636 research outputs found
Radial continuous rotation invariant valuations on star bodies
We characterize the positive radial continuous and rotation invariant
valuations defined on the star bodies of as the applications
on star bodies which admit an integral representation with respect to the
Lebesgue measure. That is, where
is a positive continuous function, is the radial function
associated to and is the Lebesgue measure on . As a corollary,
we obtain that every such valuation can be uniformly approximated on bounded
sets by a linear combination of dual quermassintegrals.Comment: Two minor gaps and several typos corrected thanks to the refere
Characterization of dual mixed volumes via polymeasures
We prove a characterization of the dual mixed volume in terms of functional
properties of the polynomial associated to it. To do this, we use tools from
the theory of multilinear operators on spaces of continuos functions. Along the
way we reprove, with these same techniques, a recently found characterization
of the dual mixed volume
Some results concerning polymeasures
We present some results concerning the general theory of polymeasures. Among them, we point out an example of a polymeasure of bounded semivariation and unbounded variation, and two different characterizations of uniform polymeasures
Testing microscopic discretization
What can we say about the spectra of a collection of microscopic variables
when only their coarse-grained sums are experimentally accessible? In this
paper, using the tools and methodology from the study of quantum nonlocality,
we develop a mathematical theory of the macroscopic fluctuations generated by
ensembles of independent microscopic discrete systems. We provide algorithms to
decide which multivariate gaussian distributions can be approximated by sums of
finitely-valued random vectors. We study non-trivial cases where the
microscopic variables have an unbounded range, as well as asymptotic scenarios
with infinitely many macroscopic variables. From a foundational point of view,
our results imply that bipartite gaussian states of light cannot be understood
as beams of independent d-dimensional particle pairs. It is also shown that the
classical description of certain macroscopic optical experiments, as opposed to
the quantum one, requires variables with infinite cardinality spectra.Comment: Proof of strong NP-hardness. Connection with random walks. New
asymptotic results. Numerous typos correcte
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