Taking traveling salesman problems (TSPs) as examples of combinatorial optimization problems, an "optimal" Hopfield network for ("optimal" neural representation of) TSPs is presented, where a vertex of state hypercube of the network is asymptotically stable if and only if it is an optimal solution. Of all the Hopfield networks for TSPs, this network most sharply distinguishes an optimal solution from other nonoptimal solutions and infeasible solutions. In this sense, we call this network "optimal" for TSPs. Whenever the network converges to a vertex, we can always obtain an optimal solution. However, we can not design such network without knowing an optimal solution to the problem. So, its approximate realization, which can be designed without a-priori knowledge of an optimal solution, is proposed. Simulations show that the "optimal" network and its approximate realization obtain optimal or good feasible solutions more frequently than familiar Hopfield networks. We can also design such "optimal" Hopfield networks for many combinatorial optimization problems as well as for TSPs.

Junction leakage current of trench isolation devices is strongly influenced by trench configuration. The origin of the leakage current is the mechanical stress that is generated by the differential thermal expansion between the Si substrate and the SiO2 filled isolation trench during the isolation forming process. A two-dimensional mechanical stress simulation was used to analyze trench-isolated devices. The simulated distribution and magnitude of stress were found to agree with Raman spectroscopic measurements of actual devices. The stress in the deeper regions between deep trenches is likely to increase greatly as the size of devices diminishes, so it is important to reduce this stress and thus suppress junction leakage current.