NATIONAL BUREAU OF STANDARDS REPORT

3795

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

A Quarterly Report

July through September 1954

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

FOR OFFICIAL USE
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. Radiometry. 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.

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

NBS PROJECT

NAML

NBS

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

July through September 1954

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication, rep

unless permission Is

i

25, D.C, Such pern

cally prepared If thi

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

n part, Is prohibited

ndards, Washington

iort has been spoclflaort for Its own use,

REPORT

3795
APPLIED MATHEMATICS DIVISION

July 1 through September 30, 1 954-

TECHNICAL ADVISORY COMMITTEE

David Blackwell, Howard University

Mark Kac , Cornell University

Philip M. Morse,

Massachusetts Institute of Technology

Mina S. Rees, Hunter College

A. H. Taub, University of Illinois

Edward Teller, University of California

DIVISION OFFICE

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

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

Olga Taussky-Todd , Ph.D., Consultant

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

**Ge rtrude Blanch, Ph.D., Mathematician.

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

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

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

Yates S. Sladen, Assistant Chief Clerk

Louis J. Meyerle, Jr., Secretary

Esther L. Turner, Secretary

NUMERICAL ANALYSIS SECTION

yenry A. Antosiewicz, Ph.D. Everett C. Dade

Philip Davis, Ph.D.

John Todd, B.S., Chief

*Ky Fan, Ph.D.

Karl Goldberg, M.A. Evelyn A. Grigg, Sec'y

Alan J ; Hoffman, Ph.D.

Morris Newman, Ph.D. Philip Rabinowitz, Ph.D.'

COMPUTATION LABORATORY Milton Abramowitz,

Irene A. Stegun, M Ph.D., Chief (Acting)

[.A., Assistant Chief

Ruth K. Anderson, M.A. Stanley D. Grant, Jr. Mary Orr

Joseph Bram, Ph.D. William G. Hall, B.S. Maxine L. Paulsen, B.S. Hans 0. Bremer, B.A. Gene vie E. Hawkins, B.S. B. Stanley Prusch

William F. Cahill, M.S. Gloria F . Holmes, B.S., Sec'y Ida Rhodes, M.A. Ruth E. Capuano Dorothea H. Jirauch, M.A. Sally Tsingou, B.S. Mary M. Dunlap, B.S. Lambert S. Joel, B.A. Bertha H. Walter

Anne F. Futterman, B.A. David S. Liepman Philip J. Walsh, B.S. Leon Gainen, B.A. Ethel C. Marden, B.A. Joseph H. Wegstein, M.S

Billie R. Gill, M.S. Kermit C . Nelson Ruth Zucker, B.A. Elizabeth F. Godefroy Peter J. O'Hara, B.S.

STATISTICAL ENGINEERING LABORATORY Churchill Eisenhart, Ph.D., Chief

Joseph M. Cameron, M.S., Assistant Chief

Marion T. Carson

Willard H. Clatworthy, Ph.D. Yvette B. Cocozzella, Sec'y

William S. Connor, Ph. D.

MATHEMATICAL PHYSICS SECTION

Anne R. Cock, B.A. Peter Henrici, Ph.D.

.Lola S. Deming , M. A.

,

Lela J. Hamilton, Sec'y

Julius Lieblein, Ph. D.

Edward W. Cannon, Ph.D., Chief

Robert F. Dressier, Ph.D., Ass John G. Hershberger, M.S. Fritz Oberhettinger , Ph.D.

Mary G. Natrella, B.A.

I-. Richard Savage, Ph.D.

Marvin Zelen, M. A.

stant Chief

Edith N. Reese, B. A. Lillian Sloane, Sec'y

Under contract with The American University ** On leave of absence
CONTENTS

Page

Status of Projects as of September 30, 195*+ ... 1 Numerical Analysis Section 1 (NBS Section 11.1)

Computation Laboratory 8

(NBS Section 11.2)

Statistical Engineering Laboratory 21 (NBS Section 11.3)

Mathematical Physics Section 27

(NBS Section 1

.k)

SE AC 30

Lectures and Symposia 32

Publication Activities 3*+

Status of Projects

September 30. 1954

I. NUMERICAL ANALYSIS SECTION

( Section 11.1)

RESEARCH IN NUMERICAL ANALYSIS AND RELATED FIELDS

Task 1 101-10-1 10V55-55

Origin and Sponsor: NBS

Managers: J. Todd, P. Davis

Authorized 8/13 /55- Revised 8/29/54

Objective: To develop new numerical techniques for the solution

of problems in mathematics, physics, chemistry, and engineering, and to

conduct research in special topics of mathematics of immediate applicability

to the techniques of computation and to the construction of mathematical

table s

Background: The use of high speed automatic digital computing

machines as tools for research, development, production, and management

is increasing rapidly. In order fully to exploit their capabilities

Comments: W

part of this task:

1102- 10- 1104/50-1

1 102- 10- 1 104/51-6

1102- 10- 1104/53-1 5

1 102-

ffe ctive

ch in all

rk on the

Re se arch

Solution

Dif f e ren

following former tasks will be continued as

10- 5116/52-33

10-5116/53-5^

in classical numerical

of Laplace equation by

:ial equation for nerve

analysis

Monte Carlo method

fiber reaction

Tables of integrals involving higher transcendental

functions

1 102- 10- 51 16/53 -

5

*+ Water waves

Also that portion of task 1 1 02- 1

- 1 1 04/50-

5 , Bibliography of Coding

Procedures, Mathematical Tables, and Numerical Analysis, that pertains to

numerical analysis, is continued under this task 0 Status: NEW. H. Antosiewicz , in a new manuscript, "Stable systems

of nonlinear differential equations," studies the differential equation

dx/dt=p(x,t) under conditions which make xeO a stable solution in the sense of Liapunov. He derives sufficient conditions on a perturbation

q(x,t) which is integrable in t over the positive half-line so that every

solution of dx/d t=p (x , t )+q ( x , t ) is stable.

In their paper, "A representation for solutions of analytic

systems of differential equations," H. Antosiewicz and M. Abramowitz show

that the general solution of the n-dimension vector differential

equation dx/d t= A( t

, z ) x , where A(t,z) satisfies certain simple conditions,

1
2 Status of Projects

may be represented as X(t,z)Y(t)c, where Y(t)c is the general solution

of some equation dy/dt=B(t)y. The usefulness of the theorem is illustrated

by obtaining convergent series representations for Coulomb wave functions

and parabolic cylinder functions.

The study of differential equations for nerve fiber reaction

(previously reported under task 1 1 02-10-1 104/53_ 1 5) by H. Antosiewicz and

P. Rabinowitz is continuing. The code is being rewritten to study a second order system of nonlinear differential equations. A report on

the mathematical aspects of the work performed so far has been prepared

and will be incorporated in a joint report with the sponsor.

M. Abramowitz and P. Henrici have begun work on an expository

account of the practical summation of series. P. Henrici has completed

a paper on the "Application of two methods of numerical analysis to the

computation of the reflected radiation of a point source." The paper

discusses the effect of applying Aitken's 6‘2-method to the summation

of the series for the hypergeometric function F ( 3/^ > 5/^+ 5

1

,z) , and then

applies the method of Davis and Rabinowitz (see publications (7) and (8)

below) to the study of the error due to numerical quadrature of an integral involving this function.

J. Todd completed a note on the work of L. F. Richardson.

J. Cameron, M. Newman, O. Taussky-Todd , and J. Todd completed the manuscript

on "Generating of random numbers on SEAC."

Publications: (1) "Coulomb wave functions along the transition

line," by M. Abramowitz and P. Rabinowitz; to appear in the Physical

Review. (2) "Coulomb wave functions in the transition region," by

M. Abramowitz and H. Antosiewicz; to appear in the Physical Review.

(3) "A representation for solutions of analytic systems of differential

equations," by H. Antosiewicz and M. Abramowitz; to appear in the Journal

of the Washington Academy of Sciences. (4-) "Some implications of Liapunov's

conditions for stability," by H. Antosiewicz and P. Davis; Journal of

Rational Mechanics and Analysis 3? 447-457 (July 19 54). (5) "Stable

systems of nonlinear differential equations with integrable forcing term,"

by H. Antosiewicz; submitted to a technical journal. ( 6 ) "Computation

of vibration modes and frequencies on SEAC," by W. Cahill and S. Levy

(NBS 6.4); submitted to a technical journal. (7) "On a problem in the

theory of mechanical quadratures," by P. Davis; to appear in Pacific

Journal of Mathematics. (8) "On the estimations of quadrature errors for

analytic functions," by P. Davis and P. Rabinowitz; to appear in Mathematical

Tables and Other Aids to Computation. (9) "A multi-purpose orthonorraalizing

code and its uses," by P. Davis and P. Rabinowitz; to appear in the Journal

of the Association for Computing Machinery. (10) "Application of two

methods of numerical analysis to the computation of the reflected radiation

of a point source," by P. Henrici; IN MANUSCRIPT. (11) "On the Lerch

zeta function," by F. Obe rhe t tinge r ; submitted to a technical journal.

(12) "Generation and testing of random numbers on SEAC," by 0. Taussky-Todd

,

J. Todd, M. Newman, and J. Cameron; IN MANUSCRIPT. (13) "Motivation for

working in numerical analysis," by J. Todd; to appear in the Transactions

of a Symposium in Applied Mathematics, held by the American Mathematical

Society and the Office of Ordnance Research in Chicago, 111., April 1954.
Status of Projects 3

TURBULENT ATMOSPHERIC CONTAMINATION

Task 1 101-10-1 104/55-59

Origin:

Sponsor:

Managers

Applied Physics

University, and

I!

Laboratory, Johns Hopkins

National Bureau of Standards

? ! f?

Authorized 7/22/5*+

E. Marden, G. Hawkins

Objective: To devise computational methods for determining the

transport of suspended particles in the atmosphere as a result of

turbulent air flow. In particular, to evaluate the total contamination

which accumulates from a number of points and finite area sources where

the amount of contamination from a single point source can be expressed as

S

( 27rx 2)3/2

exp -(x 2+v

2+z2

) 2x 2

x-direction, x

2

, is given by

t-o,)R(a)dcc,

ity

ient

t = dispersion time.

Background: The programming of this problem for solution on SEAC

arises from a proposal made by F ,N . Frenkie 1, of the Applied Physics

Laboratory, to calculate the concentration of contaminants contained in

smog by application of the statistical theory of turbulence to existing

or predicted meteorological conditions and to a known or assumed distribution

of sources. Data for the Los Angeles area are to be used for a numerical

example

and where the standard deviation in the

1 1 / x

2 = 2u2

<0 Q

= source strength

u = intensity of fluctuating veloc

R - Lagrangian correlation coeffic

Status: NEW. A code for SEAC

have been completed for several test

of unit strength and mean velocities

Los Angeles area.

has been prepared, and computations

cases using single point sources simulating prevailing winds in the

BASIC RESEARCH IN LINEAR PROGRAMMING

Task 1 102-10-5116/50-2

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

and Office of the Air Comptroller, USAF

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

Full task description: Jan-Mar 1950 issue, p. 25-

Status: CONTINUED. K. Fan, 0. Tauss ky-Todd, and J. Todd have

revised their manuscript, "Discrete analogs of inequalities of Wirtinger."

In collaboration with I. Glicksberg, K. Fan has introduced the concept

of a "fully convex" normed linear space as a generalization of the notion

of uniformly convex space. The principal results of the manuscript they

are preparing are as follows: (1) Every fully convex Banach space is

reflexive: (2) a characterization of reflexive spaces in terms of "weak

full convexity;" (3) A mean-value theorem in fully convex Banach spaces
4 Status of Projects

that generalizes G. Birkhoff's mean ergodic theorem.

K. Fan and G. Pall have determined all possible sets of n-1

complex numbers , . . . , /3n_

-j ) that may be the eigenvalues of a principal

submatrix of a normal matrix with prescribed eigenvalues (c6^ , . . . , <xn

) . This generalizes an earlier result of H. Wielandt for the Hermitian case.

A. Hoffman has pointed out that Wielandt 1 s theorem yields a simple proof

of Horn's result characterizing the set of diagonal elements of a Hermitian

matrix with prescribed eigenvalues.

