Injective and flat covers,envelopes and resolvents

Using the dual of a categorical definition of an injective envelope, injective covers can be defined. For a ringR, every leftR-module is shown to have an injective cover if and only ifR is left noetherian. Flat envelopes are defined and shown to exist for all modules over a regular local ring of dimension 2. Using injective covers, minimal injective resolvents can be defined.     

On the method of alternating resolvents

The work of Hundal [H. Hundal, An alternating projection that does not converge in norm, Nonlinear Anal. 57 (1) (2004) 35-61] has revealed that the sequence generated by the method of alternating projections converges weakly, but not strongly in general. In this paper, we present several algorithms based on alternating resolvents of two maximal monotone operators, A and B, that can be used to approximate common zeros of A and B. In particular, we prove that the sequences generated by our algorithms converge strongly. A particular case of such algorithms enables one to approximate minimum values of certain convex functionals.     

On the asymptotic exactness of Bank-Weiser's estimator

Summary In this paper we analyze an error estimator introduced by Bank and Weiser. We prove that this estimator is asymptotically exact in the energy norm for regular solutions and parallel meshes. By considering a simple example we show that this is not true for general meshes.The work was partially supported by Fundación Antorchas.The work was partially supported by the NSF under grant CCR-88-20279.     

On the decomposition theorem for meromorphic Fredholm resolvents

In this note results of B. Gramsch and W. Kaballo [8] on the decomposition of meromorphic (semi-) Fredholm resolvents are sharpened. A condition on an Orlicz function is given, under which the singular part in this decomposition can be chosen meromorphic inN , the ideal of -nuclear operators. Then the necessity of this condition is studied. Moreover, it is shown that for the rather steep Orlicz functions relevant to this question,N equalsS , the ideal of -approximable operators.Dedicated to Professor Albert Schneider on the occasion of his 60 ^th birthdayresearch supported by a grant from DAAD     

On the computation of resolvents and Galois groups

A large class of algorithms for computing resolvents of algebraic equations — so called rational transformations — is investigated and characterized group theoretically. The concept of rational transformations implies a program how to develop good methods to determine the Galois group of an equation. It is shown that some known methods are special cases of rational transformations, and a new procedure to find the group of a sextic equation is given. Moreover, all cases in which Galois resolvents can be found by means of rational transformations are classified.     

On the existence and exactness of the associated sheaf functor

Let $\text{C}$ be a category with inverse limits. A category $\text{x}$is called an $\text{A}$-topos if there is a site (?, τ), i.e. a small category ? together with a Grothendieck topology τ such that $\text{x}$is equivalent to the category Sh_τ[$\text{C}$⁰. $\text{A}$] of τ-sheaves on $\text{C}$ with values in $\text{C}$. If $\text{x}$is an $\text{C}$-topos, then so is Sh_τ'[?⁰, $\text{x}$] for any site (?', τ'). It is shown that if for every site (?,τ) the associated sheaf functor from presheaves to τ-sheaves with values in $\text{A}$ exists (and preserves finite inverse limits), then the same holds if $\text{A}$is replaced by any $\text{A}$-topos $\text{x}$. Roughly speaking, the main result is that for a site (?,τ) the associated sheaf functor [?⁰, $\text{A}$] → Sh_τ [?⁰, $\text{A}$] exists and preserves finite inverse limits, provided $\text{A}$ has filtered direct limits which commute with finite inverse limits, e.g. if $\text{A}$ is a Grothendieck category or a category of sheaves with values in a locally finitely presentable category [8. 7.1]. Analogous results hold in the additive case.     

On exactness and isoperimetric profiles of discrete groups

We introduce a quasi-isometry invariant related to Property A and explore its connections to various other invariants of finitely generated groups. This allows to establish a direct relation between asymptotic dimension on one hand and isoperimetry and random walks on the other. We apply these results to reprove sharp estimates on isoperimetric profiles of some groups and to answer some questions in dimension theory.     

On resolvents of integral and partially integral operators

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 2, pp. 88–91, March–April, 1993.     

On the exactness of a theorem of Mazur and Orlicz

For a preassigned unbounded sequence {S_n} of complex numbers, and preassigned complex numbers z₁ and z₂z₁ we construct: 1) regular matrices A=a_nk and B=b_nk such that the same bounded sequences are summable by these matrices and that , and ; 2) regular matrices A⁽¹⁾)=a _nk ⁽¹⁾ and B⁽¹⁾=b _nk ⁽¹⁾ such that B⁽¹⁾ A⁽¹⁾, and, . Our results show that the well known theorem of MazurOrlicz on the bounded consistency of two regular matrices, one of which is boundedly stronger than the other, is exact.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 431–436, April, 1972.     

On the exactness of a class of nondifferentiable penalty functions

In this paper, we consider a class of nondifferentiable penalty functions for the solution of nonlinear programming problems without convexity assumptions. Preliminarily, we introduce a notion of exactness which appears to be of relevance in connection with the solution of the constrained problem by means of unconstrained minimization methods. Then, we show that the class of penalty functions considered is exact, according to this notion. This research was partially supported by the National Research Program on “Modelli e Algoritmi per l'Ottimizzazione,” Ministero della Pubblica, Istruzione, Roma, Italy.     

Multiparameter resolvents

We give a characterization for multiparameter resolvents based upon an asymptotic condition and an analog of the resolvent equation.     

On the exactness of a theorem of G.A. Fomin

, . . [1] - . .