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We introduce a notion of a Nijenhuis operator on a pre-Lie algebra from the deformation theory. Then we explain the relationships between Nijenhuis operators and O-operators and Rota-Baxter operators. In particular, one can use Nijenhuis operators on a pre-Lie algebra to construct compatible O-operators as well as compatible L-dendriform algebras. The roles of Nijenhuis operators on pre-Lie algebras in some geometric structures are also interpreted.

题目：Minimal Integrity Bases and Irreducible Function Bases of Isotropic Invariants of Two Third Order Tensors报告人：祁力群 教授 （香港理工大学 ）地点：宁静楼110室时间：2017年12月14日(周四)下午16:00欢迎大家前往听讲

Measurement errors and outliers often arise in longitudinal data, ignoring the effects of measurement errors and outliers will lead to seriously biased estimators. Therefore, it is important to take them into account in longitudinal data analysis. In this paper, we develop a robust estimating equation method for analysis of longitudinal data with covariate measurement errors and outliers. Specifically, we eliminate the effects of measurement errors by making use of the independence of replicate measurement errors and correct the bias induced by outliers through centralizing the matrix of error-prone covariates in the estimating equation.

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.