26,800 research outputs found
Invariance of Poisson measures under random transformations
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
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
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
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
We are interested in the tail behavior of the pdf of mass density within the
one and -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 ()
or higher (). 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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