NATIONAL BUREAU OF STANDARDS REPORT

PROJECTS and PUBLICATIONS

of the

NATIONAL APPLIED MATHEMATICS LABORATORIES

A Quarterly Report

October through December 1953

3119

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

ized 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. Radioinetry. Mass Spectrometry. Solid State Physics. Electron Physics. Atomic Physics. Neutron Measurements. Infrared Spectros- copy. 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.

9Office of Basic Instrumentation ^Office of Weights and Measures.
NATIONAL BUREAU OF STANDARDS REPORT

MBS PROJECT NBS REPORT

MAMl 3119

PROJECTS and PUBLICATIONS

of the

NATIONAL APPLIED MATHEMATICS LABORATORIES

A Quarterly Report

October through December 1953

<NBS>

0. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

tha publication.! APPr°''ed for public release by the

unless permission Director of the National Institute of Standards, Washington 25,d.c, Such p Standards and Technology (NIST) report has been specifically prepared If on October 9, 2015 report for Its own use,
NATIONAL APPLIED MATHEMATICS LABORATORIES

October 1 through December 31 ? 1953

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

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. . .As sis tant Director and UCLA Liaison Officer

Gertrude Blanch, Ph.D. . .Assistant to the Director (Numerical Analysis) Research Staff Forman S. Acton, D.Sc Mathema ticiari George E. Forsythe, Ph.D Mathematician) Harry D. Huskey, Ph.D Mathematician Cornelius Lanczos, Ph.D Mathematician

Theodore S. Motzkin, 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 Walter I. Futterman, M.A. Edwin Mookini, M.A. (UCLA) Eugene Levin, M.A. Mervin Muller, M.A. Genovevo C. Lopez, B.A. (UCLA) Lloyd Philipson, M.A. Roger D. Woods,. M.A. Mathematical Services Unit

Arnold D. Hestenes, Ph.D Mathematician

Frederick H. Hollander, M.A ....Mathematician

Robert R. Reynolds, M.S Mathematician

Everett C. Yowell, Ph.D Mathematician Computing Staff Patricia B. Bremer, B.A. Frank B. Meek, M.A. Nan N. Reynolds, M.S. LaVerne E. Rickard, M.A. Albert H. Rosenthal, B.A. Louise M. Straus, M.A. Marian E. Williams Lindley S. Wilson, M.A. Machine Engineering and Development Unit Ragnar Thorensen, Ph.D Electronic Scientist

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

John Francis, B.S Electronic Engineer

Louis F. Justice, B.S Electronic Engineer

Norman F. Loretz, B.S Electronic Engineer

Robert R, Schmidt Electronic Engineer

Electronic Laboratory Staff Blanch C. Eidem Howard K. Marks Gaylord W. Jones Administrative Unit Mildred Vukmanic . . . . ..Administrative Assistant

Mildred B. Webb Administrative Clerk

Reve Vineyard Administrative Clerk

Leo Moskowitz Procurement Clerk

Secretarial Staff Vendla H. Gordanier Dorothy M. Hibbard

Elsie L. Husman Estelle H. Strauss......... Librarian AsaiS-tmU,

Leola Cutler, B.A. Lillian Forthal, B.A. Benjamin F. Handy, Jr., M.S. Ruth B. Horgan, B.A. Thomas B. Lakin, B.S. Shirley L. Marks, M.A.

MACHINE DEVELOPMENT LABORATORY S

Edward W. Cannon, Ph.D . Chief

Edith N. Reese, B.A Mathematician

COMPUTATION LABORATORY

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

Henry A. Antosiewicz, Ph.D

Philip Davis, Ph.D.. Robert F. Dressier, Ph.D.. William H. Durfee, Ph.D..., Andre N. Gleyzal, Ph.D.... Karl Goldberg, M.A

Peter Henrici, Ph.D

Alan J. Hoffman, Ph.D Morris Newman, Ph.D.......

Fritz Oberhe ttinger , 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

,AU + * Research Associate Mathematician

Mathematician

Mathematician

, Mathematician

Mathematician

,AU** Research Associate Mathematician

Mathematician

,AU++ Research Associate Mathematician

Mathematician

Phy sicis t Physicis t 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 Pruschj Donald I. Rubin, M.S. 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.JL. Notxella, -B.A,. I. Richard Savage, M.S...... Marvin Zelen, M.A

Marion T. Carson......

Yvette B. Cocozzella Leia I. Hamilton——*.- On leave of absence

, . . . Chief

Assis tant Chief

Technical Aid

.Mathematical Statistician Mathematician

Mathematician

...... ...... .Mathematician

Mathematic ian .Mathematical Statistician Computer

Secretary

.Secretary
Contents

Page

Index iv

Status of Projects as of December 31, 1953. . 1 Institute for Numerical Analysis. ..... 1 (NBS Section 11.1)

Computation Laboratory. . . . . . ..... 20

(NBS Section 11. 21

Statistical Engineering Laboratory. .... 46

(NBS Sec tion 11.3)

Machine Development Laboratory 51 (NBS Section 11.4)

Lectures and Symposia 53

Publication Activities 57
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 Applie

Mathematics Laboratories. These services are, however, fully represented

in the body of the report.

A. Research: Pure Mathematics

Miscellaneous studies in pure mathematics ......... 3? 21

B. Research: Numerical Analysis

Baker-Hausdorf f formula ................... 23

Classical numerical analysis, Research in ..... 20

Differential equations, Studies in numerical integration of . 3

Dirichlet problem for certain multiply-connected domains,

Investigation of Bergman's method for the solution of the . 20

Eigenvalues, eigenvectors, and eigenfunctions of linear

operators, Calculation of ................ . 2

Laplace equation, Solution of, by Monte Carlo method 22

Probability methods and sampling techniques . . . . 4

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

SCAMP 8

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. 25

Computing machines, Studies in the theory of digital. .... 10

*Continua, Research in mechanics of 27

Eigenvalues of a clamped plate, Generalized Ray leigh-Ritz

method for. ................... 7

Flow in supersonic nozzle ....... 44

Geomagnetic field, Analysis of ............... . 22

Hypergeometric functions. ......... .... 24

Linear programming, Basic research in ... . 24

" " , Supporting research in 25

Matrix theory, Special problems in finite . „ . 22

Nerve fiber reaction, Differential equation for 23

Painleve equation ............... 34

Program planning, Research in the mathematical theory of. . . 8

Russian mathematical progress, Study of . 7 Scattering functions. 33

Theoretical physics, Miscellaneous studies in . . . 6

Three-body problem. .......... 22

^Vibration modes and frequencies (of aircraft structures),

Computation of. 42

*Water, High temperature properties of . 4l

Water waves 25

*New task
D. Mathematical Statistics

Experiment design. Studies in the mathematics of. ...... 4-3 Non-pararaetric procedures, Procedures of. .......... 4-7 Probability and statistics, Miscellaneous studies in. .... 4-7 Propagation of error, Law of. 4-7

E „ Mathematical Tables

Airy integral., Table of the modified. ............

Antilogarithms, Table of .................. » Arcsin for complex arguments, Table of ......

Bivariate normal distribution function, Tables of the . . „ . .Collected short mathematical tables of the Computation Laboratory ..........................

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

Error function for complex arguments, Table of. .......

Exponential function for negative arguments, Extension of

tables of the .......................

Gamma function for complex arguments, Table of the.

Gases , Table of thermodynamic properties of ........ . Hyperbolic sines and cosines, Extension of the table of . . . Lagrangian coefficients for sexagesimal interpolation, Table

of. ............................

Mathieu functions, radial. .................

Power points of an a lysis-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 ..................

3128

31

12

30

28

28

32

32

28

34- 31

29

33

30

1

1

4-6 3p

34- 33

32

32

29

F . Manuals, Bibliographies, Indices, and Technical Information

A.d.c.m., Logical notation and block diagram symbolism for. . 9

Coding procedures, mathematical tables and numerical

analysis, Bibliography of ................ . 29

Fitting straight lines, Manual on ....... . 46

Statistical literature, Bibliography and guide to ..... . 4-6

G. Computing Machine Development

Air Comptroller's computing machine , ..... 51 Census computing machine, Bureau of the ...... 51 SEAC: National Bureau of Standards Eastern Automatic Computer 52

SEAC, Number-theoretical test problems for 21 SWAC : National Bureau of Standards Western Automatic Computer 9

Stains of Projects

December 31, 1953

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 1 101 -10-51 00A9-AE2

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

Origin: NBS Authorized 2/1 5/49

Sponsor: Office of Naval Research, USN Revised 9/15/49

Manager: G. E. Forsythe

Full task description appears in July-Sept 195-9 issue.

Status: CONTINUED. As part of a service problem (task 1101-40-5131/

53-6, q.v»,p.l4) it was necessary to get the inverse of a certain 43x43

minor Aj of a 45x45 matrix A. The matrix A was symmetric, and all its

eigenvalues and eigenvectors had been previously obtained for another purpose. A general technique was devised with which A^ was obtained from

the known information by matrix multiplications alone. A preliminary draft

of a note explaining the technique has been prepared.

A by-product of the above problem was experience with a standard

iteration for improving the inverse of a matrix. Let X. be an approxima-

-1 K tion to A . The iteration proceeds from the formula x

k+ i = (21 - \A)xk

•

Each element of 21-X^A, whose accuracy is crucial to the method, is obtained by a triple-precision accumulation of single-precision products.

The element is then shifted and converted to a single-precision number in

the range (0,2). The multiplication (2I-Xj,A)X^ is performed with similar

care. Without such care it was found that round-off errors made successive

X^ worse and worsel

For any symmetric positive definite matrix A with eigenvalues
2 Status of Projects

0< A,<.„.^ A , let P(A) denote the condition number A / A, of A. Let D

I n n I be any nonsingular diagonal matrix, A common practical method of scaling A

before computing A

-1 consists in forming DAD, The question arises: how

should D be chosen so that P(DAD) is minimized? Stimulated by a conjecture

of D. M. Young, the question has been studied at the Institute for Numerical

Analysis and by E. G. Straus of U.C.L.A. Matrices A are said to be "best

conditioned" when P(A)<P(DAD) for all D. Best conditioned matrices have now been characterized in terms of a certain geometrical property of the

invariant sets belonging to and A . If A and A . are simple eigenvalues,

the characterization is that the corresponding invariant lines be reflections of each other in the plane in which certain coordinates vanish. This

result proves Young's conjecture that matrices with his Property A are best

conditioned when their diagonal elements are all equal.

A new code for using the conjugate gradient method on SWAG is now being written. The major subroutines, including a triple precision

inner product, have already been written and their testing has just begun.

Several variations of the conjugate gradient method will be tried with

more use than before of the error function E(x)=lx-h| . Participants in the research on this task were G. E. Forsythe,

M. R. Hestenes, F, H. Hollander, E. H. Mookini, T. S. Motzkin, Louise Straus,

and C. B » Tompkins.

Publications: No new ones.

CALCULATION OF EIGENVALUES, EIGENVECTORS, AND EIGENFUNCTIONS

OF LINEAR OPERATORS

Task 1101-10-5100/50-3

(formerly 11.1/1-50-3)

Origin: NBS Authorized 7/1 A9

Sponsor: Office of Naval Research, USN

Manager: M. R. Hestenes

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

Status: CONTINUED. A manuscript has been completed by G. E. Forsythe

entitled "Asymptotic lower bounds for the fundamental frequency of convex membranes." This manuscript has been presented for publication.

A report has been prepared by E. E. Osborne entitled "Solution

of the matrix equation (M- XLD)X=0." This report describes a computational scheme for finding the eigenvalues of this system. It includes

a complete code for carrying out the necessary computations. The matrices

computed are 6x6 matrices with complex coefficients.

Participants in the research on this task were G. E. Forsythe,

M. R. Hestenes, and Louise Straus.

Publications: (1) "Completely continuous normal operators with

property L," by I. Kaplansky; Pac. J. Math. 3., 721-724- (Dec. 1953).

(2) "Asymptotic lower bounds for the frequencies of polygonal membranes,"

by G. E. Forsythe; submitted to a technical journal.
Status of Projects

STUDIES IN PURE MATHEMATICS

Task 1101-10-5100/50-4

(formerly 1101-11-5101/50-4)

Origin: NBS Authorized 7/1A9

Sponsor: Office of Naval Research, USN

Manager: C. B„ Tompkins

Full task description appears in July-Sept 1949 issue.

Status: CONTINUED. T. S. Motzkin studied incompatible algebraic

equations and gave a new proof of Hilbert's Nullstellensatz. He furnished bounds on the degrees and on the number of polynomials needed (also

if the coexf icients are polynomials in additional indeterminates) and developed another incompatibility criterion. Exact bounds for the degree

in a particular case and for the number of polynomials are given. The

"Helly dimension" of the set of all hypersurfaces of degree d in n-space

is shown to be less than or equal to^n °^-1, with equality for an infinite coordinate field.

Related work is reported under various other tasks. Problems

in differential geometry and in the calculus of variations gave rise to some problems involving differential equations reported under task 1101-10-

5100/51-1, (p. 3) . Discrete variable problems are reported under 1101 10-

5100/53-1, (p. 7), 1 101-40-5131/54-1 (p. 1 7) , and 1 101-40- 5 1 30/54-1 ( p.1 0 )

.

Publications: (l) "A proof of Hilbert's Nullstellensatz," by T. S.

Motzkin; submitted to a technical journal. (2) "An isope rimet ric inequality for closed curves convex in even-dimensional Euclidean space," by

I. J. Schoenberg; submitted to a technical journal.

COMPUTATION OF THE COMPLEX ZEROS OF THE KXEKANN-ZETA FUNCTION

Task 1101-10-5100/50-13

(formerly 1101-11-5101/50-13)

Origin: NBS Authorized 6/1/50

Sponsor: Office of Naval Research, USN

Manager: R. Morgan

Full task description appears in Apr-Jun 1950 issue.

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

STUDIES IN THE NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS

Task 1101-10-5100/51-1

(formerly 1 101-1 1 — 5 1 00/5 1 — 1 ) Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Manager: W. Wasow

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. The problem of embedding an intrinsically defined

surface isometrically into three-dimensional space leads to an initial

value problem for a non-linear system of three partial differential equations. C. B. Tompkins has proposed a finite difference method for the

step-by-step solution of this problem. A preliminary study by G. Blanch

of the truncation and round-off errors inherent in this method in the case
4 Status of Projects

of the sphere indicates that the procedure is practicable. It is planned

to prepare a code and to perform exploratory calculations on the SWAG.

Several- techniques for the numerical solution of Plateau's problem on the SWAG are under study. The emphasis is on the parametric case. For use in the setting-up of difference equations or equivalent

random walks, finite neighbor sets with certain properties were investigated by T. S. Motzkin (see publication 5 below). The main properties

considered are (l) every point P of a given n-dimensional lattice has s neighbors Q, s fixed; (2) Q-P is a fundamental vector of the lattice;

(3) a finite number of steps leads from P to every lattice point; (4) the

notion of neighbor is invariant under an n-dimensional discrete translation

group. Examples are given to show that ( 1 ) — ( 4) can be fulfilled for any

s>3 and s<2n. Also, a set of neighbors fulfilling (1), (2), (4) with

s=n+1 is described for which the neighbors of P form a regular simplex

with center P. The asymptotic properties for large A of differential equations

of the form „ n m

y

(n)

+ 5, »

r /

n~ p) - cb.« .,>/—> - o

(a

r ,bg ,cs constants) is being studied by L. Philipson. Particular emphasis

is given to the Stokes phenomena in the complex x-plane.

The methods of Gershgorin and Collatz for the appraisal of the

truncation error in the solution of Dirichlet's problem by finite differences has been generalized by W. Wasow so as . to apply to general types of

nets and to a general class of interpolation schemes, and also to the

"discretization error" connected with solutions of Dirichlet's problem by

random walks that are not restricted to the points of a net. With the

help of the results obtained the extrapolation technique of J. F. Richardson

for the improvement of approximations obtained by finite difference methods

can be analyzed mathematically, justifying it in certain applications, and

showing in others that it is not reliable.

