In [1], Jakubík showed that the class of $\sigma $-interpolation lattice-ordered groups forms a radical class, but left open the question of whether the class forms a torsion class. In this paper, we show that this class does indeed form a torsion class.
By analogy with the projective, injective and flat modules, in this paper we study some properties of $C$-Gorenstein projective, injective and flat modules and discuss some connections between $C$-Gorenstein injective and $C$-Gorenstein flat modules. We also investigate some connections between $C$-Gorenstein projective, injective and flat modules of change of rings.
For a nontrivial connected graph $F$, the $F$-degree of a vertex $v$ in a graph $G$ is the number of copies of $F$ in $G$ containing $v$. A graph $G$ is $F$-continuous (or $F$-degree continuous) if the $F$-degrees of every two adjacent vertices of $G$ differ by at most 1. All $P_3$-continuous graphs are determined. It is observed that if $G$ is a nontrivial connected graph that is $F$-continuous for all nontrivial connected graphs $F$, then either $G$ is regular or $G$ is a path. In the case of a 2-connected graph $F$, however, there always exists a regular graph that is not $F$-continuous. It is also shown that for every graph $H$ and every 2-connected graph $F$, there exists an $F$-continuous graph $G$ containing $H$ as an induced subgraph.
In this paper, we prove that a space $X$ is a $g$-metrizable space if and only if $X$ is a weak-open, $\pi $ and $\sigma $-image of a semi-metric space, if and only if $X$ is a strong sequence-covering, quotient, $\pi $ and $mssc$-image of a semi-metric space, where “semi-metric” can not be replaced by “metric”.
The concepts of $k$-systems, $k$-networks and $k$-covers were defined by A. Arhangel’skiǐ in 1964, P. O’Meara in 1971 and R. McCoy, I. Ntantu in 1985, respectively. In this paper the relationships among $k$-systems, $k$-networks and $k$-covers are further discussed and are established by $mk$-systems. As applications, some new characterizations of quotients or closed images of locally compact metric spaces are given by means of $mk$-systems.
Let $R$ be a commutative Noetherian ring. It is shown that the finitely generated $R$-module $M$ with finite Gorenstein dimension is reflexive if and only if $M_{\mathfrak p}$ is reflexive for ${\mathfrak p} \in {\rm Spec}(R) $ with ${\rm depth}(R_{\mathfrak p}) \leq 1$, and ${\mbox {G-{\rm dim}}}_{R_{\mathfrak p}} (M_{\mathfrak p}) \leq {\rm depth}(R_{\mathfrak p})-2 $ for ${\mathfrak p}\in {\rm Spec} (R) $ with ${\rm depth}(R_{\mathfrak p})\geq 2 $. This gives a generalization of Serre and Samuel's results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for $n\geq 2$ we give a characterization of $n$-Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown that every $R$-module has a $k$-torsionless cover provided $R$ is a $k$-Gorenstein ring.
In this paper, we consider the global existence, uniqueness and $L^{\infty }$ estimates of weak solutions to quasilinear parabolic equation of $m$-Laplacian type $u_{t}-\mathop {\rm div}(|\nabla u|^{m-2}\nabla u)=u|u|^{\beta -1}\int _{\Omega } |u|^{\alpha } {\rm d} x$ in $\Omega \times (0,\infty )$ with zero Dirichlet boundary condition in $\partial \Omega $. Further, we obtain the $L^{\infty }$ estimate of the solution $u(t)$ and $\nabla u(t)$ for $t>0$ with the initial data $u_0\in L^q(\Omega )$ $(q>1)$, and the case $\alpha +\beta < m-1$.
In this paper $LJ$-spaces are introduced and studied. They are a common generalization of Lindelöf spaces and $J$-spaces researched by E. Michael. A space $X$ is called an $LJ$-space if, whenever $\lbrace A,B\rbrace $ is a closed cover of $X$ with $A\cap B$ compact, then $A$ or $B$ is Lindelöf. Semi-strong $LJ$-spaces and strong $LJ$-spaces are also defined and investigated. It is demonstrated that the three spaces are different and have interesting properties and behaviors.