Random-Cluster Dynamics in Z2\mathbb{Z}^2

The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and electrical networks, but its dynamics have so far largely resisted analysis. In this paper we analyze the Glauber dynamics of the random-cluster model in the canonical case where the underlying graph is an $n \times n$ box in the Cartesian lattice $\mathbb{Z}^2$. Our main result is a $O(n^2\log n)$ upper bound for the mixing time at all values of the model parameter $p$ except the critical point $p=p_c(q)$, and for all values of the second model parameter $q\ge 1$. We also provide a matching lower bound proving that our result is tight. Our analysis takes as its starting point the recent breakthrough by Beffara and Duminil-Copin on the location of the random-cluster phase transition in $\mathbb{Z}^2$. It is reminiscent of similar results for spin systems such as the Ising and Potts models, but requires the reworking of several standard tools in the context of the random-cluster model, which is not a spin system in the usual sense

The Sol-Gel Process Simulated by Cluster-Cluster Aggregation

The pair-correlation function $g(r,t)$ and its Fourier transform, the structure factor $S(q,t)$, are computed during the gelation process of identical spherical particles using the diffusion-limited cluster-cluster aggregation model in a box. This numerical analysis shows that the time evolution of the characteristic cluster size $\xi$ exhibits a crossover close to the gel time $t_g$ which depends on the volumic fraction $c$. In this model $t_g$ tends to infinity when the box size $L$ tends to infinity. For systems of finite size, it is shown numerically that, when $t<t_g$, the wave vector $q_m$, at which $S(q,t)$ has a maximum, decreases as $S(q_m,t)^{-1/D}$, where $D$ is an apparent fractal dimension of clusters, as measured from the slo pe of $S(q,t)$ . The time evolution of the mean number of particles per cluster $\bar {n}$ is also investigated. Our numerical results are in qualitative agreement with small angle scattering experiments in several systems.Comment: RevTex, 13 pages + 9 postscript figures appended using "uufiles". To appear in J. of Non-Cryst. Solid

On the Effects of Subvirial Initial Conditions and the Birth Temperature of R136

We investigate the effect of different initial virial temperatures, Q, on the dynamics of star clusters. We find that the virial temperature has a strong effect on many aspects of the resulting system, including among others: the fraction of bodies escaping from the system, the depth of the collapse of the system, and the strength of the mass segregation. These differences deem the practice of using "cold" initial conditions no longer a simple choice of convenience. The choice of initial virial temperature must be carefully considered as its impact on the remainder of the simulation can be profound. We discuss the pitfalls and aim to describe the general behavior of the collapse and the resultant system as a function of the virial temperature so that a well reasoned choice of initial virial temperature can be made. We make a correction to the previous theoretical estimate for the minimum radius, $R_{min}$, of the cluster at the deepest moment of collapse to include a Q dependency, $R_{min}\approx Q + N^{(-1/3)}$, where $N$ is the number of particles. We use our numerical results to infer more about the initial conditions of the young cluster R136. Based on our analysis, we find that R136 was likely formed with a rather cool, but not cold, initial virial temperature ($Q\approx 0.13$). Using the same analysis method, we examined 15 other young clusters and found the most common initial virial temperature to be between 0.18 and 0.25.Comment: Accepted for publication in MNRA

Developing R software for simultaneous estimation of Q- and R-mode Factor Analyses using spatial and non spatial data

Simultaneous use of R- and Q-mode Factor Analysis is a powerful similarity measurement among and between variables and objects of a continuous data, but its availability is lacking in R statistical software environment. I have developed a new R package called qrfactor that can perform Factor Analysis on spatial and non spatial data. The package contains one function called qrfactor() that can perform various versions of Factor Analyses such as PCA, R-mode Factor Analysis, Q-mode Factor Analysis, Simultaneous R- and Q-mode Factor Analysis, Principal Coordinate Analysis, as wells as Multidimensional Scaling (MDS) and cluster analysis. The qrfactor() function returns values such as eigenvalues, eigenvectors, loadings, scores, and indices. Unlike other R package factor analysis functions, plot.qrfactor() offers several annotated biplots for all possible combinations of eigenvectors, loadings, and scores as well as the possibility of plotting about 60 maps in gray and full colour scales. The empirical and Eckhart–Young theorem evaluations show that ‘qrfactor’ package is mathematically correct in estimation of simultaneous R-and Q-mode Factor Analysis. The results are also in agreement with the results of other classical statistical functions and packages. Using one function to estimate various dimensions of factor analyses reduces the learning curve in R environment. Keywords: GIS, qrfactor, loadings, Multi-dimensional, R package, Factor scores, Cluster Analysis, Eckhart–Young, map