A first draft of a monograph "On systems of linear inequalities,"

has been prepared by K. Fan. Chapter I deals with purely algebraic aspects

of finite systems of linear inequalities in a linear space that is not

necessarily normed, topological, or finite-dimensional. Most of the results

in this chapter are not new, but have usually appeared in the literature

in a form which had meaning only in the finite-dimensional case. Chapter

II is devoted to results which require that the space be finite-dimensional,

and Chapter IUis a study of linear inequalities from the viewpoint of

functional analysis, requiring either that the space be normed or topological.

The results obtained are related to or generalize work of Hahn-Banach,

Rosenbloom, Hoffman, Schoenberg, Nikando, and He stene s-McShane

.

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

by H. A. Antosiewicz; to appear in Pacific Journal of Mathematics.

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

by K. Fan and A. J. Hoffman; included in Contributions to the solution of

systems of linear equations and the determination of eigenvalues, NBS

Applied Mathematics Series 39? pp. 1 1

T

7 — 1 30 . (3) "Some metric inequalities

in the space of matrices," by K. Fan and A. J. Hoffman; to appear in

Proceedings of the American Mathematical Society. (4) "Discrete analogs of

inequalities of Wirtinger," by K. Fan, 0. Taussky, and J. Todd; submitted

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

J. Todd; to appear in the Journal of the London Mathematical Society.

(6) "On the relevance of LeChate lie

r

1 s principle to linear programming,"

by A. J. Hoffman; IN MANUSCRIPT. (7) "An extremum property of sums of

eigenvalues," by H. Wielandt; to appear in the Proceedings of the American

Mathematical Society.

SUPPORTING RESEARCH IN LINEAR PROGRAMMING

Task 1102-10-5116/54-9

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, A. J. Hoffman

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

Status: CONTINUED, In collaboration with the U. S. Air Force,

the National Bureau of Standards will conduct a symposium on linear

programming to be held in Washington on January 27, 28, and 29, 1955.

Sessions are planned on computation procedures, economic theory, relations

with other branches of mathematics, and practical applications. Each

session will include a one-hour address as well as shorter papers.

Acceptances have been obtained for over half the program.

The code for solving linear programs by the so-called modified

simplex method (i.e., systematic use of the inverse of the basis leaving

the initial tableau unaltered) has been written by L. Joel and will be

tested in comparison with the original simplex method. In this connection

it has been observed that, for sparse matrices containing many unit vectors
Status of Projects 5

a good proportion of which are expected to belong to any basis, a variation of the modified simplex method will save arithmetic operations.

The observation is based on the fact that only the inverse of the submatrix

complimentary to the identity submatrix contained in the basis need be

carried

In s technique for

written as the

observed that

approach. By

may be treated

that the funct

if one formula

function which

which this res

a linear progr

functions

The

studied: Let

in a convex se

( 1 ) . Le t d be

y yielding

tudying the paper of Charnes and Lemke which developed a minimizing on a polyhedron a convex function that can be sum of convex functions of one variable, it has been

the sum condition is not an intrinsic feature of their

altering their procedure slightly, any convex function

by linear programming techniques. It is nece s sary , howe ve

ion whose minimum is to be located must be convex; i.e.,

tes the Charne s-Lemke procedure in an abstract way, the

it may handle is required to be convex. The setting in

ult is proved seems so general as to discourage hopes for

amming approach to the location of minima of non-convex

following problem (arising in an economic context) has been

(1) Ax+By>c be a polyhedron in (x,y) space, and let y vary

t K, such that for any yfK, there exists an x satisfying

a given vector in the x-space . The problem is to find

5

min max (d,x)

y€if (x,y)€L

For the p

f e asible

minimize

devising

procedure

have been

articular problem under consideration, the problem is computationally

but, more generally it leads (via the dual) to the problem:

(z,w), where zeZ, weW, Z, W given polyhedra. The problem of

an algorithm to solve this problem is under study. So far,

s for finding z , w such that (z ,w )<(z ,w) , (z ,w ) < (nz , w ) ,,

, . * o

7 o o’ o — o’ ’ o’ o — ’ o

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

method," by R. Bryce; IN MANUSCRIPT. (2) "Linear programming in bid

evaluation," by L. Gainen, D. J. Honig, and E. D. Stanley; Logistics

Research Quar. _1_, 48-54 (Mar. 1 954). (3) "On the solution of the

caterer problem," by J. W. Gaddum, A. J. Hoffman, and D. Sokolowsky;

to appear in Logistics Research Quarterly. (4) "An algorithm for solving

the transportation problem," by A. Gleyzal; to appear in Journal of

Research of the NBS . (5) "Smooth patterns of production," by A. J. Hoffman

and W. Jacobs; to appear in the Transactions of the Institute of Management

Science, (6) "On the optimal ordering of items for a two-stage process,"

by A. J. Hoffman; IN MANUSCRIPT. (7) "On block relaxation," by L.S. Joel;

IN MANUSCRIPT. ( 8 ) "A remark on the smoothing problem," by H.A. Antosiewicz

and A. J. Hoffman; to appear in the Transactions of the Institute of

Management Science.

RESEARCH IN MATHEMATICAL TOPICS APPLICABLE TO

NUMERICAL ANALYSIS

Task 1101-10-5116/55-56

Origin: Office of Naval Research

Sponsor: " " Managers: O. Taussky-Todd , M. Newman

Objective: To conduct basic and supporting research in various

branches of mathematics of particular interest to agencies of the Federal

Authorized 8/13/54

Revised 8/29/54
6 Status of Projects

government, especially in fields related to numerical analysis.

Background: Various agencies in the Federal government have problems

in mathematics, and in particular in several aspects of numerical analysis,

for solution of which the staff and facilities of the National Bureau of

Standards are uniquely fitted.

Comments: Work on the following former tasks will be continued as part of this task:

1 1 02— 1 0— 1 1 0^/50—*+ Miscellaneous studies in pure mathematics

1 1 02-1 0-1 1 04/50-5a Number-theoretical test problems for SEAC

1102-10-1104/52-34 Special problems in finite matrix theory

Status: NEW. M. Newman has completed a manuscript entitled

"The coefficients of certain modular forms," in which are studied the

coefficients Pr(u) arising in the formal expansion

tt (1-x ) = 2_ P (n)x

n-1 n=o

The principal result is the following: If p is a prime greater than

3, r(p+1)=0 mod 24, A = r(p2-1)/24, then P

p

( np+A)= (-p) ( r-2)/2p^ (n/p)

holds for r=2, 4, 6, 8, 10,14,26 and for no other values of r. Many of the

computations involved in showing the exhaustiveness of the set of r'

s

were carried out and checked on SEAC. Without the aid of the computer,

the labor would have been excessive. This is one of the few instances

in number theory where a theoretical question was definitively settled

with the aid of a high-speed computer.

In connection with this work, many values of Pr (n) have been

computed on SEAC, and it is anticipated that these will be made available

in a report soon. It was found that P^(482)=0, whereas P^(n)/0 for

all n<482 except the well-known case n=53.

M. Newman has submitted to a technical journal a manuscript

entitled "axm-by n=c." It is shown that this diophantine equation has

only a finite number of solutions for nonzero a,b,c and all positive

integral m , n except m=n=2, m=1 or n=1

.

In collaboration with J. D. Hoffman of NBS Division 7.6,

K. Goldberg is studying the eigenvalues of certain matrices arising in

the theory of dielectric relaxation. The mathematical problem is to show that, apart from exceptional cases which are physically trivial,

certain non-symme trie matrices have distinct, real eigenvalues. So far,

the reality has been proved in general, and the distinctness only for

those matrices of order 3.

0. Taussky-Todd has continued her work on the uniqueness of the

normal base in cyclic fields, and has now shown that it is not unique in

fields of degree 8. She further studied the possibility of finding a base for cyclic fields that, together with its conjugates, generates

a normal matrix (not merely a permutation of a normal matrix). She has

shown, with the assistance of K. Goldberg, that this is not possible for

fields of degree 2, 3, 4,5,6.

As an aid to a statistical problem, A. Hoffman has shown how

to construct, for every n, a latin square of order n with distinct diagonal

elements. He has further constructed, for every even n, a latin square

such that any set of n elements, no two in the same row or column, cannot

be distinct.

J. Todd has begun a study of the condition of the systems of

equatibns approximating to integral equations.

Publications: (1) "Linear functional equations and interpolation

series," by P. Davis; to appear in Pacific Journal of Mathematics.
Status of Projects 7

2 (2) "Some L aspects of Faber polynomials," by P. Davis and H. Pollack

(Bell Telephone Laboratories). (3) "Continuity in terms of connectedness,"

by K. Fan and R. A. Stuble; to appear in Proc. Nede rl. Akad . , Wetensch.

Ind. Math. (4) "Compactif ication of completely regular spaces," by

K. Fan and F. Wagner; IN MANUSCRIPT. (5) "Inequalities for eigenvalues

of Hermitian matrices," by K. Fan; included in Contributions to the

solution of systems of linear equations and the determination of eigenvalues,

NBS Applied Mathematics Series 39? pp. 131-139. (6) "Some inequalities

concerning positive definite Hermitian matrices," by K. Fan; submitted

to a technical journal. (7) "A comparison theorem for eigenvalues of

normal operators," by K. Fan; submitted to a technical journal. (8) "Power

series for log (e x .ey)", by K. Goldberg; IN MANUSCRIPT. (9) "On certain

series expansions involving Whittaker functions and Jacobi polynomials,"

by P. Henrici; to appear in Pacific Journal of Mathematics. (10) "On

generating functions of the Jacobi polynomials," by P. Henrici; to appear

in Pacific Journal of Mathematics. (11) "The number of absolute points

of a correlation," by A. Hoffman, M. Newman, E. Straus, and O. Taussky;

submitted to a technical journal. (12)"The diophantine equation

axm-by n=c," by M. Newman; submitted to a technical journal. (13) "Note

on a certain determinant," by M. Newman; IN MANUSCRIPT. (l4) "Structure

theorems for modular subgroups," by M. Newman; to appear in Duke Mathematical

Journal. (15) "The coefficients of certain modular forms," by M. Newman;

IN MANUSCRIPT. (16) "Some computational problems in algebraic number

theory," by 0. Taussky; to appear in the Proceedings of the American

Mathematical Society' s Sixth Symposium on Applied Mathematics, held at

Santa Monica, California, August 1953. (17) "Generalized commutators of

matrices and permutations of factors in a product of three matrices," by

O. Taussky; to appear in the von Mises anniversary volume. (18) "The

condition of the finite segments of the Hilbert matrix," by J. Todd;

included in Contributions to the solution of systems of linear equations

and the determination of eigenvalues, NBS Applied Mathematics Series 39?

pp. 109-116. (19) "Error bounds for eigenvalues of symmetric integral

equations," by H. Wielandt; to appear in the Proceedings of the American

Mathematical Society's Sixth Symposium on Applied Mathematics, held at

Santa Monica, August, 1953. (20) "On eigenvalues of sums of normal

matrices," by H. Wielandt; submitted to a technical journal.
II. COMPUTATION LABORATORY

(Section 11.2)

1 . Mathematical Tables

1102-40-1110/43-3 TABLES OF E

1 (z) , (z=x+iy)

Origin : Canadian National Research Council

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

Status : Inactive. For status to date, see Oct-Dec 1953 issue, p. 28.

1102-40-1110/46-1 TABLE OF THE GAMMA FUNCTION FOR COMPLEX ARGUMENTS

Origin : NBS

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

Status : Completed.

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

NBS Applied Mathematics Series 35-; available from U. S. Government

Printing Office, Washington 25 , D. C. , #2 . 00 .

1102-40-1110/47-2 TABLES OF COULOMB WAVE FUNCTIONS

Origin : NBS

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

Status: Continued. Computations o f Fq,F6

, G

0

, G^ are in progress for

L=0 , p=*\/3 5 ^2 ,l(, 2r\j 3r\? 4r\for p

varying from 0 to 50. The results

are to be included in the second volume of Coulomb wave functions.

1102-40-1110/50-7 WAVE FUNCTION FOR LITHIUM

Origin : NBS

Sponsor : Bureau of Ordnance, USN

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

Status ; Continued. The iterations for the last approximation to the wave function have been started and have been run as machine time was available. A code to compute the new approximation to the eigenvalue

has been checked out.

1102-40-1110/51-4 COLLECTED SHORT MATHEMATICAL TABLES OF THE

COMPUTATION LABORATORY

Origin : NBS

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

Status : Continued. The first volume is in press.

Publication : "Tables of functions and of zeros of functions: Collected

short tables of the Computation Laboratory," NBS Applied Mathematics

Series 37; in press, U. S. Government Printing Office.

