Continuity properties of the normal cone to the level sets of a quasiconvex function

Two main properties of the subgradient mapping of convex functions are transposed for quasiconvex ones. The continuity of the functionxf(x)^–1f(x) on the domain where it is defined is deduced from some continuity properties of the normal coneN to the level sets of the quasiconvex functionf. We also prove that, under a pseudoconvexity-type condition, the normal coneN(x) to the set {x:f(x)f(x)} can be expressed as the convex hull of the limits of type {N(x _n)}, where {x _n} is a sequence converging tox and contained in a dense subsetD. In particular, whenf is pseudoconvex,D can be taken equal to the set of points wheref is differentiable.This research was completed while the second author was on a sabbatical leave at the University of Montreal and was supported by a NSERC grant. It has its origin in the doctoral thesis of the first author (Ref. 1), prepared under the direction of the second author.The authors are grateful to an anonymous referee and C. Zalinescu for their helpful remarks on a previous version of this paper.     

水准折光及双波长激光效应

本文从“同温度层不完全平行于地面,且在变化”的“任意分层”假设出发,导出了水准折光修正公式,并利用光的色散效应在10~(-6)的精度要求下,求得r=r_T=h_1/h_2=(n_(01)-1)/(n_(02)-1)=常数最后指出,研制双波长激光水准仪的必要性和可能性,并提出了研制此种仪器的技术参数.     

红外镜反射光谱的测试与解析

红外镜反射（infrared specular reflection，简称SR）光谱与被测样品和衬底的折射指数、吸收指数以及入射角有密切关系，往往由之产生畸变。要从红外镜反射技术测得的光谱处理得到人们熟悉的吸收光谱，需要经过较复杂的计算机数学处理。本文通过几个比较特殊的镜反射例子，以说明镜反射光谱的测试和解析必须注意多种影响因素。     

双核配合物[Cu2(TS)(H2O)]·CH3OH的自组装制备和晶体结构

The binuclear complex [Cu₂(TS)(H₂O)]·CH₃OH, where TS denotes the binucleating ligand derived from the schiff base N,N′-bis(3-carboxylsalidene)trimethylenediamine, was obtained from the self-decomposition of mononuclear complex Na₂CuTS·H₂O. It crystallizes in the monoclinic system, space group P2₁/c. The lattice parameters are a=11.935(2)?,b=15.667(3)?,c=11.621(2)?,β=111.60(3)°,V=2020.2(7)?³, Z=4 with R=0.0467. The structure is made of binuclear units, in which two copper atoms are bridged by two phenolic oxygen atoms. The “inside”copper atom is coordinated by two nitrogens, two phenolic oxygens in a planar coordi-nation site, the copper atom deviates from the mean plane by 0.47?. The “outside” copper atom is five coordinated by two phenolic oxygen atoms、two equatorial carboxyl oxygen atoms and one axial water oxygen atom in a distorted tetragonal cone site, the copper atom is pulled out of the equatorial plane by 0.46?.     

Excess thermodynamic properties of the 2-propanol + dichloromethane system at 25°C

Densities, viscosities, enthalpies, vapor-liquid equilibria, and surface tensions were determined at 25°C for the 2-propanol+dichloromethane system. From the experimental results excess volumes, viscosities, enthalpies, Gibbs energies, and excess surface tensions were calculated. An attempt has been made to explain the observed deviations from ideal behavior on the basis of intermolecular interactions.     

改进的蝗虫优化算法在双目标应急物资中心选址问题中的应用

近年来世界各地频发灾情疫情等紧急事件,严重影响人民的生活物资保障。在这种情况下,急需建立应急物资中心来缓解燃眉之急。该类问题通常面临资源稀缺并且时间相对紧迫的处境,因此需要在短时间内获得合理的应急设施选址方案来提升服务的质量和效率。本文对应急物资中心选址问题展开研究,提出一种考虑后续运输成本以及有概率发生紧急事件而导致无法正常运送物资的双目标离散选址模型,并为此设计一种二进制多目标蝗虫优化算法。该算法采用模糊关联熵系数来引导迭代更新,同时为其添加外部档案,最优解选择机制和竞争决策机制来提升算法性能。多次数值实验表明该算法的计算效率和求解质量较高,可作为应急物资中心选址问题的一种可行且有效的算法。     

Convergence of an augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints

An augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints is proposed based on a Löwner operator associated with a potential function for the optimization problems with inequality constraints. The favorable properties of both the Löwner operator and the corresponding augmented Lagrangian are discussed. And under some mild assumptions, the rate of convergence of the augmented Lagrange algorithm is studied in detail.     

New Results on Generalized $c$-Distance Without Continuity in Cone $b$-Metric Spaces over Banach Algebras

In this work, some new fixed point results for generalized Lipschitz mappings on generalized $c$-distance in cone $b$-metric spaces over Banach algebras are obtained, not acquiring the condition that the underlying cone should be normal or the mappings should be continuous. Furthermore, the existence and the uniqueness of the fixed point are proven for such mappings. These results greatly improve and generalize several well-known comparable results in the literature. Moreover, some examples and an application are given to support our new results.     

Regularity of Viscosity Solutions of the Biased Infinity Laplacian Equation

In this paper, we are interested in the regularity estimates of the nonnegative viscosity super solution of the $β$−biased infinity Laplacian equation $$∆^β_∞u = 0,$$ where $β ∈ \mathbb{R}$ is a fixed constant and $∆^β_∞u := ∆^N_∞u + β|Du|,$ which arises from the random game named biased tug-of-war. By studying directly the $β$−biased infinity Laplacian equation, we construct the appropriate exponential cones as barrier functions to establish a key estimate. Based on this estimate, we obtain the Harnack inequality, Hopf boundary point lemma, Lipschitz estimate and the Liouville property etc.     

Gaussian,non-Gaussian critical fluctuations in the Curie-Weiss model

It is known that at the critical temperature the Curie-Weiss mean-field model has non-Gaussian fluctuations and that internal fluctuations can be Gaussian. Here we compute the distribution of theq-mode magnetization fluctuations as a function of the temperature, the wave vectorq, and a fading out external field. We obtain new classes of probability distributions generated by this external field as well as new critical behavior in terms of its rate of fading out. We discuss also the susceptibility as the limitq tending to zero.