SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Inhomogeneous lattice dynamical systems and the boundary effect

Jung-Chao Ban1 and Chih-Hung Chang2*

Author Affiliations

1 Department of Applied Mathematics, National Dong Hwa University, Hualien, 970003, Taiwan, ROC

2 Department of Applied Mathematics, Feng Chia University, Taichung, 40724, Taiwan, ROC

For all author emails, please log on.

Boundary Value Problems 2013, 2013:249  doi:10.1186/1687-2770-2013-249

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/249


Received:18 April 2013
Accepted:24 October 2013
Published:21 November 2013

© 2013 Ban and Chang; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This study considers the dynamics of cellular neural network-based inhomogeneous lattice dynamical systems (CNN-based ILDS). The influence of three kinds of boundary conditions, say, the periodic, Dirichlet, and Neumann boundary conditions, is elucidated. We reveal that the complete stability of CNN-based ILDS and, under some prescriptions, the topological entropies of CNN-based ILDS with/without the boundary condition are identical.

MSC: 37B10.

Keywords:
inhomogeneous lattice dynamical systems; topological entropy; boundary value problem; multiplicative shift spaces; cellular neural networks

1 Introduction

In the past few decades, the standard cellular neural networks (CNNs) introduced by Chua and Yang [1] have been one of the most investigated paradigms for neural information processing [2]. In a wide range of applications, the CNNs are required to be completely stable, i.e., each trajectory should converge toward some stationary state. In the study of stationary solution, the investigation of mosaic solutions is most essential in CNNs due to the learning algorithm and training processing. More abundant output patterns make the learning algorithm more efficient. Mathematically, the study of the mosaic solutions is reasonable due to the following two facts: (1) complete stability of a wide range of parameters, and (2) the output function of CNNs is a piecewise linear function with constant value for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M1">View MathML</a>; namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M2">View MathML</a>

The outputs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M3">View MathML</a>, called patterns, are essential for understanding CNN systems. Traditionally, the template for CNNs is homogeneous (also known as isotropic), i.e., the template is space-invariant. However, there are more and more CNNs using inhomogeneous templates to describe some of the problems that arise from the biological and ecological contexts [3-8], skeletonization [9], image processing [10,11], artificial locomotion control [12], and delayed-type CNN [13-16]. Some new and interesting phenomena of pattern formation and spatial chaos were also found in inhomogeneous multi-layer neural networks. In this paper, the entropy with/without the boundary effect for stable patterns of inhomogeneous CNN is investigated. Entropy is a quantity used for measuring the complexity of the output patterns and it plays an important role in learning algorithm. Surprisingly, such a topic reveals the deep connection with symbolic dynamical systems (SDS). In 1-d CNN, it has been proved that the space of the mosaic solutions (defined later) forms a 1-d subshift of finite type (SFT, [17]). Recently, it has also been proved that the mosaic solutions of a multi-layer CNN (MCNN) form a sofic space[18-20], which is a factor of SFT. The mosaic solutions of inhomogeneous CNN, indeed, produce new shift spaces in SDS. To clarify the investigation of inhomogeneous CNNs, we concentrate our discussion on two classes, and the methodology can be applied in a general case. More specifically, two types of inhomogeneous CNN, constant and arithmetic CNN, are presented herein. It is proved that the space of the mosaic solutions forms a new class in SDS (Theorem 2.10 and Theorem 3.5), called a multiple shift space, which was initiated from the study of the arithmetic regression property in the number theory of mathematics [21-24]. The complexity (topological entropy) can be computed due to the equivalence of the mosaic solutions and multiple shift spaces (Theorem 2.13 and Theorem 3.7). The positivity of entropy unveils the spatial chaos for given systems and pattern formation for zero entropy. Such topics, e.g., pattern formation or synchrony phenomena on LDS, have been investigated by many mathematicians and physicists [25-30].

Besides the entropy formula being established, the boundary effect for constant CNNs and arithmetic CNNs are also considered. Three types of boundary conditions, periodic, Dirichlet, and Neumann, are proposed to a given constant CNN and arithmetic CNN. Sufficient conditions are found for the preservation of entropy under the boundary constraint (Theorem 2.13 and Theorem 3.7), i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M4">View MathML</a>. This extends the results in the classical CNNs (cf.[31,32]). The preservation of entropy under the boundary constraint is unavoidable [33]; since the number of nodes in a lattice is infinite, one usually uses the finite approximation method to exploit the statistical properties of the whole lattice.

Some related topics are also addressed herein. It is known that the mosaic solution of single/multi-layer template-invariant CNNs is constrained by the so-called separation property, namely, not all but some of the patterns that satisfy this property will appear as the mosaic solution for a given CNN [34]. However, more combinations of mosaic patterns will help the learning and training process to be more efficient. It is believed that the template-variant or the multi-layer CNN will achieve this goal. In mathematical language, it means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M5">View MathML</a> will be ϵ-dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M6">View MathML</a> when parameter runs all of the parameter space, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M5">View MathML</a> denotes the entropy function according to the parameter . It is proved that constant CNNs possess the ϵ-dense property (Theorem 2.14), and it seems that arithmetic CNNs also satisfy the ϵ-dense property by numerical computation (Conjecture 3.8). We believe that further interesting applications of the results presented (or of the generalizations) can be obtained.

We organize the material in this paper as follows. Section 2 introduces the concepts of general inhomogeneous CNN-based LDS and constant-type multiple CNNs. Stability, partition of the parameter space and the equivalence of mosaic solutions with a multiple shift space are discussed therein. This together with the exact number of mosaic solutions under the boundary constraint (Lemma 2.12) is used to derive the entropy formula and entropy preservation property. Parallel discussions for arithmetic-type multiple CNNs can be found in Section 3. Some one- and two-dimensional examples are addressed in Section 4, and we leave the discussion in Section 5.

2 Constant cellular neural networks

In this section, we investigate a specified type of inhomogeneous LDS named constant-type multiple cellular neural network (constant CNN). To clarify the elucidation, Section 2.1 concentrates on the constant CNNs with nearest neighborhood. The general cases of constant CNNs and deeper architecture are investigated in the rest of this section.

