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    Coefficient convexity of divisors of x^n-1

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    The title compound, [Li3(C4F9O)3(C3H6O)3], features an open Li/O cube with an Li ion missing at one corner. Three of the four bridging O atoms of the cube carry a fluorinated tert-butyl residue, whereas the fourth is part of an acetone mol­ecule. Two of the Li atoms are further bonded to a non-bridging acetone mol­ecule. Two of the lithium ion coordination geometries are very distorted LiO4 tetra­hedra; the third could be described as a very distorted LiO3 T-shape with two distant F-atom neighbours. The Li[cdots, three dots, centered]Li contact distances for the three-coordinate Li+ ion [2.608 (14) and 2.631 (12) Å] are much shorter that the contact distance [2.940 (13) Å] between the tetra­hedrally coordinated species

    Dirac-K\"ahler particle in Riemann spherical space: boson interpretation

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    In the context of the composite boson interpretation, we construct the exact general solution of the Dirac--K\"ahler equation for the case of the spherical Riemann space of constant positive curvature, for which due to the geometry itself one may expect to have a discrete energy spectrum. In the case of the minimal value of the total angular momentum, j=0j=0, the radial equations are reduced to second-order ordinary differential equations, which are straightforwardly solved in terms of the hypergeometric functions. For non-zero values of the total angular momentum, however, the radial equations are reduced to a pair of complicated fourth-order differential equations. Employing the factorization approach, we derive the general solution of these equations involving four independent fundamental solutions written in terms of combinations of the hypergeometric functions. The corresponding discrete energy spectrum is then determined via termination of the involved hypergeometric series, resulting in quasi-polynomial wave-functions. The constructed solutions lead to notable observations when compared with those for the ordinary Dirac particle. The energy spectrum for the Dirac-K\"ahler particle in spherical space is much more complicated. Its structure substantially differs from that for the Dirac particle since it consists of two paralleled energy level series each of which is twofold degenerate. Besides, none of the two separate series coincides with the series for the Dirac particle. Thus, the Dirac--K\"ahler field cannot be interpreted as a system of four Dirac fermions. Additional arguments supporting this conclusion are discussed

    CM Values of Green Functions Associated to Special Cycles on Shimura Varieties with Applications to Siegel 3-Fold X2(2)X_2(2)

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    We generalize the definition of CM cycles beyond the small and big CM ones studied by various authors and give a uniform formula for the CM values of Green functions associated to these special cycles in general using the idea of regularized theta lifts. Finally, as an application to Siegel 3-fold X2(2)X_2(2), we can compute special values of theta functions and Rosenhain λ\lambda-invariants at a CM cycle, which is useful for genus two curve cryptography.Comment: arXiv admin note: text overlap with arXiv:1008.1669 by other author