insulator laminates.

B. Development or Selection of the Appropriate Method for

Analysis and Interpretation of Data: For example: (a) A method of

sampling of cement was developed in which the number of samples tested

for each property measured depends on the closeness of the values to the

specification limit. (b) Welch-Pitman randomization procedures were

used for the statistical evaluation of results from studies on battery

additives

.

Publications: (1) "Control charts may be all right - but...,"

by F. Proschan; Industrial Quality Control DC, 56-58 (May 1953).

(2) "Acceptance sampling of electroplated articles," by J. M. Cameron

and F. Ogburn; submitted to a technical journal. (3) "A test for statistical control applicable to a short series of observations," by

C. Eisenhart and E. P. King; submitted to a technical journal. (4) "Est

mating the standard deviation of a normal distribution," by F..- P. King;

accepted by Industrial Quality Control. (5) "Confidence and tolerance

intervals for the normal distribution," by F. Proschan; submitted to a technical journal. (6) "Rejection of outlying observations," by F. Proschan; accepted by the American Journal of Physics. (7) "The principles of experimental design," by 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.

(8) "Statistical units of measurement," by W. J. Youden; accepted by

Metals Progress. (9) "Making one measurement do the work of two," by

W. J. Youden and W. S. Connor; accepted by Chemical Engineering Progress

(10) "Performance of inspectors and gasoline pumps," by W. J. Youden

and M. Jensen; submitted to a technical journal.
Status of Projects 63

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-Oee 1951 issue.

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

APPLICATION OF THE THEORY OF STOCHASTIC PROCESSES TO

THE STUDY OF TRAJECTORIES

Task 1 103-20-51 19/52-1

Origin: U. S. Naval Ordnance Test Station, Inyokern Authorized 1/1/52

Sponsor: " " " Manager: E. Lukacs

Full task description appears in Jan-Mar 1952 issue.

Status: CONTINUED. It was decided to construct an "artificial

trajectory" to be used to demonstrate by means of an example how the

parameters of a Wiener process could be estimated from a single sample

curve. The polynomial of degree four

f(t)=31+00 + 3l0t - 2.7t 2 + ^-3 ( 1

0

-3

)

t

3 - 2.6(10

-4

',t 4

was chosen to represent the mean value curve of the fictitious process.

The values of f(n) were computed for n=0(1)100. These values would

correspond to the "true positions" at the 101 equidistant time points at

which observations are made. In order to obtain the simulated observations one has to add random numbers as "errors" to the values f(n). These

random numbers were obtained from H. Wold's tables entitled Random Normal

Deviates (Cambridge University Press, 195-8). These tables contain random

numbers representing a normal population of zero mean and unit variance.

They are arranged in columns of 50 and the sums £I(x) of the columns are also given. In view of the arrangement of Wold's tables it was decided

to choose c=50/3 as the "true value" of the variance constant. Random

numbers from a normal population with zero mean and variance c= 50/3 were

obtained by dividing the values H(x) in Wold's t8ble by «/ 3 • A set of

100 random numbers w. (i=1 , . . . ,100) was derived in this manner. The

"simulated observations" were finally obtained as

n g(n) = f (n) + XU w. . i=0 1

The parameters of the Wiener process with mean value curves were extended

on the basis of the observations. Comparisons of estimated mean value curves of various degrees were made as well as comparisons with least

square estimates. This work is reported in publication (5-) below.

Publications: (1) "On strongly continuous stochastic processes,"

by E. Lukacs; submitted to a technical journal. (2) "Tables useful in

estimating the mean value function of a fundamental random process," by

E. Lukacs and I. R. Savage; IN MANUSCRIPT. (3) "Tables of inverses of
64 Status of Projects

finite segments of the Hilbert Matrix," by I. R. Savage and E. Lukaes; to

be included in Contributions to the Solution of Systems of Linear Equations and the Determination of Eigenvalues, NBS Applied Mathematics Series

»

STATISTICAL SERVICES FOR COMMITTEE ON SHIP STEEL, NRC

Task 1 103-50-5105/52-1

Origin: Ship Structure Committee, NRC Authorized 12/1/51

Sponsor: " " Manager: W. J. Youden

Full task description appears in Oet-Dec 1951 issue.

Status: CONTINUED. The possibility of reducing failures of ship

steels by a reduction in the specified transition temperature was investigated. It was necessary to construct the distribution of transition

temperatures from the available plates which all came from ships with

f rac tures

.

