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On the Resistance of the Sierpinski Carpet
M. T. Barlow and R. F. Bass
Proceedings: Mathematical and Physical Sciences
Vol. 431, No. 1882 (Nov. 8, 1990), pp. 345-360
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/79974
Page Count: 16
You can always find the topics here!Topics: Carpets, Brownian motion, Fractals, Energy, Conductivity, Mathematical functions, Energy dissipation, Resistors, Boundary value problems, Line segments
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Let Fn be the nth stage in the construction of the Sierpinski carpet. Let Rn be the electrical resistance of Fn when the left and right sides are each short-circuited, and a voltage is applied between them. We prove that there exists a constant ρ such that 1/4ρ n≤ Rn≤ 4ρ n. The motivation for this result came from the problem of establishing (a) the existence and (b) the value of the `spectral dimension' of the Sierpinski carpet. In this and a subsequent paper, we settle (a) and give bounds for (b).
Proceedings: Mathematical and Physical Sciences © 1990 Royal Society