An $n\times n$ sign pattern $𝒜$ is said to be potentially nilpotent if there exists a nilpotent real matrix $B$ with the same sign pattern as $𝒜$. Let ${𝒟}_{n,r}$ be an $n\times n$ sign pattern with $2\le r\le n$ such that the superdiagonal and the $(n,n)$ entries are positive, the $(i,1)$ $(i=1,\cdots ,r)$ and $(i,i-r+1)$ $(i=r+1,\cdots ,n)$ entries are negative, and zeros elsewhere. We prove that for $r\ge 3$ and $n\ge 4r-2$, the sign pattern ${𝒟}_{n,r}$ is not potentially nilpotent, and so not spectrally arbitrary.

Let $k$ be a field of characteristic zero and $B$ a $k$-domain. Let $R$ be a retract of $B$ being the kernel of a locally nilpotent derivation of $B$. We show that if $B=R\oplus I$ for some principal ideal $I$ (in particular, if $B$ is a UFD), then $B=R^{\left[1\right]}$, i.e., $B$ is a polynomial algebra over $R$ in one variable. It is natural to ask that, if a retract $R$ of a $k$-UFD $B$ is the kernel of two commuting locally nilpotent derivations of $B$, then does it follow that $B\cong R^{\left[2\right]}$? We give a negative answer to this question. The interest in...

Let $𝒩$ denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra $L$ over an arbitrary field $𝔽$, there exists a smallest ideal $I$ of $L$ such that $L/I\in 𝒩$. This uniquely determined ideal of $L$ is called the nilpotent residual of $L$ and is denoted by $L^{𝒩}$. In this paper, we define the subalgebra $S\left(L\right)={\bigcap }_{H\le L}{I}_L\left(H^{𝒩}\right)$. Set $S_0\left(L\right)=0$. Define $S_{i+1}\left(L\right)/S_i\left(L\right)=S(L/S_i\left(L\right))$ for $i\ge 1$. By $S_{\infty }\left(L\right)$ denote the terminal term of the ascending series. It is proved that $L=S_{\infty }\left(L\right)$ if and only if $L^{𝒩}$ is nilpotent. In addition, we investigate the basic properties of a...

For a finite group $G$ and a fixed Sylow $p$-subgroup $P$ of $G$, Ballester-Bolinches and Guo proved in 2000 that $G$ is $p$-nilpotent if every element of $P\cap G^{\text{'}}$ with order $p$ lies in the center of $N_G\left(P\right)$ and when $p=2$, either every element of $P\cap G^{\text{'}}$ with order $4$ lies in the center of $N_G\left(P\right)$ or $P$ is quaternion-free and $N_G\left(P\right)$ is $2$-nilpotent. Asaad introduced weakly pronormal subgroup of $G$ in 2014 and proved that $G$ is $p$-nilpotent if every element of $P$ with order $p$ is weakly pronormal in $G$ and when $p=2$, every element of $P$ with...

Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $aS$-group. We study some properties of $aS$-groups. For instance, it is shown that a nilpotent group $G$ is an $aS$-group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $aS$-group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...

A theorem of Burnside asserts that a finite group $G$ is $p$-nilpotent if for some prime $p$ a Sylow $p$-subgroup of $G$ lies in the center of its normalizer. In this paper, let $G$ be a finite group and $p$ the smallest prime divisor of $\left|G\right|$, the order of $G$. Let $P\in {\mathrm{Syl}}_p\left(G\right)$. As a generalization of Burnside’s theorem, it is shown that if every non-cyclic $p$-subgroup of $G$ is self-normalizing or normal in $G$ then $G$ is solvable. In particular, if $P\ncong \langle a,b|a^{p^{n-1}}=1,b^2=1,b^{-1}ab=a^{1+p^{n-2}}\rangle$, where $n\ge 3$ for $p>2$ and $n\ge 4$ for $p=2$, then $G$ is $p$-nilpotent or $p$-closed. ...

By Descartes’ rule of signs, a real degree $d$ polynomial $P$ with all nonvanishing coefficients with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$) has $\mathrm{pos}\le c$ positive and $\neg \le p$ negative roots, where $\mathrm{pos}\equiv c(mod2)$ and $\neg \equiv p(mod2)$. For $1\le d\le 3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $(\mathrm{pos},\mathrm{neg})$ satisfying these conditions there exists a polynomial $P$ with exactly $\mathrm{pos}$ positive and exactly $\neg$ negative roots (all of them simple). For $d\ge 4$...

Let $m$ and $n$ be positive integers, and let $R=(r_1,...,r_m)$ and $S=(s_1,...,s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m\times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F\left(R\right)$ be the $m\times n$ $(0,1)$-matrix, where for each $i$, the $i$th row of $F(R,S)$ consists of $r_i$ 1’s followed by $(n-r_i)$ 0’s. Let $A\in A(R,S)$. The discrepancy of A, $\mathrm{disc}\left(A\right)$, is the number of positions in which $F\left(R\right)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m\times n$ matrices over...