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A fourth note on recent research in geometrical probability

Published online by Cambridge University Press:  01 July 2016

Adrian Baddeley
Adrian Baddeley*
Affiliation:
The Australian National University
*
∗Now at the University of Cambridge.
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Abstract

Recent research on topics related to geometrical probability is reviewed. The survey includes articles on random points, lines, line-segments and flats in Euclidean spaces, the random division of space, coverage, packing, random sets, stereology and probabilistic aspects of integral geometry.

Keywords

GEOMETRICAL PROBABILITYRANDOM POLYTOPESINTEGRAL THEOREMSPOINT PROCESSESRANDOM LINESBUFFON'S PROBLEMBUFFON-SYLVESTER PROBLEMRANDOM SECANTSRANDOM LINE-SEGMENTSPOISSON FLATSRANDOM DIVISION OF SPACEVORONOI AND DELAUNAY TESSELLATIONSRANDOM PATTERNSRANDOM COVERAGERANDOM CLUMPINGRANDOM PACKINGPARKINGMONTE CARLO SIMULATIONRANDOM SETSMATHEMATICAL MORPHOLOGYSTEREOLOGYPARTICLE SIZE DISTRIBUTIONINTEGRAL GEOMETRYGEOMETRICAL STATISTICSEDGE EFFECTS
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 4 , December 1977 , pp. 824 - 860
Copyright
Copyright © Applied Probability Trust 1977 

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References

Bibliography (A): Papers cited in previous bibliographies

Ailam, G. (1966) Moments of coverage and coverage spaces. J. Appl. Prob. 3, 550–555.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, 196–202.CrossRefGoogle Scholar
Ambartzumian, R. V. (1970a) Invariant imbedding in the theory of random lines (In Russian). Izv. Akad. Nauk. Armjan. SSR Ser. Mat. 5, 167–206.Google Scholar
Ambartzumian, R. V. (1971) Probability distributions in the geometry of clusters. Stud. Sci. Math. Hungar. 6, 235–241.Google 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. 5th Berkeley Symp. Math. Statist. Prob. 4, 349–366.Google Scholar
Bate, A. E. and Pillow, M. E. (1947) Mean free path of sound in an auditorium. Proc. Phys. Soc. 59, 535–541.CrossRefGoogle Scholar
Blaisdell, H. and Solomon, H. (1970) On random sequential packing in the plane and a conjecture of Palásti. J. Appl. Prob. 7, 667–698.Google Scholar
Borel, E. (1925) Principes et formules classiques du calcul des probabilités. Traité du calcul des probabilités et de ses applications. Gauthier-Villars, Paris.Google Scholar
Coleman, R. (1969) Random paths through convex bodies. J. Appl. Prob. 6, 430–441.Google Scholar
Crofton, M. W. (1869) Sur quelques théorèmes de calcul intégral. C. R. Acad. Sci. Paris 68, 1469–1470.Google Scholar
Davidson, R. (1969) Line-processes, roads and fibres. Bull. Inst. Internat. Statist. 43 (2), 322–324. Reproduced in Harding and Kendall (1974), 248–251.Google Scholar
Davidson, R. (1970) Construction of line-processes: second order properties. Izv. Akad. Nauk. Armjan. SSR Ser. Mat. 5, 219–234.Google Scholar
Deltheil, R. (1926) Probabilités Géométriques. Traité du calcul des probabilités et de ses applications. Gauthier-Villars, Paris.Google Scholar
Efron, B. (1965) The convex hull of a random set of points. Biometrika 52, 331–343.CrossRefGoogle Scholar
Fairthorne, D. (1964) The distances between random points in two concentric circles. Biometrika 51, 275–277.Google Scholar
Flatto, L. and Konheim, A. G. (1962) The random division of an interval and the random covering of a circle. SIAM Rev. 4, 211–222.Google Scholar
Giger, H. and Hadwiger, H. (1968) Uber Treffzahlwahrscheinlichkeiten in Eikorperfeld. Z. Wahrscheinlichkeitsth. 10, 329–334.Google Scholar
Hammersley, J. M. (1950) The distribution of distance in a hypersphere. Ann. Math. Statist. 21, 447–452.Google Scholar
Hammersley, J. M. (1951) On a certain type of integral associated with circular cylinders. Proc. R. Soc. Lond. A 210, 98–110.Google Scholar
Hammersley, J. M. (1952) Lagrangian integration coefficients for distance functions taken over right, circular cylinders. J. Math. Phys. 31, 139–150.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, 737–752 (errata 44, 448).Google Scholar
Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin, London.Google Scholar
Kingman, J. F. C. (1969) Random secants of a convex body. J. Appl. Prob. 6, 660–672.CrossRefGoogle Scholar
