We characterize the property $(W_{E})$ in terms of topological uniform descent, and give the transfer of property $(W_{E})$ from bounded linear operators to functions of operators. Moreover, the perturbation of property $(W_{E})$ under power finite rank operators is investigated. As an application, the property for hypercyclic operators and supercyclic operators is investigated.

We introduce and study topological asymptotic sensitivity, Li-Yorke topological sensitivity, pointwise topological sensitivity for general topological spaces. We study relations among these variants of topological sensitivities and $\mathcal F$ topological sensitivity, strong topological sensitivity and multi topological sensitivity. We prove that strong topological sensitivity (respectively multi topological sensitivity) on the hyperspace dynamical system is equivalent to strong topological sensitivity (respectively multi topological sensitivity) of a dynamical system for uniform compact spaces. It is shown that Li-Yorke sensitivity and Li-Yorke topological sensitivity are equivalent on compact metric spaces. We also show that on compact Hausdorff topological spaces topological sensitivity implies topological asymptotic sensitivity. Moreover, we provide necessary examples and counterexamples.

In this paper, we establish the existence of a positive ground state solution for a weighted problem under boundary Dirichlet condition in the unit ball of $\mathbb R^N,$ $N > 2$. The nonlinearity of the equation is critical or subcritical growth in view of Trudinger-Moser inequalities. In order to obtain our existence result we used minimax techniques combined with Trudinger-Moser inequalities. In the critical case, the associated energy does not satisfy the condition of compactness. We provide a new condition for growth and we stress its importance to check compactness levels.

The new Euler-Jacobi formula for double points provides an algebraic relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the planar quadratic--quartic polynomial differential systems when these systems have seven finite singular points being one of them with multiplicity two. The case with eight finite singular points has already been solved using the classical Euler-Jacobi formula.

In this short note we present a generating function computing the compactly supported Euler characteristic $\chi_c(F(X, n), K^{\boxtimes n})$ of the configuration spaces on a topologically stratified space $X$, with $K$ a constructible complex of sheaves on $X$, and we obtain as a special case a generating function for the Euler characteristic $\chi(F(X, n))$. We also recall how to use existing results to turn our computation of the Euler characteristic into a computation of the equivariant Euler characteristic.

We introduce relative CS-Rickart objects in abelian categories, as common generalizations of relative Rickart objects and extending objects. We study direct summands and (co)products of relative CS-Rickart objects as well as classes all of whose objects are self-CS-Rickart. Corresponding results for dual relative CS-Rickart objects may be automatically obtained by the duality principle. Applications are given to Grothendieck categories and, in particular, to module and comodule categories.

In 2001, Arhangel'skii proved that a topological group that is factorizable over the class of (strong) $PT$-groups, is a (strong) $PT$-group. Wegeneralize this result to Tychonoff quasitopological groups and prove that a quasitopological group $G$ that is factorizable over the class of quasi-$PT$-groups, is a quasi-$PT$-group. We define the notion of $\mathcal{P}$-sz-factorizability, generalizing $\mathcal{P}$-factorizability and prove that a (quasi)topological group $G$ that is sz-factorizable over the class of (quasi-)$PT$-groups, is a (quasi-)$PT$-group. A topological group $G$ that is sz factorizable over the class of strong $PT$-groups, is a strong $PT$-group. We also present some other results on quasi-$PT$-groups and $PT$-groups.

In this paper, we introduced the mildly version of the Hurewicz basis covering property, studied by Babinkostova, Kočinac, and Scheepers. A space $X$ is said to have mildly-Hurewicz property if for each sequence $\langle\mathcal{U}_n : n \in \omega \rangle$ of clopen covers of $X$ there is a sequence $\langle \mathcal{V}_n : n \in \omega \rangle$ such that for each $n$, $\mathcal{V}_n$ is a finite subset of $\mathcal{U}_n$ and for each $x \in X$, $x$ belongs to $\bigcup \mathcal{V}_n$ for all but finitely many $n$. Then we characterized mildly-Hurewicz property by mildly-Hurewicz Basis property and mildly-Hurewicz measure zero property for metrizable spaces.