Motivated by Stanley and Stembridge's $(\mathbf{3}+\mathbf{1})$-free conjecture on chromatic symmetric functions, Foley, Hoàng and Merkel introduced the concept of strong $e$-positivity and conjectured that a graph is strongly $e$-positive if and only if it is (claw, net)-free. In order to study strongly $e$-positive graphs, they introduced the twinning operation on a graph $G$ with respect to a vertex $v$, which adds a vertex $v'$ to $G$ such that $v$ and $v'$ are adjacent and any other vertex is adjacent to both of them or neither of them. Foley, Hoàng and Merkel conjectured that if $G$ is $e$-positive, then so is the resulting twin graph $G_v$ for any vertex $v$. By considering the twinning operation on a subclass of tadpole graphs with respect to certain vertices we disprove the latter conjecture. We further show that if $G$ is $e$-positive, the twin graph $G_v$ and more generally the clan graphs $G^{(k)}_v$ ($k \geq 1$) may not even be $s$-positive, where $G^{(k)}_v$ is obtained from $G$ by applying $k$ twinning operations to $v$.

Xu and Wu proved that if every $5$-cycle of a planar graph $G$ is not simultaneously adjacent to $3$-cycles and $4$-cycles, then $G$ is $4$-choosable. In this paper, we improve this result as follows. If $G$ is a planar graph without pairwise adjacent $3$-, $4$-, $5$-, and $6$-cycle, then $G$ is $4$-choosable.

By using the properties of trigonometric sums and the estimates of $n$-dimensional Kloosterman sums, we study the high-dimensional D. H. Lehmer problem over incomplete intervals. First we generalize the previous results in [11] by presenting some sharp asymptotic formulae. Then with the aid of a more elementary method, we improve the error terms in a straight-forward manner.

Let $(A,\mathfrak{m})$ be a noetherian local ring and $J$ an $\mathfrak{m}$-primary ideal. Elias [3] proved that $\operatorname{depth}(G(J^k))$ is constant for $k \gg 0$ and denoted this number by $\sigma(J)$. In this paper, we prove the non-positivity for the Hilbert coefficients $e_i(J)$ under some conditions for $\sigma(J)$. In case of $J = Q$ is a parameter ideal, we establish bounds for the Hilbert coefficients of $Q$ in terms of the dimension and the first Hilbert coefficient $e_1(Q)$.

Let $L$ be an even positive definite lattice with no roots, i.e., $L(2) = \{ x \in L \mid (x|x) = 2 \} = \emptyset$. Let $g \in O(L)$ be an isometry of order $4$ such that $g^2 = -1$ on $L$. In this article, we determine the full automorphism group of the orbifold vertex operator algebra $V_L^{\widehat{g}}$. As our main result, we show that $\operatorname{Aut}(V_L^{\widehat{g}})$ is isomorphic to $N_{\operatorname{Aut}(V_L)}(\langle \widehat{g} \rangle)/\langle \widehat{g} \rangle$ unless $L \cong \sqrt{2} E_8$ or $BW_{16}$.

The aim of this paper is to study the existence of solutions to the variable-order fractional Laplacian as follows: \[ (-\Delta)^{s(\cdot)}u + V(x)u = \lambda f(x,u) \quad \textrm{in $\mathbb{R}^N$}, \] where $\lambda \gt 0$ is a parameter, $N \geq 1$, $(-\Delta)^{s(\cdot)}$ is the variable-order fractional Laplacian operator with $s(\cdot) \colon \mathbb{R}^N \times \mathbb{R}^N \to (0,1)$ is continuous function with $N \gt 2s^{+} \geq 2s(x,y)$ for all $(x,y) \in \mathbb{R}^N \times \mathbb{R}^N$, and $f$ has variable growth and $V$ satisfies some suitable assumptions. Using Mountain Pass Theorem, Fountain Theorem and Genus theory, we obtain the existence of weak solutions to above problem.

We propose a new definition of preimage entropy dimension for continuous maps on compact metric spaces, investigate fundamental properties of the preimage entropy dimension, and compare the preimage entropy dimension with the topological entropy dimension. The defined preimage entropy dimension satisfies various basic properties of topological entropy dimension, for example, the preimage entropy dimension of a subsystem is bounded by that of the original system and topologically conjugated systems have the same preimage entropy dimension. Also, we discuss the relation between the preimage entropy dimension and the preimage entropy.

Serre proved that a domain $Y$ in $\mathbb{C}^n$ is Stein if and only if $H^i(Y,\mathcal{O}_Y) = 0$ for all $i \gt 0$. Laufer showed that if $Y$ is an open subset of a Stein manifold of dimension $n$ and $H^i(Y,\mathcal{O}_Y)$ is a finite dimensional complex vector space for every $i \gt 0$, then $Y$ is Stein. Vâjâitu generalized these theorems to singular Stein space of dimension $n$. In this paper, we consider singular Stein spaces $X$ with arbitrary dimension and give necessary and sufficient conditions for an open subset $Y$ in $X$ to be Stein. We show that if $Y$ is an open subset of a reduced Stein space $X$ with arbitrary dimension and singularities, then $Y$ is Stein if and only if $H^i(Y,\mathcal{O}_Y)$ is a finite dimensional complex vector space for every $i \gt 0$. Without cohomology condition, if $X-Y$ is a closed subspace of $X$, then we show that the geometric condition of the boundary $X-Y$ determines the Steinness of $Y$. More precisely, we show that if $X$ is normal and the boundary $X-Y$ is the support of an effective $\mathbb{Q}$-Cartier divisor, or $X-Y$ is of pure codimension $1$ and does not contain any singular points of $X$, then $Y$ is Stein.

In this paper, using exponential stable property, the stochastic analysis techniques and a fixed-point theorem for condensing maps, we obtain the existence results of piecewise weighted pseudo almost periodic in distribution mild solutions for a class of impulsive partial stochastic differential equations driven by Lévy noise in Hilbert spaces under non-Lipschitz conditions. Furthermore, the exponential stable of mild solution in mean square is investigated. Finally, an example is presented to illustrate the results.