Klee, V. (1969) What is the expected volume of a simplex whose vertices are chosen at random from a given convex body? Amer. Math. Monthly 76, 286–288.Google Scholar
Langford, E. (1969) Probability that a random triangle is obtuse. Biometrika 56, 689–690.Google Scholar
Lord, R. D. (1954) The distribution of distance in a hypersphere. Ann. Math. Statist. 25, 794–798.CrossRefGoogle Scholar
Matheron, G. (1972a) Ensembles fermés aléatoires, ensembles semi-markoviens et polyèdres poissoniens. Adv. Appl. Prob. 4, 508–541.Google Scholar
Miles, R. E. (1969) Poisson flats in Euclidean spaces. I. Adv. Appl. Prob. 1, 211–237.Google Scholar
Miles, R. E. (1971) Poisson flats in Euclidean spaces. II. Adv. Appl. Prob. 3, 1–43.CrossRefGoogle Scholar
Miles, R. E. (1972a) Multidimensional perspectives on stereology. J. Microsc. 95, 181–195. Also in Weibel, et al. (1972).Google Scholar
Moran, P. A. P. (1966) A note on recent research in geometrical probability. J. Appl. Prob. 3, 453–463.Google Scholar
Moran, P. A. P. (1968) Statistical theory of a high-speed photoelectric planimeter. Biometrika 55, 419–422.CrossRefGoogle Scholar
Moran, P. A. P. (1969) A second note on recent research in geometrical probability. Adv. Appl. Prob. 1, 73–89.Google Scholar
Ogston, A. G. (1958) The spaces in a uniform random suspension of fibres. Trans. Faraday Soc. 54, 1754–1757.Google Scholar
Palásti, I. (1960) On some random space filling problems. Publ. Math. Inst. Hung. Acad. Sci. 5, 353–360.Google Scholar
Rényi, A. and Sulanke, R. (1968) Zufällige konvexe Polygone in einem Ringgebiet. Z. Wahrscheinlichkeitsth. 9, 146–157.CrossRefGoogle Scholar
Roach, S. A. (1968) The Theory of Random Clumping. Metheun, London.Google Scholar
Schmidt, W. M. (1968) Some results in probabilistic geometry. Z. Wahrscheinlichkeitsth. 9, 158–162.Google Scholar
Shepp, L. A. (1972a) Covering the circle with random arcs. Israel J. Math. 11, 328–345.Google Scholar
Shepp, L. A. (1972b) Covering the line with random intervals. Z. Wahrscheinlichkeitsth. 23, 163–170.CrossRefGoogle Scholar
Varga, O. (1936) Integralgeometrie 3. Croftons Formeln für den Raum. Math. Z. 40, 387–405.Google Scholar

Bibliography (B): Papers not previously cited

Adams, D. J. and Matheson, A. J. (1972) Computation of dense random packings of hard spheres. J. Chem. Phys. 56, 1989–1994.Google Scholar
Ailam, G. (1970) The asymptotic distribution of the measure of random sets with applications to the classical occupancy problem and suggestions for curve fitting. Ann. Math. Statist. 41, 427–439.Google Scholar
Akeda, Y. and Hori, M. (1975) Numerical test of Palasti's conjecture on two-dimensional random packing density. Nature (London) 254, 318–319.CrossRefGoogle Scholar
Ambartzumian, R. V. (1970b) Random fields of segments and random mosaics on a plane. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 369–381.Google Scholar
Ambartzumian, R. V. (1973a) On the solution of the Buffon-Sylvester problem in R 3 . Sov. Math. Doklady 14, 863–866.Google Scholar
Ambartzumian, R. V. (1973b) The solution to the Buffon-Sylvester problem in R 3 . Z. Wahrscheinlichkeitsth. 27, 53–74.Google Scholar
Ambartzumian, R. V. (1974a) Combinatorial solution of the Buffon-Sylvester problem. Z. Wahrscheinlichkeitsth. 29, 25–31.Google Scholar
Ambartzumian, R. V. (1974b) The solution to the Buffon-Sylvester problem and stereology. Preprint, International Mathematical Congress, Vancouver, August 1974.Google Scholar
Ambartzumian, R. V. (1974c) Convex polygons and random tessellations. In Harding, and Kendall, (1974), 176–191.Google Scholar
Anderssen, R. S. and Bloomfield, P. (1975) Properties of the random search in global optimisation. J. Optim. Th. Applic. 6, 383–398.Google Scholar
Anderssen, R. S., Brent, R. P., Daley, D. J. and Moran, P. A. P., (1976) Concerning and a Taylor series method. SIAM J. Appl. Math. 30, 22–30.Google Scholar
Anderssen, R. S. and Jakeman, A. J. (1974) On computational stereology, Proc. 6th Austral. Comp. Conf. Sydney 2, 353–362.Google Scholar
Anderssen, R. S. and Jakeman, A. J. (1975a) Product integration for functionals of particle size distributions. Utilitas Math. 8, 111–126.Google Scholar
Anderssen, R. S. and Jakeman, A. J. (1975b) Abel type integral equations in stereology. II. Computational methods of solution and the random spheres approximation. J. Microsc. 105, 135–153.CrossRefGoogle Scholar
