NATIONAL BUREAU OF STANDARDS REPORT

3948

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

A Quarterly Report

October through December 1954

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

FOR OFFICIAL USE
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

THE NATIONAL BUREAU OF STANDARDS

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

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

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

cover of this report.

Electricity. Resistance and Reactance Measurements. Electrical Instruments. Magnetic

Measurements. Electrochemistry.

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

Technology. Length. Engineering Metrology.

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

and Lubrication. Engine Fuels. Cryogenic Engineering.

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

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

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

Spectrochemistry. Pure Substances.

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

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

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

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

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

Materials. Constitution and Microstructure.

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

Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering.

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

Instrumentation. Process Technology.

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

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

Standards. Microwave Standards.

#Office of Basic Instrumentation ^Office of Weights and Measures.
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

NAML 3948

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

October through December 1954

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication, re

unlesa permission li 25.D.C. Such per

cal ly prepared If tl

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

In part. Is prohibited

andards, Washington

iport has been speclfleport for Its own use,
APPLIED MATHEMATICS DIVISION

October 1 through December 3

1

? 195^

TECHNICAL ADVISORY COMMITTEE

David Blackwell, Howard University Mina S. Rees, Hunter College

Mack Kac , Cornell University A, H. Taub, University of Illinois

Philip M. Morse, Edward Teller, University of California

Massachusetts Institute of Technology

DIVISION OFFICE

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

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

Olga Taussky-Todd , Ph.D., Consultant

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

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

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

Yates S. Sladen , Assistant Chief Clerk

Mary E. McKinley, Secretary

Esther L. Turner, Secretary

NUMERICAL ANALYSIS SECTION

’•‘Henry A. Antosiewicz , Ph.D

.

Philip Davis, Ph.D.

Karl Goldberg, M.A.

John Todd, B.S., Chief

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 (Acting)

Ruth K. Anderson , M.A,

Irene A. Ste gun , M.A. , Assistant Chief

William G. Hall, B.S. Maxine L. Paulsen, B.S. Joseph Bram, Ph.D

.

Gene vie E . Hawkins , B.S. B. Stanley Prusch

Hans 0. Bremer, B.A. Gloria F. Holmes, B.S., Sec'y Ida Rhodes, M.A. William F. Cahill, M.S. Dorothea H. Jirauch , M.A. Sally Tsingou, B.S. Ruth E . Capuano Lambert S„ Joel, B.A. Bertha H. Walter

Mary M. Dunlap, B. S. David S. Liepman Philip J . Walsh, B.S. Anne F . Futte rman , B.A. Ethe 1 C. Warden, B.A. Joseph H. Wegstein, M.S

Leon Gainen , B.A. Kermit C . Nelson J . H. Wilkinson , Ph.D.,

Billie R. Gill, M.S. Peter J . O'Hara, B.S. Guest Worker

Elizabeth F. Godef roy Mary Orr Ruth Zucker, B.A. Stanley D . Grant , Jr.

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

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

Lola S, Doming, M.A. Mary G. Natrella, B. A. Mary L. Epling, Sec'y **I. Richard Savage , Ph.D

Le la J . Hamilton , Sec'y Marvin Zelen, M. A. Julius Lieblein, Ph.D.

Edward W. Cannon , Ph.D., Chief

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

John G. He rshbe r ge r , M.S. Edith N . Reese, B.A. *Fritz Oberhettinger, Ph.D. Lillian Sloane , Sec'y

ter Henrici, Ph.D.

Marion T. Carson

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

y

William S . Connor, Ph.D.

MATHEMATICAL PHYSICS SECTION

Anne R. Cock , B.A. Leon Feldman, B.A.

*Under contract with The American Unive r sity ** On leave of absence
CONTENTS

Page

Status of Projects as of December 31? 1954 ... 1 Numerical Analysis Section 1 (NBS Section 11.1)

Computation Laboratory 9

(NBS Section 11.2)

Statistical Engineering Laboratory 24

(NBS Section 1 1 .3)

Mathematical Physics Section 29

(NBS Section 11.4)

SE AC 32

Lectures and Symposia 34

Publication Activities 37

Status of Projects

December 31, 1954

I. NUMERICAL ANALYSIS SECTION

(Section 11.1)

RESEARCH IN NUMERICAL ANALYSIS AND RELATED FIELDS

Task 1101-10-1104/^5-55

Origin and Sponsor: NBS Authorized 8/13/54

Managers: J. Todd, P, Davis Revised 8/29/54

Full task description: July-Sept 1954 issue, p. 1 Status: CONTINUED. In planning future activity within the project

in connection with "controlled" experiments in computation, a survey of

earlier work of this type on SEAC has been made. Among the non-trivial

computational experiments carried out were the following:

( 1 ) Monte Carlo

Generation and testing of pseudo-random numbers

Inversion of a matrix

Solution of Laplace equation in various dimensions

Evaluation of multiple integrals

(

2)

*Inve r sion of matrices

( 3

)

*^onf ormal mapping

Mapping of ellipses on circle using Lichtenstein-Gerschgorin

integral equation

Computation of exterior mapping function and transfinite

diameter using orthogonal polynomials

(4)

*Game theory and linear programming

(5) Determination of characteristic roots of matrices

Reports of papers have been published on the starred items; reports on other topics are in preparation. Further experiments are planned as soon

as SEAC becomes available again.

J. Todd has been preparing an extended version of publication

(15) below, which will appear (in German) in Jahre sbe richt der Deutscher

Mathematiker Verein.

P. Davis and P. Rabinowitz have computed the first 12 orthonormal

polynomials for a certain simply connected non-convex region, in order to

study the convergence of the method of the kernel function for obtaining

the interior and exterior mapping functions of the region and its electrostatic

capacity (transfinite diameter). The computation used a general orthonormalizing

code for complex vectors written by P. Rabinowitz. The boundary was specified

at 43 points and the total computation time, including the tape movement,

was half an hour. Eleven orthonormal polynomials have apparently yielded

the electrostatic capacity to three figures. A report presenting these

results is in preparation.

The manuscript (12) by J. Cameron and others, is being revised;

further experiments will be undertaken whenever SEAC becomes available.

1
2 Status of Projects

P. Davis has undertaken experimental computations on the

evaluation of multiple integrals by sampling. For further details see task 371 1 -60-0009/55-91 , page 23 . Further studies have been made by I. Stegun with pseudo-randon

numbers of the multiplicative type. If we let ,x 2

, . .

.

5 x

n

be a sample

of n observations from a normal population and x = (xi+x 2+ . . .+x

n )/n , the

probabilities p

8? of the events x . - x x.^, have been obtained for the r r i i l+l

case when n equal 5 through 8. The results obtained in the case n = 4 were reported erroneously (in the Jan-Mar 1954 issue); the corrected

values are as follows.

Number of

n n n n n obse rvat ions n p

1 P2

p 3

p4 p

5 7000

7000

4

5 .17

.048

.65

.5-48 .17

>57 .048

7000 6 .012 .220 .535 .223 .01

1

5000 7 ,0028 .087 .408 .406 .092

5000 8 .0014 .027 .238 .468 .234

n p6

P

n

7

.0036

.031 .0004

M. G. Kendall (Biometrika 4l_, 560-564, 1944) has discussed part

of this problem theoretically. His results give the following values of

n

Pi 8 n=4, .174; n=5, .078; n=6, .040; n=7, .024; n=8, .015.

Several possible reasons for the discrepancy between the observed results

and the theoretical ones are being investigated. In particular, the

experiments will be repeated with the pseudo-random numbers of the

Fibonacci type and with different methods of obtaining normal deviates

from them.

Extensive work by H. Antosiewicz on the analysis of non-linear

differential equations simulating a war game is described in task

1 1 02-1 0-511 6/55-83 ? page 7.

H. Antosiewicz and P. Rabinowitz are continuing their work on

the differential equations of nerve fiber reactions. In the original

Hodgkin-Huxley system the first order equation for the membrane potential

has now been replaced by a second order equation which, it is hoped, will

give rise to threshold phenomena that will approximate more closely the

observed nerve fiber behavior. P. Rabinwoitz has completed a new and more flexible code for this system which will be tested as soon as SEAG is

available again.

M, Abramowitz and I. Stegun are investigating the numerical

stability of a method for generating the regular and irregular Coulomb

wave functions of arbitrary order from their recurrence relations.

Publications;

(1) Coulomb wave functions along the transition line, M. Abramowitz and

P. Rabinowitz. Phys. Rev. £6, 77~79 (Oct. 1, 1954).

(2) Coulomb wave functions in the transition region. M. Abramowitz and

H. Antosiewicz. Phys. Rev. ^6, 75~77 (Oct. 1, 1954).

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

equations. H. Antosiewicz and M. Abramowitz. J. Washington Acad.

Sci. 44, 382-384 (Dec. 1954).

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

Research Institute), H. Antosiewicz, and P. Rabinowitz. In manuscript.

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

H. Antosiewicz. Submitted to a technical journal.

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

and S. Levy (NBS 6.4). Submitted to a technical journal.
Status of Projects 3

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

To appear in the Pacific Journal of Mathematics.

( 8 ) On the estimations of quadrature errors for analytic functions.

P. Davis and P. Rabinowitz. MTAC 8, 193-203 (Oct. 195*+).

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

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

Computing Machinery.

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

of the reflected radiation of a point source. P. Henrici. To appear

in the Journal of the Washington Academy of Sciences.

(11) On the Lerch zeta function. F. Obe rhettinge r . To appear in the

Pacific Journal of Mathematics.

(12) Generation and testing of random numbers on SEAC, 0. Taussky,

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

(13) Motivation for working in numerical analysis. J. Todd. To appear

in the Transactions of a Symposium on Applied Mathematics, held by

the American Mathematical Society and the Office of Ordnance Research

in Chicago, 111., April 29, 195^.

(14) Obituary notice on L. F. Richardson. J. Todd. MTAC 8, 242-245

(Oct. 195^).

(15) Experiments in the solution of differential equations by Monte Carlo

methods. J. Todd. J. Washington Acad. Sci. 44, 377_38l (Dec. 1954).

(

1

6 ) On the solution of the Lichtenstein-Gerschgorin integral equation

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

S. E. Warschawski (University of Minnesota). To be included in

Experiments in the Computation of Conformal Maps, NBS Applied

Mathematics Series 42.

(17) The condition of matrices, II. J. Todd. Arch. Math. 249-257

(1954).

(

1

8 ) On non-linear differential equations of the second order with

integrable forcing term. H. A. Antosiewicz. To appear in the

Journal of the London Mathematical Society.

TURBULENT ATMOSPHERIC CONTAMINATION

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

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

Johns Hopkins University, and

National Bureau of Standards

Managers: E. Marden, G. Hawkins

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

Status: CONTINUED. Computations were completed for the pilot

problem, as described in the July-Sept 195*+ issue, page 3.

BASIC RESEARCH IN LINEAR PROGRAMMING

Task 1 102-10-51 16/50-2

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

ARDC, USAF, and Directorate of

Management Analysis, USAF

Managers: J. Todd, A. Hoffman

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

Status: CONTINUED. Further applications of linear programming

(more specifically, the transportation problem) to combinatorial problems
4 Status of Projects

in the spirit of the general theorem on distinct representatives of subsets

by Hoffman and Kuhn (see Apr-June 1954 issue, p, 18) have been the subject

of investigation this quarter.

