NATIONAL BUREAU OF STANDARDS REPORT

4546

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

A Quarterly Report

October through December 1 955

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 6cope 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 9 Office of Weights and Measures
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1 1.0 4546

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

October through December 1955

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

Tiw pvbiicitio*. r«i Approved for public release by the , It proWblw

iwparamioaia director of the National Institute of «uni.,WMhi»gto»

25«DtCi Such p«n Standards and Technology (NIST) iortha»beM»peeweally prepared If »l on October 9, 2015 P°rt for Ito ewe use.
APPLIED MATHEMATICS DIVISION

October 1 through December 31 ? 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? Administrative Officer

Yates S. Sladen, Administrative Assistant

Marguerite C. Hepburn, Secretary

Mary E. McKinley, Secretary

Esther L. Turner, Secretary

NUMERICAL ANALYSIS SECTION

Henry A. Antosiewicz, Ph.D.

Leila D. Bram, Ph.D.

Philip Davis, Ph.D.

John Todd, B.S., Chief

Tosio Kato, Ph.D.

Karl Goldberg, M.A

Evelyn A. Grigg, Sec'y

Alan J. Hoffman, Ph.D.

Morris Newman, Ph.D.

Philip Rabinowitz, Ph.D.

COMPUTATION LABORATORY Milton Abramowitz, Ph.D., Chief

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

Joseph Bram, Ph.D.

Hans 0. Bremer, B.A

Anne C. Bruno, B.A.

William F. Cahill, M.S.

Ruth E. Capuano

John W. Cooper, B.S.

Mary M. Dunlap, B.S.

Anne M. Futterman, B.A

Elizabeth F. Godefroy

William G. Hall, B.S.

Genevie E. Hawkins, B.S.

Emilie V. Haynsworth, Ph.D.

Urs W. Hochstrasser , Ph.D.

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

Dorothea H.Jirauch, M.A.

Lambert S. Joel, B.A

David S. Liepman

Ethel C. Warden, B.A

Kermit C. Nelson

Peter J. O'Hara, B.S.

Mary Orr

Maxine L. Paulsen, B.S.

Sally T. Peavy, B.S.

B. Stanley Prusch

Ida Rhodes, M.A

Billie G. Urban, M.S.

Philip J. Walsh, B.S.

Bertha H. Walter

Joseph H. Wegstein, M.S.

Richard G. Williams, M.S

Ruth Zucker, B.A

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.

Mary C. Dannemiller

Lola S. Deming, M.A

MATHEMATICAL PHYSICS SECTION

Peter Chiarulli, Ph.D.

Leon Feldman, B.A

Mary L. Epling, B.S.

Lela J. Hamilton, Sec'y

Julius Lieblein, Ph.D.

Mary G. Natrella, B.A

Joan R. Rosenblatt, Ph.D.

Peter Henrici, Ph.D.

Fritz Oberhettinger , Ph.D.

Edith N. Reese, B.A

Norman C. Severo, Ph.D.

Paul N. Somerville, Ph.D

Carolyn Yick, Sec'y

Marvin Zelen, M.A

Lillian Sloane, Sec'y Robert Stoneley, Ph.D.

Robert F. Dressier, Ph.D., Chief

Under contract with The American University Guest Worker On leave of absence
CONTENTS

Page

Status of projects* as of December 31 5 19 55 1

1 . Numerical analysis 1

2. Mathematical tables and programming research . . 8

3. Probability and mathematical statistics. .... 12

*+. Mathematical statistics 1 5

5. Mathematical and computational services 20

6. Statistical Engineering services 35

Application of SEAC 37

Lectures and symposia 39

Publication activities ^1

*Only unclassified projects are included in this report
.
Status of Projects

December 31, 1955

1 . NUMERICAL ANALYSIS

RESEARCH IN THEORIES OF DISCRETE STRUCTURES

Task 1100-10-51 16/26-1 £9

Origin: NBS

Sponsor: Office of Naval Research

Manager: 0. Teussky-Todd

Full task description: July-Sept 1955 issue, p. 1

Authorized 9/30/55

Status: CONTINUED. In connection with the work of this task,

K. Goldberg and M. Newman have studied certain subgroups of the modular

group G and have determined sufficient conditions for two elements of G

to generate a free subgroup of G. A manuscript on the subject has been

prepared (see item (1*+) below). This work was started through a particular

problem raised by J. L. Brenner (State College of Washington), from which

0. Taussky suggested a generalization.

The two-dimensional analog of the Wirtinger inequality was further studied by Ky Fan, 0. Taussky, and J. Todd. It was noted earlier

that in the symmetrical case (n^ variables) equality occurs for vectors

that are linear combinations of a certain set of four n^ vectors. In

the unsymme trical case of mn variables, however, there is equality only

for vectors that are linear combinations of two mn-vectors.

T. Kato and 0. Taussky have given several proofs of the theorem

that a bounded operator A in Hilbert space is normal if and only if A*A-AA*

commutes with A. For finite matrices two proofs are given without the use of characteristic roots. The theorem is established in the case when

the elements of the matrix are elements of an abstract field provided it

is formally real or that the relation H a^a^=0 implies a^=0 when is an involution in the field. This enables the theorem to be used for

matrices whose elements are polynomials in several indeterminate s , e.g.,

group matrices. Generalizations of the theorem were also found. The first

vanishing commutator of the form (A, (A, . . . (A, (A, A*) ) . . . ) ) has been examined

for some special types of finite matrices. A paper on commutators of A and

A* will appear (see (13)).

T. Kato has proved that the (infinite) Hilbert matrix (a^)

,

(i, k=1 ,2, . . . ) , with a^j

t =(i+k) -

^ has a positive eigenvector belonging

to the eigenvalue tt, thus answering in the affirmative a question raised

by 0. Taussky (Research Problem 12 , Bull. Amer. Math. Soc. 60 , 290 ( 195*+)).

The methods of proof were suggested by an examination of the actual numerical

values of the eigenvectors of the finite segments of the matrix. Some of

the material examined was published by R. A. Fairthorne and J.r.P, Miller

(MTAC 3? 399_I+00 ( 19^8-9 )) and also by 0. Taussky (Quart. J. Math., Oxford

Ser. 20 , 80-83 ( 19^9 )); relevant computations had been carried out previously

at New York University Institute of Mathematical Sciences on the UNIVAC,

and special computations were carried out by K. Goldberg on SEAC. A

manuscript has been prepared (12).

1
2 Status of Projects

Publications

:

(1) 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 (New York University).

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

of the American Mathematical Society.

(3) The formal power series for log (e x «ey). K. Goldberg. To appear in

the Duke Mathematical Journal.

(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 xey). K. Goldberg, In manuscript,

(7) Unimodular integral circulants. 0. Taussky. Appeared in Schur Memorial

Volume, Math. Zeitsch. ^j>, 286-289 (1955).

( 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. Taussky.

To appear in Transactions of the American Mathematical Society.

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

Submitted to a technical journal.

(11) An algebraic proof of the isoperime trie property of polygons. Ky Fan.

0. Taussky and J. Todd. J. Washington Acad. Sci. 339-35-2 (1955).

(12) On the Hilbert matrix. T. Kato. In manuscript.

(13) Commutators of A and A*. T. Kato and 0. Taussky. To appear in the

Journal of the Washington Academy of Sciences.

(I 1*) Some free subgroups of the modular group. K. Goldberg and M. Newman.

In manuscript.

RESEARCH IN NUMERICAL ANALYSIS AND RELATED FIELDS

Task 1101-10-1 104/55-55

Origin: NBS Authorized 8/1 3/55-

Managers: J. Todd, P. Davis Revised 8/29/55

Full task description: July-Sept 1955- issue, p. 1 Status: CONTINUED . H. A. Antosiewicz has continued his investigation

of Lyapunov’s second method. The concept of quasi-asymptotic stability, which

means the asymptotic stability of a solution that is not (simply) stable

relative to neighboring solutions, is shown to play a role similar to the

quasi-uniform asymptotic stability introduced by J. Massera. It isdistinct

from the concept of asymptotic stability in Lyapunov’s sense but implies

it for certain classes of differential equations. A number of converses to

the classical Lyapunov theorems are proved. (See publication (1^) below.)

P. Davis and K. Fan have continued their study of certain inequalities

in normed linear spaces and their application to problems of interpolation and

approximation

.

c i Completeness of order p has been introduced and studied. A sequence

l f n

l is called complete of order p, p > 1 , if g |0 (f

R

) |P < oo implies 0=0.

Making use of the space X®X P

, that is, the direct sum of X with the

sequential Banach space £p > if has been found possible to derive necessary

and sufficient conditions in order that {f

n }

he complete of order p. Such

conditions again appear as statements on the possibility of approximation with
St8tus of Projects

3

coefficients restricted in size. New theorems of Psley-Wiener type have

also been obtained.

P. Davis has been conducting numerical experiments on SEAC for

the determination of the least p-th power approximation (p ^2). The

method is an iterative one that makes use of an orthonormalizing code

(p = 2 ), already developed.

The test program for pseudo random numbers prepared by T. Kato

is being elaborated and exploited by J. M. Cameron.

J. Todd has been preparing a survey of various topics belonging

to the so-called constructive theory of functions that are of interest or

of actual or potential use in numerical analysis. Among these are the

theorems of Weierstr8ss, Bernstein, the Markoffs, Chebyshev, Zolotareff.

K. Goldberg has continued his study of generating functions and

their induced convolutions. Specific attention has been given to the

generating function |4-jf(sx) and the generalized convolution Hi(x^ ) . . .f n

( x

n

) summed for all positive integers x^ such that (x^,...,x

n

) = x for a fixed n. Several identities have been found when the functions considered are multiplicative. The formula f(m)f(n) = f (mn/(m,n) )f ( (m,n) ) , f(1) = 1,

which is an equivalent definition for multiplicity, is basic to these

considerations

.

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). J. Aeronautical Sci. £2 , 837-843 (1955).

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

J. Math. £, Suppl. 1, 669-674 (1955).

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

Journal of Mathematics.

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

appear in the Journal of the Association for Computing Machinery.

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

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

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

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

Computation

.

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

and P. Rabinowitz. To appear in the Journal of Research (NBS).

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

P. Davis and P. Rabinowitz. To appear in the Journal of the Washington

Academy of Sciences.

(10) Inequalities and approximation in function spaces. P. Davis and K. Fan.

In manuscript.

(11) A matrix with real characteristic roots. K. Goldberg. To appear in

Journal of Research (NBS).

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

(13) On asymptotic stability. H. A. Antosiewicz. In manuscript.
4 Status of Projects

RESEARCH IN MATHEMATICAL TOPICS APPLICABLE TO

NUMERICAL ANALYSIS

Task 1101-10-5116/55-56

Origin: NBS

Sponsor: Office of Naval Research

Managers: J. Todd, M. Newman

Full task description: July-Sept 1954 issue, p. 5 Status: CONTINUED. A numerical computation for the determination

of the lowest f requency^of a free square plate has been carried out on SEAC

by T. Kato in collaboration with M. Newman. The computation is based on a formula giving upper and lower bounds of eigenvalues of positive definite

operators (see T. Kato, Math. Ann. 126, 253-262 (1953)). It makes use of

many of the basic codes for matrix operations that were prepared by M. Newman,

some of which were described in a previous report and some below, and a pair of trial functions introduced by Y. Nakata and H. Fujita were used

containing 6 and 8 adjustable parameters. The result, in a suitable unit,

is 3.5257 < V < 3.5292. This is a rigorous estimate.

