Tchebycheff Systems Research Articles

Tchebycheff Systems: With Applications in Analysis and Statistics.

Tchebycheff Systems: With Applications in Analysis and Statistics.

Book Reviews

Abstract Introduction to Probability Theory and Its Applications, Vol. II, William Feller. New York: John Wiley and Sons, 1966. Pp. xviii, 626. $12.00. Reviewed by Joshua Chover Equality of Educational Opportunity. James S. Coleman, Ernest Q. Campbell, Carol J. Hobson, James McPartland, Alexander M. Mood, Frederic D. Weinfeld, and Robert L. York. Washington, D.C. U.S. Government Printing Office, 1966. pp. vi+737. $4.25. (Also a set of Correlation Tables, separately bound, is available for the use of research workers.) Reviewed by Robert J. Havighurst Federal Tax Policy. Joseph A. Pechman. Washington, The Brookings Institution, 1966. Pp. xxi, 321. Reviewed by Peter Diamond Aplied Regression Analysis. N. R. Draper and H. Smith. John Wiley and Sons, Inc., 1966, pp. ix, 407. Reviewed by C. L. Mallows Statistical Analysis in Psychology and Education. George A. Ferguson. New York: McGraw-Hill Series in Psychology. 2nd ed. 1966. 448 pp. $7.95. Reviewed by H. Fairfield Smith Research for Marketing Decisions. Paul E. Green and Donald S. Tull, Prentice-Hall 1966, pp. 532. $13.25. Reviewed by Kristian S. Palda Inventory Systems. Eliezer Naddor, John Wiley & Sons, Inc., New York. 1965, pp. 341. Reviewed by Ronald L. Gue Matrix Algebra for the Biological Sciences, S. R. Searle. New York: Wiley, 1966, Pp. 296. Reviewed by Rolf E. Bargmann The Monte Carlo Method. N. P. Buslenko, D. I. Golenko, Yu. A. Shreider, I. M. Sobel, and V. G. Sragovich. Edited by Yu. A. Shreider, Pergamon Press, 1966, $12.50, 379 pp. Reviewed by Jess, I. Marcum Statistical Association Methods for Mechanized Documentation, Symposium Proceedings, Washington 1964, edited by Mary Elizabeth Stevens, Vincent E. Giuliano and Lawrence B. Heilprin. Washington D.C.: National Bureau of Standards Miscellaneous Publication 269, 1965. Pp. vii, 261. $2.75. Reviewed by Vincent Hodgson OECD: “Techniques of Economic Forecasting.” An Account of the Methods of Short-Term Economic Forecasting used by the Governments of Canada, France, the Netherlands, Sweden, the United Kingdom and the United States with an introduction by C. W. McMahon, Paris (1965), 173 pp. Reviewed by J. Sandee Public Health and Population Change. Mindel C. Sheps and Jeanne Clare Ridley, Editors. Pittsburgh, Pennsylvania: University of Pittsburgh Press, 1965. 557+xvii pp. Reviewed by Richard G. Cornell Classical and Contagious Discrete Distributions. Edited by G. P. Patil. Statistical Publishing Society, Calcutta (Distributed outside India by Pergamon Press), 1966. Pp. xiv, 552. $21.00. Reviewed by K. Subrahmaniam Statistical Inference I and II. Jerome C. R. Li, Ann Arbor Michigan. Edwards Brothers Inc. 1964. Vol. I pp. xix, 658; Vol. II pp. xiv, 575. $10.00 Each Volume. Reviewed by Robert J. Monroe Elementary Statistics; Second Edition. Paul G. Hoel. New York: John Wiley and Sons, Inc., 1966. pp. viii, 351. Reviewed by Robert V. Hogg Tables of Bivariate Random Normal Deviates. P. V. Iyer Krishna and P. S. Simha. Defence Science Laboratory, Metcalfe House, Delhi-6. 1966, 120+iii pp. Reviewed by J. T. Webster Biometrika Tables for Statisticians, Volume 1 (Third Edition) Eds. E. S. Pearson and H. O. Hartley. Cambridge: The University Press, 1966, Pp. 264. $6.50. Tchebycheff Systems: With Applications in Analysis and Statistics, S. J. Karlin and W. J. Studden, New York: John Wiley and Sons, 1966, 586+xviii pp.

BOOK REVIEWS

Book reviewed in this article:Lamperti, J. Probability. W. A. BenjaminKarlin, S. J., and Studden, W. J. Tchebycheff Systems: with Applicatiom in Analysis and Statistics.Fiszel, H. Investment Eflcieney in a Socialist Economy.Hope, K. Elenaerttary Statistics: A Workbook.Girault, M. Stochastic Processes. Springer‐Verlag.Cotton, J. W. Elementary Statistical Themy for Behaviour Scientists.Hill, Joseph E., and Kerber, August. Models, Methods, and Analytical Procedures in Education ResearchHaight, F. A. Handbook of the Poisson Distribution. John Wiley & Sons, Inc.

Generalized absolutely monotone functions

Sufficient conditions for generalized absolutely monotone functions to possess a Taylor-type expansion in terms of the corresponding Extended Tchebycheff systems were found by Karlin and Ziegler. The question of necessary conditions, however, was left open. In this paper we solve this question by finding necessary and sufficient conditions for the validity of the expansion. The structure of the cone of generalized absolutely monotone functions and its extreme rays are also discussed.

Parallelism and equidistance in classical differential geometry

It is the purpose of this paper to discuss and compare certain concepts bearing on the relationship to one another of two families of curves on a surface in a euclidean space of three dimensions. In the first place, there are set over against one another the concept of parallelism, in the sense of LeviCivita, of one family of curves with respect to another and the concept of equidistance of the first family with respect to the second. Secondly, a measure of the deviation from parallelism of the one family of curves with respect to the other, to which is given the name is compared with a measure of the deviation from equidistance of the first family with respect to the second, which is called spread. In addition to the introduction of the concepts in question, Part I contains the fundamental relations between the spreads of each of two families with respect to the other. These relations lead to quantitative results concerning the spreads, as well as to interesting conclusions concerning the connection between parallelism and equidistance. Part II deals more in detail with the properties of the spreads. In particular, it is found that the angular spread, at a point P, of a fixed family of curves with respect to a variable family of curves is represented geometrically by a variable vector at P the locus of whose terminal point is a circle. This fact leads directly to a generalization of Liouville's formula for geodesic curvature which is not only of interest in itself but tends to show that the original formula is essentially a relationship between angular spreads. Part III bears on the connection between directional derivatives and distantial spreads. It develops an invariant form of Bonnet's formula for geodesic curvature, finds the laws of transformation of the spreads of one system of curves into those of a second, and applies these laws to the study of Tchebycheff systems. In Part IV, the relationships between the spreads of a system of curves on a surface and the spreads of the corresponding system of curves on the spherical representation of the surface are discussed and application is made of the results to the comparison of parallelism and equidistance on the sur-