Data-sparse approximation to the operator-valued functions of elliptic operator

In previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent

In the present paper, we consider various operator functions, the operator exponential negative fractional powers , the cosine operator function and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.

     

BEM with linear complexity for the classical boundary integral operators

Alternative representations of boundary integral operators corresponding to elliptic boundary value problems are developed as a starting point for numerical approximations as, e.g., Galerkin boundary elements including numerical quadrature and panel-clustering. These representations have the advantage that the integrands of the integral operators have a reduced singular behaviour allowing one to choose the order of the numerical approximations much lower than for the classical formulations. Low-order discretisations for the single layer integral equations as well as for the classical double layer potential and the hypersingular integral equation are considered. We will present fully discrete Galerkin boundary element methods where the storage amount and the CPU time grow only linearly with respect to the number of unknowns.

     

$\mathcal{F}$-Perfect Rings and Modules

Let $R$ be a ring, and let $(\mathcal{F}, C)$ be a cotorsion theory. In this article, the notion of $\mathcal{F}$-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring $R$ is said to be right $\mathcal{F}$-perfect if $F$ is projective relative to $R$ for any $F ∈ \mathcal{F}$. We give some characterizations of $\mathcal{F}$-perfect rings. For example, we show that a ring $R$ is right $\mathcal{F}$-perfect if and only if $\mathcal{F}$-covers of finitely generated modules are projective. Moreover, we define $\mathcal{F}$-perfect modules and investigate some properties of them.     

The relationship between fragmentable spaces and class spaces

In this note, we show that each fragmentable space introduced by Jayne and Rogers in 1985 is of class which was introduced by Kenderov in 1984. Our example shows that a space which is of class may not be a fragmentable space.

     

一致空间的理想导集映射

本文介绍一致空间的理想导集映射,并对这些映射的像展开研究.同时,介绍I-Hurewicz有界性的概念,研究I-Hurewicz有界性的基础拓扑运算性质.最后,得到一致空间中I-Hurewicz有界性的一个等价刻画.     

Some numerical invariants of local rings

Let be a formal power series ring over a field of characteristic zero and any ideal. The aim of this work is to introduce some numerical invariants of the local rings by using the theory of algebraic -modules. More precisely, we will prove that the multiplicities of the characteristic cycle of the local cohomology modules and , where is any prime ideal that contains , are invariants of .

     

关于余挠三元组的余星子范畴和余倾斜子范畴

设$\mathcal{A}$ 是一个Abel范畴,且 $(\mathcal{X}, \mathcal{Z},\mathcal{Y})$ 是一个完全遗传余挠三元组.介绍 $\mathcal{A}$ 的 $n$-$\mathcal{Y}$-余倾斜子范畴的定义,并给出 $n$-$\mathcal{Y}$-余倾斜子范畴的一个刻画,类似于 $n$-余倾斜模的 Bazzoni 刻画.作为应用,证明了在一个几乎 Gorenstein 环 $R$ 上, 如果 $\mathcal{GP}$ 是 $n$-$\mathcal{GI}$-余倾斜的, 那么 $R$ 是一个 $n$-Gorenstein 环, 其中 $\mathcal{GP}$ 表示 Gorenstein 投射 $R$-模组成的子范畴且 $\mathcal{GI}$ 表示 Gorenstein 内射 $R$-模组成的子范畴. 进而, 研究 任意环$R$上的$n$-余星子范畴, 以及关于余挠三元组 $(\mathcal{P}, R$-Mod, $\mathcal{I})$ 的 $n$-$\mathcal{I}$-子范畴与 $n$-余星子范畴之间的关系, 其中 $\mathcal{P}$ 表示投射左 $R$-模组成的子范畴且 $\mathcal{I}$ 表示内射左 $R$-模组成的子范畴.     

On Nonnegative Solution of Multi-Linear System with Strong $\mathcal{M}_z$-Tensors

A class of structured multi-linear system defined by strong $\mathcal{M}_z$-tensors is considered. We prove that the multi-linear system with strong $\mathcal{M}_z$-tensors always has a nonnegative solution under certain condition by the fixed point theory. We also prove that the zero solution is the only solution of the homogeneous multi-linear system for some structured tensors, such as strong $\mathcal{M}$-tensors, $\mathcal{H}^+$-tensors, strictly diagonally dominant tensors with positive diagonal elements. Numerical examples are presented to illustrate our theoretical results.     

