Pathwise Uniqueness of the Solutions toVolterra Type Stochastic DifferentialEquations in the Plane

In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the non-Lipschitz conditions. We use a martingale formula in stead of Ito formula, which leads to simplicity the process of proof and extends the result to unbounded coefficients case.     

INTEGRABILITY AND LINEARIZABILITY FOR A CLASS OF CUBIC KOLMOGOROV SYSTEMS

The integrability and linearizability for a class of cubic Kolmogorov systems are studied. A recursive formula to compute the saddle quantities of the systems is deduced firstly, and integrable conditions for the systems are obtained. Then a recursive formula to compute the coefficients of the normal form for saddle points of the systems is also applied. Finally linearizable conditions of the origin for the systems are given. Both formulas to find necessary conditions are all linear and readily done using c...     

THE POINCAR-BERTRAND FORMULA ON THE BUILDING DOMAIN OF COMPLEX BIBALLS

The Poincaré-Bertrand formula takes an important position in the study of complex singular integral.The Poincaré-Bertrand formula on the complex sphere in the multidimensional complex Euclidian spaces was given by Sheng Gong.Using the method of solid angular coefficient,the authors extend the Poincaré-Bertrand formula on the complex sphere to the building domain of the complex biballs,and obtain a more general Poincaré-Bertrand formula with the solid angular coefficients.     

THE PERMUTATION FORMULA OF SINGULAR INTEGRALS WITH BOCHNER-MARTINELLI KERNEL ON STEIN MANIFOLDS

Using the method of localization, the authors obtain the permutation formula of singular integrals with Bochner-Martinelli kernel for a relative compact domain with C(1) smooth boundary on a Stein manifold. As an application the authors discuss the regularization problem for linear singular integral equations with Bochner-Martinelli kernel and variable coefficients; using permutation formula, the singular integral equation can be reduced to a fredholm equation.     

基于切比雪夫节点的一类带显式系数的广义高斯求积公式(英文)

The goal here is to give a simple approach to a quadrature formula based on the divided diffierences of the integrand at the zeros of the nth Chebyshev polynomial of the first kind,and those of the（n-1）st Chebyshev polynomial of the second kind.Explicit expressions for the corresponding coefficients of the quadrature rule are also found after expansions of the divided diffierences,which was proposed in[14].     

Tong-type identity and the mean square of the error term for an extended Selberg class

In 1956, Tong established an asymptotic formula for the mean square of the error term of the summatory function of the Piltz divisor function d3(n). The aim of this paper is to generalize Tong's method to a class of Dirichlet series L(s) which satisfies a functional equation. Let a(n) be an arithmetical function related to a Dirichlet series L(s), and let E(x) be the error term of ′n xa(n). In this paper, after introducing a class of Diriclet series with a general functional equation(which contains the well-known Selberg class), we establish a Tong-type identity and a Tong-type truncated formula for the error term of the Riesz mean of the coefficients of this Dirichlet series L(s). This kind of Tong-type truncated formula could be used to study the mean square of E(x) under a certain assumption. In other words, we reduce the mean square of E(x) to the problem of finding a suitable constant σ*which is related to the mean square estimate of L(s). We shall represent some results of functions in the Selberg class of degrees 2–4.     

Hodge integrals with one λ-class

Hodge integrals over moduli space of stable curves play an important roles in understanding the topological properties of moduli space.ELSV formula connects the Hodge integrals with Hurwitz numbers,and the generating function of Hurwitz numbers satisfies the cut-and-join equation.Therefore,it is natural to consider how to use the cut-and-join equation for Hurwitz numbers to compute Hodge integrals which appear in ELSV formula.In this paper,at first,we will review the method introduced in Goulden et al.’s paper to get the λ g conjecture for Hodge integral.Through some variables transformation,the generating function of Hurwitz number becomes a symmetric polynomial which satisfies a symmetrized cut-and-join equation.By comparing the coefficients of the lowest degree term of both sides in this equation,we can get the λ g conjecture.Then,in a similar way,we obtain our main result in this paper:a recursive formula for Hodge integral of type contains only one λ g 1-class.We also point out that our results are closely related to the degree 0 Virasoro conjecture for a curve.     

LIMIT CYCLES NEAR A DOUBLE HOMOCLINIC LOOP

In this article,we study the expansion of the first order Melnikov function in a near-Hamiltonian system on the plane near a double homoclinic loop.We obtain an explicit formula to compute the first four coeffcients,and then identify the method of finding at least 7 limit cycles near the double homoclinic loop using these coefficients.Finally,we present some interesting applications.     

A variational formula for controlled backward stochastic partial differential equations and some application

An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to be convex, and all coefficients of the system are allowed to be random. A variational formula for the functional in a given control process direction is derived, by the Hamiltonian and associated adjoint system. As an application, a global stochastic maximum principle of Pontraygins type for the optimal controls is established.     

A BLACK-SCHOLES FORMULA FOR OPTION PRICING WITH DIVIDENDS

Abstract. We obtain a Black-Scholes formula for the arbitrage-free pricing of Eu-ropean Call options with constant coefficients when the underlylng stock generatesdividends. To hedge the Call option, we will always borrow money from bank. We seethe influence of the dividend term on the option pricing via the comparison theoremof BSDE(backward stochastic di~erential equation [5], [7]). We also consider the option pricing problem in terms of the borrowing rate R whichis not equal to the interest rate r. The corresponding Black-Sdxoles formula is given.We notice that it is in fact the borrowing rate that plays the role in the pricing formula.     

