NATIONAL BUREAU OF STANDARDS REPORT

4414

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

A Quarterly Report

July through September 1955

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

FOR OFFICIAL DISTRIBUTION
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

THE NATIONAL BUREAU OF STANDARDS

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

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

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

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

of the back cover of this report.

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

Instrumentation. Electrochemistry.

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

Technology. Length. Engineering Metrology.

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

Engines and Lubrication. Engine Fuels.

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

State Physics. Electron Physics. Atomic Physics. Nuclear Physics. Radioactivity. X-rays. Betatron. Nucleonic Instrumentation. Radiological Equipment. AEC Radiation Instruments. Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical Chemistry.

Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry. Spectrochemistry. Pure Substances.

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

Mass and Scale. Capacity, Density, and Fluid Meters. Combustion Controls.

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

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

Mineral Products. Porcelain and Pottery. Glass. Refractories. Enameled Metals. Concreting Materials. Constitution and Microstructure.

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

Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics.

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

Analogue Systems.

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

Gas Liquefaction.

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

Propagation Services.

Radio Propagation Engineering. Frequency Utilization Research. Tropospheric Propagation

Research.

Radio Standards. High Frequency Standards. Microwave Standards.

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

NBS PROJECT NBS REPORT

NAML 441 4

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

July through September 1955

<^BS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication, reprinting,

unless permission is obtained 25.D.C. Such permission li

cal ly prepared If that ageni

Approved for public release by the

director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

: prohibited

Washington

ieen speclfits own use,
APPLIED MATHEMATICS DIVISION

July 1 through September 30, 1955

TECHNICAL ADVISORY COMMITTEE

David Blackwell, Howard University

Edward U. Condon

Mark Kac , Cornell University/

Philip M. Morse, Massachusetts

Institute of Technology

Mina S. Rees, Hunter College

A. H. Taub, University of Illinois

DIVISION OFFICE

Edward W. Cannon, Ph.D., Chief

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

Olga Taussky-Todd , Ph.D., Consultant

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

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

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

Yates S. Sladen, Assistant Chief Clerk

Mary E. McKinley, Secretary

Esther L. Turner, Secretary

NUMERICAL ANALYSIS SECTION

Henry A. Antosiewicz, Ph.D.

Leila D. Bram, Ph.D.

Everett C. Dade

Philip Davis, Ph.D.

John Todd, B.S., Chief

Ky Fan, Ph.D.

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

Alan J. Hoffman, Ph.D.

Morris Newman, Ph.D.

Philip Rabinowitz, Ph.D.

COMPUTATION LABORATORY

Joseph Bram, Ph.D. Hans 0. Bremer, B.A.

Milton Abramowitz, Ph.D., Chief

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

Genevie E. Hawkins, B.S. Emilie V. Haynsworth, Ph.D.

Maxine L. Paulsen, B.S. Sally T. Peavy, B.S. William F. Cahill, M.S. Urs W. Hochstrasser, Ph.D. B. Stanley Prusch

Ruth E. Capuano Gloria F. Holmes , B ,S . , Sec'y Ida Rhodes, M.A. Anne C. Bruno, B.A. Dorothea H. Jirauch, M.A. Philip J. Walsh, B.! Mary M. Dunlap, B.S. Lambert S. Joel, B.A. Bertha H. Walter Anne M. Futterman, B.A. David S. Liepman Joseph H. Wegstein,

Jane C. Gager Ethel C. Marden, B.A. Donald F. Wehn, B.A

Billie R. Gill, M.S. Kermit C. Nelson Richard G. Williams

Elizabeth F. Godefroy Peter J. O'Hara, B.S. Ruth Zucker, B.A. William G. Hall, B.S. Mary Orr

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

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

Robert C. Burton Marion T. Carson

Willard H. Clatworthy, Ph.D.

Lola S. Deming, M. A. Mary L. Epling, B.S.

MATHEMATICAL PHYSICS SECTION

Peter Chiarulli, Ph.D.

John W. Cooper, B.S, Leon Feldman, B.A.

Seymour Geisser, Ph.D.

Lela J. Hamilton, Sec'y

Forest L. Miller, Jr. Julius Lieblein, Ph.D.

Peter Henrici, Ph.D.

Fritz Oberhettinger, Ph.D.

Mary G. Natrella, B.A. Hugh M. Pettigrew

Paul N. Somerville, Ph.D

Carolyn Yick, Sec'y

Marvin Zelen, M. A.

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

Robert F. Dressier, Ph.D., Chief

Under contract with The American University Guest worker ***0n leave of absence
CONTENTS

Page

Status of Projects* as of September 30, 1955. . . 1

Numerical Analysis 1

Mathematical Tables 10

Probability of Mathematical Statistics 15

Mathematical Physics

Mathematical and Computational Services .... 23

Statistical Engineering Services . 39

Application of SEAC 5-1

Lectures and Symposia ^3

Publication Activities ^5

*0nly unclassified projects are included in this

report 8
.
Status of Projects

September 30, 1955

1 . NUMERICAL ANALYSIS

RESEARCH IN THEORIES OF DISCRETE STRUCTURES

Task 1 100- 10-51 16/56-1 59

Origin: NBS Authorized 9/30/55

Sponsor: Office of Naval Research

Manager: O. Taussky-Todd

Objective: To conduct basic research in branches of pure and

applied mathematics, especially in problems of discrete structures,

including algebra and number theory, related to numerical analysis.

Background: Theories of discrete structures have become increasingly

important in a number of applied fields, as a result of the existence of

high-speed computers, which make it possible to perform the usually very

long calculations occurring in these areas. Status: NEW. 0. Taussky-Todd and M. Newman extended their results

concerning unimodular integral circulants from prime power orders to arbitrary orders. A by-product of this was the solution of a problem

in elementary number theory. This has been submitted to the problem

section of the Dutch journal, "Wiskundige Opgaven."

0. T. Todd, H. P. F. Swinne rton-Dye r (from Cambridge, England),

and M. Newman are using SEAC for computing units and the class number

of cubic cyclic algebraic number fields of prime conductor. This is to

check and extend a table computed by Mr. Swinner ton-Dyer previously on EDSAC , at Cambridge, England.

In connection with the work of this task, M. Newman, together with I. Reiner (presently at the Institute for Advanced Study, Princeton),

proved the following theorem: Let k be a positive integer, and let

k=r+s be some fixed partition of k into positive integers r,s. Lex M

be a kxk unimodular matrix. of rational integers, partitioned as (£®)

,

where A is an rxr matrix, B an rxs matrix, C an sxr matrix and D an sxs matrix. Let T be the multiplicative group of matrices M, and T(m,n)

the subgroup of V characterized by BsO(mod m) , C=0(mod n), where m and n are non-zero rational integers. Then if (m,n)=1 and zi is any subgroup

of F containing T(m,n), A= "T(m, ,n,) where m, | n and n,

| n. If lm,n)>1 there are subgroups of T containing r(m,n) which are not of this form

A manuscript has been prepared.

0. T. Todd continued research on the eigenvalues of positive

ma t rices

.

0. T. Todd, Ky Fan and J. Todd discussed the two-dimensional

discrete analog of the Wirtinger inequality. The only new feature occurring is the fact that equality is obtained if and only if the variable vector is a linear combination of a certain set of four (n 2

) vectors

As another by-product of their work on the Wirtinger inequality,

1
2 Status of Projects

Ky Fan, O.T.Todd and J. Todd obtained an algebraic proof of the isoperimetric

property of polygons. The usual proofs are geometrical or analytical.

The present proof, after reducing the problem to the convex equilateral

case, begins by showing that for any n (X3) complex numbers, z., ,z 2

, . . . ,z , we have

nH

j=1 z.-z. H

J J+1 > 2 ta

j=i

z

j z j+i

(where z =zj, and that there is equality if and only if

n+ 1

i

<&exp(27rij/n)+/3 , j=1 ,2, . .

.

2 ~ 1 If the z's are the vertices of an equilateral polygon this gives L >*+n tan^n F where L is the perimeter and F the area.

K. Goldberg with E. Dade investigated certain linear associative

algebras. These algebras are characterized by having a basis of incidence

matrices whose sum is the matrix of 1's. The results have been applied to the problem in graph theory mentioned in the preceding quarterly report.

K. Goldberg continued his work on bounds and asymptotic behavior

of the coefficients in the series for log exey.

Publications:

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

fields. M. Newman and 0. Taussky-Todd . To appear in Communications on Pure and Applied Mathematics (New York University).

(2) A note on group matrices. 0. Taussky-Todd. To appear in the Proceedings

of the American Mathematical Society.

(3) Power series for log(e x .ey). K. Goldberg. To appear in the Duke

Journal of Mathematics.

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

E. Straus, and 0. Taussky. To appear in the Pacific Journal of Mathematics

(5) Some computational problems in algebraic number theory. 0. Taussky.

To appear in the Proceedings of the American Mathematical Society Sixth

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

August 1953.

( 6 ) Bounds and asymptotic behavior of the coefficients in the series

log (e x

ey). K. Goldberg. In manuscript.

(7) Unimodular circulants. 0. Taussky-Todd. To appear in the I. Schur

Memorial Volume, Mathematische Zeitschrif t.

( 8 ) Algebraic equations satisfied by roots of natural numbers. E. G. Straus

(University of California at Los Angeles) and 0. Taussky. To appear

in Pacific Journal of Mathematics.

(9) Pairs of Matrices with property L,II. T. S. Motzkin and 0. Taus sky-Todd

.

To appear in Transactions of the American Mathematical Society.

(10) Generation and testing of pseudo-random numbers. 0. Taussky-Todd.

Submitted to a technical journal.

(11) An algebraic proof of the isoperimetric property of polygons. Ky Fan,

Olga Taussky and J. Todd.

Academy of Sciences.

T*o appear in the Journal of the Washington
Status of Projects

RESEARCH IN NUMERICAL ANALYSIS AND RELATED FIELDS

Task 1101-10-1101+/55_ 55

Origin: NBS

Managers: J. Todd, P. Davis

Full task description: July-Sept 195^ issue, p. 1

Authorized 8/1 3/5^

Revised 8 /29 /5^

S'tatus: CONTINUED. H. A. Antosiewicz has continued his investigation

of Lyapunov' s second method for the stability of solutions of systems of

ordinary differential equations. He has obtained necessary and sufficient

conditions for stability, simple and asymptotic, uniform and non-uniform,

which are not included in the most recent results obtained by the Russian

school.

P. Davis and Ky Fan have studied certain inequalities in normed

linear spaces and their applications to problems of interpolation and

approximation in real and complex variables. Let ff

n} be a sequence in a normed linear space X and let 0 € X*. Necessary and suf f icient conditions

that each of the following five relations hold have been derived:

I. 1 0<f„)|< %

(n=1 •> 2 ? • •

II. lim 0(f) = 0 implie

rI —

oo III.

| 0 ( f n

) | < a (n = 1 ,2,

IV.

1 0(f

n ) |> 1 (n=1 » 2 , . .

1 0(f

n )-»

n | < °n (n=1

Conditions I, II, and III have

theorems; of Paley-Wiener type

general results were mad e to

Hilbert spaces of complex analytic functions. Theorems on approximation

with coefficients restricted in size have been obtained. Applications

have also been made to the theory of conformal kernel functions of the

Bergman and Szego type and to uniqueness theory for asymptotic expansions

in the sense of Poincar€.

P. Davis carried out some further checks on the abscissae and

weights for Gaussian quadratures of high order.

E. Haynsworth continued her program of research on bounds for

determinants

U. W. Hochstrasser studied the problem of improving the speed

of convergence of over-relaxation by inverting submatrices of the coefficient

matrix of the system of linear equations.

T. Kato has prepared a code for a new SEAC demonstration problem,

which also provides a new test for random numbers. This is the classical

problem of the gambler's ruin.

J. Gager completed a code to find the unique representation of a given prime of the form ^K+l as a sum of two squares. The time for a ten

digit prime is less than a minute.

J. Todd had discussed previously, theoretically and experimentally,

the system of linear algebraic equations obtained by discretization of

equations of the form U

xx + U = kU. The case of a rectangle, in cartesian

coordinates, was the only one then discussed. Now, in collaboration with

Ky Fan and K. Goldberg, the case of a circle with polar coordinates, is

being investigated. One would expect to obtain the classical results that

the eigenvalues are j2 , where the j A are the positive zeros of J (x) O , r U , r o (as a limit of the discrete results). In particular, it is of interest to see how the largest eigenvalue of the finite matrix involved approaches the
1+ Status of Projects

limit. In view of the singularity of the coordinate system the proper

answers to various questions about the details of discretization were doubtful. Confidence in the decisions taken was given by experiments on SEAC, in the case of a 30x30 system. The matrix in question was a symmetric

triple diagonal one of the form

where A^n corresponds to an nxn discretization. Further work on this

problem is in progress.

J, Todd, with the assistance of Mrs. I. Rhodes and Miss J. Gager,

has been making a survey of recent Russian mathematical tables. Reviews

of some of these appear in Math. Reviews and in Mathematical Tables and

Other Aids to Computation,

Publications:

(1) Stable systems of differential equations with integrable perturbation

term, H. Antosiewicz. To appear in the Journal of the London

Mathematical Society.

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

and S. Levy (NBS 6.4). To appear in Journal of the Institute of

Aeronautical Sciences.

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

appear in the Pacific Journal of Mathematics.

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

Journal of Mathematics.

