NATIONAL BUREAU OF STANDARDS REPORT

3 2 8 3

PROJECTS and PUBLICATIONS

of the

NATIONAL APPLIED MATHEMATICS LABORATORIES

A Quarterly Report

January through March 1954

FOR OFFICIAL USE

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. 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 and Reactance Measurements. Electrical Instruments. Magnetic

Measurements. Electrochemistry.

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

Technology. Length. Engineering Metrology.

Heat and Power. Temperature Measurements. Thermodynamics. Cryogenic Physics. 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-Ray. Betatron. Nucleonic Instrumentation. Radiological Equipment. Atomic Energy Commission Radiation 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. Fluid Mechanics. Engineering Mechanics. Mass

and Scale. Capacity, Density, and Fluid Meters. Combustion Control.

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.

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.

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

Instrumentation. Process Technology.

Radio Propagation. Upper Atmosphere Research. Ionospheric Research. Regular Propagation

Services. Frequency Utilization Research. Tropospheric Propagation Research. High Frequency

Standards. Microwave Standards.

0Office of Basic Instrumentation 0Office of Weights and Measures.
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT

NAML

NBS REPORT

3 2 8 3

PROJECTS and PUBLICATIONS

of the

NATIONAL APPLIED MATHEMATICS LABORATORIES

January through March 1954

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication, reprl unless permission Isol 25, D.C, Such perml

cal ly prepared If tha

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

part, Is prohibited

idards, Washington

>rt has been specif)*

ort for Its own use,
NATIONAL APPLIED MATHEMATICS LABORATORIES

January 1 through March 31? 195*+

ADMINISTRATIVE OFFICE

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

*Edward W. Cannon, Ph.D., Assistant Chief Olga Taussky-Todd , Ph.D., Mathematics Consultant

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

Robert F. Dressier, 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 Billie R. Gill, M.S., Secretary

Louis J. Meyerle, Jr., Secretary

Esther L. Turner, Secretary

INSTITUTE FOR NUMERICAL ANALYSIS

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

Magnus R. Hestenes, Ph.D. . .Assistant Director and UCLA Liaison Officer Gertrude Blanch, Ph.D. . .Assistant to the Director (Numerical Analysis)

Research Staff Forman S. Acton, D.Sc Mathematician

George E. Forsythe, Ph.D.... Mathematician

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

Leopoldo Nachbin, Ph.D UCLA Research Associate (Mathematics) Thomas H. Southard, Ph.D Mathematician Daniel Teichroew, Ph.D..., Statistician

Wolfgang R. Wasow, Ph.D Mathematician

Graduate Fellows Eugene Levin, M.A. Mervin Muller, M.A. Genovevo C, Lopez, B.A, (UCLA) Lloyd Philipson, M.A,

,

Edwin Mookini, M.A. (UCLA) David A. Pope, M.A. (UCLA) Roger Woods, M.A. (UCLA) Mathematical Services Unit Frederick H. Hollander, M.A, Mathematician

Robert R. Reynolds, M.S .Mathematician Computing Staff Lillian Forthal, B.A. Frank B. Meek, M.A. Benjamin F. Handy, Jr., M.S. Nan N. Reynolds, M.S. Ruth B. Horgan, B.A. Louise M. Straus, M.A. Administrative Unit

Estelle H. Strauss Librarian Assistant

Mildred Vukmanic Administrative Assistant

Mildred B. Webb Administrative Clerk

Reve Vineyard Administrative Clerk

Leo Moskowitz Procurement Clerk Vendla H. Gordanier Secretary

Dorothy M, Hibbard Secretary

Elsie L, Husman Secretary

COMPUTATION LABORATORY

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

Henry A. Antosiewicz, Ph. D

Philip Davis, Ph.D... Karl Goldberg, M.A

Peter Henrici, Ph.D........ Alan J. Hoffman, Ph.D Morris Newman, Ph. D Fritz Oberhettinger , Ph.D.. Philip Rabinowitz, Ph.D..,.

Irene A. Stegun, M.A

Charles J. Swift, Ph.D Joseph H, Wegstein, M.S...., Computing Staff Hans 0. Bremer, B.A. William F. Cahill, M.S. Ruth E, Capuano Anne R. Cock, B.A. Mary M. Dunlap, B.S. Anne M. Futterman, B.A. Leon Gainen, B.A. Elizabeth F. Godefroy

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

** American University

Chief Assistant Chief Mathematician

All** Research Associate Mathematician

Mathematician

AU** Research Associate Mathematician

Mathematician

AU** Research Associate Mathematician

Mathematician

Physicist

Physicist

Terry W. Ledley, M.A. Norman Levine, M.S, David S. Liepman Ethel C. Marden, B.A. Kermit C. Nelson

Peter J. O'Hara, B.S. Mary Orr

Maxine L. Paulsen, B.S,

B. Stanley Prusch

Otto T. Steiner, B.A. Frank J. Stockmal, B.A. Sally Tsingou, B.S. Bertha H. Walter

Ruth Zucker, B.A.

Marjorie E. Young

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

Julius Lieblein, Ph.D Mary G. Netrella, B.A

I. Richard Savage, M.S Marvin Zelen, M.A

Marion T. Carson Yvette B, Cocozzella

Lela J. Hamilton

Chief Assistant Chief Technical Aid Mathematical Statistician

Mathematician

.Mathematician Mathematician

Mathematician

Mathematical Statistician

Computer

Secretary

Secretary

MACHINE DEVELOPMENT LABORATORY Edward W, Cannon, Ph.D Chief Edith N, Reese, B.A Mathematician On leave of absence
Contents

Page

Index iv

Status of Projects as of March 31 > 195*+. . . 1 Institute for Numerical Analysis 1 (NBS Section 11.1)

Computation Laboratory 18

(NBS Section 11.2)

Statistical Engineering Laboratory .... ^9

(NBS Section 11.3)

Machine Development Laboratory 56

(NBS Section 11.^)

Lectures and Symposia ..... 57

Publication Activities 6l
Index of Active Research and Development Projects

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

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

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

Mathematics Laboratories. These services are, however, fully represented

in the body of the report.

A. Research: Pure Mathematics

Studies in pure mathematics 3> 19

B. Research: Numerical Analysis

Baker-Hausdorf f formula 23

Classical numerical analysis, Research in. .......... 18

Differential equations, Studies in numerical integration of. . 4

Dirichlet problem for certain multiply-connected domains,

Investigation of Bergman’s method for the solution of the. . 19

Eigenvalues, eigenvectors, and eigenfunctions of linear

operators, Calculation of 2

Laplace equation, Solution of, by Monte Carlo method 21

Probability methods and sampling techniques 4

Riemann-zeta function, Computation of the complex zeros

- of . . 3

SCAMP 7

Simultaneous algebraic equations and techniques for the

inversion and iteration of matrices, Solution of sets of . . 1 Variational methods 5

C. Research: Applied Mathematics, Physics, and Astronomy

Applied mathematics, Studies in 5

Compressible flow—method of orthogonal and kernel functions . 24

Computing machines, Studies in the theory of digital ..... 8

Continua, Research in mechanics of 26

Flow in supersonic nozzle. 4l

Gases, Tablesof thermodynamic properties of. ... . 33

Geomagnetic field, Analysis of . 22

Hypergeometric functions 23

Linear programming, Basic research in 23

" ” , Supporting research in 2?

Matrix theory, Special problems in finite 22

Nerve fiber reaction, Differential equation for 23

Painleve equation 32

Program planning, Research in the mathematical theory of . . . 6

Russian mathematical progress, Study of 6

Theoretical physics, Miscellaneous studies in 6

Vibration modes and frequencies (of aircraft structures),

Computation of .... 40

Water, High temperature properties of 39

Water waves 2

7
D. Mathematical Statistics

Experiment design. Studies in the mathematics of . . Non-parametric statistics, Procedures of, . . Probability and statistics, Miscellaneous studies in

Propagation of error, Law of , E, Mathematical Tables

Airy integral, Table of the modified

Arc sin for complex arguments, Table of. .......

Collected short mathematical tables of the Computation

Laboratory, .........

Coulomb wave functions, Tables of

E.j(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

Hyperbolic sines and cosines, Extension of the table of

Lagrangian coefficients for sexagesimal interpolation, Table

Of. . . . . Mathieu functions, radial

Power points of analysis-of-variance tests, Tables of

Punched card library . , Random samples, Table to facilitate drawing . . Revision of mathematical tables , , Secants and cosecants, Table of . . Sievert's integral. . Spheroidal wave functions ....... . . . . Van der Pol equation.

Wave function for lithium

F. Manuals, Bibliographies, Indices, and Technical Information

Coding procedures, mathematical tables and numerical

analysis, Bibliography of ....... . Fitting straight lines, Manual on . . , . Statistical literature, Bibliography and guide to . . . G, Computing Machine Development

Census computing machine, Bureau of the . SEAC: National Bureau of Standards Eastern Automatic Computer . , SEAC, Number- theoretical test problems for. » , , . , . SWAC: National Bureau of Standards Western Automatic Computer , ,

51

51

50

50

30

30

29

28

27

31

31

27

30

28

32

29

10

49

29

32

32

31

31

29

28

49

49

56

56

21

8
/

. f*

’
Status of Projects

March 31 , 195^

I. INSTITUTE FOR NUMERICAL ANALYSIS

(Section 11.1)

1 . Fundamental Research

SOLUTION OF SETS OF SIMULTANEOUS ALGEBRAIC EQUATIONS AND

TECHNIQUES FOR THE INVERSION AND ITERATION OF MATRICES

Task 11 01-1 0-51 00/49-AE2

Origin: NBS Authorized 2/1 5A-9

Sponsor: Office of Naval Research, IJSN Revised 9/1 5A-9

Manager: G. E. Forsythe

Full task description appears in July-Sept 19^9 issue, p. 2.

Status: CONTINUED. Louise Straus has completed a convenient SWAC

routine for improving the inverse of a matrix by the "Newton-Schulz"

formula X^+ ^ = (2I-Xj

t A)Xjt

. The routine accepts a matrix A of orders up to

31 and an approximate inverse X^ 9 both expressed in floating binary notation. Using single-precision non-floating operation with triple-precision

accumulation of inner products, the routine improves the inverse as far

as possible and punches the answer.

In publication 2 below is a brief summary of the available SWAC

codes for solving matrix problems, as of the beginning of the quarter.

Since then one could add the above inverse-improvement code, and the Frame

routine now reported under task 1 101-10-5100/50-3, p.2.

M. R. Hestenes and E. H. Mookini have completed two SWAC codes

for solving linear systems by two variants of the conjugate-gradient

method, for orders up to 30. These employ M. R. Hestenes 1 "floating

vectors," described under task 1 101-10-5100/50-3. Publication 1 describes

these and other variants of the conjugate-gradient method, with particular

emphasis on non-hermitian matrices.

The projection method described in the July-Sept 1953 issue

turns out to have been previously discovered by E. W. Purcell. See J.

Math. Phys. vol. 32 (1953), pp. 180-183.

For work on the reduction of a matrix to Hermite normal form,

see task 1 1 01 -1 0— 5 1 50/53-1 , P. 7.

In publication 3, c . B. Tompkins and I. Heller (of George Washington University) extended a theorem of G, B. Dantzig. As a consequence,
2 Status of Projects

they prove that solutions of some realistically defined transportation

problems can be found in the ring of integers. Hence fractional oceangoing ships will not be demanded as solutionsto this abstraction of the

problem of efficient shipping!

For work on linear programming related to this task, see the

report of task 1 101-10-5102/50-1 1 , p.6.

The research of G. E. Forsythe and E. G. Straus (of U.C.L.A.)

on best conditioned matrices has been submitted in an abstract to the

International Congress of Mathematicians. The result reported in the

Oct-Dec 1953 issue has been generalized to a fairly general class of preconditioning matrices.

Publications: (l) "The conjugate gradient method for solving linear

systems," by M. R. Hestenes; in typescript. (2) "Seminar on numerical

analysis, Summary of presentation of 7 January 19

5

1+?" by G. E. Forsythe,

in typescript. (3) "Proof of an extension of a theorem of G.B. Dantzig's,"

by I. Heller and C. B. Tompkins; in typescript.

CALCULATION OF EIGENVALUES, EIGENVECTORS, AND EIGENFUNCTIONS

OF LINEAR OPERATORS

Task 1101-10-5100/50-3

Origin: NBS Authorized 7/1/*+9

Sponsor: Office of Naval Research, USN

Manager: G. E. Forsythe

Full task description appears in July-Sept 19^9 issue, p. 13.

Status: CONTINUED. SWAC codes have been prepared for obtaining

the characteristic polynomial of nonsymmetric real matrices A of order

n by direct methods. Two methods were tried:

(i) Gaussian elimination with floating binary point to find the

linear dependence between x, Ax, A2x,...,Anx, using a code prepared by J,

Pettit of the UCLA Geophysics Department.

(ii) An abridgment of the Leverrier method of calculating trace

(Ak )=2~ n /V^ (k=1,...,n) and solving Newton's identities for the coi=1 i

efficients of the characteristic polynomial. The abridgment was devised

independently by Frame, Souriau, and Sominskii and Faddeev, and is described, for example, in Dwyer's "Linear Computations," p. 225.

Method (i) seemed to lead to inconsistent results when applied

to A and AT

, and when x was varied. At best the elimination solution

probably needed improvement by a separate routine; method (i) was abandoned

in favor of method (ii).

Method (ii) was developed into a very convenient routine of

limited applicability. The matrices were carried in the "floating row"

form devised by Hestenes for the conjugate-gradient method (see task

1 101 -10-51 00/49-AE2, p. 1). For example, each row of the matrices is

written in n+1 cells, the Oth containing an integral exponent of 2 common

to each of the n components 1,...,n. In this way one obtains both the

speed of ordinary (non-floating) machine operation and the automatic scaling

of floating operation. The routine deals with nO 1* (easily modifiable to

n<30), and for n=10 will type out one coefficient of the characteristic

polynomial each 10 seconds.

Let the characteristic polynomial be written in the form

An+c.jAn 1 + c 2

An-2- ... + c n I. Let the eigenvalues of the nonsingular matrix

A be numbered so that | 1< | A

2

|<. . .< |A

n

|. Then

| c n | = | det (A)(=|A

1 A2

. . .

.

The round-off error in getting c^ with our routine has been found empiri-
3 Status of Projects

callv to be approximately 2

^ ] A 1^.

out.) Thus the least accurate coDifi

(We use 36-bit word length throughof about 2

-**01 X

icient, c^, is subject to an error e

. Hence the relative error

| e/c ! is approximately

/Ak | . Make the rough assumption that

J

has a constant

value r for k=2,3,...,n. Then r n-1 =P= | A /Aj

| , the condition-number of A.

Now, with this assumption,

| e/c

n | =2_I+^r n~^ r n-2 . . . r 2 r=2

_I+^r^T1 ) = 2“l+^pin . Suppose we want to achleve.,b binary pi

need I e/c <2“, or 2

-L|"0P2TK2- b, or

aces of accuracy in c Then we

(*) P < 2

( 80-2b)/n

Perhaps b=10 is sufficient for ordinary engineering purposes.

Thus we can state as a rough rule that our code will give the coefficients

of the characteristic polynomial of A to V.decimal accuracy for matrices

ivith P .c ondition-numbe r? not exceed ing. 2 b

-0/n . , For n=10, this limits P to be less than 2° (=64). This is a fairly severe restriction on the matrices which can be handled by the

method, and leaves one with the feeling that iterative methods are to be

preferred for most matrices.

Participants in research on this task were G. E. Forsythe and

L, Straus.

Publications: (1) "Asymptotic lower bounds for the frequencies of

polygonal membranes," by G. E. Forsythe; to appear in the Pacific Journal

of Mathematics. (2) "Asymptotic lower bounds for the fundamental frequency of convex membranes," by G. E. Forsythe; to appear in the Pacific

Journal of Mathematics.

STUDIES IN PURE MATHEMATICS

Task 1101-10-5100/50-4

Origin: NBS Authorized 7/1/49

Sponsor: Office of Naval Research, USN

Manager: C. B. Tompkins

Full task description appears in July-Sept 1949 issue, p. 16 . Status: CONTINUED. Most work is reported under other task reports.

Dr. Leopoldo Nachbin of South America has been in residence

at the INA during most of the period, and he has continued his researches

on algebra. He has addressed seminars at the University of California

at Los Angeles and the local peripatetic seminar in mathematics during

the period.

COMPUTATION OF THE COMPLEX ZEROS OF THE RIEMANN ZETA FUNCTION

Task 1101-10-5100/50-13

Origin: NBS Authorized 6/1/50

Sponsor: Office of Naval Research, USN

Manager: R. Horgan

Full task description appears in Apr-June 1950 issue, p.13.

Status: INACTIVE. For status to date see July-Sept 1953 issue,

P. 3.
1+ Status of Projects

STUDIES IN THE NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS

Task 1101-10-5100/51/1

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Manager: W. Wasow / Full t ask description appears in July-Sept 1950 issue, p. 12.

r Status: CONTINUED. Some minor work on the problem of solving differential equations that arise in the local embedding of surfaces was carried out by C. B. Tompkins. This work will continue.

The studies of L. Philipson on the asymptotic character of the

solutions of

y

(n) + <> ry

(n" r) + Zq <» s * + <= s )y

Cm_s) = °

are reaching a conclusion. Fundamental systems of explicit solutions by

Laplace contour integrals have been asymptotically evaluated for large A.

The exact sectors of validity of the various asymptotic expressions as well as the Stokes phenomena that describe their interrelations are manifest.

W. Wasow is working on a unified introduction to the asymptotic

theory of non-linear analytic differential equations.

Ur. W. F. Brown and Prof, T. Y. Thomas, who are spending the

winter at the Institute for Numerical Analysis as visitors, are contributing to the mathematical activities of the Institute.

Publications: (l) "On singular perturbation problems in the theory

of non-linear vibrations," by W. Wasow; Actes du Colloque International

des Vibrations Non Lineaires, lie de Porquerolles 1951? PP. 207-222(1 953)

.

(2) "Singular perturbation methods for nonlinear oscillations," by

W. Wasow; Proceedings of a Symposium on Nonlinear Circuit Analysis, Polytechnic Institute of Brooklyn, 1953, pp. 75“98. (3) "Discrete approximations to elliptic differential equations," by W. Wasow; IN MANUSCRIPT.

