赋范线性空间中最小时间函数的ε-次微分

在赋范线性空间中,本文讨论最小时间函数TS的ε次微分计算公式.T_S由非空闭集S和非空闭凸集U决定,并以距离函数和指示函数为其特例.本文利用新的讨论方法取消了已有结果中的重要假设：T_S满足calmness条件,建立了T_S在集合S外的点处的ε-次微分的下估计式,该估计式由相应的法锥和集合U的支撑函数的次水平集表示.     

Subdifferential properties for a class of minimal time functions with moving target sets in normed spaces

In a normed vector space, we study the minimal time function determined by a moving target set and a differential inclusion, where the set-valued mapping involved has constant values of a bounded closed convex set U. After establishing a characterization of ?-subdifferential of the minimal time function, we obtain that the limiting subdifferential of the minimal time function is representable by virtue of the corresponding normal cones of sublevel sets of the function and level or sublevel sets of the support function of U. The known results require the set U to have the origin as an interior point and the target set is a fixed set.     

Integrals and derivatives connected to the r-Lambert function

In this paper, we derive some formulas for higher order derivatives of r-Lambert functions. Moreover, an integration formula involving powers of r-Lambert function is obtained.     

On the discounted penalty function in the discrete time stationary renewal risk model

In this paper we consider the discrete time stationary renewal risk model. We express the Gerber-Shiu discounted penalty function in the stationary renewal risk model in terms of the corresponding Gerber-Shiu function in the ordinary model. In particular, we obtain a defective renewal equation for the probability generating function of ruin time. The solution of the renewal equation is then given. The explicit formulas for the discounted survival distribution of the deficit at ruin are also derived.     

一类带扰动和附索赔风险模型的破产概率

本文研究带扰动和附索赔的风险模型.利用鞅方法,得到了破产概率的指数上界及精确表达式,推广了不带扰动的风险模型相应的结论.     

The state constrained bilateral minimal time function

We generalize, to the bilateral case (that is, with variable initial and end points), the main results of Nour and Stern [C. Nour, R.J. Stern, Regularity of the state constrained minimal time function, Nonlinear Anal. 66 (1) (2007) 62–72] and Stern [R.J. Stern, Characterization of the state constrained minimal time function, SIAM J. Control Optim. 43 (2004) 697–707], where the regularity and Hamilton–Jacobi characterization of the state constrained (unilateral) minimal time function were studied.     

Numerical Analysis of a Nonlinear Singularly Perturbed Delay Volterra Integro-Differential Equation on an Adaptive Grid

In this paper, we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.     

Perturbed Recurrence Relations

In this paper, we give some new relations between two families of polynomials defined by three-term recurrence relations. These relations allow us to study how some properties of a family of orthogonal polynomials are affected when the coefficients of the recurrence relation are perturbed. In the literature some methods are already available. However, most of them are only effective for small perturbations. In order to show the sharpness of our method, we compare it with Gronwall's classical method in the case of large perturbations. Using our tool, we also give a relation between the differential equation satisfied by a family of orthogonal polynomials and its perturbed family. Some explicit results are obtained for Chebyshev polynomials of the second kind.     

分解与积分有理函数的直接方法

通过多项式分析,构造出用于确定有理函数分解式系数的两种直接公式.为了提高公式的使用效率,引入先期分解的概念,给出了公式的连环应用方法.同时,将公式的使用范围扩展到复数域,建立起适用于共轭复根的积分公式,避免了有理函数积分的递推过程.     

Bifurcation of limit cycles from a heteroclinic loop with a cusp

In this article, we study the expansion of the first Melnikov function of a near-Hamiltonian system near a heteroclinic loop with a cusp and a saddle or two cusps, obtaining formulas to compute the first coefficients of the expansion. Then we use the results to study the problem of limit cycle bifurcation for two polynomial systems.     

一类log-cosine和log-sine积分的闭形式

考虑了Beta函数偏导数的递推公式的表示问题.利用Beta函数偏导数的递推公式,给出一类log--cosine和log-sine积分的闭形式.     

Green's function of two-dimensional time-fractional diffusion equation using addition formula of Wright function

In this paper, using the Mellin transform of Wright function we derive an addition formula for the Wright function. In some special cases, addition formulas for the Hermite, Bessel and Mittag-Leffler functions are also given and the Green's function of two-dimensional time-fractional diffusion equation is presented in the whole plane.     

Numerical Quadratures for Hadamard Hypersingular Integrals

In this paper,we develop Gaussian quadrature formulas for the Hadamard fi- nite part integrals.In our formulas,the classical orthogonal polynomials such as Legendre and Chebyshev polynomials are used to approximate the density function f(x)so that the Gaussian quadrature formulas have degree n-1.The error estimates of the formulas are obtained.It is found from the numerical examples that the convergence rate and the accu- racy of the approximation results are satisfactory.Moreover,the rate and the accuracy can be improved by choosing appropriate weight functions.     

On a class of renewal risk model with random income

In this paper, we consider a renewal risk process with random premium income based on a Poisson process. Generating function for the discounted penalty function is obtained. We show that the discounted penalty function satisfies a defective renewal equation and the corresponding explicit expression can be obtained via a compound geometric tail. Finally, we consider the Laplace transform of the time to ruin, and derive the closed‐form expression for it when the claims have a discrete K_m distribution (i.e. the generating function of the distribution function is a ratio of two polynomials of order m∈?⁺). Copyright © 2008 John Wiley & Sons, Ltd.     

On the expected discounted penalty function for risk process with tax

In this paper we consider the generalized Cramér-Lundberg risk model including tax payments. We investigate how tax payments affect the behavior of a Cramér-Lundberg surplus process by defining an expected discounted penalty function at ruin. We derive an explicit expression for this function by solving a differential equation. Consequently, the explicit formulas for the discounted probability density function of the surplus immediately before ruin and the discounted joint probability density function of the surplus immediately before ruin and the deficit at ruin are obtained. We also give explicit expressions for the function for exponential claims.     

A note on asymptotic behaviour of Mittag–Leffler functions

In this paper, we give a short note on the asymptotic behaviour of the two parameter Mittag–Leffler function. Useful results are collected for the reader and also explicit estimation formulas for this function are obtained which will play a role in existence and stability theory of fractional differential equations.     

On asymptotics of the q-exponential and q-gamma functions

In this work we present a derivation for the complete asymptotic expansions of Euler?s q-exponential function and Jackson?s q-gamma function via Mellin transform. These formulas are valid everywhere, uniformly on any compact subset of the complex plane.     

Limit cycle bifurcations by perturbing a class of integrable systems with a polycycle

In this paper, we deal with the problem of limit cycle bifurcation near a 2-polycycle or 3-polycycle for a class of integrable systems by using the first order Melnikov function. We first get the formal expansion of the Melnikov function corresponding to the heteroclinic loop and then give some computational formulas for the first coefficients of the expansion. Based on the coefficients, we obtain a lower bound for the maximal number of limit cycles near the polycycle. As an application of our main results, we consider quadratic integrable polynomial systems, obtaining at least two limit cycles.     

Schwarz引理的推广及其应用

引入Schwarz引理的一个最常见的推广定理,并且作出了详细的证明.同时以引理形式介绍了一个实用的复数性质,并且利用这两个引理,给出了开圆盘内解析函数的的实部,虚部以及模的估计式.