(5) Evaluation of a definite integral. P. Henrici. An NBS report (April

1954).

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

In manuscript.

(7) A subroutine for computations with rational numbers. P. Henrici.

To appear in the Journal of the Association for Computing Machinery.

(8) A direct approach to the problem of stability in the numerical solution

of partial differential equations. J. Todd. In manuscript.

(9) Some Monte Carlo experiments for computing multiple integrals. P, Davis

and P. Rabinowitz. To appear in Mathematical Tables and Other Aids to Computation

.

(10) Abscissas and weights for Gaussian quadratures of high order. P. Davis

and P. Rabinowitz. In manuscript.

(11) Numerical experiments in potential theory using orthonormal functions.

P. Davis and P. Rabinowitz. Submit ted to a technical journal.

(12) Begrundung fur die Beschaf tigung mit numerischer analysis. J. Todd.

Jahresbericht d. D.M.V. £8, 11-38 (1955).

(13) Inequalities and approximation in function spaces. P. Davis and Ky Fan.

In manuscript.

(1 4) Matrix transformations and properties of characteristic roots. K. Goldb

In manuscript.

(15) Note on bounds for determinants. E. V. Haynsworth. In manuscript.

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

(

It has been shown that
Status of Projects 5

RESEARCH IN MATHEMATICAL TOPICS APPLICABLE TO

NUMERICAL ANALYSIS

Task 1 101-10-51 16/55-56

Origin: NBS

Sponsor: Office of Naval Research

Managers: J. Todd, M. Newman

Full task description: July-Sept 1955 issue, p. 5 Status: CONTINUED. The method of orthogonal harmonic functions

was used to solve a boundary value problem of the first kind for an annular

region bounded by two ellipses of axis ratio 2:1. The numerical work was

carried out on SEAC using a general purpose orthonormalizing code developed

by P. Davis and P. Rabinowitz. Using harmonic functions which were linear

combinations of log z and the real and imaginary parts of 1,z,z2,z -1, an average discrepancy of about 10°/o on the boundary was achieved.

Ky Fan studied systems of inequalities concerning the absolute

values of linear functionals on a complex linear space. Consistency

conditions for different types of such systems (finite and infinite) of

inequalities are obtained. In the case of a finite dimensional unitary

space or a complex Hilbert space, a characterization of the unique solution

of minimum norm by a maximum property is given. For inequalities involving

completely continuous linear transformations, one of the results is the

following: Let X be a reflexive complex normed linear space and let A

be a completely continuous linear transformation from X into itself. Let

y € X, o > 0, and let A/ 0 be a complex number. Then there exists an x°€ X satisfying llAx - Ax - y || < 5, if and only if every solution g

€ X*

of the equation A*g = Ag satisfies the relation |g (y 0

) [ < & i| <J || . In the case D = 0, this result becomes Riesz-Schauder ' s generalization of Fredholm 1 alternative theorem, except that the reflexivity of the space X is required

here. Results of this investigation have been written up as an additional

part IV of the earlier manuscript, "On systems of linear inequalities."

Ky Fan, in collaboration with I. Glicksberg (RAND Corporation)

completed a paper on fully convex normed linear spaces.

In collaboration with J. B. Kruskal (Princeton University),

A. J. Hoffman studied the question of conditions under which polyhedra admit

integral points on every face (of any dimension). Let A be an integral

matrix. In order that the convex set (1) Ax > b have an integral point on every face for every integral vector b such that (1) be non-empty it is

necessary and sufficient that (if k be the rank of A) the g.c.d. of the (")

determinants obtainable from every set of k linearly independent rows of

A be unity.

If we pose the analogous question for (1) and for (2) x > 0,

the condition becomes: every nonsingular square submatrix of A is unimodular

It has been shown further that the incidence matrix for elements vs. "full"

chains (chains which are not contained in other chains joining the same end

points) in a partially ordered set P does have this property if P is the

union of two disjoint sets L and U satisfying:

(a) no u c U precede 1 e L,

(b) if JL i L immediately precedes u e U, then X is maximal

in L and u is minimal in U,

(c) every l e L has at most one immediate successor in L,

(d) every ucU has at most one immediate predecessor in U.

T. Kato is studying the application of perturbation theory to

obtain bounds for the eigenvalues of finite matrices. It is not yet clear

whether the results obtained by this approach are better or worse than the

classical ones. A library of basic matrix codes is being written. All of the

Authorized 8/13/55

Revised 8/29/55
6

Status of Projects

routines accept data in a standard form. Routines which have been

completed by M. Newman include:

MP1

M01

MSI

Mil

MM1

MM2

MM3

M02

MAI

MF1

General purpose input routine

General purpose output routine

Computes x where Ax-b, A an nxn matrix a n<23. The computation time for n=28 is

Computes A~^ , where A is an nxn matrix,

time for n=28 is about 15 minutes.

Computes AB , where A is an mxn matrix, B

n r<40O

.

Computes A'A, where A is an mxn matrix,

Computes A' BA, where A is an mxn matrix,

and nm<3-20, m< 27 . Special output routine for MM3.

Computes A+B, where A and B are mxn matr

Computes X, where X'X=A, A an nxn symmet

and X an upper triangular matrix. Here

nd b an nxl vector. Here

about 10 minutes.

and n<28. The computation

an nxr matrix, and mn<^00

,

and n(m+n)<840.

B an mxm symmetric matrix,

ices and mn<400.

ric positive definite matrix

n<4-0.

Routines which have been completed by K. Goldberg include:

FS1-5- Floating subroutines for addition, multiplication, division and

comparison written for conciseness jnd speed and without interpretive

entry.

MP2 General purpose input routine which computes the norm of the data

and makes it less than one if it is desirable to "unfloat" the data.

MJG1 Reduces symmetric matrix of order less than or equal to 23 to triple

diagonal form by Jacobi-Givens process and stores the transformation

matrix. Unfloated.

MJG2 Checks norm and "refloats" triple diagonal form. Expandable to provide absolute check on transformation of JG1

.

MJG3 Computes roots and vectors of triple diagonal form by Sturm sequence.

Transforms and normalizes vectors to obtain those of the original

matrix.

E. Dade completed a code to find the inverse of a matrix, the

elements of which are double precision floated decimal numbers, of order

not exceeding 1*+. M. Newman continued his work on the coefficients Pr(n). Several

new identities have been derived and some further theorems concerning these

coefficients have been proved. A generalization of this problem to the

coefficients of arbitrary modular forms is under way.

Publications

:

(1) The normalizer of certain modular subgroups. M. Newman. To appear

in the Canadian Mathematical Journal.

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

forms. M. Newman. To appear in Journal of the London Mathematical

Society.

(3) Compactif ication of completely regular spaces. Ky Fan and F. Wagner

(University of Notre Dame). In manuscript.

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

Ky Fan. Proc. Cambridge Philosophical Soc. 51 .

1+1 i+-l+21 (1955).

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

To appear in the Pacific Journal of Mathematics.

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

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

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

appear in the Pacific Journal of Mathematics.
Status of Projects 7

(8) An identity for the coefficients of certain modular forms. M. Newman.

To appear in the Journal of the London Mathematical Society.

(9) Error bounds for eigenvalues of symmetric integral equations.

H. Wielandt. To appear in the Proceedings of the American Mathematical

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

California, August 1953.

(10) Fully convex normed linear spaces. Ky Fan and I. Glicksberg (RAND

Corporation). To appear in the Proceedings of the National Academy

of Sciences.

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

to a technical journal.

(12) An inclusion theorem for modular subgroups. M. Newman. In manuscript.

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

To appear in the Pacific Journal of Mathematics.

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

To appear in Annals of Mathematics Study 38.

(15) Systems of linear inequalities. Ky Fan. To appear in Annals of

Mathematics Study 38.

(16) Systems of distinct representatives and linear programming. A. J. Hoffman.

To appear in the American Mathematical Monthly.

(17) On a theorem of Dilworth. G. B. Dantzig (RAND Corporation) and

A. J. Hoffman. To appear in Annals of Mathematics Study 38.

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

manuscript

(19) Notes on some scheduling programs. A. J. Hoffman. In manuscript.

(20) On block relaxation. L. S. Joel. In manuscript.

(21) On the minimization of concave and convex functionals. G.B. Dantzig

(RAND Corporation), A. J. Hoffman, W. Hirsch (New York University).

In manuscript.

(22) How to solve a linear programming problem. A. J. Hoffman. Proceedings

of the Second Symposium in Linear Programming, held jointly by the

National Bureau of Standards and the U.S.A.F . , January 1955, pp.

397-5-24.

ANALYTIC STUDY OF WAR GAMES

Task 1101-10-51 16/55-83

Origin and Sponsor: Armament Branch, ARDC, USAF Authorized 12/29/54

Manager: H. A. Antosiewicz

Full task description: Oct-Dee 1954 issue, p. 7 Status: CONTINUED. In a recent conference initiated by the sponsor

and people working under contract on similar problems, it was decided to

hold a two-day symposium on topics in war games at the University of

Michigan sometime in November 1955. The analytical work done up to now will be reviewed by H. A. Antosiewicz.

Publication:

(1) Analytic study of war games. H. A. Antosiewicz. To appear in the

Naval Research Logistics Quarterly.
8 Status of Projects

STUDIES IN THE THEORY OF ASYMPTOTIC EXPANSIONS

Task 1101-10-5116/55-116

Origin: NBS Authorized 3/31/55

Sponsor: Office of Scientific Research, ARDC, USAF

Managers P. Davis

Full task description: Jan-Mar 1955 issue, p. 8

Status: CONTINUED. Necessary and sufficient conditions for

uniqueness classes of (Poincar^) asymptotic expansions in an arbitrary

domain have been obtained using the theory of conformal kernel functions.

For some additional- details see Apr-June 1955 issue, p. 5. The question of uniqueness classes has also been approached from

the point of view of the theory of interpolation and approximation. Using

a theorem on inequalities in Banach spaces developed by P. Davis and Ky Fan,

it has been found possible to characterize uniqueness classes by means of

the completeness properties of certain sequences of analytic functions.

(See also task 1 101—1 0-1 105-/55-55 , p. 1.)

Publication:

(1) Uniqueness theory for asymptotic expansion in arbitrary domains.

P. Davis, In manuscript.

STUDY OF DIFFERENTIAL EQUATIONS FOR NERVE EXCITATION

Task 1101-10-5116/56-148

Origin and Sponsor: National Institutes of Health, Authorized 9/30/55

Bethesda, Md.

Manager: H. A. Antosiewicz

Objective: The normal functioning of a single nerve fiber involves

the development of electrical potential differences and currents along

its length as well as across the membrane at its surface. The HodgkinHuxley differential equations, which are the basis of this study, express

the membrane current I(t) in the form

(1) I(t) - C JJ + 36n

Lf (V-12) + 120m3 h(V+1l5) + ^ (V+10.5989)

where C is the membrane capacity, V the difference of the membrane potential

from its resting values, and n,m,h are dimensionless variables related to V by the system

dn

dt a

1 (V) (1-n) -a

2

( V)n

(2) dm _ dt b

1 (V) (1-m) -t> 2

( V)m

dh

dt c

1 ( V) (1-h) -c

2 (V)h

in which the coefficients are functions of V only, too complicated to be stated explicitly here.

A previous investigation centered around the physiologically

important "all-or-none" phenomenon, i. e . , the determination of the so-called

threshold values for the current I(t).

The present study will be devoted mainly to an investigation of
Status of Projects 9

the propagated action potential whose equation was given by Hodgkin-Huxley

in the form

(3) = K ^ + £

36n

l+ (V-12) + 120m3 h(V+11?) + ^ (VM0.5989),

d t

the subsidiary variables being defined as before. The constant K depends

upon the velocity of propagation and is not known in advance. One of the

problems to be considered is the determination of K such that the value

of V becomes zero when the action potential returns to zero. Background: Previous investigations, especially of the threshold

values of I(t), were carried on under task 1 102-20-1 10h-/ 53~1 5 (see Oct-Dec

1952 issue, p. 3*+) and were continued under task 1 101—10—1 104-/55—55 (see

July-Sept 19r+ issue, p. 2, and Oct-Dec 19 5^ issue, p. 2). Both the

present investigations and the earlier ones were proposed by Dr. K. Cole,

presently of the National Institutes of Health.

Status: NEW. H. A. Antosiewicz has investigated the "linearized"

system corresponding to equations (2-) and (3) in order to obtain approximate

initial values from which numerical integrations of (2) and (3) could be

started. He has prepared a SEAC code for the solution of the linearized

system which should prove useful in determining rough approximation of

suitable values for the parameter K. Publication:

(1) Automatic computation of nerve excitation. K. S. Cole (Naval Medical

Research Institute), H. A. Antosiewicz and P. Rabinowitz. To appear

in the Journal of the Society for Industrial and Applied Mathematics.
2. MATHEMATICAL TABLES

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

Task 11 02-40-1 110A3-3

Manager: I. Stegun Authorized 7/1/47

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

Status: Continued. Preparation of an introduction to accompany

publication of the completed tables continued.

TABLES OF COULOMB WAVE FUNCTIONS

Task 1102-40-1110/47-2

Origin: NBS Authorized 7/1/47

Manager: M. Abramowitz

Full task description: Apr-June 1949 issue, p. 49

Status: Continued. A new program is being prepared to obtain G^,G£,

F

l

, Fj/ ^ or 9^ ven values of j> and

computation consists of the evaluation of G

q and G^ at the turning point

corresponding to the given value of , from asymptotic expansions recently

derived by C. E. Froberg (private communication), followed by analytic

continuation using Taylor's formula to the desired value of p. The

derivatives are obtained from recurrence relations resulting from successive

differentiation of the differential equation. Values for L > 0 are then

obtained from the recurrence relations satisfied by the Coulomb functions.

