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A second note on recent research in geometrical probability

Published online by Cambridge University Press:  01 July 2016

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

Extract

Many problems in geometrical probability have been surveyed in the monograph by Professor M. G. Kendall and myself (Kendall and Moran (1963)). In 1966 I published a note in J. Appl. Frob. 3 describing researches which had either been carried out since that book appeared, or which had escaped our notice. The present paper attempts to describe still further investigations classified roughly as to whether they are concerned with points, lines, planes, estimation of area and length, and coverage.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 

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References

Bibliography

The following bibliography is divided into two parts, the first of which contains references to papers mentioned in the present survey which are in the two previous bibliographies, and the second, which has 105 references, to papers which have either appeared since or which were missed in the previous bibliographies. These include several for which the previous references were incomplete.

Bibliography (A): Papers cited in previous bibliographies.

Gilbert, E. N. (1962) Random subdivision of space into crystals. Ann. Math. Statist. 33, 958972.Google Scholar
Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin's Statistical Monographs and Courses. No. 5. C. Griffin, London.Google 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
Rényi, A. and Sulanke, R. (1964) Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitsth. 2, 7584.Google Scholar
Solomon, H. (1967) Random packing density. Proc. Fifth Berkeley Symp. Math. Statist. and Prob. Vol. III. 119134. University of California Press.Google Scholar
Ailam, G. (1966) Moments of coverage and coverage spaces. J. Appl. Prob. 3, 550555.CrossRefGoogle Scholar
Ailam, G. (1968) On probability properties of measures of random sets and the asymptotic behaviour of empirical distribution functions. J. Appl. Prob. 5, 196202.Google Scholar
Backman, A. (1934) Bestämning av cellulosahalten i papper genom mätning av filbertätheten. Pappers och Trävarutidskrift för Finland 16, 302308.Google Scholar
Bartlett, M. S. (1964) A note on spatial patterns. Biometrics 20, 891892.CrossRefGoogle Scholar
Barton, D. E., David, F. N., Fix, E. and Merrington, M. (1967) A review of analysis of karyographs of the human cell in mitosis. Proc. Fifth Berkeley Symp. Math. Statist. and Prob. Vol. IV, 349366. University of California Press.Google Scholar
Bauersachs, F. (1942) Bestandesmassenaufnahme nach dem Mittelstammverfahren des zweitkleinsten Stammabstandes. Forstw. Zentralblatt 64, 182186.CrossRefGoogle Scholar
Boursin, J. L. (1964) Sur quelques problèmes de géometrie aléatoire. Ann. Fac. Sci. Univ. Toulouse (4) 28, 9100.CrossRefGoogle Scholar
Boursin, J. L. (1966) Etude d'ovales en probabilité géométrique. International Congress of Mathematicians, Moscow. Section 11, 6.Google Scholar
Corrsin, S. and Phillips, O. M. (1961) Contour length and surface area of multiple valued random variables. J. Soc. Indust. Appl. Math. 9, 395404.CrossRefGoogle Scholar
Corte, H. K. and Lloyd, E. H. (1965) Fluid flow through paper and sheet structure. “Consolidation of the Paper Web”. Transactions of the Cambridge Symposium 1965. British Paper and Board Makers Association, London.Google Scholar
Cover, T. (1965) Geometrical and statistical properties of systems of linear inequalities with applications to pattern recognition. Trans. Elect. Comp. IEEE. EC-14, 326334.Google Scholar
Cover, T. M. and Efron, B. (1967) Geometrical probability and random points on a hypersphere. Ann. Math. Statist. 38, 212220.Google Scholar
Cross, C. A. and Fisher, D. L. (1968) The computer simulation of lunar craters. Monthly Not. R. Astr. Soc. 139, 261272.Google Scholar
de Renna e Souza, C. (1967) Two dimensional collision in parallel paths. Operat. Res. 15, 3238.Google Scholar
Dobo, A. and Szajcz, S. (1965) On problems of random placement. Magy. Tud. Akad. Mat. Fiz. Oszt. Közl. 15, 389409.Google Scholar
