Nullspaces and frames

In this paper we give new characterizations of Riesz and conditional Riesz frames in terms of the properties of the nullspace of their synthesis operators. On the other hand, we also study the oblique dual frames whose coefficients in the reconstruction formula minimize different weighted norms.     

On angles and distances between subspaces

The angles and distances between two given subspaces of are investigated on the basis of a joint decomposition of the corresponding orthogonal projectors. Several new results are established, with the particular attention paid to the notions of inclinedness and minimal angle. To demonstrate the usefulness of the approach utilized, some results known to be valid in Hilbert space are reestablished in , either in generalized form or with considerably shorter proofs than in the original sources.     

A variational characterization of canonical angles between subspaces

Canonical angles between subspaces of a unitary space are characterized by a min-max property which involves inner products.     

The solvability of two linear matrix equations

The solvability conditions of the following two linear matrix equations (i)A₁X₁B₁ +A₂X₂B₂ +A₃X₃B₃ =C,(ii) A₁XB₁ =C₁ A₂XB₂ =C₂ are established using ranks and generalized inverses of matrices. In addition, the duality of the three types of matrix equations

(iii) A ₁ X ₁ B ₁+A ₂ X ₂ B ₂+A ₃ X ₃ B ₃+A ₄ X ₄ B ₄=C, (iv) A ₁ XB ₁=C ₁ A ₂ XB ₂=C ₂ A ₃ XB ₃=C ₃ A ₄ XB ₄=C ₄, (v) AXB+CXD=E are also considered.     

The solvability of two linear matrix equations

The solvability conditions of the following two linear matrix equations (i)A₁X₁B₁+A₂X₂B₂+A₃X₃B₃=C,(ii) A₁XB₁=C₁A₂XB₂=C₂ are established using ranks and generalized inverses of matrices. In addition, the duality of the three types of matrix equations

(iii) A₁X₁B₁+A₂X₂B₂+A₃X₃B₃+A₄X₄B₄=C, (iv) A₁XB₁=C₁A₂XB₂=C₂A₃XB₃=C₃A₄XB₄=C₄, (v) AXB+CXD=E are also considered.     

The common Re-nnd and Re-pd solutions to the matrix equations AX = C and XB = D

In this article, we consider common Re-nnd and Re-pd solutions of the matrix equations AX = C and XB = D with respect to X, where A, B, C and D are given matrices. We give necessary and sufficient conditions for the existence of common Re-nnd and Re-pd solutions to the pair of the matrix equations and derive a representation of the common Re-nnd and Re-pd solutions to these two equations when they exist. The presented examples show the advantage of the proposed approach.     

The bounds of the solutions to generalized modular equations

In this paper the authors study a monotonicity of several functions involving ma(r) and μa(r). By using these results, the authors obtain some new bounds of the solutions of Ramanujan's generalized modular equations.     

Invariant subspaces and exact solutions of a class of dispersive evolution equations

The invariant subspace method is used to classify a class of systems of nonlinear dispersive evolution equations and determine their invariant subspaces and exact solutions. A crucial step is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that systems of evolution equations admit. A few examples of presenting exact solutions with generalized separated variables illustrate the effectiveness of the invariant subspace method in solving systems of nonlinear evolution equations.     

Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization

We show that a locally Lipschitz homeomorphism function is semismooth at a given point if and only if its inverse function is semismooth at its image point. We present a sufficient condition for the semismoothness of solutions to generalized equations over cone reducible (nonpolyhedral) convex sets. We prove that the semismoothness of solutions to the Moreau-Yosida regularization of a lower semicontinuous proper convex function is implied by the semismoothness of the metric projector over the epigraph of the convex function. This paper is dedicated to Terry Rockafellar on the occasion of his seventieth birthday     

Characterizations of some classes of relatively regular operators

A bounded linear operator A on a Banach space is called relatively regular, if there is a bounded linear operator B such that ABA=A. In this case B is called a g₁-inverse of A. In this paper we characterize some classes of relatively regular operators A via the set {B₁-B₂:B₁ and B₂ are g₁-inverses of A}.     

