Numerical Results for Ground States of Spin Glasses on Bethe Lattices

The average ground state energy and entropy for +/- J spin glasses on Bethe lattices of connectivities k+1=3...,26 at T=0 are approximated numerically. To obtain sufficient accuracy for large system sizes (up to n=2048), the Extremal Optimization heuristic is employed which provides high-quality results not only for the ground state energies per spin e_{k+1} but also for their entropies s_{k+1}. The results show considerable quantitative differences between lattices of even and odd connectivities. The results for the ground state energies compare very well with recent one-step replica symmetry breaking calculations. These energies can be scaled for all even connectivities k+1 to within a fraction of a percent onto a simple functional form, e_{k+1} = E_{SK} sqrt(k+1) - {2E_{SK}+sqrt(2)} / sqrt(k+1), where E_{SK} = -0.7633 is the ground state energy for the broken replica symmetry in the Sherrington-Kirkpatrick model. But this form is in conflict with perturbative calculations at large k+1, which do not distinguish between even and odd connectivities. We find non-zero entropies s_{k+1} at small connectivities. While s_{k+1} seems to vanish asymptotically with 1/(k+1) for even connectivities, it is indistinguishable from zero already for odd k+1 >= 9.Comment: 11 pages, RevTex4, 28 ps-figures included, related papers available at http://www.physics.emory.edu/faculty/boettcher

Delineating Intra-Urban Spatial Connectivity Patterns by Travel-Activities: A Case Study of Beijing, China

Travel activities have been widely applied to quantify spatial interactions between places, regions and nations. In this paper, we model the spatial connectivities between 652 Traffic Analysis Zones (TAZs) in Beijing by a taxi OD dataset. First, we unveil the gravitational structure of intra-urban spatial connectivities of Beijing. On overall, the inter-TAZ interactions are well governed by the Gravity Model $G_{ij} = {\lambda}p_{i}p_{j}/d_{ij}$, where $p_{i}$, $p_{j}$ are degrees of TAZ $i$, $j$ and $d_{ij}$ the distance between them, with a goodness-of-fit around 0.8. Second, the network based analysis well reveals the polycentric form of Beijing. Last, we detect the semantics of inter-TAZ connectivities based on their spatiotemporal patterns. We further find that inter-TAZ connections deviating from the Gravity Model can be well explained by link semantics.Comment: 6 pages, 4 figure

Scale-free networks are not robust under neutral evolution

Recently it has been shown that a large variety of different networks have power-law (scale-free) distributions of connectivities. We investigate the robustness of such a distribution in discrete threshold networks under neutral evolution. The guiding principle for this is robustness in the resulting phenotype. The numerical results show that a power-law distribution is not stable under such an evolution, and the network approaches a homogeneous form where the overall distribution of connectivities is given by a Poisson distribution.Comment: Submitted for publicatio

Decomposition and Oxidation of the N-Extended Supersymmetric Quantum Mechanics Multiplets

We furnish an algebraic understanding of the inequivalent connectivities (computed up to $N\leq 10$) of the graphs associated to the irreducible supermultiplets of the N-extended Supersymmetric Quantum Mechanics. We prove that the inequivalent connectivities of the N=5 and N=9 irreducible supermultiplets are due to inequivalent decompositions into two sets of N=4 (respectively, N=8) supermultiplets. "Oxido-reduction" diagrams linking the irreducible supermultiplets of the N=5,6,7,8 supersymmetries are presented. We briefly discuss these results and their possible applications.Comment: 15 pages, 5 figure

Inconsistencies in the application of harmonic analysis to pulsating stars

Using ultra-precise data from space instrumentation we found that the underlying functions of stellar light curves from some AF pul- sating stars are non-analytic, and consequently their Fourier expansion is not guaranteed. This result demonstrates that periodograms do not provide a mathematically consistent estimator of the frequency content for this kind of variable stars. More importantly, this constitutes the first counterexample against the current paradigm which considers that any physical process is described by a contin- uous (band-limited) function that is infinitely differentiable.Comment: 9 pages, 8 figure

Scaling Behaviour of Developing and Decaying Networks

We find that a wide class of developing and decaying networks has scaling properties similar to those that were recently observed by Barab\'{a}si and Albert in the particular case of growing networks. The networks considered here evolve according to the following rules: (i) Each instant a new site is added, the probability of its connection to old sites is proportional to their connectivities. (ii) In addition, (a) new links between some old sites appear with probability proportional to the product of their connectivities or (b) some links between old sites are removed with equal probability.Comment: 7 pages (revtex

Refining the classification of the irreps of the 1D N-Extended Supersymmetry

The linear finite irreducible representations of the algebra of the 1D $N$-Extended Supersymmetric Quantum Mechanics are discussed in terms of their "connectivity" (a symbol encoding information on the graphs associated to the irreps). The classification of the irreducible representations with the same fields content and different connectivity is presented up to $N\leq 8$.Comment: Two extra cases added. Reply to hep-th/0611060v2 comments adde

Coarse grained description of the protein folding

We consider two- and three-dimensional lattice models of proteins which were characterized previously. We coarse grain their folding dynamics by reducing it to transitions between effective states. We consider two methods of selection of the effective states. The first method is based on the steepest descent mapping of states to underlying local energy minima and the other involves an additional projection to maximally compact conformations. Both methods generate connectivity patterns that allow to distinguish between the good and bad folders. Connectivity graphs corresponding to the folding funnel have few loops and are thus tree-like. The Arrhenius law for the median folding time of a 16-monomer sequence is established and the corresponding barrier is related to easily identifiable kinetic trap states.Comment: REVTeX, 9 pages, 15 EPS figures, to appear in Phys. Rev.