8
Status of Projects 9

1102-40- 1110/ 51-8 TABLES OF POWER POINTS OF ANALYSIS OF VARIANCE TESTS

Origin : Section 11 . 3 ? NBS

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

Status : Continued. This project has been inactive since March 1953 and

is now being resumed. Following is a summary of the status to date.

The original task was deemed too large for available funds and was,

for the present, reduced. In the notation of the full task description,

it was decided to compute nineteen (a,/3) tables based on all pairs ct,/3 derivable from oC=.01, .02, . 05 ? .20, /? = .01, .05? .10? .50? .90 with

/3 <i-oo. 1 Preliminary to the main computation it is necessary to have available

inverse tables of the incomplete beta function ratio,

Ix

( 2

f

!

> 2

f 2

} = (irf

1

» ^

^ 2 ^ ^ ^ 2'^

1 =

For <X=.01 ? .05? these had been calculated previously (C.M. Thompson,

Biometrika, vol. 32, p. 168). For OC-.02, .20, they were computed by

a code written by L. Joel. The code performed inverse interpolation

on the function Xx (p?q)? which was calculated using a well-known double

recurrence formula that allows easy estimates of the rounding error.

The main computation is being performed generally along the lines

suggested by Emma Lehmer in Annals of Mathematical Statistics . December

1944 . When fp is an even integer, the infinite series can be replaced

by a finite series, and this is worth while for f^<60. For these values

of fp and all finite f ^ , the corresponding entries in the nineteen

(&,/3) tables have been computed using a code written by S. Tsingou.

The remaining finite values of fp require computation of the infinite

series. A code which sums this series and performs the

has been written by Miss Tsingou and is being checked,

the series are computed using the recurrence

i

x

( p+ 1 ? q ) = i

x

( p ? q

)

q p

( 1-x) X

( p ? q

)

inverse interpolation

The terms of

Whether the method is feasible for fp small and odd remains to be

seen, but no difficulty is anticipated for f 9 large. When this computation

is completed, all that will remain to be don4 are the entries corresponding

to the infinite values of f^ and fp.

1 102-40-1 1 10/52-7 REVISION OF MATHEMATICAL TABLES

Origin : NBS

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

Status : Continued. Following is the status of those mathematical tables

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

planned:

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

NBS Applied Mathematics Series 32; reissued July 1954 and available

from the U. S. Government Printing Office, Washington 25? D. C., S>2.25.

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

Mathematics Series 4l ; in press, U. S. Government Printing Office.

This was originally Mathematical Table MT8, "Tables of probability

functions," vol. I (1941)

"Tables of sines and cosines for radian arguments," NBS Applied

Mathematics Series 43; in press, U. S. Government Printing Office.

This table was formerly designated as Mathematical Table MT4.

"Tables of natural logarithms," vol. IV, Mathematical Table MT12

(1941); revision in progress for reissue in the Applied Mathematics

Se rie s
10 Status of Projects

1 102-4-0-1 1 10/52-14 TABLE OF ARCSIN FOR COMPLEX ARGUMENTS

Origin : NBS

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

Status ; Inactive. For status to date, see Oct-Dec 1953 issue, p. 31.

1102-40-1110/52-18 EXTENSION OF THE TABLE OF HYPERBOLIC SINES AND COSINES

Origin : NBS

Full task description : July-Sept 1951 issue, p. 4i

Status ; Continued. Typing of the manuscript has been completed, and

checking is in progress.

1102-40-1110/52-23 TABLE OF THE MODIFIED AIRY INTEGRAL

Origin : NBS

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

Status : Inactive. For status to date, see Oct-Dec 1953 issue, p. 31

1102-40-1110/52-25 TABLE OF ERROR FUNCTION FOR COMPLEX ARGUMENTS

Origin : NBS

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

Status : Inactive. The 20-place table for x,y=0(.1)2 is available on punched cards.

1102-40-1110/52-31 EXTENSION OF TABLES OF THE EXPONENTIAL FUNCTION FOR

NEGATIVE ARGUMENTS

Origin : NBS

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

Status : Inactive. For status to date, see Oct-Dec 1953 issue, p. 32 #

1102-40-1110/52-37 SPHEROIDAL WAVE FUNCTIONS

Origin : NBS

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

Status : Inactive. For status to date, see Jan-Mar 1954 issue, p. 31

•

1102-40-1110/52-49 RADIAL MATHIEU FUNCTIONS

Origin : NBS

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

Status : Continued. Editing and differencing of the results is in progress.

1 102-40-1 1 10/52-57 SIEVERT'S INTEGRAL

Origin : NBS

Full task description : Jan-Mar 1952 issue, p. 46

Status : Inactive. For status to date, see Oct-Dec 1953 issue, p. 33.

1102-40-1110/53-35 HYPERGEOMETRIC FUNCTIONS

Origin : NBS

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

Status : Inactive. For status to date, see Apr-June 1954 issue, p. 27.

1 102-40-1 1 10/53-52 L-SHELL CONVERSION COEFFICIENTS

Origin : Oak Ridge National Laboratory

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

Status : Continued. Part III of the problem (see Apr-June 1954 issue,

PP. 27-23) involves the integration of a differential equation with
Status of Projects 1 1

radial distance r as independent variable. A singularity occurring at

r=0 is overcome by using a series expansion up to a point r=r and

numerical integration beyond that point. A number of computations were carried out in order to explore the effect of varying r

s

*

371

- 60-0009/ 55-65 RESEARCH AND DEVELOPMENT IN AUTOMATIC CODING

Origin : NBS

Ob jec tive : To investigate methods of improving programming techniques for

high speed computers and make the results generally available. 'The

immediate objectives are: (1) to improve the general purpose interpretive

floating decimal routine now in use; (2) to develop new service routines

and incorporate them in the system now in use, which facilitates rapid access to the computer; (3) to complete the improved version of the

input conversion routine.

The long-term objectives are: (1) to study current and completed

programs in order to develop special techniques for general use;

(2) to develop a general purpose pseudo-code.

Background : The present library of subroutines and service routines has

been produced largely as by-products of other projects. Their usefulness

has indicated the need for separate study of automatic programming

techniques. The experience of the staff in operating other high speed

automatic digital machines will be applied. While the object basically is

the exploitation of SEAC, nevertheless attempts will be made to keep

the work on as general a footing as possible.

Comments : That portion of the former task 1 1 02- 1

0

- 1 1 04/50-5 , Bibliography

of Coding Procedures, Mathematical Tables, and Numerical Analysis, that

pertains to coding procedures, is continued under this task.

Status : New. (1) An input conversion routine called Sebbe 3 which

composes codes, converts floating decimal numbers to floating binary

numbers, converts decimal numbers to hexadecimal numbers, and supplies

each of the above groups with appropriate read-in orders, was put into

service. ( 2 ) A course is being conducted for NBS personnel in which

programming for SEAC is being taught. (3) The sine, cosine and arctangent

subroutines are in process of being incorporated in the interpretive

floating-point routine (base 00).

2 . Mathematical Services

Note: The tasks under Mathematical Services are arranged serially

according to the digits following the slant lines in the task number.

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

in which the task was authorized.

1102-40-5126/50-13 RAY TRACING

Origin and Sponsor : NBS, Section 2.2

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

Status : Continued. Several aerial camera lens systems are being analyzed.

1102-40-5126/51-37 MOLECULAR STRUCTURE, III

Origin : Naval Research Laboratory, USN

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

Status : Continued. Computations were performed as requested.
12 Status of Projects

1 102-^0-5126/52-20 SPHERICAL BLAST

Origin and Sponsor; Naval Ordnance Laboratory

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

Status ; Inactive. For status to date, see Apr-June 195^- issue, p. 30

1 102-40-51 26/52-5-4- CALCULATIONS FOR d STAGINGS

Origin and Sponsor ; NBS , Division 9

Full task description ; Oct-Oec 1951 issue, p. Z7

Status ; Continued. Programming, coding, and checking on SEAC have been

completed. About 20 problems have been run during the past quarter.

1 102-/0-51 26/53-5- NEUTRON DIFFUSION, II

Origin and Sponsor ; Atomic Energy Commission, New York Office (NDA)

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

Status ; Terminated. The objectives of this project have been changed, and

future work growing out of this task will be performed under task

1 1 02-/0-51 26/55-78? to be established in the next quarter.

1 102-5-0-5126/53-1 1 STRENGTH OF WING COMPONENTS

Origin and Sponsor ; National Advisory Committee for Aeronautics,

Langley Field, Va.

Full task description; Oct-Dec 1952 issue, p. 5/

Status ; Inactive. For status to date, see Apr-June 1955- issue, p. 31.

1102-^0-5126/53-20 RADIANT HEATING OF SOLIDS

Origin and Sponsor ; NBS, Section 10.2

Full task description ; Oct-Dec 1952 issue, p. 55

Status ; Terminated. The objectives of this project have been changed, and

current work is being performed under tas k 1 102-5

- 0

- 51 26/5Z-Z 5, p. 16 .

1 102-/0-5126/53-27 COMPUTATION OF THERMODYNAMIC FUNCTIONS

Origin and Sponsor ; NBS, Division 5 Full task description ; Jan-Mar 1953 issue, p. 57

Status ; Continued. The SEAC code was revised to include temperatures up

to 15 > 000 °K, and thermodynamic functions were calculated for approximately

30 molecules that included these higher temperature ranges.

1102-/0-5126/53-29 DYNAMIC BEHAVIOR OF AIRCRAFT STRUCTURES

Origin and Sponsor ; NBS, Section 6./

Full task description ; Jan-Mar 1953 issue, p. 58

Status ; Inactive. For status to date, see Apr-June 195/ issue, p. 32.

1102-/0-5126/53-39 ACOUSTICAL IMPEDANCES

Origin and Sponsor ; NBS, Section 6.1

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

Status ; Inactive. For status to date, see Apr-June 1953 issue, p. 55.

1 102-/0-5126/53-/1 LORAN UNIVAC CODE

Origin and Sponsor ; Hydrographic Office, U. S. Navy

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

Status ; Continued. Code checking has been started on the Baseline

Extension and on the Skywave Correction and Trainer Codes.
Status of Projects 13

1 102-40-5126/53-51 RADIATION DIFFUSION

Origin ; NBS , Section 4.8

Sponsor : Armed Forces Special Weapons Project

Full task description; Apr-June 1953 issue, p. 57 (Neutron Diffusion III)

Status : Continued. Codes have been written and checked (a) for extension

of reflection calculations to cover the energy range from 50 to 1000

Kev., (b) for the evaluation of the energy ratios received behind a slab and behind an equal thickness of an infinite medium, and (c) for

energy and angular spectra of transmitted and reflected radiation.

A code has also been written for the determination of the -spectrum of

pulses produced by gamma rays in a sodium iodide crystal, a widely-used

type of radiation detector.

Information on the above problem is difficult to obtain experimentally

and is important for the interpretations of measurements made with

scintillation counters.

1 102-40-5126/54-4 HIGH TEMPERATURE PROPERTIES OF WATER

Origin : NBS, Division 3.2

Sponsor : Bureau of Ordnance, USN

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

Status : Continued. A code has been written to compute the thermal

functions for the one-ionic species Oyjj-j- using the formulation of

H. W. Wooley mentioned in the previous quarterly report. Most of

these functions have been computed for the temperature range between

103 and 10° degrees Kelvin at 50 intervals of temperature logarithmically

spaced. The same code will be used to compute thermal functions for

the singly ionized species H and these functions, along with the rest

of the Oyiii for the same range of temperature.

371 1-60-0009/54-7 DISTRIBUTION OF NORMAL MODES OF VIBRATION OF CUBIC

LATTICES, II

Origin and Sponsor : NBS, Division 30

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

Status : Inactive. For status to date, see Apr-June 1954 issue, p. 33.

371 1-60-0009/54-8 NORMAL VIBRATIONS IN MOLECULES

Origin and Sponsor : NBS, Section 3.2

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

Status : Inactive. For status to date, see Apr-June 1954 issue, p. 34.

371 1-60-0009/54-1 1 COMPUTATION OF VIBRATION MODES AND FREQUENCIES

Origin and Sponsor : NBS, Section 6.4

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

Status : Continued. The first five symmetric modes for three aircraft

structures (using 31 mass points) were computed.

