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Projects and Publications of the Applied Mathematics Division NATIONAL BUREAU OF STANDARDS REPORT 4658 PROJECTS and PUBLICATIONS of the APPLIED MATHEMATICS DIVISION A Quarterly Report January through March, 1956 U. S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS FOR OFFICIAL DISTRIBUTION U. S. DEPARTMENT OF COMMERCE Sinclair Weeks, Secretary NATIONAL BUREAU OF STANDARDS A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS The scope of activities of the National Bureau of Standards is suggested in the following listing of the divisions and sections engaged in technical work. In general, each section is engaged in specialized research, development, and engineering in the field indicated by its title. A brief description of the activities, and of the resultant reports and publications, appears on the inside of the back cover of this report. Electricity and Electronics. Resistance and Reactance. Electron Tubes. Electrical Instru- ments. Magnetic Measurements. Process Technology. Engineering Electronics. Electronic Instrumentation. Electrochemistry. Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic Technology. Length. Engineering Metrology. Heat and Power. Temperature Measurements. Thermodynamics. Cryogenic Physics. Engines and Lubrication. Engine Fuels. Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid State Physics. Electron Physics. Atomic Physics. Nuclear Physics. Radioactivity. X-rays. Betatron. Nucleonic Instrumentation. Radiological Equipment. AEC Radiation Instruments. Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical Chemistry. Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermo- chemistry. Spectrochemistry. Pure Substances. Mechanics. Sound. Mechanical Instruments. Fluid Mechanics. Engineering Mechanics. Mass and Scale. Capacity, Density, and Fluid Meters. Combustion Controls. Organic and Fibrous Materials. Rubber. Textiles. Paper. Leather. Testing and Specifica- tions. Polymer Structure. Organic Plastics. Dental Research. Metallurgy. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy. Corrosion. Mineral Products. Porcelain and Pottery. Glass. Refractories. Enameled Metals. Con- creting Materials. Constitution and Microstructure. Building Technology. Structural Engineering. Fire Protection. Heating and Air Con- ditioning. Floor, Roof, and Wall Coverings. Codes and Specifications. Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathe- matical Physics. Data Processing Systems. Components and Techniques. Digital Circuitry. Digital Systems. Analogue Systems. Cryogenic Engineering. Cryogenic Equipment. Cryogenic Processes. Properties of Materials. Gas Liquefaction. Radio Propagation Physics. Upper Atmosphere Research. Ionospheric Research. Regular Propagation Services. Radio Propagation Engineering. Frequency Utilization Research. Tropospheric Propagation Research. Radio Standards. High Frequency Standards. Microwave Standards. • Office of Basic Instrumentation # Office of Weights and Measures CONTENTS Page Status of Projects* as of March 31, 1956 1 1. Numerical analysis 1 2. Mathematical tables and programming research. 6 3. Probability and mathematical statistics ... 10 4. Mathematical Physics 14 5. Mathematical and computational services ... 18 6. Statistical Engineering services 31 Application of SEAC 34 Lectures and Symposia 36 Publication activities 40 Only unclassified projects are included in this report. Wm . 1 NATIONAL BUREAU OF STANDARDS REPORT NBS PROJECT NBS REPORT 1 1 .0 4658 PROJECTS and PUBLICATIONS of the APPLIED MATHEMATICS DIVISION January through March, 1956 <s> U. $. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS Tba publication, reprinting, m ^production of this report I* nay fart*, altber la whole ar la pari, la prohibits aniens panslaaloa is obtained la wiltlagfra* tbeOlaa af tba Director, National Bureau of Btaadarda, Washington IS, 9. 6, Saab penaiaslea la aat needed, however, by a flavor aweat agaaey far whieh a report baa baaa ipeoifi- cally prepared IT that agaaey wishes la reprsducs addllleaal capita aI that particular rapert far Its aw a uaa, - 'J ..L 1 -U- APPLIED MATHEMATICS DIVISION January 1 through March 31 > 1956 TECHNICAL ADVISORY COMMITTEE David Blackwell, Howard University Philip M. Morse, Massachusetts Edward U. Condon Institute of Technology Mark Kac , Cornell University Mina S. Rees, Hunter College A. H. Taub, University of Illinois DIVISION OFFICE Edward W. Cannon, Ph.D., Chief Franz L. Alt, Ph.D., Assistant Chief Olga Taussky-Todd , Ph.D., Consultant W. J. Youden, Ph.D., Consultant Myrtle R. Kellington, M.A., Technical Aid Luis 0. Rodriguez, M.A., Administrative Officer Yates S. Sladen, Administrative Assistant Marguerite C. Hepburn, Secretary Mary E. McKinley, Secretary NUMERICAL ANALYSIS