PROBABILITY METHODS AND SAMPLING TECHNIQUES

Task 1101-10-5100/51-2

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Manager: D. Teichroew

Full task description appears in July-Sept 1950 issue, p. 13.

Status: CONTINUED. Two reports (l), (2) on probability points of

the Incomplete Gamma distribution and one on the use of generalized normalization polynomials (3) have been completed.

M. Muller is continuing his investigation of procedures for

improving the efficiency of the Monte Carlo method for solving boundary

value problems by using predetermined networks or by using continuous

stochastic processes.

T. Motzkin investigated neighbor sets in n dimensions (publication (4) below)

.

Participants in the research on this task were T. Motzkin,

M. Muller, and D, Teichroew. Some of the coding was done by L. Forthal.

Publications: (l) "A table of Campbell's polynomials for the com- putation of probability points of the incomplete Gamma distribution,"
Status of Projects 5

by D. Teichroew; IN MANUSCRIPT. (2) "A table of millile probability

points of the incomplete Gamma distribution," by D. Teichroew; IN

MANUSCRIPT. (3) "Generalized normalization polynomials for the t and

incomplete Gamma distributions," by D. Teichroew; IN MANUSCRIPT. (4) "Neighbor sets for random walks and difference equations," by T. Motzkin; IN

MANUSCRIPT.

VARIATIONAL METHODS

Task 1101-10-5100/51-3

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Manager: M. R. Hestenes

Full .task description appears in July-Sept 1950 issue, p. 15.

Status: CONTINUED. Various direct numerical methods in the calculus of variations are being studied and a SWAC procedure is being prepared for the numerical computation of a minimizing curve for one dimensional fixed end point problems, using a method of "steep descent" given

by a generalized gradient. Theoretical investigations into the various

methods are being undertaken with a view to comparing their effectiveness

in multidimensional as well as one dimensional problems.

Participants in this task were D. A. Pope and C. B, Tompkins,

STUDIES IN APPLIED MATHEMATICS

Task 1101-10-5100/51-4

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Manager: T. H. Southard

Full task description appears in July-Sept 1950 issue, p. l6.

Status: CONTINUED. The seminar on numerical analysis met 20 times

during the quarter. A detailed list of sessions will be found under Lectures and Symposia in the back of this issue. Summaries of the sessions of

January 7 and March 25 were prepared by G. E. Forsythe; other summaries

are being prepared by C. B, Tompkins.

E. Levin did some work on punch indentation problems. An admissible velocity field is constructed for an arbitrary punch; hence, for

any particular case, a limit design theorem of Drucker, Prager, and

Greenberg may be used to compute an upper bound for the punch indentation

pressure. Results of the present work are used to compute an upper bound

for a punch with circular cross section.

C. B„ Tompkins and T. H. Southard have done some work on development of a routine for SWAC which will solve a 2-person, zero-sum, rectangular game whose value is zero and whose typical matrix element is A +VB

acL, where V is to be determined and -1<B <0. Such a game has applications

in quantitative studies of warfare ana elsewhere.

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

statistics from a normal population," by J. B. Rosser; submitted to a technical journal. (2) "Note on the circle theorem of hydrodynamics,"

by E. Levin; accepted by Quarterly of Applied Mathematics. (3) "Bounds

for punch indentation problems," by E. Levin; being submitted to a technical journal. (4) "Summary of January 7 numerical analysis seminar,"
6 Status of Projects

by G. E, Forsythe; multilithed typescript, 11 p. (5) "Summary of March

25 numerical analysis seminar," by G, E. Forsythe; multilithed typescript, 2 p.

MISCELLANEOUS STUDIES IN THEORETICAL PHYSICS

Task 1101-10-5100/51-5

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

Sponsor: " " Manager: R. D. Woods

Full task description appears in July-Sept 1950 issue, p. 19«

Status: CONTINUED. W. Futterman is concluding theoretical and

numerical work on calculations of the photo-disintegration of the deuteron.

R. Woods is continuing computations of the elastic scattering of protons

by various elements, based on the optical model of the nucleus.

Publications: (l) "Application of variational methods to intermediate and high energy scattering," by E. Gerjuoy and D. S. Saxon; ac- cepted by Physical Review. (2) "Variational principles for the acoustic

field," by E. Gerjuoy and D. S. Saxon; submitted to a technical journal.

STUDY OF RUSSIAN MATHEMATICAL PROGRESS

Task 1101-10-5100/52-1

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, p. 11.

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 following publication

is available in limited quantities: "Bibliographical Surveys of Russian

Mathematical Monographs, 1930 to 1951," an NBS report. A supplement to

the above is also available.

RESEARCH IN THE MATHEMATICAL THEORY OF PROGRAM PLANNING

Task 1101-10-5102/50-11

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

Sponsor: " " Manager: T. S. Motzkin

Full task description appears in Apr-June 1950 issue, p. 12.

Status: CONTINUED. In the assignment problem, various ways of

preconditioning were examined. Under certain conditions on the elements,

the expected value of a pseudo-maximum obtained by either of the two or- dinarily used first guesses was evaluated. As a simplest example, in an

n by n matrix whose elements are zero and one with independent equal

probability, the expected value of a sum obtained by first choosing a greatest element, then in its minor a greatest element, etc., is n-e
Status of Projects 7

with e f ,57 85 .*, . The question was studied when and how gradually

more accurate and elaborate methods should be used.

In the more general equidistribution problem a new characterization of the vertices of the pertinent polyhedron was given, and adjacent vertices were determined in the case where all nonzeros in a vertex are equal. Polyhedra belonging to other generalizations of the as- signment problem and to related problems of applicational interest were

also studied,

A code of D, H. Lehmer for finding a pseudo-maximum for an n by

n assignment problem with n<l6 by examining random permutations was run on the SWAC,

/ Reference is made to a theorem proved bv I. Heller and C. Tompkins

and reported under task 1 101 -10-51 00A-9-AE2 (p. 1) which has immediate

application to some transportation problems. It generalizes a theorem

first used in this connection by G, B. Dantzig.

Publications: (See also July-Sept 1953 issue, p, 9-10.) "The

assignment problem," by T. S, Motzkin (Cowles Commission Discussion Paper

Mathematics No. ^25); to appear in Proceedings of the American Mathematical Society Sixth Symposium for Applied Mathematics, held at Santa Monica

City College, August 1953.

SCAMP

Task 1101-10-5150/53-1

Origins Office of Naval Research, USN Authorized 6/10/53

Sponsor: " " Managers C, B. Tompkins

Full task description appears in Apr-June 1953 issue, p. 8.

Status: CONTINUED, The research concerning problems of discrete

variables has continued. Studies of automatic methods for removing

isomorphs have proceeded. An early paper by H, S, White, F, L, Cole, and

L, D. Cummings (Memoirs of the National Academy of Sciences, 1919) has

served as s guide in some of this work relating to Steiner triples,

A search for a pair of orthogonal Latin Squares of order 10 has

been reduced to a code, and no orthogonal squares have been found,

A report involving a problem of exhaustive search for solutions

of non-linear equations involving permutations has been submitted to the

sponsor.

Plans are continuing for the active program during the summer

of 195^.

F, B, Meek is coding a reduction of integral quadratic forms to

the so-called normal forms of Hermite,

M, Newman completed the reduction to Hermite normal form of

two matrices of orders 57 and 73? respectively, (cf, task 1 101-10-51 50/53-1 )

.

Participants in this task included F, B, Meek, J, D, Swift,

C, B, Tompkins.
8 Status of Projects

2 . Mxjiltim&ML

NATIONAL BUREAU OF STANDARDS WESTERN AUTOMATIC COMPUTER (SWAC)

Task 1101-20-5103A9-1

Origin: Aeronautical Research Laboratory, Authorized 11/1A-8

Wright Air Development Center, Air Research

and Development Command, USAF

Sponsor: M " " M Manager: H. D. Huskey

Full task description appears in Apr-June 19*+9 issue, p. 27.

Status: CONTINUED, SWAC has continued to be used on a regularly

scheduled two shift basis on both service and research problems. However, members of the University staff have occasionally operated the

machine after the end of the regular shift at midnight and on weekends,

so that since the end of January the SWAC has been used for an average

of 1.11 hours per week. During the quarter the machine was turned on for

a total of 1152 hours, 789 of which was productive computing. This is

about 68°/o of the total time. Slightly over 31 hours was unscheduled

maintenance.

During the quarter the Flexowriter input unit was replaced by

a hand keyboard on the console. This simplified the input circuitry and

removed from the computer the crystal input matrix, so that there should

be a decrease in maintenance.

STUDIES IN THE THEORY OF DIGITAL COMPUTING MACHINES

Task 1101-20-5103/53-1

Origin: NBS Authorized 9/30/52

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, Air Research and Development

Command , USAF

Manager: H. D. Huskey

Full task description appears in July-Sept 1952 issue, p. l6\

Status: CONTINUED, During this quarter there has been an effort

to consolidate information on all the various codes that are in existence

for running SWAC, In this respect a new coding manual is in the process

of preparation.

The floating point card control method of operating SWAC has

been extended to give the possibility of using repetitive routines stored

on the drum. Elementary automatic programming of formulas has been continued and the interpretative routines have been adapted for use with

floating SWACPEC, This makes it possible to automatically program a simple problem and to immediately feed in the data and obtain results

with no preliminary scaling and with a minimum of attention to the coding

process. Test routines and input and output routines have been revised

during the quarter to make the operation more efficient.
Status of Projects 9

3* Mathematical Services

COMPUTING SERVICES FOR RESEARCH STAFF OF

THE INSTITUTE FOR NUMERICAL ANALYSIS

Task 1101 -40- 51 30/5^1

Origin: NBS Authorized 9/2/48

Sponsor: Office of Naval Research, USN Revised 11/16/49

Manager: F. Hollander

Full task description appears in July-Sept 1949 issue, p» 25.

Status: CONTINUED. Research staff problems involving the use of

SWAG were as follows:

• (l) For C. B. Tompkins: (a) Studies were continued concerning

numerical embedding of surfaces. A further investigation was made of some inequalities generated in a study of organizations, and of some discrete problems. (b) A search for two orthogonal latin squares of

order 10 was continued, in connection with SCAMP , according to a code by

F. B. Meek. No pairs were found.

(2) For W. Futterman: (a) Matrix elements occurring in the problem

of the photo-electric effect for the deuteron have been computed for a central Yukawa potential up to octupole interaction. This assumed ordinary

as well as Serber force. (b) Dipole integrals for a tensor force ground

state were also calculated,

(3) For R. D. Woods: Coding was checked and some exploratory computations were made of the scattering of protons by various metals, using

an optical model for the nucleus,

(4) For the Physical Chemistry Group of the Department of Chemistry,

UCLA: The physical chemistry group carried out a series of calculations

on the SWAC applied to crystallographic structure research during the past

quarter. A crystallographic interpretation routine, SWAXTL, was coded

and first employed for structure factor calculations. These calculations

predict the intensity of an X-ray beam reflected by a plane of atoms in a given crystal. This calculation has been applied to four unique crystal

structures. A three-dimensional Fourier series calculation of the electron

density for a given crystal has been coded. It is applicable to all crystallographic space groups and has been used in calculations for four

different crystals, each with a different one of the 230 possible space

groups.

(5) For M. R, Hestenes and E. Mookini: Experiments were continued

in the conjugate gradient method for solving systems of linear equations.

Several routines involving triple precision inner products were tested on 6x6 and 8x8 systems of linear equations. For well-conditioned matrices

the results were very good. Experiments are now being carried out on ill-conditioned matrices.

(6) For G, Forsythe and L. Straus: (a) Coding was completed and

experiments made on the problem of obtaining the characteristic polynomial

of a real matrix up to order 14 by Frame’s method. Details are reported

under project 1 101-10-5100/50-3, p. 2. (b) Coding was checked, and test

cases run on the problem of improving an approximate inverse of a real

matrix, up to order 32, by the Newton-Schulz iteration: X ^ =X

Q (2I-AX^)

.

(See project 1 101 -10-51 00/49-AE2 (p. l) for further details.)

(7) For the Geophysics Department, UCLA: Calculations were continued and new codes checked out, in the problem of the high order tidal

constituents for earth and ocean tides.

(8) For the School of Business Administration, UCLA: (a) The code

was completed and checked, and some runs were made on SWAC on the problem
1

0

Status oi Projects

of the Gambling Game. The routine derives frequency distribution of the

duration of play to the basic probability game, and moments to basic

probability game. (b) A basic routine for the probability game in the

problem of the Business Game has been written but not completely checked

out on SWAC.

(9) For D. H. Lehmer and J. Selfridge: (a) The program to find

penultimate remainders of the known Mersenne primes was completed. (b)

The program for finding quadratic residues which are not squares was continued on a standby basis. This routine uses only the high speed

memory and the output typewriter. (c) Testing primes for irregularity

was completed to 2500. Kummer's criterion was checked for each irregular

found and the result is that Fermat’s last theorem has no counter-examples

for exponents less than 2500. (d) The cyclotomic numbers of order 16

were computed for all primes less than 1000, and a general cyclotomic

number routine using the drum was code checked. (e) A code was checked

out to attack the assignment problem by a Monte Carlo method. (f) A code

is checked out and ready to run which will settle the Ideal Waring Theorem

fjr exponents up to 250,000. (g) The following is a result of calculations performed: No Fermat number whose character is unknown has a divisor less than 2-^ , (10) For the Physics Department, UCLA: Computation of a proposed

wave function for the Li„ molecule using three exponential variational

parameters is proceeding. The coding for the binding energy has been

completed using the floating SWACPEC coding scheme of M. Melkanoff. Preliminary runs have been made.

(11) For the Education Department, UCLA: Detailed "item analysis"

of psychometric instruments, based on varying methods of identifying

criterion groups, has been accomplished. Scoring keys resulting from

these analyses have been developed, and differential methods of weighting

to arrive at final scores are being studied.

PUNCHED CARD LIBRARY

Task 1101-40-51 31 A9-2

Origin: NBS Authorized 7/14/48

Sponsor: Aeronautical Research Laboratory, Wright Air

Develpment Center, Air Research and Development

Command, USAF

Manager: F, Hollander

Full task description appears in Apr-June 1949 issue, p. 26 . 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, 4o5 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. A request was received from the University of

Utah for tables of sines and cosines (sin x, cos x, A, 62, x=0(.000l)l,

11D). Boeing Airplane Company requested tables of Bessel functions: Yn (x),

Y.(x), YqAx), Y. (ix) , x=0(. 01)10, 10D; and J

Q (x), x real, x=0(. 01)10,

10D. These were reproduced from our files and shipped.

A table of keypunched Bessel functions of fractional order was received from the U. S. Naval Air Missile Test Center, Point Mugu, California, and added to the punched card library. This table contains all

functions listed in the NBS "Table of Bessel Functions of Fractional

Order," vol. 1 (Columbia University Press, New York, 1948). An erratum
Status of Projects 11 was noted in the table on page 139 for v = - r, x = 20.840: the correct

6 2 is 12305) not 12299 as listed. This table was checked for accuracy

after keypunching by the staff at the Center.

SIMPLIFIED ROLLING PULLOUT EQUATIONS

Task 1101-40-5131/51-3^

Origin: Cornell Aeronautical Laboratory Authorized 6/22/51

Sponsor: Aeronautical Research Laboratory) Wright Air

Development Center, ARDC, USAF

Manager: F. Hollander

Full task description appears in Apr-June 1951 issue, p. 26.

Status: CONTINUED, The sets of data reported in the last issue

have .been used to compute the solutions of the remaining five conditions.

This essentially completes the computation on this task. A final report

remains to be written. A report describing the equations and the method

of solution has been written by E. C. Yowell.

LOW MOMENTS OF ORDER STATISTICS

Task 1101-40-5131/51-36

Origin: University of Oregon Authorized 6/22/51

Sponsor: Office of Naval Research, USN

Manager: H. D. Huskey

Full task description appears in Apr-June 1951 issue, p, 27.

Status: CONTINUED. The coding of certain routines that can be used

in this problem has been done in task 1 101-40-51 3l/5^-8, p. 16.

SIERRA WAVE PROJECT

Task 1101-40-5131/52-36

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

Sponsor: Aeronautical Research Laboratory, Wright Air Development

Center, Air Research and Development Command, USAF

Manager: T. H. Southard

Full task description appears in Jan-Mar 1952 issue, p. 33.

Status: CONTINUED. Theodolite data for several more flights have

been reduced on SWAC.
12 Status of Projects

EIGENVALUES

Task 1101-40-5131/53-6

Origin: Consolidated Vultee Aircraft Corporation Authorized 9/26/52

(Convair)

Sponsor: Aeronautical Research Laboratory, Wright Air Development

Center, Air Research and Development Command, USAF

Manager: F. Hollander

Full task description appears in July-Sept 1952 issue, p, 28.

Status: INACTIVE.

METEOROLOGICAL MEANS

Task 1101-40-5131/53-10

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

Sponsor: Aeronautical Research Laboratory, Wright Air Development

Center, Air Research and Development Command, USAF

Manager: F, H, Hollander

Full task description appears in July-Sept 1952 issue, p. 29. (See also

project 1 1 . 1/31-50-1 7» Jan-Mar 1950 issue, p. 15.)

Status: CONTINUED, Computations were suspended during January

and February. Calculations were resumed during March on the "Mountain

Term", No further work was accomplished on the 100 mb level for JanFeb 1949.

DISCRIMINANT FUNCTIONS

Task 1101-40-5131/53-18

Origin: School of Aviation Medicine, Randolph Field, Authorized 12/15/52

USAF

Manager: D. Teichroew

Full task description appears in Oct-Dec 1952 issue, p. 27.

Status: CONTINUED. Preliminary results are being examined. As

a reult of the preliminary computations the code is being modified to

permit larger values of the parameters.

BIO-ASSAY PROBLEM

Task 1101-40-5131/53-24

Origin: Stanford University Authorized 3/31/53

Sponsor: Office of Naval Research, USN

Manager: D. Teichroew

Full task description appears in Jan-Mar 1953 issue, p. 22.

Status: INACTIVE. Results are being examined by the contractor.
Status of Projects 13 - DISCRETE MINIMAL SPACES

Task 11 01 -VO-51 31/53-26

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

Sponsor: Evans Signal Laboratories, Army Signal Corps

Manager: H, D. Huskey

Full task description appears in Jan-Mar 1953 issue, p. 23.

