Linear interval inequalities

We prove that a system of linear inequalities with interval-valued data is weakly solvable (each system obtained by fixing coefficeints in the intervals prescribed has a solution) if and only if it is strongly solvable (all such systems have a solution in common) and desribe an algorithm for checking strong solvability.     

Interval systems of max-separable linear equations

An interval system of equations is weakly solvable if at least one of its subsystems is solvable, and it is strongly solvable if all its subsystems are solvable. We give necessary and sufficient conditions enabling efficient testing of weak and strong solvability over max-plus and max-min algebras.     

Relationships between discrete-time and continuous-time algebraic riccati inequalities

The bilinear transformation is used to establish a direct relationship between a discrete-time algebraic Riccati inequality (DARI) and an associated continuous-time algebraic Riccati inequality (CARI). It is shown that under mild conditions, the DARI is solvable if and only if the corresponding CARI is solvable. The relationship between the DARI and the CARI is then used to translate the general solvability conditions for a CARI given by Scherer into analogous conditions for the DARI. It is shown how such conditions can be applied to determine the solvability of a discrete-time H_∞ control problem whose solution set is characterized by two DARIs.     

The post correspondence problem over a unary alphabet

We consider the problem of finding a shortest solution for the Post correspondence problem over a unary alphabet. We show that the complexity of this problem heavily depends on the representation of the input: the problem is NP-complete if the input is given in compact (logarithmic) form, whereas it becomes polynomially solvable if the input is encoded in unary.     

Perturbation analysis of global error bounds for systems of linear inequalities

This paper studies the existence of a uniform global error bound when a system of linear inequalities is under local arbitrary perturbations. Specifically, given a possibly infinite system of linear inequalities satisfying the Slater’s condition and a certain compactness condition, it is shown that for sufficiently small arbitrary perturbations the perturbed system is solvable and there exists a uniform global error bound if and only if the original system is bounded or its homogeneous system has a strict solution. Received: April 12, 1998 / Accepted: February 11, 2000?Published online July 20, 2000     

6p阶可解对称图的分类

设ｘ是简单无向图，Ｇ是Ａｕｔ（Ｘ）的一个于群，Ｘ称为Ｇ－对称的，如果Ｇ在ｘ的１－孤（即两相邻顶点构成的有序偶）集合上的作用是传递的；ｘ称为对称图，如果Ｘ是Ａｕｔ（ｘ）－对称的；ｘ称为可解对称的，如果Ａｕｔ（Ｘ）包含可解子群Ｇ，使Ｘ是Ｇ－对称的．本文给出了具有６Ｐ个顶点的可解对称图的一个分类，这里ｐ≥５是素数．     

Can an a priori error estimate for an approximate solution of an ill-posed problem be comparable with the error in data?

For a linear operator equation of the first kind with perturbed data, it is shown that the global (on typical sets) a priori error estimate for its approximate solution can have the same order as that for the approximate data only if the operator of the problem is normally solvable. If the operator of the problem is given exactly, this is possible only if the problem is well-posed (stable).     

On the necessity of the solvable conditions of the typical boundary value problems for quasilinear hyperbolic systems

Some solvable conditions have been derived to ensure the existence and the uniqueness of the C^∞solution for the typical boundary problem on a local angular region for quasilinear hyperbolic systems in two variables[1]. These solvables conditions mean that, under the formulation of the typical boundary problem, the all order derivatives of the solution can be determined uniquely at the vertex. The main purpose of this paper is to show that these solvable conditions are also necessary. In other words, if these solvable conditions fail to hold, then the boundary value problem will either have no solution or have infinite number of solutions.     

Vector complementarity and minimal element problems

In this paper, vector complementarity problems are introduced as weak versions of vector variational inequalities in ordered Banach spaces. New dual cones are introduced and proved to be closed. In the sense of efficient point, we prove that the minimal element problem is solvable if a vector variational inequality is solvable; we also prove that any solution of a strong vector variational inequality or positive vector complementarity problem is a solution of the minimal element problem.This work was done while the author was with the Chongqing Institute of Architecture and Engineering, Chongqing, P. R. China.     

Two Generator Subsystems of Lie Triple System

For a Lie triple system T over a field of characteristic zero, some sufficient conditions for T to be two-generated are proved. We also discuss to what extent the two-generated subsystems determine the structure of the system T . One of the main results is that T is solvable if and only if every two elements generates a solvable subsystem. In fact, we give an explicit two-generated law for the two-generated subsystems.     

Constructing finitely presented monoids which have no finite complete presentation

Squier (1987) showed that if a monoid is defined by a finite complete rewriting system, then it satisfies the homological finiteness condition FP₃, and using this fact he gave monoids (groups) which have solvable word problems but cannot be presented by finite complete systems. In the present paper we show that a monoid cannot have a finite complete presentation if it contains certain special elements. This observation enables us to construct monoids without finite complete presentation in a direct and elementary way. We give a finitely presented monoid which has (1) a word problem solvable in linear time and (2) linear growth but (3) no finite complete presentation. We also give a finitely presented monoid which has (1) a word problem solvable in linear time, (2) finite derivation type in the sense of Squier and (3) the property FP_∞, but (4) no finite complete presentation.     

