A closed subspace $M$ of the Hardy space $H^2(\mathbb{D}^2)$ over the bidisk is called submodule if it is invariant under multiplication by coordinate functions $z$ and $w.$ Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem. This paper proves that every finitely generated submodule $M$ containing $θ(z)−\varphi(w)$ is Hilbert-Schmidt, where $θ(z),$ $\varphi(w)$ are two finite Blaschke products.

Poisson-Nernst-Planck systems are basic models for electrodiffusion process, particularly, for ionic flows through ion channels embedded in cell membranes. In this article, we present a brief review on a geometric singular perturbation framework for analyzing the steady-state of a quasi-one-dimensional Poisson-Nernst-Planck model. The framework is based on the general geometric singular perturbed theory from nonlinear dynamical system theory and, most crucially, on the reveal of two specific structures of Poisson-Nernst-Planck systems. As a result of the geometric framework, one obtains a governing system–an algebraic system of equations that involves all physical quantities such as protein structures of membrane channels as well as boundary conditions, and hence, provides a complete platform for studying the interplay between protein structure and boundary conditions and effects on ionic flow properties. As an illustration, we will present concrete applications of the theory to several topics of biologically significant based on collaboration works with many excellent researchers.

Model averaging is a good alternative to model selection, which can deal with the uncertainty from model selection process and make full use of the information from various candidate models. However, most of the existing model averaging criteria do not consider the influence of outliers on the estimation procedures. The purpose of this paper is to develop a robust model averaging approach based on the local outlier factor (LOF) algorithm which can downweight the outliers in the covariates. Asymptotic optimality of the proposed robust model averaging estimator is derived under some regularity conditions. Further, we prove the consistency of the LOF-based weight estimator tending to the theoretically optimal weight vector. Numerical studies including Monte Carlo simulations and a real data example are provided to illustrate our proposed methodology.

Graphical models are wildly used to describe conditional dependence relationships among interacting random variables. Among statistical inference problems of a graphical model, one particular interest is utilizing its interaction structure to reduce model complexity. As an important approach to utilizing structural information, decomposition allows a statistical inference problem to be divided into some sub-problems with lower complexities. In this paper, to investigate decomposition of covariate-dependent graphical models, we propose some useful definitions of decomposition of covariate-dependent graphical models with categorical data in the form of contingency tables. Based on such a decomposition, a covariate-dependent graphical model can be split into some sub-models, and the maximum likelihood estimation of this model can be factorized into the maximum likelihood estimations of the sub-models. Moreover, some sufficient and necessary conditions of the proposed definitions of decomposition are studied.

The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered. Under the conditions that the coefficient is quasi-monotone up to a constant and the meshes are locally refined by using the newest vertex bisection algorithm, some uniform convergence results are proved for the standard multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on new nodes and their immediate neighbours. The multigrid V-cycle algorithm uses $\mathcal{O}(N)$ operations per iteration and is optimal.

The limit distribution for homogeneous Markov processes is studied extensively and well understood, but it is not the case for inhomogeneous Markov processes. In this paper, we review some recent results on inhomogeneous Markov processes generated by non-autonomous stochastic (partial) differential equations (SDE in short). Under some suitable conditions, we show that the distribution of recurrent solutions of SDEs constitutes the limit distribution of the corresponding inhomogeneous Markov processes.