2.1 Constant cellular neural networks with nearest neighborhood

First we consider the LDS realized as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M10">View MathML</a>

(1)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M11">View MathML</a>. Denote the parameters that relate to the odd and even positions by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M13">View MathML</a>, respectively. We call <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M14">View MathML</a> the feedback template of (1), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M15">View MathML</a> is the threshold. It is seen that the templates in (1) are periodic; the prescribed model is a generalization of the classical cellular neural network and is called the constant-type multiple cellular neural network.

A system of ordinary differential equations is called completely stable if each of its solution x approaches an equilibrium state. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M17">View MathML</a> denote the collection of cells in odd and even coordinates, respectively. Express (1) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M18">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M21">View MathML</a> is a diagonal mapping (herein <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M23">View MathML</a>), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M24">View MathML</a>. The sufficient conditions for the complete stability of (1) are given as follows. The extension of Theorem 2.1 can be seen in Theorem 2.5.

Theorem 2.1A constant CNN is completely stable if, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M19">View MathML</a>, one of the following conditions is satisfied.

S1 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M26">View MathML</a>is symmetric.

S2 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M27">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M28">View MathML</a>for alli, j, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M29">View MathML</a>

The complete stability of (1) demonstrates that the investigation of the equilibrium solutions is essential. To make the discussion more clear, we focus on the mosaic solutions, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M30">View MathML</a> for all i, and study the complexity of the output space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M31">View MathML</a> of the mosaic solutions. We investigate the complexity of the output space in two aspects:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M32">View MathML</a>: The exact number of patterns of length n.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33">View MathML</a>: The topological entropy of the output space.

To achieve our target, we introduce the ordering matrix and transition matrix first. The ordering matrix is defined as

herein the pattern ‘−’ stands for the state <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M35">View MathML</a> and ‘+’ stands for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M36">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M37">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M19">View MathML</a>, define the transition matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M39">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M40">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M41">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M42">View MathML</a> consists of patterns of length n in X. Yielding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M44">View MathML</a>, we derive the formula of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33">View MathML</a>. For the general cases of constant CNNs, Theorem 2.2 is generalized by Lemma 2.11 and Theorem 2.13.

Theorem 2.2Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M47">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M49">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M43">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M44">View MathML</a>are the transition matrices of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M52">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M53">View MathML</a>, respectively. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M54">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M55">View MathML</a>for any nonnegative matrix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M56">View MathML</a>. Moreover, the topological entropy ofYis

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M57">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M58">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M59">View MathML</a>are the spectral radii of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M43">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M44">View MathML</a>, respectively.

In the meantime, it is natural to elucidate the influence of boundary conditions on the exact number of patterns of length n and topological entropy. Three types of boundary conditions, periodic, Neumann, and Dirichlet boundary conditions, are considered. To reflect the influence of the boundary conditions, we introduce three boundary matrices. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M62">View MathML</a>

The periodic boundary matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M63">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M64">View MathML</a> matrix defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M65">View MathML</a>

The Neumann boundary condition infers zero flux on both sides of the space. The left and right Neumann boundary matrices are then defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M66">View MathML</a>

respectively. Furthermore, the Dirichlet boundary condition indicates that both sides of the space are constant states and the corresponding boundary matrices are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M67">View MathML</a>

Herein <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M69">View MathML</a> relate to states ‘−’ (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M70">View MathML</a>) and ‘+’ (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M71">View MathML</a>), respectively. Before presenting the formula of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M73">View MathML</a> under the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M74">View MathML</a>, we introduce two operations of matrices.

Definition 2.3

1. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M75">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M76">View MathML</a> matrix and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M77">View MathML</a> is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M78">View MathML</a> matrix. The Kronecker product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M79">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M80">View MathML</a>

2. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M81">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M82">View MathML</a> matrices. The Hadamard product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M83">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M84">View MathML</a>

With the introduction of the boundary matrices and the Kronecker and Hadamard products, we obtain Theorem 2.4 which reveals the formulae of exact number of patterns and topological entropy under the influence of three kinds of boundary conditions. The extension of Theorem 2.4 for general constant CNNs is demonstrated by Lemma 2.12 and Theorem 2.13.

Theorem 2.4Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M47">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M49">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M43">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M44">View MathML</a>are the transition matrices of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M52">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M53">View MathML</a>, respectively. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M74">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M43">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M44">View MathML</a>are primitive matrices. Furthermore, the exact number of patterns of lengthnwith boundary condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M96">View MathML</a>are as follows:

The periodic boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M97">View MathML</a>

(3)

The Neumann boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M98">View MathML</a>

(4)

Herein<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M99">View MathML</a>.

The Dirichlet boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M100">View MathML</a>

(5)

Herein<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M101">View MathML</a>relate to the conditions that the patterns on the boundary are ‘’ and ‘+, respectively.

2.2 Stability of constant cellular neural networks

The rest of this section extends the results in Section 2.1. To make the paper compact, we introduce the general setting for multi-dimensional inhomogeneous LDS and then concentrate on the one-dimensional case. The elucidation of multi-dimensional systems will be investigated in another paper.

A D-dimensional inhomogeneous CNN-based LDS is realized as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M102">View MathML</a>

(6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M103">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M104">View MathML</a>, which is a finite subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M105">View MathML</a>, indicates the neighborhood for neuron <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M106">View MathML</a>. The piecewise linear function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M107">View MathML</a> is called the output function; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M108">View MathML</a> refers to the threshold, and the feedback template<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M109">View MathML</a> stores the weight of local interaction between neurons, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M110">View MathML</a>.