SL(2κ,C)中两类特殊可解子群的结构及其在Fuchs系统中的应用

利用SL(2,C)中可解子群的结构,给出了SL(2k,C)中两类特殊的具有两个生成元的可解子群的结构定理.由单值群的可解性与Fuchs系统可积性之间的关系,研究对应的单值群是可解的环面上只有一个正则奇点的2k阶Fuchs方程的解Riemann曲面结构,进而研究其解的大范围性质.     

对称r-循环矩阵的快速算法和并行算法

借助于快速付立叶变换(FFT),给出了一种判断对称r-循环线性系统是否有解的快速算法,并且在有解的情况下求出其解,该算法的计算复杂度为O(nlogn),且具有很好的并行性,若使用n台处理机并行处理该算法则只需要O(logn)步.当r=0时,对称r-循环矩阵变成一个上三角型Hankel矩阵,我们也给出了此类矩阵求逆的一种算法.最后将该算法推广到线性同余系统,其运算量仅为O(nlogn).     

3-part splittings for singular and rectangular linear systems

We explore iterative schemes for obtaining a solution to the linear system $\text{(}{}^1\text{) Ax = b, A ? C}{}^{\mathrm{m\; \times \; n}}$, if the system is solvable, or for obtaining an approximate solution to (¹) if the system is not solvable. Our iterative schemes are obtained via a 3-part splitting of A into A = M ? Q₁ ? Q₂. The 3-part splitting of A is, in turn, a refinement of a (2-part) subproper splitting of A into A = M ? Q. We indicate the possible usefulness of such refinements (of a 2-part splitting of A) to systems (¹) which arise from a discrete analog to the Neumann problem, where the conventional iterative schemes (i.e., iterative schemes induced by a 2-part splitting of A) are not necessarily convergent.     

An attempt to differential Galois theory of second order polynomial system and solvable subgroup of M(o)bius transformations

By introducing the conception "relativistic differential Galois group" for the second order polynomial systems, we establish the relation between the conformal relativistic differential Galois group and the subgroup of M(o)bius transformations, and prove that the system is integrable in the sense of Liouville if its conformal relativistic differential Galois group is solvable with a derived length at most 2. Some omissions on the structures of solvable subgroups of M(o)bius transformations at the first author's article published in this journal in 1996 are refreshed in this paper.     

A case of imbedding problem with a solvable kernel

Journal of Mathematical Sciences - It is shown for the solution of the imbedding problem of algebraic number fields with a finite solvable kernel one can take a field if the orders of the kernel...     

Fixed Point Methods for the Complementarity Problem

This paper is concerned with iterative procedures for the monotone complementarity problem. Our iterative methods consist of finding fixed points of appropriate continuous maps. In the case of the linear complementarity problem, it is shown that the problem is solvable if and only if the sequence of iterates is bounded in which case summability methods are used to find a solution of the problem. This procedure is then used to find a solution of the nonlinear complementarity problem satisfying certain regularity conditions for which the problem has a nonempty bounded solution set.     

On the regularization of mixed complementarity problems

A variational inequality problem (VIP) satisfying a constraint qualification can be reduced to a mixed complementarity problem (MOP). Monotonicity of the VIP implies that the MOP is also monotone. Introducing regularizing perturbations, a sequence of strictly monotone mixed complementarity problems is generated. It is shown that, if the original problem is solvable, the sequence of computable inexact solutions of the strictly monotone MCP's is bounded and every accumulation point is a solution. Under an additional condition on the precision used for solving each subproblem, the sequence converges to the minimum norm solution of the MCP.     

算子方程AX-XA=C的可解性

本文在一般Ｂａｎａｃｈ空间研究带无界算子Ａ的算子方程ＡＸ－ＸＡ＝Ｃ的可解性,利用单参数积分双半群方法,通过在算子代数Ｌ（Ｅ）上考虑间断问题弱解,证明了当算子Ａ在Ｌ（Ｅ）上诱导的算子Ａ是弱积分双半群的母元时,只要Ｃ满足一定条件,上述算子方程可解．     

四元数分析中T_Gf算子的Hlder连续性和Riemann-Hilbert边值问题

本文证明了四元数分析中的有界区域G上的非齐次Dirac方程u=f的分布解T_Gf,当f∈L_P(G),P>4时,在G上具有Holder连续性,讨论了超球和双圆柱上的方程u=f的Riemann-Hilbert边值问题,给出了可解条件和通解的积分表示,并且还证明了通解的Holder连续性。