预切空间的一致有界性，度量导数的局部常值性和点处的强右上空度
Uniform boundedness of pretangent spaces, local constancy of metric derivatives and strong right upper porosity at a point
摘要：
设$(X,d,p)$是点度量空间. $X$在$p$处的预切空间是由收敛于$p$的序列$(x_n), x_n \in X,$的一些等价类组成的度量空间，其收敛度与给定的标度序列$(r_n), r_n\downarrow 0.$相当。标度序列$(r_n)$是正规的，如果该序列最终是减的，并且存在$(x_n)$使得$\mid d(x_n,p)-r_n\mid=o(r_n)$ for $n\to\infty.$令$\mathbf{Ω_{p}^{X}(n)}$是$X$在$p$处的预切空间的集合，具有正规标度序列的证明了$\mathbf{Ω_{p}^{X}(n)}$是一致有界的当且仅当$\{d(x,p): x\in X\}$是所谓的完全强多孔集.证明了$\mathbf{Ω_{p}^{X}(n)}$的一致有界性等价于$X$上所有度量可微映射的度量导数在以预切空间的标志点为中心的固定半径的开球中的恒定性.
英文摘要：
Let $(X,d,p)$ be a pointed metric space. A pretangent space to $X$ at $p$ is a metric space consisting of some equivalence classes of convergent to $p$ sequences $(x_n), x_n \in X,$ whose degree of convergence is comparable with a given scaling sequence $(r_n), r_n\downarrow 0.$ A scaling sequence $(r_n)$ is normal if this sequence is eventually decreasing and there is $(x_n)$ such that $\mid d(x_n,p)-r_n\mid=o(r_n)$ for $n\to\infty.$ Let $\mathbf{Ω_{p}^{X}(n)}$ be the set of pretangent spaces to $X$ at $p$ with normal scaling sequences. We prove that $\mathbf{Ω_{p}^{X}(n)}$ is uniformly bounded if and only if $\{d(x,p): x\in X\}$ is a so-called completely strongly porous set. It is also proved that the uniform boundedness of $\mathbf{Ω_{p}^{X}(n)}$ is an equivalent of the constancy of metric derivatives of all metrically differentiable mappings on $X$ in the open balls of a fixed radius centered at the marked points of pretangent spaces.
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