NATIONAL BUREAU OF STANDARDS REPORT

10003

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

A Semi-Annual Report

July through December 1968

<20^-

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT

205.0

NBS REPORT

10003

PROJECTS and PUBLICATIONS

of the

APPLIED MATHEMATICS DIVISION

A Semi-Annual Report

July through December 1968

IMPORTANT NOTICE NATIONAL BUREAU OF ST/

for use within the Government, and review. For this reason, the whole or in part, is not authori Bureau of Standards, Washingtor

the Report has been specifically |

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST)

on October 9, 2015.

>s accounting documents intended

subjected to additional evaluation

listing of this Report, either in

> Office of the Director, National

iy the Government agency for which

apies tor its own use.

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
APPLIED MATHEMATICS DIVISION FUNCTIONS

Conduct research and provide consulting services to the Bureau and other Federal agencies

in various fields of mathematics important to physical and engineering sciences, automatic

data processing, and operations research with emphasis on statistical, numerical and

combinatorial analysis, and mathematical physics . Develop tools for mathematical work such

as mathematical tables, handbooks, manuals, mathematical models and computational methods,

and advise on their use. Provide training in disciplines related to these functions.

NUMERICAL ANALYSIS SECTION: The advancement of computation and the theory

of numerical analysis, particularly in the development of computing algorithms,

approximations to functions, and methods to facilitate the use of high speed

electronic computers by subject matter specialists. Design of mathematical

tables; exploratory calculations on automatic machines. Consulting services

and training, and preparation of manuals in these fields. Research in

underlying branches of pure and applied mathematics, such as matrix algebra,

•

combinatorial analysis, and number theory.

OPERATIONS RESEARCH SECTION: Development and application of mathematical and

computational techniques for the analysis, improvement or optimization of

complex systems or activity-patterns . This includes (l) research in specific

relevant areas of mathematics, such as linear programming, the theory of linear

graphs, and the theory of strategic contests, (2) investigations in the art of constructing useful mathematical models of complex systems, and of

obtaining information about the system by applying analytic or simulation

methods, and (3) application of these techniques to selected problems,

of general methodological significance, arising in the work of the Bureau

or of other Government agencies lacking specialized personnel in this field.

STATISTICAL ENGINEERING SECTION: Consulting services in the application

of mathematical statistics to physical science experiments and engineering

tests, particularly in the design of experiments and in the analysis and

interpretation of data. Research on pertinent topics in probability

and mathematical statistics . Preparation of reports, manuals, tables,

studies of computational methods and other aids to the application of

modern statistical methods

.

SYSTEMS DYNAMICS SECTION: Research and consulting in applied mathematics

basic to physics and engineering, with emphasis on analysis of the dynamic

behavior of complex physical systems . This involves, primarily, the

development and application of techniques for solving linear and non- linear systems of differential equations and integral equations, or combinations of both. Of concern also is simulation of the behavior of

physical systems by means of electronic computers using approximation

techniques and semi-analytic methods. Attention is given to problems in

plasma dynamics and the behavior of solid matter and multicomponent liquid

systems, with emphasis on developing mathematical methods of wide range

of applicability beyond the scope of the immediate problem. Investigations

are carried out on the special functions encountered in the analysis and

algorithms for their evaluation are prepared.

II
Contents

Status of Projects 0 as of December 31> 1968 1

1. Numerical analysis 1

2. Operations research 3

3 . Probability and mathematical statistics 6

4. Statistical engineering services 9

5. Systems dynamics 11

Lectures and technical meetings l6

Publications activities 18

° Only unclassified material is included in this report

.

Ill
APPLIED MATHEMATICS DIVISION

July 1 through December 31, 1968

TECHNICAL ADVISORY PANEL

John Todd, Chairman

Francis J. Anscombe, Yale University

Philip J. Davis, Brown University

Charles R. DePrima, Calif. Inst, of Technology

H. 0. Hartley, Texas ASM University

M. H. Martin, University of Maryland

Marvin L„ Minsky, Massachusetts Institute of Technology

John Riordan, The Rockefeller University

DIVISON OFFICE

Edward W. Cannon, Ph.D„, Chief

Catherine Hartsfield, Secretary Yates S. Sladen, Admin. Officer

Margaret L. Dean Irene A. Stegun, M.A. Russell A. Kirsch, M.S. Ruth Zucker, B.A.

Ida Rhodes, M.A., Consultant

NUMERICAL ANALYSIS SECTION, Morris Newman, Ph.D., Chief

Doris M. Burrell, Secretary

Karl Goldberg, Ph.D.

Seymour Haber, Ph.D.

OPERATIONS RESEARCH SECTION, Alan J.

Jack Edmonds, M.A. Judith F. Gilsinn, M.S. William G. Hall, B.S. Dolores E. Harrison, Secretary

STATISTICAL ENGINEERING LABORATORY *

Raj Chandra Bose, Ph.D „, Consultant

Eleanor S. Brown Willard H. Clatworthy, Ph.D.***

Veronica Connor, Secretary

SYSTEMS DYNAMICS SECTION, H^ns J.

Doris M. Burrell, Secretary

George B. Delancey, Ph.D.**

Joseph Lehner, Ph.D.*

Frank W. J. Olver, D.Sc.

Goldman, Ph.D., Chief

William A. Horn, Ph.D.

Lambert S. Joel, B.A. David E. Labovitz, B.A.*

David Hogben, Ph.D.

Brian L. Joiner, Ph.D.

Hsien H. Ku, Ph.D.

Mary G. Natrella, B.A.*

Sally T. Peavy, B.S.*

Oser, Ph.D., Chief

Jeffrey Tse-Wei Fong, Ph.D.

S. Richard Kraft, Ph.D.

Zella G. Ruthberg, M.A.*

Edward T. Ordman, M.A.*

Charles F. Osgood, Ph.D.**

Stephen J. Pierce, Ph.D.**

Joel Levy, M.A. Philip Meyers, Ph.D.*

Martin H. Pearl, Ph.D.*

Patsy L. Saunders, B.S.

Chief

Janace Speckman, M.S.

Ruth N. Varner, B.A.*

Roy H. Wampler, M.A. William H. Youden, Ph.D.***

Walter L. Sadowski, Ph.D.

Chan Mou Tchen, Ph.D.°

Joseph M. Cameron, M.S., Chief

Joan R. Rosenblatt, Ph.D., Ass't.

*P art- time

**Postdoctoral Resident Research Associate

***Guest Worker

°0n Leave of Absence

Joseph M. Cameron det

Dr. Joan R. Rosenblatt

August 22, 1968.

.iled to the Office of the detailed as Acting Chief,Director, Institute for Statistical EngineeringBasic Standards;

Laboratory, effective

IV
Status of Projects

1

.

NUMERICAL ANALYSIS

RESEARCH IN NUMERICAL ANALYSIS AND RELATED FIELDS

Task 20501-12-2050110/55-55

115^0

Origin: NBS Authorized 8/29/54

Manager: Morris Newman

Full task description: July - September 1954 issue, p. 1

Status: CONTINUED. M. Newman and M. Marcus have proved that an irreducible matrix group with

uniformly bounded trace is bounded, and have applied this to the study of certain matrix groups arising in investigations of singular values.

