487 research outputs found
Tameness in generalized metric structures
We broaden the framework of metric abstract elementary classes (mAECs) in
several essential ways, chiefly by allowing the metric to take values in a
well-behaved quantale. As a proof of concept we show that the result of Boney
and Zambrano on (metric) tameness under a large cardinal assumption holds in
this more general context. We briefly consider a further generalization to
partial metric spaces, and hint at connections to classes of fuzzy structures,
and structures on sheaves
The Expectation Monad in Quantum Foundations
The expectation monad is introduced abstractly via two composable
adjunctions, but concretely captures measures. It turns out to sit in between
known monads: on the one hand the distribution and ultrafilter monad, and on
the other hand the continuation monad. This expectation monad is used in two
probabilistic analogues of fundamental results of Manes and Gelfand for the
ultrafilter monad: algebras of the expectation monad are convex compact
Hausdorff spaces, and are dually equivalent to so-called Banach effect
algebras. These structures capture states and effects in quantum foundations,
and also the duality between them. Moreover, the approach leads to a new
re-formulation of Gleason's theorem, expressing that effects on a Hilbert space
are free effect modules on projections, obtained via tensoring with the unit
interval.Comment: In Proceedings QPL 2011, arXiv:1210.029
The structure of classical extensions of quantum probability theory
On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra–Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed
A decomposition theorem for fuzzy set-valued random variables and a characterization of fuzzy random translation
Let be a fuzzy set--valued random variable (\frv{}), and \huku{X} the
family of all fuzzy sets for which the Hukuhara difference X\HukuDiff B
exists --almost surely. In this paper, we prove that can be
decomposed as X(\omega)=C\Mink Y(\omega) for --almost every
, is the unique deterministic fuzzy set that minimizes
as is varying in \huku{X}, and is a centered
\frv{} (i.e. its generalized Steiner point is the origin). This decomposition
allows us to characterize all \frv{} translation (i.e. X(\omega) = M \Mink
\indicator{\xi(\omega)} for some deterministic fuzzy convex set and some
random element in \Banach). In particular, is an \frv{} translation if
and only if the Aumann expectation is equal to up to a
translation.
Examples, such as the Gaussian case, are provided.Comment: 12 pages, 1 figure. v2: minor revision. v3: minor revision;
references, affiliation and acknowledgments added. Submitted versio
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