Some inequalities have been deduced by T. Kato for estimating the

dominant eigenvalue of a positive matrix. It is believed that these improve

earlier results of various authors.

The following new programs have been prepared by M. Newman:

External floating point matrix inversion and solution of equations

routine. Computes A*"‘B, where A,B are given matrices of dimensions

not exceeding 384. Solving time for two sets of 40 equations in 40

unknowns is approximately one hour. Inversion of a 100x100 matrix

would require approximately 20 hours. A large part of the time is tape

running time. Data is in standard form.

General purpose output routine for all external matrix codes in this

collection

.

External floating point multiplication routine. Computes the product

AB, where A?B are given matrices (not necessarily square) of dimensions

not exceeding 256 . External floating point addition routine. Computes ccA +/3B, where

00 ,3 are scalars and A,B are given matrices (not necessarily square)

of dimensions not exceeding 256 .

A. J. Hoffman and J. B. Kruskal (Princeton University) have continued

their study of conditions under which a rectangular incidence matrix has

the unimodular property (every square submatrix is singular or unimodular).

Let G be a directed graph; a path in G is an ordered sequence of nodes

v

1

, . . . ,v

p such that the arc v

i v

i+ i I s positive; a loop in the associated

undirected graph |g| is said to be alternating if its arcs are traversed

alternately in positive and negative directions. Let A be the incidence

matrix of nodes vs. all paths of G. Then A has the unimodular property

if and only if all loops in | G I are alternating. This result is applied to

the study of "cross sections" of families of subsets. A family of subsets

of a given set has the "inclusive property" if, whenever two of the sets have

a non-empty intersection, one is a subset of the other. Let R be a finite

set, S = {

, . .

.

,S^

j

, T = £t.j , . . .

,

T

n ^ be two families of subsets each with

the inclusive property. Let 0 < a i (i = 1 , . . . ,m) , 0 < c^ < dj

(j = 1)...?n) be given integers. Necessary and sufficient condi tio ns are given for the existence* of a cross section CQ R such that a i ^ CAS

i < b^

(i = I)...?TM) and Cj CATj ^ ^j (j = 4 . . . ? n ). (K denotes the cardinality

of the set K. ) The proof depends on the foregoing result on incidence

matrices. A special case of the cross section theorem proves the following:

MI2.

M03.

MM4.

MA2.

Authorized 8/13/54

Revised 8/29/54
Status of Projects 5

Let a set R of nk elements be partitioned in two ways into , [T-j , . . . ,Tk } , S

± = T

i = n (i = 1 , . . . ,k) . Then for any m with 0 < m 4 n,

there exists a set C£R such that C = mk, CCiS^ = m = CHL (i = l,...,k).

M. Newman and J. Todd have considered a problem raised by

C. Stormer about the solutions in integers of certain equations of the

form m p-

^ (x) + n p_

‘ (y) = kco where p is the Weier strassian elliptic

function, with certain invariants. Some new solutions were obtained, and

effective methods for computation of all solutions are being investigated.

In connection with some work undertaken previously, a certain

class of elliptic modular functions is being studied by M. Newman. The

object of the study is to determine a polynomial basis for the entire

modular functions on T^Cp).

Professor Stiefel (Zurich) has revised his manuscript, "Kernel

Polynomials in Linear Algebra", in preparation for publication.

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) Compac tif ication of completely regular spaces. Ky Fan and F. Wagner

(University of Notre Dame). In manuscript.

(5-) On dominant eigenvalues of positive matrices. T. Kato. In manuscript.

(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. Pac. J. Math. Suppl, 1, 725-75-5- 0955).

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

appear in the Pacific Journal of Mathematics.

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

J. London Math. Soc. 3lQ, 5-88-5-93 0955).

(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). Proc. Nat. Acad. Sci. 5j_, 95-7—9 53 (1955).

(11) On eigenvalues of sums of normal matrices. H. Wielandt. Pac. J. Math.

1, 633-638 (1955).

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

Pac. J. Math. 1, 65-1-65-2 0955).

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

(Bryn Mawr College). To appear in Annals of Mathematics Study 38.

(15-) Syste of linear inequalities. Ky Fan. To appear in the Annals of

Mathematics Study 38.

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

and H. W. Kuhn (Bryn Mawr College) , To appear in the American Mathematical

Monthly.

(16) On a theorem of Dilworth. G. B. Dantzig (RAND Corporation) and A. J. Hoffman

To appear in Annals of Mathematics Study 38.

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

manuscript

.

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

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

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

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

(21) The lowest frequency of a free square plate, H. Fujita (University of Tokyo)

T. Kato, Y. Nakata (University of Tokyo) and M. Newman. In manuscript.
6 Status of Projects

(22) An inclusion theorem for modular subgroups. M. Newman. Submitted

to a technical journal.

(23) A table of the coefficients of the powers of . M. Newman.

To appear in the Proceedings of the Royal Netherlands Academy of Science.

(24) On the existence of identities for the coefficients of certain modular

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

Society.

(25) An alternative proof of a theorem on unimodular groups. M. Newman.

To appear in the Proceedings of the American Mathematical Society.

ANALYTIC STUDY OF WAR GAMES

Task 1101-10-5116/55-83

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

Manager: H. A. Antosiewicz

Full task description: Oct-Dec 1954 issue, p. 7

Status: CONTINUED. Dr. Antosiewicz attended the Michigan Conference

on War Game Simulation for Senior Air Force Officer Training, held at

Ann Arbor, November 11, 12, and participated in a panel discussion on mathematical models for strategic war games.

Publication:

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

Naval Research Logistics Quarterly.

STUDIES IN THE THEORY OF ASYMPTOTIC EXPANSIONS

Task 1101-10-5116/55-116

Origin: NBS Authorized 3/31/55

Sponsor: Office of Scientific Research, ARDC, USAF

Manager: P. Davis

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

Status: COMPLETED. A report has been submitted to the sponsor.

Publication:

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

P. Davis. In manuscript.

STUDY OF DIFFERENTIAL EQUATIONS FOR NERVE EXCITATION

Task 1101-10-51 16/56-148

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

Bethesda, Md

.

Manager: H. A. Antosiewicz

Full task description: July-Sept 1955 issue, p. 7

Status: CONTINUED. In searching for values of the parameter K for

which solutions of the basic system of differential equations have a prescribed behavior, 21 cases have been run so far. The results indicate

a great sensitivity of the system to small changes in the parameter K

as well 8 s in the integration steps used.
Status of Projects 7

Publication:

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

Research Institute), H. A. Antosiewicz and P. Rabinowitz. J. Soc. Ind.

Appl. Math. 3, 153-172 (1955).
2. MATHEMATICAL TABLES AND PROGRAMMING RESEARCH

TABLES OF E

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

Task 11 02-40-1 110A3-3

Manager: I. Stegun Authorized 7/1/4 7

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. 45

Status: CONTINUED. A code has been prepared for generating the

functions F^, F£, G^, G£ for given values of p

and and varying L.

In the range .5 4 *1 4 2 five-figure accuracy is obtained and for < .5

and n > 2 seven-figure accuracy is possible.

A table of the functions G and G' was prepared and transferred

o 0

to punched cards. The values are given for p = 0(.5)40,. 0 = 1 (.5)19.5

and = 0(. 5)50, ^

= 20(. 5)25. J v

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. Computations for the main body of the table have

been completed on SEAC. An investigation is under way to determine the

most desirable method for calculating the remainder of the table.

REVISION OF MATHEMATICAL TABLES

Task 1102-40-1110/52-7

Origin: NBS Authorized 8/10/51

Managers: W. F. Cahill, I. Stegun

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

8

Status: INACTIVE
Status of Projects 9

TABLE OF ARCSIN FOR COMPLEX ARGUMENTS

Task 1102-40-1110/52-14

Origin: NBS Authorized 10/1/51

Manager: B. Urbau

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 OF HYPERBOLIC SINES AND COSINES

Task 1102-40-1110/52-18

Origin: NBS Authorized 9/17/51

Manager: K. Nelson

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

Status: COMPLETED.

Publication: Table of hyperbolic sines and cosines (x = 2(. 001)10, 9S).

Applied Mathematics Series 45. Available from U. S. Government Printing

Office, Washington 25> D. C., 55 cents.

TABLE OF THE MODIFIED AIRY INTEGRAL

Task 1102-40-1110/52-23

Origin: NBS Authorized 10/4/51

Manager: I. Stegun

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 1102-40-1110/52-25

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.

EXTENSION OF TABLES OF THE EXPONENTIAL FUNCTION FOR

NEGATIVE ARGUMENTS

Task 1102-40-1110/52-31

Origin: NBS Authorized 10/9/51

Managers: E. Marden, S. Prusch

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

Status: COMPLETED.

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

Applied Mathematics Series 46. Available from U. S. Government Printing

Office, Washington 25> D. C., 50 cents.
10 Status of Projects

SPHEROIDAL WAVE FUNCTIONS

Task 1102-40-1110/52-37

Origin: NBS Authorized 11/28/51

Manager: D. Liepman

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

Status: INACTIVE.

S IEVERT'S INTEGRAL

Task 1102-40-1110/52-57

Origin: NBS Authorized 2/12/52

Managers: M. Paulsen, P. O’Hara

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

Status: CONTINUED. The tabular values have been typed on the cardcontrolled typewriter and checking is in progress. Preparation of the

introductory material continued.

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 L-j-jj shell, z = 3 5? were completed.

For z = 45 and z = 95> eigenvalues were determined and computations were

made for some k values. Computations for K shell, z = 65, were completed.

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 Base 01 code was placed in service containing the

following service routines and subroutines (in addition to those in the

Base 00 code): Cube Root, Equation Solver, Move Routine, Input-Output Service

Routines, Iterator. The "base code" system is facilitated by a four-address

computer and a fast input unit (magnetic wire on the SEAC) . The Base 00 and

Base 01 systems are used with the four-address SEAC entirely, in lieu of an automatic compiler system.

H. H. Howe (82.10) has completed the preparation of a "Manual for

SEAC coding in the three-address system" (in manuscript form).

The following Technical Memoranda by J. H. Wegstein are completed:

(1) Input-Output service routines for Base 00 codes

(2) Move subroutine

(3) Experimental iteration routine
Status of Projects 11

MATHEMATICAL SUBROUTINES

Ta s k 371 1 -60-0009/56-1 60

Origin: NBS Authorized 9/30/55

Managers: Staff

Full task description: July-Sept 1955 issue, p. 13

Status: CONTINUED. A Base 00 floating routine for evaluating the

modified Bessel function of the second kind K^Cx) was written, which

employs the integral representation K^Cx) = J

* 00

e x cos ^ 1 cosh nt dt.

The integral representation may be used for all values of the variable x. The routine employs the trapezoidal rule, and the integration interval

is automatically chosen to give results correct to within one or two units

in the eighth significant figure.
3. PROBABILITY AND MATHEMATICAL STATISTICS

BIBLIOGRAPHY AND GUIDE TO STATISTICAL LITERATURE

Task 1

1

03-1 0-1 1 07A9-1 a Origin: NBS Authorized 1/9A9

Manager: L. S. Deming

Full task description: Apr-June 19^9 issue, p. 75

Status: CONTINUED. For a description of the continuing activity

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

MANUAL ON FITTING STRAIGHT LINES

Task 1103-10-1107/50-2

Origin: NBS Authorized 3/1/50

Manager: F. S. Acton

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

Status: CONTINUED. The manuscript is currently with technical

advisers for comment.

