17,813 research outputs found

    Elliptic genera from multi-centers

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    I show how elliptic genera for various Calabi-Yau threefolds may be understood from supergravity localization using the quantization of the phase space of certain multi-center configurations. I present a simple procedure that allows for the enumeration of all multi-center configurations contributing to the polar sector of the elliptic genera\textemdash explicitly verifying this in the cases of the quintic in P4\mathbb{P}^4, the sextic in WP(2,1,1,1,1)\mathbb{WP}_{(2,1,1,1,1)}, the octic in WP(4,1,1,1,1)\mathbb{WP}_{(4,1,1,1,1)} and the dectic in WP(5,2,1,1,1)\mathbb{WP}_{(5,2,1,1,1)}. With an input of the corresponding `single-center' indices (Donaldson-Thomas invariants), the polar terms have been known to determine the elliptic genera completely. I argue that this multi-center approach to the low-lying spectrum of the elliptic genera is a stepping stone towards an understanding of the exact microscopic states that contribute to supersymmetric single center black hole entropy in N=2\mathcal{N}=2 supergravity.Comment: 30+1 pages, Published Versio

    Extremal families of cubic Thue equations

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    We exactly determine the integral solutions to a previously untreated infinite family of cubic Thue equations of the form F(x,y)=1F(x,y)=1 with at least 55 such solutions. Our approach combines elementary arguments, with lower bounds for linear forms in logarithms and lattice-basis reduction

    Lines Tangent to 2n-2 spheres in R^n

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    We show that there are 3 \cdot 2^(n-1) complex common tangent lines to 2n-2 general spheres in R^n and that there is a choice of spheres with all common tangents real.Comment: Minor revisions. Trans. AMer. Math. Soc., to appear. 15 pages, 3 .eps figures; also a web page with computer code verifying the computations in the paper and with additional picture

    Solution intervals for variables in spatial RCRCR linkages

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    © 2019. ElsevierAn analytic method to compute the solution intervals for the input variables of spatial RCRCR linkages and their inversions is presented. The input-output equation is formulated as the intersection of a single ellipse with a parameterized family of ellipses, both related with the possible values that certain dual angles determined by the configuration of the mechanism can take. Bounds for the angles of the input pairs of the RCRCR and RRCRC inversions are found by imposing the tangency of two ellipses, what reduces to analyzing the discriminant of a fourth degree polynomial. The bounds for the input pair of the RCRRC inversion is found as the intersection of a single ellipse with the envelope of the parameterized family of ellipses. The method provides the bounds of each of the assembly modes of the mechanism as well as the local extrema that may exist for the input variablePeer ReviewedPostprint (author's final draft
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