Publications : "Computation of vibration modes and frequencies on SEAC,"

by W. F. Cahill and S. Levy (NBS Section 6.4); submitted to a technical

journal.

1 102-40-5126/54-13 AWARD OF PROCUREMENT CONTRACTS BY LINEAR PROGRAMMING

Origin and Sponsor : New York Quartermaster Procurement Agency

Full task description : Oct-Dec 1953 issue, p. 43

Status : Continued. Contract awards for the Quartermaster Corps are being

evaluated on SEAC at the average rate of two per week. Similar UNIVAC

routines have been written and checked out in cooperation with personnel

of the Office of the Air Comptroller, USAF.
Status of Projects 14

1 102-4-0- 5 1 26/54-1 4 FLOW IN SUPERSONIC NOZZLE

Origin and Sponsor ; National Advisory Committee for Aeronautics,

Langley Field, Va.

Full task description ; Oct-Dec 1953 issue, p. 41+ Status : Inactive. For status to date, see Apr-June 195*+ issue, p. 35.

371 1 -60-0009/5^-1 7 DEPOLYMERIZATION

Origin and Sponsor : NBS , Section 7.6

Full task description : Oct-Dec 1953 issue, p. 44

Status : Inactive. For status to date, see Oct-Dec 1953 issue, p. 44.

371 1-60-0009/55-- 19 ENERGY LEVELS OF COMPLEX ATOMS

Origin and Sponsor : NBS, Section 4.1

Full task description : Jan-Mar 195*+ issue, p. 4l

Status : Continued. During this quarter the first nine characteristic

roots of a 25x25 symmetric matrix were computed.

1 102-40-5126/54-20 LOW TEMPERATURE PROPERTIES OF BORON COMPOUNDS

Origin and Sponsor : NBS, Section 3.2

Full task description : Jan-Mar 1955- issue, p. 4l

Status : Inactive. For status to date, see Jan-Mar 1955- issue, p. 4l .

1102-40-5126/54-21 EXPERIMENTAL PROGRAM FOR MULTIPLE INPUT AND OUTPUT

Origin and Sponsor : NBS, Section 12.3

Full task description : Jan-Mar 1955- issue, p. 42

Status : Inactive. For status to date, see Apr-June 1955- issue, p. 36 .

371 1 -60-0009/55-22 ENERGY DISTRIBUTIONS ON OPTICAL IMAGE

Origin and Sponsor : NBS, Section 2.2

Full task description : Jan-Mar 1955-, p. 43 2 2 Status : Continued. Computations have been completed for W=9x(x‘i+y ) for | and r\ ranging from 1 to 16. Computations for W=100(x2+y2)+200(x2+y2)2

-300(x2+y 2 )3 are being carried out for various values of | and

1102-40-5126/54-24 SOLUTION OF THE DIFFUSION EQUATION FOR ELECTRONS

Origin and Sponsor : NBS, Division 4

Full task description : Jan-Mar 1955- issue, p. 43

Status : Inactive, For status to date, see Jan-Mar 1 955- issue, p. 43.

1 102-40-5126/54-25 DEFLECTED RADIATION FROM AN INFINITE LAMBERT PLANE

Origin and Sponsor : Armed Forces Special Weapons Group

Full task description: Jan-Mar 1954 issue, p. 44

Status : Continued. Values of \jc ( u , v) for u,v = ,25(. 25)16.0 have been

computed and have been submitted to the sponsor.

371 1-60-0009/5^-27 EMF OF CELLS AT HIGH TEMPERATURE

Origin and Sponsor : NBS, Section 1.8

Full task description : Jan-Mar 1955- issue, p. 45

Status : Continued. Electromotive forces for all fluorides and iodides

have been computed.
Status of Projects 15

371 1-60-0009/55-28 LOVIBOND NETWORK FOR CIE SOURCE A

Origin and Sponsor : NBS, Section 2.1

Full task description : Jan-Mar 1955- issue, p. 46

Status : Inactive. For status to date, see Apr- June issue, p. 37.

371 1-60-0009/55-30 SPECTRUM ANALYSIS

Origin and Sponsor : NBS, Division 4

Full task description : Jan-Mar 1955- issue, p. 46

Status ; Continued. Searching of differences between all known and

predicted levels for pairs of lines which confirm known levels and predict

new levels was performed for approximate ly 20 differences for Tc I. The wave numbers (c) for over 6,000 lines were computed for ruthenium.

1 102-40-5126/54-33 AIR CONDITIONING IN UNDERGROUND STRUCTURES

Origin and Sponsor ; NBS, Section 10.3

Full task description ; Jan-Mar 1955- issue, p. 47

Status ; Continued. A supplementary code has been written to evaluate

the integral 9 _ 111

00 a2 4Gu

7T2

0 [yJ

1 (y)+aGJ0 (y) ] + [yY^y) + aGYQ

(y)]‘

dy

y

Results for this integral and the original integral have been obtained

as they were requested by the originator.

1 102-40-5126/54-34 EQUILIBRIUM CALCULATIONS FOR WATER

Origin and Sponsor ; Naval Ordnance Laboratory

Full task description ; Apr-June 1954 issue, p. 38

Status ; Continued. A set of 46 simultaneous nonlinear equations has been

solved for numerous parameters submitted by the sponsor. For some values

of the parameters the solution method used in the original program

fails to converge and consequently a technique was devised to force

convergence. The problem is being recoded to include this new technique.

1 102-40-5126/54-35 THERMAL STRESSES IN STRUCTURES

Origin and Sponsor : NBS, Section 6.4

Full task description ; Apr-June 1954 issue, p. 38

Status: Inactive. For status to date, see Apr-June 1955- issue, p. 38.

371 1-60-0009/55-36 VELOCITY OF LIGHT

Origin and Sponsor : NBS, Section 11.3

Full task description : Jan-Mar 1955- issue, p. 47

Status : Continued. Ten third degree and ten fourth degree polynomials

were obtained from data submitted by the sponsor using the orthonormalization

code previously programmed.

371 1-60-0009/54-38 COMPRESSIBILITY FACTORS OF DRY AIR

Origin and Sponsor : NBS, Section 3.2

Full task description ; Jan-Mar 1955- issue, p. 48

Status : Continued. The code has been completed and is being checked.
16 Status of Projects

1 102-40-51 26/55-39 MOLECULAR VIBRATIONS

Origin and Sponsor ; NBS , Section 3.2

Ob jjec tive : To develop four routines for solving matrix problems with

floating decimal point operations: ( 1 ) multiplication of matrices;

(2) extraction of eigenvalues; (3) extracting of eigenvectors;

(4) inversion. Routines (2), (3 )

5

and (4) are to be limited to

symmetric matrices.

Background : A problem arising from molecular vibration theory is the

prediction of frequencies of vibrations given the force constants of

the molecules, and conversely. This can be translated into an eigenvalue

problem in which the frequencies can be determined from the eigenvalues

of the matrix of force constants or the matrix of force constants can be

approximated from the eigenvalues. In the latter case the force constants

obtained are used to predict the force constants of other molecules of

a related structure.

Status : New. A routine for the evaluation of eigenvalues of symmetric

matrices using floating decimal point operation has been written and

checked out, and it will be put into use for production in the future.

1 102-40-5126/54-41 FREQUENCY CORRELATION

Origin and Sponsor : NBS, Division 6

Full task description : Apr-June 1954 issue, p. 39

Status : Inactive. For status to date, see Apr-June 195*+ issue, p. 39.

1102-40-5126/54-42 RESOLUTION CORRECTION FOR SCINTILLATION SPECTROMETER

Origin and Sponsor : NBS, Section 4.13

Full task description : Apr-June 1954 issue, p. 40

Status : Continued. Twenty systems of simultaneous equations of order 9

and 20 systems of order 10 have been solved.

1102-40-5126/54-43 CHARACTERISTICS OF CONDUCTING RESISTORS

Origin and Sponsor : NBS, Section 12.1

Full task description : Apr-June 1954 issue, p. 40

Status : Continued. The code has been completed and checked, and the

computations are being processed with data submitted by the sponsor.

1 102-40-5126/54-45 COMPUTATION OF TRANSIENT HEAT FLOW PROBLEMS

Origin and Sponsor : NBS, Section 10.2

Full task description : Apr-June 1954 issue, p. 4l

Status : Continued. The problem has been run for various dif f usivities

and heat transfer coefficients, and the results are being transmitted

to the sponsor as they are obtained.

1102-40-5126/54-46 STUDY IN OPTIMIZATION OF POWER OUTPUT

Origin : Raytheon Manufacturing Company

Sponsor : U. S. Corps of Engineers

Full task description : Apr-June 1954 issue, p. 4l

Status : Inactive. For status to date, see Apr-June 1954 issue, p. 4l„

1 102-40-5126/55-49 WIECHERT DISTRIBUTION FUNCTION

Origin and Sponsor : NBS, Section 7.1

Ob iec tive : To evaluate integrals of the following

L|,(n) =/

oo

, ( n-z) -e (n-z)

type

L n (z) d:
Status of Projects 17

Background ; The integrals arise in the study of approximation methods

in linear viscoelasticity theory.

Status : New. Programming and coding are in progress.

1 102-10-5126/55-50 AWARD OF PROCUREMENT CONTRACTS BY LINEAR PROGRAMMING

Origin and Sponsor : Navy Purchasing Office

Ob jective ; To solve problems, arising in the sponsor's work, of awarding

purchase contracts at minimum cost to the Government.

Background : The calculations to be performed are similar to those now being conducted for the Philadelphia Quartermaster Depot (see task

1 1 02-40- 5 1 26/55- 1

3

, p . 1 3 )

Status : New.

1 102-50-5126/55-52 ATMOSPHERIC WAVES

Origin and Sponsor : Weather Bureau

Objective: To solve by finite differences the system of non-linear

differential equations

M - at

-F0u au

ax -f0

c V, au

ax -F0

n ax M =_ F u at *o axu

in the region 0<u:2„4, -4.2<x<:4.2, where FQ

is a constant and u(x,0),

c(x,0) and v(x,0) are given.

Background : This problem arises in the study of non-linear atmospheric

gravity waves. The equations determine the velocity components u (east)

and v (north) and the function c, representing height, in their dependence

on time, t, and east-west distance, x. Similar computations had

previously been performed at the Weather Bureau on a small scale, using

a characteristic grid. The purpose of the present task is to extend

that work and to investigate the effect of using various fixed grids.

Status : -Completed (New). The results have been transmitted to the sponsor.

1 102-10-5126/55-62 INTEGRALS INVOLVED IN SUPERSONIC FLUTTER

Origin and Sponsor : National Advisory Committee for Aeronautics

Objective: To compute tables of the integrals

F^(M,co) = ~

Jo (f)^ e lMx

Jq

(

x)dx

r\ where A=0(1)11, M ranges from 1.2 to 5, and (T=2KM/M -1 where

K=0( .005) . 1 5( .01) ,20( .05) 1 ( . 1 )2. Results are to be tab ulated to seven decimal places.

Background : The integrals in question occur in the Possio theory of

non-stationary flow for small disturbances in a two-dimensional supersonic

flow and the application of the theory to the determination of aerodynamic

forces on an oscillating airfoil.

Status : New. The code has been written, and checking is in progress.

1 102-40-5126/55-63 VIRIAL INTEGRAL INVOLVING A MORSE POTENTIAL FUNCTION

Origin and Sponsor : NBS , Section 3.2

Objective : To evaluate the integrals

Mn

-L

-x (e 2y-2e y

) n n,

} Jy dy

^ -x(e 2y-2e y

) n,

e y dy

yo

and their first and second derivatives with respect to x for n=0,1,2,

with x ranging from .2 to 20, and yQ ranging from .9 to 5. To compute
Status of Projects

specified combinations of these functions.

Bac kqround : Previously the SEAC has been programmed to calculate ideal

gas thermal properties of molecular species up to 15,000°K, employing

the usual partition function expression. At more elevated temperatures

a different procedure is employed. Since the thermal properties at

these temperatures are sensitive to the upper regions of the potential

energy curve, and the usual partition function sum tends to diverge

when accurately summed over all the necessary energy levels, an expression

was derived for the internal partition function by considering the

equilibria between the molecular species and its dissociation products.

The equilibrium constant can be related to a virial coefficient. Thus

the internal partition function can be directly related to a virial

type integral, which explicitly includes the dependence on the potential

function

.

Status ; New. The program employing the 16-point Gaussian quadrature

formula for evaluating the integrals on SEAC has been written and code

checking is in progress.

