NATIONAL BUREAU OF STANDARDS REPORT

3417

PROJECTS and PUBLICATIONS

of the

NATIONAL APPLIED MATHEMATICS LABORATORIES

A Quarterly Report

April through June 1954

FOR OFFICIAL USE

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

THE NATIONAL BUREAU OF STANDARDS

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

the divisions and sections engaged in technical work. In general, each section is engaged in specialized research, development, and engineering in the field indicated by its title. A brief description

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

cover of this report.

Electricity. Resistance and Reactance Measurements. Electrical Instruments. Magnetic

Measurements. Electrochemistry.

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

Technology. Length. Engineering Metrology.

Heat and Power. Temperature Measurements. Thermodynamics. Cryogenic Physics. Engines

and Lubrication. Engine Fuels. Cryogenic Engineering.

Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid State Physics. Electron Physics. Atomic Physics. Neutron Measurements. Infrared Spectros- copy. Nuclear Physics. Radioactivity. X-Ray. Betatron. Nucleonic Instrumentation. Radiological Equipment. Atomic Energy Commission Radiation Instruments Branch.

Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical Chemistry.

Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry.

Spectrochemistry. Pure Substances.

Mechanics. Sound. Mechanical Instruments. Fluid Mechanics. Engineering Mechanics. Mass

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

Organic and Fibrous Materials. Rubber. Textiles. Paper. Leather. Testing and Specifications. Polymer Structure. Organic Plastics. Dental Research.

Metallurgy. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy. Corrosion.

Mineral Products. Porcelain and Pottery. Glass. Refractories. Enameled Metals. Concreting

Materials. Constitution and Microstructure.

Building Technology. Structural Engineering. Fire Protection. Heating and Air Conditioning. Floor, Roof, and Wall Coverings. Codes and Specifications.

Applied Mathematics. Numerical Analysig. Computation. Statistical Engineering.

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

Instrumentation. Process Technology.

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

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

Standards. Microwave Standards.

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

NBS PROJECT MBS REPORT

HAM. 3 4 8 7

PROJECTS and PUBLICATIONS

of fie

NATIONAL APPLIED MATHEMATICS LABORATORIES

April through June 1954

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication, rep. Approved for public release by the i part, Is prohibited

unless permission uo Director of the National Institute of ndards, Washington 25 ,d.c, Suchperm standards and Technology (NIST) ort has been specifically prepared If thg on October 9 2015 'ort for Its own use,
NATIONAL APPLIED MATHEMATICS LABORATORIES

April 1 through June 30, 195*+

ADMINISTRATIVE OFFICE

Franz L. Alt, Ph.D., Acting Thief Edward W. Cannon, Ph.D., Assistant Chief Olga Taussky-Todd , Ph.D., Mathematics Consultant

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

Robert F. Dressier, Ph.D., Mathematics Consultant Eugene Lukacs, Ph.D., Mathematics Consultant

Ida Rhodes, M.A. , Mathematician

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

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

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

Louis J. Beyerle, Jr., Secretary

Esther L. Turner, Secretary

INSTITUTE FOR NUMERICAL ANALYSIS

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

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

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

Research Staff Forman S, Acton, D.Sc Mathematician

George E. Forsythe, Ph.D Mathematician

Harry D. Huskey, Ph.D .....UCLA Research Associate (Mathematics) Theodore S. Motzkin, Ph.D UCLA Research Associate (Mathematics) Thomas H. Southard, Ph.D Mathematician

Daniel Teichroew, Ph.D Statistician

Wolfgang R. Wasow, Ph.D Mathematician

Graduate Fellows Eugene Levin, M.A. Mervin Muller, M.A. Genovevo C. Lopez, B.A. (UCLA) Lloyd Philipson, M.A. Edwin Mookini, M.A. (UCLA) David A. Pope, M.A. (UCLA) Roger Woods, M.A. (UCLA) Mathematical Services Unit Frederick H. Hollander, M.A. Mathematician

Robert R. Reynolds, M.S. Mathematician

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

Estelle H. Strauss Librarian Assistant

Mildred Vukmanic Administrative Assistant

Mildred B. Webb Administrative Clerk

Reve Vineyard Administrative Clerk

Leo Moskowitz Procurement Clerk Vendla H. Gondanier Secretary

Dorothy M. Hibbard Secretary

Elsie L. Husman.... Secretary

COMPUTATION LABORATORY

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

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

Ky Fan, Ph.D Karl Goldberg, M.A

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

Fritz Oberhettinger, Ph.D. Philip Rabinowitz, Ph.D... Irene A. Stegun, M.A...... Charles J. Swift, Ph.D.... Joseph H. W'egstein, M.S... Computing Staff

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

Anne R. Cock, B.A. Everett C. Dade

Mary M. Dunlap, B.S. Anne M. Futtermon, B.A. Leon Gainen, B.A. Billie R. Gill, M.S. Elizabeth F. Godefroy

Stanley D. Grant, Jr. William G. Hall, B.S. Genevie E. Hawkins, B.S. John G. Hershberger, M.S. Secretarial Staff Evelyn A. Grigg

Gloria F. Holmes, B.S. * American University

Chief Assistant Chief Mathematic ian

, ,.AU** Research Associate Mathematician

. ,AU + * Research Associate

. . .Mathematician

,..AU** Research Associate Mathematician

Mathematician

,..AU ++ Research Associate

Mathematic ian Mathematician

Physicist

Physicist Dorothea H. Jirauch, M.A. Lambert S. Joel, B.A. Terry W. Ledley, M.A. Norman Levine, M.S, David S. Liepman Ethel C. Marden, B.A. Kermit C. Nelson

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

Maxine L. Paulsen, B.S.

B. Stanley Prusch

Sally Tsingou, B.S. Bertha H, Walter

Ruth Zucker, B.A. Lillian Sloane Marjorie E. Young

STATISTICAL ENGINEERING LABORATORY Churchill Eisenhart, Ph.D... Joseph M. Cameron, M.S

Lola S. Deming, M.A

Willard H. Clatworthy, Ph.D. William S. Connor, Jr., Ph.D

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

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

Marion T. Carson

Yvette B. Cocozzella

Lela J. Hamilton

Chief Assistant Chief Technical Aid Mathematical Statistician

Mathematician Mathematician

. .Mathematician Mathematician Mathematical Statistician

Computer

Secretary

Secretary

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

The Institute for Numerical Analysis, In Los Angeles, California,

ceased to exist at the end of the quarter, Aprii-June 195^. However,

many of its studies and projects will be continued under the sponsorship

of the Office of Naval Research and of the Office of Ordnance Research

under the administration of the Department of Mathematics of the University

of California at Los Angeles. The physical facilities of the Institute

for Numerical Analysis including SWAC and the library will be loaned by

the National Bureau of Standards to the University of California. The

status of the projects is reported here in the usual form with the understanding that tasks designated as in a "continued" status will be continued

under the University administration. Future activity in these tasks will

not be reported in future issue of these quarterly reports, but will be

covered in communications of the University of California.

A new section, "Numerical Analysis," has been established in

Washington as of July 1, 195^. At the same time, the Machine Development

Laboratory has been replaced by a section, "Mathematical Physics".

The new organization of the National Applied Mathematics Laboratories

therefore will be as follows:

1. Numerical Analysis. J„ Todd, Chief.

2. Computation Laboratory. M. Abramowitz, Chief (Acting).

3. Statistical Engineering Laboratory. C. Eisenhart, Chief.

Mathematical Physics. E. W. Cannon, Chief.

The activities of this organization will continue to be reported in these

quarterly reports.

Contents

Page

Index. ....... ....... v Status of Projects as of June 30, 195*+ .... 1 Institute for Numerical Analysis ...... 1 (NBS Section 11.1)

Computation Laboratory ........... l4

(NBS Section 11,2)

Statistical Engineering Laboratory 42

(NBS Section 11.3)

Machine Development Laboratory . . 47

(NBS Section 11.4)

Lectures and Symposia 48

Publication Activities ... 52
Index of Active Research and Development Projects

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

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

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

Mathematics Laboratories. These services are, however, fully represented

in the body of the report.

A. Research: Pure Mathematics

Studies in pure mathematics 2, 15

8 . Research: Numerical Analysis

Baker-Hausdorf f formula

Classical numerical analysis, Research in . . . Differential equations, Studies in numerical integration of. . Dirichlet problem for certain multiply-connected domains,

Investigation of Bergman's method for the solution of the. . Eigenvalues, eigenvectors, and eigenfunctions of linear

operators, Calculation of. . . . . . Laplace equation, Solution of, by Monte Carlo method . » . . . Probability methods and sampling techniques. . . Riemann-zeta function, Computation of the complex zeros of . . SCAMP

Simultaneous algebraic equations and techniques for the

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

C. Research: Applied Mathematics, Physics and Astronomy

Applied mathematics, Studies in ...

Compressible flow—method of orthogonal and kernel functions . Computing machines, Studies in the theory of digital

Continua, Research in mechanics of „ Electromagnetic theory, Research in

Flow in supersonic nozzle . Gases, Tables of thermodynamic properties of

Geomagnetic field, Analysis of ..... . Hypergeometric functions ....

Linear programming, Basic research in. ............ " " , Supporting research in . Matrix theory, Special problems in finite

Nerve fiber reaction, Differential equation for. .......

Painleve equation.

Program planning, Research in the mathematical theory of . . . Russian mathematical progress, Study of

Theoretical physics, Miscellaneous studies in

Vibration modes and frequencies (of aircraft structures),

Computation of

Water, High temperature properties of. ............ Water waves

Waves, Diffraction of. .

18

14

3

15

2

16

4

3

7

15

5 19

8

2121

16

27

18

20

17

18

27

6

6

6

3*+ 33

20

22
D. Mathematical Statistics

Experiment design. Studies in the mathematics of. ........ 44

Won-parametric statistics, Procedures of. ... . . 43

Probability and statistics, Miscellaneous studies in. ...... 43

Propagation of error, Law of. .................. 43

E. Mathematical Tables

Airy integral, Table of the modified. .............. 26

Arcsin for complex arguments, Table of. ............. 25

Collected short mathematical tables of the Computation

Laboratory. .......................... 24

Coulomb wave functions, Tables of ................ 23

E^Cz), (z=x+iy), Tables of. ................... 23

Error function for complex arguments, Table of. ......... 26

Exponential function for negative arguments, Extension of

tables of the ......................... 26

Gamma function for complex arguments, Table of the. ....... 23

Hyperbolic sines and cosines, Extension of the table of ..... 25

Mathieu functions, radial .................... 27

Power points of analysis-of- variance tests. Tables of ...... 24

Punched card library. ...................... 9

Random samples. Table to facilitate drawing ... ... 42

Revision of mathematical tables ................. 25

Sievert's integral. ....................... 27

Spheroidal wave functions ............. 26

Van der Pol equation. ...................... 26

Wave function for lithium ........... .... 24

F. Manuals, Bibliographies, Indices, and Technical Information

Coding procedures, mathematical tables and numerical analysis,

Bibliography of ........................ 23

Fitting straight lines, Manual on , 42

Statistical literature," Bibliography and guide to ........ 42

G. Computing Machine Development

Census computing machine, Bureau of the ............. 47

SEAC; National Bureau of Standards Eastern Automatic Computer . . 47

SEAC, Number-theoretical test problems for. ........... Id

SWAC: National Bureau of Standards Western Automatic Computer . . 7

Status of Projects

June 30, 1954

1 . imilinEJF.QR. .NUMERICAL ANALYSIS

(Section 11.1)

1 . Fundamental Research

SOLUTION OF SETS OF SIMULTANEOUS ALGEBRAIC EQUATIONS AND

TECHNIQUES FOR THE INVERSION AND ITERATION OF MATRICES

Task 1 101 -10-51 00A9-AE2

Origin: NBS Authorized 2/15/49

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

Manager; G. E. Forsythe

Full task description; July-Sept 19*+9 issue, p. 2

Status: CONTINUED,, Louise Straus' (binary) SWAC routine for improving the inverse of a matrix (see Jan-Mar 195a issue) has been combined

with an elimination routine (floating binary) to get the inverse.

Some preliminary steps have been taken by B e F. Handy, Jr., and

G. E. Forsythe to assemble information on available SWAC matrix codes for

the growing SWAC manual.

U, R. Hestenes, G. Lopez, and E. H. Moddni have continued to

experiment on SWAC with various modifications of the conjugate gradient

method for solving linear systems. Running time is something like 3 or 4

minutes for 30 iterations with a matrix of order 30. This usually suffices

to solve a linear system.

B. F. Handy has finished a preliminary code for solving n general simultaneous quadratic equations in connection with tas k 1101-40-

5l31/53~42 (see p. 12). The code employs a gradient method with the

metric ]|x|| = £jx.) , and is designed for any n<l6. The first-power

metric was chosen by G. E„ Forsythe to avoid the scaling difficulties of

squaring small numbers in fixed-point operation. More work is needed on the main program before the metric can be successfully used.

Publications: (1) nThe conjugate gradient method for solving linear

systems," by M. R. Hestenes; to appear in Proceedings of the American

Mathematical Society Sixth Symposium on Applied Mathematics, in Santa Monica,

1953. (2) "Proof of an extension of a theorem of G. B. Dantzig's," by

I. Heller and C. B. Tompkins; submitted to a technical journal.
2 Status of Projects

CALCULATION OF EIGENVALUES, EIGENVECTORS, AND EIGENFUNCTIONS

OF LINEAR OPERATORS

Task 1101-10-5100/50-3

Origin; NBS Authorized 7/lA9

Sponsor: Office of Naval Research, USN

Manager: G. E. Forsythe

Full task description: July-Sept 195-9 issue, p. 13

Status: INACTIVE

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

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

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

Journal of Mathematics.

STUDIES IN PURE MATHEMATICS

Task 1101-10-5100/50-4

Origin: NBS Authorized 7/1/49

Sponsor: Office of Naval Research, USN

Manager: C. B. Tompkins

Full task description: July-Sept 1949 issue, p. Id

Status: CONTINUED. G. E. Forsythe used SWAC to generate a complete

table of semigroups of order 4. A semigroup is a set of elements with a closed and associative multiplication defined over them. Two semigroups

are here called dis tine

if they are neither isomorphic nor anti-isomorphic

Let f(n) be the number of distinct semigroups of order n, Prior tables

had shown that f(1)=1, f(2)=4, f(3)=l8, and had given 121 distinct semigroups of order 4. The present calculation, based on an exhaustive search

for associative systems among all possible multiplication tables on four

elements, established three new results: (1) The total number of semigroups with 4 given elements is 3492. Of these 126 are distinct, and 58

of the latter are commutative. (2) There is now a list on punched cards

of the 3492 semigroups. (3) Publication (1) below contains a list of all

126 distinct semigroups in lexicographic order, each semigroup multiplication table being in its normal (that is, lexicographically first) form.

T. S. Motzkin and J. L. Selfridge have proved some theorems about

the possible nature of the normal multiplication tables for semigroups

of order n, with elements 0,1,..., n-1 . They fall into n+1 classes Cq,...,C

The first row of the table is 0...0 for C^., starts with i zeros followed

by an i for C^_^, i=2,...,n-1, starts with Oil for and is 01... n-1

for C^. The class consists exactly of all groups. These results may

make possible the listing of all semigroups of order 5. The preparation

of a code for this listing has been started.

