NATIONAL BUREAU OF STANDARDS REPORT

4117

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

A Quarterly Report

January through March 1955

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

THE NATIONAL BUREAU OF STANDARDS

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

of the divisions and sections engaged in technical work. In general, each section is engaged in

specialized research, development, and engineering in the field indicated by its title. A brief

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

of the back cover of this report.

Electricity and Electronics. Resistance and Reactance. Electron Tubes. Electrical Instru- ments. Magnetic Measurements. Process Technology. Engineering Electronics. Electronic

Instrumentation. 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.

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

State Physics. Electron Physics. Atomic Physics. Nuclear Physics. Radioactivity. X-rays.

Betatron. Nucleonic Instrumentation. Radiological Equipment. AEC Radiation Instruments.

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 Controls.

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. Mathematical Physics.

Data Processing Systems. Components and Techniques. Digital Circuitry. Digital Systems.

Analogue Systems.

Cryogenic Engineering. Cryogenic Equipment. Cryogenic Processes. Properties of Materials.

Gas Liquefaction.

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

Propagation Services.

Radio Propagation Engineering. Frequency Utilization Research. Tropospheric Propagation

Research.

Radio Standards. High Frequency Standards. Microwave Standards. • Office of Basic Instrumentation # Office of Weights and Measures
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

NAML 41 17

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

January through March 1955

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication) re;

unleta permlitlon la 25.D.C. Such pen

cally prepared if th

Approved for public release by the

director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

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sort has been spedflport for Iti own ute.
APPLIED MATHEMATICS DIVISION

January 1 through March 31 , 1955

TECHNICAL ADVISORY COMMITTEE

David Blackwell, Howard University Philip M. Klorse,

Edward U. Condon Massachusetts Institute of Technology

Mark Kac *, Cornell University Mina S. Rees, Hunter College

A. H. Taub, University of Illinois

DIVISION OFFICE

Edward W. Cannon, Ph.D., Chief

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

Olga Taussky-Todd , Ph.D., Consultant

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

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

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

Yates S. Sladen, Assistant Chief Clerk

Mary E. McKinley, Secretary

Esther L. Turner, Secretary

NUMERICAL ANALYSIS SECTION

Henry A. Antosiewicz, Ph.D.

#*Leila D. Brain, Ph.D.

Philip Davis, Ph.D.

John Todd, B.S*. , Chief

Karl Goldberg, M.A.

Evelyn A. Grigg, Secretary

Alan J. Hoffman, Ph.D.

Morris Newman, Ph.D.

Philip Rabinowitz, Ph.D.

Eduard Stiefel, Ph.D.

COMPUTATION LABORATORY Milton Abramowitz, Ph.D., Chief (Acting)

Irene A. Stegun, M.A. , Assistant Chief

Ruth K. Anderson, M.A. Joseph Bram, Ph.D.

Hans 0. Bremer, B.A. William F. Cahill, M.S.

Ruth E. Capuano

Mary M. Dunlap, B.S.

Anne F. Futterman, B.A. Billie R. Gill, M.S.

Elizabeth F. Godefroy

William G. Hall, B.S.

Genevie E. Hawkins, B.S.

Urs W. Hochstrasser , Ph.D.

Gloria F. Holmes, B.S. , Sec'y

Dorothea H. Jirauch, M.A.

Lambert S. Joel, B.A. David S. Liepman

Ethel C. Marden, B.A. Kermit C. Nelson

Peter J. O'Hara, B.S.

Mary Orr

Maxine L. Paulsen, B.S.

B. Stanley Prusch

Ida Rhodes, M.A.

Sally Tsingou, B.S.

Bertha H. Walter

Philip J. Walsh, B.S.

Joseph H. Wegstein, M.S.

Donald F. Wehn, B.A. Ruth Zucker, B.A.

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

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

Marion T. Carson

Willard H. Clatworthy, Ph.D.

Lola S. Deming, M.A.

Mary L. Epling, Sec'y

Lela J. Hamilton, Sec'y

William C. Krumbein, Ph.D.

Julius Lieblein, Ph.D.

Mary G. Natrella, B.A. Marvin Zelen, M. A.

MATHEMATICAL PHYSICS SECTION

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

Under contract with The American University Guest worker On leave of absence

Edward W. Cannon, Ph.D., Chief

Robert F. Dressier, Ph.D., Assistant Chief

John G. Hershberger, M.S. Edith N. Reese, B.A. Fritz Oberhettinger , Ph.D. Lillian Sloane, Sec'y
CONTENTS

Page

Status of Projects as of March 31 > 19 55 .... 1 Numerical Analysis Section 1 (NBS Section 11,1)

Computation Laboratory, ,, ., 10

(NBS Section 11.2)

Statistical Engineering Laboratory, , , , . , 29

(NBS Section 11,3)

Mathematical Physics Section , . . 35

(NBS Section 11. 4)

SEAC 38

Lectures and Symposia 40

Publication Activities 45
\ . . • " *
Status of Projects

March 31, 1955

I. NUMERICAL ANALYSIS SECTION

(Section 11.1)

RESEARCH IN NUMERICAL ANALYSIS AND RELATED FIELDS

Task 1101 -10-1 10^/55-55

Origin and Sponsor: NBS Authorized 8/1 3/5^

Managers: J, Todd, P. Davis Revised 8/29/5^

Full task description: July-Sept 195*+ issue, p. 1 Status: CONTINUED. Further experiments were made by P. Davis to

test the versatility of the orthonormalizing code developed by him and

P. Rabinowitz.

The first 32 orthonormal polynomials for a square (in the sense

of G. Szego’) have been obtained. Using the limit of the ratio k n+ ^/k

Q

, the transfinite diameter of the square has probably been determined correctly

to 4 decimal places. This value has hitherto been known only to about

1^ decimal places. The same procedure was carried out for an arbitrarily

drawn "bean shaped" domain. For this region the first 16 orthonormal

polynomials were obtained and the transfinite diameter obtained to about

3 decimal places. A manuscript describing these matters is in preparation.

Further plans for this code consist in its application to the Dirichlet

problem by means of orthonormal harmonic functions and by the use of a method recently proposed by Z. Nehari which uses non-harmonic functions,

but for which an estimate of error is available.

P, Davis tested various methods of generating random numbers

for their suitability for the Monte Carlo evaluation of multiple integrals.

A method based upon Fibonacci sequences was rejected as being too erratic.

Successive multiplication by 5

^ 7 , 2

_L|'2 yielded results which were (experimentally)

OOA/N) where N is the number of points sampled. Using multiples of certain

irrational quantities yielded results which were 0(1/N), Tests were carried

out up to dimension 12. A manuscript entitled "Some Monte Carlo experiments

in computing multiple integrals" by P, Davis and P, Rabinowitz has been

submitted to a technical journal and contains a complete description of

these matters.

A code has been developed by P. Davis and P. Rabinowitz

which yields the abscissas and weights of Gaussian quadrature formulas

of high orders. Gaussian rules of the following orders were run off and

are available for distribution; n = 1 6, 20, 2*+, 32, 40,48,64, 80,96. The

abscissas and weights have been computed to 21 decimal places. The code

makes use of certain inequalities for the zeros of the Legendre polynomials

in terms of the zeros of J

Q (x), which were established by G. Szego in

1936. A manuscript describing the process and containing these values is

in preparation.

P. Henrici is carrying out a program of automatizing certain

formal manipulations with power series on SEAC. In order to avoid round-off

1
2 Status of Projects

difficulties and to obtain exact results, the coefficients of the series

are assumed to be rationals, and all arithmetic operations are performed

in the field of rational numbers. The two following problems have been

coded and tested:

(1) Raising a power series to an arbitrary power. Given the (rational)

coefficients a R of the power series

y - 1 + a-jX + a 2x +. .

.

and an arbitrary rational (not necessarily positive) number cb , this code

computes the coefficients b

n of the power series

oc, 2

y = 1 + b-jX + b

2x + . .

.

= (1 + a^x a 2x + . • . ) oc

(2) Inverting a power series with arbitrary leading power. Given a series

of the form

y = x* (1 + a^x + a 2x

2

+...),

where oL and the a are again arbitrary rational numbers, this code computes

the coefficients B

R

in the expansion

x = y

1//cC (1 + b.j y^ b

2y

2/'cC+ . . . ) .

The code in problem (1) is used in (2) as a subroutine. It is based on an algorithm which has been suggested by J.C.P. Miller during a recent visit.

Code (2) is based on the Lagr ange-Biirmann formula. Both codes can handle

at the present 16 coefficients. The codes can also apply to cases where

the coefficients a n and hence bn are polynomials in a variable t, say, by

computing the b

n for equidistant values of t and differencing.

Possible applications of the codes which are envisaged for the

future include combinatorial analysis (code (1)) and computation of certain

asymptotic expansions. As a first application the coefficients arising

from the inversion of

y = x

3

(1 +(3/5 )x2+(3/7)x +...)

have been computed. They arise in the asymptotic expansion of F( 1 -A; 2;

1+/3 ) for large values of /3.

' The basic operations for rationals, which had to be coded for

the routines described, can also be used independently. At the moment

their codes, are limited to rationals with numerator and denominator not

exceeding 2^ . Each result and each intermediate result is reduced to

lowest terms by the Euclidean algorithm.

A study of the numerical evaluation of the integral

f (y)

"'OO

J 0

(

J

1 (t/2)) 2

J

2 (ty)dt (0<y<2)

has been made by P. Henrici. It was found that this integral, whose

direct numerical evaluation is difficult, can be expressed for all values

of y in terms of certain hypergeometric series, which are readily computable.

To facilitate a new method of computing Coulomb wave functions

of arbitrary order devised by M. Abramowitz and I. Stegun (see publication

(13) below), P. Henrici has derived the expansion
Status of Projects

3

where

P. Henrici has studied the asymptotic location of the zeros of the so- called Bessel polynomials

of large order n. Among other things, the following approximation was found for the (only) real zero z

Q

of the polynomials of odd order n:

where

9 = 9o

(1 " •^T

Z)

>

and gQ = 3.31905 is the (only) real zero of

j q

2 +1

g = exp .

\ g - 1 Publications:

(1) Stable systems of differential equations with integrable forcing term.

H. Antosiewicz. Submitted to a technical journal,

(2) Computation of vibration modes and frequencies on SEAC, W. Cahill

and S. Levy (NBS 6,4). Accepted for publication in the Journal of

the Institute of Aeronautical Sciences.

(3) On a problem in the theory of mechanical quadratures. P, Davis.

To appear in the Pacific Journal of Mathematics.

(4) A multi-purpose orthonormalizing code and its uses. P. Davis and

P, Rabinowitz. To appear in the Journal of the Association for

Computing Machinery.

(5) Application of two methods of numerical analysis to the computation

of the reflected radiation of a point source. P. Henrici. J. Washington

Acad. Sciences 4£, 38-45 (Feb. 1955).

( 6 ) On the Lerch zeta function. F. Oberhettinger . To appear in the

Pacific Journal of Mathematics.

(7) Generation and testing of random numbers on SEAC. 0. Taussky, J, Todd,

M. Newman, and J. Cameron. In manuscript.

( 8 ) Motivation for working in numerical analysis. J. Todd. Communications

on Pure and Applied Math. 8, 97~ll6 (1955).

(9) On the solution of the Lichtenstein-Gerschgorin integral equation in

conformal mapping: II. Computational experiments. J. Todd and

S. E. Warschawski (University of Minnesota). Appeared in Experiments

in the Computation of Conformal Maps, NBS Applied Mathematics Series

42, pp. 31-44.

(10)

On non-linear differential equations of the second order with integrable

forcing term. H. A. Antosiewicz. To appear in the Journal of the

London Mathematical Society.
4

Status of Projects

(11) Evaluation of a definite integral. P. Henrici. In manuscript.

(12) Asymptotic location of the zero of the Bessel polynomials. P. Henrici.

In manuscript.

(13) The generation of Coulomb wave functions by means of recurrence

relations. I. Stegun and M. Abramowitz. Submitted to a technical

journal.

TURBULENT ATMOSPHERIC CONTAMINATION

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

Origin and Sponsor: Applied Physics Laboratory, Authorized 7/22/54

Johns Hopkins University, and

National Bureau of Standards

Managers: E. Marden, G. Hawkins

Full task description: July-Sept 1954 issue, p. 3

Status: COMPLETED. Trial computations completed on SEAC demonstrate

the feasibility of using digital computers for the speedy prediction of

the transport of atmospheric contamination, if the distribution and

intensity jf sources are known and predicted meteorological conditions

are given. The test computations deal with a single point source and

"predict'1 air pollution resulting from such a source, hour by hour, up

to 2b hours, for varying areas up to 30x30 miles, using prevailing rather

than predicted weather conditions. Computing times range from one-half

hour to several hours and allow the extrapolation that cases involving

multiple point sources or area sources can be handled in similar reasonably

short times by resorting to efficient arrangements of computations.

