臺灣博碩士論文加值系統

本論文探討二維符號動態系統上的自然測度。在二維有限子平移中，如果其有序矩陣(ordering matrix) 有不可約性(irreducible) 且非週期性(aperiodic)，我們提供一個方法去計算二維符號動態系統上的自然測度，並且將一維符號動態系統上的結果推廣至二維。最後我們應用這套方法計算出二維完全對稱系統上的自然測度的精確值。

This thesis investigates the natural measure of a symbolic dynamical system in two-dimensional lattice model. For a two-dimensional shift of finite type has irreducible ordering matrix H_2, we provided a method for the natural measure of two-dimensional lattice model. And derive all the one-dimensional model into two-dimension. Eventually apply this method to have the exact value of natural measure of totally symmetric system in the two-dimension.

摘要. . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . ii
Table of Contents . . . . . . . . . . . . . . . . . . iii
1 Introduction . . . . . . . . . . . . . . . . . . . . 1
2 The natural measure . . . . . . . . . . . . . . . . . 3
3 Pattern generation . . . . . . . . . . . . . . . . . .5
4 Codimensional systems . . . . . . . . . . . . . . . . 16
4.1 The measure of one single symbol . . . . . . . . . 16
4.2 The measure of vertical patterns . . . . . . . . . 21
4.3 The measure of arbitrary pattern . . . . . . . . . 26
5 Totally symmetric systems . . . . . . . . . . . . . . 31
6 Conclusion and feature work . . . . . .. . . . . . . 34
References . . . . . . . . . . . . . . . . . . . . . . 35

[1] P. W. Bates and A. Chmaj, “A discrete convolution model for phase transitions,” Archive for Rational Mechanics and Analysis, vol. 150, no. 4, pp. 281–368, 1999.
[2] P. W. Bates, K. Lu, and B. Wang, “Attractors for lattice dynamical systems,” International Journal of Bifurcation and Chaos, vol. 11, no. 01, pp. 143–153, 2001.
[3] J. P. Laplante and T. Erneux, “Propagation failure in arrays of coupled bistable chemical reactors,” The Journal of Physical Chemistry, vol. 96, no. 12, pp. 4931–4934, 1992.
[4] J.C. Ban and C.H. Chang, “Coloring fibonaccicayley
tree: An application to neural networks,” arXiv preprint arXiv:1707.02227, 2017.
[5] J.C. Ban, C.H. Chang, and N.Z. Huang, “Entropy bifurcation of neural networks on cayley trees,” arXiv preprint arXiv:1706.09283, 2017.
[6] J.C. Ban, C.H. Chang, and S.S. Lin, “On the structure of multilayer cellular neural networks,” Journal of Differential Equations, vol. 252, no. 8, pp. 4563–4597, 2012.
[7] J. Bell, “Some threshold results for models of myelinated nerves,” Mathematical Biosciences, vol. 54, no. 34, pp. 181–190, 1981.
[8] J. Bell and C. Cosner, “Threshold behavior and propagation for nonlinear differentialdifference systems motivated by modeling myelinated axons,” Quarterly of Applied Mathematics, vol. 42, no. 1, pp. 1–14, 1984.
[9] G. B. Ermentrout, “Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators,” SIAM Journal on Applied Mathematics, vol. 52, no. 6, pp. 1665–1687, 1992.
[10] G. B. Ermentrout and N. Kopell, “Inhibitionproduced
patterning in chains of coupled nonlinear oscillators,” SIAM Journal on Applied Mathematics, vol. 54, no. 2, pp. 478– 507, 1994. 35
[11] N. Kopell, G. Ermentrout, and T. Williams, “On chains of oscillators forced at one end,” SIAM Journal on Applied Mathematics, vol. 51, no. 5, pp. 1397–1417, 1991.
[12] J. P. Keener, “Propagation and its failure in coupled systems of discrete excitable cells,” SIAM Journal on Applied Mathematics, vol. 47, no. 3, pp. 556–572, 1987.
[13] J. P. Keener, “The effects of discrete gap junction coupling on propagation in myocardium,” Journal of theoretical biology, vol. 148, no. 1, pp. 49–82, 1991.
[14] R. L. Winslow, A. L. Kimball, A. Varghese, and D. Noble, “Simulating cardiac sinus and atrial network dynamics on the connection machine,” Physica D: Nonlinear Phenomena, vol. 64, no. 13, pp. 281–298, 1993.
[15] W.G. Hu and S.S. Lin, “The natural measure of a symbolic dynamical system,” arXiv preprint arXiv:1308.2996, 2013.
[16] J.C. Ban and S.S. Lin, “Patterns generation and transition matrices in multidimensional lattice models,” Discrete and Continuous Dynamical Systems, vol. 13, no. 3, p. 637, 2005.
[17] J.C. Ban, S.S. Lin, and Y.H. Lin, “Patterns generation and spatial entropy in twodimensional
lattice models,” Asian Journal of Mathematics, vol. 11, no. 3, pp. 497–534, 2007.
[18] D. Lind, B. Marcus, L. Douglas, and M. Brian, An introduction to symbolic dynamics and coding. Cambridge university press, 1995.