T. S. Motzkin developed asymptotic results for the number of

semigroups of order n, and noted that the equality of the (2n-2 ) !/( n-1

)

l n

meanings of a, a-... a in a general multiplication follows from but (o)

equalities. n “> T. S. Motzkin and J. L. Self ridge proved that the equation

b

c _ c a

a -b has no positive integral solutions when a/b s Many equations of a similar nature were studied. (See also Problem El 124, by L. J. Landen

and J. L. Self ridge, Amer. Math. Mo. 6l_,423 (1954).) The general solution

of a hyper-exponential diophantine equation can usually be obtained if
Status of Projects 3

there are only two variables involved, or if the equation has no higher

than second order exponentials and three variables. An exception to this

p

c g seems to be the equation a -c . Many particular solutions have been

found, including some in which a and c are not each powers of the same number, but the general solution seems rather elusive.

C. B. Tompkins, working with various colleagues, considered

several problems involving discrete variables. This work was partially

connected with project 1 101 — 1 0— 5 1 50/53-

1 ? P. 7 . Publications: (1) "SWAC computes all 126 distinct semigroups of

order 4," by G. E. Forsythe; IN MANUSCRIPT. (2) "Machine attacks on problems whose variables are permutations," by C. B. Tompkins; to appear

in the Proceedings of the American Mathematical Society Sixth Symposium

on Applied Mathematics, August 1953? Santa Monica, California.

COMPUTATION OF THE COMPLEX ZEROS OF THE RIEMANN ZETA FUNCTION

Task 1101-10-5100/50-13

Origin: NBS Authorized 6/1/50

Sponsor: Office of Naval Research, USN

Manager: R. Morgan

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

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

P. 3.

STUDIES IN THE NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS

Task 1101-10-5100/51-1

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Rssearch, USN

Manager: W. Wasow

Full task description: July-Sept 1950 issue, p. 12

Status: CONTINUED. The solution of boundary value problems for

differential equations of the form £y"=F^ (x,y , E.)y '+F 2 (x,y, e) for small

positive £ has been studied by W. Wasow. If a solution u(x) of the

"reduced" differential equation obtained for 6=0 is known which satisfies

one of the two boundary conditions y(a)=A, y(b)=B, then, under certain

additional assumptions the solution of the full equation satisfying both

boundary conditions exists and can be represented by a convergent series

of the form

y(x,e) u ( x

r co

r=1

u

r (x, e.) e r

whose coefficients satisfy certain linear differential equations and can

be approximately calculated by quadratures, if 6 is small.

M. Muller's work on the numerical solution of Dirichlet's

problem by means of random walks is described in task i 101-10-5100/51-2.

Publications; (1) "Discrete approximations to elliptic differential

equations," by W. .Vasow; to appear in Zeitschrift fur angewandte Mathematik und Physik, (2) "Singular perturbations of boundary value problems

for non-linear differential equations of the second order," by W. Wasow;

IN MANUSCRIPT.
4 Status of Projects

PROBABILITY METHODS AND SAMPLING TECHNIQUES

Task 1101-10-5100/51-2

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Manager: D. Teichroew

Full task description: July-Sept 1950 issue, p, 13

Status: CONTINUED. A table of millile probability points of the tdistribution was computed and is available on punched cards.

A special distribution problem was studied. The results are given in publication (1) below.

M. Muller has prepared a manuscript which contains the results

of his investigation of procedures for improving the efficiency of the

Monte Carlo methods for solving boundary value problems. This research

has involved two conceptually different approaches:

1. The underlying stochastic models utilize continuous parameter

processes. This will enable direct statistical estimates of the unknown

solution of the boundary value problem. This approach has been carried

out in detail for the Dirichlet problem. A proof of the strong convergence

of a very general class of processes has been given. In order that the

procedures be feasible for high speed computing machines truncation schemes

are considered. Attention is given to improving the efficiency of this

approach by using statistical design. In essence one of the major advantages of this approach is that it is not necessary to approximate the given

differential operators by difference operators. Further in all known

methods of studying estimates of the unknown solution of the Dirichlet

problem it has been necessary to introduce restrictions on the boundary

and the function defined on the boundary which are restrictions in addition to those imposed by the Dirichlet problem; this is not necessary when

using the approach introduced in this section.

2. This material follows the more classical approach of con- sidering statistical estimates for the solution of the partial difference

equations. Since almost all present numerical methods for solving partial

difference equations use some sort of network of points a new approach

is suggested to determine optimum operator approximations and admissible

networks. These results go beyond Monte Carlo solutions and could be used

for such techniques as relaxation and its many variations. This approach

looks most promising for the Dirichlet problem in three or more dimensions.

Some interesting conjectures are raised. Also, the popular belief that one always does better when the difference operator uses more neighboring

points is shown not to be true. Results are given to show that increased

efficiency is obtained by the use of statistical designs. This research

has been carried out under the valuable guidance of George W. Brown of

the University of California, Los Angeles. The contributions made by

Robert Steinberg have also been most helpful.

Publications: (1) "The mixture of normal distributions with different variances," by D. Teichroew; IN MANUSCRIPT. (2) "Studies on the Monte

Carlo method for solving boundary value problems," by M. Muller; IN

MANUSCRIPT.
Status of Projects

5 VARIATIONAL METHODS

Task 1101-10-5100/51-3

Origin: NBS Authorized 9/1/50

Sponsor: Office of Naval Research, USN

Manager: D. A. Pope

Full task description: July- Sept 1950 issue, p . 1

Status: CONTINUED. A SWAG program for computation of a minimizing

curve to a one-dimensional calculus-of-variations problem has been

completed and a test example computed. The problem of computing geodesics

of a surface by direct methods on SWAG is also being investigated, and a program is being prepared to compute this for a torus. Theoretical investigations into the effectiveness of various methods for machine calculation are being continued.

STUDIES IN APPLIED MATHEMATICS

Task 1 101-10-5100/51-4

Origin: NBS

Sponsor: Office of Naval Research, USN

Manager: T. H. Southard

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

Authorized 9/1/50

Status: CONTINUED. The seminar on numerical analysis met twenty

times during the quarter. A detailed list of sessions will be found under

Lectures and Symposia in the back of this issue. Summaries of the sessions

are being prepared under the editorship of C. B. Tompkins.

C. B. Tompkins and T. H„ Southard have done some work on development of a routine for SWAG which will solve a two-person, zero-sum,

rectangular game whose value is zero and whose typical matrix element is

A„^ + VB . where V is to be determined and -KB <0. Such a game has

aoc a<x 7 aec applicat ions in quantitative studies of warfare, and elsewhere. Some

approximate results were obtained and improvement methods are being studied,

T. S. Motzkin and J. L. Walsh (Harvard University) continued

their investigation of best approximating polynomials. This includes a special study of extremal distance polynomials (see publication 4 below).

P

n=Pn

( z )=TTi dist ZZ|, i=1,.„.,n, z and points in d-space, d>3, is a distance polynomial of degree n; on a symmetry hyperplane of the z. it is

a quasi distance polynomial Qfi . The expression P~* 2! (8P

n /9zk)2 ? where z^- is a coordinate of z, can have more than n-1 real zeros and is thus not a P

n _ 1 (nor a Qn _^), in refutation of a conjecture of G. von Sz. Nagy. The

number of real and complex "stats" (all 0P /0z =0, no dist zz. equal 0)

o i

for n>d>1 and general is conjectured, and in several cases proved to be

2

n _2n-d+2+ ( n -d+1 ) 2-d+2. For the deviation ZlyUj(P

n

( £,j))

P on a given

finite set ^ , . . . , results concerning stationary and minimizing polynomials

generalize those obtained for d<2. For continuous mass distributions jii, the behavior differs markedly from the case <K2.

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

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

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

pressure of a smooth punch," by E. Levin; accepted by the Quarterly of

Applied Mathematics. (4) "Extremal distance polynomials and the logarithmic

potential in space," by T. S. Motzkin and J.L. Walsh; IN MANUSCRIPT.
6 Status of Projects

MISCELLANEOUS STUDIES IN THEORETICAL PHYSICS

Task 1 101-10-5100/51-5

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

Sponsor: " " Manager: R. D. Woods

Full task description: July-Sept 1950 issue, p. 19

Status: CONTINUED. W. Futterman is concluding theoretical and

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

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

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

Publications: (1) "Application of variational methods to intermediate

and high energy scattering," by E. Gerjuoy and D. S. Saxon; Physical Rev.

94 , 473-491 (Apr. 15? 1954). (2) "Variational principles for the acoustic

field," by E. Gerjuoy and D. S. Saxon; accepted by The Physical Review.

(3) "On the diffuse surface optical model for nucleon-nuclei scattering,"

by R. Woods and D. S. Saxon; accepted as a Letter to the Editor of The

Physical Review.

STUDY OF RUSSIAN MATHEMATICAL PROGRESS

Task 1 101-10-5100/52-1

Origin: NBS Authorized 3/15/52

Sponsor: Office of Naval Research, USN

Manager: G. E. Forsythe

Full task description: Jan-Mar 1952 issue, p. 11 Status: CONTINUED. Accumulation of bibliographical cards on Russian mathematical monographs continues (see Jan-Mar 1952 issue, p. 11,

and Oct-Dec 1952 issue, p. 8) . A "Bibliographical Survey of Russian

Mathematical Monographs, 1930 to 1951?" as well as a supplement, is

available in limited quantities.

RESEARCH IN THE MATHEMATICAL THEORY OF PROGRAM PLANNING

Task 1 101-10-5102/50-1

Origin: Office of Air Comptroller, USAF Authorized 6/15/50

Sponsor: " " Manager: T. S. Motzkin

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

Status: TERMINATED. An approximate solution of the assignment

problem, by A. N. Gleyzal, was improved. Quadratic assignment problems

and their applications were studied, in cooperation with J. C. Koopmans

and M. Beckman (both Cowles Commission, Chicago). A special linear programming problem connected with scheduling was discussed with W. Karush,

the shortest cycle problem with G. B. Dantzig (RAND Corporation) and

I. Heller (George Washington University Logistics Project). At conferences

in Washington with J. Todd and A. J. Hoffman (both NBS), plans for a tract

or chapter on linear inequalities and (in conjunction with W. Jacobs,

USAF) for a second symposium on linear inequalities and programming were developed. Other conferences connected with the termination of the project were held with J. W. Gaddum (Michigan State College), H. W. Kuhn (Bryn

Mawr College), I. J. Schoenberg (University of Pennsylvania), R.M. Thrall
Status of Projects 7 (University of Michigan), A. W. Tucker (Princeton University).

Publications: (1) "On the relaxation method for linear inequalities,"

by T. S, Motzkin and I. J. Schoenberg; accepted by the Canadian Journal

of Mathematics. (2) "The assignment problem," by T. S. Motzkin; accepted

as a Cowles Commission Discussion Paper and by the Proceedings of the

Sixth Symposium for Applied Mathematics, held at Santa Monica, Calif.,

August 1953. (3) "The probability of solvability of linear inequalities,"

by T. S. Motzkin; IN MANUSCRIPT.

SCAMP

Task 1101-10-5150/53-1

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

Sponsor: " " Manager: C. B. Tompkins

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

Status: CONTINUED. During the end of the quarter, when most of

the participants had reached Los Angeles, the active summer phase of

the program began. Much preliminary work for the session was carried out

by M. Rees, E. H. Spanier, J. Swift, and C. B. Tompkins.

2 . Development

NATIONAL BUREAU OF STANDARDS WESTERN AUTOMATIC COMPUTER (SWAC)

Task 1 101 -20-51 03A9-1

Origin: Aeronautical Research Laboratory, Authorized 11/1/48

Wright Air Development Center, Air Research

and Development Command, USAF

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

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

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

scheduled two shift basis on both service and research problems. However,

members of the University staff have occasionally operated the machine

after the end of the regular shift at midnight and on weekends. During

the quarter the machine was turned on for a total of 1 1 89 hours, 925.2 of

which were used for productive computing. This is about 7 8 °/° of the

total time. Approximately 38 hours was unscheduled maintenance time.

During the quarter the switch which controls hexidecimal or decimal output was transferred to the console. Part of the timing circuitry

was simplified and the speed of the computer increased. Multiplication

time was decreased from 384 ytxsec. to 3^8 jxsec. , and extract from 128

to 384 jx sec. to 80 to 368 yUsec.
8 Status of Projects

STUDIES IN THE THEORY OF DIGITAL COMPUTING MACHINES

Task 1 101-20-5103/53-1

Origin: NBS Authorized 9/30/52

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, Air Research and Development

Command, USAF

Manager: H. D . Hus key

Full task description: July-Sept 1952 issue, p. 16

Status: CONTINUED. Studies on automatic coding using the interpretative floating subroutine system continued. A paper was presented on this

at the Ann Arbor meeting of the Association for Computing Machinery.

A more general interpretative subroutine which handles vector operations

with single commands has been coded and checked.

3. Mathematical Services

COMPUTING SERVICES FOR RESEARCH STAFF OF

THE INSTITUTE FOR NUMERICAL ANALYSIS

Task 1 101-40-51 30/54-

Origin: NBS

Sponsor: Office of Naval Research, USN

Manager: F. Hollander

Full task description: July-Sept 1949 issue, p. 25

Status: CONTINUED. Research staff problems involving the use of

SWAC were as follows:

(1) For C. 8„ Tompkins: A search for two orthogonal latin

squares of order 10 was continued, in connection with SCAMP, according to

a code by F. Meek. No pairs were found.

(2) For W. Futterman: The approximate, closed form integrals

occurring in the problem of the photo-electric disintegration of the

deuteron are in the process of computation.

(3) For R. D. Woods: Computations of the scattering of neutrons

and protons by various metals, using an optical model for the nucleus, were continued.

(4) For the Physical Chemistry Group of the Department of Chemistry, UCLA: Crystallographic calculations have been continued. Threedimensional Fourier summations of electron density have been computed for

four different crystalline substances; the amplitudes and phases ("structure

factors") of the diffracted waves corresponding to various possible

structures for each have also been calculated for comparison with experiment.

A significant portion of the time has been devoted to refinement of atomic

positions and of anisotropic atomic "temperature factors" by least squares

and differential Fourier syntheses.

(5) For M. R. Hestenes and E. Mookini: Experiments were continued in the conjugate gradient method for solving systems of linear

equations

(6) For G. Forsythe: G. E. Forsythe used the SWAC to compute

all semigroups of order 4 (see project 1 101-10-5100/50-4, p. 2 ). He

also did some research on machine methods of solving systems of linear

and quadratic equations. Part of this was in connection with project

1101-40-51 31/53-^2, p. 12.

Authorized 9/2/48

Revised 11/16/49
Status of Projects 9

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

tidal constituents for earth and ocean tide s

(8) For D, H. Lehmer and J „ Selfridge: (a) The Ideal Waring

Theorem code has been run up to exponent 100,000. Hence the Ideal Waring

Theorem holds for all exponents less than 100,000. (b) The program for

finding quadratic residues which are not squares was continued on a standby

basis. ^c) A recheck was made of some of the irregular prime runs between

2000 and 2550. One error was discovered. It was decided to recheck a few more values. (d) A code was checked out and run which computes the

smallest prime q=kp+1 for a given prime p. (e) The Alway routine was revised to accommodate punched card input and output.

(9) For the Physics Department of UCLA: Completed calculation

of q, the electric field gradient at the position of one of the nuclei of

a Lip molecule using a proposed variational molecular wave function.

(10) For T. Southard: SWAG was used to obtain approximate solutions of some two-person, zero-sum, warlike games called HOTSPOT games

(see 1 101-10-5100/51-4, p. 5 )„

(11) For D. A. Pope: A code for a direct method of calculation

of a one dimensional problem in the calculus of variations was checked out

and a simple problem was calculated on SWAG.

