Browsing by Author "Hu, Wen-Guei"

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NONEMPTINESS PROBLEMS OF PLANE SQUARE TILING WITH TWO COLORS
(2011-03-01)
Hu, Wen-Guei
;
Lin, Song-Sun
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應用數學系
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Department of Applied Mathematics
This investigation studies nonemptiness problems of plane square tiling. In the edge coloring (or Wang tiles) of a plane, unit squares with colored edges of p colors are arranged side by side such that adjacent tiles have the same colors. Given a set of Wang tiles B, the nonemptiness problem is to determine whether or not Sigma(B) not equal theta, where Sigma(B) is the set of all global patterns on Z(2) that can be constructed from the Wang tiles in B. When p >= 5, the problem is well known to be undecidable. This work proves that when p = 2, the problem is decidable. P(B) is the set of all periodic patterns on Z(2) that can be generated by B. If P(B) not equal empty set, then s has a subset B' of minimal cycle generator such that P(B') not equal empty set and P(B '') = empty set for B '' subset of B'. This study demonstrates that the set of all minimal cycle generators C(2) contains 38 elements. N(2) is the set of all maximal noncycle generators: if B is an element of N(2), then P(B) = empty set and (B) over tilde superset of B implies P (B) over tilde) not equal empty set. N(2) has eight elements. That Sigma (B) = theta for any B is an element of N(2) is proven, implying that if Sigma(B) not equal empty set, then P(B) not equal empty set. The problem is decidable for p = 2: Sigma(B) not equal empty set if and only if B has a subset of minimal cycle generators. The approach can be applied to corner coloring with a slight modification, and similar results hold.
Nonemptiness problems of Wang tiles with three colors
(2014-08-28)
Chen, Hung-Hsun
;
Hu, Wen-Guei
;
Lai, De-Jan
;
Lin, Song-Sun
;
應用數學系
This investigation studies nonemptiness problems of plane edge coloring with three colors. In the edge coloring (or Wang tiles) of a plane, unit squares with colored edges that have one of p colors are arranged side by side such that the touching edges of the adjacent tiles have the same colors. Given a basic set B of Wang tiles, the nonemptiness problem is to determine whether or not Sigma(B) not equal empty set , where Sigma(B) is the set of all global patterns on Z(2) that can be constructed from the Wang tiles in B. Wang's conjecture is that for any B of Wang tiles, Sigma (B) not equal empty set if and only if P(B) not equal empty set, where P(B) is the set of all periodic patterns on Z(2) that can be generated by the tiles in B. When p >= 5, Wang's conjecture is known to be wrong. When p = 2, the conjecture is true. This study proves that when p = 3, the conjecture is also true. If P(B) not equal empty set, then B has a subset B' of minimal cycle generators such that P(B') not equal empty set and P(B '') not equal empty set for B '' B'. This study demonstrates that the set C(3) of all minimal cycle generators contains 787, 605 members that can be classified into 2, 906 equivalence classes. N(3) is the set of all maximal non-cycle generators: if B is an element of N(3), then P(B) not equal empty set and P((B) over tilde) not equal empty set for (B) over tilde B. Wang's conjecture is shown to be true by proving that B is an element of N(3) implies Sigma(B) = empty set. (C) 2014 Elsevier B.V. All rights reserved.
Nonemptiness problems of Wang tiles with three colors
(2014-08-28)
Chen, Hung-Hsun
;
Hu, Wen-Guei
;
Lai, De-Jan
;
Lin, Song-Sun
;
應用數學系
This investigation studies nonemptiness problems of plane edge coloring with three colors. In the edge coloring (or Wang tiles) of a plane, unit squares with colored edges that have one of p colors are arranged side by side such that the touching edges of the adjacent tiles have the same colors. Given a basic set B of Wang tiles, the nonemptiness problem is to determine whether or not Sigma(B) not equal empty set , where Sigma(B) is the set of all global patterns on Z(2) that can be constructed from the Wang tiles in B. Wang\'s conjecture is that for any B of Wang tiles, Sigma (B) not equal empty set if and only if P(B) not equal empty set, where P(B) is the set of all periodic patterns on Z(2) that can be generated by the tiles in B. When p >= 5, Wang\'s conjecture is known to be wrong. When p = 2, the conjecture is true. This study proves that when p = 3, the conjecture is also true. If P(B) not equal empty set, then B has a subset B\' of minimal cycle generators such that P(B\') not equal empty set and P(B \'\') not equal empty set for B \'\' B\'. This study demonstrates that the set C(3) of all minimal cycle generators contains 787, 605 members that can be classified into 2, 906 equivalence classes. N(3) is the set of all maximal non-cycle generators: if B is an element of N(3), then P(B) not equal empty set and P((B) over tilde) not equal empty set for (B) over tilde B. Wang\'s conjecture is shown to be true by proving that B is an element of N(3) implies Sigma(B) = empty set. (C) 2014 Elsevier B.V. All rights reserved.