M. Newman and L. Greenberg have proved theorems on the structure of solvable groups generated by two elements of relatively prime order, and have used them to prove that the density

of such groups (in a certain well-defined sense) is 0.

M. Newman has shown that if t > 3 then no subgroup of finite index of SL(t,z) is the free product of cyclic groups.

M. Newman has derived a number of miscellaneous results on class numbers of quadratic

fields. For example if k satisfies k > 2, k = 2 mod 4, k square-free, and k = y^ - x“, where p is a prime = 1 mod 4, then the class number of the field R( v/-k) is divisible by p .

J. Lehner and A.O.L. Atkin have made a thorough study of the Hecke operators on the groups r (m), where m is a positive integer not restricted to primes. The results for m ^ prime are believed to be new. Spaces of simultaneous eigenforms are developed and are used to build up the space of cusp modular forms. Arithmetic properties of the Fourier coefficients

are studied.

K. Goldberg continued his investigation of the coefficients of the powers of the iterates of formal power series.

S. Haber continued a program of calculations designed to determine parameters for, and convergence properties of, Korobov quadrature formulas. He also found some new classes of stochastic quadrature formulas, and began a comparison of the use of various sampling schemes

in connection with these formulas.

M. Newman and S. Pierce have determined necessary and sufficient conditions that a certain class of (not necessarily irreducible) matrix groups be equivalent to unitary groups.

F.W.J. Olver has completed an error analysis for the method of stationary phase.

Applications are being made to asymptotic expansions of Bessel and other functions.

1
Publications

:

(1) Semi-groups with zeroids. K. Goldberg. To appear in the Journal of Research NBS.

(2) A recurrence related to monotone subsequences in permutations. K. Goldberg. To appear in the Journal of Research NBS.

(3) The l.u.b. of a set of determinants of order 3. K„ Goldberg. To appear in the Journal of Research NBS.

(4) On a theorem of Piatetsky-Shapiro and evaluation of multiple integrals. S. Haber and

C .F . Osgood. To appear in Math. Comp.

(5) A combination of classical and Monte Carlo methods for evaluating multiple integrals.

S. Haber. Bull. A.M.S. 74, 683-686 (1968).

(6) On the sum S < cm >

*"

. S. Haber and C .F . Osgood. Submitted to a technical journal.

(7) Maximal normal subgroups of the modular group. M. Newman. Proc. Amer. Math. Soc. 19,

1138-1144 (1968).

(8) Solving equations exactly. M. Newman. J. Res. Nat. Bur. Standards 71B, 171-179 (1967).

(9) Pairs of matrices generating discrete free groups and free products. M. Newman. Michigan Math. J. 15, 155-160 (1968).

(10) A bound for the number of conjugacy classes of a group. M. Newman. J. London Math. Soc. 43, 108-110 (1968).

(11) Bounds for the number of generators of a finite group. M. Newman. J. Res. Nat. Bur. Standards 71B, 247-248 (1967).

(12) A diophantine equation. M. Newman. J. London Math. Soc. 43, 105-107 (1968).

(13) Matrix representations of groups. M. Newman. AMS-60, of the NBS.

(14) An extension of Miller's algorithm. F.W.J. Olver. Aplikace Matematiky, vol. 13, pp. 174-176 (1968).

(15) Error bounds for the Laplace approximation for definite integrals. F.W.J. Olver.

J. of Approx. Theory (to appear).

(16) Subgroups of SL(t,z). M„ Newman. To appear in J. Res. Nat. Bur. Standards.

2
2. OPERATIONS RESEARCH

CONSULTATION IN MATHEMATICAL OPERATIONS RESEARCH

Task 205-12-2050151

11570

Origin and Sponsor: NBS Authorized 12/30/60

Manager: Alan J. Goldman

Full task description: October- December i960 issue, p.3

Status: CONTINUED.

(1) Demand for miscellaneous consulting and advisory services continued heavy. Section staff

provided such services in 104 recorded instances; of these 63 involved assistance to NBS staff. The 104instances totalled to 575 recorded man-hours. Other agencies assisted included National Institutes

of Health, Coast Guard, National Highway Safety Council, AID, U.S. Travel Service, Office of Emergency

Planning, Bureau of Public Roads and Bureau of Fish and Wildlife. Requests from universities, industry,

professional groups and journals were also met.

(2) J. Gilsinn, A.J. Goldman and W. Horn completed assistance to the Maritime Administration in

a study (with Division 431) to determine locations and sizings for inland cargo consolidation centers. A mathematical model was developed, programmed, and given illustrative application to the extent that

data permitted. The work was documented for the sponsor. (Reported here for convenience; supported

under Project 4314422). Further efforts under NBS auspices are aimed at improving this tool to aid

in locating collection-distribution facilities.

( 3 ) W.G. Hall continued assistance in developing a simulation model for a highway maintenance

depot. He also continued helping the Army Data Field Systems Command in planning, design and numericalanalysis aspects of a proposed new tactical artillery- fire control system. He continued assistance to

the National Highway Safety Bureau in planning and evaluating the data- systems elements of its activities. (Reported here for convenience; supported under Projects 4565425 and 2050625 respectively.)

(4) A.J. Goldman, W.G. Hall, W.A. Horn, J. Levy and M.H. Pearl began assistance to the Post Office

Department's Bureau of Research and Engineering on a major systems engineering project. (Reported here

for convenience; supported under Projects 4314442, 4314444, 4-314448, 4-314449*)

( 5 ) A.J. Goldman and P.R. Meyers continued the study of "contractification" of self- mappings of metric spaces by remetrization, extending previous results for the commutative case to obtain

conditions for the simultaneous contractification of a finite family of maps. W.A. Horn initiated a study of "convex homotopies" between convex real-valued functions, examining existence, monotonicity

and interpolation properties of such maps.

Publications

(1) W.F. Druckenbrod (Div. 431), W.G. Hall, et al. A simulation of a highway maintenance operational

unit. NBS Report 9938 (October, 1968 ).

(2) A.J. Goldman. Fractional container- loads and topological groups. To appear in Operations Research.

( 3 ) W.A. Horn. Some fixed point theorems for compact maps and flows in Banach spaces. Submitted to a technical journal.

(4) W.A. Horn. Determining optimal container inventory and routing. NBS Report 9936 (October, 1968 )

.

( 5 ) R.H. Jordan (Div. 431), J. Gilsinn, A.J. Goldman, W.A. Horn _et al. Systems analysis of inland

consolidation centers for the U.S. Maritime Administration. NBS Report 9892 (August, 1968 )

.

3
COMBINATORIAL METHODS

Task 205-12-2050152

11540

Origin and Sponsor: NBS

Manager: Jack Edmonds Authorized 12/30/60

Full task description: October- December 1964 issue, p. 3; April-June 1962, p. 15

Status : CONTINUED.

(1) J. Edmonds continued work on various aspects of polyhedral combinatorics, theory of

submodular functions, matching theory, and combinatorial algorithms. Edmonds and Oved Shisha prepared

a paper titled "Acute Bisections".

