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Projects and Publications of the Applied Mathematics Division NATIONAL BUREAU OF STANDARDS REPORT 4546 PROJECTS and PUBLICATIONS of the APPLIED MATHEMATICS DIVISION A Quarterly Report October through December 1 955 U. S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS FOR OFFICIAL DISTRIBUTION U. S. DEPARTMENT OF COMMERCE Sinclair Weeks, Secretary NATIONAL BUREAU OF STANDARDS A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS The 6cope of activities of the National Bureau of Standards is suggested in the following listing of the divisions and sections engaged in technical work. In general, each section is engaged in specialized research, development, and engineering in the field indicated by its title. A brief description of the activities, and of the resultant reports and publications, appears on the inside of the back cover of this report. Electricity and Electronics. Resistance and Reactance. Electron Tubes. Electrical Instru- ments. Magnetic Measurements. Process Technology. Engineering Electronics. Electronic Instrumentation. Electrochemistry. Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic Technology. Length. Engineering Metrology. Heat and Power. Temperature Measurements. Thermodynamics. Cryogenic Physics. Engines and Lubrication. Engine Fuels. Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid State Physics. Electron Physics. Atomic Physics. Nuclear Physics. Radioactivity. X-rays. Betatron. Nucleonic Instrumentation. Radiological Equipment. AEC Radiation Instruments. Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical Chemistry. Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. 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 9 Office of Weights and Measures NATIONAL BUREAU OF STANDARDS REPORT NBS PROJECT NBS REPORT 1 1.0 4546 PROJECTS and PUBLICATIONS of the APPLIED MATHEMATICS DIVISION October through December 1955 U. S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS Approved for public release by the Tiw pvbiicitio*. r«i , It proWblw iwparamioaia director of the National Institute of «uni.,WMhi»gto» 25«DtCi Such p«n Standards and Technology (NIST) iortha»beM»peew- eally prepared If »l on October 9, 2015 P° rt for Ito ewe use. APPLIED MATHEMATICS DIVISION October 1 through December 31 ? 1955 TECHNICAL ADVISORY COMMITTEE David Blackwell, Howard University 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 Esther L. Turner, Secretary NUMERICAL ANALYSIS SECTION John Todd, B.S., Chief Henry A. Antosiewicz, Ph.D. Tosio Kato, Ph.D. Alan J. Hoffman, Ph.D. Leila D. Bram, Ph.D. Karl Goldberg, M.A Morris Newman, Ph.D. Philip Davis, Ph.D. Evelyn A. Grigg, Sec'y Philip Rabinowitz, Ph.D. COMPUTATION LABORATORY Milton Abramowitz, Ph.D., Chief Irene A. Stegun, M.A. ? Assistant Chief Joseph Bram, Ph.D. Genevie E. Hawkins, B.S. Maxine L. Paulsen, B.S. Hans 0. Bremer, B.A Emilie V. Haynsworth, Ph.D. Sally T. Peavy, B.S. Anne C. Bruno, B.A. Urs W. Hochstrasser , Ph.D. B. Stanley Prusch William F. Cahill, M.S. Gloria F. Holmes, B.S., Sec'y Ida Rhodes, M.A Ruth E. Capuano Dorothea H.Jirauch, M.A. Billie G. Urban, M.S. John W. Cooper, B.S. Lambert S. Joel, B.A Philip J. Walsh, B.S. Mary M. Dunlap, B.S. David S. Liepman Bertha H. Walter Anne M. Futterman, B.A Ethel C. Warden, B.A Joseph H. Wegstein, M.S. Elizabeth F. Godefroy Kermit C. Nelson Richard G. Williams, M.S William G. Hall, B.S. Peter J. O'Hara, B.S. Ruth Zucker, B.A Mary Orr 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. Severo, Ph.D. Marion T. Carson Lela J. Hamilton, Sec'y Paul N. Somerville, Ph.D Willard H.Clatworthy , 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. Peter Henrici, Ph.D. Lillian Sloane, Sec'y Leon Feldman, B.A Fritz Oberhettinger , Ph.D. Robert Stoneley, Ph.D. Edith N. Reese, B.A Under contract with The American University Guest Worker On leave of absence CONTENTS Page Status of projects* as of December 31 5 19 55 1 1 . Numerical analysis 1 2. Mathematical tables and programming research . 8 3. Probability and mathematical statistics. .... 