Local existence of strong solutions to the generalized Boussinesq equations

This paper deals with the local existence and uniqueness of strong solutions for the generalized Boussinesq equations with fractional dissipation. As a corollary, we establish some regularity criteria to guarantee smoothness of solutions.     

Fast solving of weighted pairing least-squares systems

This paper presents a generalization of the “weighted least-squares” (WLS), named “weighted pairing least-squares” (WPLS), which uses a rectangular weight matrix and is suitable for data alignment problems. Two fast solving methods, suitable for solving full rank systems as well as rank deficient systems, are studied. Computational experiments clearly show that the best method, in terms of speed, accuracy, and numerical stability, is based on a special {1, 2, 3}-inverse, whose computation reduces to a very simple generalization of the usual “Cholesky factorization-backward substitution” method for solving linear systems.     

Symmetries and solutions to the thin film equations

We study symmetries and solutions of the generalized fourth-order nonlinear partial differential equations which arise from studies of thin liquid films. It is shown that the equations admit extended scaling and rotation symmetries and a class of generalized conditional symmetries for certain coefficient functions. Exact solutions associated to the symmetries are obtained.     

On configurations of subspaces of a Hilbert space with fixed angles between them

We investigate the set of irreducible configurations of subspaces of a Hilbert space for which the angle between every two subspaces is fixed. This is the problem of *-representations of certain algebras generated by idempotents and depending on parameters (on the set of angles). We separate the class of problems of finite and tame representation type. For these problems, we indicate conditions on angles under which the configurations of subspaces exist and describe all irreducible representations.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 606–615, May, 2004.     

Angles between infinite dimensional subspaces with applications to the Rayleigh-Ritz and alternating projectors methods

We define angles for infinite dimensional subspaces of Hilbert spaces, inspired by the work of E.J. Hannan, 1961/1962. The angles of Dixmier and Friedrichs, and the gaps are characterized. We establish connections between the angles corresponding to orthogonal complements. The sensitivity of angles with respect to subspaces is estimated. We show that the squared cosines of the angles from one subspace to another can be interpreted as Ritz values in the Rayleigh-Ritz method. The Hausdorff distance between the Ritz values, corresponding to different trial subspaces, is shown to be bounded by a constant times the gap between the subspaces. We prove a similar eigenvalue perturbation bound that involves the gap squared. An ultimate acceleration of the classical alternating projectors method is proposed. Its convergence rate is estimated in terms of the angles. We illustrate the acceleration for a domain decomposition method with a small overlap for the 1D diffusion equation.     

非线性波动方程解的Blow up

本文主要讨论高维空间非线性波动方程的Ｃａｕｃｈｙ问题整体解的非存在性，我们证明对Ｕｕ－△ｕ＝ｆ（ｕ），ｆ（ｕ）＝ｃ｜ｕ｜ｐ－１ｕ，当１＜ｐ≥时，若初始能量非正，则无论初值数据的ｃｋ－范数（连续空间范数）多么小，解按Ｃｋ－范数或按Ｈｓ（Ｒｎ）（Ｓ≥１）都在有限时间内Ｂｌｏｗｕｐ，并且有相同的生命跨度．     

Generalized inverses and Douglas equations

In this paper we study the general concept of reduced solution for a Douglas type equation and we parametrize these solutions using generalized inverses. On the other hand, we characterize different sorts of generalized inverses by means of solutions of Douglas type equations.

     

Chaplygin气体方程在不变子空间中的精确解和爆破界面

通过不变子空间方法,Chaplygin气体方程被约化为有限维动力系统.它很多有意义的精确解被构造,包括有限时间爆破解,整体解和关于时间的周期解,其中一些解的爆破界面被刻画.