n元二次齐次函数极值的判别法

<正> 在研究多元函数的极值问题中,我们经常会遇到多元二次齐次函数,本文根据这类函数的结构特点,应用实二次型的正定性,给出判定极值的一个简单方法。设实n元二次齐次函数的矩阵表达式为     

一类二元n次型的正定性问题

本讨论了一类二元n次型的正定性问题，并且获得了它正定的一个充要条件。     

矩阵的正定性与隐函数的极值

利用隐函数的导数和矩阵正定性在多元显函数极值方面的应用,给出隐函数极值存在的必要条件和充分条件,并实例说明如何根据矩阵的正定性判定隐函数的极值.     

如何判别函数的极值

提出一种判别n元函数极值的方法，当n=2时此法能解决传统法在△=0失效时的极值判别问题。     

基于矩阵初等变换的n元函数极值的快速判别法

给出了 n元函数极值的一个充分条件 ,并结合矩阵的初等变换建立了 n元函数极值的一种快速判别法 ,最后给出了一个例子     

SL(2,R)上的Jackson型正定理

本文在SL(2,R)上引入距离、光滑模、导数等概念,给出了SL(2,R)上的连续函数用Tn和Bσ,n逼近的Jackson型正定理,得到了SL(2,R)上函数的光滑性和最佳逼近阶之间的关系.     

一些单叶函数族导数的积分平均值

K(α),st_k(α),C(β,α),(α<1,β<1)分别表示α阶凸函数族,k 次对称的α阶星形函数族,α型β阶的近于凸函数族.在这篇文章中,我们分別对(?)(K(α)的闭凸包),(?),(?),(α<1,β≤1/2)决定了凸泛函L(f)=(1/(2π)(?)))~(1/p),0≤r<1,p≥1,n=1,2,…,的极值和极值函数.我们的结果表明:极值函数仅依赖于函数族而与参数 n,r,p 无关;而且对任何固定的函数族,除掉旋转外极值函数是唯一的.     

幂零Lie群上的尖锐不等式

本文首先回顾近些年关于幂零Lie群上尖锐不等式的工作,包括最佳常数、极值函数、稳定性和其他一些改进,然后证明了一个新的次临界不等式.本文讨论了共形对称性、紧性、谱估计和流方法.在具有高维中心的群上对奇异指标有正定性的限制为我们的研究带来了额外的难度.     

取代拉格朗日乘数法

给出多元函数意义,纠正现今众多文献中康托尔的一个错误.从极值(点)的几何意义逐步推导出(n×n)方程组求解等式条件下n元函数极值点的新方法;用外积得出拉格朗日乘数表达式解决了上个世纪80年代钱伟长提出的乘子是否唯一的问题.给出了不定方程(组)确定显函数组命题.本文指出对应思想方法乃处理问题的重要工具.     

紧黎曼流形上Hardy-Littlewood-Sobolev不等式的极值问题:次临界逼近法

令(M~n,g)为n维无边紧黎曼流形,0αn,qn/n-α,该文研究了下列HardyLittlewood-Sobolev (HLS)不等式‖I_αf‖_L~q(M~n)≤C‖f‖L~p(M~n),■的极值问题.首先,利用算子I_α:L~p(M~n)→L~q(M~n)在次临界情形(即p(nq)/(n+αq))时的紧致性,证明p(nq)/(n+αq)时极值函数f_p∈L~p(M~n)的存在性;进而证明函数列{f_p}为临界情形时HLS不等式的最佳常数的极值列;最后,结合极值列{f_p}在L((nq)/(n+αq))(M~n)中的一致有界性,利用文献[32]建立的集中列紧原理证明{f_p}在L((nq)/(n+αq))(M~n)中存在收敛子列,从而给出临界情形(即p=((nq)/(n+αq)))时极值函数的存在性.     

浅谈正定二次型的习题课设计

在高等代数的实二次型内容中,正定二次型占有特殊的地位.本文从概念的回顾、正定二次型与正定矩阵的判断、二次型正定及矩阵正定的性质、其它类型二次型四个方面来设计正定二次型的习题课,并通过具体例子说明例题、习题精选的原则.     