An inhomogeneous CNN-based LDS is called a constant CNN if the neighborhood , the template , and z are periodic up to shifts. More precisely, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M113">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M115">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M116">View MathML</a> satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M118">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M119">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M120">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M121">View MathML</a>

It is seen that the constant CNNs generalize the concept of the classical CNNs that were introduced in [1,35]. More precisely, a classical CNN is a constant CNN with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M122">View MathML</a>. The essential description of a one-dimensional constant CNN is presented in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M123">View MathML</a>

(7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M124">View MathML</a> and . Without loss of generality, we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M126">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M128">View MathML</a>. In this case, the feedback template of (7) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M129">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M130">View MathML</a>. A stationary solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M131">View MathML</a> is called a mosaic solution if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M30">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M133">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M134">View MathML</a> is called a mosaic pattern. A system of ordinary differential equations is said to be completely stable if every trajectory tends to an equilibrium point. Theorem 2.5 infers that a constant CNN is a completely stable system. (We remark that Theorem 2.5 is an extension of Theorem 2.1.)

Theorem 2.5Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M135">View MathML</a>is the template of (7) and the system is written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M136">View MathML</a>

Then a constant CNN is completely stable if, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M137">View MathML</a>, one of the following conditions is satisfied.

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M138">View MathML</a>is symmetric.

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M139">View MathML</a>is nonsingular and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M140">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M139">View MathML</a>is defined in (10).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M142">View MathML</a> be a finite index set. The one-dimensional lattice ℤ can be decomposed into non-overlapping subspaces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M143">View MathML</a>

Equation (7) can then be restated as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M144">View MathML</a>

(8)

(It is easily seen that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M145">View MathML</a>. We reindex the coordinates of neurons to clarify the upcoming investigation.) To prove Theorem 2.5, we consider two kinds of feedback templates separately. For the case that the feedback template of a classical CNN is symmetrical, Forti and Tesi demonstrated that it is completely stable.

Theorem 2.6 ([36])

A classical CNN with symmetric feedback template is completely stable.

For the case that the feedback template is not symmetrical, suppose that a CNN with n-neurons is described as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M146">View MathML</a>

(9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M147">View MathML</a>, A is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M148">View MathML</a> constant matrix with diagonal elements satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M149">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M150">View MathML</a> is a diagonal mapping from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M151">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M151">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M153">View MathML</a> is a constant vector. Takahashi and Chua proposed a criterion to determine whether a CNN is completely stable.

Theorem 2.7 ([37])

LetKbe an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M148">View MathML</a>matrix satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M155">View MathML</a>

(10)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M156">View MathML</a>. A classical CNN with asymmetric feedback template is completely stable ifKis nonsingular and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M157">View MathML</a>, herein a matrix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M158">View MathML</a>means that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M159">View MathML</a>for alli, j.

It comes immediately from Theorem 2.7 that if the feedback template <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M160">View MathML</a> of a CNN is asymmetric, then the system is completely stable provided there exists a positive constant r such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M161">View MathML</a>

(11)

Proof of Theorem 2.5 Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M162">View MathML</a>; in this case, a constant CNN is deduced to be a classical CNN. Theorem 2.6 infers that a constant CNN is completely stable if the feedback template is symmetrical. Whenever is asymmetric, the system is still completely stable if the matrix K defined in (10) is nonsingular and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M157">View MathML</a>. It is indicated via (8) that a constant CNN can be decomposed into independent CNN subsystems, the complete stability of a constant CNN comes from the complete stability of every subsystem. □

For a fixed template, the collection of mosaic patterns <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M166">View MathML</a> is called the output space of (7). Since the neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M167">View MathML</a> is finite for each i, the output space is determined by the so-called admissible local patterns. Suppose that y is a mosaic pattern, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M11">View MathML</a>, the necessary and sufficient condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M170">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M171">View MathML</a>

(12)

and the necessary and sufficient condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M172">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M173">View MathML</a>

(13)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M174">View MathML</a>

The set of admissible local patterns ℬ of a constant CNN is then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M175">View MathML</a>

Similar to the discussion in [17], the output space Y can be represented as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M176">View MathML</a>

(Recall that in the above equation, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M145">View MathML</a>.)

One of the important research issues in the circuit theory is the learning problem. That is to say, mathematically, for what and how many phenomena the constant CNNs are capable of exhibiting. Theorem 2.9 infers that once <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M178">View MathML</a> is fixed, there are finitely many equivalent classes of templates and z so that the basic sets of admissible local patterns <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M180">View MathML</a> are constrained. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M181">View MathML</a> be the parameter space of the classical CNNs, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M182">View MathML</a>. Theorem 2.8 indicates that the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M183">View MathML</a> can be partitioned into a finite number of subregions such that each subregion has the same mosaic patterns.

Theorem 2.8 ([34])

There is a positive integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M184">View MathML</a>and a unique set of open subregions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M185">View MathML</a>satisfying

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M186">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M187">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M188">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M189">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M190">View MathML</a>for somekif and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M191">View MathML</a>.

Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M192">View MathML</a>is the closure ofPin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M193">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M194">View MathML</a> be the parameter space of (7). The following theorem demonstrates that is also partitioned into a finite number of equivalent subregions.

Theorem 2.9 (Separation property)

There is a positive integerKand a unique set of open subregions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M185">View MathML</a>satisfying

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M197">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M187">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M188">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M189">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M190">View MathML</a>for somekif and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M191">View MathML</a>.

Proof Similar to the proof of Theorem 2.5, a constant CNN is reduced to a classical CNN whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M162">View MathML</a>, hence Theorem 2.9 is performed in this case. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M204">View MathML</a>, the basic set of admissible local patterns <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M205">View MathML</a> of (7) is the ordered union of the basic set of admissible local patterns <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M206">View MathML</a>. More specifically, is isomorphic to the direct product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M208">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M209">View MathML</a> is the parameter space of (7)j, the subsystem of (7) restricting to the cells <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M210">View MathML</a>. Since, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M211">View MathML</a>, each parameter space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M209">View MathML</a> is partitioned into a finite number of equivalent subregions by Theorem 2.8, is then the union of a unique set of open subregions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M185">View MathML</a> which satisfies conditions (i) to (iii). This derives the desired result. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M215">View MathML</a> be an integer, and let Ω be a subset of the symbolic space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M216">View MathML</a> which is invariant under the shift map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M217">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M218">View MathML</a>. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M219">View MathML</a>

which is invariant under σ. The set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M220">View MathML</a> is called a multiple subshift if Ω is a subshift. Equation (8) together with the proof of Theorem 2.9 asserts that the output space Y of a constant CNN is decomposed into subspaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M221">View MathML</a>. Observe that Y is topologically conjugated to the direct product of the output spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222">View MathML</a> of the classical CNNs, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M223">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222">View MathML</a> is determined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M225">View MathML</a>. This derives Theorem 2.10, which indicates that the output space of a constant CNN is a multiple subshift for some parameters.

