1,703 research outputs found
A Poset Connected to Artin Monoids of Simply Laced Type
Let W be a Weyl group whose type is a simply laced Dynkin diagram. On several
W-orbits of sets of mutually commuting reflections, a poset is described which
plays a role in linear representatons of the corresponding Artin group A. The
poset generalizes many properties of the usual order on positive roots of W
given by height. In this paper, a linear representation of the positive monoid
of A is defined by use of the poset
BMW algebras of simply laced type
It is known that the recently discovered representations of the Artin groups
of type A_n, the braid groups, can be constructed via BMW algebras. We
introduce similar algebras of type D_n and E_n which also lead to the newly
found faithful representations of the Artin groups of the corresponding types.
We establish finite dimensionality of these algebras. Moreover, they have
ideals I_1 and I_2 with I_2 contained in I_1 such that the quotient with
respect to I_1 is the Hecke algebra and I_1/I_2 is a module for the
corresponding Artin group generalizing the Lawrence-Krammer representation.
Finally we give conjectures on the structure, the dimension and parabolic
subalgebras of the BMW algebra, as well as on a generalization of deformations
to Brauer algebras for simply laced spherical type other than A_n.Comment: 39 page
Tangle and Brauer Diagram Algebras of Type Dn
A generalization of the Kauffman tangle algebra is given for Coxeter type Dn.
The tangles involve a pole or order 2. The algebra is shown to be isomorphic to
the Birman-Murakami-Wenzl algebra of the same type. This result extends the
isomorphism between the two algebras in the classical case, which in our
set-up, occurs when the Coxeter type is of type A with index n-1. The proof
involves a diagrammatic version of the Brauer algebra of type Dn in which the
Temperley-Lieb algebra of type Dn is a subalgebra.Comment: 33 page
NP-hardness of the cluster minimization problem revisited
The computational complexity of the "cluster minimization problem" is
revisited [L. T. Wille and J. Vennik, J. Phys. A 18, L419 (1985)]. It is argued
that the original NP-hardness proof does not apply to pairwise potentials of
physical interest, such as those that depend on the geometric distance between
the particles. A geometric analog of the original problem is formulated, and a
new proof for such potentials is provided by polynomial time transformation
from the independent set problem for unit disk graphs. Limitations of this
formulation are pointed out, and new subproblems that bear more direct
consequences to the numerical study of clusters are suggested.Comment: 8 pages, 2 figures, accepted to J. Phys. A: Math. and Ge
Network of inherent structures in spin glasses: scaling and scale-free distributions
The local minima (inherent structures) of a system and their associated
transition links give rise to a network. Here we consider the topological and
distance properties of such a network in the context of spin glasses. We use
steepest descent dynamics, determining for each disorder sample the transition
links appearing within a given barrier height. We find that differences between
linked inherent structures are typically associated with local clusters of
spins; we interpret this within a framework based on droplets in which the
characteristic ``length scale'' grows with the barrier height. We also consider
the network connectivity and the degrees of its nodes. Interestingly, for spin
glasses based on random graphs, the degree distribution of the network of
inherent structures exhibits a non-trivial scale-free tail.Comment: minor changes and references adde
Phase Transitions from Saddles of the Potential Energy Landscape
The relation between saddle points of the potential of a classical
many-particle system and the analyticity properties of its thermodynamic
functions is studied. For finite systems, each saddle point is found to cause a
nonanalyticity in the Boltzmann entropy, and the functional form of this
nonanalytic term is derived. For large systems, the order of the nonanalytic
term increases unboundedly, leading to an increasing differentiability of the
entropy. Analyzing the contribution of the saddle points to the density of
states in the thermodynamic limit, our results provide an explanation of how,
and under which circumstances, saddle points of the potential energy landscape
may (or may not) be at the origin of a phase transition in the thermodynamic
limit. As an application, the puzzling observations by Risau-Gusman et al. on
topological signatures of the spherical model are elucidated.Comment: 5 pages, no figure
Energy Landscape and Global Optimization for a Frustrated Model Protein
The three-color (BLN) 69-residue model protein was designed to exhibit frustrated folding. We investigate the energy landscape of this protein using disconnectivity graphs and compare it to a Go model, which is designed to reduce the frustration by removing all non-native attractive interactions. Finding the global minimum on a frustrated energy landscape is a good test of global optimization techniques, and we present calculations evaluating the performance of basin-hopping and genetic algorithms for this system.Comparisons are made with the widely studied 46-residue BLN protein.We show that the energy landscape of the 69-residue BLN protein contains several deep funnels, each of which corresponds to a different β-barrel structure
Power-law distributions for the areas of the basins of attraction on a potential energy landscape
Energy landscape approaches have become increasingly popular for analysing a
wide variety of chemical physics phenomena. Basic to many of these applications
has been the inherent structure mapping, which divides up the potential energy
landscape into basins of attraction surrounding the minima. Here, we probe the
nature of this division by introducing a method to compute the basin area
distribution and applying it to some archetypal supercooled liquids. We find
that this probability distribution is a power law over a large number of
decades with the lower-energy minima having larger basins of attraction.
Interestingly, the exponent for this power law is approximately the same as
that for a high-dimensional Apollonian packing, providing further support for
the suggestion that there is a strong analogy between the way the energy
landscape is divided into basins, and the way that space is packed in
self-similar, space-filling hypersphere packings, such as the Apollonian
packing. These results suggest that the basins of attraction provide a
fractal-like tiling of the energy landscape, and that a scale-free pattern of
connections between the minima is a general property of energy landscapes.Comment: 4 pages, 3 figure
Topological methods for searching barriers and reaction paths
We present a family of algorithms for the fast determination of reaction
paths and barriers in phase space and the computation of the corresponding
rates. The method requires the reaction times be large compared to the
microscopic time, irrespective of the origin - energetic, entropic, cooperative
- of the timescale separation. It lends itself to temperature cycling as in
simulated annealing and to activation-relaxation routines. The dynamics is
ultimately based on supersymmetry methods used years ago to derive Morse
theory. Thus, the formalism automatically incorporates all relevant topological
information.Comment: 4 pages, 4 figures, RevTex
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