SECTION John Todd, B.S., Chief Henry A. Antosiewicz, Ph.D. Karl Goldberg, M.A. Alan J. Hoffman, Ph.D. N. Aronszajn, D.Sc. Evelyn A. Grigg, Sec'y Morris Newman, Ph.D. Philip Davis, Ph.D. COMPUTATION LABORATORY Milton Abramowitz, Ph.D., Chief Irene A. Stegun, M.A., Assistant Chief Joseph Bram, Ph.D. Urs W. Hochstrasser , Ph.D. Sally T. Peavy, B.S. Hans 0. Bremer, B.A. Gloria F. Holmes, B.S., Sec'y B. Stanley Prusch William F. Cahill, M.S. Dorothea H.Jirauch, M.A. Elizabeth A. Radcliffe Ruth E. Capuano Lambert S. Joel, B.A. Ida Rhodes, M.A. John W. Cooper, B.S. David S. Liepman Billie G. Urban, M.S. Mary M. Dunlap, B.S. Ethel C. Marden, B.A. Philip J. Walsh, B.S. Robert L. Durrah, M.A. Kermit C. Nelson Bertha H. Walter Elizabeth F. Godefroy Peter J. O'Hara, B.S. Joseph H. Wegstein, M.S. William G. Hall, B.S. Mary Orr Richard G. Williams, M.S Genevie E. Hawkins, B.S. Maxine L. Paulsen, B.S. Ruth Zucker , B.A. Emilie V. Haynsworth, Ph.D. STATISTICAL ENGINEERING LABORATORY Churchill Eisenhart, Ph.D., Chief Joseph M. Cameron, M.S., Assistant Chief Robert C. Burton Mary L. Epling, B.S. Norman C. Severn, Ph.D. Marion T. Carson Lela J. Hamilton, Sec'y Paul N. Somerville, Ph.D William S. Connor, Ph.D. Julius Lieblein, Ph.D. Carolyn Yick, Sec'y Mary C, Dannemiller Mary G. Natrella, B.A. Marvin Zelen, M.A. Lola S. Deming, M.A. Joan R. Rosenblatt, Ph.D. MATHEMATICAL PHYSICS SECTION Robert F. Dressier, Ph.D., Chief Peter Chiarulli, Ph.D. Fritz Oberhettinger , Ph.D. **Robert Stoneley, Sc*D. Leon Feldman, B.A. Edith N. Reese, B.A. *John L. Synge, Sc.D, Peter Henrici, Ph.D. Lillian Sloane, Sec'y Under contract with The American University **Guest Worker ***0n leave of absence Status of Projects March 31 ? 1956 1 . NUMERICAL ANALYSIS RESEARCH IN THEORIES OF DISCRETE STRUCTURES Task 1100-10-5116/56-159 Origin: NBS Authorized 9/30/55 Sponsor: Office of Naval Research Manager: 0. Taussky-Todd Full task description: July-Sept 1955 issue, p. 1 Status: CONTINUED. 0. Taussky continued her investigations concerning the higher commutators of a matrix A and its transposed and conjugate A*. For 3x3 matrices examples were constructed which show that the second, fourth, fifth, or sixth order commutator (A, A, . ,A,A*) of A and A* can be the first one to vanish. If none of these vanishes, then no commutator of any order will vanish. Group matrices with vanishing fourth order commutators were also studied. M. Newman and 0. Taussky had proved earlier that a unimodular circulant of integers of the form AA* is also of the form CC 1 when C is again a circulant of integers. They now have shown that the weaker condition of positive definiteness does not suffice for circulants of order n ^ 8. A manuscript has been prepared. M. Newman and 0. Taussky are studying hermitian matrices and their decomposition into the form AA* where A may be rectangular. Publications: (1) On a generalization of the normal basis in abelian algebraic number fields. M. Newman and 0. Taussky. Comm. Pure Appl. Math. (New York University) .2, 85-91 (1956). (2) A note on group matrices. 0. Taussky. Proc. Amer. Math. Soc. 6, 98^-986 (1955). x (3) The formal power series for log (e »ey). K. Goldberg. To appear in the Duke Mathematical Journal. (4) The number of absolute points of a correlation. A. Hoffman, M. Newman, E. Straus, 0. Taussky. To appear in the Pacific Journal of Mathematics. (5) Some computational problems in algebraic number theory. 0. Taussky. To appear in the Proceedings of the American Mathematical Society Sixth Symposium on Applied Mathematics, held at Santa Monica, California, 1953 (6) Bounds and asymptotic behavior of the coefficients in the series log (e x ey). K. Goldberg. In manuscript. (7) Algebraic equations satisfied by roots of natural numbers. E. G. Straus (University of California at Los Angeles) and 0. Taussky. To appear in Pacific Journal of Mathematics. (8) Pairs of matrices with property L, II. T. S. Motzkin and 0. Taussky. Trans. Amer. Math. Soc. 80, 387_1+01 (1955). (9) Generation and testing of pseudo-random numbers. 0. Taussky and J. Todd To appear in the Proceedings of a Symposium on Monte Carlo Methods, held at Gainesville, Florida, 1 9 5^+, 1 . 2 Status of Projects ( 10 ) On the Hilbert matrix, T, Kato. Submitted to a technical journal. ( 11 ) Commutators of A and A*. T. Kato and 0. Taussky. J. Wash. Acad. Sci. 1+6, 38-i+O (1956). ( 12 ) Some free subgroups of the modular group. K. Goldberg and M„ Newman. In manuscript. ( 13 ) Classes of positive definite unimodular circulants. M. Newman and 0. Taussky. In manuscript. RESEARCH IN NUMERICAL ANALYSIS AND RELATED FIELDS Task 1 101 -10-1 10V55-55 Origin: NBS Authorized 8/1 3/5*+ Managers: J. Todd , P. Davis Revised 8/29/5*+ Full task description: July-Sept 195*+
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