BASIC RESEARCH IN LINEAR PROGRAMMING

Task 1102-10-5116/50-2

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

ARDC, USAF, and Directorate of

Management Analysis, USAF

Managers: J. Todd, A. Hoffman

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

Status: CONTINUED. In the study of some simple scheduling problems,

it has been observed that the principal item of importance is a partial

ordering among a given set of jobs, i.e., a { b if an individual can complete

job .a and begin job b. In this connection, A. J. Hoffman has found a simple necessary and sufficient condition for a given partially ordered

set to be isomorphic to a set of intervals on the real line with their

natural partial ordering. A further result on interval sets, relevant

to scheduling problems, is a linear programming algorithm for finding

the subcollection of a given collection of intervals (whose union covers

(0,1)) that also covers (0,1) and minimizes the sum of weights assigned

to the intervals. The algorithm depends on the lemma that the absolute

value of the determinant of an incidence matrix in which the units of any

column appear consecutively cannot exceed one. As a special case one can derive a linear programming proof via duality of the fact that the smallest

cardinal number of a covering subcollection is the largest number of points

in (0,1) no two of which are in the same subinterval. Similarly, for any

collection of intervals, covering or not, the subcollection of largest

cardinal number whose subintervals do not overlap is the smallest number

of points containing a representative of each subinterval of the original
Status of Projects 5

collection. These "duality" statements are, however, directly accessible.

Using a result of A. W. Tucker the following generalization of a result of Dantzig and Hirsch has been proved: If K is a non-empty

intersection of half-spaces in Euclidean n-space, and if f is a concave

function defined on K and bounded from below, then (i) the greatest lower

bound f is attained, and (ii) if K has extreme points, then the greatest

lower bound f is attained at an extreme point.

Publications:

(1) A theorem on alternatives for pairs of matrices. H. A. Antosiewicz.

To appear in the Pacific Journal of Mathematics,

(2) Some metric inequalities in the space of matrices. Ky Fan and

A. J. Hoffman. Proc. Am. Math. Soc. 6, 111-116 (1955).

(3) Discrete analogs of inequalities of Wirtinger. Ky Fan, 0. Taussky,

and J. Todd. To appear in Monatshefte fur Mathematik.

(4) A determinantal inequality. Ky Fan and J. Todd. J. London Math. Soc.

30, 58— 6M- (Jan. 1955).

(5) On distinct systems of representatives. A. J. Hoffman and H. W. Kuhn

(Bryn Mawr College). In manuscript.

(6) On the relevance of LeChatelier 1 s principle to linear programming.

A. J. Hoffman. In manuscript.

(7) An extremum property of sums of eigenvalues. H. Wielandt. Proc. Am.

Math. Soc. 6, 106-110 (Feb. 1955).

SUPPORTING RESEARCH IN LINEAR PROGRAMMING

Task 1102-10-5116/5^-9

Origin: Directorate of Management Analysis, 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: COMPLETED. A Linear Programming Symposium, sponsored by

the Office of Scientific Research of the Air Research and Development

Command, was presented by the National Bureau of Standards and the

Directorate of Management Analysis, DCS/Comptroller , USAF, in Washington,

D. C., on January 27>28,29. The sessions at the Pentagon and the National

Bureau of Standards were attended by 364 registrants. The full program

is listed under Lectures and Symposia, page 42, of this report.

Editorial work on the Proceedings of the symposium is now

under way, and it is anticipated that the Proceedings will be published

sometime this fall.

A final report on the work of this project is being prepared.

Publications:

(1) On "overshoot" in the "furthest hyperplane" method. R. Bryce. In

manuscript.

(2) On the solution of the caterer problem. J. W. Gaddum, A. J. Hoffman,

and D. Sokolowsky. Logistics Res. Quar. 1_, 223~229 (1954).

(3) An algorithm for solving the transportation problem. A. Gleyzal.

To appear in the Journal of Research of the NBS.

(4) On the optimal ordering of items for a two-stage process. A. J. Hoffman.

In manuscript.

(5) On block relaxation. L. S. Joel. In manuscript.
6 Status of Projects

RESEARCH IN MATHEMATICAL TOPICS APPLICABLE TO

NUMERICAL ANALYSIS

Task 1101-10-5116/55-56

Origin and Sponsor: Office of Naval Research Authorized 8/13/5*+

Managers: 0. Taussky-Todd, M. Newman Revised 8/29/5*+

Full task description: July-Sept 195*+ issue, p. 5

Status: CONTINUED. 0. Taussky-Todd and M. Newman prepared a manuscript on their results concerning the uniqueness of the normal basis,

and a generalization of the concept of normal basis, of algebraic number

fields. They can now show that the normal basis in a cyclic field is

not unique unless the degree is 2,3,*+, 6. Also Mrs. Todd prepared a manuscript concerning unimodular circulants. Mrs. Todd investigated some properties of group matrices, - in particular the cases where the group

matrix is a normal matrix. She prepared a manuscript on these results.

M. Newman continued his investigations on the coefficients P (n)

defined by

TT (1-xV = Z P_ (n)x n n=1 n=0 r

Identities for these coefficients which are generalizations of previously

proved identities have been discovered, and a general theorem concerning

the existence of identities for all positive integral r has been proved.

In connection with this work, a table of the coefficients P

r (n) has been

prepared on SEAC which gives the first 800 values of P

r (n) for 1<r<13?

and correspondingly fewer values of ?

r (n) for larger values of r. It is

believed that this table will be of fundamental importance to workers in

fields dealing with the elliptic modular functions , and plans for its

publication are under way.

J. Todd is making further experiments in connection with the

discrepancies found in the Youden problem. In particular, a better

approximation to the normal deviates is being used.

Mr. Todd is planning experiments on conformal mapping of regions

defined numerically by an (r,0) equation. As a control, the case of

ellipses, which have already been examined using their equation in parametric

form, will be handled first. (See J. Todd and S. E. Warschawski in NBS

Applied Mathematics Series M-2, p. 31.)

H. A. Antosiewicz is studying the various definitions of

asymptotic stability for solutions of ordinary differential equations, viz.,

asymptotic stability in Lyapunov's sense, equiasymptotic stability, and

uniform asymptotic stability, with a view toward unification of the great

number of results presently available. For instance, asymptotic stability

in the case of linear systems with constant coefficients implies uniform

asymptotic stability; asymptotic stability in the case of linear systems

with variable coefficients or autonomous systems implies equiasymptotic

stability. In turn, these imply the existence of a so-called Lyapunov

function and conversely the existence of a Lyapunov function for such

systems implies these types of stability. A report (see publication (20)

below is being prepared.

H. A. Antosiewicz and P. Rabinowitz are continuing their

investigation of the differential equations of nerve fiber excitation.

A new code, prepared by P. Rabinowitz, has been tested and is now being

used in all computations on the original Hodgkin and Huxley system of

first order equations as well as on the new system containing a second

order equation for the membrane potential. While the study of the original

system centered around the search for so-called threshold values of the

input current, the investigations of the modified system concern the search
Status of Projects 7

for parameter values which yield a desired behavior of the membrane

potential. A comprehensive report (see publication (19) below on the

results obtained to date in the investigation is being prepared jointly

with K. S. Cole (Naval Medical Research Institute).

Publications:

(1) On a generalization of the normal basis in abelian algebraic number

fields. M. Newman and 0. Taussky-Todd . Accepted for publication in

Communications on Pure and Applied Mathematics (New York University).

(2) A note on group matrices. 0, Taussky-Todd. Submitted to a technical

journal.

(3) The normalizer of certain modular subgroups. M. Newman. In manuscript.

(^) Generalizations of identities for the coefficients of certain modular

forms. M. Newman, In manuscript. t (5) Compactif ication of completely regular spaces. Ky Fan and F. Wagner

(University of Notre Dame). In manuscript.

(6) Some inequalities concerning positive-definite Hermitian matrices.

Ky Fan. To appear in the Proceedings of the Cambridge Philosophical

Society.

(7) A comparison theorem for eigenvalues of normal matrices. Ky Fan.

Submitted to a technical journal.

(8) Power series for log(e x »e^). K. Goldberg. Submitted to a technical

journal.

(9) On certain series expansions involving Whittaker functions and Jacobi

polynomials. P. Henrici. To appear in the Pacific Journal of

Mathematics

.

(10) On generating functions of the Jacobi polynomials, P. Henrici. To

appear in the Pacific Journal of Mathematics.

(11) The number of absolute points of a correlation. A. Hoffman, M. Newman,

E. Straus, and 0. Taussky. Submitted to a technical journal.

(12) Structure theorems for modular subgroups. M. Newman. To appear in

the Duke Journal of Mathematics, March 1955.

(13) The coefficients of certain modular forms. M. Newman. To appear in

the Journal of the London Mathematical Society.

(14) 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,

California, August 1953.

(15) Error bounds for 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,

California, August 1953.

(16) On eigenvalues of sums of normal matrices. H. Wielandt. Submitted

to a technical journal.

(17) Automatic computations of nerve fiber excitation. K. S. Cole (Naval

Medical Research Institute), H. A. Antosiewicz, and P. Rabinowitz.

In manuscript.

(18) On asymptotic stability. H. A. Antosiewicz. In manuscript.

ANALYTIC STUDY OF WAR GAMES

Task 1101-10-5116/55-83

Origin and Sponsor: Armament Branch, ARDC , USAF Authorized 12/29/5^

Manager: H. A. Antosiewicz

Full task description: Oct-Dec 195^ issue, p. 7

Status: CONTINUED. H. A. Antosiewicz has prepared a report containing

the results that have been obtained to date and has submitted it, to the
8 Status of Projects

sponsor. The majority of results concern the model as proposed under the

assumption that defensive forces are absent. A necessary and sufficient

condition has been proved for victory of one party over its opponent in

which victory is described by the magnitudes of the ultimate values of

the offensive forces and the rates of production. This condition, as might be expected, is satisfied, for example, when one party’s initial

offensive force and initial rate of production exceeds those of its

opponent. Also, relations among the ultimate values of the strategic

variables (including defensive forces) have been derived, which yield

necessary conditions for victory in terms of the strategic parameters.

Some of these relations can be used to obtain sufficient conditions so that one party’s strategic variables do not ultimately exceed those of

its opponent; that is, if these conditions are fulfilled for one party,

its chances to defeat its opponent will be improved, but its victory

is not guaranteed.

Publication:

(1) Analytic study of war games. H. A. Antosiewicz. In manuscript.

STUDIES IN THE THEORY OF ASYMPTOTIC EXPANSIONS

Task 1101-10-5116/55-116

Origin: NBS Authorized 3/31/55

Sponsor: Office of Scientific Research, ARDC, USAF

Manager: P. Davis

Objective: To study the class of functions f that are regular in a fixed domain B, that possess finite norm

JT f ( z ) 2dA < 00

B

and that, in addition, possess zero asymptotic expansions (in the sense

of Poincare) at fixed boundary points of B. Designate such a class by A.

It is proposed to develop for A, representation theory, kernel function

theory, and estimates of growth, and to study in this way the class of

functions that possess preassigned asymptotic expansions at n fixed

boundary points and which possess finite norm. The fundamental question

to be answered in this direction is the following. Given a domain B and

n fixed points z^,...,z on its boundary. Given also n sequences of

complex constants aj k ( j=0 ,1 ,2,...; k=1 ,2, . . . ,n) . What are the necessary

and sufficient conditions on a^ in order that there exist a regular

function of finite norm over B and which at the points z^ possesses the

asymptotic expansion

f (z) a jk (z 'z

k

: ( k— 1 j 2 , . . . ,n)

.

These questions may be considered as a special case of a more

general class of problems which can be outlined as follows. If, at a boundary point z* of a domain B, an analytical function f(z) possesses an asymptotic expansion,
Status of Projects

9

each coefficient a^ may be considered as a certain linear functional L

k

over the class of functions regular in D: a^ = L

k (f). In view of the

fact that the point z* lies on the boundary of D, becomes an unbounded

operator. The general problem therefore is one of the existence, the

uniqueness, and the representation of solutions of the interpolation

problem.

0) L

k (f) = a

k (k = 0,1,2,...)

for a preassigned sequence of unbounded linear functionals and for a preassigned sequence of constants a^.

Background: The theory of asymptotic expansions (in the sense of

Poincare) of analytic functions lacks thus far both an interpolatory and

an aspect. In reference [1] below, a method was developed which

handled problems of type (1 ) where the linear functionals L

k

were all

bounded. This development was aided materially by the introduction of a certain set of orthogonal functions that possess, in addition, a property

of biorthogonality with respect to L^. These functions may be considered

as a generalization of the set of doubly orthogonal functions introduced

by S. Bergman (see, e.g., [*+]). In reference [2] this method was further

extended to the class of analytic functions in one (or in several) variables

that satisfy a linear differential equation, and a satisfactory interpolatory theory was developed.

In passing from bounded to unbounded linear functionals, it may

be anticipated that a number of questions of considerable difficulty will

arise. It is proposed to study the problem (1), making use of the technique

of references [1], [2], [3]» and appropriate modifications wherever

possible, and to determine the extent to which a coherent theory can be

developed

.

As stressed above, such an approach to the subject of asymptotic

series appears to be new, although in reference [5], we can already

discern the beginnings of it. It would be of great value to have a purely

function-theoretic approach to this subject where development has hitherto

been attached principally to special function theory.

References: [1] J. L. Walsh and P. Davis, Interpolation and

orthonormal systems, J. d'Analyse Mathematique (Jerusalem) 2., 1-28 (1952).

[2] P. Davis and J. L. Walsh, Representations and extensions of bounded

linear functionals defined on classes of analytic functions, Trans. Amer.

Math, Soc. 26, 190-206 (195^). [3] P. Davis, Linear functional equations

and interpolation series, Pacific Journal of Mathematics (to appear).

[4] S. Bergman, The kernel function and conformal mapping (American

Mathematical Society, 1950). [5] T. Carleman, Les fonctions quasianalytiques (Paris, 1926).