Spot checks made for ^ > 2 indicate that results are correct to at least

five decimal places or significant figures.

and varying values of L. The method of

TABLES OF POWER POINTS OF ANALYSIS OF VARIANCE TESTS

Task 1102-40-1110/51-8

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

Manager: S. Peavy

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

Status: Continued. More runs have been made in order to expand the

tables for the same values of and /3 (which are listed in the Inn-Mar 1953

issue, p. 48). >

10
REVISION OF MATHEMATICAL TABLES

Task 1 1 02-40-1 1 1 0/52-7

Origin: NBS Authorized 8/10/51

Managers: W. F. Cahill, I. Stegun

Full task description: July-Sept 1951 issue, p. 4-1

Status: CONTINUED. Revision of "Tables of natural logarithms," vol. IV,

Mathematical Tables MT1 2 (1941), for reissue in the Applied Mathematics

Series continued.

TABLE OF ARCSIN FOR COMPLEX ARGUMENTS

Task 1102-40-1110/52-14

Origin: NBS Authorized 10/1/51

Manager: B. Gill

Full task description: July-Sept 1951 issue, p. 4l Status: INACTIVE. For status to date, see Jan-Mar 1955 issue, p. 11.

EXTENSION OF THE TABLE (S' HYPERBOLIC SINES AND COSINES

Task 1 102-40-1 1 10/52-1

Origin: NBS Authorized 9/1 7/51

Manager: K. Nelson

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

Status: CONTINUED. Proofs have been checked.

Publication: Table of hyperbolic sines and cosines (x=2 to x=10, 9S).

Applied Mathematics Series 45. In press.

TABLE OF THE MODIFIED AIRY INTEGRAL

Task 11 02-40- 1110/52-23

Origin: NBS Authorized 10/4/51

Manager: P. Rabinovvitz

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

Status: CONTINUED. Preparation of the manuscript for publication

continued.

TABLE OF ERROR FUNCTION FOR COMPLEX ARGUMENTS

Task 1 102-40-1 1 10/52-25

Origin: NBS Authorized 10/5/51

Manager: W. Hall

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

Status: INACTIVE. For status to date, see Oct-Dec 1954 issue, p. 11.
12 Status of Projects

EXTENSION OF TABLES OF THE EXPONENTIAL FUNCTION FOR

NEGATIVE ARGUMENTS

Task 1102-40-1110/52-31

Origin: NBS Authorized 10/9/51

Managers: E. Mardeo, S. Prusch

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

Status: CONTINUED. Proofs have been checked.

Publication: Table of the descending exponential (x=2.5 to x=10, 20D).

Applied Mathematics Series 46. In press.

SPHEROIDAL WAVE FUNCTIONS

Task 1102-40-1110/52-37

Origin: NBS Authorized 11/28/51

Manager: D. Liepman

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

Status: INACTIVE.

RADIAL MATHIEU FUNCTIONS

Task 1102-40-1110/52-49

Origin: NBS Authorized 2/1/52

Manager: I. Rhodes

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

Status: COMPLETED.

Publication: Table of characteristic values of Mathieu's equation for

large values of the parameter. G. Blanch and I. Rhodes. J. Wash. Acad.

Sci. 45, 166-196 (June 1955); reprints available.

SIE VERT'S INTEGRAL

Task 1102-40-1 1 10/52-57

Origin: NBS Authorized 2/12/52

Managers: M. Paulsen, P. O'Hara

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

Status: INACTIVE. For status to date, see Apr-June 1955 issue, p. 9.
Status of Projects 13

L-SHELL CONVERSION COEFFICIENTS

Task 1102-40-1110/53-52

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

Manager: W. Hall

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

Status: CONTINUED. Computations for the shell have been

completed. Computations for the Ljjj shell were completed for z=25,65,75,85;

this is approximately half of the Ljjj shell runs.

AUTOMATIC CODING

Task 3711-60-0009/55-65

Origin: NBS Authorized 9/29/54

Manager: J. Wegstein

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

Status: CONTINUED. A subroutine of P. Rabinowitz for evaluating

integrals of the form f(x)dx by Gaussian quadrature, using 2-, 4-, 3-,

and l6-point formulas, was rewritten by J. Cooper for incorporation into

a special copy of Base 00 and is being checked out. It is intended to serve as a tool for the evaluation of integrals in small problems with a minimum of coding. Entry, exit and all necessary information for using

the routine will be specified by a two-word "order" or set of code words

giving the integration limits, location of subroutine for f(x), degree of

integration formula, etc.

The possibility of generating this code to perform two-, three-,

or four-fold integrations has been considered. Plans have been made to

generalize the scheme to (two-fold or three-fold ) integrations . Some

preliminary analysis on the probable accuracy of such a systecr. for two-fold

integrals has been done. A report has been written describing a new threeaddress compiler code which was described in the April-June issue (see p. 9).

With its aid several programs have been written.

MATHEMATICAL SUBROUTINES

Task 371 1-60-0009/56-160

Origin: NBS Authorized 9/30/55

Managers: Staff

Objective: To develop new general-purpose subroutines and service

routines for addition to the SEAC library, and to assemble and record such

subroutines developed in connection with other tasks.

Background: An extensive library of often used subroutines and service

routines is an essential part of the Computation Laboratory. As such new routines become available, they will be listed here for general information.

Status: NEW. A routine for solving algebraic equations was written

by U. Hochstrasser . The procedure employs an iterative method due to

Laguerre for improving an assumed approximation to a root. Complex roots

are also obtained.
14- Status of Projects

A routine for evaluating integrals by Filon' s method, originally

started by P, Rabinowitz, was completed by U. Hochstrasser . This method

is useful for evaluating Fourier transforms, cf. L. N. G. Filon, Proc. Roy.

Soc. Edinburgh 42, 38 (1928-1929).

A routine was developed for computing P(x) for real x from the

formula

r(x) = P(x+n+1 )[x(x+1 ) * * * (x+n) ]

^

where P(x+n+1) is obtained from the asymptotic expansion (see British

Association Tables, vol. 1, Introduction) with n=10.

A routine was written by E. Haynsworth for factorizing a symmetric matrix A into the product CC • of a triangular matrix C and its transpose. One application of this technique is the evaluation of the characteristic roots of the product of two symmetric matrices AB where A is

positive definite. While AB is in general not symmetric, CBG 1 is and has

the same roots as AB , which can then be determined by means of the existing

programs for symmetric matrices. Matrices up to order 28 can be factorized.
3. PROBABILITY OF MATHEMATICAL STATISTICS

BIBLIOGRAPHY AND GUIDE TO STATISTICAL LITERATURE

Task 1103-10-1 107A9-1 a

Origin: NBS Authorized 1/9/49

Manager: L. S. Deming

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

Status: CONTINUED. For a description of the continuing activity

on this task, see the Jan-Mar 1 9 A issue, p. 49.

MANUAL ON FITTING STRAIGHT LINES

Task 1103-10-1107/50-2

Origin: NBS Authorized 3/1/50

Manager: F. S. Acton

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

Status: CONTINUED. Review of the manuscript by technical advisors

continued

TABLES TO FACILITATE DRAWING RANDOM SAMPLES

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

Origin: NBS Authorized 7/1/50

Managers: C. Eisenhart, L. S. Deming

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

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

P. 64.

MISCELLANEOUS STUDIES IN PROBABILITY AND STATISTICS

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

Origin: NBS Authorized 7/1/50

Manager: C. Eisenhart

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

Status: CONTINUED. N. Severo prepared a manuscript on the derivation of the characteristic function of the multi-variate distribution. J. Lieblein prepared a manuscript on evaluating integrals when the parameters

take on complex values.

Preliminary steps were undertaken in the development of a compilation of distribution functions and their characteristic functions, both

known and new, for use in mathematical statistics. To this end two

15
16 Status of Projects

standard mathematical statistics texts, by M. G. Kendall and by H. Cramer, were examined, and a survey was made of the first of the Bateman volumes

of integral transforms. The texts yielded about fifty entries, and the

Bateman volume about two hundred.

Publications;

(1) Time-discrete stochastic processes in arbitrary sets, with applications

to processes with absorbing regions and to the problem of loops in

Markoff chains. D. van Dantzig. Accepted for publication (in French)

in Annales de 1’Institut Henri Poincare (Paris).

(2) A comparison between the mean successive differences and the root mean square successive difference. S. Geisser. Submitted to a technical journal.

(3) A note on the normal distribution. S. Geisser. Submitted to a technical journal.

(4) Inequalities for probabilities associated with the multivariate normal

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

PROCEDURES OF NON-PARAMETRIC STATISTICS

Task 1103-10-1107/52-2

Origin: MBS Authorized 9/17/51

Manager: I. R. Savage

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

Status: INACTIVE. For the latest report on this task see July-Sept

195^ issue, p. 22.

Publications:

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

Submitted to a technical journal.

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

Submitted to a technical journal.

(3) Simultaneous test's of randomness and other hypotheses. I. R. Savage.

In manuscript.

STUDIES IN THE MATHEMATICS OF EXPERIMENT DESIGN

Task 1103-10-1107/53-1

Origin: NBS Authorized 10/15/52

Manager: W. H. Clatworthy

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

Status: CONTINUED. Two summer trainees, H„ Pettigrew and F. L. Miller,

Jr., (under the supervision of M. Zelen) have constructed fractional

factorial designs of the 2

n series for fractions 1/6*+, 1/128, and 1/256 and for

factors up to n=l6. This is an extension of the previous work of Clatworthy,

Connor, and Zelen. Work is now in progress to combine the two compilations

of fractional factorial designs into on© large monograph suitable for

publication in the NBS Applied Mathematics Series.

W. H. Clatworthy has enumerated and constructed a large number

of partially balanced incomplete block (PBIB) designs thus considerably

extending the list given in Bose, Clatworthy, and Shrikhande, "Tables of

Partially Balanced Designs with Two Associate Classes." C. Burton and

Dr. Clatworthy have worked out the duals of a number of these designs and

Mr. Burton has proved some theorems regarding the existence of such
Status of Projects 17

duals. All this work on PBIB designs is being readied for publication.

Publications:

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

(American Cyanamid Co.). Accepted for publication in Biometrics.

(2) On the analysis of covariance. M. Zelen. Submitted to a technical

journal.

(3) On the enumeration of partially balanced designs with two associate

classes. W. H. Clatworthy. In manuscript.

(4) Exact tests of significance for combining inter- and intra-block

information. M. Zelen. In manuscript.

RESEARCH ON MATHEMATICAL ASPECTS OF ORDER

STATISTICS METHODS

Task 1103-10-1 107/55-1 10

Origin: NBS Authorized 3/3/55

Manager: J. Lieblein

Full task description: Jan-Mar 1955 issue, p. 31

Status: CONTINUED. Correspondence was exchanged with Professor

Greenberg of the University of North Carolina concerning the possibility

of a monograph on the subject of order statistics, and suggestions for

contents were considered. Discussion of extreme-value methods, in

particular the publication below, with J. T. Hack of the U. S. Geological

Survey, led to a decision to undertake application of these methods to a study of stream-bed deposits. To facilitate the understanding of extremevalue concepts, graphical aids were constructed reducing to straight lines

the relationships between return period, number of years, and probability

of occurrence. Advice was furnished to W. H. Sammons of the Kentucky

Department of Highways regarding interpretation of rainfall records of

different durations from a previously neglected aspect of extreme-value

theory

Publication:

(1) Application of extreme-value theory to cobbles and boulders in gravel

deposits. W. C. Krumbein (Northwestern University) and J. Lieblein.

Submitted to a technical journal.
b. MATHEMATICAL PHYSICS

RESEARCH IN MATHEMATICAL PHYSICS AND RELATED FIELDS

Task 1 104-10-1 1 1 5/55-57

Origin: NBS Authorized 9/1/55-

Manager: R. F. Dressier

Full task description: July-Sept 1955- issue, p. 27

Status: CONTINUED. P. Henrici studied the problem of numerical

evaluation of

f (a,b,c) JL(at) J^(bt)

<r M

J^(ct)dt

where Re (jt+v+ A) > -1 and a,b,c are positive. This can be represented by

a hypergeometric series if two of the numbers a,b,c are different.

Evaluation of this integral was required for an elasticity problem for

the case a = b, /i -if- 1, \- 2.

Some theoretical aspects of Rutishauser 1 s quotient-difference

algorithm for computation of eigenvalues of a matrix were studied. This

method yields non-dominant eigenvalues as well as the dominant one. Instead

of the use of continued fractions as in the original method, it has now been shown that the algorithm can be related to a classical determinantal

identity and to theorems in function theory.

Another study of eigenvalue computation was devoted to consideration

of the effectiveness of Aitken's t>2 method applied to the vibrating membrane

for an irregular L-shaped domain. Results indicated that one application

is quite effective, saving about 30 percent in computing time, but repeated

applications decrease the effectiveness. A three-fold application to this

problem showed definite signs of instability.

In a continuation of the study concerning addition theorems for

special functions, it has been shown that the classical theorems for Legendre

polynomials, and Bessel and Mathieu functions, as well as the recently

derived theorem for the general Legendre functions, fit into the theory

relating to addition theorems for solutions of partial differential equations.

