96 research outputs found
Perimeter ring currents in benzenoids from Pauling bond orders
It is shown that the ring currents in perimeter hexagonal rings of Kekulean benzenoids, as estimated
within the RandiÄ conjugated-circuit model, can be calculated directly without tedious pairwise
comparison of Kekulé structures or Kekulé counting for cycle-deleted subgraphs. Required are only
the Pauling bond orders of perimeter bonds and the number of Kekulé structures of the benzenoid,
both readily available from the adjacency matrix of the carbon skeleton. This approach provides easy
calculation of complete current maps for benzenoids in which every face has at least one bond on the
perimeter (as in the example of cata-condensed benzenoids), and allows qualitative evaluation of the
main ring-current contributions to 1H chemical shifts in general benzenoids. A combined RandiÄ-
Pauling model for correlation of ring current and bond length through bond order is derived and shown
to be consistent with resilience of current under bond alternation
Entangled networks, synchronization, and optimal network topology
A new family of graphs, {\it entangled networks}, with optimal properties in
many respects, is introduced. By definition, their topology is such that
optimizes synchronizability for many dynamical processes. These networks are
shown to have an extremely homogeneous structure: degree, node-distance,
betweenness, and loop distributions are all very narrow. Also, they are
characterized by a very interwoven (entangled) structure with short average
distances, large loops, and no well-defined community-structure. This family of
nets exhibits an excellent performance with respect to other flow properties
such as robustness against errors and attacks, minimal first-passage time of
random walks, efficient communication, etc. These remarkable features convert
entangled networks in a useful concept, optimal or almost-optimal in many
senses, and with plenty of potential applications computer science or
neuroscience.Comment: Slightly modified version, as accepted in Phys. Rev. Let
Distributed curvature and stability of fullerenes
Energies of non-planar conjugated Ï systems are typically described qualitatively in terms of the balance of Ï stabilisation and the steric strain associated with geometric curvature. Curvature also has a purely graph-theoretical description: combinatorial curvature at a vertex of a polyhedral graph is defined as one minus half the vertex degree plus the sum of reciprocal sizes of the faces meeting at that vertex. Prisms and antiprisms have positive combinatorial vertex curvature at every vertex. Excluding these two infinite families, we call any other polyhedron with everywhere positive combinatorial curvature a PCC polyhedron. Cubic PCC polyhedra are initially common, but must eventually die out with increasing vertex count; the largest example constructed so far has 132 vertices. The fullerenes Cn have cubic polyhedral molecular graphs with n vertices, 12 pentagonal and (n/2 â 10) hexagonal faces. We show that there are exactly 39 PCC fullerenes, all in the range 20 †n †60. In this range, there is only partial correlation between PCC status and stability as defined by minimum pentagon adjacency. The sum of vertex curvatures is 2 for any polyhedron; for fullerenes the sum of squared vertex curvatures is linearly related to the number of pentagon adjacencies and hence is a direct measure of relative stability of the lower (n †60) fullerenes. For n â„ 62, non-PCC fullerenes with a minimum number of pentagon adjacencies minimise mean-square curvature. For n â„ 70, minimum mean-square curvature implies isolation of pentagons, which is the strongest indicator of stability for a bare fullerene
Random and exhaustive generation of permutations and cycles
In 1986 S. Sattolo introduced a simple algorithm for uniform random
generation of cyclic permutations on a fixed number of symbols. This algorithm
is very similar to the standard method for generating a random permutation, but
is less well known.
We consider both methods in a unified way, and discuss their relation with
exhaustive generation methods. We analyse several random variables associated
with the algorithms and find their grand probability generating functions,
which gives easy access to moments and limit laws.Comment: 9 page
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-Ï model. Graph-based models include a family of conjugated-circuit (CC) models and the molecular-orbital HĂŒckel-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-Ï grid maps highlights their overall similarity to HL maps, but also reveals systematic differences. These are ascribed to unavoidable pseudo-Ï proximity limitations for benzenoids with short nonbonded distances, and to poor continuity of pseudo-Ï 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 Lâ-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 18âŻ360 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)
Inclusive One Jet Production With Multiple Interactions in the Regge Limit of pQCD
DIS on a two nucleon system in the regge limit is considered. In this
framework a review is given of a pQCD approach for the computation of the
corrections to the inclusive one jet production cross section at finite number
of colors and discuss the general results.Comment: 4 pages, latex, aicproc format, Contribution to the proceedings of
"Diffraction 2008", 9-14 Sep. 2008, La Londe-les-Maures, Franc
Efficient indexing of necklaces and irreducible polynomials over finite fields
We study the problem of indexing irreducible polynomials over finite fields,
and give the first efficient algorithm for this problem. Specifically, we show
the existence of poly(n, log q)-size circuits that compute a bijection between
{1, ... , |S|} and the set S of all irreducible, monic, univariate polynomials
of degree n over a finite field F_q. This has applications in pseudorandomness,
and answers an open question of Alon, Goldreich, H{\aa}stad and Peralta[AGHP].
Our approach uses a connection between irreducible polynomials and necklaces
( equivalence classes of strings under cyclic rotation). Along the way, we give
the first efficient algorithm for indexing necklaces of a given length over a
given alphabet, which may be of independent interest
Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that
We report on some recent developments in the search for optimal network
topologies. First we review some basic concepts on spectral graph theory,
including adjacency and Laplacian matrices, and paying special attention to the
topological implications of having large spectral gaps. We also introduce
related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we
discuss two different dynamical feautures of networks: synchronizability and
flow of random walkers and so that they are optimized if the corresponding
Laplacian matrix have a large spectral gap. From this, we show, by developing a
numerical optimization algorithm that maximum synchronizability and fast random
walk spreading are obtained for a particular type of extremely homogeneous
regular networks, with long loops and poor modular structure, that we call
entangled networks. These turn out to be related to Ramanujan and Cage graphs.
We argue also that these graphs are very good finite-size approximations to
Bethe lattices, and provide almost or almost optimal solutions to many other
problems as, for instance, searchability in the presence of congestion or
performance of neural networks. Finally, we study how these results are
modified when studying dynamical processes controlled by a normalized (weighted
and directed) dynamics; much more heterogeneous graphs are optimal in this
case. Finally, a critical discussion of the limitations and possible extensions
of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted
for pub. in JSTA
An Alternative Interpretation of Statistical Mechanics
In this paper I propose an interpretation of classical statistical mechanics that centers on taking seriously the idea that probability measures represent complete states of statistical mechanical systems. I show how this leads naturally to the idea that the stochasticity of statistical mechanics is associated directly with the observables of the theory rather than with the microstates (as traditional accounts would have it). The usual assumption that microstates are representationally significant in the theory is therefore dispensable, a consequence which suggests interesting possibilities for developing non-equilibrium statistical mechanics and investigating inter-theoretic answers to the foundational questions of statistical mechanics
Solving the measurement problem: de Broglie-Bohm loses out to Everett
The quantum theory of de Broglie and Bohm solves the measurement problem, but
the hypothetical corpuscles play no role in the argument. The solution finds a
more natural home in the Everett interpretation.Comment: 20 pages; submitted to special issue of Foundations of Physics, in
honour of James T. Cushin
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