6,915 research outputs found

    Huygens' principle and Dirac-Weyl equation

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    We investigate the validity of Huygens' principle for forward propagation in the massless Dirac-Weyl equation. The principle holds for odd space dimension n, while it is invalid for even n. We explicitly solve the cases n=1,2 and 3 and discuss generic nn. We compare with the massless Klein-Gordon equation and comment on possible generalizations and applications.Comment: 7 pages, 1 figur

    The cost of continuity: performance of iterative solvers on isogeometric finite elements

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    In this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized Galerkin weak form. Specifically, we compare performance of linear solvers when discretizing using C0C^0 B-splines, which span traditional finite element spaces, and Cp1C^{p-1} B-splines, which represent maximum continuity. We provide theoretical estimates for the increase in cost of the matrix-vector product as well as for the construction and application of black-box preconditioners. We accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size hh and polynomial order of approximation pp. Finally, we present timing results for a range of preconditioning options for the Laplace problem. We conclude that the matrix-vector product operation is at most \slfrac{33p^2}{8} times more expensive for the more continuous space, although for moderately low pp, this number is significantly reduced. Moreover, if static condensation is not employed, this number further reduces to at most a value of 8, even for high pp. Preconditioning options can be up to p3p^3 times more expensive to setup, although this difference significantly decreases for some popular preconditioners such as Incomplete LU factorization

    Electronic structure of V4_4O7_7: charge ordering, metal-insulator transition and magnetism

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    The low and high-temperature phases of V4_4O7_7 have been studied by \textit{ab initio} calculations. At high temperature, all V atoms are electronically equivalent and the material is metallic. Charge and orbital ordering, associated with the distortions in the V pseudo-rutile chains, occur below the metal-insulator transition. Orbital ordering in the low-temperature phase, different in V3+^{3+} and V4+^{4+} chains, allows to explain the distortion pattern in the insulating phase of V4_4O7_7. The in-chain magnetic couplings in the low-temperature phase turn out to be antiferromagnetic, but very different in the various V4+^{4+} and V3+^{3+} bonds. The V4+^{4+} dimers formed below the transition temperature form spin singlets, but V3+^{3+} ions, despite dimerization, apparently participate in magnetic ordering.Comment: 10 pages, 6 figures, 2 table
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