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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
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On the Resistance of the Sierpinski Carpet
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Abstract

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).

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