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 Array

Sponsor: " 11 " Manager: C. Eisenhart

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. Final reports on two special problems were

being prepared for the sponsor.
IV. Machine Development. Laboratory

(Section 11.4)

in cooperation with Electronic Computer Section

(Section 12.3)

1 . Development; Design and Construction of Automatic Digital

Computing Machines

THE BUREAU OF THE CENSUS COMPUTING MACHINE

Task 1 104-3^-5107/47-1

(formerly 11,4/21-47-1)

Origin; The Bureau of the Census

Sponsor; " ” Full task description appears in Apr-Jun 1949 issue-.

Authorized 7/1/47

Status; CONTINUED, The installation of the No, 1 UNXVAC System at

its pes'manent site in Federal Office Building No, 3? Suitland, Maryland,

was completed during this period, and the machine is again in useful service. Maintenance is being provided primarily by engineers from the ERA

Division of Remington Rand provided under extension of the installation

contract between Remington Rand, Inc., and the Bureau of the Census. NBS

personnel assisted in debugging the system following the move and were

also made available to assist with two special problems that arose

and have been solved during this period. One problem involved air flow

through the cooling system, and the other was a problem of vibration of

a heavy blower.

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, ISAF Authorized 7/1/47

Sponsor: " " Full task description appears in Apr-Jun 1949 issue.

Status: CONTINUED. The second UNIVAC System has continued to be

operated and maintained by personnel of the Office of the Air Comptroller

on around-the-clock seven-days-a-week schedule. The NBS has as well as the individual owner-agencies concerned continued its cooperative endeavor

to procure parts for the first three UNIVAC Systems. The stock piling of

replacement parts continues to be a slow process due to other production

priorities

.

65
66

Status of Projects

WRIGHT DEVELOPMENT CENTER COMPUTING MACHINE

(formerly AIR MATERIEL COMMAND COMPUTING MACHINE)

Task 1 10^-3^-5107/^9-10

(formerly 1

1

.4/23-49-1

)

Origin; Aeronautical Research Laboratory, Authorized 3/21/49

Wright Air Development Center, USAF

Sponsor; " 11 Full task description appears in Apr-Jan 1949 issue.

Status; CONTINUED, The GARAC (Office of Air Research Automatic

Computer) was moved to its permanent site at Wright Air Development Center,

Wright-Patterson Air Force Base and is now in operation by their Aeronautical Research Laboratory. Only a few small items such as manuals and

diagrams remain to be completed prior to final termination of th§ contract

between the National Bureau of Standards and the General Electric Company,

NATIONAL BUREAU OF STANDARDS EASTERN AUTOMATIC COMPUTER (SEAC)

Task 1 104-34-5107/49-1

(formerly 11 .4/24-49-1

)

Origin; MBS Authorized 12/15/48

Sponsor; Office of the Air Comptroller, ISAF

Full task description appears in Apr-Jun 1949 issue.

Status: CONTINUED. During the quarter from March through June 1953

approximately 90 hours a week have been used for computation. Operating

efficiency for the period, i.e., total productive time versus total time

scheduled, was 8l percent. Installation of the regulated d-c power supplies

and a-c voltage stabilizers has now been completed. Engineering work on and modification of the experimental Williams type memory are nearing completion, and an extended problem which utilizes both the acoustic and

electrostatic memories of SEAC has been prepared and scheduled. Mechanical

tests of the cavity-type wire drive unit are in progress before its installation in SEAC for evaluation as part of the input-output system, and

circuitry for adapting it to the SEAC has been assembled.

Additional technical memoranda issued are as follows:

Technical

Memorandum

No. 42: Four subroutines S30, S31 , S32, and S33 for the step-by-step

integration of the first order differential equation y'=f(x,y).

No. 43: Subroutine for 1 6-point Gaussian quadrature.

No. 4/: Subroutine for conversion of decimal degrees to binary radians

and vice-versa.

No. 50: Automonitor processing routine (designated as BOXE 0 80IE).

No. 51 : Matrix inversion routine; von Neumann-Goldstine method (modified).

No. 52: Subroutine for Log N; single precision, fixed binary point.

No. 53s Matrix inversion routine: Laoczos-Hes tenes method (for symmetric

matrices )

.

No. 54: Matrix multiplication routine.

No. 55: Subroutine for aresin x and arceos x, single precision; fixed

binary point.