Anderssen, R. S. and Jakeman, A. J. (1976) Computational methods in stereology. In Underwood (1976), 13–18.Google Scholar
Artstein, Z. and Vitale, R. A. (1975) A strong law of large numbers for random compact sets. Ann. Prob. 3, 879–882.Google Scholar
Bach, G. (1959) Über die Größenverteilung von Kugelschnitten in durchsichtigen Schnitten endlicher Dicke. Z. Wiss. Mikrosk. 64, 265–270.Google Scholar
Bach, G. (1963) Über die Bestimmung von characteristichen Größen einer Kugelverteilung aus der Verteilung der Schnittkreise. Z. Wiss. Mikrosk. Also in International Society for Stereology (1963).Google Scholar
Bach, G. (1964) Über die Bestimmung der Anzahl dreiachsiger Ellipsoide aus der Anzahl ihrer Schittellipsen in zufälliger Schnittebener. Z. Angew. Math. Phys. 15, 205–209.Google Scholar
Bach, G. (1965) Über die Bestimmung von characteristichen Größen einen Kugelverteilung aus der unvollständige Verteilung der Schnittkreise. Metrika 9, 228–233.Google Scholar
Baddeley, A. (1977) Integrals on a moving manifold and geometrical probability. Adv. Appl. Prob. 9, 588–603.CrossRefGoogle Scholar
Bartlett, M. S. (1975) The Statistical Analysis of Spatial Pattern. Chapman and Hall, London.Google Scholar
Bartlett, M. S. (1967) The spectral analysis of line-processes. Proc. 5th Berkeley Symp. Math. Statist. Prob. 3, 135–153.Google Scholar
Boots, B. N. (1974) Delaunay triangles: an alternative approach to point pattern analysis. Proc. Assoc. Amer. Geographers 6, 26–29.Google Scholar
Carlsson, S. and Grenander, U. (1967) Statistical approximation of plane convex sets. Skand. Aktuarietidskr. 50, 113–127.Google Scholar
Chakerian, G. D. (1972) The mean volume of boxes and cylinders circumscribed about a convex body. Israel J. Math. 12, 249–256.Google Scholar
Choquet, G. (1955) Theory of capacities. Ann. Inst. Fourier 5, 131–295.Google Scholar
Clement, C. F. (1972) The distribution of path lengths through a porous material. J. Phys. D (Appl. Phys.) 5, 793–798.Google Scholar
Coleman, R. (1972) Sampling procedures for the lengths of random straight lines. Biometrika 59, 415–426.Google Scholar
Coleman, R. (1973) Procedures for sampling the lengths of random straight lines. Edit. Acad. Republ. Soc. Romania 1973, 265–268. Also in Proc. 4th Conf. Prob. Theory, Brasov (1971).Google Scholar
Coleman, R. (1974) The distance from a given point to the nearest end of one member of a random process of linear segments. In Harding, and Kendall, (1974), 192–201.Google Scholar
Coleman, R. (to appear) Random paths through rectangles and cubes.Google Scholar
Cooke, P. J. (1974) Bounds for coverage probabilities with applications to sequential coverage problems. J. Appl. Prob. 11, 281–293.Google Scholar
Crain, I. K. and Miles, R. E. (1976) Monte Carlo estimates of the distributions of the random polygons determined by random lines in the plane. J. Statist. Comp. Simul. 4, 293–325.Google Scholar
Cruz Orive, L.–M. (1976) Correction of stereological parameters from biased samples on nucleated particle phases. In Underwood, (1976), 79–82.Google Scholar
d'Athis, P. (1976) Analysis of a set of spherical cells relative to their volume. In Underwood, (1976), 507–508.Google Scholar
Davidson, R. (1974a) Stochastic processes of flats and exchangeability. In Harding, and Kendall, (1974), 13–45.Google Scholar
Davidson, R. (1974b) Positive-definiteness of product-moment densities of line-processes. In Harding, and Kendall, (1974).Google Scholar
Davidson, R. (1974c) Exchangeable point-processes. In Harding, and Kendall, (1974).Google Scholar
Davies, A. G. (1973) Estimation of number and diameter of isodiametric spherical particles in microtome sections. J. Microsc. 98, 79–83.Google Scholar
Davy, P. J. (1976a) Math. Rev. 52, No 968.Google Scholar
Davy, P. J. (1976b) Projected thick sections through multidimensional particle aggregates. J. Appl. Prob. 13, 714–722.CrossRefGoogle Scholar
Davy, P. J. and Miles, R. E. (to appear) Sampling theory for opaque spatial specimens.Google Scholar
De Hoff, R. T. and Gehl, S. M. (1976) Quantitative microscopy of lineal features in three dimensions. In Underwood, (1976), 29–40.Google Scholar
Delfiner, P. (1972) A generalisation of the concept of size. In Weibel, et al. (1972), 203–216.Google Scholar