Status: New
II. COMPUTATION LABORATORY

(Section 11.2)

1 . Mathematical Tables

11 02-40-1 110A3-3 TABLES OF E^z), (z=x+iy)

Origin : Canadian National Research Council

Manager : I. Stegun

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

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

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

Origin : NBS

Manager : P. Rabinowitz

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

Status : Continued. Values of the regular and irregular functions and

their derivatives were computed for = L=0, .5 < < 1 4 , 5 •

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

Origin : NBS

Sponsor : Bureau of Ordnance, USN

Manager : W. F. Cahill

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

Status : Inactive. For status to date, see Oct-Dec 1954 issue, p. 9.

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

Origin : Section 11.3? NBS

Managers : A. J. Hoffman, S. Tsingou

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

Status : Continued. Results for 19 values of (x,/3) have been computed

to date. Some test cases were run on SEAC for very large values of

f-j and in order to determine the feasibility of obtaining the

results with the original code.

1102-40-1110/52-7 REVISION OF MATHEMATICAL TABLES

Origin : NBS

Managers : W. F. Cahill, I. Stegun

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 sines and cosines for radian arguments," NBS

Applied Mathematics Series 43; available from U. S. Government Printing

Office, Washington 25, D. C., #3.00. This table was formerly designated

as Mathematical Tables MT4.

10
Status of Projects 11

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

MT1 2 (1941 ) ; revision is in progress for reissue in the Applied

Mathematics Series.

1102-40-1110/52-14 TABLE OF ARCS IN FOR COMPLEX ARGUMENTS

Origin : NBS

Manager; D. Liepman

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

Status : Continued. Checking of the tables is in process.

1 1 02-40-1 1 10/52-1 8 EXTENSION OF THE TABLE OF HYPERBOLIC SINES AND COSINES

Origin : NBS

Manage rs : W. F. Cahill, K. Nelson

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

Status : Continued. Checking of the manuscript of the tables has been

completed, and the introduction and. preface have been prepared.

1 1 02-40-1 1 1 0/52-23 TABLE OF THE MODIFIED AIRY INTEGRAL

Origin : NBS

Manager : P. Rabinowitz

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

Status : Continued. The preface has been written, and preparation of

the manuscript for publication is in process.

11 02-40-1 110/52-r2 5 TABLE OF ERROR FUNCTION FOR COMPLEX ARGUMENTS

Origin : NBS

Manager : W. Hall

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

Status : Inactive. FolTM status to date, see Oct-Dec 1954 issue, p. 11.

See also the report under "Hypergeometric Functions," task 1 1 02-40-1 1 1 0/53~3

5

p. 12 of this issue.

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

NEGATIVE ARGUMENTS

Origin : NBS

Managers : E. Marden, S. Prusch

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

Status : Inactive. For status to date see Oct-Dec 1954 issue, p, 11,

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

Origin : NBS

Manager : D. Liepman

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

Status : Continued. Checking of computations has been resumed.

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

Origin : NBS

Manager : I. Rhodes

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

Status : Inactive. For status to date, see July-Sept 1954 issue, p. 10.
12 Status of Projects

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

Origin : NBS

Managers : M. Paulsen, P. O'Hara

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

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

1 1 02-40-1 1 1 0/53-35 HYPERGEOMETRIC FUNCTIONS

Origin : NBS

Managers : P. Rabinowitz, W. F. Cahill

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

Status : Completed. A code to compute the hypergeometric function for

complex parameters and arguments has been written and checked out.

The code computes this function by its power series expansion,

F(a, b;c;z) u

ab z_

‘ c I' aCa+1 ) b( b+1 ) z

2

c ( c+ 1 ) 21

The code uses double precision operations and in general produces a result accurate to 10*

1

_s significant figures where F(a,b;c;z) and

its largest term are both less than 10 s

. The code has been modified to compute the confluent hypergeometric

function. Using this modified code co=1. 35218 220099 4 has been found

as a zero of the function

^ F-| Q ; 1 ; 2c4 . Tabulation of these functions is not contemplated at this time.

The codes are kept on file for computation of values as needed.

A skeleton table of the complex error function (see publication

1 below)

Erf(z) = ~

was completed for x,y=0( .2) 1 ,2( .4)3.2, where z=x+iy, by use of the

relation

Erf (z) = 2Z F (1/2;3/2;-z 2

).

'IT ' * Publications : (1 ) A short table of the error function for complex arguments.

W. F. Cahill. An NBS report. (2) Programs for computing the hypergeometric

series. W. F. Cahill. MTAC VIII, 36-37 (Jan. 1954).

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

Origin : Oak Ridge National Laboratory

Managers : W. Hall, J. Wegstein

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

Status : Continued. Computations for z =75) z=65, and z=45 were completed

in the case of the L-j shell.

371 1 -60-0009/ 55-65 automatic coding

Origin : NBS

Manager : J. Wegstein

Full task description : July-Sept 1954 issue, p. 11

Status : Continued. The routines for raising a number to an integral power

and for computing sines or cosines of numbers in floating point

representation have been written, code-checked, and incorporated into

the Base 00 system. On an experimental basis the following subroutines

and supervisory routines have also been included in the Base 00 system:
Status of Projects 13

(1) a subroutine initiated by a single word that moves

a block of information from one place in the memory

to another.

( 2 ) a supervisory routine for accelerating the convergence

of certain iterative processes

(3) a supervisory routine for quadratic interpolation

(4) a service routine that enables the machine operator

to print out or type in a series of decimal numbers by

inserting a single word into the computer.

A technical memorandum describing the operation and use of Base

00 as a computing system is being written.

A service routine which will find all referands to any specific

memory address has been written and is being code-checked. This routine

uses only standard memory locations and thus ipay be used for code-checking.

A general purpose routine for generating iterative loops was developed, which appears to bfc useful in many codes. The n code words

which are to be repeated with suitable modifications are preceded by a single word whose first three addresses are n, m, and p. This word

indicates that the n succeeding words (instructions) are to be repeated

either m times or until the number in cell p is less than the number in

cell m. The addresses in each instruction being repeated can be modified

each time the instruction is repeated, and the specifications for these

modifications are carried by each instruction itself. As an example,

a table of values y^ax^+z^, i=1(1)m, can be computed from tables of x^

and z

^ using the iterative loop generator and only three computer words.

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

Manager : E. Marden

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

Status : inactive.

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

Origin : Naval Research Laboratory, USN

Manager : P. O'Hara

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

Status : Continued. Preliminary calculations for phase determination

have been completed for the aureomycin crystal. Refinement procedures

are being carried out to improve the coordinates of the 19 atoms of

the spurite crystal.

1 102-40-51 26/52-20 SPHERICAL BLAST

Origin and Sponsor : Naval Ordnance Laboratory

Manager : D. Jirauch

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

Status : Continued. A preliminary report on the method used in coding

the problem, together with a brief description of the problem, is being

revised in accordance with comments from readers.
I

1* Stat us of Projects

1 102-40-5126/52-44 CALCULATIONS FOR d SPACINGS

Origin and Sponsor ; NBS, Division 9

Manager : A. Futterman

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

Status : Continued. Calculations for d-spacings were performed for the

following cases: (1) Tetragonal crystals CaWC^, ZnF2

, Ba02, Sr0

2

; (2) Hexagonal crystals Zn

2SiO^, Be 2SiO^, K

2PfF^, Rb2PfF^, (NH^.) 2Pf F^.

1102-40-5126/53-27 COMPUTATION OF THERMODYNAMIC FUNCTIONS

Origin and Sponsor : NBS, Section 3.2

Manager : E. Marden

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

Status : Continued. Thermodynamic functions were calculated for a number

of additional molecules. At present, two different codes are in use.

One, corresponding to the original formulation of this task (see

1 1 .2/33-49-5, Oct-Dec 1948 issue, p. 49) includes only the rigid rotator

harmonic oscillator terms (cf. D. E. Mann, N. Acquista, and E.K. Plyler,

J. Chem. Phys. 21 . 1949 (1953)). The other includes first-order correction

terms for an harmonicity rotation-vibration interaction, and centrifugal

stretching, as well as certain second-order low-temperature correction

terms (cf. A. S. Friedman and L. Haar, J. Chem. Phys. 22, 2051 (1954)).

Comparisons were made in order to evaluate the importance of the added

correction terms. To date such calculations have been carried out for over one hundred diatomic hydrides, denterides and tritides; the results

are being prepared for issuance as a Bureau circular. More recently

several polyatomic molecules have been treated, including HpS, DpS,

TpS, HDS, HTS, DTS, and their isotopes.

1102-40-5126/53-51 RADIATION DIFFUSION

Origin :

- NBS , Section 4.8

Sponsor : Armed Forces Special Weapons Project

Manager : A. Futterman

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

Status : Continued. The boundary problem mentioned in the previous report

is being pursued. The results of several Monte Carlo calculations

(response functions for scintillation spectrometers, transmission and

reflection of gamma radiation through finite barriers) have been analyzed.

3711-60-0009/54-11 COMPUTATION OF VIBRATION MODES AND FREQUENCIES

Origin : NBS, Section 6.4

Managers : I. Rhodes, W. F. Cahill

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

Status : Completed. The first five antisymmetric modes of an aircraft

structure (using 26 mass points) were computed during this period. In

all, the first five symmetric and antisymmetric modes of eight aircraft

structures have been completed.

Publication : Computation of vibration modes and frequencies on SEAC.

W. F. Cahill and S. Levy (NBS Section 6,4). To appear in the Journal

of the Institute of Aeronautical Sciences.

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

Origin and Sponsor : New York Quartermaster Procurement Agency

Manager : H. Bremer

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

Status : Continued, During the past quarter the Quartermaster Corps

submitted three procurement problems. The analyses of these problems

were done on SEAC, using the simplex method for solving the general

transportation problem.
15 Status of Projects

371 1 -60-0009/5^-1 7 DE POLYMERIZATION

Origin : NBS, Section 7.6

Manager : J. Bram

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

S tatus : Continued. Preparation of the code continued. It is being modified

by an addition requested by the originator.

3711-60-0009/55-19 ENERGY LEVELS OF COMPLEX ATOMS

Origin : NBS, Section 4.1

Manager : W. Hall

Full task description : Jan-Mar 1955- issue, p. 5-1 Status : Inactive. For status to date, see July-Sept 1955- issue, page 14.

1 102-40- 5126/55-20 LOW TEMPERATURE PROPERTIES OF BORON COMPOUNDS

Origin and Sponsor : NBS, Section 3.2

Manager : J. Wegstein

Full task description : Jan-Mar 1955- issue, p. 5-1 Status : Terminated.

1 102-5

- 0- 5126/55-21 EXPERIMENTAL PROGRAM FOR MULTIPLE INPUT AND OUTPUT

Origin and Sponsor : NBS, Section 12.3

Manager : I. Rhodes

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

Status : Inactive.

3711 - 60-0009/54-22 ENERGY DISTRIBUTIONS ON OPTICAL IMAGE

Origin : NBS, Section 2.2

Manager : L. S. Joel

Full task description : Jan-Mar 1954 issue, p. 43

Status : Inactive.

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

Origin and Sponsor : Armed Forces Special Weapons Group

Managers : P. Henrici, L. S. Joel

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

Status : Completed. The portion of the task dealing with reflected

radiation from a horizontal surface has been completed. Computations

for the vertical cases are being considered; if and when they are to be

carried out they will appear as a new task.

A general subroutine for Gaussian quadrature, developed for

this problem, is on file. The numerical analysis underlying the computation

has been presented in the publication below.

Publication : Application of two methods of numerical analysis to the

computation of the reflected radiation of a point source. P. HenriciJ. Wash. Acad. Sci. 4£, 38-45 (Feb. 1955).

371 1 -60-0009/ 54-2.7 EMF OF CELLS AT HIGH TEMPERATURE

Origin : NBS, Section 1.8

Managers : E. Marden, G. Hawkins

Full task description : Jan-Mar 1954 issue, p. 45

Status : Completed. Computation of the electromotive forces for all iodides,

chlorides, fluorides, bromides, and for several extraneous compounds

has been completed.
16 Status of Projects

371 1-60-0009/5^-28 LOVIBOND NETWORK FOR CIE SOURCE A

Origin : NBS, Section 2.1

Managers : E. Marden, G. Hawkins

Full task description : Jan-Mar 1954 issue, p. 46

Status : Completed. Computations were made on SEAC relating the Lovibond

network to CIE sources A and C using the new data supplied from England

and the results for both sets of computations have been transmitted

to the sponsor.

3711-60-0009/54-30 SPECTRUM ANALYSIS

Origin : NBS, Division 4

Manager : S. Prusch

Full task description : Jan-Mar 1954 issue, p. 46

Status : Continued. Preparation for publication of the table of arc and

spark spectra for ruthenium on the card-controlled typewriter was completed

.

1102-40-5126/54-33 AIR CONDITIONING IN UNDERGROUND STRUCTURES

Origin and Sponsor : NBS, Section 10.3

Manager : S. Tsingou

Full task description : Jan-Mar 1954 issue, p. 47

Status : Continued. The original code has been changed so that the

following integral can be evaluated. _ s£

y

2

2_ ak 2 r°° L d*

J0 [Y^y )]

2 + [J,(y )]2

y

3

O

for ak = 1 , and oct/a =.005(various) 50.