PUNCHED CARD LIBRARY

Task noi-40-51 31 A9-

Origin: NBS Authorized 7/14/48

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, Air Research and Development

Command, USAF

Manager: F. Hollander

Full task description: Apr-June 1949 issue, p. 26

Status: CONTINUED. Tables of random digits and random normal

deviates were reproduced and shipped to the Naval Electronics Laboratory,

San Diego, California. No new tables were added this quarter.

SIMPLIFIED ROLLING PULLOUT EQUATIONS

Task 1 101-40-5131/51-34

Origin: Cornell Aeronautical Laboratory Authorized 6/22/51

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, ARDC, USAF

Manager: F. Hollander

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

Status: COMPLETED, All data received have been processed and

results submitted to originator. "Report on the solution of the equations

of flight during a rolling pull-out maneuver," by E. C. Yowell, will

serve as a final report on this problem.
10 Status of Projects

LOW MOMENTS OF ORDER STATISTICS

Task 1101-^0-5131/51-36

Origin: University of Oregon Authorized 6/22/51

Sponsor: Office of Naval Research, USN

Manager: H. D. Huskey

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

Status: INACTIVE,, For status to date, see Jan-Mar 1955- issue, p.11.

SIERRA WAVE PROJECT

Task 1 101-5-0-51 31/52-36

Origin: Department of Meteorology, U.C.L.A. -Authorized 5-/1/52

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, Air Research and Development

Command, USAF

Manager: T. H, Southard

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

Status: COMPLETED. More theodolite data were reduced on SWAC.

A routine for reduction of three-dimensional Raydist data was coded.

EIGENVALUES

Task 1101-40-5131/53-6

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

(Convair)

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, Air Research and Development

Command, USAF

Manager: F. Hollander

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

Status: TERMINATED.

METEOROLOGICAL MEANS

Task 1101-40-5131/53-10

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

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, Air Research and Development

Command, USAF

Manager: F. H. Hollander

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

project 11.1/31 — 50— 17* Jan-Mar, 1950 issue, p. 15.)

Status: CONTINUED. Work on the '"Mountain Term" for the summer months of July-Aug 1949 was completed. Work is in progress on an evaluation of the non-geostropic flux of angular momentum for both the winter

and summer months. This evaluation makes use of some previously obtained

results, namely, the flux of total kinetic energy and the flux of the

kinetic energy of zonal motion.
DISCRIMINANT FUNCTIONS

Task 1101-^0-5131/53-18

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

~~ USAF

Manager: D„ Teichroew

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

Status: COMPLETED, The SWAC code was extended to cover the range

of p, the number of dimensions, from one to fifteen. Results have been

submitted to the contractor.

BIO-ASSAY PROBLEM

Task 1 101-40-51 31/53-2^

Origin: Stanford University Authorized 3/31/53

Sponsor: Office of Naval Research, USN

Manager: D. Teichroew

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

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

DISCRETE MINIMAL SPACES

Task 1101-40-5131/53-26

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

Sponsor: Evans Signal Laboratories, Army Signal Corps

Manager: H. D. Huskey

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

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

P. 23.

LINEAR EQUATIONS (CQNVAIR)

Task 1101-40-5131/53-35

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

Sponsor: Bureau of Aeronautics, USN

Manager: F. Hollander

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

Status; TERMINATED. For status to date, see Oct-Dec 1953 issue,

P. 16.
12 Status of Projects

ACCEPTANCE TESTS

Task 1101-40-51 31/53-41

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

Institute of Technology

Sponsor: Ordnance rorps, U. S. Army

Manager: H. D. Huskey

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

Status: TERMINATED. The computer will be delivered after the end

of this quarter, and the test routines developed under this project will

be used.

B.P.A. Studies

Task 1101-40-5131/53-42

Origin: Bonneville Power Administration Authorized 6/29/53

Sponsor: " " Manager: G. E. Forsythe

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

Status: CONTINUED. The Bonneville Power Administration asked for a gradient solution of three simultaneous quadratics in three complex

variables—that is, six real quadratics. To solve this, a general quadratic

code has been prepared, discussed under tas k 1101-10-5100/49-AE2, p. 1.

HELICOPTER STABILITY STUDIES

Task 1 101-40-51 31/53-1+4

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

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, Air Research and Development

Command, USAF

Manager: G. E. Forsythe

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

Status: CONTINUED. Almost all the real matrices of order 16

referred to in the Jan-Mar 1954 report have been successfully inverted.

Another determinant of type det( TvF+G) of order 13 is being evaluated

with the Frame code. See task 1 101-10-5100/50-3 in Jan-Mar 1954 issue,

P. 2.

COMPUTATIONS FOR SCAMP

Task 1 101-40-5131/54-1

Origins Office of Naval Research, USN Authorized 9/29/53

Sponsor: Office of Naval Research, USN

Manager: C. B, Tompkins

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

Status: CONTINUED. Computations continued on many discrete variable

problems under the general supervision of J. Swift and C. B. Tompkins.

F. B. Meek prepared a code for the reduction of a matrix of integers to a
Status of Projects 13

triangular form (of Hermite), and reduced several small matrices to this

form. This code was prepared primarily to facilitate some studies which

are to be undertaken by G. Pall in connection with finite projective planes.

PARACHUTE DATA ANALYSIS

Task 1 101 -40-51 31 / 9+-1

Origin; U. S. Naval Auxiliary Air Station, Authorized 2/25/51+ E1 Centro, Calif.

Sponsor: U. S. Navy

Manager: D. Teichroew

Full task description: Jan-Mar 195^ issue, p. 16

Status: INACTIVE.

GALACTIC DISTRIBUTION STUDIES

Task 1 101-40-5131/5^-8

Origin: Statistical Laboratory, University of Authorized 2/2 5/51+ California, Berkeley

Manager: B. F. Handy, Jr.

Full task description: Jan-Mar 195^ issue, p. 16

Status: TERMINATED.

MATRIX INVERSIONS

Task 1101-Lf0-5l31/5l+-9

Origin: Northrop Aircraft, Inc. Authorized 2/25/5^

Manager: F. H. Hollander

Full task description: Jan-Mar 195^ issue, p. 17

Status: TERMINATED.

SKEW RAYS

Task 1 101-40-51 31/5^-10

Origin: Pacific Optical Company Authorized 2/25/5^

Sponsor: Aeronautical Research Laboratory, Wright Air

Development Center, Air Research and Development

Command, USAF

Manager: F. H. Hollander

Full task description: Jan-Mar 195^ issue, p. 17

Status: CONTINUED. A small amount of code checking was done with

data supplied by the originator, but no results were obtained.
II. COMPUTATION LABORATORY

(Section 11.2)

1 . Research

RESEARCH IN CLASSICAL NUMERICAL ANALYSIS

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

Origins MBS Authorized 1/1/50

Managers: J. Todd, M. Abramowitz, H. A. An to siewicz

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

Status: CONTINUED . J. Todd completed an expository paper (see

American Mathematical Society and Office of Ordnance Research Symposium

on Applied Mathematics (Chicago, April 1954). In addition to classical

numerical analysis, it contained material on modern and ultra-modern

numerical analysis.

American University, was continued. Participants during this quarter were

W. F. Cahill, K. Goldberg, P. Henrici, M. Hyman (Westinghouse Corporation),

L. S. Joel, N. Levine., and P. C. Rosenbloom (Minnesota University and

Institute for Advanced Study).

M. Abramowitz; J. Math. Phys. J2, 188-192 ( July-Oc t . 1 953) . (2) "The

practical evaluation of integrals," by M. Abramowitz; J. Soc. Indust. App.

Math. 2, 20-35 (Mar. 1954). (3) "Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clif ford functions',' by M. Abramowitz;

J. Math. Phys. 33, 111-116 (Apr. 1954). (4) "Coulomb wave functions

along the transition line," by M. Abramowitz and P. Rabinowitz; to appear

in The Physical Review. (5) "Some implications of Liapunov's conditions

for stability," by H. A. Antosiewicz and P. Davis; accepted for publication

in the Journal of Rational Mechanics and Analysis. (6) "Computation of

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

MANUSCRIPT. (7) "On a problem in the theory of mechanical quadratures,"

by P. Davis; accepted by the Pacific Journal of Mathematics. (8) "On the

estimation of quadrature errors for analytic functions," by P. Davis and

P. Rabinowitz; submitted to a technical journal, (9) "The evaluation of

the exponential integral for large complex arguments," J. Todd; J. Res.

NBS 32, 31 3-31 8 (June 1954). (10) "Motivation for working in numerical

analysis," by J. Todd; to appear in the Transactions of a Symposium in

Applied Mathematics, held by the American Mathematical Society and the

Office of Ordnance Research in Chicago, 111., April 1954.

publication (10) below), which was based on an invited address before the

The seminar on Numerical Analysis, held in conjunction with

Publications: ( 1

)

14
Status of Projects 15

RESEARCH IN MODERN NUMERICAL ANALYSIS; INVESTIGATION

OF BERGMAN'S METHOD FOR THE SOLUTION OF THE DXRICHLET

PROBLEM FOR CERTAIN MULTIPLY CONNECTED DOMAINS

Task 1102-10-1104/50-2

Origin: NBS Authorized 3/1/50

Manager: P. Davis

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

Status: TERMINATED,

Oct-Bec 1951 issue, p. 26).

The computations have been completed (see

There are no present plans for publication.

MISCELLANEOUS STUDIES IN PURE MATHEMATICS

Task 1102-10-1104/50-4

Origin: NBS Authorized 1/1/50

Managers: O a Taussky-Todd, J. Todd, M. Abraraowitz, A. Hoffman

Full task descriptions Jan-Mar 1950 issue, p, 23

Status: CONTINUED. K. Fan has studied a class of Banach spaces

closely related to uniformly convex spaces. He has proved for this class

of spaces a generalization, not restricted to linear operators, of

G. Birkhoff's mean er-godic theorem.

0. Taussky-Todd continued her investigation of normal matrices

in certain problems of algebraic number theory. In this connection Mrs.

Todd also studied the following question suggested to her by S. Gorn from

Aberdeen Proving Ground: For what degrees is the normal basis for the

integers of an algebraic number field unique, provided there exists one?

She studied this question so far for cyclic fields of squarefree degrees

and for n=4 only, and it was found that for n=2, 3>4}6 the normal basis

is unique, but not for the other degrees. Also, for non- cyclic abelian

fields of degree 4, the normal basis is unique.

K. Goldberg has revised his manuscript on the Baker-Hausdorf

formula (previously reported under task 1102-10-1104/53-16) to exploit

newly-discovered relations with Faber polynomials.

Publications? (1) "Linear functional equations and interpolation

series," by P. Davis; to appear in the Pacific Journal of Mathematics.

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

to appear in Duke Mathematical Journal. (3) "On representations and ex- tensions of bounded linear functionals defined on classes of analytic

functions," by P. Davis and J, L. Walsh; Trans. Amer. Math. Soc. 76 , 190-206 (1954). (4) "Continuity in terms of connectedness," by K. Fan

and R. A. Stuble; to appear in Proc. Nederl. Akad., Wetensch, Indagationes

Math. (5) "Compact!! ication of completely regular spaces," by K. Fan and

F. Wagner; IN MANUSCRIPT. (6) "The representation of integers by binary

quadratic rational forms," by K. Goldberg, M. Newman, E. Straus, J.D. Swift

Arch. Math. 12-18 (1954), (7) "Ueber die Fssnktionen von Gegenbauer,"

by P. Henrici; Arch. Math. 92-98 (1954). (8) "On certain series ex- pansions involving Whittaker functions and Jacobi polynomials," by

P. Henrici; accepted by the Pacific Journal of Mathematics. (9) "On

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

by the Pacific Journal of Mathematics. (10) "The number of absolute

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

submitted to a technical journal. (11) "A conjecture of Erdos" by

M, Newman; submitted to a technical journal. (12) "Note on a certain

determinant," by M. Newman; IN MANUSCRIPT. 03) "Structure theorems for

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

Journal. ( 1 4 ) "Some computational problems in algebraic number theory,"
16

Status of Projects

by 0. Taussky; submitted for inclusion in the Proceedings of the American

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

Santa Monica, California, August 1953. 05) "Power series for log (

x . eY ),

by K. Goldberg; IN MANUSCRIPT.

NUMBER THEORETICAL TEST PROBLEMS FOR SEAC

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

Origin: NBS Authorized 6/1/50

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

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

Status: INACTIVE. For status to date, see Jan-Mar 195*+ issue,

P. 21.

SOLUTION OF LAPLACE EQUATION BY MONTE CARLO METHOD

Task 1 1 02-10-1 10V51 -6

Origin: NBS Authorized 9/28/50

Manager: M. Abramowitz

Full task descriptions Juiy-Se.pt 1950 issue, p. 38

Status: CONTINUED. J. Todd and 0. Taussky-Todd completed the

manuscript of the paper on generation and testing of pseudo-random numbers.

Further experiments and tests on the Fibonacci sequence have been carried

out by J. M. Cameron and M. Newman.

I. Stegun continued the study of the location of the mean in

samples from a normal population. J. Todd completed his paper on the

solution of the Laplace equation by Monte Carlo methods.

Publications: (1 ) "Generation and testing of pseudo-random numbers,"

by 0. Taussky and J. Todd; IN MANUSCRIPT. (2) "Experiments in the solution

of differential equations by Monte Carlo methods," by J. Todd; IN MANUSCRIPT.

ANALYSIS OF GEOMAGNETIC FIELD

Task 1 102-10-1 10V52-8

Origins NBS Authorized 8/10/51

Managers C. J. Swift

Full task description: July-Sept 1951 issue, p. 35-

Status: TERMINATED. For status to date see Oct-Dec 1951 issue,

P. 29.
Status of Projects 17

SPECIAL PROBLEMS IN FINITE MATRIX THEORY

Task 1102-10-1104/52-34

©rigin: NBS Authorized 11/6/51

Manager: 0. Taus sky-Todd

Full task description: Oct-Dec 1951 issue, p„ 30

Status: CONTINUED. 0. Taus sky-Todd and T. S. Motzkin prepared a new and enlarged draft of their paper, "Classes of matrices with property

L, II". Mrs. Todd studied a connection between sums of positive definite

Hermitian matrices and sums of real (not necessarily symmetric) matrices

with positive dominant main diagonal elements. It was known that, for sums of matrices of either type, | A+B | |A | + |B|, and it is now shown

that the inequality for matrices of the former type implies the inequality

for matrices of the latter type.

Reference is also made here to the work by K. Fan, 0. Taussky-Todd,

and J. Todd in publication (5) of task 1 102-10-51 16/50-2, p. 18.

K. Fan has prepared manuscripts on Hermitian and normal matrices

(see publications (3) and (4) below). It is shown in (4) that if the

(r+1)th singular value of the difference of two normal matrices is not

greater than £ , and if a circular disk contains at least p eigenvalues

of one of the matrices, the concentric disk with radius increased by £

contains at least p-r eigenvalues of the second matrix. This generalizes

earlier results of Ledermann and Wielandt. Results given in (3) for a positive definite Hermitian matrix H include: Minimum characterizations

of det H/det C, where € is a principal submatrix of H; new inequalities

for the eigenvalues of H in terms of its principal minors; and upper bounds

for det H in terms of its principal minors.

A. J. Hoffman continued! his study of minimax characterizations

of functions of the eigenvalues of a positive definite Hermitian matrix,

obtaining extensions of earlier results on sums and products.

