Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-28T08:32:01.294Z Has data issue: false hasContentIssue false
  • English
  • Français

The horizon problem for random surfaces

Published online by Cambridge University Press:  24 October 2008

K. J. Falconer
K. J. Falconer
Affiliation:
School of Mathematics, University Walk, Bristol BS8 1TW
Get access Rights & Permissions [Opens in a new window]

Extract

Computer simulation of landscapes and skylines has recently attracted a great deal of interest: see [6, 7]. Specification of a ‘landscape’ requires a function f: D → ℝ on a subset D of ℝ2, selected so that the apparent irregularity and randomness of the surface {(t,f(t)): t ∈ D} corresponds to what might be observed in nature. It is natural to look to random fields (that is, stochastic processes in two variables), and in particular to Gaussian fields, for functions with such properties. Even when an appropriate random field has been selected, determination of a typical sample function is far from easy [7].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 1 , January 1991 , pp. 211 - 219
Copyright
Copyright © Cambridge Philosophical Society 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Adler, R. J.. The Geometry of Random Fields (John Wiley, 1981).Google Scholar
[2]Dym, H. and McKean, H. P.. Fourier Series and Integrals (Academic Press, 1972).Google Scholar
[3]Falconer, K. J.. Fractal Geometry – Mathematical Foundations and Applications (John Wiley, 1990).CrossRefGoogle Scholar
[4]German, D. and Horowitz, J.. Occupation densities. Ann. Probab. 8 (1980), 1–67.Google Scholar
[5]Kahane, J.-P.. Some Random Series of Functions (Cambridge University Press, 1985).Google Scholar
[6]Mandelbrot, B. B.. The Fractal Geometry of Nature (W. H. Freeman, 1982).Google Scholar
[7]Peitgen, H.-O. and Saupe, D. (editors). The Science of Fractal Images (Springer-Verlag, 1988).Google Scholar
[8]Pitt, L. D.. Local times for Gaussian vector fields. Indiana Univ. Math. J. 27 (1978), 309–330.CrossRefGoogle Scholar