Topological and random spread models with frozen symbols.

When a symbol or a type has been "frozen" (namely, a type of which an individual only produces one individual of the same type), its spread pattern will be changed and this change will affect the long-term behavior of the whole system. However, in a frozen system, the ξ-matrix and the offspring mean matrix are no longer primitive so that the Perron-Frobenius theorem cannot be applied directly when predicting the spread rates. In this paper, our goal is to characterize these key matrices and analyze the spread rate under more general settings both in the topological and random spread models with frozen symbols. More specifically, we propose an algorithm for explicitly computing the spread rate and relate the rate with the eigenvectors of the ξ-matrix or offspring mean matrix. In addition, we reveal that the growth of the population is exponential and that the composition of the population is asymptotically periodic. Furthermore, numerical experiments are provided as supporting evidence for the theory.

Mathematical analysis of topological and random m-order spread models.

This paper focuses on the analysis of two particular models, from deterministic and random perspective respectively, for spreading processes. With a proper encoding of propagation patterns, the spread rate of each pattern is discussed for both models by virtue of the substitution dynamical systems and branching process. In view of this, we are empowered to draw a comparison between two spreading processes according to their spreading models, based on which explanations are proposed on a higher frequency of a pattern in one model than the other. These results are then supported by the numerical evidence later in the article.

Spread rates of spread models with frozen symbols.

This article aims to compare the long-term behavior of a spread model before and after a type in the model becomes frozen, namely, a type of which an individual only produces individuals of the same type. By means of substitution dynamical systems and matrix analysis, the first result of this work gives the spread rates of a 1-spread model with one frozen symbol. Later, in the work, this is shown to hold under more general settings, which include generalized frozen symbols and frozen symbols in m-spread models. Numerical experiments are provided as supporting evidence for the theory.

When are two multi-layer cellular neural networks the same?

This paper aims to characterize whether a multi-layer cellular neural network is of deep architecture; namely, when can an n-layer cellular neural network be replaced by an m-layer cellular neural network for m

Realization problem of multi-layer cellular neural networks.

This paper investigates whether the output space of a multi-layer cellular neural network can be realized via a single layer cellular neural network in the sense of the existence of finite-to-one map from one output space to the other. Whenever such realization exists, the phenomena exhibited in the output space of the revealed single layer cellular neural network is at most a constant multiple of the phenomena exhibited in the output space of the original multi-layer cellular neural network. Meanwhile, the computation complexity of a single layer system is much less than the complexity of a multi-layer system. Namely, one can trade the precision of the results for the execution time. We remark that a routine extension of the proposed methodology in this paper can be applied to the substitution of hidden spaces although the detailed illustration is omitted.

The learning problem of multi-layer neural networks.

This manuscript considers the learning problem of multi-layer neural networks (MNNs) with an activation function which comes from cellular neural networks. A systematic investigation of the partition of the parameter space is provided. Furthermore, the recursive formula of the transition matrix of an MNN is obtained. By implementing the well-developed tools in the symbolic dynamical systems, the topological entropy of an MNN can be computed explicitly. A novel phenomenon, the asymmetry of a topological diagram that was seen in Ban, Chang, Lin, and Lin (2009) [J. Differential Equations 246, pp. 552-580, 2009], is revealed.