1 102-4-0-51 26/55-64 HIGH- TEMPERATURE THERMODYNAMIC TABLES (FMT)

Origin and Sponsor ; NBS, Section 3.2

Ob jjec tive ; To prepare a table of ten generalized thermodynamic properties

(i.e. enthalpy, Helmholtz free energy, etc.) for the elements at high

temperatures. Reduced temperature and atomic volume are to be used as arguments so that the properties are independent of atomic number.

Background : The method to be followed is that of Feynman, Metropolis,

and Teller, Phys. Rev. 23.1 1561 (195-9) using their exact treatment

of high temperature effects. The solutions of the temperature dependent

Fe rmi- Thomas equation are to be extended beyond those obtained in the

FMT paper in the course of these new calculations.

Status : New.

371 1-60-0009/55-66 RECONSTITUTION OF MONOCHROMATIC LIGHT INTENSITIES

Origin and Sponsor : NBS, Division 30

Objective: To solve integral equations of the type

( 1 ) E(x) = I J

P(y) K(x,y)dy = IP,

where the domain of integration is a suitably defined neighborhood

of x. Specifically, to investigate the convergence of iterative methods

of the form

( 2 ) or

(3)

p i+ , = Pi (p/Pi)

p

i+1 = Pi <?/v

Bac kqround : Integral equations of the form (1) arise in the problem of

deblurring of photographic negatives. E(x) is the light incident on the

point x of a photographic negative, P(x) the light emanating from the

original. The kernel K(x,y) represents the effect of light from the

point y on the original on the point x of the negative; in other words,

it represents the degree of imperfection of the photographic system.

If the iterations (2) or (3) converged, then it would be possible to

design automatic equipment for carrying them out. Such equipment could

carry out the operation P. by means of the original photographic system,

and therefore would not require explicit knowledge of the kernel K(x,y).

Comment : This problem was proposed specifically by C. H. Page (NBS, Div.30)

Status s New. First trials have been made on the SEAC at the numerical
Status of Projects 19

solution using an assumed rectangular kernel of the following form:

1 4 1 (each element normalized by the sum 4 16 1 of all elements of the kernel)

1 4 1 For the runs that have been completed, and cU=2 have been used.

Behavior of the iterated values shows marked instability for slight

scaling variations of the original kernel and investigation will

continue

1 102-40-5126/55-67 DOSAGE integral

Origin and Sponsor : U. S. Weather Bureau

Ob je c tive : To tabulate D(d) for various values of the parameters,

D(d)

BQ0 . 4^

5/2

c

3

whe re

oo oo ® oo

' r P r

v j -00-00-00

Exp{-(ut) n 2

j- x

2

+y

2+z

2

j - /[ (d-ut-x) 2

+y

2+z2

]

t a (ut) 3/2(2-n)|- (d-ut-x) 2 + y^+z 2

]

dxdydzdt

The parameters are: the initial source strength (Qq)

,

the Sutton virtual

diffusion coefficient (C) , the average wind speed Cu) , the Sutton

stability index (n), the gamma absorption coefficient in air (/), the

decay parameter (a), the conversion factor to roentgen units (B), and

the distance of receptor at ground downwind of point of release (d).

Background : The problem arises in calculating the potential hazard to

surrounding communities in the event of an accidental release of

radioactive fission products from a nuclear reactor. The problem is

complicated by the fact that during their travel in the atmosphere

these products would be subject to radioactive decay as well as atmospheric dilution through the action of turbulent eddies. In

addition the exposure received by a person on the ground is further

affected by the absorption and scattering of the rays on route from

various parts of the cloud.

Status : New. The quadruple integral has been reduced to a double

integral and D(d) has been computed on the SEAC for one set of

parameters and two values of d.

371 1-60-0009/55-69 THEORY OF DIELECTRIC RELAXATION

Origin and Sponsor : NBS, Section 7.6

Ob jec tive : To investigate the properties of the characteristic roots of

certain matrices arising in dielectric relaxation theory.

Bac kqround : A problem in dielectric relaxation theory is to determine the

possible relaxation times of the dipoles in a crystal when an electric

field which has been applied to the crystal is removed. These relaxation

times can be determined as the eigenvalues of a matrix of transition

probabilities depending on the crystal model.

Status : New. It has been proved that all of the matrices arising in

this connection have real eigenvalues. It remains to be shown that the

roots are distinct in non-trivial cases.
20 Status of Projects

1 102-40-5126/55-72 MARYLAND IN TC R- INDUS TRY STUDY

Origin and Sponsor : Office of Scientific Research, ARDC, USAF

Ob jective : To undertake linear algebraic computation for a State of

Maryland inter-industry study.

Bac kqround : This analysis of the economy of the State of Maryland is

based on the reduction of all Maryland economic activity to 24 basic

industrial categories and the study of the interrelationships among

these classifications in terms of input and output of the industries

in dollars.

Status : New. The Leontieff matrix ll-Al where A is an input-output

matrix of order 24 was inverted; also 1 1—A | X=Y was solved for six

vectors Y.
III. STATISTICAL ENGINEERING LABORATORY

(Section 11.3)

1 . Fundamental Research in Mathematical Statistics

BIBLIOGRAPHY AND GUIDE TO STATISTICAL LITERATURE

Task 1 103-10-1 107A9-1 a Origin: NBS Authorized 1/9/49

Manager: L. S. Deming

Full task description: Apr-June 1949 issue, p. 75

Status: CONTINUED. For a description of the continuing activity

on this task, see the Jan-Mar 195*+ issue, p. 49.

MANUAL ON FITTING STRAIGHT LINES

Task 1 103-10-1 107/50-2

Origin: NBS Authorized 3/1/50

Manager: F. S. Acton

Full task description: Jan-Mar 1950 issue, p. 42

Status: CONTINUED. F. S. Acton has very nearly completed the

remaining sections of the first draft. About half of the manuscript is

currently in the hands of technical advisors for comment.

TABLES TO FACILITATE DRAWING RANDOM SAMPLES

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

Origin: NBS Authorized 7/1/50

Managers: C. Eisenhart, L. S. Deming

Full task description: July-Sept 1950 issue, p. 57

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

P. 64.

MISCELLANEOUS STUDIES IN PROBABILITY AND STATISTICS

Task 1 103-10-1 107/51-2

Origin: NBS Authorized 7/1/50

Manager: C. Eisenhart

Full task description: July-Sept 1950 issue, p. 58

Status: CONTINUED. Mary G. Natrella prepared a composite table of

21
22 Status of Projects

the 5 ? 2 , and 1 percent significance levels of the smaller rank sum for

the Wilcoxon test, by compilation from various published tables, with

correction of all known errata in the originals . A copy ©f the table

has been deposited in the Unpublished Mathematical Tables file maintained

by the editor of Mathematical Tables and Other Aids to Computation.

Publications: (1) "Optimum grouping in one-criterion variance

components analysis," by E. P. King; J. Am. Stat. Assoc. 42 , No. 267 , 637“639 (Sept. 1954 ). ( 2 ) "Time-discrete stochastic processes in

arbitrary sets, with applications to processes with absorbing regions

and to the problem of loops in Markoff chains," by D. van Dantzig;

accepted for publication (in French) in Annales de I'Institut Henri Poincare

(Paris). ( 3 ) "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;

accepted for publication in Biometrika. ( 4 ) "Tables of the inverses of

finite segments of the Hilbert matrix," by I. R. Savage and E. Lukacs;

included in Contributions to the solution of systems of linear equations

and the determination of eigenvalues, NBS Applied Mathematics Series 39 , pp. 105- 108 . ( 5 ) "Bounds on a distribution function which are functions

of moments to order four," by M. Zelen; accepted for publication in the

Journal of Research of the NBS. (6) "On the variances and covariances

of order statistics from the Weibull distribution," by J. Lieblein;

submitted to a technical journal.

LAW OF PROPAGATION OF ERROR

Task 1 103- 10- 1 107/52-1

Origin: NBS Authorized 6/23/51

Managers: C. Eisenhart, I. R. Savage

Full task description: July-Sept 1951 issue, p. 65

Status: TERMINATED. For the last report on this task see July-Sept

1951 issue, p. 65.

PROCEDURES OF NON-PARAMETRIC STATISTICS

Task 1 103- 10- 1 107/52-2

Origin: NBS Authorized 9/17/51

Manager: I. R. Savage

Full task description: July-Sept 1951 issue, p. 66

Status: CONTINUED. I. R. Savage completed an extensive manuscript

entitled, "Contributions to the theory of rank order statistics," which

provides a unified treatment of his research under this project to date.

The problem of the construction of optimum nonparame trie tests

and the evaluation of their power functions is considered. All of the

situations treated involve mutually independent random variables

X1 ,. . . ,Xm

, Y^,...,Y

n where the X

1 s have a continuous cumulative distribution

function F (x) and the Y

1 s have a continuous cumulative distribution

function G(x) . When the observed values of the random variables are arranged in an ascending sequence the rank order is defined as z=(z^ , . . . ,z^

,

. . .

,

z m+n) whe re z^-0(1) if the i-th observation in the ascending sequence

is from the F(G) distribution. A rank order test is formed by assigning
Status of Projects 23

a probability a(z) to each rank order and if a rank order z occurs HqS

F = G is rejected with probability a(z).

If for the rank order z and z' we have P(Z-z ) > P(Z=z') for

all alternatives under consideration then the criterion of admissibility

requires that a = 1 whenever a

1 > 0. When enough relationships of the

above type fail to exist the construction of best tests becomes extremely

complicated. It is shown that for such alternatives as HgZ F(x) > G(x)

,

where the inequality holds for some x, there does not exist an adequate

number of the desired relationships. For the alternative H^: F and G

have density functions f(x) = A(x)B(9^) exp [C(x)D(9^)J and g(x) = A(x)B( 02) exp[ C( x)D( 02) ] where C(x) and D(9) are increasing functions,

it is shown that if two rank orders z and z' are identical except for the

i and j (i<j) elements which are (0,1) for z and (1,0) for z' then

P(Z-z) > P(Z-z'). It is then shown that the Wilcoxon statistic and the

c^ statistic of Terry satisfy the above criterion of admissibility but

that the one-sided Smirnov test is inadmissible

Next the hypothesis H^: G(x) = [F(x)]° where b > 1 is considered.

is a special case of when F(x) has a density function, but even

if there is no density function the previously mentioned result regarding

the ordering of the probabilities of the rank orders is applicable.

However, in this case more can be done towards the construction of

optimum tests. It is shown that

/m+n i

P(Z=z) = m'. n\ o n/ TT ill (l-z.+ bz,)] . / i=1 j=1 J J

Thus for a particular value of o under Hj it is possible to compute the

probabilities of the rank orders and to construct the most powerful

rank order test. For small sample sizes it is possible to examine the

expressions given for the probabilities of the rank orders and to find

other pairs z and z' than those given above such that P(Z-z) > P(Z=z').

For very small sample sizes enough ordering relationships are obtained

for constructing uniformly most powerful rank order tests. For slightly

larger sample sizes where, for some levels of significance, uniformly

most powerful rank order tests fail to exist, it is possible using the

above expression for the probability of a rank order to construct most

stringent rank order tests. Under it is shown that tests based on small values of m+n

T - XI z,DN

-, where

i=1 1 N1 N i

m+n

j=1

.-1

J

are locally most powerful and that T is approximately normally distributed

for large samples. It is indicated that decisions resulting from the T

and the Wilcoxon tests will frequently be the same; but the decisions of

the Wilcoxon and Terry tests will be in agreement even more often.

Various tables are given of probabilities of the rank orders

under H., of the distribution of T, and of the power functions of

proposed tests. These tables enable one to conduct tests of H. against

Hq and to estimate their power for all combinations of sample sizes. An

important empirical conclusion which can be drawn from these tables is

that for all practical purposes the test based on T will give results

almost as powerful as the best test even for very small sample sizes.

Publications: (1) "Easily used simultaneous confidence limits

for a line," by W. S. Connor; submitted to a technical journal. (2)

"Contributions to the theory of rank order statistics," by I. R. Savage;

submitted to a technical journal.
Status of Projects 24-

STUDIES IN THE MATHEMATICS OF EXPERIMENT DESIGN

Task 1103-10-1107/53-1

Origin: NBS Authorized 10/15/52

Manager: W, S. Connor

Full task description: Oct-Dec 1952 issue, p. 60

Status: CONTINUED. 1 » An alternate procedure which results in a more simplified computational technique has .been worked out for analyzing

chain block designs.