TABLES TO FACILITATE DRAWING RANDOM SAMPLES

Task 1103-10-1107/51-1

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; 1103-10-1107/51-2

Origin: NBS Authorized 7/1/50

Manager: C. Eisenhart

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

Status: CONTINUED. A first draft of a paper on M The probability of

non-failure of several components" by Joan R. Rosenblatt was prepared.

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).

12
Status of Projects 13

(2) A comparison between the mean successive difference and the root

mean square successive difference. S. Geisser. Submitted to a technical journal.

(3) A note on the normal distribution. S. Geisser, To appear in the

&mals of Mathematical Statistics.

(l+) 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: NBS Authorized 9/17/51

Manager: I. R. Savage

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

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

195^ issue, p. 22. Further work under this task will be reported under

task 1 103-10-1 107/56- 1 70 (see p.14 ).

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.

STUDIES IN TOE 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. Readying of the work on partially balanced

incomplete block (PBIB) designs for publication continued. The publication

will include (1) W. H. Clatworthy* s enumeration and construction of a large number of PBIB designs, which considerably extends the list given

in Bose, Clatworthy, and Shrikhande, (2) the duals of a number of these

designs that were worked out by C, Burton and Dr. Clatworthy, and (3)

proofs by Mr. Burton of some theorems regarding the existence of such duals.

Publications:

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

(American Cyanamid Co.). To appear in Biometrics.

(2) On partially balanced incomplete block designs with two associate

classes. W. H. Clatworthy. In manuscript.

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

information. M. Zelen. Submitted to a technical journal.

(*+') On the analysis of covariance. M. Zelen. Submitted to a technical

journal.
14 Status of Projects

RESEARCH ON MATHEMATICAL ASPECTS OF ORDER

STATISTICS METHODS

Task 1103-10-1107/55-110

Origin: NBS Authorized 3/3/55

Manager: J. Lieblein

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

Status: CONTINUED. Attention was given to several types of problems

in the asymptotic theory of extreme values that had arisen in practical

applications. One problem involved the relationship of alternative

limiting processes to the selection of an asymptotic type for fitting

a given set of extreme data. A second problem was to investigate the

behavior of an extreme event as a function of duration, for example,

the relationship between the peak five-minute wind intensity and the

considerably higher peak one-minute intensity. The results of this work

were included in notes for a talk on developments in the application of

extreme value theory presented before the Annual Meeting of the Institute

of Mathematical Statistics in New York City on December 27.

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.

STUDY OF NON- PARAMETRIC STATISTICAL TECHNIQUES

Task 1103-10-1107/56-170

Origin: NBS Authorized 12/15/55

Manager: Joan R. Rosenblatt

Objective: To conduct research on the properties and interpretation

of so-called non-parame trie statistical methods. This includes, in

particular, "distribution-free" tests and the analysis of rankings.

Background: There is great need for better understanding of

techniques of statistical analysis which may be used when only very general

assumptions can be made about the underlying stochastic process which

generates data. Criteria are needed for choosing among various distributionfree methods which have been proposed for similar experimental situations.

Many experimental situations yield data in the form of paired comparisons

or rankings; there is great need for work on models which will lead to the

efficient design and analysis of such experiments.

Comment: This task supersedes task 1 1 03-1 0-1 1 07/52-2 (Procedures of

non-parametric statistics), terminated this quarter* which had a somewhat

different objective.

Status: NEW. An investigation is being made, in collaboration with

G. L. Burrows of the Department of Agriculture, of the distribution properties

of certain statistics which measure agreement among judges, when m judges

have assigned rankings to n objects.

A paper on the asymptotic efficiency of certain types of nonparametric tests is nearly completed.
MATHEMATICAL PHYSICS

RESEARCH IN MATHEMATICAL PHYSICS AND RELATED FIELDS

Task 1 104-1 0-1 11 5/55-57

Origin: NBS Authorized 9/1/5*+

Manager: R. F. Dressier

Full task description: July-Sept 195^ issue, p. 27

Status: CONTINUED. At the invitation of the Research Committee on Pressure Elements of the American Society of Mechanical Engineers,

R. F. Dressier prepared a discussion consisting of an analysis and evaluation

of the Haringx linear and non-linear theories of corrugated elastic diaphragms.

P. Henrici solved a problem concerning generalized axially

symmetric potentials. A. Weinstein and others have shown that every solution

u(x,y) of the equation

ii!a + Jifa + ^ m = o ( v ¥ -1,-2, -3,...),

ax ay y ax

which is regular in a domain containing a segment of the axis y = 0,

is uniquely determined by its values on y = 0, The present study answers

the question to what extent the domain of regularity of the function

g(z)=u(z,0) (considered as a function of complex z) determines the domain

of regularity of u(x,y). This generalized earlier results of A. Erdelyi

and G. Szego.

P. Henrici also continued his investigation of the QuotientDifference algorithm of H. Rutishauser. Basically a method for determining

the poles of a meromorphis function from the coefficients of one of its

Taylor expansions, the algorithm can be applied to problems such as the

computation of the eigenvalues and eigenvectors of a matrix, factorization

of polynomials, interpolation by sums of exponentials, and others, A

comprehensive report on this subject has now been prepared,

A new addition theorem has been derived by P. Henrici for the

product of two Jacobi polynomials, and a useful series expansion has been

derived for the integral

f n (x,y,z) J„(xt) t dt

4.2 n 2

-y

which for n = 0 is related to Sommerfeld's integral.

P. Chiarulli has initiated a study on axially symmetric propagation

of strong shocks in gases. A similarity analysis has been started to explore

the feasibility of considering, as an approximation to the actual shock, a shock of ellipsoidal shape with the center of the ellipsoid in motion.

Publications:

(1 ) A discussion of the Haringx diaphragm theories. R. F. Dressier. To be

published in the Transactions of the American Society of Mechanical

Engineers

.

(2) Addition theorems for general Legendre and Gegenbauer functions.

P. Henrici. J. Rational Mech. Anal. 4, 983-1018 (1955).

15
16 Status of Projects

(3) On the representation of a certain integral involving Bessel functions

of hyper geometric series. P, Henrici. Submitted to a technical journal.

(h-) Review by P. Henrici of "Kernel functions and elliptic differential

equations in mathematical physics, " by S. Bergman and M. Schiffer,

Bull. Amer. Math. Soc. 6l_, 596-600 (19 55).

(5) Automatic computations with power series. P. Henrici. To appear in

the Journal of the Association for Computing Machinery.

(6) On the domain of regularity of generalized axially symmetric potentials.

P. Henrici. Submitted to a technical journal.

(7) The quotient-difference algorithm. P. Henrici. In manuscript.

RESEARCH IN ELECTROMAGNETIC THEORY

Task 11 04-1 0-51 60/55-5-7

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

Department of the Army Revised 9/29/5*+

Manager: F. Oberhettinger

Full task description: July-Sept 195*+ issue, p. 28

Status: INACTIVE.

RESEARCH IN MATHEMATICAL ELASTICITY

Task 1 1 0^-1 0-51 60/55-85

Origin: NBS Authorized 12/27/5*+

Sponsor: Office of Scientific Research, ARDC, USAF

Manager: R. F. Dressier

Full task description: Oct-Dec 195*+ issue, p. 30

Status: CONTINUED. The most difficult phase of this project has

been successfully completed, namely, the derivation of a practical and

accurate analysis and accompanying computational procedure for determining

all stresses under combined stretching and bending, and all displacements, for any corrugated pressure elements of arbitrary profile and

thickness, under any type of lateral loading and boundary conditions. Full

results for the clamped edge and semi-free edge problems together with a full description of the method have been presented in a paper, "Properties

of corrugated diaphragms," by R. F. Dressier, at the American Society of

Mechanical Engineers National Meeting, Chicago, 111., November 13-18.

(See publication (1) below.) The second phase of the project is now in

progress, — to solve a sequence of problems for a diaphragm of fixed shape

for different values of the thickness parameter, in order to settle the

important question concerning the relative importance of stretching versus

bending effects. Two thicknesses corresponding to h (= .007 in.) and 3h

have now been computed, and cases h/3 and h/9 remain to be done. A third

phase of the project will then be an analysis of the effect on stress

concentrations by addition of a central stiffening disk, frequently used

in instrumentation applications. All of these results should follow easily

from the basic procedure now available.

On the vibrating triangular wing problem a method for determining

the lowest mode and frequency has been successfully completed. Several cases

have been computed for different values of Poisson's ratio, which for this

application to plates with free edges remains an essential parameter since

it appears explicitly in the boundary conditions. It has been shown that a matrix of order *+5 is sufficiently large for obtaining the lowest mode.
Status of Projects 17

For the second mode the method has not yet been successful. Here our use

of the hyper geometric polynomial matrix function to spread the eigenvalues

for faster convergence produces values which converge rapidly, but not to

the correct result. Likewise the results for the bi-orthogonal lef Xr and righthand eigenvectors are not now correct. Intensive checking of the code

has therefore been in progress by I. Rhodes and U. Hochstrasser by studying

small matrices having known spectra.

An investigation has been started by R. F. Dressier on the theory

of the edge effect in generalized plane stress, utilizing some of the

stretching methods for the boundary layer domain already successfully

applied to the bending problem for plates. The basic three-dimensional

equations have been set up in the interior space variables and in the

transformed variables for the edge domain half-space. One object is to

obtain the correct simplified differential equations and associated boundary

conditions for describing stresses near the edge in thin and thick plates.

Publication:

(1) Properties of corrugated diaphragms. R. F. Dressier. To appear in

the Transactions of the American Society of Mechanical Engineers (now

available as ASME Preprint No. 55—A— 1 81 )

.

FOURIER TRANSFORMS OF PROBABILITY DISTRIBUTION FUNCTIONS

Task 11 0*4-1 0-5160/56-15^

Origin: NBS Authorized 9/30/55

Sponsor: Office of Naval Research

Managers: F. Oberhettinger , J, Lieblein

Full task description: July-Sept 1955 issue, p. 20

Status: CONTINUED. Slightly more than 600 Fourier transforms for

distribution functions have been compiled so far. The compilation includes

some previously underived results that have been generated to increase its

scope. The results are now arranged in a systematic card index, which is

immediately applicable for publication in book form.

RESEARCH IN FLUID DYNAMICS OF TWO- PHASE FLOWS

Task 1 1 0^—1 0-51 60/56-1 55

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

Sponsor: " ” Manager: R. F. Dressier

Full task description: July-Sept 1955 issue, p. 21

Status: CONTINUED. Appropriate discontinuity relations have been

derived that are valid for the states on either side of an idealized plane

interface with a condensible gas on one side and condensed liquid on the other.

The theory has been set up for a general case where discontinuities are permitted in all thermodynamic and kinetic quantities. Some typical calculations

for the specific application to steam and water have been carried out. An

investigation is now being made to find typical initial conditions which would

lead to such a condensation interface, both with and without heat conduction

through the duct walls. The question of the extent of the analogy between

this phenomena and flame-fronts is also under investigation.

