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Abstract: System identification consists in finding a model of an unknown system starting from a finite set of noise-corrupted data. A fundamental problem in this context is to asses the accuracy of the identified model. In this paper, the problem is investigated for the case of nonlinear systems within the Set Membership—Information Based Complexity framework of [M. Milanese, C. Novara, Set membership identification of nonlinear systems, Automatica 40(6) (2004) 957–975]. In that paper, a (locally) optimal algorithm has been derived, giving (locally) optimal models in nonlinear regression form. The corresponding (local) radius of information, providing the worst-case identification error, can be consequently used to measure the quality of the identified model. In the present paper, two algorithms are proposed for the computation of the local radius of information: The first provides the exact value but requires a computational complexity exponential in the dimension of the regressor space. The second is approximate but involves a polynomial (quadratic) complexity. [Copyright &y& Elsevier]

Abstract: We study the problem, initiated by Kacewicz [Randomized and quantum algorithms yield a speed-up for initial-value problems, J. Complexity 20 (2004) 821–834; see also http://arXiv.org/abs/quant-ph/0311148], of finding randomized and quantum complexity of initial-value problems. We showed in Kacewicz (2004) that a speed-up in both settings over the worst-case deterministic complexity is possible. In the present paper we prove, by defining new algorithms, that further improvement in upper bounds on the randomized and quantum complexity can be achieved. In the Hölder class of right-hand side functions with r continuous bounded partial derivatives, with rth derivative being a Hölder function with exponent , the -complexity is shown to be in the randomized setting, and on a quantum computer (up to logarithmic factors). This is an improvement for the general problem over the results from Kacewicz (2004). The gap still remaining between upper and lower bounds on the complexity is further discussed for a special problem. We consider scalar autonomous problems, with the aim of computing the solution at the end point of the interval of integration. For this problem, we fill up the gap by establishing (essentially) matching upper and lower complexity bounds. We show that the complexity in this case is in the randomized setting, and in the quantum setting (again up to logarithmic factors). Hence, this problem is essentially as hard as the integration problem. [Copyright &y& Elsevier]

Abstract: Let V be a closed algebraic subvariety of the n-dimensional projective space over the complex or real numbers and suppose that V is non-empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motivated in Bank et al. (Kybernetika 40 (2004), to appear). As particular instances of this notion of a generalized polar variety one reobtains the classic one and an alternative type of a polar variety, called dual. As main result of the present paper we show that for a generic choice of their parameters the generalized polar varieties of V are empty or equidimensional and smooth in any regular point of V. In the case that the variety V is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of V by explicit equations. Finally, we indicate how this description may be used in order to design in the context of algorithmic elimination theory a highly efficient, probabilistic elimination procedure for the following task: In case, that the variety V is -definable and affine, having a complete intersection ideal of definition, and that the real trace of V is non-empty and smooth, find for each connected component of the real trace of V an algebraic sample point. [Copyright &y& Elsevier]

Quantum algorithms and complexity have recently been studied not only for discrete, but also for some numerical problems. Most attention has been paid so far to the integration and approximation problems, for which a speed-up is shown in many important cases by quantum computers with respect to deterministic and randomized algorithms on a classical computer. In this paper, we deal with the randomized and quantum complexity of initial-value problems. For this nonlinear problem, we show that both randomized and quantum algorithms yield a speed-up over deterministic algorithms. Upper bounds on the complexity in the randomized and quantum settings are shown by constructing algorithms with a suitable cost, where the construction is based on integral information. Lower bounds result from the respective bounds for the integration problem. [Copyright &y& Elsevier]

For a Viterbi-like algorithm over a sectionalized trellis of a linear block code, the decoding procedure consists of three parts: computing the metrics of the edges, selecting the survivor edge between each pair of adjacent vertices and determining the survivor path from the origin to each vertex. In this paper, some new methods for computing the metrics of the edges are proposed. Our method of “partition of index set” for computing the metrics is shown to be near-optimal. The proposed methods are then applied to Reed–Muller (RM) codes. For some RM codes, the computational complexity of decoding is significantly reduced in comparison to the best-known ones. For the RM codes, a direct method for constructing their trellis-oriented-generator-matrices is proposed and some shift invariances are deduced. [Copyright &y& Elsevier]