Status: INACTIVE. For status to date see Apr-June 1953 issue, p. 23.

LINEAR EQUATIONS (CONVAIR)

Task 11 01 -VO-51 31/53-35

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

Sponsor: Bureau of Aeronautics, USN

Manager: F. Hollander

Full task description appears in Jan-Mar 1953 issue, p. 28 .

' Status: INACTIVE. For status to date, see Qct-Dec 1953 issue,

P. 16.

MONTE CARLO STUDIES

Task 11 01 -VO-51 31/53-39

Origin: RAND Corporation Authorized 3/31/53

Sponsor: Aeronautical Research Laboratory, Wright Terminated 3/31/5V

Air Development Center, Air Research and Development

Command, USAF

Manager: H. D. Huskey

Objective: To study the Ising problem and order-disorder phenomena

using the Monte Carlo method.

Comments: This task originates with RAND’s Air Force contract.

RAND is coding the problem for SWAC, and the NBS will furnish SWAG time

and such operator and consulting time as is required.

Status: TERMINATED.

RANKING PROBABILITIES

Task 1 101-V0-5l31/53-V0

Origin: Cornell University Authorized 3/31/53

Sponsor: Office of Naval Research Completed 3/31/5V

Manager: D. Teichroew

Objective: To compute a table of the function

P(h I r , k) = r f [F(x + h)] k r

[ 1 - F(x)] r 1 £(x)dx

J -oo where
14 Status of Projects

F(x) *

e dt and f(x) = ^ e-^*2

J -oo for k=4(l)l0, r=l(l)[^] where [ m] is the largest integer less than or equal to m, and h= 0(.0l)S

r ,k where S

r ,k is the smallest two-decimal-place

number for which P(S

p ,k

| r,k)>.99. Certain related tables will be obtained by inverse interpolation in the P(h I r,k) table.

Background: The P(h I r,k) integral arises in the study of power

functions of tests based on order statistics in samples from a normal

distribution. The present computations are designed as an aid in applying the theory developed by R. Bechhofer. "The probability of a correct ranking (Preliminary Report)," Annals of Mathematical Statistics,

1 39(Mar. 1952).

Status: COMPLETED. The final report below has been submitted for

publication.

Publication: "A table giving a probability associated with order

statistics in samples from two normal populations which have the same variance but different means," by D. Teichroew; IN MANUSCRIPT.

ACCEPTANCE TESTS

Task 1 101-40-51 31/53-^1

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

Institute of Technology

Sponsor: Ordnance Corps, U. S. Army

Manager: H. D. Huskey

Full task description appears in Apr-June 1953 issue, p. 27.

Status: CONTINUED. Miss L. Forthal of the Institute and Mr. Carr

of the Jet Propulsion Laboratory have completed the coding of the test

procedure and are checking it. At the present time delivery of the computer to the JPL has been delayed and it is expected that the test routines

will be run during the next quarter.

B.P.A. Studies

Task 1101-40-51 31/53-^2

% Origin: Bonneville Power Administration Authorized 6/29/53

Sponsor: " " Manager: R. R. Reynolds

Full task description appears in Apr-June 1953 issue, p. 27.

Status: INACTIVE. For status to date, see Oct-Dec 1953 issue,

P. 17.
Status of Projects 15

HELICOPTER STABILITY STUDIES

Task 1101-40-5131/53-44

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

Sponsor: Aeronautical Research Laboratory, Wright Air Development

Center, Air Research and Development Command, USAF

Manager: G. E. Forsythe

Full task description appears in July-Sept 1953 issue, p, 23.

Status: CONTINUED, The main effort has been to obtain two polynomials in A of type det(AF-t-G), which govern the stability of a helicopter

control system. The approach has been to calculate AfGr1 F very carefully,

and then to try to get the characteristic polynomial of A by one of the

two methods currently reported under task 1 101- 10-5100/50-3, p. 2. The

approach was successful for one of the two polynomials. The other proved

too badly conditioned, however, and has not been evaluated.

A second series of problems, representing a different approach

to a similar stability problem, requires the inversion of some forty-odd

real matrices of order l6. The input cards are all prepared, and running

on SWAC will commence soon.

COMPUTATIONS FOR SCAMP

Task 1101-40-5131/5^-1

Origin: Office of Naval Research, USN Authorized 9/29/53

Sponsor: Office of Naval Research, USN

Manager: C. B, Tompkins

Full task description appears in July-Sept 1953 issue, p. 24.

Status: CONTINUED, The computations carried out during the period

concerned a search for orthogonal Latin Squares of order 10 and continued

computation of Steiner triples of order 15.

CRUISE CONTROLLER

Task 1101-40-5131/54-3

Origin: J. B. Rea Company Authorized 9/29/5,3

Sponsor: Aeronautical Research Laboratory, Wright Terminated 3/31/54

Air Development Center, ARDC , USAF

Manager: F. H. Hollander

Objective: A cruise controller is an instrument which is designed

to determine the appropriate throttle setting to obtain the maximum miles

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

input which results in the smallest hunting period.

Comments: The originator is analyzing this problem to determine

its appropriate mathematical form.

Status: TERMINATED
16 Status of Projects

PARACHUTE DATA ANALYSIS

Task 1101-40-5131/54-7

Origin: U. S. Naval Auxiliary Air Station, Authorized 2/25/54

El Centro, Calif.

Sponsor: U. S. Navy

Manager: D. Teichroew

Objective: To provide services to assist the Navy Parachute Unit

by conducting a study designed to recommend statistical procedures and

modern computing techniques applicable to the parachute test activity.

Status: NEW. A report containing a statistical analysis of several

experiments has been submitted to the contractor.

GALACTIC DISTRIBUTION STUDIES

Task 1101-40-5131/54-8

Origin: Statistical Laboratory, University of Authorized 2/25/54

California, Berkeley

Manager: B. F. Handy, Jr.

Objective: Given parameters: r,s, t,0(a,b)

A

<r,/S,e

(r ’ s,t) =

00 00 00 U

P xljjx^x^OCc’x.j )0(fl’X2)©(tf,x^)e ndx

1 dx2dx^

Jo Jo Jo

12 2 2 where h = + x

2 + x^ - x.jx 2 - (x^+x2

) x-^ cos 2/# ]

yf -i 2

and 0(y) = f e du

J-oo

with w=a-5log-j 0

y - 5.l5x10~^ylb .

Background: This computation originated as a result of studies

being carried on at the Lick Observatory, University of California, Berkeley, in conjunction with the Statistical Laboratory, University of California, Berkeley, to find a mathematical model of the distribution of

galaxies in a theoretically non-expanding universe. By comparing the computed results with observed phenomena it is hoped to be able to make certain statistical observations concerning the make-up of the universe. As

a result of the present computations certain changes in the statistical

model can be made and new computations performed.

Status: NEW. A code, written by S. Marks, for evaluating

to 30 decimal places, for x= -12(.02)0 was checked out, and the table

computed. Methods of interpolation in this table are being studied, and

the code for computation of 0(y) is being written.
Status of Projects 17

MATRIX INVERSIONS

Task ItOI -40-5131/54-9

Origin: Northrop Aircraft, Inc, Authorized 2/25/54

Manager: F, H, Hollander

Objective: To invert matrices up to order 32, using an existing

floating point routine.

Background: The originator occasionally will submit for solution

sets' of equations which arise in aircraft design. Results are to be

transmitted as obtained.

Status: NEW. The inverse of a 15x15 matrix was calculated using

a routine coded by J . Pettit of the UCLA Department of Geophysics,

SKEW RAYS

Task 1101-40-5131/54-10

Origin? Pacific Optical Company Authorized 2/25/54

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, Air Research and Development

Command , USAF

Manager: F. H, Hollander

Objective: To trace bundles of skew rays through a centered lens

system using the equations developed by D. Feder in J 0 Opt, Soc, Amer,

630-635 (Sept. 195l). The code will trace a pattern of rays over

half of the entrance pupil. Vignetting will be cpmputed as part of the

results.

Background: A centered lens system with 12 spherical air to glass

surfaces was designed. Skew ray traces are needed to determine improvements to the system. The development of formulae suited to automatic

computing systems makes the determination of skew rays a relatively

simple problem.

Status: NEW, The coding has been completed for the tracing of

skew rays through a system of six components, with 12 sir-to-glass surface

Data ere to be supplied as soon as possible, so that the computations

can be started.
II. CQMPUTATION~TABORATORY

(Section 11.2)

1 . Rgjjg.ar.ch

RESEARCH IN CLASSICAL NUMERICAL ANALYSIS

Task 1 102-10-1 10V50-1

Origins NBS Authorized 1/1/50

Managers? J. Todd, M. Abramowitz, H. A. Antosiewicz

Full task description appears in Jan-Mar 1950 issue, p. 21.

Status? CONTINUED. M. Abramowitz and W. F. Cahill made a study

of the application of the converging factor to the asymptotic expansion

of the Coulomb wave functions,

H, A. Antosiewicz completed his investigation of the boundedness of every solution of equations of the type

» • 9 0 x + p(x,x) x + h(x) = e(t)

with integrable forcing term e(t). (See publication (l> below.)

M. Abramowitz and P, Rabinowitz have prepared a paper on the

Coulomb wave function along the transition line. (See publication (2)

below.)

In connection with the forthcoming table of E 1 (z) (task 1102-10-

1 10^A3-3i P» 2?)? J® Todd prepared a paper on the use 6t the Laguerre

approximate quadrature formula to evaluate (z) in the complex domain,

either for large values of the arguments where the function is not tabulated, or within the table instead of using a two-dimensional interpolation.

This method is more efficient than the use of the asymptotic series.

A seminar in Numerical Analysis held in conjunction with American University, is being conducted by J. Todd. The general topic is

stability considerations in connection with the numerical solution of

differential equations. Among those who have participated to date are

M. Abramowitz, F. L. Alt, S. Corn (Ballistics Research Laboratory),

P. Rabinowitz, and J. Todd.

Publications: (l) n0n nonlinear differential equations of the

second order with integrable forcing term'* t by H. A, Antosiewicz; submitted to a technical journal. (2) ’’Coulomb wave functions along the

transition line," by M, Abramowitz and P. Rabinowitz; IN MANUSCRIPT.

(3) "Computation of vibration modes and frequencies on SEAC," by W, Cahill

and S. Levy; IN MANUSCRIPT. (4) "On a problem in the theory of mechanical

Quadratures," by P. Davis; accepted by the Pacific Journal of Mathematics,

(5) "On the estimation of quadrature errors for analytic functions," by

P. Davis and P. Rabinowitz; submitted to a technical journal. (6) "The

practical evaluation of integrals," by M, Abramowitz; submitted to a technical journal. (7) "The evaluation of the exponential integral for

large complex arguments," J. Todd; to appear in NBS Journal of Research,

(8) "Evaluation of thej0 exp(-u2-(x/u) ) du , " by M. Abramowitz; accepted

18
Status of Projects

19

by the Journal of Mathematics and Physics. (9) "Regular and irregular

Coulomb wave functions expressed in terms of Bessel-Clif f ord functions,"

by M. Abramowitz; accepted by the Journal of Mathematics and Physics.

(10) "Asymptotic solution of linear differential equations with a parameter

by H. Antosiewicz; submitted to a technical journal. (11) "On the differential equation x + k(f(x) + g(x)x)x = ke(t)," by H. Antosiewicz; submitted to a technical journal. (12) "On a certain integral involving

Bessel functions," by H. Antosiewicz; submitted to a technical journal.

(13) "Some implications of Liapunov's conditions for stability," by

H. Antosiewicz and P. Davis; submitted to a technical journal.

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

Origin: NBS Authorized 3/1/50

Manager: P. Davis

Full task description appears in Jan-Mar 1950 issue, p. 22.

Status: INACTIVE. For status to date, see Jan-Mar 1952 issue, p. 35.

MISCELLANEOUS STUDIES IN PURE MATHEMATICS

Task 1102-10-1104/50-4

Origin: NBS Authorized 1/1/50

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

Full task description appears in Jan-Mar 1950 issue, p. 23.

Status: CONTINUED. 0. Taussky-Todd continued her study of the

significance of normal matrices in certain aspects of algebraic number

theory; in particular numerical examples in the case of cubic fields

(normal or not normal) were investigated.

M. Newman completed two papers entitled "A conjecture of Erdos"

and "Note on a certain determinant". In the first it is shown that for a large class of pairs m,n the diophantine equation x^y0 =1 has only a finite number of rational integral solutions x,y. Later 0. Taussky-Todd

showed how to establish the result for all pairs m,nj further generalizations are under study by M. Newman. In the second paper a determinant

related to the Vandermonde determinant is evaluated.

In this period a seminar in algebra, held in conjunction with

the American University and directed by 0. Taussky-Todd, was concluded.

The general topic was computational problems in algebraic number theory

and the speakers included N. C, Ankony (Johns Hopkins University),,

B, V. Dean (Natio al Security Agency). K, Goldberg, A. J. Hoffman, M;Newman

J. Todd (National Bureau of Standards).

K. Fan continued work on topology. In particular he completed

a paper with R. A. Stuble (Illinois Institute of Technology) on "Continuity

in terms of connectedness" and is working on another with F. Wagner concerning "Compactif ication of completely regular spaces".

We report here, some of the minor consulting jobs carried out

by P. Henrici for other workers in NBS. Similar activities have been

carried out by other members of the Computation Laboratory staff. The

purpose of this record is to indicate types of problems arising elsewhere

in NBS that can be solved painlessly by CL staff members.
20 Status of Projects

1 . Evaluation of

k m 2T a^Tm-aT* + qr)(N-m-qr)

m=1 q=0 q * vm q; *

2. Estimation of error caused by truncating the range in the integral

oo J—~

S- j\y

sLy. 0 [y Y

Q

(y) - nY^y)] 2 + [y J

Q

(y) - nJ^y)] 2 y

3.

Estimation of integrals of the form

F(p) =

PCD

sin px f(x)dx,

where f(x) > 0, f(x) monotonically decreasing.

4,

Evaluation of the integral

CD

J a

'OO

J-00

-/3(x

2

+y

2

) (x

2 + 2\2

y ) dx dy

5. Asymptotic evaluations of the solution of the differential equation

= q( ae_q - 1 ) (a > 1

)

q(0) = qQ

(0 < qQ

< log a)

6. Determination of the complete solution of a system of four linear

differential equations of the first order with constant coefficients.

oo / p ^

\ 2

7. Reduction of 2Z

^ n to - functions.

n=0

Publications: Cl) "Some computational problems in algebraic

number theory," by 0. Taussky; submitted for inclusion in the Proceedings of the American Mathematical Society's Sixth Symposium on Applied

Mathematics, held at Santa Monica, California, August 1953. (2) "The

number of absolute points of a correlation," by A. Hoffman, M. Newman,

E. Straus, 0. Taussky; IN MANUSCRIPT. (3) "The representation of integers by binary quadratic rational forms," by K. Goldberg, M. Newman,

E. Straus, and D. Swift; accepted by Archiv der Mathematik. (4) "Ueber

die Funktionen von Gegenbauer," by P. Henrici; accepted by Archiv der

Mathematik. (5) M0n certain series expansions involving Whittaker functions and Jacobi polynomials," by P. Henrici; accepted by Pacific Journal

of Mathematics. (6) "Linear functional equations and interpolation

series," by P. Davis; accepted for publication in the Pacific Journal

of Mathematics. (7) "On generating functions of the Jacobi polynomials,"

by P. Henrici; IN MANUSCRIPT. ( 8 ) "A conjecture of Erdos" by M. Newman;

submitted to a technical journal. (9) "Note on a certain determinant,"
Status of Projects 21

by M, Newman; IN MANUSCRIPT, ( 1 0 ) "Continuity in terms of connectedness,"

by K. Fan and R. A, Stuble; IN MANUSCRIPT. ( 1 1 ) "Compac tif ication of

completely regular spaces," by K. Fan and F. Wagner; IN MANUSCRIPT.

(12; "Some L2 aspects of Faber polynomials," by P. Davis and H. Pollack;

accepted by the Duke Mathematical Journal. (13) "On representations and

extensions of bounded linear functionals defined on classes of analytic

functions," by P. Davis and J. L. Walsh; accepted by the Transactions of

the American Mathematical Society. (1^) "Structure theorems for modular

subgroups," by M. Newman; submitted to a technical journal.

NUMBER THEORETICAL TEST PROBLEMS FOR SEAC

Task 1 102-10-1 10V50-5a

Origin: NBS Authorized 6/1/50

Managers: 0. Taus sky-Todd , K, Goldberg

Full task description appears in Apr-June 1950 issue, p. 31.

Status: CONTINUED. The routine to determine primes p < 100,000

for which 2P-1 -1 is divisible by p

2 is being run as time is available

and is now extended to p<83, LK)0 with no positive results other than the

well-known cases p=1093 and p=35l1. _

The routine to compute ji(mod p) , where (t + uv

/ p)/2 is the

fundamental unit in the field R(i/p), p a prime, has been run for all primes

less_than 18,000 and for all primes of the form ^n+1<100,000. No cases

of u=0 (mod p) have been found.

The work previously reported on consecutive !-th power residues

(for April-June 1950 and January-March 1951) has recently been successfully

used by H. S. Vandiver. Further extension of this work is desirable.

SOLUTION OF LAPLACE EQUATION BY MONTE CARLO METHOD

Task 1102-10-1 10^/51-6

Origin: NBS Authorized 9/28/50

Manager: M. Abramowitz

Full task description appears in July-oept 1950 issue, p. 36.

Status: CONTINUED, A study is in progress of new methods for

generating pseudo-random numbers.

' We report here a study begun by I. A. Stegun at the request of

W. J. Youden. Let x.| ,Xp, . . . , x

n be a sample of n observations from a normal population. Let x=(x..+Xp+. . ,+x )/n. Determine the probabilities

p^ of the events x^<x<x^+ ^

. The following results have been obtained by

taking 8000 observations in the case n=4:

P^ = .2665, Pj = .4723, P2 = .2612.

Further work is in progress.
22 Status of Projects

ANALYSIS OF GEOMAGNETIC FIELD

Task 1 102-10-1 104/52-8

Origin: NBS Authorized 8/10/51

Manager: C, J. Swift

Full task description appears in July-Sept 1951 issue, p. 34.