Necessary conditions for their existence are (a) that the equations must

possess a complete system of solutions, and (b) that the equations must

admit a group of transformations.

Publications:

(1) Addition theorems for Legendre and Gegenbauer functions. P. Henrici.

Accepted for publication in the Journal of Rational Mechanics and

Analysis

.

(2) On the vibration of a square clamped plate. M. Abramowitz and

W. Cahill. J. Assoc. Comp. Mach. 2, 162-168 (July 1955).

(3) Kleine Bemerkung zur asymptotischen Entwicklung des Fehlerintegrals

.

P. Henrici. Zeitsch. angew. Math. Phys. 6, 1 4-5—1 M-6 (1955).

(5-) The representation of an integral involving Bessel functions by

hypergeometric series. P. Henrici. In manuscript.

13
Status of Projects 19

RESEARCH IN ELECTROMAGNETIC THEORY

Task 1104-10-51 60/54-47

Origin and Sponsor: Diamond Ordnance Fuze Authorized 6/29/54

Laboratory, Department of Revised 9/29/54

the Army

Manager: F. Oberhettinger

Full task description: July-Sept 1954 issue, p. 28

Status: INACTIVE.

RESEARCH IN MATHEMATICAL ELASTICITY

Task 1104-10-5160/55-85

Origin: NBS Authorized 12/27/54

Sponsor: Office of Scientific Research, ARDC, USAF

Manager: R. F. Dressier

Full task description: Oct-Dec 1954 issue, p. 30

Status: CONTINUED. The project to derive a practical method for

analyzing all stresses and displacements in a typical corrugated diaphragm

has been successfully completed this quarter, and a paper has been written

by R. F. Dressier which describes in detail the methods used and the

quantitative results obtained. For a capsule of Battelle shape consisting

of 2^ corrugations and a flat central section, the tangential, normal,

vertical, and horizontal displacement components have been derived, along

with the normal shear stresses and the circumferential and radial stresses,

for constant air pressure loading on the outer face. The separate

contributions causing membrane effects and bending effects have also been

obtained for the radial and circumferential stresses.

Maximum stresses producing radial bending and circumferential

stretching are approximately equal, although the maxima occur in different

regions. The largest total stress is a circumferential tension, occurring

on the inner face at the bottom of the corrugation nearest to the edge.

Its magnitude represents a stress magnification of 4130 times the loading

pressure

On SEAC using the code composed by W. Cahill, all of these

quantities have been computed for two different boundary-value problems:

(a) the clamped edge, and (b) the semi-free edge corresponding to applications in aneroid capsules. The central deflection is about six

percent greater for the semi-free case than for the clamped case, both

results being in very close agreement with experimental data. All other

quantities are almost identical for the two cases except in a narrow

region near the outer edge where very significant differences occur. Results on stresses for the semi-free case can be compared to the Grover

and Bell approximate results. Good agreement occurs except near the

outer boundary and in the central flat section. The computed results

for the shear force can be compared with known exact values, obtained

from static equilibrium considerations, at the five horizontal positions

of the corrugations; the maximum computational error at these points is

less than three percent. This order of accuracy was produced with 190

normal-size mesh intervals plus about 300 smaller intervals in the transition

zones surrounding the points of discontinuous meridial curvature. Including

the mid-points which must be used, this totals almost 1000 integration

points over the range.

For a given diaphragm shape, the general solution by the RungeKutta method applied four times on SEAC (with floating decimal point

routine) required six hours total, plus another hour to form the linear

combination satisfying the outer boundary conditions and to compute

stresses from displacements.
20

Status of Projects

The method is equally applicable to any shapes and patterns of

corrugations and is now available for practical design calculations for

various shapes, thicknesses, and loading conditions. These results will

be presented in a talk by R. F. Dressier at the national meeting of the

American Society of Mechanical Engineers in Chicago, November l4-l8.

For the vibrating triangular wing with two free edges, R.F. Dressier

and W. Cahill have now formulated this eigenvalue problem as an order 7

8

unsymraetric real matrix for the cases: aspect ratio r = 2, Poisson's ratio

(T = . 25 , .29, and .3?, These formulations were each based upon 20 different

stencil patterns defining the biharmonic operator in finite difference

form, modified in various unsymmetric ways by the free edge and free corner boundary conditions. The lengthy calculations to evaluate these stencils were done by D. Liepman and A. C. Bruno. A SEAC code for the lowest

eigenvalue for unsymmetric matrices up to order 96 has been programmed

and written by I. Rhodes and U. Hochstrasser „ This method computes at each step with a 19th-degree hypergeometric polynomial function of the

matrix in order to improve eigenvalue ratios to accelerate convergence.

The calculation produces the bi-orthogonal families of left-hand and righthand eigenvectors. In some respects this code may be considered as one

of the most complicated and difficult ones thus far successfully used

on SEAC.

Results for the lowest frequency and mode obtained thus far

for the case r = 2, <T = .29, are as follows: In scaled dimensionless

notation the frequencies (rounded) are as follows:

computed from order 28 matrix .085

computed from order 45 matrix .079

computed from order 78 matrix .074

experimental result of Gustafson, et al. .066

The order 78 result is tentative since the computation has not completely

converged yet. The present value differs by 12 percent from the experimental,

but the latter is too low since it contains air loading and imperfect

clamping effects which might easily explain the full discrepancy. The

computation requires 30 minutes per iteration and produced the .074 value

after six iterations. The previous unaccelerated code required several

hundred iterations for the order 45 matrix compared with 18 iterations

by the new method. In spite of the unsymmetric formulation, it is seen

that order 45 is clearly large enough for the first mode, but probably

order 78 will be necessary for accurate determination of higher modes.

I. Rhodes is now working on the code extension to obtain higher

modes. Also, results for other values of r and 9" have yet to be calculated.

Publication:

(1) Properties of currugated diaphragms. R. F. Dressier. In manuscript.

FOURIER TRANSFORMS OF PROBABILITY DISTRIBUTION FUNCTIONS

Task 1104-10-5160/56-154

Origin: NBS Authorized 9/30/55

Sponsor: Office of Naval Research

Managers: F. Oberhettinger , J. Lieblein

Objective: To compile a comprehensive table containing a number

(about 1000) of Fourier transforms for a variety of different probability

distribution functions. This will include not only the more commonly

used material but also a large number of hitherto unknown results.

Background: Although the book "Tables of integral transforms,
Status of Projects 21

vol. I, by A. Erdelyi, W. Magnus, F. Ober hettinger and F. G. Tricomi,

(McGraw-Hill Book Co., New York, 1954), containing a few transforms

of probability functions is available, there is need for a more adequate

and complete listing in a separate publication pertaining only to those

functions which occur as probability distribution functions.

Status: NEW. F. Ober hettinger has generated some transforms of

probability functions which were hitherto underived, and the systematic

compilation is in progress. J. Lieblein is engaged on an extensive

literature search for other transforms.

RESEARCH IN FLUID DYNAMICS OF TWO-PHASE FLOWS

Task 1104-10-5160/56-155

Origin: Office of Naval Research Authorized 9/30/55

Sponsor: H " " Manager: R. F. Dressier

Objective: To attempt the construction of mathematical models

to isolate the basic phenomena related to the issuance of a steam jet

into water, and to solve the equations governing each particular model.

In particular, the problem will be considered from a one-dimensional and

from a two-dimensional point of view. In the one-dimensional case,

attention will first be focused on the following: (1) The derivation at

a condensation interface of discontinuity conditions based upon continuity,

momentum, and energy considerations, and their application to the problem

of quasi-one-dimensional steady discontinuous flows of vapors in ducts

and jets. (2) The study of shocks in steam, taking into account the

non-polytropic variation of the thermodynamic properties of steam, and

the question as to whether separate steam shocks must exist in a supersonic

steam flow before a condensation interface can occur. (3) The study of

the stability of such condensation interfaces. (4) The study of steam

shocks and condensation interfaces considered as zones of finite width.

This treatment would include an analysis from the viewpoint of kinetic

theory and statistical mechanics.

Background: A better understanding of the essential physical

phenomena governing condensible jet flows is desirable. Also the study

of basic shock phenomena in steam has hitherto received comparatively

little attention. The investigation described above is designed to

combine both these objectives.

Status: NEW.

RESEARCH IN MATHEMATICAL GEOPHYSICS

Task 1104-10-5160/56-156

Origin: The American University Authorized 9/30/55

Sponsor: Office of Naval Research

Manager: R. Stoneley

Objective: To investigate in detail the static yielding of the

earth to forces of tidal type, and also to determine the modes of

oscillation and the corresponding periods for the earth using the known

distributions for its density and elastic properties.
22 Status of Projects

A related problem to be investigated is the stability of

gravitating elastic spheres, especially applied to the problem of the

inner planets.

Background: A knowledge of the deformations is required, for

instance, in order to estimate the elastic recovery of the earth after

the melting of snow caps. The periods of oscillation are of particular

interest in the interpretation of earth-tide observations and the longperiod oscillations shown in seismograms.

This research is being conducted by Dr. R. Stoneley of the

American University, on leave from the University of Cambridge, England,

with the sponsorship of the Office of Naval Research. The National Bureau

of Standards is according guest worker status to Dr. Stoneley and is

making its facilities available for the performance of this task.

Status: NEW. A monograph on elastic surface waves is in preparation

by R . Stoneley.
5. MATHEMATICAL AND COMPUTATIONAL SERVICES

1102-40-5126/50-13 RAY TRACING

Origin and Sponsor : NBS, Section 2.2

Manager : E. Marden

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

Status: Inactive

1 102-40-51 26/51-37 MOLECULAR STRUCTURE, III

Origin and Sponsor : Naval Research Laboratory, USN

Manager : P. O'Hara

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

Status : Continued. The following calculations were made for the

dimethoxybenzophenone crystal: The observed intensities were corrected

for length and obliquity, resolution and polarization. After making

a correction for vibrational motion the normalized structure factors

(E) were computed. The next portion of the problem involves the

determination of the sign of E.

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

Origin and Sponsor : NBS, Division 9

Manager : A. Futterman, R. Zucker

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

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

following cases: (1) tetragonal crystals: Hg(CN) 2

, PbWO^, (NH^)

2

, PdCi-L,.? BaMoO^, CdMoO^, Ce^MoO^)^, BaWO^, SrWO^; (2) hexagonal crystals:

Si2Ta, Ti^Si^, Na^iF^, VaSi

2

, NbSi

2

, Sbl-^; (3) orthorhombic crystals:

humite, TlNO,, MgSiO-, Cs^rO^, NH^CLO^, ZrSi

2

, CaC^O^H^, (NH^SO^,

KCNS , Rb 2 SOl+ . Redetermination of unit cell constants by least squares fitting

to measured d-spacings was carried out for the following crystals:

(1) Tetragonal crystals: BaWO^, SrWO^; (2) hexagonal crystals: TaSi2

, Ti^Si^, Sbl^, Bil^, Na

2 SiF^, Asl^; (3) orthorhombic crystals: NH^C^O^,

MgSiO^ , ZrSi2

.

1 102-40-5126/53-27 COMPUTATION OF THERMODYNAMIC FUNCTIONS

Origin and Sponsor : NBS, Section 3.2

Manager : E. Marden

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

Status : Continued. Calculations were performed using new physical constants

for approximately 12 molecules for which thermodynamic functions have

previously been computed; among them are the isotopic hydrogen cyanides

and dichlorodif luoromethane.

23
24 Status of Projects

1102-40-5126/53-51 RADIATION DIFFUSION

Origin : NBS, Section 4.8

Sponsor : Atomic Energy Commission

Manager : J. Doggett

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

Status : Continued. An exact expression for the elastic scattering of

electrons by nuclei was derived by N. F. Mott in 1929. Because of the

complexity of the expresion, it has been evaluated for only a limited

number of cases by hand computation to date. The formula involves

series in Legendre (associated) polynomials, the coefficients of which

depend on the Gamma functions with complex arguments. The existing

tabulations of the Gamma function are not sufficiently fine in argument

to avoid interpolation difficulties. So a code accurate to 8 significant

figures was designed. At present the entire numerical process has been

coded, and results for some light elements have been obtained.

1 102-40-51 26/54-13 AWARD OF PROCUREMENT CONTRACTS BY LINEAR PROGRAMMING

Origin and Sponsor : New York Quartermaster Procurement Agency

Manager : H. Bremer

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

Status : Continued. Analysis of contract award problems continued to be

processed on SEAC as submitted.

371 1 -60-0009/54-1 7 DE POLYMERIZATION

Origin : NBS, Section 7.6

Manager : J. Bram

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

Status : Continued. A program has been prepared for the solution of the

system of N linear differential equations ^ = Aq where A =

^ a jj^

is

triangular, q = [qlj ; i,j - 1,2,...,N. Here,

a ii

= ~(1 -(r) ( 2 i— 1 ) = \

a

i , i+P

- ^d) * i''- 1 , 2 , . . . ,N 1. and q^(0) - 0 for i < N, q n (0) “ 1. The program produces the required

functions n

2 (i+1 )q ± (t) N

B(t) =

Pi

> C(t) =.1 -jjly £ (i+ 1 ) q i

( t ) , and & , Z q,(t) 1-1

i=1 1 but not the functions q^.