Dodds, J. A. (1975) Simplest statistical geometric model of the simplest version of the multicomponent random packing problem. Nature (London) 256, 187–189.Google Scholar
Dodson, C. T. J. (1971) Spatial variability and the theory of sampling in random fibrous networks. J. R. Statist. Soc. B 33, 88–94.Google Scholar
Dolby, J. L. and Solomon, H. (1975) Information density phenomena and random loose packing. J. Appl. Prob. 12, 364–370.Google Scholar
Downs, T. D. (1972) Orientation statistics. Biometrika 59, 665–676.Google Scholar
Dullien, F. A. L. and Chang, K. S. (1976) On size distribution methods. In Underwood, (1976), 499–502.Google Scholar
Eberl, W. and Hafner, R. (1971) Die Asymptotische Verteilung von Koinzidenzen. Z. Wahrscheinlichkeitsth. 18, 322–332.Google Scholar
Elias, H., (ed.) (1967) Proc. 2nd Internat. Congress for Stereology, Chicago 1967. Springer-Verlag, Berlin.Google Scholar
Evans, D. A. and Clarke, K. R. (1975) Estimation of embedded particle properties from plane section intercepts. Adv. Appl. Prob. 7, 542–560.CrossRefGoogle Scholar
Filipescu, D. (1971) Formule integrale pentru sisteme de figuri convexe aleatoare în spatii riemanniene. Bull. Inst. Politehn. Bucuresti 33, (5), 25–30.Google Scholar
Filipescu, D. (1972a) Formule integrale pentru sisteme de figuri convexe aleatoare în spatii euclidiene. Bul. Inst. Politehn. Bucuresti 34 (1), 21–36.Google Scholar
Filipescu, D. (1972b) Multimi convexe aleatoare în planul euclidian. Bul. Inst. Politehn. Bucuresti 34 (5), 21–31.Google Scholar
Firey, W. J. (1974) Kinematic measures for sets of support figures. Mathematika 21, 270–281.Google Scholar
Fischer, R. A. and Miles, R. E. (1973) The role of spatial pattern in the competition between crop plants and weeds. A theoretical analysis. Math. Biosci. 18, 335–350.Google Scholar
Fisher, L. (1972) A survey of the mathematical theory of multidimensional point processes. In Lewis, (1972), 468–513.Google Scholar
Flatto, L. (1973) A limit theorem for random coverings of a circle. Israel J. Math. 15, 167–184.Google Scholar
Fratila, E. (1968) O metoda statistica pentru evaluera lungimii unui arc de curba plana. Studii si Cercetari Matem. 20 (8), 1153–1157.Google Scholar
Fratila, E. (1971) Une méthode statistique d'évaluer la longueur d'un arc de courbe spatiale. Rev. Roum. Math. Pures Appl. 16, 493–498.Google Scholar
Gamba, A. (1975) Random packing of equal spheres. Nature 256, 521–522.Google Scholar
Garwood, F. (1974) The vectorial representation of the frequency of encounters of freely flowing vehicles. J. Appl. Prob. 11, 797–808.Google Scholar
Gečiauskas, E. (1970) On linear and plane searches. Selected Trans. Math. Statist. Prob. 9, 311–315.Google Scholar
Goldman, A. S., Lewis, H. D. and Visscher, W. M. (1974) Random packing of particles: simulation with a one dimensional parking model. Technometrics 16, 301–309.CrossRefGoogle Scholar
Gotoh, K. and Finney, J. L. (1974) Statistical geometrical approach to random packing density of equal spheres. Nature (London) 252, 202–205.Google Scholar
Gotoh, K. and Finney, J. L. (1975) Reply to A. Gamba. Nature (London) 256, 522.Google Scholar
Grenander, U. (1973) Statistical geometry: a tool for pattern analysis. Bull Amer. Math. Soc. 79, 829–856.CrossRefGoogle Scholar
Groemer, H. (1973) On some mean values associated with a randomly selected simplex in a convex set. Pacific J. Math. 45, 525–533.Google Scholar
Groemer, H. (1974) On the mean value of the volume of a random polytope in a convex set. Arch. Math. (Basel) 25, 86–90.Google Scholar
Guggenheimer, H. (1973) A formula of Furstenberg–Tzkoni type. Israel J. Math. 14, 281–282.Google Scholar
Hafner, R. (1972a) Die asymptotische Verteilung von mehrfachen Koinzidenzen. Z. Wahrscheinlichkeitsth. 21, 96–108.Google Scholar
Hafner, R. (1972b) The asymptotic distribution of random clumps. Computing 10, 335–351.Google Scholar
Halasz, S. and Kleitman, D. J. (1974) A note on random triangles. Studies in Appl. Math. 53, 225–237.Google Scholar
Hamilton, J. F., Lawton, W. H. and Trabka, E. A. (1972) Some spatial and temporal point processes in photographic science. In Lewis, (1972), 817–867.Google Scholar
Hamilton, W. D. (1971) Geometry for the selfish herd. J. Theoret. Biol. 31, 295–311.Google Scholar