Publications: (1) "Some remarks on commutators of matrices," by

K. Fan, Arch. Math. 102-107 (1954). (2) "Inequalities for eigenvalues

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

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

NBS Applied Mathematics Series 39. (3) "Some inequalities concerning

positive definite Hermitian matrices," by K. Fan; IN MANUSCRIPT. (4) "A

comparison theorem for eigenvalues of normal operators," by X. Fan; IN

MANUSCRIPT. (5) "Some metric inequalities in the space of matrices," by

K. Fan and A. J. Hoffman; to appear in the Proceedings of the American

Mathematical Society. (6) "Characteristic roots of quaternion matrices,"

by 0. Taussky; Arch. Math, j?, 99~101 (1954). (7) "Generalized commutators

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

by. 0. Taussky; to appear in the von Mises Anniversary volume. (8) "The

condition of the finite segments of the Hilbert matrix," by J. Todd; to be included in Contributions to the systems of linear equations and the

determination of eigenvalues, NBS Applied Mathematics Series 39. (9) "The

conditions of certain matrice s , II , " by J. Todd; to appear in Archiv der

Mathematik. (10) "Error bounds for eigenvalues of symmetric integral

equations," by H. Wielandt; submitted for inclusion in the Proceedings

'f the American Mathematical Society Sixth Symposium on Applied Mathematics,

aeld at Santa Monica, California, August 1953. (11) "Einschlie ssung von Eigenwerten he rmite scher Matrizen nach dem Abschnittsverf ahren, " by

H. Wielandt; Arch. Math. 108-114 (1954). (12) "On eigenvalues of sums

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

DIFFERENTIAL EQUATION FOR NERVE FIBER REACTION

Task 1 102-10-1 10V53-1 5 Origin; National Naval Medical Institute Authorized 12/8/52

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

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

Status: • CONTINUED. A joint report with K„ S. Cole of the Naval

Medical Research Institute is being prepared.

BAKER-HAUSDORFF FORMULA.

Task 1102-10-1105-/53-16

Origin: NBS Authorized 12/8/52

Managers: 0. Taussky-Todd , K. Goldberg

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

Status: TERMINATED s See task 1102-10-1104/50-5-, p. 15, for work

performed during this quarter. Future work will be reported as part of

that task.

BASIC RESEARCH IN LINEAR PROGRAMMING

Task 1102-10-5116/50-2

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

and Office of the Air Comptroller, USAF

Sponsor: Office of Scientific Research, ARDC, USAF

Managers: J. Todd, A. Hoffman

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

Status: CONTINUED. During this quarter, K. Fan continued his work

on systems of linear inequalities, emphasizing infinite systems in a normed

linear space. The results thus far developed concern existence theorems

and characterizations of the solutions of minimum norm. He is also working

on a monograph on the theory and application of convex sets.

K. Fan, 0. Taussky-Todd and J. Todd studied the discrete analogs

of a set of integral inequalities involving a function and its derivatives

due to W. Wirtinger. New inequalities, involving a function and its second

derivative, were established in the continuous and discrete cases. Discrete

analogs were obtained for inequalities of Bellman and Northcott, and a complete solution was obtained for a geometrical extremal problem suggested

by these results. The general tools used in deriving these results were extremal properties of eigenvalues of finite matrices. Best possible

constants and characterizations of equality cases were also obtained. The

investigation of additional problems to which the matrix techniques do

not apply is proceeding.

A. J» Hoffman, in collabration with H. W. Kuhn (Bryn Mawr

College), has applied the transportation problem of linear programming to

the study of systems of distinct representatives. It is shown how old

and new results on this combinational problem are special cases of the

duality theorem applied to the transportation problem. New results include necessary and sufficient conditions that a system of distinct re- presentatives for given subsets of a set intersect each summand of a partition of the set in a number of elements whose upper and lower bounds

are prescribed.
Status of Projects 19

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

matrices," by H. A, Antosiewicz; to appear in Pacific Journal of Mathematics. (2) "On a theorem of Ostrowski and Taussky," by R. Bellman and

A. J. Hoffman; Arch. Math. 123-127 (195*+). (3) "Lower bounds for the

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

equations and the determination of eigenvalues, NBS Applied Mathematics

Series 39. (^) "Some metric inequalities in the space of matrices," by

K. Fan and A. J„ Hoffman; to appear in Proceedings of the American Mathematical Society. (5) "Discrete analogs of inequalities of Wirtinger,"

by K. Fan, 0. Taussky, and J.Todd; IN MANUSCRIPT. (6) "A dete rminantal

inequality," by K. Fan and J. Todd; to appear in the journal of the London

Mathematical Society. (7) "On the relevance of LeChatelier ' s principle

to linear programming," by A. J. Hoffman; IN MANUSCRIPT. (8) "An extremum

property of sums of eigenvalues," by H. Wielandt; to appear in the Proceedings of the American Mathematical Society.

COMPRESSIBLE FLOW — METHOD OF ORTHOGONAL AND KERNEL FUNCTIONS

Task 1102-10-5116/52-16

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

Air Development Center, USAF, and Harvard

University

Sponsors Aeronautical Research Laboratory, WADC, USAF

Managers; P. Davis, F. L. Alt * Full task description: July-Sept 1951 issue, p. 36

Status: COMPLETED. Publication (1) has been transmitted to the

sponsor.

Publications: (1) "Some SEAC computations of subsonic flow by

Bergman's method of integral operators," by P. Davis and P. Rabinowitz;

an NBS report. (2) "A multi-purpose orthono realizing code and its uses,"

by P. Davis and P. Rabinowitz; submitted to a technical journal.

TABLES OF INTEGRALS INVOLVING THE HIGHER TRANSCENDENTAL FUNCTIONS

Task 1102-10-5116/52-33

Origin: NBS Authorized 10/11/51

Manager: F. Oberhettinger

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

Status: CONTINUED. The article concerning Mathieu and Lame wave functions for the Bateman project handbook is in preparation.

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

submitted to a technical journal.
20 Status of Projects

WATER WAVES

Task 1102-10-51 16/53-5^

Origin; NBS Authorized 6/1/53

Sponsor; Off-ice of Naval Research, USN

Manager; P. Davis

Fall task description: Apr-June 1953 issue, p. 38

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

p. 25,

SUPPORTING RESEARCH IN LINEAR PROGRAMMING

Task 1102-10-51 16/5^9

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

Sponsor: Office of Scientific Research, Air Research and

Development Command, USAF

Managers: J. Todd, A. J„ Hoffman

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

Status; CONTINUED 0 A seminar in linear programming was held fortnightly at American University and will be resumed in the fall. The

speakers and topics are listed under "Lectures and Symposia", p. 49.

Thirteen linear programming problems were solved using the

simplex method. It was found convenient to obtain the answers by doing

the dual rather than the primal problem in each case. A feature of interest was the fact that 12 of these grouped naturally into three sets,

in each of which the dual problem varied only in the objective function.

For each set the four objective functions were carried and transformed,

with the criteria for changing bases and arriving at a solution being

applied to each objective function in turn. Less than half the number

of iterations required to solve for the first objective function were needed in going from the first to second, and so forth. . A code for solving the personnel problem by the "Hungarian"

method of H„ W. Kuhn is being written.

Publications: (1) "On 'overshoot* on the furthest hyperplane method,

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

by L. Gainen, D. J. Honig, and E. D. Stanley; to appear in Logistics

Research Quarterly. (3) "On the solution of the caterer problem," by

J. W. Gaddum, A. J. Hoffman, D. Sokolowsky; submitted to a technical

journal. (4) "An algorithm for solving the transportation problem," by

A. Gieyzal; IN MANUSCRIPT. (5) "Smooth patterns of production," by

A. J. Hoffman and W. Jacobs; submitted to a technical journal. (6) "On

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

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

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

A. J. Hoffman; submitted to a technical journal.
Status of Projects 21 RESEARCH IN MECHANICS OF CONTINUA

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

Origin: NBS Authorized 12/29/53

Sponsor: Office of Naval Research, USN

Manager: R. F. Dressier

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

Status: CONTINUED, The analysis concerning thermodynamical effects

on dissipative flow in rarefaction waves has been completed by R, F. Dressier,

and a manuscript has been prepared. Solution is possible, with the aid

of the Riemann relations along the three Mach lines, for first-order effects on entropy, density, and velocity. Results are expressible as seventh degree polynomials plus second order pole terms. Evaluation indicates that only a small difference results for the back half of the wave compared with results from a simpler flow model in which heat generation accompanying frictional retardation is ignored.

R. F. Dressier is completing work on a manuscript comparing

various theoretical and experimental results on the dam-break problem.

For large t, this includes comparison of previous mathematical results

due to Dressier and more recent formulas by G. B. Whitham with experiments

done at NBS. For small t, an analysis by F, V, Pohle is considered,

using Green's functions applied to Lagrangian coordinates to predict the

sudden collapse of a free vertical wall of water. The necessary experimental work to check this theory has been completed this quarter using

high-speed motion picture photography.

P. Henrici has completed an investigation on slightly rotational

ax i symmetrical flow of a viscous fluid through a cylindrical tube. This

is an extension of the work of Collatz and Gortler, who obtained an eigenvalue problem for a function proportional to the angular velocity component. The present results produce explicit asymptotic expressions for

the large eigenvalues and eigenfunctions, and derive the low ones by an alternative c alcula-tion . A manuscript describing this work has been

completed

A compilation of Green's functions for the heat equation, the wave equation, and the potential equation, for various regions is being

continued by F„ Qberhettinger.

Publications: (1) "Turbulent flow in shock tubes of varying cross- section," by R, F. Dressier; to appear in the Journal of Research of the

NBS. (2) "Entropy changes in rarefaction waves," by R. F. Dressier; IN

MANUSCRIPT. (3) "Comparison of theories and experiments for the hydraulic

dam-break wave," by R. F. Dressier; IN MANUSCRIPT. (*+) "On helical

elastic springs of finite cross-section," by P. Henrici; to appear in the

Quarterly of Applied Mathematics. (5) "On rotational viscous flow through

a tube," by P. Henrici; IN MANUSCRIPT.

RESEARCH IN ELECTROMAGNETIC THEORY

Task 1 102- 10- 51 26/55-5-7

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

Department of Defense

Sponsors: " " " Manager: F. Oberhettinqer

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

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

function representations for the electromagnetic field in homogeneous
22 Status of Projects

and inhomogeneous enclosed media.

Background: (1) This research treats the problem of the diffraction

of electromagnetic waves by a wedge at large distances from the diffracting

edge of the wedge. It adapts the results of task 1 102-40- 5 1 26/54—51 to the problems of the sponsor. (2) This research is directed toward media

having tensor permeabilities, such as the ferrite rotator. It is hoped

that results will facilitate the prediction of the amount of rotation,

the propagation constants, and the energy distribution.

Status: NEW. The problems are under study.

DIFFRACTION OF WAVES

Task 1102-40-5126/54-51

Origin: NBS , Division 14 Authorized 6/29/54

Sponsor: " " Manager: F. Oberhettinger

Objective: To investigate the diffraction of waves by a wedge.

Background: This problem considers the diffraction of acoustic or electromagnetic plane, cylindrical, or spherical waves due to an infinite

wedge

Status: NEW. The problem was reduced (see publication (1) be.low)

to the determination of an infinite integral of the form

r*i Kb, a.) = 2ir sin ( vr a) / '-'O whe re

r e ~ikp cosh x

H^^[k(p2 + p

2 + 2 on' cosh x) 2

] H(x) = < ° ^ ^ C

e ~ik [(0 2

+(d 2+2 pp* cosh x + (z-z')^J 2

[p

2

+ (>' 2 + 2^^'cosh x + (z-z 1 ) 2

] 2

L according as the incident field is a plane, a cylindrical, or a spherical

wave. An asymptotic expansion for I(b,oc) for large values of kp (large

distances from the edge of the wedge) which is valid also for b=0 (which

defines the boundaries of one of the geometric shadows) was derived (see

publication (2) below). This expression consists of a sum of a Fresnel

integral as leading term and an ordinary asymptotic series with elementary

coefficients.

Publications: (1) "Diffraction of waves by a wedge," by F. Oberhettinger; to appear in Communications on Pure and Applied Mathematics.

(2) "On asymptotic series for functions occurring in the theory of diffraction of waves by wedges," by F. Oberhettinger; IN MANUSCRIPT.

H(x)dx

cosh(^ x) - cos(5 b)
Status of Projects

2. Mathematical Tables and Experimental Computations

23

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

Task 11 02-1 0-11 04A3-3

©riqin: Canadian National Research Council Authorized 7/lA7

Manager: I. A. Stequn

Full task description: Apr-June 1 9^+9 issue, p. 4l

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

P. 28.

TABLE OF THE GAMMA FUNCTIONS FOR COMPLEX ARGUMENTS

Task 1 102-10-1 10M+6-1

Origin: NBS Authorized 7/1 A

Manaaer: J. Todd

Full task description: Apr-J une 19^9 issue, p. 43

Status: CONTINUED. The volume is in press. The final proofs

have been checked.

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

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

Office

TABLES OF COULOMB WAVE FUNCTIONS

Task 1102-10-1104A7-2

Origin: NBS Authorized 7/1/47

Managers: M. Abramowitz, P. Rabinowitz

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

Status: INACTIVE. For status to date, see Jan-Mar 1954 issue.

P. 28.

BIBLIOGRAPHY OF CODING PROCEDURES, MATHEMATICAL TABLES

AND NUMERICAL ANALYSIS

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

Origin: NBS Authorized 3/1/50

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

Full task description: Oct-Dec 1952 issue, p. 40

Status: CONTINUED. Preparation of the bibliographies continues.

The following Technical Memoranda were issued:
24 Status of Projects

47 ' Subroutine for conversion of decimal degrees to binary radians and

vice versa.

50 Automonitor processing routine.

51 Matrix inversion routine (internal storage, maximum order 19):

modified Von Neumann-Goldstine method.

52 Subroutine for log N; single precision, fixed binary point.

53 Subroutine for evaluation of Bessel functions of the first kind of

integral order, fixed binary point.

54 Matrix multiplication routine (external storage, maximum order 104).

55 Subroutine for arcsin x arid arccos x; single precision, fixed binary

point

59 Matrix inversion routine (external storage, maximum order 50):

modified elimination method.

61 Base codes 1,2,3 (libraries of subroutines)

62 Binary to decimal conversion and transfer routine; single precision,

fixed binary point.

63 Subroutine for Arc tan x; single precision, fixed binary point.

64 Routine for transforming quadratic forms.

65 Routine for accelerating convergence of sequences (Aitken's method).

WAVE FUNCTION FOR LITHIUM

Task 1102-10-1104/50-7

Origin: NBS Authorized 6/1/50

Sponsor; Bureau of Ordnance, USN

Manager: W. F. Cahill

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

Status: CONTINUED. The iteration for the second approximation

to the wave function has converged. A code to compute a new approximation

to the eigenvalue is being checked out.

COLLECTED SHORT MATHEMATICAL TABLES OF THE COMPUTATION LABORATORY

Task 1102-10-1104/51-4

Origin: NBS Authorized 9/28/50

Manager: J. Todd

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

Status: CONTINUED. The first volume is in press.

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

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

Applied Mathematics Series 37$ in press, U. S. Government Printing Office.

TABLES OF POWER POINTS OF ANALYSIS OF VARIANCE TESTS

Task 1304-34-6351/51-8

Origin: Section 11.3, NBS Authorized 3/26/51

Managers: A. Hoffman, S. Tsingou

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

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

P. 48
Status of Projects

25

REVISION OF MATHEMATICAL TABLES

Task 1102-10-1105-/52-7

Origins NBS Authorized 8/10/51

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

Full task description: July-Sept 1951 issue, p, 5-1

Status: CONTINUED, .Following is the status of those mathematical

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

is planned:

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

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

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

Table Mil 3 (1952).

