ON AN EXTENSION OF ABEL-GONTSCHAROFF''''S EXPANSION FORMULA

We present a constructive generalization of Abel-Gontscharoff's series expansion to higher dimensions. A constructive application to a problem of multivariate interpolation is also investigated. In addition, two algorithms for constructing the basis functions of the interpolants are given.     

Exact Boundary Controllability and Exact Boundary Observability for a Coupled System of Quasilinear Wave Equations

Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the authors apply a unified constructive method to establish the local exact boundary(null) controllability and the local boundary(weak) observability for a coupled system of 1-D quasilinear wave equations with various types of boundary conditions.     

ON CRACK PROBLEMS IN AN INFINITE PLANE CONSISTING OF THREE DIFFERENT MEDIA

The mathematical problem of an infinite elastic plane consisting of three different media with an arbitrary number of cracks is considered. It is reduced to singular integral equations along the interfaces and the cracks by a constructive method. Those along the interfaces are further reduced to Fredholm ones.     

数学年刊 第37卷B辑第6期(2016) 目次和提要（英文）

正Exact Boundary Observability for a Kind of Second-Order Quasilinear Hyperbolic Systems Ke WANG Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems,the local exact boundary observability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method.     

Reconstruction of convection coefficients of an elliptic equation in the plane by Dirichlet to Neumann map

The inverse problem of determining two convection coefficients of an ellipticpartial differential equation by Dirichlet to Neumann map is discussed.It is well knownthat this is a severely ill-posed problem with high nonlinearity.By the inverse scatteringtechnique for first order elliptic system in the plane and the theory of generalized analyticfunctions,we give a constructive method for this inverse problem.     

The spectrum sequences of periodic frame multiresolution analysis

The periodic multiresolution analysis （PMRA） and the periodic frame multiresolution analysis （PFMRA） provide a general recipe for the construction of periodic wavelets and periodic wavelet frames, respectively. This paper addresses PFMRAs by the introduction of the notion of spectrum sequence. In terms of spectrum sequences, the scaling function sequences generating a normalized PFMRA are characterized; a characterization of the spectrum sequences of PFMRAs is obtained, which provides a method to construct PFMRAs since its proof is constructive; a necessary and sufficient condition for a PFMRA to admit a single wavelet frame sequence is obtained; a necessary and sufficient condition for a PFMRA to be contained in a given PMRA is also obtained. What is more, it is proved that an arbitrary PFMRA must be contained in some PMRA. In the meanwhile, some examples are provided to illustrate the general theory.     

Exact Boundary Controllability of Weak Solutions for a Kind of First Order Hyperbolic System — the HUM Method*

The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work [Lu, X. and Li, T. T., Exact boundary controllability of weak solutions for a kind of first order hyperbolic system — the constructive method, Chin. Ann. Math. Ser. B, 42(5), 2021, 643–676]. In this paper, in order to study these problems from the viewpoint of duality, the authors establish ...     

Standard tri-point transition function

It is usually difficult to express a family of tri-point transition function (TTF) by a transition matrix as Markov processes with one parameter. In this paper, we define three kinds of connection matrixes on the states of standard tri-point transition function (STTF) and study their essential character, give a constructive method on the constant-value standard tri-point transition function and a general expression of the state-symmetric standard tri-point transition function by a sequence of the transition matrixes of special and simple Markov processes with one parameter.     

Construction of two-direction tight wavelet frames

We investigate the construction of two-direction tight wavelet frames First, a sufficient condition for a two-direction refinable function generating two-direction tight wavelet frames is derived. Second, a simple constructive method of two-direction tight wavelet frames is given. Third, based on the obtained two-direction tight wavelet frames, one can construct a symmetric multiwavelet frame easily. Finally, some examples are given to illustrate the results.     

关于局部紧域的加法特征的一点注记

In this note, we give an elementary and constructive proof for that the additive character group of a locally compact field is isomorphic to itself as an additive topological group.     

Stabilization of fractional bilinear systems with multiple inputs

In this paper, we study in a constructive way the stabilization problem of fractional bilinear systems with multiple inputs. Using the quadratic Lyapunov functions and some additional hypotheses on the unit sphere, we construct stabilizing feedback laws for the considered fractional bilinear system. A numerical example is given to illustrate the efficiency of the obtained result.     