These memoranda are collected and issued as NBS reports under the

title "SEAC operating and programming notes." Two additional reports in

this series have been Issued which contain "Technical Memoranda Nos. 32-36

and 37-41 .
States of Projects 67

ARMY MAP SERVICE COMPUTING MACHINE

Task 1

1

04-34-51 07A9-1 b

(formerly 11.4/25-49-1)

Origin: Army Map Service,, IB A Authorized 12/15/48

Sponsor: " " Full task description appears in Apr-Jun 1949 issue.

Status: CONTINUED. This period saw the completion of the cooperative

effort on the part of MBS to procure replacement parts for the third

UNXVAC System which has been in regular operation at its Army Map Service

site. Maintenance has been provided under contractual agreement with the

ERA Division of Remington Rand, Inc. Because operating efficiency for the

quarter has been very low, an experimental program of large-scale tub®

replacement has been initiated by the Army Map Service technician.

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 S ecurity Agency Authorized 12/31/50

Sponsor: " Full task description appears in Oct-Dee 1950 issue.

Status: INACTIVE. For status to date see July-Sept 1951 issue
Lectures and Symposia

Not©; 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.

Numerical Analysis Colloquium Series

”TLos Angeles, California)

MINORSKY, N. (Stanford University) On the stroboscopic method. April 13.

DE VOGEIAERE, Ft. (Universite Laval, Quebec, Canada) On nonlinear differential equations connected with conservative dynamical

problems of two degrees of freedom. May 25.

BQCHNER, S, (Princeton University and University of California) Structure

of infinitely subdivisible stochastic processes. June 22.

Seminar on Numerical and Computational Aspects of Linear Problems;

Games, Linear Equalities, " Linear Inequalities, Programming, ...

"(Los Angeles, California)"

~

TOMPKINS, C.B. A description of a two-person zero-sum game resembling a military battle with probabilistic attrition and a preliminary discussion of computational problems involved in its

solution. April 20, 22, 24.

MORGAN, R. B. SWAC coding of a search for a solution to a large set

( 128 ) of simple linear inequalities in many ( 80 ) variables.

May 5. M0TZKIN, T. S. Elements of a theory of sequential projection. May 7

and 13 . HANDY, B. (1) Coding the assignment problem. May 11. (2) Frequencies

of vibrating membranes, II. May 20.

FORSYTHE, G. E. (1) Frequencies of vibrating membranes —comparison between lowest eigenvalues of a differential equation and approximating difference equations, I. May 1 8 , (2) Matrix

eigenvalues —Chebyshev polynomials or Liebmann methods.

June 1 . WEBER, M. SWAC experience on eigenvalues of a matrix of 45 rows and

columns, I. May 25.

68
Lectures and Symposia 69

YOWELL, E. C. SWAC experience on eigenvalues of a matrix of ^5 rows and columns, II* May 27.

HESTENES , M. R. Matrix inversion by the conjugate gradient method, I. Background and development of a program, June 3»

WILSON, L. Matrix inversion by the conjugate gradient method, II. Coding

for SWAC and experience with computations 0 June 8.

TEXCHKOEW, D. A game theory approach to the search for a global maximum

of a function. June 1 5. BARANKIN, E. W. Stochastic research for the global maximum of a function.

June 1 7.

Game Theory and Its Applications ; A Series of Weekly Lectures Sponsored by The American University

in Cooperation with the National Bureau of Standards

HOFFMAN, A. J. (National Bureau of Standards) Computational methods for

solving games. April 7.

BROW, G. W. (International Telemeter Corporation) Modelling of applications. April 1 *+. HAYWOOD, Colonel 0. G. (Air Research and Development Command, USAF) The

applications of game theory to the military doctrine of

decision. April 21

.

Statistical Engineering Seminar

SAVAGE, I. R. Nonpa rametric methods in the analysis of variance. April 17.

Papers and Invited Talks

Presented by Members of the Staff

at Meetings of Outside Organizations

Papers presented at the meeting of the American Mathematical Society,

New York, N. Y. April 23-25:

ABRAMOWITZ, M. (1) On regular and irregular Coulomb wave functions in

terms of Bes sel-Clif f ord functions. (2) On the differential

equation in a problem of heat convection of laminar flow

through a tube. Both presented by title.

ANTOS XEWXCZ , H. A. A boundedness theorem for a nonlinear differential

equation.

LUKACS , E, On strongly continuous stochastic processes.