1102-40-5126/54-34 EQUILIBRIUM CALCULATIONS FOR WATER

Origin and Sponsor : Naval Ordnance Laboratory

Manager : J. Wegstein

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

Status : Continued. During the period that SEAC was being moved, the

tables of thermodynamic properties that were previously calculated with

the SEAC were used to calculate entropy tables for water at high

temperatures on an IBM 701 calculator. The results were transmitted

to the sponsor.

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

Origin and Sponsor : NBS, Section 6,4

Manager : W. Hall

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

Status : Continued. Two problems were run using boundary conditions

corresponding to those observed in tests conducted in the Engineering

Mechanics Section. The observed conditions were approximated by a sequence of linear functions of time.

3711-60-0009/54-36 VELOCITY OF LIGHT

Origin : NBS, Section 11.3

Manage

r

: P. Rabinowitz

Full task description : Jan-Mar 1954 issue, p. 47

Status : Continued. Several third degree polynomials were fitted to data

submitted by the sponsor.
Status of Projects 17

371 1 -60-0009/5^-38 COMPRESSIBILITY FACTORS OF DRY AIR

Origin : NBS, Section 3.2

Manager : M. Paulsen

Full task description : Jan-Mar 1954 issue, p. 48

Status : Continued. Because of the slowness of convergence of the iterative

method of computation, the code is being revised to accelerate the

iteration process.

1 102-40-5126/55-39 MOLECULAR VIBRATIONS

Origin and Sponsor : NBS , Section 3.2

Manager : K. Goldberg

Full task description : July-Sept 1954 issue, p. 16

Status : Continued. A new code is being prepared to obtain the eigenvectors

as well as the eigenvalues of a symmetric matrix up to order 40 by

the Jacobi-Givens method.

1 102-40-5126/54-43 CHARACTERISTICS OF CONDUCTING RESISTORS

Origin and Sponsor : NBS, Section 12.1

Manager : B. Gill

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

Status : Inactive.

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

Origin : Raytheon Manufacturing Company

Sponso r: U. S. Corps of Engineers

Manager : L. S. Joel

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

Status : Completed. The problem described in the Apr-June 1954 issue,

p. 41 , have been solved. *

1 102-40-5126/55-58 ELECTROMAGNETIC RADIATION FROM LIGHTNING

Origin and Sponsor : NBS, Section 82.1

Manager : B. R. Gill

Objective : To perform Fourier analyses on time records of lightning

flashes

.

Background : The intense electrical discharge of a lightning flash

produces a brief but intense electromagnetic radiation. It momentarily

blots out radio reception in the near vicinity; the addition of many

lightning flashes produces much of the steady noise at a distance that

limits the receivable signal level. This task, however, is concerned

with the vertical component of the electrical field of individual

lightning flashes.

It is the purpose of these computations to obtain information

about the appearance of the record of the field of lightning flashes,

such as the distribution of the energy as a function of frequency at

various distances and the propagation of waves at very low frequencies

(1 to 50 kc). In particular, it is desired to study the attenuation

and distortion of the frequency functions with respect to distance.

Status : New. Twenty-seven curves were scaled on SEAC for nine sets of

data observed simultaneously at Fort Belvoir, Va. , Boulder, Colo., and

Stanford, Calif.; ten curves were scaled for five sets cf data from

Fort Belvoir and Stanford.
18

Status of Projects

1102-40-5126/55-61 ELASTIC CROSS SECTION FOR NEUTRON SCATTERING

Origin and Sponsor : Naval Research Laboratory

Managers : I. Stegun, R. Zucker

Full task description : Oct-Dec 1954 issue, p. 18

Status : Continued. The code for carrying out the computations has been

written and checked. Theoretical results are being matched against

experimental results for lead, copper, carbon, tin, and aluminum.

1 102-40-5126/55-62 INTEGRALS INVOLVED IN SUPERSONIC FLUTTER

Origin and Sponsor : National Advisory Committee for Aeronautics

Manager : S. Tsingou

Full task description : July-Sept 1954 issue, p. 17

Status : Continued. The results are in the process of being checked.

1102-40-5126/55-64 HIGH-TEMPERATURE THERMODYNAMIC TABLES (FMT)

Origin and Sponsor : NBS, Section 3.2

Manager : J. Wegstein

Full task description : July-Sept 1954 issue, p. 18

Status : Inactive.

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

Origin : NBS, Division 30

Manager : H. Bremer

Full task description : July-Sept 1954 issue, p. 18

Status : Inactive. For status to date, see Oct-Dec 1954 issue, p. 19.

1102-40-5126/55-67 DOSAGE INTEGRAL

Origin and Sponsor : U. S. Weather Bureau

Manager : W. F. Cahill

Full task description : July-Sept 1954 issue, p. 19

Status : Completed. The original four-fold integral was reduced to

(t-d) 2

+r

2

c 2t b

+ (AC t (d-t)r/c 2

t b

r

l — drdt

and was computed on SEAC for six sets of the parameters. A similar

integral where the original integrand was multiplied by (d-ut-x)2+y2+z2

] was likewise reduced to a double integral and was computed on SEAC for

one set of parameters. The results have been transmitted to the sponsor.

3711-60-0009/55-68 CRYSTAL STRUCTURE CALCULATIONS

Origin : NBS, Section 9

Manager : A. Futterman

Objective : To perform calculations necessary for the determination of

crystal structures from X-ray diffraction patterns such as

(a) Fourier summations

y?(x,y,z) = y £jr£F (h,k, 1 ) exp[-27ri( hx+ky+ lz )] , (b) structure factor calculations,

F(h,k,l) - f n exp 27ri (hx

n+hyn+lzn

)

,

n-1

(c) least squares refinement of approximate structure factors, and

(d) methods to obtain approximate structure factors.
Status of Projects 19

Background ; Let />(x,y,z) be the electron density at the point (x,y,z)

in the unit cell of a crystal. An X-ray diffraction pattern obtained

from the crystal consists of a number of spots which can be characterized

by means of the "indices" h,k,l. Let F(h,k,l) be the complex number

denoting the amplitude and phase of the diffracted ray. Then the

relation (a) above holds. Of the "structure factors" F, the amplitudes

for a finite number of h,k,l may be obtained from measurements of the

intensity of the diffracted rays, while the phases are unknown. If the

N atoms in the crystal are idealized to points ( x

n

>yn> z

n

) > the structure

factors are approximated by (b) above, where the coefficient f n (h,k,l)

is the "form factor" of the n-th atom. Form factors have been tabulated

for most types of atoms. The locations ( x

n ?yn

: z n

) are sometimes approximately known.

The goal of X-ray diffraction analysis is to determine the

electron density and structure factors completely. Each of these, if

completely known, would determine the other by virtue of (a) or (b).

The problem is to fit together the incomplete pieces of information on these two functions in such a way that (a) and (b) are satisfied.

Comment : This task is related to tasks 1 1 02-40-51 26/51 -37? 1102-40-5126/52-44

and 1102-40-5126/55-87.

Status : New. A general code for the calculation of d-spacings for all

crystals has been written and is in the process of being checked.

3711-60-0009/55-69 THEORY OF DIELECTRIC RELAXATION

Origin : NBS , Section 7.6

Manager : K. Goldberg

Full task description : July-Sept 1954 issue, p. 19

Status ; Inactive, For status to date, see Oct-Dec 1954 issue, p. 19.

1102-40-5126/55-72 MARYLAND INTER- INDUS TRY STUDY

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

Manager : L. S. Joel

Full task description : July-Sept 1954 issue, p. 20

Status : Continued. Problems relating to allocation of factors of production

in an idealized economic model are being solved as linear programs by

the simplex method.

1102-40-5126/55-73 HEAT CONVECTION

Origin : University of Minnesota

Sponsor : Atomic Energy Commission

Manager : G. Hawkins

Full task description : Oct-Dec 1954 issue, p. 20

Status : Inactive. For status to date, see Oct-Dec 1954 issue, p. 20,

1 102-40-5126/55-74 LIQUID-VAPOR TRANSITION, II,

Origin and Sponsor : Naval Medical Research Institute

Managers : R, Zucker, S. Prusch

Objective : To solve by numerical methods an integral equation arising

in the statistical-mechanical theory of the transition zone between

the liquid and vapor phases according to two approximate theories of

liquid state.

Background : The problem was proposed by T. L. Hill of the Naval Medical

Research Institute. It is a continuation of task 1102-40-1106/51-22,

the principal difference being in the choice of the function 0 and the

parameters

.

Status : New. The integral equation was solved on punched card machines

for one value of the parameter together with one given set of functional

values. The solution for several similar equations arising in the same theory is in progress.
20 Status of Projects

3711-60-0009/55-75 PARAMETER OF THE DISPERSION EQUATION FOR OPTICAL GLASS

Origin ; NBS, Section 1.6

Manager ; R. Zucker

Objective ; To evaluate constants k^ of the equation

n

2-1 =

j

5*1 - Ai2

by the method of least squares from given values of n, A and A^; to

compute n and the residuals for the specified values of n and A : and

to compute n by the formula for various values of the wavelength A.

Background ; The results will enable one to express more accurately the

ref rac tivities of glass for use in optical design throughout a longer

interval of the spectrum than is now used in optical glass catalogues.

Status ; New. The problem was coded for SEAC, and many cases for various

values of the parameters were run.

1102-40-5126/55-78 NEUTRON TRANSPORT

Origin and Sponsor ; Atomic Energy Commission, New York

Manager ; S. Tsingou

Full task description ; Oct-Dec 195*+ issue, p. 21

Status ; Inactive.

1102-40-5126/55-79 COMPLETE DEGRADATION IN THE NEUTRON

Origin and Sponsor ; Atomic Energy Commission, New York

Manager ; A. Futterman

Full task description ; Oct-Dec 1954 issue, p. 21

Status ; Continued. Several cases for m<2 were run. The case where

m=-2

“ has given erroneous answers, and the code is under study for possible

revision.

1 102-40-51 26/55-80 ALPHA-ANALYSIS

Origin and Sponsor ; Atomic Energy Commission, New York

Manager ; W. G. Hall

Full task description ; Oct-Dec 1954 issue, p. 22

Status ; Inactive.

371 1-60-5126/55-81 COMBINING TESTS FOR SIGNIFICANCE

Origin ; NBS, Section 11.3

Manager ; L. S, Joel

Full task description ; Oct-Dec 1954, p. 23

Status ; Continued. The code for computing the power points of combining

the probabilities of two independent variance-ratio tests into a single

test of significance has been written and is now being checked.

Preliminary results have been obtained for the percentage points (under

the null hypothesis) for the two probability points .05 and .01.

371 1-60-0009/55-82 THERMOMETER CALIBRATIONS

Origin ; NBS, Section 3.1

Manager ; S. Prusch

Objective ; To prepare tables of R

t /RQ versus temperature for each

thermometer tested. The temperature range is t = -183(10)630 degrees

Centigrade,
Status of Projects

21

The problem in calibrating a standard platinum resistance

thermometer on the international scale is to determine the constants eC, 6

, ^3, and Rq in the equation

_ 1

R

t Ro

"

oc Ro

+ 6 ( 100 ft ( 100 1 )

(

10o)

The problem is attacked by determining the resistance R at four specified

temperatures, namely, the normal boiling points of oxygen, water, sulfur,

and the ice point.

Background : The results previously have been obtained by hand computing.

A study is being made to see if the work can be systematized and results

obtained by machine methods.

Status : New. Punched-card procedures for preparing calibration tables

for various types of standard platinum resistance thermometers, excluding

low temperature ones, have been set up.

371 1 -60-0009/55-86 FLOW COEFFICIENTS FOR FLUIDS

Origin : NBS, Section 6.7

Manager : P. J. Walsh

Objective : To compute the flow coefficient K=KQ+bA of an orifice from

empirical formulae for Kq and b such as

( 1 )

KrrC. 6014-.01352D 1/4

) + (.376-.07257d

1/4

) ^5 +/3 V5/3

16 l

b = (.0002 + -2^ + (.0038 + fe^+( 16.5+5d)/3

iu

],

D

2

/3

2+.002?D

0004, r „2 16

for a large number of values of the parameters D, y3, and A. D is the

inside diameter of the pipe in which the orifice is installed concentrically.

the diameter of the orifice

the inside diameter of the pipe diameter ratio,

Rp = the Reynolds number referred to the diameter of the pipe, and

= .1 P.A is a convenient independent variable for use in coordinating

VRd

observed values of K.

Background : These formulae were developed empirically from the original

test data gathered by the ASME Research Committee on Fluid Meters and

the A.G.A. Gas Measurement Committee at Ohio State University during

1931-1933. In addition to these two groups, the Committee on Flow

Measurement of the International Standards Organization is now interested

in the test data. The formulas are used for computing the flow coefficient

of the orifices for various pipe diameters D when pressure taps are situated at assigned positions preceding and following the orifice plate.

Comment : This problem was proposed by H. S. Bean (Section 5.7).

Status : New. Using formula (1), the values K have been computed for

various pipe diameters (2.067<D<1 5.25) with ft ranging from .1 to .8

(*+1 values )

.
22 Status of Projects

1 102-40-5126/55-87 "ZERO" METHOD DETERMINATION OF CRYSTAL STRUCTURE

Origin and Sponsor : NBS, Section 9.7

Manager : J. Bram

Objective : To calculate the best (least squares) solution ,y^ ,z^ , . . .

,

x^,y^,z^ of the equations

5H

j=1

27ri(h^x j+k

±y j+liz

j

) F(h

i ,k

i ,li

), i ,2 ? • • • ? 85.