Defining the triangular matrix P by the requirement that

q(t) = P {

e

1

}

,

we have PL = AP, where L is the diagonal matrix,

L = j/U

)

• Letting a and b denote the constant vectors

a — (1,1,1,...,1),

h - (2,3»4,.„., N+ 1 )

,

N r A i t

] N r A.t

XI (i+1 )q . (t) = bP { e

}

; XZ q.: (t) = aP e i=1 1 J 1 l

we have
Status of Projects 25

The code was written to compute the vectors aP and bP; these

vectors were then used in a supplementary code to get the final answers.

As many as N=200 equations can be solved with this code. The running

time is one and three quarters hours. The arithmetic is done in fixed

point notation, with tests (and possible descaling) at various states

for overflow.

371 1-60-0009/5^-19 ENERGY LEVELS OF COMPLEX ATOMS

Origin : NBS, Section 4.1

Manager : W. F. Cahill

Full task description : Jan-Mar 1955- issue, p. 5-1 Status : Completed. The energy levels of complex spectra in intermediate

coupling including configuration interaction were computed for the low even configurations of the spectrum of singly ionized Tantalum (Tall).

Publication : Computation of atomic energy levels, the spectrum of singly

ionized tantalum. R. E. Trees (NBS Section 4.1), W. F. Cahill, and

P. Rabinowitz. To appear in the Journal of Research of the NBS.

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

Origin : NBS, Section 2.2

Manager : L. S. Joel

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

Status : Continued. Because of severe oscillations of the integrand

occurring in this problem, a new code is being written utilizing Filon's

method for quadrature of an oscillating function.

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

Origin : NBS, Division 4

Manager : S. Prusch

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

Status : Continued. The SEAC code for searching for pairs of lines having

a common difference was completed. Checking of the code is in progress.

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

Origin and Sponsor : Naval Ordnance Laboratory

Manager : J. Wegstein

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

Status : Terminated. No further calculations are contemplated at present.

The methods developed in this task are being applied in several problems

underway for the Thermodynamics Section (NBS 3.2).

1102-40-5126/54-35 THERMAL STRESSES IN STRUCTURES

Origin and Sponsor : NBS, Section 6.4

Manager : W. Hall

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

Status : Terminated. Continued work on this task has been absorbed under

task 1 102-40-51 26/55-123 (see p. 31 ).

371 1-60-0009/54-36 VELOCITY OF LIGHT

Origin : NBS, Section 11.3

Manager : J. M. Cameron

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

Status : Terminated. Additional third degree polynomials were fitted to data

submitted by the sponsor to determine rotational constants of OCS and 0^2*

In the future, this work will be reported under task 3737-60-0002/51 -1 of

the Statistical Engineering Laboratory (see p.39).
26 Status of Projects

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

Origin : NBS, Section 3.2

Manager : M. Paulsen

Full task description : Jan-Mar 195*+ issue, p. 48

Status : Inactive. For status to date, see Apr-Juoe 1955 issue, p. 12.

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

Origin and Sponsor : NBS, Section 3.2

Manager ; K. Goldberg

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

Status : laactive

1 102-40-51 26/54-43 CHARACTERISTICS OF CONDUCTING RESISTORS

Origin and Sponsor : NBS, Section 12.1

Manager : B. Gill

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

Status : Inactive. For status to date, see Apr-June 1955 issue, p. 12.

371 1-60-0009/55-53 ELECTRONIC FUNCTIONS

Origin : NBS, Section 4.5

Manager : W. G. Hall

Full task description : Apr-June 1955 issue, p. 12

Status : Continued. Computations have been completed, and the results were transferred to punched cards.

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

Or igin and Sponsor : NBS, Section 82.1

Man a qe r ; B. Gill

Full task description : Jan-Mar 1955 issue, p. 17

Status : Inactive. For status to date, see Jan-Mar 1955 issue, p. 17,

1 102-40-51 26/55-61 ELASTIC CROSS SECTION FOR NEUTRON SCATTERING

Origin and Sponsor : Naval Research Laboratory

Manager : I. Stegun

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

Status s Inactive. For status to date, see Apr-June 1955 issue, p. 13.

1 102-40-51 26/55-64 HIGH- TEMPERATURE THERMODYNAMIC TABLES (FMT)

Origin and Sponsor : NBS, Section 3.2

Manager : J. Wegstein

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

Status : Terminated. A flow diagram has been prepared; no further work on programming or computing is contemplated at this time. The task was established originally with the intention of affording a check on parts

of task 1 1 02-40-51 26/54-4, High-Temperature Properties of Water

With the completion of the latter task, the present one has lost its urgency.

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

Origin : NBS, Division 30

Manager : H. Bremer

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

Status : Inactive. For status to date, see Oct-Dec 1954 issue, p. 19.
Status of Projects 27

371 1-60-0009/55-68 CRYSTAL structure calculations

Origin : NBS, Division 9

Manager : A. Futterman, I. Stegun

Full task description : Jan-Mar 1955 issue, p. 18

Status : Continued. Work on this task is continuing in cooperation with

D. R. Prosen of the Mineral Products Division, who is pursuing a National Research Council-National Bureau of Standards Research

Associateship. Dr. Prosen has developed a code for the calculation

of structure factors for space groups of any symmetry, which includes

anistropic or isotropic temperature factors.

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

Origin : NBS, Section 7.6

Manager : K. Goldberg

Full task description : July-Sept 1955- issue, p. 19

Status : Completed. The paper explaining the method used to prove the

reality of the roots of certain matrices and applying this method to the

matrices arising in the theory of dielectric relaxation was completed.

Publication : Matrix transformations and properties of characteristic

roots. K. Goldberg. In manuscript.

1 1 02-40-51 26/55-75- LIQUID-VAPOR TRANSITION, II

Origin and Sponsor : Naval Medical Research Institute

Manager : S. Prusch

Full task description : Jan-Mar 1955 issue, p. 19

Status : Continued. Solution of the integral equation on punched card

machines continued for two more values of the parameter and two given

sets of functional values.

371 1-60-0009/55-75 PARAMETER OF THE DISPERSION EQUATION FOR OPTICAL GLASS

Origin : NBS, Section 1 .6

Manager : R. Zucker

Full task description : Jan-Mar 1955 issue, p. 20

Status : Continued. Computations were performed to determine the refractivity

of KRS-5.

3711-60-5126/55-81 COMBINING TESTS FOR SIGNIFICANCE

Origin : NBS, Section 11.3

Manager : L. S. Joel

Full task description : Oct-Dec 1955- issue, p. 23.

Status : Continued. Checking of the code continued.

371

-60-0009/55-82 THERMOMETER CALIBRATIONS

Origin : NBS, Section 3.1

Manager : S. Prusch

Full task description : Jan-Mar 1955 issue, p. 20

Status : Continued. Calibration tables were run on punched card machines

for approximately 48 thermometers under test.

371

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

Origin : NBS, Section 6.7

Manager : P. J. Walsh

Full task description : Jan-Mar 1955 issue, p. 21 Status : Completed. Using empirical formulae similar to the following:
28

Status of Projects

Ko=(.60l4-.01352D~ lA ) + (375-.07257D~ lA

) J g ‘?°02 ^ +/3 4

+ 1.5,3

[D (i + .00250 / /

b = .0002 + + (.0038 + *^--)//32 + (16.5+5D) /0

l6

j

,

a code was written and checked to evaluate the flow coefficients

K-K + bA for both throat and vena-contracta taps. The range of the

parameters was 2.067 < D < 15.25 (10 values),

.1 < [i < .82 (42 va lue s )

,

0 < A < 44.7214 (21 values).

The results have been submitted t© the sponsor.

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

Origin and Sponsor ; NBS, Section 9.7

Manager ; J. Brans Full task description ; Jan-Mar 1955 issue, p. 22

Status ; Inactive. For status to date, see Jan-Mar 1955 issue, p. 22.

1102-40-5126/55-88 STRESSES IN A WALL FOUNDATION

Origin and Sponsor ; NBS, Section 10.1

Managers ; E. Marden, G. Hawkins

Full task description ; I an-Mar 1955 issue, p, 22

Status : Inactive. For status to date, see Apr-June 1955 issue, p. 15.

371 1-60-0009/55-89 TEMPERATURE DISTRIBUTION IN SOLID WINGS HEATED

AER(DYNAMICALLY

Origin ; NBS, Section 6.4

Manager s P. Davis, W. F. Cahill

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

Status : Terminated, in favor of tasks of higher priority.

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

Origin s NBS, Section 4.5

Manager s W. G. Hall

Full task description : Jan-Mar 1955 issue, p. 24

Status i Completed. Four additional trajectories were computed, and the

results were transmitted to the sponsor.

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

Origin and Sponsor ; Bureau of Supplies and Accounts, Dept, of the Navy

Managers : G. Hawkins, E, Marden

EH-LL-t ask description ; Jan-Mar 1955 issue, p. 25

Status : Completed. Data for the month of April were run on UNIVAC and

results were transmitted to the sponsor. A similar but new problem

superseding this one has been formulated and is reported as task

1102-40-5126/56-152, p.38.
Status of Projects 29

11 02-^0-51 26/55-1 03 MISSOURI RIVER PROBLEM

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

Manager : H. Bremer

Full task description : Jan-Mar 1955 issue, p. 26

gtatus : Continued. The codes have been rewritten, for shorter machine time,

and checked out.

1102-40-5126/55-105 AERONOMIC TIDAL WINDS OF THERMAL ORIGIN

Origin and Sponsor : NBS, Section 82.1

Managers : B. Gill, H. Howe

Full task description : Jan-Mar 1955 issue, p. 26

Status : Inactive. For status to date, see Jan-Mar 1955 issue, p. 26.

1 102-40-51 26/55-1 13 REACTOR DESIOSI

Origin : Westinghouse Atomic Power Division

Sponsor : Atomic Energy Commission

Manager : U. Hochstrasser

Full task description : Jan-Mar 1955 issue, p. 28

Status : Continued. The code for finding the solution of the difference

equations of the test problem mentioned in the April-June 1955 issue,

p. 1 7? has been completed and sections of it have been checked out.

In addition, several smaller codes have been written to load initial

flux data, to convert and read out the final flux distribution, and to

set up the iteration process of solution.

The coding of the iteration has been done in a general fashion

so that comparisons between different methods of solution may be made

easily and accurately. Accordingly, it is possible to "sweep" the mesh

in any direction whatsoever, without changing the code, and to change

the method of solving the difference equations with a minimum of coding.

1102-40-5126/55-115 ADSORPTION INTEGRALS

Origin and Sponsor : NBS, Section 3.1

Manager : J. Cooper

Full task description : Apr-June 1955 issue, p. 17

S’tatus : Completed. The code used to compute the integral

rv+Av

T0(V) = -2.2 / SO//7T,.

y sin y exp (— -

p ? )dx

x+ar oiJ-AV

was modified to allow computation of the following integrals:

'V+AV

T

1 (v) =

T

2

(l/) =

-2 2

y sin y exp [-a ^

Si 04A

V-Lir u+Av

i=1 (x-x

± ) + oU

;] dx

2

3 S. OC./TT

exp [— 22.670''— x) -a

1/-AV i=1 (x-x

±

) 2+ O^2 ] dx

where S^, ot

, Av , x^ are input parameters, and y=(i'-x)7r/A v . .2 T

3

(J/) =

1 v +Av

V-AV

(x-x )' exp (-22. 67 -b — — ln2) dx
30

CP+AP

Vi/) =

J P~AP

rp +AP

T^(p) =

<< io

Status of Projects

JA2L2

(p-x) 2+(a y) 2 exp (-

x

2+ .16

2-2 -14,0 x In exp (_ 2CJUC, dx

(l'-x)>(4)l^ .16

dx

where A/ and Al* are input parameters.

T-j (p) was evaluated for one set of input parameters for

P - -2. 4( .04)2.4. T

2

(l') was evaluated for seven sets of parameters

for P- -2. 4( .04)2.4. was evaluated for nine sets of parameters

for P = -2, 4(. 04)2.4. T^Ci^) was evaluated for three sets of parameters

for V = -2. 4(.04)2.0(. 25)5.0. was evaluated for one set of

parameters for P- 0(„04)2.0(. 25)5.0. In addition, T

0 (V) was evaluated

tor U = 0(.04)2.0(. 25)5.0 for eight new sets of the parameters a and AP

All calculations were multiplied by a normalizing factor which was either a constant or a constant times the parameter AV . The results were submitted to the sponsor and have been used

by H. J. Koskowskl of the Temperature Measurements Section (3.1) in a study of slit function effects in determining infrared widths and int' isities

.

1102-40-5126/55-117 ATTENUATION OF PRESSURE PULSES OF FINITE AMPLITUDE

Origin ; NBS, Section 3.2

Sponsor ; Bureau of Aeronautics, U. S. Navy

Manager : M. Paulsen

Full task description : Apr-June 1955 issue, p. 18

Status : Continued. Four subroutines for computing the particle velocity

p, and sound velocity a at the boundaries were completed and checked out.

1102-40-5126/55-118 THERMOMETER CALIBRATION

Origin and Sponsor : NBS, Section 3.1

Manager : B. Gill

Full task description : Apr-June 1955 issue, p. 18

Status : Continued. It has been established that a sixth degree polynomial

is satisfactory for predicting temperature-resistance differences. The

code for computing calibration tables has been further refined and

checked out.