Finding Periodic Solutions of High Order Duffing Equations via Homotopy Method

This paper presents a detailed analysis of finding the periodic solutions for the high order Duffing equation x^（2n） ＋ g（x） = e（t） （n ≥ 1）. Firstly, we give a constructive proof for the existence of periodic solutions via the homotopy method. Then we establish an efficient and global convergence method to find periodic solutions numerically.     

球面上的几何连续插值

In this paper, a new method for geometrically continuous interpolation in spheres is proposed. The method is entirely based on the spherical B′ezier curves defined by the generalized de Casteljau algorithm. Firstly we compute the tangent directions and curvature vectors at the endpoints of a spherical B′ezier curve. Then, based on the above results, we design a piecewise spherical B′ezier curve with G 1 and G 2 continuity. In order to get the optimal piecewise curve according to two different criteria, we also give a constructive method to determine the shape parameters of the curve. According to the method, any given spherical points can be directly interpolated in the sphere. Experimental results also demonstrate that the method performs well both in uniform speed and magnitude of covariant acceleration.     

THE POISSON FORMULA FOR HARMONIC (1, 0)-FORMS BALL OF C~(n*)

More than thirty five years ago I learned the eoncept of manifolds and the proof of de Rham theorem from a series of lectures given by Prof. Wen-tsün Wu.Prof. Wu' s lectures are duo to the original paper of de Rham. Though nowadays the proof of de Rham theorem is much shorter by using the abstract sheaf theory, but the constructive proof I learned gave mo much stronger impression. This made me     

Global Exact Boundary Controllability for Cubic Semi-linear Wave Equations and Klein-Gordon Equations

The authors prove the global exact boundary controllability for the cubic semi-linear wave equation in three space dimensions, subject to Dirichlet, Neumann, or any other kind of boundary controls which result in the well-posedness of the corresponding initial-boundary value problem. The exponential decay of energy is first established for the cubic semi-linear wave equation with some boundary condition by the multiplier method, which reduces the global exact boundary controllability problem to a local one. The proof is carried out in line with [2, 15]. Then a constructive method that has been developed in [13] is used to study the local problem. Especially when the region is star-complemented, it is obtained that the control function only need to be applied on a relatively open subset of the boundary. For the cubic Klein-Gordon equation, similar results of the global exact boundary controllability are proved by such an idea.     

SHARED VALUE IN A FINITE-DIMENSIONAL BANACH MANIFOLD

Sufficient conditions are given to assert that two C1-mappings share only one value in a connected compact Banach manifold modelled over Rn. The proof of the result, which is based upon continuation methods, is constructive.     

内点同伦方法求解更一般非凸集上的不动点问题(英文)

In this paper,we are mainly devoted to solving fixed point problems in more general nonconvex sets via an interior point homotopy method.Under suitable conditions,a constructive proof is given to prove the existence of fixed points,which can lead to an implementable globally convergent algorithm.     

Inverse Problems for the Dirac Operator on a Star Graph

Following the previous work, we shall study some inverse problems for the Dirac operator on an equilateral star graph. It is proven that the so-called Weyl function uniquely determines the potentials. Furthermore, we pay attention to the inverse problem of recovering the potentials from the spectral data, which consists of the eigenvalues and weight matrices, and present a constructive algorithm. The basic tool in this paper is the method of spectral mappings developed by Yurko.     

Tree wavelet approximations with applications

We construct a tree wavelet approximation by using a constructive greedy scheme (CGS). We define a function class which contains the functions whose piecewise polynomial approximations generated by the CGS have a prescribed global convergence rate and establish embedding properties of this class. We provide sufficient conditions on a tree index set and on bi-orthogonal wavelet bases which ensure optimal order of convergence for the wavelet approximations encoded on the tree index set using the bi-orthogonal wavelet bases. We then show that if we use the tree index set associated with the partition generated by the CGS to encode a wavelet approximation, it gives optimal order of convergence.     

Generalized Mbius Inversion with Applications to Integral Equations and Interpolation Process

The object of this expository paper is to establish a kind of generalized M(?)bius-Rota inversion formula by the aid of non-standard analysis. Certain constructive applications to Volterra type integral equations and toequidistant smooth interpolation process are expounded in some detail. Moreover, some of our earlier results (cf. [1], [2], [4]) are refined and improved.