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

Series 5-1

; in press, U„ S, Government Printing Office, This was originally

Mathematical Table MT8, "Tables of probability functions," vol. I (1951).

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

Series 53; in press, U. S. ' Government Printing Office. This table was formerly designated as Mathematical Table MT5,

"Tables of natural logarithms," vol. IV, Mathematical Table MT12 (1951);

revision in progress for reissue in the Applied Mathematics Series.

TABLE OF ARCS IN FOR COMPLEX ARGUMENTS

Task 1102-10-1105/52-15

Origin: NBS Authorized 10/1/51

Manager: D„ Liepman

Full task description: July-Sept 1951 issue, p. 5l

Status: INACTIVE. JFor status to date, see Oet-Dec 1953 issue,

P= 31.

EXTENSION OF THE TABLE OF HYPERBOLIC SINES AND COSINES

Task 1102-10-1105/52-18

Origin: NBS Authorized 9/17/51

Manager: W. F. Cahill, K. Nelson

Full task description: July-Sept 1951 issue, p. 5l

Status: INACTIVE. For status to date, see ©ct-Dee 1953 issue,

P. 31
26 Status of Projects

TABLE OF THE MODIFIED AIRY INTEGRAL

Task 1102-10-1104/52-23

Origin; NBS Authorized 10/4/51

Manager; P. Rabinowitz

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

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

p. 31 .

TABLE OF ERROR FUNCTION FOR COMPLEX ARGUMENTS

Task 1102-10-1104/52-25

Origin; NBS Authorized 10/5/51

Manager; W. G. Hall

Full task descriptions July-Sept 1951 issue, p. 42

Status; INACTIVE. The 20-place table for x,y=0(.1)2 is available

on punched cards.

EXTENSION OF TABLES OF THE EXPONENTIAL FUNCTION FOR

NEGATIVE ARGUMENTS

Task 1102-10-1104/52-31

Origin; NBS Authorized 10/9/51

Manager; E. Marden, S. B. Prusch

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

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

P. 32.

SPHEROIDAL WAVE FUNCTIONS

Task 1102-10-1104/52-37

Origin; NBS • Authorized 11/28/51

Manager; 0. Liepman

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

Status; INACTIVE. For status to date, see Jan-Mar 1954 issue,

P. 31

VAN DER POL EQUATION

Task 1102-10-1104/52-43

Origin; NBS Authorized 11/28/51

Manager; W. F. Cahill

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

P. 49.

Status: TERMINATED. For status to date, see Apr-June 1952 issue,
Status of Projects 27

RADIAL MATHXEU FUNCTIONS

Task 1102-10-1104/52-49

Origin: MBS Authorized 2/1/52

Managers: J. Todd, I. Rhodes

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

Status: INACTIVE, For status to date, see Jan-Mar 1954 issue.

P. 32.

SIEVE RT'S INTEGRAL

Task 1 102-10-1 104/52-57

Origin: NBS Authorized 2/12/52

Managers: M. L. Paulsen, P. J, O’Hara

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

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

p. 33.

PAINLEVf EQUATION

Task 1102-10-1104/53-3

Origin: NBS Authorized 8/11/52

Managers: J. Todd, H, A. Antosiewicz

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

Status: TERMINATED,

HYPERGEOMETRIC FUNCTIONS

Task 1102-10-1104/53-35

Origin; NBS Authorized 3/25/53

Managers: P, Rabinowitz, W„ Cahill

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

Status: CONTINUED. The first three roots of the confluent hypergeometric function F(1-B§2;4b) were computed for P.pHenrici.

The hypergeometric function F(26,26;53j p ), p

-0.05,0, 1 (0 , 1 )0. 8,

10S, was computed as exploratory work for a problem from the Air Force

Cambridge Research Center,

L-SHELL CONVERSION COEFFICIENTS

Task 1102-10-5110/53-52

‘

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

Manager: W. G„ Hall, J . H. Wegstein

Full task descriptions Apr-June 1953 issue, p. 45

Status: CONTINUED. Machine codes for ail required calculations
28 Status of Projects

have been completed.

, Present plans call for the calculation of conversion coefficients

for the ,Lp and shells for atomic numbers Z=5(10)95? for photon

energies K=0.05( .05) .2 ( . 2) 1 ,0( . 5)2.0, for both electric and magnetic

multipoles of multiplicities 2X

, x=1(1)5. In addition, a certain number

of calculations for the K-shell, especially for low energies, are to be

included. The computations proceed in three parts, of which Part I re- sults in the screening function for the Thomas-Fermi-Dirac statistical

model of the atom, and Part II in the eigenvalues for the ground state wave functions. Part III obtains the wave functions both for the ground

state and for the disturbed ‘state, based on the previously obtained eigenvalue, as well as the wave function of the photon, and evaluates the

convolution integrals resulting in the conversion coefficients.

Of this program, the following portions have been completed to date:

Part I has been carried out for all required atomic numbers.

The resulting screening functions are on file in manuscript form.

Part II was completed for the and shells for all desired

Z, and for the K shell for all except Z=95. It was subsequently found

possible to improve these values by a simple adaptation of the code for

part III; this is currently being done simultaneously with the computations

under Part III. For the shell, the method first employed in Part II

failed; these eigenvalues will be obtained by means of the adapted code

of Part III.

Part III has been completed for the shell for

Z=1 5,25,55,75,85, and for the L

p shell for Z=25, 55,85. Values for the K

shell have been completed for Z=25, 35, 55,75,85, for comparison with previously published result's. Each set of coefficients requires almost three

hours of automatic SEAC operation for its computation. During that time,

1 8 distinct codes, totalling 7,526 SEAC words, are read into the machine

from wire cartridges.

3. Mathematical Services

Note:- The tasks under Mathematical Services are arranged serially ac- cording 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.

TABLES OF THERMODYNAMIC PROPERTIES OF GASES

Task 0302-40-2606A9-5

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

Sponsor: National Advisory Committee for Aeronautics

Manager: F. L. Alt

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

Status: TERMINATED. This task is superseded by task 1102-40-5126/

53-27, p.. 31 .
Status of Projects 29

KAY TRACING

Task 0202-10-2308/50-13

Origin; MBS , Section 2.2 Authorised 3/1/50

Sponsor; " " Manager; R. K. Anderson

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

Status: CONTINUED. Completed ray tracing and spot diagram codes

have been "debugged" and improvements have been incorporated. The

Metrogon and Pianigon aerial camera lenses, two triplet lenses, and a heliar lens have been analyzed.

MOLECULAR STRUCTURE CALCULATIONS, II

Task 1102-40-5126/50-16

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

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

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

Status: TERMINATED. This task is superseded by task 1102-40-5126/

51-37.

LIQUID-VAPOR TRANSITION

Task 1 102-40-5126/51-22

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

Sponsor: " " Manager: I. Stegun

Full task descriptions Jan-Mar 1951 issue, p. 44

Status; TERMINATED. The formulation of new work is being considered

by the sponsor and will be reported under a new task when it is decided

upon.

MOLECULAR STRUCTURE, III

Task 1 102-40-5126/51-37

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

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

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

Status: CONTINUED. Computations were performed as requested
30 Status of Projects

SHOCK WAVE PARAMETERS, II

Task 1102-40-5126/51-38

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

Sponsor; " " Manager: I. A. Stegun

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

Status.: TERMINATED. Current work growing out of this task is being

performed under task 1102-40-5126/54-34, p. 38 .

PRECISE DETERMINATION OF THE PARAMETER OF DISPERSION

EQUATION FOR SEVERAL TYPES OF OPTICAL GLASS

Task 1102-40-5126/52-17

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

Sponsor: " " Manager: I. A. Stegun

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

Status: TERMINATED. No further work is currently contemplated.

SPHERICAL BLAST

Task 1102-40-5126/52-20

Origin: Naval Ordnance Laboratory Authorized 9/27/51

Sponsor: " " Manager: D. H. Jirauch

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

Status: CONTINUED. A final report is in preparation.

CALCULATIONS FOR d SPACINGS

Task 1102-40-5126/52-44

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

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

Full task description: Qct-Oec 1951 issue, p. 47

Status: CONTINUED. Programming for SEAC continued. Test cases

are being checked in.

GAS ADSORPTION BY HIGH POLYMERS

Task 1 102-40-5126/52-70

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

Sponsor: " " Manager: I. Stegun

Full task description: Jan-Mar 1952 issue, p. 59
Status of Projects 31 Status: TERMINATED,, The formulation of future work is being

considered by the sponsor and will be reported under a new task when it

is decided upon.

NEUTRON DIFFUSION, II

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

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

Office, (NBA)

Sponsor: " " " Manager: W. G. Hall

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

Status: CONTINUED, Computations are being performed on SEAC as requested, and the results are being sent to the sponsor as they are com- pleted.

STRENGTH OF WING COMPONENTS

Task 1102-40-5126/53-11

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

Aeronautics, Langley Field, Va.

Sponsor: " " " Manager: E. Marden

Full task description: Oct-Dec 1 952 issue, p. 54

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

RADIANT HEATING OF SOLIDS

Task 1102-40-5126/53-20

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

Sponsor: " 11 Manager: W. F, Cahill

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

Status: INACTIVE.

COMPUTATION OF THERMODYNAMIC FUNCTIONS

Task 1102-40-5126/53-2

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

Sponsor: " " Manager: E. Marden

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

Status: CONTINUED. Calculations were performed with revised codes

and the results were transmitted to the sponsor.
32 Status of Projects

STUDY OF TRICALCIUM ALUMINATE

Task 1102-40-5126/53-28

Origin: NBS , Division 9 Authorized 3/30/53

Sponsor: " " Manager: R. Anderson

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

Status:- TERMINATED, pending suitable formulation by the problem

originator.

DYNAMIC BEHAVIOR OF AIRCRAFT STRUCTURES

Task 1102-40-5126/53-29

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

Sponsor: " " Manager: I. Rhodes

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

Status: CONTINUED. Computations are being performed as requested,

and results are being transmitted to the sponsor.

ACOUSTICAL IMPEDANCES

Task 1102-40-5126/53-39

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

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

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

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

p. 55.

LORAN UNIVAC CODE

Task 1102-40-5126/53-41

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

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

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

Status: CONTINUED. The three routines are coded. The station

pair code has been checked and at present is being run by the sponsor.

The baseline code and the skywave correction and trainer code are awaiting

time on the UNIVAC for checking.
Status of Projects 33

RADIATION DIFFUSION

Task 1102-40-5126/53-51

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

Sponsors Armed Forces Special Weapons Project

Manager: W. G, Hall

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

Diffusion III)

Status; CONTINUED. The code for the first phase of this problem

has been modified to incorporate a recent development. A code treating

the spatial distribution of photons analytically is near completion.

HIGH TEMPERATURE PROPERTIES OF WATER

Task 1102-40-5126/54-4

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

Sponsor: Bureau of Ordnance, USM

Manager: J. H, Wegstein

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

Status: CONTINUED. K. W. Woolley has prepared a complete set of

formulas for the computation of the thermal functions Q,-(F^-EQ)/RT,

(H°-Eg)/RT, SO/R, and CO/R for one ionic species Oyxil* Theseu equations

,

apart from changes in constants, can be applied to any ion which has

only one bound electron. These rather elaborate formulas are being coded

for evaluation with the SEAC using the interpretive floating point rou- tines (Base 00).

Similar formulas are being prepared by Mr, Woolley for each

of the other ionic species of oxygen.

DISTRIBUTION OF NORMAL MODES OF VIBRATION OF CUBIC LATTICES, II

Task 371 1-60-0009/54-7

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

Sponsor: " Managers: W. F. Cahill

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

Status: CONTINUED. Two codes to compute the distribution of

normal modes of vibration for body-centered and face-centered cubic lattices have been checked out. Each was run for 13*824 points (N=48)

This involved generating 27,648 3x3 matrices, and finding their characteristic values.

NORMAL VIBRATIONS IN MOLECULES

Task 3711-60-0009/54-8

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

Sponsor: " " Manager: M. Newman

Full task description: Oct-Dec 1953 issue, p. 42
34 Status of Projects

Status: CONTINUED. During this quarter the eigenvaTues of a symmetric matrix of order 14- were obtained for the originator.

COMPUTATION OF VIBRATION MODES AND FREQUENCIES

Task 3711-60-0009/54-11

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

Sponsor: " " Managers: I. Rhodes, W. Cahill

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

Status: INACTIVE. For status to date, see Jan-Mar 1954 issue,

p. 40.

Publication: "Computation of vibration modes and frequencies on SEAC , "by W. F. Cahill and S. Levy (NBS Section 6.4); IN MANUSCRIPT.

SPACE AVERAGE OF TEMPERATURE DISTRIBUTION WITH A VOLUME DISTRIBUTION

OF SOURCES

Task 371 1-60-0009/54-12

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

Sponsor: " " Manager: E. Marden

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

Status: COMPLETED. The results have been transmitted to the

originator

AWARD OF PROCUREMENT CONTRACTS BY LINEAR PROGRAMMING

Task 1 102-40-5126/54-13

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

Sponsor: " " " Manager: H. Bremer

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

Status: CONTINUED. Ten bid evaluation problems were submitted by the

NYQMPA during the quarter. Thp answers were obtained on SEAC and sent back

usually within two days after receipt of the problem. One problem had to be solved using the general simplex method, all others used the special

"transportation problem" routine. The code for UNIVAC is now being

checked in cooperation with personnel at the Office of the Air Comptroller,

USAF.
Status of Projects 35

FLOW IN SUPERSONIC NOZZLE

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

Origins National Advisory Committee for Aeronautics, Authorized 12/16/53

Langley Field, Va.

Sponsor: " " " Managers: P. Henrici, E, Marden, G„ Hawkins

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

Status: CONTINUED, Computations for supersonic flow in nozzles

for four additional Mach numbers have been completed and transmitted to

the originator.

DEPOLYMERIZATION

Task 371 1-60-0009/54-17

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

Sponsor: " " Manager: R. Anderson

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

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

P. 5-4.

ENERGY LEVELS OF COMPLEX ATOMS

Task 371 1-60-0009/55-19

Origin: NBS, Section 4.1 Authorized 3/31/54

Sponsor: ” " Manager: W. G. Hall

Full task description: Jan-Mar 1954 issue, p. 4l

Status: CONTINUED. Coding is now being performed by the sponsor.

LOW TEMPERATURE PROPERTIES OF BORON COMPOUNDS

Task 1 102-40-5126/54-20

Origin: NBS, Section 3.2 Authorized 2/24/54

Sponsor: " " Manager: J. H. Wegstein

Full task description: Jan-Mar 1954 issue, p. 4l

Status: INACTIVE
36 Status of Projects

EXPERIMENTAL PROGRAM FOR MULTIPLE INPUT AND OUTPUT

Task 1102-40-5126/54-21

Origin; NBS , Section 12.3 Authorized 2/24/54

Sponsor; " " Manager; I; Rhodes

Full task description: Jan-Mar 1954 issue, p. 42

Status: CONTINUED. The combined sorting-file merging program was adapted for use in an experimental interconnection of SEAC and DYSEAC

and was demonstrated for the Bureau of Supplies and Accounts during the

quarter.

ENERGY DISTRIBUTIONS ON OPTICAL IMAGE

Task 371 1-60-0009/54-22

Origin: MBS, Section 2.2 Authorized 2/9/54

Sponsor: " n Revised 3/30/54

Manager: L. S. Joel

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

Status: CONTINUED. A code for the computation of I(j;,^) has been

completed and checked for (_§, r^)=( 1 , 1 ) , ( 1 ,6) , (o, 1 ) , (6,6) .