2. Let an observation y. . have the mathematical model

constants and

that E € jj=0,

€ij and 5

j E6 .=0

J

„.,v; j=1 ? ... ? b where and t. are fixed

are mutually independent random variables such

var € i j - CT

, cov(ei j, ej_, j

1 )=o for i/i

x _ ~2 var b.- (T b

, cov(0j,6j,) =0 for j/j

or j/j'

,

If the observations are taken using an the restriction b>v, then methods have

confidence limits on the ratio

P S

a-2

*

incomplete block design satisfying

been worked out for placing

3. The present theory of factorial arrangements is useful only where

the levels of the different factors are all at the same prime number

.(i.e„, p n series) or are all powers of the same prime number. Also,

the present theory only allows the construction of designs having block

sizes which are multiples of the level of the factors. Methods have

recently been found for adapting incomplete block designs to mixed

factorials where the restriction on the size of the block is less severe. Publications: (1) "Analysis for some partially balanced incomplete

block designs having a missing block," by M. Zelen; Biometrics J_0, No. 2,

273-281 (June 1954-). (2) "A note on partially balanced designs," by

M. Zelen; Ann. Math. Stat. .25, No. 3* 599“602 (Sept. 1954-). (3) "Partially

balanced incomplete block designs with two associate classes and two

treatments per block," by W. H. Clatworthy; accepted for publication in

the Journal of Research of the NBS. (4) "New experimental designs for

paired observations," by W. S. Connor and W. J. Youden; accepted for

publication in the Journal of Research of the NBS. (5) "On the enumeration

of partially balanced designs with two associate classes," by W„H. Clatworthy

submitted to a technical journal.
Status of Projects

2 . Applied Research in Mathematical Statistics

25

COLLABORATION ON STATISTICAL ASPECTS OF NBS

RESEARCH AND TESTING

Task 3737-60-0002/51-1

Origin: NBS Authorized 7/1/50

Managers: W. J. Youden, J. Cameron

Full task description: July-Sept 1950 issue, p. 60

Status: CONTINUED. Personnel of the Section are collaborating

with G. Conrad (Section 1.6, Engineering Electronics) on the statistical

aspects of his studies of resistor noise. Resistor noise varies from

resistor to resistor (and even for repeat measurements on the same resistor). It is necessary to characterize the behavior of resistors

by statistical techniques which describe the behavior of groups of

resistors. The statistical aspects of this investigation have been

concerned with:

(a) the underlying distribution of resistor noise

(b) the statistical design of experiments to determine the

behavior of noise with respect to changes in frequency, current, and

resistance

(c) the behavior of individual resistors over a long period

of time

(d) comparison of noise characteristics of resistors from

different manufacturers.

Publications: (1) "Instrumental drift," by W. J. Youden;

accepted for publication in Science. (2) "A comparison of four national

radium standards. Part II: Statistical procedures and survey," by

W. J. Youden and W. S. Connor; J. Res. NBS jQ, 191-196 (Sept. 1955-).

STATISTICAL ASPECTS OF NBS ADMINISTRATIVE OPERATIONS

Task 3737-60-0002/52-1

Origin: NBS Authorized 10/1/51

Manager: I. R. Savage

Full task description: Oct-Dec 1951 issue, p. 56

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

p. 68 .

STATISTICAL SERVICES FOR COMMITTEE ON SHIP STEEL, NRC

Task 1 103-^0-5105/52-1

Origin: Ship Structure Committee, NRC Authorized 12/1/51

Sponsor: " " " Manager: W. J. Youden

Full task description: Oct-Dec 1951 issue, p. 58

Status: CONTINUED. Work was begun on the construction of a program

of sampling of production of ship steel that would give information on sources of variability in manufacturing.
26 Status of Projects

RESEARCH IN APPLICATIONS OF MATHEMATICAL STATISTICS TO

PROBLEMS OF THE CHEMICAL CORPS

Task 1103-*+0-5l 18/52-1

Origin: Biological Laboratories, Chemical Corps Authorized 10/1/51

Dept, of the Army

Sponsor: " " " Manager: C. Eisenhart

Full task description: Oct-Dec 1951 issue, p. 57

Status: CONTINUED. A report giving a tabulation of fractional

replications arranged in blocks of the 2

n series of factorial designs

was submitted. Some work was done on methods of programming analysis

of variance computations for large scale computers.

STATISTICAL ANALYSIS AND DESIGN OF EXPERIMENTS

FOR THE U. S. GEOLOGICAL SURVEY

Task 1 103-*+0-5l*+0/5*+-1

Origin: U. S. Geological Survey, Department of Authorized 10/9/53

Xnte rior

Sponsor; " " " Managers: C. Eisenhart, W. J. Youden

Full task description: Oct-Dec 1953 issue, p. 50

Status: CONTINUED. Studies are being made of the variation in

lithology of collections of pebbles taken at different sites, and the

variations of samples taken at the same site.

STATISTICAL ANALYSIS OF BALL BEARING FATIGUE DATA

Task 1 103-^0- 5l*+ 5/5*+-

1

Origin: American Standards Association Authorized 1/13/5*+

Sponsor: 11 ,! Managers J. Lieblein

Full task description: Jan-Mar 195*+ issue, p. 5*+

Status: CONTINUED. The method aimed at obtaining improved life

estimates referred to in the previous report has been worked out. This

method is in process of being applied to the test groups of ball bearing

data. Programming of the necessary computations for the electronic

computer (SEAC) has been completed and is undergoing a trial run. The

revised estimates of L^q and L^-q will form the basis of a regression

study of the various parameters in the life formula including the

exponent p

.
IV. MATHEMATICAL PHYSICS SECTION

(Section 1 1 .4)

RESEARCH IN MATHEMATICAL PHYSICS AND RELATED FIELDS

Task 1104-10-1115/55-57

Origin and Sponsor: NBS Authorized 9/1/54

Manager: E. W. Cannon

Objective: To perform research in various fields of applied

mathematics important in theoretical physics and engineering science,

with special emphasis on problems connected with the program of other

divisions of the Bureau.

Background: During the fiscal year 1954 this program was carried

out under project 1102-10-1104. This project covers the basic formulation

of mathematical theories in new and currently developing fields, the

derivation of solutions to specific problems either in analytic form or by reformulation in a manner suitable for computational procedures, and

the study of mathematical disciplies underlying these problems, such as differential equations, special functions, stochastic processes, etc.

Status: NEW.

RESEARCH IN MECHANICS OF CONTINUA

" Task 1104-10-5160/54-23

Origin: NBS Authorized 12/29/53

Sponsor: Office of Naval Research, USN

Manager: R. F. Dressier

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

Status: CONTINUED. A talk on "Entropy changes in the equations

for rarefaction waves" was presented by R. F. Dressier at the International

Congress of Mathematicians, held in Amsterdam, in September. The manuscript

incorporating these results is now being brought into final form for

publication. Another talk on the "Comparison of theories and experiments

for the hydraulic dam-break wave" was given by Dr. Dressier at the meeting

of the International Union of Geodesy and Geophysics, Rome, Italy, in

September, and the manuscript presenting this material has been accepted

for publication in the Proceedings of the I.U.G.G., Hydrology Section.

Results on an aerodynamic model for unsteady flow which considers

frictional forces and varying shock tube cross-section, but neglects heat

generation, was completed; the paper describing these results appears in the

Journal of Research of the NBS, October 1954 (RP2541).

P. Henrici, in collaboration with P. Chaix, completed a paper

concerning the design of two-dimensional supersonic nozzles. The authors

use the Mach line originating from the throat (the S-line) as carrier

of the initial data. The geometrical data of the S-line are expressed

in terms of three geometrical parameters of the nozzle. Velocity

distribution on the center line is not assumed to be linear. A study

of the transonic region in nozzles using T. M. Cherry's hodograph method

27
28 Status of Projects

has also been started.

M. Abraroowitz has resumed a study of heat transfer in laminar

cylindrical flow. In this investigation the axial temperature gradients

are not suppressed in comparison with radial gradiants.

Publications: (1) "Turbulent flow in shock tubes of varying

cross-section," by R. F. Dressier; accepted for publication in the Journal

of Research of the NBS. (2) "Comparison of theories and experiments for

the hydraulic dam-break wave," by R. F. Dressier; to appear in Proceedings

of the International Union of Geodesy and Geophysics, Rome, Italy, 1 9

.

(3) "Entropy changes in rarefaction waves," by R. F. Dressier; IN MANUSCRIPT.

(4) "On helical elastic springs of finite cross-section," by P. Henrici;

to appear in the Quarterly of Applied Mathematics. (5) "On rotational

viscous flow through a tube" by P. Henrici; to appear in Zeitschrift fur

angewandte Mathematik und Physik. (6) "On the design of two-dimensional

supersonic nozzles" by B„ Chaix and P. Henrici; IN MANUSCRIPT.

RESEARCH IN ELECTROMAGNETIC THEORY

Task 1 104-1 0-51 60/5b-b7

Origin and Sponsor: Diamond Ordnance Fuze Laboratory, Authorized 6/29/5^+

Department of the Army Revised 9/29/5^

Manager: F, Obe rhe ttinge

r

Objective: (1) To investigate the asymptotic expansion of certain

integrals. (2) To investigate the convergence of various Green's

function representations for the electromagnetic field in homogeneous and

inhomogeneous enclosed media. (3) To investigate the problem of the

diffraction of electric and acoustic waves by certain obstacles which

consist of an array of half planes.

Background: (1) The solution of a number of diffraction problems

can be reduced to integral representations which are of the form of a Kontorovich transform. The behavior of such an integral transform for

large values of the variable is of importance for the knowledge of the

behavior of certain asymptotic properties of the diffracted field.

(2) Green's function of the wave equation for the three-dimensional case

of a rectangular cross section (rectangular wave guide) can be represented

in the form of an infinite double series (representation by modes).

This representation becomes useless when the cross section which contains

the source point is approached. Alternate expressions which do not show

this behavior are therefore desired. (3) This research is directed

towards an extension and generalization of previously obtained results

(task 1102-10-5160/54-51, p.29 ).

Status: CONTINUED. F. Obe rhe ttinge r did preliminary work on the

problem of diffraction of a wave field generated by a dipole with

arbitrary orientation on an ideally conducting half-plane.
Status of Projects 29

DIFFRACTION OF WAVES

Task 11 04-10-51 60/54-51

Origin and Sponsor: NBS , Division l4 Authorized 6/29/5*+

Manager: F. Obe rhe ttinge

Full task description: Apr-June 1954 issue, p. 22

Status: COMPLETED.

Publications: (1) "Diffraction of waves by a wedge," by F. Obe rhe ttinge

Communications on Pure and Applied Mathematics VII, 551-563 (Aug. 1954).

(2) "On asymptotic series for functions occurring in the theory of

diffraction of waves by wedges," by F. Oberhettinger; submitted to a technical journal.
NATIONAL BUREAU OF STANDARDS EASTERN AUTOMATIC COMPUTER (SEAC)

As a result of recent reorganizations it has become necessary to move SEAC to another building on the Bureau grounds. During this quarter the new site has been prepared. It is anticipated that the installation will

be shut down on November 15 for the move and will be out of operation

for an estimated period of six to eight weeks. In preparation for the move a new chassis layout for the computer proper has been made, and a signal lead harness identifying every lead has been designed and is being

installed. A new frame and base have been erected in the new location,

and a contractor is now in process of installing a redesigned primary

power distribution system. Also, a new clock distribution system has been

installed so that as soon as SEAC is shut down each chassis can be taken

from the old to the new location and immediately plugged in. This advance

engineering has been done in the hope that it will minimize the length of

time required for putting SEAC back into operation.

During this quarter (the third quarter of its operation as a general

Bureau facility), the SEAC has been operated with an over-all efficiency

of 89°/° during scheduled operation. This compares quite favorably with

its average of 75°/° for its first four years of operation (May 1950 to

May 1 955-) . Scheduled computation time was distributed as follows:

Total scheduled time available for computation . . 843 hours

Good operating time ............... 742 " Idle-in-orde r time ................ 9

" Time lost during or following malfunction .... 92 "

In addition, the SEAC was utilized for 363 hours of nonscheduled computation,

i.e., operation without a duty engineer in attendance for machine

maintenance, and 379 hours for engineering and preventive maintenance.

