最佳L2局部逼近存在唯一的充分必要条件

本文给出了最佳Ｌ２局部逼近的存在唯一性定理，设ｆ∈Ｌ２（０，δ），Ｓｎ＝ｓｐａｎ（ｕ０，ｕ１，．．．Ｕｎ－１）Ｃ＾ｎ－１（０，δ），且ｄｅｔＷｎ（ｕ０，ｕ１，．．．ｕｎ－１；０）≠０，那么，当ｘ→０时，网（Ｐｘ（ｆ，Ｓｎ）收敛于Ｓｎ中某元素Ｐ０（ｆ，Ｓｎ）的充要条件为：ｆ＝Ｐｎ－１＋ｈ，其中Ｐｎ－１（ｔ）＝ｎ－１∑ｉ＝１ａｉｔｉ（ｈ，１）ｘ＝０（Ｘ＾ｎ），ｘ→０，且Ｐ０（ｆ，Ｓｎ）＝ＵＷ＾－１ｎＡ     

一类非线性中立型差分方程解的振动性

本文我们证明了下列两个差分方程Δ（ｘｎ － ｘｎ－ ｋ）α＋ ｑｎｆ（ｘｎ－ Ｔ） ＝ ０Δ（Δｙｎ－ １）α＋ ｋ－ αｑｎｆ（ｙｎ） ＝ ０振动的等价性．其中ｑｎ０,ｋ,Ｔ为正整数,α为两奇数之商,ｆ∈Ｃ（Ｒ,Ｒ）是非减的并满足ｘｆ（ｘ）＞ ０（ｘ≠０）．获得了这些方程振动的一些充分条件．     

离散系统的随机作用随机扰动

（Ｘ,Ａ,ｕ）为σ－有限测度空间,Ｓ：Ｘ→Ｘ为非奇异映射,Ｐｓ为Ｓ所对应的Ｆｒｏｂｅｎｉｕｓ－Ｐｅｒｒｏｎ算子。Ｓ在随机作用随机扰动下的演化过程由方程ｘｎ＋１＝ηｎＳ（ｘｎ）＋（１－ηｎ）ξｎ ｎ＝０,１,２,…所描述,其中ξ０,η０,ξ１,η１,…为相互独立随机变量,ｐｒｏｂ（ηｎ＝１）＝１－ε,ｐｒｏｂ（ηｎ＝０）＝ε,ξｎ具有相同概率密度ｇ（ｘ）。本文借助于弱预紧性和遍历平均Ａｎｇ＝１ｎ∑ｎ     

Jacobi多项式零点为结点的Lagrange插值多项式之逼近

对于可微函数ｆ∈Ｃｑ［－１，１］，本文研究以Ｊａｃｏｂｉ多项式Ｊ（α，β）ｎ（ｘ）的零点为结点组之Ｌａｇｒａｎｇｅ插值多项式对ｆ及其导数的同时逼近，证明不等式Ｌ（ｓ）ｎ（ｆ，α，β，ｘ）－ｆ（ｓ）（ｘ）＝Ｏ（１）Δ－ｓｎ（ｘ）Δｑｎ（ｘ）ω（ｆ（ｑ），Δｎ（ｘ））ｌｏｇｎ｛＋（１－ｘ＋ｎ－１）－α－１２ｎ－ｑω（ｆ（ｑ），ｎ－１）｝，在［０，１］上对于ｓ＝０，１，２，…，ｑ一致成立，其中Δｎ（ｘ）＝ｎ－１１－ｘ２＋ｎ－２     

强相依高斯序列超过数点过程与部分和的联合渐近分布

（Ｘｎ）为标准化平稳高斯序列,ｐｎ＝ＥＸ１Ｘｎ＋１,Ｎｎ为Ｘ１,Ｘ２,…,Ｘｎ对水平ｕｎ＝ｘ／ａｎ＋ｂｎ的超过数形成的点过程,Ｍｎ＾（ｋ）为Ｘ１,Ｘ２,…,Ｘｎ的第ｋ个最大值,Ｓｎ＝（ｎ）∑（ｉ＝１）Ｘｉ,ｐｎｌｏｇｎ→ｒ∈（０,∞）时,得到Ｎｎ与Ｓｎ、Ｍｎ＾（ｋ）与Ｓｎ的联合渐近分布。     

Dirichlet多项式翻转公式的一个新结果

设ｆ（ｘ）是一个实函数，ｆ（ｘ＋ｉｙ）在某个包含区间［ａ，ｂ］的某区域内解析，则Σ↓α〈ｎ≤βｅ（ｆ（ｎ））＝ｅ（－１／８）Σ↓α〈ｎ≤β│ｆ″（ｘｎ）│＾－１／２ｅ（ｆ（ｘ）－ｎｘｎ）＋△（ｆ，ａ，ｂ）其中α，β，ｘｎ的定义是（１），余项△是（９），它改进了文［１］，［２］的结果。     

多项式翻转公式的一个新结果(英文)