In collaboration with G. Dantzig (RAND Corporation), A. Hoffman

has shown how the smallest number C of disjunct chains covering a finite

partially ordered set P can be computed (and the chains found) as a special case of the transportation problem. Consideration of the dual

yields the result that C is also the largest number of mutually indifferent

elements of P, a result originally due to R. P. Dilworth,

A variation of this device can be used to compute and locate

the largest number p of mutually disjunct paths joining vertices of two

disjunct sets of vertices in a finite graph. The dual yields Menger's

theorem that p is also the smallest number of vertices that intersect

each such path. The corresponding situation for arcs rather than vertices

also is amenable to the same treatment. The analogue of Menger's theorem

in this case had recently been established by Fulkerson and Ford (RAND

Corporation). These theorems hold whether the graph is oriented or not.

Finally, a new method of finding the distance between two disjunct

sets of vertices in a graph has been developed. The dual of this problem

does not seem to have a natural formulation.

Publications:

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

To appear in the Pacific Journal of Mathematics.

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

A. J. Hoffman. To appear in the Proceedings of the American Mathematical

Socle ty

.

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

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

(4) A de te rminantai inequality. Ky Fan and J. Todd. To appear in the

Journal of the London Mathematical Society.

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

(Bryn Mawr College). In manuscript.

( 6 ) On the relevance of LeChate lie r ' s principle to linear programming.

A. J. Hoffman. In manuscript,

(7) An extremum property of sums of eigenvalues. H. Wielandt. To appear

in the Proceedings of the American Mathematical Society.

SUPPORTING RESEARCH IN LINEAR PROGRAMMING

Task 1102-10-5116/54-9

Origin: Directorate of Management Analysis, USAF Authorized 9/25/53

Sponsor: Office of Scientific Research, Air Research

and Development Command, USAF

Managers: J. Todd, A. J. Hoffman

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

Status: CONTINUED. The principal activity of the quarter has been

organization of the Linear Programming Symposium to be held in Washington

on January 27, 28, 29, 1955. The formal program consists of five onehour addresses and 26 shorter talks.

In preparation for his address on computation procedures

A. Hoffman has considered the relative merits of the inverse of the basis

and the "product" form of the inverse for performing simplex computations.

The study, while not yet complete, seems to favor the product form.

During this quarter the fortnightly Linear Programming Seminar

conducted by interested personnel in the Washington area was reactivated.

The first lecture, given on December 15? was delivered by A. Hoffman on
Status of Projects 5 "The solution of the generalized caterer problem."

Publications

:

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

manuscript

.

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

and D. Sokolowsky. To appear in the Logistics Research Quarterly.

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

To appear in the Journal of Research of the NBS.

(4) Smooth patterns of production. A. J. Hoffman and W. Jacobs.

Management Science X, 86-91 095*+).

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

In manuscript.

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

(7) A remark on the smoothing problem. H. A. Antosiewicz and A.J. Hoffman.

Management Science _1_i 92-95 (1955-).

RESEARCH IN MATHEMATICAL TOPICS APPLICABLE TO

NUMERICAL ANALYSIS

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

Origin and Sponsor: Office of Naval Research Authorized 8/13/54

Managers: 0. Taussky-Todd , M. Newman Revised 8/29/55-

Full task description: July-Sept 1955- issue, p. 5 Status: CONTINUED. 0, Taussky-Todd, in collaboration with

M. Newman, continued an investigation of the uniqueness and generalization

of the normal basis of algebraic number fields. In particular, it was shown that if an n x n unimodular circulant (n a power of a prime) is of

the form AA' , then AA'=BB’ , where B is an integral circulant. This had

previously been shown for n - 6. Some known results on circulants were extended to general group matrices.

M. Newman has derived a simple proof of his result that, if

H is a subgroup of the modular group G which contains the subgroup GQ(n)

(the totality of elements (^ c ^*)), then H=GQ(d), d/n. He has also shown

that the normalizer of GQ(n) is GQ(n/6), where 6/24 and is precisely

determined by the highest powers of 2 and 3 dividing n. Further work by

Dr. Newman, in support of an investigation in statistical mechanics,

is reported under task 371 1 -60-0009/55_ 90 (see p. 23 ). E. C. Dade completed a routine to compute the coefficients of

the polynomials 2 n_+n

2 JJ (1 - x

r

) = A" x

j r- 1 j=0

by means of the recursion formula

A

n-1 s-n •

All the coefficients were calculated for n=1(1)8, and the least

and greatest coefficients together with their locations were calculated

for the first 100 values of n. In addition the central terms were computed and some interesting empirical results obtained, some of which
6 Status of Projects

were later verified by Mr. Dade by means of an integral formula for A^

which he found

.

A. Hoffman has applied linear programming to combinatorial

problems in partially ordered sets and graphs. For further details, see task 1102-10-5116/^0-2, pa ge 3.

K, Goldberg has continued his investigations of the coefficients

in the formal power series expansion of log (e x .ey). New results include

bounds for the coefficients and asymptotic information,

J, Bram has derived an alternative proof of the bounded case of

the well-known theorem of Fuglede that an operator commuting with a normal

operator A on a Hilbert space commutes with A*, The new proof is based

on his previous result that, in the closed subspace S generated by the

images of a fixed vector f under polynomials in A and A*, there is a vector

g such that S is the closure of the images of g under polynomials in

A above,

P, Davis and H, Poliak (Bell Telephone Laboratories) are preparing a manuscript "On the analytic continuation of mapping functions".

The object of the investigation is to relate the continuation properties

of a region B with an analytic boundary to the geometry of the region as expressed through the sequence of moments JgJ x m

y

n dxdy

.

Two seminars were conducted. One was in Algebra, held in

conjunction with American University, under the direction of M. Newman and

0. Taussky-Todd; for details see page 35. The other was concerned with

miscellaneous topics in mathematics of current interest to the staff of

the division; a list of topics appears on page 3

1+. A book on combinatorial analysis is in preparation by M. Newman.

The book is the outgrowth of a series of lectures in this subject that were given by him at American University in the spring of 1953. The object

of the book is to present the various methods which are of use in combinatorial

analysis, together with important examples to illustrate these methods.

The planned table of contents is as follows:

Binomial coefficients, permutations, combinations

Methods of enumeration by recurrence formulas

Theorems in logic and number theory

The Sylvester sieve formula

The Mdbius function

Ramsey's theorem

Symbolic methods of Kaplansky

Partitions

Selected Topics

Systems of distinct representatives

Finite projective geometries

Incidence matrices

Hadamard matrices

Difference sets

Latin squares

Triple systems.

Chapter 1 . Chapter 2.

Chapter 3.

Chapter 4.

Chapter 5.

Publications:

(1) Linear functional equations and interpolation series. P. Davis

Pac. J. Math. 4, 503^532 (1954).

(2) Complex biorthogonality for certain sets of polynomials. P. Davis

and H. Poliak (Bell Telephone Laboratories). Duke Math. J, 21,

653-668 (Dec. 1954).

(3) Continuity in terms of connectedness. Ky Fan and R. A. Stuble

(University of Notre Dame). Proc. Nederl. Akad., Wetensch. 457,

1 61-164.

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

(University of Notre Dame). In manuscript.

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

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

Socie ty

,
Status of Projects 7

(6) A comparison theorem for eigenvalues of normal operators. Ky Fan.

Submitted to a technical journal.

(7)

' Power series for log (e x *eY). K. Goldberg. In manuscript.

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

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

Mathematic s

.

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

appear in the Pacific Journal of Mathematics.

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

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

(11) The diophantine equation ax^by^c. M. Newman. Submitted to a technical journal.

(12) Note on a certain determinant. M. Newman. In manuscript.

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

the Duke Journal of Mathematics.

(14) The coefficients of certain modular forms. M. Newman. Submitted

to a technical journal.

(15) 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.

(

1

6 ) Generalized commutators of matrices and permutations of factors in

a product of three matrices. 0. Taussky. Studies in Mathematics

and Mechanics Presented to R. von Mises, pp. 67-68 (Academic Press,

Inc., New York, 19 5

L0 . (17) 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.

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

to a technical journal.

ANALYTIC STUDY OF WAR GAMES

Task 1101-10-511.6/55-83

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

Manager: H. A. Antosiewicz

Objective: To study a mathematical model of a war game between two

opposing parties involving a system of six ordinary non-linear differential

equations in which the dependent variables are the strengths of the

opposing forces and the rates of each party's production. The choice of

coefficients in this system of equations together with the initial values

of the dependent variables determines the dynamic evolution of the war

model. The central problem is the determination of coefficients in the

system of differential equations in such a way as to guarantee victory

of one party over its opponent. This amounts essentially to a decision

of how best to divide one party's attacking effort between the opponent's

actual force-in-being and his productive capacity. A victory is to be

characterized by analytic conditions upon the coefficients of the system.

Although the concept of victory is well understood intuitively, it is

necessary to formulate it here precisely in analytic terms.

To simplify matters, once a choice of coefficients has been

made, the same coefficients are used throughout for a particular war game.

Thus, the finial outcome of the game is completely determined by the

coefficients of the system and the initial values of the dependent variables.

Numerical evidence obtained from previous investigations on SEAC

has pointed up the rather insignificant role played by the defensive forces
Status of Projects

in this model. Therefo

assumptions that defens

reduced to a system of

six essential parameter

investigation to the fu

ten parameters.

re, the present study is initially based upon the

ive forces are absent. The system is thereby

four non-linear differential equations containing

s. In the future it is planned to extend this

11 system of six equations which is governed by

Background: The model under investigation was proposed by General

L, I. Davis, USAF. Earlier numerical investigations were carried out on REAC and on SEAC; also, this model of a war game was used for demonstration

purposes on DYSEAC , at its dedication. Results obtained in this way were somewhat inconclusive as regards the optimum choice of coefficients to

guarantee victory for one party and pointed up the desirability of further

analytic investigation.

Status: NEW. Numerical evidence has pointed up the very insignificant

role played by the defensive forces in this model. Therefore the analytic

investigation is presently based upon the assumption that defensive forces

are absent. The system is thereby reduced to four non-linear differential

equations containing six essential parameters.

The following results have been obtained to date. If the

strength of A' s offensive forces tends to a non-zero finite limit as time

increases indefinitely so does the strength of B' s offensive forces, and

the rates of production of A and B approach finite limits whose ratio is

equal to the ratio of the constants determining the fraction of one party's

offensive forces sent against his opponents offensive forces. If these

constants are equal, victory is determined solely by the initial values

of the strengths of offensive forces and the initial rates of production.

In particular, if A and B attack only one another's offensive forces,

their production rates both approach the same (finite) limit, and the

strengths of the offensive forces approach (finite) limits, which are given

explicitly in terms of the initial values.
II . COMPUTATION LABORATORY

(Section 11.2)

1 . Mathematical Tables

1102-40-1110/43-3 TABLES OF E.(z), (z=x+iy)

Origin : Canadian National Research Council

Manage

r

: I. Stegun

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

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

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

Origin : NBS

Manage r : P. Rabinowitz

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

Status : Inactive. For status to date, see Juiy-Sept 1954 issue, page 8.

1 102-40-1 1 10/50-7 WAVE FUNCTION FOR LITHIUM

Origin : NBS

Sponsor : Bureau of Ordnance, USN

Manage r : W. F. Cahill

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

Status : Continued. The computation of the atomic wave function of the

Li+ 1 s2s state has been completed, and the results are being checked.