Drapal, S. and Horalek, V. (1959) Some relations between parameters of structure in the plane of metallographic specimen surface and in the space of metal specimen. Acta Tech. 6 (in English).Google Scholar
Duncan, R. L. (1967) A variation of the Buffon needle problem. Math. Mag. 40, 3638.Google Scholar
Essed, F. E. (1956) A quick, simple and at the same time accurate method for estimating the total volume and the incremental percent of evenaged stands. Indian Forester 82, 260263.Google Scholar
Finney, J. L. and Bernal, J. D. (1967) Random close packing and the heats of fusion of simple liquids. Nature 213, 10791082.CrossRefGoogle Scholar
Garwood, F. and Holroyd, E. M. (1966) The distance of a “random chord” of a circle from the centre. Math. Gaz. 50, 283286.Google Scholar
Geffroy, J. (1959) Contribution à la théorie des valeurs extrêmes. II. Publ. Inst. Statist. Univ. Paris 8, 365.Google Scholar
Geffroy, J. (1961) Localisation asymptotique du polyèdre d'appui d'un échantillon Laplacien à k dimensions. Publ. Inst. Statist. Univ. Paris 10, 213218.Google Scholar
Geffroy, J. (1964) Sur un problème d'estimation géométrique. Publ. Inst. Statist. Univ. Paris 13, 191210.Google Scholar
Gilbert, E. N. (1961) Random plane networks. J. Soc. Indust. Appl. Math. 9, 533543.CrossRefGoogle Scholar
Gilliland, D. C. (1966) Some bombing problems. Amer. Math. Monthly 73, 713716.Google Scholar
Gjacjauskas, E. (1965) On linear and plane searches. Litovsk. Mat. Sb. 5, 227231.Google Scholar
Gjacjauskas, E. (1966a) Uniform scan in space. Litovsk. Mat. Sb. 6, 3740.Google Scholar
Gjacjauskas, E. P. (1966b) Distribution of the distance between two points in an oval. Litovsk. Mat. Sb. 6, 245248.Google Scholar
Halperin, M. (1960) Some asymptotic results for a coverage problem. Ann. Math. Statist. 31, 10631076.Google Scholar
Hemmer, P. C. (1959) A problem of geometrical probabilities. Det. Kong. Nors. Vid. Selsk. Forh. Trondheim 32, 117120.Google Scholar
Hilliard, J E. (1962a) The counting and sizing of particles in transmission microscopy-Trans. Metallurgical Soc. A.I.M.E. 224, 906–117.Google Scholar
Hilliard, J. E. (1962b) Specification and measurement of microstructural anisotropy. Trans. Metallurgical Soc. A.I.M.E. 224, 12011211.Google Scholar
Hlawka, E. (1950) Über Integrale auf konvexen Körpern. I. Monatsh. Math. 54, 136.CrossRefGoogle Scholar
Holgate, P. (1967) The angle-count method. Biometrika 54, 615623.Google Scholar
Kallmes, O. and Corte, H. (1960) The structure of paper. I. The statistical geometry of an ideal two dimensional fiber network. Tappi. 43, 737752. (errata 44, 448).Google Scholar
Kane, M. W. (1956) The determination of average fibre length. Tappi. 39, 478480.Google Scholar
Kelvin, Lord (Thompson, W. Sir) (1887) On the division of space with minimal partitional area. Phil. Mag. 5th series 24, 503514. (Reprinted in Acta Math. 11, (1888) 121–134.) Google Scholar
Kiang, T. (1966) Random fragmentation in two and three dimensions. Z. Astrophys. 64, 433439.Google Scholar
Kilpper, W. (1949) Entwicklung einer Schnellmethode zur Bestimmung der Faserlänge von Fasergemischen für betriebliche Zwecke. Wochenblatt für Papierfabrikation 75, 160164.Google Scholar
Krengel, U. (1967) A problem on random points in a triangle. Amer. Math. Monthly 74, 814.Google Scholar
Kuusela, K. (1966) A basal mean area tree method in forest inventory. Comm. Inst. For. Fenn. 61, No. 2.Google Scholar
Lukaszewicz, J. (1967) Obituary notice. J. Perkal. Colloq. Math. 17, 147159.Google Scholar
Mack, C. (1956) On clumps formed when convex laminae are placed at random in two or three dimensions. Proc. Camb. Phil. Soc. 52, 246250.Google Scholar
Marks, E. S. (1948) A lower bound for the expected travel among m random points. Ann. Math. Statist. 19, 419422.CrossRefGoogle Scholar
Masuyama, M. (1953) A rapid method of estimating basal area in timber survey — an application of integral geometry to areal sampling problems. Sankhya 12, 291302.Google Scholar
Masuyama, M. and Sengupta, J. M. (1955) On a bias in a crop-cutting experiment (application of integral geometry to areal sampling problems — part V). Sankhya 15, 373376.Google Scholar
