Absolute Value Equation Solution Via Linear Programming

By utilizing a dual complementarity property, we propose a new linear programming method for solving the NP-hard absolute value equation (AVE): Ax?|x|=b, where A is an n×n square matrix. The algorithm makes no assumptions on the AVE other than solvability and consists of solving a few linear programs, typically less than four. The algorithm was tested on 500 consecutively generated random solvable instances of the AVE with n=10, 50, 100, 500 and 1000. The algorithm solved 100 % of the test problems to an accuracy of 10^?8 by solving an average of 3.3 linear programs per AVE problem.     

On the Minimum Norm Solution of Linear Programs

This paper describes a new technique to find the minimum norm solution of a linear program. The main idea is to reformulate this problem as an unconstrained minimization problem with a convex and smooth objective function. The minimization of this objective function can be carried out by a Newton-type method which is shown to be globally convergent. Furthermore, under certain assumptions, this Newton-type method converges in a finite number of iterations to the minimum norm solution of the underlying linear program.     

Sylvester矩阵方程极小范数最小二乘解的迭代解法

研究了Sylvester矩阵方程最小二乘解以及极小范数最小二乘解的迭代解法,首先利用递阶辨识原理,得到了求解矩阵方程AX+YB=C的极小范数最小二乘解的一种迭代算法,进而,将这种算法推广到一般线性矩阵方程A_iX_iB_i=C的情形,最后,数值例子验证了算法的有效性.     

线性流形上矩阵方程AXAT=B的极小Frobenius范数对称正交对称解

利用矩阵对的商奇异值分解,给出了线性流形上矩阵方程AXAT=B存在极小Frobe- nius范数对称正交对称解的充要条件及其解的表达式．     

绝对值方程算例分析

分别给出了绝对值方程存在唯一解的算例、存在2~n个解的算例、存在无穷多个解的算例以及无解的算例,并分析了这些算例中矩阵奇异值的特点.     

残差绝对值之和最小准则下回归方程的求法

本应用线性规划方法,解决了“残差绝对值之和最小”,“最大偏差最小”准则下回归方程的求法,这种方法对于建立多元线性回归方程、多项式回归方程和可以线性化的回归方程都是有效的。     

广义绝对值方程算例分析

分别给出了广义绝对值方程存在唯一解的算例、存在多个解的算例以及无解的算例,并分析了这些算例中矩阵奇异值的特点.     

Uniqueness of the Weak Extremal Solution to Biharmonic Equation with Logarithmically Convex Nonlinearities

In this note, we investigate the existence of the minimal solution and the uniqueness of the weak extremal （probably singular） solution to the biharmonic equation
△2ω=λg（ω）
with Dirichlet boundary condition in the unit ball in Rn, where the source term is logarithmically convex. An example is also given to illustrate that the logarithmical convexity is not a necessary condition to ensure the uniqueness of the extremal solution.     

一类矩阵方程的极小Frobenius范数双对称解

利用矩阵的广义奇异值分解,给出了实矩阵方程ATXA=B存在极小Frobenius范数双对称解的充要条件及其解的表达式.     

特定条件下广义伯努利方程的求解方法及其应用

将伯努利方程进行推广,给出了广义伯努利方程的一般形式,进一步分别利用全微分法和变量替换法推导了在特定条件下求广义伯努利方程的隐式解公式.同时将该公式用于求解Gompterz模型和Riccati模型.     

Derivative swift-Hohenberg方程的初边值问题解的存在性

运用压缩影射原理证明了DS-H方程的初边值问题的局部解的存在性和唯一性.     

(?)-方程的Cｐ，ｑｋ＋α-形式解及其上确界范数估计

文中研究了(?)－方程在拟凸域上的积分解算子的性质，得到了(?)－方程在具Ｃ^ｋ（ｋ≥０）类边界的拟凸开集上的Ｃ_ｐ，ｑ^ｋ＋α－形式的积分算子解及其解的上确界范数估计，它包含了文［１］仅对Ｃ^∞的情形     

Functional Equation Related to Quadratic and Norm Forms

Let T(a ₁, a ₂,..., a _n ) be a norm form of some finite proper extension of ℚ in a certain fixed integral basis, or even some integer form satisfying certain conditions. We are interested in two problems. One is finding all functions f : ℤ → ℂ satisfying the integer functional equation f(T(a ₁, a ₂,..., a _n )) = T(f(a ₁), f(a ₂),..., f(a _n )). Another closely related problem is finding all functions f : ℤ → ℂ such that T(f(a ₁, f(a ₂,..., f(a _n )) depends only on the value of T(a ₁, a ₂,..., a _n ).The second question was studied before for special quadratic forms. We extend these investigations to other types of quadratic forms and, thus, partially solve the second problem for them. The solution of the first problem for one cubic field is also presented. Finally, we give the corresponding conjecture for the first problem and, additionally, several remarks concerning the choice of norm forms.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 153–172, April–June, 2005.     

Lower Semicontinuity of the Minimum in Parametric Convex Programs

Recently, Best and Ding (Ref. 1) established a result on the lower semicontinuity of the infimum value function of a parametric convex quadratic program. In this paper, we extend this result to general convex programs. The case of semi-infinite convex optimization is included.     

Constructing the Solution to Robin's Equation

AMS (MOS): 45B05, 45C05

The standard method for obtaining a nontrivial solution to Robin's equation is by successive approximation. The usual proof of the convergence of the sequence obtained by this iteration procedure is based on Neumann's method of arithmetic mean, and requires that the surface over which the integral operator is defined be smooth and convex. Without assuming that the surface is convex, in this paper we give a new and simpler proof of the convergence of this sequence.     

Entropy of Absolute Convex Hulls in Hilbert Spaces

The metric entropy of absolute convex hulls of sets in Hilbertspaces is studied for the general case when the metric entropyof the sets is arbitrary. Under some regularity assumptions,the results are sharp. 2000 Mathematical Subject Classification41A46 (primary), 60G15 (secondary).     

On the Norm Equation Over Function Fields

If K is an algebraic function field of one variable over analgebraically closed field k and F is a finite extension ofK, then any element a of K can be written as a norm of someb in F by Tsen's theorem. All zeros and poles of a lead to zerosand poles of b, but in general additional zeros and poles occur.The paper shows how this number of additional zeros and polesof b can be restricted in terms of the genus of K, respectivelyF. If k is the field of all complex numbers, then we use Abel'stheorem concerning the existence of meromorphic functions ona compact Riemann surface. From this, the general case of characteristic0 can be derived by means of principles from model theory, sincethe theory of algebraically closed fields is model-complete.Some of these results also carry over to the case of characteristicp>0 using standard arguments from valuation theory.