Let $\sum_{n=1}^\infty x_n$ be a conditionally convergent series in a Banach space and let $τ$ be a permutation of natural numbers. We study the set $\operatorname{LIM}(\sum_{n=1}^\infty x_{τ(n)})$ of all limit points of a sequence $(\sum_{n=1}^p x_{τ(n)})_{p=1}^\infty$ of partial sums of a rearranged series $\sum_{n=1}^\infty x_{τ(n)}$. We give full characterization of limit sets in finite dimensional spaces. Namely, a limit set in $\mathbb{R}^m$ is either compact and connected or it is closed and all its connected components are unbounded. On the other hand each set of one of these types is a limit set of some rearranged conditionally convergent series. Moreover, this characterization does not hold in infinite dimensional spaces.
We show that if $\sum_{n=1}^\infty x_n$ has the Rearrangement Property and $A$ is a closed subset of the closure of the $\sum_{n=1}^\infty x_n$ sum range and it is $\varepsilon$-chainable for every $\varepsilon>0$, then there is a permutation $τ$ such that $A=\operatorname{LIM}(\sum_{n=1}^\infty x_{τ(n)})$. As a byproduct of this observation we obtain that series having the Rearrangement Property have closed sum ranges.
The joint modulus of variation of metric space valued functions and pointwise selection principles
给定$T\subset\mathbb{R}$和一个度量空间$M$，我们在$M^T$（从$T$到$M$的所有函数的集合）上引入一个非减伪度量序列$\{ν_n\}$，称为\emph{joint modulus of variation} .证明了如果$M^T$的两个函数序列$\{f_j\}$和$\{g_j\}$使得$\{f_j\}$是逐点预紧的，$\{g_j\}$是逐点收敛的，且$ν_n(f_j,g_j)$作为$j\to\infty$的上极限是$o(n)$作为$n\to\infty$，则$\{f_j\}$存在一个逐点收敛的子序列，其极限是一个条件调节函数.我们通过例子说明了这个结果的尖锐性（特别是，关于$\limsup$的假设对于一致收敛序列$\{f_j\}$和$\{g_j\}$是必要的，并且当它们逐点收敛时是“几乎必要的”），并表明大多数已知的Helly型逐点选择定理都是其特例。
Achievement sets and sum ranges with ideal supports
我们引入了一系列实数的理想支撑成就集的概念。我们分析了它们的复杂性和拓扑性质。我们比较的概念，理想的成就集的概念，理想的支持和范围的真正的系列，认为Filipów和Szuca。我们完成了Filipów和Szuca关于理想和值域的刻画，[R. Filipów，P. Szuca，Rearrangement of conditional收敛级数on a small set，J. Math. Anal. Appl.362（2010），no. 1，64-71.]，并得到了黎曼定理的一些推广。
