Differentiable spaces that are subcartesian

We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is weakly reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differential forms, that satisfies Smith’s version of de Rham’s theorem. Because the orbit space is a locally closed subcartesian space, it has vector fields and their flows.     

The full basis theorem does not imply analytic wellordering

A finite support product of ${\omega }_1$ clones of Jensen's minimal ${\Pi }_2^1$ singleton forcing is used to define a model in which any non-empty analytically definable set of reals contains an analytically definable real (the full basis theorem), but there is no analytically definable wellordering of the reals.     

Boundedness of semilinear Duffing equations at resonance in a critical situation

In this paper, we propose a sufficient and necessary condition for the boundedness of all the solutions for the equation $\ddot{x}+n^{2}x+g(x)=p(t)$ with the critical situation that $|{\int }_0^{2\pi }p(t)e^{?int}dt|=2|g(+\infty )?g(?\infty )|$ on g and p, where $n\in {\mathbb{N}}^{+}$, $p(t)$ is periodic and $g(x)$ is bounded.     

The Chang-Łoś-Suszko theorem in a topological setting

The Chang-Łoś-Suszko theorem of first-order model theory characterizes universal-existential classes of models as just those elementary classes that are closed under unions of chains. This theorem can then be used to equate two model-theoretic closure conditions for elementary classes; namely unions of chains and existential substructures. In the present paper we prove a topological analogue and indicate some applications.     

Necessary Conditions for Differentiability of Distance Functions

Necessary conditions for the Gâteaux differentiability of the distance function to a set are considered. A series of characterizing results is obtained.     

凸域内单叶函数的导数

设函数f（w）为凸区域D内的单叶解析函数,对于2≤n≤8和所有w∈D,本文得到估计式|f（n）（w）/f＇（w）|的精确上界.这个结果推广了一些已知的结论.     

On the duration of the total eclipse of a satellite of a body

This paper solves the problem of the duration of the total eclipseof a satellite of a body, provided that (a) the body is sphericalin shape and the mass distribution inside the body is sphericallysymmetrical; (b) the satellite is a very small object; (c) thesatellite's orbit around a body is circular and the satellitecan pass through the centre of its shadow; (d) a source of lightis the source of the conical body's shadow (umbra) and it doesnot change its apparent position in the sky; (e) there are noperturbations on the satellite's orbit. The solution presentedrepresents the general discussion of the problem based on utilizationof some parts of the theory of contraction mappings.     

系数变号的半线性椭圆方程的正解

本文研究半线性椭圆方程Dirichlet问题-△u=α（x）f(u），x∈Ω， u（x）=0，x∈ЭΩ，正解的存在性，其中Ω为R^n中有界的带光滑边界的区域，α(x)可以变号。     

线性分式规划最优解集的求法

本文使用多面集的表示定理，导出了线性分式规划最优解集的结构，并给出确定全部最优解的计算步骤。     

Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.