Wen-Guei Hu

ZETA FUNCTIONS FOR HIGHER-DIMENSIONAL SHIFTS OF FINITE TYPE

This work investigates zeta functions for d-dimensional shifts of finite type, d ≥ 3. First, the three-dimensional case is studied. The trace operator Ta1,a2;b12 and rotational matrices Rx;a1,a2;b12 and Ry;a1,a2;b12 are introduced to study

Spatial chaos of Wang tiles with two symbols

{\scriptsize\left[\begin{array}{@{}c@{\quad}c@{\quad}c@{}} a_{1} & b_{12} & b_{23}\\[1pt] 0 & a_{2} & b_{23} \\[1pt] 0 & 0 & a_{3} \end{array}\right]}

Complexity of two dimensional multi-layer cellular neural networks

-periodic patterns. The rotational symmetry of Ta1,a2;b12 induces the reduced trace operator τa1,a2;b12 and then the associated zeta function ζa1,a2;b12 = (det(I-sa1a2τa1,a2;b12))-1. The zeta function ζ is then expressed as

Cellular neural networks and zeta functions

\zeta=\prod_{a_{1}=1}^{\infty}\prod_{a_{2}=1}^{\infty} \prod_{b_{12}=0}^{a_{1}-1}\zeta_{a_{1},a_{2};b_{12}}

NONEMPTINESS PROBLEMS OF PLANE SQUARE TILING WITH TWO COLORS

, a reciprocal of an infinite product of polynomials. The results hold for any inclined coordinates, determined by unimodular transformation in GL3(ℤ). Hence, a family of zeta functions exists with the same integer coefficients in their Taylor series expansions at the origin, and yields a fami...

Zeta functions for two-dimensional shifts of finite type

This investigation completely classifies the spatial chaos problem in plane edge coloring (Wang tiles) with two symbols. For a set of Wang tiles B, spatial chaos occurs when the spatial entropy h(B) is positive. B is called a minimal cycle generator if P(B)≠0 and P(B′)=0 whenever B′⫋B, where P(B) is the set of all periodic patterns on ℤ2 generated by B. Given a set of Wang tiles B, write B=C1∪C2∪⋯∪Ck∪N, where Cj, 1 ≤ j ≤ k, are minimal cycle generators and B contains no minimal cycle generator except those contained in C1∪C2∪⋯∪Ck. Then, the positivity of spatial entropy h(B) is completely determined by C1∪C2∪⋯∪Ck. Furthermore, there are 39 equivalence classes of marginal positive-entropy sets of Wang tiles and 18 equivalence classes of saturated zero-entropy sets of Wang tiles. For a set of Wang tiles B, h(B) is positive if and only if B contains a MPE set, and h(B) is zero if and only if B is a subset of a SZE set.

Verification of mixing properties in two-dimensional shifts of finite type

This study investigates the complexity of the global set of output patterns for two-dimensional multi-layer cellular neural networks. Applying labeling to the output space produces a two-dimensional sofic shift space. The ordering matrices and symbolic transition matrices are introduced to study the spatial entropy of the output space.

Pattern generation problems arising in multiplicative integer systems

This talk is concerned with zeta functions of two-dimensional shifts of finite type. The zeta function is an important invariant, which combines information of all periodic patterns. The zeta function can be explicitly expressed as a reciprocal of an infinite product of polynomials by patterns generation approaches. The methods can apply to two-dimensional cellular neural networks.