Experimental Mathematics

On the Dimension of the Space of Harmonic Functions on a Discrete Torus

Masato Goshima and Masakazu Yamagishi

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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.

Article information

Source
Experiment. Math., Volume 19, Issue 4 (2010), 421-429.

Dates
First available in Project Euclid: 4 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.em/1317758102

Mathematical Reviews number (MathSciNet)
MR2778655

Zentralblatt MATH identifier
1292.11134

Keywords
Lights out puzzle graph Laplacian discrete torus elliptic curve Chebyshev-Dickson polynomials

Citation

Goshima, Masato; Yamagishi, Masakazu. On the Dimension of the Space of Harmonic Functions on a Discrete Torus. Experiment. Math. 19 (2010), no. 4, 421--429. https://projecteuclid.org/euclid.em/1317758102


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