A study is also in progress on the quasi-one-dimensional steady flow

of a submerged steam jet. The model is as follows: Steam flows from an
18 Status of Projects

infinite reservoir of steam at given state into another water reservoir,

thus forming a steam pocket surrounded by water. All the steam is assumed

to condense on a vertical end wall of water with given pressure and

temperature. The side walls of the jet therefore consist of a non-condensing

water interface. The equations of motion and energy balance including

friction at the side walls, heat-conduction through the side walls, and

heat conduction in the steam have been studied. The boundary conditions

at the end wall will depend upon the outcome of the discontinuity study

outlined above. This particular model has been chosen to study the flow

which occurs with various combinations of the initial parameters, and to

ascertain whether such a flow is possible in a continuous manner, or whether it would be necessary to include shock or condensation phenomena

within the steam pocket itself.

RESEARCH IN MATHEMATICAL GEOPHYSICS

Task 11 0^-1 0-51 60/56-1 56

Origin: The American University Authorized 9/30/55

Sponsor: Office of Naval Research

Manager: R. Stoneley

Full task description: July-Sept 1955 issue, p. 21

Status: CONTINUED. A problem on the variation of amplitude of

Rayleigh waves with depth in an elastic medium with a double surface layer

has been formulated by R. Stoneley and programmed for SEAC by U. Hochstr asser

.

This is now awaiting computation.

The mathematical part of an investigation of the propagation of

Rayleigh waves in an elastic medium with a double surface layer covered

with uniform thickness of water has been completed by Dr. Stoneley and

has been written up for publication. The equation giving the wave velocity

as a function of wave length appears as an 11-row determinant, which is

now being programmed for numerical solution on the SEAC.

An investigation has begun of the hyd r ©dynamic a 1 problem

concerning the turbidity current initiated by a submarine earthquake.

Dr. Stoneley is continuing the preparation of a monograph on elastic surface waves.

Public ations:

(1) The transmission of Rayleigh waves across an ocean floor with twosurface layers. R. Stoneley. To appear in the Bulletin of the

Seismological Society of America.

RESEARCH IN RADIATION THEORY

Task 1 1 04-1 0-51 60/56-1 75

Origin: NBS Authorized 12/28/55

Sponsor: Office of Naval Research

Manager: F. Oberhettinger

Objective: (a) To study the rigorous solutions to certain important

problems in electromagnetics, such as diffraction from a linear antenna in

front of a semi-infinite reflecting screen or a semi-infinite conducting

wedge having arbitrary angle oC. It is aimed to obtain valid representations

for rigorous solutions to such problems to serve as a standard for evaluating

the usual approximate results, which neglect the full exact boundary conditions

in the formulation. (b) To attempt to obtain an improved representation for
Status of Projects 19

the Green's function G(P) associated with a rectangular wave guide to

overcome the disadvantage of the present double series expansion, which

is not convergent when the radiation point source Q is in the same cross

sectional plane as P. (c) To determine modes in ferrite elements.

Background: (a) The excitation problem for an infinite wedge of

arbitrary angle has appeared in F. Oberhe ttinger ' s paper, "Diffraction of

waves by a wedge," (Comm. Pure Appl. Math. 2 ? ( 1955))? but the semi-infinite

case appears to present more difficulty and should be studied in general.

Only the case oc = tt has so far been treated (see A. E. Heins, Carnegie

Institute of Technology Tech. Report No. 17 ( 1955 )). Results concerning

the edge effects for reflecting screens would be useful. (b) To determine

the interior and exterior field of a slotted wave guide antenna, a representation of Green' s function is needed that would hold for all relative

positions of the points P and Q. (c) The ferrites, because of their high

resistivities, exhibit new and useful magnetic properties at frequencies

well above those at which the usual magnetic alloys are applicable. A

basic problem is that of a cylindrical ferrite rod in free space.

Status: NEW.
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

1102-40-5126/51-37 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. Signs of the normalized structure factor E were

determined for 300 reflections of the dimethoxybenzophenone crystal.

A three-dimensional Fourier synthesis was computed for this crystal

using E as amplitude.

1102-40-5126/52-44 CALCULATIONS FOR d SPACINGS

Origin and Sponsor : NBS, Division 9

Manager : R. Zucker

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

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

for various tetragonal, hexagonal and orthorhombic crystals; about 20

were carried out during the past quarter. Also redetermination of

unit cell constants by least squares fitting to measured d-spacings was performed for 8bout 20 crystals.

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

Origin and Sponsor : NBS, Section 3.2

Manager : E. Marden

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

Status : Continued. Using different physical constants, thermodynamic

functions were re-computed for several molecules.

1102-40-5126/53-51 RADIATION DIFFUSION

Origin : NBS, Section 4.8

Sponsor : Atomic Energy Commission

Manager : J. Doggett (*+. 8 ) Full task description : Apr-June 1953 issue, p. 57 (Neutron Diffusion III)

Status : Continued. The Mott formula for the elastic scattering of electrons

and positrons by nuclei has been evaluated for Al, Cu, Sn, Pb and U at

various energies in the range .05 < E < 10 Mev.

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

Origin and Sponsor : New York Quartermaster Procurement Agency

Manager : H. Bremer

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

Status : Continued. Analysis of contract award problems are processed on SEAC

as they are submitted by the sponsor.

20
Status of Projects 21

371 1 -60-0009/55-1 7 DEPOLYMERIZATION

Origin : NBS, Section 7.6

Manager : J. Bram

Full task description : Oct-Dec 1953 issue, p. 5-4- Status : Continued. Several cases were run, and the results transmitted

to the originator.

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

Origin : NBS, Section 2.2

Manager : L. S. Joel

Full task description : Jan-Mar 195*+ issue, p. 5-3 Status : Continued. The new code, which uses Filons method for quadrature

of an oscillating function, W8S completed and is being checked.

3711-60-0009/55-30 SPECTRUM ANALYSIS

Origin : NBS, Division 5- Manaoer : S. Prusch

Full task description : Jan-Mar 1955- issue, p. 5-6 Status : Continued. SEAC codes have been written and checked out for

computing wave numbers from given wavelengths and for searching for

pairs of lines having a common difference. A code is being prepared to

search for differences between all known and predicted levels for pairs

of lines which confirm known levels and predict new levels. Searching

for pairs of lines for approximately 7& common differences was carried

out for ruthenium I.

3711-60-0009/55-38 COMPRESSIBILITY FACTORS OF DRY AIR

Origin : NBS, Section 3.2

Manager : M. Paulsen

Full task description : Jan-Mar 1955- issue, p. 5-8 Status : Inactive. For status to date, see Apr-June 1955 issue, p. 12,

1 102-5-0-5126/55-39 MOLECULAR VIBRATIONS

Origin and Sponsor : NBS, Section 3.2

Manager : K. Goldberg

Full task description : July-Sept 1955- issue, p. 16

Status: Inactive.

1 102-5-0-5126/5^5-3 CHARACTERISTICS OF CONDUCTING RESISTORS

Origin and Sponsor : NBS, Section 1.6 , Manager : B. G. Urban

Full task description : Apr-June 1955- issue, p. 5-0 Status : Inactive. For status to date, see Apr-June 1955 issue, p. 12 ,

3711-60-0009/55-53 ELECTRONIC FUNCTIONS

Origin : NBS, Section 5-. 5

Manager : W. G. Hall

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

Status : Continued. Editing of the card deck into format suitable for

printing was completed. An introduction to the tables, by L. L. Marton,

C. Marton, and W. G. Hall, has been prepared.
22 Status of Projects

1102-40-5126/55*58 ELECTROMAGNETIC RADIATION FROM LIGHTNING

Origin and Sponsor ; NBS, Section 82.1

Manager : B, G. Urban

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

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

1102-^0-5126/55-61 ELASTIC CROSS SECTION FOR NEUTRON SCATTERING

Origin and Sponsor : Naval Research Laboratory

Manager : I. Stegun

Full task description : Oct-Dec 1955- issue, p. 18

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

3711-60-0009/55-66 RECONSTITUTION OF MONOCHROMATIC LIGHT INTENSITIES

Origin : NBS, Division 30

Manager : H. Bremer

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

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

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

Origin : NBS, Division 9

Manager : I. Stegun

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

Status : Continued. A general code has been written by D. R. Prosen that

will calculate two- and three-dimensional Fourier and Patterson functions.

This routine can be used for all crystallographic space groups and

has the further advantage that it can calculate selectively certain

Fourier or Harker sections.

11 02-40-51 26/55-71+ 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 one more value of the parameter and one given set

of functional values.

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

Origin : NBS, Section 2.2

Manager : R. Zucker

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

Status : Continued. Further calculations for the refractive index of

ICRS-5 were carried out for n at 19°C, 25°C, and 31°C.

1 102-40-5126/55-81 COMBINING TESTS FOR SIGNIFICANCE

Origin : NBS, Section 11.3

Manager : L. S. Joel

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

Status : Continued. The code is complete, and results are being transmitted

to the originator as they are computed (about 5000 values to date).

3711-60-0009/55-82 THERMOMETER CALIBRATIONS

Origin : NBS, Section 3.1

Manager : S. Prusch

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

Status: Continued. Calibrations were completed for Constants 06 , 0 and ft for

for approximately 25 thermometers under test. /
Status of Projects 23

1 102-4-0-51 26/55-87 "ZERO" METHOD DETERMINATION OF CRYSTAL STRUCTURE

Origin and Sponsor : NBS, Section 9.7

Manager : J. Bram

Full task description : Jan-Mar 1955 issue, p. 22

Status : Terminated. Trial runs had indicated the inadequacy of the

present method of computation.

1 102-40-5126/55-88 STRESSES IN A WALL FOUNDATION

Origin and Sponsor : NBS, Section 10,1

Managers : E. Marden, G. Hawkins

Full task description : Jan-Mar 1955 issue, p. 22

Status : Continued. Using new coefficients, stresses for another wall

foundation were computed, and results were transmitted to the sponsor.

1102-40-5126/55-103 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

Status : Continued. Several cases have been run on UNIVAC, and the routines

have been rewritten to accommodate larger amounts of data.

1 102-40-5126/55-105 AERONOMIC TIDAL WINDS OF THERMAL ORIGIN

Origin and Sponsor : NBS, Section 82.1

Managers : B. G. Urban, H. H. Howe (82.1)

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

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

1 102-40-5126/55-1 13 REACTOR DESIGN

Origin : Westinghouse Atomic Power Division

Sponsor : Atomic Energy Commission

Manager : U. Hochstrasser

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

Status : Continued. Checking of the codes mentioned in the previous issue

has been completed. Several runs have been made with various overrelaxation factors in order to determine the influence of this factor on the convergence of the approximations to the solution of the system of

difference equations. Further runs are planned for investigating the

effect of scanning in different ways through the points in the grid.

1 102-40-5126/55-117 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 : Inactive. For status to date, see July-Sept 1955 issue, p. 30.

1 102-40-5126/55-118 THERMOMETER CALIBRATION, II

Origin and Sponsor : NBS, Section 3.1

Manager : B. G. Urban

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

Status : Continued. The code is being expanded to compute certain constants

and differences formerly computed by hand; then it will be revised to

produce calibration tables at one-fifth the former interval.
24

Status of Projects

1102-40-5126/55-121 ELECTRON PENETRATION

Origin ; NBS, Section 4.8

Sponsor : Atomic Energy Commission

Manager : S. Peavy

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

Status : Continued. A new code is in the process of being written for a simpler method devised by L. V. Spencer (4.8).

