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Creators/Authors contains: "Khesin, Boris"

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  1. ; ; (, International Mathematics Research Notices)
    Abstract We consider pairs of point vortices having circulations $$\Gamma _{1}$$ and $$\Gamma _{2}$$ and confined to a two-dimensional surface $$S$$. In the limit of zero initial separation $$\varepsilon $$, we prove that they follow a magnetic geodesic in unison, if properly renormalized. Specifically, the “singular vortex pair” moves as a single-charged particle on the surface with a charge of order $$1/\varepsilon ^{2}$$ in an magnetic field $$B$$ that is everywhere normal to the surface and of strength $$|B|=\Gamma _{1} +\Gamma _{2}$$. In the case $$\Gamma _{1}=-\Gamma _{2}$$, this gives another proof of Kimura’s conjecture [11] that singular dipoles follow geodesics. 
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  2. ; (, Calculus of Variations and Partial Differential Equations)
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  3. ; (, Bulletin of the London Mathematical Society)
    Abstract The pentagram map on polygons in the projective plane was introduced by R. Schwartz in 1992 and is by now one of the most popular and classical discrete integrable systems. In the present paper we introduce and prove integrability of long‐diagonal pentagram maps on polygons in , by now the most universal pentagram‐type map encompassing all known integrable cases. We also establish an equivalence of long‐diagonal and bi‐diagonal maps and present a simple self‐contained construction of the Lax form for both. Finally, we prove that the continuous limit of all these maps is equivalent to the ‐KdV equation, generalizing the Boussinesq equation for . 
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