(2) A.J. Goldman considered the problem of locating n "centers" in a network so as to minimize

transportation costs. He showed S. Hakimi's result, that only vertex-locations for the centers need

be considered, remains valid when shipped material is to pass through one or two centers in its movement from source to destination, the applicable unit transport cost changing with each such passage

.

(3) W. A. Horn established the extendibility, of a geodesic metric on the 1-skeleton of a simplicial complex (e.g., a graph imbedded in the plane), to the whole complex.

Publications

(l) J. Edmonds and D.R. Fulkerson (RAND Corporation). Bottleneck extrema. To appear in

J. Combinatorial Theory.

LINEAR AND NON-LINEAR PROGRAMMING

Task 205-12-2050153

11540

Origin and Sponsor: NBS Authorized 12/30/60

Manager: Jack Edmonds

Full task description: October- December i960 issue, p. 3

Status: CONTINUED.

A.J. Goldman and P.R. Meyers continued studies on the minimax error selection of incompletely

specified discrete probability distributions, with error given by the L norm. Attention focussed

on the bivariate case with given margins; this situation was shown to admit a linear programming

formulation which may permit a "tailored" solution algorithm.

Publications

(1) A.J. Goldman. The minimax transportation problem. To appear in Transportation Science

(Letters to Editor).

(2) A.J. Goldman. Minimax error selection of a discrete univariate distribution with prescribed

componentwise ranking. Journal of Research NBS, 72B (1968), pp. 273-277-

(3) A.J. Goldman and P.R. Meyers. Minimax error selection of a discrete univariate distribution

with prescribed componentwise bounds. Journal of Research NBS 72B (1968), pp. 263-271.

MATHEMATICAL METHODS FOR HIGH SPEED GROUND TRANSPORTATION STUDY

Task 205-12-2058456

51550

Origin: Technical Analysis Division, NBS

Sponsor: Northeast Corridor Transportation Project, Dept, of Transportation

Manager: A.J. Goldman

Full task description: January-June 1966 issue, p. 14

(l) J. Gilsinn, A.J. Goldman, W.A. Horn and P. Saunders prepared memoranda on various phases of the Northeast Corridor Transportation Project, including reviews of several contractors' reports.

4
(2) P. Saunders continued work on improving and implementing mathematical scheduling methods.

Several versions for high-speed trains are now available on the NBS computer. An adaptation to a VTOL/STOL context was made, and is now in routine use by MITRE in its study of this mode for the Corridor Project.

(3) J. Gilsinn, A.J. Goldman and L.S. Joel (with members of Division 431) are involved in the development of improved models for predicting demand by mode for intercity passenger movement. An important aspect of this work is the identification of appropriate criteria for testing such models.

Estimation of parameters is being carried out using (a) OMNITAB for linearized models, (b) a program

written by W.G. Hall and P. Saunders for non-linear least-squares estimation in log-linear models,

and (c) a general non-linear regression procedure (Marquardt) made operational at NBS by B.L. Joiner

(205.03).

(4) J. Levy and M.H. Pearl continued the development of methods for evaluating feedback vs. non- feedback methods for regulating flow at a merge point in a transport network. Several working

papers on this subject have been prepared.

(5) J. Gilsinn continued computational experiments for the comparative evaluation of shortestpath algorithms. Three of the methods tested, on both real and synthesized graphs have performed

significantly better than the others.

(6) W.A. Horn studied the problem of optimally providing cross-ties between roughly parallel

routes to achieve a connected network.

Publications

(1) W.A. Horn. Optimal networks connecting N points in the plane. To appear in Proceedings of

Fourth International Symposium (Karlsruhe, Germany; June, 1968) on Traffic Flow.

(2) W.A. Horn. Minimum- length covering by intersecting intervals. To appear in Journal of Research

NBS, 73B (1969).

5
3- PROBABILITY AND MATHEMATICAL STATISTICS

RESEARCH IN PROBABILITY AND MATHEMATICAL STATISTICS

Task 20503-12-2050131/63-1259

11550

Origin: NBS Authorized 10/1/62

Manager: Joan Raup Rosenblatt

Full task description: July - December 1962

Status: CONTINUED. Two linear least squares test problems, both fifth degree polynomials, have

been run by Roy H. Wampler on more than twenty different computer programs in order to assess their

numerical accuracy. Among the programs tested were representatives from various widely used

statistical packages,- some of these were highly inaccurate in comparison to good programs (such as NBS's ORTHO) that have been available for many years. Mr. Wampler's paper reports a number of programs that yielded results (coefficients) that were completely erroneous, containing not even one correct significant digit.

Essentially five different algorithms were used in the various programs to obtain the > coefficients

of the least squares fits. The tests were run on several different computers, in double precision

as well as single precision. By comparing the coefficients reported, it was found that those programs using orthogonal Householder transformations or Gram-Schmidt orthonormalization were much more

accurate than those using elimination algorithms. Programs using orthogonal polynomials (suitable

only for polynomial fits) also proved to be superior to those using elimination algorithms.

Morris Newman's program, using congruential methods and integer arithmetic, was used to obtain

exact solutions to approximate problems with rounded input. The paper includes detailed documentation

of the results, including the sources of programs and the computers used.

H. H. Ku has served as editor of Volume I, Statistical Concepts and Procedures, now in press as part

of the twelve-volume NBS Special Publication 300, Precision Measurement and Calibration . This

44l-page volume collects, for easy reference, reprints of 40 selected articles, mainly by Bureau

authors, on topics of special importance to metrologists. Each of six sections of the volume is

preceded by a foreword pointing up the purpose for which the papers are included. Section titles

are: (l) The measurement process, precision, systematic error, and accuracy; (2) Design of experiments in calibration; (3) Interlaboratory test; (4) Functional relationships; (5) Statistical

treatment of measurement data; ( 6 ) Miscellaneous topics. A list of references and an extensive

subject index are provided.

Ruth Varner prepared two FORTRAN V programs, KKV68 -A and KKV68 -B, for the analysis of four-way contingency tables using the information approach (Ku and Kullback 1968 ). The dimension limitations

of the table and memory location requirements of the computer are as follows; Memory

T location

4 48818

3 21257

R _C_ D

KKV68 A <9 19 9 KXV68 B <7 9 4

Publications:

(1) The distribution of the sample variance from a two-point binomial population. D. Hogben.

American Statistician , Vol. 22, No. 5> December 1968 .

(2) The mean deviation and range for n = 3. B. L. Joiner and J. R. Rosenblatt. American

Statistician , Vol. 22, No. 4, October 1968 .

(3) Expressions of imprecision, systematic error, and uncertainty associated with a reported value.

H. H. Ku. M and D Measurements and Data , Vol. 2, No. 4, July-August 1968, pp. 72-77-

(4) Interaction in multidimensional contingency tables: An information theoretic approach. H. H. Ku. and S. Kullback (G.W.U.). J. Res. NBS - Math. Sciences, Vol. 72B, No. 3, July-Dee. 1968, pp. 159-199.

6
(5) The median significance level and other small sample measures of test efficacy. B. L. Joiner.

Submitted to a technical journal.

(6) Precision Measurement and Calibration , Vol. 1 - Statistical Concepts and Procedures. H. H. Ku,

ed. NBS Spec. Pub. 300, in press.

