Extremal eigenvalue intervals of symmetric tridiagonal interval matrices

Computing the extremal eigenvalue bounds of interval matrices is non‐deterministic polynomial‐time (NP)‐hard. We investigate bounds on real eigenvalues of real symmetric tridiagonal interval matrices and prove that for a given real symmetric tridiagonal interval matrices, we can achieve its exact range of the smallest and largest eigenvalues just by computing extremal eigenvalues of four symmetric tridiagonal matrices.     

Extremal spectral properties of Otsuki tori

Otsuki tori form a countable family of immersed minimal two‐dimensional tori in the unitary three‐dimensional sphere. According to the El Soufi‐Ilias theorem, the metrics on the Otsuki tori are extremal for some unknown eigenvalues of the Laplace‐Beltrami operator. Despite the fact that the Otsuki tori are defined in quite an implicit way, we find explicitly the numbers of the corresponding extremal eigenvalues. In particular we provide an extremal metric for the third eigenvalue of the torus.     

Extremal eigenvalues of measure differential equations with fixed variation

In this paper we will study eigenvalues of measure differential equations which are motivated by physical problems when physical quantities are not absolutely continuous. By taking Neumann eigenvalues of measure differential equations as an example, we will show how the extremal problems can be completely solved by exploiting the continuity results of eigenvalues in weak* topology of measures and the Lagrange multiplier rule for nonsmooth functionals. These results can give another explanation for extremal eigenvalues of Sturm-Liouville operators with integrable potentials.     

Extremal problems for Fredholm eigenvalues

The study of extremal problems for Fredholm eigenvalues was initiated by Schiffer in the context of the existence of conformal maps onto canonical domains. We present a different approach to solving rather general extremal problems for Fredholm eigenvalues related to appropriate univalent functions with quasiconformal extensions. It involves the complex geometry of the universal Teichmüller space.     

An extremal property of positive operators and its spectral implications

The Perron-Frobenius theory of a positive operator T (defined on an ordered space E) is developed from an extremal characterization of its largest eigenvalue. This characterization has manifest intrinsic interest. Additionally, it is used to give a particularly revealing derivation of the basic results concerning the existence, multiplicity, and distribution of the eigenvalues of T of maximum modulus. A significant feature of this derivation is that the customary assumptions that the space E be complete and/or that its positive cone have a nonvoid interior are often unnecessary or can be replaced by weaker hypotheses more amenable to practical applications (see 1, 3). The extremal characterization proof of the distribution properties of the eigenvalues of maximum modulus is new in the infinite dimensional case. Also, several new results on the extent of applicability of the extremal characterization are given     

A spectral trichotomy method for symplectic matrices

This paper presents an algorithm for the numerical approximation of spectral projectors onto the invariant subspaces corresponding to the eigenvalues inside, on, and outside the unit circle of a symplectic matrix. The algorithm constructs iteratively three matrix sequences from which the projectors are obtained. The convergence depends essentially on the gap between the unit circle and the eigenvalues inside it. A larger gap leads to faster convergence. Theoretical and algorithmic aspects of the algorithm are developed. Numerical results are reported.     

A theorem on the extremum of a nonnegative bounded function

Using perturbation formulas for eigenvalues of linear operators, an equation is obtained for the extremal values of certain nonnegative bounded functions. Formulas are obtained for the derivative of regularizing maximal eigenvalues of matrices which depend on a parameter.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 3–7, 1989.     

拟共形映射与调和函数

本文利用调和函数的Ｄｉｒｉｃｈｌｅｔ积分来估计具有给定边界值的拟共形映射的极值伸缩商,回答Ｄ．Ｐａｒｔｙｋａ关于拟对称函数谱值和特征值的一个问题。     

Circular trichotomy of the spectrum of regular matrix pencils

We introduce a method for approximating the right and left deflating subspaces of a regular matrix pencil corresponding to the eigenvalues inside, on and outside the unit circle. The method extends the iteration used in the context of spectral dichotomy, where the assumption on the absence of eigenvalues on the unit circle is removed. It constructs two matrix sequences whose null spaces and the null space of their sum lead to approximations of the deflating subspaces corresponding to the eigenvalues of modulus less than or equal to 1, equal to 1 and larger than or equal to 1. An orthogonalization process is then used to extract the desired delating subspaces. The resulting algorithm is an inverse free, easy to implement, and sufficiently fast. The derived convergence estimates reveal the key parameters, which determine the rate of convergence. The method is tested on several numerical examples.     

k＜1时Cartan-Hartogs域上的极值问题

本文讨论两种类型的极值问题,其中一种类型的极值问题可以认为是复平面上经典的Schwarz引理在高维的一个推广;另一种类型的极值是某空间上的度量,可以用来考虑域的双全纯等价分类问题．在本文中,k＜1时Cartan-Hartogs域与单位超球间的极值与极值映照被得到。     

Spectrum of One-Dimensional p-Laplacian with an Indefinite Integrable Weight

Motivated by extremal problems of weighted Dirichlet or Neumann eigenvalues, we will establish two fundamental results on the dependence of weighted eigenvalues of the one-dimensional p-Laplacian on indefinite integrable weights. One is the continuous differentiability of eigenvalues in weights in the Lebesgue spaces L ^γ with the usual norms. Another is the continuity of eigenvalues in weights with respect to the weak topologies in L ^γ spaces. Here 1 ≤ γ ≤ ∞. In doing so, we will give a simpler explanation to the corresponding spectrum problems, with the help of several typical techniques in nonlinear analysis such as the Fréchet derivative and weak* convergence.     