TODD , J. The condition of the finite segments of the Hilbert matrix.

Presented by title.
70 Lectures and Symposia

Papers presented at a joint meeting of the Institute of Mathematical

Statistics and the Biometric Society, Eastern North American Region,

Washington, D. C . , April 29 - May Is CAMERON, J. M® Control and measurement of experimental error,

CLATWORTHY, W. H„ Necessary conditions for the existence of partially

balanced incomplete block designs with two associate classes

CONNOR, W, S, The correspondence between two classes ©f balanced incomplete block designs,

KING, E. P, A property of the normal distribution related to a theorem

of S . Bernstein,

ZELEM, M, An analysis of some incomplete block designs with a missing

block.

Talks presented to the Applied Mathematics Division, Naval Ordnance

Laboratory, White Oak, Md

.

DAVIS, P. Linear functional equations and interpolation series. June *+. HOFFMAN, A. J, Bounds for the rank and location of eigenvalues of a matrix. May 1*+. WEGSTEIN, J. H. Automatic coding techniques. April 30.

Papers presented elsewhere;

DAVIS, P« Linear functional equations and interpolation series. Presented at the Ballistics Research Laboratories, Aberdeen,

Md . , May 7

.

EISENHART, C. Why mathematical statistics. Presented to the staff of

the Logistics Research Project, George Washington University

Washington, D. C., June 17.

FORSYTHE, G. E. Arbitrarily close bounds for the fundamental frequency

of certain vibrating membranes. Presented at a meeting Of

the American Mathematical Society, Palo Alto, Cal., May 2.

HOFFMAN, A. J. Some problems in the foundations of geometry. Presented

to the Columbian Mathematics Club, George Washington University, Washington, D. € . , June 1.

LEHMER, D. H. Number theoretic studies with a high speed computer.

Presented at a Peripatetic Seminar, California Institute of

Technology, held on the UCLA campus, Los Angeles, Calif.,

May b.

LIEBLEIN, J. New statistical methods for the study of wind data. Presented at a Conference of Utility Commission Engineers,

St. Paul, Minn., May 28.

LUKACS , E. On some stochastic processes. Presented at the Admiralty

Research Laboratory, Teddington, England, Jurne 25.
Lectures and Symposia 71 MOTZKIN, T

TEICHROEW,

THORENSEN,

TODD, J.

WAS OW , W.

YQUDEN, W.

WIE LAND!,

. S. and C . B. TOMPKINS. Boundedness of sequential projections.

Presented at a meeting of the American Mathematical Society,

Palo Alto, Cal., May 2.

D. Distribution sampling with high-speed computers. Presented

at a Seminar of the Institute of Statistics, North Carolina

State College, Raleigh, N. C., May 8.

R. Electrostatic storage systems. Presented at a meeting of

the IRE Professional Group on Electronic Computers, Detroit,

Mich., April 24.

(1) The condition of the finite segments of the Hilbert matrix.

Presented at a meeting of the Mathematical Association of

America, Dahlgren, Va . , May 2. (2) Linear equations and

computing machines. Presented at a National Science Foundation Conference in Collegiate Mathematics, University of

Colorado, Boulder, Colo., June 22.

Singular perturbation methods. Presented at a Symposium on Nonlinear Circuit Analysis, Microwave Research Institute,

Polytechnic Institute of Brooklyn, N. Y. , April 23, 24.

J. (1) Design of experiments. Presented at Johns Hopkins

University, Baltimore, Md . , April 1. Also presented to the

technical staff of Rath and Strong, Inc,, Pittsfield, Mass.,

April 22. (2) New experimental designs for research and

development. Presented to the Hampton Roads Section of the

American Chemical Society, Norfolk, Va . , May 23.

H. (1) Inclusion of eigenvalues. Presented to the Department

of Mathematics, University of Pennsylvania, Philadelphia, Pa.

April 16. (2) Proofs of impossibility. Presented to the

Mathematics Club, Wilson College, Chambersburg, Pa., April 17

(3) On eigenvalues of matrices. Presented to the Department

of Mathematics, University of North Carolina, Chapel Hill,

N. C., May 6. (4) On eigenvalues of matrices arising in the

numerical solution of integral equations. Presented at the

NBS , June 12.
Publication Activities

1 . PUBLICATIONS WHICH APPEARED DURING THE QUARTER

1.1 Mathematical Tables

(1) Tables for rocket and comet orbits. S. Herrick, NBS Applied

Mathematics Series 20. Available from U. S. Government Printing

Office, Washington 25? D. C., gf 1 .75.

