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Original Articles

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

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Pages 421-429 | Received 29 Jan 2010, Accepted 03 Feb 2010, Published online: 23 Feb 2011

REFERENCES

  • Barua , Rana and Ramakrishnan , S. 1996 . “Σ-Game, Σ+-Game and Two-Dimensional Additive Cellular Automata.” . Theoret. Comput. Sci. , 154 ( 2 ) : 349 – 366 . [Barua and Ramakrishnan 96]
  • Brouwer , Andries E. 2008 . “Button Madness and Lights Out on Rectangles.” Available online http://www.win.tue.nl/∼aeb/ca/madness/madrect.html[Brouwer 08]
  • Cartier , P. 1972 . “Fonctions harmoniques sur un arbre.” . In Symposia Mathematica, Vol. IX (Convegno di Calcolo delle Probabilitá, INDAM, Rome, 1971) 203 – 270 . London : Academic Press. . [Cartier 72]
  • Cartier , Pierre . 1973 . “Géométrie et analyse sur les arbres.” 123 – 140 . Berlin : Springer. . Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 407, Lecture Notes in Math. 317, [Cartier 73]
  • Cremona , J. E. 1997 . Algorithms for Modular Elliptic Curves, , second edition Cambridge, , UK : Cambridge University Press. . [Cremona 97]
  • Goldwasser , John , Klostermeyer , William and Ware , Henry . 2002 . “Fibonacci Polynomials and Parity Domination in Grid Graphs.” . Graphs Combin. , 18 ( 2 ) : 271 – 283 . [Goldwasser et al. 02]
  • Goshima , Masato and Yamagishi , Masakazu . 2009 . “Two Remarks on Torus Lights Out Puzzle.” . Adv. Appl. Discrete Math. , 4 ( 2 ) : 115 – 126 . [Goshima and Yamagishi 09]
  • Hunziker , Markus , Machiavelo , António and Park , Jihun . 2004 . “Chebyshev Polynomials over Finite Fields and Reversibility of Σ-Automata on Square Grids.” . Theoret. Comput. Sci. , 320 ( 2–3 ) : 465 – 483 . [Hunziker et al. 04]
  • Joyner , David . 2002 . Adventures in Group Theory: Rubik's Cube, Merlin's Machine and Other Mathematical Toys Baltimore : Johns Hopkins University Press. . [Joyner 02]
  • Silverman , Joseph H. 1986 . The Arithmetic of Elliptic Curves New York : Springer. . Graduate Texts in Mathematics 106, [Silverman 86]
  • Sutner , Klaus . 1989 . “Linear Cellular Automata and the Garden-of-Eden.” . Math. Intelligencer , 11 ( 2 ) : 49 – 53 . [Sutner 89]
  • Sutner , Klaus . 1990 . “The Σ-Game and Cellular Automata.” . Amer. Math. Monthly , 97 ( 1 ) : 24 – 34 . [Sutner 90]
  • Sutner , Klaus . 2000 . “Σ-Automata and Chebyshev Polynomials.” . Theoret. Comput. Sci. , 230 ( 1–2 ) : 49 – 73 . [Sutner 00]
  • Zaidenberg , Mikhail . 2008 . “Convolution Equations on Lattices: Periodic Solutions with Values in a Prime Characteristic Field.” . In Geometry and Dynamics of Groups and Spaces 721 – 742 . Basel : Birkhäuser. . Progr. Math. 265, [Zaidenberg 08a]
  • Zaidenberg , Mikhail . 2008 . “Periodic Binary Harmonic Functions on Lattices.” . Adv. in Appl. Math. , 40 ( 2 ) : 225 – 265 . [Zaidenberg 08b]
  • Zaidenberg , Mikhail . 2009 . “Periodic Harmonic Functions on Lattices and Points Count in Positive Characteristic.” . Cent. Eur. J. Math. , 7 ( 3 ) : 365 – 381 . [Zaidenberg 09]

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