Periodic harmonic functions on lattices and points count in positive characteristic

Zaidenberg, Mikhail

doi:10.2478/s11533-009-0029-0

Cited by 6 publications

(4 citation statements)

References 14 publications

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“…Variants of Lights Out with more than one toggle mode have been considered by many authors. [4] [22] It is possible to extend our results in this direction.…”

Section: Future Workmentioning

confidence: 60%

Lights Out On A Random Graph

Forrest¹,

Nicole²

2021

Preprint

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We consider the generalized game Lights Out played on a graph and investigate the following question: for a given positive integer n, what is the probability that a graph chosen uniformly at random from the set of graphs with n vertices yields a universally solvable game of Lights Out? When n ≤ 11, we compute this probability exactly by determining if the game is universally solvable for each graph with n vertices. We approximate this probability for each positive integer n with n ≤ 87 by applying a Monte Carlo method using 1,000,000 trials. We also perform the analogous computations for connected graphs.

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“…Variants of Lights Out with more than one toggle mode have been considered by many authors. [4] [22] It is possible to extend our results in this direction.…”

Section: Future Workmentioning

confidence: 60%

Lights Out On A Random Graph

Forrest¹,

Nicole²

2021

Preprint

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“…(i) This is proved in [5,Corollary 5]. See also [14,Proposition 2.1]. We give an alternative proof using Chebyshev polynomials.…”

Section: Lemma 22mentioning

confidence: 82%

“…have been extensively investigated. See, for example, [1,3,4,6,[9][10][11][12][13][14]. For these graphs, the σ + -game is mathematically much deeper than the σ -game.…”

Section: Examplementioning

confidence: 97%

A singular quartic curve over a finite field and the Trisentis game

Yamagishi

2011

Finite Fields and Their Applications

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Communicated by Simeon Ball MSC: 05C50 11G20 14G15 31C20 91A46 Keywords: σ -game σ + -game Graph Laplacian Lights Out Strong product Kronecker product Chebyshev polynomialsThe Trisentis game consists of a rectangular array of lights each of which also functions as a toggle switch for its (up to eight) neighboring lights. The lights are OFF at the start, and the object is to turn them all ON. We give explicit formulas for the dimension of the kernel of the Laplacian associated to this game as well as some variants, in some cases, by counting rational points of the singular quartic curve (x + x −1 + 1)( y + y −1 + 1) = 1 over finite fields. As a corollary, we have an affirmative answer to a question of Clausing whether the n × n Trisentis game has a unique solution if n = 2 · 4 k or n = 2 · 4 k − 2.

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“…It looks interesting if we can also adapt that research to the study of countable weighted digraphs. In the more algebraic direction, one can consider harmonic functions with values in a field of positive characteristic [60]. We conclude the paper with two simple examples about harmonic functions of finite Laplacians and invite the readers to see if it is possible to find a generalization to some infinite settings.…”

Section: Accepted Manuscriptmentioning

confidence: 96%

Three conjectures of Ostrander on digraph Laplacian eigenvectors

Zhao

2021

Art Discrete Appl. Math.

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Ostrander proposed three conjectures on the connections between topological properties of a weighted digraph and combinatorial properties of its Laplacian eigenvectors. We verify one of his conjectures, give counterexamples to the other two, and then seek for related valid connections and generalizations to Schrödinger operators on countable digraphs.We suggest the open question of deciding if the countability assumption can be dropped from our main results.

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Periodic harmonic functions on lattices and points count in positive characteristic

Abstract: Abstract. This survey addresses pluri-periodic harmonic functions on lattices with values in a positive characteristic field. We mention, as a motivation, the game "Lights Out" following the work of Sutner

Cited by 6 publications

References 14 publications

Lights Out On A Random Graph

Lights Out On A Random Graph

A singular quartic curve over a finite field and the Trisentis game

Three conjectures of Ostrander on digraph Laplacian eigenvectors

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