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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: [email protected]
KENNETH FALCONER
Affiliation:
Mathematical Institute, University of St Andrews, St Andrews, Fife KY16 9SS. e-mail: [email protected]
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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

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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