(2) Tables of Aretan x. NBS Applied Mathematics Series 26. (Supersedes NBS Mathematical Table Mil 6). Available from U. S, Government Printing Office, Washington 25, D. C., $1 . 75.

(3) A table of Wilson quotients and the third Wilson prime. K. Goldberg. J, London Math. S oc . , 28, Pi. 2, 252-256 (Apr. 1953) * Reprints available,

1.2 Manuals, bibliographies, indices

(1)

Addendum to a guide to tables on punched cards. G. Blanch and

E. C .. Yowell. RfTAC VII, 1-6(Jan. 1953) • Reprints available.

1.3 Technical Papers

(1) Analyzing straight line data, F. S. Acton. J. Chem. Ed. 30 , 128-133 (Mar. 1953). Reprints available.

( 2 ) On a recursion formula and on some Tauberian theorems, N. G.

de Bruijn and P. Erdos. .J. Res. NBS 50, 161-164. (Mar. 1953).

(3) Contribution to the theory of Markov chains. Kai Lai Chung.

J. Res. NBS 50, 203-208 (Apr. 1953).

(4) Elements of a mathematical theory of probability. J. H. Curtiss.

Math. Mag. 26, 233-254 (May-June 1953).

(5) Sequential decision problems for processes with continuous time

parameter. Testing hypotheses. A. Dvoretsky, J. Kiefer, and

1. Wolfowitz. Ann. Math. Stat. 24, No. 2, 254-264 (1953). Reprints available.

(6) A Neumann series for the product of two Whittaker functions.

P. Henrici. Proc. Am. Math. Soc. 4, 331-334 (Apr. 1953). Reprints available.

(7) On the exact evaluation of the variances and covariances of order

statistics in samples from the extreme-value distribution. J. Lieblein. Ann. Math. Stat. 24, 282-287 (June 1953). Reprints

ava liable

.

(8) The torsion of anisotropic elastic cylinders by forces applied on the lateral surface. H. Luxenberg. J. Res. NBS 50, 263-276

(May 1953).
(9) Publication Activities 73

The double description method. T. S. Motzkin, H. Raiffa, G. L. Thompson, R, M. Thrall. Contributions to the theory of games II,

edited by H. W. Kuhn and A. W. Tucker, Annals of Mathematics Study

28, chapter 3 (Princeton, 1953). No reprints.

(10) On the derivative of a polynomial and Chebyshev approximation.

T. S„ Motzkin and J. L. Walsh. Proc. Am. Math. S oc . 4, 76-87

(Feb. 1953). Reprints available.

(11) Two explicit formulae for the distribution function of the sums

of n uniformly distributed variables. A. M. Ostrowski. Arehiv

Math. Ill, 3-11 (1952). Reprints available.

(12) On smoothing operations and their generating functions. I. J. Schoenberg. Bui, Am. Math. Soc. 52, 199-230 (1953). Reprints

available

.

(13) On Polya frequency functions III: The positivity of translation

determinants with an application to the interpolation problem by

spline curves. I. J. Schoenberg and A. Whitney. Trans. Am. Math.

Soc, 2!+? 246-259 (Mar. 1953). Reprints available.

(1^) On methods for obtaining solutions of fixed end-point problems

in the calculus of variations. M. L, Stein. J. Res. NBS 50,

277-297 (May 1953).

( 1 5) On the Gibbs phenomenon for a class of linear transforms. 0. Szasz.

Acad. Serb© Sei. Publ. Inst. Math. 4, 135-144 (1952). Rep.rints

available.

(16) Systems of equations, matrices and determinants. 0. Taussky. and

J. Todd. Math. Mag. 26, 9-20 (Sept-Oct 1952); 71-88 (Nov-Dec.

1952). Reprints available.

(17) The tabulation of an integral arising in the theory of cooperative

phenomena. M. Tikson. J. Res. NBS jfo, 1 77-1 78 (Mar. 1953).

(18) Metodi probabilistic! per la soluzione numerica di alcuni problerai

di analisi. W. Wasow. Rend. Mat. App. {vj , XI, 336-346 (Roma

1952); also issued separately as Pubblicazioni dell 1 Is tituto per

le Applieazioni del Calcolo N. 354 (Roma 1953).

