Geometric critical exponent inequalities for general random cluster models

A set of new critical exponent inequalities,d(1 –1 /)2 –, dv(1 – 1/), andd> 1, is proved for a general class of random cluster models, which includes (independent or dependent) percolations, lattice animals (with any interactions), and various stochastic cluster growth models. The inequalities imply that the critical phenomena in the models are inevitably not mean-field-like in the dimensions one, two, and three.The present work was reported at the 56th Statistical Mechanics Meeting (Rutgers, December 1986).     

Consistency of a linear system of inequalities

We show that a consistency check of a linear system of inequalitiesAxb reduces to check whetherQb0 for a certain matrixQ. It is a direct consequence of the Farkas-Minkowski theorem. Thus, when one has to check consistency for different values ofb, one need not run a numerical algorithm for eachb.On leave at the Electronics Research Laboratory of the University of California at Berkeley in a CNRS/NSF Exchange Program.     

Selecting parameters and limits for equipment operational qualification

While operational qualification (OQ) is a well-established term within equipment qualification, users of equipment often become unsure when it comes to implementation. The biggest problem is how to select procedures and acceptance criteria. Should these be the vendor's specifications or should the users define their own limits, and, if so, how? Should all instruments of the same type have the same values or should these be optimized for each individual instrument? This article will provide an overall strategy and specific examples for HPLC on how to select procedures and acceptance limits that are based on efficient use of resources, on practicality and on the intended use of the equipment.     

最小一乘稳健多元分析校正

本文论述最小一乘求解的多元分析校正算法，探讨了最小一乘较常规最小二乘法及其他隐健算法的优点。用计算机数值模拟及实际多组分光谱体系对方法进行了检验，展示了最小一乘法在分析化学计量学中实际应用的可行性。     

Formulation–composition map of a lecithin-based emulsion

Formulation–composition map is an interesting tool to predict the nature of an emulsion, stability, viscosity and nevertheless to decide the mixing protocol of its ingredients. Information based on optimum formulation (environmental conditions at which the affinity of an emulsifier for oil and for aqueous phase is same), which is depicted through hydrophilic–lipophilic deviation (HLD) concept, is necessary to make a formulation–composition map of an emulsion. In order to apply this concept in food emulsions, it is necessary to determine characteristic constants of each component of the system, i.e. the aqueous phase, the oil phase and the emulsifier at equilibrium. In this work formulation–composition map of a sunflower oil–water–lecithin system, based on the knowledge of phase behavior of lecithin at equilibrium and emulsification, was made. The shape of inversion line on formulation–composition map was not the classical stair type rather an almost vertical inversion line at water-fraction (f_w) near 0.20 was observed. It was supposed to be linked to the viscosity of oil phase which was 50 times the viscosity of aqueous phase. Additionally, emulsions were of oil-in-water (O/W) type for f_w higher than 0.20, but their viscosity and the drop size behavior with respect to salt concentration as formulation variable did not show the existence of transitional inversion line on formulation–composition map. Such map in advance can certainly facilitate the guidelines for dynamic emulsification.     

考虑订单取件时间和柔性时间窗的取送货车辆路径问题

研究了同城配送中考虑订单取货时间和柔性时间窗的取送货车辆路径问题,考虑同城配送中订单起终点,订单取货时间和订单配送的柔性时间窗,车容量限制等因素。首先构建以配送成本与超时惩罚成本之和最小化为目标的混合整数线性模型。其次,设计了含多种有效不等式及其对应分离算法的改进分支切割算法对该模型进行精确求解。最后通过实验测试分析了不等式的性能,验证了算法的有效性,实验表明适当的减少车辆数和增大装载能力能够有效的减少成本。     

考虑偏差值的中立型区间交叉效率研究

在不确定性环境下,当决策单元(DMU)的投入产出数据为区间数形式时,为解决决策单元之间既不是合作也不是竞争关系时的交叉评价问题,本文提出一种中立型区间交叉效率模型。从所有被评价者的角度出发解决评价权重的选取问题,以决策单元投入得分的平均偏差与产出得分的平均偏差之和最小化为目标,建立决策单元在最佳和最差两种生产状态下的中立型区间交叉效率模型。在本文提出的中立型模型视角下,DMU的投入得分平均偏差和产出得分平均偏差之和达到最小。算例结果表明该中立型区间交叉效率模型的有效性,解决了不确定性环境下的交叉评价问题,保证评价的客观公正,更加符合现实。     

Smoothing methods for convex inequalities and linear complementarity problems

A smooth approximationp (x, ) to the plus function max{x, 0} is obtained by integrating the sigmoid function 1/(1 + e^–x ), commonly used in neural networks. By means of this approximation, linear and convex inequalities are converted into smooth, convex unconstrained minimization problems, the solution of which approximates the solution of the original problem to a high degree of accuracy for sufficiently large. In the special case when a Slater constraint qualification is satisfied, an exact solution can be obtained for finite. Speedup over MINOS 5.4 was as high as 1142 times for linear inequalities of size 2000 × 1000, and 580 times for convex inequalities with 400 variables. Linear complementarity problems are converted into a system of smooth nonlinear equations and are solved by a quadratically convergent Newton method. For monotone LCPs with as many as 10 000 variables, the proposed approach was as much as 63 times faster than Lemke's method.This material is based on research supported by Air Force Office of Scientific Research Grant F49620-94-1-0036 and National Science Foundation Grants CCR-9101801 and CCR-9322479.     

Large deviation analysis of the single server queue

We establish the large deviation principle (LDP) for the virtual waiting time and queue length processes in the GI/GI/1 queue. The rate functions are found explicitly. As an application, we obtain the logarithmic asymptotics of the probabilities that the virtual waiting time and queue length exceed high levels at large times. Additional new results deal with the LDP for renewal processes and with the derivation of unconditional LDPs for conditional ones. Our approach applies in large deviations ideas and methods of weak convergence theory.This work was supported in part by AT&T Bell Labs.     

The covariance matrix of the Potts model: A random cluster analysis

We consider the covariance matrix,G ^mm =q ²<(_x,m);(_y,m)>, of thed-dimensionalq-states Potts model, rewriting it in the random cluster representation of Fortuin and Kasteleyn. In any of theq ordered phases, we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances, thereby attributing algebraic significance to these stochastic geometric quantities. We also show that the correlation length corresponding to the decay rate of one of the eigenvalues is the same as the inverse decay rate of the diameter of finite clusers. For dimensiond=2, we show that this correlation length and the correlation length of the two-point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two. For systems with first-order transitions, this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self-dual point _o. For systems with second order transitions, this relation implies the equality of the correlation length exponents from above and below threshold, as well as an amplitude ratio of two. In the course of proving the above results, we establish several properties of independent interest, including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions, and left continuity of the two-dimensional free boundary condition percolation probability at _o. We also introduce DLR equations for the random cluster model and use them to establish ergodicity of the free measure. In order to prove these results, we introduce a new class of events which we call decoupling events and two inequalities for these events. The first is similar to the FKG inequality, but holds for events which are neither increasing nor decreasing; the second is similar to the van den Berg-Kesten inequality in standard percolation. Both inequalities hold for an arbitrary FKG measure.