有限区间上具有高阶消失矩的四阶样条小波

本文构造了具有讥阶消失矩的样条小波，通过B一样条函数和小波消失矩公式的相结合，得到了具有任意阶消失矩的样条小波函数，这种小波可以有效控制工程计算中得时间和复杂度。     

Integral Representations for Harmonic Functions of Infinite Order in a Cone

A harmonic function of infinite order defined in an n-dimensional cone and continuous in the closure can be represented in terms of the modified Poisson integral and an infinite sum of harmonic polynomials vanishing on the boundary.     

A spectral solution method for a boundary-value problem for a mixed-type equation with two singularity lines

We consider a boundary-value problem for a mixed-type equation with two perpendicular singularity lines given in a domain whose elliptic part is a rectangle, while the hyperbolic one is a vertical half-strip. This problem differs from the Dirichlet one by the fact that at the left boundary of the rectangle and the half-strip we specify the vanishing order of the desired function rather than its value. We find a solution to the problem by a spectral method with the use of the Fourier–Bessel series and prove the uniqueness of the solution. We substantiate the uniform convergence of the corresponding series under certain requirements to the problem statement.     

Optimal Production Planning in a Stochastic Manufacturing System with Long-Run Average Cost

This paper is concerned with the optimal production planning in a dynamic stochastic manufacturing system consisting of a single machine that is failure prone and facing a constant demand. The objective is to choose the rate of production over time in order to minimize the long-run average cost of production and surplus. The analysis proceeds with a study of the corresponding problem with a discounted cost. It is shown using the vanishing discount approach that the Hamilton–Jacobi–Bellman equation for the average cost problem has a solution giving rise to the minimal average cost and the so-called potential function. The result helps in establishing a verification theorem. Finally, the optimal control policy is specified in terms of the potential function.     

Quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients

We investigate the quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients. Quantitative unique continuation described by the vanishing order is a quantitative form of strong unique continuation property. We characterize the vanishing order of solutions for higher order elliptic equations in terms of the norms of coefficient functions in their respective Lebesgue spaces. New versions of quantitative Carleman estimates are established.     

On construction of multivariate wavelets with vanishing moments

Wavelets with matrix dilation are studied. An explicit formula for masks providing vanishing moments is found. The class of interpolatory masks providing vanishing moments is also described. For an interpolatory mask, formulas for a dual mask which also provides vanishing moments of the same order and for wavelet masks are given explicitly. An example of construction of symmetric and antisymmetric wavelets for a concrete matrix dilation is presented.     

Evaluation of a continuous wavelet transform by solving the Cauchy problem for a system of partial differential equations

It is shown that the problem of evaluating the continuous Morlet wavelet transform can be stated as the Cauchy problem for a system of two partial differential equations. The initial conditions for the desired functions, i.e., for the real and imaginary parts of the wavelet transform, are the analyzed function and a vanishing function, respectively. Numerical examples are given.     

Positive harmonic functions of finite order in a Denjoy type domain

We introduce a Denjoy type domain and prove that the dimension of the cone of positive harmonic functions of finite order in the domain with vanishing boundary values is one or two, whenever the boundary is included in a certain set.

     

Harmonic Functions in a Cone Which Vanish on the Boundary

A harmonic function defined in a cone and vanishing on the boundary is expanded into an infinite sum of certain fundamental harmonic functions. The growth conditions under which it is reduced to a finite sum of them are discussed.     

Regularity of Viscous Solutions for a Degenerate Non-linear Cauchy Problem

We consider the Cauchy problem for a class of nonlinear degenerate parabolic equation with forcing. By using the vanishing viscosity method it is possible to construct a generalized solution. Moreover, this solution is a Lipschitz function on the spatial variable and Hölder continuous with exponent 1/2 on the temporal variable.     

An Identity related to Wilson functions

We proved in [2] that a relatively general even function on the real line satisfying a vanishing condition can be expanded in terms of a certain function series closely related to the Wilson functions. In this paper we show that this vanishing condition was necessary by proving that every term of the function series satisfies this condition. In fact we prove a more general identity which implies these vanishing properties.     