The f- and F(h^,k^,l^) are known; all but three of the F’s are taken

to be^zero. 1

Background : The data obtained by X-ray diffraction studies consist of

intensity values proportional to

27ri(hx .+ ky .+ lz .) F(h,k,l) = e

J J J

j

J

where (x,y,z) are the coordinates of an individual atom. However,

only | F | is known. The present problem arises in taking those nonspace group

reflections which are so small that they can be assumed equal to zero;

one must then solve the above system for the x.,y^,z^. The actual

equations used in this project are those that arise in the study of

napthalene

.

For other approaches to the problem of determining crystal

structure from X-ray diffraction patterns, see tasks 1 102-40-51 26/51-37

and 1102-40-5126/52-44.

Status : New. A code for computation on SEAC has been prepared and checked.

1 102-40-5126/55-88 STRESSES in a wall foundation

Origin and Sponsor : NBS, Section 10.1

Managers : E. Marden, G. Hawkins

Ob iec tive : To compute on SEAC stresses, <r , <T, CT , in a wall foundation

x y xy

for given values of the length and height of the wall and the unit

shortening due to shrinkage.

Background : If the interior of the wall is considered to be an elastic

slab, the stress function 0 satisfies the biharmonic equation

2

. 0

ax

2

ay

2

In a system of rectangular coordinates, x and y, in the plane of

the wall,

0~ 2 2 x = a 0/3y = stress in x-direction

2 2

CT = 3 0/ax = stress in y-direction

J 2 0~ = -3 0/axay = sheer stress.

The boundary conditions at the top of the wall, y=b, are

(CT ) _. = 0; (X ) . = 0 y'y-b 5 v xy'y=b

and along the bottom of the wall, y=b/2, the conditions are
Status of Projects 23

(' 6x

) y=b/2 S/E; (v )y=b/2 “ °’ where S/E is the unit shortening due to shrinkage, E is Young's modulus,

and v is the vertical displacement. Along the sides x=0 and x=a, the

boundary conditions are

^ CJx^x=0, a = o; (rxy'x=0,a ) _ n 0 .

It is assumed that the wall shrinks uniformly as it dries out,

and that the footing restrains the shrinkage of the bottom of the wall.

The solution is to be obtained in the form of series expansions.

Status : New. A general code has been written for the SEAC to obtain

solutions for walls whose lengths and heights will be specified by the

input data in each case. Cdde checking is now in progress.

3711-60-0009/55-89 TEMPERATURE DISTRIBUTION IN SOLID WINGS HEATED

AERODYNAMICALLY

Origin : NBS, Section 6.4

Manager : P. Davis, W. F. Cahill

Full task description : Oct-Dec 1954 issue, p, 22

Status : Continued. A program to compute the roots B

R

of the expression

„ -

2

and to evaluate the series

00 -B t 2 = 1 - H e n R (AB cos B

1 n n=1 n n n + sin B

n ^ ) where

R

n = 2(AB

n sin Bn -cos B

n+1

)

B

n {

A\

2 + 2A + 1 }

has been coded and is being checked out.

These computations will cover the simplified case of time

independent boundary conditions. The more complicated case in which

the boundary conditions are time varying is being studied to arrive at

a computation scheme.

371 1-60-0009/55-90 STUDY OF A DIFFERENCE EQUATION ARISING IN STATISTICAL

MECHANICS

Origin : NBS, Section 3.2

Manager : M. Newman

Full task description : Oct-Dec 1954 issue, p. 23

Status : Inactive.

371 1-60-0009/55-91 COMPUTATION OF VIRIAL COEFFICIENTS

Origin : NBS, 9ection 3.2

Manager : P. Davis

Full task description : Oct-Dec 1954 issue, p. 23

Status : Continued. Experimentation with the Monte Carlo method for
2b Status of Projects

numerical integration continued. A discussion of the findings is

included under task 1 101 -10-1 104/55_

55> p. 1 . A preliminary code

has been written for evaluation, by the Monte Carlo method, of the ninefold integral defined in the preceding report (Oct-Dec 195^ issue, p. 23).

371 1 -60-0009/ 55-92 CYLINDRICAL ELECTRON LENS CALCULATIONS

Origin : NBS, Section *+.5

Manage

r

: W. G. Hall

Objective : To compute on SEAC the trajectories for an electron from the

equations of motion

= -eE

x = e my = -eE = e J y ay

where 0 is the electric potential distribution given by

0(x,y) 2Acos x cosh y

cos2x + cosh 2y *

E x = x-directed field,

E = y-directed field,

y

A = a parameter determined by the geometry of the lens.

The problem is to solve this system given the initial values

x0

,y0

,X 0

,y o

, and to compare trajectories having small changes in the

initial conditions, such as when the initial velocity differs by one part in 10,000 or less, or when velocity direction differs by 3 » 10~5

radians. It is necessary to obtain the results to at least six

significant figures.

Background : These calculations are being performed to investigate the

possibilities of obtaining better resolution in an electron lens than

is possible by the use of current procedures.

Comment : This computation was suggested by L. Marton (NBS Section H-.5)

and relates to a problem of W. E. Waters (DOFL)

.

Status : New. A number of trajectories have been computed. A form of the

Runge-Kutta method was used. To give an indication of the accuracy

obtained, tj,he difference in velocity as given by v = (x^+y2)2 and

v [(e/m) 0]2 was computed at each step and the maximum difference was printed along with the minimum v. In general, the maximum difference

in velocity has been of the order of 10~10 and the minimum velocity

has been of the order of 10~^.

For the bulk of computation done so far, an interval At=.l6

has been used. In the test case run using the intervals of . 08 , . 04 , and .02, results differed only in the eighth place from the largest

interval to the smallest.

1 102-^0

- 5126/55-95 HEAT TRANSFER IN AN EXTENSIVE MEDIUM

Origin and Sponsor : NBS, Section 10.3

Manager : D. Liepman _

Objective : To evaluate e n erf bn for various values of n. Bac kgr ound : This function is encountered in the theoretical analysis

of the heat transfer in an extensive medium internally bounded by a hollow sphere. The surface conditions of the sphere are desired.

Status : Completed (New). The function has been evaluated and the

results have been transmitted to the sponsor.
Status of Projects 25

1102-40-5126/55-96 DISTRIBUTION of light output from neutron spectrometers

Origin and Sponsor : NBS, Section 4.11

Manager : I. Rhodes

Objective : To calculate by random sampling, on SEAC, the slowing down

of a fast neutron in a restricted volume of liquid scintillator. From

this to calculate the light output for each incident neutron that

remains in the volume. By following the paths of many neutrons the

probability distribution of the light output for a given incident neutron

may be obtained.

Background : It is the object of the calculation to design a spectrometer,

based upon the principle of total absorption, in which the distribution

of light output is as narrow as possible, at the same time maintaining

a reasonably high detection efficiency. Hand Monte-Carlo calculations

indicated that such a spectrometer was promising and would have quite

high detection efficiency. All previous fast neutron detectors which

have good energy resolution have efficiencies of about 10~4 # The

proposed spectrometer may increase this efficiency by a factor of 100,

Comments : This problem was proposed by J. E, Leiss (NBS, Section 4,11),

Status : New, The problem has been coded for SEAC, and runs have been made

for several different sizes of spectrometers at neutron energies of 7

mev and 14 mev.

1102-40-5126/55-98 THERMAL DIFFUSIVITY

Origin and Sponsor : NBS, 9ection 3.2

Manager : P. Davis

Objective : To obtain the temperature distribution in a cylinder and in an annular cylinder to which sinusoidal heating has been applied.

Background : This problem arises in the experimental determination of the

thermal dif f usivities of various specimens.

Status : New, The simplified case has been completely evaluated. Consideration

of the complete case awaits further experimental work.

1102-40-5126/55-99 INTEGRAL OF a PRODUCT OF BESSEL FUNCTIONS

Origin and Sponsor : NBS, Section 6,4

Managers : P, Henrici, D. Jirauch

Objective : To evaluate the integral

for x = 0( ,01 )2.00.

Background : The evaluation of this integral is required for the solution

of a problem in structural mechanics. Direct evaluation of the integral

by numerical quadrature methods is difficult since for x large the

integrand approaches zero only as x“3/2 # The integral can be transformed

however, by complex variable methods into a product of hypergeometric

series, which can be composed rapidly.

Status : Completed (New), The numerical results obtained on SEAC,

accurate to eight significant digits, have been transmitted to the

sponsor, A report describing the method of computation is in preparation.

1 102-40-5126/55-102 COST ACCOUNTING OPERATIONS FOR HIGH SPEED COMPUTERS

Origin and Sponsor : Bureau of Supplies and Accounts, Department of the Navy

Managers : G. Hawkins, E. Marden

Objective : To program certain accounting operations of the originating

agency for performance on electronic computers; in particular, to

distribute the monthly financial obligations for various U. S. Naval

installations in this country and abroad according to different types
26 Status of Projects

of supply suballotment functions carried on at these installations;

to convert these data into cost per unit within each of the suballotment

functions and combinations of functions; to compare identical functions

between similar activities and to summarize obligations for all activities

combined

.

Background : The Bureau of Supplies and Accounts of the Navy Department in

Washington receives monthly recapitulations of financial obligations for

supply operations from more than 60 U. S. Naval supply depots, U, S.

Naval supply centers, U. S, Naval shipyards, U. S. Naval stations, U.S.

Naval submarine bases, U. S. Naval fuel depots, construction battalions,

and other supply activities. At the same time there is reported the number

of man hours employed in the activities connected with the Naval installations

to account for the obligations to expend funds. These obligations fall

into more than 30 different categories, depending upon the nature of the

activities conducted at the Naval installation. These data are used for

the administration of current funds and as a basis for projecting budget

requests for future requirements.

Status

:

New. Coding the problem for UNIVAC is in progress.

1 102-40-51 26A5-1 03 MISSOURI RIVER PROBLEM

Origin and Sponsor : Missouri River Division, Corps of Engineers, U.S. Army

Manage

r

: H. Bremer

Objective : To attempt to develop multiple regression equations to predict

the amount of run-off in the Missouri River based upon such influencing

factors as snow cover, snow-water content, precipitation, and possibly

such other factors as spring temperature and precipitation from the

previous years.

Background : The Army Corps of Engineers is interested in predicting the

amount of water in the Missouri River flowing past a specific point

for a fixed period of time (this is known as the run-off). Such data

may aid in the determination of possible dam sites and sizes and in

the prediction of spring floods. The method of multiple regression

has been used commonly in forecasting seasonal run-off for other rivers.

This approach has been shown to work well for data collected for the

other river basins. In this particular problem, because of the

insufficient length of reliable records, the method will have to include

a considerable amount of preliminary independent decisions to be made

on the basis of the physical characteristics of the basin. The regression

analysis itself is to be performed on a high speed computer such as UNIVAC, which is most readily available to the Corps of Engineers,

Status : New. Two codes have been written for UNIVAC. One obtains the

means, standard deviations, and simple cross correlations for any set

of observations that may occur; the other does the major part of the

computations involved in obtaining the multiple regression coefficients

for 9 or less independent variables. The first code has been checked

out and with it the following was obtained: ( 1 ) means, standard

deviations and simple cross correlations for snow-water content for

37 observation stations, each with 20 observations for the 20 years over which records have been kept; (2) means, standard deviations, and

simple cross correlations for precipitation taken at 33 stations each

with eight different sets of observations extending also over 20 years.

1102-^-0-5126/55-105 AERONOMIC TIDAL WINDS OF THERMAL ORIGIN

Origin and Sponsor : NBS , Section 82.1

Managers : B. Gill, H. Howe

Objective : To find on SEAC the speed and direction of the horizontal

tidal winds and the tidal pressure oscillations for several periods as

a function of altitude above the earth’s surface using (1) two atmospheric

models (NACA and Rocket Panel), and (2) three different boundary
Status of Projects 27

conditions at the top of the atmosphere.

Background : The earth's atmosphere exhibits tidal oscillations just as do

the earth's oceans, with the difference that the atmospheric tides are strongly affected by solar radiation. These atmospheric tides manifest

themselves as barometric variations, mainly of a semi-diurnal nature;

the variations depend on the altitude.

Status : New. The code was written and checked out; trial values were

computed

,

1 1 02-40-51 26/55-1 06 REFLEX KLYSTRON

Origin and Sponsor : Diamond Ordnance Fuze Laboratory, Department of Defens

Manager : J. Bram

Ob iec tive : To evaluate on SEAC the efficiency of a klystron

n = XL_

k 277V,

27

r

cos (9 +coT(9)) d9

where T(9)

3T

q 2V(V1 /Vq

) cos 9 ^

2 n/2 4c

dx

J:

;

X

X2 + c Vx+(v

1 /v0 )cos 9

3T,

if 0 < 9 < ? or < 9 < 2tt,

dx

2^2 J(V

1 /V0 )cos 9 Jx

-

:§- + c v

/ x+ ( V_

] /Vo ) cos Q

if f< 9 < |^ ,

for various values of the parameters.

Bac kground : For the given klystron, cjTq is the transit angle associated

with the transit time Tq that is required for an electron to leave and

return to the gap; c is a parameter associated with the amount of space

charge; V^/Vq is the ratio of peak RF voltage to the voltage of the d.c.

power supply; is the efficiency of the klystron.