(7) An application of minimum discrimination information estimation to a problem of Grizzle and

Berkson on th_- test of "no interaction" hypothesis. H. H. Ku and S. Kullback (G.W.U.).

Submitted to a technical journal.

(8) Approximating discrete probability distributions. H. H. Ku and S. Kullback (G.W.U.). Submitted

to a technical journal.

DEVELOPMENT OF "OMNITAB"

Task 20503-12-2050131

11550

Origin and Sponsor: NBS Authorized ll/l/68

Managers: David Hogben, Sally T. Peavy

Objective: To develop and conduct research on, techniques of improving OMNITAB, for the purpose

of adding efficiency to computing effort of scientists at NBS and elsewhere. This .ncludes, in particular, development and addition of new instructions; action on user's comments; expanding the

size of the worksheet; adding capability for handling tape, double precision and user's own sub- routines; revision and refinement of existing instructions.

Background: There is a great need for easy, fast and effective access to the computer by

scientists who are not programmers. OMNITAB is a general-purpose computer program for data,

statistical, and numerical analysis which is designed to meet this need. The value of OMNITAB has

been demonstrated by its adoption at various government agencies, universities and private concerns, but further development is highly desireable. Standards ("tentative") of computing practice are needed and OMNITAB has attempted to set these standards.

The first version of OMNITAB was written specifically for the IBM 7090/709^, partly in machine

language. The guiding philosophy was conceived and developed by Joseph Hilsenrath of the NBS Equation of State Section, and most of the programming was done by members of the NBS Computation

Laboratory. The operation of the program is described in NBS Handbook 101, OMNITAB : A Computer

Program for Statistical and Numerical Analysis (1966) prepared by Hilsenrath together with

Guy G. Ziegler, Carla G. Messina, Philip J. Walsh, and Robert J. Herbold.

Major innovations and improvements were developed by Walter J. Gilbert, NBS Computer Services

Division, who initiated the conversion of OMNITAB to ASA FORTRAN for the UNIVAC 1108. The present

relatively machine -independent version is largely the work of Sally T. Peavy and Ruth N. Varner

of the Statistical Engineering Laboratory.

Status: NEW. Primary responsibility for development and maintenance of the OMNITAB program now

resides in the Statistical Engineering Laboratory. Earlier work on this task has been reported

under Task 63-1259* Research in Probability and Mathematical Statistics.

Sally T. Peavy and Ruth N. Varner have finished rewriting OMNITAB in ASA standard FORTRAN to make

it virtually machine independent. The NBS version was put in the absolute mode giving a sub- stantial reduction in computing cost. Several instructions have been added to the program. To

increase the size of the worksheet from 5*000 to 12,500 and to make possible the addition of more instructions, overlay and segmentation features have been developed. The number of instructions

that can now be added is essentially unlimited.

J. M. Cameron, S. T. Peavy and R. N. Varner made major revisions in the instructions FIT and P0LYFIT. Several additional statistics can now be stored automatically and the amount of storage can be controlled. Items added to the automatic printout include, in particular, the standard deviation

of predicted values, analysis of variance and an assessment of computing accuracy. The last item

represents an important step forward in computer practice.

7
Also, the set of matrix instructions was substantially improved and increased in number. In particular, a new instruction was written which gives about twenty properties of a matrix (or

array) including the trace, sum of squares and orthogonality.

To keep users informed of latest developments, notices are now printed at the end of each OMNITAB computer run giving corrections, additions and improvements incorporated in the current version.

Thirty-five user's comments were carefully studied and many of them resulted in improvements which

were incorporated in the OMNITAB program.

David Hogben wrote two subroutines which enable the printing of numbers in a form more readable than

heretofore available. Unlike existing methods, these subroutines make it possible for the data to determine the "best" format. An initial version of a new instruction, RPRINT, with several options,

incorporates these subroutines. In addition, a new instruction, TWOWAY, was constructed which

easily gives an analysis of variance for a balanced or unbalanced two-way table, including Tukey's

test for non -additivity in the balanced case.

8
4. STATISTICAL ENGINEERING SERVICES

COLLABORATION ON STATISTICAL ASPECTS OF

NBS RESEARCH AND TESTING

Task 13911-61-2050950/51-1

99500

Origin: NBS Authorized 7/1/50

Managers: J. M. Cameron, H. H. Ku, J. R. Rosenblatt

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

Status: CONTINUED. This is a continuing project involving cooperation with other Bureau scientists

on the statistical aspects of their investigations. These services vary from short (one-hour)

sessions to extended collaborations involving several man-months; and are concerned primarily with

statistical design of experiments, analysis and interpretation of data, and the use of computers in statistical analysis of data. Typical examples of the services performed are the following.

Janace A. Speckman is collaborating with Thomas E. Wells of the Resistance and Reactance Section on studies leading to identification of components of variance in the data generated by intercomparisons

of standard resistors. Estimates of these variance components are needed for comparative evaluation

of alternative experiment designs that have been proposed for use in routine intercomparisons.

Brian L. Joiner participated in an international Workshop on Mass Spectrometric Analysis of Solids,

sponsored by the NBS Spectrochemical Analysis Section, on November l8 and 19, 1968. Topic 1 (of

5 discussed) was "The Accuracy and Precision Problem," and the discussion of this topic was opened

by an informal talk on "The role of statistics," presented by Dr. Joiner. There were about 100 participants in the workshop, including 10 from 6 foreign countries.

Joan R. Rosenblatt is collaborating with H. S. Lew and E. 0. Pfrang of the Structures Section (NBS

Building Research Division) on development of methods for analysis of data from fatigue-test

experiments on composite steel-and -concrete beams.

The NBS Office of Standard Reference Materials, and a number of sections in the Analytical Chemistry

Division, are being assisted by Brian L. Joiner in the development of experimental designs and

statistical analyses appropriate for evaluating the extent of any inhomogeneity in the material.

Similar statistical problems arise in a number of areas where a nominal value and some uncertainty

limits must be reported for the material, e.g. on the certificate that accompanies a Standard

Reference Material issued by the Bureau.

Publications:

(1) The statistical consultant in a scientific laboratory. J. M. Cameron. To appear in Technometrics.

(2) Interlaboratory comparison of the potential heat test method. D. Gross (421.02) and

M. G. Natrella. Submitted to a technical journal.

(3) A survey of blemishes on government microfilm. C. S. McCamy (212.13), R. S. Wiley (Army

Map Service), and J. A. Speckman. To appear in J. Res. NBS - Phys. and Chem. , Jan. -Feb. 1969.

9
STATISTICAL SERVICES

Task 20503-40-2050132/58-346

11550

Origin and Sponsors: Various Agencies Authorized 3/31/58

Manager: J. M. Cameron, J. R. Rosenblatt

Full task description: January - March 1958 issue, p. 45

Status: CONTINUED. This is a continuing project which involves providing, upon request, statistical

services to other governmental agencies, universities, industrial organizations, and other non- governmental agencies. Approximately 30 such. requests are handled per month ranging from short conferences to collaboration involving several days work.