Participants in the research on this project were G. Blanch,

T. S. Motzkin, L. Philipson, C. B. Tompkins, W. Wasow.

Publications: (1) "Asymptotic solution of the differential equation

of hydrodynamic stability in a domain containing a transition point," by

W. Wasow; Ann. Math. 3L8? 222-252(Sept. 1953). (2) "An expansion method

for parabolic partial differential equations," by J. W . Green; J. Res. NBS

51 , 1 27-1 32( Sept . 1953). (3) "On small disturbances of plane Couette flow,"

by W. Wasow; J. Res. NBS 5l? 195~202(0ct. 1953) 0 (4) "On the convergence

of asymptotic solutions of linear differential equations," by R. M. Redheffer

and W. Wasow; submitted to a technical journal. (5) "Neighbor sets for

random walks and difference equations," by T. S. Motzkin; IN MANUSCRIPT.

PROBABILITY METHODS AND SAMPLING TECHNIQUES

Task 1101-10-5100/51-2

(formerly 1101-11-5100/51-2)

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Manager: D. Teichroew

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. The asymptotic expansion for expressing t- and

chi-square variates in terms of normal deviates with the same probability

value has been generalized by the use of an auxiliary parameter. The
Status of Projects

5 generalized expansion permits approximation with greater accuracy than was possible before. A code has been prepared to compute the necessary

coefficients

.

The SWAG has been used to compute two tables of the Incomplete

Gamma Function

1 T

x

? -t a-1

p ~ ~ruy e 1 dt •

•Jo

The first table gives p to 10 decimals for a= 1 ( 1 ) 20 and the second gives

x for p= .001 ( .001 ) .999 for selected values of a between 5 and 100.

p Initial runs on the SWAG have been made for determining the

Power Function and the Expected Sample Size for Sequential Tests of Linear

Hypotheses. The Efficiency of this procedure is being compared to that

of the classical fixed sample size approach for the case of Analysis of

Variance

.

M. Muller is investigating procedures for improving the efficiency of the Monte Carlo method for solving boundary value problems.

A SWAG code was prepared by G, Forsythe for the solution of a certain Dirichlet problem in a square by random walks.

Two new procedures for generating random numbers were outlined

by T. S. Motzkin.

Participants in this task research were G. Forsythe, T. Motzkin,

M. Muller, D. Teichroew, and W, Wasow. Some of the coding was done by

L. Forthal.

Publications: "On the asymptotic transformation of certain distributions into the normal distribution," by W. asow; to appear in the Proceedings of the American Mathematical Society Sixth Symposium on Applied

Mathematics, held at Santa Monica City College, August 26-23, 1953.

VARIATIONAL METHODS

Task 1 101-10-5100/51-3

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Manager: M. R . Hestenes

Full task description appears in July-Sept 1950 issue.

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

STUDIES IN APPLIED MATHEMATICS

Task 1 101 -1 0-51 00/51-5-

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.

Status: CONTINUED. The seminar on numerical analysis met 22 times

during the quarter. A detailed list of sessions will be found under Lectures and Symposia in the back of this issue. The third summary of presentations, covering the period from June 1 5 to August 1 3 > was prepared

by T. S. Motzkin.
6 Status of Projects

As a preliminary to coding of numerically or analytically given

functions, T. S. Motzkin investigated the efficiency of operational se- quences, measured by the number k of parameters of the family obtained.

It was shown that for a sequence of n two-entry operations Ck of any kind

other than addition and an unlimited number of additions, k<2n+1 . If all

CK are divis ions , the algorithm of recurrent fractions gives, even for one variable, k=2n+1; the same is true for partial fractions, with a limitation

on the resulting family of rational functions by the reality of the roots

of the denominator. If all CK are multiplications, the bound for one variable is k=2n, an improvement on the Newton-Horner synthetic division

procedure. If a number is "forgotten" when used, the bound for m multiplications and n divisions is 2(m+n)-[ (m-1 )/(n+1 ) ] , where [ ] denotes

the largest integer function. The determination, by Tchebyshev methods,

of the best coefficients to be used for these approximating rational functions was also studied,

E, Levin did some work on the Circle Theorem of Hydrodynamics.

One of the hypotheses of the Circle Theorem was that there be no rigid

boundaries in the plane. This restriction may be somewhat relaxed and one can obtain a necessary and sufficient condition for admissible rigid

boundaries

.

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

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

by P. Erdos and G„ Hunt; Pac . J. Math. 3? 673-637 (Dec. 1953). (3) "A

numerical analyst's fifteen-foot shelf," by G. E. Forsythe; MTAC VII,

221-228 (Oct. 1953). (5) "Seminar on numerical analysis—summary of

presentations between April 20 and May 13? 1953?" by C. B. Tompkins;

multilithed typescript, 10 p. (5) "Seminar on numerical analysis—suaaaary

of presentations between May 18 and June 8, 1953?" by G. E, Forsythe;

multilithed typescript, 38 p. (6) "Third Summary—Seminar on Numerical

Analysis, June 1 5~Augu s t 13? 1953?" by T. S, Motzkin; multilithed typescript? 38 p. (7) "Note on the circle theorem of hydrodynamics," by

E. Levin; submitted to a technical journal.

MISCELLANEOUS STUDIES IN THEORETICAL PHYSICS

Task 1 101-10-5100/51-5

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

Sponsor: " " Manager: R. D. Woods

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. W. Futterman is concluding theoretical and

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

R. Woods is continuing programming and preliminary computations of the

elastic scattering of protons by various elements, based on the optical

model of the nucleus.

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

D. S. Saxon and A. S. Cahn; Qu. J. Mech. App. Math. (.Oxford) VI, Ft. 3?

273_ 285( 1 953) • (2) "Application of variational methods to intermediate • and high energy scattering," by E. Gerjuoy and D. S. Saxon; submitted

to a technical journal. (3) "Variational principles for the acoustic

field," by E. Gerjuoy and D. S. Saxon; submitted to a technical journal.
Status of Projects 7

STUDY OF RUSSIAN MATHEMATICAL PROGRESS

Task 1 101-10-5100/52-1

Origin: NBS Authorized 3/15/52

Sponsor: Office of Naval Research, USN

Manager: G. E. Forsythe

Full task description appears in Jan-Mar 1952 issue.

Status: CONTINUED. The task manager is continuing to accumulate

bibliographical cards on Russian mathematical monographs (see Jan-Mar 1952

issue, p. 11, and Oct-Dec 1952 issue, p. 8). The translations formerly

reported under this task are now reported in connection with the pertinent

tasks of the Institute for Numerical Analysis.

The following report has been re-run and is available in limited

quantities: "Bibliographical survey of Russian mathematical monographs,

1930 to 1951." A supplement to the above is also available.

GENERALIZED RAYLEIGH- RITE METHOD FOR EIGENVALUES OF A CLAMPED PLATE

Task 1101-10-5100/53-1

Origin: NBS Authorized 12/29/5

Sponsor: Office of Naval Research, USN Terminated 12/31/5

Manager: C. B. Tompkins

Objective: To test the method developed by N. Aronszajn, by use of

the theory of Hilbert space, for eigenvalue problems. Specifically, let

there be given a square plate, clamped at the edges, and defined by

|xj <1 ; [y|<1. Let u be a function satisfying

(I) = -Stt = 0

3n on the boundary.

The objective is to find

has nontrivial solutions

that

approximations to the eigenvalues jx for which

In the above A is the Laplacian operator so

4- 4- ,2 _ 9 u . 9 u A u =

3x

b 3y"

(1 )

Background: Upper bounds for the eigenvalues //can be obtained by

the Rayleigh-Ritz method, which involves the solution of a system of

algebraic equations. The disadvantage of the method is that at any stage

upper bounds are obtained only for the first n eigenvalues, where n is

the order of the determinant associated with the approximation. In the

present study, an attempt will be made to apply Aronszajn' s generalized

Rayleigh-Ritz method for differential problems. The advantage of the

method is that at every stage upper bounds can be obtained for all

eigenvalues. Moreover, it is hoped that at the n-th stage, the approximation obtained from the generalized method is better than the corresponding approximation obtainable from the n-th stage of the ordinary

Rayleigh-Ritz method. However, the generalized method involves a com- puting process which is considerably more involved than the ordinary

method, and one of the prime objects of the study will be to test whether

programming of the process for high-speed equipment is feasible. The

method involves using as auxiliary functions the known solutions for the

clamped circular plate (for the circle inscribed in the square).

Status: TERMINATED.

C\J

nO
Status of Projects

RESEARCH IN THE MATHEMATICAL THEORY OF PROGRAM PLANNING

Task 1101-10-5102/50-11

(formerly 11.1/1-50-11)

Origin: Office of Air Comptroller, USAF Authorized 6/

1

5/50

Sponsor: " " Manager: T. S. Motzkin

Full task description appears in Apr-Jane 1950 issue.

Status: CONTINUED. The fundamental problem of the existence of a solution of a problem of linear programming or linear inequalities was investigated from a probabilistic point of view. Concerning a system of

n linear inequalities in m unknowns, it is shown, under rather general

suppositions concerning the probability distribution of the coefficients,

that the probability p for compatibility is

It follows readily that p=1/2 as n=2m+1

.

The analogy was studied between the theory of linear inequalities

(in particular, theorems on implication, incompatibility, duality, and

Helly dimension) and those of linear disphantine equations and of algebraic

equations (see task 1 101-10-5100/50-4, p. 3 ).

Continued attention was given to the assignment problem (see

July-Sept 1953 issue, p. 7_ 8) and its ramifications.

Progress was made in the write-up of the Bibliography on Linear

Inequalities and Programming, and of the chapter on linear inequalities

for the Handbook of Numerical Analysis.

Reference is made to related work of R. Morgan and C. B. Tompkins

reported under tas k 1101 -40- 5130/54- 1

, p. 10 . Publications: Same as given in July-Sept 1953 issue, p. 9-10.

SCAMP

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

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

Sponsor: " " Manager: C. B. Tompkins

Full task description appears in Apr-June 1953 issue.

Status: CONTINUED. Research concerning problems with discrete

variables continued. Attention was still directed toward problems in

which variables are permutations. M. Hall, Jr., and J. D. Swift made

suggestions which have led to a variation in the code which will probably

permit determination of all Steiner triples of order 15 without inacceptable repetition of isomo rphs . A study of results of a problem of

exhaustive search for solutions of non-linear equations involving permutations has been made and a report will be submitted to the sponsor.

F. B. Meek, J. D. Swift, and C. B„ Tompkins have initiated a systematic

search for a pair of orthogonal Latin Squares of order 10. Computations

for SCAMP are reported as tas k 1 101-40-5131/54-1 , p. 17 . Prospective participants in the more active phase of the program

during the summer of 1954 have received letters requesting them to es- tablish clearance and to take care of other formalities involved in their

participation

.

M. Newman has completed and checked a SEAC code for the com-
Status of Projects 9

putation of the Hermite normal form of a matrix with rational integral

elements, of order not exceeding 200. Several small matrices have been

run, and the calculation of the Hermite normal form of a 91x91 incidence

matrix for a non-De s arguian finite plane projective geometry of order 9

is in progress.

2 . Development

NATIONAL BUREAU OF STANDARDS WESTERN AUTOMATIC COMPUTER (SWAC)

Task 1101-20-5103A9-1

(formerly 1 101-34-5103A9-1 ) Origins Aeronautical Research Laboratory, Authorized 11/1 A3

Wright Air Development Center, Air Research

and Development Command, USAF

Sponsor: " " " Full task description appears in Apr-June 19^9 issue.

Status: CONTINUED, The SWAC has been used on a two-shift basis

on both service and research problems. In recent weeks approximately 55

hours out of an available 60 have been used for productive computing.

Approximately three hours per day are used for preventative maintenance.

There have been no engineers on the staff at the Institute since

November 1 , all maintenance is being done by contract with Magnavox

Company

.

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

Task 1 101-20-5103A9-2

(formerly 11.1/22-/9-2)

Origin: NBS

Sponsor: Aeronautical Research Laboratory, Wright

Air Development Center, Air Research and

Development Command, USAF

Manager: H. D. Huskey

Objective: To establish a suitable and consistent terminology,

notation, and symbolism for the automatic computing machine (SWAC) being

constructed under task 1 101-20-5103A9-1 (formerly 11.1/22-49-1), with

a view to developing thereby an acceptance proposal for a standard languagefor describing logical aspects of a.d.c.m. in general.

Background: Because of the diversity of groups working on computing

machines a number of different sets of conventions regarding terminology

and symbolism are currently in use. Criticism has developed among the

mathematicians of the Institute for Numerical Analysis regarding the consistency, compactness, and suitability of some of the systems of notation

and symbolism now in use. It has seemed to be definitely worthwhile from

the viewpoint of the research program of the Institute to develop a no- tation and symbolism for the Institute machine which will be acceptable

to the research staff, and it is hoped that if this task is accomplished

Authorized 2/1 5A9

Terminated 12/31/53
10

thoughtfully

Status

:

Status of Projects

a standard of wide applicability may result.

TERMINATED.

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.

Status: CONTINUED. New systems of coding are being studied. A

floating point card controlled method of using the SWAG was perfected

by Melkanoff of UCLA. Elementary automatic programming of formulae was started and more elaborate interpretive routines are being developed by

Huskey

.

A simple foundation of automatic coding, outlined by T. S.Motzkin,

involves the standardized use of closing brackets as evaluation command.

Techniques for expediting code checking are being studied by

several staff members.

3 . Mathematical Service s

COMPUTING SERVICES FOR RESEARCH STAFF OF

THE INSTITUTE FOR NUMERICAL ANALYSIS

Task 1 101-40-5130/55-1

(formerly 1101-40-1111/59-1 a)

Origin: NBS Authorized 9/2/48

Sponsor: Office of Naval Research, USN Revised i1/l6/‘+9

Manager: F. Hollander

Full task description appears in July-Sept 1949 issue (see task 11.1/32-49-1).

Status: CONTINUED. Research staff problems involving the use of

SWAC were:

(1) For D. H. Lehmer and J. Self ridge: (a) Runs were made of? several

Mersenne primes of large size to obtain the penultimate remainders

of the numbers. The routine was a modification of the one originally coded to determine primality of numbers of form 2 n -1 by a theorem of Lucas. (b) A difference ^et of order 127 wa,s checked.

(c) All solutions were found for: a"1 °-b * = 1 (mod p), a2'F-b24=i ( mo d p)

where p=nk+1 , n is the exponent, and k is an arbitrary

integer equal to 1,2,..., for all primes less than 1000. (d) Certain prime numbers were tested by Alway's method. (e) A backlog

problem has been set up which consists of tests of irregular primes

in the investigation of Fermat's last theorem.

For R. Woods: Code checking and calculations were continued on an optical model for calculating the scattering of protons by heavy

nuclei.

( 2 )
Status of Projects 1 1

(3)

(4)

(5)

( 6 )

(7)

( 8 )

(9)

( 10 )

( 11 )

For the Geophysics Department of U.C.L.A.: Data processing was begun on the problem of the high order tidal constituents for

earth and ocean tides.

For the Physics Department of U.C.L.A.: (a) Computation has been

started on the solution of the molecular wave function for the Li„

molecule. (b) Computations have been done to obtain the eigenvalues of a system of non-linear differential equations in a problem of nuclear scattering. A method involving a "corrector-predictor" was used. (See MTAC VI, 253-254, Oct. 1952.) Coding was done by P. Kaus and B. F. Handy.

For the Education Department of U.C.L.A.: Computations were continued on item analysis of certain psychological instruments.

For the Business Administration Department of U.C.L.A.: Determination of working capital needs by stochastic methods, on various

simplified models of business organization.

For the Chemistry Department of U.C.L.A.: Calculations in the

problem of crystal structure, involving a three-dimensional Fourier

analysis, and calculations on Fourier transforms.