Status: INACTIVE. For status to date see Oct-Dec 1951 issue, p.29.

SPECIAL PROBLEMS IN FINITE MATRIX THEORY

Task 1102-10-1104/52-34

Origin: NBS Authorized 11/6/51

Manager: 0. Taussky-Todd

Full task description appears in Oct-Dec 1951 issue, p. 30.

Status: CONTINUED,

replacement of differential equations such as u =u , „ - t

J. Todd studied the matrices that arise by the „„ „ uxx=u

tt

b y sijaP le

finite difference equations. The characteristic roots of these matrices

can be found by methods similar to those used for equations of the form

u + u =ku.

0. Taussky-Todd began a revision of her bibliography on bounds

for characteristic roots of finite matrices.

K. Fan considered the following problem. For a positive-definite

Hermitian matrix H of order n, it is well known that the inequality

det H< ( 1, 2, . . . , p) (p+1 , p+2, , . . ,n) holds, where (l,2,...,p) denotes the

principal minor of H corresponding to the first p unit vectors e^e^,...^.

A more complete result than this inequality is obtained:

det H

(p+1, p+2, . . . ,n)

is the minimum of p-rowed det ( (Hx^ ,x

j

) ) , when the p vectors x.j ,x.p, . . . ,x , (l^i, j<p), vary under the biorthonormal condition (x^,

e

j )=6^ j (Ki, j<p)

,

Other inequalities concerning positive definite Hermitian matrices are also

obtained, among which we have the inequality

-d-S-t.. H < tj Xis

p/. l j p+ 2 , , n)

(p+1 ,p+2, . . . ,n) x ^ (p+1 ,p+2, . . . ,n)

Publications: (l) "Some remarks on commutators of matrices," by

K, Fan; accepted by Archiv der Mathematik. (2) "Inequalities for eigenvalues of Hermitian matrices," by K. Fan; to appear in "Contributions to

the solutions of systems of linear equations and the determination of

eigenvalues," NBS Applied Mathematics Series 39. (3) "Some metric inequalities in the space of matrices," by K. Fan and A. Hoffman; submitted

to a technical journal. (4) "The conditions of certain matrices, II,"

by J. Todd; accepted by Archiv der Mathematik. (5) "Error bounds for

eigenvalues of symmetric integral equations," by H. Wielandt; submitted

for inclusion in the Proceedings of the American Mathematical Society

Sixth Symposium on Applied Mathematics, held at Santa Monica, California,

August 1953. (6) "Einschliessung von Eigenwerten hermitescher Matrizen

nach dem Abschnitt sverf ahren, " by H. Wielandt; accepted by Archiv der

Mathematik. (7) "Characteristic roots of quaternion matrices," by 0.

Taussky; accepted by Archiv der Mathematik. (8) "On eigenvalues of sums

of normal matrices," by H e Wielandt; submitted to a technical journal.
Status of Projects 23

(9) "Generalized commutators of matrices and permutations of factors in a product of three matrices," by 0. Taussky; to appear in the von Mises Anniversary volume. (10) "The condition of the finite segments of the Hilbert

matrix," by J. Todd; to be included in Contributions to the systems of

linear equations and the determination of eigenvalues, NBS Applied Mathematics Series 39.

DIFFERENTIAL EQUATION FOR NERVE FIBER REACTION

Task 1102-10-1104/53-15

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

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

Full task description appears in Oct-Dec 1952 issue, p. 34.

' Status: INACTIVE.

BAKER-HAUSDORFF FORMULA

Task 1102-10-1104/53-16

Origin: NBS Authorized 12/8/52

Managers: 0, Taussky- Todd , K, Goldberg

Full task description appears in Oct-Dec 1952 issue, p, 35.

Status: CONTINUED. Work on the manuscript, "The equation e z=e

x ,e^

in a free associative ring," by K. Goldberg, continued.

HYPERGEOMETRIC FUNCTIONS

Task 1102-10-1104/53-35

Origin: NBS Authorized 3/25/53

Managers: P. Rabinowitz, W. Cahill

Full task description appears in Jan-Mar 1953 issue, p. 36.

Status: INACTIVE.

Publication: "Programs for computing hypergeometric series," by

W. F. Cahill; MTAC fi, 36-37 (Jan. 1954).

BASIC RESEARCH IN LINEAR PROGRAMMING

Task 1102-10-5116/50-2

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

and Office of the Air Comptroller, USAF

Sponsor: Office of Scientific Research, ARDC, USAF

Managers: J. Todd, A. Hoffman

Full task description appears in Jan-Mar 1950 issue, p. 24.

Status: CONTINUED. Using theorems on convex sets, K. Fan has derived various classical results on Chebyshev approximation. The method
24 Status of Projects

employed is a generalization of Rademacher and Schoenberg's use of Helly's

theorem to attack Chebyshev's problem. He has also studied the intersection properties of a linear manifold of arbitrary dimension with the

intersection of a finite number of closed half-spaces. From these properties, he derives existence theorems (some of which are known results of

Tchernikow and Blumenthal) for systems of linear inequalities in a very

simple way. Both the geometric part and the applications of this study

are continuing.

K. Fan and J. Todd have generalized an inequality of J.B.Chassan,

concerned with the estimation of the position of an object given by lineof-sight observations. The generalization (see publication (5) below) is

based on geometric considerations, and the proof is algebraic, unlike

Chassan's statistical proof.

H. Antosiewicz is studying methods for the systematic simplification and solution of systems of linear inequalities.

A. J. Hoffman has given simple proofs of some theorems of Frobeniu

and Ullman on properties of matrices with non-negative elements. The

theorems of Frobenius are relevant to stability problems in various econometric studies, such as inter-industry analysis.

A seminar in linear programming and related problems was organized by A. J. Hoffman. The first meeting took place on March 31

.

Publications: (1) "A theorem on alternatives for pairs of matrices,"

by H. A. Antosiewiczj submitted to a technical journal. (2) "On a theorem

of Ostrowski and Tausskv," by R. Bellman and A. J. Hoffman; accepted by

Archiv der 'Mathematik, (3) "Lower bounds for rank and location of the

eigenvalues of a matrix," by K. Fan and A. J. Hoffman; to appear in "Contributions to the solution of systems of linear equations and the determination

of eigenvalues," NBS Applied Mathematics Series 39. (4) "Some metric inequalities in the space of matrices," by K. Fan and A. J. Hoffman; submitted

to a technical journal. (5) "A de terminantal inequality," by K. Fan and

J. Todd; IN MANUSCRIPT. (6) "A characterization of normal matrices," by

A. J. Hoffman and 0, Taussky; J. Res. NBS 52? 17~19 (Jan. 1954). (7) "On

the relevance of LeChatelier 1 s principle to linear programming," by

A. J. Hoffman; IN MANUSCRIPT. (8) "An extremum property of sums of eigenvalues," by H. Wielandt; submitted to a technical journal. (9) "On a combinatorial theorem," by A. J. Hoffman; submitted to a technical journal.

COMPRESSIBLE FLOW—METHOD OF ORTHOGONAL AND KERNEL FUNCTIONS

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

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

Air Development Center, USAF, and Harvard

University

Sponsor: Aeronautical Research Laboratory, WADC, USAF

Managers: P. Davis, F. L. Alt

Full task description appears in July-Sept 1951 issue, p. 36 . Status: CONTINUED. A report entitled "Some SEAC computations of

subsonic fluid flows by Bergman's method of integral operators" is in

manuscript. This report describes and discusses some computations that

were made for a boundary value problem corresponding to a univalent flow

of type S. These computations serve to indicate the degree of approximation which can be achieved with the first few terms of the various infinite processes occuring in the theory. The report contains a detailed

description of the boundary value problem solved and extensive discussion

of the numerical computation of the particular solutions and the singular

solutions

.
Status o£ Projects 25

Publication: "A multiple-purpose ortho-normalizing code and its

uses," by P. Davis and P. Rabinowitz; IN MANUSCRIPT.

TABLES OF INTEGRALS INVOLVING THE HIGHER TRANSCENDENTAL FUNCTIONS

Task 1102-10-5116/52-33

Origin: NBS Authorized 10/11/51

Manager: F. Oberhettinger

Full task description appears in July-Sept 1951 issue, p. 37.

Status: CONTINUED. Some additional material for the second Volume

of the tables of integrals, which is in the process of being varityped,

has been furnished. Rechecking and correcting of a number of samples for

this volume turned out to be necessary.

[We note that the first volume of this collection has now appeared: "Tables of Integral Transforms, Vol. I," edited By A. Erdelyi with

W. Magnus, F Oberhettinger, and F. G. Tricomi, has been published by

McGraw-Hill.

]

Publication: "On the Lerch zeta function," by F. Oberhettinger;

submitted to a technical journal.

SUPPORTING RESEARCH IN LINEAR PROGRAMMING

Task 1102-10-5116/5^-9

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

Sponsor: Office of ""Scientific Research, Air Research and

Development Command, USAF

Managers: J. Todd, A. J. Hoffman

Full task description appears in July-Sept 1953 issue, p. 35.

Status: CONTINUED. A rapid method of solving the modification of the

caterer problem described in the Oct-Dec 1953 issue, p. 26 , second paragraph, has been found. The essential device consists of (i) a transformation reducing the number of variables, and (ii) a permutation of the var- iables so that the appearance of the matrix is sufficiently similar to that

of the unmodified caterer problem for the techniques of publication (3)

to apply. Although (ii) is not always possible for the most general problem of this type that could be proposed, it is possible for the specific

case of interest to the sponsor. A noteworthy feature of the algorithmic

solution is that there is no penalty for failure to estimate requirements

in the distant future accurately, provided the correct requirements are known in time.

The construction of an optimal purchase pattern for an item for

which there is limited storage space has been under study. The case in

which requirements and prices are precisely known for the various time

periods has been solved, and work is proceeding on situations in which

less information is available.

The experiment to try to compute the expected value of the value

of a game whose coefficients are only known probabilistically, but will

be known exactly at the time the game is played, has thus far been un- successful. The tools used were various modifications of Brown's method,

and machine trials of a small problem did not converge to the known answer. Trials of the Brown-Koopmans method of solving a linear program

revealed a difficulty when an iterate is on an intersection of two or more
26 Status of Projects

bounding hyperplanes. A modification has been proved valid theoretically,

and will be coded.

The code for performing the simplex method by systematic use of

the inverse of the basic matrix has been completed and is undergoing tests.

Publications* (l) "A remark on the smoothing problem,” by H. A.

Antosiewicz and A. J. Hoffman* an NBS report. (2) "On ’overshoot’ in the

furthest hyperplane method," by R. Bryce, IN MANUSCRIPT. (3) "Linear

programming in bid evaluation," by L. Gainen, D. J. Honig, and B. D.

Stanley; to appear in Logistics Research Quarterly. (4) "A method of

solving the transportation problem," by A, Gleyzal; submitted to a technical journal. (5) "Smooth patterns of production," by A. J. Hoffman and

W. Jacobs; an NBS report. (o) "On the optimal ordering of items for a twostage process," by A. J. Hoffman, IN MANUSCRIPT. (7) "On block relaxation,"

by L. S. Joel; IN MANUSCRIPT.

RESEARCH IN MECHANICS OF CONTINUA

Task 1102-10-51 16/5^-23

Origin* NBS Authorized 12/29/53

Sponsor: Office of Naval Research, USN

Manager: R. F. Dressier

Full task description appears in Oct-Dec 1953 issue, p. 27.

Status: CONTINUED. The study of resistive action and cross-section

variations on shock tube flow for a purely mechanical model has been

completed by R. F. Dressier, and a paper submitted for publication. Exact

results for the wave front have been compared with first order effects for

the full wave. These first order effects are obtained analytically by

solving a mixed-type boundary value problem for a pair of linear partial

differential equations with variable coefficients. Exponential ducts

create an antisymmetrical effect on the sound speed; the resistance raises

densities in the back wave and lowers densities in the forward portion.

An analysis involving the full thermodynamical effects on a dissipative aerodynamic flow has been initiated by R, F. Dressier. This

study involves the energy equation with variable entropy as well as modified momentum and continuity equations. Using the Riemann relations

along the three Mach line families, it has been found possible to solve

explicitly for first order effects on entropy, velocity, density, and

temperature for the back portion of a rarefaction wave. Research is

almost completed on this problem and a manuscript will be prepared next

quarter, "Entropy and temperature changes in rarefaction waves."

An improved method for obtaining higher approximations in the

elastic problem of helical springs has been worked out by P. Henrici.

A revised manuscript incorporating these new results is now being prepared.

Results represent an improvement over the recent work of Freiberger on this problem.

A study has been started by M. Abramowitz and H.A. Antosiewicz

concerning the behavior of solutions to a type of second order differential

equation y"+0(x)y=O connected with the mechanics of particles in a Coulomb

field of force. The case of interest concerns the solutions in the neighborhood of a point Xq where the sign of 0(x) changes. Some representations

for these solutions nave already been obtained in terms of Airy integrals.

Two problems in heat conduction proposed by P. Haas ( 1

Lj- . 8) have

been solved by F. Oberhettinger . These, and similar requests, have suggested that a compilation of Green’s function for the heat equation,

the wave equation, and for the potential equation, for various regions be

made. This has been started by F, Oberhettinger.
Status of Projects 2

7

Publications: (l) "Turbulent flow in shock tubes of varying crosssection," by R. F. Dressier; to appear in the NBS Journal of Research.

(2) "Helical elastic springs of finite cross-section," by P. Henrici;

IN MANUSCRIPT.

WATER WAVES

Task 1.102-10-5116/53-5^

Origin: NBS Authorized 6/1/53

Sponsor: Office of Naval Research, USN

Manager: P. Davis

Full task description appears in Apr-June 1953 issue, p. 38.

Status: INACTIVE. For status to date, see Oct-Dec 1953 issue,

p. 25.

2 . Mathematical Tables and Experimental Computations

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

Task 11 02-1 0-1 104A3-3

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

Manager: I. A. Stegun

Full task description appears in Apr-June 1949 issue, p. 4l

.

Status: INACTIVE. For status to date, see Oct-Dec 1953 issue,

P. 28.

TABLE OF THE GAMMA FUNCTIONS FOR COMPLEX ARGUMENTS

Task 1102-10-1 104A6-1

Origin: NBS Authorized 7/1/47

Manager: J. Todd

Full task description appears in Apr-June 1949 issue, p. 43.

Status: CONTINUED, The volume is in press.

Publication: "Table of the gamma function for complex arguments,"

NBS Applied Mathematics Series 34; in press, U. S. Government Printing

Office

.
28 Status of Projects

TABLES OF COULOMB WAVE FUNCTIONS

Task 11 02-1 0-1104A7-

2

Origin: NBS Authorized 7/1 A

7

Managers: M. Abramowitz, P. Rabinowitz

Full task description appears in Apr-June 1949 issue, p. 45.

Status: CONTINUED. Another short table has been issued as an NBS

report. Computations were carried out for values at the turning point >>= 2 ^.

TABLE OF LAGRANGIAN COEFFICIENTS

FOR SEXAGESIMAL INTERPOLATION

Task 1 102-10-1 104A8-2

Origin: NBS

Manager: J. Todd

Authorized 5/25A8

Completed 3/31/5*+

Objective: Preparation of a volume of interpolation coefficients

for the argument p at intervals of 1/3600 for 3-

> 4-, 5~ , 6-, 7~point

interpolation and at intervals of 1/60 for 8-, 9~? 10-, 11-point interpolation.

Background: Functions tabulated with arguments in degrees occur

in every branch of applied science. The present tables will simplify and

reduce the labor of interpolation in such tables, whenever the functions

are desired for minutes and seconds.

Status: COMPLETED.

Publication: "Tables of Lagrangian coefficients for sexag

interpolation," NBS Applied Mathematics Series 35 (Jan. 4, 195*+}

Available from U. S. Government Printing Office, Washington 25,

$2 , 00 .

esimal

D. C.,

BIBLIOGRAPHY OF CODING PROCEDURES, MATHEMATICAL TABLES

AND NUMERICAL ANALYSIS

Task 1102-10-1104/50-5

Origin: NBS Authorized 3/1/50

Managers: J. Todd, J. H. Wegstein, P. Rabinowitz Revised 1/9/53

Full task description appears in the Oct-Dec 1952 issue, p. 40.

Status: CONTINUED. Preparation of the bibliographies continues.

The following Technical Memoranda were issued:

44 Subroutine for binary to decimal conversion of a double precision

number with fixed binary point.

45 Subroutine for binary to decimal conversion of a single precision

number with floating binary point.

46 Subroutine for decimal to binary conversion of a single precision

number with floating binary point.

48 Subroutine for the evaluation of polynomials, single precision, fixed

binary point.

49 Automatic transfer routine.
Status of Projects 29

56 Fast square root subroutine, single precision, fixed binary point.

57 Decimal to binary and. binary to decimal conversion subroutines for

integers and fractions.

WAVE FUNCTION FOR LITHIUM

Task 11 02- 10-11 04/50-7 • Origin: NBS - Authorized 6/1/50

Sponsor: Bureau of Ordnance, USN

Manager: W, F. Cahill

Full task description appears in Apr-June 1950 issue, p. 36.

Status: CONTINUED. The iteration for the first approximation to

the wave function is continued as machine time is available. A code to

compute a new approximation to the eigenvalue is being written; this includes a double integration involving the first approximation.

COLLECTED SHORT MATHEMATICAL TABLES OF THE COMPUTATION LABORATORY

Task 1102-10-1104/51-4

Origin: NBS Authorized 9/28/50

Manager: J. Todd

Full task description appears in July-Sept 1950 issue, p. 43.

Status: CONTINUED. The first 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, U. S. Government Printing Office.

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, S. Tsingou

Full task description appears in Apr-June 1951 issue, p. 49.

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

P. 45.

REVISION OF MATHEMATICAL TABLES

Task 1102-10-1104/52-7

Origin: NBS Authorized 8/10/51

Managers: J. Todd, W. F, Cahill, I. Stegun

Full task description appears in July-Sept 1951 issue, p. 4l

.

Status: CONTINUED. Following is the status of those mathematical

tables the sales stock of which has been exhausted and for which reissue

is planned:
30 Status of Projects

"Table of sine and cosine integrals for arguments from 10 to 100," NBS

Applied Mathematics Series 32; in press, U. S. Government Printing Office.

