The problem of reconstructing a quasihomogeneous string from its part

We consider quasihomogeneous strings with piecewise constant density and derive the quasihomogeneity condition. For a small number of string components, we present explicit formulas for reconstructing a string from its part. For an arbitrary number of string components, we construct a theory of matrices of special form (that we call distinguished) and present an algorithm for string reconstruction from its part.     

Tensor invariants of quasihomogeneous systems of differential equations,and the Kovalevskaya-Lyapunov asymptotic method

Translated from Matematicheskie Zametki, Vol. 51, No. 2, pp. 46–52, February, 1992.     

On the Milnor K-theory of the function fields of quasihomogeneous cones

Let V be a quasihomogeneous normal variety. The aim of this paper is to describe the Milnor K-theory of the function field of V in terms of the second residue homomorphisms associated with subvarieties and resolution data of V.Supported by KBN, 2 P301 010 06.     

The de Rham complex of a quasihomogeneous complete intersection

Functional Analysis and Its Applications -     

凸二次函数优化问题在快速多极边界元法中的应用

利用快速多极边界元法(FMM-BEM)求解大规模工程问题最终结为稀疏线性方程组的求解,因此,采用更好的方法求解线性方程组可以提高边界元法的计算效率.本文利用最优化数值技术处理,将稀疏线性方程组的求解等价为求解一个凸二次函数极小化的问题,并利用最优化理论及相关数学理论证明了其解的存在唯一性,为该理论的形成和发展奠定了理论基础.     

A penalty function method for solving inverse optimal value problem

In order to consider the inverse optimal value problem under more general conditions, we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently. Using the Kuhn–Tucker optimality condition of the lower level problem, we transform the nonlinear bilevel programming into a normal nonlinear programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty. Then we give via an exact penalty method an existence theorem of solutions and propose an algorithm for the inverse optimal value problem, also analysis the convergence of the proposed algorithm. The numerical result shows that the algorithm can solve a wider class of inverse optimal value problem.     

基于光滑化方法求解非线性l1问题

讨论了求解非线性l1问题的一种新的光滑函数法.通过对非线性l1问题模型的转化,将该问题化为一个不可微优化问题,据此提出了基于BFGS迭代的非线性l1问题的光滑函数法,介绍了非线性l1问题的光滑函数的有关性质、算法步骤及其收敛性.数值仿真显示了提出的光滑函数方法可以避免数值计算的溢出,具有一定的有效性.     

The combinatorics of weight systems and characteristic polynomials of isolated quasihomogeneous singularities

Journal of Algebraic Combinatorics - A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated...     

The center problem via averaging method

The averaging method has been used to study the problem of the determination of the number of hyperbolic limit cycles that can bifurcate from the period annulus of a center. In this paper we use the averaging method up to any order in the perturbation parameter to determine the center conditions of monodromic singular points of analytic planar vector fields.     

Solving nonlinearly constrained global optimization problem via an auxiliary function method

This paper considers the nonlinearly constrained continuous global minimization problem. Based on the idea of the penalty function method, an auxiliary function, which has approximately the same global minimizers as the original problem, is constructed. An algorithm is developed to minimize the auxiliary function to find an approximate constrained global minimizer of the constrained global minimization problem. The algorithm can escape from the previously converged local minimizers, and can converge to an approximate global minimizer of the problem asymptotically with probability one. Numerical experiments show that it is better than some other well known recent methods for constrained global minimization problems.     

The value function of a transportation problem

We investigate the value of an optimal transportation problem with the maximization objective as a function of costs and vectors of production and consumption. The value is concave in production. For generic costs, the numbers of linearity domains and peak points are independent of costs and consumption. The peak points are determined by an auxiliary assignment problem. The volumes of the linearity domains are independent of costs while their dependence on consumption can be expressed via the multinomial distribution.     

A potential function method for the Cauchy problem of elliptic operators

In this paper, we deal with the Cauchy problem of elliptic operators. Through the use of a single-layer potential function, we devise a numerical method for approximating the solution of the Cauchy problem of elliptic operators, which are well known to be highly ill-posed in nature. The method is based on the denseness of single-layer potential functions. Convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method. It has been observed that the use of more Cauchy data will greatly improve the accuracy of the approximate solutions.     

Solonnikov parabolic systems with quasihomogeneous structure

We consider a new class of systems of equations that combine the structures of Solonnikov and éidel’man parabolic systems. We prove a theorem on the reduction of a general initial-value problem to a problem with zero initial data and a theorem on the correct solvability of an initial-value problem in a model case. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1501–1510, November, 2006.     

The Perron method and the non-linear Plateau problem

We describe a novel technique for solving the Plateau problem for constant curvature hypersurfaces based on recent work of Harvey and Lawson. This is illustrated by an existence theorem for hypersurfaces of constant Gaussian curvature in ${\mathbb{R}^{n+1}}$ .