Status : New. The program has been prepared and is ready to be code checked

1 102-40-5126/55-109 ELECTRON PULSE HEIGHT DISTRIBUTION

Origin and Sponsor : NBS, Section 4.11

Manager : R. Zucker

Objective: For given distributions f(x^) and f(xj), to find the sum f(x.)f(x.) for assigned values of x. + x.. This represents a x

ix

j

1

folding together of different mono-energetic electron pulse height

distributions obtained from the NBS high energy X-ray spectrometer.

Background : This folding procedure is being performed to synthesize the

response of the spectrometer to mono-energetic X-rays.

Comment : This problem was proposed by H. W. Koch (NBS Section 4.11),

Status : New. Calculations are performed as data are received from

originator

.
28 Status of Projects

11 02-40-5126/55-1 13 REACTOR DESIGN

Origin : Westinghouse Atomic Power Division , Sponsor : Atomic Energy Commission

Manager : U. Hochstrasser

Objective : To find the lowest eigenvalue A and the corresponding solutions

0j(x,y ?z) [j = 0,1,2,...,n] of the n+1 partial differential equations:

~ V {

h

0 (x ’ y ’ z) ^0

O

(x ’ y,z)

) + f

0

^ x ’ y ? z) 0

O

(x ’ y ’ z)

n = A YL g 0k (x,y,z ^ 0

k (x,y ’ z ^ k=0

“ v

|

h

j( x ? y > z ) V 0j(x?y5 z

)] + f

J (x,y ,z ) 0j (x,y,z)

= A gj k (x J y»z)0

k (x,y,z)+dj(x,y,z)0j_

1 (x,y,z),

The x,y,z are the coordinates of any point lying in the interior R of

a closed surface SB , but not on given closed surf aces. S .[ i=1 ,... ? m ]

,

which are distributed in some way in R. hj(x,y,z), fj(x,y,z),

gjkC x 5 y ? z ) ? dj(x,y,z) are given functions in R, which have discontinuities

only on S . . The boundary conditions on SD are given by Cl it 0j(x,y,z)=O: j—0 5

1

j 2 j . . .

,

n

5 (x,y,z) £ S^.

In addition the "currents" normal to the surfaces S . have to be

continuous when passing through S

c

^:

|j[hj(x,y,z) 0j(x,y,z)] = h

j

(x,y,z) 0j(x,y,z) ] x,y,z —

s

ci+ x,y,z—

s

ci_

where S

c ^+ means the outside of the closed surface $

c ^, S the inside.

Background : The problem arises in reactor design. The function 0j is

the j-th energy level flux. The coefficients hj(x,y,z), fj(x,y,z),

g-i.( x 5 y, z ), d .(x,y,z) are determined by measurements.

J K J Status : New.
III. STATISTICAL ENGINEERING LABORATORY

(Section 11.3)

1 . Fundamental Research in Mathematical Statistics

BIBLIOGRAPHY AND GUIDE TO STATISTICAL LITERATURE

Task 1103-10-1107A9-1a

Origin: NBS Authorized 1 /9/*+9

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 1954 issue, page 49.

MANUAL ON FITTING STRAIGHT LINES

Task 1103-10-1107/50-2

Origin: NBS Authorized 3/1/50

Manager: F. S. Acton

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

Status: CONTINUED. The manuscript is currently in the hands of

technical advisors, for comment.

TABLES TO FACILITATE DRAWING RANDOM SAMPLES

Task 1 1 03-10-1 107/51-1

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,

page 64.

29
30 Status of Projects

MISCELLANEOUS STUDIES IN PROBABILITY AND STATISTICS

Task 1103-10-1107/51-2

Origin: NBS Authorized 7/1/50

Manager: C. Eisenhart

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

Status: CONTINUED. For a description of the continuing activity

on this task, see July-Sept 195^ issue, page 21.

Publications:

(1) 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 l'Institut Henri Poincare (Paris).

(2) Inequalities for probabilities associated with the multivariate

normal distribution. I. R . Savage. Submitted to a technical journal.

(3) On the variances and covariances of order statistics from the Weibull

distribution. J. Lieblein. Accepted for publication in the Annals

of Mathematical Statistics.

PROCEDURES OF NON-PARAMETRIC STATISTICS

Task 1103-10-1107/52-2

Origin: NBS Authorized 9/17/51

Manager: I. R. Savage

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

Status: INACTIVE. For the latest report on this task see July-Sept 1 955- issue, page 22.

Public ations

:

(1) Easily used simultaneous confidence limits for a line. W. S. Connor.

Submitted to a technical journal.

(2) Contributions to the theory of rank order statistics. I. R. Savage.

Submitted to a technical journal.

STUDIES IN THE MATHEMATICS OF EXPERIMENT DESIGN

Task 1103-10-1107/53-1

Origin: NBS Authorized 10/15/52

Manager: W. H. Clatworthy

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

Status: CONTINUED. Work on the analysis of covariance applied

to incomplete block designs is being continued. For details see Oct-Dec

195*+ issue, page 25.

The material of publication (3) listed below is being reworked

to include two-way elimination of heterogeneity.

Publications:

(1) 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.

(2) Partially replicated latin squares. W. J. Youden and J, S. Hunter

(American Cyanamid Co.). Submitted to a technical journal.
Status of Projects 31

(3) Some fractional factorial arrangements for factors at two levels.

W. H. Clatworthy, W. S. Connor, and M. Zelen. In manuscript.

RESEARCH ON MATHEMATICAL ASPECTS OF ORDER

STATISTICS METHODS

Task 1103-10-1107/55-110

Origin: NBS Authorized 3/3/55

Manager: J. Lieblein

Objective: To conduct mathematical research on problems involving

the use of order statistics.

Background: Modern applications of statistical method in physical

science experimentation make use of an increasing variety of "order

statistics" procedures, i.e., those involving the size-order of observed

quantities. In many cases this has made possible useful short cuts in

the analysis of data compared with older methods which compute moments

and other statistics of the observations without regard to order.

Examples are the use of linear functions of order statistics to estimate

the mean and standard deviation of a population, use of range in place

of standard deviation in quality control, and use of criteria for

rejection of outlying observations based on sample statistics such as x

n-x ’ x

n x ’ ( xn'xn-i^ //(' x

n _x

1 ^ ’ where *n ? x-j , xn_i> are respectively

the largest, smallest, and (i+1)-st largest observations and x and x are the sample mean and median. This project is intended to investigate

mathematical problems that arise in the development and application of new methods concerned with the use of order statistics. It is also

planned to provide an aid for the use of available methods by means of an up-to-date compilation of theoretical and tabular information on various

types of order statistics.

Comments: Distributions of extremes such as the largest (smallest),

second largest (second smallest), etc., sample values often approach

limiting distributions when the sample size becomes indefinitely large.

Such asymptotic distributions are generally referred to as "extreme value"

distributions, and greatly facilitate the study of extreme values in

large samples. Thus, as an outgrowth of the order statistics approach,

there has arisen a wide area of statistical application concerned with

extreme values. Illustrations are cases of breaking strength of materials

and fatigue problems where a ’weakest-link’ argument may be applied

that rupture is caused by failure of the weakest of the many hypothetical

small ’elements’ of which the material is considered composed, —and

maximum wind velocities, used as a factor in building codes. Two recent

publications of the Bureau, Applied Mathematics Series 22, "Probability

Tables for the Analysis of Extreme-Value Data", and Applied Mathematics

Series 33? "Statistical Theory of Extreme Values and Some Practical

Applications", a series of lectures by E. J. Gumbel for which an additional

printing became necessary, have stimulated widie-spread interest in the

subject. The numerous inquiries on the part of users have indicated the

desirability of further attention to various mathematical and statistical

aspects, such as investigation of assumptions underlying the limiting

distribution of extreme values.

Status: New. Correspondence has been exchanged with several

workers in the field of extreme values concerning their application in

fatigue problems and in breakdown voltage of capacitors.
32 Status of Projects

2. Applied Research in Mathematical Statistics

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. W. J. Youden has been consulting with J. B. Wachtman

Section 9.1 (Porcelain and Pottery) on a study of the modulus of rupture

of alumina briquettes as a function of temperature. An incomplete block

design having a novel feature was employed. The material was available in

blocks of sufficient size to furnish three specimens. A preliminary run was made at four different temperatures using 6 pairs of specimens, each

pair from a different block. When the results from this incomplete block

design were obtained, three additional test temperatures, were chosen and

the remaining test specimens so assigned that a second incomplete block

design was obtained. The novel feature of the design was that all the

results of the first design using blocks of size two were incorporated

in the second design having blocks of size three. Thus at both stages

of the investigation any differences between bricks were eliminated on the comparison of results at different temperatures.

Publication:

(1) Velocity of light from the molecular constants of carbon monoxide.

E. K. Plyler, L. R. Blaine (4.2), and W. S. Connor. J. Opt. Soc.

Amer. 45, 102-106 (Feb. 1955).

STATISTICAL ASPECTS OF NBS ADMINISTRATIVE OPERATIONS

Task 3737-60-0002/52-1

Origin: NBS Authorized 10/1/51

Manager: C. Eisenhart

Full task description: Oct-Dec 1951 issue, p. 5.6 Status: INACTIVE. For status to date see July-Sept 1952 issue,

page 68 .

STATISTICAL SERVICES FOR COMMITTEE ON SHIP STEEL, NRC

Task 1103-40-5105/52-1

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

Manager: W. J. Youden

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

Status: INACTIVE. For status to date see Oct-Dec 1954 issue,

page 27.
Status of Projects 33

RESEARCH IN APPLICATIONS OF MATHEMATICAL STATISTICS TO

PROBLEMS OF THE CHEMICAL CORPS

Task 1103 -^0- 5118/52-1

Origin and Sponsor: Biological Laboratories, Authorized 10/1/51

Chemical Corps, Dept, of the Army

Manager: C. Eisenhart

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

Status: CONTINUED. A report on a method of programming the analysis

of variance for factorial experiments was submitted. Trial runs using

the method have been made on SEAC and on punched card machines.

Extensive calculations have been made on SEAC of a function

arising in the work of the Chemical Corps.

STATISTICAL ANALYSIS AND DESIGN OF EXPERIMENTS

FOR THE U. S. GEOLOGICAL SURVEY

Task 11 03-40-51 40/54-1

Origin and Sponsor: U. S. Geological Survey, Authorized 10/9/53

Department of Interior

Managers: C. Eisenhart, W. J. Youden

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

Status: CONTINUED. W. C. Krumbein of Northwestern University was

a Guest Worker in the Statistical Engineering Laboratory during the first

several months of 1955 . He was WAE with the Geochemistry and Petrology

Branch of the U. S. Geological Survey as consultant with SEL staff members

on geological problems requiring statistical advice or review. Among

those availing themselves of consulting services during this quarter were

W. C. Irving and A. Homka of the U.S.G.S. Division staff on publications,

P. F. Narten and D. C. Alvord on biogeochemical exploration, R. Gulbrandson

on phosphatic deposits, J. T. Hack on general geology of the Potomac Basin,

Dorothy Carroll on Shenandoah Valley stream deposits, and W. H. Bradley

on Maine tidal flat organisms.

STATISTICAL ANALYSIS OF BALL BEARING FATIGUE DATA

Task 1103-40- 5145/54-1

Origin and Sponsor: American Standards Association Authorized 1 /13/54

Manager: J. Lieblein

Full task description: Jan-Mar 1954 issue, p. 54

Status: COMPLETED. The final report on this project, covering the

methods of analysis and interpretation of the data, has been completed

and transmitted to the sponsor.
3^ Status of Projects

MANUAL ON EXPERIMENTAL STATISTICS

FOR ORDNANCE ENGINEERS

Task 1 1 03-^0-51 ^6/55-93

Origin and Sponsor: Office of Ordnance Research Authorized 1 2/29/ 5

1+ Manager: C. Eisenhart

Full task description: Oct-Dec 19

5

^ issue, p. 28

Status: CONTINUED. Conferences were held with a number of

interested persons on the form and content of the proposed manual. Active

work on the manual is expected to get underway early in April, as soon

as Paul N. Somerville, who is to h e one of its authors, reports for duty

in Washington.
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

Full task description: July-Sept 1954 issue, p. 27

Status: CONTINUED. E. W. Cannon is continuing some basic investigations

in the theory of high polymer structure. R. F. Dressier has resumed an investigation on the effect of local domain curvature on edge stresses in

bent plates, for the second order approximation equations.

M. Abramowitz and W. Cahill have completed the preparation of a manuscript concerning the role of proper choice of difference expressions

for boundary derivatives applied to the elastic clamped square plate problem.

Publications:

(1) Addition theorems for general Legendre and Gegenbauer functions.

P. Henrici. Submitted to a technical journal.

(2) Kleine Bemerkung in asymptotischen Entwicklung des Fehlerintegr als

.

P. Henrici, Accepted for publication by Zeitschrift fur angewandte

Mathematik und Physik,

(3) On the vibration of a square clamped plate. M. Abramowitz and W. Cahill.

Submitted to a technical journal.

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. 2

7

Status: TERMINATED.

Publications

:

(1) Comparison of theories and experiments for the hydraulic dam-break wave.

R. F. Dressier. To appear in the Proceedings of the Tenth General

Assembly of the International Union of Geodesy and Geophysics, held

in Rome, Italy, September 1954.

(2) Entropy changes in the equations for rarefaction waves. R. F. Dressier.

In manuscript.

(3) On helical springs of finite thickness, P. Henrici. Qu. App. Math.

XIII, 106-110 (Apr. 1955).