1102-40-5126/55-121 ELECTRON PENETRATION

Origin : NBS, Section 4.8

Sponsor : Atomic Energy Commission

Managers : P. Walsh, A. Fatter man Full task description : Apr-June 1955 issue, p. 19

Status : Continued. The code which computes the electron depth distributions

of elements for isotropic cases has been checked, and results for polystyrene

for source energies ranging from .1 to 2.2 have been submitted to the NBS

Nuclear Physics Section.

1 102-40-5126/55-122 SOLUTION OF NORMAL EQUATIONS

Origin : Advisory Committee on Weather Control

Sponsor : U. §. Weather Bureau

Manager : M. Newman

Full task description s Apr-June 1955 issue, p. 20

Status : Continued. Eight new rues were completed with revised data from

the sponsor.
Status of Projects 31

1102-40- 5126/55-123 TEMPERATURE DISTRIBUTION

Origin : NBS, Section 6.4

Sponsor : NBS, Section 6.4

Manager : W. G. Hall

Full task description : Apr-June 1955 issue, p. 19

Status : Continued. The code was completed and checked.

1 102-40- 5126/55-124 EVALUATION OF INTEGRALS RELATED TO GAS FLOW

Origin : NBS, Section 3.2

Sponsor : Bureau of Aeronautics, U. S. Navy

Manager : J. Cooper

Full task description : Apr-June 1955 issue, p. 20

Status : Completed. The results as reported in the previous issue, p. 20, were transmitted to the originator.

1102-40- 5126/55-126 AERODYNAMIC HEATING

Origin and Sponsor : Diamond Ordnance Fuze Laboratory, Dept.

Manager : B. Walter, R. Capuano

Objective : To solve the difference equation

ATW

TX AW

Bh At

e

T

W

+ *

of the Army

as a function of time, where T. Ii are given functions of time

e Solutions are desired

for Tyy ao a luutiiuu ui txuic , nucic A AW*

and B depends on initial conditions and on T^

for different intervals At and initial conditions.

Background ; The unknown function Tw

represents the temperature of a thin plate moving through air.

from

¥he recovery temperature T^ is determined

taw = 75.0 r * V

where V, the velocity, and T

Q

, the ambient temperature, are given functions

of time. B is the reciprocal of the heat capacity per unit surface area of

the plate and is given by the equation

B =

1 ycT ’

where <f = density,

function h , the

e

7

c = specific heat, = thickness of the plate. The

convective heat transfer coefficient, is calculated from

the last factor beingthe effective convective heat transfer coefficient at

V = 1 . The symbol over T^w

, B and h

g indicates use of the mean of the

values at the beginning and end of the time interval At.

Status : New. The aerodynamic heating T

w was calculated for four missile

trajectories, with each trajectory calculated for four different values

of heat capacity (1/B).
32 Status of Projects

1 102-40-5126/55-127 VIBRATIONS OF CIRCULAR DISC

Origin and Sponsor ; Evans Signal Laboratory

Manager : U. Hochstrasser

Objective : To determine the solutions y of the equation

o-/>tt2 + -|(3 + '-? « 2 - f

- ^

2

>t> 5?r2r3

-<* 'p - ^)r2 -ao + -pr, + /

2

> o

-T—i J

0 (Ci7)

where

V

i (/) , i J

1 (C.,7) = 1,2,3; c

1 = 1

; C

2=/3; C

3 = /3

a (1 -v)

/ 2 2 2

a = V"^1- +/3- ^

5 b -

^ — ; e = Q - ft

; J , J 1 are Bessel functions

(i-v)(i + g) 1+g l + g

0 1

of the first kind; g,v are given constants; and is given by the relation

7 = [

1 - /3 2 where & is a parameter 1 < & < 1.1.

e g(i + /3 2

) 2

Also, the solutions of the corresponding equation have to be found for

the case where £ < 1, and therefore ft is imaginary. Only those solutions

are required for which

0 < i

= 1 a ^< 200.

Background : The above equation is the frequency equation for the vibrations

of circumferential order unity of a circular disc. & is the dimensionless

resonant frequency, d the diameter, and h the thickness of the disc.

Status : New. Two subroutines for computing have been written and tested

on the SEAC. A code has been set up for solving the equation 2c 3 y Ty* - 7

2 for 30 values of 6.

1102-40-5126/55-128 GROUND REFLECTION COEFFICIENTS

Origin and Sponsor : NBS, Section 82.1

Managers : B. Gill, H. H. Howe (82.1)

Full task description : Apr-June 1955 issue, p. 21 Status : Completed. Extensive tables of ground reflection coefficients for

electromagnetic waves, totalling two hundred ten pages, have been calculated

and were transmitted to the Boulder Laboratories (82.1).

1 102-40-5126/55-129 PROCESSING OF PUBLIC HOUSING DATA

Origin : NBS, Section 12.1

Sponsor : U. S. Public Housing Authority

Manager : I. Rhodes

Full task description : Apr-June 1955 issue, p. 20

Status : Completed. The results of trial runs have been transmitted to the

sponsoring agency where they are used as a basis for decisions on future

use of automatic computers.
Status of Projects 33

3711-60-0009/56-131 RAY TRACING

Origin : NBS. , Section 2.2

Manager : E. Marden

Objective : To compute the wave front and the vignetted entrance pupil

of an optical system.

Background : This is an extension of the ray-tracing computations referred

to in the Jan-Mar 1950 issue, p. 33 (task 11. 2/3

— 50—13). In that work,

individual rays were arbitrarily selected and traced through the system.

The principal purpose of the extension is to determine the size and

shape of the vignetted entrance pupil and the wave front formed by the

ensemble of such rays. A polynomial is determined which is physically

interpreted as the distance between the actual wave front and the ideal

reference sphere.

Status : New. The code falls naturally into three sections. A code has

been written for the third part and is in the process of being debugged;

it uses test data instead of calculated results from the first two parts

of the code.

1102-^0-5126/56-133 COMPLEX EIGENVALUES

Origin : Moore School of Engineering

Sponsor : Office of Naval Research

Manager : I. Stegun

Objective : To obtain on SEAC the largest complex eigenvalues for various

order matrices.

Background : The problem arises in a research program conducted by the

University of Pennsylvania for the Special Devices Center, U. S. Navy.

The matrices are the coefficients of linearized versions of the equations

of flight of a particular airplane when that plane is in certain extreme

modes of flight. These modes of flight represent the worst conditions

under which the plane is expected to fly, and the eigenvalues represent

the "natural resonant frequencies" (both damping and oscillation) of the

plane under those conditions. The stability chart criterion, developed

at the Moore School of Electrical Engineering of the University of Pennsylvania,

enables one to choose a numerical method and a computation interval for

solving differential equations, provided one knows the largest eigenvalues

of the linearized system. Thus, the computations of eigenvalues will be

used to justify the choice of a particular method and interval.

Status : New. Largest complex eigenvalues have been obtained for four

matrices of order 10 and two matrices of order 6. The eigenvalues are found from the limiting form of high powers of the matrix. If E represents the value of an element in some arbitrary fixed position in

yu

r (where y is an arbitrary initial vector), then if the roots of greatest

modulus are Aj = u+iv and Ag = u-iv, then is found by solving the equation

t 2 2 \ (u + v )E 2 uE _^ 1 + E _0 = 0

nH- 1 m+2

for two successive values of m. As a check, the equation is solved for the same m using different elements.
3^ Status of Projects

1 102-40- 51 26/56-1 3^ TRANSMISSION-DELAY TIMES

Origin : NBS, Section 82.1

Sponsor : NBS, Section 82.1

Managers : B. Gill, H. Howe (82.1)

Ob iective : To compute a three-argument table, showing the transmission-delay

time as a function of n, the number of hops; of H, the height of the

reflecting layer; and of D, the surface distance between transmitting

and receiving points. Specular ray-reflection is assumed; it is also

assumed that the speed of the wave is constant throughout its path and

that the earth is spherical. TWo other tables are also to be computed,

one showing 0 , the angle between the ray and the vertical at the point

of reflection; and one showing /? , the angle between the ray and the

horizontal at point of transmission or reception. The formulas involved are

Delay time = (2R n/c) [\/2(vers y)(1+Q) + Q

2 -y ]

,

0

o tan sin y vers y + Q

P = - 0o - y,

wnere

y - D/2nR, Q

= H/R.

Here R is the radius of the earth, and c is the speed of light.

Background : If a short pulse of electromagnetic energy is transmitted, it

may sometimes be received at a distant station by reflection from the

ionosphere one or more times, or by a ground wave traveling around the

surface of the earth. The sky wave will arrive later than the ground wave

by an amount called the transmission-delay time. If certain assumptions

are made, the transmission-delay time can be computed, or, conversely,

an observed time can be used to check assumptions about the transmission.

Status : New.

1 102-40-51 26/56-136 CALCULATION OF WAVE FUNCTIONS BY HARTREE METHOD

Origin : Naval Research Laboratory

Sponsor : " " " Manager : S. Peavy

Ob iective : To solve the Schroedinger equation

[E - V(r) ] l|r = 0

V(r) = |

(e

-,t(r+ z)_

e -/i I r-z | ) z d2>

The problem can be simplified by the substitution ^ = R/r. The only

desired solutions are those for which lim R/r - 0.

Background : The problem arises from an EtTer^t to calculate the best one- particle wave function for various nuclei. It was proposed by R. Jastrow

and J. Aviles of N.R.L.

Status : New. The code is completed and checked out.
Status of Projects 35

1102-40-5126/56-137 STABILITY OF SUPPORTED PLATES

Origin and Sponsor : National Advisory Committee on Aeronautics

Manager : R. Zucker , > Objective : To evaluate the stiffness parameter "'l/ b^/^D by solving for z in the equation

z(

2 ' An S

n nn .

V d ') 5=

Vn

d

V- An U„ X

Y n n t— d

V *nV

d

z

V.

d Z

S

n

d

v vd y Vd Z W

d E V

d y

Unxy

d

z W

d Z V

d

y- un*y

d

A

n = 1 +1(*)’. z = 4

r . " D

2 C^+Yg) 1"3 4 ’

S

n = An

2

d = S

n

2 -«„

2

U

n = 1

k

,8 /3 3

» x = sin nj

r Z

.00

n=1 ?3 j 5, .

1 • y = cos

n7r r

Values are required for various given combinations of k

c ,k

g ,/3, and

r=2,3. When r=2, ^^=0, and only the first three rows and columns of the

determinants are used; when r=3

= ^2 and tbe ent;*- re determinant is used.

Background : The computations arise in connection with the stability analysis

of supported plates under direct compression and shear loads. \|r^ b3/7r +D is

a non-dimensional deflection restraint parameter. For the theory, see NACA

Technical Note 2987, "Charts Relating the Compressive Buckling Stress of

Longitudinal Supported Plates to the Effective Deflectional and Rotational

Stiffness of the Supports," by R. A. Anderson and J. W. Semorian.

Status : New. Computations for the case r=3 were completed; those for r=2

are in progress.

1 102-40-5126/56-138 CRYSTAL COUNTER EFFICIENCY

Origin and Sponsor : Naval Research Laboratory

Manager : L. Joel

Objective : Given a cylindrical sodium-iodide counter of radius R and length b placed on the polar axis of the coordinate system (r,9,0), with an isotropic

y

source S placed at the origin, a distance a from the near end of the

counter. To compute the counter efficiency t for every combination of 18

crystal sizes; 21 values of <T, the gamma absorption cross section in cm^/g,

and for 12 distances a:

6 = (1 - cos 0-,)" 1 jl - cos 0

1 -J

2 sin 0 e _M<rb sec G d0

sin 0 e -Mcr(R esc 0 - a sec 0) d0
36

9-j _ t® 11 ' where g?

Status of Projects

tan Ra+b ?

M is the crystal density in q/cw? = 3.66.

BackgrouEd s These computations are being made in connection with a study to obtain more reliable standards needed in the evaluation of

crystals as radiation counters. The problem was suggested by R. Jastrow,

F. Brooks, and A. Wolicki of the NRL.

Status : Completed (New). The problem was coded and the computations were

carried out—yielding a total of 4536 values. The results have been

transmitted to the sponsor.

1102-40-5126/56-139 STUDY OF 3NTERNUCLEAR POTENTIAL FOR H,

Origin and Sponsor s NBS, Section 3.2 ^ Manager : E. Haynsworth

Objective : To find solutions C. and £. of the simultaneous secular

equations

(K+G^C^ = £* S' Cf

(H+G^K-f = £f S cf ’ (1)

( H+ G*)

C

2 ~ S C

2 where H and S are given 3x3 symmetric matrices, G

a and are 3x3 symmetric

matrices whose elements are functions of the vector components C4 and . These vector components obey two scaling equations and in addition, the

equation:

C

1

1

C

1 2+Sab^ C

1

1

C

22+C

21 C12 ')+C 21 C

22+S bc^ C

31 C

22+C 21 C

32 ^

+C

31 C32+s

ac

(c

31

C

12+c11 c32 ) 0 ( 2 )

Trial vectors C(f and which are properly scaled and obey equations (2) are assumed, and from these the G'*' and G''3 matrices are calculated and the

secular equations (1) are solved. This gives new vectors C, and the process

is repeated until the vector components assumed are equal to the vector

components obtained.

Background : The equations arise from a project involving the study of the

binding energy; the internuclear potential of the system is obtained as

a function of internuclear distance and angle by the solecular orbital selfconsistent field approximation. The activation energies of the exchange

reactions involving atomic and molecular hydrogen and its isotopes will then

be computed. A more realistic internuclear potential function for the

calculation of the second virial coefficients and transport coefficients of

hydrogen and its isotopes may result from the calculations.

