Multiple phase transitions in the generalized Curie-Weiss model

The generalized Curie-Weiss model is an extension of the classical Curie-Weiss model in which the quadratic interaction function of the mean spin value is replaced by a more general interaction function. It is shown that the generalized Curie-Weiss model can have a sequence of phase transitions at different critical temperatures. Both first-order and second-order phase transitions can occur, and explicit criteria for the two types are given. Three examples of generalized Curie-Weiss models are worked out in detail, including one example with infinitely many phase transitions. A number of results are derived using large-deviation techniques.     

Positive solutions of second-order differential systems with nonhomogeneous boundary conditions

This paper is to investigate the positive solutions of the systems of second-order ordinary differential equations with nonhomogeneous multi-point boundary conditions. By the lower and upper solutions method, Schauder fixed point theorem and fixed point index theory, under certain conditions, it is proved that there exist appropriate regions of parameters in which the problem has at least two, at least one or no positive solution.     

三级相互作用对聚亚甲基链末端距四次矩的影响

本文用三级相互作用近似理论导出了高分子链末端距的四次矩的计算公式。计算聚亚甲基链的末端距四次矩,并同二级相互作用的结果进行了比较。对于聚亚甲基链,三级相互作用近似的四次矩小于二级相互作用近似的四次矩。     

Design and synthesis of interpenetrating polymer networks for second-order nonlinear optics

There has been a tremendous recent interest in the development of second-order nonlinear optical (NLO) polymeric materials for photonic applications. However, a major drawback of second-order NLO polymers that prevents them from being used in device applications is the instability of their electric field induced dipolar alignment. The randomization of the dipole orientation leads to the decay of second-order optical nonlinearities. Numerous efforts have been made to increase the stability of the second-order NLO properties of polymers. The search for new approaches to develop NLO polymers with optimal properties has been an active research area since the past decade. A novel approach, combining the hybrid properties of high glass transition temperatures, extensively extensively crosslinked networks and permanent entanglements, based on interpenetrating polymer networks (IPN) is introduced to develop stable second-order NLO materials. Two types of IPN systems are prepared and their properties are investigated. The designing criteria and the rationale for the selection of polymers are discussed. The IPN samples show excellent temporal stability at elevated temperatures. Long-term stability of the optical nonlinearity at 100°C has been observed in these materials. Temporal stability of the NLO properties of these IPNs is synergistically enhanced. Relaxation behavior of the optical nonlinearity of an IPN system has been studied and compared with that of a typical guest/host system. The improved temporal stability of the second-order NLO properties of this IPN system is a result of the combination of the high rigidity of the polymer backbones, crosslinked matrices and permanent entanglements of the polymer networks. A slight modification of the chemical structure resulted in an improvement of the optical quality of the sample.     

Solution thermodynamics near the liquid–liquid critical point. II. Excess second-order derivatives

We present a comprehensive study of the behaviour of excess second-order derivatives of binary mixtures near the liquid–liquid critical point. Specifically, excess (isobaric and isochoric) molar heat capacities ( and ), excess (isothermal and isentropic) compressibilities ( and ), and excess isobaric thermal expansivities () have been determined over the whole composition (x) range at atmospheric pressure and in the homogeneous region within (293.15–323.15) K. Results are consistent with the predictions of the current theory of critical phenomena. Remarkably, anomalous and curves in the critical region are observed. Such anomalies, which are reported here for the first time, are of significant size for and very mild in the case of , thereby indicating that volumetric effects at near-liquid–liquid criticality are, as expected, very small.     

洛汾碱衍生物的电子结构及光谱性质的理论研究

采用密度泛函理论(DFT)的B3LYP/6-31G*方法,对4种洛汾碱类化合物的几何构型进行了优化,在此基础上计算分子的电子结构,并结合有限场FF方法研究了二阶非线性光学(NLO)性质.用含时密度泛函理论(TD-DFT)对上述化合物分子进行吸收光谱的研究.研究表明在4,5-二-苯基-2-对甲酰苯基咪唑生色团中4,5苯环上引入硝基和3位N原子引入苄基改变分子的共轭平面,使二阶非线性极化率总有效值(βtot)减小,吸收峰总体蓝移.同时还发现,在CH2Cl2溶剂中a和c分子的λmax主要来源于HOMO→LUMO的π一π*跃迁,b和d分子的λmax主要来源于HOMO→LUMO+2的π→π*跃迁.     

Convergence of the Augmented Lagrangian Method for Nonlinear Optimization Problems over Second-Order Cones

The paper analyzes the rate of local convergence of the augmented Lagrangian method for nonlinear second-order cone optimization problems. Under the constraint nondegeneracy condition and the strong second order sufficient condition, we demonstrate that the sequence of iterate points generated by the augmented Lagrangian method locally converges to a local minimizer at a linear rate, whose ratio constant is proportional to 1/τ with penalty parameter τ not less than a threshold . Importantly and interestingly enough, the analysis does not require the strict complementarity condition. Supported by the National Natural Science Foundation of China under Project 10771026 and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.     

Error Estimates for Semi-Discrete and Fully Discrete Galerkin Finite Element Approximations of the General Linear Second-Order Hyperbolic Equation

In this article, we derive error estimates for the semi-discrete and fully discrete Galerkin approximations of a general linear second-order hyperbolic partial differential equation with general damping (which includes boundary damping). The results can be applied to a variety of cases (e.g. vibrating systems of linked elastic bodies). The results generalize pioneering work of Dupont and complement a recent article by Basson and Van Rensburg.