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The results obtained in classical 1-D rational approximation are extended in this paper to rational approximation of M-D functions. A full analog of classical Montessus de Ballore theorem for the convergence of the rows of Pad´e’s tables is obtained. It is shown that the appropriate theorem for uniform convergence in C2 will really take place only in the case of choosing the necessary determinative (interpolation) sets Ij(n,m), j = 1, 2. This theorem allows us to handle the problem of deriving the transfer function of a M-D digital system, that is described by its state-space representation1. The results of computer modelling by using MATLAB software are presented. Both the convergence theorem and results of modelling show that from theoretical and practical points of view the proposed approach is promising.

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In this paper, we present the theoretical as well as experimental results on resonantly enhanced tunneling of Bose-Einstein condensates in optical lattices both in the linear case and for small nonlinearities. Our results demonstrate the usefulness of condensates in optical lattices for simulating Hamiltonians originally used for describing solid-state phenomena.