Browsing by Author "Ban, Jung-Chao"

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Decidability of Irreducible Tree Shifts of Finite Type
(1970-01-01)
Ban, Jung-Chao
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Chang, Chih-Hung
;
Huang, Nai-Zhu
;
Wu, Yu-Liang
;
應用數學系
We reveal an algorithm for determining the complete prefix code irreducibility (CPC-irreducibility) of dyadic trees labeled by a finite alphabet. By introducing an extended directed graph representation of tree shift of finite type (TSFT), we show that the CPC-irreducibility of TSFTs is related to the connectivity of its graph representation, which is a similar result to one-dimensional shifts of finite type.
Entropy Bifurcation of Neural Networks on Cayley Trees
(2020-01-01)
Ban, Jung-Chao
;
Chang, Chih-Hung
;
Huang, Nai-Zhu
;
應用數學系
;
Department of Applied Mathematics
It has been demonstrated that excitable media with a tree structure performed better than other network topologies, therefore it is natural to consider neural networks defined on Cayley trees. The investigation of a symbolic space called tree-shift of finite type is important when it comes to the discussion of the equilibrium solutions of neural networks on Cayley trees. Entropy is a frequently used invariant for measuring the complexity of a system, and constant entropy for an open set of coupling weights between neurons means that the specific network is stable. This paper gives a complete characterization of entropy spectrum of neural networks on Cayley trees and reveals whether the entropy bifurcates when the coupling weights change.
Entropy dimension of shift spaces on monoids
(2020-07-01)
Ban, Jung-Chao
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Chang, Chih-Hung
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Huang, Nai-Zhu
;
應用數學系
;
Department of Applied Mathematics
We consider the entropy dimension of G-shifts of finite type for the case where G is a non-Abelian monoid. Entropy dimension tells us whether a shift space has zero topological entropy. Suppose the Cayley graph C-G of G has a finite representation (that is, {C-gG : g is an element of G} is a finite set up to graph isomorphism), and relations among generators of G are determined by a matrix A. We reveal an association between the characteristic polynomial of A and the finite representation of the Cayley graph. After introducing an algorithm for the computation of the entropy dimension, the set of entropy dimensions is related to a collection of matrices in which the sum of each row of every matrix is bounded by the number of leaves of the graph. Furthermore, the algorithm extends to G having finitely many free-followers.
ON THE DENSE ENTROPY OF TWO-DIMENSIONAL INHOMOGENEOUS CELLULAR NEURAL NETWORKS
(2008-11-01)
Ban, Jung-Chao
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Chang, Chih-Hung
;
應用數學系
;
Department of Applied Mathematics
This investigation elucidates the dense entropy of two-dimensional inhomogeneous cellular neural networks (ICNN) with/without input. It is strongly related to the learning problem (or inverse problem); the necessary and sufficient conditions for the admissibility of local patterns must be characterized. For ICNN with/without input, the entropy function is dense in [ 0, log 2] with respect to the parameter space and the radius of the interacting cells, indicating that, in some sense, ICNN exhibit a wide range of phenomena.
On the structure of multi-layer cellular neural networks
(2012-04-15)
Ban, Jung-Chao
;
Chang, Chih-Hung
;
Lin, Song-Sun
;
應用數學系
;
Department of Applied Mathematics
Let Y subset of {-1, 1}(Z infinity xn) be the mosaic solution space of an n-layer cellular neural network. We decouple Y into n subspaces, say Y-(1), Y-(2), ... , Y-(n), and give a necessary and sufficient condition for the existence of factor maps between them. In such a case, Y-(i) is a sofic shift for 1 <= i <= n. This investigation is equivalent to study the existence of factor maps between, two sofic shifts. Moreover, we investigate whether Y-(i) and Y-(j) are topological conjugate, strongly shift equivalent, shift equivalent, or finitely equivalent via the well-developed theory in symbolic dynamical systems. This clarifies, in a multi-layer cellular neural network, each layer's structure. As an extension, we can decouple Y into arbitrary k-subspaces. where 2 <= k <= n, and demonstrates each subspace's structure. (C) 2012 Elsevier Inc. All rights reserved.