Fixed point theorems for multivalued non-linear $F$-contractions on quasi metric spaces with an application

In this paper, we introduce $(\alpha,\beta)$-type $F-\tau$ contraction and utilize the same to prove some fixed point results for multivalued mappings in quasi metric spaces. Furthermore, we furnish with some examples to exhibit the utility of our results. As an application, we establish the existence of a solution for a non-linear integral equation.     

An ultrafilter with property

Let be a -supercompact cardinal. We show that carries a normal ultrafilter with a property introduced by Menas. With it we give a transparent proof of Kamo's theorem that carries a normal ultrafilter with the partition property.

     

Spaces of type and the Hankel convolution

In this paper we introduce new function spaces that are denoted by , -1/2$"> and and that are spaces of type where the Hankel convolution and the Hankel transformation are defined. The spaces will play the same role in the Hankel setting that the spaces play in the theory of Fourier transformation.     

Perturbations and Weyl's theorem

A Banach space operator is completely hereditarily normaloid, , if either every part, and (also) for every invertible part , of is normaloid or if for every complex number every part of is normaloid. Sufficient conditions for the perturbation of by an algebraic operator to satisfy Weyl's theorem are proved. Our sufficient conditions lead us to the conclusion that the conjugate operator satisfies -Weyl's theorem.

     

The best approximation to a class of functions of several variables by another class and related extremum problems

We study the relationship between several extremum problems for unbounded linear operators of convolution type in the spaces , m ≥ 1, 1 ≤ γ ≤ ∞. For the problem of calculating the modulus of continuity of the convolution operatorA on the function classQ defined by a similar operator and for the Stechkin problem on the best approximation of the operatorA on the classQ by bounded linear operators, we construct dual problems in dual spaces, which are the problems on, respectively, the best and the worst approximation to a class of functions by another class. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 323–340, September, 1998. This research was supported by INTAS under grant No. 94-4070.     

Singularities of the hypergeometric system associated with a monomial curve

We compute, using -module restrictions, the slopes of the irregular hypergeometric system associated with a monomial curve. We also study rational solutions and reducibility of such systems.

     

On decompositions of continuity and some weaker forms of continuity via idealization

Summary We give decompositions of continuity and some weaker forms of continuity via idealization using the concepts of <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mathcal{A_{I}}$-sets, $\alpha \mathcal{A_{I}}$-sets, $B_{1I}$-sets, $B_{2I}$-sets, $B_{3I}$-sets, $\alpha C_{I}$-sets and $WLC_{I}$-sets.     

A separable non-remainder of

We prove that there is a compact separable continuum that (consistently) is not a remainder of the real line.

     

A Fejér type theorem to determine jumps in terms of the Abel-Poisson mean of double Fourier series

We extend from single to double Fourier series a theorem of Zygmund to determine the generalized jumps of a periodic integrable function at a simple discontinuity point. As a by-product of the proof, we obtain an estimate of the fourth mixed partial derivative of the Abel-Poisson mean of any integrable function at such a point where is smooth. We also consider the extension of the Zygmund classes and to the two-dimensional torus .

     

On irreducible p,q‐representations of Lie algebras \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {G} (0,1)$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {G} (0,0)$\end{document}

In this paper, the irreducible p, q‐representations of the Lie algebras $\mathcal {G}(0,1)$ and $\mathcal {G}(0,0)$ are discussed. We prove two theorems that classify certain irreducible p, q‐representations of these Lie algebras and construct their one variable models in terms of p, q‐derivative and dilation operators. As an application, we derive a p, q‐special function identity based on one such model.     

Extensions of modules over Weyl algebras

In this paper we calculate some groups of singular modules over the complex Weyl algebra . In particular we determine conditions under which is an infinite dimensional vector space when or .

     

Fixed Point Theorems for Weakly Contractive Mappings in Ordered Metric Spaces with an Application

In this paper, we prove fixed point theorem for weakly contractive mappings using locally $T$-transitivity of binary relation and presenting an analogous version of Harjani and Sadarangani theorem involving more general relation theoretic metrical notions. Our fixed point results under universal relation reduces to Harjani and Sadarangani [Nonlinear Anal., 71 (2009), 3403–3410] fixed point theorems. In this way we also generalize some of the recent fixed point theorems for weak contraction in the existing literature.