73 research outputs found
Localization in the Discrete Non-linear Schrodinger Equation and Geometric Properties of the Microcanonical Surface
It is well known that, if the initial conditions have sufficiently high energy density, the dynamics of the classical Discrete Non-Linear Schrodinger Equation (DNLSE) on a lattice shows a form of breaking of ergodicity, with a finite fraction of the total charge accumulating on a few sites and residing there for times that diverge quickly in the thermodynamic limit. In this paper we show that this kind of localization can be attributed to some geometric properties of the microcanonical potential energy surface, and that it can be associated to a phase transition in the lowest eigenvalue of the Laplacian on said surface. We also show that the approximation of considering the phase space motion on the potential energy surface only, with effective decoupling of the potential and kinetic partition functions, is justified in the large connectivity limit, or fully connected model. In this model we further observe a synchronization transition, with a synchronized phase at low temperatures
Noncyclic covers of knot complements
Hempel has shown that the fundamental groups of knot complements are
residually finite. This implies that every nontrivial knot must have a
finite-sheeted, noncyclic cover. We give an explicit bound, , such
that if is a nontrivial knot in the three-sphere with a diagram with
crossings and a particularly simple JSJ decomposition then the complement of
has a finite-sheeted, noncyclic cover with at most sheets.Comment: 29 pages, 8 figures, from Ph.D. thesis at Columbia University;
Acknowledgments added; Content correcte
Spin Foams and Noncommutative Geometry
We extend the formalism of embedded spin networks and spin foams to include
topological data that encode the underlying three-manifold or four-manifold as
a branched cover. These data are expressed as monodromies, in a way similar to
the encoding of the gravitational field via holonomies. We then describe
convolution algebras of spin networks and spin foams, based on the different
ways in which the same topology can be realized as a branched covering via
covering moves, and on possible composition operations on spin foams. We
illustrate the case of the groupoid algebra of the equivalence relation
determined by covering moves and a 2-semigroupoid algebra arising from a
2-category of spin foams with composition operations corresponding to a fibered
product of the branched coverings and the gluing of cobordisms. The spin foam
amplitudes then give rise to dynamical flows on these algebras, and the
existence of low temperature equilibrium states of Gibbs form is related to
questions on the existence of topological invariants of embedded graphs and
embedded two-complexes with given properties. We end by sketching a possible
approach to combining the spin network and spin foam formalism with matter
within the framework of spectral triples in noncommutative geometry.Comment: 48 pages LaTeX, 30 PDF figure
Compact 3-manifolds via 4-colored graphs
We introduce a representation of compact 3-manifolds without spherical
boundary components via (regular) 4-colored graphs, which turns out to be very
convenient for computer aided study and tabulation. Our construction is a
direct generalization of the one given in the eighties by S. Lins for closed
3-manifolds, which is in turn dual to the earlier construction introduced by
Pezzana's school in Modena. In this context we establish some results
concerning fundamental groups, connected sums, moves between graphs
representing the same manifold, Heegaard genus and complexity, as well as an
enumeration and classification of compact 3-manifolds representable by graphs
with few vertices ( in the non-orientable case and in the
orientable one).Comment: 25 pages, 11 figures; changes suggested by referee: references added,
figure 2 modified, results about classification of the manifolds in
Proposition 17 announced at the end of section 9. Accepted for publication in
RACSAM. The final publication is available at Springer (see DOI
Standard moves for standard polyhedra and spines
A finite 2-dimensional CW-complex P is called a standard (or special) polyhedron if the link of any vertex of P is homeomorphic to a circle with three radii and the link of any other point of its l-skeleton is homeomorphic to a circle with one diameter. Three transformations of standard polyhedra are defined, called standard moves. Moves I and III change a small neighbourhood of a vertex, move II changes a small neighbourhood of an edge. Let P, Q be two standard polyhedra. The following two results are proved: 1) P can be 3-deformed (in the sense of J. H. C. Whitehead) to Q if and only if P can be obtained from Q by a finite sequence of moves I, II, III and its inverses; 2) If P is a spine of a 3-manifold then Q is a spine of the same manifold if and only if P and Q can be related by moves I, II and its inverses only
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