15,560 research outputs found
Blow up and Blur constructions in Algebraic Logic
The idea in the title is to blow up a finite structure, replacing each
'colour or atom' by infinitely many, using blurs to represent the resulting
term algebra, but the blurs are not enough to blur the structure of the finite
structure in the complex algebra. Then, the latter cannot be representable due
to a {finite- infinite} contradiction. This structure can be a finite clique in
a graph or a finite relation algebra or a finite cylindric algebra. This theme
gives examples of weakly representable atom structures that are not strongly
representable. Many constructions existing in the literature are placed in a
rigorous way in such a framework, properly defined.
This is the essence too of construction of Monk like-algebras, one constructs
graphs with finite colouring (finitely many blurs), converging to one with
infinitely many, so that the original algebra is also blurred at the complex
algebra level, and the term algebra is completey representable, yielding a
representation of its completion the complex algebra.
A reverse of this process exists in the literature, it builds algebras with
infinite blurs converging to one with finite blurs. This idea due to Hirsch and
Hodkinson, uses probabilistic methods of Erdos to construct a sequence of
graphs with infinite chromatic number one that is 2 colourable. This
construction, which works for both relation and cylindric algebras, further
shows that the class of strongly representable atom structures is not
elementary.Comment: arXiv admin note: text overlap with arXiv:1304.114
Two Proofs of Fine's Theorem
Fine's theorem concerns the question of determining the conditions under
which a certain set of probabilities for pairs of four bivalent quantities may
be taken to be the marginals of an underlying probability distribution. The
eight CHSH inequalities are well-known to be necessary conditions, but Fine's
theorem is the striking result that they are also a sufficient condition. It
has application to the question of finding a local hidden variables theory for
measurements of pairs of spins for a system in an EPRB state. Here we present
two simple and self-contained proofs of Fine's theorem in which the origins of
this non-obvious result can be easily seen. The first is a physically motivated
proof which simply notes that this matching problem is solved using a local
hidden variables model given by Peres. The second is a straightforward
algebraic proof which uses a representation of the probabilities in terms of
correlation functions and takes advantage of certain simplifications naturally
arising in that representation. A third, unsuccessful attempt at a proof,
involving the maximum entropy technique is also briefly describedComment: 17 pages, latex. Revised argument for setting average spins to zero.
References added. Corrected figur
Reaction-diffusion processes and non-perturbative renormalisation group
This paper is devoted to investigating non-equilibrium phase transitions to
an absorbing state, which are generically encountered in reaction-diffusion
processes. It is a review, based on [Phys. Rev. Lett. 92, 195703; Phys. Rev.
Lett. 92, 255703; Phys. Rev. Lett. 95, 100601], of recent progress in this
field that has been allowed by a non-perturbative renormalisation group
approach. We mainly focus on branching and annihilating random walks and show
that their critical properties strongly rely on non-perturbative features and
that hence the use of a non-perturbative method turns out to be crucial to get
a correct picture of the physics of these models.Comment: 14 pages, submitted to J. Phys. A for the proceedings of the
conference 'Renormalization Group 2005', Helsink
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