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On the Mean Geometric Densities of Random Closed Sets, and Their Estimation: Application to the Estimation of the Mean Density of Inhomogeneous Fibre Processes

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Part of the book series: Mathematics in Industry ((TECMI,volume 12))

It has been a great honour for me to deliver the “Alan Tayler Lecture” in this ECMI Conference, to honour one of the leading founders and Presidents of ECMI. I have collaborated with Alan for many years, especially during my term as Chairman of the Educational Committee, and later during the first ECMI-HCM Project. While he was already very ill, he found the way to participate (even though only for a couple of days) in a workshop in Milan, opening ECMI to the Italian academic and industrial community, and highly supported the birth of MIRIAM (the Milan Research Centre for Industrial and Applied Mathematics).

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References

  1. Ambrosio, L., Capasso, V., Villa, E.: On the approximation of geometric densities of random closed sets. RICAM Report N. 2006-14, 2006.

    Google Scholar 

  2. Ambrosio, L., Colesanti, A., Villa, E.: First order Steiner formulas for some classes of closed sets. An application to stochastic geometry. In preparation, 2006.

    Google Scholar 

  3. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000)

    MATH  Google Scholar 

  4. Araujo, A., Giné, E.: The Central Limit Theorem for Real and Banach Valued Random Variables. John Wiley & Sons, New York (1980)

    MATH  Google Scholar 

  5. Baddeley, A.J., Molchanov, I.S.: On the expected measure of a random set. In: Proceedings of the International Symposium on Advances in Theory and Applications of Random Sets (Fontainebleau, 1996). World Sci. Publishing, River Edge, NJ, 1, 3-20 (1997)

    Google Scholar 

  6. Benes, V., Rataj, J.: Stochastic Geometry: Selected Topics. Kluwer Academic Publishers, Norwell (2004).

    MATH  Google Scholar 

  7. Bosq, D.: Linear Processes in Function Spaces. Theory and Applications. Lecture Notes in Statistics 149. Springer-Verlag, New York (2000).

    Google Scholar 

  8. Capasso, V. (ed): Mathematical Modelling for Polymer Processing. Polymerization, Crystallization, Manufacturing. Mathematics in Industry Vol 2. Springer Verlag, Heidelberg (2003)

    Google Scholar 

  9. Capasso, V., Micheletti, A.: The local mean volume and surface densities for inhomogeneous random sets. Rend Circ. Mat. Palermo Suppl., 65,49-66 (2000).

    MathSciNet  Google Scholar 

  10. Capasso, V., Micheletti, A.: Local spherical contact distribution function and local mean densities for inhomogeneous random sets. Stochastics and Stoch. Rep., 71, 51-67, (2000).

    Article  MATH  MathSciNet  Google Scholar 

  11. Capasso, V., Micheletti, A.: Stochastic geometry and related statistical problems in biomedicine. In: A. Quarteroniet al. (Eds), Complex Systems in Biomedicine. Springer, Milano (2005).

    Google Scholar 

  12. Capasso, V., Micheletti, A., Kernel-like estimators of the intensity of inhomogeneous fibre processes. Preprint (2006).

    Google Scholar 

  13. Capasso, V., Tayler, A. (Eds): ECMI Brochure, Bari (1994)

    Google Scholar 

  14. Capasso, V., Villa, E.: On the continuity and absolute continuity of random closed sets. Stoch. An. Appl. 24, 381-397, (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Capasso, V., Villa, E.: Some remarks on the continuity of random closed sets. In: Lechnerová R., Saxl I., Beneš V. (eds), Proceedings of the Intenational Conference in Stereology, Spatial Statistics and Stochastic Geometry, UCMP, Prague, 69-74, (2006)

    Google Scholar 

  16. Capasso, V., Villa, E.: On the geometric densities of random closed sets. RICAM Report 13/2006, Linz, (2006).

    Google Scholar 

  17. Capasso, V., Villa, E.: On mean densities of inhomogeneous geometric processes arising in material science and medicine. Preprint (2006)

    Google Scholar 

  18. Chaplain, M.A.J., Anderson, A.R.A.: Modelling the growth and form of capillary networks. In: Chaplain, M.A.J.et al. (eds) On Growth and Form. Spatio-temporal Pattern Formation in Biology. John Wiley & Sons, Chichester (1999).

    Google Scholar 

  19. Falconer, K.J.: The Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985).

    MATH  Google Scholar 

  20. Federer, H.: Geometric Measure Theory. Springer, Berlin (1996)

    MATH  Google Scholar 

  21. Friedman, L.H., Chrzan, D.G.: Scaling theory of the Hall-Petch relation for multilayers. Phys. Rev. Letters, 81, 2715-2718 (1998).

    Article  Google Scholar 

  22. Hardle, W.: Smoothing techniques. With Implementation in S, Springer-Verlag, New York (1991).

    Google Scholar 

  23. Kolmogorov, A.N., Fomin S.V.: Introductory Real Analysis. Prentice-Hall, Englewood Cliffs (N.J.), (1970).

    MATH  Google Scholar 

  24. Jain, R.K., Carmeliet, P.F.: Vessels of Death or Life. Scientific American 285,38-45 (2001).

    Article  Google Scholar 

  25. Jones, D.S.: The Theory of Generalised Functions. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  26. Matheron, G.: Random sets and integral geometry. John Wiley & Sons, New York (1975)

    MATH  Google Scholar 

  27. McDougall, S.R., Anderson, A.R.A., Chaplain, M.A.J.: Mathematical modelling of dynamic tumour-induced angiogenesis: Clinical implications and therapeutic targeting strategies. J. Theor. Biology, 241, 564-589 (2006)

    Article  MathSciNet  Google Scholar 

  28. Møller, J.: Random Johnson-Mehl tessellations. Adv. Appl. Prob., 24, 814-844 (1992)

    Article  Google Scholar 

  29. Møller, J.: Lectures on Random Voronoi Tessellations. Lecture Notes in Statistics 87. Springer-Verlag, New York, Berlin, Heidelberg (1994)

    Google Scholar 

  30. Pestman, W.R.: Mathematical Statistics. An Introduction. Walter de Gruyter, Berlin (1998).

    Google Scholar 

  31. Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and its Application, John Wiley & Sons, New York (1995)

    Google Scholar 

  32. Vladimirov, V.S.: Generalized Functions in Mathematical Physics. Mir Publishers, Moscow (1979).

    Google Scholar 

  33. Zähle, M.: Random processes of Hausdorff rectifiable closed sets. Math. Nachr. 108, 49-72 (1982)

    Article  MathSciNet  Google Scholar 

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Capasso, V., Micheletti, A. (2008). On the Mean Geometric Densities of Random Closed Sets, and Their Estimation: Application to the Estimation of the Mean Density of Inhomogeneous Fibre Processes. In: Bonilla, L.L., Moscoso, M., Platero, G., Vega, J.M. (eds) Progress in Industrial Mathematics at ECMI 2006. Mathematics in Industry, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71992-2_1

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