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A note on recent research in geometrical probability

Published online by Cambridge University Press:  14 July 2016

P. A. P. Moran*
Affiliation:
Australian National University

Extract

The subject of geometrical probability has recently shown signs of renewed vigour which is partly due to the attractive nature of the many unsolved problems which it contains, and partly the ever widening field of applications. These applications are so diverse and so scattered in the literatures of different branches of science that when Professor M.G. Kendall and I published a monograph on the subject in 1963 some interesting papers were ignored. Others have since appeared and the purpose of the present paper is to survey, in a very brief manner, work not described in that book. In fact the present bibliography, which contains 94 items, is disjoint from the bibliography in our joint monograph.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 

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References

Ambartsumian, P. (1963) Proceedings of the 7th All-Union Conference on Mathematical Statistics and Probability, Tbilissi.Google Scholar
Bankovi, G. (1962) On gaps generated by a random space filling procedure. Publ. Math. Inst. Hung. Acad Sci. 7, 395407.Google Scholar
Barton, D. E. and David, F. N. (1961) The analysis of chromosome patterns in the normal cell. Ann. Hum. Genet. 25, 323329.CrossRefGoogle Scholar
Barton, D. E. and David, F. N. (1962) Randomization bases for multivariate tests. I. The bivariate case. Randomness of N points in a plane. Bull. Inst. Internat. Statist. 39, 2e livraison, 455463.Google Scholar
Barton, D. E., and David, F. N. (1963) The analysis of chromosome patterns in the abnormal cell. Ann. Hum. Genet. 26, 347348.CrossRefGoogle ScholarPubMed
Barton, D. E., David, F. N. and Fix, E. (1963) Random points in a circle and the analysis of chromosome patterns, Biometrika 50, 2329.CrossRefGoogle Scholar
Barton, D. E., David, F. N. and Merrington, M. (1963) Numerical analysis of chromosome patterns. Ann. Hum. Genet. 26, 349353.CrossRefGoogle ScholarPubMed
Bernal, J. D. (1959) A geometrical approach to the structure of fluids. Nature 183, 141147.CrossRefGoogle Scholar
Bernal, J. D. (1960) Geometry of the structure of monatomic liquids. Nature 185, 6870.CrossRefGoogle Scholar
Bernal, J. D. (1964) The structure of liquids. Proc. Roy. Soc. A. 280, 299322.Google Scholar
Bernal, J. D. and Mason, J. (1960) Coordination of randomly packed spheres. Nature 188, 910911.CrossRefGoogle Scholar
Blascke, W. (1923) Vorlesungen über Differential-geometrie. II. Affine Differential-geometrie. Springer, Berlin.Google Scholar
Boerdijk, A. H. (1952) Some remarks concerning close packing of equal spheres. Philips Research Reports 7, 303313.Google Scholar
Chayes, F. (1956) Petrographic Model Analysis. Wiley, New York.Google Scholar
Chernoff, H. and Daly, J. F. (1957) The distribution of shadows. J. Math. Mech. 6, 567584.Google Scholar
Corssin, S. (1955) A measure of the area of a homogeneous random surface in space. Quart. Appl. Math. 12, 404408.CrossRefGoogle Scholar
Cover, T. and Efron, B. The division of space by hyperplanes with applications to geometric probability. Canad. J. Math. (to appear).Google Scholar
Coxeter, H. S. M. (1958) Close-packing and froth. Illinois J. Math. 2, 746758.CrossRefGoogle Scholar
Coxeter, H. S. M. (1961) An Introduction to Geometry. Wiley. New York & London.Google Scholar
Dalenius, T., Hájek, J. and Zubryzcki, S. (1961) On plane sampling and related geometrical problems. Proc. Fourth Berkeley Symposium on Mathematical Statistics and Probability I, 125150.Google Scholar
Daniels, H. E. (1952) The covering circle of a sample from a circular distribution. Biometrika 39, 137143.CrossRefGoogle Scholar
David, F. N. and Fix, E. (1964) Intersections of random chords of a circle. Biometrika 51, 373379.CrossRefGoogle Scholar
David, F. N. and Moore, P. T. (1957) A bivariate test for the clumping of supposedly random individuals. Ann. Bot. 21 (N.S.), 315320.CrossRefGoogle Scholar
De Bruin, N. G. (1965) Asymptotic distribution of lattice points in a rectangle. SIAM Review. 7, 274275.CrossRefGoogle Scholar
Drápel, S., Horálek, V. and ResžNy, Z. (1957) Quantitative metallographic lattice analysis. Hutnické Listy 12, 485491.Google Scholar