Hammersley, J. M. (1972) Stochastic models for the distribution of particles in space. Suppl. Adv. Appl. Prob. 1972, 47–68.Google Scholar
Harding, E. F. and Kendall, D. G. (1974) Stochastic Geometry: A Tribute to the Memory of Rollo Davidson. Wiley, London.Google Scholar
Hasofer, A. M. (1974) A probability model for the fusion of nucleoli in metabolic cells. J. Theoret. Biol. 45, 305–310.Google Scholar
Haughey, D. P. and Beveridge, G. S. G. (1969) Structural properties of packed beds—a review. Canad. J. Chem. Eng. 47, 130–140.Google Scholar
Hawkes, J. (1973) On the covering of small sets by random intervals. Quart. J. Math. Oxford (2) 24, 427–432.Google Scholar
Hennig, A. (1972) The problem of paired spheres and of straight chains of spheres in slices or polished sections. J. Microsc. 96, 271–283.Google Scholar
Hilliard, J. E. (1972) Stereology: an experimental viewpoint. Suppl. Adv. Appl. Prob. 1972, 92–111.Google Scholar
Hilliard, J. E. (1976) Assessment of sampling errors in stereological analyses. In Underwood, (1976), 59–67.Google Scholar
Hilliard, J. E. and Anacker, D. C. (1974) Estimation of the size and orientation distribution of filamentary features from measurements on a two-dimensional section. J. Microsc. 102, 41–48.Google Scholar
Hoffmann-Jørgensen, J. (1973) Coverings of metric spaces with randomly placed balls. Math. Scand. 32, 169–186.Google Scholar
Holgate, P. (1972) The use of distance methods for the analysis of spatial distribution of points. In Lewis, (1972), 122–135.Google Scholar
International Society for Stereology (1963) Proceedings of the First International Congress for Stereology (Vienna, April 1963).Google Scholar
Iwata, H. and Homma, T. (1974) Distribution of coordination numbers in random packing of homogeneous spheres. Powder Tech. 10, 79–83.Google Scholar
Jakeman, A. J. and Anderssen, R. S. (1974) A note on numerical methods for the thin-section model. In Saunders, and Goodchild, (1974) 2, 4–5.Google Scholar
Jakeman, A. J. and Anderssen, R. S. (1975) Abel type integral equations in stereology. I. General discussion. J. Microsc. 105, 121–133.Google Scholar
Jakeman, A. J. and Anderssen, R. S. (1976) On optimal forms for stereological data. In Underwood, (1976), 69–74.Google Scholar
Joyce, W. B. (1974a) Classical-particle description of photons and phonons. Phys. Rev. 9, 3234–3256.Google Scholar
Joyce, W. B. (1974b) Geometrical properties of random particles and the extraction of photons from electroluminescent diodes. J. Appl. Phys. 45, 2229–2253.Google Scholar
Joyce, W. B. (1975) Sabine's reverberation time and ergodic auditoriums. J. Acoust. Soc. Amer. 58, 643–655.Google Scholar
Kahane, J.-P. (1968) Some Random Series of Functions. Heath, Lexington, Mass. Google Scholar
Kallenberg, O. (1976) On the structure of stationary flat processes. Publication 1976–4, Department of Mathematics, University of Göteborg, Sweden.Google Scholar
Kallmes, O., Corte, H. and Bernier, G. (1961) The structure of paper, II. The statistical geometry of a multiplanar fibre network. Techn. Assn. Pulp and Paper Ind. 44, 519–528.Google Scholar
Kendall, D. G. (1974a) An introduction to stochastic geometry. In Harding, and Kendall, (1974), 3–9.Google Scholar
Kendall, D. G. (1974b) Foundations of a theory of random sets. In Harding, and Kendall, (1974), 322–376.Google Scholar
Kendall, D. G. and Harding, E. F. (1974) Stochastic Analysis: A Tribute to the Memory of Rollo Davidson. Wiley, London.Google Scholar
Kester, A. (1975) Asymptotic normality of the number of small distances between random points in a cube. Stoc. Proc. Appl. 3, 45–54.Google Scholar
Kingman, J. F. C. (1975) Review of Random Sets and Integral Geometry. Bull. Amer. Math. Soc. 81, 844–847.Google Scholar
Klahn, D., Austin, D., Mukherjee, A. K. and Dorn, J. E. (1972) The importance of geometric statistics to dislocation motion. Suppl. Adv. Appl. Prob. 1972, 112–150.Google Scholar
Klein, J. C. and Serra, J. (1972) The texture analyser. In Weibel, et al. (1972), 349–356.Google Scholar
Kneser, H. (1963) Schnitte durch Tetraeder. In International Society for Stereology (1963).Google Scholar
Krickeberg, K. (1972) Theory of hyperplane processes. In Lewis, (1972).Google Scholar
Krickeberg, K. (1973) Moments of point processes. Lecture Notes is Mathematics 296, Springer-Verlag, Berlin; 70–101; also in Harding and Kendall (1974), 89–113.Google Scholar