1 1 02—40—1 1 1 0/51—4 COLLECTED SHORT MATHEMATICAL TABLES OF THE

COMPUTATION LABORATORY

Origin : NBS

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

Status : Completed. This volume brings together 18 short tables of functions

and of zeros of functions. Seven of these tables were published

originally in the Journal of Mathematics and Physics; others appeared

in the Bulletin of the American Mathematical Society, the Quarterly

of Applied Mathematics, the Philosophical Magazine, and Mathematical

Tables and Other Aids to Computation; and three (marked by asterisks)

are published here for the first time. The tables included are as follows

On the function H(m, a,x) = exp(-ix)F (m+1 -ia , 2m+2; 2ix) . A. N. Lowan and

W. Horenstein.

Table of the integrals P x J~(t)dt and f X Yn (t)dt. A. N. Lowan and

M. Abramowitz. J0

Table of Jig(x) = f

00

[ jQ(t)/t]dt and related functions. A. N. Lowan,

* \

G. Blanch, and M. Abramowitz.

Table of f (x )- (x/2)~ n

n \ J (x )

.

The functions E

n (x)=J^ e xu

u

n du. G. Placek.

9
10 Status of Projects

Table of the Struve functions Lj/x) and H^(x) . Table of Fourier coefficients. A. N. Lowan and J. Laderman.

*Table of sines and cosines for radian arguments between 100 and 1,000.

*Radix table for finding logarithms to 25 decimal places. H. E. Salzer.

*Table of x

n /n[.

Table of the zeros of the Legendre polynomials of order 1 — 1 6 and the

weight coefficients for Gauss' mechanical quadrature formula.

A. N. Lowan, N. Davids, and A. Levinson.

Table of the zeros and weight factors of the first fifteen Laguerre

polynomials. H. E. Salzer and R. Zucker.

A short table of the first five zeros of the transcendental equation

jQ(x)YQ(kx)-jQ( kx)YQ(x) :=0. A. N. Lowan and A. Hillman.

Zeros of certain Bessel functions of fractional order. M. Abr amowitz

.

More zeros of certain Bessel functions of fractional order.

Zeros of the derivative of Bessel functions of fractional order.

H. E. Salzer.

Complex zeros of Yq(z), Y-j (z) , and Yj(z). A. Hillman and I. Sherman.

Roots of sin z = z. A. P. Hillman and H. E. Salzer.

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

short tables of the Computation Laboratory," NBS Applied Mathematics

Series 37; available from U. S. Government Printing Office, Washington

25, D. C., #2.25.

1 102-40-1 1 10/51-8 TABLES OF POWER POINTS OF ANALYSIS OF VARIANCE TESTS

Origin : Section 11.3, NBS

Managers : A. J. Hoffman, S. Tsingou

Full task description : Apr-June 1951 issue, p. 5-9 Status : Inactive. For status to date, see July-Sept 1954 issue, page 9.

1 102-5-0-1 1 10/52-7 REVISION OF MATHEMATICAL TABLES

Origin : NBS

Manage rs : W. F. Cahill, I. Stegun

Full task description : July-Sept 1951 issue, p. 5-1 Status : Continued. Following is the status of those mathematical tables

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

planned

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

Mathematics Series 5-1; available from U. S. Government Printing Office,

Washington 25, D. C. , #3.25. This was originally Mathematical Table MT8,

"Tables of probability functions," vol. I (1941).

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

Mathematics Series 5-3; in press, U. S. Government Printing Office. This

table was formerly designated as Mathematical Table MT4.

"Tables of natural logarithms," vol. IV, Mathematical Table MT1

2

(19^1); revision in progress for reissue in the Applied Mathematics Series.

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

Origin : NBS

Manager : D. Liepman

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

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

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

Origin : NBS

Managers : W. F. Cahill, K. Nelson

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

1 1 Status : Continued. The manuscript of the table has been typed and

checked. An introduction has been written, and the tables are being

prepared for publication.

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

Origin : NBS

Manage r : P. Rabinowitz

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

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

1 102-40-1 1 10/52-25 TABLE OF ERROR FUNCTION FOR COMPLEX ARGUMENTS

Origin ; NBS

Manage

r

: W. Hall

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

Status ; Inactive. The 20-place table for x,y=0(.1)2 is available on punched cards. In view of the publication of the Russian Academy of

Science tables of this function, the program of this task is being

reconside red

.

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

NEGATIVE ARGUMENTS

Origin : NBS

Manage r

s

: E. Marden, S. Prusch

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

Status : Continued. The manuscript has been typed and is in the process of

being checked.

1 102-40-1 1 10/52-37 SPHEROIDAL WAVE FUNCTIONS

Origin : NBS

Manager : D. Liepman

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

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

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

Origin : NBS

Manager : I. Rhodes

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

Status : Inactive. For status to date, see July-Sept 1954 issue, page 10.

1 102-40-1 1 10/52-57 SIE VERT'S INTEGRAL

Origin : NBS

Managers : M. Paulsen, P. O'Hara

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

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

1 102-40-1 1 10/53-35 HYPERGEOMETRIC FUNCTIONS

Origin : NBS

Manage rs : P. Rabinowitz, W. F. Cahill

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

Status : Inactive. For status to date, see Apr-June 1954 issue, page 27.
12 Status of Projects

1102-40-1110/53-52 L-SHELL CONVERSION COEFFICIENTS

Origin ; Oak Ridge National Laboratory

Manage r

: W. Hall, J. Wegstein

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

Status ; Continued. Further exploratory computations (see July-Sept 1 9

5^+ issue, p. 10) were made to overcome singularities at r=0, and a combination of series expansions and stepwise integrations, which

appears satisfactory, was arrived at. Computations for the shell

of z=45 and z=65 were completed for several energy levels.

371 1-60-0009/55-65 AUTOMATIC CODING

Origin : NBS

Manager ; J„ Wegstein

Full task description ; July-Sept 195“+ issue, p. 11 Status ; Continued. The general purpose interpretive floating decimal

point routine (Base 00) now in use requires only one instruction word

to execute each of the following on floating decimal point numbers;

addition, subtraction, multiplication, division, logarithm, exponential,

square root, binary to decimal conversion, decimal to binary conversion,

and comparison of absolute values. The interpretive routine also permits

treating numbers in their usual SEAC form as well as automonitoring.

This routine is being modified to include the following operations:

sine, cosine, raising numbers to integral powers. A set of instructions

for its use is also being prepared.

"2'. Mathematical Services

Note: The tasks under Mathematical Services are arranged serially

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

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

in which the task was authorized.

1 102-40-5126/50-13 RAY TRACING

Origin and Sponsor : NBS, Section 2.2

Manage r : E. Marden

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

Status : Continued. Seven additional aerial camera lens systems were

analyzed, and spot diagrams were computed for them. Several hundred

rays were traced through each such lens system.

1 102-40-5126/51-37 MOLECULAR STRUCTURE, III

Origin : Naval Research Laboratory, USN

Manage r : P. O'Hara

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

Status : Continued. Computations to aid in the determination of the

molecular structure of the spurite crystal were performed, as requested.

1 102-40-5126/52-20 SPHERICAL BLAST

Origin and Sponsor :Naval Ordnance Laboratory

Manage r : D. Jirauch

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

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

problem, together with a brief description of the problem, was drafted.
Status of Projects 13

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

Origin and Sponsor : NBS, Division 9

Manager : A. Futterman

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

Status : Continued. Computations for several different crystals were performed in accordance with sets of parameters furnished by the sponsor.

1 1 02-40-51 26/53-1 1 STRENGTH OF WING COMPONENTS

Origin and Sponsor : National Advisory Committee for Aeronautics,

Langley Field, Va.

Manager : G. Hawkins

Full task description : Oct-Dec 1952 issue, p. 54

Status : Completed. The computations were completed according to the code

set up by the originator and were transmitted to that agency.

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

Origin and Sponsor : NBS, Section 3.2

Manage

r

: E. Marden

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

Status : Continued. Computations were performed for several additional

molecules. In addition, thermodynamic functions were re-calculated for

for several molecules for which improved physical constants were obtained.

1 102-40-5126/53-29 DYNAMIC BEHAVIOR OF AIRCRAFT STRUCTURES

Origin and Sponsor : NBS, Section 6.4

Manage

r

: I. Rhodes

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

Status : Completed. The results have been transmitted to the sponsor.

1 102-40-5126/53-39 ACOUSTICAL impedances

O rigin and Sponsor : NBS, Section 6.1

Manager : S. Prusch

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

Status : Completed. The computations were performed as requested by the

originator.

1 102-40-5126/53-41 LORAN UNIVAC CODE

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

Managers : I. Rhodes, D. Jirauch

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

Status : Completed. The Loran UNIVAC codes have been turned over to the

sponsor.

1102-40-5126/53-51 RADIATION DIFFUSION

Origin : NBS, Section 4.8

Sponsor : Armed Forces Special Weapons Project

Manage r : A. Futterman

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

Status : Continued. The response functions for several sodium iodide

crystals to 5"*9") have been obtained over an energy range .279

to 4.45 Mev. by means of random sampling techniques. Approximately

50,000 photon histories have been generated to date.

Monte Carlo methods are being applied to investigate photon scattering

in non-'homogeneous media. Present interest is being fixed on photon

crossing the plane interface of two media with different densities.
14- Status

1 102-40-5126/54-4 HIGH TEMPERATURE

Origin ; NBS ? Division 3.2

Sponsor ; Bureau of Ordnance, USN

Manage rs ; J„ Wegstein, J. Cooper

Full task description ; Oct-Dec 1953

Status ; Completed. The calculations

ionized 0,,TTT and H were completed

- VJLXJL range 10° to 10° degrees Kelvin.

The code was modified to compute

for all of the other polyelectric

modified formulation developed by

thermal functions was completed,

ionic species for another element

f Projects

PROPERTIES OF WATER

sue, p. 4l

f the rmal functions for s ingly

0 r 50 poin ts in the tempe nature

he s ame functions in1 the same range

ns of oxygen except 0

+ us ing a Woo ley • The comput ation for these

In addition all of the polyelectric

were completed using the same procedure.

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

LATTICES, II

Origin ; NBS, Division 30

Manage; W. F. Cahill

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

Status : Completed. The distribution of normal modes of vibration in

cubic lattices for various crystal structures were computed using a 48x48x48 grid.

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

Origin : NBS, Section 3.2

Manager : M. Newman

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

Status : Completed. A symmetric matrix was calculated, and its eigenvalues

and eigenvectors were computed. The results have been transmitted to the sponsor.

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

Origin : NBS, Section 6.4

Manage r

; I. Rhodes, W. F. Cahill

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

Status : Continued. The first five symmetric modes for a fourth aircraft

structure (using 31 mass points) were computed. Also, the first five

antisymmetric modes for three aircraft structures (using 26 mass points) were computed.

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

by W. F. Cahill and S. Levy (NBS Section 6.4); to appear in the Journal

of the Institute of Aeronautical Sciences.

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

Origin and Sponsor : New York Quartermaster Procurement Agency

Manager : H. Bremer

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

Status : Continued. At the present time contract awards are- being

evaluated at the rate of about two per week using UNIVAC routines. The

UNIVAC codes that are used for this work were written by M. DeCarlo-Cottone

in the Office of the Directorate of Management Analysis, U.S.A.F.

Computations on SEAC will be resumed as soon as the machine is available

(see p. 32).
I

Status of Projects 15

1 102-40-5126/54-1 4 FLOW IN SUPERSONIC NOZZLE

Origin and Sponsor : National Advisory Committee for Aeronautics,

Langley Field, Va.

Managers : E. Marden, G. Hawkins

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

Status : Completed. The boundary contours of supersonic nozzles have

been calculated for given Mach numbers. The increment used in the

angle of expansion between successive Mach lines was .25 degree.

37 1 1 -60-0009/54- 1 7 DE POLYME RIZATION

Origin : NBS, Section 7.6

Manager : J. Bram

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

Status : Continued. The coding for SEAC has been completed and is ready

to be code-checked.

