10,934 research outputs found
Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature
We study the rate of concentration of a Brownian bridge in time one around
the corresponding geodesical segment on a Cartan-Hadamard manifold with pinched
negative sectional curvature, when the distance between the two extremities
tends to infinity. This improves on previous results by A. Eberle, and one of
us. Along the way, we derive a new asymptotic estimate for the logarithmic
derivative of the heat kernel on such manifolds, in bounded time and with one
space parameter tending to infinity, which can be viewed as a counterpart to
Bismut's asymptotic formula in small time
The OS* Algorithm: a Joint Approach to Exact Optimization and Sampling
Most current sampling algorithms for high-dimensional distributions are based
on MCMC techniques and are approximate in the sense that they are valid only
asymptotically. Rejection sampling, on the other hand, produces valid samples,
but is unrealistically slow in high-dimension spaces. The OS* algorithm that we
propose is a unified approach to exact optimization and sampling, based on
incremental refinements of a functional upper bound, which combines ideas of
adaptive rejection sampling and of A* optimization search. We show that the
choice of the refinement can be done in a way that ensures tractability in
high-dimension spaces, and we present first experiments in two different
settings: inference in high-order HMMs and in large discrete graphical models.Comment: 21 page
Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes
The present work extends our short communication Phys. Rev. Lett. 95, 111102
(2005). For smooth marginally outer trapped surfaces (MOTS) in a smooth
spacetime we define stability with respect to variations along arbitrary
vectors v normal to the MOTS. After giving some introductory material about
linear non self-adjoint elliptic operators, we introduce the stability operator
L_v and we characterize stable MOTS in terms of sign conditions on the
principal eigenvalue of L_v. The main result shows that given a strictly stable
MOTS S contained in one leaf of a given reference foliation in a spacetime,
there is an open marginally outer trapped tube (MOTT), adapted to the reference
foliation, which contains S. We give conditions under which the MOTT can be
completed. Finally, we show that under standard energy conditions on the
spacetime, the MOTT must be either locally achronal, spacelike or null.Comment: 33 pages, no figures, typos corrected, minor changes in presentatio
Solving Polynomial Systems via a Stabilized Representation of Quotient Algebras
We consider the problem of finding the isolated common roots of a set of
polynomial functions defining a zero-dimensional ideal I in a ring R of
polynomials over C. We propose a general algebraic framework to find the
solutions and to compute the structure of the quotient ring R/I from the null
space of a Macaulay-type matrix. The affine dense, affine sparse, homogeneous
and multi-homogeneous cases are treated. In the presented framework, the
concept of a border basis is generalized by relaxing the conditions on the set
of basis elements. This allows for algorithms to adapt the choice of basis in
order to enhance the numerical stability. We present such an algorithm and show
numerical results
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