Equivalent operator preconditioning for elliptic problems

The numerical solution of linear elliptic partial differential equations most often involves a finite element or finite difference discretization. To preserve sparsity, the arising system is normally solved using an iterative solution method, commonly a preconditioned conjugate gradient method. Preconditioning is a crucial part of such a solution process. In order to enable the solution of very large-scale systems, it is desirable that the total computational cost will be of optimal order, i.e. proportional to the degrees of freedom of the approximation used, which also induces mesh independent convergence of the iteration. This paper surveys the equivalent operator approach, which has proven to provide an efficient general framework to construct such preconditioners. Hereby one first approximates the given differential operator by some simpler differential operator, and then chooses as preconditioner the discretization of this operator for the same mesh. In this survey we give a uniform presentation of this approach, including theoretical foundation and several practically important applications for both symmetric and nonsymmetric equations and systems, and some nonlinear examples in the context of Newton linearization. Dedicated to the memory of Gene Golub for his friendly manner and for his broad interest and significant impact on numerical analysis.     

Absolutely summing multipliers on abstract Hardy spaces

We study summing multipliers from Banach spaces of analytic functions on the unit disc of the complex plane to the complex Banach sequence lattices. The domain spaces are abstract variants of the classical Hardy spaces generated by the complex symmetric spaces. Applying interpolation methods, we prove the Hausdorff Young and Hardy-Littlewood type theorems. We show applications of these results to study summing multipliers from the Hardy-Orlicz spaces to the Orlicz sequence lattices. The obtained results extend the well-known results for the Hp spaces.     

Hardy Spaces and bmo on Manifolds with Bounded Geometry

We develop the theory of the “local” Hardy space and John-Nirenberg space when M is a Riemannian manifold with bounded geometry, building on the classic work of Fefferman-Stein and subsequent material, particularly of Goldberg and Ionescu. Results include – duality, L ^p estimates on an appropriate variant of the sharp maximal function, and bmo-Sobolev spaces, and action of a natural class of pseudodifferential operators, including a natural class of functions of the Laplace operator, in a setting that unifies these results with results on L ^p -Sobolev spaces. We apply results on these topics to some interpolation theorems, motivated in part by the search for dispersive estimates for wave equations.      

Some properties of Lagrangian sequences

Here we discuss a sequence of Lagrangians and corresponding Euler–Lagrange equations and point out some interesting properties that this particular sequence holds. This is an extension on recent results obtained on sequences of differential equations to Lagrangians as it leads to differential equations on the application of the Variational Principle.     

The p-adic quantum plane algebras and quantum Weyl algebra

Let q be a principal unit of the ring of valuation of a complete valued field K, extension of the field of p-adic numbers. Generalizing Mahler basis, K. Conrad has constructed orthonormal basis, depending on q, of the space of continuous functions on the ring of p-adic integers with values in K. Attached to q there are two models of the quantum plane and a model of the quantum Weyl algebra, as algebras of bounded linear operators on the space of p-adic continuous functions. For q not a root of unit, interesting orthonormal (orthogonal) families of these algebras are exhibited and providing p-adic completion of quantum plane and quantum Weyl algebras. The text was submitted by the authors in English.     

Addition theorem and matrix elements of radiative multipole operators

The addition theorem for radiative multipole operators, i.e., electric-dipole, electric-quadropole, or magnetic-dipole, etc., is derived through a translational transformation. The addition theorem of μth component of the angular momentum operator, L _μ (r), is also derived as a simple expression that represents a general translation of the angular momentum operator along an arbitrary orientation of a displacement vector and when this displacement is along the Z-axis. The addition theorem of the multipole operators is then used to analytically evaluate the matrix elements of the electric and magnetic multipole operators over the basis functions, the spherical Laguerre Gaussian-type function (LGTF), . The explicit and simple formulas obtained for the matrix elements of these operators are in terms of vector-coupling coefficients and LGTFs of the internuclear coordinates. The matrix element of the magnetic multipole operator is shown to be a linear combination of the matrix element of the electric multipole operator     

多方参与且评价指标集有差异的方案筛选方法

针对多方给出评价指标集存在差异情形的方案筛选问题,提出了一种基于软集理论的方案筛选方法。在该方法中,首先依据各方给出关于备选方案符合评价指标要求的可接受水平,构建各方相应的评价指标子集,进而针对各备选方案符合评价指标要求的结果构建针对各方评价指标子集的软集;然后构建所有方案均符合全部指标要求的软集,进而通过各评价指标子集的软集与所有方案均符合全部指标要求的软集的积运算和算子运算,得到对应评价指标子集的每方可接受的备选方案集;进一步地,通过各方可接受的备选方案集间的并集运算及交集运算,可得到最终的方案筛选结果。最后,通过一个算例说明了本文给出方法的可行性和有效性。     

von Neumann群代数

设G是带左正则表示λ的一个离散群, VN（G）是λ（g）生成的von Neumann群代数。我们刻画了von Neu-mann群代数上的Hardy空间。     

New formulas for normalizing photon-added（-subtracted） two-mode squeezed thermal states

For the first time,we derive the compact forms of normalization factors for photon-added(-subtracted) two-mode squeezed thermal states by using the P-representation and the integration within an ordered product of operators(IWOP) technique.It is found that these two factors are related to the Jacobi polynomials.In addition,some new relationships for Jacobi polynomials are presented.     

单位球上βP空间到Za型空间的加权Cesàro算子

利用泛函分析多复变的方法,研究了单位球上βP空间到Za空间的加权Cesàro算子的有界性和紧性问题.获得了单位球上βP空间到Za空间的加权Cesàro算子为有界算子和紧算子的充要条件.