32,983 research outputs found
The maximum forcing number of polyomino
The forcing number of a perfect matching of a graph is the
cardinality of the smallest subset of that is contained in no other perfect
matchings of . For a planar embedding of a 2-connected bipartite planar
graph which has a perfect matching, the concept of Clar number of hexagonal
system had been extended by Abeledo and Atkinson as follows: a spanning
subgraph of is called a Clar cover of if each of its components is
either an even face or an edge, the maximum number of even faces in Clar covers
of is called Clar number of , and the Clar cover with the maximum number
of even faces is called the maximum Clar cover. It was proved that if is a
hexagonal system with a perfect matching and is a set of hexagons in a
maximum Clar cover of , then has a unique 1-factor. Using this
result, Xu {\it et. at.} proved that the maximum forcing number of the
elementary hexagonal system are equal to their Clar numbers, and then the
maximum forcing number of the elementary hexagonal system can be computed in
polynomial time. In this paper, we show that an elementary polyomino has a
unique perfect matching when removing the set of tetragons from its maximum
Clar cover. Thus the maximum forcing number of elementary polyomino equals to
its Clar number and can be computed in polynomial time. Also, we have extended
our result to the non-elementary polyomino and hexagonal system
Ring-current maps for benzenoids : comparisons, contradictions, and a versatile combinatorial model
As a key diagnostic property of benzenoids and other polycyclic hydrocarbons, induced ring current has inspired diverse approaches for calculation, modeling, and interpretation. Grid-based methods include the ipsocentric ab initio calculation of current maps, and its surrogate, the pseudo-π model. Graph-based models include a family of conjugated-circuit (CC) models and the molecular-orbital Hückel-London (HL) model. To assess competing claims for physical relevance of derived current maps for benzenoids, a protocol for graph-reduction and comparison was devised. Graph reduction of pseudo-π grid maps highlights their overall similarity to HL maps, but also reveals systematic differences. These are ascribed to unavoidable pseudo-π proximity limitations for benzenoids with short nonbonded distances, and to poor continuity of pseudo-π current for classes of benzenoids with fixed bonds, where single-reference methods can be unreliable. Comparison between graph-based approaches shows that the published CC models all shadow HL maps reasonably well for most benzenoids (as judged by L1-, L2-, and L∞-error norms on scaled bond currents), though all exhibit physically implausible currents for systems with fixed bonds. These comparisons inspire a new combinatorial model (Model W) based on cycle decomposition of current, taking into account the two terms of lowest order that occur in the characteristic polynomial. This improves on all pure-CC models within their range of applicability, giving excellent adherence to HL maps for all Kekulean benzenoids, including those with fixed bonds (halving the rms discrepancy against scaled HL bond currents, from 11% in the best CC model, to 5% for the set of 18 360 Kekulean benzenoids on up to 10 hexagonal rings). Model W also has excellent performance for open-shell systems, where currents cannot be described at all by pure CC models (4% rms discrepancy against scaled HL bond currents for the 20112 non-Kekulean benzenoids on up to 10 hexagonal rings). Consideration of largest and next-to-largest matchings is a useful strategy for modeling and interpretation of currents in Kekulean and non-Kekulean benzenoids (nanographenes)
Generic model for tunable colloidal aggregation in multidirectional fields
Based on Brownian Dynamics computer simulations in two dimensions we
investigate aggregation scenarios of colloidal particles with directional
interactions induced by multiple external fields. To this end we propose a
model which allows continuous change in the particle interactions from
point-dipole-like to patchy-like (with four patches). We show that, as a result
of this change, the non-equilibrium aggregation occurring at low densities and
temperatures transforms from conventional diffusion-limited cluster aggregation
(DLCA) to slippery DLCA involving rotating bonds; this is accompanied by a
pronounced change of the underlying lattice structure of the aggregates from
square-like to hexagonal ordering. Increasing the temperature we find a
transformation to a fluid phase, consistent with results of a simple mean-field
density functional theory
Relation between directed polymers in random media and random bond dimer models
We reassess the relation between classical lattice dimer models and the
continuum elastic description of a lattice of fluctuating polymers. In the
absence of randomness we determine the density and line tension of the polymers
in terms of the bond weights of hard-core dimers on the square and the
hexagonal lattice. For the latter, we demonstrate the equivalence of the
canonical ensemble for the dimer model and the grand-canonical description for
polymers by performing explicitly the continuum limit. Using this equivalence
for the random bond dimer model on a square lattice, we resolve a previously
observed discrepancy between numerical results for the random dimer model and a
replica approach for polymers in random media. Further potential applications
of the equivalence are briefly discussed.Comment: 6 pages, 3 figure
Order-by-disorder in classical oscillator systems
We consider classical nonlinear oscillators on hexagonal lattices. When the
coupling between the elements is repulsive, we observe coexisting states, each
one with its own basin of attraction. These states differ by their degree of
synchronization and by patterns of phase-locked motion. When disorder is
introduced into the system by additive or multiplicative Gaussian noise, we
observe a non-monotonic dependence of the degree of order in the system as a
function of the noise intensity: intervals of noise intensity with low
synchronization between the oscillators alternate with intervals where more
oscillators are synchronized. In the latter case, noise induces a higher degree
of order in the sense of a larger number of nearly coinciding phases. This
order-by-disorder effect is reminiscent to the analogous phenomenon known from
spin systems. Surprisingly, this non-monotonic evolution of the degree of order
is found not only for a single interval of intermediate noise strength, but
repeatedly as a function of increasing noise intensity. We observe noise-driven
migration of oscillator phases in a rough potential landscape.Comment: 12 pages, 13 figures; comments are welcom
Locations of multicritical points for spin glasses on regular lattices
We present an analysis leading to precise locations of the multicritical
points for spin glasses on regular lattices. The conventional technique for
determination of the location of the multicritical point was previously derived
using a hypothesis emerging from duality and the replica method. In the present
study, we propose a systematic technique, by an improved technique, giving more
precise locations of the multicritical points on the square, triangular, and
hexagonal lattices by carefully examining relationship between two partition
functions related with each other by the duality. We can find that the
multicritical points of the Ising model are located at
on the square lattice, where means the probability of ,
at on the triangular lattice, and at on the
hexagonal lattice. These results are in excellent agreement with recent
numerical estimations.Comment: 17pages, this is the published version with some minnor corrections.
Previous title was "Precise locations of multicritical points for spin
glasses on regular lattices
A challenge for critical point of spin glass in ground state
We show several calculations to identify the critical point in the ground
state in random spin systems including spin glasses on the basis of the duality
analysis. The duality analysis is a profound method to obtain the precise
location of the critical point in finite temperature even for spin glasses. We
propose a single equality for identifying the critical point in the ground
state from several speculations. The equality can indeed give the exact
location of the critical points for the bond-dilution Ising model on several
lattices and provides insight on further analysis on the ground state in spin
glasses.Comment: 7 pages, 2 figures, to appear in Proceedings of 4th YSM-SPIP (Sendai,
14-16 December 2012
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