Matern, B. (1964) A method of estimating the total lengths of roads by means of a line survey. Studia Forestalia Suecica 18, 6870. (Appendix to von Segebaden (1964)).Google Scholar
Matern, B. On the geometry of the cross-section of a stem. Meddelanden från statens skogsforskningsinstitut 46, No. 11.Google Scholar
Matern, B. Maximum distance between the seedlings in scarified spots. Meddelanden från statens skogsforskningsinstitut 52, No. 4, 237239, 259–261.Google Scholar
Matern, B. and Persson, O. (1965) On the extremum properties of the equilateral triangular lattice and the regular hexagonal network. Research Note No. 7. Dept of Forest Biometry, Royal College of Forestry, Stockholm.Google Scholar
Matzke, E. B. (1945) The three dimensional shapes of bubbles in foams. Proc. Nat. Acad. Sci. U.S.A. 31, 281289.Google Scholar
Matzke, E. B. (1946) The three dimensional shape of bubbles in foam — an analysis of the role of surface forces in three dimensional cell shape determination. Amer. J. Bot. 33, 5880.Google Scholar
Matzke, E. B. and Nestler, J. (1946) Volume-shape relationships in variant foams. A further study of the role of surface forces in three dimensional cell shape determination. Amer. J. Bot. 33, 130144.Google Scholar
Melhuish, F. M. and Lang, A. R. G. (1968) Length and diameters of cotton roots in a clay-loam soil by analysis of surface-ground blocks of resin-impregnated soil. Soil Science 106, 1622.Google Scholar
Miles, R. E. (1969) Poisson flats in Euclidean space. Adv. Appl. Prob. 1 (to appear).Google Scholar
Miller, J. B. (1967) A formula for average foliage density. Aust. J. Bot. 15, 141144.CrossRefGoogle Scholar
Moran, P. A. P. (1968) Statistical theory of a high-speed photoelectric planimeter. Biometrika 55, 419422.Google Scholar
Moran, P. A. P. (1966) A note on recent research in geometric probability. J. Appl. Prob. 3, 453463.Google Scholar
Morisita, M. (1957) A new method for the estimation of density by the spacing method applicable to non-randomly distributed populations. Physiology and Ecology 7, 134144.Google Scholar
Morton, R. R. A. (1966) The expected number and angle of intersections between random curves in a plane. J. Appl. Prob. 3, 559562.Google Scholar
Morton, V. M. (1967) The determination of angular distribution of planes in space. Proc. Roy. Soc. (A). 302, 5168.Google Scholar
Newman, E. I. (1966) A method of estimating the total length of root in a sample. J. Appl. Ecol. 3, 139145.Google Scholar
Ogston, A. G. (1959) The spaces in a uniform random suspension of fibres. Trans. Faraday Soc. 54, 17541757.Google Scholar
Ohotomo, E. (1966) A study on angle count method. Proc. Inst. Statist. Math. 14, 315.Google Scholar
Olson, F. C. W. (1950) Quantitative estimates of filamentous algae. Trans. Amer. Micros. Soc. 69, 272279.Google Scholar
Persson, O. (1965) Distance Methods. II. Research Note No. 6. Dept Forest Biometry, Royal College of Forestry, Stockholm.Google Scholar
Philip, J. R. (1966) The use of point quadrats, with special reference to stem-like organs. Aust. J. Bot. 14, 105125.Google Scholar
Pielou, E. C. (1959) The use of plant-to-plant distances in the study of pattern of plant populations. J. Ecology 47, 603613.Google Scholar
Pielou, E. C. (1960) A single mechanism to account for regular random and aggregated populations. J. Ecology 48, 575584.CrossRefGoogle Scholar
Pielou, E. C. (1964) The spatial pattern of two-phase patchworks of vegetation. Biometrics 20, 156167.Google Scholar
Raynaud, H. (1965) Sur le comportement asymptotique de l'enveloppe convexe d'une image de points tirés au hasard dans R n . C.R. Acad. Sci. Paris 261, 627629.Google Scholar
Rényi, A. and Sulanke, R. (1968) Zufällige konvexe Polygone in einem Ringgebiet. Z. Wahrscheinlichkeitsth. 9, 146157.Google Scholar
Roberts, F. D. K. (1967) A Monte Carlo solution of a two-dimensional unstructured cluster problem. Biometrika 54, 625628.Google Scholar
Roberts, F. D. K. (1968) Random minimal trees. Biometrika 55, 255258.Google Scholar
Roberts, F. D. K. and Storey, S. H. (1968) A three dimensional cluster problem. Biometrika 55, 258260.Google Scholar
