Let $G$ be a finite group. The automorphism $sigma$ of a group $G$ is said to be an absolute central automorphism, if for all $xin G$, $x^{-1}x^{sigma}in L(G)$, where $L(G)$ be the absolute centre of $G$. In this paper, we study  some properties of absolute central automorphisms of a given finite $p$-group.

Let $M$ be a right module over a ring $R$, $tau_M$ a preradical on $sigma[M]$, and$Ninsigma[M]$. In this note we show that if $N_1, N_2in sigma[M]$ are two$tau_M$-lifting modules such that $N_i$ is $N_j$-projective ($i,j=1,2$), then $N=N_1oplusN_2$ is $tau_M$-lifting. We investigate when homomorphic image of a $tau_M$-lifting moduleis $tau_M$-lifting.

let $m$ be a right module over a ring $r$, $tau_m$ a preradical on $sigma[m]$, and$ninsigma[m]$. in this note we show that if $n_1, n_2in sigma[m]$ are two$tau_m$-lifting modules such that $n_i$ is $n_j$-projective ($i,j=1,2$), then $n=n_1oplusn_2$ is $tau_m$-lifting. we investigate when homomorphic image of a $tau_m$-lifting moduleis $tau_m$-lifting.

The rapid scaling in modern CMOS technology has motivated the researchers to design new analog-to-digital converter (ADC) architectures that can properly work in lower supply voltage. An exchanging the data quantization procedure from the amplitude to the time domain, can be a promising alternative well adapt with the technology scaling. This paper is going to review the recent development in t...

in proposition 2.6 in (g‎. ‎gruenhage‎, ‎a‎. ‎lutzer‎, ‎baire and volterra spaces‎, ‎textit{proc‎. ‎amer‎. ‎math‎. ‎soc.} {128} (2000)‎, ‎no‎. ‎10‎, ‎3115--3124) a condition that‎ ‎every point of $d$ is $g_delta$ in $x$ was overlooked‎. ‎so we‎ ‎proved some conditions by which a baire space is equivalent to a‎ ‎volterra space‎. ‎in this note we show that if $x$ is a‎ ‎monotonically normal $t_1$...

let $m_r$ be a non-zero‎ ‎module and ${mathcal f}‎: ‎sigma[m_r]times sigma[m_r]‎ ‎rightarrow$ mod-$bbb{z}$ a bifunctor‎. ‎the‎ ‎$mathcal{f}$-reversibility of $m$ is defined by ${mathcal‎ ‎f}(x,y)=0 rightarrow {mathcal f}(y,x)=0$ for all non-zero $x,y$‎ ‎in $sigma[m_r]$‎. ‎hom (resp‎. ‎rej)-reversibility of $m$ is‎ ‎characterized in different ways‎. ‎among other things‎, ‎it is shown‎ ‎th...