设ｆ（ｘ）是一个实函数，ｆ（ｘ＋ｉｙ）在某个包含区间［ａ，ｂ］的某区域内解析，则∑ａ＜ｎ≤ｂｅ（ｆ（ｎ））＝ｅ（－１８）∑α＜ｎ≤β｜ｆ″（ｘｎ）｜－１２ｅ（ｆ（ｘｎ）－ｎｘｎ）＋△（ｆ，ａ，ｂ）其中α，β，ｘｎ的定义是（１），余项△是（９），它改进了文［１］，［２］的结果．     

非自治时滞差分方程的线性化渐近稳定性

本文证明了，在某种条件下，非自治非线性时滞差分方程ｘｎ＋１－ｘｎ＋ｐｎｆ（ｘｎ－ｋｎ）＝０，ｎ∈Ｎ零解的渐近稳定性可由其线性化方程ｙｎ＋１－ｙｎ＋ｐｎｙｎ－ｋｎ＝０，ｎ∈Ｎ零解的渐近稳定性来确定．作为应用，也研究了离散时滞ｌｏｇｉｓｔｉｃ方程的稳定性．     

关于Thue方程的解数

设ｍ是正整数，ｆ（Ｘ，Ｙ）＝ａ０Ｘｎ＋ａ１Ｘ（ｎ－１）Ｙ＋．．．＋ａｎＹｎ∈Ｚ［Ｘ，Ｙ］是Ｑ上不可约化的叫ｎ（ｎ≥３）次齐次多项式。本文证明了：当ｇｃｄ（ｍ，ａ０）＝１，ｎ≥４００且ｍ≥１０（３５）时，方程｜ｆ（ｘ，ｙ）｜＝ｍ，ｘ，ｙ∈ｚ，ｇｃｄ（ｘ，ｙ）＝１，至多有６ｎｖ（ｍ）组解（ｘ，ｙ），其中ｖ（ｍ）是同余式Ｆ（ｚ）＝ｆ（ｚ，１）≡０（ｍｏｄｍ）的解数。特别是当ｇｃｄ（ｍ，ＤＦ）＝１时，该方程至多有６ｎ（ω（ｍ）＋１）组解（ｘ，ｙ），其中ＤＦ是多项式Ｆ的判别式，ω（ｍ）是ｍ的不同素因数的个数．     

一类非线性椭圆方程组正解的存在性定理

本文考虑如下的椭圆方程组△ｙ＋ｆ（ｘ,ｕ）＋Эｕ＝０,ｘ∈Ω △ｕ＋ｕ－ｖ＝０,ｘ∈Ω ｕ＝ｖ＝０,ｘ∈ЭΩ 其中,Ω∈Ｒ＾Ｎ（Ｎ≥３）是带光滑边界的有界区域,ｆ（ｘ,ｕ）＝ｈ（ｘ）ｕ＾α＋ｕ＾β＋λｕ＾ｐ,ｈ（ｘ）∈Ｃ＾ｒ（Ω）（０〈ｒ〈１）,α,β,ｐ是正常数且０〈β〈α〈１〈ｐ〈（Ｎ＋２）／（Ｎ－２）,λ,δ是正参数,由临界点理论证明了该方程组至少存在二对正解。     

The motion of a body with a cavity partly filled with a viscous liquid

Many papers are concerned with the dynamics of a rigid body with a cavity filled with liquid (see the bibliography in [1]). The present paper deals with the motion of a rigid body having a cavity partly filled with a viscous incompressible liquid, and having a free surface. The shape of the cavity is arbitrary. The problem is considered in a linear formulation. The oscillations of the body with respect to its center of inertia and the motion of the liquid in the cavity are assumed small. The viscosity of the liquid is considered low. The solution of the problem of the oscillations of a body with a cavity partly filled with an ideal liquid is used as an initial approximation [1 to 6]. The viscosity is taken into consideration by the boundary layer method used before in similar problems [1 and 7 to 10). General equations are derived for the dynamics of a body filled with a liquid, for an arbitrary form of cavity. The coefficients of those integro-differential equations depend only on the solution of the problem of the oscillations of a body with a cavity of the given form filled with an ideal liquid. Since the corresponding problem has been solved for cavities of many forms [1 to 6, 11 and 12] in the case of an ideal liquid, the determination of the characteristic coefficients is reduced to the evaluation of quadratures. Several particular cases of motion are considered.     

Interaction of a breather with a magnetization wave in a ferromagnet with light-axis anisotropy

We use the dressing method to find exact solutions of the Landau-Lifshitz equation for a ferromagnet with light-axis anisotropy. These solutions describe the interaction of a nonlinear precession wave of arbitrary amplitude with solitons. We analyze the change of the internal structure and the physical parameters of the solitons as a result of their interaction with the magnetization wave. We find an infinite series of integrals of motion that stabilize the soliton on the background of the pumping wave.     

