Critical exponents,hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponentsv and 2 ₄ – as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relationdv = 2 ₄ –. In two dimensions, we confirm the predicted exponentv=3/4 and the hyperscaling relation; we estimate the universal ratios <R _g ² >/<R _e ² >=0.14026±0.00007, <R _m ² >/<R _e ² >=0.43961±0.00034, and ^*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimatev=0.5877±0.0006 with a correctionto-scaling exponent ₁=0.56±0.03 (subjective 68% confidence limits). This value forv agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for ₁. Earlier Monte Carlo estimates ofv, which were 0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios <R _g ² >/<R _e ² >=0.1599±0.0002 and ^*=0.2471±0.0003; since ^*>0, hyperscaling holds. The approach to ^* is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. In an appendix, we prove rigorously (modulo some standard scaling assumptions) the hyperscaling relationdv = 2 ₄ – for two-dimensional SAWs.     

On position-space renormalization group approach to percolation

In a position-space renormalization group (PSRG) approach to percolation one calculates the probabilityR(p,b) that a finite lattice of linear sizeb percolates, wherep is the occupation probability of a site or bond. A sequence of percolation thresholdsp _c (b) is then estimated fromR(p _c ,b)=p _c (b) and extrapolated to the limitb to obtainp _c =p _c (). Recently, it was shown that for a certain spanning rule and boundary condition,R(p _c ,)=R _c is universal, and sincep _c is not universal, the validity of PSRG approaches was questioned. We suggest that the equationR(p _c ,b)=, where isany number in (0,1), provides a sequence ofp _c (b)'s thatalways converges top _c asb. Thus, there is anenvelope from any point inside of which one can converge top _c . However, the convergence is optimal if =R _c . By calculating the fractal dimension of the sample-spanning cluster atp _c , we show that the same is true aboutany critical exponent of percolation that is calculated by a PSRG method. Thus PSRG methods are still a useful tool for investigating percolation properties of disordered systems.     

Self-avoiding walks on random fractal environments

Self-avoiding random walks (SAWs) are studied on several hierarchical lattices in a randomly disordered environment. An analytical method to determine whether their fractal dimensionD _saw is affected by disorder is introduced. Using this method, it is found that for some lattices,D _saw is unaffected by weak disorder; while for othersD _saw changes even for infinitestimal disorder. A weak disorder exponent is defined and calculated analytically [ measures the dependence of the variance in the partition function (or in the effective fugacity per step)vL on the end-to-end distance of the SAW,L]. For lattices which are stable against weak disorder (<0) a phase transition exists at a critical valuev=v ^* which separates weak- and strong-disorder phases. The geometrical properties which contribute to the value of are discussed.     

CHAOS FOR MIXING TRANSFORMATIONS

ＣＨＡＯＳＦＯＲＭＩＸＩＮＧＴＲＡＮＳＦＯＲＭＡＴＩＯＮＳ￥ＬＩＡＯＧＯＮＧＦＵ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ,ＪｉｌｉｎＵｎｉｖｅｒｓｉｔｙ,Ｃｈａｎｇｃｈｕｎ１３００２３,Ｃｈｉｎａ．）Ａｂｓｔｒａｃｔ：Ｆｏｒａｃｌａｓｓｏｆｍ...     

Generalized Aluthge transformation on -hyponormal operators

We shall introduce a generalized Aluthge transformation on -
hyponormal operators and also, by using the Furuta inequality, we shall give several properties on this generalized Aluthge transformation as further extensions of some results of Aluthge.

     

不共轴光学系统自动优化中的有关问题

描述不共轴光学系统计算机优化中坐标变换与逆变换、“入瞳”与光线坐标、像面基准与质量函数定义的一致性问题。     

Explicit inertial range renormalization theory in a model for turbulent diffusion

The inertial range for a statistical turbulent velocity field consists of those scales that are larger than the dissipation scale but smaller than the integral scale. Here the complete scale-invariant explicit inertial range renormalization theory for all the higher-order statistics of a diffusing passive scalar is developed in a model which, despite its simplicity, involves turbulent diffusion by statistical velocity fields with arbitrarily many scales, infrared divergence, long-range spatial correlations, and rapid fluctuations in time-such velocity fields retain several characteristic features of those in fully developed turbulence. The main tool in the development of this explicit renormalization theory for the model is an exact quantum mechanical analogy which relates higher-order statistics of the diffusing scalar to the properties of solutions of a family ofN- body parabolic quantum problems. The canonical inertial range renormalized statistical fixed point is developed explicitly here as a function of the velocity spectral parameter, which measures the strength of the infrared divergence: for<2, mean-field behavior in the inertial range occurs with Gaussian statistical behavior for the scalar and standard diffusive scaling laws; for>2 a phase transition occurs to a fixed point with anomalous inertial range scaling laws and a non-Gaussian renormalized statistical fixed point. Several explicit connections between the renormalization theory in the model and intermediate asymptotics are developed explicitly as well as links between anomalous turbulent decay and explicit spectral properties of Schrödinger operators. The differences between this inertial range renormalization theory and the earlier theories for large-scale eddy diffusivity developed by Avellaneda and the author in such models are also discussed here.     

Renormalization group for Markov chains and application to metastability

In this paper we introduce a new renormalization group method for the study of the long-time behavior of Markov chains with finite state space and with transition probabilities exponentially small in an external parameter. A general approach of metastability problems emerges from this analysis and is discussed in detail in the case of a two-dimensional Ising system at low temperature,     

硼酸溶液中氯柱硼镁石的溶解及相转化动力学

对氯柱硼镁石（2MgO•2B2O3•MgCl2•14H2O）在4.5％H3BO3水溶液40 ℃的溶解及相转化过程进行了动力学研究.对不同时间取出的液相的化学分析及不同阶段分离固相的鉴定结果表明,该复盐在溶解阶段出现与前所报导不同的同步溶解，最终产物是库水硼镁石（2MgO•3B2O3•15H2O）.提出了库水硼镁石的形成条件和溶解及相转化动力学机理，利用单纯形优化法配合Runge-Kutta微分方程组数值解法对实验数据进行处理，给出了转化结晶动力学方程.     

Formation of sol-gel nanostructured mullite by additions of fluoride ion

Summary The subject of this study was to investigate the effect of fluoride ions addition on the temperature of sol gel mullite formation based on the hypotheses that the presence of fluoride ions can decrease the temperature of mullite formation (in respect to common 980&deg;C, in sol-gel processing). Polymeric sols were prepared by mixing TEOS and aluminum nitrate nanohydrate and by adding fluoride ions (from 2 to 5 mass&percnt;). DTA, TG, XRD and SEM were used for characterisation of mullite gel and crystalline mullite. The experimental results confirmed that the addition of fluoride ions decrease the temperature of mullite formation up to 890&deg;C for the fluorine concentration of 3.5 mass&percnt;. Experimental results showed that the temperature of mullite formation is not a simple function of the fluoride ion content. The mechanism of fluorine effect was discussed in terms of the gelling process, gel structure and the phase separation before the mullite formation.