The first part of this article is an overview on some recent major developments in the field of analysis and partial different equations. It is a brief presentation given by the author at a round table discussion. The second part is a supplement of various details provided by several outstanding researchers on subjects.

The goal of this article is to provide a lower bound for the lifespan of smooth solutions to 3-D anisotropic incompressible Navier-Stokes system, which in particular extends a similar type of result for the classical 3-D incompressible Navier-Stokes system.

In this paper, an equivalence relation between the $ω$-limit set of initial values and the $ω$-limit set of solutions is established for the Cauchy problem of evolution $p$-Laplacian equation in the unbounded space $\mathcal{Y}$_$σ$($ℝ$^$N$). To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions, we establish the propagation estimate and the growth estimate for the solutions. It will be demonstrated that the equivalence relation can be used to study the asymptotic behavior of solutions.

We investigate the structure of a large precision matrix in Gaussian graphical models by decomposing it into a low rank component and a remainder part with sparse precision matrix. Based on the decomposition, we propose to estimate the large precision matrix by inverting a principal orthogonal decomposition (IPOD). The IPOD approach has appealing practical interpretations in conditional graphical models given the low rank component, and it connects to Gaussian graphical models with latent variables. Specifically, we show that the low rank component in the decomposition of the large precision matrix can be viewed as the contribution from the latent variables in a Gaussian graphical model. Compared with existing approaches for latent variable graphical models, the IPOD is conveniently feasible in practice where only inverting a low-dimensional matrix is required. To identify the number of latent variables, which is an objective of its own interest, we investigate and justify an approach by examining the ratios of adjacent eigenvalues of the sample covariance matrix. Theoretical properties, numerical examples, and a real data application demonstrate the merits of the IPOD approach in its convenience, performance, and interpretability.

A hybrid method is presented for determining maximal eigenvalue and its eigenvector (called eigenpair) of a large, dense, symmetric matrix. Many problems require finding only a small part of the eigenpairs, and some require only the maximal one. In a series of papers, efficient algorithms have been developed by Mufa Chen for computing the maximal eigenpairs of tridiagonal matrices with positive off-diagonal elements. The key idea is to explicitly construct effective initial guess of the maximal eigenpair and then to employ a self-closed iterative algorithm. In this paper, we will extend Mufa Chen's algorithm to find maximal eigenpair for a large scale, dense, symmetric matrix. Our strategy is to first convert the underlying matrix into the tridiagonal form by using similarity transformations. We then handle the cases that prevent us from applying Chen's algorithm directly, e.g., the cases with zero or negative super- or sub-diagonal elements. Serval numerical experiments are carried out to demonstrate the efficiency of the proposed hybrid method.