代数特征值反问题之解的稳定性分析

考虑如下代数特征值反问题: 问题 G(A;{A_k}_1~n;λ).设 A=(a_(ij)),A_k=(a_(ij)~((k))),k=1,…,n是n+1个n×n的实对称矩阵,λ=(λ_1,…,λ_n)是n维实向量且λ_i≠λ_j,i≠j.求n维实向量c=(c_1,…,c_n)~T,使矩阵A(c)=A+sum from k=1 to n (c_kA_k)的特征值是λ_1,…,λ_n. 这一问题是经典加法问题的推广.当A_k-e_ke_k~~T(e_k是n阶单位阵的第k列)时,     

加法与乘法逆特征值问题的可解性

1.引言 本文讨论如下代数特征值反问题可解的充分条件: 问题A(加法逆特征值问题)。给定一Hermite矩阵A=(a_(ij))_(n×n)及n个实数λ_1,…,λ_n,求一实对角阵D=diag(c_1…,c_n),使得A+D的特征值为λ_1,…,λ_n。 问题M(乘法逆特征值问题)。给定一正定Hermite矩阵A=(a_(ij))_(n×n)和n个正实数     

代数特征值反问题可解的充分条件

本文讨论如下代数特征值反问题的可解性:问题G.设A=(a_(ij))和A_k=(a_(ij)~((k)))(k=1,…,n)是一组n+1个n×n实矩     

由谱数据数值稳定地构造实对称带状矩阵

§1.引言 设r,n是正整数并且0r有a_(ij)=0.     

p.n.p.矩阵的一些性质

一个n阶实方阵若其各阶主子式皆非正,则称为部分非正阵,简写作p.n.p.矩阵.特别地,各阶主子式皆负的p.n.p.矩阵称为部分负矩阵,简写为p.n.矩阵。文[1]、[5]讨论了p.n.p.矩阵的谱性质。本文在[5]的基础上讨论了p.n.p.矩阵的若干性质,并给出p.n.p.矩阵特征值的某些估计式。 引理1 设A=(A_(ij)_n×n为一p.n.p.矩阵,则A的特征值之实部不全为负(n≥2)。 证 设λ_1,λ_2,…,λ_n为A的全部特征值。假定A的每一特征值之实部皆为负。分两种情     

广义对角占优矩阵的判定

本文给出了广义严格对角占优矩阵的几个判定条件以及等价表征,这些结论分别推广了[3]与[4]的一些结果。作为约定本文总假设;A=(a_(ij))_n×n 表示复矩阵,∧_k=(?)|a_(kj)|当|a(kk)|≠0时,σ_k=(∧_k)/|α_(kk)|,θ_A={s||a_(ss)|≤∧_s,s∈N={1,2,…,n}},J_A={k||a_(kk)>∧_k,k∈N}     

布尔矩阵广义逆的若干判定定理

本文所论的矩阵均指 n 阶布尔方阵。A=(a_(ij)),B=(b_(ij)),若 a_(ij)≤b_(ij),i,j=1,2,…,n,则称 A≤B.对 A=(a_(ij)),若存在矩阵 G,使 AGA=A,称 G 是 A 的广义逆(g 逆),又令(?)称矩阵 A_0=(g_(ij))为 A 的相伴阵。A_0的转置阵为 A_0~T=(g_(ij)~T).     

矩阵方幂序列{A^k}的通项公式

文[1]、[2]对特征根满足λ_1~m=λ_2~m=…=λ_n~m的n阶矩阵方幂序列给出了通项公式,但未能给出一般矩阵方幂序列的通项公式,本文试图解决这一问题。 本文约定:A=(a_(ij))是实方阵,E是与A同     

解广义特征值反问题的同伦方法

1.引言 我们讨论下列广义特征值反问题: (G)已知B是n×n阶对称半正定矩阵,λ=(λ_1,…,λ_(2n-1))~T∈R~(2n-1),且{λ_i}~(n_3),和{λ_i}_(n+1)~(2n-1)严格交错。问题是欲求一个实对称三对角n×n阶矩阵A,使得λ_1…,λ_n是Ax=λBx的特征值,λ_(n+1),…,λ_(2n-1)是A_(n-1)x=λB_(n-1)x的特征值,其中A_(n-1),B_(n-1)分别是矩阵A,B的前n-1阶主子阵。     

关于二次型教学的浅见

<正> 任给二次型f=sum from i,i=1 to n (a_(ij)x_ix_j(a_(ij)=a(j1)),总有正交变换X=PY,使f化成标准形:f=λ_1y_1~2=…=λ_ny_n~2,其中λ_1,…,λ_n是f的矩阵A=(a_(ij))的特征值。这里我们只在实数范围内进行讨论。用正交变换化二次型为标准形的问题,也就是用正交矩阵P化实对称矩阵A为对角矩阵A的问题。这历来是教学中的重点和难点。一则由于这方面的内容有着广泛的应用,因而     

On the local semiconcavity of the solutions of the eikonal equation

We show that if the Hamiltonian is locally semiconvex with respect to the state variables and strictly convex with respect to the gradient then every viscosity solution of the eikonal equation is locally semiconcave. Furthermore, in the 1D case, we show that every viscosity solution of the eikonal equation is semiconcave if and only if the Hamiltonian is Lipschitz continuous with respect to the state variable.     