1102-40-5126/55-122 SOLUTION OF NORMAL EQUATIONS

Origin : Advisory Committee on Weather Control

Sponsor : U. S. Weather Bureau

Manager : M. Newman

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

Status : Inactive.

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 used on a test problem especially devised

to try out all parts of the code. The results were as expected.

1102-40-5126/55-126 AERODYNAMIC HEATING

Origin and Sponsor : Diamond Ordnance Fuze Laboratory, Dept, of the Army

Managers : B. Walter, R. Capuano

Full task description : July-Sept 1955 issue, p. 31

Status : Continued. The aerodynamic heating T^ has been calculated for

four missile trajectories, with each trajectory calculated for four

different values of missile shell heat capacity (1/B),

at = TAW

~ TW

W '^X_ + 1

Bh At 2

e

Then T^ was calculated for the above with the following modifications:

(1) Bh

eAt increased by + 1.6°/°, (2) T increased by + 8°F, and Bh

e At

increased by ,4°/o. In each case the modifications were originated at

various positions of the missile trajectory.

1102-40-5126/55-127 VIBRATIONS OF CIRCULAR DISC

Origin and Sponsor : Evans Signal Laboratory

Manager : U. Hochstrasser

Full task description : July-Sept 1955 issue, p. 32

Status : Continued. The solutions for the preliminary problem mentioned

in the July-September 1955 issue were computed on SEAC and transmitted

to the sponsor. Codes for solving the main equations have been written

and checked out.
Status of Projects 25

3711-60-0009/56-131 RAY TRACING

Origin : NBS, Section 2.2

Manager : E. Marden

Full task description : July-Sept 1955 issue, p. 33

Sta is : Inactive. For status to date, see July-Sept 1955 issue, p. 33.

1102-40-5126/56-133 complex eigenvalues

Origin : Moore School of Engineering

Sponsor : Office of Naval Research

Manager : I. Stegun

Full task description : July-Sept 1955 issue, p. 33

Status: Continued. The largest eigenvalues have been obtained for seven matrices of order 10 and one matrix of order 6.

11 02-40-51 26/56-1 34 TRANSMISSION-DELAY TIMES

Origin : NBS, Section 82.1

Sponsor : NBS, Section 82,1

Managers : B, G. Urban, H. H. Howe (82,1)

Full task description : July-Sept 1955 issue, p. 34

Status : Inactive.

1102-40- Origin : Sponsor : Manager : Full tas

5126/56-136 CALCULATION OF WAVE FUNCTIONS BY HARTREE METHOD

Naval Research Laboratory

tt I! II S. Peavy

k description : July-Sept 1955 issue, p. 34

Status : Continued. After the code was checked out, several runs were made.

However, since this code used considerable time on SEAC, new approximations

for V(r) were derived by the sponsor and a new code is to be written.

The new equations for V(r) are:

For r <

V(r) = 31 l-[P

f

IrCz) 1 2 dz + 4- /Vr lR(z)| 2 [z4(r2- i

5 )-2r]dz];

for r -y

V(r ) = R(z) [z + i

(r‘ - 2r] dz.

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

Origin and Sponsor : National Advisory Committee on Aeronautics

Manager : R. Zucker

Full task description : July-Sept 1955 issue, p. 35

Status : Continued. Computations for r=3 and r=2 were completed for various

combinations of k

c ,kg , ^

3 .- As answers are obtained, they are submitted to

the sponsor and graphed. From time to time additional sets of parameters

are calculated around maxima and minima*
26 Status of Projects

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

Origin and Sponsor : Naval Research Laboratory

Manager : L. Joel

Full task description : July-Sept 1955 issue, p. 36

Status : Continued (Reactivated). The sponsor requested data for two

additional distances: 0 and 3 cm. These were computed.

1 1 02-5-0-51 26/56-1 39 STUDY OF INTERNUCLEAR POTENTIAL FOR H.

Origin and Sponsor : NBS, Section 3,2 ^ Manager : E. Haynsworth

Full task description : July-Sept 1955 issue, p. 36

Status : Continued. The main code was checked out and several problems

were run. An additional code was written to compute the total electronic

energy of the system, which is a function of the vector coefficients

c^j obtained by the previous calculation. That is,

E = T

r

[ (f

l0U + + p2oU

)H

p,r,q,t

2

=1 prqt Ifrt (P

lo° +

/ pq

l+ />pq/5t -P pr * qt J *

a r The 3x3 matrices p- (P. •) are found as products C'C of the vectors

0. A J C

1 ,C^,C

2

, and the J

prq t are given two-electron integrals over atomic

orbitals. a?

1102-50-5126/56-150 MULTIPLE CORRELATION ROUTINES

Origin and Sponsor : Agricultural Economics Division, Department of

Agriculture

Managers : H. Bremer, M. Paulsen

Objective: To analyze on SEAC economic data on natural commodities.

This involves the preparation of routines to obtain the normal equations,

the regression coefficients and their standard errors, the multiple

correlation coefficients, the standard error of estimate and the constant

term for sets of commodity data, and the development of a progran/ toward

fitting systems of equations relating to the data by the limited information

approach.

Background : The Agricultural Marketing Service of the Agricultural

Economics Division undertakes a program to determine the extent to which

economic factors influence the utilization and prices of natural commodities

in the nation, such as cotton, corn, wheat, hogs, beef cattle, and others.

The analysis for each commodity will be used to predict estimates of

future utilization and price, or for analyzing the effects of alternative

Government policies with respect to these commodities.

Status : New. The codes were written, and checking is in progress.

1 102-50-51 26/56-141 X-RAY SPECTRA CALIBRATION

Origin and Sponsor : NBS, Section 4.11

Manager : I. A. Stegun

Ob iective : To obtain energy and angular distributions for the X-rays

produced in thin targets by high energy electrons. The electron energies

are 2.5, 5 5 and 10 Mev, and the target materials are beryllium, aluminum,

and gold. A theoretical expression for this distribution, given by

Gluckstern and Hull (Physical Review ^0, 1030 (1953)) 5 is to be used.
Status of Projects 27

Background : The X-radiation from all high-energy electron accelerators

has a continuous distribution in energy. Every experiment performed

with such a machine depends upon the shape of this distribution.

Although these experiments have been in progress for the last fifteen

years, no rigorous interpretation of them has been possible because

the shape was not known accurately. Recently techniques have been

developed at NBS with which to measure high-energy X-ray energies and

a well-controlled experiment has been performed to measure the shape

and to compare with theory. The experiment was performed in considerable

detail at a variety of angles and energies. Therefore, in order to make

a complete comparison with theory, the theoretical expression must be

evaluated at the same energies and angles.

The problem was requested by H. W. Koch of the NBS Betatron

Section (4.1).

Status : Completed (New). The results were transmitted to the sponsor.

1 1 02-40-51 26/56-1 43 CHARGED PARTICLE TRAJECTORIES IN THE MAGNETIC FIELD

OF THE EARTH

Origin and Sponsor : Naval Research Laboratory

Manager : J. H. Wegstein

Objective : To integrate numerically the differential

determine the trajectories of a charged particle in

of the earth.

The particles are primarily protons and the

is represented as that of a magnetic dipole. Using

cartesian coordinate system, the equations of motion

equations which

the magnetic field

field of the earth

Stormer units in a for the particles are

_ / 2 2 , 2s-5/2rro 2 2 2, . x = -(x +y +z ) ' L(2z -x -y ) y - 3yzzJ

y = -(x2

+y

2

+z

2

) ^

2

[ (x

2

+y

2-2z 2

) x + 3xzz]

z = 3z[h(x 2

+y

2+z

2 )"^//2+(x 2

+y

2

) (x

2

+y

2+z2

) _If

]

where h = (xy-yx) - (x

2

+y

2

) (x

2

+y

2

+z

2

) ^

2

. These equations are to be numerically integrated from given

initial conditions, using the Runge-Kutta method for simultaneous

differential equations. The accuracy of the calculation is to be

checked periodically by observing the values of x

2+y 2+z^ which should

equal 1 . The integration interval is determined automatically from the

2 ..2 ..2 2 curvature C obtainable by the formula C = x +y +z . Background : Observations on the time and space variation of the earth's

magnetic field suggest the presence of large electrical currents in the

upper atmosphere. The existence of these currents is also indicated

by recent experiments of W. H. Bennett on the Stormertron, a device which

may be described as an analogue computer for the determination of

trajectories in a dipole field. In the present calculations those

trajectories will be investigated which appear favorable for the production

of large currents. The Stormertron results and the earlier calculations

of Stdrmer will be used as guides in the choice of initial conditions for

these trajectories.

This problem is being programmed for solution with the NORC.

Status : New. The code has been written and checked out on the NORC. The

trajectories for 111 cases have been computed, and the results were sent

to the sponsor.
28 Status of Projects

1102-40-51 26/56-145 GAMMA-RAY SPECTROSCOPY

Origin and Sponsor : NBS, Section 4.1

Manager : M. Newman

Ob iective : To compute the inverse of a 56x56 matrix arising in gamma-ray

spectroscopy, and to find the solutiois of several systems of equations

in 56 unknowns with this matrix as ground matrix.

Background : The problem arises in connection with scintillation spectrometers.

A recent experiment performed in the Betratron Laboratory resulted in a large number of pulse height distributions obtained with the scintillation

spectrometer. The conversion of these distributions into Bremstrahlung

spectra that can be compared with theory requires the use of the inverse

of a 56x56 matrix. The computation was requested by M. Starfeld (4.1).

Status : Completed (New). The inverse of the 56x56 matrix and the solutions

of the corresponding equations (about 15) were computed. The results

were transmitted to the sponsor.

1102-40-51 26/56-146 SPORADIC-E THEORY BY TURBULENCE

Origin and Sponsor : NBS, Section 82.1

Managers : B. G. Urban, H. H. Howe, (82.1)

Objective: To compute tables showing the transmission, t, of a parabolic

E-region of thickness 2Y, subject to uniform turbulence, as a function

of three parameters , k, and P. Given f c = critical frequency of

E-region, and f = frequency of wave under study, then - f/f

c

, which is

greater than 1 . Given i = linear dimension of the turbulence (definable

only on a statistical basis) and Xc = wavelength in free space of a wave of the critical frequency, then k = bir A/X , and k is of the order

of unity. The mean ion-density N varies as 1-z 2 be%ween z=-1 and z=1

,

and is zero elsewhere; z is the altitude, suitably modified to make it

dimensionless. Turbulence causes the ion-density to change from N to

N+AN; AN varies with position and time, but we here assume that the

mean squ are of AN/N is constant, of the order of .0001. P is defined

as (AN/N) 2 • (Y/2i), and is of the order of .01. The transmission t of

the layer is computed by the following formulas in reverse order:

1 + z

2

)

.

The transmission is tabulated both as t and in decibels, where the

latter is computed as 10 Mt*. The integration is performed by the ninepoint Gaussian formula.

Background : Sometimes the E-region of the ionosphere reflects radio waves

of a frequency much higher than normal. This is called Sporadic E and

implies a much greater ion-density than normal. The theory that is here

being studied assumes that sporadic E occurs because of irregularities

in the ion-density caused by turbulence. This investigation covers only

the fipst of a number of models: It is assumed that the general level

of ion-density is a parabolic function of height, and that the turbulence

is uniform throughout the region (i.e., percent fluctuation from general

level is the same throughout the region, and linear dimensions of fluctuations

8re the same throughout). At this time, only the amount of radiation

that passes through the E-region to the F-region is computed.