Krickeberg, K. (1974) L'estimation du spectre de processus de droites. Ann. Sci. Univ. Clermont 51, fasc. 9, 35–42.Google Scholar
Langford, E. (1970) A problem in geometric probability. Math. Mag. 43, 237–244.Google Scholar
Lehman, R. L. and Brisbane, R. W. (1968) Random-drift sampling—a study by computer simulation. Nucl. Instr. Meth. 64, 269–277.Google Scholar
Lewis, P. A. W., (ed.) (1972) Stochastic Point Processes: Statistical Analysis, Theory and Applications. Wiley, New York.Google Scholar
Ling, K. D. (1971) Small sample distribution of the measure of a random linear set. Nanta Math. 5, 41–48.Google Scholar
Little, D. V. (1974) A third note on recent research in geometrical probability. Adv. Appl. Prob. 6, 103–130.Google Scholar
Lomnicki, Z. A. and Zaremba, S. K. (1957) A further instance of the Central Limit Theorem for dependent random variables. Math. Z. 66, 490–494.Google Scholar
Marcus, A. (1964) A stochastic model of the formation and survival of lunar craters, I. Icarus 3, 460–472.Google Scholar
Marcus, A. (1966a) II. Icarus 5, 165–177.Google Scholar
Marcus, A. (1966b) III. Icarus 5, 178–189.Google Scholar
Marcus, A. (1966c) IV. Icarus 5, 190–200.Google Scholar
Marcus, A. (1966d) V. Icarus 5, 590–605.Google Scholar
Marcus, A. (1967a) VI. Icarus 6, 56–74.Google Scholar
Marcus, A. (1967b) Further interpretations of crater depth statistics and lunar history. Icarus 7, 407–409.Google Scholar
Marcus, A. (1967c) Statistical theories of lunar and Martian craters. In Mantles of the Earth and Terrestrial Planets, ed. Runcorn, S. K. Wiley-Interscience, London, 417–424.Google Scholar
Marcus, A. (1972) Some point process models of lunar and planetary surfaces. In Lewis, (1972), 682–699.Google Scholar
Mardia, K. V. (1972) Statistics of Directional Data. Academic Press, London.Google Scholar
Marsaglia, G. (1972) Choosing a point from the surface of a sphere. Ann. Math. Statist. 43, 645–646.Google Scholar
Mathematical Foundations of Stereology (1976) Proceedings of a workshop in Underwood (1976), 439–464.Google Scholar
Matheron, G. (1972b) Random sets theory and its applications to stereology. In Weibel, et al. (1972).Google Scholar
Matheron, G. (1974a) Un théorème d'unicité pour les hyperplans poissoniens. J. Appl. Prob. 11, 184–189.Google Scholar
Matheron, G. (1974b) Hyperplans poissoniens et compacts de Steiner. Adv. Appl. Prob. 6, 563–579.Google Scholar
Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Matheson, A. J. (1974) Computation of a random packing of hard spheres. J. Phys. C 7, 2569–2576.Google Scholar
Mayhew, T. M. and Cruz Orive, L.-M. (1973) Stereological correction procedures for estimating true volume proportions from biased samples. J. Microsc. 99, 287–299.Google Scholar
Mayhew, T. M. and Cruz Orive, L.-M. (1974) Caveat on the use of the Delesse principle of areal analysis for estimating component volume densities. J. Microsc. 102, 195–207.Google Scholar
McMullen, P. (1974) A dice probability problem. Mathematika 21, 193–198.Google Scholar
Melnyk, T. W. and Rowlinson, J. S. (1971) The statistics of the volumes covered by systems of penetrating spheres. J. Comput. Phys. 7, 383–393.Google Scholar
Miles, R. E. (1972b) The random division of space. Suppl. Adv. Appl. Prob. 1972, 243–266.Google Scholar
Miles, R. E. (1972c) Appendix to ‘The migration of lymphocytes across the vascular endothelium in lymphoid tissue. A reexamination’ by G. I. Schoefl. J. Exp. Medicine 136, 584–588.Google Scholar
Miles, R. E. (1973a) A simple derivation of a formula of Furstenberg and Tzkoni. Israel J. Math. 14, 278–280.Google Scholar
Miles, R. E. (1973b) The various aggregates of random polygons determined by random lines in a plane. Adv. Math. 10, 256–290.Google Scholar
Miles, R. E. (1973c) On the information derivable from random plane and line sections of an aggregate of convex particles embedded in an opaque medium. Proc. Fourth Conf. on Prob. Theory, Brasov 1971; Edit. Acad. Republ. Soc. România 1973, 305–317.Google Scholar
Miles, R. E. (1974a) On the elimination of edge effects in planar sampling. In Harding, and Kendall, (1974), 227–247.Google Scholar
Miles, R. E. (1974b) The estimation of aggregate and overall characteristics from thick sections by transmission microscopy. In Saunders, and Goodchild, (1974), 6–7.Google Scholar