(4) On rotational viscous flow through a tube. P. Henrici. ZAMP j?,

511-516 (1951*).

35
36 Status of Projects

(5) On the design of two-dimensional supersonic nozzles. B. Chaix

(Federal Institute of Technology, Zurich, Switzerland) and P. Henrici.

Submitted to a technical journal.

(6) The generation of Coulomb wave functions by means of recurrence

relations. I. Stegun and M. Abramowitz. Submitted to a technical

journal.

(7) Forced heat convection in laminar flow through a tube. M. Abramowitz.

Submitted to a technical journal.

RESEARCH IN ELECTROMAGNETIC THEORY

Task 1104-1 0-51 60/54-47

Origin and Sponsor: Diamond Ordnance Fuze Authorized 6/29/54

Laboratory, Department of Revised 9/29/54

the Army

Manager: F. Obe rhe ttinger

Full task description: July-Sept 1954 issue, p. 28

Status: CONTINUED. F. Oberhe ttinger is continuing an investigation

on the defraction theory for the electromagnetic field generated by a dipole with arbitrary orientation.

RESEARCH IN MATHEMATICAL ELASTICITY

Task 1104-10-5160/55-85

Origin and Sponsor: Office of Scientific Research, Authorized 12/27/54

ARDC , USAF

Manager: R. F. Dressier

Full task description: Oct-Dec 1954 issue, p. 30

Status: CONTINUED. The first studies on vibrations of wings with

free edges using finite difference methods were begun with a consideration

of right triangular shapes (half delta wings) for various aspect ratios.

One side of the wing is elastically clamped; the other side, the hypotenuse,

and the included sharp corner are free. The problem has been transformed so that one fixed domain can be used for all wing shapes. This permits use

of the same square mesh for all problems. The form for the special sharp

corner boundary condition for corner angles not equal to 90° has been

obtained for this fixed domain and utilized in the elimination of exterior

mesh points near the corner.

R, Dressier, M. Abramowitz, and W. Cahill have completed a detailed study concerning the best methods for eliminating the exterior

mesh points in terms of the free boundary conditions. These finite

difference formulations present considerable practical difficulty for

several reasons: (1) the differential operator is of high order (fourth),

(2) the free edge conditions are of high order (second and third),

(3) all possible third order cross derivatives occur in the shear conditions

on the hypotenuse. About twelve difference patterns have been considered,

for both square and oblique meshes, and with the differential equation

expressed both on the boundary and off. Many of the schemes would lead

to an excess of exterior points over available boundary conditions.

A pattern has been chosen using slightly unsymmetric approximations for

the boundary conditions and the differential equation on the hypotenuse.

This has the great advantage of permitting elimination of exterior points

locally without the use of the computing machine. At the free corner it
Status of Projects 37

has been necessary to solve separately one ninth order algebraic system.

This is now being done for special cases.

W. Cahill has completed the problem formulation for aspect

ratio equal to 2 and CT = .29 for a mesh with ten points per side. This

will serve as a preliminary test by solving its matrix of order m- 5 for

the lowest eigenvalue and eigenvector using a code for SEAC which is

already available. For the main results, however, U. Hochstrasser and

I. Rhodes have begun preparation of a new machine code for larger,

unsymmetric matrices and for higher eigenvalues and vectors.

For the linear deflections of corrugated diaphragms with

rotational symmetry, R. F. Dressier has completed a review of existing

theories. It is felt that a more rigorous formulation in terms of shell

theory (combined bending and stretching) should prove more dependable

than engineering-type approaches. The problem is being formulated for

the case of constant normal loading, e . g. , air pressure, for diaphragms of

arbitrary meridial shape for (a) constant thickness and (b) with a thick

stiffening disk of arbitrary radius at the center.

F. Obe rhe ttinger is studying available analytic techniques

for certain types of boundary value problems for the non-homogeneous

biharmonic equation. Such problems arise in the stretching theory for

the elastic edge layer. If practical, such methods will be applied to the

solution of edge layer stresses for specific plate shapes where the

interior solution is now known. Simultaneously, alternative approaches

of numerical type, such as use of finite differences or particular

orthonormal solutions, are being considered.
NATIONAL BUREAU OF STANDARDS AUTOMATIC COMPUTER (SEAC)

Reassembly of the SEAC computer and auxiliary units continued through

the month of January in the new location. The first engineering test

routines were run on the machine January 7> and scheduled computation was resumed January 22 for evening and early morning operation. Full threeshift computation was resumed February 21.

During the remainder of the quarter completely reliable operation

was hampered by a number of intermittent troubles occurring in the central

machine and in both the acoustic and electrostatic memories. This trouble

was largely attributed to breakage of the solder connections during the

move. There were also a number of cracked crystals in the cells of the

acoustic memory.

During the portion of the quarter that SEAC was in operation, its

overall efficiency was 73 per cent during scheduled operation compared

with 81 per cent for the last quarter reported. Scheduled computation

time was distributed as follows:

Total scheduled time available for computation . , 535 hours

Good operating time during scheduled time 38*+ " Idle-in-order time 7

M Time lost during or following malfunction 144 "

In addition, SEAC was utilized for 172 hours of nonscheduled computation,

ihat is, good operation without a duty engineer in attendance for

machine maintenance, — and 434 hours for engineering and preventive

maintenance

.

The record of SEAC operations from January 22 through March 31 is

as follows:

Hours Used:

Task No. Title Code Productive

NBS

:

Checkinq ODeration

TT64/55-55 Research in Numerical Analysis 9 12

1 1 1 0A7-2 Tables of Coulomb wave functions 5 8

nio/50-7 Wave function for lithium 1 mo/51-8 Table of power points of analysis-ofvariance tests 1

1 110/52-25 Table of error function for complex

arguments 2 4

1110/53-52 L-shell conversion coefficients 35

5126/52-34 Molecular vibrations 1 5126/52-44 Calculations for d spacings 1 5126/53-15 Nerve fiber reaction 1 7

5126/53-27 Thermodynamic functions 2

5126/53-29 Dynamic behavior of aircraft structure 2 10

5126/53-51 Radiation diffusion 4 25

5126/53-53 Heterogeneity model 45

0009/54-1

5

Matrix reduction 41 46

0009/54-17 Depo lymerization 1 0009/54-28 Lovibond network for CIE source A 3

5126/54-33 Air conditioning in underground structures 1 5126/54-35 Thermal stresses in structures 3 0009/54-38 Compressibility factors of dry air 5

5126/54-45 Transient flow 2

38
39

Task No . NBS:

1126/5^-36

5126/55-39

5126/55-63

5126/55-65

0009/55-68

5126/55-75

5126/55-81

5160/55-85

0009/55-86

5126/55-87

5126/55-88

0009/55-89

0009/55-92

5126/55-96

5126/55-97

5126/55-99

5126/55-115

0002

Other : 5T2S751-3

5126/54-13

51 26/54-44

5126/55-61

5126/55-62

5126/55-73

5126/55-79

5126/55-101

5126/55-104

5126/55-105

5126/55-106

5126/55-107

Hours Used:

Title Code Productive

Checkinq Operation

Velocity of light 1 Molecular vibrations 2

Morse integrals 4

Automatic coding 12 1 Crystal structures 3

Dispersion equation for optical glass 4 1 Combining tests for significance 1 Research in mathematical elasticity 1 5

Flow coefficients for fluids

"Zero" method determination of crystal

6 3

structures 3 7

Stresses in a wall foundation 1 Temperature distribution in solid wings 1 Cylindrical electron lens calculations

Distribution of light output from neutron

6 23

spectrometers 1 4l

High temperature properties of air 5 67

Integral products of Bessel functions 1 Adsorptions integrals 1 Statistical engineering 1 Meteorology

Award of procurement contracts by linear

6

programming 2

Flight performance

4

4

Elastic neutron scattering 17

Integrals involved in supersonic flutter 3

Heat convection 1 0t -analysis 1 Transcendental equations 1 Calculations of various fuzing systems 1 2

Aeronomic tidal winds of thermal origin 1 Reflex klystron 1 3

Missile trajectory 2

Totals: 131 397
Lectures and Symposia

Note: In general, copies of papers or talks listed

not available from the National Bureau of Standards,

is to be published, it will be listed in the section

Publication Activities.

in this section are

If and when a paper

of this report on

Applied Mathematics Division Lecture

BRENNER, J. (State College of Washington and Ballistics Research Laboratory)

Bounds for determinants and new methods for locating roots

of matrices. March 28.

Seminar in Functional Analysis and Applications

ANTOSIEWICZ, H. (1) Linear spaces. February 16. (2) Normed linear spaces.

February 23. (3) Banach-Steinhaus theorem and some of its applications.

March 2.

HENRICI, P. (1) Iteration methods: Linear equations. March 9.

(2) Iteration methods: Nonlinear equations. March 16.

DAVIS, P. (1) Approximate methods of analysis. March 23.

(2) Infinite systems of linear equations: Integral equations. March 30.

Applied Statistics Seminar

TUKEY, J. W. (Princeton University and Bell Telephone Laboratories).

Some basic problems in planning and interpreting experiments.

January 20.

Algebra Seminar

(Offered by The American University in cooperation with the National

Bureau of Standards)

JACKSON, I. (Bureau of Ships) A paper of Shoda (I). January 5 . MARSH, R. (Department of Defense) A paper of Shoda (II). January 12.

bO
Lectures and Symposia 4-1 BRAM, Leila D. On a method of Todd and Coxeter for determining the

subgroups of a given finite group. January 17.

GOLDBERG, K. On a paper of Et6. January 26.

TAUSSKY-TODD , 0. Commutator sub-groups of p-groups. January 26.

Papers and Invited Talks

Presented by Members of the Staff

at Meetings of Outside Organizations

Papers presented at the meeting of the American Mathematical Society,

Brooklyn, N. Y. , April I5~l6, 1955:

ABRAMOWITZ, M. Forced heat convection in laminar flow through a tube.

GOLDBERG, K. Bounds and asymptotic behavior of coefficients in the series

log e xe y.

NEWMAN, M. , and 0. TAUSSKY. Unimodular circulants of rational numbers.

STEGUN, I., and M. ABRAMOWITZ. The generation of Coulomb wave functions

by means of recurrence relations. (Presented by title.)

* * * * * ALT, F. On the equation of burning. Presented at a meeting of the

Institute of Fluid Dynamics and Applied Mathematics,

University of Maryland, College Park, Md., January 1*+. EISENHART, C. Review of some of the recent work of the Statistical

Engineering Laboratory, NBS. Presented at a Biostatistics

Seminar, Johns Hopkins University, Baltimore, Md . , January 2

7

TODD, J. Controlled experiments in computing and the coding program at

the NBS. Presented to the AEC Computing Facility, New York

University, N. Y., March 10.

YOUDEN, W. J. (1) The gun problem. Presented before the Chicago Section,

American Society for Quality Control, Chicago, Illinois,

January 18. (2) Design of experiments in the physical

sciences. Presented before the Committee on Statistics,

University of Chicago, Chicago, Illinois, January 18, and

at the Eleventh Annual Quality Control Clinic, American

Society for Quality Control, Rochester, New York, February 15

(3) Statistical techniques light the way. Presented before

the Chicago Section, American Society for Quality Control,

Chicago, Illinois, January 19. (*+) Series of nine lectures

on: Evaluation and control of experimental error; Principles

of statistical design analysis of variance; and Factorial

experiments and concept of interaction. Given as part of a short course in statistics at the Institute of Statistics,

North Carolina State College, Raleigh, N.C., January 22-30.

(5) Laboratory control and experimentation. Presented before

the Eleventh Annual Quality Control Clinic, American Society
42 Lectures and Symposia

for Quality Control, Rochester, New York, February 15.

(6) Design of experiments. Presented before the Esso

Manufacturing Technical Committee, Buck Hill Falls, Pennsylvania,

February 17; and before the Washington Section, American

Society for Quality Control, Washington, D. C. , February 23.

(7) Statistical design in the development of test methods.

Given at a meeting of Committee D-1 , the American Society

for Testing Materials, Roanoke, Virginia, February 22.

(8) Interlaboratory evaluation of analytical methods.

Presented at the Pittsburgh Conference on Analytical Chemistry

and Applied Spectroscopy Exposition of Modern Laboratory

Equipment, sponsored by the Analytical Chemistry Group,

Pittsburgh Section, American Chemistry Society and the Spectroscopy

Society of Pittsburgh, March 2. (9) Experimental design for

problems in physics. Presented at a Nuclear Physics Seminar

at the NBS, March 1o, (10) Statistical treatment of data.

Presented before the Akron Section, American Chemical Society,

Akron, Ohio, March 17. (11) Opportunities of mathematics in

science. Presented at Paul Junior High School, Washington,

D. C. , March 25.

Linear Programming Symposium

January 2 7? 28,29? 1955

The National Bureau of Standards and the Directorate of Management

Analysis, DCS/Comptroller , USAF, presented a Linear Programming Symposium,

which was sponsored by the Office of Scientific Research of the Air Research

and Development Command, USAF. The program follows:

Thursday, January 27

Address of Welcome

Brigadier General Harold Q. Huglin, Directorate of Management Analysis,

Headquarters, USAF

Session on Applications, I.

Chairman: Merle M. Andrew, Office of Scientific Research, Air Research

and Development Command

Military applications of linear programming.