第四类超Cartan域上的极值问题

讨论了第四类超Cartan域Y_(Ⅳ)(N;n;k)上的极值问题,得到了第四类超Car- tan域与单位超球间的极值和极值映照.     

A new matrix characterization of fredholm eigenvalues of quasicircles

We give a new characterization of the Fredholm eigenvalues of a quasicircle or of a quasisymmetric transformation. This leads to a matrix eigenvalue problem for a suitable Hermitian matrix. There are connections to extremal quasiconformal mappings and reflections.     

A Note on the Calculation of Step-Lengths in Interior-Point Methods for Semidefinite Programming

In each iteration of an interior-point method for semidefinite programming, the maximum step-length that can be taken by the iterate while maintaining the positive semidefiniteness constraint needs to be estimated. In this note, we show how the maximum step-length can be estimated via the Lanczos iteration, a standard iterative method for estimating the extremal eigenvalues of a matrix. We also give a posteriori error bounds for the estimate. Numerical results on the performance of the proposed method against two commonly used methods for calculating step-lengths (backtracking via Cholesky factorizations and exact eigenvalues computations) are included.     

等参函数及相关问题研究

著名的Yau 猜想断言单位球面中的紧致嵌入极小超曲面的Laplace 算子的第一特征值等于其维数. 近年来有许多几何学家致力于对Yau 猜想的研究, 但是到目前为止, 已有的结论只是一些关于第一特征值估计的不等式. 作为本文的一个主要结果, 本文证明了对于单位球面中的等参极小超曲面,Yau 猜想是正确的. 进一步地, 对于等参超曲面的焦流形(实际上是球面的极小子流形), 本文还证明了在一定维数条件下, 它的第一特征值也是其维数.
作为本文的第二个主要结果, 以著名的Schoen-Yau-Gromov-Lawson 的关于数量曲率的手术理论为出发点, 本文在一个Riemann 流形的嵌入超曲面处作手术, 构造了一个新的具有丰富几何性质的流形, 称为double 流形. 特别地, 本文在单位球面的极小等参超曲面处实行了这一手术, 发现得到的double 流形不仅有很复杂的拓扑(但其示性类有精确描述), 还存在数量曲率为正的度量, 更重要的是保持了等参叶状结构.
比Willmore 曲面更广泛的定义是Willmore 子流形, 即Willmore 泛函在球面中的的极值子流形.单位球面中的Willmore 子流形的例子在已有文献中是非常罕见的. 作为本文的另外两个主要结果, 通过深入挖掘单位球面上的OT-FKM- 型等参函数的焦流形的性质, 本文发现其极大值对应的焦流形是单位球面的一系列Willmore 子流形; 之后, 本文用几何办法统一证明了单位球面中具有4 个不同主曲率的等参超曲面的焦流形都是单位球面的Willmore 子流形. 这些新的Willmore 子流形是极小的,但一般不是Einstein 的.     

A bound for ratios of eigenvalues of Schrödinger operators with single-well potentials

For Schrödinger operators with nonnegative single-well potentials ratios of eigenvalues are extremal only in the case of zero potential. To prove this, we investigate some monotonicity properties of Prüfer-type variables.

     

Two-sided harmonic subspace extractions for the generalized eigenvalue problem

One crucial step of the solution of large-scale generalized eigenvalue problems with iterative subspace methods, e.g. Arnoldi, Jacobi-Davidson, is a projection of the original large-scale problem onto a low dimensional subspaces. Here we investigate two-sided methods, where approximate eigenvalues together with their right and left eigenvectors of the full-size problem are extracted from the resulting small eigenproblem. The two-sided Ritz-Galerkin projection can be seen as the most basic form of this approach. It usually provides a good convergence towards the extremal eigenvalues of the spectrum. For improving the convergence towards interior eigenvalues, we investigate two approaches based on harmonic subspace extractions for the generalized eigenvalue problem. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal a priori error bounds for the Rayleigh-Ritz method

We derive error bounds for the Rayleigh-Ritz method for the approximation to extremal eigenpairs of a symmetric matrix. The bounds are expressed in terms of the eigenvalues of the matrix and the angle between the subspace and the eigenvector. We also present a sharp bound.

     

单圈图的零度的一个注记

The number of zero eigenvalues in the spectrum of the graph G is called its nullity and is denoted by η(G).In this paper,we determine the all extremal unicyclic graphs achieving the fifth upper bound n-6 and the sixth upperbound n-7.     

Dependence of solutions and eigenvalues of measure differential equations on measures

It is well known that solutions of ordinary differential equations are continuously dependent on finite-dimensional parameters in equations. In this paper we study the dependence of solutions and eigenvalues of second-order linear measure differential equations on measures as an infinitely dimensional parameter. We will provide two fundamental results, which are the continuity and continuous Fréchet differentiability in measures when the weak^? topology and the norm topology of total variations for measures are considered respectively. In some sense the continuity result obtained in this paper is the strongest one. As an application, we will give a natural, simple explanation to extremal problems of eigenvalues of Sturm–Liouville operators with integrable potentials.