On the Stochastic Geometry of Growth
The pioneering book by D’Arcy Thompson, entitled “On Growth and Form” , was perhaps the first to consider applying (deterministic) mathematics to problems in biology, in particular those problems associated with the growth of biological objects. However most people nowadays are aware of the fact that we cannot ignore stochasticity in real biological phenomena. The scope of this chapter is to introduce relevant nomenclature and mathematical methods for the analysis of geometries related to stochastic birth-and-growth processes, thus providing a guided tour in a selected bibliography. First of all let us introduce the current terminology in this framework. The most important chapters of mathematical interest in this context are the following, for which we refer to the relevant literature: While MORPHOGENESIS (PATTERN FORMATION) deals with the mathematical modelling of the causal description of a pattern (a direct problem) [11, 14] (see Fig. 4.1), STOCHASTIC GEOMETRY deals with the analysis of geometric aspects of “patterns” subject to stochastic fluctuations (direct and inverse problems) [1, 12] (see Figs. 4.2, 4.3).
KeywordsHausdorff Dimension Boolean Model Stochastic Fluctuation Stochastic Geometry Random Object
Unable to display preview. Download preview PDF.
- 1.Barndorff-Nielsen, O.E., Kendall, W.S., van Lieshout, M. N. A. Eds. Stochastic Geometry. Likelihood and Computation,Chapman andHall-CRC, Boca Raton,.Google Scholar
- 2.Capasso, V. Ed. (2003). Mathematical Modelling for Polymer Processing. Polymerization. Crystallization; Manufacturing, Springer-Verlag. Heidelberg. In press.Google Scholar
- 4.Capasso, V., Micheletti, A. (2003). Stochastic geometry of spatially structured birth-and-growth processes. Application to crystallization processes. In Spatial Stochastic Processes (E.:Merzbach, Ed.). Lecture Notes in Mathematics - CIME Subseries, Springer-Verlag, Heidelberg. In press.CrossRefGoogle Scholar
- 5.Capasso, V., Micheletti, A., Burger, M. (2001). Densities of n-facets of incomplete Johnson-Mehl tessellations generated by inhomogeneous birth-and-growth processes. Preprint,.Google Scholar
- 6.Cressie, N. (1993). Statistics for Spatial Data, Wiley, New York.Google Scholar
- 7.Johnson, W.A., Mehl, R.F. (1939). Reaction kinetics in processes of nucleation and growth, Trans. A.I.M.M.E., 135, 416–458.Google Scholar
- 8.Kolmogorov, A.N. (1956). Foundations of the Theory of Probability, Chelsea Pub. Co., New York.Google Scholar
- 12.Stoyan, D., Kendall, W.S., Mecke, J. (1995). Stochastic Geometry and its Application., John Wiley & Sons, New York.Google Scholar
- 13.Thompson, D.W. (1970). On Growth and Forrn (1917), Cambridge University Press, Cambridge,.Google Scholar