Asymptotic expansion for the resistance between two maximally separated nodes on an M by N resistor network.

We analyze the exact formulas for the resistance between two arbitrary notes in a rectangular network of resistors under free, periodic and cylindrical boundary conditions obtained by Wu [J. Phys. A 37, 6653 (2004)]. Based on such expression, we then apply the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive the exact asymptotic expansions of the resistance between two maximally separated nodes on an M×N rectangular network of resistors with resistors r and s in the two spatial directions. Our results is 1/s (R(M×N))(r,s) = c(ρ)ln S + c(0)(ρ,ξ) + ∑(p=1)(∞) (c(2p)(ρ,ξ))/S(p) with S = MN, ρ = r/s and ξ = M/N. The all coefficients in this expansion are expressed through analytical functions. We have introduced the effective aspect ratio ξeff = square root(ρ)ξ for free and periodic boundary conditions and ξeff = square root(ρ)ξ/2 for cylindrical boundary condition and show that all finite-size correction terms are invariant under transformation ξeff→1/ξeff.

Fluctuations in gene regulatory networks as Gaussian colored noise.

The study of fluctuations in gene regulatory networks is extended to the case of Gaussian colored noise. First, the solution of the corresponding Langevin equation with colored noise is expressed in terms of an Ito integral. Then, two important lemmas concerning the variance of an Ito integral and the covariance of two Ito integrals are shown. Based on the lemmas, we give the general formulas for the variances and covariance of molecular concentrations for a regulatory network near a stable equilibrium explicitly. Two examples, the gene autoregulatory network and the toggle switch, are presented in details. In general, it is found that the finite correlation time of noise reduces the fluctuations and enhances the correlation between the fluctuations of the molecular components.

Self-similarity in the classification of finite-size scaling functions for toroidal boundary conditions.

The conventional periodic boundary conditions in two dimensions are extended to general boundary conditions, prescribed by primitive vector pairs that may not coincide with the coordinate axes. This extension is shown to be unambiguously specified by the twisting scheme. Equivalent relations between different twist settings are constructed explicitly. The classification of finite-size scaling functions is discussed based on the equivalent relations. A self-similar pattern for distinct classes of finite-size scaling functions is shown to appear on the plane that parametrizes the toroidal geometry.

Master equation approach to folding kinetics of lattice polymers based on conformation networks.

Based on the master equation with the inherent structure of conformation network, the authors investigate some important issues in the folding kinetics of lattice polymers. First, the topologies of conformation networks are discussed. Moreover, a new scheme of implementing Metropolis algorithm, which fulfills the condition of detailed balance, is proposed. Then, upon incorporating this new scheme into the geometric structure of conformation network the authors provide a theorem which can be used to place an upper bound on relaxation time. To effectively identify the kinetic traps of folding, the authors also introduce a new quantity, which is employed from the continuous time Monte Carlo method, called rigidity factor. Throughout the discussions, the authors analyze the results for different move sets to demonstrate the methods and to study the features of the kinetics of folding.

Partition functions and finite-size scalings of Ising model on helical tori.

The exact closed forms of the partition functions of a two-dimensional Ising model on square lattices with twisted boundary conditions are given. The constructions of helical tori are unambiguously related to the twisted boundary conditions by virtue of the SL(2, Z) transforms. The numerical analyses on the deviations of the specific-heat peaks away from the bulk critical temperature reveal that the finite-size effect of herical tori is independent of the chirality.

Phase distribution and phase correlation of financial time series.

The scaling, phase distribution, and phase correlation of financial time series are investigated based on the Dow Jones Industry Average and NASDAQ 10-min intraday data for a period from 1 Aug. 1997 to 31 Dec. 2003. The returns of the two indices are shown to have nice scaling behaviors and belong to stable distributions according to the criterion of Lévy's alpha stable distribution condition. An approach catching characteristic features of financial time series based on the concept of instantaneous phase is further proposed to study the phase distribution and correlation. Analysis of the phase distribution concludes that return time series fall into a class which is different from other nonstationary time series. The correlation between returns of the two indices probed by the distribution of phase difference indicates that there was a remarkable change of trading activities after the event of the 9/11 attack, and this change persisted in later trading activities.

Distribution and density of the partition function zeros for the diamond-decorated Ising model.

Exact renormalization map of temperature between two successive decorated lattices is given, and the distribution of the partition function zeros in the complex temperature plane is obtained for any decoration level. The rule governing the variation of the distribution pattern as the decoration level changes is given. The densities of the zeros for the first two decoration levels are calculated explicitly, and the qualitative features about the densities of higher decoration levels are given by conjecture. The Julia set associated with the renormalization map is contained in the distribution of the zeros in the limit of infinite decoration level, and the formation of the Julia set in the course of increasing the decoration level is given in terms of the variations of the zero density.

Evolution and structure formation of the distribution of partition function zeros: triangular type Ising lattices with cell decoration.

The distribution of partition function zeros of the two-dimensional Ising model in the complex temperature plane is studied within the context of triangular decorated lattices and their triangle-star transformations. Exact recursion relations for the zeros are deduced for the description of the evolution of the distribution of the zeros subject to the change of decoration level. In the limit of infinite decoration level, the decorated lattices essentially possess the Sierpinski gasket or its triangle-star transformation as the inherent structure. The positions of the zeros for the infinite decorated lattices are shown to coincide with the ones for the Sierpinski gasket or its triangle-star transformation, and the distributions of zeros all appear to be a union of infinite scattered points and a Jordan curve, which is the limit of the scattered points.