57 research outputs found
Integral geometry of complex space forms
We show how Alesker's theory of valuations on manifolds gives rise to an
algebraic picture of the integral geometry of any Riemannian isotropic space.
We then apply this method to give a thorough account of the integral geometry
of the complex space forms, i.e. complex projective space, complex hyperbolic
space and complex euclidean space. In particular, we compute the family of
kinematic formulas for invariant valuations and invariant curvature measures in
these spaces. In addition to new and more efficient framings of the tube
formulas of Gray and the kinematic formulas of Shifrin, this approach yields a
new formula expressing the volumes of the tubes about a totally real
submanifold in terms of its intrinsic Riemannian structure. We also show by
direct calculation that the Lipschitz-Killing valuations stabilize the subspace
of invariant angular curvature measures, suggesting the possibility that a
similar phenomenon holds for all Riemannian manifolds. We conclude with a
number of open questions and conjectures.Comment: 68 pages; minor change
Extended morphometric analysis of neuronal cells with Minkowski valuations
Minkowski valuations provide a systematic framework for quantifying different
aspects of morphology. In this paper we apply vector- and tensor-valued
Minkowski valuations to neuronal cells from the cat's retina in order to
describe their morphological structure in a comprehensive way. We introduce the
framework of Minkowski valuations, discuss their implementation for neuronal
cells and show how they can discriminate between cells of different types.Comment: 14 pages, 18 postscript figure
Logarithmically-concave moment measures I
We discuss a certain Riemannian metric, related to the toric Kahler-Einstein
equation, that is associated in a linearly-invariant manner with a given
log-concave measure in R^n. We use this metric in order to bound the second
derivatives of the solution to the toric Kahler-Einstein equation, and in order
to obtain spectral-gap estimates similar to those of Payne and Weinberger.Comment: 27 page
Optimal measures and Markov transition kernels
We study optimal solutions to an abstract optimization problem for measures,
which is a generalization of classical variational problems in information
theory and statistical physics. In the classical problems, information and
relative entropy are defined using the Kullback-Leibler divergence, and for
this reason optimal measures belong to a one-parameter exponential family.
Measures within such a family have the property of mutual absolute continuity.
Here we show that this property characterizes other families of optimal
positive measures if a functional representing information has a strictly
convex dual. Mutual absolute continuity of optimal probability measures allows
us to strictly separate deterministic and non-deterministic Markov transition
kernels, which play an important role in theories of decisions, estimation,
control, communication and computation. We show that deterministic transitions
are strictly sub-optimal, unless information resource with a strictly convex
dual is unconstrained. For illustration, we construct an example where, unlike
non-deterministic, any deterministic kernel either has negatively infinite
expected utility (unbounded expected error) or communicates infinite
information.Comment: Replaced with a final and accepted draft; Journal of Global
Optimization, Springer, Jan 1, 201
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