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    Quantum Doubles from a Class of Noncocommutative Weak Hopf Algebras

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    The concept of biperfect (noncocommutative) weak Hopf algebras is introduced and their properties are discussed. A new type of quasi-bicrossed products are constructed by means of weak Hopf skew-pairs of the weak Hopf algebras which are generalizations of the Hopf pairs introduced by Takeuchi. As a special case, the quantum double of a finite dimensional biperfect (noncocommutative) weak Hopf algebra is built. Examples of quantum doubles from a Clifford monoid as well as a noncommutative and noncocommutative weak Hopf algebra are given, generalizing quantum doubles from a group and a noncommutative and noncocommutative Hopf algebra, respectively. Moreover, some characterisations of quantum doubles of finite dimensional biperfect weak Hopf algebras are obtained.Comment: LaTex 18 pages, to appear in J. Math. Phys. (To compile, need pb-diagram.sty, pb-lams.sty, pb-xy.sty and lamsarrow.sty

    Empirical risk minimization as parameter choice rule for general linear regularization methods.

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    We consider the statistical inverse problem to recover f from noisy measurements Y = Tf + sigma xi where xi is Gaussian white noise and T a compact operator between Hilbert spaces. Considering general reconstruction methods of the form (f) over cap (alpha) = q(alpha) (T*T)T*Y with an ordered filter q(alpha), we investigate the choice of the regularization parameter alpha by minimizing an unbiased estiate of the predictive risk E[parallel to T f - T (f) over cap (alpha)parallel to(2)]. The corresponding parameter alpha(pred) and its usage are well-known in the literature, but oracle inequalities and optimality results in this general setting are unknown. We prove a (generalized) oracle inequality, which relates the direct risk E[parallel to f - (f) over cap (alpha pred)parallel to(2)] with the oracle prediction risk inf(alpha>0) E[parallel to T f - T (f) over cap (alpha)parallel to(2)]. From this oracle inequality we are then able to conclude that the investigated parameter choice rule is of optimal order in the minimax sense. Finally we also present numerical simulations, which support the order optimality of the method and the quality of the parameter choice in finite sample situations

    Matter Effects in Active-Sterile Solar Neutrino Oscillations

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    The matter effects for solar neutrino oscillations are studied in a general scheme with an arbitrary number of sterile neutrinos, without any constraint on the mixing, assuming only a realistic hierarchy of neutrino squared-mass differences in which the smallest squared-mass difference is effective in solar neutrino oscillations. The validity of the analytic results are illustrated with a numerical solution of the evolution equation in three examples of the possible mixing matrix in the simplest case of four-neutrino mixing.Comment: 26 pages. Final version published in Phys. Rev. D80 (2009) 11300
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