A three-day Seminar on Precision and Accuracy in Measurement and Calibration was presented at the

West Coast University in Orange, California, to 37 metrologists from industrial and government

laboratories. The seminar had been presented four times in Washington (to a total of 136 participants) and its presentation on the West Coast was in response to requests for a presentation

there in the interest of savings in travel expense. The presentation was sponsored by the National

Conference of Standards Laboratories with the cooperation of the West Coast University which made

facilities available at no charge.

The seminar dealt with the concept of a National Measurement System as developed by R. D. Huntcon

and the role of the National Bureau of Standards in the system (presented by J. M. Cameron) and with the approach to measurement as a production process and the description of precision and accuracy in terms of parameters of that process. (R. C. Raybold).

Lectures were presented on Statistics of Control of a Measurement Process (B. L. Joiner), Problems

of Interlaboratory Measurement Agreement (j. M. Cameron), Propagation of Error and Representation

of Precision and Accuracy (B. L. Joiner), and Design of Experiments (j. M. Cameron).

The participation of personnel from Bureau laboratories apart from the Statistical Engineering

Laboratory added considerably to the value of the program. Woodward G. Eicke of the Electrochemistry

section presented some new methods for intercomparing groups of voltage standards and maintaining

surveillance over the measurement process involved. Robert C. Raybold of the Office of Measurement

Services presented an account of measurement problems in mass and in photometry and steps in the

evaluation of parameters of those processes. Paul E. Pontius of the Mass and Volume Section presented a paper on Measurement from a Systems Viewpoint and the concluding lecture of the seminar was his account of the Pilot Program in mass and the demonstration of compatible processes in 17 major laboratories.

Joan R. Rosenblatt has completed a three-year term of office as Program Secretary of the Institute

of Mathematical Statistics.

10
5. SYSTEMS DYNAMICS SECTION

RESEARCH IN MATHEMATICAL PHYSICS

AND RELATED FIELDS

Task 20540-12-2050141/55-57

11540

Origin: NBS Authorized 9/1/54

Manager: H. J. Oser

Full Task description: July-September 1954 issue, p. 27

Research has been conducte^ in the area of multicomponent diffusion and heat transfer. Analytical solutions have been formulated for the system of equations describing these phenomena that include i) the coupled flux expressions postulated in irreversible thermodynamics, ii) chemical reactions described by

linear rate laws, and iii) time dependent diffusion and kinetic parameters. A stable numerical tech- nique for the transport contribution has been developed that steps each equation independently through

time according to the associated eigenvalue in the transport coefficient matrix.

The role of thermodynamic coupling in gas absorption systems has been investigated by studying the characteristics of a physical model of a quinary system that include i) the coupled flux expressions, ii) heats of absorption released at the gas-liquid interface, and iii) an equilibrium

relation between the surface concentrations and temperature.

Multicomponent fgiction coefficients have been completely defined on the basis of the entropy generation

due to diffusion . Symmetry in these coefficients has been shown to be both necessary and sufficient

for the Onsager reciprocal relation to be valid. A method for calculating the coefficients from diffusion

and thermodynamic data obtained in arbitrary composition scales and velocity frames has been derived.

Two methods of analyzing diaphragm cell data have been devised that do not impose additional mathematical

or theoretical restrictions on the experimental technique for measuring multicomponent diffusion coefficients. The methods are not restricted by the number of species in the mixture and a general analysis

of the experimental error has been developed.

A discussion of thermodynamic coupling was presented at the National Meeting of the American Institute

of Chemical Engineers, Tampa, Florida, May 19-22, 1968.

In collaboration with Dr. R. deWit of the Lattice Defects and Microstructure Section (312.03), J. T. Fong continued to study the geometric foundation of a class of continuum theories of dislocations. In particular, a manuscript entitled "On existence of shifters in a general geometric manifold and their

role in a class of continuum theory of dislocations" is being prepared for submittal to a technical

journal

.

In collaboration with L. Zapas of the Rheology Section (213.05), J. T. Fong began a study of the propa- gation of viscoelastic waves in a semi-infinite medium obeying the constitutive equations of a BernsteinKearsley-Zapas fluid. The possibility of deriving the elastic potential for a real material from a molecular theory of polymer solution with some measure of polymer interactions is being explored.

A paper by J. C. Smith of the Polymer Interface Section (311.07) and J. T. Fong entitled "On the coupling

of longitudinal and transverse waves in a linear three-element viscoelastic string subjected to trans- verse impact” appears in the July-September issue of the NBS J. Research, Vol. 72B, No. 3 (1968).

R. S. Kraft directed his research towards the following three areas: (a) A technique due to Garabedian

of investigating elliptic systems of partial differential equations by converting them into equations

of hyperbolic type has been simplified and generalized. The generalized technique was used to obtain

analyticity and reflectivity properties of first order elliptic systems. A paper containing these re- sults is being submitted for publication. (b) A formal procedure for constructing Riemann functions by the method of generalized progressing wave expansions has been obtained. The convergence properties of

the wave expansions are under investigation. (c) Numerical solution of the Nernst-Planck equations are being studied in collaboration with Professor David Goldman of Womens Medical College, Philadelphia,

Pennsylvania

.

11
W. L. Sadowski and Z. Ruthberg continued the analysis of Hermite polynomial series for 1) the two- stream initial condition and 2) a temperature rise in a plasma. A computer program was written to determine the number of terms needed in each of these cases to express the function in the form of a truncated series within a specified error.

An asymptotic expansion for the Hermite polynomials yielded closed forms for the expansion coefficients,

and an estimate for the neglected terms as a function of N in each case. A spot check of the computer

results indicated that the error bounds found analytically are consistent with the numerical results.

Routine consultations by other members of the staff involved the computation of special functions, the solution of a specified problem in the theory of helium lasers and continuation of the cooperation with

the Rheology Section on a problem in propagation of acoustic waves.

Publications

(1) An Analysis of Nonisothermal Multicomponent Diffusion with Chemical Reaction. George B. DeLancey

(with S. H. Chiang, University of Pittsburgh). Submitted to Chem. Eng. Sci.

(2) Analysis of Multicomponent Diaphragm Cell Data. George B. DeLancey. Submitted to J. Phys . Chem.

(3) Friction Coefficients in Multicomponent Systems. George B. DeLancey. Submitted to Xnd . Eng. Chem. Fundam.

(4) On the Coupling of Longitudinal and Transverse Waves in a Linear Three-Element Viscoelastic String

Subjected to Transverse Impact. J. C. Smith (311.07, NBS) and J. T. Fong. July-Sept. 1968, NBS J. Res., Vol . 72B , No. 3.

12
5. SYSTEMS DYNAMICS SECTION

RESEARCH IN MATHEMATICAL PHYSICS

AND RELATED FIELDS

Task 20540-12-2050141/55-57

11540

Origin: NBS Authorized 9/1/54

Manager: H. J. Oser

Full Task description: July-September 1954 issue, p. 27

Research has been conducte^ in the area of multicomponent diffusion and heat transfer. Analytical solutions have been formulated for the system of equations describing these phenomena that include i) the coupled flux expressions postulated in irreversible thermodynamics, ii) chemical reactions described by

linear rate laws, and iii) time dependent diffusion and kinetic parameters. A stable numerical tech- nique for the transport contribution has been developed that steps each equation independently through

time according to the associated eigenvalue in the transport coefficient matrix.