Walter W. Jacobs, Headquarters, USAF

Linear programming in bid evaluation

Leon Gainen, National Bureau of Standards

Linear programming under uncertainty: An example of a failure

John M. Danskin, Operations Evaluation Group

The assembly line balancing problem

Melvin Salveson, General Electric Company

A commercial use of linear programming

James L. Batchelor, Management Consultant

Session on Computation

Chairman: Max A. Woodbury, Logistics Research Project, George Washington

University

How to solve a linear programming problem

Alan J. Hoffman, National Bureau of Standards
Lectures and Symposia ^3

Cyclic projections

Charles B. Tompkins, University of California at Los Angele

Reduction of systems of linear relations

Philip Wolfe, Princeton University

Optimizing a function of additively separated variables subject to

simple restrictions

Andrew Vazsonyi, Ramo-Wooldridge Corporation

Some results in nonlinear programming

Robert M. Thrall, University of Michigan

The first feasible solution to the linear programming problem

Saul Gass, Headquarters, USAF

Concepts and computing procedures for certain x. . Harry Markowitz, RAND Corporation

Friday, January 28

Address of Welcome

Allen V. Astin, Director, National Bureau of Standards

Session on Applications, II

Chairman: Franz L. Alt, National Bureau of Standards

Application of linear programming to optical filter design

Alex Orden, Burroughs Corporation

Linear programming and structural design

John Foulkes, Brown University

A production smoothing problem

George B. Dantzig and Selmer Johnson, RAND Corporation

A model for optimizing production by reference to cost surrogates

Abraham Charnes and William Cooper, Carnegie Institute

of Technology

Programming under conditions involving uncertainty

D. F. Votaw, Jr # , Yale University

Stochastic linear programming with applications to agricultural

economics

Gerhard Tintner, Iowa State College

Session on Economic Theory

Chairman: Oscar Morgenstern, Princeton University

Linear programming and economic theory

Paul Samuelson, Massachusetts Institute of Technology

On the Wald existence theorem for general economic equilibrium

Harold W. Kuhn, Bryn Mawr College and Princeton University

The linear team: An example of linear programming under uncertainty

Roy Radner, Cowles Commission

On competetive equilibrium with a finite linear technology and

dependent consumer preferences

Lionel Mackenzie, Duke University
kk Lectures and Symposia

Limitationality and foreign trade

Nicholas Geor gescu-Roe gen , Vanderbilt University

An activity analysis approach to location

Martin Beckmann, Cowles Commission

Tom Marschak, RAND Corporation

Saturday, January 29

Session on Theory of Linear Inequalities

Chairman: Edward W. Cannon, National Bureau of Standards

Linear inequalities and convex polyhedral sets

Albert W. Tucker, Princeton University

Consistency conditions for finite and infinite systems of linear

inequalities

Ky Fan, University of Notre Dame and American University

The probability for the existence of a solution of a system of

linear inequalities

Theodore S. Motzkin, University of California at Los Angele

Optimal rays for linear programs

Alan J. Goldman, Princeton University

The multi-dimensional transportation problem

Emil Schell, Ford Motor Company

The present st'atus of the traveling salesman problem

Isidor Heller, Logistics Research Project, George

Washington University

Linear programming activities in England

S. Vajda, Admiralty Research Laboratory, England

Linear programming 1 s future

George B. Dantzig, RAND Corporation
Publication Activities

1 . PUBLICATIONS THAT APPEARED DURING THE QUARTER

1.1 Mathematical Tables

(1)

Tables of sines and cosines for radian arguments. NBS Applied

Mathematics Series 43. (A reissue of NBS Mathematical Table 4.)

Available from U. S. Government Printing Office, Washington 2 5,

D. C., £3.00.

1.3 Technical Papers

(1) A determinantal inequality. K. Fan and J. Todd. J. Lond. Math,

Soc., 30, 58-64 (Jan. 1955.).

(2) Some metric inequalities in the space of matrices. K. Fan and

A. J. Hoffman. Proc. Am. Math. Soc. 6, 111-116 (1955).

(3) On the solution of the caterer problem. J. W. Gaddum, A. J. Hoffman,

and D. Sokolowsky. Log. Res. Quart. 1, 223-229 (1954).

(4) Application of two methods of numerical analysis to the computation

of the reflected radiation of a point source. P. Henrici.

J. Wash. Acad. Sci. , 45, 38-45 (Feb. 1955).

(5) On helical springs ctf finite thickness. P. Henrici. Qu. App.

Math. XIII, 106-110 (Apr. 1955).

(6) On rotational viscous flow through a tube. P. Henrici. ZAMP,

I, 511-516 (1954).

(7) Least p-th power polynomials on a real finite point set. T.S. Motzkin

and J. L. Walsh. Trans. Am. Math. Soc. ZQi 67-81 (Jan. 1955).

(8) On the linear iteration procedures for symmetric matrices,

A. M. Ostrowski. Rend. Mat. App.{v], VIII, 3-4 (1954).

(9) On the spectrum of a one parametric family of matrices.

A. M. Ostrowski. J. Reine Angew. Math. 193 * 143-160 (1954).

(10) Motivation for working in numerical analysis. J. Todd. Communications

on Pure and Applied Math, VIII, 97_ 1 1 6 (1955).

(11) A method for the numerical integration of differential equations

of second order without explicit first derivatives. R. de Vogelaere.

J. Res. NBS 54, 119-125 (Mar. 1.955) , RP2572.

(12) An extremum property of sums of eigenvalues, H. Wielandt. Proc.

Am. Math. Soc. 6, 106-110 (1955).

1.5 Miscellaneous Publications

(1) Experiments in the computation of conformal maps. NBS Applied

Mathematics Series 42. Available from U. S. Government Printing

Office, Washington 25, D. C., 40 cents.

45
L+6 Publication Activities

2. MANUSCRIPTS IN THE PROCESS OF PUBLICATION MARCH 31, 1955.

2.1 Mathematical Tables

(1)

Table of characteristic values of Mathieu's equation for large

values of the parameter. G. Blanch and I. Rhodes. Accepted for

publication in the Journal of the Washington Academy of Sciences.

2.3 Technical Papers

(1) Forced heat convection in laminar flow through a tube. M. Abramowitz.

Submitted to a technical journal.

(2) On the vibration of a square clamped plate. M. Abramowitz and

W. F. Cahill. Submitted to a technical journal.

(3) Approximate method for rapid Loran computation. M. Abramowitz,

D. H. Call, and J. C. Mathews. Submitted to a technical journal.

(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 of differential equations with integrable forcing

term. H. A. Antosiewicz. Submitted to a technical journal.

(10) Computation of vibration modes and frequencies on SEAC. W.F.Cahill

and S. Levy (NBS 6.4). Accepted for publication in the Journal

of the Institute of Aeronautical Sciences.

(11) On the design of two-dimensional nozzles by the method of

characteristics. B. Chaix (Federal Institute of Technology,

Zurich) and P. Henrici. Submitted to a technical journal.

(1

2)

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.

(13) Easily used simultaneous confidence limits for a line. W.S. Connor.

Submitted to a technical journal.

(14) 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 l’Institut Henri PoincarS

(Paris )

.

(1 5) On a problem in the theory of mechanical quadratures. P. Davis.

Accepted for publication in the Pacific Journal of Mathematics.
Publication Activities 47

(16) Some Monte Carlo experiments in computing multiple integrals.

P. Davis and P. Rabinowitz. Submitted to a technical journal.

(17) 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.

(18) Some sampling results on the power of nonparametric tests against

normal alternatives. W. J. Dixon (University of Oregon) and

D. Teichroew. Submitted to a technical journal,

(19) Comparison of theories and experiments for the hydraulic dam-break

wave. R. F. Dressier. To appear in the Proceedings of the Tenth

General Assembly, of the International Union of Geodesy and

Geophysics, held in Rome, Italy, September 1954.

(20) Heat flow in a fluid with eddying flow. W. H. Durfee (now with

Operations Research Office, Johns Hopkins University). Submitted

to a technical journal'.

(21) On the optimal character of the (s,S) policy in inventory theory.

A. Dvoretzky, J. Kiefer, and J. Wolfowitz. Submitted to a technical journal.

(22) A comparison theorem for eigenvalues of normal matrices, K. Fan.

Submitted to a technical journal.

(23) On systems of linear inequalities. K. Fan, Submitted to a technical journal.

(24) Some inequalities concerning positive-definite Hermitian matrices.

K. Fan. Accepted for publication in the Proceedings of the

Cambridge Philosophical Society.

(25) Discrete analogs of inequalities of Wirtinger. K. Fan, 0. Taussky

and J. Todd. To appear in Monatschefte fur Mathematik.

(26) Asymptotic lower bounds for the fundamental frequency of convex

membranes. G. E. Forsythe. Accepted for publication in the

Pacific Journal of Mathematics.

(27) An algorithm for solving the transportation problem. A. Gleyzal.

Accepted for publication in the Journal of Research of the NBS.

(28) The formal power series for log e xe^. K. Goldberg. Submitted

to a technical journal.

(29) Addition theorems for general Legendre and Gegenbauer functions.

P, Henrici. Submitted to a technical journal.

(30) On certain series expansions involving Whittaker functions and

Jacobi polynomials. P. Henrici. Accepted for publication in the

Pacific Journal of Mathematics.

(31) On generating functions of the Jacobi polynomials. P. Henrici.

Accepted for publication in the Pacific Journal of Mathematics,

(32) The number of absolute points of a correlation. A. J. Hoffman,

M. Newman, E. G. Straus, and 0. Taussky. Submitted to a technical

journal.
48 Publication Activities

(33) Indentation pressure of a smooth punch. E. Levin. Accepted for

publication in the Quarterly of Applied Mathematics.

(34) 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.

(35) On moments of order statistics from the Weibull distribution.

J. Lieblein. Accepted for publication in the Annals of Mathematical

Statistics

.

(3^) A proof of Hilbert’s Nullstellensatz. T. S. Motzkin. Submitted

to a technical journal.

(37) 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.

(38) On Fejer sets in linear and spherical spaces. T. S. Motzkin and

I. J. Schoenberg. Accepted for publication in Annals of Mathematics.

(39) Pairs of matrices with property L, II. T. S. Motzkin and 0. Taussky.

Submitted to a technical journal.

(40) An alternative proof of a theorem on unimodular groups. M. Newman.

Submitted to a technical journal.

(41 ) The coefficients of certain modular forms. M. Newman. Submitted

to a technical journal.

(42) Structure theorems for modular subgroups. M. Newman. Accepted

for publication in the Duke Mathematical Journal.

(43) The diophantine equation axm-byn=c, M. Newman. Submitted to a technical journal,

(44) On a generalization of the normal basis in abelian algebraic

number fields. M. Newman and 0. Taussky. To appear in

Communications on Pure and Applied Mathematics of New York University.

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

diffraction of waves by wedges. F. Oberhettinger . Submitted to

a technical journal.

(46) On the Lerch zeta function, F. Oberhettinger. Accepted for

publication in the Pacific Journal of Mathematics.

(47) Determinanten mit ueberwiegender Hauptdiagonale und die absolute

Konvergenz von linearen Iterationsprozessen, A. M. Ostrowski,

Accepted for publication in ComentariiMathematici Helvetici.

(48) On Gauss' speeding up device in the theory of single step iteration,

A. M. Ostrowski. Submitted to a technical journal,

(49) On the convergence of Gauss’ alternating procedure in the method

of the least squares, I. A. M. Ostrowski, Submitted to a technical journal.

(50) On spectra of second-order differential operators. D. Ray,

Submitted to a technical journal.
Publication Activities 1+9 (51) 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.

(52) Osculatory interpolation in the complex plane. H. E. Salzer

(Department of the Army). Accepted for publication in the Journal

of Research of the NBS.

(53) Contributions to the theory of rank order statistics. I. R. Savage.

Submitted to a technical journal.

(5*+) Inequalities for probabilities associated with the multivariate

normal distribution. I. R. Savage. Submitted to a technical

journal.

(55) The generation of Coulomb wave functions by means of recurrence

relations. I. A. Stegun and M. Abramowitz. Submitted to a technical journal.

(56) A note on group matrices. 0. Taussky. Submitted to a technical

journal.

(57) Some computational problems in algebraic number theory. 0. Taussky.

To appear in the Proceedings of the Amer ican Mathematical Society

Sixth Symposium on Applied Mathematics, held at Santa Monica City

College, August 1953.

(58) Generation and testing of pseudo-random numbers. 0. Taussky and

J. Todd. Submitted to a technical journal.

(59) 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 1953.

(60) Machine attacks on problems whose variables are permutations.

C. B. Tompkins. To appear in the Proceedings of the American

Mathematical Society Sixth Symposium on Numerical Analysis, held

in Santa Monica, Calif., August 1953.

( 61) An extension of a theorem of Dantzig's. C. Tompkins and I. Heller

(George Washington University). Submitted to a technical journal.

( 62 ) Best approximation polynomials of given degree, J, Walsh. To

appear in the Proceedings of the American Mathematical Society

Sixth Symposium on Applied Mathematics, held in Santa Monica, Calif.,

August 1953.

( 63 ) Discrete approximations to elliptic dif f erential e quations . W. Wasow,

Accepted for publication in Zeitschrift fur angewandte Mathematik

und Physik.

(64) 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, Calif., August 1953.

(65) Error bounds for 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, Calif., August 1953.
50 Publication Activities

(66) On eigenvalues of sums of normal matrices. H. Wielandt,

to a technical journal.

(67) Partially replicated latin squares. W. J. Youden and J.

Submitted to a technical journal.

Submitted

S. Hunter.
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