Status : Completed (New), The required tables were computed and transmitted

to the originator.
Status of Projects 29

1 102-l+0- 51 26/56-1^-7 GROUND-WAVE ATTENUATION

Origin and Sponsor : NBS, Section 82.1

Managers : B.G;Urban, H. H. Howe (82.1)

Ob iective : To compute the intensity f* of a radio ground-wave as a function

of four parameters D, f , (J“, and k. Here D is the distance in km; f

is the frequency in kc/sec; (Tis the conductivity of the ground (assumed

the same throughout the path) in mhos/meter; k is the ratio by which

the radius of the earth is increased to allow for atmospheric refraction.

The result is a complex number and is to be given in both polar and

rectangular forms, T=P+iQ=Re ie

. The formulas involve double summation and involve certain

auxiliary parameters; certain peculiarities of form arise from the fact

that they are only approximations to other more involved formulas.

The formulas are:

K = 830,45 jr/Jl ,

6 = (KA1/3 )e i(135°)

t % = .0008022 D f 1/3/k2/3

, T = 42iri}C

oo if X e s S^O 2t - (1/0)-

5

7TS

OO = ^ A

»,j J=0

or J-o

£ J

where the A-series is used for large b and the B-series for small b.

The terms of the series are found by iteration formulas:

If j > 1,

A

s,j " ( 2A 0 [ j-1

s,o n=j-2

A

s,n

A s,j-n V ( 1 2>

‘s,j ^ B

s,n

B s,j-n-2^

For j=1 , A

s,i =

».,i

='b •

For j = 0, the formula if s is large is:

A

s,o = i [i1r(s + ^

)]2/3 l>

ir/3

]’

and the formula for B,, ^ is identical, except that the 1/4 is replaced

S j o by 3A. But for small s, a more accurate value is to be obtained from

the following table:

a: 0 1 2 3 4 5

A

s,o: °* 808 2 - 577 3 - 824 4 - 892 ^.851 6 - 737

B

’ i 1.856 3.245 4.382 5.386 6.305 7.161

5 5 O

where all values are multiplied by e*

77

"73

.
30 Status of Projects

one The dividing point between the A

converges, occurs where = i.

and B series

When this

, at which neither

equation is nearly

satisfied convergence is so slow that it is impracticable, since all

values of A or B must be retained in the memory until they become

negligibly small. Hence, for small D, where many values of s must be

used, there is quite a range of values of 6 (and hence of f) for which

the computations cannot be carried out by these formulas. It is probable

that they could be computed by analytic continuation.

Background : In connection with studies oi low-frequency radio propagation,

information is needed relating the field strength broadcast by a source

to th8t of the ground wave reaching a distant point. The tables we

prepared in this task express the theoretical relation between them for

various values of four parameters.

Status : New. A code was prepared to a suitably scaled parameter D, and

computations were made for 12 values of f and for three values each of

D and 5.

1102-^0-5126/56-1^9 LOGICAL MATRIX MULTIPLICATION

Origin and Sponsor : Signal Corps Engineering Laboratories

Manager : D. Jirauch

Ob iective : To multiply logically according to certain rules the rows of an incidence matrix of order 170. The results desired are (a) a certain

symmetric incidence matrix, and (b) certain n-tuples of numbers, where n should be as large as 1

6

, if possible.

Background : The problem arises in the assignment of broadcast frequencies

to transmitters in a given area so that interference is at a minimum.

The elements (0 or 1) of the symmetric incidence matrix indicate the

presence or absence of interference between transmitters operating on two frequencies; the desired results are sets of 1

6

frequencies no two

of which interfere with each other.

Status : Completed (New). The results were transmitted to the sponsor.

1 102- 10- 5126/56-151 MULTIPLE SCATTERING IN CROSS SECTION MEASUREMENTS

Origin and Sponsor : Naval Research Laboratory

Manager : I. Rhodes

Objective : To compute corrections for multiple scattering of neutrons in

thick target measurements by Monte Carlo techniques.

Background : In cross section measurements with thick targets, allowance

must be made for the possibility that the incident particle will scatter

more than once before emerging from the target. These multiple scattering

events m8y make a major contribution to the detected flux. Their magnitude

may be determined by an iterative Monte Carlo computation, using the

single scattering results as the starting point in the iteration. The

program has been developed for the geometry of a ring target, and for

specific application to measurements on the angular distribution of

e last ically scattered neutrons in the medium energy region. The problem

was proposed by R. Jastrow and H. Wagman of the Naval Research Laboratory.

Status : New.

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

Full task description : July-Sept 1955 issue, p. 38

Status : Continued. A code to distribute expenditures and obligations was

written, and code-checking of this on UNIVAC was started. A code to

handle special shipyard installations was also written. The code for

presenting and editing the output is almost completed.
Status of Projects 31

1102-40-51 26/56-1 53 VIBRATIONAL EIGENVALUES OF HYDROGEN, DEUTERIUM AND

TRITIUM COMPOUNDS

Origin and Sponsor : NBS, Section 3.2

Manager : E. Haynsworth

Ob iec tive : To determine the eigenvalues of the non- symmetric product, FG,

of positive definite, symmetric, nxn matrices, F end G, which arise in

the study of molecular spectra.

Background : The eigenvalues obtained from this routine are related simply

to the normal vibrations of the 8toms in the molecule. The group of

molecules for which the frequencies are presently being calculated are ammonia, phosphine, arsine, and stibine and all of the possible isotopes

formed by substitutions of hydrogen, deuterium and tritium. This constitutes

a group of 40 molecules now under consideration. The secular equations

are sixth degree corresponding to the six vibrational degrees of freedom

which exist for these molecules. The normal frequencies are unique in

that any vibration of 8 system can be expressed as a superposition of

normal frequencies. The ultimate goal is to use these normal frequencies

to compute thermodynamic function^ namely, specific heats, free energy,

enthalpy, and entropy, for all 40 molecules. The plan is to continue

this type of calculation for molecules associated with 18th degree secular

equations and perhaps with equations of even higher degree.

Status : Completed (New). Some of the matrices had enough elements 8like to

satisfy the conditions of a theorem by E. Haynsworth (not yet published)

by use of which the matrices could be reduced by inspection to matrices

of smaller order which had the same roots. In this particular problem

n=6, and these matrices were reduced to three 2x2 matrices. So the

problem of finding the roots was trivial.

For the non-reducible matrices, a general routine was set up to

factor the matrix F into the product C’C, where C is a real upper

triangular matrix, and C is its transpose. Then the symmetric matrix

CGC* was formed, and its eigenvalues found, as CGC’ had the same roots

as FG.

The completed computations were transmitted to the sponsor.

1 1 02-4-0-51 26/56-1 57 DAMAGE ASSESSMENT

Origin and Sponsor : Office of the Assistant Secretary of Defense (Supply

and Logistics)

Managers : A. J. Hoffman, H. Bremer

Objective

:

To develop programs for obtaining, by means of high-speed

computers, estimates of the damage to the country from enemy air attack.

Background : For purposes of analysis, it is vital that the Department of

Defense be able to deduce, from a knowledge of the weapon characteristics

and ground zeros of an enemy air attack, estimates of the damage inflicted

on the population and productive facilities of the country. Previous

work on this problem has produced a set of UNIVAC codes, and it is the

object of the present task to reformulate the procedures in order to

obtain programs for UNIVAC that will (1 ) permit a more realistic assessment

of the damage and (2) produce output in a more usable form, yet (3) keep

the computing time within reasonable bounds.

Status : New. Some conferences on weapon effects have been held, and

preliminary flow charting for part of the program has been started.

1 102-40-5126/56-158 FEDERAL RESERVE DATA

Origin and Sponsor ; Federal Reserve Board

Manager : E. Marden

Objective : To program and code for UNIVAC a series of tabulations based on data collected from the 12 Federal Reserve districts. The tabulations

will furnish such information as numbers of loans, amounts outstanding,

etc., and will make use of cross classifications according to such
32 Status of Projects

characteristics of the loan data as business of borrower, assets of

borrower, maturity dates, geographical location of borrower, and others.

Background : The processing of data for this survey, which occurs from time

to time 8t the Federal Reserve Board, is being carried out on a digital

computing machine for the first time. This tabulation will be used to

assess the application of digital computing machines to the Guard's work.

Status : New.

1102-40-51 26/56-1 63 ANGULAR DISTRIBUTIONS AND POLARIZATION EFFECTS IN

NUCLEAR SCATTERING

Origin and Sponsor : Naval Research Laboratory

Manager : I. Stegun

Ob iective : To calculate differential cross sections and polarization effects

in the nuclear scattering of neutrons and protons by numerical integration

of the wave equation and exact computation of asymptotic phase shifts.

Background : Experiments have been performed on the angular distribution

of neutrons and protons elastically scattered by nuclei at energies up

to 300 million volts. Polarization effects in double scattering have

also been measured, primarily at high energies. The theoretical

interpretation of these measurements provides information regarding

several aspects of nuclear structure, among these the mean radius and

density profile of the nucleus, the absorption coefficient and refractive

index of nucleons in nuclear matter, and the strength of the spin-orbit

coupling.

In the present calculations the nucleus is represented by a complex potential well of arbitrary radial form, with the coulomb field

and a Thomas spin-orbit term included. The phase shifts which determine

the cross section are computed exactly by numerical integration of the

wave equation in this potential. The calculations will include a survey

of cross sections for neutrons and protons over the full range of energies

and atomic weights.

Calculations have been carried out at energies of 1 50 and 3^0

million volts, in which the same nuclear model is assumed, but phase

shifts are computed in WKB approximation. Below 100 million volts the

WKB approximation is inadequate and the present analysis based on exaet

computation of the phase shifts is required. Exact calculations will also

be carried out at the higher energies, however, for comparison with the

WKB results.

The problem was proposed by Dr. Robert Jastrow of the Naval Research

Laboratory.

Status : New. The code was written and is being checked for a sample problem

describing the scattering of protons by aluminum at 300 Mev.

371 1-60-0009/56-16^ TRANSIENT HEAT FLOW PROBLEM, II

Origin : NBS, Section 10.2

Manager : W. F. Cahill

Objective : To obtain data on transient heat flow through various geometrical

configurations which closely approximate structures tested by the NBS

Fire Protection Section (10.2).

Background : This task is a continuation of task 1 1 02-4-0-51 26/54-:45 for

additional values of the parameters.

Status : Completed (New). The cases for the new parameters were computed and

were sent to the originator.

1102-^0-5126/56-165 INTEGRALS FOR SCATTERING FUNCTIONS

Origin and Sponsor : Naval Research Laboratory

Manager : R. Zucker

Objective » To evaluate 11 mathematical expressions involving integrals that

are functions of six parameters. The integrands involve the function
Status of Projects 33

I =

oo = f(^,p),

(n+1) sin (6

n - b^,)

where

tan 6_ =

tan

T .1 Jn'<*>

tan ~

! J

n

( *)

~V!^ tan ^ ^ vq^)

1

' 4 1^5? ' “"S’* * ("*’ '

F^n+2 2 P | 9n+ 9 I 0(n~t^

'1n+1 - 2(p-»

B

2(p- )

| 2n+2

| 2(p-^j

1 | 2n+2

l 2(P-J)*j

Jn is a spherical Bessel function and Y

n is a spherical Neumann function.

