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T he World Wide Web is a large, distributed hypertext repository of information, which users navigate through links and view with browsers. The heavy Internet traffic resulting from the Web's popularity has significantly increased userperceived latency. The obvious solution-to increase the bandwidth-is not viable, because we cannot easily change the Web's infrastructure (the Internet) without significant economic cost. However, if we could predict future user requests, we could put those pages into the clientside cache when the browser is free. When a user requests one of the pages, the browser could retrieve it directly from cache.Much current research has examined modeling and predicting user access behavior on the Web to improve Web prefetching, 1,2 enhance search engines, 3 and understand and influence buying patterns. 4 To predict Web access, we need a method for modeling and analyzing Web access sequences. With this information, we can deduce future user requests.Some researchers have used traditional Markov models, which are often employed to study stochastic processes and predict user access behavior. 5,6 In general, they use the sequence of Web pages a user has accessed as input, with the goal of building Markov models with which they can predict the page the user will most likely access next. Venkata N. Padmanabhan and Jeffrey C. Mogul used N-hop Markov models to improve prefetching strategies for Web caches; 2 Ramesh R. Sarukkai used Markov models to predict the next page accessed by the user; 7 and Igor V. Cadez and colleagues used Markov models to categorize user sessions. 8 Peter Pirolli and colleagues, however, tested the performance of different-order Markov models for Web access prediction and found traditional Markov models to be inadequate for this purpose. 5 Therefore, we need a new Markov model for Web access prediction.Our hybrid-order tree-like Markov model can predict Web access precisely, providing high coverage and good scalability. HTMM intelligently merges two methods: a tree-like structure Markov model method that aggregates the access sequences by pattern matching and a hybrid-order method that combines varying-order Markov models. Performance evaluations comparing our HTMM with traditional Markov models confirm its usefulness.Accurately predicting Web user access behavior can minimize user-perceived latency, which is crucial in the rapidly growing World Wide Web. Although traditional Markov models have helped predict user access behavior, they have serious limitations. Hybrid-order treelike Markov models predict Web access precisely while providing high coverage and scalability.

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A symplectic pseudospectral time-domain (SPSTD) scheme is developed to solve Schrödinger equation. Instead of spatial finite differences in conventional finite-difference time-domain (FDTD) methods, the fast Fourier transform is used to calculate the spatial derivatives. In time domain, the scheme adopts high-order symplectic integrators to simulate time evolution of Schrödinger equation. A detailed numerical study on the eigenvalue problems of 1D quantum well and 3D harmonic oscillator is carried out. The simulation results strongly confirm the advantages of the SPSTD scheme over the traditional PSTD method and FDTD approach. Furthermore, by comparing to the traditional PSTD method and the non-symplectic Runge-Kutta (RK) method, the explicit SPSTD scheme which is an infinite order of accuracy in space domain and energy-conserving in time domain, is well suited for a long-term simulation.