Hostname: page-component-788cddb947-tr9hg Total loading time: 0 Render date: 2024-10-19T05:56:29.764Z Has data issue: false hasContentIssue false
  • English
  • Français

The Hausdorff dimension of pulse-sum graphs

Published online by Cambridge University Press:  01 July 2007

YANN DEMICHEL  and
KENNETH FALCONER
YANN DEMICHEL
Affiliation:
Department of Mathematics, University Blaise Pascal, 63177 Aubière Cedex, France. e-mail: demichel.yann@tiscali.fr
KENNETH FALCONER
Affiliation:
Mathematical Institute, University of St Andrews, St Andrews, Fife KY16 9SS. e-mail: kjf@st-and.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider random functions formed as sums of pulseswhere Xn are independent random vectors, 0<α<1, and G is an elementary “pulse” or “bump”. Typically such functions have fractal graphs and we find the Hausdorff dimension of these graphs using a novel variant on the potential theoretic method.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 1 , July 2007 , pp. 145 - 155
Copyright
Copyright © Cambridge Philosophical Society 2007

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]Demichel, Y. and Tricot, C.. Analysis of the Fractal Sum of Pulses. Math. Proc. Camb. Phil. Soc. 141 (2006), 355–370.CrossRefGoogle Scholar
[2]Falconer, K.. Fractal Geometry, Mathematical Foundations and Applications, 2nd Ed. (John Wiley and Sons, 2003).CrossRefGoogle Scholar
[3]Hunt, B. R.. The Hausdorff dimension of graphs of Weierstrass functions. Proc. Amer. Math. Soc. 126 (1998), 791–800.CrossRefGoogle Scholar
[4]Kahane, J.-P.. Some Random Series of Functions (Cambridge University Press 1985).Google Scholar
[5]Mandelbrot, B.. Introduction to fractal sums of pulses. In Lévy Processes and Related Phenomena in Physics. Lecture Notes in Phys. vol. 450 (Springer–Verlag, 1995), pp. 110–123.CrossRefGoogle Scholar
[6]Mandelbrot, B.. Fractal sums of pulses and a practical challenge to the distinction between local and global dependence. In Lévy Processes and Related Phenomena in Physics, Lecture Notes in Phys. vol. 621 (Springer–Verlag, 2003), pp. 118–135.CrossRefGoogle Scholar
[7]Mandelbrot, B. and Barral, J.. Multifractal products of cylindrical pulses. Probab. Theory Related Fields 124 (2002), 409–430.Google Scholar
[8]Tricot, C.. A model for rough surfaces. Composites Science and Technology 63 (2003), 1089–1096.CrossRefGoogle Scholar