7,679 research outputs found
On the Connection Between 2d Topological Gravity and the Reduced Hermitian Matrix Model
We discuss how concepts such as geodesic length and the volume of space-time
can appear in 2d topological gravity. We then construct a detailed mapping
between the reduced Hermitian matrix model and 2d topological gravity at genus
zero. This leads to a complete solution of the counting problem for planar
graphs with vertices of even coordination number. The connection between
multi-critical matrix models and multi-critical topological gravity at genus
zero is studied in some detail.Comment: 29 pages, LaTe
Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach
We study the continuity of an abstract generalization of the maximum-entropy
inference - a maximizer. It is defined as a right-inverse of a linear map
restricted to a convex body which uniquely maximizes on each fiber of the
linear map a continuous function on the convex body. Using convex geometry we
prove, amongst others, the existence of discontinuities of the maximizer at
limits of extremal points not being extremal points themselves and apply the
result to quantum correlations. Further, we use numerical range methods in the
case of quantum inference which refers to two observables. One result is a
complete characterization of points of discontinuity for matrices.Comment: 27 page
Entropy Distance: New Quantum Phenomena
We study a curve of Gibbsian families of complex 3x3-matrices and point out
new features, absent in commutative finite-dimensional algebras: a
discontinuous maximum-entropy inference, a discontinuous entropy distance and
non-exposed faces of the mean value set. We analyze these problems from various
aspects including convex geometry, topology and information geometry. This
research is motivated by a theory of info-max principles, where we contribute
by computing first order optimality conditions of the entropy distance.Comment: 34 pages, 5 figure
Stochastic integration in UMD Banach spaces
In this paper we construct a theory of stochastic integration of processes
with values in , where is a separable Hilbert space and
is a UMD Banach space (i.e., a space in which martingale differences are
unconditional). The integrator is an -cylindrical Brownian motion. Our
approach is based on a two-sided -decoupling inequality for UMD spaces due
to Garling, which is combined with the theory of stochastic integration of
-valued functions introduced recently by two of the authors.
We obtain various characterizations of the stochastic integral and prove
versions of the It\^{o} isometry, the Burkholder--Davis--Gundy inequalities,
and the representation theorem for Brownian martingales.Comment: Published at http://dx.doi.org/10.1214/009117906000001006 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stochastic evolution equations in UMD Banach spaces
We discuss existence, uniqueness, and space-time H\"older regularity for
solutions of the parabolic stochastic evolution equation dU(t) = (AU(t) +
F(t,U(t))) dt + B(t,U(t)) dW_H(t), t\in [0,\Tend], U(0) = u_0, where
generates an analytic -semigroup on a UMD Banach space and is a
cylindrical Brownian motion with values in a Hilbert space . We prove that
if the mappings and satisfy suitable Lipschitz conditions and is
\F_0-measurable and bounded, then this problem has a unique mild solution,
which has trajectories in C^\l([0,T];\D((-A)^\theta) provided
and satisfy \l+\theta<\frac12. Various extensions of this
result are given and the results are applied to parabolic stochastic partial
differential equations.Comment: Accepted for publication in Journal of Functional Analysi
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