This will be a reissue of the table formerly designated as Mathematical

Table BffT.13 (1942).

"Tables of the error function and its derivative," NBS Applied Mathematics

Series 41 ; in press, U. S. Government Printing Office. This was originally

Mathematical Table MT8, "Tables of probability functions," vol. I (1941).

"Tables of sines and cosines for radian arguments," to be reissued in the

Applied Mathematics Series; formerly designated as Mathematical Table MT4.

"Tables of natural logarithms," vol. IV, Mathematical Table MT12 (1941 );

revision in progress for reissue in the Applied Mathematics Series.

"Miscellaneous Physical Tables," Mathematical Table MT17. It has been

decided not to reissue this. A revision and extension of the tables of

Planck’s radiation functions is under consideration by Tunis Wentink, Jr.,

General Electric Co. A revision of the second part, Tables of electronic

functions, suggested by Dr. L. L. Marton, will be carried out on SEAC;

publication of the tables will be handled by Division 4 of the NBS.

TABLE OF ARCS IN FOR COMPLEX ARGUMENTS

Task 1102-10-1 104/52-1

4

Origin: NBS Authorized 10/1/51

Manager: A. A. Goldstein

Full task description appears in July- Sept 1951 issue, p. 4l

.

Status: INACTIVE. For status to date, see Oct-Dec 1953 issue,

P. 31.

EXTENSION OF THE TABLE OF HYPERBOLIC SINES AND COSINES

Task 1102-10-1104/52-18

Origin: NBS Authorized 9/17/51

Manager: W. F. Cahill

Full task description appears in July-Sept 1951 issue, p. 4l

.

Status: INACTIVE. For status to date, see Oct-Dec 1953 issue,

P. 31

.

TABLE OF THE MODIFIED AIRY INTEGRAL

Task 1102-10-1104/52-23

Origin: NBS Authorized 10/4/51

Manager: P. Rabinowitz

Full task description appears in July-Sept 1951 issue, p. 42.

Status: INACTIVE. For status to date, see Oct-Dec 1953 issue,

P. 31.
Status of Projects

31

TABLE OF ERROR FUNCTION FOR COMPLEX ARGUMENTS

Task 1102-10-1104/52-25

Origin: NBS Authorized 10/5/51

Manager: M. Abramowitz

Full task description appears in July-Sept 1951 issue, p. 42.

Status: INACTIVE. The 20-place table for x,y=0(.l)2 is available

on punched cards.

EXTENSION OF TABLES OF THE EXPONENTIAL FUNCTION FOR

NEGATIVE ARGUMENTS

Task 1102-10-1104/52-31

Origin: NBS Authorized 10/9/51

Manager: E. Marden

Full task description appears in July-Sept 1951 issue, p, 43.

Status: INACTIVE. For status to date, see Oct-Dec 1953 issue,

p. 32.

SPHEROIDAL WAVE FUNCTIONS

Task 1102-10-1104/52-37

Origin: NBS Authorized 11/28/51

Manager: T. Ledley

Full task description appears in Oct-Dec 1951 issue, p. 38.

Status: CONTINUED. Computations of the characteristic values for

the prolate and oblate spheroidal wave equations for m=0(l)5, 1=0(1 )10,

c^=0(.0l)l, i/c=0( .005)1 have been completed except for a few values of

1/c for m=5, !=9?10. The values have been transferred to punched cards.

VAN DER POL EQUATION

Task 1102-10-1104/52-43

(formerly 1102-10-1110/52-43)

Origin: NBS Authorized 11/28/51

Manager: W. F. Cahill

Full task description appears in Oct-Dec 1951 issue, p. 38.

Status: INACTIVE. Fo^r status to date, see Apr-June 1952 issue,

p. 49.
32 Status of Projects

RADIAL MAfHIEU FUNCTIONS

Task 1102-10-1104/52-49

Origin: NBS Authorized 2/1/52

Managers: J # Todd, I. Rhodes

Full task description appears in Jan-Mar 1952 issue, p. 45.

Status: CONTINUED, Checking of the values by E. Aparo and

D. Dainelli at the Istituto Nazionale per le Applicazioni del Calcolo

(Rome) is continuing.

S IEVERT'S INTEGRAL

Task 1 102-10-1 10V52-57

Origin: NBS Authorized 2/12/52

Managers: M. L. Paulsen, P. J. O'Hara

Full task description appears in Jan-Mar 1952 issue, p. 1+6.

Status: INACTIVE. For status to date, see Oct-Dec 1953 issue,

p. 33.

TABLE OF SECANTS AND COSECANTS

Task 1102-10-1104/52-81

(formerly 1102-10-1110/52-81)

Origin: NBS Authorized 7/1/52

Managers: K. C. Nelson, I. A. Stegun Completed 3/31/54

Objective: To prepare a table of secant x and cosecant x to nine

significant figures for x = 0(.01°)90°.

Background: The tables have been in possession of the Computation

Laboratory for several years and were calculated from the previously

published tables of sines and cosines. The proposed volume is intended

as a companion to the earlier volume. Comparison with Gifford's Table

of Secants has disclosed a large number of errors in Gifford.

Status: COMPLETED.

Publication: "Table of secants and cosecants to nine significant

figures at hundredths of a degree," NBS, Applied Mathematics Series 40

(Mar. 26, 1954). Available from U. S. Government Printing Office, Washington 25, D. C., 35 cents.

PAINLEVE EQUATION

Task 1102-10-1104/53-3.

Origin: NBS Authorized 8/11/52

Managers: J. Todd, H. A. Antosiewicz

Full task description appears in July-Sept 1952 issue, p. 49.

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

p. 48.
Status of Projects 33

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, p. 58.

Status: CONTINUED. Computations are being performed as requested,

and results are being transmitted to the sponsor.

L-SHELL CONVERSION COEFFICIENTS

Task 1102-10-5110/53-52

Origin: Oak Ridge National Laboratory Authorized 5/20/53

Manager: C. J. Swift

Full task description appears in Apr-June 1953 issue, p. 45.

Status: CONTINUED. Production is continued as SEAC time is available. The K-shell is about half done and the L

1 -shell is about one-eighth

done

.

3. M_axhe.nia.lAC.aJL Sejyj.c.e_s

Note: The tasks under Mathematical Services are arranged serially ac- cording to the digits following the slant lines in the task numbers. The

first two digits following the slant line designate the fiscal year in

which the task was authorized.

TABLES OF THERMODYNAMIC PROPERTIES OF GASES

Task 0302-40-2606A9-5

Origin: NBS, Section 3.2 Authorized 11/29/48

Sponsor: National Advisory Committee for Aeronautics

Manager: F. L. Alt

Full task description appears in Apr-June 1949 issue, p. 53.

% Status: INACTIVE.

RAY TRACING

Task 0202-10-2308/50-13

Origin: NBS, Section 2.2 Authorized 3/1/50

Sponsor: " " Manager: R. K. Anderson

Full task description appears in Jan-Mar 1950 issue, p. 33.

Status: CONTINUED. SEAC codes for generating and tracing rays
34 Status of Projects

through spheric and aspheric surfaces have been completed. Two optical

systems have been analyzed with the new code.

MOLECULAR STRUCTURE CALCULATIONS, II

Task 11 02-40-51 26/50-1

6

Origin: Naval Research Laboratory, USN Authorized 3/31/50

Sponsor: M " M M Manager: P. J. O'Hara

Full task description appears in Jan-Mar 1950 issue, p. 34.

Status: CONTINUED. Computations were performed as requested.

LIQUID-VAPOR TRANSITION

Task 1102-40-5126/51-22

Origin: Naval Medical Research Institute Authorized 2/1/51

Sponsor: " " " " Manager: I. Stegun

Full task description appears in Jan-Mar 1951 issue, p. 44.

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

P. 52.

MOLECULAR STRUCTURE, III

Task 11 02-40- 5 1 26/51 -3 7

Origin: Naval Research Laboratory, USN Authorized 8/10/51

Managers: P. J. O'Hara, I. A. Stegun

Full task description appears in July-Sept 1951 issue, p. 50.

Status: CONTINUED. Computations were performed as requested.

SHOCK WAVE PARAMETERS, II

Task 1102-40-5126/51-38

Origin: Bureau of Ordnance, USN Authorized 6/28/51

Sponsor: " " " Manager: I. A. Stegun

Full task description appears in Apr-June 1951 issue, p. 46.

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

P. 48.
Status of Projects 35

POWDER DIFFRACTION

Task n 02-40-51 26/52-6

Origin: NBS, Section 9.7 Authorized 9/17/51

Manager: E. Marden Terminated 3/31/54

Objective: To calculate interplanar spacings in crystals for postulated lattice parameters, in order to index x-ray powder diffraction

patterns

.

Background: The indexing of powder diffraction patterns is checked

by comparison of observed and calculated values of d. . j> . The computation

is done on SEAC, by means of a routine prepared by F, Ordway and E. C.

Marden as a part of task 1 102—21—1 104/51—3 (see Jan-Mar 1951 issue, p. 37).

Tapes containing lists of indices, suitable for requested use in this

task, have been prepared by E, Tatge (section 9.7).

Status: TERMINATED, in favor of task 1 102-40-51 26/52-44 (see p. 35).

PRECISE DETERMINATION OF THE PARAMETER OF DISPERSION

EQUATION FOR SEVERAL TYPES OF OPTICAL GLASS

Task 1102-40-5126/52-17

Origin: NBS, Division 2 Authorized 9/29/51

Sponsor: " " Manager: I. A. Stegun

Full task description appears in July-Sept 1951 issue, p. 56.

Status: INACTIVE.

SPHERICAL BLAST

Task 1102-40-5126/52-20

Origin: Naval Ordnance Laboratory Authorized 9/27/51

Sponsor: M " Manager: D, H. Jirauch

Full task description appears in July-Sept 1951 issue, p. 56.

Status: CONTINUED. Some work continued to reorganize the material

prior to test runs on SEAC,

CALCULATIONS FOR d SPACINGS

Task 1102-40-5126/52-44

Origin: NBS, Div. 9 Authorized 12/7/51

Sponsor: " " Managers: R. K. Anderson, A, M. Futterman

Full task description appears in Oct-Dec 1951 issue, p. 47.

Status: CONTINUED. Programming for SEAC is now in progress.
36 Status of Projects

GAS ADSORPTION BY HIGH POLYMERS

Task 1 102-^0-5126/52-70

Origin: Naval Medical Center, Be the sd a, Md. Authorized 1+/1/52

Sponsor: " " M " Manager: I. Stegun

Full task description appears in Jan-Mar 1952 issue, p. 59.

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

P. 59.

NEUTRON DIFFUSION, II

Task 11 02-40-51 26/53-4

Origin: Atomic Energy Commission, New York Authorized 9/30/52

Office, (NDA)

Sponsor: " " " Managers: 0. Steiner, N. Levine

Full task description appears in July- Sept 1952 issue, p. 60.

Status: CONTINUED. Results are being sent to the sponsor as they

are completed. New codes are being written and code-checked as required

to cover various source geometries. Computations have been carried out

for a number of materials for different sets of data.

STRENGTH OF WING COMPONENTS

Task 1102-40-5126/53-11

Origin: National Advisory Committee for Authorized 12/8/52

Aeronautics, Langley Field, Va.

Sponsor: " " Manager: E. Marden

Full task description appears in Oct-Bec 1952 issue, p. 54.

Status: CONTINUED. Computations are being performed on SEAC as requested.

RADIANT HEATING OF SOLIDS

Task 1 102-40-5126/53-20

Origin: NBS, Section 10.2 Authorized 12/15/52

Sponsor: " M Manager: W. F. Cahill

Full t ask description appears in Oct-Dec 1952 issue, p. 55.

Status: INACTIVE
Status of Projects 37

COMPUTATION OF THERMODYNAMIC FUNCTIONS

Task 1102-^0-5126/53-27

Origin: NBS, Division 5 Authorized 3/18/53

Sponsor: " " Manager: E, Marden

Full task description appears in Jan-Mar 1953 issue, p. 57.

Status: CONTINUED, Computations have continued on SEAC, and

thermodynamic functions for several molecules have been calculated using

the two previous codes. In addition, a new code has been written to

compute polynomials in powers of 1/T (where T is temperature) for a range

of temperatures from 50° to 5000°K,

STUDY OF TRICALCIUM ALUMINATE

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, p, 58,

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

P. 58,

REDUCTION OF ECLIPSE DATA

Task 1102-40-5126/53-34

Origin: Air Photographing and Charting Services, USAF Authorized 3/30/53

Sponsor: " M M " Completed 3/31/54

Manager: A, A, Goldstein

Objective: (l) To obtain all possible information from the photoelectric observation of the 1952 eclipse, particularly with reference to

geodetic position, (2) From the results of the above to propose specific

observational methods to determine geodetic position from the eclipse

of 195*.

Background: Measurements of previous eclipses were directed toward

obtaining the two times of inner contact of the sun and moon. This gave

two equations of condition which would allow the correction of two parameters at each station. Photo-electric measurements provide a continuous

curve of light intensity during the eclipse, with the consequent possibility of many equations of condition.

Status: COMPLETED, Measurements of light intensity taken during

the 1952 total eclipse from three stations in Africa (Bangui, Khartoum,

and Port Sudan) have been reduced and analyzed with the following results,

1, Information is available for certain corrections to the

geodetic positions of the three stations.

2. A correction to the difference between the solar aqd lunar

diameter has been found. The accuracy of the measurement as indicated

by the probable error is roughly one part in a thousand. If the known

solar diameter is taken as exact, we obtain for the geocentric lunar

radius S=( ,272374±, 00001 1 )tt, where tt is the lunar parallax. The values
38 Status of Projects

in general use are S=. 272274^ for eclipses, and

S=,27250tt for occultations

.

3. The Solar Limb Darkening (for light of wave length 8000

angstroms), has been determined. The results obtained are over the last

one percent of . the solar edge. They are believed to be more consistent than

previously published results,

4. The method developed and used here appears practical for

large-scale applications and indicates the desirability of conducting

light intensity measurements during the next total eclipse on a larger

scale.

ACOUSTICAL IMPEDANCES

Task 1102-40-5126/53-39

Origin: NBS, Section 6.1 Authorized 6/30/53

Sponsbr: M M Managers:,S. Prusch, K. Nelson

Full task description appears in Apr-June 1953 issue, p. 55.

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

P. 55.

LORAN UNIVAC CODE

Task 1102-40-5126/53-41

Origin: Hydrographic Office, U. S. Navy Authorized 4/20/53

Sponsor: " " Managers: I. Rhodes, £>. H, Jirauch

Full task description appears in Apr-June 1953 issue, p. 56,

Status: CONTINUED, The baseline code and the skywave correction

and trainer code are ready for code checking.

RADIATION DIFFUSION

Task 1102-40-5126/53-51

Origin: NBS, Section 4,8 Authorized 6/9/53

Sponsor: Armed Forces Special Weapons Project

Managers: F, Stockmal, W, F. Cahill

Full task description appears in Apr-June 1953 issue, p. 57 (Neutron

Diffusion III),

Status: CONTINUED. Computations on SEAC continue as time is

available.
Status of Projects 39

HIGH TEMPERATURE PROPERTIES OF WATER

Task 1102-40-5126/54-4

Origin: NBS, Division 3.2 Authorized 12/7/53

Sponsor: Bureau of Ordnance, USN

Manager: «F. H. Wegstein

Full task description appears in Oct-Dec 1953 issue, p. 4l

.

Status: CONTINUED* Preliminary to finding formulas to represent

the thermal functions of water -at high temperatures such as specific heat

and the Gibbs free energy, Mr, Woolley (Section 3,2) has written formulas

for the partition function describing the high temperature properties of

hydrogen. These formulas involve physical conditions and a parameter

which affect the computed properties in an unknown manner. It is planned

to evaluate these formulas with the SEAC in order to check them. If

these formulas give a satisfactory description of hydrogen at high temperatures it should then be possible to extend the method to the complete

system of fifteen or twenty molecules, atoms, and ions representing water

at high temperatures.

The SEAC interpretive floating-point routines are being perfected in preparation for evaluating these formulas*

DISTRIBUTION OF NORMAL MODES OF VIBRATION OF CUBIC LATTICES, II

Task 371 1-60-0009/54-7

Origin: NBS, Division 30 Authorized 12/29/53

Sponsor: " " Managers: F, Stockmal, W. F. Cahill

Full task description appears in Oct-Dec 1953 issue, p. 42,

Status: CONTINUED. A new code to compute the distribution of

normal modes of vibration for body centered cubic lattices has been com- posed and cheeked out.

NORMAL VIBRATIONS IN MOLECULES

Task 3711-60-0009/54-8

Origin: NBS, Section 3.2 Authorized 11/27/53

Sponsor: ft " Managers M, Newman

Full task description appears in Oct-Dec 1953 issue, p, 42*

Status: CONTINUED. Seven eight by eight symmetric matrices were

inverted. The programming for the computation of the eigenvalues and

eigenvectors of a symmetric matrix by the method of Jacobi is underway.
4-0 Status of Projects

COMPUTATION OF VIBRATION MODES AND FREQUENCIES

Task 3711-60-0009/5^-11

Origin: NBS, Section 6.4 Authorized 11/27/53

Sponsor: " M Managers: I. Rhodes, W. Cahill

Full task description appears in Oct-Dec 1953 issue, p. 42.

Status: CONTINUED. The first five antisymmetric and first four

symmetric vibration modes and frequencies of an aircraft structure were

computed using a 24x24 matrix of influence coefficients for the symmetric

case and a 31x31 matrix of influence coefficients for the antisymmetric

case.

SPACE AVERAGE OF TEMPERATURE DISTRIBUTION WITH A VOLUME DISTRIBUTION

OF SOURCES

Task 371 1 -60-0009/5^-1

2

Origin: NBS, Division 6 Authorized 12/11/53

Sponsor: " M Manager: E. Warden

Full task description appears in Oct-Dec 1953 issue, p. 43.

Status? INACTIVE, For status to date, see Oct-Dec 1953 issue,

P. *+3,

AWARD OF PROCUREMENT CONTRACTS BY LINEAR PROGRAMMING

Task 1102-40-5126/54-13

Origin: New York Quartermaster Procurement Agency Authorized 11/27/53

Sponsor: " " " " Manager: H, Bremer

Full task description appears in Oct-Dec 1953 issue, p, 43.