SOLUTION OF THE DIFFUSION EQUATION FOR ELECTRONS

Task 1102-40-5126/54-24

Origin: NBS , Division 4 Authorized 3/5/54

Sponsor: " 11 Managers: M. Abramowitz, A. Futterman

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

Status: INACTIVE. This problem awaits further specifications from

the originator. For status to date see Jan-Mar 1954 issue, p. 43.

DEFLECTED RADIATION FROM AN INFINITE LAMBERT PLANE

Task 1102-40-5126/54-25

Origin: Armed Forced Special Weapons Group Authorized 2/9/54

Sponsor: " " " Managers: P. Henrici, L. S. Joel

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

Status: CONTINUED. Values of \Wu, v) are being computed for

u,v=.25(. 25)16.0.
Status of Projects 37

EMF OF CELLS AT HIGH TEMPERATURE

Task 371 1-60-0009/54-27

Origin; NBS, Section 1.8 Authorized 3/5/54

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

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

Status; CONTINUED. The coding of this problem has been completed

and the electromotive forces of all chlorides and bromides have been

computed

,

LOVIBOND NETWORK TOR CIE SOURCE A

Task 3711-60-0009/54-28

Origin; NBS, Section 2.1 Authorized 2/28/54

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

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

Status: CONTINUED. The computation of the Lovibond networks for

CIE sources A and C has been completed. In addition, slight modifications

were made in the code to do special weekly calculations for the originator.

SPECTRUM ANALYSIS

Task 371 1-60-0009/54-30

Origin: NBS, Division 4 Authorized 3A/54

Sponsors- " " Manager: S, Prusch

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

Status: CONTINUED. Differences between all known and predicted

odd levels were each used to search for pairs of lines which confirm odd

levels and predicted new even levels. This was done for Tc I. Data for over 6,000 wave lengths were punched for the element ruthenium.

AIM CONDITIONING IN UNDERGROUND STRUCTURES

Task 1102-40-5126/54-3

3

Origin: NBS, Section 10.3 Authorized 2/24/54

Sponsor: " " Manager: S. Tsingou

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

Status: CONTINUED. The coding has been completed, and the com- putations are being performed in accordance with specifications of the

problem originator.
38 Status of Projects

EQUILIBRIUM CALCULATIONS FOR WATER

Task 1102-40-5126/54-34

Origin: Naval Ordnance Laboratory Authorized 5/31/54

Sponsor: '! " Manager: J. Wegstein

Objective: To solve systems of simultaneous transcendental equations

containing 40 or more unknowns describing the equilibrium among the, species

resulting from high-temperature decomposition of water.

Background: Thermodynamic properties of various gaseous mixtures

are needed for temperatures up to 1,000, 000°K and for gaseous densities

of 0.5 to 2.0. For this range of temperatures and densities the familiar

ideal-gas treatment is not applicable. It is necessary to account for the

imperfect gas behavior and for the long distance forces between ionic

particles and electronic excitation. In neither case a satisfactory theoretical approach is known, but various approximation procedures may be

applied. A flexible procedure is necessary to account for the different

approximation methods.

Status: NEW. The problem has been coded and a number of cases

have been solved and submitted to the originator. Modifications of the

original formulation are now being considered.

THERMAL STRESSES IN STRUCTURES

Task 1102-40-5126/54-35

Origin: NBS , Section 6.4 Authorized 6/29/54

Sponsors " " Manager: W. Hall

Objective: To consider the effect of thermal transients on stress

distribution in metal beams, especially beams of materials used in modern

aircraft structures. A method for predicting temperature distribution due

to conduction, radiation, and convection must first be found. Temperature dependent changes in elastic and thermal properties of the material

and in the thermal resistance of joints are to be taken into account. The

investigation is aimed at constructing mathematical models which will

conform realistically to the physical phenomena.

Background: The large thermal stresses developed in high speed

aircraft and missiles make it desirable to develop methods of thermal

stress and temperature distribution analysis which will be useful as a basis for rational engineering design. It is believed that the use of

numerical methods in conjunction with high-speed automatic computing

machines may be successfully used in such analyses.

Status: NEW. Some results have been submitted to the sponsor.
Status of Projects

39

VELOCITY OF LIGHT

Task 3711-60-0009/5^-36

Origin”. NBS , Section 11.3 Authorized 3/16/5*+

Sponsors " " Manager: P. Rabinowitz

Full task descriptions Jan-Mar 195*+ issue, p. 47

Statuss CONTINUED. Results are transmitted to the originator

whenever required. A double precision floating matrix inversion routine,

handling matrices of order up to 85, was coded for checking purposes.

CORRELATION OF IONOSPHERIC DATA

Task 3711-60-0009/54-37

Origin: NBS, Section 1 4 . 1 Authorized 3/12/54

Sponsor: " " Managers: J. H. Wegsteio, H. H. Howe

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

Status: COMPLETED. The computations were made successfully with

the three-address SEAC. Besides developing several useful subroutines,

H. H» Howe made several suggestions for improving the logic of the threeaddress system. The results of the computations were transmitted to the

originator.

COMPRESSIBILITY FACTORS OF DRY AIR

Task 371 1-60-0009/54-38

Origin: NBS, Section 3.2 Authorized 3/29/54

Sponsor: " " Manager: M. Paulsen

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

Status: INACTIVE. For status to date, see Jan-Mar 1954 issue,

p. 48.

FREQUENCY CORRELATION

Task 1 102-40-5126/54-41

Origin: NBS, Division 6 Authorized 6/29/51

Sponsor: " " Manager: L. S. Joel

Objective: To evaluate

R = 0.200 + 0.800 2Z cos ¥ [ (k«-k)r. .] i=1 j<i A

ij where k ' , k, are vectors:
4o Status of Projects

k' - (cos O', sin O'), k = (cos 0, sin 0)

r ij- r i“ r j> r r (x

i >y i

);

input data: k or 0, k' or 0', A, r^.

Background: The problem is concerned with the evaluation of cor- relation coefficients for an array of microphones in a plane exposed to a sound wave. Status: The first problem has been completed except for two values

out of 1100. ’For the initial run R was evaluated for n=5* 0=-w/6,

0 ,= O(a/5O)2 tt

, for 11 values of A.

RESOLUTION CORRECTION FOR SCINTILLATION SPECTROMETER

Task 1102-40-5126/54-42

Origin: NBS , Section 4.13 Authorized 6A/54

Sponsor: " " Manager: M. Newman

Objective: To invert a number of matrices of orders 9* 10 and 19}

and to solve systems of equations having these matrices as coefficient

matrices.

Background: The energy resolution of a scintillation spectrometer

is strongly dependent on the quantum energy of the radiation. This fact causes a distortion of the continuous spectra measured by this instrument

and necessitates a correction. This correction is made by solving systems

of linear equations relating the true spectral distribution to the measured spectral distribution by means of a correction matrix embodying ex- perimental resolution data.

Comment: This task was proposed by M. Ehrlich, NBS Section 4.13.

Status: NEW. Thirteen systems of equations of order 10, three of

order 9} and two of order 19 have been solved. In addition, the coefficient matrices for these systems have been inverted.

CHARACTERISTICS OF CONDUCTING RESISTORS

Task 1 102-40-5126/54-43

Origin: NBS, Section 12.1 Authorized 6/29/54

Sponsor: " " Manager: R. K„ Anderson

Objective: For each test resistor, experimental data gives values

of I (direct current in milliampe res) and corresponding jp (average of

open circuit noise voltage squared per cycle per second). It is required

to compute values of constants a and b in order to determine the best

least-squares fit of

log P = a + b log I,
Status of Projects 4i

and to obtain further quantities needed for the analysis of variance.

Background: It is generally known that composition resistors,

when conducting, generate a fluctuating potential which under normal

conditions is greater than that due to thermal agitation. It is the purpose of this task to study commercial and experimental resistors and to

develop standard test procedures and specifications for military use. Tomment: This task was proposed by G. T. Gonrad (NBS Section 12.1).

Status: NEW. Programming is in progress.

COMPUTATION OF TRANSIENT HEAT FLOW PROBLEMS

Task 1 102-40-5126/54-45

Origin: NBS, Section 10.2 Authorized 6/4/54

Sponsors: " " Manager: W. F. Cahill

Objective: To obtain data on transient heat flow through various

geometrical configurations which closely approximate structures tested

by the NBS Fire Protection Section (10.2).

Background: For many years the Fire Protection Section has been

testing the heat resistance of structures such as columns, floors, and

panels. The necessity for theoretical studies of heat flow through these

structures has been recognized by the staff of the section for a number

of years, but many important calculations were beyond the realm of possibility until the invention of the high speed digital computer.

Status: NEW. A code, using finite differences, to compute twodimensional heat flow has been composed and checked out.

STUDY IN OPTIMIZATION OF POWER OUTPUT

Task 1102-40-5126/54-46

Origin: Raytheon Manufacturing Company Authorized 5/18/54

Sponsor: U. S. Corps of Engineers

Manager: L. Gainen

Objectives To solve several linear programming problems by the

simplex method.

Background: These problems are based on a linear approximation to the question of determining desirable levels of storage and rates of

change of storage of water in a sequence of dams for the purpose of

optimizing the output of electrical power.

Status: NEW. Two problems were submitted by the originator for

solution, one involving the use of four dams and a second assuming only

three existing dams. The first of these problems, solved on the SEAC by

the simplex method as a set of 33 equations in 1 52 variables, is complete

and the results have been transmitted to the originator. The second problem is being computed as a set of 22 equations in 1 1 4 variables. These

problems are being solved in dualized form because of a limitation on the column size imposed by the SEAC coding, and because the dual allows

use of a natural basis. Forty simplex iterations were required to compute

the 33 by 1 52 set.
III. STATISTICAL ENGINEERING LABORATORY

(Section 11.3)

1 . Fundamental Research in Mathematical Statistics

BIBLIOGRAPHY AND GUIDE TO STATISTICAL LITERATURE

Task 1 103-10-1 1 07A9- la

Origin: NBS Authorized 1/9 A-9

Manager: L. S. Denning

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 1955- issue, p. 49.

MANUAL ON FITTING STRAIGHT LINES

Task 1103-10-1107/50-2

Origins NBS Authorized 3/1/50

Manager: F. S. Acton

Full task descriptions Jam-Mar 1950 issue, p. 42

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

P. 58.

TABLES TO FACILITATE DRAWING RANDOM SAMPLES

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

Origin: NBS Authorized 7/1/50

Managers: C. Eisenhart and L.-S. Deming

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

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

p, 64.
Status of Projects 5-3 MISCELLANEOUS STUDIES IN PROBABILITY AND STATISTICS

Task 1103-10-1107/51-2

Origin: NBS Authorized 7/1/50

Manager: C. Eisenhart

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

Status: CONTINUED,

Publications: (1) "On- strongly continuous stochastic processes,"

by E. Lukacs; Sankhya, JJi, 219-228 (Mar, 195*+). (2) "A property of the

normal distribution," by E„ Lukacs and E„ P. King; Ann, Math. Stat. 25,

389-391+ (June 1955-). (3) "Nonnegative trigonometric polynomials and certain

rational characteristic functions," by E. Lukacs and 0. Szasz; J. Res. NBS

52, 153-160 (Mar. 1955-). (5-) "Certain Fourier transforms of distribution,

II,"' by E. Lukacs and 0. Sz^sz; Canadian J, Math. 6, 1 86-1 89 (Mar, 1955-).

(5) "On optimum grouping in one-criterion variance components analysis,"

by E. P. King; accepted for publication in the Journal of the American

Statistical Association. (6) "Tables of inverses of finite segments of

the Hilbert matrix," by I. R. Savage and E. Lukacs; to be included in NBS

Applied Mathematics Series 39, "Contributions to the Solution of Systems

of Linear Equations and the Determination of Eigenvalues." (7) "Bounds

on a distribution function which are functions of moments to order four,"

by M. Zelen; accepted for publication in the Journal of Research of the NBS.

(8) "Time-discrete stochastic processes in arbitrary sets, with applications

to processes with absorbing regions and to the problem of loops in Markoff

chains," by D, van Damtzig; accepted for publication (in French) in Annales

de I'Institut Henri Poincar^ (Paris). (9) "The use of extreme-value

methods in engineering problems," by J. Lieblein; submitted to a technical

journal. (10) "On the variances and covariances of order statistics from

the Weibull distribution," by J. Lieblein; submitted to a technical journal.

(11) "Two early papers on the relation between extreme-values and tensile

strength," by J. Lieblein; submitted to a technical journal.

LAW OF PROPAGATION OF ERROR

Task 1103-10-1107/52-1

Origin: NBS Authorized 6/23/51

Managers: C. Eisenhart and I. R. Savage

Full task description; Juiy-Sept 1951 issue, p. 65

Status: INACTIVE. For status to date see Juiy-Sept 1951 issue,

P. 65.

PROCEDURES OF NON-PARAMETRIC STATISTICS

Task 1103-10-1107/52-2

Origin: NBS Authorized 9/17/51

Manager: I. R. Savage

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

Status: CONTINUED. The work of I. R. Savage described in the preceding issue has been completed and has been prepared as an NBS report.

After suitable revision it will be submitted to a technical journal.
44 Status of Projects

Publications: (1) "Most probable rank orders," by I. R. Savage;

withdrawn from publication „ (2) "Easily used simultaneous confidence

limits for a line," by W. S. Connor; submitted to a technical journal.

STUDIES IN THE MATHEMATICS OF EXPERIMENT DESIGN

Task 1103-10-1107/53-1

Origin: NBS Authorized 10/15/52

Manager: W. S. Connor

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

Status: CONTINUED.

Publications: (1) "New experimental designs for paired observations,

by W. S. Connor and W. J. Youden; accepted for publication by Journal of

Research of the NBS. (2) "Analysis for some partially balanced incomplete

block designs having a missing block," by M. Zelen; accepted for publication in Biometrics. (3) "A note on partially balanced designs," by

ML Zelen; accepted for publication in the Annals of Mathematical Statistics

(4) "Partially balanced incomplete block designs with two associate

classes and two treatments per block," by W. H. Clatworthy; submitted to

a technical journal. (5) "On the enumeration of partially balanced designs with two associate classes," by W. H. Clatworthy; submitted to a technical journal. (6) "Covariance analysis for incomplete block designs,"

by M. Zelen; IN MANUSCRIPT.

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 and I. Caaeron

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

Status: CONTINUED. Activity under this project fell into two main

categories:

A. Design of Experiments: In an experiment to determine the

relative performance of 120 samples of cement on a gain in moisture test,

only 15 samples could be accommodated in a run. The problem was to arrange

samples to be assigned to each run so that the statistical analysis would

largely remove the effects of different conditions from run-to-run from

the average observed rates of gain for the samples. This was achieved

by the use of a linked block design.

B. Development or Selection of the Appropriate Method for

Analysis and Interpretation of Data: A method for providing mutually

consistent conversion ratios for the activities of four national radium

standards was devised. A further check upon the work was provided by

comparing these adjusted ratios with those obtained from the weights of

the samples to obtain the best estimates of a linear law relating activity

to the mass of the sample in a situation where both measurements are
Status of Projects 45

subject to errors. Publications: (1) "Probability limits for the average chart when

process standards are unspecified," by E. P. King; Industrial Quality

Control 62-64 (May 1954). (2) "A comparison of four National Radium

Standards, II," by W. J. Youden and W, S. Connor; IN MANUSCRIPT.

