Variable Selection and Regularization in Quantile Regression via Minimum Covariance Determinant Based Weights

The importance of variable selection and regularization procedures in multiple regression analysis cannot be overemphasized. These procedures are adversely affected by predictor space data aberrations as well as outliers in the response space. To counter the latter, robust statistical procedures such as quantile regression which generalizes the well-known least absolute deviation procedure to all quantile levels have been proposed in the literature. Quantile regression is robust to response variable outliers but very susceptible to outliers in the predictor space (high leverage points) which may alter the eigen-structure of the predictor matrix. High leverage points that alter the eigen-structure of the predictor matrix by creating or hiding collinearity are referred to as collinearity influential points. In this paper, we suggest generalizing the penalized weighted least absolute deviation to all quantile levels, i.e., to penalized weighted quantile regression using the RIDGE, LASSO, and elastic net penalties as a remedy against collinearity influential points and high leverage points in general. To maintain robustness, we make use of very robust weights based on the computationally intensive high breakdown minimum covariance determinant. Simulations and applications to well-known data sets from the literature show an improvement in variable selection and regularization due to the robust weighting formulation.     

Multi-solitonic solutions for the variable-coefficient variant Boussinesq model of the nonlinear water waves

For the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth of the water, we consider a variable-coefficient variant Boussinesq (vcvB) model with symbolic computation. We construct the connection between the vcvB model and a variable-coefficient Ablowitz-Kaup-Newell-Segur (vcAKNS) system under certain constraints. Using the N-fold Darboux transformation of the vcAKNS system, we present two sets of multi-solitonic solutions for the vcvB model, which are expressed in terms of the Vandermonde-like and double Wronskian determinants, respectively. Dynamics of those solutions are analyzed and graphically discussed, such as the parallel solitonic waves, shape-changing collision, head-on collision, fusion-fission behavior and elastic-fusion coupled interaction.     

求条件极值的一种新方法

利用线性代数的理论方法，对多元函数求条件极值的拉格朗日乘数法加以改进，建立了求条件极值的一种新方法     

Vector bundles with a fixed determinant on an irreducible nodal curve

LetM be the moduli space of generalized parabolic bundles (GPBs) of rankr and degree dona smooth curveX. LetM _−L be the closure of its subset consisting of GPBs with fixed determinant− L. We define a moduli functor for whichM _−L is the coarse moduli scheme. Using the correspondence between GPBs onX and torsion-free sheaves on a nodal curveY of whichX is a desingularization, we show thatM _−L can be regarded as the compactified moduli scheme of vector bundles onY with fixed determinant. We get a natural scheme structure on the closure of the subset consisting of torsion-free sheaves with a fixed determinant in the moduli space of torsion-free sheaves onY. The relation to Seshadri-Nagaraj conjecture is studied.     

Newton’s method as a tool for finding the eigenvalues of certain two-parameter (multiparameter) spectral problems

An iterative algorithm is examined for finding the eigenvalues of the two-parameter (multiparameter) algebraic eigenvalue problem. This algorithm uses Newton’s method and an efficient numerical procedure for differentiating determinants. Some numerical examples are given.     

二类广义Vandermonde行列式的计算

给出了二类广义Vandermonde行列式计算的显式表示式.     

复函数Cauchy微分中值定理的推广

以分割区域D为基础将解析函数与共轭解析函数的微分中值定理推广到高阶形式.     

广义实正定矩阵的几个不等式

推广了文献[3]中的广义实正定矩阵的行列式不等式,同时给出了广义实正定矩阵的凸性不等式.     

复正定矩阵的Minkowski不等式

建立了复正定矩阵的几个行列式不等式,将正定Ｈｅｒｍｉｔｅ阵的Ｍｉｎｋｏｗｓｋｉ不等式、 Ｏｓｔｒｏｗｓｋｉ－Ｔａｕｓｓｋｙ不等式推广到了复正定矩阵上,推广改进了一些文献的结果．     

Hill-Type Formula and Krein-Type Trace Formula for Hamiltonian Systems

In this paper, we give a survey on the Hill-type formula and its applications.Moreover, we generalize the Hill-type formula for linear Hamiltonian systems and Sturm-Liouville systems with any self-adjoint boundary conditions, which include the standard Neumann, Dirichlet and periodic boundary conditions. The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions. Further, based on the Hill-type formula, we derive the Krein-type trace formula. As applications, we give nontrivial estimations for the eigenvalue problem and the relative Morse index.