Analytic Representation of CR Functions on Hypersurfaces with Singularities

We prove a theorem on analytic representation of integrable CR functions on hypersurfaces with singular points. Moreover, the behaviour of representing analytic functions near singular points is investigated. We are aimed at explaining the new effect caused by the presence of a singularity rather than at treating the problem in full generality.     

The Spherical Harmonic Spectrum of a Function with Algebraic Singularities

The asymptotic behaviour of the spectral coefficients of a function provides a useful diagnostic of its smoothness. On a spherical surface, we consider the coefficients $a_{l}^{m}$ of fully normalised spherical harmonics of a function that is smooth except either at a point or on a line of colatitude, at which it has an algebraic singularity taking the form ?? ^p or |????? ₀| ^p respectively, where ?? is the co-latitude and p>?1. It is proven that each type of singularity has a signature on the rotationally invariant energy spectrum, $E(l) = \sqrt{\sum_{m} (a_{l}^{m})^{2}}$ where l and m are the spherical harmonic degree and order, of l ^?(p+3/2) or l ^?(p+1) respectively. This result is extended to any collection of finitely many point or (possibly intersecting) line singularities of arbitrary orientation: in such a case, it is shown that the overall behaviour of E(l) is controlled by the gravest singularity. Several numerical examples are presented to illustrate the results. We discuss the generalisation of singularities on lines of colatitude to those on any closed curve on a spherical surface.     

半平面中调和函数的积分表示

In this article, we consider the integral representation of harmonic functions. Using a property of the modified Poisson kernel in a half plane, we prove that a harmonic function u（z） in a half plane with its positive part u^＋（z） = max{u（z）, 0} satisfying a slowly growing condition can be represented by its integral of a measure on the boundary of the half plan.     

The Singularities of the Distance Function Near Convex Boundary Points

In an open bounded set Ω, we consider the distance function from ∂Ω associated to a Riemannian metric with C ^1,1 coefficients. Assuming that Ω is convex near a boundary point x ₀, we show that the distance function is differentiable at x ₀ if and only if there exists the tangent space to ∂Ω at x ₀. Furthermore, if the distance function is not differentiable at x ₀ then there exists a Lipschitz continuous curve, with initial point at x ₀, such that the distance function is not differentiable along such a curve.      

On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels

We consider the three-dimensional Schrödinger operators and where , A is a magnetic potential generating a constant magnetic field of strength , and where decays fast enough at infinity. Then, A. Pushnitskis representation of the spectral shift function (SSF) for the pair of operators is well defined for energies We study the behaviour of the associated representative of the equivalence class determined by the SSF, in a neighbourhood of the Landau levels Reducing our analysis to the study of the eigenvalue asymptotics for a family of compact operators of Toeplitz type, we establish a relation between the type of the singularities of the SSF at the Landau levels and the decay rate of V at infinity. Communicated by Bernard HelfferSubmitted 23/09/03, accepted 15/01/04     

Approximation of the Singularities of a Bounded Function by the Partial Sums of Its Differentiated Fourier Series

In our earlier work we developed an algorithm for approximating the locations of discontinuities and the magnitudes of jumps of a bounded function by means of its truncated Fourier series. The algorithm is based on some asymptotic expansion formulas. In the present paper we give proofs for those formulas.     

分位点函数的光滑非参数估计的BAHADUR表示

文中对分位函数给出了具有更广泛应用的光滑分位估计,证明了该光滑分位估计的逐点和一致的Bahadur强表示定理;并由此结果推导了估计的重对数律,强逼近等深刻结果。     

Representation Formulas for Analytic Function Non－exponential in a Half Plane

ＲｅｐｒｅｓｅｎｔａｔｉｏｎＦｏｒｍｕｌａｓｆｏｒＡｎａｌｙｔｉｃＦｕｎｃｔｉｏｎＮｏｎ－ｅｘｐｏｎｅｎｔｉａｌｉｎａＨａｌｆＰｌａｎｅＤｅｎｇＧｕａｎｔｉｅ（邓冠铁）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｕａｚｈｏｎｇＵｎｉｖｅｒｓｉ...     

模糊牛顿插值及模糊样条函数的表示

本文在模糊Ｌａｇｒａｎｇｅ插值的基础上，引进了模糊牛顿插值公式及其适定性定理和求法。并针对“非结点”型边界条件给出了模糊样条函数的具体表示。     

Integral Representation of Some Functions Related to the Gamma Function

We prove that the functions and are Stieltjes transforms.     

A-函数关于Weyl函数m的反表示

考虑Simon反谱理论新方法中引入的A-函数,根据Weyl函数m关于A-函数的表示关系,利用广义函数和Fourier变换的方法求出A-函数关于Weyl函数m的反表示,该结论表明A-函数的本质是广义函数.     

Singularities of the Modular Curve

Let X₀(ℓ) be the modular curve, parameterizing cyclic isogenies of degree ℓ, and Z₀(ℓ) be its plane model, given by the classical modular equation Φ_ℓ(X, Y)=0. We prove that all singularities of Z₀(ℓ), except two cusps, are intersections of smooth branches, and evaluate the order of contact of these branches.     

The Fredholm Determinant Representation for the Partition Function of the Six-Vertex Model

The six-vertex model with domain-wall boundary conditions is considered. The Fredholm determinant representation for the partition function of the model is obtained. The kernel of the corresponding integral operator depends on Laguerre polynomials. Bibliography: 13 titles.     

Singularities of the secant variety

We give positivity conditions on the embedding of a smooth variety which guarantee the normality of the secant variety, generalizing earlier results of the author and others. We also give classes of secant varieties satisfying the Hodge conjecture as well as a result on the singular locus of degenerate secant varieties.     

Singularities of the hypergeometric system associated with a monomial curve

We compute, using -module restrictions, the slopes of the irregular hypergeometric system associated with a monomial curve. We also study rational solutions and reducibility of such systems.

     

Resolving Singularities in a Ramsey-type Growth Model

The aim of this paper is to discuss a mathematical solution procedure to solve a Ramsay-type growth model that explains the fundamentals of consumption and capital accumula-tion in a dynamic equilibrium setting. The problem is formulated as a system of recursive equations and studied through some numerical experiments for the time path of the different variables of the model under some alternative assumption for the steady-state equilibrium of the labour market conditioning the possible singularity of the model.