371 1-60-0009/54-19 ENERGY LEVELS OF COMPLEX ATOMS

Origin : NBS, Section 4.1

Manage

r

: W. Hall

Full task description : Jan-Mar 1954 issue, p. 4l

Status : Inactive. For status to date, see July-Sept 1964 issue, page 14.

1102-40-5126/54-20 LOW TEMPERATURE PROPERTIES OF BORON COMPOUNDS

Origin and Sponsor : NBS, Section 3.2

Manager : J. Wegstein

Full task description : Jan-Mar 1954 issue, p. 4l

Status : Inactive. For status to date, see Jan-Mar 1954 issue, page 4l .

1 102-40-5126/54-21 EXPERIMENTAL PROGRAM FOR MULTIPLE INPUT AND OUTPUT

Origin and Sponsor : NBS, Section 12.3

Manager : I. Rhodes

F ull task description : Jan-Mar 1954 issue, p. 42

Status : Continued. Further studies are being made in the application

to chemical warfare.

371 1-60-0009/54-22 ENERGY DISTRIBUTIONS ON OPTICAL IMAGE

Origin : NBS, Section 2.2

Manager : L. S. Joel

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

Status : Continued. Studies are being undertaken to improve the speed

and precision of the integrations.

1 102-40-5126/54-24 SOLUTION OF THE DIFFUSION EQUATION FOR ELECTRONS

Origin and Sponsor : NBS, Division 4

Manage r : A. Futterman

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

Status : Completed. The results have been sent to the sponsor.

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

Origin and Sponsor : Armed Forces Special Weapons Group

Manage rs : P. Henrici, L. S. Joel

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

Status : Inactive. For status to date, see July-Sept 1954 issue, page l4.
Status of Projects

371

1

- 60-0009/ 54- 2 7 EMF of cells at high temperature

Origin : NBS, Section 1.8

Managers : E. Marden, G. Hawkins

Full task description : Jan-Mar 195^ issue, p. 45

Status : Continued. Electromotive forces for several additional compounds

were computed.

3711-60-0009/54-28 LOVIBOND NETWORK FOR CIE SOURCE A

Origin : NBS, Section 2.1

Managers : E. Marden, G. Hawkins

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

Status : Continued. Changes were made in the SEAC code to recompute the

network for CIE sources A and C using new data supplied from England.

371 1-60-0009/5^-30 SPECTRUM ANALYSIS

Origin : NBS, Division 4

Manager : S. Prusch

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

Status : Continued. Approximately 15 differences for Tc I were used.

A table of arc and spark spectra of ruthenium is being prepared for

publication. Differences between all known and predicted levels were

used to search for pairs of lines which confirm known levels and predict new levels.

A rectangular array of known levels and corresponding pairs of lines

was prepared for Ru I.

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

Origin and Sponsor : NBS, Section 10.3

Manage

r

: S. Tsingou

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

Status : Continued. Results for aG=2,3,4,5, and t/a 2= .05 , . 1 , . 2 , . 3 , .4 , . 5?

1 ,2,5,10,30,40 h ave been computed with the supplementary code.

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

Origin and Sponsor : Naval Ordnance Laboratory

Manager : J. Wegstein

Full task description : Apr-June 1955- issue, p. 38

Status : Continued. Convergence was obtained in the iterative solution

of 88 sets of 46 simultaneous transcendental equations. The results were transmitted to the sponsor.

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

Origin and Sponsor : NBS, Section 6.4

Manager : W. Hall

Full task description : Apr-June 1955- issue, p. 38

Status : Continued. The code was modified to change boundary conditions,

and it will be run as soon as machine time becomes available.

371 1-60-0009/5^-36 VELOCITY OF LIGHT

Origin : NBS, Section 11.3

Manager : P. Rabinowitz

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

Status : Continued. Six polynomials were fitted to data submitted by

sponsor. At the same time the experimental errors were computed.
Status of Projects 17

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

Origin : NBS, Section 3.2

Manage

r

: M. Paulsen

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

Status : Continued. Several test cases have been run in checking the codu.

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

Origin and Sponsor : NBS, Section 3.2

Manage

r

: K. Goldberg

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

Status : Continued. The routine for obtaining the eigenvalues of a symmetric matrix up to order 30 ? internally and floating, using the

"Givens Variation" on the Jacobi method, was put into production in

the month before SEAC was moved. It takes five minutes to find all

the eigenvalues of a matrix of order 12 .

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

Origin and Sponsor : NBS, Division 6

Manager : L.S. Joel

Full task description : Apr-June 1955- issue, p. 39

Status : Completed. The results have been sent to the sponsor.

1 102-40-5126/54-42 RESOLUTION CORRECTION FOR SCINTILLATION SPECTROMETER

Origin and Sponsor : NBS, Section 4.13

Manage

r

: M. Newman

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

Status : Completed. Matrices of orders 9? 10, and 19 were inverted, and

systems of equations having these matrices as coefficients were solved.

The results have been transmitted to the sponsor.

1 102-40-5126/54-43 characteristics of conducting resistors

Origin and Sponsor : NBS, Section 12.1

Manager : B. Gill

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

Status : Continued. Sixty cases were run with revisions of former data.

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

Origin and Sponsor : NBS, Section 10.2

Manage

r

: W. F. Cahill

Full task description : Apr-June 195*+ issue, p. 4l

Status : Completed. The problem has been run for all the heat dif f usivities

and heat transfer coefficients requested, and the results have been

transmitted to the sponsor.

1 102-40-5126/54-46 STUDY IN OPTIMIZATION OF POWER OUTPUT

Origin : Raytheon Manufacturing Company

Sponso

r

: U. S. Corps of Engineers

Manager : L. S. Joel

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

Status : Inactive. For status to date, see Apr-June 1955- issue, page 4l .

1 102-40-5126/55-49 wiechert distribution function

Origin and Sponsor : NBS, Section 7.1

Manager : 1. A. Stegun

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

Status : Completed. The integrals have been evaluated, and the results

have been transmitted to the sponsor.
Status of Projects

1 102-40-5126/55-^0 AWARD OF PROCUREMENT CONTRACTS BY LINEAR PROGRAMMING

Origin and Sponsor : Navy Purchasing Office

Manager : H. Bremer

Full task description : July-Sept 195*+ issue, p. 17

Status : Terminated.

1 102-40-5126/55-61 ELASTIC CROSS SECTION FOR NEUTRON SCATTERING

Origin and Sponsor : Naval Research Laboratory

Managers : I. A. Stegun, R. Zucker

Ob jec tive : To determine values of kR, Vq/E and ^such that the function

lf(0)l^ best approximates given curves.

i <*L f( e) = 2 ik ZI (2X+1) P^(cos 0) [r^ -1]

— 0

tan 6j>(kr)[tan 0C^( kR)-Z^( pR) ]-i[ Z^ ( pR)-tan ( k R ) ] ~\l

= tan ( kR)[ tan oc^ (kR)-Z^( pR) ] + i[Z^(pR)-tan /3^(kR)]

v l_ Z£ ( pR)= 1 -pR cr^ (pR) ; pR=kR[1+ ^(1 + i^ )] 2=a+ib

where for 1 = 0, 0/,(a+ib) =

S ^ n iu^oh^b

’ 0 cosh 2b - cns2a

x

2+l[ xcrn 1^1, x<rA

(x) =

cos2a

(x) ] -lc

1 (x)]

P^ is the Legendre function and cLp [3^ and be are defined in terms of

spherical Bessel functions. (Scattering ana Radiation from Circular

Cylinder and Spheres - U. S. Navy Dept. O.R.I. July 196 .)

Background : The function arises in the study of the scattering of neutrons.

The functions to be calculated represent the differential elastic cross section for neutron scattering; an experimental determination of this

function, to be matched against the theoretical results obtained here,

is under way at the Naval Research Laboratory.

Status : New.

11 02-^0-51 26/55-62 INTEGRALS INVOLVED IN SUPERSONIC FLUTTER

Origin and Sponsor : National Advisory Committee for Aeronautics

Manager : S. Tsingou

Full task description : July-Sept 195/ issue, p. 17

Status : Continued. All the results necessary to complete the tables

have been computed and printed. A code to difference the results has

been completed and will be checked as soon as SEAC time becomes

available

1 102-/0-5126/55-63 VIRIAL INTEGRAL INVOLVING A MORSE POTENTIAL FUNCTION

Origin and Sponsor : NBS, Section 3.2

Manager : J. Cooper

Full task description : July-Sept 195/ issue, p. 17

Status : Completed. The integrals listed have been computed for values

of the parameters corresponding to the diatomic species OH, H„ , 09 ,

_j ©2 , OH , OH , H Q , 0 , for temperatures from 10 to 10' degrees

Kelvin. The results have been transmitted to the sponsor.
Status of Projects 19

1 1 02-40-51 26/55-64 HIGH- TEMPERATURE THERMODYNAMIC TABLES ( FMT)

Origin and Sponsor : NBS, Section 3.2

Manager : J. Wegstein

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

Status : Inactive.

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

Origin : NBS, Division 30

Manager : H. Bremer

Full task description : July-Sept 1955- issue, p. 18

Status : Continued. Several trials have been run on the SEAC, as explained

in the previous issue, and the results are being studied.

1 102-5-0-5126/55-67 DOSAGE INTEGRAL

Origin and Sponsor : U. S. Weather Bureau

Manager : W. F, Cahill

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

Status : Continued. The dosage integral was evaluated for three sets of

parameters. A new integral, in which the original integrand was multiplied by [ (d-ut-x) 2+y2+ x

2

] ? was evaluated for one set of parameters

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

Origin : NBS, Section 7.6

Manage

r

: K. Goldberg

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

Status : Continued. No new results on the distinctness of the roots have

been obtained, but work towards this end continues.

1 102-40-5126/55-70 BALL BEARING FATIGUE DATA

Origin : NBS, Section 11.3

Sponsor : American Standards Association

Manager : A. Futterman

Ob jec tive : To compute

( 1 ) T • ( k)_

nj (k)

. - y~ w. . J iti u

T

i,

in x ij’ k= 1 ,2,3,5-; j= 1 ( 1 ) 260 ,

and to find e

’’ I/T

2

? e t

3

t5- , x

, e . Here \x^jj , i=1,2,...,n, represent

260 sets of data submitted by the sponsor under task 1 1 03-5-0-515-5/55—

1

(see p. 28) and then randomized by SEAC. The w.(k) are given weights.

1J (k) After the results (1) are obtained, the variances (Tj J

) are to be

found by the usual formula for the variance of a sum of independent

variable s

.

Bac kground

:

This task furnishes one phase of the computations necessary

in connection with task 1 1 03—5-0—5 1 ^5/55— 1 (see Jan-Mar 1954 issue, p.54)

Status : Completed (New). The code was prepared and checked, and the value

f k

)

Tj 1 were obtained for the 260 sets of data.

1 102-40-5126/55-71 LORAN BASE LINE COMPUTATIONS

Origin and Sponsor : Hydrographic Office, U. S. N.

Manager : D. Jirauch

Objective : To complete baseline extension and skywave correction tables

for three rates.
20 Status of Projects

Background : Baseline data and skywave correction tables are necessary

for use with the standard Loran navigation tables to enable navigators

to determine their positions with the aid of special electronic equipment.

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

1 102-4C - 5126/55-72 MARYLAND INTER-INDUSTRY STUDY

Origin and Sponsor : Office of Scientific Research, ARDC, US AF

Manage r : L. S . Joel

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

Status : Continued. The matrix (I-AK) where K is scalar was inverted,

and (I-AK)x=y was solved for five vectors y.