Theorem 2.10Given a set of templates<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M226">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M227">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M228">View MathML</a>. LetYbe the solution space of the constant CNN with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M135">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M230">View MathML</a>

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M231">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M232">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M233">View MathML</a>, where Ω is a SFT that comes from the output space of the classical CNN with respect to template<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M234">View MathML</a>.

2.3 Boundary effect on constant cellular neural networks

This subsection elucidates the influence of the boundary condition on the exact number of mosaic patterns of finite length and on the growth rate as the length increases. The investigation starts with formulating the number of patterns. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M235">View MathML</a> the coordinates of the neurons. In this case, the boundary sites are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M236">View MathML</a>. For the constant CNNs on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M237">View MathML</a>, the following three types of boundary conditions are considered:

(i) (7)n-N: constant CNNs with Neumann boundary condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M237">View MathML</a>;

(ii) (7)n-P: constant CNNs with periodic boundary condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M237">View MathML</a>;

(iii) (7)n-D: constant CNNs with Dirichlet boundary condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M237">View MathML</a>.

These boundary conditions are discrete analogues of the ones in PDEs; to be specific, a pattern <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M241">View MathML</a> satisfies: (i) the Neumann boundary condition if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M242">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M243">View MathML</a>; (ii) the periodic boundary condition if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M244">View MathML</a>; (iii) the Dirichlet boundary condition if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M245">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M246">View MathML</a> are prescribed.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M247">View MathML</a>, the total number of patterns of finite length in a constant CNN relates to the number of patterns in the subspaces. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248">View MathML</a>, there is a transition matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M249">View MathML</a> that is implemented for the investigation of the subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222">View MathML</a>a (cf.[17] and Section 4). Lemma 2.11 elucidates the exact number of mosaic patterns of length n of a constant CNN without the influence of the boundary condition. The verification is straightforward and is omitted.

Lemma 2.11For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M252">View MathML</a>, write<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M253">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M254">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M255">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M256">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M258">View MathML</a>denotes the number of patterns of lengthqinX.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M259">View MathML</a> denote the collection of output patterns of length n with boundary condition B, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M260">View MathML</a>, and D stands for the periodic, Neumann, and Dirichlet boundary conditions, respectively. To find the exact number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M261">View MathML</a>, we introduce the following boundary matrices.

(i) Periodic boundary matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M262">View MathML</a>. More precisely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M263">View MathML</a>

(ii) Dirichlet boundary matrices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M266">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M267">View MathML</a> stands for the left/right Dirichlet boundary condition that is given by ‘−’ and ‘+’, respectively.

(iii) Neumann boundary matrices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M268">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M269">View MathML</a>. More precisely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M270">View MathML</a>

Here ⊗ is the Kronecker product, E is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M271">View MathML</a> matrix with entries being 1’s, I is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M271">View MathML</a> identity matrix, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M273">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M274">View MathML</a> column vector with entries being 1’s. Suppose that M is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M275">View MathML</a> matrix. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M276">View MathML</a> by letting all the even/odd columns be zero vectors. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M277">View MathML</a> indicates the matrix obtained from M by setting each of the lower-/upper-half rows as a zero vector.

Recall that a set function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M278">View MathML</a> is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M279">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M280">View MathML</a> for E being a nonempty subset of ℝ. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M252">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M282">View MathML</a>

(14)

It is seen that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M283">View MathML</a> is a nonnegative integer. To clarify the formulae of the exact number of patterns of length n of constant CNNs with boundary conditions, we introduce some notations first. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M284">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M285">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248">View MathML</a>, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M287">View MathML</a>

(15)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M288">View MathML</a>

(16)

Herein <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M289">View MathML</a> refers to the 1-norm of the matrix M. Lemma 2.12 demonstrates the explicit formulae of the number of patterns of length n with boundary conditions.

Lemma 2.12Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M284">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M291">View MathML</a>. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M292">View MathML</a>, then the exact number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M293">View MathML</a>with boundary condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M294">View MathML</a>are as follows:

(i) The periodic boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M295">View MathML</a>

(17)

(ii) The Dirichlet boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M296">View MathML</a>

(18)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M297">View MathML</a>means the pattern on the boundary is ‘<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M298">View MathML</a>.

(iii) The Neumann boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M299">View MathML</a>

(19)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M300">View MathML</a>

(20)

otherwise.

Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M301">View MathML</a>is a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M302">View MathML</a>matrix with entries being 1’s, andmeans the Hadamard product.

Proof We address the proof of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M303">View MathML</a>, where the other cases can be verified in an analogous method.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M304">View MathML</a>. It is seen from Lemma 2.11 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M305">View MathML</a>

At the same time, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M292">View MathML</a> indicates that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M307">View MathML</a> for all j. A straightforward examination demonstrates that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M308">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M309">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M310">View MathML</a>. Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M311">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M312">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M244">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M314">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M315">View MathML</a> refers to the number of patterns

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M316">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M317">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M318">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M319">View MathML</a> are patterns of length <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M320">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M321">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M322">View MathML</a>, respectively. It is verified that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M323">View MathML</a>

This derives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M324">View MathML</a>

and completes the proof. □

Next, to study the influence of boundary conditions on the exact number of patterns of finite length, we consider the effect on the growth rate of the number of patterns; more specifically, the topological entropy of the output space Y. The topological entropy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M325">View MathML</a> of a space X is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M326">View MathML</a>

(21)

The existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33">View MathML</a> comes immediately from the submultiplicativity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M328">View MathML</a>, which can be verified by applying Lemma 2.11. Theorem 2.13 declares the formula of the topological entropies of the constant CNNs, and the relation between the topological entropies of the constant CNNs and the classical CNNs.