Status: CONTINUED, Approximately 17 contracts (for socks, raincoats, wiping cloth, soap, etc.) have been awarded in accordance with

answers obtained on the SEAC, One of these problems required use of the

general simplex method; all others were solved using the special version

of the simplex method applicable to the so-called transportation problem.

New codes to facilitate the computation of optimum shipping schedules

have been completed and checked and will soon be put into operation. A

code for UNIVAC is being prepared in collaboration with personnel of the

Office of the Air Comptroller, USAF,

At present problems are arriving at the rate of about three

every two weeks, and the sponsor receives the answers in one or two days.
Status of Projects 4l

FLOW IN SUPERSONIC NOZZLE

Task 11 02-40-5126/54-1

4

Origin: National Advisory Committee for Aeronautics Authorized 12/16/53

Langley Field, Va.

Sponsor: M " " Managers: P. Henrici, E. Marden, G. Hawkins

Full task description appears in Oct-Dec 1953 issue, p. 44.

Status: CONTINUED^ The SEAC code has been completed and computations for supersonic flow in nozzles for two different Mach numbers have

been completed.

DE POLYMERIZATION

Task 3711-60-0009/5^-17

Origin: NBS , Section 7.6 Authorized 12/16/53

Sponsor: " ,r Manager: R. Anderson

Full task description appears in Oct-Dec 1953 issue, p. 44.

Status: INACTIVE. For status to date, see Oct-Dec 1953 issue,

P. 44.

ENERGY LEVELS OF COMPLEX ATOMS

Task 3711-60-0009/54-19

Origin: NBS, Section 4.1 Authorized 3/31/54

Sponsor: M M Manager: W. G. Hall

Objective: To determine the characteristic values and characteristic

vectors of matrices arising in the determination of energy levels of com- plex atoms. The orders of the matrices range from 5x5 to 30x30.

Background: Energy levels of complex atoms are obtainable as eigenvalues of a Schrodinger equation. By use of a perturbation procedure,

these energies can be expressed as eigenvalues of matrices. The elements

of these matrices are linear forms in several adjustable parameters. The

eigenvalues are calculated using estimated values of the parameters, the

estimate of parameters are improved by using a least squares calculation

with observed values, and eigenvalues of the new matrix calculated.

Status: NEW. Some of the characteristic roots of several matrices

have been computed using a code prepared by W. F. Cahill, which uses the

power method.

LOW TEMPERATURE PROPERTIES OF BORON COMPOUNDS

Task 1102-40-5126/54-20

Origin: NBS, Section 3.2 Authorized 2/24/54

Sponsor: " M Manager: J. H. Wegstein

Objective: (a) To obtain specific heat from primary electrical
42 Status of Projects

data by utilizing- electronic computers. (b) To correlate and compute

thermodynamic properties (enthalpy, entropy, and free energy functions)

from specific heat data using semi-empirical or theoretical methods such

as the inversion of the partition function to obtain the frequency distribution of the solid from the measured specific heats. This also includes

the smoothing of heat capacity data and integration of such to obtain the

other thermodynamic properties.

Background: The present computation of thermodynamic properties by

hand methods permits treatment of about five compounds a year. It is desired to analyze more compounds and to make additional computations on their thermodynamic properties.

Status: NEW.

EXPERIMENTAL PROGRAM FOR MULTIPLE INPUT AND OUTPUT

Task 11 02-40-51 26/5^-21

Origin: NBS, Section 12.3 Authorized 2/24/54

Sponsor: " " Manager: I. Rhodes

Objective: To provide expert programming assistance for experimental

machine studies to evaluate new data-processing auxiliary storage equipment and to advise on desirable selection features for multiple inputoutput equipment which is under development, such as notched magnetic

disks, wire storage arrays, and high-speed tape cartridges.

Background: The design of DYSEAC provides for input-output operations, which are concurrent with internal computations, to as many as 256 input-output units and for special buffering so that information

transfers may be made automatically to a variety of input-output devices

operating at varying speeds. These design features provide opportunities

to evaluate new devices which have been proposed for use as permanent

or auxiliary data-storage and to try out theoretical programs for sorting

and searching operations which require many outputs or inputs operable

in parallel. The experimental studies will include the programming of

sample data-processing problems involving multiple input-output operations

to simulate the use of new multiple input-output devices and to provide

data for design decisions as to external selection equipment, desired

switching speeds, and similar considerations, for improvements to SEAC

and improved computer systems design.

Status: NEW. Several subroutines were written on a sorting-file

merging problem using SEAC to simulate the use of 16 outputs for coarse

sorting. A prelimina ry report of results has been issued, entitled "Preliminary report on a combined sorting file merging method for electronic

data processing," by Ida Rhodes and M. Stevens.
Status of Projects 43

ENERGY DISTRIBUTIONS ON OPTICAL IMAGE

Task 371 1-60-0009/54-22

Origin: NBS, Section 2,2

Sponsor: " " Manager: L, S. Joel

Authorized 2/9/54

Revised 3/30/54

Objective:

2

To compute

exp i (-10Z2 + 200Z^ 200Z6

) Z J

Q 0\Z)dZ

for n=0(.5)9(l)40.

Background: The light intensity of a point source formed by an optical system with aberration can be expressed as a double integral

exp l‘{i x + i\y + W(x,y) dx dy,

where W(x,y) is the aberration expressed as an error of phase in the

integral for I(|,>\) and are coordinates in the image plane. Taking

the x,y coordinates in the pupil, the integral is to be evaluated over

the pupil. In the most general case W(x,y) would be obtained by fitting

a polynomial to ray tracing results. If we assume that W(x,y) can be

expressed as a polynomial of the sixth degree and that the pupil is circular the double integral reduces to a simple integral involving the

Bessel function Jq(z).

Comments: This problem was originated by Dr. Weinstein, guest

worker in Section 2.2, from the Technical Optics Section, Imperial College

of Science and Technology, London, England.

Status: NEW. Quadratures were carried out on SEAC, using a standard

subroutine which divides the integration interval into k equal subintervals and applies 16-point Gaussian quadrature to each. Calculations were

performed with k=6 and, for checking, k=7 for all r|,and k=l6 for V\=0(l)9.

The standard subroutine for J

fi (X) was used for r\Z<9. This subroutine

computes J

n (X) by means of the series expansion. For values of \Z<9 the

asymptotic expansion was employed with three terms in the cosine factor

and two in the sine factor. Initial examination of results indicates that

for rp-^25 it will be necessary to recompute using k=l6, as the number of

oscillations in the integrand is quite large.

SOLUTION OF THE DIFFUSION EQUATION FOR ELECTRONS

Task 1102-40-5126/54-24

Origin: NBS, Division 4 Authorized 3/5/54

Sponsor: " " Managers: M. Abramowitz, A. Futterman

Objective: To calculate electron depth distributions by solving

an integro-dif f erential diffusion equation which describes fairly accurately the penetration of electrons. _ Background: The diffusion equation governing electron penetration

and diffusion in plane geometry contains three independent variables describing an electron's energy, location, and direction. This equation has
Mf Status of Projects

resisted solution largely because (a) the scattering cross section is

strongly peaked in angle and (b) an asymptotic description of the spatial

distribution has not been available. Integral methods have recently been

developed and tested for calculating spatial moments of fairly high order.

Partly with the help of these, the desired asymptotic description has been

obtained. These integral methods, which reduce the calculation of spatial

moments to a recursive system, are ideally suited to the capabilities of

high speed computers. What is desired, then, is to calculate spatial

moments up to the 30th or 4-Oth order with the SEAC and to make a Legendre

polynomial representation of the spatial distribution. This can be used

to test the accuracy of simpler methods based on the knowledge of a spatial

weight function. Following this, spatial moments or distributions for a number of interesting and practical problems may be obtained using the SEAC.

Status: NEW. The code is finished. The problem will be run on SEAC according to specifications given by the originator.

REFLECTED RADIATION FROM AN INFINITE LAMBERT PLANE

Task 1102-MD-5l26/51+-25

Origin: Armed Forces Special Weapons Group Authorized 2/9/5l+ Sponsor: M n Managers: P. Henrici, L. S. Joel

Objective: To evaluate the integral

§(h,r,d) =

7T

00

0 J

2TT J>£j> d£L

(h

2

+r

2

+y>

2

-2r^> cos 0)^^2

(f>

2+ d2

) 2

for various values of the parameters.

Background: Let the plane z=0 in an (x,y ,z)-space be a Lambert

plane, and let a monochromatic source of light of intensity I be fixed at

the point (0,0, h). Let a small horizontal plane p be situated at (x,y,d).

Then the illumination of p due to reflection at the Lambert plane is

yu.I$ (h,r,d), r= vx^+y^, where yU. is a constant depending only on the

physical properties of the Lambert plane.

Status: NEW. After some transformations, the double integral can

be expressed as a finite simple integral, which depends only on the two

variables u=r/h, v=d/h. Putting

$(h,r,d) = "4; ^(u,v),

one finds

>|r(u, v)

where

n —jl.

(i+u2 )V2

/ d+n2

) 372 (q+n) 2 d+qq) 2

F (£,f’l5 ; pn -2

--2

-) d

>x.

2 2

u v

P “ O »

<I — O’

1+u^ 1+u^

and F denotes the hypergeometric function. This integral is being evaluated
Status of Projects 45

by 1 6-point Gaussian quadratures. The hypergeometric function is computed

from the power series, the convergence of which is sped up by Aitken’s

62-method. The integration error has been estimated by a method due to

P. Davis (J. Rational Mech. Anal. 2(1953)? 303-313). T The code for the problem has been completed and y*(u,v) has been

computed for u=0( .05)1 .60, v=.05( .05) 1 .60.

PHASE DETERMINATION FOR COLEMANITE BY PROBABILITY METHODS

Task 1 102-40-51 26/54-26

Origin: U. S. Geological Survey Authorized 3A/54

Sponsor: M M " Manager: P. J. O'Hara

Objective: To determine the phases (or signs) of the structure

factors of colemanite for later use in computing the electron density

function.

Background: This project deals with the application of probability

methods to the study of the structure factor equations for a centrosymmetric

crystal. The magnitude of the crystal structure factor is computed using

the observed X-ray intensities and known quantities involving the chemical

composition and symmetry of the crystal. The phases (or signs) of the

structure factors are regarded as unknown.

Status: COMPLETED (NEW). The results have been transmitted to the

sponsor

.

EMF OF CELLS AT HIGH TEMPERATURE

Task 3711-60-0009/54-27

Origin: NBS, Section 1.8 Authorized 3/5/54

Sponsor: M " Managers: E. Marden, G. Hawkins

Objective: To compute the electromotive force of galvanic cells

of type M/MX/X, where M=metal, MX=compound, X=non-metal, for various

temperatures, from the relationship E^, = - AFr^/n^ , where E^ = emf at

temperature T, AFT

= free energy of formation at temperature T, n = small

integer, #= 23060.5.

Background: AF,p is computed from temperature by means of the following equation:

AFt= Hq - AaT'XnT - - |f + *PT.

The equations are computed over a given temperature range, which will vary

with each problem. At the limits of the temperature ranges, Up.-, . m . t must be modified by a transition factor, and the temperatures of transition

(allotropic change, fusion and vaporization) will vary from one problem

to another. The computations are to be performed for all known fluorides,

chlorides, bromides, iodides, oxides, sulfides, and possibly nitrides,

phosphides, hydrides, silicides, and carbides.

Status: NEW. Coding of this problem for SEAC is in progress.
46 Status of Projects

LOVIBOND NETWORK FOR CIE SOURCE A

Task 3711-60-0009/54-28

Origin: NBS, Section 2.1 Authorized 2/28/54

Sponsor: " " Managers: E. Marden, G. Hawkins

Objective: To compute the Lovibond network for CIE source A, using

the equations as noted below. (The Lovibond network for CIE sources B and

C have been computed in England.)

Background: The numbers to be computed are tristimulus values X,Y,

and Z; and chromaticity coordinates x', y

1 , and z', defined as follows:

X = TRT

Y

T

B

(E* )A ^ ’ Y = ZT

RTY

I B

CE^)AA i Z = HT

R

T

YTB (E^)Zi ^ > X . Y . Z

x

’ ! =

(X + Y + Z) 5 y

' = (X + Y + Z) 5 z' = (X + Y + Z) •

One of the exponents r,y,b is always zero, and the other two are integers as follows: r = 0( 1 )20(2j4o( 5) 80( 10) 100,

y = 0(1)6,8,10,15,20,30,40,60,100, b= 0(1)10(2)20,25,30,40,50 . Tp, Ty, and Tg are the spectral internal transmittances of the

Lovibond unit red, yellow, and blue glasses, respectively, and the products Ex, Ey, and Ez &re functions of the wavelength. _ Status: NEW. Coding of this problem for SEAC is in progress.

' SPECTRUM ANALYSIS

Task 3711-60-0009/54-30

Origin: NBS, Division 4 Authorized 3/4/54

Sponsor: " " Manager: S. Prusch

Objective: To analyse the atomic structure of Tc I and Tc II. Since

the energy levels of atoms can be expressed as a function of the number of

waves per cm(cr) of the light emitted by excited atoms, the wave numbers (c)

of over 4500 lines for experimentally measured values of wavelengths (A) are to be computed to eight places using the formula of Edlen,

(„-!)• 10

8 = 6432.8 + 2&281G + , 1 46— T

2 4l-<r

where n is the index of refraction of air, and

vac n A

Jair

Many energy levels of an atom can be discovered due to the fact that

transitions between levels give rise to groups of lines which have differences in common. Differences between all known and predicted even levels

can be used to search for pairs of lines which would confirm the even levels and predict new odd levels. The differences between odd levels can
Status of Projects 47

be used to search for pairs of lines to confirm the odd levels and predict new even levels.

Background: The calculations of wave numbers have previously been

done from Kayser's Schwingungszahlen; however, a double interpolation

has to be made, then a correction applied to bring the values up to date.

The formula of Edlen gives the up to date values directly.

The searching for energy levels has formerly been done manually,

requiring hundreds of hours combing a list of spectral lines for pairs of

lines having a common difference.

Status: NEW. Differences between all known and predicted even levels were each used to search for pairs of lines which confirm even levels and predict new odd levels. This was done for Tc I.

AIR CONDITIONING IN UNDERGROUND STRUCTURES

Task 1102-40- 5126/54-33

Origin: NBS, Section 10.3 Authorized 2/24/54

Sponsor: " " Manager: S. Tsingou

Objective: To evaluate the integrals

oo

1 - exp(- y

2

)

[ yY

Q (y)-ak Y^y)] 2 + [yJQ (y)-ak J^y)]^ 3

*

and

"oo

1 -exp(~ ^

y

2

) dx

[yY

o (y)-ak Y^y)]2

+ [yJ

0 (y)-ak J.,(y)] 2 y

p for various values of the parameters oct/a = .5,1 ,2,5,10,50, ak=.5,1,1.5,

2 , 3 . Background: The integrals in question arise in the problem of heat

transfer from water to an underground reservoir.

Status: NEW. The code has been completed and ehecked. One case

has been run, and others are to be run as requested.

VELOCITY OF LIGHT

Task 371

1

-60-0009/54-36

Origin: NBS, Section 11.3 Authorized 3/16/54

Sponsor: " " Manager: P. Rabinowitz

Objective: To fit, in the sense of least squares, polynomials of

the third and fourth degree to a set of data, to compute the experimental

error, and to compute the variances and covariances of the coefficients
Status of Projects

of the polynomials.

Background: These computations are required by the Radiometry Section (4.2) for reducing experimental observations in connection with the

determination of the velocity of light.

Comments: This problem was proposed by E. K. Plyler (4.2) in cooperation with W. S. Connor (11.3).

Status: NEW. Least square polynomials of third and fourth degrees

have been fitted to data submitted by the sponsor.

CORRELATION OF IONOSPHERIC DATA

Task 3711-60-0009/54-37

Origin: NBS , Section 1 4. 1 Authorized 3/12/54

Sponsor: " " Managers: J. H. Wegstein, H. H. Howe

Objective: To correlate certain ionospheric data (magnetic K-figures,

maximum usable frequency, hours of fading) with sunspot number, both

simultaneous and with time lag, in order to determine whether the other

phenomena show any significant lag behind sunspots, and how much.

Background: This problem was tackled previously on the CPC from a different point of view.

Status: NEW. A code has been written and proved in on the threeaddress SEAC. About one hour of SEAC time is required to complete the

calculations. This task has afforded an opportunity to test the performance of the three-address SEAC.

COMPRESSIBILITY FACTORS OF DRY AIR

Task 3711-60-0009/54-38

Origin: NBS, Section 3.2 Authorized 3/29/54

Sponsor: " " Manager: M. Paulsen

Objective: To prepare SEAC routines for evaluating the dimensionless compressibility factor z and the density j> from the relations

z = P/RTp

z = A + Bp + Cf? + Djd . .

.

for prescribed values of the constants, P = pressure, T = temperature, and

the virial coefficients A,B,C,D,... . When the programs have been checked,

tables will be prepared as desired.

Background: The desired tables were computed heretofore either by

hand or IBM equipment. It is hoped that the successful accomplishment of

this task will make available a more comprehensive tabulation of those factors of importance in explaining the influence of imperfections on the

thermodynamic properties of gases.

Status: NEW. A flow diagram has been prepared.
III. STATISTICAL ENGINEERING LABORATORY

(Section 11.3)

1 . Fundamental Research in Mathematical Statistics

BIBLIOGRAPHY AND GUIDE TO STATISTICAL LITERATURE

Task 1 103-10-1 107A9-1 a Origin: NBS Authorized 1/9A9

Manager: L. S. Deming

Full task description appears in Apr-June 19^9 issue, p. 75.

Status: CONTINUED. Abstracts of statistical literature have been

taken from the two main reviewing services of mathematical and statistical

publications, namely, Zentralblatt fur Mathematik (for the years 1931-19^1)

and Mathematical Reviews (beginning with 1939 and continuing with the

current issues as they arrive). These abstracts are mounted individually

on cards, are classified by subject, and are arranged in an alphabetical

file by author. Because of the almost complete overlap of the two abstracting journals since 19^1? only abstracts from Mathematical Reviews

will be added to the file on a current basis.