1 102-40-5126/55-73 HEAT convection

O rigin : University of Minnesota

Sponsor : Atomic Energy Commission

Manage r : G. Hawkins

Objective : To consider the unsteady state performance of a cooling system

having the form of a simple natural circulation loop, where boiling

occurs in the hot portion of the loop. The goal of the investigation

is the development of a mathematical model which conforms to transient

one-phase or two-phase flow and which uses partial differential equations

that can be solved numerically.

Bac kqround : Natural circulation loops are taking an added importance as the use of atomic reactors for power generation increases. The analysis

of transient flows occurring in a simple natural circulation loop where

no boiling is permitted to occur has been successfully treated, using

a system of difference equations which were developed from the equation

of motion, the continuity equation, and the energy equation. The

analysis is now being expanded to include the case in which some boiling

occurs in the hot portion of the loop.

Status : New.

1 102-40-5126/55-77 longitudinal waves in a string

Origin and Sponsor : NBS, Section 7.2

Manager : J. Br am Ob jec tive : To evaluate the expression

_t /2 i? r -^2

P(^,r) = e 5/2 (T-£) 11^ 1 12

2 (T-t) &

d t

for various values of the arguments with , Bac kground : The above expression is the solution of the polygraph line

equation

- pr r- pr ~ 0

under particular bounda*y and initial conditions. This equation

describes the longitudinal waves that are set up in a semi-infinite

string, made of an elastic, viscous material, when one end begins

moving at a uniform velocity from an initial position of rest.

Status: Completed (New). The expression P(|,r) was evaluated on SEAC

for the various values of the arguments requested, and the results

were transmitted to the sponsor.
Status of Projects 21

1 102-40-51 26/55-78 NEUTRON TRANSPORT

Origin and Sponsor : Atomic Energy Commission, New York

Manager : S. Tsingou

Ob iec tive : To solve the integral equations

<’> P

t {ei <u '“' ) ]

du' + aG(u)6n0 6l0

J u

rJ +

2 1 +1 S h (u) ( ^+1) Dn-1,l+1 (u) + Dn-1,I-1 (u)

where G(u) and F^(u) are given,

(2) F|(u) D

nl^ u ^ = M

Dn |(u') P^ (e 2<' u u )] du' -B|(u)D n ^(u+h)

/y + oCG(u) + ( 2 X +i ) £h

( u )

( ^ +1 )Dn-1 ,1+1 (u)+^ D

n-1 ,1-1 (u)

B|(u) and F^(u) are given.

Bac kground : These equations arise through transformation of the neutron

transport equation by the moments method. Scattering from all elements

other than hydrogen is calculated by assuming in (1) a zero order and

in (2) a first order expansion of the associated scattering integrals

in powers of the reciprocal of the atomic mass. For further details

see, "A solution of the neutron transport equation," by J. Certaine,

NYO 3081 (Nuclear Development Associates 15C-43).

Status : New. Computations were performed for approximate ly 1 00 combinations

of values of the parameters and were transmitted to the sponsor.

1 102-40-5126/55-79 COMPLETE DEGRADATION IN THE NEUTRON

Origin and Sponsor : Atomic Energy Commission, New York

Manage

r

: A. Futterman

Ob iec tive : To compute

pk

p k 21 P Di G

ik j=0 PJ J

for 0^p<P, 0<k< N+1 = [-^ ln(y^) ]+1 , where the quantities

P PJJand G-. are obtained recursively from relations

K

( I, \ P

p+ 1,j+ 1 = < 2 »+ 3) ^j£i)

J + 3 J

p

k 2p+1

po p

11 _ £—1

' k p-1,0 2p-3 P-2,0

00

’•> k So J+s

s = 2n for j even

s = 2n+1 for j odd.

- 0«n-1 i+2_

n-J

and d 0Q-1, d Qn

- ^n+2

' ’ d jn " (n+lf(n+2) ZI

A

d

j-1 ,n-1-A

A=o

for j > 0.
22 Status of Projects

The quantities A^ and yU^ are functions of the parameters h and m. The calculations are to be done

Bac kqround ; The matrix \J/'q=

(tjr a new treatment of the neutPon

have the property that if f (

1 ) at u'k - u+kh, and u

1 = u+2

for assigned values of these parameters,

computed above is the initial stage for

transport equation. The elements \Jf . is a polygonal function with values

ln( 1 + m)-ln( 1 + 2myU +m^) , then

N+1

f (u« )P

p

(/A)d/x - II

o

f ke -kh

pk*

Status : New. The code was revised and tested, and several runs made.

Continued runs will be made as soon as SEAC is available.

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

Origin and Sponsor : Atomic Energy Commission, New York

Manager : W. G. Hall

Ob jec tive : To compute the function

F(r,u;ct) = e *jT D

Q

( u) (^) n+1

Q^J (ocr)

n=0 nU 0

n

N where oLq is a previously determined constant, Qn is a polynomial of

degree N, and the coefficients D

n Q have been calculated in the neutron

transport problem (task 1 1 02-40-51 26/55~78) . Calculations are to be

done for assigned values of the parameters oC, r and u. Bac kground : It is desired to compute a function with quasi-exponential

behavior whose first N+1 even moments are given by D q(u). Specifically,

this procedure is used to reconstruct the point isotropic neutron flux

density from the previously computed moments D q.

Status : New. A number of cases were run, and the results were sent to the sponsor.

3711-60-0009/55-81 COMBINING TESTS FOR SIGNIFICANCE

Origin : NBS, Section 11.3

Manager : L. S. Joel

Ob jective : To compute tables of the power function for combining two

independent tests of significance when the underlying distribution

of both tests is a variance ratio distribution.

Bac kground : In the analysis of incomplete block designs it has recently

been found that it is possible to obtain two independent tests of

significance for testing the null hypothesis that all treatment effects

are the same. Since the power of the individual tests are different,

both tests of significance are combined into a single test by weighting

each test in proportion to its power.

Status : New. The code is being prepared. Test runs will be made as soon

as SEAC is available.

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

AERODYNAMICALLY

Origin : NBS, Section 6.4

Manaqe

: P. Davis, W. F. Cahill

Ob jec tive : To determine the temperature distributions in solid wings

heated ae rodynamically for a range of possible heating environments

assuming heat flow in thickness directions only.
Status of Projects 23

Bac kqround : The study will show the extent to which it is necessary to

know the flight conditions in order to predict the operating range of

a solid wing.

Status : New.

3711-60-0009/55-90 STUDY OF A DIFFERENCE EQUATION ARISING IN STATISTICAL

MECHANICS

Origin : NBS, Section 3.2

Manager : M. Newman

Objective : To study the solutions of the difference equation

(1) V2+a Vi+bVc Vi+d V2=eAn 9n

> n^-3 ’ 1^°’

which arises in the study of counting problems of the "Ising" type.

The problem differs from other problems of this type in that the quantities

9^ 5 ^2 ’

9

3 ? 94 are not arbitrary but depend on the solution. Thus a circular type of reasoning is involved.

Bac kground : The problem is of application in the theory of ferromagnetism,

order-disorder problems, statistical mechanics, etc.

Status : New. The solution of the difference equation has been obtained

as a certain rational function of several hyper-elliptic functions.

Further study is under way.

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

Origin : NBS, Section 3.2

Manager : P. Davis

Objective : To calculate

,^3 = J

fi+3 f 32 f 21f 4i

^ + f 3l + E

f 3if 42^ dr 2 dr

3

diV

ufhp v*P fjj = exp [-/3V(lr i - r

^ O ] - 1

and

V(r) = 4 (r 12 - r ^), r^ e 0.

Since r^ are vectors in three-dimensional space, /

3^ is a nine-fold

integral. It is intended to evaluate it by means of the Monte Carlo

method

.

Background : This is the fourth virial coefficient of a Lennard-Jones gas.

Status : New. The fourth virial coefficient of a Lennard-Jones gas

involves a 9 _ fold integral and the possibility of evaluating it by

Monte Carlo methods is being studied. Preliminary tests to gain

experience with the method were begun on the computation of the volume

of an n-dimensional sphere. Values have been obtained for n=2 , 3 ,4 , 5, 6 ,

9

and indicate the need for more tests.
III. STATISTICAL ENGINEERING LABORATORY

(Section 11.3)

1 . Fundamental Research in Mathematical Statistics

BIBLIOGRAPHY AND GUIDE TO STATISTICAL LITERATURE

Tas k 1103-10-11 07/49A1

a

Origin: NBS Authorized 1/9/49

Manager: L. S. Deming

Full task descriptions Apr-June 1949 issue, p. 75

Status: CONTINUED. For a description of the continuing activity

on this task, see the Jan-Mar 1954 issue, page 49.

MANUAL ON FITTING STRAIGHT LINES

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

Origin: NBS Authorized 3/1/50

Manager: F. S. Acton

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

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

technical advisors, for comment.

TABLES TO FACILITATE DRAWING RANDOM SAMPLES

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

Origin: NBS Authorized 7/1/50

Managers: C. Eisenhart, L. S. Deming

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

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

page 64.

24
Status of Projects 25

MISCELLANEOUS STUDIES IN PROBABILITY AND STATISTICS

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

Origin: NBS Authorized 7/1/50

Manager: C. Eisenhart

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

Status: CONTINUED. For a description of the continuing activity

on this task, see July-Sept 1955- issue, page 21.

Publications

:

(1) Some applications of extreme-value theory. E. J. Gumbel and

J. Lieblein. Am. Stat. 8, 14-17 (Dec. 1954).

(2) Two early papers on the relation between extreme values and tensile

strength. J. Lieblein. Biometrika 4l_, Pts. 3 and 4, 559~560

(Dec. 1954).

(3) Bounds on a distribution function that are functions of moments to

order four. M. Zelen. J. Res. NBS 53? 377-381 (Dec. 195^).

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

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

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

in Annales de l'lnstitut Henri Poincare (Paris).

(5) Inequalities for probabilities associated with the multivariate normal

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

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

distribution. J. Lieblein. Submitted to a technical journal.

PROCEDURES OF NON- PARAMETRIC STATISTICS

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

Origin: NBS Authorized 9/17/51

Manager: I. R. Savage

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

Status: INACTIVE. For the latest report on this task see July-Sept 1954 issue, page 22.

Publications

:

(1) Easily used simultaneous confidence limits for a line. W. 0. Connor.

Submitted to a technical journal.

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

Submitted to a technical journal.

STUDIES IN THE MATHEMATICS OF EXPERIMENT DESIGN

Task 1 103-10-1 107/53-1

Origin: NBS Authorized 10/15/52

Manager: W. S. Connor

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

Status: CONTINUED. Work is continuing on the analysis of

covariance applied to incomplete block designs. The general inter-block

analysis has been derived and seems to be quite useful as it gives

additional estimates not only for the treatments but for the unknown

regression coefficients as well. Another aspect of the covariance
26 Status of Projects

investigation is development of the analysis for observations having the

underlying mathematical model

b

/A + t • + b . + V” ftx . . + € . . 7 i J jT-j / J iJ ij ( i-1 ,2 v; j=1,2,...,b)

where ju., t^, b^ and (5j are fixed constants and are identically distributed normal variates with E(€^j)=0,

This model differs from the usual covariance model in

for the existence of different regression coefficients

independent

var( j)= CT . that it allows

for each block.

Publications

(1) New experimental designs for paired observations. W. J. Youden and

W. S. Connor. J. Res. NBS 53, 191-196 (Sept. 1954).

(2) Partially balanced incomplete block designs with two associate

classes and two treatments per block. W. H. Clatworthy. Accepted

for publication in the Journal of Research of the NBS.