The role of thermodynamic coupling in gas absorption systems has been investigated by studying the characteristics of a physical model of a quinary system that include i) the coupled flux ex- pressions, ii) heats of absorption released at the gas-liquid interface, and iii) an equilibrium

relation between the surface concentrations and temperature.

Multicomponent fgiction coefficients have been completely defined on the basis of the entropy generation

due to diffusion . Symmetry in these coefficients has been shown to be both necessary and sufficient

for the Onsager reciprocal relation to be valid. A method for calculating the coefficients from diffusion

and thermodynamic data obtained in arbitrary composition scales and velocity frames has been derived.

Two methods of analyzing diaphragm cell data have been devised that do not impose additional mathematical

or theoretical restrictions on the experimental technique for measuring multicomponent diffusion coefficients. The methods are not restricted by the number of species in the mixture and a general analysis

of the experimental error has been developed.

A discussion of thermodynamic coupling was presented at the National Meeting of the American Institute

of Chemical Engineers, Tampa, Florida, May 19-22, 1968.

In collaboration with Dr. R. deWit of the Lattice Defects and Microstructure Section (312.03), J. T. Fong continued to study the geometric foundation of a class of continuum theories of dislocations. In particular, a manuscript entitled "On existence of shifters in a general geometric manifold and their

role in a class of continuum theory of dislocations" is being prepared for submittal to a technical

journal

.

In collaboration with L. Zapas of the Rheology Section (213.05), J. T. Fong began a study of the propa- gation of viscoelastic waves in a semi-infinite medium obeying the constitutive equations of a BernsteinKearsley-Zapas fluid. The possibility of deriving the elastic potential for a real material from a molecular theory of polymer solution with some measure, of polymer interactions is being explored.

A paper by J. C. Smith of the Polymer Interface Section (311.07) and J. T. Fong entitled "On the coupling

of longitudinal and transverse waves in a linear three-element viscoelastic string subjected to trans- verse impact" appears in the July-September issue of the NBS J. Research, Vol. 72B, No. 3 (1968).

R. S. Kraft directed his research towards the following three areas: (a) A technique due to Garabedian

of investigating elliptic systems of partial differential equations by converting them into equations

of hyperbolic type has been simplified and generalized. The generalized technique was used to obtain

analyticity and reflectivity properties of first order elliptic systems. A paper containing these re- sults is being submitted for publication. (b) A formal procedure for constructing Riemann functions by the method of generalized progressing wave expansions has been obtained. The convergence properties of the wave expansions are under investigation. (c) Numerical solution of the Nernst-Planck equations are being studied in collaboration with Professor David Goldman of Womens Medical College, Philadelphia,

Pennsylvania

.

11
W. L. Sadowski and Z. Ruthberg continued the analysis of Hermite polynomial series for 1) the two- stream initial condition and 2) a temperature rise in a plasma. A computer program was written to determine the number of terms needed in each of these cases to express the function in the form of a truncated series within a specified error.

An asymptotic expansion for the Hermite polynomials yielded closed forms for the expansion coefficients,

and an estimate for the neglected terms as a function of N in each case. A spot check of the computer

results indicated that the error bounds found analytically are consistent with the numerical results.

Routine consultations by other members of the staff involved the computation of special functions, the solution of a specified problem in the theory of helium lasers and continuation of the cooperation with

the Rheology Section on a problem in propagation of acoustic waves.

Publications

(1) An Analysis of Nonisothermal Multicomponent Diffusion with Chemical Reaction. George B. DeLancey

(with S. H. Chiang, University of Pittsburgh). Submitted to Chem. Eng. Sci.

(2) Analysis of Multicomponent Diaphragm Cell Data. George B. DeLancey. Submitted to J. Phys . Chem.

(3) Friction Coefficients in Multicomponent Systems. George B. DeLancey. Submitted to Ind . Eng.

Chem. Fund am.

(4) On the Coupling of Longitudinal and Transverse Waves in a Linear Three-Element Viscoelastic String

Subjected to Transverse Impact. J. C. Smith (311.07, NBS) and J. T. Fong. July-Sept. 1968, NBS J. Res., Vol . 72B, No. 3.

12
PLASMA PHYSICS

Task 20504-12-2050140/59-442

11540

Origin: NBS Authorized 10/13/61

Manager: H. J. Oser

Full task description: October-December 1961 issue, p. 12

The work on the combinatorial approach to the solution of nonlinear partial differential equations was continued. A manuscript has been completed on general combinatorial properties of binomial nonlinear

differential operators. Determination of the coefficients with the aid of the LISP language (a list

processor) indicated that this language is too cumbersome to handle a general combinatorial problem

of this kind. An analytic approach permits calculating the combinatorial weights starting from a much

higher level than was possible with a straightforward repetitive LISP program.

The combinatorial method was specifically applied to the nonlinear Vlasov equation and the problem is now in process of being programmed for the computer. The advantage of the special method is that only

a few physically significant vectors are calculated as functions of time in the eigenfunction solution.

A manuscript has been completed on the possibility of selective excitation of harmonics by counterstreaming beams of electrons travelling through a plasma. The necessary collimation in velocity space

can be obtained by a truncated set of Hermite polynomials. The coefficients of this expansion were obtained in closed form by the use of transformation properties of Hermite polynomials. Computer

solutions of the expansion are in good agreement with the analytical result. The analytically obtained

truncation error is a realistic upper bound to the one obtained numerically. The possibility of con- tinuous transformation of the base function in velocity space was examined and the transformation

coefficients derived. The results show, however, that truncation errors in the expansion would not be significantly reduced for the values of parameters used in the study.

13
BIOMEDICAL IMAGE AND LANGUAGE PATTERN PROCESSING

Task 20500-12-2050404

51560

Origin: NBS Authorized l/2l/64

Sponsor: National Institutes of Health

Manager: Russell A. Kirsch

Full task description: January-June 1964 issue, p. 19

Status: CONTINUED. A general procedure was developed on the Q32 computer for automatic segmentation

of quantized images. This procedure yields as output a partial ordering (induced by the inclusion

relation) of all morphological objects in an image. The elements of this partial ordering serve as inputs for a higher level syntactic analysis . This procedure was applied to a cortex section photograph and to a sequence of optical serial section photographs of a single neuron. A serial section

reconstruction was also made for the single neuron. Considerable expansion of the LISPAX image

processing system took place during this period. The programmable microscope stage and focus

control was connected to the LINC-PDP-8 computer and used in conjunction with a spectrophotometer

to control the stage and focus while doing photometric measurements

.

SCANNING MICROSCOPE PROCESSING

Task 20500-2050L08

51560

Origin: NBS Authorized 6/18/68

Sponsor: National Institutes of Health

Manager: Russell A. Kirsch

Objective: To study ways of using a programmable scanning microscope with a general purpose digital

computer. The computer is to have facilities for image processing and for scanner control. The use

of the scanner-computer complex is in automating auto-radiographic analysis of tritium-labelled blood

cells

.

Status . NEW . Five kinds of leucocytes were scanned from photographs made at several optical

wavelengths, l'nese quantized images were analyzed on the Q32 computer. The optical frequency selection was made on the programmable microscope using computer control of a monochromator light source. Some modifications were made to the electronics of the microscope scanner controls

.