For D. Teichroew: Tables of the incomplete 'T'-function were calculated.

For T. Thomas: Several algebraic and exponential equations were

so Ived

.

For C. B. Tompkins: (a) Two large sets of linear inequalities derived from a linear programming problem have been attacked by

cyclic projections with acceleration. A stable cycle indicating

no possible solution was found for one and a cycle probably stable

within roundoff error was found for the other. These sets each

involved 128 inequalities in 80 unknowns, and the solution was coded for SWAC by R. B. Horgan. (b) Solution of games has been

attempted in which the payoff matrix has a parameter which must be

set to make the value of the game zero. It is hoped to solve games

up to ten by ten in size derived from studies of attrition pro- cesses. The method is the method of fictitious play due to Brown

and von Neumann accelerated by a scheme of cyclic under-relaxed

projections

.

For W. Futterman: Code checking was begun on a routine to calculate the matrix elements occurring in the photoelectric disintegration of the deuteron.

PUNCHED CARD LIBRARY

Task 1101-40-5131/49-2

(formerly 1 101—53— 1 101/49—2)

Origin: NBS Authorized 7/14/43

Sponsor: Aeronautical Research Laboratory, Wright Air Development

Center, Air Research and Development Command, USAF

Manager: F. .Hollander

Full task description appears in Apr-June 1949 issue.

Comments: A catalog of tables on punched cards which are on file at the Institute may be obtained by addressing the Institute for Numerical

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

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

will be duplicated upon request, provided the requester furnishes the

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

Status: CONTINUED. Requests were received for tables of Bessel

functions Y (x), Y^ (x) and K

q (x), K^x), 6 2

, from Stanford University; and
Status of Projects

12

for E„(x) =

^°°

(e~~ xu/u 2 )du from Iowa State. These were reproduced from

2 J1 our files and shipped.

The table of the volume under the bivariate normal surface and

auxiliary tables relating to this function were added to the library. The

functions tabulated ares

(1) L(h,k,r) = V -

3—/

' exp ( ^—

5

- (x 2+y

2-2rxy) l dxdy

2 irJ^ l J k |

2( 1-r ) j

for h= 0 (. 1)4.0, k=0 (.l)h, and r= 0 ( .05)

.

9 5 (

.

01 ) 1 . 00 ; values are given to

at least 6d

,

for positive and negative values of r

.

( 2 ) V(h,k)

and

(3) V(k,h), where k/h= A. , for h=0 (. 01)4.0, A=0. 1 ( . 1 ) 1 . 0 .

STATISTICAL SMOOTHING

Task 1 101-40-5131/51-19

(formerly 1 101—5 3~ 1 101/51 — 19)

Origins Stanford Research Institute, Stanford Authorized 1/15/51

University Completed 12/31/53

Sponsors Office of Research Operations, U.S.Army

Managers N. Reynolds

Objectives To perform the statistical smoothing of empirical twodimensional data.

Statuss COMPLETED. Three cases were completed this quarter.

TABLES OF THE BIVARIATE NORMAL DISTRIBUTION FUNCTION

Task 1101-40-5131/51-32

(formerly 1 101 - 53- 1 101 /51 -32 ) Origins Division 13, N8S Authorized 5/31/51

Sponsors Office of Chief of Ordnance, U.S. Army Completed 12/31/53

Managers C. B. Tompkins

Objectives To compile a table of the bivariate normal distribution

function, defined by

L(h,k,r) :JjL

2

i-. y.2-

2 r^y)

2(1 - r 2

) dx dy.

Backgrounds The function L(h,k,r) has been tabulated and published

in Karl Pearson’s Tables for Sta ti sticians and Biometricians , part II, (1931)

p. 78-137, Biometric Laboratory, University College, London, for the

following range of parameters + r=0(.05)lj h ,k=0( .1 )2.6. This table has
Status of Projects 13

been extended by Evelyn Fix to cover a range of h and k up to 4, for

the same range of r . This extension is unpublished, but available in typewritten form at the University of California Statistical Laboratory, Berkeley. The major portion of the compilation will comprise these two basic

tables. They will be unified and perhaps augmented by suitable auxiliary

tables to increase their usefulness. Also the values will be keypunched,

so as to be useful in conjunction with IBM computations.

Status: COMPLETED. The tables have been completed and some distribution in preliminary draft and by means of punched cards has been completed. Dr. Gertrude Blanch was the chief participant in the accomplishment of this task.

SIMPLIFIED ROLLING PULLOUT EQUATIONS

Task 1 101-40-51 31/51-34

(formerly 1 101 — 53— 1 101/51 -3*0

Origin: Cornell Aeronautical Laboratory Authorised 6/22/5'i

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, ARDC, USAF

Manager: F. Hollander

Full task description appears in Apr-June 1951 issue.

Status: CONTINUED. Six sets of observed data were received, and

modifications of the code are being made to integrate the equations using

this information. Some computation on the first set of data has been done.

LOW MOMENTS OF ORDER STATISTICS

Task 1101-40-5131/51-36

(formerly 1 101 — 53 — 1 101 /5

1

- 36)

Origin: University of Oregon

Sponsor: Office of Naval Research, USN

Manager: H. D. Huskey

Full task description appears in Apr-June 1951

Status: CONTINUED. The routine for the

Authorized

issud

.

integral (1/2TF)

6/22/51

-t^/2

e

z 7 ^dt

u-oo [x = -12. 00 ( .02)12.00] has been coded by S. Marks and the code has been

checked out.* Conversion routines have been coded by B. F. Handy, Jr.

SIERRA WAVE PROJECT

Task 1 101-40-5131/52-36

(formerly 1101-53-1101/52-36)

Origin: Department of Meteorology, U.C.L.A. 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.

Status: CONTINUED. A code is being developed for the reduction on SWAG of theodolite data for several more flights.
l4 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.

Status: CONTINUED. The 45x45 matrix, whose eigenvalues and eigenvectors were determined previously was inverted after deletion of two

rows and columns to remove two singularities. An approximation to the

inverse, with an accuracy of four decimals, was forwarded to the proposer

of the problem.

METEOROLOGICAL MEANS

Task 1 101-40-51 31/53-10

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. (See also project

1 1 .1/31-50-17, Jan-Mar 1950 issue, p. 15.)

Status: CONTINUED. All computations for the 850 mb level have been

completed. Coding for the "Mountain Term" is finished and checked, and some computing has been done. ("Mountain Term" represents the equations

used to compute the pressure differences across mountain ranges, where

these intersect a given plane of elevation above sea level, at a given

latitude .

)

Coding for and computation of the time averages of the "Mountain

Term" remain to be done. Vertical integrations of these time averages, as well as of the pressure differences for each day and latitude, are also

not yet computed. The 100 mb level for Jan-Feb 1949 is still to be done.

Computation on this problem has been stopped until further

funds are available.

DISCRIMINANT FUNCTIONS

Task 1 101-40-5131/53-13

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

USAF

Sponsor: " Manager: D. Teichroew

Full task description appears in Oct-Dee 1952 issue.

Status: CONTINUED. Preliminary results are being examined.
Status of Projects 15

BIO ASSAY PROBLEM

Task 1 10 1 -40-51 31 / 53- 24-

Origin: Stanford University Authorized 3/31/53

Sponsor: Office of Naval Research, USN

Manager: D. Teichroew

Full task description appears in J an-Msr 1953 issue.

Status: INACTIVE. Results are being examined by the contractor.

SINGLE SHOT PROBABILITIES

Task 1 101 -1+0- 5131 /53-25

Origin: Naval Air Missile Test Center, Point Mugu Authorized 3/31/5

3

Sponsor: Bureau of Aeronautics, USN Terminated 12/31/53

Manager: D. Teichroew

Objective: To investigate the distribution of random variables

in populations in which parameters themselves have a probability distribution and in particular to determine the effect of such distributions

on single shot hit probabilities.

Background: In testing a hypothesis by means of a sample of individuals one usually assumes that the individuals are a random saraole from

a polulati on with fixed values for its parameters. Sometimes it is more

realistic to assume that the values of the parameters may be different

for the different individuals, in particular, that in the whole population

the parameter values have a certain given distribution. For example,

it has been found that accident frequencies sometimes resemble a negative

binomial distribution more closely than they do a Poisson distribution;

the negative binomial distribution would be expected if one assumed that

the number of accidents happening to one person is a Poisson variate and

that over a number of individuals the mean has a Gamma distribution.

Status: TERMINATED.

DISCRETE MINIMAL SPACES

Task 1 101 -40- 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 J an-Mar 1953 issue.

Status*: INACTIVE. For status to date see Apr-June 1953 issue
1

6

Status of Projects

ROOTS OF DETERMINANTS

Task 1 101-40-51 31/53-32

Origin: North American Aviation Company Authorized 3/31/53

Sponsor: Atomic Energy Commission Completed 12/31/53

Manager: H. D, Huskey

Objective: To determine the value of a variable such that a determinant whose elements are functions of this variable should vanish.

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

AEC group at North American Aviation Company. Several sets of these determinants are to be submitted for solution.

Status: COMPLETED.

LINEAR EQUATIONS (CONVAIR)

Task 1101-^0-5131/53-35

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.

Status: CONTINUED. The double precision code was completed but

has not been checked out. The system of equations was withdrawn by the

originator of the problem. A new set of equations will be submitted for

solution.

MONTE CARLO STUDIES

Task 1101-40-5131/53-39

Origin: RAND Corporation Authorized 3/31/53

Sponsor: Aeronautical Research Laboratory, Wright Air Development

Center, Air Research and Development Command, USAF

Manager: H. D. Huskey

Full task description appears in Jan-Mar 1953 issue.

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

RANKING PROBABILITIES

Task 1 101-40-5131/53-40

Origin: Cornell University Authorized 3/3 1/53

Sponsor: Office of Naval Research, USN

Manager: D. Teichroew

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. A final report is being prepared.
Status of Projects 17

ACCEPTANCE TESTS

Task 1101-40-5131/53-^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.

Status: CONTINUED. E. C. Yowell prepared a report on acceptance

testing entitled "Recommendations for an Acceptance Test for a Consolidated

Engineering Corporation Model 30-201 Computer." L. Forthal of INA and

M. Carr of JPL have begun coding the procedure.

B. P. A. Studies

Task 1101-40-5131/53-42

Origin: Bonneville Power Administration Authorized 6/29/53

Sponsor: " Manager: R. R. Reynolds

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

Status: CONTINUED. Reports 1, 3^ and 5, by R. J. Cypser, P, R. Johannessen, and J. J. Carey of the Economy Loading Research Group, Massachusetts Institute of Technology, were reviewed . A report was submitted

to the sponsor.

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.

Status: CONTINUED. The first phase of the problem is to compute

two polynomials in /.of type det (AF+G) . Here F, G are of orders 9 or 10,

real, non- symmetric . Since F is singular, we are going to compute

det(G~lF+ A.-1l) as a polynomial in A~1 . Let A=G-

'F. It is hoped to find the characteristic polynomial of A by elimination from a sequence of

vectors x, Ax, A2x,...,An x, where n is the order of A. Participants in the solution of this problem are G. E. Forsythe,

F. H. Hollander, Louise Straus.

COMPUTATIONS FOR SCAMP

Task 1101-40-5131/54-1

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.
13 Status of Projects

Status; CONTINUED. E. Lehmer and M. Hall, Jr . > studied the use of

multipliers in seeking difference sets. E. Lehmer coded the problems and

ran several cases of interest; no new difference sets were found, but the

possibilities were exhausted in some cases where there was doubt as to whether a set exists. L. E. Rickard computed some probability tables upon

which rough estimates of the extent of some problems involving search

through all permutations can be based. The code of L. E. Rickard for the

computation of Steiner triples was modified to decrease the number of

isomorphic outputs, and computation of Steiner triples of order 1 5 was undertaken by R„ B. Horgan and J. D. Swift. A problem involving exhaustive

search through permutations was solved by C. B. Tompkins with special

attention being paid to experience concerning the amount of data needed

and the size of batches of permutations which could be rejected from com- petition on the basis of one piece of information. B. F. Handy, Jr.,

coded a continuous variable approach to a problem closely related to the one computed by C. B. Tompkins, but experience with computing showed that

time requirements were excessive and a new code is being prepared.

CRUISE CONTROLLER

Task 1101-40-5131/5^-3

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

Sponsor: Aeronautical Research Laboratory, Wright Air Development Center,

Air Research and Development Command, USAF

Manager: F. Hollander

Full task description appears in July-Sept 1953 issue.

Status: INACTIVE. The originator is continuing analysis of the

problem

.

RAM ROCKET TRAJECTORIES - PARAMETER STUDY

Task 1 101-40-5131/54-4

Origin: Marquardt Aircraft Company Authorized 9/29/53

Sponsor: Aeronautical Research Laboratory, Wright Air Development

Center, Air Research and Development Command, USAF

Manager: F. H. Hollander

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

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

and coding consultation as is necessary.

Status: COMPLETED. All work was performed by Marquardt Aircraft

Company employees, except for necessary coding assistance and operating

assistance on SWAC

.
Status of Projects 19

LINEAR EQUATIONS (BERKELEY)

Task 1101-40-5131/54-6

Origin: Statistical Laboratory, University of Authorized 12/29/53

California, Berkeley

Manager: H. D. Huskey

Objective: To utilize existing codes to solve sets of linear

equations

„

Background: Sixteen sets of 10 linear equations in 10 unknowns, and

1

6

sets of 11 linear equations in 11 unknowns, were submitted for solution

on the SWAC.

Status: COMPLETED (NEW). The equations were solved using a routine

coded by J. Pettit of the Geophysics Department of U.C.L.A. This code

operates in floating binary notation, and utilizes a method of direct

elimination of the unknowns. Computing time was approximately three hours.
II. COMPUTATION LABORATORY

(Section 11.2)

1 . Research

RESEARCH IN CLASSICAL NUMERICAL ANALYSIS

Task 1102-10-1104/50-1

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

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

Full task description appears in Jan-Mar 1950 issue.

Status: CONTINUED. H. Antosiewicz investigated the boundedness of

every solution of equations of the type x+0(x,x)x+h(x)=e(t) whenever the

forcing term e(t) is integrable over the entire positive half line.

M. Abramowitz and W. F . Cahill are continuing their work on the

vibration of plates. A preliminary report on the use of difference equations is being prepared.

M. Abramowitz prepared a paper on the practical evaluation of

integrals.

In continuation of earlier work of Gohner (Ing. Arch. 1, 6 1

9

(1930)) and Freiberger (Australian J. Sc. Res. [A] 2, 355 (195-9)),

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

finite circular cross-section. It is shown that Freiberger's solution

of the problem can be developed in terms of powers of the spring index.

The first three terms of this power series reproduces Gohner' s earlier

approximation

.

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

by H. A. Antosiewicz; submitted to a technical journal. (2) "Forced heat

convection in laminar flow through a tube" by M. Abramowitz; IN MANUSCRIPT

(3) "On helical springs of finite thickness" by P. Henrici; IN MANUSCRIPT.

(4) "Computa tion of vibration modes and frequencies on SEAC" by W. Cahill

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

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

"On the estimation of quadrature errors for analytic functions" by

P. Davis and P. Rabinowitz; submitted to a technical journal. (7) "The

practical evaluation of integrals" by M. Abramowitz; IN MANUSCRIPT.

RESEARCH IN MODERN NUMERICAL ANALYSIS: INVESTIGATION

OF BERGMAN'S METHOD FOR THE SOLUTION OF THE DIRICHLET

PROBLEM FOR CERTAIN MULTIPLY CONNECTED DOMAINS

Task 1102-10-1104/50-2

(formerly 11.2/11-50-2)

Manager: P. Davis

Full task description appears in Jan-Mar 1950 issue.

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

Origin: NBS Authorized 3/1/50

20
Status of Projects 21 MISCELLANEOUS STUDIES IN PURE MATHEMATICS

Task 1102-10-1104/50-4

(formerly 11.2/11-50-4)

Origin: NBS Authorized 1/1/50

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

Full task description appears in Jan-Mar 1950 issue.

