Higher order optimality conditions with an arbitrary non-differentiable function

In this paper, we introduce a higher order directional derivative and higher order subdifferential of Hadamard type of a given proper extended real function. We obtain necessary and sufficient optimality conditions of order n (n is a positive integer) for unconstrained problems in terms of them. We do not require any restrictions on the function in our results. In contrast to the most known directional derivatives, our derivative is harmonized with the classical higher order Fréchet directional derivative of the same order in the sense that both of them coincide, provided that the last one exists. A notion of a higher order critical direction is introduced. It is applied in the characterizations of the isolated local minimum of order n. Higher order invex functions are defined. They are the largest class such that the necessary conditions for a local minimum are sufficient for global one. We compare our results with some previous ones. As an application, we improve a result due to V. F. Demyanov, showing that the condition introduced by this author is a complete characterization of isolated local minimizers of order n.     

The growth order of solutions of systems of complex difference equations

Using Nevanlinna theory of the value distribution of meromorphic functions and Wiman-Valiron theory of entire functions, we investigate the problem of growth order of solutions of a type of systems of difference equations, and extend some results of the growth order of solutions of systems of differential equations to systems of difference equations.     

Approximation order of bivariate spline interpolation for arbitrary smoothness

By using the algorithm of Nürnberger and Riessinger (1995), we construct Hermite interpolation sets for spaces of bivariate splines S_q^r(Δ¹) of arbitrary smoothness defined on the uniform type triangulations. It is shown that our Hermite interpolation method yields optimal approximation order for q 3.5r + 1. In order to prove this, we use the concept of weak interpolation and arguments of Birkhoff interpolation.     

含任意次正幂项的广义五阶KdV方程的精确解

利用齐次平衡原理,通过引进含非线性辅助微分方程（sub-ODE）,获得了含任意次正幂项的广义五阶KdV方程的精确解,包括钟状孤波解,扭状孤波解和三角函数表示的周期波解.所得精确解与前人用其它方法所获得一致,并包含了以往文献未提供的部分解,扩充并完善了以往文献的相关结果.     

Conservative arbitrary order finite difference schemes for shallow-water flows

The classical nonlinear shallow-water model (SWM) of an ideal fluid is considered. For the model, a new method for the construction of mass and total energy conserving finite difference schemes is suggested. In fact, it produces an infinite family of finite difference schemes, which are either linear or nonlinear depending on the choice of certain parameters. The developed schemes can be applied in a variety of domains on the plane and on the sphere. The method essentially involves splitting of the model operator by geometric coordinates and by physical processes, which provides substantial benefits in the computational cost of solution. Besides, in case of the whole sphere it allows applying the same algorithms as in a doubly periodic domain on the plane and constructing finite difference schemes of arbitrary approximation order in space. Results of numerical experiments illustrate the skillfulness of the schemes in describing the shallow-water dynamics.     

A difference method for solving parabolic equations of order 2n

In this paper, a finite difference method is used to approximate for the solution of the parabolic partial differential equation of order 2n and error of the method is determined. The resulting system is solved by efficient implicit iterations. In some numerical examples, MAPLE modules are designed for the purpose of testing and using the method.     

Identities for the Chebyshev functional involving derivatives of arbitrary order and applications

We generalize the identities of J. Pe?ari? and A.M. Fink for the Chebyshev functional. The identities are shown to be very useful in establishing various bounds for the Chebyshev functional.     

High accuracy nonconforming finite elements for fourth order problems

The approach of nonconforming finite element method admits users to solve the partial differential equations with lower complexity,but the accuracy is usually low.In this paper,we present a family of highaccuracy nonconforming finite element methods for fourth order problems in arbitrary dimensions.The finite element methods are given in a unified way with respect to the dimension.This is an effort to reveal the balance between the accuracy and the complexity of finite element methods.     

On the construction of arbitrary order schemes for the many dimensional wave equation

We show that it is possible to construct arbitrary order stable schemes for the homogeneous and heterogeneous wave equation in any dimension. The construction is elementary and uses formal power series techniques. We shall also calculate exact stability limits in various cases, and apparently this limit depends only on the dimension of the space.     

局部平方可积鞅Rosenthal}不等式的另一形式

本文研究局部平方可积鞅的一种Rosenthal不等式中常数的性态,证明了其系数与离散参数鞅情形有相同的增长阶.     

Laws of the iterated logarithm for locally square integrable martingales

Three types of laws of the iterated logarithm （LIL） for locally square integrable martingales with continuous parameter are considered by a discretization approach. By this approach, a lower bound of LIL and a number of FLIL are obtained, and Chung LIL is extended.     

Locally one-dimensional difference schemes for the fractional order diffusion equation

Locally-one-dimensional difference schemes for the fractional diffusion equation in multidimensional domains are considered. Stability and convergence of locally one-dimensional schemes for this equation are proved.     

The Simplest Tauberian Theorem

The following problem is considered: obtain the asymptotic properties of a function u from the asymptotic properties of the integral ₀ ^r u(t),dt. As is well known, this can be done under additional constraints on the function u(t). In this paper, we obtain a theorem in which these constraints are weaker than in other well-known versions of such theorems.     

多尺度紧支撑向量值正交小波的构造

引入了多尺度向量值多分辨分析．利用矩阵理论,给出多尺度紧支撑向量值正交小波存在的必要条件及其构造方法．     

Optimality conditions for an isolated minimum of order two in C constrained optimization

In this paper we obtain first and second-order optimality conditions for an isolated minimum of order two for the problem with inequality constraints and a set constraint. First-order sufficient conditions are derived in terms of generalized convex functions. In the necessary conditions we suppose that the data are continuously differentiable. A notion of strongly KT invex inequality constrained problem is introduced. It is shown that each Kuhn-Tucker point is an isolated global minimizer of order two if and only if the problem is strongly KT invex. The article could be considered as a continuation of [I. Ginchev, V.I. Ivanov, Second-order optimality conditions for problems with C¹ data, J. Math. Anal. Appl. 340 (2008) 646-657].     

Multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities

This paper is devoted to the study of nonlinear singular and non-singular fourth order difference equations coupled with periodic boundary value conditions. In the paper some existence and nonexistence results are given. The results are deduced from Krasnoselskii's fixed point theorems in cones. An exhaustive study of the Green's function related to a linear fourth order problem is done.     

The nearest adjoining order method for pairwise comparisons in the form of difference of ranks

The problem of ranking of elements from some finite set on the basis of nearest adjoining order method for pairwise comparisons is investigated in this paper. It is assumed that in the set under consideration there exists a weak preference relation, which is to be identified (estimated) on the basis of pairwise comparisons in the form of difference of ranks. Moreover, the results of comparisons may be disturbed with random errors; the assumptions made about error distributions are not restrictive. The paper comprises: the problem formulation (definitions, assumptions, and optimisation problem, which provides the NAO solution) and the theoretical background – the form of distributions of random variables which make it possible to determine the properties of NAO solution, in particular, evaluation of the probability, that the NAO solution is equivalent to the errorless one. The approach presented in the paper can be extended to the case of more than one comparison for each pair of elements, i.e., completely formalised multi-experts ranking procedure. This revised version was published online in June 2006 with corrections to the Cover Date.