Let $\mathbb{P}=\{p_1,p_2,\cdots ,p_i,\cdots \}$ be the set of prime numbers (or more generally a set of pairwise co-prime elements). Let us denote $A_p^{a,b}=\{p^{an+b}m\mid n\in \mathbb{N}\cup \left\{0\right\};m\in \mathbb{N},p\mathrm{does}\mathrm{not}\mathrm{divide}m\}$, where $a\in \mathbb{N},b\in \mathbb{N}\cup \left\{0\right\}$. Then for arbitrary finite set $B$, $B\subset \mathbb{P}$ holds $$d\left(\bigcap _{p_i\in B}{A}_{p_i}^{a_i,b_i}\right)=\prod _{p_i\in B}d\left(A_{p_i}^{a_i,b_i}\right),$$ and $$d\left(A_{p_i}^{a_i,b_i}\right)=\frac{\frac{1}{p_i^{b_i}}\left(1-\frac{1}{p_i}\right)}{1-\frac{1}{p_i^{a_i}}}.$$ If we denote $$A=\left\{\frac{\frac{1}{p^b}\left(1-\frac{1}{p}\right)}{1-\frac{1}{p^a}}\mid p\in \mathbb{P},a\in \mathbb{N},b\in \mathbb{N}\cup \left\{0\right\}\right\},$$ where $\mathbb{P}$ is the set of all prime numbers, then for closure of set $A$ holds $$\mathrm{cl}A=A\cup B\cup \{0,1\},$$ where $B=\left\{\frac{1}{p^b}\left(1-\frac{1}{p}\right)\mid p\in \mathbb{P},b\in \mathbb{N}\cup \left\{0\right\}\right\}$.

Let $\parallel ·\parallel$ be a norm on the algebra ${ℳ}_n$ of all $n\times n$ matrices over $ℂ$. An interesting problem in matrix theory is that “Are there two norms ${\parallel ·\parallel }_1$ and ${\parallel ·\parallel }_2$ on ${ℂ}^n$ such that $\parallel A\parallel =max\{\parallel Ax{\parallel }_2\parallel x{\parallel }_1=1\}$ for all $A\in {ℳ}_n$?” We will investigate this problem and its various aspects and will discuss some conditions under which ${\parallel ·\parallel }_1={\parallel ·\parallel }_2$.

Let $r\ge 3$, $n\ge r$ and $\pi =(d_1,d_2,...,d_n)$ be a non-increasing sequence of nonnegative integers. If $\pi$ has a realization $G$ with vertex set $V\left(G\right)=\{v_1,v_2,...,v_n\}$ such that $d_G\left(v_i\right)=d_i$ for $i=1,2,...,n$ and $v_1{v}_2\cdots v_r{v}_1$ is a cycle of length $r$ in $G$, then $\pi$ is said to be potentially $C_r^{\text{'}\text{'}}$-graphic. In this paper, we give a characterization for $\pi$ to be potentially $C_r^{\text{'}\text{'}}$-graphic.

Let $n>m\ge 2$ be positive integers and $n=(m+1)\ell +r$, where $0\le r\le m.$ Let $C$ be a subset of $\{0,1,\cdots ,n\}$. We prove that if $$\left|C\right|>\left\{\begin{array}{cc}\lfloor n/2\rfloor +1\hfill & \text{if}m\text{is}\text{odd},\hfill \\ m\ell /2+\delta \hfill & \text{if}m\text{is}\text{even},\hfill \end{array}\right.$$ where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ and $\delta$ denotes the cardinality of even numbers in the interval $[0,min\{r,m-2\left\}\right]$, then $C-C$ contains a power of $m$. We also show that these lower bounds are best possible.