Long-Range Dependence in Financial Markets: a Moving Average Cluster Entropy Approach

A perspective is taken on the intangible complexity of economic and social systems by investigating the underlying dynamical processes that produce, store and transmit information in financial time series in terms of the \textit{moving average cluster entropy}. An extensive analysis has evidenced market and horizon dependence of the \textit{moving average cluster entropy} in real world financial assets. The origin of the behavior is scrutinized by applying the \textit{moving average cluster entropy} approach to long-range correlated stochastic processes as the Autoregressive Fractionally Integrated Moving Average (ARFIMA) and Fractional Brownian motion (FBM). To that end, an extensive set of series is generated with a broad range of values of the Hurst exponent $H$ and of the autoregressive, differencing and moving average parameters $p,d,q$. A systematic relation between \textit{moving average cluster entropy}, \textit{Market Dynamic Index} and long-range correlation parameters $H$, $d$ is observed. This study shows that the characteristic behaviour exhibited by the horizon dependence of the cluster entropy is related to long-range positive correlation in financial markets. Specifically, long range positively correlated ARFIMA processes with differencing parameter $d\simeq 0.05$, $d\simeq 0.15$ and $d\simeq 0.25$ are consistent with \textit{moving average cluster entropy} results obtained in time series of DJIA, S\&P500 and NASDAQ

Folding of the triangular lattice in a discrete three-dimensional space: Crumpling transitions in the negative-bending-rigidity regime

Folding of the triangular lattice in a discrete three-dimensional space is studied numerically. Such ``discrete folding'' was introduced by Bowick and co-workers as a simplified version of the polymerized membrane in thermal equilibrium. According to their cluster-variation method (CVM) analysis, there appear various types of phases as the bending rigidity K changes in the range -infty < K < infty. In this paper, we investigate the K<0 regime, for which the CVM analysis with the single-hexagon-cluster approximation predicts two types of (crumpling) transitions of both continuous and discontinuous characters. We diagonalized the transfer matrix for the strip widths up to L=26 with the aid of the density-matrix renormalization group. Thereby, we found that discontinuous transitions occur successively at K=-0.76(1) and -0.32(1). Actually, these transitions are accompanied with distinct hysteresis effects. On the contrary, the latent-heat releases are suppressed considerably as Q=0.03(2) and 0.04(2) for respective transitions. These results indicate that the singularity of crumpling transition can turn into a weak-first-order type by appreciating the fluctuations beyond a meanfield level

Heterogeneity issues in the meta-analysis of cluster randomization trials.

An increasing number of systematic reviews summarize results from cluster randomization trials. Applying existing meta-analysis methods to such trials is problematic because responses of subjects within clusters are likely correlated. The aim of this thesis is to evaluate heterogeneity in the context of fixed effects models providing guidance for conducting a meta-analysis of such trials. The approaches include the adjusted Q statistic, adjusted heterogeneity variance estimators and their corresponding confidence intervals and adjusted measures of heterogeneity and their corresponding confidence intervals. Attention is limited to meta-analyses of completely randomized trials having a binary outcome. An analytic expression for power of Q test is derived, which may be useful in planning a meta-analysis. The Type I error and power for the Q statistic, bias and mean square errors for the estimators and the coverage, tail errors and interval width for the confidence interval methods are investigated using Monte Carlo simulation. Simulation results show that the adjusted Q statistic has a Type I error close to the nominal level of 0.05 as compared to the unadjusted Q statistic which has a highly inflated Type I error. Power estimated using the algebraic formula had similar results to empirical power. For the heterogeneity variance estimators, the iterative REML estimator consistently had little bias. However, the noniterative MVVC and DLVC estimators with relatively low bias may also be recommended for small and large heterogeneity, respectively. The Q profile confidence interval approach for heterogeneity variance had generally nominal coverage for large heterogeneity. The measures of heterogeneity had generally low bias for large number of trials. For confidence interval approaches, the MOVER consistently maintained nominal coverage for \u27low\u27 to \u27moderate\u27 heterogeneity. For the absence of heterogeneity, the approach based on the Q statistic is preferred. Data from four cluster randomization trials are used to illustrate methods of analysis