Theorem 2.13<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M329">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M92">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M294">View MathML</a>provided<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222">View MathML</a>is mixing for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M252">View MathML</a>, there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M335">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M336">View MathML</a>

Lemma 2.11 infers that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M337">View MathML</a>

Applying the squeeze theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M338">View MathML</a>

This completes the first part of the proof.

To evaluate the boundary effect on the topological entropy of Y, we demonstrate that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M339">View MathML</a>. The other cases can be done analogously. Let τ denote the smallest integer such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M340">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248">View MathML</a>, restated, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M342">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M343">View MathML</a>. According to the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M63">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M345">View MathML</a>

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M284">View MathML</a>. Lemma 2.12 implements

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M347">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M304">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M349">View MathML</a>

otherwise. On the other hand, it is easily checked that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M350">View MathML</a>

The above observation derives that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M351">View MathML</a>

and thus we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M92">View MathML</a>. □

The following theorem comes immediately from Theorem 2.13, the proof is omitted.

Theorem 2.14The set of topological entropies of the constant CNNs is dense in the closed interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M353">View MathML</a>. More precisely, given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M354">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M355">View MathML</a>, there exists a constant CNN such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M356">View MathML</a>.

3 Arithmetic cellular neural networks

This section elucidates another kind of inhomogeneous CNN-based LDS named arithmetic-type multiple cellular neural network (arithmetic CNN). It is seen that the templates of a constant CNN are periodic; in other words, the number of distinct templates is finite. This section investigates inhomogeneous CNNs whose number of distinct templates is infinite. First we consider a one-dimensional LDS with nearest neighborhood to interpret the idea of our methodology, then the derived results are generalized to general cases in the rest of this section.

3.1 Arithmetic cellular neural networks with nearest neighborhood

To clarify the study of an inhomogeneous LDS with nearest neighborhood, we consider the following system,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M357">View MathML</a>

(22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M358">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M359">View MathML</a> is odd. The feedback template <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M360">View MathML</a> consists of infinitely many subtemplates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M361">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M362">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M363">View MathML</a>, and the threshold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M364">View MathML</a> is an infinite vector. An inhomogeneous CNN realized as (22) is called the arithmetic CNN.

Similar to the discussion in the previous subsection, we demonstrate that arithmetic CNNs are completely stable. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M365">View MathML</a> denote the collection of cells related to initial coordinate q. Express (22) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M366">View MathML</a>

(23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M367">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M368">View MathML</a> are similar to those defined in the previous subsection. A sufficient condition for the complete stability of (22) is presented as Theorem 3.1, which is a special case of Theorem 3.4.

Theorem 3.1An arithmetic CNN is completely stable if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M369">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M370">View MathML</a>for allq, i, j, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M371">View MathML</a>

Following the complete stability of an arithmetic CNN is the spatial complexity of the output space and the influence of boundary conditions. Note that the output space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M372">View MathML</a> is different from the one in the previous subsection. The ordering matrix is then defined as

After redefining the ordering matrix, we obtain a sequence of transition matrices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M374">View MathML</a> corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M375">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M362">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M363">View MathML</a>. The following theorem exhibits the computation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33">View MathML</a>. Furthermore, Theorem 3.2 is generalized to Theorem 3.7 for general arithmetic CNNs.

Theorem 3.2Suppose thatYis the output space of an arithmetic CNN. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M380">View MathML</a>

whereqis odd and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M381">View MathML</a>is the Gauss function. Furthermore, the topological entropy ofYis

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M382">View MathML</a>

For the influence of the boundary conditions, we define the boundary matrices as follows. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M383">View MathML</a>

The periodic boundary matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M63">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M385">View MathML</a> matrix defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M386">View MathML</a>

and the left and right Neumann boundary matrices are then defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M387">View MathML</a>

respectively. Furthermore, the Dirichlet boundary matrices are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M388">View MathML</a>

To simplify the formulae of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M389">View MathML</a>, the following theorem presents the specific case. The general case is postponed to Lemma 3.6 and Theorem 3.7.

Theorem 3.3Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M390">View MathML</a>for somekandYis the output space of an arithmetic CNN. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M392">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M374">View MathML</a>are primitive for allq. Furthermore, the exact number of patterns of lengthnwith boundary condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M392">View MathML</a>are as follows:

The periodic boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M395">View MathML</a>

(24)

The Neumann boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M396">View MathML</a>

(25)

The Dirichlet boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M397">View MathML</a>

(26)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M101">View MathML</a>.

3.2 Stability of arithmetic cellular neural networks

The rest of this section considers the inhomogeneous CNN-based LDS with the neighborhood consisting of infinitely many elements. A D-dimensional inhomogeneous CNN-based LDS is called an arithmetic CNN if the neighborhood , the template , and the threshold z are periodic up to a multiplication. More precisely, there exists a positive integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M402">View MathML</a> such thatb

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M404">View MathML</a>

Herein

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M405">View MathML</a>

The essential description of a one-dimensional arithmetic CNN is that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M406">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M407">View MathML</a>. More precisely, a one-dimensional arithmetic CNN is realized as the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M408">View MathML</a>

(27)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M409">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M410">View MathML</a>, , and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M412">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M413">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M414">View MathML</a> be an infinite index set. The set of positive integers ℕ is then decomposed into the disjoint union of infinitely many subsets by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M415">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M416">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M358">View MathML</a>. Equation (27) can then be represented as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M419">View MathML</a>

(28)

In this case, the feedback template <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M420">View MathML</a> consists of infinitely many smaller templates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M421">View MathML</a>, and the threshold is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M422">View MathML</a>. Similar to the discussion in the previous section, Theorem 3.4 asserts that an arithmetic CNN is completely stable. The proof is omitted.