Status: CONTINUED. 0. Taussky-Todd prepared a written report of her

talk at the Santa Monica Symposium 1953 (see item 1 under publications

below)

.

The authors of publication (2) have extended their work to include a comparisori of the number of absolute points of a correlation p

with the number of absolute points of , where k is prime to the order

of p . 0. Taussky-Todd is studying the significance of normal matrices

in certain aspects of algebraic number theory where only symmetric matrices

had been considered previously.

Publications: (l) "Some computational problems in algebraic number

theory" by 0. Taussky-Todd; submitted for inclusion in the Proceedings

of the American Mathematical Society 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. Kenrici ; accepted by Archiv der Mathematik. (5) "On

certain series expansions involving Whittaker functions and Jacobi polynomials" by P. Henrici; submitted to a technical journal. (6) "Linear

functional equations end interpolation series" by P. Davis; submitted to a technical journal.

NUMBER THEORETICAL TEST PROBLEMS FOR SEAC

Task 1 102-10-1 104/50-5a

(formerly 1

1

. 2/ 1 1-50-5)

Origin: NBS Authorized 6/1/50

Managers: 0. Taussky-Todd and K. Goldberg

Full task description appears in Apr-Jun 1950 issue.

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

for which 2P_

^-1 is divisible by p

1- is being run as time is available

and is now extended to p<75?000 with no positive results other than the

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

The routine to determine certain congruence properties of the

fundamental unit t+u/p in the field R(/p) where p is a prime has been

used to verify the conjecture that u^0(mod p)for all primes of the form

4n+1 less than 100,000. (See N. C. Ankeny, E. Artin, S. Chowla, "The

class number of real quadratic fields," Ann. Math. 59, 480, 1952.)

The table of least positive primitive roots is being extended

as time is available.
Status of Projects 22

SOLUTION OF LAPLACE EQUATION BY MONTE CARLO METHOD

Task 1 102-1 0-11 04/51-6

Origin: NBS Authorized 9/20/50

Manager: M. Abramowitz

Full task description appears in July-Sept 1950 issue.

Status: INACTIVE. For status to date see Apr-June 1952 issue, p.31+

.

THREE- BODY PROBLEM

Task 1102-21-1104/52-4

Origin: NBS Authorized 10/5/51

Manager: A. Goldstein Terminated 12/31/53

Objective: To develop and test a machine method for calculations

of particular orbits in the restricted planar three- body problem.

Background: The

motion of an infinites

bodies of finite size

.

be solved in closed fo

ferential equations of

of the problem is that

and moon; particular o starting at the earth

maining between earth

will be tabulated.

"restricted" three-body problem is that of the

imally small body in the gravitational field of two

It has long been known that the problem cannot

rm; therefore, numerical integration of the difmotion is appropriate. Among the many applications

to a missile moving under the influence of earth

rbits of interest are, for instance, (1) orbits

and ending at the moon, or (2) stable orbits reand moon. In this project some orbits of type (l)

Status: TERMINATED.

ANALYSIS OF GEOMAGNETIC FIELD

Task 1102-10-1104/52-8

Origin: NBS Authorized 8/10/51

Manager: C. J. Swift

Full task description appears in July-Sept 1951 issue.

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

SPECIAL PROBLEMS IN FINITE MATRIX THEORY

Task 1102-10-1104/52-34

Origin: NBS Authorized 11/6/51

Manager: 0. Taus sky-Todd

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. H. Wielandt prepared two reports on his work

concerning (a) the location of the eigenvalues of A+B, where A and B are normal matrices with known spectra; and (b) a maximum-minimum characteristic of sums of eigenvalues leading to linear inequalities for the eigenvalues of three hermitian matrices A,B,C satisfying A+B=C.
Status of Projects 23

A. Hoffman has shown that one may adapt the methods of Wielandt

to derive minimax characterizations of the product of singular values of

a matrix or the product of eigenvalues of a hermitian positive semidefinite matrix. One such characterization yields inequalities relating

the singular values of matrices A,B,C where C=AB„

Publications: ( 1 ) "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 be included 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 condition 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) "Einschlies sung von Eigenwerten hermitesc her Matrizen

nach dem Abschnittsverf 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. Wielandt; submitted to a technical journal.

DIFFERENTIAL EQUATION FOR NERVE FIBER REACTION

Task 1102-10-1104/53-15

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

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

Full task description appears in Oct-Dec 1952 issue.

Status: CONTINUED. Computations are performed as requested by the

sponsor.

BAKER-HAUSDORFF FORMULA

Task 1102-10-1104/53-16

Origin: NBS Authorized 12/8/52

Manager: 0. Taussky—Todd and K. Goldberg

Full task description appears in Oct-Dec 1952 issue.

Status: CONTINUED. Work on the publication mentioned below continued

Publication: "The equation e z=e

x *e^ in a free associative ring" by

K. Goldberg; IN MANUSCRIPT.
24 Status of P rejects

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.

Status: CONTINUED.

Publication: "Programs for computing hypergeometric series" by

W. F. Cahill; to appear in Mathematical Tables and Other Aids to Computation .

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 and A. Hoffman

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

Status: CONTINUED. A new proof of equivalence of two theorems of

the alternative for pairs of matrices has been obtained (see publication

item 1 below). The proof is purely algebraic, making no use of the

hyperplane separation principle. It is based, instead, on the relation

A-*--4- = A, where A is a submanifold of a linear space.

The relevance of the second part of the Le Chatelier Principle

to problems in linear programming is considered in publication item 7

below. Of the four reasonable interpretations in the linear programming

context of this part of the principle, three are shown to be false, and

a fourth is true, but essentially trivial. These results clarify a heuristic discussion given elsewhere.

The method used in proving an extension of an inequality of

Hardy, Littlewood and Polya (see publication 7 below) has led to the

following problem. Let jTj, ..., r be given constants. Determine the

vertices of the set of all vectors x=(x.,...,x ) satisfying x.<x..,,

k k

I n 1 1+1 ’ ZL H r., k= 1 , . . .

, n . A geometric characterization has been obi=1 i=1 1 tained

.

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

by H. A. Antosiewicz; IN MANUSCRIPT. (2) "On a theorem of Ostrowski and

Taussky" by R. Bellman and A. J. Hoffman; accepted by Archiv der Mathematik

(3) "Lower bounds for the rank and location of the eigenvalues of a matrix,

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

NBS Applied Mathematics Series 39. (4) "Some metric inequalities in the

space of matrices," by K. Fan and A, J. Hoffman; submitted to a technical

journal. (5) "A characterization of normal matrices," by A. J. Hoffman

and 0. Taussky; to appear in the Journal of Research of the NBS. (6) "On

an inequality of Hardy, Littlewood and Polya," by A. J. Hoffman; an NBS

report. (7) "On the relevance of Le Chatelier's principle to linear programming," by A. J „ Hoffman; IN MANUSCRIPT. (8) "An extremum property

of eigenvalues," by H. Wielandt; submitted to a technical journal.
Status of Projects 25

COMPRESSIBLE FLOW - METHOD OF ORTHOGONAL AND KERNEL FUNCTIONS

Task 1102-10-5116/52-16

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 and F. L. Alt

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. Particular solutions \|r* of the equation

A\|/*+4F(Z) \J/"*=0 have now been computed in the range - . 2 5 ( .01 ) (- . 1 5) for A

and 06,01 ). 20 for 9. Computation of the singular solution L, 2 )

? q- ^ ,©©) over the above range is nearly completed. Coding for the solution

of the boundary value problem in the pseudo-logarithmic plane using

least square procedures with particular solutions is completed and this

boundary value problem is being run. A report, discussing the computations carried out, is being

prepared.

TABLES OF INTEGRALS INVOLVING THE HIGHER TRANSCENDENTAL FUNCTIONS

Task 1102-10-5116/52-33

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

Origin: NBS Authorized 10/11/51

Manager: F. Oberhettinger

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. F. Oberhettinger has prepared a chapter on Mathieu functions for inclusion in a future volume of "Higher Transcendental Functions" of the Bateman Manuscript Project.

WATER WAVES

Task 1 102-10-51 16/53-5^

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

Sponsor: Office of Naval Research, USN

Manager: P. Davis

Full task description appears in Apr-June 1953 issue.

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

entitled "Numerical integration of a non-linear equation of motion of a fluid bounded by -a free surface," has been prepared in draft form.

SUPPORTING RESEARCH IN LINEAR PROGRAMMING

Task 1 102-10-51 16/54-9

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 and A. J. Hoffman

Full task description appears in July-Sept 1953 issue.
26 Status of Projects

Status? CONTINUED. The problem of contract awards, described in

the July-Sept 1953 issue, was further studied for the purpose of solving

the difficulties arising from the necessity of distributing the sizes and

monthly schedules of the various bidders to meet the depot requirements.

It was decided to follow a procedure which (1) ignored the difficulty for

the purpose of deciding the total each bidder was to receive, then (2)

distributed these amounts proportionately and optimally for each sizemonth combination. Other alternatives considered involved an inordinate

amount of computing for what appeared to be small savings to the Government. Experiments so far have disclosed that this method will result in some net financial gain to the Government, possibly small, apart from the

collateral advantages of speed and confidence. The work prior to the

latest experiments is described in publication (4).

Further work on the caterer problem—principally a recasting of

the demonstrations, which shed new light not only on this problem but on

the properties of solutions of the general multiparameter linear programming problem—was incorporated in publication (3). Work has now begun

on a modification of the caterer problem, involving a caterer with several

establishments but only one laundry.

Publication (6) presents what is apparently the only known ex- ample to illustrate the disturbing phenomenon of "cycling" in a degenerate

simplex problem. It is shown that cycling may arise in any one of these

circumstances? (a) the minimum has been attained, (b) there is no minimum,

(c) there is a minimum not yet attained. In all three cases, the algorithm

cycles, and thus thwarts discovery of the facts.

Some moderately general circumstances under which a dynamic linear

program breaks up into smaller problems have been shown to be illustrated

by the smoothing problem. These are reported in publication (7), along

with a formula for the solution in the special case that the monthly require

ments increase. Publication (1) derives a formula in the case that the

requirements increase up to a given month, then decrease.

A perspicuous proof of the result of S. Johnson on the optimal

ordering of items for a two-machine production process is presented in

publication ( 8 ). The result is shown to be a consequence of the fact that

the time it takes to process a set of items is not less than the time it

takes to process a subset.

A code has been written for the purpose of studying linear programs when some of the coefficients—either of the inequalities or the

objective function—are only known probabilistically, a situation arising

in most military applications. Several aspects are to be considered. For

example, one may wish to know the distribution of the optimal value of the

objective function if, at the time of actual application, the coefficients

are known exactly. Or they may never have been known exactly, and one may

still wish to discover the vector which will optimize the expected value

of the objective. The code provides for various modifications of Brown's

method, which seems to be a suitable tool.

Public atio

siewicz and A. J furthest hyperpl

ution of the cat

sky; an NBS repo

L. Gainen, D. J. Quarterly. (5)

A. Gleyzal; subm

method , " by A. J duction," by A.

optimal ordering

MANUSCRIPT. (9)

(10) "The double

report

.

ns? (l) "A remark on the smoothing problem," by H. A. Anto-

. Hoffman; IN MANUSCRIPT. (2) "On 'overshoot' in the

ane method," by R. Bryce; IN MANUSCRIPT. (3) "On the solerer problem," by J. W. Gaddum, A. J. Hoffman and D. Sokolow

rt. (4) "Linear programming in bid evaluations," by

Honig and E. D. Stanley; to appear in Logistics Research

"A method of solving the transportation problem," by

itted to a technical journal. (6) "Cycling in the simplex

. Hoffman; IN MANUSCRIPT. (7) "Smooth patterns of proJ. Hoffman and W. Jacobs; IN MANUSCRIPT. (8) "On the

of items for a two-stage process," by A. J. Hoffman; IN

"On block relaxation," by L. S. Joel; IN MANUSCRIPT,

description method on the SEAC," by S. Pollack; an NBS
27

Status of Projects

RESEARCH IN MECHANICS OF CONTINUA

Task 1 102-10-51 16/5^-23

Origin; NBS Authorized 12/29/53

Sponsor: Office of Naval Research, USN

Manager: R. F. Dressier

Objective: To investigate a class of problems arising from the

systems of partial differential equations describing the Newtonian mechanics

of continuous media. The main emphasis will be upon the fields of linear

and non-linear elasticity, aerodynamics, hydrodynamics, and theoretical hydraulics. Where possible, it is desired to obtain results first in closed

analytic form rather than by numerical integration of specific cases. Activities will embrace numerical analysis, however, when results contain

expressions difficult to evaluate or include types of problems with novel

features in computational technique, such as free-boundary problems in

hydrodynamics. In the initial phases of this program, topics will be selected from among those listed below; subsequent activity may branch out,

depending upon implications of early results.

Elasticity: The elastic boundary layer ("edge effect") in generalized plane stress; edge effect for non-linear boundary of plates; vibrations of delta wings by combined Ravleigh-Ritz-Weinstein methods; best

computational techniques for V

r0f=f (x,y) in semi-infinite strip domains,

connected with problems of edge stress in plates; stress distributions in

helical springs of finite cross-section.

Aerodynamics: Rarefaction- and compression-wave propagation

subject to dissipation and cross-section variations, from standpoint of

Mach line theory.

Hydrodynamics and Hydraulics: Stability criteria for interface

behavior of two liquids in density-current flow; behavior of resistance

controlled wave tips by developing better fitting procedures and by a study

of the asymptotic tip equation 0->+ \0f=O; the use of the asymptotic fitting

X i- procedure in other classes of wave problems; development of feasible methods

for numerical computation of free-boundary water wave problems arising from

the exact non-linear theory.

Background: Elasticity: The boundary layer effect for bent plates

has been handled by Friedrichs and Dressier; the problem now proposed is

the supplementary one in the linear theory. An attempt should also be made

to extend this technique to the vonKarman non-linear theory for bent plates.

Although the Rayleigh-Ritz-Weins tein methods have been broadly

generalized in theory, the application to vibration modes has been accom- plished thus far only in a few isolated instances; shapes for fixed and

fixed-free edges, and to ascertain the feasibility of employing high-speed

computers to carry it out.

Although the helical spring problem is a classical one, it is

believed that the present results of Gohner and Freiberger can be extended

and improved.

Aerodynamics: It is highly desirable to know quantitatively how

the Mach number and pressure are actually affected by frictional dissipation, and how they might be controlled by varying cross-section. This

problem applies not only to shock tubes and wind tunnels, but to various

types of pneumatic control systems and air brakes; experimental investigations on such controls are now active in the Thermodynamics Section of NBS.

Hydrodynamics: The problem of attaching a wave tip to a centered simple wave has been handled by Whitham and Dressier; in a manner

which is satisfactory except in the transition region. A better fitting

procedure is needed for this problem and the method should be applied to other types of waves featuring in practical engineering applications.

The interfacial stability problem is one of significance in

rivers and oceans, where such underwater wave movement may exist between

different thermal strata or between layers of different salinity.

Status: NEW.
23 Status of Projects

2 . Mathematical Tables and Exper imental .Computation s

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

Task 1 102-10-1 10M+3-3

(formerly 1102-10-1110/43-3)

Origins Canadian National Research Council Authorized 7/1 A7

Managers I. A. Stegun

Full task description appears in Apr-June 1 9*+9 issue, see task 11.2/2-43-3.

Statuss CONTINUED. The final manuscript has been completed and the

introduction is in process of revision.

TABLE OF THE GAMMA FUNCTIONS FOR COMPLEX ARGUMENTS

Task 1102-1 0-1 104A6-1

(formerly 1102-10-1110/46-1)

Origins NBS Authorized 7/1/47

Managers J. Todd

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

Statuss CONTINUED. The page proofs have been received for this

volume

.