ON SPATIAL ENTROPY OF MULTI-DIMENSIONAL SYMBOLIC DYNAMICAL SYSTEMS
(2016-07)
Hu, Wen-Guei
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Lin, Song-Sun
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應用數學系
;
Department of Applied Mathematics
The commonly used topological entropy h(top) (U) of the multidimensional shift space U is the rectangular spatial entropy h(r) (U) which is the limit of growth rate of admissible local patterns on finite rectangular sub-lattices which expands to whole space Z(d), d >= 2. This work studies spatial entropy h(Omega)(U) of shift space U on general expanding system Omega = {Omega(n)}(n=1)(infinity) where Omega(n) is increasing fi nite sublattices and expands to Z(d). Omega is called genuinely d-dimensional if Omega(n) contains no lower-dimensional part whose size is comparable to that of its d-dimensional part. We show that h(r)(U) is the supremum of h(Omega)(U) for all genuinely d-dimensional Omega. Furthermore, when Omega is genuinely d-dimensional and satis fi es certain conditions, then h(Omega) (U) = h(r)(U). On the contrary, when Omega(n) contains a lower-dimensional part which is comparable to its d-dimensional part, then h r (U) < h(Omega)(U) for some U. Therefore, h(r)(U) is appropriate to be the d-dimensional spatial entropy.
Pattern generation problems arising in multiplicative integer systems
(2019-05-01)
Ban, Jung-Chao
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Hu, Wen-Guei
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Lin, Song-Sun
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應用數學系
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Department of Applied Mathematics
This study investigates a multiplicative integer system, an invariant subset of the full shift under the action of the semigroup of multiplicative integers, by using a method that was developed for studying pattern generation problems. The spatial entropy and the Minkowski dimensions of general multiplicative systems can thus be computed. A coupled system is the intersection of a multiplicative integer system and the golden mean shift, which can be decoupled by removing the multiplicative relation set and then performing procedures similar to those applied to a decoupled system. The spatial entropy can be obtained after the remaining error term is shown to approach zero.
Spatial chaos of Wang tiles with two symbols
(2016-02)
Chen, Jin-Yu
;
Chen, Yu-Jie
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Hu, Wen-Guei
;
Lin, Song-Sun
;
應用數學系
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) not equal empty set and P(B\') = empty set whenever B\' not subset of B, where P(B) is the set of all periodic patterns on Z(2) generated by B. Given a set of Wang tiles B, write B = C-1 boolean OR C-2 boolean OR ... boolean OR C-k boolean OR N, where C-j, 1 <= j <= k, are minimal cycle generators and B contains no minimal cycle generator except those contained in C-1 boolean OR C-2 boolean OR ... boolean OR C-k. Then, the positivity of spatial entropy h(B) is completely determined by C-1 boolean OR C-2 boolean OR ... boolean OR C-k. 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. (C) 2016 AIP Publishing LLC.
ZETA FUNCTIONS FOR HIGHER-DIMENSIONAL SHIFTS OF FINITE TYPE
(2009-11-01)
Hu, Wen-Guei
;
Lin, Song-Sun
;
應用數學系
;
Department of Applied Mathematics
This work investigates zeta functions for d-dimensional shifts of finite type, d >= 3. First, the three-dimensional case is studied. The trace operator T(a1,a2;b12) and rotational matrices R(x;a1,a2;b12) and R(y;a1,a2;b12) are introduced to study [GRAPHICS] -periodic patterns. The rotational symmetry of T(a1,a2;b12) induces the reduced trace operator tau(a1,a2;b12) and then the associated zeta function zeta(a1,a2;b12) = (det(I - s(a1a2)tau(a1,a2;b12)))(-1). The zeta function zeta is then expressed as zeta = Pi(infinity)(a1=1) Pi(infinity)(a2=1) Pi(a1-1)(b12=0) zeta(a1,a2;b12), a reciprocal of an infinite product of polynomials. The results hold for any inclined coordinates, determined by unimodular transformation in GL(3)(Z). Hence, a family of zeta functions exists with the same integer coefficients in their Taylor series expansions at the origin, and yields a family of identities in number theory. The methods used herein are also valid for d-dimensional cases, d >= 4, and can be applied to thermodynamic zeta functions for the three-dimensional Ising model with finite range interactions.