5,785 research outputs found

    An extension of heat hierarchy

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    We propose a formally completely integrable extension of heat hierarchy based on the space of symmetries isomorphic to the Weyl algebra A1\mathcal{A}_1. The extended heat hierarchy will be the basic model for the analysis of the extension of KP hierarchy, and other integrable equations.Comment: This note is incorporated into arXiv:1409.7024, arXiv:1408.324

    Legendrian Gronwall conjecture

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    The Gronwall conjecture states that a planar 3-web of foliations which admits more than one distinct linearizations is locally equivalent to an algebraic web. We propose an analogue of the Gronwall conjecture for the 3-web of foliations by Legendrian curves in a contact three manifold. The Legendrian Gronwall conjecture states that a Legendrian 3-web admits at most one distinct local linearization, with the only exception when it is locally equivalent to the dual linear Legendrian 3-web of the Legendrian twisted cubic in \,\PP^3. We give a partial answer to the conjecture in the affirmative for the class of Legendrian 3-webs of maximum rank. We also show that a linear Legendrian 3-web which is sufficiently flat at a reference point is rigid under local linear Legendrian deformation.Comment: 15 page

    Projectively deformable Legendrian surfaces

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    Consider an immersed Legendrian surface in the five dimensional complex projective space equipped with the standard homogeneous contact structure. We introduce a class of fourth order projective Legendrian deformation called \emph{ Ψ\,\Psi-deformation}, and give a differential geometric characterization of surfaces admitting maximum three parameter family of such deformations. Two explicit examples of maximally  Ψ\, \Psi-deformable surfaces are constructed; the first one is given by a Legendrian map from \, \PP^2 blown up at three distinct collinear points, which is an embedding away from the -2-curve and degenerates to a point along the -2-curve. The second one is a Legendrian embedding of the degree 6 del Pezzo surface, \, \PP^2 blown up at three non-collinear points. In both cases, the Legendrian map is given by a system of cubics through the three points, which is a subsystem of the anti-canonical system.Comment: 33 page

    In vivo evidence for NMDA receptor mediated excitotoxicity in a murine genetic model of Huntington Disease

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    N-methyl-D-aspartate receptor (NMDAR) mediated excitotoxicity is implicated as a proximate cause of neurodegeneration in Huntington Disease (HD). However, this hypothesis has not been tested rigorously in vivo. NMDAR NR2B-subunits are the predominant NR2 subunit expressed by the striatal medium spiny neurons that degenerate in HD. To test this hypothesis, we crossed a well validated murine genetic model of HD (Hdh(CAG)150) with a transgenic line overexpressing NMDAR NR2B-subunits. In the resulting double mutant line, we show exacerbation of selective striatal neuron degeneration. These results provide the first direct in vivo evidence of NR2B-NMDAR mediated excitotoxicity in the context of HD. Our results are consistent with prior suggestions that direct and/or indirect interactions of mutant huntingtin with NMDARs are a proximate cause of neurodegeneration in HD
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