Russel, A. M. (1956) Statistical approach to spatial measurement. Amer. J. Phys. 24, 562567.Google Scholar
Santaló, L. A. (1936a) Tntegralgeometrie. 5. Über das kinematische Mass im Raum. Actualités Sci. Indust. No. 357. Hermann, Paris.Google Scholar
Santaló, L. A. (1936b) A problem requiring geometric probability. Rev. Mat. Hisp.-Amer. (2). 11, 8797. (Spanish.).Google Scholar
Santaló, L. A. (1936c) Integral geometry 4. On the kinematic measure in the plane. Abh. Mat. Sem. Univ. Hamburg 11, 222236.Google Scholar
Santaló, L. A. (1936d) Integral geometry 7. New applications of the concept of kinematic measure in the plane and in space. Rev. Acad. Ci. Madrid 33, 451477, 481–504. (Spanish).Google Scholar
Santaló, L. A. (1937) Integral geometry 15. Fundamental formulae for kinematic measure for cylinders and parallel planes. Abh. Math. Sem. Univ. Hamburg 12. (Spanish).Google Scholar
Santaló, L. A. (1939) Integral geometry of unbounded figures. Publ. Inst. Mat. Univ. Nac. Litoral. 1, No. 2. 50 pp. (Spanish).Google Scholar
Santaló, L. A. (1940a) Integral geometry 32. Some integral formulae in the plane and in space. Abh. Math. Sem. Univ. Hamburg 13, 344356. (French).Google Scholar
Santaló, L. A. (1940b) Integral geometry 31. On mean values and geometric probabilities. Abh. Math. Sem. Univ. Hamburg 13. 284294. (Spanish).Google Scholar
Santaló, L. A. (1941) A system of mean values in the theory of geometric probabilities. Rev. Ci. Lima 43, 147154. (Spanish).Google Scholar
Santaló, L. A. (1948) Sobre la distribución de pianos en el espacio. Rev. Un. Mat. Argentina 13, 120124.Google Scholar
Santaló, L. A. (1950) On some integral formulae and mean values concerning movable convex figures in the plane. Univ. Buenos Aires. Contrib. Ci. Ser. A. 1, 2345. (Spanish).Google Scholar
Santaló, L. A. (1952) Some mean values on the hemisphere. Math. Notae 1213, 32–37.Google Scholar
Santaló, L. A. (1966) Valores medios para poligonos formados por rectas al azar en el piano hiperbolico. Universidad Nat. de Tucuman. Revista Ser. A. 16, 2943.Google Scholar
Schmidt, W. M. (1968) Some results in probabilistic geometry. Z. Wahrscheinlichkeitsth. 9, 158162.Google Scholar
Scriven, R. A. and Williams, H. D. (1965) The derivation of angular distributions of planes by sectioning methods. Trans. Metall. Soc. A.I.M.E. 233, 15931602.Google Scholar
Stoffels, A. (1955) Die Genauigkeit der Bestimmung der Stammzahl pro Hektar durch Messung von Stammabständen. Forstw. Zentralblatt 74, 211218.Google Scholar
Stoka, M. I. (1967) Geometrie Integrala. Editura Acad. Repub. Soc. Romania.Google Scholar
Strand, L. (1954) Mâl for forderlingen av individuer over ett omrâde. Medd. fra det norske skogsforsoksvesen. 12, 191207.Google Scholar
Sulanke, R. (1961) Der Verteilung der Sehnenlängen an ebenen und raümlichen Figuren. Math. Nachr. 23, 5174.Google Scholar
Sulanke, R. (1965) Schnittpunkte zufälliger Geraden. Arch. Math. 16, 320324.Google Scholar
Sutherland, D. N. (1966) Comments on Void's simulation of floe formation. J. Colloid Interface Science 22, 300302.Google Scholar
Sutherland, D. N. (1967) A theoretical model of floe structure. J. Colloid Interface Science 25, o373380.Google Scholar
Switzer, P. (1967) Reconstructing patterns from sample data. Ann. Math. Statist. 38, 138154.Google Scholar
Switzer, P. (1965) A random set process in the plane with a Markovian property. Ann. Math. Statist. 36, 18591863.Google Scholar
Vold, M. J. (1963) Computer simulation of floc formation in a colloidal suspension. J. Colloid. Sci. 18, 684695.Google Scholar
von Segebaden, G. (1964) Studies of cross country transport distances and road net extension. Studia Forestalia Suecia No. 18. Stockholm.Google Scholar
Warren Wilson, J. (1963) Estimation of foliage denseness and foliage angle by inclined point quadrats. Aust. J. Bot. 11, 95105.Google Scholar
Warren Wilson, J. (1965) Point quadrat analysis of foliage distribution for plants growing singly or in rows. Aust. J. Bot. 13, 405409.Google Scholar
Williams, R. E. (1968) Space-filling polyhedron: its relation to aggregates of soap bubbles, plant cells, and metal crystalites. Science 161, 276277.Google Scholar