On the Structure of Two-Layer Cellular Neural Networks
(2013)
Ban, Jung-Chao
;
Chang, Chih-Hung
;
Lin, Song-Sun
;
應用數學系
;
Department of Applied Mathematics
Let Y subset of{-1,1}(Z infinity x2) be the mosaic solution space of a two-layer cellular neural network (TCNN). We decouple Y into two subspaces, say Y-(1) and Y-(2), and give a necessary and sufficient condition for the existence of factor maps between them. In such a case, Y-(i) is a sofic shift for i = 1,2. This investigation is equivalent to study the existence of factor maps between two sofic shifts. Moreover, we investigate whether Y-(1) and Y-(2) are topological conjugate, strongly shift equivalent, shift equivalent, or finitely equivalent via the well-developed theory in symbolic dynamical systems. This clarifies, in a TCNN, each layer\'s structure.
Pattern generation problems arising in multiplicative integer systems
(2019-05-01)
Ban, Jung-Chao
;
Hu, Wen-Guei
;
Lin, Song-Sun
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應用數學系
;
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.
Patterns generation and spatial entropy in two-dimensional lattice models
(2007-09-01)
Ban, Jung-Chao
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Lin, Song-Sun
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Lin, Yin-Heng
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應用數學系
;
Department of Applied Mathematics
Patterns generation problems in two-dimensional lattice models are studied. Let S be the set of p symbols and Z(2lx2l), l >= 1, be a fixed finite square sublattice of Z(2). Function U : Z(2lx2l) -> S is called local pattern. Given a basic set B of local patterns, a unique transition matrix A(2) which is a q(2)xq(2) matrix, q = p(l2), can be defined. The recursive formulae of higher transition matrix A(n) on Z(2lxnl) have already been derived [4]. Now A(n)(m), m >= 1, contains all admissible patterns on Z((m+1)lxnl) which can be generated by B. In this paper, the connecting operator C-m, which comprises all admissible patterns on Z((m+1)lx2l), is carefully arranged. C-m can be used to extend A(n)(m) to A(n+1)(m) recursively for n >= 2. Furthermore, the lower bound of spatial entropy h(A(2)) can be derived through the diagonal part of C-m. This yields a powerful method for verifying the positivity of spatial entropy which is important in examining the complexity of the set of admissible global patterns. The trace operator T-m of C-m can also be introduced. In the case of symmetric A(2), T-2m gives a good estimate of the upper bound on spatial entropy. Combining C-m with T-m helps to understand the patterns generation problems more systematically.
Spatial complexity in multi-layer cellular neural networks
(2009-01-15)
Ban, Jung-Chao
;
Chang, Chih-Hung
;
Lin, Song-Sun
;
Lin, Yin-Heng
;
應用數學系
This study investigates the complexity of the global set of output patterns for one-dimensional multi-layer cellular neural networks with input. Applying labeling to the output space produces a sofic shift space. Two invariants, namely spatial entropy and dynamical zeta function, can be exactly computed by studying the induced sofic shift space. This study gives sofic shift a realization through a realistic model. Furthermore, a new phenomenon, the broken of symmetry of entropy, is discovered in multi-layer cellular neural networks with input. (C) 2008 Elsevier Inc. All rights reserved.
Spatial complexity in multi-layer cellular neural networks
(2009-01-15)
Ban, Jung-Chao
;
Chang, Chih-Hung
;
Lin, Song-Sun
;
Lin, Yin-Heng
;
應用數學系
This study investigates the complexity of the global set of output patterns for one-dimensional multi-layer cellular neural networks with input. Applying labeling to the output space produces a sofic shift space. Two invariants, namely spatial entropy and dynamical zeta function, can be exactly computed by studying the induced sofic shift space. This study gives sofic shift a realization through a realistic model. Furthermore, a new phenomenon, the broken of symmetry of entropy, is discovered in multi-layer cellular neural networks with input. (C) 2008 Elsevier Inc. All rights reserved.
Three-dimensional Cellular Neural Networks and pattern generation problems
(2008-04-01)
Ban, Jung-Chao
;
Lin, Song-Sun
;
Lin, Yin-Heng
;
應用數學系
;
Department of Applied Mathematics
This work investigates three-dimensional pattern generation problems and their applications to three-dimensional Cellular Neural Networks (3DCNN). An ordering matrix for the set of all local patterns is established to derive a recursive formula for the ordering matrix of a larger finite lattice. For a given admissible set of local patterns, the transition matrix is defined and the recursive formula of high order transition matrix is presented. Then, the spatial entropy is obtained by computing the maximum eigenvalues of a sequence of transition matrices. The connecting operators are used to verify the positivity of the spatial entropy, which is important in determining the complexity of the set of admissible global patterns. The results are useful in studying a set of global stationary solutions in various Lattice Dynamical Systems and Cellular Neural Networks.