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

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

Office

»

TABLES OF COULOMB WAVE FUNCTIONS

Task 1 102-10-1 104A7-2

(formerly 1102-10-1110/47-2)

Origins NBS Authorized 7/1A7

Managers: M. Abramowitz and P. Rabinowitz

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

Status: CONTINUED. A short table of the regular and irregular solutions for L=0,5,10 and p

and in the range from 0 to 10 has been computed.

An introduction to these tables is being prepared.

TABLE OF ANTI LOGARITHMS

Task 1 102-10-1 104A7-3

(formerly 1102-10-1110/47-3)

Origin: NBS Authorized 7/1 A

7

Manager: J. Todd Completed 12/31/53

Objective: To prepare a table of 10x

to ten decimal places for

x — 0 ( .00001 )1 .
Status of Projects 29

Backgrounds The function 10x is of basic importance. The best

existing table is that of J. Dodson, "An tilogarithmic Canon," London (1742),

which is out of print. The proposed publication, which was suggested by

H. E. Salzer of the Computation Laboratory, will be an improvement over Dodson's table from the standpoint of accuracy and format.

Status: COMPLETED.

Publication: "Tables of 10

x

," NBS Applied Mathematics Series 27

(U. S. Government Printing Office, Washington 2 5? D. C. , S3. 50).

TABLE OF LAGRANGIAN COEFFICIENTS

FOR SEXAGESIMAL INTERPOLATION

Task 1102-10-1104/48-2

(formerly 1102-10-1110/48-2)

Origin: NBS Authorized 5/25/48

Manager: J, Todd

Full task description appears in Apr-June 1949 issue, see task 11.2/2-48-2.

Status: CONTINUED. Checking of the page proof for this volume continued .Publication: "Tables of Lagrangian coefficients for sexagesimal

interpolation," NBS Applied Mathematics Series 35? in press, U.S, Government Printing Office.

BIBLIOGRAPHY OF CODING PROCEDURES, MATHEMATICAL TABLES

AND NUMERICAL ANALYSIS

Task 1 102-10-1 104/50-5

(formerly 1102-10-1110/50-5)

Origin: NBS Authorized 3/1/50

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

P. Rabinowitz

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

Status: CONTINUED. Preparation of the bibliographies continues.

WAVE FUNCTION FOR LITHIUM

Task 1 102-10-1 104/50-7

(formerly 1 102-10-1 1 10/50-7)

Origin: NBS Authorized 6/1/50

Sponsor: Bureau of Ordnance, USN

Manager: J. H. Wegstein

Full task description appears in Apr-June 1950 issue.

Status: CONTINUED. Computation on SEAC continues as time is

available

.
30 Status of Projects

COLLECTED SHORT MATHEMATICAL TABLES OF THE COMPUTATION LABORATORY

Task 1102-10-1104/51-1+

(formerly 1102-10-1110/51-4)

Origin: NBS Authorized 9/23/50

Manager: J . Todd

Full task description appears in July-Sept 1950 issue.

Status: CONTINUED. The first volume of such tables is in press.

Publication: "Tables of functions and of zeros of functions," Volume

I of Collected Short Tables of the Computation Laboratory, NBS Applied

Mathematics Series 37? in press, 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 and L. Joel

Full task description appears in Apr-June 1951 issue.

Status: INACTIVE, For status to date see Jan-Mar 1953 issue, p. 45.

REVISION OF MATHEMATICAL TABLES

Task 1102-10-1104/52-7

(formerly 1 102-10-1 1 10/52-7)

Origin: NBS Authorized 8/10/51

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

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. Following is the status of those mathematical

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

is planned:

"Table of natural logarithms for arguments between zero

and five to sixteen decimal places," NBS Applied

Mathematics Series 31 (1953)? available from U. S.

Government Printing Office, Washington 25? D. C.,

$3.25. This is a reissue of NBS Mathematical Table

MT10, "Table of natural logarithms," vol. Ill

(1941 ).

"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 MT13.(1942).

"Tables of circular and hyperbolic sines and cosines

for radian arguments," NBS Applied Mathematics Series 36 ; available from U. S, Government Printing

Office, Washington 25? D. C., $3.00. This is a reissue of the tables formerly designated as Mathematical Table MT3 (1939? 2d ed . 1949).

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

Applied Mathematics Series 4l ; in press, U. S. Government Printing Office. This was originally Mathe-
Status of Projects 31

matical Table MT8, "Tables of probability functions,"

vo 1 , I (1941).

"Tables of sines and cosines for radian arguments to be reissued in the Applied Mathematics Series; formerly designated as Mathematical Table MT4.

"Table of natural logarithms," vol. IV, Mathematical

Table MT12 (194-1) ; revision in progress for reissue

in the Applied Mathematics Series.

"Tables of sine, cosine, and exponential integrals,"

vol. I, Mathematical Table MT5 (1940); out of print,

mot to be reissued.

TABLE OF ARCSIN FOR COMPLEX ARGUMENTS

Task 1 102-10-1 104-/52-14-

(formerly 1102-10-1110/52-14)

Origin; NBS Authorized 10/1/51

Manager: A. A. Goldstein

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The information has been put on punched cards

for checking and editing.

EXTENSION OF THE TABLE OF HYPERBOLIC SINES AND COSINES

Task 1102-10-1104/52-18

(formerly 1102-10-1110/52-18)

Origin: NP-S Authorized 9/17/51

Manager: W. F. Cahill

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. Processing of the punched cards for typing

continued

.

TABLE OF THE MODIFIED AIRY INTEGRAL

Task 1102-10-1104/52-23

(formerly 1102-10-1110/52-23)

Origin: NBS Authorized 10/4/51

Manager: P, Rabinowitz

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The manuscript has been typed and is in process

of being checked. An introduction to the volume has been prepared.
32 Status of Projects

TABLE OF EEROM FUNCTION FOR COMPLEX ARGUMENTS

Task 1102-10-1104/52-25

(formerly 1102-10-1110/52-25)

Origin: NBS Authorized 10/5/51

Managers: M. Abramowitz and F. J. Stockmal

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. The table for x,y=0( s 1)2 has been computed and

is available on punch cards. In addition a short table of the error function for complex arguments for x,y=0( .2)2( .4)3,2 to 9 significant figures

is in preparation as an NBS report.

EXTENSION OF TABLES OF THE EXPONENTIAL FUNCTION FOR

NEGATIVE ARGUMENTS

Task 1102-10-1104/52-31

(formerly 1102-10-1110/52-31)

Origin: NBS Authorized 10/9/51

Manager: E, Marden

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. Functions have been differenced and punched

cards prepared for typing of the final manuscript.

SPHEROIDAL WAVE FUNCTIONS

Task 1 102-10-1 104/52-37

(formerly 1 102-10-1 1 10/52-37)

Origin: NBS Authorized 11/28/51

Manager: T. Ledley

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. The machine computation is practically completed

for m=0(l)5, 1=0(1)10. A complete description is being written up.

VAN DEM 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.

Status: INACTIVE. For status to date see Apr-June 1952 issue, p. 49.
Status of Projects 33

RADIAL MATHIEU FUNCTIONS

Task 1102-10-1104/52-49

(formerly 1102-10-1110/52-49)

Origin? NBS Authorized 2/1/52

Managers: J. Todd and I. Rhodes

Full task description appears in Jan-Mar 1952 issue.

Status: CONTINUED. All tabulations have been computed and are being

checked at the Istituto Nazionale per le Applicazioni del Calcolo (Rome)

by E. Aparo and D. Dainelli, who assisted in the analysis and codingprogramming during their stay at the Computation Laboratory as guest

workers.

SIEVE RT ' S INTEGRAL

Task 1102-10-1104/52-57

(formerly 1 102-10-1 1 10/52-57)

Origin: NBS Authorized 2/12/52

Managers: M.L. Paulsen and P. J. O’Hara

Full task description appears in Jan-Mar 1952 issue.

Status: CONTINUED. Computations have been completed and differencing

of the results, which have been transferred to punch cards, is nearly

completed.

SCATTERING FUNCTIONS

Task 1102-10-1104/52-63

’(formerly 1102-10-1110/52-63)

Origin: NBS Authorized 3/10/52

Manager: A. Goldstein Terminated 12/31/53

Objective: To develop practical numerical and analytical methods

for determining the scattering of plane waves by obstacles (or force

centers in quantum mechanical cases). Mathematically this requires the

solution of the wave equation

V

2x + k 2 (r)x = 0

with proper behavior at large distances from the scatterer. The possibility of using high-speed digital computers for this problem will be

investigated

.

Background: There is considerable technological interest in the

scattering of electromagnetic waves by large molecules, fogs, dust particles, etc. The interpretation of experimental results in the scattering

of high energy particles by nuclei is one of the central problems in

nuclear physics.

Status: TERMINATED, in favor of tasks of higher priority.
34 Status of Projects

TABLE OF SECANTS AND COSECANTS

Task 1102-10-1104/52-81

(formerly 1 102—10—1 1 10/52—81

)

Origin: NBS Authorized 7/1/52

Managers: K. C. Nelson and I. A. Stegun

Full task description appears in July-Sept 1952 issue.

Status: CONTINUED. In press.

Publication: "Table of secants and cosecants to nine significant

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

in press, U. S . Government Printing Office.

PAINLEVE EQUATION

Task 1102-10-1104/53-3

(formerly 1102-10-1110/53-3)

Origin: NBS Authorized 8/11/52

Managers: J. Todd and H, A. Antosiewicz

Full task description appears in July-Sept 1952 issue.

Status: INACTIVE. For status to date see Jan-Mar 1953 issue, p. 48.

L-SHELL CONVERSION COEFFICIENTS

Task 1 102-10-51 10/53-52

(formerly 1 102—10—1 1 10/53“ 52)

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

Manager: C. J. Swift

Full task description appears in Apr-June 1953 issue.

Status: CONTINUED. Sonne answers have been obtained and the code

is in production.

3. Mathematical Services

Note: The tasks under Mathematical Services are arranged serially according

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 authorised.

TABLES OF THERMODYNAMIC PROPERTIES OF GASES

Task 0302-40-2606A9-5

(formerly 11.2/33-49-5)

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

Sponsor: National Advisory Committee for Aeronautics

Manager: F. L. Alt

Full task description appears in Apr-June 1949 issue.

Status: INACTIVE.
Status of Projects 35

RAY TRACING

Task 0202-10-2308/50-13

(formerly 11.2/33-50-13)

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

Sponsor: " Managers R. K. Anderson

Full task description appears in Jan-Mar 1950 issue.

Status: CONTINUED. Computations are being performed as requested.

MOLECULAR STRUCTURE CALCULATIONS, II

Task 1102-40-5126/50-16

(formerly 1 102-53-1 106/50-1 6)

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

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

Full task description appears in Jan-Mar 1950 issue, see task 1 1 . 2/33- 50-1 6

.

Status: CONTINUED. Computations were performed as requested.

LIQUID-VAPOR TRANSITION

Task 1102-40-5126/51-22

(formerly 1102-53-1106/51-22)

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

Sponsor: " " Manager: I. Stegun

Full task description appears in Jan-Mar 1951 issue.

Status: INACTIVE. For status to date see July-Sept 1952 issue, p. 52.

MOLECULAR STRUCTURE, III

Task 1102-40-5126/51-37

(formerly 1 102—53—1 106/51—37)

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

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

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. Computations were performed as requested.

SHOCK WAVE PARAMETERS, II

Task 1102-40-5126/51-38

(formerly 1 102-53-1106/51-38)

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

Sponsor: " " Manager: I. A. Stegun

Full task description appears in Apr-June 1951 issue.

Status INACTIVE. For status to date see Apr-June 1953 issue, p.48.

:
36 Status of Projects

POWDER DIFFRACTION

Task 1102-40-5126/52-6

(formerly 1 102~53_ 1 106/52-6)

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

Manager: E. Marden

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

Status: INACTIVE. For status to date see July-Sept 1951 issue, p. 53.

PRECISE DETERMINATION OF THE PARAMETER OF DISPERSION

EQUATION FOR SEVERAL TYPES OF OPTICAL GLASS

Task 1102-40-5126/52-17

(formerly 1 102—53—1 106/52—1 7)

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

Sponsor: " Manager: I. A. Stegun

Full task description appears in July-Sept 1951 issue.

Status: INACTIVE.

SPHERICAL BLAST

Task 1102-40-5126/52-20

(formerly 1102-53-1106/52-20)

Origin: Naval Ordnance Laboratory Authorized 9/27/51

Sponsor: " " Manager: D. H. Jirauch

Full task description appears in July-Sept 1951 issue.

Status: CONTINUED. Some work was done to reorganize the material'

prior to test runs on SEAC.

CALCULATIONS FOR d S PACINGS

Task 1102-40-5126/52-44

(formerly 1 102-53-1 106/52-44)

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

Sponsor: " Manager: I. Stegun

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. During this quarter d-spacing calculations were computed for various compounds and least squares calculations were made.
37 States of Projects

GAS ADSORPTION BY HIGH POLYMERS

Task 1 102-40-5126/52-70

(formerly 1102-53-1106/52-70)

Origin: Naval Medical Center, Bethesda, Md. Authorized 4/1/52

Sponsor: " " Manager: I. Stegun

Full task description appears in Jan-Mar 1952 issue.

Status: INACTIVE. For status to date see Apr-June 1952 issue, p. 59.

CHEMICAL TRANSITION PROBABILITIES

Task 1102-40-5126/52-82

(formerly 1 102-53-1 106/52-82)

Origin: NBS Section 3.2 and Hydrocarbon Authorized 6/1/52

Research Corp. Completed 12/31/53

Sponsor: Office of Naval Research, USN

Manager: A. A. Goldstein

Objective: To compute quantum-mechanical transition probabilities

related to the chemical reactions: + 0 — OH + H, 0,^ + H —*OH + 0.

The detailed computations involve the evaluation of integrals, frequently

referred to as matrix elements, that appear in the time-dependent theory

of quantum mechanics which has been applied to chemical kinetics. (See

for example. S. Golden and A. M. Peiser, J. Chem. Phys. J_Z, 630 (1949)).

In addition, the computations involve averaging the squared magnitudes of

the matrix elements over distributions of the parameters upon which the

former depend. The distributions are related to the probability of finding

the initial reactant molecules in specified energy and momentum states.

Background: The results of these computations are needed to provide

a theoretical description of certain chemical reactions which are presumed

to be important hydrogen-oxygen flames. Of special interest is the possibility that the distribution of the OH radical may be determined in its

various rotational and vibrational energy states. The results of these

computations will enable the correlation to be made of the experimental

results which may be obtained from spectroscopic studies of these flames.

This task was proposed and suggested by Drs. Peiser and Golden, Hydrocarbon

Research Corpo ration, and by H. P. Broida, NBS 3.2.

Status: COMPLETED.

NEUTRON DIFFUSION, II

Task 1 102-40-51 26/53-^

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

Office, (NDA)

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

Full task description appears in July-Sept 1952 issue.

Status: CONTINUED. Results are being sent to the sponsor as they are completed. New codes are being written to cover various source geometries,

including pulse sources, rectangular sources, fission sources, and conical

sources. Computations have been carried out for a number of materials including hydrogen, lithium, oxygen, water.
38 Status of Projects

STRENGTH OF WING COMPONENTS

Task 1102-40-5126/53-11

Origins National Advisory Committee for Authorized 12/8/52

Aeronautics, Langley Field, Va.

Sponsors " " Manager; E. Warden

Full task description appears in Oct-Dec 1952 issue.

Status; CONTINUED.

RADIANT HEATING OF SOLIDS

Task 1102-40-5126/53-20

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

Sponsor: " Manager: W. F. Cahill

Full task description appears in Oct-Dec 1952 issue.

Status: INACTIVE.

COMPUTATION OF THERMODYNAMIC FUNCTIONS

Task 1102-40-5126/53-2

7

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

Sponsor: " Manager: E. Warden

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. The previous code was improved to obtain more significant figures in the final results, and thermodynamic functions

(heat capacity, enthalpy, free energy, and entropy) and their differences

were computed for several additional molecules.

