Narrow Results

For locally compact quantum groups $𝔾$, we initiate an investigation of stable states with respect to unitary co-representations $U$ of $𝔾$ on Hilbert spaces $H_U$; in particular, we study the subject on the multiplicative unitary operator $W_{𝔾}$ of $𝔾$ with some examples on locally compact quantum groups arising from discrete groups and compact groups. As the main result, we consider the one co-dimensional Hilbert subspace of $H_U$ associated to a suitable vector $\eta$, to present an operator theoretic characterization of stable states with respect to a related unitary co-representation $U_{\eta }$. This provides a quantum version of an interesting result on unitary representations of locally compact groups given by Lau and Paterson in 1991.

This paper investigates the convergence theorems that are associated with a Discrete Hopfield Neural Network (DHNN) with delay. We present two updating rules, one for serial mode and the other for parallel mode. The speed of convergence of these proposed updating rules is faster than all of the existing updating rules. It has been proved in this paper that a DHNN with delay will converge to a stable state when operating in a serial mode if the matrix of weights of the no-delay term is symmetric. In addition, it has been proved that they will converge to a stable state when operating in a parallel mode if the matrix of weights of the no-delay term is a symmetric and non-negative definite matrix. The condition for convergence of a DHNN without delay can been relaxed from the need to have a symmetric matrix to an even weaker condition of having a quasi-symmetric matrix. The results in this paper extend both the existing results concerning the convergence of a DHNN without delay and our previous findings. By means of the new network structure and its convergence theorems, we propose a local searching algorithm for combinatorial optimization. We also relate the maximum value of a bivariate energy function to the stable states of a DHNN with delay, which generalizes Hopfield's energy function. Moreover, for the serial model we give the relationship between the convergence of the energy function and the convergence of the corresponding network. One application is presented to demonstrate the higher rate of convergence and the accuracy of the classification using our algorithm.

In this paper, the Hopfield neural network with delay (HNND) is studied from the standpoint of regarding it as an optimizing computational model. Two general updating rules for networks with delay (GURD) are given based on Hopfield-type neural networks with delay for optimization problems and characterized by dynamic thresholds. It is proved that in any sequence of updating rule modes, the GURD monotonously converges to a stable state of the network. The diagonal elements of the connection matrix are shown to have an important influence on the convergence process, and they represent the relationship of the local maximum value of the energy function to the stable states of the networks. All the ordinary discrete Hopfield neural network (DHNN) algorithms are instances of the GURD. It can be shown that the convergence conditions of the GURD may be relaxed in the context of applications, for instance, the condition of nonnegative diagonal elements of the connection matrix can be removed from the original convergence theorem. A new updating rule mode and restrictive conditions can guarantee the network to achieve a local maximum of the energy function with a step-by-step algorithm. The convergence rate improves evidently when compared with other methods. For a delay item considered as a noise disturbance item, the step-by-step algorithm demonstrates its efficiency and a high convergence rate. Experimental results support our proposed algorithm.