A critical case of Rallis inner product formula

Letπbe a genuine cuspidal representation of the metaplectic group of rank n.We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension 2n+1.We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands L-function ofπtwisted by a character.The bulk of this paper focuses on proving a case of regularised Siegel-Weil formula,on which the Rallis inner product formula is based and whose proof is missing in the literature.     

On Radon-Penrose transformation and k-Cauchy-Fueter operator

It is known that there is a 1-1 correspondence between the first cohomology of the sheaf O(-k-2) over the projective space and the solutions to the k-Cauchy-Fueter equations on the quaternionic space Hn.We find an explicit Radon-Penrose type integral formula to realize this correspondence:given a -closed(0,1)form f with coefficients in the(-k-2)th power of the hyperplane section bundle H-k-2,there is an integral representation Pf such that ι*(Pf) is a solution to the k-Cauchy-Fueter equations,where ι is an embedding of the quaternionic space Hn into C4n.     

HIGHER ORDER SINGULAR INTEGRAL EQUATIONS ON COMPLEX HYPERSPHERE

A theory of a class of higher order singular integral under the operator(Lf)(u)=1/(ū [ū1 f u 1(u) 1 f ū1(u)+f(u)] is given.We transform the higher order singular integral to a usual Cauchy integral,extend the permutation formula of the higher order singular integral deduced by Qian and Zhong in [4] to a general case,and discuss the regularization problem of the higher order singular integral equations with Cauchy kernel and variable coefficients on complex hypersphere.     

Constrained LQ Problem with a Random Jump and Application to Portfolio Selection

This paper deals with a constrained stochastic linear-quadratic(LQ for short)optimal control problem where the control is constrained in a closed cone. The state process is governed by a controlled SDE with random coefficients. Moreover, there is a random jump of the state process. In mathematical finance, the random jump often represents the default of a counter party. Thanks to the It-Tanaka formula, optimal control and optimal value can be obtained by solutions of a system of backward stochastic differential equations(BSDEs for short). The solvability of the BSDEs is obtained by solving a recursive system of BSDEs driven by the Brownian motions. The author also applies the result to the mean variance portfolio selection problem in which the stock price can be affected by the default of a counterparty.     

针对一维空间分数阶扩散方程的一个新的最大模近似公式

Based on the maximum principle,the difference formula defined on a non-integral node is given to approximate the fractional Riemann-Liouville derivative and the finite difference scheme for solving one-dimensional space fractional diffusion equations(FDEs) with variable coefficients is presented.Furthermore,using the maximum principle the scheme is proved unconditionally stable and secondorder accuracy in spatial grid size.Several numerical examples are given to verify the efficiency of the scheme.     

D''''ALEMBERT FORMULA AND THE DIRECT OR INVERSE PROBLEMS FOR WAVE EQUATION

In this paper, a new proof of D'Alembert formula for Cauchy problems of wave equation is proposed. And some D'Alembert-type formulas for Cauchy problem with discontinuous coefficients on semi-infinite initial boundary value problem are proposed. Based on above discussion, the c(?)racteristics methods for the direct and inverse problems of the stress wave equation for uniform layerad elastic medium are investigated. The aim of this paper is to stress the significance and application of the characteristics concept and approach.     

Basic functions and unramified local L-factors for split groups

According to a program of Braverman, Kazhdan and Ng, for a large class of split unramified reductive groups G and representations ρ of the dual group G, the unramified local L-factor L(s, π, ρ) can be expressed as the trace of π(f_(ρ,s)) for a function f_(ρ,s) with non-compact support whenever Re(s)0. Such a function should have useful interpretations in terms of geometry or combinatorics, and it can be plugged into the trace formula to study certain sums of automorphic L-functions. It also fits into the conjectural framework of Schwartz spaces for reductive monoids due to Sakellaridis, who coined the term basic functions; this is supposed to lead to a generalized Tamagawa-Godement-Jacquet theory for(G, ρ). In this paper, we derive some basic properties for the basic functions f_(ρ,s) and interpret them via invariant theory. In particular, their coefficients are interpreted as certain generalized Kostka-Foulkes polynomials defined by Panyushev. These coefficients can be encoded into a rational generating function.     

论一种用到Howard广义斯特灵数的求和公式

It is shown that Howard＇s degenerate weighted Stirling numbers can be used to construct a fruitful summation formula for a class of formal power series involving generalized factorials.This is achieved with the aid of a series-transformation formula due to He,Hsu and Shiue,and several identities involving generalized Stirling numbers and Bell numbers are given to illustrate the application of the formula obtained.     

A Kind of Discrete Non-Reflecting Boundary Conditions for Varieties of Wave Equations

Abstract In this paper, a new kind of discrete non-reflecting boundary conditions is developed.It can be usedfor a variety of wave equations such as the acoustic wave equation, the isotropic and anisotropic elastic waveequations and the equations for wave propagation in multi-phase media and so on.In this kind of boundaryconditions,the composition of all artifical reflected waves,but not the individual reflected ones,is consideredand eliminated.Thus, it has a uniform formula for different wave equations.The velocity C_A of the composedreflected wave is determined in the way to make the reflection coefficients minimal,the value of which depends onequations.In this psper,the construction of the boundary conditions illustrated and C_A is found,numericalresults are presented to illustrate the effectiveness of the boundary conditions.     

CONVERGENCE AND GROWTH OF MULTIPLE DIRICHLET SERIES

This article investigates the convergence and growth of multiple Dirichlet series. The Valiron formula of Dirichlet series is extended to n-tuple Dirichlet series and an equivalence relation between the order of n-tuple Dirichlet series and its coefficients and exponents is obtained.