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

classes. W. H. Clatworthy. Submitted to a technical journal.

(4) Partially replicated Latin squares. W. J. Youden and J, S. Hunter;

submitted to a technical journal.

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

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

2 . Applied Research in Mathematical Statistics

COLLABORATION ON STATISTICAL ASPECTS OF NBS

RESEARCH AND TESTING

Task 3737-60-0002/51-1

Origin: NBS Authorized 7/1/50

Managers: W. J. Youden, J. Cameron

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

Status: CONTINUED. W. H. Clatworthy and W. J. Youden are collaborating with D. C. Schubert (Section 4.5, Electron Physics) on

the statistical aspects of his study of magnetron space charge density.

Repeat determinations of the several quantities of interest display

variability and consequently require statistical analysis and interpretation

The statistical parts of the magnetron space charge experiment involved

(1) an investigation of drift in the experimental equipment, and (2) the

determination of the least squares equation of the electrostatic field

within the magnetron by use of orthogonal polynomials.

In the investigation of drift in the experimental apparatus

two approaches were used. In the first approach statistical techniques

were applied to test whether different "runs" were significantly

different. In the second approach the data were recorded according to a predetermined schedule that permits estimation of the trend, provided

it exists, and adjustment for it before proceeding with the curve fitting.

Publications

(1) Acceptance sampling of electroplated articles. J. M. Cameron and

F. Ogburn (NBS Section 5.6). Proc. Am. Electroplater/* Soc . 4l , 19-22 (1954); also appeared in Plating 4j_, 43-46 (Jan. 1954).

(2) Instrumental drift. W. J. Youden. Science 1 20 . 627-631 (Oct. 22, 1954)
Status of Projects 27

STATISTICAL ASPECTS

Task

OF NBS ADMINISTRATIVE OPERATIONS

3737-60-0002/52-1

Origin: NBS Authorized 10/1/51

Manager: C. Eisenhart

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

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

page 68.

STATISTICAL SERVICES FOR COMMITTEE ON SHIP STEEL, NRC

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

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 program of sampling of production of ship

steel to give information on sources of variability in manufacturing

was devised. Methods of selection of specimens for v-notch tests are being investigated.

RESEARCH IN APPLICATIONS OF MATHEMATICAL STATISTICS TO

PROBLEMS OF THE CHEMICAL CORPS

Task 1 103-^0-51 18/52-1

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

Chemical Corps, Dept, of the Army

Manager: C. Eisenhart

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

Status: CONTINUED. Work was continued on methods of programming

analysis of variance computations for large scale computers.

STATISTICAL ANALYSIS AND DESIGN OF EXPERIMENTS

FOR THE U. S. GEOLOGICAL SURVEY

Task 1 103-50- 51 40/55--

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. Studies are continuing on data on the variation

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

the variations of samples taken at the same site. Work was done for

L. R. Stieff and T. W. Stern (of the Geochemistry and Petrology Branch,

U.S.G.S. Naval Gun Factory Laboratory, Washington, D. C.) on statistical

aspects of a problem on radioactive emission. Conferences were held with

F. Flanigan (of the Geochemistry and Petrology Branch, U.S.G.S. Naval

Gun Factory Laboratory, Washington, D. C.) on evaluation of precision

of three types of sample dividers, and with 0. Raup (of the U.S.G.S.

Geologic Studies Group, Grand Junction, Colorado) on his double-sampling

technique for achieving preassigned precision of results in orientation

studie s

.
28 Status of Projects

STATISTICAL ANALYSIS OF BALL BEARING FATIGUE DATA

Task 11 03-^0- 5 145/ 54-1

Origin and Sponsor: American Standards Association Authorized 1 / 13/54

Manager: J. Lieblein

Full task description: Jan-Mar 1954 issue, p. 54

Status: CONTINUED. Improved estimates for the L^q and L^q life

values were calculated on the SEAC for all the sets of data. These new estimates served as a basis for a series of analyses designed to obtain

values for the unknown parameters f

c

, , a

9 , a in the life equation

L =

Oh 8 p 3 ^ f ( i cos a ) Z D

c

nP

Most of the results from these analyses have been obtained and it is

expected that all results will be known shortly.

MANUAL ON EXPERIMENTAL STATISTICS

FOR ORDNANCE ENGINEERS

Task 1 103-40-5146/55-93

Origin and Sponsor: Office of Ordnance Research Authorized 12/29/54

Manager: C. Eisenhart

Objective: To prepare a section on Experimental Statistics for

the Ordnance Engineers Handbook.

Background: The Office of Ordnance Research is preparing an Ordnance Engineering Handbook. An important section of this handbook

will be the section on Experimental Statistics in which it is proposed

to include descriptions, and criteria for the selection of statistical

techniques useful in the programming and analysis of data from experiments.

The contents and level at which the manual should be written

should be based on a survey of the current practices and needs of

Ordnance engineers and should have realistic examples from the field.

No similar manual is in existence and the available handbooks on statistics do not have criteria to assist the non-statistican in the

selection of the appropriate methods and on this account their usefulness

is seriously limited. In addition nearly all presentations of statistical

methods in the literature are directed towards the needs of the agricultural

and biological sciences and there is a real need for translation of the

concepts and methods into the language of the physical sciences.

The Statistical Engineering Laboratory was requested to prepare

this section on Experimental Statistics because there is no comparable

statistical group working in the field of application of statistics in

the physical sciences.

Status: NEW.
IV. MATHEMATICAL PHYSICS SECTION

(Section 11.4)

RESEARCH IN MATHEMATICAL PHYSICS AND RELATED FIELDS

Task 1 104-10-11 15/55-57

Origin and Sponsor: NBS Authorized 9/1/54

Manager: E. VV. Cannon

Full task description: July-Sept 1954 issue, p. 27

Status: CONTINUED. E. W. Cannon continued exploratory investigations

in the structure of high polymers.

P. Henrici completed a paper on the generalization of the

classical addition theorems for Legendre functions and Gegenbauer polynomials.

A study of the application of the results to new potential problems

involving tores and spheres is under way.

P, Henrici also wrote a note on the remainder term in the

expression of the error integral for complex values of the argument. It was shown that the remainder can be made smaller in magnitude if the last

considered term of the asymptotic formula is multiplied by a suitable

complex constant, which is given explicitly.

Publications:

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

P. Henrici. Submitted to a technical journal.

(2) Kleine Bemerkung in asymptotischen Entivichlung des Fehlerintegrals

.

P. Henrici. In manuscript.

RESEARCH IN MECHANICS OF CONTINUA

Task 1 104-10-5160/54-23

Origin: NBS Authorized 12/29/53

Sponsor: Office of Naval Research, USN

Manager: R. F. Dressier

Full task description: Oct-Dec 1953 issue, p. 2

7

Status: CONTINUED. R. F. Dressier has begun the study of the

behavior in rarefied regions of the local sound speed of a polytropic gas

in unsteady flow, subject to frictional dissipation and entropy increase.

Methods used in a previous study are not applicable here to domains

neighboring a vacuum, since energy and momentum equations are singular on such wave fronts. It is hoped that a modified type of boundary layer

treatment will reveal the implications of the dynamical model used for

these low density regions.

The manuscript on "Entropy changes in rarefaction waves" by

R. F. Dressier has been rewritten.

M. Abramowitz has completed a study of heat convection in

laminar flow. The usual Graetz assumption is not used, i.e. assuming heat

conduction in the axial direction to be negligible. Results are obtained

29
30 Status of Projects

for small and large ranges of a significant parameter.

M. Abramowitz and I. Stegun have completed a study concerning

some solutions of the Coulomb wave equation, and a manuscript has been

pre pared

(1 )

( 2 )

(3)

( 4 )

(5)

( 6 )

(7)

( 8 )

Publications

Comparison of theories and experiments for the hydrau

wave. R . F. Dressier. To appear in the Proceedings

General Assembly of the International Union of Geodes

held in Rome, Italy, September 195^.

Turbulent flow in shock tubes of varying cross sectio

J. Res. NBS 53, 253-260 (Oct. 195^), RP25^1

Entropy changes in the equations for rarefaction wave

In manuscript.

On helical springs of finite thickness. P. Henrici.

the Quarterly of Applied Mathematics.

On rotational viscous flow through a tube. P. Henric

Zeitschrift fur angewandte Mathematik und Physik.

On the design of two-dimensional supersonic nozzles.

Institute of Technology, Zurich, Switzerland) and P. manuscript

Generation of solutions of the Coulomb wave equation,

and I. Stegun. In manuscript.

Forced heat convection in laminar flow through a tube

In manuscript.

lie dam-break

of the Tenth

y and Geophysics,

n. R. F. Dressier,

s. R. F. Dressier,

To appear in

i. To appear in

R. Chaix (Fede ral

Henrici. In

M. Abramowitz

. M. Abramowitz.

RESEARCH IN ELECTROMAGNETIC THEORY

Task 1 104-1 0-51 60/54-47

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: CONTINUED. F. Oberhettinger continued work on the problem

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

orientation on an ideally conducting half-plane.

RESEARCH IN MATHEMATICAL ELASTICITY

Task 1 10^-10-5160/55-85

Origin and Sponsor: Office of Scientific Research, Authorized 12/27/51! ARDC , USAF

Manager: R. F. Dressier

Objective: To carry on research in the fields of elastic equilibrium

and vibrations, including investigations of a basic research nature and

solution of problems of a more specific character. Analytical approaches are envisaged for the basic problems, chosen from among such topics as edgelayers in generalized plane stress and shell theory, and nonlinear edge

effects in plate bending; whereas the other problems will require convenient

mathematical formulation for use in high-speed computational procedures.

In the latter category it is hoped to obtain specific numerical results

for problems such as vibrations of delta wings, edge layer stress distributions

in bent plates of the more common shapes, and design data for corrugated

diaphragms in aeronautical instruments. Topics mentioned represent a list
Status of Projects 31

already approved by the sponsor, and related or alternative topics may

be investigated as subsequent progress dictates.

Background: Earlier studies concerning the role of boundary condition

approximations on convergence rapidity using finite differences for plate

vibrations are described in M. Abramowitz and W. Cahill, "On the vibration

of a square clamped plate" (1953? an NBS Report). This will be amplified

and applied to delta wings, and compared with results to be obtained by

variational techniques. Edge layers stress computations will be based

upon a method described in K. 0. Friedrichs and R. F. Dressier, "A boundary

layer theory for elastic bending of plates" (Proceedings Eighth Internationa]

Congress on Mechanics, Istanbul, 1952), which leads to biharmonic and

related equations in semi-infinite strips. Plane stress edge-layer effect

represents the complementary problems to bent plates, already investigated

by the stretching technique. Suitable extensions may also be possible

for shell theory and nonlinear plate bending.

Status

:

NEW.
NATIONAL BUREAU OF STANDARDS EASTERN AUTOMATIC COMPUTER (SEAC)

SEAC continued xn productive operation for the first half of the

second quarter of the fiscal year 1955 while the new site for the machine

was still in preparation. From mid-November through the end of the

quarter the entire installation was moved to its new location. By

December 23 all of the signal leads had been connected and the signals were being checked. Debugging of the computer proper and the acoustic

and Williams memories was well under way and was progressing satisfactorily

as of the end of the quarter.*

During the approximately seven weeks that SEAC was in operation its

overall efficiency was 8l°/o during scheduled operation, which compared

quite favorably with 89°/° for the first quarter of fiscal year 1955

and was considerably better than the 75°/° average of the first four

years of operation. Scheduled computation time was distributed as follows

Total scheduled time available for computation . . 5*+5 hours

Good operating time. *+5-6 Id le-in-orde r time 4

Time lost during or following malfunction 101

In addition, SEAC was utilized for 3*+8 hours of nonscheduled computation

(non scheduled computation being defined as operation without a duty

engineer in attendance for machine maintenance) and 182 hours for

engineering and preventive maintenance.

