Addition and subspace theorems for asymptotic large inductive dimension

粗近邻空间的归纳维数

Inductive dimensions of coarse proximity spaces

本文研究了粗近邻空间的渐近归纳维数，证明其大于等于边界的归纳维数，并提出了完全可追溯性的条件，以及具有特定边界的空间的性质。

Universal spaces for asymptotic dimension

We construct a universal space for the class of proper metric spaces of bounded geometry and of given asymptotic dimension. As a consequence of this result, we establish coincidence of the asymptotic dimension with the asymptotic inductive dimensions.

A simple inductive proof of Levy-Steinitz theorem

We present a relatively simple inductive proof of the classical Levy-Steinitz Theorem saying that for a sequence $(x_n)_{n=1}^\infty$ in a finite-dimensional Banach space $X$ the set of all sums of rearranged series $\sum_{n=1}^\infty x_{σ(n)}$ is an affine subspace of $X$. This affine subspace is not empty if and only if for any linear functional $f:X\to \mathbb R$ the series $\sum_{n=1}^\infty f(x_{σ(n)})$ is convergent for some permutation $σ$ of $\mathbb N$. This gives an answer to a problem of Vaja Tarieladze, posed in Lviv Scottish Book in September, 2017. Also we construct a sequence $(…