Miles, R. E. (1974c) The fundamental formula of Blaschke in integral geometry and geometrical probability, and its iteration, for domains with fixed orientations. Austral. J. Statist. 16, 111–118.CrossRefGoogle Scholar
Miles, R. E. (1975a) Direct derivations of certain surface integral formulae for the mean projections of a convex set. Adv. Appl. Prob. 7, 818–829.Google Scholar
Miles, R. E. (1975b) On estimating aggregate and overall characteristics from thick sections by transmission microscopy. In Underwood, (1976).Google Scholar
Miles, R. E. and Davy, P. J. (1976) Precise and general conditions for the validity of a comprehensive set of stereological fundamental formulae. J. Microsc. 107, 211–226.Google Scholar
Miles, R. E. and Davy, P. J. (to appear) On the choice of quadrats in stereology.Google Scholar
Moore, M. (1972) Consistency in the reconstruction of patterns from sample data. Canad. Math. Bull. 15, 305–307.Google Scholar
Moore, M. (1973) Non-stationary random set processes with application to pattern reconstruction. J. Appl. Prob. 10, 857–863.Google Scholar
Moore, M. (1974) Anisotropically random mosaics. J. Appl. Prob. 11, 374–376.Google Scholar
Moran, P. A. P. (1972) The probabilistic basis of stereology. Suppl. Adv. Appl. Prob. (1972) 69–91.Google Scholar
Moran, P. A. P. (1973a) The random volume of interpenetrating spheres in space. J. Appl. Prob. 10, 483–490.Google Scholar
Moran, P. A. P. (1973b) A central limit theorem for exchangeable variates with geometric applications. J. Appl. Prob. 10, 837–846.Google Scholar
Moran, P. A. P. (1974) The volume occupied by normally distributed spheres. Acta. Math. 133, 273–286.Google Scholar
Moran, P. A. P. (1975) Quaternions, Haar measure and the estimation of a palaeomagnetic rotation. In Perspectives in Probability and Statistics, ed. Gani, J. Applied Probability Trust, Sheffield, 295–301.Google Scholar
Myers, E. J. (1963a) Sectioning of polyhedrons. In International Socicty for Stereology (1963).Google Scholar
Myers, E. J. (1963b) Quantitative metallography of cylinders, cubes and other polyhedrons. In International Society for Stereology (1963).Google Scholar
Neuts, M. F. and Purdue, P. (1971) Buffon in the round. Math. Mag. 44, 81–89.Google Scholar
Neyman, J. and Scott, E. L. (1972) Processes of clustering and applications. In Lewis, (1972), 646–681.Google Scholar
Nicholson, W. L. (Ed.) (1972) Proc. Symp. on Statist. Prob. Problems in Metallurgy, Seattle, Washington August 1971. Supplement to Adv. Appl. Prob., December 1972.Google Scholar
Nicholson, W. L. (1976) Estimation of linear functionals by maximum likelihood. In Underwood, (1976), 19–24.Google Scholar
Ogawa, T. and Tanemura, M. (1974) Geometrical considerations on hard core problems. Prog. Theoret. Phys. 51, 399–417.Google Scholar
Ogston, A. G., Preston, B. N. and Wells, J. D. (1973) On the transport of compact particles through solutions of chain polymers. Proc. R. Soc. London A 333, 297–316.Google Scholar
Papangelou, F. (1972) Summary of some results on point and line processes. In Lewis, (1972), 522–532.Google Scholar
Papangelou, F. (1974a) On the Palm probabilities of processes of points and processes of lines. In Harding and Kendall (1974), 114–147.Google Scholar
Papangelou, F. (1974b) On an unfinished manuscript of R. Davidson's. In Harding and Kendall (1974), 252–255.Google Scholar
Parker, P. R. and Cowan, R. J. (1976) Some properties of line segment processes. J. Appl. Prob. 13, 96–107.Google Scholar
Piefke, F. (1976) Estimation of linear properties of spherical bodies in thin foils from their projections. In Underwood (1976), 497–498.Google Scholar
Pleijel, A. (1956a) Zwei kurze Beweise der isoperimetrischen Ungleichung. Arch. Math. 7, 317–319.Google Scholar
Pleijel, A. (1956b) Zwei kennzeichnende Kreisergenschaften. Arch. Math. 7, 420–424.Google Scholar
Prékopa, A. (1972) On the number of vertices of random convex polyhedra. Period. Math. Hungar. 2, 259–282.Google Scholar
Quickenden, T. I. and Tan, G. K. (1974) Random packing in two dimensions and the structure of monolayers. J. Colloid and Interface Sci. 48, 382–393.Google Scholar
Reed, W. J. (1974) Random points in a simplex. Pacific J. Math. 54, 183–198.Google Scholar
