A universal differentiability set in Banach spaces with separable dual

We show that any non-zero Banach space with a separable dual contains a totally disconnected, closed and bounded subset S of Hausdorff dimension 1 such that every Lipschitz function on the space is Fréchet differentiable somewhere in S.     

多元凸函数的一个定理

本文在［１］的基础上．给出了与多元函数中值定理结论类似的一个定理．     

Nonequivalence of Various Definitions of Differentiability Directions for Vector Measures

It is proved that the definitions of differentiability directions for vector measures in various topologies, namely, the topology of convergence on a system measurable sets, the topology of convergence with respect to semivariation, and the topology of convergence in variation, are generally pairwise nonequivalent. It is also proved that, for measures with values in a Banach space with the Radon--Nikodym property, these definitions are equivalent.     

Analysis of Symmetric Matrix Valued Functions

For any symmetric function f: ? ⁿ  → ? ⁿ , one can define a corresponding function on the space of n × n real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Fréchet differentiability, and continuous differentiability.     

Melnikov函数在中心处的可微性问题

本文证明（一阶）Melnikov函数在初等中心处关于Hamilton量至少为二次可微，井且得到二阶Melnikov函数为二次可微的充要条件，最后举例说明文[3,4]所讨论的一类扰动系统的后继函数在中心处不是二阶可微的．     

Lateral boundary differentiability of solutions of parabolic equations in nondivergence form

The lateral boundary differentiability is shown for solutions of parabolic differential equations in nondivergence form under the assumptions that the parabolic boundary satisfies the exterior Dini condition and is punctually C¹$C^1$ differentiable one-sided in t-direction. The classical barrier technique, the maximum principle, the interior Harnack inequality and an iteration procedure are the main analytical tools.     

Some Remarks About Heisenberg Groups

In this paper, we give two inequalities and another characterizations about Heisenberg groups which do not coincide with the related results in classical case.     

An elementary proof of a converse mean-value theorem

We present a new converse mean value theorem, with a rather elementary proof.     

Porosity and differentiability in smooth Banach spaces

We improve a result of Preiss, Phelps and Namioka, showing that every submonotone mapping in a Gateaux smooth Banach space is single-valued on the complement of a -cone porous subset. If a Banach space has a uniformly -differentiable Lipschitz bump function (with respect to some bornology ), then we show with a much simpler argument (localization of -minimum of a perturbed function) that every continuous convex function on is -differentiable on the complement of a -uniformly porous set.

     

Radon-Nikodym性质与范数的Gteaux可微性

本文研究了Banach空间中集合的RNP与范数的G^ateaux可微性之间的关系 ,给出了关于Banach空间中集合具RNP的一些充要条件     

关于Banach空间的k-光滑性

本文研究了k-光滑的Banach空间,引入了k-G可微性的概念.得到了k-光滑的Banach空间的几个等价条件.     

函数的逼近及其在下半连续函数可微性中的应用

通过函数的下卷积函数列的逼近方法,在变分原理中从扰动最小值点集的“大小”入手,研究了下半连续函数的可微性。     

A selection of convex-compact-valued multi-functions with remarkable properties: the steiner selection

This paper shows how the Steiner selection has very nice properties in view of differentiability, measurability, and integrability of multi-functions.     

A Course of Statistical Treatment of Experimental Data for Technical People with the Use of a Time‐Sharing Digital Computer in the Classroom

After a short examination of the problem of training in industry with the aim of integrating the knowledge of young scientists and technical people, and of overcoming the obsolescence of personnel charged with technical work, this paper stresses the importance of the statistical treatment of experimental data. The paper describes the programme of a course on this subject, developed through the extensive use of a time‐sharing digital computer with a terminal in the classroom: the computer being programmed to simulate sources of experimental results and to verify the essential theorems of statistics.     

Dependence of eigenvalues of Dirac system on the parameters

The dependence of eigenvalues of Dirac system with general boundary conditions is studied. It is shown that the eigenvalues of Dirac operators depend not only continuously but also smoothly on the coefficients, the boundary conditions, and the endpoints of the problem. Furthermore, the differential expressions of the eigenvalues as regards these parameters are given. The results obtained in this paper would provide theoretical support for the numerical calculations of eigenvalues of the corresponding problems.     

关于初值问题解的稳定性

本文利用解对初值和参数的可微性,提出一种不同于Liapunov直接法的方法,用它可以判定初值问题非驻定解的稳定性.而该非驻定解无需事先求出.     

Singularities of minima: a walk on the wild side of the Calculus of Variations

I will report on some recent developments concerning the problem of estimating the Hausdorff dimension of the singular sets of solutions to elliptic and variational problems. Emphasis will be given on some open issues. Connections with measure data problems will be outlined.     

Fractal and smoothness properties of space-time Gaussian models

Spatio-temporal models are widely used for inference in statistics and many applied areas. In such contexts, interests are often in the fractal nature of the sample surfaces and in the rate of change of the spatial surface at a given location in a given direction. In this paper, we apply the theory of Yaglom (1957) to construct a large class of space-time Gaussian models with stationary increments, establish bounds on the prediction errors, and determine the smoothness properties and fractal properties of this class of Gaussian models. Our results can be applied directly to analyze the stationary spacetime models introduced by Cressie and Huang (1999), Gneiting (2002), and Stein (2005), respectively.     

DIMENSION AND DIFFERENTIABILIIY OF A CLASS OF FRACTAL INTERPOLATION FUNCTIONS

In this paper, a new iterated function system consisting of non-linear affinemaps is constructed. We investigate the fractal interpolation functions generated bysuch a system and get its differentiability, its box dimension, its packing dimension,and a lower bound of its Hansdorff dimension.