MANUAL ON FITTING STRAIGHT LINES

Task 1103-10-1107/50-2

Origin: NBS Authorized 3/1/50

Manager: F. S. Acton

Full task description appears in Jan-Mar 1950 issue, p. *+2. Status: INACTIVE. For status to date see Oct-Dec 1952 issue,

P. 58.

TABLE TO FACILITATE DRAWING RANDOM SAMPLES

Task 1103-10-1107/51-1

Origin: NBS Authorized 7/1/50

Managers: C. Eisenhart and L, S, Deming

Full task description appears in July-Sept 1950 issue, p. 57.

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

p. 6^-.

^9
50 Status of Projects

MISCELLANEOUS STUDIES IN PROBABILITY AND STATISTICS

Task 1103-10-1107/51-2

Origin: NBS Authorized 7/1/50

Manager: C. Eisenhart

Full task description appears in July-Sept 1950 issue, p. 58.

Status: CONTINUED,

Publications: (l) "Tables of expected values of 1/X for positive

Bernoulli and Poisson variables," by E, Grab and I. R, Savage; J. Amer.

Stat. Assoc., 1+2, 1 69—1 73 (Mar. 195*+). (2) "A new method for analyzing

extreme-value data," by J. Lieblein; NACA Technical Note 3053 (Jan. 195*+).

(3) "Tables of the inverses of finite segments of the Hilbert matrix, " by

I. R. Savage and E. Lukacs; to be included in Contributions to the solution of systems of linear equations and the determination of eigenvalues,

NBS Applied Mathematics Series 39. (*+) "A property of the normal distribution," by E. P. King and E. Lukacs; accepted for publication in the

Annals of Mathematical Statistics. (5) "On strongly continuous stochastic

processes," by E, Lukacs; accepted for publication in Sankhya. (6) "Certain Fourier transforms to distributions (II)," by E, Lukacs and 0. Szasz;

accepted for publication in the Canadian Journal of Mathematics. (7) "Non

negative trigonometric polynomials and certain rational characteristic

functions," by E. Lukacs and 0. Szasz; accepted for publication in the

Journal of Research of the NBS, (8) "Time-discrete stochastic processes

in arbitrary sets, with applications to processes with absorbing regions

and to the problem of loops in Markoff chains," by D. van Dantzig; submitted to a technical journal. (9) "On the variances and covariances of

order statistics from the Weibull distribution," by J. Lieblein; submitted

to a technical journal. (10) "Two early papers on the relation between

extreme values and tensile strength," by J # Lieblein; submitted to a technical journal. (11) "On optimum grouping in one-criterion variance

components analysis," by E. P. King; accepted for publication in the

Journal of the American Statistical Association. (12) "The use of extreme

value methods in engineering problems," by E. J. Gumbel and J. Lieblein;

IN MANUSCRIPT.

LAW OF PROPAGATION OF ERROR

Task 1103-10-1107/52-1

Origin: NBS Authorized 6/23/51

Managers: C. Eisenhart and I. R. Savage

Full task description appears in July-Sept 1951 issue, p. 65.

Status: INACTIVE. For status to date see July-Sept 1951 issue,

P. 65.
Status of Projects 51

PROCEDURES OF NON-PARAMETRIC STATISTICS

Task 1103-10-1107/52-2

Origin: NBS Authorized 9/17/51

Manager: I. R. Savage

Full task description appears in July-Sept 1951 issue, p. 66.

Status: CONTINUED. The work of I. R. Savage on most probable rank

orders and alternatives of the Lehmann type has been continued. In particular, the following testing situation has been studied intensively:

Hq — X.j , . . . ,Xm

,Y.j , . . . ,Y

n are mutually independent and identically

random variables.

H. — X1 , . . . ,X , Y. , . . .

, Y are mutually independent, and there exists

1 I 1x2 1 n

a continuous cumulative distribution function F(x) and positive

constants and A^ (A^>A^ ) such that the X's are identically

distributed with the c.d.f. F^ and the Y’s are identically distriAo buted with the c.d.f. F

This situation has statistical interest since special cases involve the

testing of the scale parameter for the exponential distribution and the

location parameter for the extreme value distribution. For specific

values of the parameters it is possible to find the exact probabilities

for each rank order without the necessity of numerical integration. For

all values of the parameters the construction of optimum test procedures

for small samples is possible due to the fact that the probabilities of

rank orders are partially ordered. Finally, the limiting test has been

derived and it is interesting to note that this procedure gives results

almost identical with those obtained by the best procedures for- small

samples

.

W. S. Connor has studied the function y=a+bx in the case when

x is without error but y can be observed only through the random variable

Y which has a continuous distribution function with a unique median y.

When observations on y are taken at only two values of x, a simple non- parametric technique is available for determining simultaneous confidence

intervals for all y's.

Publications: (1) "Bibliography of nonparametric statistics and

related topics, " by I. R, Savage; J. Amer. Stat. Assoc. V8, 8

1+4—906

(Dec. 1953). (2) "Most probable rank orders," by I. R. Savage; submitted

to a technical journal.

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, p. 60.

Status: CONTINUED. M. Zelen has continued the study of a new

method for combining information from inter- and intra-block analyses

for exact tests of significance. Preliminary results indicate that the

new method is more powerful than any existing method.

M. Zelen (with John Mandel, Division 7? Section 5) has analyzed

new designs for the two-way elimination of heterogeneity.

Publication: (l) "Some theorems for partially balanced designs,"

hy W. S. Connor and W.. H. Clatworthy; Ann. Math. Stat. 25? 100-112
52 Status of Projects

(Mar. 195*+). (2) "An embedding theorem for balanced incomplete block

designs," by M, Hall, Jr. (Ohio State University) and W. S. Connor;

Canadian J. Math. VI , No. 1, 35-1+1 (195k) » (3) "Analysis for some incomplete block designs having a missing block," by M. Zelen, accepted

for publication in Biometrics. (4) "A note on partially balanced designs,"

by M. Zelen? accepted for publication in the Annals of Mathematical Statistics. (5) "Partially balanced incomplete block designs with two asso- ciate classes and two treatments per block," by W. H. Clatworthy; submitted to a technical journal. (6) "On the enumeration of partially balanced designs with two associate daises," by W. H. Clatworthy; submitted

to a technical journal. (7) "New experimental designs for paired observations," by W. S. Connor and W. J. Youden; IN MANUSCRIPT.

2. Applied Research in Mathematical Statistics

COLLABORATION ON STATISTICAL ASPECTS OF NBS

RESEARCH AND TESTING

Task 3737-60-0002/51-1

Origins NBS Authorized 7/1/50

Managers: W. J. Youden and J. Cameron

Full task description appears in July-Sept 1950 issue, p. 60.

Status: CONTINUED. Activity under this project fell into two main

categories:

A. Design of Experiments: (a) A partially balanced design was constructed for an investigation of hygroscopic properties of cement in

which 120 kinds of cement were to be tested in groups of 8. An experimental arrangement involving two replications was constructed from a balanced incomplete block design by means of its incidence matrix. (b) A 1/3

replicate of a 3x5x6 factorial design was developed for testing the effect

of various factors on the warping of plastic laminates used in printed

circuits.

B. Development or Selection of the Appropriate Method for Analysis and Interpretation of Data: (a) In corrosion studies of pitting of

metal pipes the average maximum pit depth was shown by extreme-value

methods to be a logarithmic function of the length of pipe. This result

provides a theoretical treatment for a problem for which only empirical

methods have previously been used. (b) An optimum assignment of a given

fixed number of observations in order to minimize the variance of the

intercept of a straight line was worked out for the case where two x values

were used. This problem arose in connection with work on resistence

thermometers.

Publications: (l) "Acceptance sampling of electroplated articles,"

by J. M. Cameron and F. Ogburn (5.6); Plating kl, k3-k6 (Jan. 195^).

(2) "Performance of inspectors and gasoline pumps," by W. J. Youden and

M. Jensen (32.0); NBS Miscellaneous Publication 209, 82-88 (available

from U. S. Government Printing Office, Washington 25, D. C., *+0 cents).

(3) "Probability limits for the average chart when process standards are unspecified," by E. P. King; submitted to a technical journal.
Status of Projects 53

STATISTICAL ASPECTS OF NBS ADMINISTRATIVE OPERATIONS

Task 3737-60-0002/52-1

Origin: NBS Authorized 10/1/51

Manager: I. R. Savage

Full task description appears in Oct-Dec 1951 issue, p. 56.

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

P. 68.

STATISTICAL SERVICES FOR COMMITTEE ON SHIP STEEL, NRC

Task 1103-40-5105/52-1

Origin: Ship Structure Committee, NRC Authorized 12/1/51

Sponsor: M " Manager: W. J. Youden

Full task description appears in Oct-Dec 1951 issue, p. 58.

Status: CONTINUED. The method of estimating the transition temperature of a steel plate by linear interpolation between two test temperatures was evaluated by estimating the transition temperature from a second degree polynomial fitted to observations at several test temperatures, This was done for two each of "source", "through," and "end"

plates.

RESEARCH IN APPLICATIONS OF MATHEMATICAL STATISTICS TO

PROBLEMS OF THE CHEMICAL CORPS

Task 1103-40-5118/52-1

Origin: Biological Laboratories, Chemical Corps Authorized 10/1/51

Dept, of the Army

Sponsor: " " " Manager: C, Eisenhart

Full task description appears in Oct-Dec 1951 issue, p. 57.

Status: CONTINUED. A study is in progress to construct and compile

fractional replications of 2n and 3n factorial designs in incomplete blocks

A classified meeting was held at the Bureau on March 16 on

the subject of Area Weapons. About 4o experts from various scientific

fields and from various Government agencies and universities attended.

The purposes of the meeting were to exchange current research information,

to improve the exchange of information in the future, and to organize

research in such a manner that the results will be of the greatest use.
54 Status of Projects

STATISTICAL ANALYSIS AND DESIGN OF EXPERIMENTS

FOR THE U. S. GEOLOGICAL SURVEY

Task 1103-40-5140/54-1

Origin: U. S. Geological Survey, Department of Authorized 10/9/53

Interior

Sponsor: M " " Managers: C. Eisenhart and W. J. Youden

Full task description appears in Oct-Dec 1953 issue, p. 50.

Status: CONTINUED. Attention was devoted during this quarter

principally to study and evaluation of methods advocated in earth-sciences

literature for the analysis of orientation data, with a view to developing

simplified procedures (a) for testing whether an observed distribution of

points and a circumference departs sufficiently from a uniform distribution to warrant the conclusion that one or more "preferred directions"

exist, (b) for estimating the azimuth of the "preferred direction" and

determining the statistical uncertainty of the result, and (c) for testing

whether two or more sets of orientation data may be regarded as random

samples from distributions having one and the same preferred direction.

A preliminary draft version of a note giving "A quick-and-easy test for

the existence of preferred directions" was completed for limited circulation for comment as the quarter ended; another, giving a numerical evaluation of the bias of the so-called "empty-test" as commonly used, was essentially completed; and a third, giving "A quick-and-easy method for

determining 'preferred direction 1 , "was nearing completion in preliminary

draft

.

In addition, some correspondence was exchanged with certain

individual members of the U. S. Geological Survey on such topics as random

subsampling versus replicate subsampling, precision of chemical analyses,

use of statistical methods of correlation and regression as tools in the

study of geochemical associates, and the application of the log-normal

distribution to problems of ore sampling.

STATISTICAL ANALYSIS OF BALL BEARING FATIGUE DATA

Task 1103-40-5145/54-1

Origin: American Standards Association Authorized 1/13/54

Sponsor: " " Manager: J. Lieblein

Objective: To study and evaluate available methods for analysis of

data on fatigue-life of ball bearings; and to determine the load-life

curve of ball bearings indicated by the test data obtained and made available by the cooperating ball bearing manufacturers.

Background: Sub-Committee 7 of the American Standards Association

Committee B3 has assigned to it the task of analyzing a mass of test data

compiled by the laboratories of prominent ball bearing manufacturers for

the purpose of determining load-life relations. (The sub-committee has

discussed various methods of analysis and there is some question as to

the correct method. Variation in method would lead to radically different

conclusions resulting in confusion and disagreement as to exactly what the

data indicated.) It is the unanimous opinion of all concerned that the

Statistical Engineering Laboratory of the National Bureau of Standards is

best fitted to resolve the issue, in that it has the necessary competence

and impartiality and is acceptable to all parties involved. The results

are expected to be of widespread engineering importance and economic value
Status of Projects 55

to ball bearing users representing almost every engineering field.

Status: NEW, A meeting between the Sub-Committee and Bureau

representatives was held early in the quarter at which time various aspects

of the project were discussed and available data were turned over to the

Statistical Engineering Laboratory for study and analysis. A report of the

meeting embodying background material for the project was drafted and submitted to the Sub-Committee for comments. A revised report embodying the

suggestions received is in preparation. The endurance data for bell

bearings that were received from the Sub-Committee were examined on an overall, exploratory basis as regards the suitability of various types of

load-life relationships. The wide scatter in fatigue lives present re- quired further study to be made of the best methods for treating the data.
IV. MACHINE DEVELOPMENT LABORATORY

(Section 1 1 .4)

in cooperation with

ELECTRONIC COMPUTER SECTION

(Section 12.3)

THE BUREAU OF THE CENSUS COMPUTING MACHINE

Task 1 104-34- 5107A7~1

Origin? The Bureau of the Census Authorized 7/1/47

Sponsors M " Full task description appears in Apr-June 1949 issue, p. 58.

Status: CONTINUED. Under contract between the NBS and the EckertMauchly Division of Remington Rand Inc., stockpiling of replacement parts

for UNIVAG System No. 1 has been continued.

NATIONAL BUREAU OF STANDARDS EASTERN AUTOMATIC COMPUTER (SEAC)

Task 11 04-34-51 07A9-1

Origin: NBS Authorized 12/15A8

Sponsor: Office of the Air Comptroller, USAF

Full task description appears in Apr-June 1949 issue, p, 59.

Status: CONTINUED. Since January 1, 1954, the SEAC installation

has been operated as an item of general Bureau equipment.

During this quarter SEAC was operated with an over-all efficiency

of 85°/° during scheduled operation. The computation time was distributed

as follows: Total time available — 721 hours; Good operating time —

612 hours; Down-time— 109 hours. In addition, 1 86 hours were devoted

to engineering and 137 hours to scheduled maintenance.

56
Lectures and Symposia

Note: In general, copies of papers or talks listed in this section are not available from the National Bureau of Standards. If and when a paper

is to be published, it will be listed in the section of this report on Publication Activities.

Numerical Analysis Colloquium

(Los Angeles, California)

NEYMAN, J. (University of California, Berkeley). Studies of the spatial

distribution of galaxies with particular reference to the hypothesis of the expanding universe. February 1

.

LANCZOS, C. (North American Aviation Corporation). Spectroscopic eigenvalue analysis. March 1.

Numer ica l An alysis S eminar

(Los Angeles, California)

HOLMBOE, J. (University of California, Los Angeles). On the behavior of

nonsteady two-dimensional rotational flow. January 5.

FORSYTHE, G.E. A survey of SWAC codes for solving matrix problems. January 7 . LANCZOS, C. (North American Aviation Corporation). On a quadrature method

of solving eigenvalue problems for ordinary differential equations. January 12.

HUSKEY, H.D. Diagnosing SWAC failures. January l4,

WASOW, W.R. On improving approximate solutions of partial differential

equations by h

2-extrapolation. January 19.

KAUS , P. (University of California, Los Angeles; graduate student). Machine

solution of a non-linear eigenvalue problem of elementary particle theory. January 21.

MELKANOFF, M.A. (University of California, Los Angeles; graduate student).

Machine computation of a proposed wave function for the Li^ molecule. January 26.

MOTZKIN, T.S. Minimal number of multiplications for evaluating a polynomial.

January 28.

HOLLANDER, F.H. Diagnosing SWAC failures, II. February 16.

57
5 8 Lectures and Symposia

TOMPKINS, C.B. SWAG results on isometric embedding of surfaces of re- volution. February 18,

FUTTEftMAN, W. A machine calculation of the photo-electric effect for

the deuteron. February 23.

ROGERS, T. A. (University of California, Los Angeles). Electronic analogue

computers. March 2.

LEBELL, D. (University of California, Los Angeles). Survey of UCLA analogue

computers. March

SELFRIDGE, J.L, (University of California, Los Angeles). Some applications

of SWAC to number-theoretic computations. March 9.

MOOKINI, E, SWAC codes for solving systems of linear equations by the

conjugate-gradient method. March 11.

MacNEAL, R. (California Institute of Technology). Solution of partial

differential equations on C.I.T. analogue equipment. March 1 6 , KING, G.W. (International Telemeter Corporation). Three-dimensional

random walks with excluded volume. March 18.

MULLER, M, Use of SWAC to study the behavior of such statistical theories

as sequential analysis. March 23,

PETTIT, J.T, (University of California, Los Angeles), A matrix inversion

code and its application to earth tides, March 25.

KRUSE, F.H. and SPARKS, R, (University of California, Los Angeles). Computing problems in chemical crystallography. March 30.

SPECIAL SEMINAR, sponsored jointly by the Department of Mathematics, University of California, Los Angeles, and the National Bureau of Standards.

Los Angeles:

COURANT, R, (New York University). Problems of stability and

soap film demonstrations. January 12.

Applied Mathematics Colloquim

(Washington)

BUSK, T.H. (University of Copenhagen), A class of polynomials. January 22.

Statis tical Enginee ring Co lloquia

ZELEN, M. Inter-block analysis for incomplete block designs. January 8.

CONNOR, W.S, Simultaneous confidence interval estimation. January 29.
Lectures and Symposia

Applied Statistics Colloquium

59

EISENHART, C. Statistical method in experimentation, II. January 15.

Papers end Invited Ta lks

Presented by Memb ers of the Staff

at Meetings of Outside Organizations

ALT, F.L, A survey of computers and mathematical methods in management.

Presented at the Computation Laboratory, Wayne University,

Detroit, Mich., March 3.

ANTOSIEWICZ, H. Asymptotic solution of ordinary differential equations.

Presented at the David Taylor Model Basin, Washington, D. C,

,

January 15.

FORSYTHE, G.E. What are relaxation methods? Presented as part of the

Lecture Series, "Mathematics for Modern Engineering," at the

University of California: Los Angeles, February 1, Corona,

February 2; Berkeley, February h,

HENRICI, P, On certain equations related to the theory of generalized

axially symmetric potentials. Presented at the Institute for

Applied Mathematics and Fluid Dynamics, University of Maryland,

College Park, Md., January 8.

