Let Y be an open subset of a reduced compact complex space X such that X - Y is the support of an effective divisor D. If X is a surface and D is an effective Weil divisor, we give sufficient conditions so that Y is Stein. If X is of pure dimension d ≥ 1 and X - Y is the support of an effective Cartier divisor D, we show that Y is Stein if Y contains no compact curves, ${H}^{i}\left(Y{,}_{Y}\right)=0$ for all i > 0, and for every point x₀ ∈ X-Y there is an n ∈ ℕ such that ${\Phi}_{\left|nD\right|}^{-1}\left({\Phi}_{\left|nD\right|}\left(x\u2080\right)\right)\cap Y$ is empty or has dimension 0, where ${\Phi}_{\left|nD\right|}$ is the map from...

Let $(M,g)$ be a complete noncompact Riemannian manifold. We consider gradient estimates on positive solutions to the following nonlinear equation ${\Delta}_{f}u+c{u}^{-\alpha}=0$ in $M$, where $\alpha $, $c$ are two real constants and $\alpha >0$, $f$ is a smooth real valued function on $M$ and ${\Delta}_{f}=\Delta -\nabla f\nabla $. When $N$ is finite and the $N$-Bakry-Emery Ricci tensor is bounded from below, we obtain a gradient estimate for positive solutions of the above equation. Moreover, under the assumption that $\infty $-Bakry-Emery Ricci tensor is bounded from below and $\left|\nabla f\right|$ is bounded from above,...

In this paper we observe that the minimal signless Laplacian spectral radius is obtained uniquely at the kite graph PKn−ω,ω among all connected graphs with n vertices and clique number ω. In addition, we show that the spectral radius μ of PKm,ω (m ≥ 1) satisfies [...] More precisely, for m > 1, μ satisfies the equation [...] where [...] and [...] . At last the spectral radius μ(PK∞,ω) of the infinite graph PK∞,ω is also discussed.

In this note, we show how the determinant of the q-distance matrix Dq(T) of a weighted directed graph G can be expressed in terms of the corresponding determinants for the blocks of G, and thus generalize the results obtained by Graham et al. [R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85-88]. Further, by means of the result, we determine the determinant of the q-distance matrix of the graph obtained from a connected weighted graph...

Let ${\mathcal{L}}_{1}=-\Delta +V$ be a Schrödinger operator and let ${\mathcal{L}}_{2}={(-\Delta )}^{2}+{V}^{2}$ be a Schrödinger type operator on ${\mathbb{R}}^{n}$
$(n\ge 5)$, where $V\ne 0$ is a nonnegative potential belonging to certain reverse Hölder class ${B}_{s}$ for $s\ge n/2$. The Hardy type space ${H}_{{\mathcal{L}}_{2}}^{1}$ is defined in terms of the maximal function with respect to the semigroup $\left\{{\mathrm{e}}^{-t{\mathcal{L}}_{2}}\right\}$ and it is identical to the Hardy space ${H}_{{\mathcal{L}}_{1}}^{1}$ established by Dziubański and Zienkiewicz. In this article, we prove the ${L}^{p}$-boundedness of the commutator ${\mathcal{R}}_{b}=b\mathcal{R}f-\mathcal{R}\left(bf\right)$ generated by the Riesz transform $\mathcal{R}={\nabla}^{2}{\mathcal{L}}_{2}^{-1/2}$, where $b\in {\mathrm{BMO}}_{\theta}\left(\rho \right)$, which is larger than the...

The connected dominating set (CDS) has become a well-known approach for constructing a virtual backbone in wireless sensor networks. Then traffic can forwarded by the virtual backbone and other nodes turn off their radios to save energy. Furthermore, a smaller CDS incurs fewer interference problems. However, constructing a minimum CDS is an NP-hard problem, and thus most researchers concentrate on how to derive approximate algorithms. In this paper, a novel algorithm based on the induced tree of...

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