The record of SEAC operations from October 1 through November 20 is

as follows:

Task No . NBS

TToi+/55-55

51 16/54-9

1 1 1 0A7-2

1 1 10/50-7

1110/52-37

1 110/53-52

1 110/55-65

2308/50-13

0009/52-44

5126/53-27

5126/53-29

5126/53-51

5126/54-4

0009/54-8

0009/54-15

0009/54-17

0009/54-19

0009/54-22

5126/54-33

0009/54-36

5126/74-43

5126/54-49

5125/55-39

32

Hours Used ; Title Code

Chec king

Research in numerical analysis

Supporting research in linear programming

Tables of Coulomb wave functions

Wave function for lithium

Spheroidal wave functions

Internal conversion coefficients

Automatic coding

Ray tracing

Calculations for d spacing

Thermodynamic functions

Dynamic behavior of aircraft structures

Radiation diffusion

High temperature properties of water

Normal vibrations in molecules

Matrix reduction

De polymerization

Energy levels of complex atoms

Energy distribution on optical image

Air conditioning in underground structures

Velocity of light

Characteristics of conducting resistors

Wiechert distributing function

Molecular vibrations

1 13

Productive

Operation

3

3

27

1 51

1

1

1 28

91 105

56

1

1

1

1

17

27
Task No . NBS

:

5125/55-63

5126/55-65-

0009/55-66

5126/55-70

5126/55-77

0009/55-81

Other :

1 104/55-59

5126/53-15

5126/53-55

5126/55-13

5126/55-25

5126/54-34

5126/54-40

5126/54-44

5126/55-52

5126/55-60

5126/55-61

5126/55-62

5126/55-67

5126/55-72

5126/55-73

5126/55-79

Title Hours Used

:

Code Productive

Checkinq Operation

Morse integrals 1 High-tempe rature thermodynamic tables 2 Reconstitution of light intensities 2 4

Ball bearing fatigue data 8 21 Evaluation of integrals 7 Combining tests for significance 1 Totals : 69- TO

Turbulent atmospheric contamination 5 Neutron diffusion 3 18

Theory of games 2 1

1

Award of procurement contracts by 1 45

linear programming

Retlected radiation from an infinite 1 3

Lambert plane

Equilibrium calculations for water 8 60

Flight simulator 1 l4

Flight performance 1 22

Atmospheric waves 2 9

Detonator voltage and current oscillograms 3 Elastic neutron scatterings 1 5 Integrals involved in supersonic flutter 6 53

Dosage integrals 1 7

Maryland inter-industry study 1 Heat Convection 16 3

cL -Analysis - NDA . 5 Totals : 43 264

on a part-time basis on January 22,

*Ope ration of SEAC was resumed 1955.
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.

Applied Mathematics Division Lectures

WILKINSON, J. (National Physical Laboratory, Teddington, England)

(1) Linear algebra on the pilot A.C.E. October 19.

(2) Latent roots of matrices. October 26.

MORSE, P. M. (Massachusetts Institute of Technology) Computing needs

in operations research. December 3.

Mathematics Seminar

NEWMAN, M. The coefficients of certain modular forms. October 25.

ANTOSIEWICZ, H. Stability of differential equations. November 1. HENRICI, P. The quotient difference algorithm of H. Rutishauser, and

a remark on Aitken's o2_me thod. November 8 . RABINOWITZ, P. Paracompact spaces. November 22.

DAVIS, P. How shall we evaluate multiple integrals? December 6 . HOFFMAN, A. Incidence matrices, linear programming, and combinatorial

problems. December 13.

ABRAMOWITZ, M. Mathieu functions. December 20.

Statistical Engineering Seminars

BERGER, M.J. (Nuclear Physics Section, NBS) A study of Gamma ray

diffusion by random sampling. November 24.

34
Lectures and Symposia 35

LUKACS, E. (Office of Naval Research) On the characterization of

populations by properties of suitable statistics. December 16.

Algebra Seminar

(Offered by The American University in cooperation with the National

Bureau of Standards

NEWMAN, M. (1) Commutator subgroups. October 13.

(2) Modular groups. October 27, 29.

(3) Automorphisms of the modular group. November 17, 24.

MARSH, R. (Department of Defense) p-groups. November 3.

GOLDBERG, K. p-adic numbers. November 10.

JOEL, L. Commutators of matrices. December 1. JACKSON, I. (Bureau of Ships) Commutator subgroups of infinite permutation

matrices. December 8, 15.

Papers and Invited Talks

Presented by Members of the Staff

at Meetings of Outside Organizations

ALT, F. L. Scientific computation on high speed machines. Presented at

a Seminar on Automatic Computation at the Battelle Memorial

Institute, Columbus, Ohio, November 17.

EISENHART, C. (1-3) Statistical engineering at the National Bureau of

Standards. Presented to the Graduate Seminar Group,

Department of Mathematics, Princeton University, October 19;

also at a Seminar at Stanford University, November 3; and at

a Seminar at the University of Chicago, November 7. (4-5)

Scales, scale readings, and information. Presented before

the Central New Jersey Chapter, of the American Statistical

Association, Princeton, N. J., October 19; and at a Seminar

at the Magnolia Petroleum Company, Dallas, Texas, October 29.

FAN, K., 0. TAUSSKY, and J. TODD. Discrete analogs of inequalities of

Wirtinger. Presented at the annual meeting of the American

Mathematical Society, Pittsburgh, Pennsylvania, December 27-28.

LIEBLEIN, J, Estimation of 'censored' samples of extreme data. Presented

at the Third Berkeley Symposium on Mathematical Statistics

and Probability, Institute of Mathematical Statistics, Berkeley,

California, December 29.

TAUSSKY, 0. Estimation of characteristic values of finite matrices.

Presented at a Physics Seminar at the University College,

London, October 21.
36 Lectures and Symposia

TODD, J. (1) The theory and practice of the solution of large systems

of linear equations. Presented at a Physics Seminar at the

University College, London, October 21. (2) The applied

mathematics program of the National Bureau of Standards. Presented

at the Admiral Research Laboratory, Teddington, England, November 1. YOUDEN, W. J. (1,2) Principles of experimental design. Presented before

the Madison Section of the American Chemical Society, Madison,

Wisconsin, October 4; and at the U. S. Forest Products Laboratory,

Madison, Wisconsin, October 5. (3~11) Statistical design.

A series of nine lectures presented before the New York Section,

American Chemical Society, New York, N. Y., October 6, 13, 20,

27, November 3, 10, 17, 23, and December 8. (12) Statistical

methods in the design of experiments and interpretation of data.

Presented before a meeting of the Association of Research

Directors, Kearney, N. J., October 21. (13) Analysis with

experimental data. Presented before the Notional Printing Ink

Research Institute, Bethlehem, Pa., October 22. (l4) Getting

the most out of your experiments. Presented before the Sixteenth

Annual Forum, Packaging Institute, New York, N. Y. , October 27.

(15) Sophisticated chemistry. Presented before the Student

Chemical Society, Lehigh University, Bethlehem, Pa., December 9.

(

6 ) The gun problem. Presented at the Princeton Conference,

American Society for Quality Control, Princeton, N. J., December

10. (17) Opportunities of mathematics in science. Presented

before the Science Club, Woodrow Wilson High School, Washington,

D. C. , December 1*+.
Publication Activities

1 . PUBLICATIONS THAT APPEARED DURING THE QUARTER

1.1 Mathematical Tables

(1) Tables of functions and of zeros of functions. Collected short

tables of the Computation Laboratory. NBS Applied Mathematics

Series 37. Available from U. S. Government Printing Office,

Washington 25, D. C., #2.25.

(2) Tables of the error function and its derivative. NBS Applied

Mathematics Series 4l . (A reissue of NBS Mathematical Table 8,

Tables of probability functions, vol. I.) Available from U. S.

Government Printing Office, Washington 25, D. C., #3.25.

(3) Table of salvo kill probabilities for square targets. NBS Applied

Mathematics Series 43. Available from U. S. Government Printing

Office, Washington 25, D. C., 30 cents.

1.3 Technical Papers

(1) Coulomb wave functions in the transition region. M. Abramowitz

and H. Antosiewicz. Phys. Rev. 5*6, 75_ 77 (Oct. 1, 195*+).

(2) Coulomb wave functions along the transition line. M. Abramowitz

and P. Rabinowitz. Phys. Rev. 26, 77~79 (Oct. 1, 195*0.

(3) A representation for solutions of analytic systems of linear

differential equations. H. A. Antosiewicz and M. Abramowitz.

J. Wash. Acad. Sci. 44, 382-384 (Dec. 195*+).

(4)

.A remark on the smoothing problem. H. Antosiewicz and A. J. Hoffman.

Management Science 1_, 92-95 (Oct. 1954).

(5) Acceptance sampling of electroplated articles. J. M. Cameron and

F. Ogburn (NBS Section 5.6). Proc. Am. Electroplaters Soc. 4l_,

19-22 (1954). Also appeared in Plating 4l , 43-46 (Jan. 1954).

(6)

A comparison of four national radium standards. Part II: Statistical

procedures and survey. W. S. Connor and W. J. Youden. J. Res.

NBS 23, 273-275 (Nov. 1954).

(7) New experimental designs for paired observations. W. S. Connor

and W. J. Youden. J. Res. NBS 23, 191-196 (Sept. 1954).

(8) Linear functional equations and interpolation series. P. Davis.

Pac. J. Math. 4, 503-532 (195*+).

(9) Complex biorthogonality for certain sets of polynomials. P. Davis

and H. Poliak (Bell Telephone Laboratories). Duke Math. J. 21 , 653-668 (Dec. 195*+).

(10)

On the estimation of quadrature errors for analytic functions.

P. Davis and P. Rabinowitz. MTAC 48, 193-203 (Oct. 1954).

37
38 Publication Activities

(11) Turbulent flow in shock tubes of varying cross-section. R.F .Dressier

J. Res. NBS 13, 253-260 (Oct. 1955), RP2551

.

(12) Some applications of extreme-value theory. E. J. Gumbe 1 and

J. Lieblein. Amer. Stat. 8 , 15-17 (Dec. 1955).

(13) Smooth patterns of production. A. J. Hoffman and W. Jacobs.

Management Science _1_, 86-91 (1955).

(15)

Contrac tibility and convexity. H. W. Kuhn (Bryn Mawr College).

Proc. Am. Math. Soc. 1, 777~779 (Oct. 1955).

(15) 0° two problems in abstract algebra connected with Horner's rule.

A. M. Ostrowski. Studies in Mathematics and Mechanics Presented

to Richard von Mises, pp. 50-58 (Academic Press Inc., New York,

1955).

(16) An isope rime trie inequality for closed curves convex in even- dimensional Euclidean space. I. J. Schoenberg. Acta Math. 91 <

1 53-1 65 (1955).

(17) Generalized commutators of matrices and permutations of factors

in a product of three matrices. 0. Taussky. Studies in Mathematics

and Mechanics presented to Richard von Mises, pp. 67-68 (Academic

Press Inc., New York, 1955).

(18) Experiments in the solution of differential equations by Monte

Carlo methods. J. Todd. J. Wash. Acad. Sci. 55? 37 7—3 8 1 (Dec. 1955)

(19) The condition of certain matrices, II. J. Todd. Arch. Math. V,

259-257 (1955).