CHEMICAL BIOLOGICAL INFORMATION PROCESSING

Task 20500-20504-10

51560

Origin: NBS Authorized 6/27/68

Sponsor: National Institutes of Health

Manager: Russell A. Kirsch

Objective: To consult on the use of remote computers in communicating pharmacological information

regarding chemical structure and biological activity among a group of collaborating pharmacologists.

Status: NEW . Advisory work continued on Chemical Biological Information Handling Techniques.

The use of the Q32 facility was provided to the sponsor for file handling experiments.

14
MATHEMATICAL TABLES

Task 20500-40-2050121/57-216

11550

Origin and Sponsor: UBS Manager: I. A. Stegun

Full task description: July-December 1964 issue, p. 4.

Status. CONTINUED. The updating of "bibliographic material and correction lists continued for all

published volumes . A seventh printing of AMS 55 - "the Handbook of Mathematical Functions was re- leased in May 1968 .

Assistance at the rate of ten consultations per month to NBS, other government agencies, industry

and universities was offered in the evaluation or modification of computing techniques, identification

of functions and their tabulation.

In response to a request for obtaining eigenvalues -eigenvectors of a non-symmetric matrix a FORTRAN program published in the Communications of the Association for Computing Machinery was explored. It was found satisfactory for the given cases but failed for other test cases. I. A. Stegun and

R . Zucker have continued updating a file of test matrices . Routines recommended for general library usage must supply satisfactory results for the present file.

I. A. Stegun and R. Zucker have started on a more systematic evaluation of methods of computing

special functions . Concentration at present is centered on accuracy, and feasible range of power

series and asymptotic expansions.

15
Lectures and Technical Meetings

HOADLEY , A. B.

EDMONDS, J.

ROSENBLATT, J. R.

HABER, S.

CLATWORTHY, W. H.

BERMAN, Mones

PLOTKIN, A.

HEARON, John Z.

OLVER, F. W. J.

GOLDBERG, Karl

HALL, William G.

NATRELLA, Mary G.

NEWMAN, Morris

VARNER, Ruth N.

CAMERON, J. M.

EDMONDS, J. R.

EDMONDS, J. R.

Mathematical Statistics Seminar

(Bell Telephone Laboratories) A Bayesian approach to the inverse

regression problem. September 30.

Mathematics Division Expository Seminar

Optimum branchings. July 24.

Distribution-free statistical tests for comparative experiments.

August 14.

Multiple integration using classical and Monte Carlo approaches.

September 25.

(SUNY Buffalo and Univ. of No. Carolina) Isomorphisms and non- isomorphisms in balanced incomplete block designs. November 13.

Systems Dynamics Section Seminar

(National Institutes of Health) SAAM program and its uses in the study

of metabolic and kinetic systems. October 9.

(University of Maryland) A numerical solution for the laminar wake

behind a finite flat plate. November 13.

(National Institutes of Health) Generalized inverses and linear

systems of equations. December 11.

Colloquia - Special Topics in Mathematics

Asymptotic Expansions.

In-Hours Courses. Fall Semester

Introduction to Abstract Algebra

Analysis II

Statistics of Measurement

Selected Topics in Group Theory and Group Representations

FORTRAN Programming for Beginners

Papers and Invited Talks

Presented by Members of the Staff

At Meetings of Outside Organizations

The statistical consultant in a scientific laboratory. American

Statistical Assoc., Pittsburgh, August 22.

Matroid intersection. Cornell University, November 26.

Matroid intersection. University of Michigan, December 9.

l6
GOLDMAN, A. J. Minimaximax selection of incompletely specified probability distributions. Graduate School of Business, Information, Decision and

Control Workshop. University of Chicago, Chicago, 111., October 24.

The minimax transportation problem. Operations Research Society of America. Philadelphia, Pennsylvania, November 6.

GOLDMAN, A. J. MEYERS, P. R. Minimax error selection of a discrete univariate distribution.

Operations Research of America. Philadelphia, Pennsylvania, November 6

HOGBEN, D. Approximations to control chart constants for electronic computers.

Fourteenth Conference on the Design of Experiments in Army Research,

Development and Testing, Edgewood Arsenal, Md., October 23.

JOINER, B. L. Some properties of the range in samples from Tukey's lambda-distributions. American Statistical Association, Pittsburgh, August 22.

JOINER, B. L. OMNITAB in 3 lectures:

1. Introduction, Sept. 16.

2. Intermediate, Sept. 26.

3. Advanced, October 4.

The role of statistics. Workshop on Mass Spectrometric Analysis of Solids, NBS, November 18.

KU, H. H. Minimum discrimination information estimation of cell probabilities

in a contingency table - applications. Institute of Mathematical

Statistics, Madison, August 29.

Analysis of multidimensional contingency tables. Fourteenth Conference

on the Design of Experiments in Army Research, Development and Testing,

Edgewood Arsenal, Md., October 23.

WAMPLER, R. H. An evaluation of linear least squares computer codes. Fourteenth

Conference on the Design of Experiments in Army Research, Development

and Testing, Edgewood Arsenal, Md., October 23.

17
PUBLICATIONS ACTIVITIES

1.0 PUBLICATIONS THAT APPEARED DURING THIS PERIOD

1.1 Mathematical Tables

Handbook of Mathematical Functions, AMS 55, Seventh Printing (with corrections), May 1968.

1.2 Monograph

Matrix representations of groups. M. Newman. AMS 60, July 1968.

1.3 Technical Papers

On the coupling of longitudinal and transverse waves in a linear three-element viscoelastic

string subjected to transverse impact. J. C. Smith (Polymer interface Section, 311.07) and

J. T. Fong. J. of Research NBS , 72B , No. 3 (1968).

Minimax error selection of a discrete univariate distribution with prescribed componentwise

ranking. A. J. Goldman. J. of Research NBS, 72B . pp. 273-277 (1968).

Minimax error selection of a discrete univariate distribution with prescribed componentwise

bounds. A. J. Goldman and P. R. Meyers. J. of Research IBS, 72B . pp . 263-271 (1968).

A combination of classical and Monte Carlo methods for evaluating multiple integrals.

S. Haber. Bull. A.M.S. 74, 683-686 (1968).

The distribution of the sample variance from a two-point binomial population. D. Hogben.

American Statistician. Vol. 22, No. 5, December 1968.

The mean deviation and range for n = 3. B. L. Joiner and J. R. Rosenblatt. American

Statistician , Vol. 22, No. 4, October 1968.

Citation searching and bibliographic coupling with remote on-line computer access. F. L. Alt (AIP) and R. A. Kirsch. J. of Research NBS, 72B , pp. 61-78 (1968).

Expressions of imprecision, systematic error, and uncertainty associated with a reported

value. H. H. Ku. M and D Measurements and Data . Vol. 2, No, 4, July-August 1968, pp

.

72-77.

Interaction in multidimensional contingency tables: An information theoretic approach.

H. H. Ku and S. Kullback (G.W.U.). J. of Research NBS, 72B, No. 3, pp . 159-199,

(December 1968)

.

