A new subspace iteration method for the algebraic Riccati equation

We consider the numerical solution of the continuous algebraic Riccati equation A*X + XA ? XFX + G = 0, with F = F*,G = G* of low rank and A large and sparse. We develop an algorithm for the low‐rank approximation of X by means of an invariant subspace iteration on a function of the associated Hamiltonian matrix. We show that the sought‐after approximation can be obtained by a low‐rank update, in the style of the well known Alternating Direction Implicit (ADI) iteration for the linear equation, from which the new method inherits many algebraic properties. Moreover, we establish new insightful matrix relations with emerging projection‐type methods, which will help increase our understanding of this latter class of solution strategies. Copyright © 2014 John Wiley & Sons, Ltd.     

Extremally disconnected spaces, subspaces and retracts

A characterization of compact subspaces of extremally disconnected spaces is given which is similar to Gleason's characterization that the extremally disconnected spaces are projective for the class of compact spaces. An equivalence is established between the questions of whether every basically disconnected space is embeddable into an extremally disconnected space and the Borel lifting problems for the category algebras of generalized Cantor cubes. We study an inductive method of strengthening the product topology on the Cantor cubes to extremally disconnected topologies and use this to establish, from CH, that every compact F-space of weight c⁺ embeds into an extremally disconnected space.     

RICE CONDITION NUMBERS OF CERTAIN CHARACTERISTIC SUBSPACES

This paper proposes the Rice condition numbers for invariant subspace, singular sub-spaces of a matrix and deflating subspaces of a regular matrix pair. The first-order perturbation estimations for these subspaces are derived by applying perturbation expansions of orthogonal projection operators.     

On the Vector ε-Algorithm for Solving Linear Systems of Equations

The four vector extrapolation methods, minimal polynomial extrapolation, reduced rank extrapolation, modified minimal polynomial extrapolation and the topological epsilon algorithm, when applied to linearly generated vector sequences are Krylov subspace methods and it is known that they are equivalent to some well-known conjugate gradient type methods. However, the vector -algorithm is an extrapolation method, older than the four extrapolation methods above, and no similar results are known for it. In this paper, a determinantal formula for the vector -algorithm is given. Then it is shown that, when applied to a linearly generated vector sequence, the algorithm is also a Krylov subspace method and for a class of matrices the method is equivalent to a preconditioned Lanczos method. A new determinantal formula for the CGS is given, and an algebraic comparison between the vector -algorithm for linear systems and CGS is also given.     

Perturbation Analysis for the Eigenvalue Problem of a Formal Product of Matrices

We study the perturbation theory for the eigenvalue problem of a formal matrix product A ₁ ^{s 1} ··· A _p ^{s p}, where all A _k are square and s _k {–1, 1}. We generalize the classical perturbation results for matrices and matrix pencils to perturbation results for generalized deflating subspaces and eigenvalues of such formal matrix products. As an application we then extend the structured perturbation theory for the eigenvalue problem of Hamiltonian matrices to Hamiltonian/skew-Hamiltonian pencils.     

Linear Relations, Differential Inclusions, and Degenerate Semigroups

A generalized linear differential equation in a Banach space is studied. The construction of a phase space and solutions with the help of the spectral theory of linear operators, ergodic theorems, and degenerate semigroups of linear operators is carried out.     

可变抽样区间的非参数控制图

最近几年一些学者研究了可变抽样区间的质量控制图。Amin等提出了可变抽样区间(VSI)的非参数控制图———符号 (Sign)统计量图〔1〕。本文在此基础上研究位置VSI符号控制图的制定方法 ,并设计离散VSI符号控制图。符号控制图的优点是对非正态总体亦可应用 ,并且不需要过程方差的信息。本文将所设计的VSI符号控制图同固定抽样区间 (FSI)的常规图作比较 ,并举实例说明符号控制图的应用     

Beurling Type Theorem on the Hilbert Space Generated by a Positive Sequence

Let H²(γ) be the Hilbert space over the bidisk D² generated by a positive sequence γ={γ_nm}_n,m ≥ 0. In this paper, we prove that the Beurling type theorem holds for the shift operator on H²(γ) with γ={γ_nm}_n,m ≥ 0 satisfying certain series of inequalities. As a corollary, we give several applications to a class of classical analytic reproducing kernel Hilbert spaces over the bidisk D².     

Dual Lie Bialgebra Structures of W -algebra W (2, 2) Type

In the present paper, we investigate the dual Lie coalgebras of the centerless W(2, 2) algebra by studying the maximal good subspaces. Based on this, we construct the dual Lie bialgebra structures of the centerless W(2, 2) Lie bialgebra. As by-products, four new infinite dimensional Lie algebras are obtained.     

A new class of conformable spectral observers for signal reconstruction

In this work, the design of spectral observers for signal reconstruction based on Kalman filters is performed and evaluated. The conformable derivative and the beta‐derivative were used to design the Kalman filters. Both derivatives satisfy the same formulas of the classical derivation, eg, the chain rule. The derivative order, the Ricatti equation parameters, and the observers tuning parameters were optimized using an optimization algorithm based on the bat's echolocation behavior (Bat algorithm). The simulation results showed the advantages of using the proposed observers for the signal reconstruction.