A general vectorial Ekeland’s variational principle with a P-distance

In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle （in short EVP）, where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space （which is a uniform space） has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi＇s fixed point theorem and a general vectorial Takahashi＇s nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results.     

广义Ekeland变分原理的应用

研究了广义Ekeland变分原理在拟度量空间中的一些重要应用.利用广义Ekeland变分原理证明了函数f满足关于α的Takahashi ε-条件当且仅当f满足关于相同α的Hamel ε-条件.此外,利用关于α的Takahashi ε-条件得到了一些重要结论.     

An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures

We study hypersurfaces in Euclidean space whose position vector x satisfies the condition L _k x = Ax + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed , is a constant matrix and is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form , with . This extends a previous classification for hypersurfaces in satisfying , where is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A 53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].      

Weight of the comprehension axiom in a theory based on logic without contractions

A set-theoretic LST system based on a logic without rules of contraction of identical formulas in the antecedent or succedent of sequents is introduced. The set-theoretic axioms used are the comprehension principle,

多腔薄膜梳状滤波器中心波长温度稳定性分析

利用高桥模型详细分析了多腔薄膜梳状滤波器的中心波长温度稳定性问题。发现由于多腔薄膜梳状滤波器的固体腔较厚而反射镜膜系的厚度相对较薄,因此膜系与固体腔之间的温度引起的弹性应力导致的膜系光学常数的变化已经不是主要影响因素,而固体腔材料的折射率温度系数引起的中心波长漂移对整个器件的影响较大。计算表明对于一个密集波分复用中使用的100 G信道间隔的三腔薄膜梳状滤波器而言,当固体腔材料(熔融石英)的折射率温度系数为1×10-5/℃时,器件的中心波长漂移在-40 ℃~85 ℃的温度范围内可达±1 nm。     

P-distances, q-distances and a generalized Ekeland’s variational principle in uniform spaces

In this paper, we attempt to give a unified approach to the existing several versions of Ekeland’s variational principle. In the framework of uniform spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance (particularly, a p-distance), which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized Takahashi’s nonconvex minimization theorem, a generalized Ekeland’s variational principle and a generalized Caristi’s fixed point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland’s variational principle, we deduce a number of particular versions of Ekeland’s principle, which include many known versions of the principle and their improvements.     

波分复用薄膜带通滤光片的中心波长温度稳定性

讨论了影响波分复用薄膜滤光片中心波长漂移的因素 ,着重分析了滤光片的温度稳定性。根据实验结果 ,借助于高桥模型 ,分析计算了滤光片的折射率温度系数、线膨胀系数和泊松比以及它们对温度漂移的影响。得到了Ta2 O5/SiO2 滤光片薄膜的折射率温度系数、线膨胀系数、泊松比分别为 1× 10 -5℃-1,5× 10 -7℃-1和 0 12 ,指出了这三个参量是影响温度稳定性最重要的因素 ,特别是薄膜的折射率温度系数。对特定的基板热膨胀系数 ,通过调节滤光片的干涉级次和间隔层材料 ,可望得到零温度漂移的稳定滤光片。     

On the equivalence of the Mizoguchi–Takahashi fixed point theorem to Nadler’s theorem

In this work, we prove that on a metrically convex complete metric space, the Mizoguchi–Takahashi theorem is equivalent to Nadler’s theorem. Also, we obtain its equivalence on a compact metric space.     

On Ha＇s Version of Set-valued Ekeland＇s Variational Principle

By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124, 187–206 (2005)] established a new version of Ekeland’s variational principle for set-valued maps, which is expressed by the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement of Ha’s version of set-valued Ekeland’s variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev theorem. From the improved Ha’s version, we deduce a Caristi-Kirk’s fixed point theorem and a Takahashi’s nonconvex minimization theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other.     

Current-conserving nonlinear response theory in driven systems

Employing the closed-time path 2PI effective action (CTP 2PI EA) approach, we study the response of an open interacting electronic system to time-dependent external electromagnetic fields. We show that the 2PI EA provides a systematic way of calculating the propagator and response functions of the system. Due to the invariance of the 2PI EA under external gauge transformations, the response functions calculated from it are such that the Ward–Takahashi hierarchy, which ensures current conservation beyond the expectation value level, is satisfied. These findings may be useful in the study of interacting electronic pumping devices, and serve to clarify the connection between current conservation (beyond the mean value level) and real-time nonlinear response theory.