(20) Instrumental drift. W. J. Youden. Science 1 20 , 627— 63 1 (Oct. 22, 1955).

(21) Bounds on a distribution function which are functions of moments

to order four. M. Zelen. J. Res. NBS 13? 3 7

7

—3 8 1 (Dec. 1955).

1.5 Reviews, Notes

(1) Note on the circle theorem of hydrodynamics. E. Levin. Qu.

App. Math. XII, 315-316 (Oct. 1955).

(2) Two early papers on the relation between extreme values and

tensile strength. J. Lieblein. Biometrika 5l_, 559-560 (Dec. 1955).

(3) Obituary notice on L. F. Richardson. J. Todd. MTAC 8, 252-255

(1955)

.

2. MANUSCRIPTS IN THE PROCESS OF PUBLICATION DECEMBER 31, 1955

2.1 Mathematical Tables

(1) Tables of sines and cosines for radian arguments. NBS Applied

Mathematics Series 53. (A reissue of NBS Mathematical Table 5.)

In press, Government Printing Office.

2.3 Technical Papers

(1) Approximate method for rapid Loran computation. M. Abramowitz,

D. H. Call, and J„ C. Mathews. Submitted to a technical journal.
Publication Activities 39

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

Accepted for publication in the Pacific Journal of Mathematics.

(3) Asymptotic solution of linear differential equations with a parameter. H. Antosiewicz. Submitted to a technical journal.

(4) On a cert ain integral involving Bessel functions. H. Antosiewicz.

Submitted to a technical journal.

(5) On nonlinear differential equations of the second order with

integrable forcing term. H. Antosiewicz. Accepted for publication

in the Journal of the London Mathematical Society.

. . • • (6) On the differential equation x + k(f(x) + g(x)x)x = ke(t).

H. A. Antosiewicz. Submitted to a technical journal.

(7) Stable systems of differential equations with integrable forcing

term. H. Antosiewicz. Submitted to a technical journal.

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

and S. Levy (NBS 6.4). Accepted for publication in the Journal

of the Institute of Aeronautical Sciences.

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

classes. W. H. Clatworthy. Submitted to a technical journal.

(10) Partially balanced incomplete block designs with two associate

classes and two treatments per block. W. H. Clatworthy. Accepted

for publication in the Journal of Research of the NBS.

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

Submitted to a technical journal.

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

applications to processes with absorbing regions and to the

problem of loops in Markoff chains. D. van Dantzig. Accepted

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

(Paris)

.

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

Accepted for publication in the Pacific Journal of Mathematics.

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

P. Rabinowitz. Accepted for publication in the Journal of the

Association for Computing Machinery.

(15) Some sampling results on the power of nonparame trie tests against

normal alternatives. W. J. Dixon (University of Oregon) and

D. Teichroew. Submitted to a technical journal.

(16) Comparison of theories and experiments for the hydraulic dam-break

wave. R. F. Dressier. To appear in the Proceedings of the Tenth

General Assembly, of the International Union of Geodesy and

Geophysics, held in Rome, Italy, September 1954.

(17) Heat flow in a fluid with eddying flow. W. H. Durfee (now with

Operations Research Office, Johns Hopkins University). Submitted

to a technical journal.

(18) On the optimal character of the (s,S) policy in inventory theory.

A. Dvoretzky, J. Kiefer, and J. Wolfowitz. Submitted to a technical journal.
4o Publication Activities

(19) A comparison theorem ior eigenvalues of normal matrices. K. Fan,

Submitted to a technical journal.

(20) On systems of linear inequalities. K. Fan. Submitted to a technical journal.

(21) Some inequalities concerning positive-definite He rmitian matrices.

K. Fan. Accepted for publication in the Proceedings of the

Cambridge Philosophical Society.

(22) Some metric inequalities in the space of matrices. K. Fan and

A. J. Hoffman. Accepted for publication in the Proceedings of the

American Mathematical Society.

(23) Discrete analogs of inequalities of Wirtinger, K. Fan, 0. Taussky,

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

(24) A determinantal inequality. K. Fan and J. Todd. Accepted for

publication in the Journal of the London Mathematical Society.

(2^) Asymptotic lower bounds for the fundamental frequency of convex membranes. G. E. Forsythe. Accepted for publication in the

Pacific Journal of Mathematics.

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

and D. Sokolowsky. To appear in the Logistics Research Quarterly.

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

Accepted for publication in the Journal of Research of the NBS.

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

P. Henrici. Submitted to a technical journal.

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

of the reflected radiation of a point source. P. Henrici. Accepted

for publication in the Journal of the Washington Academy of Sciences

(30) On certain series expansions involving Whittaker functions and

Jacobi polynomials, P. Henrici. Accepted for publication in the

Pacific Journal of Mathematics.

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

Accepted for publication in the Pacific Journal, of Mathematics.

(32) On helical springs of finite thickness. P. Henrici. Accepted

for publication in the Quarterly of Applied Mathematics.

(33) On rotational viscous flow through a tube, P. Henrici. Accepted

for publication in ZAMP.

(34) The number of absolute points of a correlation. A. J. Hoffman,

M. Newman, E. G. Straus, and 0. Taussky. Submitted to a technical

journal.

(35) Indentation pressure of a smooth punch. E. Levin. Accepted for

publication in the Quarterly of Applied Mathematics.

(36) Acous tic radiation pressure on a circular disk. H. Levine. To

appear in the Proceedings of the Fifth Symposium on Applied

Mathematics of the American Mathematical Society, held in

Pittsburgh, Pa., June 1952.
Publication Activities 4i

(37) On the variances and covariances of order statistics from the

Weibull distribution. J. Lieblein. Submitted to a technical

journal.

(38) A proof of Hilbert's Nulls te llensatz . T. S. Motzkin. Submitted

to a technical journal.

(39) The assignment problem. M. Motzkin. To appear in the Proceedings

of the American Mathematical Society's Sixth Symposium on Applied

Mathematics, held at Santa Monica City College, August 1953.

(40) On Fejer sets in linear and spherical spaces. T. S. Motzkin and

I. J. Schoenberg. Accepted for publication in Annals of Mathematics.

(4l ) Pairs of matrices with property L, II. T. S. Motzkin and 0. Taussky.

Submitted to a technical journal.

(42) Least p-th power polynomials on a real finite point set. T. S. Motzkin

and J. L. Walsh. To appear in the Transactions of the American

Mathematical Society.

(43) The coefficients of certain modular forms. M. Newman. Submitted

to a technical journal.

(44) The d iophantine equation ax

D,-byn=c. M. Newman. Submitted to a technical journal.

(45) Structure theorems for modular subgroups. M. Newman. Accepted

for publication in the Duke Mathematical Journal.

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

diffraction of waves by a wedge. F. Oberhe ttinge r . Submitted

to a technical journal.

(47) On the Lerch zeta function. F. Obe rhe ttinge r . Accepted for

publication in the Pacific Journal of Mathematics.

(48) De te rminanten mit ueberwiegender Hauptdiagonale und die absolute

Konvergenz von linearen Iterationsprozessen. A. M. Ostrowski.

Submitted to a technical journal.

(49) On the convergence of Gauss' alternating procedure in the method

of the least squares, I. A. M. Ostrowski. Submitted to a technical journal,

(50) On the linear iteration procedures for symmetric matrices.

A. M. Ostrowski. Accepted for publication in Rendiconti di

Matematico, Rome.

(51) On the spectrum of a one parametric family of matrices.

A. M. Ostrowski. Accepted for publication in Journal fur die reine

und angewandte Mathematik.

(52) On Gauss' speeding up device in. the theory of single step

iteration. A. M. Ostrowski. Submitted to a technical journal.

(53) On absolute convergence of linear iteration processes. A. M. Ostrowski

.

Submitted to a technical journal.

(54) On spectre of second-order differential operators. D. Ray.

Submitted to a technical journal.
42 Publication Activities

(55) On the convergence of asymptotic solutions of linear differential

equations. R . M. Redheffer (U.C.L.A.) and W. Wasow. Submitted

to a technical journal.

( 56) Osculatory interpolation in the complex plane. H. E. Salzer

(Department of the Army). Accepted for publication in the Journal

of Research of the NRS.

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

Submitted to a technical journal.

(58) Inequalities for probabilities associated with the multivariate

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

journal

.

(59) 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,

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

J. Todd. Submitted to a technical journal,

(61) An improved cathode ray tube storage system. R. Thorensen. To

appear in the Proceedings of the Western Computer Conference of

the AIEE-IRE-ACM held in Los Angeles, Calif., February 4,5,6, 1953.

(62) Motivation for working in numerical analysis. J. Todd. To

appear in the Transactions of a Symposium on Applied Mathematics

sponsored by the Office of Ordnance Research and held in Chicago,

111., April 29, 1954.

( 63 ) An extension of a theorem of Dantzig's. C. Tompkins and I. Heller.

(George Washington University). Submitted to a technical journal.

(64) A method for the numerical integration of differential equations

of second order without explicit first derivatives. R, de Vogelaere,

Accepted for the Journal of Research of the NRS.

( 65 ) Discrete approximations to elliptic differential equations.

W. Wasow. Accepted for publication in Zeitschrift fiir angewandte

Mathematik und Physik,

(66) On the asymptotic transformation of certain distributions into

the normal distribution. W. Wasow. To appear in the Proceedings

of the American Mathematical Society Sixth Symposium on Applied

Mathematics, held at Santa Monica City College, August 26-28, 1953.

( 67 ) An extremum property of sums of eigenvalues. H. Wielandt.

Accepted for publication in the Proceedings of the American

Mathematical Society.

(68) E rror 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 City College, August 1953.

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

to a technical journal.

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

Submitted to a technical journal.

2.5 Miscellaneous Publications

(1) Experiments in the computation of conformal maps. MBS Applied

Mathematics Series 42. In press, Government Printing Office.
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and

maintenance of the national standards of measurement and the provision of means and methods

for making measurements consistent with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials,

devices, and structures; advisory services to Government Agencies on scientific and technical

problems; invention and development of devices to serve special needs of the Government; and the

development of standard practices, codes, and specifications. The work includes basic and applied

research, development, engineering, instrumentation, testing, evaluation, calibration services, and

various consultation and information services. A major portion of the Bureau’s work is performed

for other Government Agencies, particularly the Department of Defense and the Atomic Energy

Commission. The scope of activities is suggested by the listing of divisions and sections on the

inside of the front cover.

Reports and Publications

The results of the Bureau’s work take the form of either actual equipment and devices or published papers and reports. Reports are issued to the sponsoring agency of a particular project or program. Published papers appear either in the Bureau’s own series of publications or in the

journals of professional and scientific societies. The Bureau itself publishes three monthly periodicals, available from the Government Printing Office: The Journal of Research, which presents

complete papers reporting technical investigations; the Technical News Bulletin, which presents

summary and preliminary reports on work in progress; and Basic Radio Propagation Predictions,

which provides data for determining the best frequencies to use for radio communications throughout

the world. There are also five series of nonperiodical publications: The Applied Mathematics

Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous

Publications.

Information on the Bureau’s publications can be found in NBS Circular 460, Publications of

the National Bureau of Standards ($1.00). Information on calibration services and fees can be

found in NBS Circular 483, Testing by the National Bureau of Standards (25 cents). Both are

available from the Government Printing Office. Inquiries regarding the Bureau’s reports and

publications should be addressed to the Office of Scientific Publications, National Bureau of Standards, Washington 25, D. C.
ii