The record of SEAC operations from July 1 through September 30 is as follows:

Task No . NBS

:

TTOV55-55

5116/54-9

1 110A7-2

1 110/50-7

1 110/ 51-8

1110/52-49

1 110/53-52

1 1 10/55-65

5126/50-13

0009/52-44

5126/53-27

5126/53-29

5126/53-51

5126/54-4

0009/54-8

0009/54-15

0009/54-19

30

Hours Used:

Title Code Productive

Checkinq Operation

Research in numerical analysis 61 25

Supporting research in linear programming 5 10

Tables of Coulomb wave function 13

Wave function for lithium 6 50

Power points of analysis of variance tests 1 Mathieu functions 7

Internal conversion coefficients 23

Automatic coding 3 3

Ray tracing 9 33

Calculations for d spacings 6 8

Thermodynamic functions 4

Dynamic behavior of aircraft structures 1 43

Radiation diffusion 24 143

High temperature properties of water 27 8

Normal vibrations in molecules 2

Matrix reduction 10 12

Energy levels of complex atoms 2
31

Task No . NBS

0009/55-22

0009/55-27

5126/55-33

0009/55-36

5126/55-53

5126/55-55

5126/55-59

5126/55-39

5126/55-63

5126/55-65

0009/55-66

5126/55-67

5126/55-70

Hours Used : Title Code Productive

Checking Ope ration

Energy distribution on optical image 3 25

EMF of cells at high temperature 2

Air conditioning in underground structures 3 6

Velocity of light 3

Characteristics of conducting resistors 1

1 Computation of transient heat flow 35

Wiechert distribution function 20

Molecular vibrations 12 1 Morse integrals 9 55

High-tempe rature thermodynamic tables 1 Reconstitution of light intensities 12 2 Dosage integrals 1 1 Ball bearing fatigue data 3

Totals

22& 505

Other:

1 105/55-59

5126/53-5

5126/53-55

5126/53-53

5126/55-13

5126/55-25

5126/55-35

5126/55-50

5126/55-55

5126/55-52

5126/55-60

5126/55-61

5126/55-62

5126/55-71

5126/55-72

Turbulent atmospheric contamination 7

Neutron diffusion 2

Theory of games 17

Computing services to Chemical Corps 2

Award of procurement contracts by

linear programming 5

Reflected radiation from an infinite

Lambert plane 2

Equilibrium calculations for water 2

Flight simulator

Flight performance 2

Atmospheric waves 1 Detonator voltage and current on oscillograms 9

Elastic neutron scattering 6

Integrals involved in supersonic flutter

Loran tables

Maryland inter-industry study

^5

23

30

2

132

57

6

2

7

55

15

6

19

3

Totals 35^
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.

Papers and Invited Talks

Presented by Members of the Staff

at Meetings of Outside Organizations

Papers presented at the International Congress of Mathematicians,

Amsterdam, Netherlands, September 2-9

:

ALT, F. L. Recent results on the equation of burning.

ORESSLER, R. F. Entropy changes in the equation for rarefaction waves. TAUSSKY-TODD, 0. Normal matrices in some problems in algebraic number

theory

.

TODD, J. The condition of matrices.

Papers presented at the meeting of the American Mathematical Society,

Laramie, Wyoming, August 31 - September 3: ANTOSIEWICZ, H. A. Stable systems of differential equations with integrable

forcing term,

DAVIS, P. , and 0. 0. POLLACK (Bell Telephone Laboratories). Complex

biorthogonality for certain sets of polynomials.

Papers presented at the annual meeting of the American Statistical

Association, Montreal, Canada, September 1 0—1 3

s

EXSENHART, C. The joint effects of reading errors and grouping on standard methods of statistical inference. Presented at

the session sponsored by the Committee on Statistics in

the Physical Sciences.

ZELEN, M. (1) The use of incomplete block designs for factorial experiments.

Presented at a session sponsored by the Committee on Statistics in the Physical Sciences. (2) Exact tests of

significance for combining inter- and intra-block information

in incomplete block designs.

32
Lectures and Symposia 33

Papers presented at other meetings:

ABRAMOWITZ, M. Program for mathematical tables at the National Bureau of

Standards. Presented at the Symposium on Mathematical

Tables held at the Massachusetts Institute of Technology,

Cambridge, Mass., September I5_ l6.

CONNOR, W. S. Statistical procedures. Presented at the AAAS Gordon

Research Conference on Coal, held in New Hampton, N. H.

,

August 25.

DRESSLCR, R. F. Comparison of theories and experiments for the hydraulic

dam-break wave. Presented at the International Geophysics

Meeting, Rome, Italy, September 195*+.

TAU.SSKY-TODD, 0. Bounds for characteristic roots of finite matrices,

and their application in numerical analysis. (1) Presented

at the Institute of Mathematics, Mechanics, and Theoretical

Physics, University of Lund, Sweden, September 23. (2)

Presented at the Mathematical Institute, University of

Stockholm, Sweden, September 27.

TODD, J. (1) Relative difficulty of the solution of differential equations

of different orders and dimensions. Presented at the

Institute of Mathematics, Mechanics, and Theoretical

Physics, University of Lund, Sweden, September 23.

(2) The solution of large systems of linear equations —

theory and practice. Presented at the Mathematical

Institute, University of Stockholm, Sweden, September 2b,

WEGSTEIN, J. H. (1) Automatic programming and a general purpose pseudocode. Presented at the David Taylor Model Basin,

Washington, D. C. , August 23. (2) Automatic coding.

Presented to the Logistics Research Project, George

Washington University, Washington, D. C. , September 30.

YOUDEN, W. J. (1) Control of experimental error by statistical design,

and (2) New statistical designs useful in the physical

sciences. Presented at the Shell Oil Company Seminar on Statistical Methods, Houston, Texas, July 1-2. (3) The

statistical part of the planning of chemical experiments.

Presented at the American Association for the Advancement

of Science, Gordon Research Conference on Statistics in

Chemistry, New Hampton, N. H. , July 26. (4) Statistical

design as an aid to improving the precision of instruments.

Presented at the American Association for the Advancement

of Science, Gordon Conference on Instrumentation, New

London, N. H. , July 30. (5) Interpretation of chemical

data. Presented to the Lehigh Valley Section of the

American Chemical Society, Palmerton, Pa., September 17.

(6) Marbles, measurements, and mathematics. Presented

to the Washington Junior Academy of Science, Washington, D.C.,

September 18. (7) The gun problem. Presented to the

staff of Logistics Research Project, George Washington

University, Washington, D. C. , September 23.
Publication Activities

1 . PUBLICATIONS WHICH APPEARED DURING THE QUARTER

1.1 Mathematical Tables

(1) Table of sine and cosine integrals for arguments from 10 to 100,

NB5 Applied Mathematics Series 32. (A reissue of NBS Mathematical

Table MT13). Available from the U. S. Government Printing Office,

Washington 2 5, D. C. , $2.25.

(2) Table of the Gamma function for complex arguments. NBS Applied

Mathematics Series 3*+. Available from the U. S. Government

Printing Office, Washington 25? D. C. , $2.00.

(3) Tables of the inverses of finite segments of the Hilbert matrix.

I. R. Savage and E. Lukacs. Included in Contributions to the

solution of systems of linear equations and the determination of

eigenvalues, NBS Applied Mathematics Series 39? pp. 105-108;

available from the U. S. Government Printing Office, Washington,

25, D. C., $2.00.

1.3 Technical Papers

The first six papers listed below are included in "Contributions

to the Solution of Linear Equations and the Determination of Eigenvalues

NBS Applied Mathematics Series 39, available from the U. S. Government

Printing Office, Washington 25, D. C., $2.00.

(1) Inequalities for eigenvalues of Hermitian matrices. K. Fan.

Pp, 131-139.

(2) Lower bounds for the rank and location of the eigenvalues of a matrix. K. Fan and A. J. Hoffman. Pp. 117-130.

(3) Punched-card experiments with accelerated gradient methods for

linear equations. A. I. Forsythe and G. E. Forsythe. Pp. 55~69.

(4) Practical solution of linear equations and inversion of matrices.

L. Fox. Pp. 1-55-.

(5) Iterative methods of solving linear problems on Hilbert space.

R. M. Hayes. Pp. 71-103.

(6) The condition of the finite segment of the Hilbert matrix.

J. Todd. Pp. 109-116.

(7) The relaxation method for linear inequalities. S. Agmon. Can. J. Math. 6, 382-392 (1955-)

.

(8) Some implications of Liapunov's conditions for stability.

H. A. Antosiewicz and P. Davis. Journal of Rational Mechanics

and Analysis 3, 5

- 5

- 7-5-57 (July 19 5*+) .
Publication Activities 35 (9)

Asymptotic lower bounds for the frequencies of polygonal membranes.

G. E. Forsythe. Pac. J. Math. 4} 467-480 (Sept. 195*+).

(10) Linear programming in bid evaluation. L. Gainen, D. J. Honig,

and E. D. Stanley. Logistics Research Quar. J_, 48-55- (Mar. 1954).

(11) Variational principles for the acoustic field. E. Gerjuoy and

D. S. Saxon. The Physical Review 2ii? 1445 (June 15? 195*+).

(12) Optimum grouping in one-criterion variance components analysis.

E. P. King. J. Am. Stat. Assoc. 42? 637~639 (Sept. 1955-).

(13) On the relaxation method for linear inequalities. T. S. Motzkin

and I. J. Schoenberg. Can. J. Math. 6, 393-5-04 (195*+).

( 14) Diffraction of waves by a wedge. F. Oberhettinger. Communications

on Pure and Applied Mathematics 2? 551-563 (Aug. 1955-).

0 5) The evaluation of the exponential integral for large complex

arguments. J. Todd. J. Res. NBS 22? 313-318 (June 1955-).

( 16 ) Statistics and planning tests at elevated temperatures. W.J.Youden.

Proceedings of the Society for Experimental Stress Analysis XI?

No. 2, 219-222 (Nov. 1950).

(17) A comparison of four national radium standards. Part II: Statistical procedures and survey. W. J. Youden and W. S. Connor.

J. Res. NBS 23, 191-196 (Sept. 1955-).

( 1 8) A note on partially balanced designs. M. Zelen. Ann. Math. Stat.

22? 599-602 (Sept. 1955-).

(19) Analysis for some partially balanced incomplete block designs

having a missing block. M. Zelen. Biometrics 10, 273-281

(June 1 955-) .

1.5 Miscellaneous Publications

(1)

Contributions to the solution of systems of linear equations

and the determination of eigenvalues. NBS Applied Mathematics

Series 39. Available from U. S. Government Printing Office,

Washington 25, D. C. , #2.00.

2. MANUSCRIPTS IN THE PROCESS OF PUBLICATION SEPTEMBER 30, 1 955-

2.1 Mathematical Tables

(1) Tables of functions and of zeros of functions.

Collected short tables of the Computation Laboratory. NBS

Applied Mathematics Series 37. In press, Government Printing

Office

(2) 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.

(3) Tables of sines and cosines for radian arguments. NBS Applied

Mathematics Series 43. (A reissue of NBS Mathematical Table 4.)

In press, Government Printing Office.

(4) Table of salvo kill probabilities for square targets. NBS Applied

Mathematics Series 43. In press, Government Printing Office.
36

2.3 Technical Papers

Publication Activities

(1) Coulomb wave functions in the transition region. M. Abramowitz

and H. Antosiewicz. Accepted for publication in The Physical Review.

(2) Approximate method for rapid Loran computation. M. Abramowitz,

D. H. Call, and J. C. Mathews. Submitted to a technical journal.

(3) Coulomb wave functions along the transition line. M.. Abramowitz

and P. Rabiraowitz. Accepted for publication in The Physical Review.

(4) A theorem on alternatives for pairs of matrices. H. Antosiewicz.

Accepted for publication in the Pacific Journal of Mathematics.

(5) Asymptotic solution of linear differential equations with a parameter. H. Antosiewicz. Submitted to a technical journal,

(6) On a certain integral involving Bessel functions. H. Antosiewicz.

Submitted to a technical journal.

(7) On nonlinear differential equations of the second order with

integrable forcing term. H. Antosiewicz. Accepted for publication

in the Journal of the London Mathematical Society.

(8) On the differential equation x + k(f(x) + g(x)x)x = ke(t).

H.-A. Antosiewicz. Submitted to a technical journal,

(9) Stable systems o.f differential equations with integrable forcing

term. H. Antosiewicz. Submitted to a technical journal.

