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2010 On the Dimension of the Space of Harmonic Functions on a Discrete Torus
Masato Goshima, Masakazu Yamagishi
Experiment. Math. 19(4): 421-429 (2010).

Abstract

Let $d(n)$ denote the corank of $I + A$ over the field with two elements, where $A$ is the adjacency matrix of the discrete torus $C_n × C_n$, and $I$ is the identity matrix. We shall prove that $d(2n) = 2d(n)$ and $d(2^r + 1) = d(2^r − 1) + 4$. For the proof of the latter result, we use an elliptic curve. Our motivation for this study is the “lights out” puzzle.

Citation

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Masato Goshima. Masakazu Yamagishi. "On the Dimension of the Space of Harmonic Functions on a Discrete Torus." Experiment. Math. 19 (4) 421 - 429, 2010.

Information

Published: 2010
First available in Project Euclid: 4 October 2011

zbMATH: 1292.11134
MathSciNet: MR2778655

Rights: Copyright © 2010 A K Peters, Ltd.

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Vol.19 • No. 4 • 2010
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