Theorem 3.4Suppose that an arithmetic CNN is presented as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M423">View MathML</a>

Then the system is completely stable if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M424">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M168">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M139">View MathML</a>comes from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M427">View MathML</a>defined in (10).

Suppose that y is a mosaic pattern; for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M429">View MathML</a>, the necessary and sufficient condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M430">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M431">View MathML</a>

(29)

and the necessary and sufficient condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M172">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M433">View MathML</a>

(30)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M434">View MathML</a>

The set of admissible local patterns ℬ of an arithmetic CNN is then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M435">View MathML</a>

The output space Y is then represented as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M436">View MathML</a>

Recall that the output space Y of a constant CNN can be decomposed into finitely many subspaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222">View MathML</a> is a SFT for each j. In other words, the output space of a constant CNN extends the concept of SFTs. The output space of an arithmetic CNN is decomposed into countable subspaces; more precisely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M439">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222">View MathML</a> is determined by the basic set of admissible local patterns <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M441">View MathML</a>. Theorem 3.5 demonstrates that the output space of an arithmetic CNN is a generalization of the so-called multiplicative shifts.

In [22], the authors introduced the concept of multiplicative subshifts in the context of symbolic dynamical systems. Let Ω be a subshift of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M442">View MathML</a>. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M443">View MathML</a>

which is invariant under the action of multiplicative integers:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M444">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M220">View MathML</a> is called a multiplicative subshift. We define a semigroup action on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M446">View MathML</a> by the following. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M447">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M448">View MathML</a>, the action <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M449">View MathML</a> is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M450">View MathML</a>. It is seen that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M220">View MathML</a> is invariant under the action. In other words, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M452">View MathML</a> defines a multiplicative subshift.

A straightforward examination indicates that the output space Y of an arithmetic CNN is a multiplicative subshift if the neighborhood and the templates of (27) are invariant; restated, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M453">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M231">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M455">View MathML</a>. The proof is omitted.

Theorem 3.5Given a set of templates<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M456">View MathML</a>. LetYbe the solution space of the arithmetic CNN with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M135">View MathML</a>. ThenYis a multiplicative subshift if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M458">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M459">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M460">View MathML</a>. More precisely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M461">View MathML</a>

where Ω is the SFT that comes from the output space of the classical CNN with respect to the template<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M234">View MathML</a>.

3.3 Boundary effects on arithmetic cellular neural networks

Recall that a set function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M278">View MathML</a> is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M279">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M280">View MathML</a> for E is a nonempty subset of ℝ. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M168">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M468">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M469">View MathML</a>

(31)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M470">View MathML</a>

(32)

It is seen that both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M471">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M472">View MathML</a> are nonnegative integers. To clarify the formulae of the exact number of patterns of length n of an arithmetic CNN with boundary condition, we introduce some notations first. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M473">View MathML</a>

(33)

Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M249">View MathML</a> is the transition matrix of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222">View MathML</a>.

The exact number of patterns of the arithmetic CNNs with boundary condition is obtained via a small modification of the discussion in the proof of Lemma 2.12. Before presenting the formulae, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M476">View MathML</a> for all j and redefine the boundary matrices as follows. Suppose that E is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M477">View MathML</a> matrix with all entries being 1’s. The periodic boundary matrix, left and right Neumann boundary matrices are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M478">View MathML</a> matrices given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M479">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M480">View MathML</a>

respectively. The left and right Dirichlet boundary matrices are defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M481">View MathML</a>

where I is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M477">View MathML</a> identity matrix. The formulae of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M72">View MathML</a> are addressed as follows and the demonstration is omitted.

Lemma 3.6Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M484">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M485">View MathML</a>

(34)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M486">View MathML</a>

(35)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M487">View MathML</a>

(36)

Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M488">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M489">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M490">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M491">View MathML</a>. Then:

(i) The periodic boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M492">View MathML</a>

(37)

(ii) The Dirichlet boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M493">View MathML</a>

(38)

(iii) The Neumann boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M494">View MathML</a>

(39)

Theorem 3.7 formulates the topological entropy of the output space of an arithmetic CNN with/without boundary conditions.

Theorem 3.7Suppose that there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M495">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M496">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M498">View MathML</a>

(40)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M499">View MathML</a>is defined in (33). Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M92">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M294">View MathML</a>provided<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222">View MathML</a>is mixing for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M248">View MathML</a>.

Proof The calculation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33">View MathML</a> is presented; the effect of the boundary condition on the topological entropy can be elucidated via similar discussion, and as with the proof of Theorem 2.13, thus is omitted.

Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M505">View MathML</a>

Hence we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M506">View MathML</a>

This completes the proof. □

The numerical experiment asserts that, similar to Theorem 2.14, the set of topological entropies of the arithmetic CNNs is dense in the closed interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M353">View MathML</a>. The theoretical proof of the following conjecture is not complete yet.

Conjecture 3.8Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M354">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M509">View MathML</a>, there exists an arithmetic CNN such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M510">View MathML</a>.

4 Examples

4.1 One-dimensional cellular neural networks

Example 4.1 Consider a constant CNN with templates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M511">View MathML</a> and z being given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M512">View MathML</a>

(Notably, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M513">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M514">View MathML</a> in this case.) The transition matrices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M515">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M516">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M321">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M518">View MathML</a> are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M519">View MathML</a>

respectively. Theorem 2.13 infers that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M520">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M521">View MathML</a> is the golden mean and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M522">View MathML</a> is the maximal root of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M523">View MathML</a>.