On multivariate approximation by integer translates of a basis function

Approximation properties of the dilations of the integer translates of a smooth function, with some derivatives vanishing at infinity, are studied. The results apply to fundamental solutions of homogeneous elliptic operators and to “shifted” fundamental solutions of the iterated Laplacian. Following the approach from spline theory, the question of polynomial reproduction by quasi-interpolation is addressed first. The analysis makes an essential use of the structure of the generalized Fourier transform of the basis function. In contrast with spline theory, polynomial reproduction is not sufficient for the derivation of exact order of convergence by dilated quasi-interpolants. These convergence orders are established by a careful and quite involved examination of the decay rates of the basis function. Furthermore, it is shown that the same approximation orders are obtained with quasi-interpolants defined on a bounded domain. Supported in part by the United States under contract No. DAAL-87-K-0030, and by Carl de Boor’s Steenbock Professorship, University of Wisconsin-Madison.     

Vanishing theorems for ach Kähler manifolds and harmonic maps

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L ² cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincaré inequality case and establish a vanishing theorem provided that the weighted function ρ is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincaré inequality.     

A bound on the multiplicity of horizontal sections for a connection on a Riemann surface

We address the question of bounding the multiplicity of the solutions of a linear differential system, setting the problem in invariant terms. A meromorphic connection is considered on a holomorphic vector bundle over a compact Riemann surface. We produce an upper bound on the order of vanishing of an arbitrary horizontal section, which depends only on global data, provided the connection has only regular singularities or the underlying monodromy is irreducible.     

On certain families of Drinfeld quasi-modular forms

This paper deals with two problems arising in the study of Drinfeld quasi-modular forms. The first problem is to find the maximal order of vanishing at infinity of a non-zero Drinfeld quasi-modular form and leads to the notion of “extremal” quasi-modular form (highest possible order of vanishing for fixed weight and depth). The second problem is determining differential properties of extremal forms, leading to the notion of “differentially extremal” form. From our investigations, we will obtain an upper bound for the order of vanishing at infinity of non-zero Drinfeld quasi-modular forms of small depths. The paper ends with a collection of tools used in the previous parts. The notion of “extremal” form is similar to one introduced by Kaneko and Koike in [M. Kaneko, M. Koike, On extremal quasimodular forms, Kyushu J. Math. 60 (2006) 457-470].     

Note on convergence and orders of vanishing along curves

LetX be a complex space andC a family of curves through ξ ∈X. Many conditions on the size ofC are known to be sufficient for the following statement: if a formal function f∈O _X ^{^} , ξ, converges (resp. has high order of vanishing) along all curves H∈C, then f converges (resp. has high order of vanishing) on X. We improve these known results using Gabrielov's theorem (resp. Lemma 1.4 below) on pullbacks of formal functions. Dedicated to Professor Masahisa Adachi on his 60th birthday     

Doubling Property and Vanishing Order of Steklov Eigenfunctions

The paper is concerned with the doubling estimates and vanishing order of the Steklov eigenfunctions on the boundary of a smooth domain in ? ⁿ . The eigenfunction is given by a Dirichlet-to-Neumann map. We improve the doubling property shown by Bellova and Lin. Furthermore, we show that the optimal vanishing order of Steklov eigenfunction is everywhere less than Cλ where λ is the Steklov eigenvalue and C depends only on Ω.     

Nonexistence of positive solutions of nonlinear elliptic systems with potentials vanishing at infinity

We study the nonexistence of positive solutions for nonlinear elliptic systems with potentials vanishing at infinity, and establish the optimal vanishing order of potentials for the nonexistence of positive supersolutions in exterior domains.     

Variable-step variable-order 3-stage Hermite-Birkhoff-Obrechkoff DDE solver of order 4 to 14

This article presents a solver for delay differential equations (DDEs) called HBO414DDE based on a hybrid variable-step variable-order 3-stage Hermite-Birkhoff-Obrechkoff ODE solver of order 4 to 14. The current version of our method solves DDEs with state dependent, non-vanishing, small, vanishing and asymptotically vanishing delays, except neutral type and initial value DDEs. Delayed values are computed using Hermite interpolation, small delays are dealt with by extrapolation, and discontinuities are located by a bisection method. HBO414DDE was tested on several problems and results were compared with those of known solvers like SYSDEL and the recent Matlab DDE solver ddesd and statistics show that it gives, most of the time, a smaller relative error than the other solvers for the same number of function evaluations.     

Uniqueness of solutions for Keller–Segel system of porous medium type coupled to fluid equations

We prove the uniqueness of Hölder continuous weak solutions via duality argument and vanishing viscosity method for the Keller–Segel system of porous medium type equations coupled to the Stokes system in dimensions three. An important step is the estimate of the Green function of parabolic equations with lower order terms of variable coefficients, which seems to be of independent interest.