Dvoretzky, A. and Robbins, H. (1964) On the “parking” problem. Publ. Math. Inst. Hung. Acad. Sci. 9, A 209226.Google Scholar
Efron, B. (1965) The convex hull of a random set of points. Biometrika 52, 331343.CrossRefGoogle Scholar
Efron, B. Gaussian distributions for random hyperplanes (to appear).Google Scholar
Eggleton, P. and Kermack, W. O. (1944) A problem in the random distribution of particles. Proc. Roy. Soc. Edin. A. 62, 103115.Google Scholar
Evans, U. R. (1945) The laws of expanding circles and spheres in relation to the lateral growth of surface films and the grain size of metals. Trans. Faraday Soc. 41, 365374.CrossRefGoogle Scholar
Fairthorne, D. (1964) The distance between random points in two concentric circles. Biometrika 51, 275277.CrossRefGoogle Scholar
Gilbert, E. N. (1962) Random subdivision of space into crystals. Ann. Math. Statist. 33, 958–72.CrossRefGoogle Scholar
Gilbert, E. N. (1963) Random minimal trees. J. Soc. Indust. Appl. Math. 13, 376387.CrossRefGoogle Scholar
Gilbert, E. N. (1965) The probability of covering a sphere with n circular caps. Biometrika 52, 323330.CrossRefGoogle Scholar
Griffiths, J. S. (1962) Packing of equal O-spheres. Nature. 196, 764.CrossRefGoogle Scholar
Guter, R. S. (1964) On the problem of detecting a region by a linear search. Theor. Probability Appl. 9, 331333.CrossRefGoogle Scholar
Gyachyauskas, E. (1964a) Search by an oval. Theor. Probability Appl. 9, 634637.CrossRefGoogle Scholar
Gyachyauskas, E. (1964b) On statistical quadratures. Theor. Probability Appl. 9, 637640.CrossRefGoogle Scholar
Hasofer, A. M. (1963) On the reliability of the point-counter method in petrography. Aust. J. Appl. Sci. 14, 168179.Google Scholar
Hawskley, P. G. W. (1954) Theory of particle sizing and counting by track scanning. The Physics of Particle Size Analysis. Brit. Jour. Appl. Physics. Suppl. No. 3.CrossRefGoogle Scholar
Higuti, I. (1961) A statistical study of random packing of unequal spheres. Ann. Inst. Statist. Math. 12, 257271.CrossRefGoogle Scholar
Hogendijk, M. J. (1963) Random dense packing of spheres with a discrete distribution of the radii. Phillips Research Reports 18, 109126.Google Scholar
Holgate, P. (1965) Tests of randomness based on distance methods. Biometrika 52, 345353.CrossRefGoogle Scholar
Holgate, P. (1965) The distance from a random point to the nearest point of a closely packed lattice. Biometrika 52, 261263.CrossRefGoogle Scholar
Horowitz, M. (1965) Probability of random paths across elementary geometrical shapes. J. Appl. Prob. 2, 169177.CrossRefGoogle Scholar
Jackson, J. L. and Montroll, E. W. (1958) Free radical statistics. J. Chem. Phys. 28, 11011109.CrossRefGoogle Scholar
Johnson, W. A. and Mehl, R. F. (1939) Reaction kinetics in processes of nucleation and growth. Trans. Amer. Inst. Min. Met. Eng. 135, 416458.Google Scholar
Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin's Statistical Monographs and Courses. No. 5 London. C. Griffin.Google Scholar
Keuls, M., Over, H. J. and De Wit, C. T. (1963) The distance method for estimating densities. Stat. Neerl. 17, 7191.CrossRefGoogle Scholar
Kingman, J. F. C. (1965) Mean free paths in a convex reflecting region. J. Appl. Prob. 2, 162168.CrossRefGoogle Scholar
Kosten, C. W. (1960) The mean free path in room acoustics. Acustica 10, 245250.Google Scholar
Krengel, U. A problem on random points in a triangle. Amer. Math. Monthly (to appear).Google Scholar
Mannion, D. (1964) Random space-filling in one dimension. Publ. Math. Inst. Hung. Acad. Sci. 9 (A) 143154.Google Scholar
Matern, B. (1959) Nagra tillampningar av teorin för geometriska sannolikheter. Svenska skogsvardsforeningens tidskr. 3, 453458.Google Scholar
Matzke, E. B. (1950) In the twinkling of an eye. Bull. Torrey Botanical Club. 77, 222227.CrossRefGoogle Scholar
Meijering, J. L. (1953) Interface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Research Reports. 8, 270290.Google Scholar
Miles, R. E. A wide class of distributions in geometric probability (to appear).Google Scholar
Miles, R. E. (1964a) Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. 52, 902907.Google ScholarPubMed
Miles, R. E. (1964b) Random polygons determined by random lines in a plane. II. Proc. Nat. Acad. Sci. 52, 11571160.CrossRefGoogle Scholar
Miles, R. E. (1965) On random rotations in R3. Biometrika 52, 636639.Google Scholar