Background : In the measuring of the transport properties of semiconductors,

it has become necessary to evaluate the influence of the scattering of

free charge carriers by ionized impurities. The existing theoretical

treatments amount to calculations in Born approximations and are thus

restricted to the condition that the wavelength of the charge carrier

be much smaller than the size of the scattering center. This excludes

from the allowed range most of the concentrations of impurities and

regions of temperature where ionized impurity scattering is dominant.

In order to get the correct picture over the range realizable by

experiment, it is necessary to obtain the above exact calculations.

Status : New. The code was written,% snd checking was started.

1102-^0-5126/56-171 COLLISION INTEGRALS USED IN TRANSPORT THEORY

Origin and Sponsor : NBS, Section 3.2

Manager : J. Cooper

Ob iec tive : To compute a table of transport properties as a function of

temperature. This involves three steps:

(a) To calculate the angle of deflection of a particle scattered by a central force,

where pC is the angular deviation measured from the initial line of approac

of the particle,

g is the relative velocity of the particle and the force center,

s is the impact parameter or distance of the line of approach from

a line drawn parallel to it through the force center,

r m

is the distance of closest approach,

0(r) is the potential of the force center.
3 *+ Status of Projects

This involves solution by an iterative process of the equation

(*r ) 2

- ^ = o

to find the largest root and then integration to find the value of

as a function of g and b for various potential functions.

(b) To evaluate the collision integrals for parameters t end s (integers < 5 )> end T (renging from .1 to 1),

(c) To combine the collision integrels for eech velue of T to compute

the trensport properties.

Background : The problem is communiceted by Dr. Melville S. Green, NBS

Thermodynamics Section.

Status : New.

( 1 )
j6. 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. During this quarter members of the Section

provided statistical assistance and advice to a number of Bureau personnel.

The following are representative examples:

(1) Fitting of polynomials to data on atomic spectra was carried

out with the aid of SEAC, for E. K. Plyler and H. C. Allen, Jr., of Section

4 2

(2) Least squares determination of the virial coefficients of

air was done on SEAC, for A. Friedman of Section 3.2,

(3) Experimental data were analyzed for evaluating the modulus

of rupture of 0° sapphire rods over a range of temperatures, for J.B. Wachtman

of Section 9.1.

(4) The experimental results of full-scale field trials, which

made use of a 1/4 fractional replication experimental design, were analyzed

for P. Haas of Section 33.1? DOFL, with the aid of SEAC.

(5) An investigation of the precision of measuring the sound

absorption coefficient was carried out for R. Waterhouse of Section 6.1.

(6) Two exploratory experiments were planned and analyzed for

evaluating how well judges can discriminate distinctness of image and

brightness characteristics for standard wood samples, for I. Nimeroff of

Section 2.1.

(7) Some exploratory work on the distribution of properties of

fuels which are mixtures of two other fuels was done for R. S. Jessup of

Section 3.2.

Publications:

(1) National physical standards and design of experiments. W. J. Youden.

To appear in Revue de L’Institut International de Statistique (The

Hague)

.

(2) More scientists. W. J. Youden. Submitted to a technical journal,

(3) A simplified calibration of platinum versus platinum-rhodium

thermocouples. B. E. Squires (3.1) and M. Zelen, In manuscript.

STATISTICAL SERVICES FOR COMMITTEE ON SHIP STEEL, NRC

Task 1103-40-5105/52-1

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

Manager: W. J, Youden

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

Status: CONTINUED. A multiple regression study of the effect of

chemical composition on ship steel is nearly completed.

35
36

RESEARCH IN APPLICATIONS OF MATHEMATICAL STATISTICS TO

PROBLEMS OF THE CHEMICAL CORPS

Task 1 103-40-51 18/52-1

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

Chemical Corps, Slept, of the Army

Manager: C. Eisenhart

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

Status: CONTINUED. A report giving fractional replicated designs

of the 2

n series for n = 13, 14, 15, 16 factors was submitted.

The SEAL code for evaluating the power function of combining

two independent tests of significance has been written and code checked.

The computation of the power points is now underway.

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. Some additional work was done on the use of

extreme value techniques in the sampling and analysis of data on pebble

deposits. Correspondence was carried on with Helen Cannon of Colorado

Plateau District on the problem of the association between the incidence

of certain plants and uranium deposits.

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 progress has been made in the writing of

the manual. An important section, "Statistical Tests Concerning Averages,"

was written and revised, and copies were forwarded to the Office of

Ordnance Research. Other sections are in various stages of development.
APPLICATION

of

NATIONAL BUREAU OF STANDARDS AUTOMATIC COMPUTER (SEAC)

The record of SEAC operations for tasks of the Applied Mathematics

Division for the period October 1 through December 31 is as follows:

Task No . NBS : 1104/55-55

5116/55-56

51 16/56-148

1110/47-2

1110/52-31

1110/52-57

1110/53-51

1110/53-52

1110/55-65

5126/51-8

51 26/52-44

5126/53-27

5126/53-29

0009/5^-17

0009/5^-19

5126/5^-30

0009/55-53

0009/55-75

5126/55-81

5126/55-88

5126/55-96

5126/5^-97

5126/55-105

5126/55-115

51 26/55-1 18

5126/56-121

5126/55-123

5126/56-129

0009/56-131

5126/56-13^

5126/56-139

5126/56-141

5126/56-145

5126/56-146

5126/56-147

5126/56-150

5126/56-153

51 26/56- 1 60

0009/56-164

Hours Used:

Code Productive

Title Checking Ooeration

Research in numerical analysis 22 1 Research in mathematical topics applicable 1 to numerical analysis

Nerve fiber reaction 4 13

Tables of Coulomb wave functions 30 16

Exponential function for negative 1 argument

Sieverts integral 1 Radiation diffusion III 13 56

L-Shell conversion coefficients 1 62

Automatic coding 7 10

Tables of power points of analysis of

variance tests 9

Calculations for d-spacings 1 8

Thermodynamics functions 1 Dynamic behavior of aircraft structure 2 12

De polymerization 2 22

Energy levels of complex atoms 2

Spectrometer analysis 8 17

Electronic functions 2

Parameter of the dispersion equation for

optical glass 1 Combining tests for significance 3 25

Stresses in a wall foundation 4 7

Monte Carlo 7

High temperature properties of air 7 103

Aeronomic tidal winds of thermal origin 3 1 Adsorption integrals 1 Thermometer calibrations 1 Electron penetration 3

Temperature distribution 2

Processing of Public Housing data 1 Ray tracing 18 22

Transmission delay times 1 Study of internuclear potential- for H^ 6 1 Spectrometer calibration 3 14

Gamma-ray spectroscopy 1 21

Sporadic E-Theory by turbulence 8 2

Ground-wave attenuation 12 33

Mathematical expression 2 3 Vibrational eigenvalues 6 4

Matrix factoring 9 2

Transient heat flow problem 13

37
38

Task No . NBS

:

TT25/56-167

Miscellaneous

0002/ 52-1

51 60/55-85

Title

Code

Checking

Hours Used : Productive

Operation

Multiple regression

Training 3

Statistical aspects of NBS administrative

operations 6

Research in mathematical elasticity 1

1

47

24

Other:

5126/53-45

5126/54-13

5126/55-104

5126/55-113

5126/55-119

5126/55-122

5126/55-127

5126/56-130

5126/56-133

5126/56-136

5126/56-137

5126/56-138

5126/56-140

5126/56-151

5126/56-163

Miscellaneous

Application of game theory

Award of procurement contracts for

linear programming

"Fuse" problem

Reactor design

Field rocket problem

Solution of normal equation

Vibrations of a circular disc

Aircraft responses

Complex eigenvalues

Calculation of wave functions

Stability of supported plates

Crystal counter efficiency

Multiple correlation routines

Multiple scattering in cross section

measurements

Angular distributions and polarization

effects in nuclear scattering

2

24

13

2

13

3

10

15

11

4

5

2

3

11

11

32

9

19

3

21

Totals: 258 740
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.

Seminar: Special Topics in Numerical Analysis

(Offered by the Department of Mathematics of The American

University in collaboration with the Graduate School of the

National Bureau of Standards)

KATO, T. (University of Tokyo) (1) Perturbation theory of eigenvalues:

1. Operators in Banach space. October 17 . 2, Finite Matrices.

October 2k m (2) On helium wave functions. October 31, (3) On

eigenvalues of free plates. November 7.

TODD, J. Approximation theory: 1. The theorems of Weierstrass, Bernstein

and Chebyshev. November 15. 2. Theorems of Bernstein,

Voronowska ja , Chebyshev and Markoff. December 6.

DAVIS, P. Extremal problems and criteria for closure: Three lectures,

November 22, December 13, December 20.

Papers and Invited Talks

Presented by Members of the Staff

at Meetings of Outside Organizations

Papers presented at the Office of Ordnance Research Conference on the Design

and Experiments in Army Research, Deve lopment , and Testing held at the

National Bureau of Standards, Washington, D. C. , October 19-21.

CAMERON, J. M. Some examples of the use of high speed computers in statistics.

EISENHART, C. The principle of randomization in the design of experiments.

YOUDEN, W. J. Design of experiments in industrial research and development.

s(c 4s

Papers presented at the Joint Meeting of the American Statistical Association

and the Institute of Mathematical Statistics, New York, N.Y. , December 27~30:

BURTON, R. C. Some combinatorial relationships arising in the dualization of

incomplete block designs.

LIEBLEIN, J. Recent developments in the application of extreme value theory.

MILLER, F, L., Jr. The construction of fractional factorial designs for the

2“ series.

NATRELLA, M. G. Unpublished tables prepared in the Statistical Engineering

Laboratory, National Bureau of Standards.

39
kO Lectures and Symposia

PETTIGREW, H. The use of the 2

n fractional factorial designs for factors

at 4- levels.

ROSENBLATT, J. R. On the efficiency of certain classes of tests based on U-s tatis tics

.

YOUDEN, W. J. (1) Special training arranged by Local Sections of the

American Chemical Society. (2) Who makes designs?

ZELEN, M. On the analysis of incomplete block designs.

ALT, F. L. Development of automatic computers. Presented at the Institute

on Electronics in Management held by the Department of Government

and Public Administration of The American University, Washington,

D. C. , November 1*+. ANTOSIEWICZ, H.A. Stability theory of nonlinear differential equations.

Presented at the Department of Mathematics, Princeton University,

N. J. , December 9.

DAVIS, P. Orthonormalizing codes and their uses. Presented at the Moore

School of Engineering, University of Pennsylvania, Philadelphia,

Pa . , November 10.

EISENHART, C. Random notes. Presented before the Washington Chapter,

American Statistical Society, Washington, D. C. , November 28.

HOFFMAN) A. J. Mathematical approach to production scheduling. Presented

before the American Institute of Industrial Engineers, Washington,

D. C., October 11.

KATO, T. (University of Tokyo) and 0. TAUSSKY- TODD . Normal matrices.

Presented by title at the meeting of the American Mathematical

Society, Houston, Texas, December 30.

NEWMAN, M. The coefficients of modular form. Presented at the meeting of

the American Mathematical Society, University of Maryland, College

Park, Md., October 29.