Ruben, H. (1967) An intrinsic formula for volume. J. Reine Angew. Math. 226, 116–119.Google Scholar
Ruben, H. (1970) On a class of double space integrals with applications in mensuration, statistical physics and geometrical probability. In Proc. 12th Biennial Seminar Canad. Math. Congress, 209–230.Google Scholar
Ruben, H. and Reed, W. J. (1973) A more general form of a theorem of Crofton. J. Appl. Prob. 10, 479–482.Google Scholar
Santaló, L. A. (1970) Mean values and curvatures. Izv. Akad. Nauk. Armjan. SSR Ser. Mat. 5, 286–295. Also in Harding and Kendall (1974), 165–175.Google Scholar
Santaló, L. A. (1970/71) Probabilities on convex bodies and cylinders. (In Spanish) Rev. Un. Mat. Argentina 25, 95–104.Google Scholar
Santaló, L. A. and Yanez, I. (1972) Averages for polygons formed by random lines in Euclidean and hyperbolic planes. J. Appl. Prob. 9, 140–157.Google Scholar
Saunders, J. V. and Goodchild, D. J., (eds.) (1974) Electron Microscopy 1974. Australian Academy of Sciences, Canberra.Google Scholar
Scheaffer, R. L. (1973) Tests for uniform clustering and randomness. Comm. Stats. 2, 479–492.Google Scholar
Schneider, R. (1972) The mean surface area of the boxes circumscribed about a convex body. Ann. Polon. Math. 25, 325–328.Google Scholar
Schuster, E. F. (1974) Buffon's needle experiment. Amer. Math. Monthly 81, 26–29.Google Scholar
Serra, J. (1969) Introduction à la morphologie mathématique. Cahiers Centre Morph. Math., fasc. 3. Fontainebleau (France.) Google Scholar
Serra, J. (1972) Stereology and structuring elements. In Weibel, et al. (1972), 93–103.Google Scholar
Serra, J. (1976) Stochastic models in stereology: strength and weaknesses. In Underwood, (1976), 83–86.Google Scholar
Solomon, H. and Wang, P. C. C. (1972) Non-homogeneous Poisson fields of random lines with applications to traffic flow. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 383–400.Google Scholar
Spitzer, F. and Widom, H. (1961) The circumference of a convex polygon. Proc. Amer. Math. Soc. 12, 506–509.Google Scholar
Stickels, C. A. and Hucke, E. E. (1964) Measurement of dihedral angles. Trans. Metall. Soc. AIME 230, 795–801.Google Scholar
Störmer, H. (1971) Zufällige Überdeckungen auf dem Kreis. Z. Angew. Math. Mech. 51, 91–96.Google Scholar
Streit, F. (1973) Mean value formulae for a class of random sets. J. R. Statist. Soc. B 35, 437–444.Google Scholar
Streit, F. (1975) Results on the intersection of randomly located sets. J. Appl. Prob. 12, 817–823.Google Scholar
Sulanke, R. and Wintgen, P. (1972) Zufällige konvexe Polyeder im N-dimensionalen euklidischen Raum. Period. Math. Hungar. 2, 215–221.Google Scholar
Underwood, E. E. (1972) The stereology of projected images. In Weibel, et al. (1972), 25–44.Google Scholar
Underwood, E. E., (Ed.) (1976) Stereology 4. Proceedings of the 4th International Congress for Stereology. U.S. National Bureau of Standards Special Publication No. 431.Google Scholar
Visscher, W. M. and Bolsterli, M. (1972) Random packing of equal and unequal spheres in two and three dimensions. Nature (London) 239, 504–507.Google Scholar
Watson, G. S. (1974) Geometrical Statistics: a report of the Satellite Symposium. Rev. Internat. Statist. Inst. 42, 211–212.Google Scholar
Weibel, E. R. (1974) Selection of the best method in stereology. J. Microsc. 100, 261–269.Google Scholar
Weibel, E. R., Meek, G., Ralph, B., Echlin, P. and Ross, R. (1972) Stereology 3: Proceedings of the 3rd International Congress for Stereology. Published for the Royal Microscopical Soc. by Blackwell, Oxford. Articles are reproduced in J. Microsc. 95.Google Scholar
‘What is Random Packing?’ (1972) Nature (London) 239, 488–489.Google Scholar
Whitehouse, W. J. (1974) A stereological method for calculating internal surface areas in structures which have become anisotropic as the result of linear expansions or contractions. J. Microsc. 101, 169–176.Google Scholar
Widom, B. and Rowlinson, J. S. (1970) New model for the study of liquid-vapour phase transitions. J. Chem. Phys. 52, 1670–1684.Google Scholar
Wschebor, M. (1973a) Sur le recouvrement du cercle par des ensembles placées au hasard. Israel J. Math. 15, 1–11.Google Scholar
Wschebor, M. (1973b) Sur un théorème de Leonard Shepp. Z. Wahrscheinlichkeitsth. 27, 179–184.Google Scholar