Proximinal subspaces of 2-normed spaces

In 1965 Ga¨hler introduced 2-normed spaces and since then, this topic have been intensively studied and developed. We shall introduce the notion of 1-type proximinal subspaces of 2-normed spaces and give some results in this field.     

On tridiagonal matrices unitarily equivalent to normal matrices

In this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied.It is well-known that any matrix is unitarily equivalent to a tridiagonal matrix. In case of a normal matrix the resulting tridiagonal inherits a strong relation between its super- and subdiagonal elements. The corresponding elements of the super- and subdiagonal will have the same absolute value.In this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied. Both an iterative reduction based on Krylov sequences as a direct tridiagonalization procedure via Householder transformations are reconsidered. This equivalence transformation is then applied to the normal case and equality of the absolute value between the super- and subdiagonals is proved. Self-adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is proved. Properties when applying the reduction on symmetric, skew-symmetric, Hermitian, skew-Hermitian and unitary matrices and their relations with, e.g., complex symmetric and pseudo-symmetric matrices are presented.It is shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the Takagi factorization. Finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given.     

Hypercyclic algebras for convolution and composition operators

We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator D of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras for many convolution operators not induced by polynomials, such as $\text{cos}(D)$, $D{e}^D$, or $e^D?aI$, where $0<a\le 1$. In contrast, weighted composition operators on function algebras of analytic functions on a plane domain fail to support supercyclic algebras.     

Block‐triangular preconditioners for PDE‐constrained optimization

In this paper we investigate the possibility of using a block‐triangular preconditioner for saddle point problems arising in PDE‐constrained optimization. In particular, we focus on a conjugate gradient‐type method introduced by Bramble and Pasciak that uses self‐adjointness of the preconditioned system in a non‐standard inner product. We show when the Chebyshev semi‐iteration is used as a preconditioner for the relevant matrix blocks involving the finite element mass matrix that the main drawback of the Bramble–Pasciak method—the appropriate scaling of the preconditioners—is easily overcome. We present an eigenvalue analysis for the block‐triangular preconditioners that gives convergence bounds in the non‐standard inner product and illustrates their competitiveness on a number of computed examples. Copyright © 2010 John Wiley & Sons, Ltd.     

多元统计描述中随机误差与变量空间角的关系

多元分析的误差传递需要一种简单、准确、数值化的表达方法.向量空间中,线性多元混合信号的随机误差可表述成真值子空间中随机向量的表现;由体系多元变量对应的向量构成的真值子空间中,被关注向量和其他向量子空间的空间角θ是描述多元体系的重要参数.如果被关注向量和其他向量子空间关系确定,体系总体误差呈正态分布,那么,被关注向量上误差也是正态分布,其多元统计分析结果的标准差与体系误差标准差的比值为1/(2·sin(θ/2),结论在构造算例和邻、间、对苯二酚混合体系的紫外光度分析中得到验证.     

Computational Krylov‐based methods for large‐scale differential Sylvester matrix problems

In the present paper, we propose Krylov‐based methods for solving large‐scale differential Sylvester matrix equations having a low‐rank constant term. We present two new approaches for solving such differential matrix equations. The first approach is based on the integral expression of the exact solution and a Krylov method for the computation of the exponential of a matrix times a block of vectors. In the second approach, we first project the initial problem onto a block (or extended block) Krylov subspace and get a low‐dimensional differential Sylvester matrix equation. The latter problem is then solved by some integration numerical methods such as the backward differentiation formula or Rosenbrock method, and the obtained solution is used to build the low‐rank approximate solution of the original problem. We give some new theoretical results such as a simple expression of the residual norm and upper bounds for the norm of the error. Some numerical experiments are given in order to compare the two approaches.     

Continuation of invariant subspaces

In this work we consider implementation and testing of an algorithm for continuation of invariant subspaces. Copyright © 2001 John Wiley & Sons, Ltd.     

Maximal Projective Subspaces in the Variety of Planar Normal Sections of a Flag Manifold

We continue the study of the variety X[M] of planar normal sections on a natural embedding of a flag manifold M. Here we consider those subvarieties of X[M] that are projective spaces. When M=G/T is the manifold of complete flags of a compact simple Lie group G, we obtain our main results. The first one characterizes those subspaces of the tangent space T[T] (M), invariant by the torus action and which give rise to real projective spaces in X[M]. The other one is the following. Let be the tangent space of the inner symmetric space G/K at [K] . Then RP ( ) is maximal in X[M] if and only if 2(G/K) does not vanish.     

Shift Invariant Subspaces with Arbitrary Indices in ? Spaces

We construct right shift invariant subspaces of index n, 1?n?∞, in ?^p spaces, 2<p<∞, and in weighted ?^p spaces.