生态系统的参数确定问题

通过不同观测数据研究捕食者—被捕食者生态系统参数确定问题.研究了四种情形1.观察数据无误差,并已知一个参数值.这种情况下,参数可由其相轨线和最小二乘法精确确定.2.观察数据无误差,但所有参数未知.此时仅靠相轨线的研究,无论给出多少组精确数据,都无法精确确定这些参数.通过原非线性模型的数值计算和网格搜索法,至少需要4组数据,同样得到了精度较高的参数值.3.当观测数据有误差时,根据解的周期性,引入标准周期的概念,在一个标准周期里讨论参数的确定问题,并利用标准周期内的捕食者与被捕食者的数量均值与系统的平衡点的关系对参数进行修正,然后使用网格法进行搜索,进一步提高了参数的精度.4.当观测时间也有误差时,先选取相对最优的随机正态数对观测时刻进行修正,然后再利用3.的方法估计参数.     

Optimization of the chemotherapy process on the base of the maximum principle

We consider an optimal control problems for the chemotherapy process on the base of a known dynamical model. The solution technique is connected with a combination of control modes determined in accordance with the Pontryagin maximum principle. We propose extremal control procedures with particular components supplementing the known therapy strategies within the considered model.     

On the Gap in the Spectrum of the Translations

The spectrum of the translations in local quantum field theory will be analyzed in order to show that in a positive energy representation without vacuum vector and with lowest mass m₁ there is no gap in the spectrum which is larger than 2m₁. In particular in a zero mass representation there is no hole at all. These results are obtained with methods of analytic functions of several complex variables.     

一类上、下限互为倒数的定积分计算方法

本文利用对数函数变换简便地解决了积分上、下限互为倒数的一类定积分计算问题．     

On the influence of initial stresses on the velocities of the stoneley waves

Wave propagation along a plane boundary separating compressible, previously deformed bodies with elastic potential of arbitrary form, is studied. The linearized theory of wave propagation in bodies with finite initial deformation is used. A case in which one of the bodies is a liquid, is studied. It is shown that in the case of the Murnaghan and harmonic type potentials the wave velocities depend linearly on the initial stresses. In contrast with the case of an unbounded isotropic body /1/, here the character of the dependence is not influenced by the choice of the form of the potential. In the absence of the initial stresses the relations obtained coincide with the results of /2/.     

Beyond the Mean-Field Model of the Ring Cavity

The ring cavity device with large diffraction path in the free-space of the cavity cannotbe described within the mean-field model. It is shown to generate a large variety of monoconicaland multiconical patterns with wave and/or Türing modes, for anonlinear medium either made of two-level atoms or with a χ ⁽²⁾ crystal.Even in the limit of a single-longitudinal mode operation, monoconical structures can be differentfrom those predicted by the mean-field model. For instance, chaotic localized structures with anatomic medium and square patterns with a DOPO are presented.     

On the nonuniqueness of the solution of the Tricomi problem with the generalized Frankl matching condition

We obtain an integral representation of the solution of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions in the elliptic part of the domain and with zero posed on one characteristic of the equation. The gradient of the solution is not continuous but satisfies some condition referred to as the “generalized Frankl matching condition.” We state theorems implying that the inhomogeneous Tricomi problem either has a unique solution or is determined modulo a solution of the homogeneous Tricomi problem.     

用常微分方程模型分析预防和隔离措施对SARS发病率的影响

通过建立常微分方程模型 ,分析了预防和隔离措施对 SARS发病率的影响 ,并把计算结果与实际统计数据进行了比较 ,结果表明 ,及时高效的预防和隔离措施能够有效地控制 SARS的传播 .     

On the problem of the finiteness of the number of the combinatorial types of infinite convex polyhedra with equiangular faces

An infinite (complete) convex polyhedron with equiangular faces, that is, such that all the angles of each of its faces are equal to one another, is called irreducible if the number of monogonal faces belonging to it cannot be decreased (by identifying their sides) while preserving the equiangularity of all of the other faces and the convexity of the polyhedron itself (a lack of conditional edges). Any infinite convex polyhedron with equiangular faces can be obtained from the corresponding irreducible one by pasting in the missing number of monogons. It is proved that the number of combinatorially different irreducible polyhedra is finite, not counting three infinite series of frusta of cones with finite or infinite bases and right prisms with infinite bases. It is also established that, without exception, all infinite convex polyhedra with equiangular faces and total curvature 2are the derivatives of closed convex polyhedra with equiangular faces. The proof is carried out with the help of A. D. Milka's method from Ross. Zh. Mat., 1988, 3A830. Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 75–83, 1992.