(10) A representation for solutions of analytic systems of linear

differential equations. H. A. Antosiewicz and M. Abramowitz.

Accepted for publication in the Journal of the Washington Academy

of Sciences.

(11) A remark on the smoothing problem. H. Antosiewicz and A.J. Hoffman,

Accepted for publication in the Transactions of the Institute of

Management Science.

(12) Computation of vibration modes and frequencies on SEAC.

W. F. Cahill and S. Levy (NBS 6.4). Submitted to a technical

journal.

(13) On the enumeration of partially balanced designs with two associate

classes. W. H. Clatworthy. Submitted to a technical journal.

(14) Partially balanced incomplete block designs with two associate

classes and two treatments per block. W. H. Clatworthy. Accepted

for publication in the Journal of Research of the NBS.

(15) Easily used simultaneous confidence limits for a line. W.S. Connor.

Submitted to a technical journal.

(16) Experimental designs for paired observations, W. S. Connor and

W. J. Youden. Accepted for publication in the Journal of Research

of the NBS,

(17) Time-discrete stochastic processes in arbitrary sets, with

applications to processes with absorbing regions and to the problem

of loops in Markoff chains. D. van Dantzig. Accepted for

publication (in French) in Annales de I'Institut Henri Poincare

(Paris)

.
Publication Activities 37

(

8 ) Linear functional equations and interpolation series. P. Davis.

Accepted for publication in the Pacific Journal of Mathematics.

( 19 ) On a problem in the theory of mechanical quadratures. P. Davis.

Accepted for publication in the Pacific Journal of Mathematics.

2

( 20 ) Some L“~ aspects of Faber polynomials. P. Davis and H. Pollack

(Bell Telephone Laboratories). Accepted for publication in the

Duke Mathematical Journal.

( 21 ) A multi-purpose orthonormalizing code and its uses. P. Davis

and P. Rabinowitz. Accepted for publication in the Journal of

the Association for Computing Machinery.

( 22 ) On the estimation of quadrature errors for analytic functions.

P. Davis and P. Rabinowitz. Accepted for publication in

Mathematical Tables and Other Aids to Computation.

( 23 ) Some SEAC computations of subsonic fluid flows by Bergman's

method of integral operators. P. Davis and P. Rabinowitz.

To be an appendix in the volume, "Bergman's Operator Method,"

by M.Z. v.Krzywoblocki (University of Illinois).

( 24 ) Some sampling results on the power of nonparame trie tests against

normal alternatives. W. J. Dixon (University of Oregon) and

D. Teichroew. Submitted to a technical journal.

( 25 ) Turbulent flow in shock tubes of varying cross-section. R. F. Dressier

Accepted by the Journal of Research of the NBS.

( 26 ) On the optimal character of the (s,S) policy in inventory theory.

A. Dvoretzky, J. Kiefer, and J. Wolfowitz-. Submitted to a technical journal.

( 27 ) A comparison theorem for eigenvalues of normal matrices. K. Fan.

Submitted to a technical journal.

( 28 ) Some inequalities concerning positive-definite Hermitian matrices.

K. Fan. Submitted to a technical journal.

( 29 ) Some metric inequalities in the space of matrices. K. Fan and

A. J. Hoffman. Accepted for publication in the Proceedings of

the American Mathematical Society.

( 30 ) Discrete analogs of inequalities of Wirtinger. K. Fan, O.Taussky,

and J. Todd. Submitted to a technical journal.

( 31 ) A de te rminantal inequality. K. Fan and J. Todd. Accepted for

publication in the Journal of the London Mathematical Society.

( 32 ) Asymptotic lower bounds for the fundamental frequency of convex membranes. G. E. Forsythe. Accepted for publication in the

Pacific Journal of Mathematics.

(33) On the solution of the caterer problem. J. W. Gaddum, A.J. Hoffman,

and D. Sokolowsky. To appear in the Logistics Research Quarterly.

( 34 ) An algorithm for solving the transportation problem. A. Gleyzal.

Accepted for publication in the Journal of Research of the NBS.

(35) Equations of physics in general Newtonian space-time. A. N. Gleyzal.

Submitted to a technical journal.
38 Public ation Activities

(36) The use of extreme-value methods in engineering problems,

E, J. Gumbel and J. Lieblein, Accepted for publication in the

American Statistician,

(37) On certain series expansions involving Whittaker functions and

Jacobi polynomials, P, Henrici, Accepted for publication in

the Pacific Journal of Mathematics,

(38) On generating functions of the Jacobi polynomials, P. Henrici.

Accepted for publication in the Pacific Journal of Mathematics.

(39) On helical springs of finite thickness. P, Henrici, Accepted

for publication in the Quarterly of Applied Mathematics.

(40) On rotational viscous flow through a tube. P. Henrici. Accepted

for publication in ZAMP.

(4l ) Smooth patterns of production, A. J. Hoffman and W. Jacobs.

Accepted for publication in the Transactions of the Institute

of Management Science.

(42) The number of absolute points of a correlation. A. J. Hoffman,

M. Newman, E„ G. Straus, and 0. Taussky. Submitted to a technical

journal,

(43) Contractibility and convexity. H, W. Kuhn (Bryn Mawr College).

Accepted for publication in the Proceedings of the American

Mathematical Society.

(44) On certain character matrices. D. H. Lehmer. Submitted to a technical journal,

(45) Indentation pressure of a smooth punch. E. Levin. Accepted for

publication in the Quarterly of Applied Mathematics.

(46) 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, held in

Pittsburgh, Pa., June 1952 .

(47) On the variances and covariances of order statistics from the

Weibull distribution. J. Lieblein. Submitted to a technical

journal.

(48) A proof of Hilbert’s Nullste llensatz. T. S. Motzkin. Submitted

to a technical journal.

(49) The assignment problem. M. Motzkin. To appear in the Proceedings

of the American Mathematical Society’s Sixth Symposium on Applied

Mathematics, held at Santa Monica City College, August 1953.

(50) On Fejer sets in linear and spherical spaces. T. S. Motzkin and

I. J. Schoenberg, Accepted for publication in Annals of

Mathematic s

.

(51) Pairs of matrices with property L, II. T. S. Motzkin and 0. Taussky.

Submitted to a technical journal.

( 52 ) Least p-th power polynomials on a real finite point set. T.S. Motzkin

and J. L. Walsh. Submitted to a technical journal.

(53) The diophantine equation axm byn-c . M. Newman. Submitted to a technical journal.
Publication Activities 39

(5^0 Structure theorems for modular subgroups. M. Newman. Accepted

for publication in the Duke Mathematical Journal.

(55) On asymptotic series for functions occurring in the theory of

diffraction of waves by a wedge. F. Oberhettinger. Submitted

to a technical journal.

(56) On the Lerch zeta function. F. Oberhettinger. Submitted to a technical journal.

(57) On two problems in abstract algebra connected with Horner's rule.

A. M. Ostrowski. To appear in the von Mises Anniversary Volume.

(58) De te rrainanten mit ueberwiegender Hauptdiagonale und die absolute

Konvergenz von linearen Iterationsprozessen. A. M. Ostrowski.

Submitted to a technical journal.

(59) On the convergence of Gauss' alternating procedure in the method

of the least squares, I. A. M. Ostrowski. Submitted to a technical journal.

(60) On the linear iteration procedures for symmetric matrices.

A. M. Ostrowski. Accepted for publication in Rendiconti di

Matematico, Rome.

(61 ) On the spectrum of a one parametric family of matrices.

A. M. Ostrowski. Submitted to a technical journal.

(62) On Gauss' speeding up device in the theory of single step

iteration. A. M. Ostrowski. Submitted to a technical journal.

( 63 ) On absolute convergence of linear iteration processes.

A. M. Ostrowski. Submitted to a technical journal.

(6*+) On spectra of second-order differential operators. D. Ray.

Submitted to a technical journal.

(65) On the convergence of asymptotic solutions of linear differential

equations. R. M. Redheffer (U.C.L.A.) and W. Wasow. Submitted

to a technical journal.

(66) Numerical computation of low moments of order statistics from a normal population. J. B. Rosser. Submitted to a technical journal.

( 67 ) Osculatory interpolation in the complex plane. H. E. Salzer

(Department of the Army). Accepted for publication in the Journal

of Research of the NBS

(68) Contributions to the theory of rank order statistics. I. R. Savage.

Submitted to a technical journal.

( 69 ) An isope rime trie inequality for closed curves convex in even- dimensional Euclidean space. I. J. Schoenberg. Accepted for

publication in Acta Mathematica.

(70) Generalized commutators of matrices and permutations of factors

in a product of three matrices. 0. Taussky. For publication in

the von Mises Anniversary volume.

(71) Some computational problems in algebraic number theory. 0. 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.
4-0 Publication Activities

(72) Generation and testing of pseudo-random numbers. 0. Tauss ky

and J. Todd. Submitted to a technical journal.

(73) 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»

(74-) Experiments in the solution of differential equations by Monte Carlo

methods. J. Todd. Submitted to a technical journal.

(75) Motivation for working in numerical analysis. J. Todd. To appear

in the Transactions of a Symposium on Applied Mathematics sponsored

by the Office of Ordnance Research and held in Chicago, 111.,

April 29> 195*+.

(76) The condition of certain matrices, II. J. Todd. To appear in

Archiv der Mathematik.

(77) An extension of a theorem of Dantzig's. C. Tompkins and I. Heller

(George Washington University). Submitted to a technical journal.

(78) A method for the numerical integration of differential equations

of second order without explicit first derivatives. R. de Vogelaere.

Accepted for the Journal of Research of the NBS.

(79) Discrete approximations to elliptic differential equations.

W. Wasow. Accepted for publication in Zeitschrift fur angewandte

Mathematik und Physik.

(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.

(81) An extremum property of sums of eigenvalues. H. Wielandt.

Accepted for publication in the Proceedings of the American

Mathematical Society.

(82) 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.

(83) On eigenvalues of sums of normal matrices. H. Wielandt. Submitted

to a technical journal,

(84-) Instrumental drift. W. J. Youden. To appear in Science.

(85) Bounds on a distribution function which are functions of moments

to order four. M. Zelen. Accepted by the Journal of Research

of the NBS.

2.4- Reviews, Notes

(1) Note on the circle theorem of hydrodynamics. E. Levin. Accepted

for publication by the Quarterly of Applied Mathematics.

(2) Two early papers on the relation between extreme values and tensile

strength. (Formerly "A historical note on the application of the

'weakest link' idea to tensile strengths.") J. Lieblein.

To appear in Biomet rika.
Publication Activities 4l

2.5 Miscellaneous Publications

(1) Experiments in the computation of conformal maps. NBS Applied

Mathematics Series 5-2. In press, Government Printing Office.

THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and

maintenance of the national standards of measurement and the provision of means and methods

for making measurements consistent with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials,

devices, and structures; advisory services to Government Agencies on scientific and technical

problems; invention and development of devices to serve special needs of the Government; and the

development of standard practices, codes, and specifications. The work includes basic and applied

research, development, engineering, instrumentation, testing, evaluation, calibration services, and

various consultation and information services. A major portion of the Bureau’s work is performed

for other Government Agencies, particularly the Department of Defense and the Atomic Energy

Commission. The scope of activities is suggested by the listing of divisions and sections on the

inside of the front cover.

Reports and Publications

The results of the Bureau’s work take the form of either actual equipment and devices or published papers and reports. Reports are issued to the sponsoring agency of a particular project or program. Published papers appear either in the Bureau’s own series of publications or in the

journals of professional and scientific societies. The Bureau itself publishes three monthly periodicals, available from the Government Printing Office: The Journal of Research, which presents

complete papers reporting technical investigations; the Technical News Bulletin, which presents

summary and preliminary reports on work in progress; and Basic Radio Propagation Predictions,

which provides data for determining the best frequencies to use for radio communications throughout

the world. There are also five series of nonperiodical publications: The Applied Mathematics

Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous

Publications.

Information on the Bureau’s publications can be found in NBS Circular 460, Publications of

the National Bureau of Standards ($1.00). Information on calibration services and fees can be

found in NBS Circular 483, Testing by the National Bureau of Standards (25 cents). Both are

available from the Government Printing Office. Inquiries regarding the Bureau’s reports and

publications should be addressed to the Office of Scientific Publications, National Bureau of Standards, Washington 25, D. C.
NBS