HOFFMAN, A.J. The use of high-speed computers in awarding contracts.

Presented to the Mobilization Designation Unit No. 1, Office of

the Quartermaster General, Ft. Myer, Arlington, Va., March 1 5,

HUSKEY, H.D. A solution for automatic unit control. Presented at the

IRE Computer Conference, Los Angeles, Calif., February 11.

SOUTHARD, T.H. Elements of nomography. Presented to the Pi Mu Epsilon

Fraternity, University of California, Los Angeles, March 29.

TAUSSKY-TODD, 0. Generation of pseudo-random numbers on high speed automatic digital computing machines. Presented at the Monte Carlo

Symposium sponsored by the Wright Air Development Command, USAF,

and held at Gainesville, Fla., March 15.

TODD, J. (l) What is numerical analysis? Presented at the Mathematics

Colloquium, University of Maryland, College Park, Md., January 7.

(2) Some tools of the numerical analyst. Presented at the NBS

Scientific Staff meeting, Washington, March 12. (3) Tests of

new systems of pseudo-random numbers. Presented at the Monte

Carlo Symposium, sponsored by the Wright Air Development Command,

USAF, and held at Gainesville, Fla e , March 15.

TOMPKINS, C.B. (l) Some general objectives of a numerical analysis curriculum. Presented at the meeting of the Southern California

Section of the Mathematical Association of America, held at the

George Pepperdine College, Los Angeles, Calif., March 13.

(2) Methods of steepest descent. Presented as pert of the

Lecture Series, "Mathematics for Modern Engineering," at the

University of California: Los Angeles, March 29; Corona, March 30;

Berkeley, April 1

.

YOUDEN, W.J. (1) Control and measurement of experimental error. Presented

to the local chapter of the American Statistical Association,
60 Lectures and Symposia

University of Illinois, Urbana, 111., January 14. (2) Behavior

of measurements. Special lecture presented at the University of

Illinois Short Course on Quality Control, Urbana, 111.^ January

I

1*. (3) New experimental designs. Presented to the Central

Illinois Section of the American Society for Quality Control,

University of Illinois, Urbana, 111., January 15. (*+) Scheduling

project work for greater returns. Presented to the Engineering

Group, Bureau of Ships, Washington, D. C„, February 10. (5~13)

Interpretation of chemical data. Presented to the local sections

of the American Chemical Society located at Kingsport, Knoxville,

Nashville, Memphis, and Chattanooga, Tenn, and at Sheffield,

Birmingham, and Auburn, Ala. and at Atlanta, Ga., February 16-26.

(l4) Design of experiments in operations research. Operations

Research Office, The Johns Hopkins University, Md., March 9.
Publication Activities

1 , PUBLICATIONS WHICH APPEARED DURING THE QUARTER

1.1 Mathematical Tables

(1) Tables of Lagrangian coefficients for sexagesimal interpolation*

NBS Applied Mathematics Series 35. Available from U.S. Government

Printing Office, $2. GO.

(2) Table of secants and cosecants to nine significant figures at

hundredths of a degree. NBS Applied Mathematics Series 40 s Available from U. S. Government Printing Office, Washington 25, D,C,,

35 cents.

(3) Tables of expected values of 1/X for positive Bernoulli and Poisson

variables. E. Grab and I. R. Savage. J. Amer. Stat. Assoc. 49 . 169-173 (Mar. 1954).

1.2 Manuals, Bibliographies, Indices

(1) The statistical theory of extreme values and some practical applications (A series of lectures by E. J. Gumbel) . NBS Applied

Mathematics Series 33. Available from U. S. Government Printing

Office, Washington 25, D. C., 40 cents.

(2) Bibliography of nonparametric statistics and related topics,

I. R. Savage. J. Amer. Stat. Assoc. 48, 844-906 (Dec, 1953).

1.3 Technical Papers

(1) On mildly nonlinear partial difference equations of elliptic

type. L. Bers. J. Res. NBS 5l, 229-236 (Nov. 1953).

(2) On modified divided differences. G. Blanch. MTAC 8, 1-1 1 ( Jan . 1 954)

,

(3) Programs for computing hypergeometric series. W. F. Cahill.

MTAC 8, 36-37 (Ian. 1954).

(4) Acceptance sampling of electroplated articles. J. M. Cameron and

F . Ogburn. Plating 4l, 43-46 (Jan. 1954).

(5) Some theorems for partially balanced designs. W, S. Connor and

W, H. Clatworthy. Ann. Math. Stat. 254 100-112 (Mar, 1954).

(6) An embedding theorem for balanced incomplete block designs.

Ms Hall, Jr., and W. S, Connor, Canadian J. Math. 6, No. 1,

35-41 (1954).

(7) A characterization of normal matrices. A. J. Hoffman and

0 e Taussky. J. Res. NBS 52, 17-19 (Jan. 1954).

(8) A new method of analyzing extreme-value data. J. Lieblein.

NACA Technical Note 3053 (Jan. 1954, Washington, D. C. )

.

Available only from the NACA.
62 (9) Publication Activities

On the accuracy of the numerical solution of the Dirichlet problem

by finite differences. J.L. Walsh and D. Young. J. Res. NBS 51 . 343-363 (Dec. 1953).

(10) Singular perturbation methods for nonlinear oscillations. W.Wasow

Proceedings of a Symposium on Nonlinear Circuit Analysis, held by

the Polytechnic Institute of Brooklyn, N. Y. , 1953? pp. 75-98.

(11) Performance of inspectors and gasoline pumps. W.J. Youden and

M. W. Jensen. Report of the Thirty-Eighth National Conference

on Weights and Measures 1953? National Bureau of Standards Miscellaneous Publication 209 (Dec. 21, 1953)? p. 82-88. Available

from U. S. Government Printing Office? 40 cents.

2. MANUSCRIPTS IN THE PROCESS OF PUBLICATION MARCH 31, 1954

2.X Mathematical Tables

(1) Table of sine and cosine integrals for arguments from 10 to 100.

NBS Applied Mathematics Series 32. (A reissue of NBS Mathematical

Table MT13.) In press, Government Printing Office.

(2) Table of the Gamma function for complex arguments. NBS Applied

Mathematics Series 34. In press, Government Printing Office.

(3) 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.

(4) Tables of the error function and its derivative. NBS Applied

Mathematics Series 4l . (A reissue of NBS Mathematical Table 8 , Tables of probability functions, vol. I.) In press, Government

Printing Office.

(5) Tables of sines and cosines for radian arguments. To appear in

the NBS Applied Mathematics Series. (A reissue of NBS Mathematical Table 4.

)

(6) 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 39.

2.3 Technical Papers

r oo _ 2_/ / \ (1) Evaluation of the integral e u x u ^du. M. Abramowitz.

J 0

Accepted for publication in the Journal of Mathematics and Physics

(2) 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.

(3) The practical evaluation of integrals. M. Abramowitz. Submitted

to a technical journal.

(4) Approximate method for rapid Loran computation. M. Abramowitz?

D.H. Call, and J. C. Mathews. Submitted to a technical journal,

(5) The relaxation method for linear inequalities. S. Agmon. Accepted for publication in the Canadian Journal of Mathematics.

(6) A theorem on alternatives for pairs of matrices. H. Antosiewicz.

Submitted to a technical journal.
Publication Activities 63

(7) Asymptotic solution of linear differential equations with a parameter. H. Antosiewicz. Submitted to a technical journal,

(8) On nonlinear differential equations of the second order with integrable forcing term. H. Antosiewicz. Submitted to a technical

journal.

(9) On the differential equation x + k(f(x) + g(x)x)x = ke(t).

H. A e Antosiewicz, Submitted to a technical journal.

(10) On a certain integral involving Bessel functions. H. Antosiewicz.

Submitted to a technical journal.

(11) Some implications of Liapunov's conditions for stability. H. A.

Antosiewicz and P. Davis, Submitted to a technical journal.

(12) On a theorem of Ostrowski and Taussky. R. Bellman and A. J.

Hoffman. Accepted for publication in Archiv der Mathematik,

(13) On the enumeration of partially balanced designs with two associate

classes. W. H. Clatworthy. Submitted to 8 technical journal.

(14) Partially balanced incomplete block designs with two associate

classes and two treatments per block. W. H. Clatworthy. Submitted to a technical journal.

(15) 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.

(16) Linear functional equations and interpolation series. P. Davis.

Accepted for publication in the Pacific Journal of Mathematics.

(17) On a problem in the theory of mechanical quadratures. P. Davis.

Accepted for publication in the Pacific Journal of Mathematics.

2

(18) Some L aspects of Faber polynomials, P. Davis and H. Pollack.

Accepted for publication in the Duke Mathematical Journal,

(19) On the estimation of quadrature errors for analytic functions.

P. Davis and P e Rabinowitz. 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.

Accepted for publication in the Transactions of the American

Mathematical Society.

(21) Some sampling results on the power of nonparametric tests against

normal alternatives. W. J. Dixon and D. Teichroew. Submitted

to a technical journal,

(22) Turbulent flow in shock tubes of varying cross-section. R. F.

Dressier. Accepted by the Journal of Research of the NBS

.

(23) On the optimal character of the (s,S) policy in inventory theory.

A. Dvoretzky, J. Kiefer, and J. Wolfowitz. Submitted to a technical journal.

(24) Inequalities for eigenvalues of Hermitian matrices. K e Fan. To

be included in Contributions to the solution of systems of linear
Publication Activities

equations and the determination of eigenvalues, NBS Applied

Mathematics Series 39.

(25) Some remarks on commutators of matrices. K. Fan. Accepted for

publication in Archiv der Mathematik.

(26) 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 39.

(27) Some metric inequalities in the space of matrices. K. Fan and

A, J. Hoffman. Submitted to a technical journal.

( 28 ) Punched-card experiments with accelerated gradient methods for

linear equations. A. I. and G. E. Forsythe. To be included in

Contributions to the solution of systems of linear equations and

the determination of eigenvalues, NBS Applied Mathematics Series 39

(29) Asymptotic lower bounds for the frequencies of polynomial membranes

G. E. Forsythe. Accepted for publication in the Pacific Journal

of Mathematics.

(30) Asymptotic lower bounds for the fundamental frequency of convex

membranes. G, E. Forsythe. Accepted for publication in the

Pacific Journal of Mathematics.

(31) SWAC. G. E. Forsythe. 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 39.

(33) A numerical solution of Schroedinger 1 s equation in the continuum.

W. Futterman, E. Osborne, and D. S. Saxon. Accepted for publican

tion in the Journal of Research of the NBS.

(3*0 Linear programming in bid evaluation. L. Gainen, D. J. Honig,

and E. D. Stanley. Accepted for publication in Logistics Research

Quarterly.

(35) A nonlinear model for the composite pi-meson. S. G. Gasiorowicz.

Submitted to a technical journal.

(36) Application of variational methods to intermediate and high energy

scattering. E. Gerjuoy and D. S. Saxon. Accepted for publication

in Physical Review.

(37) Variational principles for the acoustic field. E. Gerjuoy and

D. S. Saxon. Submitted to a technical journal.

( 38) A method of solving the transportation problem. A. N. Gleyzal,

Submitted to a technical journal.

(39) Equations of physics in general Newtonian space-time. A. N. Gleyzal.

Submitted to a technical journal.

(40) The representation of integers by binary quadratic rational forms.

K. Goldberg, M. Newman, E. G. Straus, and J. D. Swift. Accepted

for publication in Archiv der Mathematik.
Publication Activities 65

(41) 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.

(42) On certain series expansions involving Whittaker functions and

Jacobi polynomials. P 3 Henrici* Accepted for publication in the

Canadian Journal of Mathematics,

(43) Ueber die Funktionen von Gegenbauer. P, Henrici. To appear in

Archiv der Mathematik.

(44) On a combinatorial theorem, A. J. Hoffman. Submitted to a technical journal.

(45) On optimum grouping in one-criterion variance components analysis

E. P. King, Accepted for publication in the Journal of the American Statistical Association.

(46) Probability limits for the average chart when process standards

are unspecified. E e P, King, Submitted to a technical journal.

(47) A property of the normal distribution. E, P, King and E. Lukacs.

Accepted for publication in the Annals of Mathematical Statistics

(48) Contrac tibility and convexity, H, W. Kuhn, Accepted for publication in the Proceedings of the American Mathematical Society,

(49) On certain character matrices. D. H. Lehmer. Submitted to a technical journal,

(50) Bounds for punch indentation problems, E, Levin, Submitted to

a technical journal.

(5D Acoustic radiation pressure on a circular disk. H. Levine. To

appear in the Proceedings of the Fifth Symposium on Applied Mathe

matics of the .American Mathematical Society.

(52) On the variances and covariances of order statistics from the

Weibull distribution. J, Lieblein, Submitted to a technical

journal.

(53) On strongly continuous stochastic processes. E. Lukas. Accepted

for publication in Saakhya,

(5^) Certain Fourier transforms of distributions (II), E, Lukacs and

0. Szasz. Accepted for publication in the Canadian Journal of

Mathematics.

(55) Nonnegative trigonometric polynomials and certain rational characteristic functions, E. Lukacs and 0, Szasz. Accepted for publication in the Journal of Research of the NBS,

(56) A proof of Hilbert's Nullstellensatz. T, S. Motzkin. Submitted

to a technical journal,

(57) On Fejer sets in linear and spherical spaces. T. S. Motzkin and

1. J, Schoenberg, Accepted for publication in Annals of Mathematics,

(58) On the relaxation method for linear inequalities. T, S. Motzkin

and I. J, Schoenberg. Submitted to a technical journal.
66 Publication Activities

(59) Least p-th power polynomials on a real finite point set. T. S.

Motzkin and 3. L. Walsh. Submitted to a technical journal.

(60) A conjecture of Erdos. M. Newman. Submitted to a technical

journal.

(61) Structure theorems for modular subgroups. M. Newman. Submitted

to a technical journal.

( 62 ) On the Lerch zete function. F. Oberhettinger . Submitted to

a technical journal.

( 63 ) On two problems in abstract algebra connected with Horner's rule,,

A, M. Ostrowski, Submitted to a technical journal.

(6*+) On nearly triangular matrices. A. Ostrowski. Submitted to a technical journal.

(65) Oeterminanten mit ueberwiegender Hauptdiagonale und die absolute

Konvergenz von linearen Iterationsprczessen, A. M. Ostrowski,.

Submitted to a technical journal.

(66) On the convergence of Gauss' alternating procedure in the method

of the least squares, I. A. M, Ostrowski. Submitted to a technical journal.

( 67 ) On the linear iteration procedures for symmetric matrices. A. M.

Ostrowski. Submitted to a technical journal.

(68) On the spectrum of a one parametric family of matrices. A. M.

Ostrowski. Submitted to a technical journal.

( 69 ) On Gauss' speeding up device in the theory of single step iteration, A, M, Ostrowski. Submitted to a technical journal.

(70) On absolute convergence of linear iteration processes, A. M.

Ostrowski, Submitted to a technical journal,

(71) On spectra of second-order differential operators. D. Ray,

Submitted to a technical journal,

(72) On the convergence of asymptotic solutions of linear differential

equations. R. M„ Redheffer and W, Wesow, Submitted to a technical journal.

(73) Numerical computation of low moments of order statistics from a normal population. J. B, Rosser, Submitted to a technical

journal.

(7*+) Most probable rank orders, I. R, Savage. Submitted to a technical journal.

(75) An isoperimetric inequality for closed curves convex in evendimensional Euclidean space, I. J, Schoenberg. Submitted to a technical journal,

( 76 ) Characteristic roots of quaternion matrices, 0. Taussky. Accepted for publication in Archiv der Mathematik.

(77) Generalized commutators of matrices and permutations of factors

in a product of three matrices. 0. Taussky. For publication in

the von Mises Anniversary volume.
Publication Activities 67

( 78) Some computational problems in algebraic number theory. O.Taussky.

To appear in the Proceedings of the American Mathematical Society

Sixth Symposium on Applied Mathematics, held at Santa Monica City

College, August 26-28, 1953.

(79) An improved cathode ray tube storage system. 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.

(80) The condition of certain matrices, II. J. Todd. To appear in

Archiv der Mathematik.

(81) 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 39.

(82) The evaluation of the exponential integral for large complex arguments. J. Todd. Accepted for publication in the Journal of

Research of the NBS.

( 83 ) Discrete approximations to elliptic differential equations.

W. Wasow. Submitted to a technical journal.

(84) On the asymptotic transformation of certain distributions into

the normal distribution. W. Wasow. To appear in the Proceedings

of the American Mathematical Society Sixth Symposium on Applied

Mathematics, held at Santa Monica City College, August 26-28,1953.

(85) An extremum property of sums of eigenvalues. H. Wielandt. Submitted to a technical journal.

(86) Einschliessung von Eigenwerten hermitescher Matrizen nach dem

Abschnittsverf ahren. H. Wielandt. To appear in Archiv der Mathematik.

( 87 ) Error bounds for the eigenvalues of symmetric integral equations.

H. Wielandt. To appear in the Proceedings of the American Mathematical Society Sixth Symposium on Applied Mathematics, held at

Santa Monica City College, August 26- 28 , 1953.

(88) On eigenvalues of sums of normal matrices. H. Wielandt. Submitted to a technical journal.

( 89 ) A note on partially balanced designs. M. Zelen. Accepted for

publication in Annals of Mathematical Statistics.

(90) Analysis for some incomplete block designs having a missing block.

M. Zelen. Accepted for publication in Biometrics.

2.4 Reviews, Notes

(1) Note on the circle theorem of hydrodynamics. E. Levin. Accepted

for publication by the Quarterly of Applied Mathematics.

(2) Two early papers on the relation between extreme values and tensile strength. (Formerly "A historical note on the application

of the ‘weakest link' idea to tensile strengths.”) J. Lieblein.

Submitted to a technical journal.
68 Publication Activities

2.5 Miscellaneous Publications

(1) Contributions to the solution of systems of linear equations and

the determination of eigenvalues. NBS Applied Mathematics Series

39 . In press, Government Printing Office.

(2) Experiments in the computation of conformal maps. To appear in

the NBS 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 ($1.25) and its Supplement ($0.75), available from the Superintendent of Documents, 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.