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.

Status: INACTIVE, For status to date see Jan-Mar 1953 issue, p. 58.
Status of Projects 39

DYNAMIC BEHAVIOR OF AIRCRAFT STRUCTURES

Task 1 102-40-51 26/53-29

Origin: NBS, Section 6.4 Authorized 1/23/53

Sponsor: " Manager: I. Rhodes

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. Computations are being performed as requested,

and results are being transmitted to the sponsor.

REDUCTION OF ECLIPSE DATA

Task 1102-40-5126/53-34

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

Sponsor: " " Manager: A. Goldstein

Full task description appears in Jan-Mar 1953 issue.

Status: CONTINUED. The final report is being prepared.

SPECTRAL ANALYSIS OF STATIONARY TIME SERIES

Task 1 102-40-5126/53-37

Origin: Statistical Research Center, University Authorized 4/20/53

of Chicago Completed 12/31/53

Sponsor: Office of Naval Research, USN

Manager: I. Stegun

Objective: To compute the serial products

N-M N- | si

c,,- ) xx, and Y - ') x^x^, v L— n n+v 0 s L— n n+|s|

n= 1 n=

1

which are used in estimating the spectral distribution function and its

kurtosis, respectively.

Background: The present statistical theory of analysis of stationary

time series has assumed complete knowledge of the covariance sequence or, equivalently, of the spectrum of the process. It is, therefore, important

to be able to estimate one of these. Knowledge of the spectrum seems to yield greater immediate insight into the structure of the process.

Comments: This problem was proposed by M. Rosenblatt and U.Grenander,

SRC, University of Chicago.

Status: COMPLETED.

Publication: "Tables of autoregressive series" by Computation

Laboratory; an NBS report.
40 Status of Projects

ACOUSTICAL IMPEDANCES

Task 1102-^0-5126/53-39

Origins NBS, Section 6.1 Authorized 6/30/53

Sponsor: " Managers: S. Prusch and K. Nelson

Full task description appears in Apr-June 1953 issue.

Status: INACTIVE. For status to date see Apr-June 1953 issue, p. 55.

LORAN UNIVAC CODE

Task 1 102-40- 5 126/53~4l

Origin: Hydrographic Office, U. S. Navy Authorized 5-/20/53

Sponsor: " " Managers: I. Rhodes and D. H. Jirauch

Full task description appears in Apr-June 1953 issue.

Status: CONTINUED. The computation of Loran tables has been coded

and successfully proved in on the UNIVAC. The Skywave correction and

trainer code was coded and is awaiting code checking. The baseline routine

is in process of being coded for UNIVAC.

OPTIMUM SECTIONS FOR DELTA WING

Task 1 102-5-0-51 26/53-59

Origin: National Advisory Committee for Authorized 5-/29/53

Aeronautics, Langley Field, Va. Completed 12/31/53

Sponsor: " " Manager: 0. Steiner

Objective: To perform computations to determine optimum sections,

for delta wings at supersonic speeds.

Status: COMPLETED. Results were transmitted to the sponsor.

RADIATION DIFFUSION

Task 1 102-40-5126/53-51

Origin: NBS, Section 4.8 Authorized 6/9/53

Sponsor: Armed Forces Special Weapons Project

Manager: F. Stockmal

Full task description appears in Apr-June 1953 issue.

Status: CONTINUED. The code for part 1 (Photon random walk in an infinite Compton scattering medium) has been completely checked and used

to generate 1000 random walks.

These random walks have been used for the calculation of 3 problems in gamma-ray diffusion: (l) the reflection and transmission of radiation by finite plane parallel barriers; (2) the albedo of semi-infinite

media; (3) the penetration of radiation in an infinite medium.
Status of Projects * 4-1

TRANSPORTATION PROBLEM II

Task 1102-^0-5126/53-55

Origin: Logistics Research Project,

George Washington University

Sponsor: Office of Naval Research

Managers: A. J. Hoffman and L. Gainen

Objective: To solve systems of equations arising in the problem of

allocating shipments from bidders to depots minimizing the total cost

of the operation. To investigate problems arising from conditions imposed

by bidders by time phasing of requirements, etc., and to formulate these

conditions for computation.

Background: The technique of G. B. Dantzig in applying the simplex

method in finding solutions of the transportation problem of HitchcockKoopmans is used in solving standard allocation problems.

Status: TERMINATED. This work is now being carried on as part of

task 1102-5-0-5126/55—13 (see p.H-3).

Authorized 6/9/53

Terminated 12/31/53

STANDARD LORAN TABLES - Rates 1L4, 1 L5

Task 1102-40-5126/55-2

Origin: Hydrographic Office, U. S. Navy Authorized 9/16/53

Sponsor: " " Completed 12/31/53

Manager: D. H. Jirauch

Objective: To prepare tables giving coordinates of hyperbolic lines

of position.

Background: Standard Loran navigation tables are necessary for

preparation of charts used by navigators in determining their positions

with the aid of certain electronic equipment.

Status: COMPLETED.

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: J. H. Wegstein

Objective: To derive equations describing the high temperature

thermodynamic properties of water and to solve these equations numerically

over a very large temperature range and over the maximum pressure range

that can be treated theoretically.

Background: The properties of water at high temperature are of

interest to the Naval Ordnance Laboratory. Such properties are not now

known in the range under consideration.

Status: NEW.
42 Status of Projects

DISTRIBUTION OF NORMAL MODES OF VIBRATION OF CUBIC LATTICES, II

Task 3711-60-0009/5^-7

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

Sponsor: " Managers F. Stockmal

Objective: To compute the distribution of normal modes of vibration

in cubic lattices, on which depends the vibrational contribution to the

thermodynamic properties of polyatomic molecules and crystals.

Background: This task is a continuation of task number 1102-21-1104/

52-62, a complete description of which appeared in the Jan-Mar 1952 issue.

The code for the earlier problem had been run on the SEAC, and distribution

densities obtained for cubic lattices of various sizes.

Status: NEW. Evaluation was made of earlier programs in order to modify the procedures for suitability in the newly formulated problem.

NORMAL VIBRATIONS IN MOLECULES

Task 371 1-60-0009/54-8

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

Sponsor: " Manager: M. Newman

Objective: Given the nxn+3 matrix S and the nxn symmetric matrix F,

to calculate the symmetric matrix A = S'FS, and then to compute the eigenvalues and eigenvectors of A.

Background: This problem arises in molecular spectroscopy. The

object of the problem is the calculation of the best force constants between the atoms of a molecule. With these constants the normal frequencies of vibrations are calculated.

Status: NEW. Fifty-eight 6 x6 matrices and four l4xl4 matrices M were computed from M^S'FS, where S is an nx( n+3) matrix, F an nxn symmetric

matrix. The eigenvalues and eigenvectors of these matrices M were then

computed. In addition three 6 x 0 matrices and one 7x7 matrix were computed,

and then inverted.

COMPUTATION OF VIBRATION MODES AND FREQUENCIES

Task 3711-60-0009/55-11

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

Sponsor: NBS, Section 6.4

Managers: I. Rhodes and W. Cahill

Objective: To compute the free vibration modes and frequencies

of aircraft structures. The codes are to be general in nature and apply

without modification to any aircraft structure whose influence coefficients,

mass distribution, and geometry are specified.

Background: A knowledge of the vibration modes and frequencies of

aircraft is of primary importance in determining their flutter speed and in

making dynamic analyses of the response of the structure to landing impact

and other shock loads.

Status: NEW.
Status of Projects 5-3 SPACE AVERAGE OF TEMPERATURE DISTRIBUTION WITH A VOLUME DISTRIBUTION

OF SOURCES

Task 3711-60-0009/5^-12

Origin: NBS, Division 6 Authorized 12/11/53

Sponsor: " Manager: E. Marden

Objective: To evaluate

A = 32 *2 co 00

,

T

2 r n n» X

B =_ OD ®

1 (2» +1 ) 2

{[(x n /a) 2+ ( {2v +1 )f)

2

] 2 + (3 2

}

(x

r/a)

2 + [ (2j/+ 1 )j]

2

( 2 v + 1 ) 2 {[(x

n/a) 2 + ((2V +1)j) 2

] 2+/3 2

}

J>- (A2+B2 )^

, a - B tan 9 = -jwhere x

fl

, n-1 , 2 , . . „ runs over the zeros of Jq (the Bessel functions of order

0), V =0, 1 , . . . , and /3 assumes values from .20 to 5-500. Several values are taken for the parameters .a and JL. Background: This problem arises in measuring the pressure response

of condenser microphones.

Status: NEW. Computations were carried out to evaluate the performance of several condenser microphones from data furnished defining

their shapes. A SEAC code is available for running of additional cases.

AWARD OF PROCUREMENT CONTRACTS BY LINEAR PROGRAMMING

Task 1102-40-5126/55-13

Origin: New York Quartermaster Procurement Agency Authorized 11/27/53

Sponsor: " " " Manager: H. Bremer

Objective: To solve the problems, arising in the sponsor's work, of

awarding purchase contracts at minimum cost to the Government.

Background: This task is a generalization of the investigations

that were undertaken in task 1 102-40-51 26/53~55.

Status^ NEW. The work of task 1 1 02-40-51 26/53_ 55 (see p.4i),now

terminated, is being applied to actual problems arising in the sponsor's

operations. The results of three computations have been transmitted to

the sponsor for use in awarding contracts. A new code is being written

to expedite the machine handling of the problem of phasing the successful

contractor's shipments to the various depots.
44 Status of Projects

FLOW IN SUPERSONIC NOZZLE

Task 1 102-40-51 26/54-14

Qrigin: National Advisory Committee for Aeronautics, Authorized 12/16/53

Langley Field, Va,

Sponsor: " " " Managers: E. Marden, G. Hawkins

Objective: To calculate the boundary contour of a supersonic nozzle

for a given Mach number.

Background: It is assumed that the sonic surface is a plane normal

to the axis of the nozzle. The Mach net is then computed by the PrandtlBusemann method (see L, Prandtl and A. Busemann: Stodala-Festschrif t,

Zurich, 1929? p. 5-99)? starting from the first Mach line and from the given convex part of the boundary. In order to compute the slopes of the two

Mach lines at each point of the net it is necessary to solve for £ the

following equation:

where 9 is given and ^ = 1.4 is the adiabatic constant. This could be done

by making a table of the functions and using inverse interpolation? but

for SEAC computation it is simpler to solve the equation at each step by

Newton’s method using the solution of the previous point as a first approximation. The net is then built up progressively by computing the

intersection of the left-running and right-running Mach lines.

After the boundary has reached its maximum slope it must be

modified in such a way that uniform flow is obtained in the downstream

region of the nozzle.

This problem has been solved for several nozzles using the Bell

relay computer at Langley Field. However, an increment in the angle of

expansion between two subsequent Mach lines of one degree was used there.

It is proposed in the SEAC calculation to use increments of twenty-five

hundredths of a degree.

Status: NEW. Programming and coding of this problem for SEAC have

been completed and checking in of the code has begun.

DEPOLYMERIZATION

Task 3711-60-0009/54-17

Origin: NBS, Section 7.6 Authorized 12/16/53

Sponsor: " Manager: R. Anderson

Objective: To obtain numerical solutions for the system of differential equations

dQN-1 = -(1 + <T)(N - 4)QN-1

dQ

n K0 1 df = -(^)("-3)<V2(^ - i ) Q„+ 2+ 2

N-n

-

3 K

2 _i

: n+2i+2 ; i=1

4<n<N-3,
Status of Projects 45

with boundary conditions at T=0,

where is defined in terms of the parameters <T and £ by the equation

K . f

. . , ^

= 2(1+£ <t)

[ i- ^[l+3(1-£)](l + £)

1

) + o’.

Backgrounds This system arises in the theory of chain reactions

which describes the depolymerization of long polymer molecules as developed

by R. Siaiha, L. A. Wall, and P. J. Blatzj J. Polymer Science, 1950. (N-1 ) represents the initial number of units in a chain, Q

r the fraction of

molecules containing n units, and T is a dimensionless variable related

to time. 1/£ is the average kinetic chain length of reaction and tr the

ratio between the probabilities of transfer and initiation of a free

radical chain.

Status: NEW. Analysis of the problem for computation on SEAC has

been started.
III. STATISTICAL ENGINEERING LABORATORY

(Section 11.3)

1 . Fundamental Research in Mathematic a l Statistics

BIBLIOGRAPHY AND GUIDE TO STATISTICAL LITERATURE

Task 1 103-10-1 107A9-1 a (formerly 11.3/2-49-1)

Origin: NBS Authorized 1/9/49

Manager: L. S. Doming

Full task description appears in Jan-Mar 1949 issue.

Status: CONTINUED. The file of abstracts of statistical literature

taken from Mathematical Reviews is on a current basis. The preparation

of abstracts from Zentralblatt fur Mathematik is continuing. These

Zentralblatt abstracts start at 1 940 and work back through the years;

1933 is the earliest volume at this date from which material for this file

has been drawn.

MANUAL ON FITTING STRAIGHT LINES

Task 1103-10-1107/50-2

(formerly 11.3/2-50-2)

Origin: NBS Authorized 3/1/50

Manager: F. S. Acton

Full task description appears in Jan-Mar 1950 issue.

Status: INACTIVE. For status to date see Oct-Dec 1952 issue, p.58.

TABLE TO FACILITATE DRAWING RANDOM SAMPLES

Task 1 103-10-1 107/51-1

Origin: NBS

Managers: C. Eisenhart and L. Full task description appears

Authorized 7/1/50

S. Deming

in July-Sept 1950 issue.

Status: INACTIVE. For status to date see July-Sept 1952 issue, p.64.

46
Status of Projects

MISCELLANEOUS STUDIES IN PROBABILITY AND STATISTICS

Task 1103-10-1 107/51-2

47

Origin: NBS Authorized 7/1/50

Manager: C. Eisenhart

Full task description appears in July- Sept 1950 issue.

Status: CONTINUED, J. Lieblein prepared a working paper dealing

with a probability problem arising in connection with counter data.

Publications: (1) "A property of the normal distribution," by

E. P„ King and E. Lukacs; submitted to a technical journal. (2 ) "A new method of analyzing extreme-value data," by J. Lieblein; accepted for

publication as a NACA Technical Note. (3) "On strongly continuous

stochastic processes," by E. Lukacs; accepted for publication in Sankhya.

(4) "Certain Fourier transforms of distributions (II)," by E. Lukacs

and 0. Szasz; accepted for publication in the Canadian Journal of Mathematics. (5) "Nonnegative trigonometric polynomials and certain rational

characteristic functions," by E. Lukacs and 0. Szasz; accepted for publication in the Journal of Research of the NBS. (6) "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") by J. Lieblein; submitted to a technical journal. ( 7 ) "Tables

of expected values of 1/X for positive Bernoulli and Poisson variables,"

by E. Grab and I. R. Savage; accepted for publication in the Journal of

the American Statistical Association. (8) "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. (9) "On optimum grouping in one-criterion variance components

analysis," by E. P. King; submitted to a technical journal. (10) "On the

variances and covariances of order statistics from the Weibull distribution," by J. Lieblein; submitted to a technical journal. (11 ) "Timediscrete 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. (12) "Bounds

on a distribution function which are functions of moments to order four,"

by M. Zelen; submitted to a technical journal.

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.

Status: INACTIVE. For status to date see July-Sept 1951 issue, p.6'5.

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.

Status: CONTINUED. An investigation has been started to find the

most probable rank orders in various sampling situations arising in
5-8 Status of Projects

nonparametric problems. These results are of use in the construction

of optimum nonparametric test procedures. Preliminary work suggests

that it will be possible to construct uniformly most powerful rank order

tests for a nonparametric situation discussed recently by Lehman.

Publications: (1) "Bibliography of nonparametric statistics and

related topics," by I. R. Savage; accepted for publication by the Journal

of the American Statistical Association. (2) "Most probable rank orders,"

by I. R. Savage; IN MANUSCRIPT.

