Surface Tension Effect on Harmonics of Rayleigh-Taylor Instability

Using the method of the parameter expansion up to the third order, explicitly investigates surface tension effect on harmonics at weakly nonlinear stage in Rayleigh-Taylor instability (RTI) for arbitrary Atwood numbers and compares the results with those of classical RTI within the framework of the third-order weakly nonlinear theory. It is found that surface tension strongly reduces the linear growth rate of time, resulting in mild growth of the amplitude of the fundamental mode, and changes amplitudes of the second and third harmonics, as is expressed as a tension factor coupling in amplitudes of the harmonics. On the one hand, surface tension can either decrease or increase the space amplitude; on the other hand, surface tension can also change their phases for some conditions which are explicitly determined.     

关于双特征Beltrami方程

该文研究空间Ｂｅｌｔｒａｍｉ方程的推广形式，即双特征Ｂｅｌｔｒａｍｉ方程．利用外微分形式与矩阵的外代数等工具，将双特征Ｂｅｌｔｒａｍｉ方程转化为一个非齐次的狆 调和方程，转化过程中只用到加于特征矩阵的一致椭圆型条件．然后验证了算子犃满足的条件：Ｌｉｐｓｃｈｉｔｚ型条件、单调不等式、齐次性条件以及算子犅满足的控制增长条件．并利用得到的狆 调和方程，给出了双特征Ｂｅｌｔｒａｍｉ方程广义解分量函数的弱单调性结果．     

The decay rate for a fractional differential equation

We consider the fractional differential equation

     

Semistrictly quasiconvex mappings and non-convex vector optimization

This paper introduces a new class of non-convex vector functions strictly larger than that of P-quasiconvexity, with P ^m being the underlying ordering cone, called semistrictly ( ^m\ –int P)-quasiconvex functions. This notion allows us to unify various results on existence of weakly efficient (weakly Pareto) optima. By imposing a coercivity condition we establish also the compactness of the set of weakly Pareto solutions. In addition, we provide various characterizations for the non-emptiness, convexity and compactness of the solution set for a subclass of quasiconvex vector optimization problems on the real-line. Finally, it is also introduced the notion of explicit ( ^m\ –int P)-quasiconvexity (equivalently explicit (int P)-quasiconvexity) which plays the role of explicit quasiconvexity (quasiconvexity and semistrict quasiconvexity) of real-valued functions.Acknowldegements.The author wishes to thank both referees for their careful reading of the paper, their comments, remarks, helped to improve the presentation of some results. One of the referee provided the references [5, 6] and indirectly [20].     

Regularity Results for the Generalized Beltrami System

Abstract For the generalized Beltrami system with two characteristic matrices, we deal with the regularity of its very weak solutions in the Sobolev class (1 < r < n). By changing the generalized Beltrami system into a class of a divergent elliptic system with nonhomogeneous items, we obtain that each of its very weak solutions is essentially a classical weak solution of a usual Sobolev class. Furthermore, we also establish a higher regularity of its weak solution if the regularity hypotheses of two characteristic matrices are improved. Supported by the National Natural Science Foundation of China (49805005) and by the research foundation of Northern Jiaotong University (2002SM061)     

Weak Uniform Normal Structure and Fixed Points of Asymptotically Regular Semigroups

Let X be a Banach space with a weak uniform normal structure and C a non–empty convexweakly compact subset of X. Under some suitable restriction, we prove that every asymptoticallyregular semigroup T = {T(t) : t ∈¸ S} of selfmappings on C satisfying

${\mathop {\lim \inf }\limits_{S \mathrel\backepsilon t \to \infty } }{\left| {{\left\| {T(t)} \right\|}} \right|} < {\text{WCS}}(X)$

has a common fixed point, where WCS(X) is the weakly convergent sequence coefficient of X, and\({\left| {{\left\| {T(t)} \right\|}} \right|}\) is the exact Lipschitz constant of T(t).     

Order Environments of Topological Spaces

In this paper it is proved that a space may be realized as the set of the maximal elements in a continuous poset if and only if it is Tychonoff.     

Boolean algebras and uniform convergence of series

Several classical results on uniform convergence of unconditionally Cauchy series are generalized to weakly unconditionally Cauchy series. This uniform convergence is characterized through subalgebras and subfamilies of . A generalization of the Orlicz-Pettis theorem is also proved by mean of subalgebras of .     

Vector Equilibrium Problems Under Asymptotic Analysis

Given a closed convex set K in Rⁿ; a vector function F:K×K R^m; a closed convex (not necessarily pointed) cone P(x) in ^m with non-empty interior, PP(x) Ø, various existence results to the problemfind xK such that F(x,y)- int P(x) y K under P(x)-convexity/lower semicontinuity of F(x,) and pseudomonotonicity on F, are established. Moreover, under a stronger pseudomonotonicity assumption on F (which reduces to the previous one in case m=1), some characterizations of the non-emptiness of the solution set are given. Also, several alternative necessary and/or sufficient conditions for the solution set to be non-empty and compact are presented. However, the solution set fails to be convex in general. A sufficient condition to the solution set to be a singleton is also stated. The classical case P(x)=^m ₊ is specially discussed by assuming semi-strict quasiconvexity. The results are then applied to vector variational inequalities and minimization problems. Our approach is based upon the computing of certain cones containing particular recession directions of K and F.