100 research outputs found
Phase transition in the assignment problem for random matrices
We report an analytic and numerical study of a phase transition in a P
problem (the assignment problem) that separates two phases whose
representatives are the simple matching problem (an easy P problem) and the
traveling salesman problem (a NP-complete problem). Like other phase
transitions found in combinatoric problems (K-satisfiability, number
partitioning) this can help to understand the nature of the difficulties in
solving NP problems an to find more accurate algorithms for them.Comment: 7 pages, 5 figures; accepted for publication in Europhys. Lett.
http://www.edpsciences.org/journal/index.cfm?edpsname=ep
Finite entanglement entropy from the zero-point-area of spacetime
The calculation of entanglement entropy S of quantum fields in spacetimes
with horizon shows that, quite generically, S (a) is proportional to the area A
of the horizon and (b) is divergent. I argue that this divergence, which arises
even in the case of Rindler horizon in flat spacetime, is yet another
indication of a deep connection between horizon thermodynamics and
gravitational dynamics. In an emergent perspective of gravity, which
accommodates this connection, the fluctuations around the equipartition value
in the area elements will lead to a minimal quantum of area, of the order of
L_P^2, which will act as a regulator for this divergence. In a particular
prescription for incorporating L_P^2 as zero-point-area of spacetime, this does
happen and the divergence in entanglement entropy is regularized, leading to S
proportional to (A/L_P^2) in Einstein gravity. In more general models of
gravity, the surface density of microscopic degrees of freedom is different
which leads to a modified regularisation procedure and the possibility that the
entanglement entropy - when appropriately regularised - matches the Wald
entropy.Comment: ver 2: minor clarifications added; reformatted with Sections; 11
page
Reduction of Lie--Jordan algebras: Quantum
In this paper we present a theory of reduction of quantum systems in the
presence of symmetries and constraints. The language used is that of
Lie--Jordan Banach algebras, which are discussed in some detail together with
spectrum properties and the space of states. The reduced Lie--Jordan Banach
algebra is characterized together with the Dirac states on the physical algebra
of observables
Revisiting Lie integrability by quadratures from a geometric perspective
After a short review of the classical Lie theorem, a finite dimensional Lie
algebra of vector fields is considered and the most general conditions under
which the integral curves of one of the fields can be obtained by quadratures
in a prescribed way will be discussed, determining also the number of
quadratures needed to integrate the system. The theory will be illustrated with
examples andbn an extension of the theorem where the Lie algebras are replaced
by some distributions will also be presented.Comment: 14 pages, proceedings of the conference "50th Seminar Sophus Lie", 25
September - 1 October 2016, Bedlewo, Polan
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