Manager: W. S. Connor

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

Status: CONTINUED. (1) W. H. Clatworthy prepared a working paper

which presents some new partially balanced designs with two associate

classes. (2) M. Zeien has obtained preliminary results on exact tests

of significance for combining information from both inter and intra block

analysis of incomplete block designs. The power functions of different

critical regions are under investigation. (3) W. S. Connor has begun

a study of the combinational properties of triangular designs.

Publications: (1) "Partially balanced incomplete block designs

with two associate classes and two treatments per block," by W. H. Clatworthy; submitted to a technical journal. (2) "Some theorems for

partially balanced designs," by W. S. Connor and W. H. Clatworthy; ac- cepted for publication in Annals of Mathematical Statistics. (3) "An

embedding theorem for balanced incomplete block designs," by M. Hall, Jr.

(Ohio State University) and W. S. Connor: accepted for publication in

the Canadian Journal of Mathematics. (b) "A note on partially balanced

designs," by M„ Zeien; accepted for publication in the Annals of Mathematical Statistics. (5) "Analysis for some incomplete block designs

having a missing block," by M. Zeien; accepted for publication in Biometrics. (6) "Designs for paired observations," by W. S. Connor and

W. J. Youden; IN MANUSCRIPT. (7) "On the enumeration of partially balanced designs with two associate classes," by W. H. Clatworthy; IN MANUSCRIPT.

STUDIES IN THE MATHEMATICS OF EXPERIMENT DESIGN

Task 1 103-10-1 107/53-1

Origin: NBS Authorized 10/15/52

2. Applied Research in Ma thematical Statistics

COLLABORATION ON STATISTICAL ASPECTS OF NBS

RESEARCH AND TESTING

Task 3737-60-0002/51-1

(formerly 3011-60-0002/51-1)

Origin: NBS

Managers:W. J. Youden and J. Cameron

Full task description appears in July-Sept 1950 issue.

Authorized 7/1/50

Status: CONTINUED. Activity under this task fell into two main
Status of Projects 49

categories:

A. Design of Experiments: A special partially balanced design

was constructed for use in calibrating thermocouple wires. Because of

the construction of the apparatus, only differences between wires could

be measured. This necessitated a modification of the usual incomplete

block design to account for the fact that within a block only a subset of

the usual treatment comparisons could be measured at any one time.

B. Development or Selection of the Appropriate Method for

Analysis and Interpretation of Data: A sampling plan for insuring that

inversion thermometers meet a desired tolerance specification (i.e., in

repeat determinations, a given proportion must repeat within .01° C of

the resulting mean) was constructed. This plan is implemented by using

the range of these repeat determinations.

Publications: (1) "Rejection of outlying observations," by F.Proschan

Am. J. Phys. 21: 520-525 (Oct. 1953). (2) "The principles of experimental

design," by W. J. Youden; appeared as a chapter in Research Operations in

Industry , o. 31 7~329(Kings Crown Press, Columbia University, New York,

1953). (3) "Making one measurement do the work of two," by W. J. Youden

and W. S. Connor*, Chem. Eng. Prog. 549~552 (Oct. 1953). (4) "Acceptance sampling of electroplated articles," by J. M. Cameron and F. Ogburn;

accepted for publication in Plating. (5) "Probability limits for the

average chart when process standards are unspecified," by E. P. King;

submitted to a technical journal. (6) "Performance of inspectors and

gasoline pumps," by W. J. Youden and M. W„ Jensen; submitted to a technical journal.

STATISTICAL ASPECTS OF NBS ADMINISTRATIVE OPERATIONS

Task 3737-60-0002/52-1

(formerly 3011-60-0002/52-1)

Origin: NBS Authorized 10/1/51

Manager: I. R. Savage

Full task description appears in Oct-Dec 1951 issue.

Status: INACTIVE. For status to date see July-Sept 1952 issue, p.68

STATISTICAL SERVICES FOR COMMITTEE ON SHIP STEEL, NSC

Task 1103-40-5105/52-1

Origin: Ship Structure Committee, NRC Authorized 12/1/51

Sponsors " " Manager: W. J. Youden

Full task description appears in Oct-Dec 1951 issue.

Status:^ CONTINUED. Work in progress is directed towards improving the discriminant function for predicting plate performance, and

establishing confidence limits associated with coefficients in the formula

relating transition temperature and plate composition.
50 Status of Projects

RESEARCH IN APPLICATIONS OF MATHEMATICAL STATISTICS TO

PROBLEMS OF THE CHEMICAL CORPS

Task 1103-^0-5118/52-1

Origin: Biological Laboratories, Chemical Corps Authorized 10/1/51

Oept. of the Army

Sponsor: " " " Manager: C. Eisenhart

Full task description appears in Oct-Dec 1951 issue.

Status: CONTINUED. Plans for special computations were worked out.

STATISTICAL ANALYSIS AND DESIGN OF EXPERIMENTS

FOR THE U. S, GEOLOGICAL SURVEY

Task 1 103-^0-51 1+0/51+-1

Origin: U. S. Geological Survey, Department of Authorized 10/9/53

Interior

Sponsor: " " " Managers: C. Eisenhart and W. J . Youden

Objective: The survey of problems in field geology in AEC-sponsored

projects at the U. S. Geological Survey offices in Grand Junction, Colorado

the application of statistical methods to field sampling and field observations in geology; and the application of statistical methods to laboratory work in connection with field geology.

Background: Problems encountered in large and complex projects of

the U. S. Geological Survey necessitate the application of statistical

techniques to ensure that the observations are taken properly and interpreted correctly for significant conclusions to be drawn.

Status: NEW. C. Eisenhart spent a day at the Trace Elements Laboratory of the U. S. Geological Survey at Denver, Colorado, and W. J. Youden and C. Eisenhart spent four days with personnel of the Grand Junction District, U. S. Geological Survey, making a survey of the statistical

problems in AEC-sponsored Geological studies of the Colorado Plateau.

After their return to Washington, conferences were held with

those Geological Survey personnel in Washington concerned with the geologic studies of the Colorado Plateau, in which the trip to Colorado was reviewed and further activities outlined.

As a member of the Panel on Earth Sciences of the Committee

on Statistics in the Physical Sciences of the American Statistics Association, Dr. Eisenhart participated as a discusser at a Symposium on Statistics in Geology held in conjunction with the Annual Meeting of the

American Statistical Association, Washington, D. C., December 29, 1953?

and then met informally to discuss the application of statistics in geology with a number of geologists assembled in Washington for this Symposium .
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 1 104-3^-5107/^7-1

(formerly 11.4/21-47-1)

Origin: The Bureau of the Census Authorized 7/1/4?

Sponsor: " " Full task description appears in Apr-June 1949 issue.

Status: CONTINUED. Under contract between the NBS and the EckertMauchly Division of Remington Rand Inc., stockpiling of replacement parts

for UNIVAC System No. 1 has been continued. Additional auxiliaries,

particularly high-speed printing equipment, have been also investigated

and evaluated at the request of the Bureau of the Census.

THE AIR COMPTROLLER’S COMPUTING MACHINE

Task 1 104-34-51 07A7-3

(formerly 11.4/24-47-3)

Origin: Office of the Air Comptroller, USAF Authorized 7/1/47

Sponsor: " " Completed 12/31/53

Objective: To develop specifications for, and to construct an automatic-sequenced electronic digital computing machine suitable for use

by the U. S. Air Force in program planning and control.

Background: The Air Comptroller's Office requires a high-speed and

flexible computing machine to calculate detailed programs consistent with

general policy decisions, and to facilitate rapid recomputation of programs to meet budgetary and other limitations. The problems involved are

of wide applicability, and a part of the present project consists in

formulating them mathematically. It is expected that the primary computation problem to be solved by the machine will consist of finding rapidly

the solutions of large systems of simultaneous equations containing up to 1000 unknowns. The computer must be able to store and classify large

quantities of data, and to refer rapidly for needed items to huge tables

of organization, equipment, supply and other similar data. These tables

will contain millions of items. It is required, in addition, that the

printing devices associated with the computer will be capable of extremely

high-speed printing of the complete details of the Air Force's programs

that have been computed.

Status: COMPLETED. With the continued successful operation of

the machine by the Air Comptroller's Office, this Bureau's responsibility

is considered to have been discharged.
52 Status of Projects

NATIONAL BUREAU OF STANDARDS EASTERN AUTOMATIC COMPUTER (SEAC)

Task 1 104-34-51 07A9~1

(formerly 1 1 .4/24-*+9-1 ) Origin: NBS Authorized 12/15/48

Sponsor: Office of the Air Comptroller, USAF

Full task description appears in Apr-June 1949 issue.

Status: CONTINUED, During the past quarter, SEAC was operated

with an over-all efficiency of 7

1+.5°/° during scheduled computation, as compared with 8l°/° for the previous quarter. With the completion of

debugging of the electrostatic memory, regular computation using 1,024

words of storage was begun. It is believed that the requirement to return

the full memory, 512 words of acoustic and 5 1 2 words of electrostatic,

to operation following a failure has been at least partially responsible

for the drop in operating efficiency from the earlier level of around 80°/o.

The policy has been adopted that beginning January 1, 1954, the

SEAC be utilized principally as an item of general Bureau equipment rather

than as a service facility for other Federal agencies. Maintenance costs

will be distributed to all technical projects at the Bureau, which benefit

from the constant availability of such equipment.
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)

FEKETE, M. (Hebrew University). Asymptotic behavior of certain polynomials

of least deviation on point sets of given transfinite diameter.

October 20.

Numerical Analysis Seminar

(Los Angeles, California)

TOMPKINS, C. B. Numerical local embedding of a surface with riemannian

metric into euclidean 3~space . I. Geometrical introduction

and classical formulas. September 29.

BLANCH, G. Numerical local embedding of a surface with riemannian metric

into euclidean 3- space. II. Some analysis of the partial differential equations involved. October 1

.

SWIFT, J. D. Steiner triples and their calculation on SWAC. October 6.

TEICHROEW, D. The generation of random variables on SWAC. October 8 . CODDINGTON, E. A. (University of California, Los Angeles). On perturbations

of linear systems possessing periodic solutions. October 13.

RUTHRAUFF, R. (Douglas Aircraft Company). I. Symbolic coding for an IBM 701 computer. II. Experience with the conjugate-gradient

and elimination methods for linear systems. October 1 5. REYNOLDS, R. R. Review of Caratheodory ' s 1937 paper on sweeping-out process for Dirichlet's problem. October 20.

WASOW, W. R % On the truncation error for Dirichlet's problem, I. October

°22. II. November 19.

HUSKEY, H. D. New topics in coding for SWAC. October 27.

HESTENES, M. R. (University of California, Los Angeles). A variational

approach to linear partial differential equations of elliptic

type, I. October 29. II. November 3. III. November 5.

SOUTHARD, T. H. On the military evaluation function for the Hotspot

game (a war game of attrition). November 10.

TOMPKINS, C. B. A computer looks at Plateau's problem. I. November 12.

II. November 17.
Lectures and Symposia

REDHEFFER, R. M. (University of California, Los Angeles). Operators and

initial-value problems. November 24.

MOTZKIN, T. S. (University of California, Los Angeles). Preliminaries

to the over-all coding of functions. December 1. TEICHROEW. D. Use of "asymptotic" normalization polynomials in the

computation of probability functions and probability points.

December 3.

BUSK, T. (University of Copenhagen). Aspects of a generalized difference

calculus. December 8 . MILES, J. W. (University of California, Los Angeles). Structural fatigue

under random loading. December 10.

KAHN, L. B. (Naval Supply Center, Oakland, California). Forecasting ac- tivities and managing inventories at the U. 5. Naval Supply

Center, Oakland. December 15.

LEVIN, E. Review of Douglas' 1927 paper on numerical solution of Plateau's

problem. December 17*

St a.U. s.t

i

p. .a l._Ep

g

i q e e

r

ijag, , CpJl&fljiia

EISENHART, C. Some notes on the evolution of the statistical method as

a research tool. October 16.

ZELEN, M, Methods of measuring useful life of equipment under operational

conditions. October 30.

CONNOR, W. S. On the experimental attainment of optimum conditions, a discussion of the methods of G.E.P.Box of the Imperial

Chemical Industries, Manchester, England. December 11.

SAVAGE, I. R. Most probable rank orders. December 18.

Papers and Invited Talks

Pre sented by Member s of the Staff

at Meetings of Outside O rganization s DAVIS, P. Program in the theory of mechanical quadrature. Presented at

the Mathematics Colloquium, University of Maryland,

College Park, Md., December 13.

EISENHART, C. (l) Statistical method in research and development. A

series of three lectures presented as part of "Mathematics

for Model n Engineering" at the University of California,

October 26-29. (2) A new look at the probable error—it never was what it used to be. Presented at a meeting of

the Mathematics Division, Naval Ordnance Test Station,

China Lake, California, October 28; at a meeting of Sigma

Xi, Oregon State College, Corvallis, Oregon, November 3?

and at a Mathematics Seminar, University of Washington,
Lectures and Symposia 55

Seattle, Washington, November 5-. (3) The role of statistical

method in scientific research. Presented at a Mathematics

Seminar, University of Oregon, Eugene, Oregon, November 3.

(4) Statis tical method for the chemical, physical, and

engineering sciences. Presented at the Engineering Institute, University of Wisconsin, Madison, Wisconsin, November 1 6. (5) Precision and accuracy. Presented to the

Basic Science Division Study Group of the American Institute

of Electrical Engineers, New York, N. Y. , November 1 8

.

(6) Statistical method in experimentation. Presented at a Scientific Staff Meeting at the NBS , Washington, D. C .

,

December 18.

FEKETE, M. The radii of plane domains and their applications. Presented

by The American University in cooperation with the National

Bureau of Standards at the Bureau, Washington, D. C.

,

December 1 1

.

FORSYTHE, G. E. (1) Best conditioned matrices. Presented at a meeting of

the Mathematics Department, University of California, Los

Angeles, November 25. (2) How should we solve a system of

linear equations? Presented to the Institute for Certificated

Personnel of the Los Angeles City Schools, Los Angeles City

College, December 4.

SAVAGE, I. R. Statistics without assumptions. Presented to the Columbian

Society, George Washington University, Washington, D. C.,

December 7.

TODD, J. Comparative theory of numerical solutions of differential equations. Presented at the Applied Physics Laboratory Colloquium, Johns Hopkins University, Silver Spring, Md., October 30.

TOMPKINS, C. B. The mathematician in Government establishments. Presented at the meeting of the American Mathematical Society

cosponsored by the National Research Council, New York, N.Y.,

October 23.

YOUDEN, W. J. (l) Experimental design. Presented at a staff meeting of

the Logistics Research Project, George Washington University,

Washington, D. C., October 1; at a Symposium of the D-13

Committee of the American Society for Testing Materials,

New York, N. Y. , October 1 5 j to the Study Group of the

Basic Science Division of the American Institute of Electrical Engineers, New York, N. Y., November 25? and at a meeting of the local chapter of the American Chemical Society, York, Pa., December 3. (2) Behavior of measurements.

Presented at a meeting of the Washington Area Section of

the American Society for Stress Analysis, Washington, D. C.

,

October 26; and at a Seminar at the Naval Ordnance Laboratory, White Oak, Silver Spring, Md., December 10. (3) Statistics and experimental work. Presented at the Squibb

Institute for Medical Research, New Brunswick, N. J.,

November 4.
56 Lectures and Symposia

Papers presented at the meeting of the American Mathematical Society,

Baltimore, Md . , December 28-31:

HOFFMAN, A.J. (l) On the singular values of the product of two operators.

(2) On an inequality of Hardy, Littlewood, and Polya. Both

presented by title.

HOFFMAN, A. J., M. NEWMAN, E, G. STRAUS, 0 O TAUSSKY. The number of absolute

points of a correlation.

Papers presented at the joint annual meetings of the American Statistical

Association, the Biometric Society, and the Institute of Mathematical

Statistics, Washington, D. C., December 27-30:

CONNOR, W. S. New experimental designs for paired observations.