1.5 Miscellaneous Publications

(1) Scientific teamwork in a Computation Laboratory. J. H. Curtiss.

Teamwork in research, edited by G. P. Bush and L. H. Hattery,

chap. 17 (American University Press, 1953). No reprints.

(2) Control charts may be all right, but—. F. Proschan. Industrial

Quality Control IX, 56-58 (May 1953).

2. MANUSCRIPTS IN THE PROCESS OF PUBLICATION JUNE 30, 1953.

2.1 Mathematical Tables

(1) Probability tables for analysis of extreme-value data. NBS Applied Mathematics Series 22. In press, Government Printing

Office

.

(2) Tables of normal probability functions. NBS Applied Mathematics

Series 23. (Supersedes NBS Mathematical Table MTl4, Tables of
74 Publication Activities

probability functions, Volume II.) In press, Government Printing Office.

(3) Tables of 10x

. NBS Applied Mathematics Series 27. In press,

Government Printing Office.

(4) Tables of coefficients for the numerical calculation of Laplace

transforms. H, E. Salzer. NBS Applied Mathematics Series 30.

In press, Government Printing Office.

(5) Table of natural logarithms for decimal numbers from 0.0001 to

5.0000. (A reissue of NBS Mathematical Table MT10, Table of

natural logarithms, vol . III.) NBS Applied Mathematics Series 31.

In press, Government Printing Office.

( 6 ) 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.

(7) Table of the Gamma function for complex arguments. NBS Applied

Mathematics Series 34. In press, Government Printing Office.

(8) Tables of Lagrangian coefficients for sexagesimal interpolation.

NBS Applied Mathematics Series 35 . In press, Government Printing

Office.

(9) 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

.

(10) 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.

(11) 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.

(12) Tables of expected values of 1/X for positive Bernoulli and

Poisson variables. I. R. Savage and E. Grab. Submitted to a technical journal.

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.

2.3 Technical Papers

(1) Evaluation of the integral

oo

0

5 u (x/u)^.

M. Abramowitz.

Accepted for publication in the Journal of Mathematics and Physics.

(2) On the solution of the differential equation occurring in the

problem of heat convection in laminar flow through a tube. M

Abramowitz. Submitted to a technical journal.
Publication Activities 75

(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.

(5-) Approximate method for rapid Loran computation. M. Abramowitz,

D. H. Call, and J. C . Mathews. Submitted to a technical journal.

(5) Note on the simulation of autoregressive series. F. L. Alt.

Submitted to a technical journal.

(6) The relaxation method for linear inequalities. S. Agmon. Submitted to a technical journal.

(7) On the differential equation x + k(f(x) + g(x)x)x = ke(t).

H. A. Antosiewicz. Submitted to a technical journal.

(8) Some implications of Liapunov's conditions for stability. H. A. Antosiewicz and P. Davis. Submitted to a technical journal.

(9) The convergence of numerical iteration. H. A. Antosiewicz and

J. M. Hammersley. Submitted to a technical journal.

(10) On mildly nonlinear partial difference equations of elliptic

type. L. Bers. Accepted for publication in the Journal of

Research of the MBS

.

(11) On the numerical solution of parabolic partial differential equations. G. Blanch. Accepted for publication in the Journal of

Research of the NBS

.

(12) Boolean geometry I. L. M. Blumenthal. To appear in Rendiconti

del Circolo Matematico di Palermo.

(13) 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.

(14) Programs for computing the hypergeometric series. W. F. Cahill.

Submitted to a technical journal.

(15) Acceptance sampling of electroplated articles. J, M. Cameron

and F. Ogburn. Submitted to a technical journal.

( 1 6 ) The embedding theorem for balanced incomplete block designs.

W. S. Connor and M. Hall, Jr. Accepted for publication in the

Canadian Journal of Mathematics.

(17) 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.

2

( 1 8) Some L aspects of Faber polynomials. P. Davis and H. Pollack.

Submitted to a technical journal.

(19) Linear functional equations and interpolation series. P. Davis.

Submitted to a technical journal.

(20) 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.
Publication Activities

(21) On absolute measurement . N. E. Dorsey and C. Eisenha rt . Accepted for publication in Scientific Monthly.

(22) On the optimal character of the (s,S) policy in inventory theory.

A. Dvoretzky, J. Kiefer, and J. Wolfowitz. Submitted to a technical journal.

(23) A test for statistical control applicable to a short series of

observations. C. Eisenhart and E. P. King. Submitted to a technical journal.

