222,588 research outputs found
Generating all finite modular lattices of a given size
Modular lattices, introduced by R. Dedekind, are an important subvariety of
lattices that includes all distributive lattices. Heitzig and Reinhold
developed an algorithm to enumerate, up to isomorphism, all finite lattices up
to size 18. Here we adapt and improve this algorithm to construct and count
modular lattices up to size 24, semimodular lattices up to size 22, and
lattices of size 19. We also show that is a lower bound for the
number of nonisomorphic modular lattices of size .Comment: Preprint, 12 pages, 2 figures, 1 tabl
Distributive Lattices have the Intersection Property
Distributive lattices form an important, well-behaved class of lattices. They
are instances of two larger classes of lattices: congruence-uniform and
semidistributive lattices. Congruence-uniform lattices allow for a remarkable
second order of their elements: the core label order; semidistributive lattices
naturally possess an associated flag simplicial complex: the canonical join
complex. In this article we present a characterization of finite distributive
lattices in terms of the core label order and the canonical join complex, and
we show that the core label order of a finite distributive lattice is always a
meet-semilattice.Comment: 9 pages, 3 figures. Final version. Comments are very welcom
Stability of discrete dark solitons in nonlinear Schrodinger lattices
We obtain new results on the stability of discrete dark solitons bifurcating
from the anti-continuum limit of the discrete nonlinear Schrodinger equation,
following the analysis of our previous paper [Physica D 212, 1-19 (2005)]. We
derive a criterion for stability or instability of dark solitons from the
limiting configuration of the discrete dark soliton and confirm this criterion
numerically. We also develop detailed calculations of the relevant eigenvalues
for a number of prototypical configurations and obtain very good agreement of
asymptotic predictions with the numerical data.Comment: 11 pages, 5 figure
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