On the critical exponents of random k‐SAT

There has been much recent interest in the satisfiability of random Boolean formulas. A random k‐SAT formula is the conjunction of m random clauses, each of which is the disjunction of k literals (a variable or its negation). It is known that when the number of variables n is large, there is a sharp transition from satisfiability to unsatisfiability; in the case of 2‐SAT this happens when m/n → 1, for 3‐SAT the critical ratio is thought to be m/n ≈ 4.2. The sharpness of this transition is characterized by a critical exponent, sometimes called ν = ν_k (the smaller the value of ν the sharper the transition). Experiments have suggested that ν₃ = 1.5 ± 0.1. ν₄ = 1.25 ± 0.05, ν₅ = 1.1 ± 0.05, ν₆ = 1.05 ± 0.05, and heuristics have suggested that ν_k → 1 as k → ∞. We give here a simple proof that each of these exponents is at least 2 (provided the exponent is well defined). This result holds for each of the three standard ensembles of random k‐SAT formulas: m clauses selected uniformly at random without replacement, m clauses selected uniformly at random with replacement, and each clause selected with probability p independent of the other clauses. We also obtain similar results for q‐colorability and the appearance of a q‐core in a random graph. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 182–195, 2002     

8 mm回旋行波管和返波管的三折螺旋波纹波导色散分析

 三折螺旋波纹波导使TE11模和TE21模相互耦合，其色散曲线能够在宽频带内与电子回旋共振，因此需要对色散方程进行研究。提出了适合工程实用的行波模式和返波模式的色散方程，并对方程中的耦合系数进行了简化，误差在1%左右。利用CST软件的VBA语言对螺旋波纹波导进行建模和计算，根据模拟得到的传输特性曲线的特点，提出一种模拟方法，将模拟与理论计算得到的色散曲线作对比，误差在5%左右。     

Analysis of two-sample truncated data using generalized logistic models

Parallel to Cox's [JRSS B34 (1972) 187-230] proportional hazards model, generalized logistic models have been discussed by Anderson [Bull. Int. Statist. Inst. 48 (1979) 35-53] and others. The essential assumption is that the two densities ratio has a known parametric form. A nice property of this model is that it naturally relates to the logistic regression model for categorical data. In astronomic, demographic, epidemiological, and other studies the variable of interest is often truncated by an associated variable. This paper studies generalized logistic models for the two-sample truncated data problem, where the two lifetime densities ratio is assumed to have the form exp{α+φ(x;β)}. Here φ is a known function of x and β, and the baseline density is unspecified. We develop a semiparametric maximum likelihood method for the case where the two samples have a common truncation distribution. It is shown that inferences for β do not depend the nonparametric components. We also derive an iterative algorithm to maximize the semiparametric likelihood for the general case where different truncation distributions are allowed. We further discuss how to check goodness of fit of the generalized logistic model. The developed methods are illustrated and evaluated using both simulated and real data.     

A triangular map with homoclinic orbits and no infinite ω-limit set containing periodic points

Recently, Forti, Paganoni and Smítal constructed an example of a triangular map of the unite square, F(x,y)=(f(x),g(x,y)), possessing periodic orbits of all periods and such that no infinite ω-limit set of F contains a periodic point. In this note we show that the above quoted map F has a homoclinic orbit. As a consequence, we answer in the negative the problem presented by A.N. Sharkovsky in the eighties whether, for a triangular map of the square, existence of a homoclinic orbit implies the existence of an infinite ω-limit set containing a periodic point. It is well known that, for a continuous map of the interval, the answer is positive.     

Stability of steady nonisothermal flow in a vertical layer with permeable boundaries in the microconvection model

The problem of the stability of steady convective viscous incompressible fluid flow in a vertical layer with boundaries at different temperatures is considered in the presence of transverse blowing through the layer. The complete spectral problem is solved for a silicon melt. The neutral curve is constructed and the critical Grashof number is found. The numerical calculations show that the presence of transverse blowing significantly affects the flow stability. As compared with the Oberbeck-Boussinesq model, in the microconvection model the instability develops at lower wavenumbers.     

双面加热环形通道内干涸点的确定

根据环形管通道内流体流动和换热的特点,以Kirillov和Smogalev提出的干涸点理论模型假设为基础,从最基本的质量守恒方程出发,并引入临界液膜厚度等相应的辅助模型,得到了双面加热环形通道内流动沸腾干涸点的理论模型。同时针对间隙为1.0mm和1.5mm的环形窄缝进行了低压低质量流速工况下干涸点的实验研究。比较发现理论模型预测值与实验结果基本相符。说明本文提出的理论模型适用于低压低流量条件下的窄环形通道。实验同时发现:环状流临界热流密度在系统压力为2.2MPa达到最大值,临界含汽量随质量流速的增大呈缓慢下降趋势。     

Liquid–gas critical phenomena under confinement: small-angle neutron scattering studies of CO2 in aerogel

Small angle neutron scattering (SANS) is a well-established technique for investigating the behavior of confined binary liquid solutions, as it can probe the correlation length and susceptibility in pores on length scales 1 – 100 nm. We applied SANS to explore the influence of confinement on critical behavior of an individual fluid carbon dioxide (CO₂) in a highly porous aerogel. The results demonstrate that quenched disorder induced by aerogel significantly depresses density fluctuations. Despite the negligible volume occupied by aerogel (< 4%), the macroscopic phase separation of confined CO₂ into coexisting liquid and gaseous phases is suppressed and below the critical temperature of the bulk fluid frozen methastable microdomains are formed. Experimental data show that critical adsorption is as important as the effect of confinement in defining the behavior of confined fluids.     

广义Sierpinski集

本文考察了包括平面上的各种广义 Cantor集 ,Sierpinski集和包括某些连续不可微曲线在内的广义 Sierpinski集 .由相似变换 ,导出了它们的级数表达式 ,并利用它和字符串空间的对应关系 ,计算出它们的Hausdorff维数     

一类混合型微分差分方程的周期解

利用Fenchel变换,我们推出一类微分差分方程存在周期解等价于某泛函具有临界点,并求出方程具有周期解的充分条件.