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     

Sobolev-Hardy不等式与临界双重调和问题

该文讨论一类带有奇异系数的双重调和方程{△^2u-μu/|x|^s=f(x,u),x∈Ω,u=δu/δv=0,x∈δΩ，这里Ω包含R^N是包含0的有界光滑区域，u∈H0^2(Ω），μ∈R是参数，0≤s≤2,△^2=△△表示双重拉普拉斯算子，当f(x,u)=u^p,p=2N/N-4时，上述问题就是一个临界双重调和问题，该文运用Sobolev-Hardy不等式和变分方法，得到它的解的存在性的一些结果。     

具有控制系数的高阶线性微分方程复振荡的一个结果

该文考虑具有控制系数 A\-0 和系数仅有有限个极点的高阶线性齐次微分方程（1.1）。得到了一个复振荡结果，该结果是J. K. Langley［11］等作者在整系数下相应结果的推广。     

An inner product lemma

Given the operator product BA in which both A and B are symmetric positive‐definite operators, for which symmetric positive‐definite operators C is BA symmetric positive‐definite in the C inner product 〈x, y〉C? This question arises naturally in preconditioned iterative solution methods, and will be answered completely here. Copyright © 2004 John Wiley & Sons, Ltd.     

稀疏过程在破产问题中的应用

本讨论一类人寿保险的风险过程，其中保单到达服从齐次Poisson过程。而描述退保及索赔发生的计数过程分别为这一过程的q-稀疏与p-稀疏．对此模型给出其破产概率的具体上界，并与其它一类风险模型进行比较．     

Critical Behavior of Two Interacting Linear Polymer Chains in a Good Solvent

A model of two interacting (chemically different) linear polymer chains is solved exactly using the real-space renormalization group transformation on a family of Sierpinski gasket type fractals and on a truncated 4-simplex lattice. The members of the family of the Sierpinski gasket-type fractals are characterized by an integer scale factorb which runs from 2 to ∞. The Hausdorff dimensiond _F of these fractals tends to 2 from below asb → ∞. We calculate the contact exponenty for the transition from the State of segregation to a State in which the two chains are entangled forb = 2-5. Using arguments based on the finite-size scaling theory, we show that forb→∞, y = 2 - v(b) d _F, wherev is the end-toend distance exponent of a chain. For a truncated 4-simplex lattice it is shown that the system of two chains either remains in a State in which these chains are intermingled in such a way that they cannot be told apart, in the sense that the chemical difference between the polymer chains completely drop out of the thermodynamics of the system, or in a State in which they are either zipped or entangled. We show the region of existence of these different phases separated by tricritical lines. The value of the contact exponenty is calculated at the tricritical points.     

环面上具有间断梯度的势函数的模拟退火

本文证明了环面上具有间断梯度的势函数的模拟退火过程：dXt=-VU(Xt)dt √2dWt概率收敛到势函数的全局极小集附近。     

Exceptional set of a representation with fractional powers

We prove that for any given c, 1 < c < 17/11, almost all natural numbers are representable in the form [x ^c] + [p ^c], where x is a natural number and p is a prime.     

Sobolev-Type Classes of Functions with Values in a Metric Space. II

We prove equivalence of the definitions by the author and by Korevaar and Schoen of the Sobolev classes of mappings of a domain of an arithmetic n-dimensional space to a metric space.     

PV型HgCdTe光电探测器中的混沌及其诊断

 对激光辐照半导体光电探测器的实验研究，发现当一束不太强的稳定连续波激光照射在PV型HgCdTe光电探测器上时，可以引起探测器出现混沌行为。利用光电探测器的光电压信号随时间变化的实验数据，通过求功率谱、计算Lyapunov指数对混沌进行了诊断。