To estimate the exact number of the mosaic patterns of length n with boundary conditions, we consider the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M524">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M525">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M526">View MathML</a>

The periodic and Neumann boundary matrices are then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M527">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M528">View MathML</a>

respectively. Then the exact number of the mosaic patterns of length 20 with periodic boundary condition is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M529">View MathML</a>

the exact number of the mosaic patterns of length 20 with Neumann boundary condition is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M530">View MathML</a>

Furthermore, the Dirichlet boundary matrices are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M531">View MathML</a>

The exact number of the mosaic patterns of length 20 enclosed by the pattern ‘−’ is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M532">View MathML</a>

the exact number of the mosaic patterns of length 20 enclosed by the pattern ‘+’ is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M533">View MathML</a>

Suppose that the template <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M534">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M535">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M536">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M537">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M538">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M539">View MathML</a> are unknown. It is known that (cf.[17,32]) there are only finite possibilities of topological entropies for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M321">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M518">View MathML</a> as the parameters vary. More precisely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M542">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M543">View MathML</a> is the maximal root of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M544">View MathML</a>. The topological entropies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33">View MathML</a> with the parameters varying are seen in Table 1.

Table 1. The topological entropy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M33">View MathML</a>of constant CNNs with2-components and templates being given by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M547">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M548">View MathML</a>

Example 4.2 Consider an arithmetic CNN with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M514">View MathML</a> and an invariant template <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M558">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M559">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M560">View MathML</a>. Suppose that the transition matrix of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M562">View MathML</a>

In other words, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222">View MathML</a> is a golden mean shift with topological entropy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M564">View MathML</a> for all j. We remark that Fan et al.[23] investigated the Minkowski dimension of Y. To compute the topological entropy of Y, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M252">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M566">View MathML</a>

be the sets of integers such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M567">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M568">View MathML</a>. A straightforward verification infers that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M569">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M570">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M571">View MathML</a>. In other words, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M572">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M573">View MathML</a> is decreasing with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M574">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M575">View MathML</a> is the Fibonacci sequence defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M576">View MathML</a>

By induction we derive that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M577">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M578">View MathML</a>

4.2 Two-dimensional constant cellular neural networks

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M579">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M580">View MathML</a>, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M581">View MathML</a>

The two-dimensional lattice <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M582">View MathML</a> is written as the union of non-overlapping subspaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M583">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M584">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M585">View MathML</a>, we index the entries in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M586">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M587">View MathML</a>. Consider the two-dimensional constant CNNs of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M588">View MathML</a>

(41)

with template <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M589">View MathML</a> being given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M590">View MathML</a>

(42)

Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M580">View MathML</a>, Juang and Lin [17] studied (41) systematically and estimated the lower bound of the topological entropy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M592">View MathML</a>. More precisely, the lower bound of the topological entropy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M592">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M594">View MathML</a>

Suppose that the template <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M595">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M596">View MathML</a> are chosen so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M597">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M598">View MathML</a>

A detailed and complete investigation is postponed to the upcoming manuscript.

5 Conclusions

The present paper studies two types of one-dimensional inhomogeneous CNN-based LDS, say, constant- and arithmetic-type multiple CNNs, which are a generalization of the classical CNNs. Sufficient conditions for the complete stability of constant and arithmetic CNNs are revealed. Since there is a wide range of parameters making the system completely stable, it is essential to investigate the complexity of mosaic patterns of the given system. A systematic methodology is proposed to interpret the exact number of mosaic patterns of inquired length and the topological entropy of the output space. Furthermore, the exact number of mosaic patterns and the topological entropy of the output space under the influence of three boundary conditions, say, the periodic, Neumann, and Dirichlet boundary conditions, are obtained. Remarkably, the boundary condition does not influence the topological entropy under some presumption.

The reveal that the set of topological entropies of the output spaces of constant CNNs is dense in the close interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M353">View MathML</a> indicates how rich phenomena constant CNNs could exhibit. Although there is a lack of rigorous proof for the density of the set of topological entropies of arithmetic CNNs, numerical experiments assert an affirmative result.

The methodology we propose in this investigation can be applied to multi-dimensional cases. A detailed discussion is on-going.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

J-CB and C-HC contributed equally. All authors read and approved the final manuscript.

Acknowledgements

We thank the anonymous referees for their valuable comments that helped improve the quality and readability of the paper. Ban is partially supported by the National Science Council, ROC (Contract No. NSC 102-2628-M-259-001-MY3). Chang is grateful for the partial support of the National Science Council, ROC (Contract No. NSC 102-2115-M-035-004-).

End notes

  1. Notably every subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M222">View MathML</a> is a subshift of finite type (SFT) in symbolic dynamical systems, and thus can be studied via the graph theory and matrix theory. The reader is referred to [17] and [38] for more details.

  2. An arithmetic CNN is a classical CNN for the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/249/mathml/M403">View MathML</a>, this makes the requirement essential.

References

  1. Chua, LO, Yang, L: Cellular neural networks: theory. IEEE Trans. Circuits Syst.. 35, 1257–1272 (1988). Publisher Full Text OpenURL

  2. Chua, LO: CNN: A Paradigm for Complexity, World Scientific, Singapore (1998)

  3. Killingback, T, Loftus, G, Sundaram, B: Competitively coupled maps and spatial pattern formation (2012). arXiv:1204.2463 OpenURL

  4. Yokozawa, M, Kubota, Y, Hara, T: Effects of competition mode on the spatial pattern dynamics of wave regeneration in subalpine tree stands. Ecol. Model.. 118, 73–86 (1999). Publisher Full Text OpenURL

  5. Yokozama, M, Kubota, Y, Hara, T: Effects of competition mode on spatial pattern dynamics in plant communities. Ecol. Model.. 106, 1–16 (1998). Publisher Full Text OpenURL

  6. Doebeli, M, Killingback, T: Metapopulation dynamics with quasi-local competition. Theor. Popul. Biol.. 64, 397–416 (2003). PubMed Abstract | Publisher Full Text OpenURL

  7. Doebeli, M, Hauert, C, Killingback, T: The evolutionary origin of cooperators and defectors. Science. 306, 859–862 (2004). PubMed Abstract | Publisher Full Text OpenURL

  8. Hauert, C, Doebeli, M: Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature. 428, 643–646 (2004). PubMed Abstract | Publisher Full Text OpenURL

