let $mathcal {a} $ and $mathcal {b} $ be c$^*$-algebras. assume that $mathcal {a}$ is of real rank zero and unital with unit $i$ and $k>0$ is a real number. it is shown that if $phi:mathcal{a} tomathcal{b}$ is an additive map preserving $|cdot|^k$ for all normal elements; that is, $phi(|a|^k)=|phi(a)|^k $ for all normal elements $ainmathcal a$, $phi(i)$ is a projection, and there exists a posit...

Let $mathcal{A}$ be a $C^*$-algebra and $Z(mathcal{A})$ the‎ ‎center of $mathcal{A}$‎. ‎A sequence ${L_{n}}_{n=0}^{infty}$ of‎ ‎linear mappings on $mathcal{A}$ with $L_{0}=I$‎, ‎where $I$ is the‎ ‎identity mapping‎ ‎on $mathcal{A}$‎, ‎is called a Lie higher derivation if‎ ‎$L_{n}[x,y]=sum_{i+j=n} [L_{i}x,L_{j}y]$ for all $x,y in  ‎mathcal{A}$ and all $ngeqslant0$‎. ‎We show that‎ ‎${L_{n}}_{n...

let $mathcal{a}$ be a $c^*$-algebra and $z(mathcal{a})$ the‎ ‎center of $mathcal{a}$‎. ‎a sequence ${l_{n}}_{n=0}^{infty}$ of‎ ‎linear mappings on $mathcal{a}$ with $l_{0}=i$‎, ‎where $i$ is the‎ ‎identity mapping‎ ‎on $mathcal{a}$‎, ‎is called a lie higher derivation if‎ ‎$l_{n}[x,y]=sum_{i+j=n} [l_{i}x,l_{j}y]$ for all $x,y in  ‎mathcal{a}$ and all $ngeqslant0$‎. ‎we show that‎ ‎${l_{n}}_{n...

‎let $g$ be a finite group and let $n$ be a normal subgroup of $g$‎. ‎suppose that ${rm{irr}} (g | n)$ is the set of the irreducible characters of $g$ that contain $n$ in their kernels‎. ‎in this paper‎, ‎we classify solvable groups $g$ in which the set $mathcal{c} (g) = {{rm{irr}} (g | n) | 1 ne n trianglelefteq g }$ has at most three elements‎. ‎we also compute the set $mathcal{c}(g)$ for suc...

Let $ \pi:X \rightarrow X_{0}$ be a projective morphism of schemes, such that $ X_{0}$ is Noetherian and essentially of finite type over a field $ K$. Let $ i \in \mathbb{N}_{0}$, let $ {\mathcal{F}}$ be a coherent sheaf of $ {\mathcal{O}}_{X}$-modules and let $ {\mathcal{L}}$ be an ample invertible sheaf over $ X$. Let $ Z_{0} \subseteq X_{0}$ be a closed set. We show that the depth of the hig...

emph{The (undirected) power graph on the conjugacy classes} $mathcal{P_C}(G)$ of a group $G$ is a simple graph in which the vertices are the conjugacy classes of $G$ and two distinct vertices $C$ and $C'$ are adjacent in $mathcal{P_C}(G)$ if one is a subset of a power of the other. In this paper, we describe groups whose associated graphs are $k$-regular for $k=5,6$.

Let $mathcal {A} $ and $mathcal {B} $ be C$^*$-algebras. Assume that $mathcal {A}$ is of real rank zero and unital with unit $I$ and $k>0$ is a real number. It is shown that if $Phi:mathcal{A} tomathcal{B}$ is an additive map preserving $|cdot|^k$ for all normal elements; that is, $Phi(|A|^k)=|Phi(A)|^k $ for all normal elements $Ainmathcal A$, $Phi(I)$ is a projection, and there exists a posit...