BOUNDARY BEHAVIOR OF CAUCHY-TYPE INTEGRALS IN CLIFFORD ANALYSIS

In this article,the authors discussed the boundary behavior for the Cauchy- type integrals with values in a Clifford algebra,obtained some Sochocki–Plemelj formulae and Privalov–Muskhelishvili theorems for the Cauchy-type integral taken over a smooth surface by rather simple method.     

Wall Rational Functions and Khrushchev’s Formula for Orthogonal Rational Functions

The Nevalinna–Pick algorithm yields a continued fraction expansion of every Schur function, whose approximants are identified. These approximants are quotients of rational functions which can be understood as the rational analogs of the Wall polynomials. The properties of these Wall rational functions and the corresponding approximants permit us to obtain a Khrushchev’s formula for orthogonal rational functions. An introduction to the convergence of the Wall approximants in the indeterminate case is presented. This work was partially realized during two stays of the second author at the Norwegian University of Science and Technology (NTNU) financed respectively by Secretaría de Estado de Universidades e Investigación from the Ministry of Education and Science of Spain and by the Department of Mathematical Sciences of NTNU. The work of the second author was also partially supported by the Spanish grants from the Ministry of Education and Science, project code MTM2005-08648-C02-01, and the Ministry of Science and Innovation, project code MTM2008-06689-C02-01, and by Project E-64 of Diputación General de Aragón (Spain).     

A Boundedness Criterion via Atoms for Linear Operators in Hardy Spaces

Let p∈(0,1] and s≥[n(1/p−1)], where [n(1/p−1)] denotes the maximal integer no more than n(1/p−1). In this paper, the authors prove that a linear operator T extends to a bounded linear operator from the Hardy space H ^p (ℝ ⁿ ) to some quasi-Banach space ℬ if and only if T maps all (p,2,s)-atoms into uniformly bounded elements of ℬ.      

Quadrature formula for approximating the singular integral of Cauchy type with unbounded weight function on the edges

New quadrature formulas (QFs) for evaluating the singular integral (SI) of Cauchy type with unbounded weight function on the edges is constructed. The construction of the QFs is based on the modification of discrete vortices method (MMDV) and linear spline interpolation over the finite interval [−1,1]. It is proved that the constructed QFs converge for any singular point x not coinciding with the end points of the interval [−1,1]. Numerical results are given to validate the accuracy of the QFs. The error bounds are found to be of order O(h^α|lnh|) and O(h|lnh|) in the classes of functions H^α([−1,1]) and C¹([−1,1]), respectively.     

Simpson校正公式

给出了Simpson校正公式的截断误差,分析了复化Simpson校正公式的收敛阶.数值算例验证了理论分析的正确性.     

Feynman-Kac Formula for Left Continuous Additive Functionals and Extended Kato Class Measures

The perturbation of the generator of a Borel right process by a signed measure is investigated, using probabilistic and analytic potential theoretical methods. We establish a Feynman-Kac formula associated with measures charging no polar set and belonging to an extended Kato class. A main tool of this approach is the validity of a Khas’minskii Lemma for Stieltjes exponentials of positive left continuous additive functionals.      

常息力更新场合有限时间破产概率对负相依索赔额的不敏感性

设索赔来到过程为具有常数利息力度的更新风险模型.在索赔额分布为负相依的次指数分布假定下,建立了有限时间破产概率的一个渐近等价公式.所得结果显示,在独立同分布索赔额情形,有限时间破产概率的有关渐近等价公式,在负相依场合依然成立.这表明有限时间破产概率对于索赔额的负相依结构是不敏感的.     

Rational approximations of the Arrhenius integral using Jacobi fractions and gaussian quadrature

The aim of this work is to find approaches for the Arrhenius integral by using the n-th convergent of the Jacobi fractions. The n-th convergent is a rational function whose numerator and denominator are polynomials which can be easily computed from three-term recurrence relations. It is noticed that such approaches are equivalent to the one established by the Gauss quadrature formula and it can be seen that the coefficients in the quadrature formula can be given as a function of the coefficients in the recurrence relations. An analysis of the relative error percentages in the approximations is also presented.     

The Vlasov–Poisson–Fokker–Planck equation in an interval with kinetic absorbing boundary conditions

We study the initial–boundary value problem for the Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions. We first prove the existence of weak solutions of the linearized equation in an interval with absorbing boundary conditions. Moreover, the weak solution converges to zero exponentially in time. Then we extend the above results to the fully nonlinear Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions; the existence and the longtime behavior of weak solutions. Finally, we prove that the weak solution is actually a classical solution by showing the hypoellipticity of the solution away from the grazing set and the Hölder continuity of the solution up to the grazing set.     

Schur positivity and log-concavity related to longest increasing subsequences

Chen proposed a conjecture on the log-concavity of the generating function for the symmetric group with respect to the length of longest increasing subsequences of permutations. Motivated by Chen’s log-concavity conjecture, Bóna, Lackner and Sagan further studied similar problems by restricting the whole symmetric group to certain of its subsets. They obtained the log-concavity of the corresponding generating functions for these subsets by using the hook-length formula. In this paper, we generalize and prove their results by establishing the Schur positivity of certain symmetric functions. This also enables us to propose a new approach to Chen’s original conjecture.