  9. Harrer, H, Nossek, J: Skeletonization: a new application for discrete-time cellular neural networks using time-variant templates. Circuits and Systems, 1992. ISCAS ’92. Proceedings., 1992 IEEE International Symposium on. 2897–2900 (1992)

  10. Costantini, G, Casali, D, Carota, M: A pattern classification method based on a space-variant CNN template. Cellular Neural Networks and Their Applications, 2006. CNNA ’06. 10th International Workshop on. 1–5 (2006)

  11. Wu, CY, Cheng, CH: A learnable cellular neural network (CNN) structure with ratio memory for image processing. IEEE Trans. Circuits Syst. I. 49, 37–40 (2002)

  12. Arena, P, Fortuna, L, Frasca, M, Marchese, C: Multi-template approach to artificial locomotion control. Circuits and Systems, 2001. ISCAS 2001. The 2001 IEEE International Symposium on. 37–40 (2001)

  13. Roska, T, Chua, L: Cellular neural networks with nonlinear and delay-type template elements. Cellular Neural Networks and Their Applications, 1990. CNNA-90 Proceedings., 1990 IEEE International Workshop on. 12–25 (1990)

  14. Kim, H, Son, HH, Roska, T, Chua, LO: Optimal path finding with space- and time-variant metric weights with multi-layer CNN. Int. J. Circuit Theory Appl.. 30, 247–270 (2002). Publisher Full Text OpenURL

  15. Cao, J, Liang, J: Boundedness and stability for Cohen-Grossberg neural network with time-varying delays. J. Math. Anal. Appl.. 296, 665–685 (2004). Publisher Full Text OpenURL

  16. Li, Y: Existence and stability of periodic solutions for Cohen-Grossberg neural networks with multiple delays. Chaos Solitons Fractals. 20, 459–466 (2004). Publisher Full Text OpenURL

  17. Juang, J, Lin, SS: Cellular neural networks: mosaic pattern and spatial chaos. SIAM J. Appl. Math.. 60, 891–915 (2000). Publisher Full Text OpenURL

  18. Ban, JC, Chang, CH, Lin, SS, Lin, YH: Spatial complexity in multi-layer cellular neural networks. J. Differ. Equ.. 246, 552–580 (2009). Publisher Full Text OpenURL

  19. Ban, JC, Chang, CH, Lin, SS: The structure of multi-layer cellular neural networks. J. Differ. Equ.. 252, 4563–4597 (2012). Publisher Full Text OpenURL

  20. Ban, JC, Chang, CH: On the structure of multi-layer cellular neural networks. Part II: the complexity between two layers (2012, submitted)

  21. Furstenberg, H: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory. 1, 1–49 (1967). Publisher Full Text OpenURL

  22. Kenyon, R, Peres, Y, Solomyak, B: Hausdorff dimension for fractals invariant under multiplicative integers. Ergod. Theory Dyn. Syst.. 32(5), 1567–1584 (2012). Publisher Full Text OpenURL

  23. Fan, AH, Liao, L, Ma, JH: Level sets of multiple ergodic averages. Monatshefte Math.. 168, 17–26 (2012). Publisher Full Text OpenURL

  24. Fan, A, Schmeling, J, Wu, M: Multifractal analysis of multiple ergodic averages. C. R. Math. Acad. Sci. Paris. 349, 961–964 (2011). Publisher Full Text OpenURL

  25. Golubitsky, M, Stewart, I: The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Birkhäuser, Basel (2002)

  26. Golubitsky, M, Stewart, I, Török, A: Patterns of symmetry in coupled cell networks with multiple arrows. SIAM J. Appl. Dyn. Syst.. 4, 78–100 (2005). Publisher Full Text OpenURL

  27. Stewart, I: Networking opportunity. Nature. 427, 601–604 (2004). PubMed Abstract | Publisher Full Text OpenURL

  28. Stewart, I, Golubitsky, M, Pivato, M: Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM J. Appl. Dyn. Syst.. 2, 609–646 (2003). Publisher Full Text OpenURL

  29. Dawes, J: Localised pattern formation with a large-scale mode: slanted snaking. SIAM J. Appl. Dyn. Syst.. 7, 186–206 (2008). Publisher Full Text OpenURL

  30. Dawes, J, Lilley, S: Localized states in a model of pattern formation in a vertically vibrated layer. SIAM J. Appl. Dyn. Syst.. 9, 238–260 (2010). Publisher Full Text OpenURL

  31. Ban, JC, Lin, SS, Shih, CW: Exact number of mosaic patterns in cellular neural networks. Int. J. Bifurc. Chaos Appl. Sci. Eng.. 11, 1645–1653 (2001). Publisher Full Text OpenURL

  32. Shih, CW: Influence of boundary conditions on pattern formation and spatial chaos in lattice systems. SIAM J. Appl. Math.. 61, 335–368 (2000). Publisher Full Text OpenURL

  33. Afraimovich, VS, Hsu, SB: Lectures on Chaotic Dynamical Systems, Am. Math. Soc., Providence (2003)

  34. Hsu, CH, Juang, J, Lin, SS, Lin, WW: Cellular neural networks: local patterns for general template. Int. J. Bifurc. Chaos Appl. Sci. Eng.. 10, 1645–1659 (2000). Publisher Full Text OpenURL

  35. Chua, LO, Yang, L: Cellular neural networks: applications. IEEE Trans. Circuits Syst.. 35, 1273–1290 (1988). Publisher Full Text OpenURL

  36. Forti, M, Tesi, A: A new method to analyze complete stability of PWL cellular neural networks. Int. J. Bifurc. Chaos Appl. Sci. Eng.. 11, 655–676 (2001). Publisher Full Text OpenURL

  37. Takahashi, N, Chua, LO: On the complete stability of nonsymmetric cellular neural networks. IEEE Trans. Circuits Syst. I. 45, 754–758 (1998)

  38. Lind, D, Marcus, B: An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge (1995)