In this paper, recursive equations are obtained for compound distribution with the number of claims belonging to (a, b)-family and the severity distribution of the mixed type. Numerical methods to solve these equation...In this paper, recursive equations are obtained for compound distribution with the number of claims belonging to (a, b)-family and the severity distribution of the mixed type. Numerical methods to solve these equations are presented, and some numerical results are given.展开更多
In this paper, we study the local asymptotic behavior of the regression spline estimator in the framework of marginal semiparametric model. Similarly to Zhu, Fung and He (2008), we give explicit expression for the asy...In this paper, we study the local asymptotic behavior of the regression spline estimator in the framework of marginal semiparametric model. Similarly to Zhu, Fung and He (2008), we give explicit expression for the asymptotic bias of regression spline estimator for nonparametric function f. Our results also show that the asymptotic bias of the regression spline estimator does not depend on the working covariance matrix, which distinguishes the regression splines from the smoothing splines and the seemingly unrelated kernel. To understand the local bias result of the regression spline estimator, we show that the regression spline estimator can be obtained iteratively by applying the standard weighted least squares regression spline estimator to pseudo-observations. At each iteration, the bias of the estimator is unchanged and only the variance is updated.展开更多
In this paper, for b ∈ （-∞,∞） and b ≠ -1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation ut ＋ 2kux - uxxt ＋ （1 ＋ b）u2ux =buxuxx ＋ uuxxx, which contains the generaliz...In this paper, for b ∈ （-∞,∞） and b ≠ -1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation ut ＋ 2kux - uxxt ＋ （1 ＋ b）u2ux =buxuxx ＋ uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable a. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for a. When a tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = -1 or b = -2, we guess that the equation does not have any one of above solutions.展开更多
The authors prove the Hyers-Ulam-Rassias stability of the quadratic mapping in Banachmodules over a unital C*-algebra, and prove the Hyers-Ulam-Rassias stability of the quadraticmapping in Banach modules over a unital...The authors prove the Hyers-Ulam-Rassias stability of the quadratic mapping in Banachmodules over a unital C*-algebra, and prove the Hyers-Ulam-Rassias stability of the quadraticmapping in Banach modules over a unital Banach algebra.展开更多
In this article, we study the blow-up solutions for a case of b-family equations.Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up ...In this article, we study the blow-up solutions for a case of b-family equations.Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up solutions: the hyperbolic blow-up solution, the fractional blow-up solution, the trigonometric blow-up solution, the first elliptic blow-up solution, and the second elliptic blow-up solution. Not only are the expressions of these blow-up solutions given, but also their relationships are discovered. In particular, it is found that two bounded solitary solutions are bifurcated from an elliptic blow-up solution.展开更多
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.19831020&10071003).
文摘In this paper, recursive equations are obtained for compound distribution with the number of claims belonging to (a, b)-family and the severity distribution of the mixed type. Numerical methods to solve these equations are presented, and some numerical results are given.
基金supported by National Natural Science Foundation of China (Grant Nos.10671038,10801039)Youth Science Foundation of Fudan University (Grant No.08FQ29)Shanghai Leading Academic Discipline Project (Grant No.B118)
文摘In this paper, we study the local asymptotic behavior of the regression spline estimator in the framework of marginal semiparametric model. Similarly to Zhu, Fung and He (2008), we give explicit expression for the asymptotic bias of regression spline estimator for nonparametric function f. Our results also show that the asymptotic bias of the regression spline estimator does not depend on the working covariance matrix, which distinguishes the regression splines from the smoothing splines and the seemingly unrelated kernel. To understand the local bias result of the regression spline estimator, we show that the regression spline estimator can be obtained iteratively by applying the standard weighted least squares regression spline estimator to pseudo-observations. At each iteration, the bias of the estimator is unchanged and only the variance is updated.
基金Supported by the National Natural Science Foundation of China(No.11401222)Natural Science Foundation of Guangdong Province(No.S2012040007959)+1 种基金The Fundamental Research Funds for the Central Universities(No.2014ZZ0064)Pearl River Science and Technology Nova Program of Guangzhou
文摘In this paper, for b ∈ （-∞,∞） and b ≠ -1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation ut ＋ 2kux - uxxt ＋ （1 ＋ b）u2ux =buxuxx ＋ uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable a. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for a. When a tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = -1 or b = -2, we guess that the equation does not have any one of above solutions.
基金Project supported by grant No.KRF-2000-015-DP0038.
文摘The authors prove the Hyers-Ulam-Rassias stability of the quadratic mapping in Banachmodules over a unital C*-algebra, and prove the Hyers-Ulam-Rassias stability of the quadraticmapping in Banach modules over a unital Banach algebra.
基金partially supported by the National Natural Science Foundation of China(11771152,11971176)Guangdong Basic and Applied Basic Research Foundation(2019B151502062)the Fundamental Research Founds for the Central Universities(2019MS111)。
文摘In this article, we study the blow-up solutions for a case of b-family equations.Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up solutions: the hyperbolic blow-up solution, the fractional blow-up solution, the trigonometric blow-up solution, the first elliptic blow-up solution, and the second elliptic blow-up solution. Not only are the expressions of these blow-up solutions given, but also their relationships are discovered. In particular, it is found that two bounded solitary solutions are bifurcated from an elliptic blow-up solution.