We consider the interaction of three hydrogen atoms separated by

the distances R,Jt' where (1) S=R 1 or (2) R/R 1 . We want to calculate the

binding energy of the triatomic hydrogen system. To do this we assume that

the ground state energy of the system is given by the Schroedinger wave

equation, HiJr= , where H is the Hamiltonian operator of the system, S'

- is

the wave functions describing the system and E is the energy corresponding to

\lf„ We assume that is formed as the antisymmetrized product of molecular

spin orbitals:
Status of Projects 37

'l'] \lfj ^3

\|/ = -7= vlr* \|f |

vir|

•T3T 1 2 3 ^

3 ^

3 ^3

where = 0] ( 00) , tlr2 = 0^), vl/3 = 02 (a');

0c and /3 are orthonormal spin functions,

0i* t c ji*r

and the j*s range over all atomic orbitals. The X3 are IS atomic functions

on the hydrogen atoms a,b and c, and the 0

^ are linear combinations of

these functions forming orthonormal sets of molecular orbitals.

h=;thu + £ y ~r~ + v(r),

U UFV U^

H" = -iv 2

a - EP

p ranges over all nuclei.

On substitution of \Jf into the Schroedinger wave equation and by taking

advantage of the orthonormality properties of the 0^ and yj s, the energy

of the system is found to be

E

E

* H \J/- dr

<*»/3 H. + £ L.

i) j ij t ) K

ij

<M-/3 where ®®®®s summation over molecular orbitals associated with spin functions

<*» and yd, where and are functions of the coefficients, and where * denotes the conjugate transpose. The energy corresponding to the ground

state is obtained by varying the parameters of E to obtain minimum energy.

This corresponds to solving equation (1) for molecular coefficients until

self-consistency is obtained. The H-,J { and K. are then calculated and E is

found

References: (1) R. S. Mulliken, ONR Report 1947, University of Chicago.

(2) C. C. J. Roothaan, Modern developments in MO theory, Rev. Modern Physics

23 . 69(1951). (3) V. Griffing et al. J. Chem. Phys. 23 , (1955). (*+) G.Berthier,

J. Chem. Phys. (July-August 195^). (5) Pople and Nesbitt, J. Chem. Phys. 22,

(Letter to Editor) 195*+. The problem was proposed and formulated by Dr. B. Ransil, who is

pursuing a National Research Council-National Bureau of Standards Postdoctoral

Research Associateship in the NBS Thermodynamics Section (3.2).

Status : New. Preliminary matrix subroutines have been coded and checked out.

Coding of the overall problem is in progress.
33 Status of Projects

1 102-40-51 26/56-1 52 COST ACCOUNTING OPERATIONS FOR HIGH SPEED COMPUTERS, II Origin and Sponsor : Bureau of Supplies and Accounts, Department of the Navy

Manager : G. Hawkins

Objective : To program cost accounting operations of the originating agency

for performance on electronic computers; in particular, to distribute

monthly expenditures, unliquidated obligations, man hours, and work

measurement data to categorized accounts and suballotment functions; to

compute cost of labor per man hour; to determine, of the total Naval

Industrial Funds, the amount of such funds to be paid by the Bureau of

Supplies and Accounts, using as a basis for the computations certain

supplementary data reported by shipyard installations; to audit incorrect

expenditure accounts; and to compute quarterly summary reports.

Background : This problem is an outgrowth of task 1102-40-5126/55-102 (see

Jan-Mar 1955 issue, p. 25). After evaluating the information obtained

from the afore mentioned task, it was decided to increase both the number

of the different types of input data and the resultant output, much of

which could be secured through the accumulation of by-products.

The Bureau of Supplies and Accounts of the Navy Department in

Washington received monthly recapitulations of financial obligations and

expenditures for supply operations from more than 60 U. S. Naval supply

depots, U. S. Naval supply centers, U. S. Naval shipyards, U. 5. Naval

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

battalions, and other supply activities. There is also reported the

number of man hours employed and work measurement units produced at these

activities. These data are used for the administration of current funds

and as a basis for projecting budget requests for future requirements.

Status : New. Coding the problem for UNIVAC is in progress.
6. STATISTICAL ENGINEERING SERVICES

COLLABORATION ON STATISTICAL ASPECTS OF NBS

RESEARCH AND TESTING

Task 3737-60-0002/51-1

Origin: NBS Authorized 7/1/50

Managers: W„ J. Youden, J. Cameron

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

Status: CONTINUED. The following are representative examples of

statistical assistance^ and advice provided by members of the Statistical

Engineering Laboratory to a number of Bureau personnel during this quarter:

(1) Fitting of polynomials to atomic spectra data, using SEAC,

for E„ K. Plyler and H. C. Allen, Jr., of Section 5-. 2. (2) Analysis of data to determine effects of size and shape

on noise of composition type resistors, for G. T. Conrad, Jr., of Section

1 .6.

(3) Least squares determination of constants of the curve relating velocity of sound in water with temperature, and statistical

analysis to determine error of measurement process and appropriate degree

of a polynomial, using SEA£, for M„ Greenspan of Section 6.1.

(5-) Multiple regression analysis of data on amplifier circuits^

for J. H. Muncy, of Section 6.1.

(5) Design of experiment for intercomparing standards of

luminosity, for R. Teele, of Section 2,1.

(6) An experimental plan was devised for the testing of a new device by personnel in DOFL. This plan involved seven factors at varying

levels totaling 5185 different combinations. The final experimental plan

required only one-fourth of these combinations. (This work was performed

on a reimbursable basis.)

(7) An analysis was carried out for J. Richmond of Section 9.5,

to determine how well experienced judges agree among themselves as compared

to a group of inexperienced judges with respect to qualitative ratings of

color samples.

(8) An experiment was planned to evaluate the effects of various

temperatures on Young's Modulus, for J. B. Wachtman of Section 9.1.

Publications:

(1) Variability of wool content in part wool blankets. E. Horowitz (7.5)

and W. S. Connor. ASTM Bulletin 208, 5-2-55 (Sept. 1955).

(2) More scientists. W, J. Youden. Submitted to a technical journal.

(3) National physical standards and design of experiments. W. J. Youden.

Submitted to a technical journal.

(5) A simplified calibration of platinum versus platinum-rhodium

thermocouples. B. E. Squires (3.1) and M. Zelea. In manuscript.

STATISTICAL SERVICES FOR COMMITTEE ON SHIP STEEL, NRC

Task 1 103-50-5105/52-1

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

Manager: W. J. Youden

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

Status: CONTINUED. A multiple regression study of the effect of

chemical composition on ship steel was begun.

39
40 Status of Projects

RESEARCH IN APPLICATIONS OF MATHEMATICAL STATISTICS TO

PROBLEMS OF THE CHEMICAL CORPS

Task 1103-40-51 18/52-1

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

Chemical Corps, Dept, of the Army

Manager: C. Eisenhart

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

Status: CONTINUED. The catalogue of fractional replicated designs

of the 2

n series has been extended to n=13> 14, 15, 16 factors, and to

fractional replicates of 1/64, 1/128, 1/25© of a full factorial. These

results are being combined with previous results for publication.

A code for computing

oo -h„ d f a* +

u(V'h) £ —L

' log de ~£ x

c

42 TT dt

has been written for SEAC by I. Stegun and R. Zucker (11,2) and the code was checked. A note describing the fitness of the code is being written.

The usual formulas for computing p{T > T }, where T is an observed value of some statistic such as t, F, X^, $tc., are nBt in the

best form for computation. Some work has been done on seeking an alternate

formula for the mechanization of such computations.

STATISTICAL ANALYSIS AND DESIGN OF EXPERIMENTS

FOR THE U. S. GEOLOGICAL SURVEY

Task 1103-40-5140/54-1

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

Department of Interior

Managers: C. Eisenhart, W. J. Youden

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

Status: CONTINUED. (1) Statistical studies were continued on the

sampling of fine-grained deposits in streams for use in a study of mineral

content for Dorothy Carroll of the Sedimentary Petrology Project of the

Geochemistry and Petrology Branch. (2) Some work was done on the statistical

aspects of the problem of splitting samples of crushed rock for F.J. Flanagan.

MANUAL ON EXPERIMENTAL STATISTICS

FOR ORDNANCE ENGINEERS

Task 1103-40-5146/55-93

Origin and Sponsor: Office of Ordnance Research Authorized 12/29/54

Manager: C. Eisenhart

Full task description: Oct-Dec 1954 issue, p. 28

Status: CONTINUED. Further conferences at Ordnance establishments

were held to determine the kind of manual which would be most useful to

Ordnance Engineers and to discuss the type of statistical problems that occur. A tentative outline of the manual has been prepared and drafts

have been made of some sections.
AP PLICATION

of

NATIONAL BUREAU OF STANDARDS AUTOMATIC COMPUTER (SEAC)

The record of SEAC operations for tasks of the Applied Mathematics

Division for the period July 1 through September 30 is as follows:

Task No . NBS : 1104/55-55

5116/55-56

51 1 6/56—1 4-8

1 110A7-2

1 1 10/55-9*+

5126/51-8

5126/52-44

5126/53-25

5126/53-27

5126/53-29

5126/53-48

5126/53-51

0009/54-1

0009/54-17

0009/54-19

0009/55-53

0009/55-65

5126/55-68

5126/55-81

5126/55-82

0009/55-86

5126/55-87

5126/55-92

5126/55-97

0009/55-99

5126/55-115

5126/55-1 17

0009/55-1 18

5126/55-121

5126/55-126

5126/56-128

5126/56-129

5126/56-134

5126/56-135

5126/56-139

5126/56-141

5126/56-145

0002/52-1

Title

Hours

Code

Checking

Used : Productive

Operations

Research in numerical analysis 8 3

Research in mathematical topics applicable

to numerical analysis 4 1 Nerve fiber reaction 1 Tables of Coulomb wave functions 16 3 Tide tables 9

Tables of power points of analysis of

variance tests 1 24

Calculations for d-spacings 3 9

Legendre function 1 Thermodynamics functions 1 3 Dynamic behavior of aircraft structure 5 6

Analysis of ionospheric data 19 33

Radiation diffusion l4 72

Matrix reduction 5 Depolymerization 2 3 Energy levels of complex atoms 1 8

Electronic functions 5 Automatic coding 8 3

Crystal structure calculations 9

Combining tests for significance 1 Thermometer calibrations 1 Flow coefficients for fluids 5 2 "Zero" method determination of crystal

structures 3 13

Cylindrical electron lens calculations 2 High temperature properties of air 3 59

Integrals of products of Bessel functions 1 Adsorption integrals 9 31 Attenuation of pressure pulses of finite

amplitude 8 1 Thermometer calibrations 2 1 Electron penetration 3 Aerodynamic heating 2

Ground reflection coefficients 3 2 Processing of Public Housing data 9

Transmission delay times 4 4

Evans hyperbolic charts 2 1 Study of internuclear potential for H 1 Spectro calibration ^2 1 Gamma-ray spectroscopy 9 2

Statistical aspects of NBS administrative

operations 7 37
42

Hours Used:

Task No. Title

Code

Checking

Productive

Operations

NBS

:

51 60/5 5-8

5

Research in mathematical elasticity 12 109

Misc . Least squares 4 1 n Determinants 6

n Matrix factoring, subroutines, etc. 2

ii Lommel function 1 n Roots of polynomials 3 Other:

1110/53-52 L-Shell conversion coefficients 2 76

5126/51-3 Equations of heat transfer type 1 5126/53-^5 Applications of game theory 4 2 5126/54-13

5126/54-44

Award of procurement contracts for linear

programming

Flight performance computations

20

4

5126/55-61 Elastic cross section for neutron scattering 2 2 5126/55-104 Fuse problem 9 20

5126/55-113 Reactor design 14

5126/55-119 Field rocket problem 10

5126/55-122 Solution of normal equation 3 5126/55-125 Matrix multiplication 3 34

5126/55-127 Vibrations of circular disc 10

5126/56-130 Aircraft responses 21 13

5126/56-133 Complex eigenvalues 3

5126/56-136 Calculation of wave functions 2 1 5126/55-137 Stability of supported plates 10 1 5126/56-138 Crystal counter efficiency 4 28

5126/56-142 Matrix problem 1 Totals: 288 £67
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.

Statistical Engineering Laboratory Seminar

HAMMERSLEY, J. M. (University of Oxford) Antithetic variate method in

Monte Carlo computations. September 1 6 .

Papers and Invited Talks

Presented by Members of the Staff

at Meetings of Outside Organizations

Papers presented at the meeting of the Association for Computing Machinery,

Philadelphia, Pa., September 1*+-l6:

TODD, J., and P. DAVIS. Conformal mapping.

WEGSTEIN , J. Accelerating convergence of iterative processes.

Talks presented at the Summer Seminar and Colloquium Series on Numerical

Analysis of the Numerical Analysis Research Group, University of

California at Los Angeles:

HENRICI, P. Evaluation of some definite integrals on SEAC: 1, July 15;

II, July 19. The quotient-difference algorithm of H. Rutishauser:

I, July 20; II, July 21.

TAUSSKY-TODD, 0. Integral group matrices. June 27.

TOE®, J. Topics in numerical analysis. A series of three lectures.

July 5,6,7.