STATISTICAL ASPECTS OF NBS ADMINISTRATIVE OPERATIONS

Task 3737-60-0002/52-1

Origin: NBS Authorized 10/1/51

Manager: I. R. Savage

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

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

P. 68.

STATISTICAL SERVICES FOR COMMITTEE ON SHIP STEEL, NRC

Task 1103-40-5105/52-1

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

Sponsor: " " Manager: W. J. Youden

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

Status: CONTINUED. The discriminant function for classifying steel

pi ates as source, through, or end plates is being recomputed because of

the redetermination of ferrite grain size by a conventional method of

measurement

The statistical analysis carried out on data obtained by thirteen

cooperating laboratories in an interlaboratory study of notching and

testing effects is being described for publication in a suitable technical

journal.

RESEARCH IN APPLICATIONS OF MATHEMATICAL STATISTICS TO

PROBLEMS OF THE CHEMICAL CORPS

Task 1103-40-5118/52-1

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

Dept, of the Army

Sponsor: " " " Manager: C. Eisenhart

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

Status: CONTINUED. A tabulation of fractional replications of the

2

n series of factorial designs arranged in incomplete blocks was nearing

completion at the end of this quarter. A report containing papers presented at the Area Weapons Conference was prepared.
46 Status of Projects

STATISTICAL ANALYSIS AND DESIGN OF EXPERIMENTS

FOR THE U. S. GEOLOGICAL SURVEY

Task 1 103-40-5140/54-1

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

Interior

Sponsor: " " " Managers: C. Eisenhart and W. J. Youden

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

Status: CONTINUED. The study of a method of analysis of orientation

data was continued. In addition correspondence was exchanged with

individual members of the U. S. Geological Survey on such topics as design

of experiments for determining precision and accuracy of certain chemical

laboratory test methods and method of sampling for some lithology studies.

STATISTICAL ANALYSIS OF BALL BEARING FATIGUE DATA

Task 1 1 03—40— 5 1 4 5/54—

Origin: American Standards Association Authorized 1/13/54

Sponsor: " " Pdanager: J. Lieblein

Full task description: Jan-Mar 1954 issue, p. 54

Status: CONTINUED. (1) At a meeting held May 28, 1954, in New

York City between the Sub-Committee and Bureau representatives certain

aspects of the nature of the ball bearing data at hand were clarified,

which made it possible to identify statistically homogeneous subsets for

study. (2) Research is being carried out on a method of deriving improved

life estimates from the test groups of ball bearing data. The method is

designed for use with "censored" data, in which only the lowest k out of

n observations are available. For the case of ball bearing fatigue data

this corresponds to a test group where the test is discontinued before

all the bearings in the group have failed, so that the endurance lives of

those remaining intact are necessarily unavailable. The method based on order statistics is derived from that developed in National Advisory

Committee for Aeronautics Technical Note 3053? by Julius Lieblein, entitled

"A new method of analyzing extreme-value data". After the necessary

computations are completed the applicability of the method will be tested

on the data at hand.
iv„ MACHINE DEVELOPMENT LABORATORY

(Section 1 1

, 4

)

in cooperation with

ELECTRONIC COMPUTER SECTION

(Section 12.3)

THE BUREAU OF THE CENSUS COMPUTING MACHINE

Task 11 0*4-34-51 07A7-1

Origin: The Bureau of the Census Authorized 7/1/47

Sponsor: " " Full task description: Apr-June 1949 issue, p. 58

Status: TERMINATED. Stockpiling of replacement parts for UNIVAC

System No. 1, under Contract CST-10279, has been completed with the exception of a few minor items to be delivered within the next quarter. With

the delivery of the latest versions of the maintenance and operating instruction books, the technical manual requirements of the contract have

been met. It is expected that the contract will be terminated by September,

NATIONAL BUREAU OF STANDARDS EASTERN AUTOMATIC COMPUTER (SEAC)

Task 1 104-34- 5107A9-1

Origin: NBS

Sponsor: Office of the Air Comptrolle

Full task description: Apr-June 1949

Status: CONTINUED. During the

has been operated as a general Bureau

of 7

8

°/o during scheduled operation,

as follows:

Authorized 12/1 5A8

USAF

.ssue, p. 59

past quarter, the SEAC installation

facility with an overall efficiency

The computation time was distributed

Total time available for computation 1 ,239 hours

Good operating time 1 ,005 " Idle in order. 5

" Down-time. ............. 230 "

In addition the SEAC was utilized 360 hours for engineering and preventive

maintenai$©e . "Down-time" is defined as the time during which the computer

was under repair or was rerunning a previous ly-run problem found to be in error as a result of machine malfunction. Preliminary planning and en- gineering layout have been under way in preparation for moving the SEAC

installation to a permanent location, which is scheduled to occur in the

fall.

47
Lectures and Symposia

Note: In general, copies of papers or talks listed in this section are not available from the National Bureau of Standards. If and when a paper

is to be published, it will be listed in the section of this report on Publication Activities.

Numerical Analysis Colloquium

(Los Angeles, California)

MORREY, C. B. (University of California, Berkeley). On the derivation

of the equations of hydrodynamics from statistical mechanics.

July 12.

Informal Seminar on Problems Relating to Numerical Analysis

(Los Angeles, California)

ZIEGLER, J.R. (U.C.L.A.) Use of SWA! for item analysis of psychological

instruments. April 1. WOODS, R. SWAC computations of the scattering of protons, using the

optical model of the nucleus. April 6.

FULLER, F.E. (Lockheed Aircraft Corporation). Some considerations of the

flutter problem (with 10-minute movie of rocket flutter). April 8.

0RC HARD-HAYS , W. (RAND Corporation). Computational experience in solving

linear programming problems. April 13.

PHILIPSON, L. Asymptotic character of the solutions of a class of or- dinary linear differential equations depending on a parameter.

April 1 5. FORSYTHE, G.E. On direct methods of obtaining the characteristic values

of a matrix. April 20.

GROSCH, H.R.J. (General Electric Company, Cincinnati). Description of

General Electric computing activities. April 21.

SOKOLNIKOFF, I.S. (U.C.L.A.) Boundary value problems in elasticity, I. May 4.

POPE, D.A. On direct methods of solving problems in the calculus of var- iations. May 6.

HASSIG, H.J. (Lockheed Aircraft Corporation). Related digital and analog

methods of flutter analysis. May 11.

SOKOLNIKOFF, I.S. (U.C.L.A.) Boundary value problems in elasticity, II.

May 13.

48
Lectures and Symposia 49

TOMPKINS, C. B e Problems in numerical analysis related to the Morse theory

in the calculus of variations. May 18,

UELLMAN, Re (RAND Corporation). Some problems in the theory of dynamic

programming. May 20.

KLIEFORTH, H. (U.C.L.A.) SWAC computation of flight paths from phototheodolite measurements. May 25.

LEVIN, E. Can .a computing machine play chess? May 27.

SALZER, J.M. (Magnavox Company). Convergence and error studies by transform

methods. June 1.

SPROWLS, R.C. (U.C.L.A.) Use of SWAC in the stochastic determination’ of

cash balances. June 3.

FORSYTHE, G.E. SWAC computation of all semi-groups of order 4. June 22,

LOPEZ, G. Review of a new method by von Neumann for solving linear games.

June 24.

THOMAS, T. Y. (Indiana University), A discussion of Luder's bands,

June 29.

Special Lectures

(Los Angeles, California)

HALL, M.„Jr. (Ohio State University) The Burnside problem. April 21.

RHODES, Ida (National Bureau of Standards, Washington). Recent developments in high speed computing. April 2.

HOLLANDER, F. li. Ten lectures on coding for the SWAC. May 1954.

SOUTHARD, T.H., F. H. HOLLANDER, aid R. MORGAN. SWAC Coding Course,

beginning June 21, 1954.

Linear Programming Seminar

HOFFMAN, A.J. An application of linear programming to a combinatorial

problem. March 31

,

JACOBS, W. (U. S. Air Force), The caterer problem. April l4.

FAN, K. Existence theorem for linear inequalities. April 28.

MOTZKIN, T. S. Linear inequalities in Statu Nascendi. May 5. JOSEPH, J. (U. S, Air Force). Some military applications of linear programming. May 12.

GASS, S. (U. S. Air Force). Parametric minimization. May 12 r GAINEN, L., H. FISCHER (Captain, U. S. Air Force), and J. NATRELLA (U.S.

Air Force). Symposium on computational experience, I. May 26.
Lectures and Symposia 50

KELLEY, J. (George Washington University). An extended transportation

problem. June 9.

GAINEN, L. , A. J. HOFFMAN, and L. S. JOEL. Symposium on computational

experience, II. June 23.

Papers and Invited Talks

Presented by Members of the Staff

at Meetings of Outside Organizations

ANTOSIEWICZ, H. On nonlinear difference equations. Presented at the

meeting of the American Mathematical Society, Chicago, 111.,

April 21 -May 1.

EISENHAKT, C. The Student-Fischer revolution in modern statistics. Presented at a Colloquium in Statistical Design, sponsored by the

Naval Ordnance Laboratory in collaboration with the Office of

Naval Research and the Bureau of Ordnance, held at the Naval

Ordnance Laboratory, White Oak, Md. , April 1*+.

FORSYTHE, G. E. SWAC computes all 126 distinct semigroups of order 4.

Presented at a meeting of the American Mathematical Society,

Portland, Oregon, June 19.

HUSKEY, H. D. (1) Undergraduate curriculum in machine computation.

Presented at a Conference on Training Personnel for the Computing

Machine Field, held at Wayne University, Detroit, Michigan,

June 22-23. (2) Automatic coding, a progress report. Presented

at the meeting of the Association for Computing Machinery,

Ann Arbor, Mich. June 2b, 195*+.

TAUSSKY-TODD, 0. Group theory and football forecasting. Presented at a meeting of Sigma Xi, Washington, D. C. , May 24.

TODD, J. (1) Motivation for working in numerical analysis. Presented

at a Symposium on Applied Mathematics held by the Office of

Ordnance Research, held at Chicago, 111., April 29. (2) Effect

of high speed computers on the training of mathematicians.

Presented at a Conference on Training Personnel for the Computing

Machine Field, Wayne University, Detroit, Mich., June 22-23.

TOMPKINS, C. B. (1) Considerations involved in the formulation of a numerical attack on a game problem to be solved on a high speed

automatic computer. Presented at a Mathematic® Seminar, University of Rochester, Rochester, N. Y. , April 27. (2) A theorem

relating to the Kaczmarz Projection Scheme for solving linear

equations and inequalities. Presented at a Logistics Research

Project Seminar, Washington, D. C., April 29. (3) Some small

matrix games of attrition and their numerical solution. Presented at the Princeton University Games Seminar, Princeton, N.J.,

May 7. (4) Computers in industrial research. Presented at the

Fifth Anneal Conference on Industrial Research under the auspices

of Columbia University at Harriman, N. Y., June 11.

WASOW, W. R. Discrete approximations to elliptic differential equations.

Presented at a meeting of the American Mathematical Society,

Yosemite, California, May 1.
Lectures and Symposia cj-j YOUDEN, W. J. Principles of experimental design. Presented at a meeting

of the Milwaukee Chapter of the American Chemical Society,

Milwaukee, Wis . , April 23.

Papers presented at the meeting of the Mathematical Association of America,

held at the University of Maryland, College Park, Md., May 1: CAMERON, J., and M. NEWMAN. Pseudo random numbers for high speed computers.

HOFFMAN, J. Some practical and theoretical applications of the transportation problem.

Papers presented at the Annual Meeting of the Virginia Academy of Science,

held at the University of Virginia, Charlottesville, Va„, May 7

’

CONNOR, W. S. On a problem in fitting a straight line.

LIEBLE1N, J, On a new method of analyzing extreme-value data.

Papers presented at the meeting of the Association for Computing Machinery,

held at the University of Michigan, Ann Arbor, Michigan, June 22-25s

ABRAMOWITZ, M. Operation of a computation laboratory.

DAVIS, P „5 and P. RABIN0WITZ. A multiple purpose orthonormalizing code

and its use. WEGSTE1N, J, H. A general purpose pseudo-code.

Papers presented at the meeting of the American Mathematical Society,

New York, N. Y„, April 23~24;

ABRAMOWITZ, M, , and H. ANTOSIEWICZ. Coulomb wave functions in the transition region,

GOLDBERG, K. Log(e xy

) in a free associative ring.
Publication Activities

1 . PUBLICATIONS WHICH APPEARED DURING THE QUARTER

1.3 Technical Papers

“oo _ 2 / / \

(1)

Evaluation oi the integral e

51 u,; du.

Jo

J. Math. Phys. 3£? 188-192 (July-Oct. 1953).

M. Abramowitz.

(2) Regular and irregular Coulomb wave functions expressed in terms

of Bessel-Clif f ord functions. M. Abramowitz. J. Math. Phys. 33 « 111-116 (Apr. 195*+)

(3) On the practical evaluation of integrals. M. Abramowitz. J. Soc,

Indust, App. Math. 2, 20-35 (Mar. 19 54),

(4) On a theorem of Qstrowski and Taussky. R. Bellman (RAND Corporation) and A. J. Hoffman. Arch, Math. £? 123-127 (195*+).

(5) On representations and extensions of bounded linear functionals

defined on classes of analytic functions. P. Davis and J. L. Walsh.

Trans. Amer. Math. Soc. 2*2? 190-206 (195*+).

(6) Some remarks ojn commutators of matrices, K. Fan. Arch. Math.

102-107 095*+)

(7) A numerical solution of Schroedinger 1 s equation in the continuum.

W. Futterman? E. Osborne? and D. S. Saxon. J. Res. NBS 52 . 259-264 (May 1954) , MP24-98.

(8) Application of variational methods to intermediate and high energy

scattering. E. Gerjuoy and D. S. Saxon. Physical Rev. 94 , 478-491 (Apr. 15, 1954).

(9) The representation of integers by binary quadratic rational forms.

K. Goldberg? M. Newman? E. G. Straus? and J. D. Swift. Arch. Math.

It 12-18 (1954).

(10) Ueber die Funktionen von Gegenbauer. P. Henrici. Arch. Math. 5?

92-98 (1954).

(11) Probability limits for -the average chart when process standards

are unspecified,. E, P. King. Xnd. Dual. Control 10? 62-64

(May 1954).

(12) On strongly continuous stochastic processes. E. Lukaes, Sankhya

13, 219-228 (Mar. 1954),

(13) A property of the normal distribution. E. Lukaes and E. P. King.

Ann. Math. Stat. 2£, 389-394 (June 1954).

(14) Certain Fourier transforms of distributions (II). E. Lukaes and

0. Szasz, Canadian J. Math. 6? 1 86—1 89 (1954).

52
Publication Activities 53

0 5) Nonnegative trigonometric polynomials and certain rational characteristic functions. E. Lukacs and 0. Szasz. J. Res. NBS 52,

153-160 (Mar. 1954), MP2484.

(16) On nearly triangular matrices. A. Ostrowski. J. Res. NBS 52 . 319-345 (June 1954), RP2509.

(17) Characteristic roots of quaternion matrices. 0. Taussky, Arch.

Math. £, 99-101 (1954).

(18) Einschliessung von Eigenwerten hermitescher Matrizen nach dem

Abschirittsverf ahren. H. Wielandt. Arch. Math. jj, 108-114(1954).

2. MANUSCRIPTS IN THE PROCESS OF PUBLICATION JUNE 30, 1954

2.1 Mathematical Tables

(1) Table of sine and cosine integrals for arguments from 10 to 100.

