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This paper considers the problems of global exponential stability and exponential convergence rate for impulsive high-order Hopfield-type neural networks with time-varying delays. By using the method of Lyapunov functions, some sufficient conditions for ensuring global exponential stability of these networks are derived, and the estimated exponential convergence rate is also obtained. As an illustration, an numerical example is worked out using the results obtained. Index Terms--Exponential stability, impulsive high-order Hopfield-type Lyapunov function, neural networks.

Finding theK-shortest paths in timetable-based public transportation systems is an important problem in practice. It has three typical variants: theK-earliest arrival problem (K-EAP), theK-shortest travel time problem (K-STTP), and theK-minimum number of transfers problem (K-MNTP). In this article we show that these problems can be solved efficiently by first modeling the timetable information with the time-expanded approach, then applying the Martins and Santos (MS) algorithm. Then we model the timetable information with the time-dependent approach and propose a modified version of the MS algorithm for solving theK-EAP. Experimental results on real-world data show that forKsmaller than 100, which is enough for most applications, the execution times of the MS algorithm for the problems in the time-expanded model are less than 100 ms on a server with a 1.86-GHz central processing unit (CPU) and 4 GB of memory. For solving theK-EAP the modified MS algorithm in the time-dependent model is even more efficient (about three times faster forK≤ 100) than the original algorithm in the time-expanded model. Our results imply the great potential of the MS algorithms in practical transportation service systems. [ABSTRACT FROM PUBLISHER]