* * * * *

CLATWORTHY, W. Incomplete block designs. Presented at the Department of

Agriculture, Washington, D. C. , September 1 6 . DAVIS, P. On the analytic continuation of mapping functions. Presented

before the meeting of the American Mathematical Society, Ann Arbor,

Michigan, August 30-September 3.

^3
kk Lectures and Symposia

HENRir.I, P. Addition theorems for solutions of partial differential

equations. Presented at the Seminar for Applied Mathematics and

Statistics, Stanford University, California, August 13.

Automatic computations with power series. Presented before the

meeting of the Association for Computing Machinery, Philadelphia, Pa.,

September 13-14.

HOFFMAN, A. and H. W. KUHN (Bryn Mawr College). Systems of distinct

representatives and other combinatorial applications of linear programming.

Presented at the meeting of the Institute of Mathematical Statistics,

Ann Arbor, Michigan, August 29.

YOUDEN, W. J. National physical standards and design of experiments.

Presented before the 29th Session of the International Statistical

Institute, Rio de Janeiro, Brazil, June 30.

Design of experiments in the physical sciences. Presented at a Biometric Meeting, Pampinas, Brazil, July 5. Dice, data, and deductions. Talk given to NBS summer trainees,

August 3.

Making good measurements. Presented before the Gordon Research

Conference of the AAAS, Meriden, New Hampshire, August 8. What is a statistician? Presented before the Lions Club,

Washington, D. C. , August 25.

Making good measurements. Presented at the Naval Gun Factory,

Washington, D. C., August 30.

Design of experiments in the physical sciences. Presented at

the Naval Gun Factory, Washington, D. C., September 7.
Publication Activities

1 . PUBLICATIONS THAT APPEARED DURING THE QUARTER

1.3 Technical Papers

(1) On the vibration of a square clamped plate. M. Abramowitz and

W. F. Cahill. J. Assoc. Comp. Mach. 2, 1 62—1 68 (July 1955).

(2) On the design of two-dimensional nozzles by the method of

characteristics. B. Chaix (Federal Institute of Technology, Zurich) and

P. Henrici. J. Aeronaut. Sci. 22, l4o-l42 (1955),

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

K. Fan. Proc. Cambridge Philosophical Soc. £l_, *+l4-h-21 (1955).

(4) How to solve a linear programming problem. A. J. Hoffman.

Proceedings of the Second Symposium in Linear Programming, held

jointly by the National Bureau of Standards and the U.S.A.F.,

Washington, D. C., January 1955, pp. 397-1+2It.

(5) Variability of wool content in part wool blankets. E. Horowitz

(NBS Section 7.5) and W. S. Connor. ASTM Bulletin 208, *+2-^5

(Sept. 1955).

(6) Begriindung fur die Beschaf tigung rait numerischer Analysis. J. Todd.

Jahresbericht d. Deutschen Mathematike r- Ver einigung £8, 1 1— 38(1955)

(Translation of "Motivation for working in numerical analysis.")

(7) Discrete approximations to elliptic differential equations.

W. Wasow. Zeit. angew. Math. Phys. 6, 81-97 (1955).

1.5 Miscellaneous

(1) Proceedings of the Second Symposium in Linear Programming, held

jointly by the National Bureau of Standards and the United States

Air Force, January 1955.

2. MANUSCRIPTS IN THE PROCESS OF PUBLICATION SEPTEMBER 30, 1955

2.1 Mathematical Tables

(1) Table of hyperbolic sines and cosines, x=2 to x=10, 9S. Applied

Mathematics Series 45. In press, U. S, Government Printing Office.

(2) Table of the descending exponential, x=2.5 to x=10. Applied

Mathematics Series 46. In press, U. S. Government Printing Office.

2.3 Technical Papers

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

Accepted, Pacific Journal of Mathematics.

(2) Analytic study of war games. H. Antosiewicz. Accepted, Naval

Research Logistics Quarterly.

v
46 Publication Activities

(3) Asymptotic solution of linear differential equations with a parameter. H. Antosiewicz . Submitted to a technical journal.

(4) On a certain integral involving Bessel functions. H. Antosiewicz.

Submitted to a technical journal.

(5) On the differential equation x + k(f(x) + g(x)x)x = ke(t).

H. A. Antosiewicz. Submitted to a technical journal.

(6) Stable systems of differential equations with integrable forcing

term. H. A. Antosiewicz. Accepted, Journal of the London

Mathematical Society.

(7) Computation of vibration modes and frequencies on SEAC. W. F. Cahill

and S. Levy (NBS 6.4), Accepted, Journal of the Institute of

Aeronautical Sciences.

(8) Automatic computation of nerve excitation. K. Cole (National

Institutes of Health), H. Antosiewicz, and P. Rabinowitz. Accepted,

Journal of the Society for Industrial and Applied Mathematics.

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

Submitted to a technical journal.

(10) Time-discrete stochastic processes in arbitrary sets, with

applications to processes with absorbing regions and to the problem

of loops in Markoff chains, D. van Dantzig. Accepted for publication

(in French) in Annales de 1'Institut Henri Poincard (Paris).

(11) On the theorem of Dilworth. G. B, Dantzig (RAND) and A. J. Hoffman.

To appear in "Papers on linear inequalities and related systems,"

Annals of Mathematics Study 38.

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

Accepted, Pacific Journal of Mathematics.

(13) Numerical experiments in potential theory using orthonormal

functions, P. Davis and P, Rabinowitz. Submitted to a technical

journal.

( 1 4 ) Some Monte Carlo experiments in computing multiple integrals.

P. Davis and P. Rabinowitz. Accepted, Mathematical Tables and

Other Aids to Computation.

0 5) Some sampling results on the power of raonparametric tests against

normal alternatives. W. J. Dixon (University of Oregon) and

D. Teichroew. Submitted to a technical journal. (Abstract appeared

in Ann. Math, Stat. (Mar, 195*+), p. 175.

(

1

6 ) Comparison of theories and experiments for the hydraulic dam-break

wave. R. F. Dressier. To appear in the Proceedings of the Tenth

General Assembly, of the International Union of Geodesy and

Geophysics, held in Rome, Italy, September 1955-.

(17) Heat flow in a fluid with eddying flow. W. H. Durfee (now with

Operations Research Office, Johns Hopkins University). Submitted

to a technical journal.

(18) A comparison theorem for eigenvalues of normal matrices. K. Fan.

Accepted, Pacific Journal of Mathematics.
Publication Activities

*+7

(19) On systems of linear inequalities. K. Fan. To appear in "Papers

on linear inequalities and related systems," Annals of Mathematics

Study 38, ed. by H. W. Kuhn and A. W. Tucker.

( 20 ) Systems of inequalities involving convex functions. Ky Fan.

Submitted to a technical journal.

(21) Fully convex normed linear spaces. K. Fan and I. Glicksburg

(RAND Corporation). Accepted, Proceedings of the National Academy

of Sciences.

( 22 ) An algebraic proof of the isoperimetric inequality for polygons.

K. Fan, 0, Taussky and J. Todd. Accepted, Journal of the Washington

Academy of Sciences.

(23) Asymptotic lower bounds for the fundamental frequency of convex membranes. G. E. Forsythe. Accepted, Pacific Journal of Mathematics.

(2k) A comparison between the mean successive difference and the root mean square successive difference, S. Geisser. Submitted to a technical journal.

(25) A note on the normal distribution. S. Geisser. Submitted to a technical journal.

( 26 ) Power series for log e x e^„ K. Goldberg. Accepted, Duke Journal

of Mathematics.

( 27 ) A subroutine for computations with rational numbers. P. Henrici.

Accepted, Journal of the Association for Computing Machinery.

(28) Addition theorems for general Legendre and Gegenbauer functions.

P. Henrici. Accepted, Journal of Rational Mechanics and Analysis,

( 29 ) On certain series expansions involving Whittaker functions and

Jacobi polynomials. P, Henrici. Accepted, Pacific Journal of

Mathematics

( 30 ) On generating functions of the Jacobi polynomials. P. Henrici.

Accepted, Pacific Journal of Mathematics.

(31) On systems of distinct representatives. A. J. Hoffman and

H. W. Kuhn (Bryn Mawr College). To appear in "Papers on linear

inequalities and related systems," Annals of Mathematics Study 38.

(32) Systems of distinct representatives and linear programming.

A. J. Hoffman and H s W. Kuhn (Bryn Mawr College). To appear in

the American Mathematical Monthly.

(33) The number of absolute points of a correlation. A. J. Hoffman,

M. Newman, E. G. Straus, 0. Taussky. Accepted, Pacific Journal

of Mathematics.

(3*+) Applications of extreme value theory to cobbles and boulders in

gravel pits. W. C. Krumbein and J. Lieblein. Submitted to a technical journal.

(35) A proof of Hilbert's Nullstellensatz. T. S. Motzkin. Accepted,

Mathematische Zeitschrift.

(36) 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.
43 Publication Activities

(37) On Fejer sets in linear and spherical spaces. T. S. Motzkin and

I. J. Schoenberg. Accepted, Annals of Mathematics.

(38) Pairs of matrices with property L, II. r. S. Motzkin and

0. Taussky. Accepted, Transactions of the American Mathematical

Society

.

(39) An alternative proof of a theorem on unimodular groups. M. Newman.

Submitted to a technical journal.

(^0) An identity for the coefficients of certain modular forms,

M. Newman. Accepted, Journal of the London Mathematical Society.

(4l) An inclusion theorem for modular groups. M. Newman. Submitted to

a technical journal.

(^2) Generalizations of identities for the coefficients of certain

modular forms. M. Newman. Accepted, Journal of the London

Mathematical Society.

(*+3) to the existence of identities for the coefficients of certain

modular forms. M. Newman. Submitted to a technical journal,

(^4) The diophantine equation axm-byn=c. M. Newman. Submitted to a technical journal.

(45) The normalizer of certain modular subgroups. M. Newman. Accepted, Canadian Journal of Mathematics.

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

number fields. M. Newman and 0. Taussky. To appear in Communications

on Pure and Applied Mathematics of New York University.

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

diffraction of waves by wedges. F. Oberhettinger. Submitted to

a technical journal.

(48) On the Lerch zeta function. F. Oberhettinger. Accepted, Pacific

Journal of Mathematics.

(49) Det erminanten mit ueberwiegender Hauptdiagonale und die absolute

Konvergenz von linearen Iterationsprozessen. A. M. Ostrowski.

Accepted, Comentarii Mathematic! Helvetici.

(50) On Gauss' speeding up device in the theory of single step iteration.

A. M. Ostrowski. Submitted to a technical journal.

(51) On the convergence of Gauss' alternating procedure in the method

of the least squares, I. A. M. Ostrowski. Submitted to a technical journal.

(52) On the convergence of asymptotic solutions of linear differential

equations. R. M. Redheffer (U.C.L.A.) and W. Wssow. Submitted to

a technical journal.

(53) Contributions to the theory of rank order statistics. I. R. Savage. Submitted to a technical journal.

(54) Inequalities for probabilities associated with the multivariate

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

journal.
Publication Activities 49

(55) Algebraic equations satisfied by roots of natural numbers.

E. G. Straus (University of California at Los Angeles) and

O. Taussky. Accepted, Pacific Journal of Mathematics.

(56) A note on group matrices. 0. Taussky. Accepted, Proceedings of

the American Mathematical Society.

(57) 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 1953.

(58) Unimodular circulants. 0. Taussky. To appear in the I. Schur

Memorial Volume to be published by Mathematische Zeitschrift.

(59) Generation and testing of pseudo-random numbers. 0. Taussky and

J. Todd. Submitted to a technical journal.

(60) A direct approach to the problem of stability in the numerical

solution of partial differential equations. J. Todd. To appear

in the Proceedings of a Symposium on Partial Differential Equations,

held at Berkeley, California, June 1955.

(61 ) Machine attacks on problems whose variables are permutations.

C. B. Tompkins. To appear in the Proceedings of the American

Mathematical Society Sixth Symposium on Numerical Analysis, held

in Santa Monica, Calif., August 1953.

(62) An extension of a theorem of Dantzig's. C. Tompkins and I. Heller

(George Washington University). To appear in "Papers on linear

inequalities and related systems," Annals of Mathematics Study 38.

( 63 ) Computation of atomic levels: the spectrum of singly ionized

tantulum. R. E. Trees (NBS Section 4.1), W. F. Cahill, and

P. Rabinowitz . Accepted, Journal of Research, NBS.

(64) Best approximation polynomials of given degree. J. Walsh. To

appear in the Proceedings of the American Mathematical Society

Sixth Symposium on Applied Mathematics, held in Santa Monica,

Calif., August 1953.

(65) 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 Applied

Mathematics, held at Santa Monica, Calif., August 1953.

(66) Accelerating convergence of iterative processes. J. Wegstein.

Submitted to a technical journal.

(67) Er ror bounds for eigenvalues of symmetric integral equations.

H. Wielandt. To appear in the Proceedings of the American Mathematical

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

Calif., August 1953.

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

to a technical journal.

( 69 ) National physical standards and design of experiment. W. J. Youden.

To appear in Revue de L'Institut International de Statistique

(The Hague).
50 Publication Activities

( 70 ) Partially replicated latin squares. W. J. Youden and J. S. Hunte

(American Cyanimid Co.). Accepted, Biometrics.

( 71 ) On the analysis of covariance. M, Zeien. Submitted to a technical journal.

2.5 Miscellaneous

(1) More scientists. W. J. Youden. Submitted to a technical journal

COMM-DC-49131^
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

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