We survey some unsolvable conjectures in finite p-groups and their research progress.

Suppose that H is a subgroup of a finite group G. We call H is semipermutable in G if HK = KH for any subgroup K of G such that (|H|, |K|) = 1; H is s-semipermutable in G if HG_p = G_pH, for any Sylow p-subgroup G_p of G such that (|H|, p) = 1. These two concepts have been received the attention of many scholars in group theory since they were introduced by Professor Zhongmu Chen in 1987. In recent decades, there are a lot of papers published via the application of these concepts. Here we summarize the results in this area and gives some thoughts in the research process.

This paper is a survey for development of linear distributed parameter system. At first we point out some questions existing in current study of control theory for the L^p linear system with an unbounded control operator and an unbounded observation operator, such as stabilization problem and observer theory that are closely relevant to state feedback operator. After then we survey briefly some results on relevant problems that are related to solvability of linear differential equations in general Banach space and semigroup perturbations. As a principle, we propose a concept of admissible state feedback operator for system (A, B). Finally we give an existence result of admissible state feedback operators, including semigroup generation and the equivalent conditions of admissibility of state feedback operators, for an L^p well-posed system.

The core of the nonparametric/semiparametric Bayesian analysis is to relax the particular parametric assumptions on the distributions of interest to be unknown and random, and assign them a prior. Selecting a suitable prior therefore is especially critical in the nonparametric Bayesian fitting. As the distribution of distribution, Dirichlet process (DP) is the most appreciated nonparametric prior due to its nice theoretical proprieties, modeling flexibility and computational feasibility. In this paper, we review and summarize some developments of DP during the past decades. Our focus is mainly concentrated upon its theoretical properties, various extensions, statistical modeling and applications to the latent variable models.

In this paper, we consider the weighted local polynomial calibration estimation and imputation estimation of a non-parametric function when the data are right censored and the censoring indicators are missing at random, and establish the asymptotic normality of these estimators. As their applications, we derive the weighted local linear calibration estimators and imputation estimations of the conditional distribution function, the conditional density function and the conditional quantile function, and investigate the asymptotic normality of these estimators. Finally, the simulation studies are conducted to illustrate the finite sample performance of the estimators.

The random weighting method is an emerging computing method in statistics. In this paper, we propose a novel estimation of the survival function for right censored data based on the random weighting method. Under some regularity conditions, we prove the strong consistency of this estimation.

An augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints is proposed based on a Löwner operator associated with a potential function for the optimization problems with inequality constraints. The favorable properties of both the Löwner operator and the corresponding augmented Lagrangian are discussed. And under some mild assumptions, the rate of convergence of the augmented Lagrange algorithm is studied in detail.