Motivated by the recent works of Bringmann, Guerzhoy, Kent, and Ono [4] and Bringmann, Fricke, and Kent [3], we introduce $L$-series for vector-valued weakly holomorphic cusp forms, and mock modular period polynomials for vector-valued harmonic weak Maass forms. In particular, we will discuss an integral representation of this new $L$-series and the limiting behavior of special values. Moreover, we also give relations between mock modular periods and $L$-series for vector-valued harmonic weak Maass forms.

Let $A$ be a real quadratic order of discriminant $p$ or $4p$ with a prime $p$. In this paper we classify all proper totally imaginary quadratic $A$-orders $B$ with index $w(B) = [B^\times : A^\times] \gt 1$. We also calculate numerical invariants of these orders including the class number, the index $w(B)$ and the numbers of local optimal embeddings of these orders into quaternion orders. These numerical invariants are useful for computing the class numbers of totally definite quaternion algebras.

We study the finiteness of the sets $\operatorname{Ass}(H^d_{I, J}(M))$ and $\operatorname{Ass}(\operatorname{Hom}_R(R/I, H^d_{I, J}(M)))$ concerning Grothendieck's conjecture. We also show some properties of local cohomology modules $H^i_{I, J}(M)$ from the point of view of Serre subcategories.

We study Zariski decomposition with support in a negative definite cycle, a variation of Zariski decomposition introduced by Miyaoka [4]: given a negative definite cycle $G$, any $\mathbb{Q}$-divisor $D$ decomposes into the sum of a $G$-nef and a rigid $\mathbb{Q}$-divisor. We prove that such a decomposition actually exists for an arbitrary $\mathbb{Q}$-divisor. Moreover, we show that, under the hypothesis that $D$ is pseudo-effective, we can drop the assumption of $G$ being negative definite, and obtain decompositions of $D$ with respect to arbitrary cycles. Our methods are inspired by a work of Bauer [1], in which he gives a simpler proof of Zariski's original result [5], and by adapting his proof to other cases, we are able to provide an alternative approach to this circle of ideas.

We prove the sharp bound of at most $64$ lines on projective quartic surfaces $S \subset \mathbb{P}^3(\mathbb{C})$ (resp. affine quartics $S \subset \mathbb{C}^3$) that are not ruled by lines. We study configurations of lines on certain non-$K3$ surfaces of degree four and give various examples of singular quartics with many lines.

We give a construction which is Lie theoretic of anti-invariant Riemannian submersions from almost Hermitian manifolds, from quaternion manifolds, from para-Hermitian manifolds, from para-quaternion manifolds, and from octonian manifolds. This yields many compact Einstein examples.

In this paper, we consider with the stability of traveling wave fronts for the nonlocal diffusion equation with delay and global response. We first establish the existence and comparison theorem of solutions for the nonlocal reaction-diffusion equation by appealing to the theory of abstract functional differential equation. Then we further show that the traveling wave fronts are asymptotical stability with phase shift. Our main technique is the super and subsolution method coupled with the comparison principle and squeezing method.

In this paper, a generalized KdV-Caudrey-Dodd-Gibbon (KdV-CDG) equation is investigated, which describes certain situations in the fluid mechanics, ocean dynamics and plasma physics. By using Bell polynomials, a lucid and systematic approach is proposed to systematically study its Hirota's bilinear form and $N$-soliton solution, respectively. Furthermore, based on the Riemann theta function, the one-quasi- and two-quasi-periodic wave solutions are also constructed. Finally, an asymptotic relation of the quasi-periodic wave solutions are strictly analyzed to reveal the relations between quasi-periodic wave solutions and soliton solutions.

In this paper, we consider a basic $N$-dimensional competition model in the unstirred chemostat with Beddington-DeAngelis functional response. The bifurcation solutions from a simple eigenvalue and a double eigenvalue are obtained respectively. In particular, for the double eigenvalue, we establish the existence and stability of coexistence solutions by the techniques of space decomposition and Lyapunov-Schmidt procedure. Moreover, we describe the global structure of these bifurcation solutions.

In this paper, we consider a class of nonlocal dispersal equation with nonlocal time-delayed reaction. We prove that all noncritical wavefronts are globally exponentially stable by the weighted energy method and comparison principle. However, for the critical wavefronts, we prove that they are globally asymptotically stable.

Let $(\mathcal{X}, d, \mu)$ be a complete RD-space. Let $\rho$ be an admissible function on $\mathcal{X}$, which means that $\rho$ is a positive function on $\mathcal{X}$ and there exist positive constants $C_0$ and $k_0$ such that, for any $x, y \in \mathcal{X}$,\[ \rho(y) \leq C_0 [\rho(x)]^{1/(1+k_0)} [\rho(x) + d(x, y)]^{k_0/(1+k_0)}.\]In this paper, we define a space $\operatorname{VMO}_\rho(\mathcal{X})$ and show that it is the predual of the localized Hardy space $H^1_\rho(\mathcal{X})$ introduced by Yang and Zhou [14]. Then we prove a version of the classical theorem of Jones and Journé [7] on weak$^*$-convergence in $H^1_\rho(\mathcal{X})$. As an application, we give an atomic characterization of $H^1_\rho(\mathcal{X})$.

In this article, we establish the dual log-Brunn-Minkowski inequality and the dual log-Minkowski inequality. Moreover, the equivalence between the dual log-Brunn-Minkowski inequality and the dual log-Minkowski inequality is demonstrated.

This paper considers a stochastic logistic model with infinite delay and impulsive perturbation. Firstly, with the space $C_{g}$ as phase space, the definition of solution to a stochastic functional differential equation with infinite delay and impulsive perturbation is established. According to this definition, we show that our model has an unique global positive solution. Then we establish the sufficient and necessary conditions for extinction and stochastic permanence of the model. In addition, the effects of impulsive perturbation and delay on extinction and stochastic permanence are discussed, respectively.

Lanczos bidiagonalization method is the most popular method for computing some largest singular triplets of large matrices. In this method, $2m+1$ base vectors are generated from the $m$-step Lanczos bidiagonalization process, but only $2m$ of them are used to form the approximate singular vectors and one of them is not used. In this paper, we make two improvements on the classical Lanczos bidiagonalization method. Firstly, following Jia and Elsner's idea for eigenproblems [9], we form the new approximate singular vectors by minimizing the corresponding residual norms in subspaces generated by $2m+1$ base vectors to replace the old approximate singular vectors. Secondly, in the process of implicit restarting, we replace the classical exact shifts by new shifts based on the information of the new approximate singular vectors. The total extra cost of the new method can be neglected. Numerical experiments show that, after two improvements, the new method proposed in this paper performs much better than the classical Lanczos bidiagonalization method. It uses less restarts and CPU time to reach the desired convergence.