12 *+. Mathematical statistics 1 5 5. Mathematical and computational services 20 6. Statistical Engineering services 35 Application of SEAC 37 Lectures and symposia 39 Publication activities ^1 *Only unclassified projects are included in this report . , Status of Projects December 31, 1955 1 . NUMERICAL ANALYSIS RESEARCH IN THEORIES OF DISCRETE STRUCTURES Task 1100-10-51 16/26-1 £9 Origin: NBS Authorized 9/30/55 Sponsor: Office of Naval Research Manager: 0. Teussky-Todd Full task description: July-Sept 1955 issue, p. 1 Status: CONTINUED. In connection with the work of this task, K. Goldberg and M. Newman have studied certain subgroups of the modular group G and have determined sufficient conditions for two elements of G to generate a free subgroup of G. A manuscript on the subject has been prepared (see item (1*+) below). This work was started through a particular problem raised by J. L. Brenner (State College of Washington), from which 0. Taussky suggested a generalization. The two-dimensional analog of the Wirtinger inequality was further studied by Ky Fan, 0. Taussky, and J. Todd. It was noted earlier that in the symmetrical case (n^ variables) equality occurs for vectors that are linear combinations of a certain set of four n^ vectors. In the unsymme trical case of mn variables, however, there is equality only for vectors that are linear combinations of two mn-vectors. T. Kato and 0. Taussky have given several proofs of the theorem that a bounded operator A in Hilbert space is normal if and only if A*A-AA* commutes with A. For finite matrices two proofs are given without the use of characteristic roots. The theorem is established in the case when the elements of the matrix are elements of an abstract field provided it is formally real or that the relation H a^a^=0 implies a^=0 when is an involution in the field. This enables the theorem to be used for matrices whose elements are polynomials in several indeterminate s , e.g., group matrices. Generalizations of the theorem were also found. The first vanishing commutator of the form (A, (A, . (A, (A, A*) ) . ) ) has been examined for some special types of finite matrices. A paper on commutators of A and A* will appear (see (13)). T. Kato has proved that the (infinite) Hilbert matrix (a^) - ^ (i, k=1 ,2, . ) , with a^j =(i+k) has a positive eigenvector belonging t to the eigenvalue tt, thus answering in the affirmative a question raised by 0. Taussky (Research Problem 12 , Bull. Amer. Math. Soc. 60 , 290 ( 195*+)). The methods of proof were suggested by an examination of the actual numerical values of the eigenvectors of the finite segments of the matrix. Some of the material examined was published by R. A. Fairthorne and J.r.P, Miller _I - (MTAC 3? 399 +00 ( 19 ^8 9 )) and also by 0. Taussky (Quart. J. Math., Oxford Ser. 20 , 80-83 ( 19^9 )); relevant computations had been carried out previously at New York University Institute of Mathematical Sciences on the UNIVAC, and special computations were carried out by K. Goldberg on SEAC. A manuscript has been prepared (12). 1 . : 2 Status of Projects Publications (1) On a generalization of the normal basis in abelian algebraic number fields. M. Newman and 0. Taussky. To appear in Communications on Pure and Applied Mathematics (New York University). (2) A note on group matrices. 0. Taussky. To appear in the Proceedings of the American Mathematical Society. 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, and 0. Taussky. To appear in the Pacific Journal of Mathematics, (5) Some computational problems in algebraic number theory. 0. Taussky. To appear in the Proceedings of the American Mathematical Society Sixth Symposium on Applied Mathematics, held at Santa Monica, California, August 1953, ( 6 ) Bounds and asymptotic behavior of the coefficients in the series log(e x ey). K. Goldberg, In manuscript, (7) Unimodular integral circulants. 0. Taussky. Appeared in Schur Memorial Volume, Math. Zeitsch. ^j>, 286-289 (1955). ( 8 ) Algebraic equations satisfied by roots of natural numbers.
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