26,800 research outputs found

    Invariance of Poisson measures under random transformations

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    We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method applied in [22] on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples of application include transformations conditioned by random sets such as the convex hull of a Poisson random measure

    Theta Bodies for Polynomial Ideals

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    Inspired by a question of Lov\'asz, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lov\'asz's theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre's relaxations for real solutions to ideals, and that they can be computed explicitly using combinatorial moment matrices. Theta bodies provide a new canonical set of semidefinite relaxations for the max cut problem. For vanishing ideals of finite point sets, we give several equivalent characterizations of when the first theta body equals the convex hull of the points. We also determine the structure of the first theta body for all ideals.Comment: 26 pages, 3 figure

    On fine differentiability properties of horizons and applications to Riemannian geometry

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    We study fine differentiability properties of horizons. We show that the set of end points of generators of a n-dimensional horizon H (which is included in a (n+1)-dimensional space-time M) has vanishing n-dimensional Hausdorff measure. This is proved by showing that the set of end points of generators at which the horizon is differentiable has the same property. For 1\le k\le n+1 we show (using deep results of Alberti) that the set of points where the convex hull of the set of generators leaving the horizon has dimension k is ``almost a C^2 manifold of dimension n+1-k'': it can be covered, up to a set of vanishing (n+1-k)-dimensional Hausdorff measure, by a countable number of C^2 manifolds. We use our Lorentzian geometry results to derive information about the fine differentiability properties of the distance function and the structure of cut loci in Riemannian geometry.Comment: Latex2e, 13 pages in A4 forma

    New Twistor String Theories revisited

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    A gauged version of Berkovits twistor string theory featuring the particle content of N=8 supergravity was suggested by Abou-Zeid, Hull and Mason. The equations of motion for a particular multiplet in the modified theory are examined on the level of basic twistor fields and thereby shown to imply the vanishing of the negative helicity graviton on-shell. Additionally, the restrictions emerging from the equation of motion for the new gauge field \bar{B} reveal the chiral nature of interactions in theories constructed in this manner. Moreover, a particular amplitude in Berkovits open string theory is shown to be in agreement with the corresponding result in Einstein gravity.Comment: 15 pages, v2: typos corrected, replaced with published versio

    Singularities and the distribution of density in the Burgers/adhesion model

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    We are interested in the tail behavior of the pdf of mass density within the one and dd-dimensional Burgers/adhesion model used, e.g., to model the formation of large-scale structures in the Universe after baryon-photon decoupling. We show that large densities are localized near ``kurtoparabolic'' singularities residing on space-time manifolds of codimension two (d≤2d \le 2) or higher (d≥3d \ge 3). For smooth initial conditions, such singularities are obtained from the convex hull of the Lagrangian potential (the initial velocity potential minus a parabolic term). The singularities contribute {\em \hbox{universal} power-law tails} to the density pdf when the initial conditions are random. In one dimension the singularities are preshocks (nascent shocks), whereas in two and three dimensions they persist in time and correspond to boundaries of shocks; in all cases the corresponding density pdf has the exponent -7/2, originally proposed by E, Khanin, Mazel and Sinai (1997 Phys. Rev. Lett. 78, 1904) for the pdf of velocity gradients in one-dimensional forced Burgers turbulence. We also briefly consider models permitting particle crossings and thus multi-stream solutions, such as the Zel'dovich approximation and the (Jeans)--Vlasov--Poisson equation with single-stream initial data: they have singularities of codimension one, yielding power-law tails with exponent -3.Comment: LATEX 11 pages, 6 figures, revised; Physica D, in pres
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