(24) Changes of sign of sums of random variables. P. Erdos and G. A.

Hunt. 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.

Submitted to a technical journal.

(27) Asymptotic lower bounds for the frequencies of polygonal membranes.

G. E. Forsythe. Submitted to a technical journal.

(28) Solving linear algebraic equations can be interesting. G. E.

Forsythe. Accepted for publication in Bulletin of the American

Mathematical Society.

(29) Tentative classification of methods and bibliography on solving

systems of linear equations. G. E. Forsythe. To appear in Simultaneous linear equations and the determination of eigenvalues.

Proceedings of an NBS Symposium held in Los Angeles, August 1951.

NBS Applied Mathematics Series 29. In press, Government Printing

Office.

(30) 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

S eries

.

(31) Additive functionals of a Markoff process. R. Fortet. Submitted

to a technical journal.

(32) 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.

(33) A numerical solution of Schroedinger 1 s equation in the continuum.

W. Futterman, E. Osborne, and D. S. Saxon. Accepted for publication in the Journal of Research of the NBS.

(34) A nonlinear model for the composite pi-meson. S. G. Gasiorowicz.

Submitted to a technical journal.

(35) A 2-basic set of density zero. K. Goldberg. Submitted to a technical journal.
Publication Activities 77

(36) An expansion method for parabolic partial differential equations.

J. W, Green, Accepted for publication in the Journal of Research

of the MBS ,

(37) A non-ha rraonic Fourier series. J. M. Hammersley. Accepted for

publication in Acta Matematica.

(38) 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,

(39) On a combinatorial theorem. A. J. Hoffman. Submitted to a technical journal.

(40) Computational experience in solving linear programs. A. J. Hoffman, M. Mannos, D. Sokolowsky, and N. Weigmann. Accepted

for publication in the Journal of the Society for Industrial

and Applied Mathematics,

(4-1) The SWAC—design features and operating experience. H. Huskey,

R. Thorensen, B, F. Arabrosio, and E. C. Yowell . Submitted to

a technical journal.

(4-2) Completely continuous normal operators with property L. I. Kaplansky. Submitted to a technical journal.

(4-3) Estimating the standard deviation of a normal population.

E. P. King, Accepted for publication in Industrial Quality

Cont rol

,

(44-) On some procedures for the rejection of suspected data. E. P. King. Accepted for publication in the Journal of the American

Statistical Association.

(45) Probability limits for the average chart when process standards

are unspecified, E. P. King. Submitted to a technical journal.

(46) On certain character matrices, D. H. Lehrner. Submitted to a technical journal.

(47) 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.

(48) On s trongly continuous stochastic processes. E. Lukacs. Submitted to a technical journal.

(49) A property of the normal distribution related to a theorem of

S. Bernstein. E. Lukacs and E. P. King. Submitted to a technical journal.

(50) Certain Fourier transforms of distribution (II). E. Lukacs and

0. Szasz. Submitted to a technical journal.

(51) Nonnegative trigonometric polynomials and certain rational

characteristic functions. E. Lukacs and 0. Szasz. Accepted for

publication in the Journal of Research of the NBS.

(52) Eigenvectors of matric polynomials. M. Mannos. Accepted for

publication in the Journal of Research of the NBS.
78 Publication Activities

(53) On the relaxation method for linear inequalities, T. S. Motzkin

and I. J. Schoenberg. Submitted to a technical journal.

(5*+) On Fejer sets in linear and spherical spaces. T. S. Motzkin

and I. J. Schoenberg. Accepted for publication in Annals of

Ma thema tics ..

(55) Pairs of matrices with property L, II, (summary) by T. S.

Motzkin and 0. Taussky. Accepted for publication in the Proceedings of the National Academy of Science.

(56) The arithmetic structure of certain modular subgroups. M.

Newman. Submitted to a technical journal.

(57) The coefficients of certain infinite products. M. Newman.

Accepted for publication in Proceedings of the American Mathematical Society.

(58) On the Lerch zeta function. F. Oberhettinger. Submitted to a technical journal.

(59) On two problems in abstract algebra connected with Horner’s

rule. A. M. Ostrowski. Submitted to a technical journal.

(60) On nearly triangular matrices. A. Ostrowski. Submitted to a technical journal.

