This paper considers an impulsive stochastic logistic model with infinite delay at the phase space $C_{g}$. Firstly, the definition of solution to an impulsive stochastic functional differential equation with infinite delay is established. Based on this definition, we show that our model has a unique global positive solution. Then we establish the sufficient conditions for extinction, nonpersistence in the mean, weak persistence and stochastic permanence of the solution. The threshold between weak persistence and extinction is obtained. In addition, the effects of impulsive perturbation and delay on persistence and extinction are discussed, respectively. Finally, numerical simulations are introduced to support the theoretical analysis results.

Corresponding to every finite simplicial complex $K$, there is a moment-angle complex $\mathcal{Z}_{K}$; if $K$ is a triangulation of a sphere, $\mathcal{Z}_{K}$ is a compact manifold. The question of whether $\mathcal{Z}_{K}$ is a connected sum of sphere products was considered in [3, Section 11]. So far, all known examples of moment-angle manifolds which are homeomorphic to connected sums of sphere products have the property that every product is of exactly two spheres. In this paper, we give a example whose cohomology ring is isomorphic to that of a connected sum of sphere products with one product of three spheres. We also give some general properties of this kind of moment-angle manifolds.

We will give an explicit description of the center of the De Concini--Kac type specialization of a quantized enveloping algebra at an even root of unity. The case of an odd root of unity was already dealt with by De Concini--Kac--Procesi. Our description in the even case is similar to but a little more complicated than the odd case.

Let $f\colon S \to \mathbb{P}^{1}$ be a family of genus $g \geq 2$ curves with two singular fibers $F_{1}$ and $F_{2}$. We show that $F_{1} = {F_{2}}^{*}$ and $F_{2} = {F_{1}}^{*}$ are dual to each other, $S$ is a ruled surface, the geometric genera of the singular fibers are equal to the irregularity of the surface, and the virtual Mordell--Weil rank of $f$ is zero. We prove also that $c_{1}^{2}(S) \leq -2$ if $g = 2$, and $c_{1}^{2}(S) \leq -4$ if $g > 2$. As an application, we will classify all such fibrations of genus $g = 2$.

In this paper, we give a criterion for $4$-noids to have nullity greater than $3$ and its applications. We also compute the indices and the nullities of some families of $\mathbf{Z}_{N}$-invariant $n$-noids, and analyze the correspondence between nullity and a flux map.

Spectral measures of Wigner matrices are investigated. The Wigner semicircle law for spectral measures is proved. Regard this as the law of large number, the central limit theorem moments of spectral measures is also derived. The proof is based on moment method and combinatorial method.

We consider the substitution $\sigma_{a, b}$ defined by \begin{align*} \sigma_{a, b}\colon & 1 \mapsto \underbrace{1\ldots 1}_{a}2{,} \\ & 2 \mapsto \underbrace{1\ldots 1}_{b}3{,} \\ & 3 \mapsto 1 \end{align*} with $a \geq b \geq 1$. The shift dynamical system induced by $\sigma_{a, b}$ is measure theoretically isomorphic to an exchange of three domains on a compact tile $\mathcal{T}_{a, b}$ with fractal boundary. We prove that $\mathcal{T}_{a, b}$ is homeomorphic to the closed disk iff $2b-a\leq 3$. This solves a conjecture of Shigeki Akiyama posed in 1997. To this effect, we construct a Hölder continuous parametrization $C_{a, b}\colon \mathbb{S}^{1} \to \partial\mathcal{T}_{a, b}$ of the boundary of $\mathcal{T}_{a, b}$. As a by-product, this parametrization gives rise to an increasing sequence of polygonal approximations of $\partial\mathcal{T}_{a, b}$, whose vertices lye on $\partial\mathcal{T}_{a, b}$ and have algebraic pre-images in the parametrization.

When two Radon measures on the half line are given, the harmonic mean of their Stieltjes transforms is again the Stieltjes transform of a Radon measure. We study the relationship between the asymptotic behavior of the resulting measure and those of the original ones. The problem comes from the spectral theory of second--order differential operators and the results are applied to linear diffusions neither boundaries of which is regular.

By a theorem of A'Campo, the eigenvalues of certain Coxeter transformations are positive real or lie on the unit circle. By optimally bounding the signature of tree-like positive Hopf plumbings from below by the genus, we prove that at least two thirds of them lie on the unit circle. In contrast, we show that for divide links, the signature cannot be linearly bounded from below by the genus.

Let $r$ be a positive integer. An $r$-set is a pair $X= (V(X), R(X))$ consisting of a set $V(X)$ with a subset $R(X)$ of the direct product $V(X)^{r}$. The object of this paper is to investigate the Hom complexes of $r$-sets, which were introduced for graphs in the context of the graph coloring problem. In the first part, we introduce simplicial sets which we call singular complexes, and show that singular complexes and Hom complexes are naturally homotopy equivalent. The second part is devoted to the generalization of $\times$-homotopy theory established by Dochtermann. We show the folding theorem for hypergraphs which was partly proved by Iriye and Kishimoto.