258 research outputs found
A saturation property of structures obtained by forcing with a compact family of random variables
A method how to construct Boolean-valued models of some fragments of
arithmetic was developed in Krajicek (2011), with the intended applications in
bounded arithmetic and proof complexity. Such a model is formed by a family of
random variables defined on a pseudo-finite sample space. We show that under a
fairly natural condition on the family (called compactness in K.(2011)) the
resulting structure has a property that is naturally interpreted as saturation
for existential types. We also give an example showing that this cannot be
extended to universal types.Comment: preprint February 201
Generalizing Tsirelson's bound on Bell inequalities using a min-max principle
Bounds on the norm of quantum operators associated with classical Bell-type
inequalities can be derived from their maximal eigenvalues. This quantitative
method enables detailed predictions of the maximal violations of Bell-type
inequalities.Comment: 4 pages, 2 figures, RevTeX4, replaced with published versio
New optimal tests of quantum nonlocality
We explore correlation polytopes to derive a set of all Boole-Bell type
conditions of possible classical experience which are both maximal and
complete. These are compared with the respective quantum expressions for the
Greenberger-Horne-Zeilinger (GHZ) case and for two particles with spin state
measurements along three directions.Comment: 10 page
Testing the bounds on quantum probabilities
Bounds on quantum probabilities and expectation values are derived for
experimental setups associated with Bell-type inequalities. In analogy to the
classical bounds, the quantum limits are experimentally testable and therefore
serve as criteria for the validity of quantum mechanics.Comment: 9 pages, Revte
Bell inequalities as constraints on unmeasurable correlations
The interpretation of the violation of Bell-Clauser-Horne inequalities is
revisited, in relation with the notion of extension of QM predictions to
unmeasurable correlations. Such extensions are compatible with QM predictions
in many cases, in particular for observables with compatibility relations
described by tree graphs. This implies classical representability of any set of
correlations , , , and the equivalence of the
Bell-Clauser-Horne inequalities to a non void intersection between the ranges
of values for the unmeasurable correlation associated to different
choices for B. The same analysis applies to the Hardy model and to the "perfect
correlations" discussed by Greenberger, Horne, Shimony and Zeilinger. In all
the cases, the dependence of an unmeasurable correlation on a set of variables
allowing for a classical representation is the only basis for arguments about
violations of locality and causality.Comment: Some modifications have been done in order to improve clarity of
presentation and comparison with other approache
Pointwise convergence of Birkhoff averages for global observables
It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in
infinite ergodic theory is trivial; it states that for any
infinite-measure-preserving ergodic system the Birkhoff average of every
integrable function is almost everywhere zero. Nor does a different rescaling
of the Birkhoff sum that leads to a non-degenerate pointwise limit exist. In
this paper we give a version of Birkhoff's theorem for conservative, ergodic,
infinite-measure-preserving dynamical systems where instead of integrable
functions we use certain elements of , which we generically call
global observables. Our main theorem applies to general systems but requires an
hypothesis of "approximate partial averaging" on the observables. The idea
behind the result, however, applies to more general situations, as we show with
an example. Finally, by means of counterexamples and numerical simulations, we
discuss the question of finding the optimal class of observables for which a
Birkhoff theorem holds for infinite-measure-preserving systems.Comment: Final version. 33 pages, 10 figure
Localization Bounds for Multiparticle Systems
We consider the spectral and dynamical properties of quantum systems of
particles on the lattice , of arbitrary dimension, with a Hamiltonian
which in addition to the kinetic term includes a random potential with iid
values at the lattice sites and a finite-range interaction. Two basic
parameters of the model are the strength of the disorder and the strength of
the interparticle interaction. It is established here that for all there
are regimes of high disorder, and/or weak enough interactions, for which the
system exhibits spectral and dynamical localization. The localization is
expressed through bounds on the transition amplitudes, which are uniform in
time and decay exponentially in the Hausdorff distance in the configuration
space. The results are derived through the analysis of fractional moments of
the -particle Green function, and related bounds on the eigenfunction
correlators
How much contextuality?
The amount of contextuality is quantified in terms of the probability of the
necessary violations of noncontextual assignments to counterfactual elements of
physical reality.Comment: 5 pages, 3 figure
Hahn's Symmetric Quantum Variational Calculus
We introduce and develop the Hahn symmetric quantum calculus with
applications to the calculus of variations. Namely, we obtain a necessary
optimality condition of Euler-Lagrange type and a sufficient optimality
condition for variational problems within the context of Hahn's symmetric
calculus. Moreover, we show the effectiveness of Leitmann's direct method when
applied to Hahn's symmetric variational calculus. Illustrative examples are
provided.Comment: This is a preprint of a paper whose final and definite form will
appear in the international journal Numerical Algebra, Control and
Optimization (NACO). Paper accepted for publication 06-Sept-201
The Schr\"oder functional equation and its relation to the invariant measures of chaotic maps
The aim of this paper is to show that the invariant measure for a class of
one dimensional chaotic maps, , is an extended solution of the Schr\"oder
functional equation, , induced by them. Hence, we give an
unified treatment of a collection of exactly solved examples worked out in the
current literature. In particular, we show that these examples belongs to a
class of functions introduced by Mira, (see text). Moreover, as a new example,
we compute the invariant densities for a class of rational maps having the
Weierstrass functions as an invariant one. Also, we study the relation
between that equation and the well known Frobenius-Perron and Koopman's
operators.Comment: 9 page
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