Motivated by the analysis of quality of life data from a clinical trial on early breast cancer, we propose in this generalized partially linear mean-covariance regression model for longitudinal proportional data, which are bounded in a closed interval. Cholesky decomposition of the covariance matrix for within-subject responses and generalized estimation equations are used to estimate unknown parameters and the nonlinear function in the model. Simulation studies are performed to evaluate the performance of the proposed estimation procedures. Our new model is also applied to analyze the data from the cancer clinical trial that motivated this research.

In this paper, we consider a high-dimensional quantile regression model for panel data with multiple structural breaks. We develop a penalzied estimation method with both slope coefficients and individual fixed effects by combining the informations over multiple quantile levels. We show that with probability tending to one our proposed method can correctly determine the unknown number of the the breaks and estimate the common break dates consistently. The asymptotic distributions of the Lasso estimators of the regression coefficients and the post Lasso versions are also established. Simulation results demostrate that the proposed method works well in the finite samples. The perfomance of the the proposed method is futher illustrated by the analysis of a environmental Kuznets curves data.

We study the resolvent and spectral measure of certain doubly infinite Jacobi matrices via asymptotic solutions of two-sided difference equations. By finding the subdominant (or minimal) solutions or calculating the continued fractions for the difference equations, we derive explicit formulas for the matrix entries of resolvent of doubly infinite Jacobi matrices corresponding to Lommel polynomials, associated ultraspherical polynomials, and Al-Salam-Ismail polynomials. The spectral measures are then obtained by inverting Stieltjes transformations.

The sterile insect technique (SIT) is an effective weapon to prevent transmission of mosquito-borne diseases, in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population. To study the impact of the sterile insect technique on the disease transmission, we formulate stage-structured discrete-time models for the interactive dynamics of the wild and sterile mosquitoes using Beverton-Holt-type of survivability, based on difference equations. We incorporate different strategies for releasing sterile mosquitoes, and investigate the model dynamics. Numerical simulations are also provided to demonstrate dynamical features of the models.

The irreducible modules for the $/Z_{2}$-orbifold vertex operator subalgebra of the parafermion vertex operator algebra associated to the integrable highest weight modules for the affine Kac-Moody algebra $A_1^{(1)}$ of level $k$ are classified and constructed. This is a joint work with Cuipo Jiang.

In this talk we provide a new method to study global dynamics of planar quasi homogeneous differential systems. We first prove that all planar quasi-homogeneous polynomial differential systems can be translated into homogeneous differential systems and show that all quintic quasi-homogeneous but non-homogeneous systems can be reduced to four homogeneous ones. Then we present some properties of homogeneous systems, which can be used to discuss the dynamics of quasi-homogeneous systems.

This talk concerns with a reaction-diffusion system modeling the population dynamics of two predators and one prey with nonlinear prey-taxis. We first investigate the global existence and boundedness of solution for the general model. Then we study the global stabilities of nonnegative spatially homogeneous equilibria for an explicit system with type I functional responses and density-dependent death rates for the predators and logistic growth for the prey. Moreover, the convergence rates are established.

In this talk, we will talk about the rotational symmetry of gradient steady ricci solitons with linear dacay. Perelmann conjectured that any 3-d nonflat noncollapsed steady ricci solitons must be rotationally symmetric. This conjecture has been solved by Brendle aroung 2012. Recently, we generalize this result to noncollapsed steady ricci solitons with nonnegative curvature operator in high dimensions in addition that the scalar curvature has linear decay. This is a joint work with Prof. Xiaohua Zhu.