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 E be a Banach space. We consider a Cauchy problem of the type ⎧ $D_t^{k}u+{\sum }_{j=0}^{k-1}{\sum }_{\left|\alpha \right|\le m}{A}_{j,\alpha }\left(D_t^j{D}_x^{\alpha }u\right)=f$ in ${ℝ}^{n+1}$, ⎨ ⎩ $D_t^{j}u(0,x)={\phi }_j\left(x\right)$ in ${ℝ}^n$, j=0,...,k-1, where each $A_{j,\alpha }$ is a given continuous linear operator from E into itself. We prove that if the operators $A_{j,\alpha }$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u\in C^{\infty }({ℝ}^{n+1},E)$ whose derivatives are equi-bounded on each bounded subset of ${ℝ}^{n+1}$.

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.

It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form $F(x₁,...,x_n)=x-H\left(x\right):=(x₁-H₁(x₁,...,x_n),...,x_n-H_n(x₁,...,x_n))$, where $H_j$ are homogeneous polynomials of degree 3 with real coefficients (or $H_j=0$), j = 1,...,n and H’(x) is a nilpotent matrix for each $x=(x₁,...,x_n)\in {ℝ}^n$. We give another proof of Yu’s theorem that in the case of non-negative coefficients of H the mapping F is a polynomial automorphism, and we moreover prove that in that case $deg{F}^{-1}\le {\left(degF\right)}^{indF-1}$, where $indF:=maxind{H}^{\text{'}}\left(x\right):x\in {ℝ}^n$. Note that the above inequality is not true when the coefficients...

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...

An $n\times n$ ray pattern $𝒜$ is called a spectrally arbitrary ray pattern if the complex matrices in $Q\left(𝒜\right)$ give rise to all possible complex polynomials of degree $n$. In a paper of Mei, Gao, Shao, and Wang (2014) was proved that the minimum number of nonzeros in an $n\times n$ irreducible spectrally arbitrary ray pattern is $3n-1$. In this paper, we introduce a new family of spectrally arbitrary ray patterns of order $n$ with exactly $3n-1$ nonzeros.

Let $𝔐$ be the class of groups satisfying the minimal condition on normal subgroups and let $\Omega$ be the class of groups of finite lower central depth, that is groups $G$ such that ${\gamma }_i\left(G\right)={\gamma }_{i+1}\left(G\right)$ for some positive integer $i$. The main result states that if $G$ is a finitely generated hyper-(Abelian-by-finite) group such that for every $x\in G$, there exists a normal subgroup $H_x$ of finite index in $G$ satisfying $\langle x,x^h\rangle \in 𝔐\Omega$ for every $h\in H_x$, then $G$ is finite-by-nilpotent. As a consequence of this result, we prove that a finitely generated...

Denote by $C$ the commutator $AB-BA$ of two bounded operators $A$ and $B$ acting on a locally convex topological vector space. If $AC-CA=0$, we show that $C$ is a quasinilpotent operator and we prove that if $AC-CA$ is a compact operator, then $C$ is a Riesz operator.

We prove that the study of the category C-Comod of left comodules over a K-coalgebra C reduces to the study of K-linear representations of a quiver with relations if K is an algebraically closed field, and to the study of K-linear representations of a K-species with relations if K is a perfect field. Given a field K and a quiver Q = (Q₀,Q₁), we show that any subcoalgebra C of the path K-coalgebra K◻Q containing $K^{◻}Q₀\oplus K^{◻}Q₁$ is the path coalgebra $K^{◻}(Q,)$ of a profinite bound quiver (Q,), and the category...

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. ...

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 (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: $Pu(x,y,z):=(y{\partial }_x-z{\partial }_y)u(x,y,z)=f(x,y,z){\in }_{x,y,z}$, where $P=y{\partial }_x-z{\partial }_y{:}_{x,y,z}{\to }_{x,y,z}$. For this equation, our aim is to characterize the solvability on ${}_{x,y,z}$ by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.