(61 ) On over and under relaxation in the theory of the cyclic single

step iteration. A. Ostrowski. Submitted to a technical journal.

(62) Determinanten mit ueberwiegender Hauptdiagonale und die absolute

Konvergenz von linearen Iterationsprozessen. A. M. Ostrowski.

Submitted to a technical journal.

(63) On the convergence of Gauss’ alternating procedure in the method

of the least squares, I, A. M. Ostrowski. Submitted to a technical journal.

(6h-) On the linear iteration procedures for symmetric matrices. A.

M. Ostrowski. Submitted to a technical journal.

(65) On the spectrum of a one parametric family of matrices. A. M.

Ostrowski. Submitted to a technical journal.

(66) On Gauss' speeding up device in the theory of single step iteration. A. M. Ostrowski. Submitted to a technical journal.

( 67 ) On absolute convergence of linear iteration processes. A. M.

Ostrowski. Submitted to a technical journal.

(68) Confidence and tolerance intervals for the normal distribution.

F. Proschan. Submitted to a technical joqrnal.

( 69 ) Rejec tion of outlying observations. F. Proschan. Accepted for

publication in the American Journal of Physics.

(70) On spectra of second-order differential operators. D. Ray.

Submitted to a technical journal.

(71) Numerical computation of low moments of order statistics from

a normal population. J. B. Rosser. Submitted to a technical

journal

.
Publication Activities 79

(7 2) 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,

(73) An is ope rimet ric inequality for closed curves convex in even- dimensional Euclidean space. I. J. Schoenberg. Submitted to a technical journal.

(7*0 Generalized commutators of matrices and permutations of factors

in a product of three matrices. 0. T8ussky. For publication

in the von Mises Anniversary volume.

(75) An improved cathode ray tube storage system. R„ Thorensen. To

appear in the Proceedings of the Western Computer Conference of

the A3EE-IRE-ACM held in Los Angeles, Calif., February 4,5,6,

1953.

(76 ) 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.

(77) On the accuracy of the numerical solution of the Diriehlet

problem by finite differences. J. L. Walsh and D. Young. Accepted for publication in the Journal of Research of the NBS.

(78) On singular perturbation problems in the theory of nonlinear

vibrations. W. Wasow. To be published in the Proceedings of

the Symposium on Nonlinear Vibrations, held at Isle de Porquerolles, France, September 18-22, 1951 .

(79) Asymptotic solution of the differential equation of hydrc:-

dynamic stability in a domain containing a transition point.

W. Wasow. Accepted for publication in the Annals of Mathematics.

(80) On small disturbances of plane Couette flow. W. Wasow. Accepted for publication in the NBS Journal of Research.

(81) Singular perturbation methods for nonlinear oscillations.

W. Wasow. To appear in the Proceedings of a Symposium on Nonlinear Circuit Analysis, held by the Polytechnic Institute

of Brooklyn, N. Y.

(82) On the eigenvalues of A+B and AB . H. Wielandt. Submitted to

a technical journal.

(83) Pairs of normal matrices with property L. H. Wielandt. Accepted for publication in the Journal of Research of the NBS.

(84) Statistical units of measurement. W. J. Youden. Accepted for

publication in Metals Progress.

(85) 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.

(86) The chain block design. W. J. Youden and W. S. Connor. Accepted for publication in Biometrics.
80 Publication Activities

( 87 ) Making one measurement do the work of two® W» J. Youden and

W. S, Connor® Accepted for publication in Chemical Engineering

Progress

.

( 88 ) Performance of inspectors and gasoline pumps « W„ J. Youden

and Mo W. Jensen. Submitted to a technical journal®

2.5 Miscellaneous Publications

(1) Simultaneous linear equations and the- determination of eigenvalues. Proceedings of an MBS Symposium held in Los Angeles,

August 1951. MBS Applied Mathematics Series 29. In press,

Government Printing Office.

(2) Mathematical services useful in industry. A. S„ Cahn, Jr.

To appear in the Proceedings of & Symposium ©n Industrial

Applications of Automatic Computing Equipment held by the

Midwest Research Institute, Kansas City, Mo., January 8 , 1953.

(3) A historical note on the application of the "weakest-link"

idea to tensile strengths. J, Lieblein. Submitted to 0 technical journal.

(5) Contributions to the solution of systems of linear equations

and the determination ©£ eigenvalues. To be issued in the MBS

Applied Mathematics Series.
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.

Reports and Publications

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.
fX