Interaction of a hollow cylinder of finite length and a plate with a cylindrical cavity with a rigid insert

Two problems of the interaction of a hollow circular cylinder with load-free ends and an unbounded plate with a cylindrical cavity and a symmetrically imbedded rigid insert are considered. Homogeneous solutions are found and the generalized orthogonality of these solutions is used when the modified boundary conditions are satisfied. As a result, we have a system of two integral equations in functions of the displacements of the outer and inner surfaces of the hollow cylinder. These functions are sought in the form of sums of a trigonometric series and a power function with a root singularity. The ill-posed infinite systems of linear algebraic equations obtained are regularized by the introduction of small positive parameters. Since the elements of the matrices of the systems as well as the contact stresses are defined by poorly converging numerical and functional series, an efficient method for calculating of the remainders of the above-mentioned series is developed. Formulae are found for the contact pressure distribution function and the integral characteristic. Examples of the calculation of the interaction of the cylinder and the plate with an insert are given.The method of solving contact problems described here has been used earlier1, 2 and the generalized orthogonality of the solutions found for bodies of finite dimensions, that is, for a rectangle and cylinders of finite length, is its basis. Problems for hollow cylinders with a band ² and an insert reduce to a system of two integral equations, and the problem for a rectangle¹ reduces to one integral equation. Solving these integral equations, ill-posed systems of linear algebraic equations are obtained which are subject to regularization³.     

Homogenization of the heat equation for a domain with a network of pipes with a well-mixed fluid

We consider the linear heat equation in a domain occupied by a solid material with a network of pipes in which a well-mixed fluid is circulating. The temperature of the fluid in the pipe is uniform and its time variation is determined by the thermal flux on the wall of the pipe, plus a given internal source; continuity of the temperature across the pipe is also assumed. We suppose that we deal with a periodic geometry, with cells of size with inclusions of size r_g; we study in detail in the case r, referring to a previous paper for the case r In the limit »0 we get a homogenized equation. The limit depends strongly on the ratio between the time variation of the temperature in the inclusions and the thermal flux through the interface. The homogenized equation has a new specific heat, which depends on the porosity and the constant of proportionality between the time variation of temperature and the flux on the boundary of the pipe. We also have a new thermal conductivity depending on the microstructure, and volume sources appear. The main tool is the energy method and we generalize the classical results for the more standard boundary conditions for parabolic equations. Finally, we consider the network of pipes forming a random ball structure. We prove convergence for this case. The homogenized equation is of the same form as in the periodic case but auxiliary problems are stochastic.     

On a variational inequality associated with a stopping game combined with a control

We study a non-linear elliptic variational inequality which corresponds to a zero-sum stopping game (Dynkin game) combined with a control. Our result is a generalization of the existing works by Bensoussan [ Stochastic Control by Functional Analysis Methods (North-Holland, Amsterdam), 1982], Bensoussan and Lions [ Applications des Inéquations Variationnelles en Contrôle Stochastique (Dunod, Paris), 1978] and Friedman [ Stochastic Differential Equations and Applications (Academic Press, New York), 1976] in the sense that a non-linear term appears in the variational inequality, or equivalently, that the underlying process for the corresponding stopping game is subject to a control. By using the dynamic programming principle and the method of penalization, we show the existence and uniqueness of a viscosity solution of the variational inequality and describe it as the value function of the corresponding combined-stochastic game problem.     

Calculating a function of a matrix with a real spectrum

Let T be a square matrix with a real spectrum, and let f be an analytic function. The problem of the approximate calculation of f(T) is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that T is triangular and its diagonal entries t_ii are arranged in increasing order. To avoid calculations using the differences t_ii ? t_jj with close (including equal) t_ii and t_jj, it is proposed to represent T in a block form and calculate the two main block diagonals using interpolating polynomials. The rest of the f(T) entries can be calculated using the Parlett recurrence algorithm. It is also proposed to perform some scalar operations (such as the building of interpolating polynomials) with an enlarged number of significant decimal digits.

     

Interaction of a free wave with a semicrystal

The Schrodinger operator with semiperiodic potential in L₂(Rⁿ), n=2,3, is studied. Incident and reflected waves in a free half-space are considered. An asymptotic expansion of the reflection matrix for a high energy region is constructed and its relationship with the potential is established. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 107–114, 1990. Translated by Yu. E. Karpeshina.     

Interaction of a rigid punch with a base-secured elastic rectangle with stress-free sides

The plane contact problem of the indentation of a rigid punch into a base-sucured elastic rectangle with stress-free sides is considered. The problem is solved by a method tested earlier and reduces to a system of two integral equations in functions describing the displacement of the surface of the rectangle outside the punch and the normal or shear stress on its base. These functions are sought in the form of the sum of trigonometric series and an exponential function with a root singularity. The ill-posed infinite systems of algebraic equations obtained as a result of this are regularized by introducing small positive parameters. Because the matrix elements of the systems, and also the contact stresses, are defined by poorly converging numerical and functional series, the previously developed method of summation of these series is used. The contact pressure distribution and the dimensionless indenting force are found. Examples of a plane punch calculation are given.