Dixon, W. J., and D. TEICHROEW. Some sampling results on the power of

non-parametric tests against normal alternatives.

LIEBLEXN, J. A historical note on the relation between extreme values

and tensile strength. Presented by title.

SAVAGE, I. R. TEICHROEW, D.Optimum nonparametric tests for small samples.

A method for generating random variates on electronic

computers

.
Publication Activities

1 . PUBLICATIONS WHICH APPEARED DURING THE QUARTER

1.1 Mathematical Tables

(1) Tables of 10x

. N8S Applied Mathematics Series 27. Available

from U. S. Government Printing Office, Washington 25* D. C., 03.50.

(2) Table of natural logarithms for arguments between zero and five

to sixteen decimal places. NBS Applied Mathematics Series 31.

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

D. C . , 03.25. (This is a reissue of NBS Mathematical Table MT10,

Table of natural logarithms, vol. Ill (195-1)).

(3) Tables of circular and hyperbolic sines and cosines for radian

arguments. NBS Applied Mathematics Series 36. Available from U.S.

Government Printing Office, Washington 25, D. C., 03.00. (This

is a reissue of NBS Mathematical Table MT3 (1939, 2d ed. 195-9)).

1.3 Technical Papers

(1) On the solution of the differential equation occurring in the problem of heat convection in laminar flow through a tube. M. Abramowitz.

J. Math. Phys. XXXII, 1 85-1 87(July-Oct . 1 953)

.

(2) The convergence of numerical iteration. H. A. Antosiewicz and

J. M. Hammersley . Am. Math. Mo. LX, 6o5~607(Nov. 1953).

(3) Changes of sign of sums of random variables. P. Erdos and G. A.

Hunt. Pac. J. Math. 2, 673-687(Dec. 1953).

(4) A numerical analyst's fifteen-foot shelf. G. E. Forsythe. M.T.A.C.

VII, 221-228(0ct. 1953).

(5) An expansion method for parabolic partial differential equations.

J. W. Green. J. Res. NBS 5l, 1 27-1 32(Sept. 1953).

(6) The SWAC—design features and operating experience. H. Huskey,

R. Tfyorensen, B. F. Ambrosio, and E. C. Yowell. Proc . IRE 4l , 1294-l299(Oct. 1953).

(7) Completely continuous normal operators with property L. I. Kaplamsky. Pac. J. Math. 3? 721-724(Oec, 1953).

(8) Pairs of matrices with property L, II ( summary) „ T. S. Motzkin

and 0. Taussky. Proc. Nat. Acad. Sci. 32? 961 -963(Sept . 1953).

(9) Rejection of outlying observations, F. Proschan. Am. J. Phys.

21, 520-525(Oct. 1953).

(10)

Modes of vibration of a suspended chain, D. S. Saxon and A, S. Cahn.

Qu. J. Mech. App. Math. VI, Pt. 3 , 273~285( 1 953)

.

57
58 Publication Activities

(11) The magnetic field of cylindrical coils. C. Snow. NBS Applied

Mathematics Series 38. Available from U. S. Government Printing

Office, Washington 25? D. €., 25 cents.

(12) Asymptotic solution of the differential equation of hydrodynamic

stability in a domain containing a transition point. W. Wasow.

Ann. Math. 18, 222-252(Sept . 1953).

(13) On small disturbances of plane Couette flow. W. Wasow. J. Res.

NBS H, 195-202(Qct. 1953).

(14) The principles of experimental design. W. J, Youden. Appeared

as a chapter in "Research Operations in Industry," (Papers delivered at the Third Annual Conference on Industrial Research,

June 1953)? pp. 31 7“329(K.ings Crown Press, Columbia University,

New York, N. ¥., 1953).

( 15) Making one measurement do the work of two. W. J. Youden and

W. S. Connor. Chem. Eng. Prog, 42, 549~552(Oct. 1953).

1 .4 Notes

(l) On a computation of the capacity of a cube. P. Davis and

W. F. Cahill. M.T.A.C. VII, 272-273(Oct. 1953).

1.5 Miscellaneous Publications

(1) Mathematical services useful in industry. A. S. Cahn, Jr. Proceedings of Symposium on Industrial Applications of Automatic

Computing Equipment held by the Midwest Research Institute, Kansas

City, Mo., January 8, 1953? pp. 19-27.

2. MANUSCRIPTS IN THE PROCESS OF PUBLICATION DECEMBER 31, 1953

2,1 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 Lagrangian coefficients for sexagesimal interpolation.

NBS Applied Mathematics Series 35. In press, Government Printing

Office

.

(4) Tables of f unctions 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.

(5) Table of secants and cosecants to nine significant figures at

hundredths of a degree. NBS Applied Mathematics Series 40. In

press, Government Printing Office.

(6) 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.
Publication Activities 59

(7) Tables of expected values of 1/X for positive Bernoulli and

Poisson variables* E. Grab and I. R. Savage. Accepted for

publication in the Journal of the American Statistical Association.

(8) 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.

2.2 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; in press, Government Printing Office.

(2) Bibliography of nonparametrie statistics and related topics.

I. R. Savage. Accepted for publication in the Journal of the

American Statistical Association.

2.3 Technical Papers

(1) Evaluation of the integral

poo 2/ / \ e

n x ,J 'du„ M. Abramowitz.

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) Approximate method for rapid Loran computation. M. Abramowitz,

0. H. Call, and J. C. Mathews. Submitted to a technical journal.

(4) The relaxation method for linear inequalities. S. Agmon . Accepted

for publication in the Canadian Journal of Mathematics.

(5) On the differential equation x + k(f(x) + g(x)x)x = ke(t).

H. A. Antosiewicz. Submitted to a technical journal.

(6) On a certain integral involving Bessel functions. H. Antosiewicz.

Submitted to a technical journal.

(7)

Some implications of Liapunov's conditions for stability. H. A.

Antosiewicz and P» Davis . Submitted to a technical journal.

(8) On a theorem of Ostrowski and Taussky. R. Bellman and A. J.

Hoffman. Accepted for publication in Archiv der Mathematik.

(9) On mildly nonlinear partial difference equations of elliptic iype.

L. Bers. Accepted for publication in the Journal of Research of

the NBS.

(10) On modified divided differences. G. Blanch. Accepted for publication in Mathematical Tables and Other Aids to Computation.

( 1 1 ) A general-purpose control panel for a model II CPC. P. B. Bremer,

D. Teichroerw, and E. C. Yowell. To be published in the IBM Newsletter.

(12) Programs for computing hypergeometric series. W. F. Cahill. Accepted for publication in Mathematical Tables and Other Aids to Computation

.
60 Publication Activities

(13) Acceptance. sampling of electroplated articles. J. M. Cameron and

F. Ogburn. Accepted for publication in Plating.

(1*0 Partially balanced incomplete block designs with two associate

classes and two treatments per block. W. H. Clatworthy. Submitted to a technical journal.

(15) Some theorems for partially balanced designs. W. S. Connor and

W. H. Clatworthy. Accepted for publication in the Annals of

Mathematical Statistics.

( 16 ) Time-discrete stochastic processes in arbitrary sets, with applications to processes with absorbing regions and to the problem

of loops in Markoff chains. D. van Dantzig. Submitted to a technical journal.

(17) Linear functional equations and interpolation series. P. Davis.

Submitted to a technical journal.

( 18 ) On a problem in the theory of mechanical quadratures. P. Davis.

Accepted for publication in the Pacific Journal of Mathematics.

(19)

p Some L aspects of Faber polynomials. P. Davis and H. Pollack.

Submitted to a technical journal.

(20) On representations and extensions of bounded linear functionals

defined on classes of analytic functions. P. Davis and J. L. Walsh

Submitted to a technical journal.

(21 ) On the optimal character of the (s,S) policy in inventory theory.

A. Dvoretzky, J. Kiefer, and J. Wolfowitz. Submitted to a technical journal.

(22) A test for statistical control applicable to a short series of

observations. C. Eisenhart and E. P. King. Submitted to a technical journal.

(23) Inequalities for eigenvalues of Hermitian matrices. K. Fan. To

be included in Contributions to the solution of systems of linear

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

CMJr Some remarks on commutators of matrices. K, Fan. Accepted for

publication in Archiv der Mathematik

.

(25) Lower bounds for the rank and location of the eigenvalues of a matrix. K. Fan and A. J. Hoffman. To be included in Contributions to the solution of systems of linear equations and the determination of eigenvalues, NBS Applied Mathematics Series 39.

( 26 ) Some metric inequalities in the space of matrices. K. Fan and

A. J. Hoffman. Submitted to a technical journal.

(27) 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

(28) Asymptotic lower bounds for the frequencies of polynomial membranes

G. E. Forsythe. Submitted to a technical journal.
Publication Activities 61

(29) SWAC. G. E. Forsythe. Submitted to a technical journal.

(30) 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.

(31) A numerical solution of Schroedinger 1 s equation in the continuum.

W. Futterman, E. Osborne, and D. S. Saxon. Accepted for publication in the Journal of Research of the NBS.

(32) Linear programming in bid evaluation. L.Gainen, D. J. Honig, and

E. D. Stanley. Accepted for publication in Logistics Research

Quarterly

.

(33) A nonlinear model for the composite pi-meson. S. G. Gasiorowicz.

Submitted to a technical journal.

(34) Application of variational methods to intermediate and high energy

scattering. E. Gerjuoy and D. S. Saxon. Submitted to a technical

journal

.

(35) Variational principles for the acoustic field. E. Gerjuoy and

D, S. Saxon. Submitted to a technical journal.

(36)A method of solving the transportation problem. A. N. Gleyzal.

Submitted to a technical journal.

(37) Equations of physics in general Newtonian space-time. A. N. Gleyzal. Submitted to a technical journal.

(38) The representation of integer^ by binary quadratic rational forms,

K. Goldberg, M. Newman, E. G. Straus, and J. D. Swift. Accepted

for publication in Archiv der Mathematik.

(39) An embedding theorem for balanced incomplete block designs.

M. Hall, Jr., and W. S. Connor. Accepted for publication in the

Canadian Journal of Mathematics.

(40) 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.

(4l ) On certain series expansions involving Whittaker functions and

Jacobi polynomials. P. Henrici. Submitted to a technical journal.

(42) Ueber die Funktionen von Gegenbauer. P. Henrici. To appear in

Archiv der Mathematik.

(43) On a combinatorial theorem. A. J. Hoffman. Submitted to a technical journal.

(44) A characterization of normal matrices. A. J. Hoffman and 0.

Taussky. Accepted for publication in the Journal of Research

of the NBS,

(45) On optimum grouping in one-criterion variance components analysis.

E, P. King. Submitted to a technical journal.

(46) Probability limits for the average chart when process standards

are unspecified. E. P, King. Submitted to a technical journal.
62 Publication Activities

(47) A property of the normal dis tribution . E. P. King and E. Lukacs.

Submitted to a technical journal,,

(48) On certain character matrices. D. H. Lehmer. Submitted to a technical journal.

(49) Acoustic radiation pressure on a circular disk. H. Levine. To

appear in the Proceedings of the Fifth Symposium on Applied

Mathematics of the American Mathematical Society.

(50) On the variances and covariances of order statistics from the

Weibull distribution. J. Lieblein. Submitted to a technical

journal

.

(51 ) On strongly continuous stochastic processes. E. Lukacs. Accepted

for publication in Sankhya.

(52) Certain Fourier transforms of distributions (II). E. Lukacs and

0. Szasz. Accepted for publication in the Canadian Journal of

Mathematics

.

( 53 ) Nonnegative trigonometric polynomials and certain rational characteristic functions. E. Lukacs and 0. Szasz. Accepted for phblication in the Journal of Research of the NBS.

(54) A proof of Hilbert's Nulls tellensatz . T. S. Motzkin. Submitted

to a technical journal.

( 55 ) On Fejer sets in linear and spherical spaces. . T. S. Motzkin and

1. J. Schoenberg. Accepted for publication in Annals of Mathematics .

(56) On the relaxation method for linear inequalities. T. S. Motzkin

and I. J. Schoenberg. Submitted to a technical journal.

( 57 ) Least p-th power polynomials on a real finite point set. T. S. Motzkin and J. L. Walsh. Submitted to a technical journal.

(58) Structure theorems and identities for modular subgroups. M. Newman.

Submitted to a technical journal.

( 59 ) The arithmetic structure of certain modular subgroups. M. Newman.

Submitted to a technical journal.

(60) On the Lerch zeta function. F. Oberhettinger . Submitted to a technical journal.

(61 ) On two problems in abstract algebra connected with Horner's rule.

A. M. Ostrowski. Submitted to a technical journal.

(62) On nearly triangular matrices. A. Ostrowski. Submitted to a technical journal.

( 63 ) Det erminanten mit ueberwiegender Hauptdiagonale und die absolute

Konvergenz von linearen Iterationsprozessen . A. M. Ostrowski.

Submitted to a technical journal.

(64) On the convergence of Gauss' alternating procedure in the method

of the least squares, I. A. M. Ostrowski. Submitted to a technical journal.
Publication Activities 63

(65) On the linear iteration procedures for symmetric matrices. A.M.

©strowski. Submitted to a technical journal.

( 66 ) On the spectrum of a one parametric family of matrices. A. M.

Ostrowski. Submitted to a technical journal.

( 67 ) On Gauss' speeding up device in the theory of single step iteration. A. M. Ostrowski. Submitted to a technical journal.

(68) On absolute convergence of linear iteration processes. A. M.

Ostrowski. Submitted to a technical journal.

( 69 ) On spectra of second-order differential operators. D. Ray. Submitted to a technical journal.

(70) On the convergence of asymptotic solutions of linear differential

equations. I. M. Kedheffer and W. Wasow. Submitted to a technical

journal

.

(71) Numerical computation of low moments of order statistics from a normal population. J. B. Rosser. Submitted to a technical

journal.

(72) An isoperimetric inequality for closed curves convex in even- dimensional Euclidean space. I, J. Schoenberg. Submitted to a technical journal.

(73) Characteristic roots of quaternion matrices. 0. Taussky. Accepted for publication in Archiv der Mathematik.

(7*+) Generalized commutators 'of matrices and permutations of factors

in a product, of three matrices. 0. Taussky. For publication in

the von Mises Anniversary volume.

(75) Some computational problems in algebraic number theory. ©.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*

( 76 ) 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.

(77) The condition of certain matrices, II. J. Todd. To appear in

Archiv der Mathematik.

( 78 ) 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.

(79) On"the accuracy of the numerical solution of the Dirichlet problem

by finite differences. J, L. Walsh and D. Young. Accepted for

publication in the Journal of Research of the NBS. #

(80) 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.
64 Publication Activities

(81 ) Singular perturbation methods for nonlinear oscillations. W. Wasow.

To appear in the Proceedings of a Symposium on Nonlinear Circuit

Analysis, held by the Polytechnic Institute of Brooklyn, N. Y.

(82) An extremum property of sums of eigenvalues. 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.

(83) Einschliessung von Eigenwerten hermitescher Matrizen nach dem

Abschnittsverf ahren . H, Wielandt. To appear in Archiv der

Mathematik.

(84) 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.

(85) On eigenvalues of sums of normal matrices. H. Wielandt. Submitted

to a technical journal.

( 86 ) On the eigenvalues of A+B and AB. H. Wielandt. Submitted to a technical journal.

(87) Performance of inspectors and gasoline pumps. W. I. Youden and

M. W. Jensen. Submitted to a technical journal.

(88) A note on partially balanced designs. M, Zelen. Accepted for

publication in Annals of Mathematical Statistics.

(89) Analysis for some incomplete block designs having a missing block.

M. Zelen. Accepted for publication in Biometrics.

(90) Bounds on a distribution function, which are functions of moments

to order four. M. Zelen. Submitted to a technical journal.

2.4 Reviews, Notes

(1) Note on the circle theorem of hydrodynamics, E. Levin. Submitted

to a technical journal.

(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.

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.
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