TODD, J. The problem of stability in the numerical solution of ordinary

and partial differential equations. Presented at a Seminar of

the Moore School of Engineering, University of Pennsylvania,

Philadelphia, Pa., December 1*+. YOUDEN, W. J. Making good measurements. (1 ) Presented before the RESA

Society, Camp Detrick, Frederick, Maryland, November 17, and (2)

before the St. Louis Section of the American Chemical Society, St.

Louis, Missouri, December 5.

Programming measurements in experimentation. Presented before

the Washington Chapter, Institute of Radio Engineers, Washington,

D. C. , November 21

.

Marbles, measurements, and mathematics. Presented at Northwestern

High School, Hyattsville, Maryland, December 1*f. ZELEN, M. Exact tests of significance for the recovery of inter-block

information. Presented at the Department of Agriculture, Washington,

D. C. , October 1 4-.
Publication Activities

1 . PUBLICATIONS THAT APPEARED DURING THE QUARTER

1.1 Mathematical Tables

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

Mathematics Series 45. Available from U. S. Government Printing

Office, Washington, D. C. , 55 cents.

(2) Table of the descending exponential, x=2.5 to x=10. Applied

Mathematics Series 46. Available from U. S. Government Printing

Office, Washington, D. C. , 50 cents.

1.3 Technical Papers

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

Pac. J. Math. Suppl. 1, 641-642 (1955).

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

and S. Levy (NBS 6.4). J. Aeronautical Sciences 22, 837-843

(Dec. 1955).

(3) Automatic computation of nerve excitation. K. Cole (National

Institutes of Health), H. Antosiewicz, and P. Rabinowitz. J. Soc.

Ind. Appl. Math, 3, 153-172 (1955).

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

Pac. J. Math. 5? Suppl. 1, 669-674 (1955).

(5) Fully convex normed linear spaces. K. Fan and I. Glicksburg

(RAND Corporation). Proc. Nat. Acad. Sci. 4l . 947-953 (1955).

(6) An algebraic proof of the isoperimetric inequality for polygons.

K. Fan, 0. Taussky and J. Todd. Washington Acad. Sci. 4£, 334—342

(1955).

(7) Addition theorems for general Legendre and Gegenbauer functions.

P. Henrici. J. Rational Mech. Anal. 4, 983-1018 (1955).

( 8 ) On certain series expansions involving Whittaker functions and

Jacobi polynomials. P. Henrici. Pac. J. Math. Suppl. 1,

725-744 (1955).

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

J. London Math. Soc. 30, 488-493 (1955).

(10)

Unimodular circulants. 0. Taussky. Math. Zeitschr. 63, 286-289

(1955).

(11)

Computation of atomic levels: the spectrum of singly ionized

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

P. Rabinowitz. J. Res. NBS 335—34l (Dec. 1955), RP2639.

(12)

On eigenvalues of sums of normal matrices.

Math. 5, 633-638 (1955).

H. Wielandt. Pac. J.

41
k-2

1 .*+ Note s

Publication Activities

(1)

Kleine Bemerkung zur asymptotischen Entwicklunq des Fehlerintegrals.

P. Henrici. Zeitschr. angew. Math. Phys. 6, 1*+5-1*+6 (1955).

2. MANUSCRIPTS IN THE PROCESS OF PUBLICATION DECEMBER 31, 1955

2.1 Mathematical Tables

(1) A table of the coefficients of the powers r(_('C9 . M. Newman.

To appear in the Proceedings of the Royal Netherlands Academy

of Sciences.

2.3 Technical Papers

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

Naval Research Logistics Quarterly.

(2) On the differential equation x + k(f(x) + g(x)x)x = ke(t).

H. A. Antosiewicz. Submitted to a technical journal.

(3) Stable systems of differential equations with integrable forcing

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

Mathematical Society.

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

Submitted to a technical journal.

(5) 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. To appear (in French) in Annales

de l'Institut Henri Poincar€ (Paris).

(6) 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.

(7) Uniqueness theory for asymptotic expansions in general regions.

P. Davis. Submitted to a technical journal.

( 8 ) Abscissas and weights for Gaussian quadratures of high order.

P. Davis and P. Rabinowitz. To appear in the Journal of Research

(NBS) . (9) Numerical experiments in potential theory using orthonormal functions.

P. Davis and P. Rabinowitz. To appear in the Journal of the Washington

Academy of Sciences.

(10) Some Monte Carlo experiments in computing multiple integrals.

P. Davis and P. Rabinowitz. To appear in Mathematical fables and

Other Aids to Computation.

(11) 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 195*+.

(12) Properties of currugated diaphragms. R. F. Dressier. To appear in

the Transactions of the American Society of Mechanical Engineers.
Publication Activities 43

(13) Heat flow in a fluid with eddying flow. W. H. Durfee (now at

Mt. Holyoke College). Submitted to a technical journal.

(1*+) A comparison theorem for eigenvalues of normal matrices. K. Fan.

To appear in Pacific Journal of Mathematics.

(15) 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.

(1 6 ) Systems of inequalities involving convex functions. Ky Fan.

Submitted to a technical journal.

(17) A comparison between the mean successive difference and the root

mean square successive difference. S. Geisser. Submitted to a technical journal.

(18) A note on the normal distribution. S. Geisser. To appear in Annals

of Mathematical Statistics.

(19) A matrix with real characteristic roots. K. Goldberg. To appear

in the Journal of Research (NBS).

(20) The formal power series for log e xe^. K. Goldberg. To appear in

the Duke Journal of Mathematics.

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

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

(22) Automatic computations with power series. P. Henrici. To appear

in the Journal of the Association for Computing Machinery.

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

To appear in Pacific Journal of Mathematics.

(24) On the domain of regularity of generalized axially symmetric

potentials. P. Henrici. Submitted to a technical journal.

(25) Chi the representation of a certain integral involving Bessel

functions by hyper geometric series. Submitted to a technical

journal.

(26) 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.

(27) Systems of distinct representatives and linear programming.

A. J. Hoffman and H. W. Kuhn (Bryn Mawr College). To appear in

the American Mathematical Monthly.

(28) The number of absolute points of a correlation. A. J. Hoffman,

M. Newman, E. G. Straus, 0, Taussky, To appear in the Pacific

Journal of Mathematics.

(29) Commutators of A and A*. T. Kato and 0. Taussky, To appear in the

Journal of the Washington Academy of Sciences.

(30) Applications of extreme value theory to cobbles and boulders in

gravel pits. W. C. Krumbein (Northwestern University) and

J. Lieblein. Submitted to a technical journal.
Publication Activities

(31) Solvability and consistency for linear equations and inequalities.

H. W. Kuhn (Bryn Mawr College). To appear in the American Mathematical

Monthly.

(32) Pairs of matrices with property L, II. T. S. Motzkin and 0. Taussky.

To appear in the Transactions of the American Mathematical Society.

( 33 ) An alternative proof of a theorem on unimodular groups. M. Newman.

To appear in the Proceedings of the America* Mathematical Society.

(3*0 An inclusion theorem for modular groups. M. Newman. Submitted to

a technical journal.

( 35 ) Generalizations of identities for the coefficients of certain

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

Mathematical Society.

(36) On the existence of identities for the coefficients of certain

modular forms. M. Newman. Submitted to a technical journal.

( 37 ) The diophantine equation axm-byn=c . M. Newman. Submitted to a technical journal.

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

appear in Canadian Journal of Mathematics.

( 39 ) 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 (New York University).

(5-0) On asymptotic series for functions occurring in the theory of

diffraction of waves by wedges. F. Oberhettinger . To appear in

the Journal of Mathematics and Physics.

( 5-1 ) On the Lerch zeta function. F. Oberhettinger. To appear in the

Pacific Journal of Mathematics.

( 5

- 2 ) Determinanten mit ueberwiegender Hauptdiagonale und die absolute

Konvergenz von linearen Iterationsprozessen. A. M. Ostrowski.

To appear in Comentarii Mathematici Helvetici.

( 5

- 3 ) On Gauss' speeding up device in the theory of single step iteration.

A. M. Ostrowski. Submitted to a technical journal.

( 5

- 5-) On the convergence of Gauss' alternating procedure in the method of

the least squares, I. A. M. Ostrowski. Submitted to a technical

journal.

( 5

- 5 ) Contributions for inverse osculatory interpolation. H. E. Salzer.

(Department of the Army). To appear in the Journal of Research of the

NBS,

( 5-60 Contributions to the theory of rank order statistics. I. R. Savage.

Submitted to a technical journal.

( 5

- 7 ) Inequalities for probabilities associated with the multivariate

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

journal.

( 5

- 8 ) Simultaneous tests of randomness and other hypotheses. I. R. Savage.

Submitted to a technical journal.
Publication Activities 25

(^9) Algebraic equations satisfied by roots of natural numbers.

E. G. Straus (University of California at Los Angeles) and 0. Taussky.

To appear in the Pacific Journal of Mathematics.

( 50 ) A note on group matrices. 0. Taussky. To appear in the Proceedings

of the American Mathematical Society.

(51 ) 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 .

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

J. Todd. Submitted to a technical journal.

( 53 ) 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 . (20 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 .

( 55 ) 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 .

( 56 ) 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 .

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

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

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

Calif., August 1953 .

( 58 ) National physical standards and design of experiment. W. J. Youden.

To appear in Revue de L'Institut International de Statistique

(The Hague)

.

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

(American Cyanimid Co.). To appear in Biometrics.

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

information. M. Zelen. Submitted to a technical journal.

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

journal.

.5 Miscellaneous

(1) More scientists. W. J. Youden. Submitted to a technical journal.

00MM-DC-U9 ,223
"

i

.

’
THE NATIONAL BUREAU OF STANDARDS

Functions and Actirities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and

maintenance of the national standards of measurement and the provision of means and methods

for making measurements consistent with these standards; the determination of physical constants

and properties of materials; the development of methods and instruments for testing materials,

devices, and structures; advisory services to Government Agencies on scientific and technical

problems; invention and development of devices to serve special needs of the Government; and the

development of standard practices, codes, and specifications. The work includes basic and applied

research, development, engineering, instrumentation, testing, evaluation, calibration services, and

various consultation and information services. A major portion of the Bureau’s work is performed

for other Government Agencies, particularly the Department of Defense and the Atomic Energy

Commission. The scope of activities is suggested by the listing of divisions and sections on the

inside of the front cover.

Reports and Publications

The results of the Bureau’s work take the form of either actual equipment and devices or published papers and reports. Reports are issued to the sponsoring agency of a particular project

or program. Published papers appear either in the Bureau’s own series of publications or in the

journals of professional and scientific societies. The Bureau itself publishes three monthly periodicals, available from the Government Printing Office: The Journal of Research, which presents

complete papers reporting technical investigations; the Technical News Bulletin, which presents

summary and preliminary reports on work in progress; and Basic Radio Propagation Predictions,

which provides data for determining the best frequencies to use for radio communications throughout

the world. There are also five series of nonperiodical publications: The Applied Mathematics

Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous

Publications.

Information on the Bureau’s publications can be found in NBS Circular 460, Publications of

the National Bureau of Standards ($1.25) and its Supplement (S0.75), available from the Superintendent of Documents, Government Printing Office. Inquiries regarding the Bureau’s reports and

publications should be addressed to the Office of Scientific Publications, National Bureau of

Standards, Washington 25, D. C.