The Extremism of Two Variable Function Based on the Positive Definite Property

In this paper we give out a sufficient and solution of quadratic function's maximum with theory of quadratic form and give out the definition of the positive definite property of the following homogeneous polynomials of degree In two variables function, based on the definite of local maximum of two variables function.     

An improved Perry conjugate gradient method with adaptive parameter choice

In this paper, we present a conjugate gradient method for solving unconstrained optimization problems. Motivated by Perry conjugate gradient method and Dai-Liao method, an improved Perry update matrix is proposed to overcome the non-symmetric positive definite property of the Perry matrix. The parameter in the update matrix is determined by minimizing the condition number of the iterative matrix which can ensure the positive definite property. The obtained method can also be considered as a modified form of CG-DESCENT method with an adjusted term. Under some mild conditions, the presented method is global convergent. Numerical experiments under CUTEst environment show that the proposed algorithm is promising.     

一类中值问题的另证

通过考察微分中值定理中的中值点的性质,利用积分中值定理,得到一类有关定积分的中值问题的一种新证明．     

Skew-Hermitian Triangular Splitting Iteration Methods for Non-Hermitian Positive Definite Linear Systems of Strong Skew-Hermitian Parts

By further generalizing the skew-symmetric triangular splitting iteration method studied by Krukier, Chikina and Belokon (Applied Numerical Mathematics, 41 (2002), pp. 89–105), in this paper, we present a new iteration scheme, called the modified skew-Hermitian triangular splitting iteration method, for solving the strongly non-Hermitian systems of linear equations with positive definite coefficient matrices. We discuss the convergence property and the optimal parameters of this new method in depth. Moreover, when it is applied to precondition the Krylov subspace methods like GMRES, the preconditioning property of the modified skew-Hermitian triangular splitting iteration is analyzed in detail. Numerical results show that, as both solver and preconditioner, the modified skew-Hermitian triangular splitting iteration method is very effective for solving large sparse positive definite systems of linear equations of strong skew-Hermitian parts.     

2n阶Lidstone边值问题正解的逐次迭代

研究了2n阶Lidstone边值问题正解的逐次迭代,其中非线性项依赖于所有偶数阶导数.通过考察非线性项在某些有国介集合上的“高度”并利用单调迭代方法构造了一个逐次迭代程序.这个迭代程序从一个多项式开始并且是可行的.使用这个结论获得了m个正解的迭代方法,其中m是一个任意的自然数.     

Method for nonlocal improvement of extreme controls in the maximization of the terminal state norm

A numerical approach is applied to the maximization of a positive definite quadratic form on the set of terminal states of a linear system with interval constraints on the control. An optimality criterion is used to develop a method for nonlocal improvement of controls satisfying the maximum principle (extreme points of the reachable set). The iterative procedure of the method is proved to converge. Numerical results are presented.     

Algebraic multilevel iteration method for Stieltjes matrices

The numerical solution of elliptic selfadjoint second-order boundary value problems leads to a class of linear systems of equations with symmetric, positive definite, large and sparse matrices which can be solved iteratively using a preconditioned version of some algorithm. Such differential equations originate from various applications such as heat conducting and electromagnetics. Systems of equations of similar type can also arise in the finite element analysis of structures. We discuss a recursive method constructing preconditioners to a symmetric, positive definite matrix. An algebraic multilevel technique based on partitioning of the matrix in two by two matrix block form, approximating some of these by other matrices with more simple sparsity structure and using the corresponding Schur complement as a matrix on the lower level, is considered. The quality of the preconditioners is improved by special matrix polynomials which recursively connect the preconditioners on every two adjoining levels. Upper and lower bounds for the degree of the polynomials are derived as conditions for a computational complexity of optimal order for each level and for an optimal rate of convergence, respectively. The method is an extended and more accurate algebraic formulation of a method for nine-point and mixed five- and nine-point difference matrices, presented in some previous papers.