Maximal normal subgroups of the modular group. M. Newman. Proc. Amer. Math. Soc. 19,

1138-1144 (1968).

2.0 MANUSCRIPTS IN THE PROCESS OF PUBLICATION

2.3 Technical Papers

The statistical consultant in a scientific laboratory. J. M. Cameron. To appear in Technometrics

.

Friction coefficients in multicomponent systems. George B. Delancey. Submitted to Industrial and Engineering Chemistry, Fundamentals.

Role of coupling in nonisothermal diffusion: Gas ^absorption. George B. Delancey and

S. H. Chiang (Univ. of Pittsburg.) To be submitted to a technical journal.

An analysis of nonisothermal multicomponent diffusion with chemical reaction. George B. Delancey and S. H. Chiang (Univ. of Pittsburgh). Submitted to a technical journal.

Bottleneck extrema. J. Edmonds and D. R. Fulkerson (RAND Corporation). To appear in J. Combinatorial Theory.

Stochastic quadrature formulas. S. Haber. To appear in Math. Comp.

18
On a theorem of Piatetsky-Shapiro and evaluation of multiple integrals. S. Haber and C. F. Osgood. To appear in Math. Comp.

On the sum 2)Can> S. Haber and C. F. Osgood. Submitted to a technical journal.

Optimal networks connecting N points in the plane. W. A. Horn. To appear in Proceedings

of Fourth International Symposium on Traffic Flow (Karlsruhe, Germany^ June, 1968).

Minimum- length covering by intersecting intervals. W. A. Horn. To appear in J. of Research NBS , 73B (1969).

The median significance level and other small sample measures of test efficacy. B. L. Joiner. Submitted to a technical journal.

An unusual mixed initial boundary problem in two independent variables. Richard Kraft.

Submitted to J. Math. Analysis and Applications, Academic Press, Inc.

Precision Measurement and Calibration, Vol. 1 - Statistical Concepts and Procedures.

H. H. Ku, ed. NBS Spec. Pub. 300, in press.

An application of minimum discrimination information estimation to a problem of Grizzle

and Berkson on the test of "no interaction" hypothesis. H. H. Ku and S. Kullback (G.W.U.).

Submitted to a technical journal.

Approximating discrete probability distributions. H. H. Ku and S. Kullback (G.W.U.).

Submitted to a technical journal.

On the multipliers of the Dedekind modular function. J. Lehner. To appear in J. of Research NBS.

Interlaboratory comparison of the potential heat test method. D. Gross (421.02) and M. G. Natrella. Submitted to a technical journal.

Some results on unitary matrix groups. M. Newman and M. Marcus. To appear in J. of Linear Algebra.

Subgroups of SL(t,z). M. Newman. To appear in J. of Research NBS.

Error bounds for the Laplace approximation for definite integrals. F. W. J. Olver. To appear in J. of Approx. Theory.

A method in diophantine approximation IV. C. F. Osgood. To appear in Acta Mathematics

(Warsaw, Poland).

A survey of blemishes on government microfilm. C. S. McCamy (212.13), R. S„ Wiley (Army

Map Service), and J. A. Speckman. To appear in J. of Research NBS - Phys. and Chem.,

Jan. -Feb. 1969.

2.4 Notes

Analysis of multicomponent diaphragm cell data. George B. Delancey. Submitted to Journal

of Physical Chemistry.

The minimax transportation problem. A. J. Goldman. To appear in Transportation Science

(Letters to Editor).

Fractional container-loads and topological groups. A. J. Goldman. To appear in Operations

Research.

2.5 Books

Boundary Layers of Flow and Temperature, by Alfred Walz , Translated by Hans J. Oser. M.I.T.

Press, (in press).

19

NBS TECHNICAL PUBLICATIONS

PERIODICALS

JOURNAL OF RESEARCH reports National

Bureau of Standards research and development in physics, mathematics, chemistry, and engineering.

Comprehensive scientific papers give complete details of the work, including laboratory data, experimental

procedures, and theoretical and mathematical analyses. Illustrated with photographs, drawings, and

charts.

Published in three sections , available separately: • Physics and Chemistry

Papers of interest primarily to scientists working in these fields. This section covers a broad range of physical and chemical research, with major emphasis on standards of physical measurement, fundamental

constants, and properties of matter. Issued six times

a year. Annual subscription: Domestic, $6.00; for- eign, $7.25*.

• Mathematical Sciences

Studies and compilations designed mainly for the

mathematician and theoretical physicist. Topics in mathematical statistics, theory of experiment design,

numerical analysis, theoretical physics and chemistry, logical design and programming of computers

and computer systems. Short numerical tables. Issued quarterly. Annual subscription: Domestic,

$2.25; foreign, $2.75*. • Engineering and Instrumentation

Reporting results of interest chiefly to the engineer

and the applied scientist. This section includes many

of the new developments in instrumentation resulting

from the Bureau’s work in physical measurement,

data processing, and development of test methods.

It will also cover some of the work in acoustics,

applied mechanics, building research, and cryogenic

engineering. Issued quarterly. Annual subscription:

Domestic, $2.75; foreign, $3.50*.

TECHNICAL NEWS BULLETIN

The best single source of information concerning

the Bureau’s research, developmental, cooperative

and publication activities, 1 this monthly publication

is designed for the industry-oriented individual whose

daily work involves intimate contact with science

and technology

for engineers, chemists, physicists , research managers, product-development managers, and

company executives. Annual subscription: Domestic,

$3.00; foreign, $4.00*.

•Difference in price is due to extra cost of foreign mailing. Order NBS publications from:

NONPERIODICALS

Applied Mathematics Series. Mathematical

tables, manuals, and studies. Building Science Series. Research results, test methods, and performance criteria of building materials, components, systems, and structures. Handbooks. Recommended codes of engineering

and industrial practice (including safety codes) developed in cooperation with interested industries,

professional organizations, and regulatory bodies.

Special Publications. Proceedings of NBS conferences, bibliographies, annual reports, wall charts,

pamphlets, etc. Monographs. Major contributions to the technical literature on various subjects related to the Bureau’s scientific and technical activities. National Standard Reference Data Series.

NSRDS provides quantitative data on the physical

and chemical properties of materials, compiled from

the world’s literature and critically evaluated.

Product Standards. Provide requirements for

sizes, types, quality and methods for testing various

industrial products. These standards are developed

cooperatively with interested Government and in- dustry groups and provide the basis for common

understanding of product characteristics for both

buyers and sellers. Their use is voluntary.

Technical Notes. This series consists of communications and reports (covering both other agency

and NBS-sponsored work) of limited or transitory

interest. Federal Information Processing Standards

Publications. This series is the official publication

within the Federal Government for information on

standards adopted and promulgated under the Public Law 89-306, and Bureau of the Budget Circular

A-86 entitled, Standardization of Data Elements

and Codes in Data Systems.

CLEARINGHOUSE

The Clearinghouse for Federal Scientific and

Technical Information, operated by NBS, supplies

unclassified information related to Governmentgenerated science and technology in defense, space, atomic energy, and other national programs. For

further information on Clearinghouse services, write:

Clearinghouse

U.S. Department of Commerce

Springfield, Virginia 22151

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Government Printing Office Washington, D.C. 20402