92 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
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
On the M\"obius transformation in the entanglement entropy of fermionic chains
There is an intimate relation between entanglement entropy and Riemann
surfaces. This fact is explicitly noticed for the case of quadratic fermionic
Hamiltonians with finite range couplings. After recollecting this fact, we make
a comprehensive analysis of the action of the M\"obius transformations on the
Riemann surface. We are then able to uncover the origin of some symmetries and
dualities of the entanglement entropy already noticed recently in the
literature. These results give further support for the use of entanglement
entropy to analyse phase transition.Comment: 29 pages, 5 figures. Final version published in JSTAT. Two new
figures. Some comments and references added. Typos correcte
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