Miller, J. B. (1964) An integral equation from phytology. J. Aust. Math. Soc. 4, 397402.CrossRefGoogle Scholar
Moon, J. W. (1965) On the distribution of crossings in random complete graphs. J. Soc. Indust. Appl. Math. 13, 506510.CrossRefGoogle Scholar
Moran, P. A. P. Measuring the length of a curve. Biometrika (to appear).Google Scholar
Moran, P. A. P., de St Groth, S. Fazekas (1962) Random circles on a sphere. Biometrika 49, 389396.CrossRefGoogle Scholar
Mountford, M. D. (1961) On E. C. Pielou's Index of non-randomness. J. Ecol. 49, 271275.CrossRefGoogle Scholar
Naus, J. I. (1965a) The distribution of the size of the maximum cluster of points on a line. J. Amer. Statist. Ass. 60, 532538.CrossRefGoogle Scholar
Naus, J. I. (1965b) Clustering of random points in two dimensions. Biometrika 52, 263267.CrossRefGoogle Scholar
Ney, P. E. (1962) A random interval filling problem. Ann. Math. Statist. 33, 702718.CrossRefGoogle Scholar
Palasti, I. (1960) On some random space filling problems. Publ. Math. Inst. Hung. Acad. Sci. 5, 353360.Google Scholar
Persson, O. (1964) Distance methods. Studia Forestallia Sueccia. No. 15.Google Scholar
Philip, J. The recovery of radial distributions. Aust. J. Phys. (to appear).Google Scholar
Philip, J. Some integral equations in geometric probability Biometrika. (to appear).Google Scholar
Philip, J. (1966) The distribution of foliage density with foliage angle estimated from inclined point quadrat observations. Aust. J. Bot. 13, 357–66.Google Scholar
Philip, J. R. (1965). The distribution of foilage density on single plants. Aust. J. Bot. 13 411418.CrossRefGoogle Scholar
Phillips, J. W. (1954) Some fundamental aspects of particle counting and sizing by line scans. The Physics of Particles Size Analysis. Brit. J. Appl. Physics. Suppl. No. 3.CrossRefGoogle Scholar
Rényi, A. and Sulanke, R. (1963) Uber die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheorie. 2, 7584.CrossRefGoogle Scholar
Rényi, A. and Sulanke, R. (1964) Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheorie. 3, 138147.CrossRefGoogle Scholar
Richards, P. I. (1964) Averages for polygons formed by random lines. Proc. Nat. Acad. Sci. 52, 11601164.CrossRefGoogle ScholarPubMed
Russell, A. M. and Josephson, N. S. (1965) Measurement of area by counting. J. Appl. Prob. 2, 339351.CrossRefGoogle Scholar
Scott, G. D. (1962) Radial distribution of the random close packing of equal spheres. Nature 194, 956958.CrossRefGoogle Scholar
Silverman, D. L. (1964) Problem E1658. Amer. Math. Monthly 71, 11351136.CrossRefGoogle Scholar
Smalley, I. J. (1962) Packing of O-spheres. Nature. 194, 1271.CrossRefGoogle Scholar
Smith, C. A. B. (1957) Relazione fra i valori reali e quelli osservati di cellule a forma di sferoide. La Ricerca Scientifica. Supplemento. 27.Google Scholar
Smith, C. S. (1952) Grain shape and other metallurgical applications of topology. Metal Interfaces. Amer. Soc. for Metals, Cleveland 65108.Google Scholar
Smith, C. S. (1953) Further notes on the shape of metal grains: space-filling polyhedra with unlimited sharing of corners and faces. Acta Metallurgica 1, 259300.CrossRefGoogle Scholar
Smith, C. S. and Guttman, L. (1953) Measurement of internal boundaries in three dimensional structure by random sectioning. J. Metals 5, 8187.Google Scholar
Solomon, H. (1966) Random packing density. Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability. University of California Press (to appear).Google Scholar
Ulam, S. (1961) Monte Carlo calculations in problems of mathematical physics. Modern Mathematcs for the Engineer (ed. Beckenbach, E. F.). Second Series. McGraw Hill.Google Scholar
Warren, W. G. (1962) Contributions to the Study of Spatial Processes. Ph. D. Thesis. Univ. of North Carolina.Google Scholar
Wilson, J. Warren (1960) Inclined point quadrats. New Phytologist. 59, 18.CrossRefGoogle Scholar
Wendel, J. G. (1962) A problem in geometric probability. Math. Scand. 11, 109111.CrossRefGoogle Scholar
Wise, M. E. (1952) Dense random packing of unequal spheres. Philips Research Reports 7, 321343.Google Scholar
Wright, G. H. (1954) A geometrical factor in the variability of sensations of warmth evoked by radiation. Proc. Camb. Phil. Soc. 50, 474484.Google Scholar
Yerazunis, S., Cornell, S. W. and Wintner, B. (1965) Dense random packing of binary mixtures of spheres. Nature. 207, 835837.CrossRefGoogle Scholar