NBS Applied Mathematics Series 32. (A reissue of NBS Mathematical

Table MT13.) In press, Government Printing Office.

(2) Table of the Gamma function for complex arguments. NBS Applied

Mathematics Series 34. In press, Government Printing Office.

(3) Tables of functions and of zeros of functions. Volume I of Collected short tables of the Computation Laboratory. NBS Applied

Mathematics Series 37. In press, Government Printing Office.

(4) Tables of the error function and its derivative. NBS Applied

Mathematics Series 4l . (A reissue of NBS Mathematical Table 8,

Tables of probability functions, vol. I.) In press, Government

Printing Office.

(5) Tables of sines and cosines for radian arguments. NBS Applied

Mathematics Series 43. (A reissue of NBS Mathematical Table 4.)

In press, Government Printing Office,

(6) Tables of the inverses of finite segments of the Hilbert matrix.

I. R. Savage and E. Lukacs. To be included in Contributions to the solution of systems of linear equations and the determination

of eigenvalues, NBS Applied Mathematics Series 39.

(7) Table of salvo kill probabilities for square targets, NBS Applied

Mathematics Series 43, In press, Government Printing Office.

2.3 Technical Papers

(1) Coulomb wave functions in the transition region. M, Abramowitz

and H. Antosiewicz. Submitted to a technical journal.

(2) Approximate method for rapid Lor an computation. M. Abramowitz,

D. H. Call, and J. C. Mathews. Submitted to a technical journal.

(3) Coulomb wave functions along the transition line. M, Abramowitz

and P, Rabinowitz. Accepted for publication in The Physical

Review.

( 4 ) The relaxation method for linear inequalities. S. Aqmon. Accepted for publication in the Canadian Journal of Mathematics.

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

Accepted for publication in the Pacific Journal of Mathematics.
54 Publication Activities

(6) Asymptotic solution of linear differential equations with a parameter. H. Antosiewicz. Submitted to a technical journal.

(7) On nonlinear differential equations of the second order with

integrabie forcing term. H. Antosiewicz. Submitted to a technical

journal.

(8) On the differential equation x + k(f(x) + g(x)x)x = ke(t).

H. A. Antosiewicz. Submitted to a technical journal.

(9) On a certain integral involving Bessel functions. H. Antosiewicz.

Submitted to a technical journal.

(10) A representation for solutions of analytic systems of linear

differential equations. H. A. Antosiewicz and M. Abramowitz ., Accepted for publication in the Journal of the Washington Academy

of Sciences.

(11) Some implications of Liapunov's conditions for stability. H. A. Antosiewicz and P. Davis. Accepted for publication in the Journal

of Rational Mechanics and Analysis.

(12) On the enumeration of partially balanced designs with two associate classes. W. H. Clatworthy. Submitted to a technical

journal.

(13) 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

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

Submitted to a technical journal.

(15) Experimental designs for paired observations. W. S. Connor and

W. J. Youden. Accepted for publication in the Journal of Research

of the NBS.

(

6 ) 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 Anmales de I'Institut Henri Poincare

(Paris )

(17) Linear functional equations and interpolation series. P. Davis.

Accepted for publication in the Pacific Journal of Mathematics.

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

Accepted for publication in the Pacific Journal of Mathematics.

p

(19) Some L aspects of Faber polynomials. P. Davis and H. Pollack

(Bell Telephone Laboratories). Accepted for publication in the

Duke Mathematical Journal.

(20) A multiple purpose orthonormalizing code and its uses. P. Davis,

and P. Rabinowitz. Submitted to a technical journal.

(21) On the estimation of quadrature errors for analytic functions.

P. Davis and P. Rabinowitz. Submitted to a technical journal.

(22) Some SEAC computations of subsonic fluid flows by Bergman's

method of integral operators. P. Davis and P. Rabinowitz. To be

an appendix in the volume, "Bergman's Operator Method," by

M. Z. v. Krzywoblocki (University of Illinois).
Publication Activities 55

(23) Some sampling results on the power ol nonparametric tests

against normal alternatives. W. J, Dixon (University oi Oregon)

and D. Teichroew. Submitted to a technical journal.

(24) Turbulent flow in shock tubes of varying cross-section. R. F. Dressier. Accepted by the Journal of Research of the NBS.

(25) On the optimal character of the (s,S) policy in inventory theory.

A. Dvoretzky, J. Kiefer, and J. Wolfowitz. Submitted to a technical journal.

(26) Inequalities for eigenvalues of Hermitian matrices. K, Fan.

To be included in Contributions to the solution of systems of

linear equations and the determination of eigenvalues, NBS

Applied Mathematics Series 39.

(27) Lower bounds for the rank and location of the eigenvalues of a matrix. K. Fan and A. J. Hoffman. To be included in Contributions to the solution of systems of linear equations and the

determination of eigenvalues, NBS Applied Mathematics Series 39.

(28) Some metric inequalities in the space of matrices. K. Fan and

A. J. Hoffman. Accepted for publication in the Proceedings of

the American Mathematical Society.

(29) A determinants! inequality. K. Fan and 0. Taussky. Submitted

to a technical journal.

(30) Punched-card experiments with accelerated gradient methods for

linear equations. A, I. Forsythe and G. E. Forsythe. To be

included in Contributions to the solution of systems of linear

equations and the determination of eigenvalues, NBS Applied

Mathematics Series 39.

(31) Asymptotic lower bounds for the frequencies of polygonal membranes. G. E. Forsythe. Accepted for publication in the Pacific

Journal of Mathematics.

(32) Asymptotic lower bounds for the fundamental frequency of convex membranes, G. E. Forsythe. Accepted for publication in the

Pacific Journal of Mathematics,

(33) Practical solution of linear equations and inversion of matrices.

L. Fox. To be included in Contributions to the solution of

systems of linear equations and the determination of eigenvalues,

NBS Applied Mathematics Series 39.

(3^-) On the solution of the caterer problem. J. W. Gaddum, A. J. Hoffman, and D. Sokolowsky, Submitted to a technical journal.

(35) Linear programming in bid evaluation. L. Gainen, D. J. Honig,

and E. D, Stanley. Accepted for publication in Logistics Research Quarterly.

( 36 ) Variational principles for the acoustic field. E, Gerjuoy and

D. S. Saxon. Accepted for publication in The Physical Review.

(37) Equations of physics in general Newtonian space-time. A. N.

Gleyzal. Submitted to a technical journal.

(38) The use of extreme-value methods in engineering problems. E, J. GumbeX and J. Lieblein. Submitted to a technical journal.
56 Publication Activities

(39) Iterative methods of solving linear problems on Hilbert space.

R, M. Hayes. To be included in Contributions to the solution of

systems of linear equations and the determination of eigenvalues,

NBS Applied Mathematics Series 39.

(40) On certain series expansions involving Whittaker functions and

Jacobi polynomials. P. Henrici,. Accepted for publication in the

Pacific Journal of Mathematics.

(41 ) On generating functions of the Jacobi polynomials. P. Henrici.

Accepted for publication in the Pacific Journal of Mathematics.

(42) On helical springs of finite thickness. P. Henrici. Accepted

for publication in the Quarterly Journal of Applied Mathematics.

(43) On a combinatorial theorem. A. J. Hoffman. Submitted to a technical journal.

(44) Smooth patterns of production. A. J. Hoffman and W. Jacobs.

Submitted to a technical journal.

(45) On the number of absolute points of a correlation. A. J. Hoffman,

M. Newman, E. G. Straus, and 0. Taussky. Submitted to a technical

journal.

(46) On optimum grouping in one-criterion variance components analysis.

E. P. King. Accepted for publication in the Journal of the

American Statistical Association.

(47) Contractibility and convexity. H. W. Kuhn. Accepted for publication in the Proceedings of the American Mathematical Society.

(48) On certain character matrices. D. H. Lehmer. Submitted to a technical journal.

(49) Indentation pressure of a smooth punch. E. Levin. Accepted for

publication in the Quarterly of Applied Mathematics.

(50) Acoustic radiation pressure on a circular disk. H. Levine. To

appear in the P roceedings. of the Fifth Symposium on Applied

Mathematics of the American Mathematical Society, held in Pittsburgh, Pa., June 1952.

(51) On the variances and covariances of order statistics from the

Weibull distribution. J. Lieblein. Submitted to a technical

journal.

(52) A proof of Hilbert’s Nulls tellensatz . T. S. Motzkin. Submitted

to a technical journal.

(53) 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.

(54) On Fejer sets in linear and spherical spaces. T. S. Motzkin and

I. J. Schoenberg. Accepted for publication in Annals of Mathematics.

(55) On the relaxation method for linear inequalities. T. S. Motzkin

and I. J. Schoenberg. Accepted for publication in the Canadian

Journal of Mathematics.
Publication Activities 57

(56) Least p-th power polynomials on a real finite point set. T. S. Motzkin and J. L. Walsh. Submitted to a technical journal.

(57) A conjecture of Erdos. M. Newman. Submitted to a technical

journal.

(58) st ructure theorems for modular subgroups. M. Newman. Accepted

for publication in the Duke Mathematical Journal.

(59) Diffraction of waves by a wedge. F, Oberhettinger. Submitted

to a technical journal.

(60) On the Lerch zeta function. F. Oberhettinger. Submitted to a technical journal.

(61 ) On two problems in abstract algebra connected with Horner's pule.

A. M. Ostrowski. To appear in the von Mises Anniversary Volume.

(62) Det erminanten mit ueberwiegender Hauptdiagonale und die absolute

Konvergenz von linearen Iterationsprozessen . A. M, Ostrowski.

Submitted to a technical journal.

( 63 ) On the convergence of Gauss' alternating procedure in the method

of the least squares, I. A. M. Ostrowski. Submitted to a technical journal.

(64) On the linear iteration procedures for symmetric matrices. A. M.

Ostrowski. Submitted to a technical journal.

(65) On the spectrum of a one parametric family of matrices. A. M.

Ostrowski. Submitted to a technical journal.

(66) On Gauss* speeding up device in the theory of single step iteration. A. M. Ostrowski. Submitted to a technical journal.

( 67 ) On absolute convergence of linear iteration processes. A. M.

Ostrowski. Submitted to a technical journal.

(68) On spectra of second-order differential operators. D. Ray.

Submitted to a technical journal.

( 69 ) 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.

(70) Numerical computation of low moments of order statistics from a normal population. J. B. Rosser. Submitted to a technical

journal.

(71) Most probable rank orders. I. R. Savage. Submitted to a technical journal.

(72) An isoperimetric inequality for closed curves convex in even- dimensional Euclidean space, I. J. Schoenberg. Accepted for

publication in Acta Mathematica

(73) Generalized commutators of matrices and permutations of factors

in a product of three matrices. 0. Taussky. For publication in

the von Mises Anniversary volume.
58 Publication Activities

(7*+) Some computational problems in algebraic number theory. 0 . Taussky.

To appear in the Proceedings of the American Mathematical Society

Sixth Symposium on Applied Mathematics, held at Santa Monica City

College, August 26-28, 1953.

(75)

An improved cathode ray tube storage system. R. Thorensen. To

appear in the Proceedings of the Western Computer Conference of

the AIEE-IRE-ACM held in Los Angeles, Calif., February 4, 5,6, 1 953.

(76)Motivation for working in numerical analysis. J. Todd. To appear

in the Transactions of a Symposium on Applied Mathematics sponsored

by the Office of Ordnance Research and held in Chicago, 111.,

April 29, 1954.

(77) The condition of certain matrices, II. J. Todd. To appear in

Archiv der Mathematik.

(7 8 ) The condition of the finite segments of the Hilbert matrix.

J. Todd. To be included in Contributions to the solution of

systems of linear equations and the determination of eigenvalues,

NBS Applied Mathematics Series 39.

(79) The evaluation of the exponential integral for large complex

arguments. J. Todd. Accepted for publication in the Journal of

Research of the NBS.

(80) A method for the numerical integration of differential equations

of second order without explicit first derivatives. R. de Vogelaere.

Accepted for the Journal of Research of the NBS.

(81) Discrete approximations to elliptic differential equations.

W. Wasow. Accepted for publication in Zeitschrift fur angewandte

Mathematik und Physik.

(82) On the asymptotic transformation of certain distributions into

the normal distribution. W. Wasow. To appear in the Proceedings

of the American Mathematical Society Sixth Symposium on Applied

Mathematics, held at Santa Monica City College, August 26-28,1953.

( 83 ) An extremum property of sums of eigenvalues. H. Wielandt. Accepted for publication in the Proceedings of the American Mathematical Society.

(84) E rror bounds for the eigenvalues of symmetric integral equations.

H. Wielandt. To appear in the Proceedings of the American Mathematical Society Sixth Symposium on Applied Mathematics, held at

Santa Monica City College, August 26-28, 1953.

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

to a technical journal.

(86) Statistics and planning tests at elevated temperatures. W. J. Youden. To appear in the Proceedings of the Society for Experimental Stress Analysis.

(87) A note on partially balanced designs. M. Zelen. Accepted for

publication in Annals of Mathematical Statistics.

(88) Analysis for some incomplete block designs having a missing block.

M. Zelen. Accepted for publication in Biometrics.

( 89 ) Bounds on a distribution function which are functions of moments

to order four. M. Zelen. Accepted by the Journal of Research

of the NBS.
Publication Activities 59

2.4 Reviews, Notes

(1) Note on the circle theorem of hydrodynamics. E. Levin. Accepted

for publication by the Quarterly of Applied Mathematics.

(2) Two early papers on the relation between extreme values and tensile

strength. (Formerly "A historical note on the application of the

•weakest Irnk' idea to tensile strengths.") J. Lieblein. Submitted to a technical journal.

2.5 Miscellaneous Publications

(1) Contributions to the solution of systems of linear equations and

the determination of eigenvalues. NBS Applied Mathematics Series

39. In pr@ 2 S, Government Printing Office.

(2) Experiments in the computation of conformal maps. NBS Applied

Mathematics Series 42. In press, Government Printing Office.

THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and

maintenance of the national standards of measurement and the provision of means and methods

for making measurements consistent with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials,

devices, and structures; advisory services to Government Agencies on scientific and technical

problems; invention and development of devices to serve special needs of the Government; and the

development of standard practices, codes, and specifications. The work includes basic and applied

research, development, engineering, instrumentation, testing, evaluation, calibration services, and

various consultation and information services. A major portion of the Bureau’s work is performed

for other Government Agencies, particularly the Department of Defense and the Atomic Energy

Commission. The scope of activities is suggested by the listing of divisions and sections on the

inside of the front cover.

Reports and Publications

The results of the Bureau’s work take the form of either actual equipment and devices or published papers and reports. Reports are issued to the sponsoring agency of a particular project or program. Published papers appear either in the Bureau’s own series of publications or in the

journals of professional and scientific societies. The Bureau itself publishes three monthly periodicals, available from the Government Printing Office: The Journal of Research, which presents

complete papers reporting technical investigations; the Technical News Bulletin, which presents

summary and preliminary reports on work in progress; and Basic Radio Propagation Predictions,

which provides data for determining the best frequencies to use for radio communications throughout

the world. There are also five series of nonperiodical publications: The Applied Mathematics

Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous

Publications.

Information on the Bureau’s publications can be found in NBS Circular 460, Publications of

the National Bureau of Standards ($1.25) and its Supplement ($0.75), available from the Superintendent of Documents, Government Printing Office. Inquiries regarding the Bureau’s reports and

publications should be addressed to the Office of Scientific Publications, National Bureau of

Standards, Washington 25, D. C.