The strong approximation property

We study the weak bounded approximation property (weak BAP) of pairs and show that each of the spaces c0 and 1 has a subspace having the approximation property but failing the weak BAP.

Let G be a countable exact discrete group. G has the strong invariant approximation property(SIAP) if and only if C � u(G,S) G = C � � (G) ⊗ S for any Hilbert space H and closed subspace S ⊆ H. We shall use results of Haagerup and Kraus on the approximation property (AP) to investigate some permanence properties of the SIAP for discrete groups. This can be done most efficiently for exact groups. In this paper we describe that the stability properties of the SIAP property pass to semi direct products, and extensions for discrete exact groups.

Let G be a locally compact group. We prove that every L∞(G)-comodule is nondegenerate and saturated, whereas every L(G)-comodule is nondegenerate if and only if every L(G)-comodule is saturated if and only if G has the approximation property of Haagerup and Kraus. This allows us to extend known results from the duality theory of crossed products of von Neumann algebras to the recent theory of crossed products of dual operator spaces. Also, we obtain a characterization of groups with the approximation property involving crossed products improving a recent result of Crann and Neufang and we generalize a theorem of Anoussis, Katavolos and Todorov providing a less technical proof of it.

The injective and surjective hulls of the ideal of (p,q)-compact operators and their approximation properties

High-order neural networks can be considered as an expansion of Hopfield neural networks, and have stronger approximation property and faster convergence rate. However, in practice high order network is seldom to be used to solve combinatorial optimization problem. In this paper crossbar switch problem, which is an NP-complete problem, is used as an example to demonstrate how to use high order discrete Hopfield neural network to solve engineering optimization problems. The construction method of energy function and the neural computing algorithm are presented. It is also discussed the method how to speed the convergence and escape from local minima. Experimental results show that high order network has a quick convergence speed, and outperforms the traditional discrete Hopfield network.KeywordsHopfield networkconstraint satisfactioncrossbar switch problem

In this paper, approximation properties in Felbin-fuzzy normed spaces are studied. These approximation properties have been recently introduced in Felbin-fuzzy normed spaces. We make topological tools to analyze such approximation properties. We especially develop the representation of dual spaces related to our contexts. By using this representation, we establish characterizations of approximation properties in terms of infinite sequences. Finally, we provide dual problems for approximation properties and their results in our contexts.

The Dual Complexity Space as the Dual of a Normed Cone

The characterization of bounded approximation properties defined by arbitrary operator ideals due to Oja is extended to bounded convex approximation properties. As an application, it is shown that the unique extension property of a Banach space X enables to lift the metric convex approximation property from a Banach space X to its dual space X*.

It is shown how to construct, given a Banach space which does not have the approximation property, another Banach space which does not have the approximation property but which does have the compact approximation property.

We show that any super-reflexive Banach space is a $\Lambda$-space (i.e., the weak-polynomial convergence for sequences implies the norm convergence). We introduce the notion of $\kappa$-space (i.e., a Banach space where the weak-polynomial convergence for sequences is different from the weak convergence) and we prove that if a dual Banach space $Z$ is a $\kappa$-space with the approximation property, then the uniform algebra $A(B)$ on the unit ball of $Z$ generated by the weak-star continuous polynomials is not tight.

Throughout this section let X be a separable Banach space. We investigate certain approximation properties for X and deal with the question under which additional condition then X has a basis. In this context we also give a sufficient criterion for X to be a dual Banach space.

We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Frechet space with its dual having the approximation property and with E′b locally Asplund we show that the space of n-homogeneous polynomials on (E′b)′b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Frechet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous.

A remark on the approximation of p-compact operators by finite-rank operators

Nested Artin strong approximation property

While the picture of approximation complexity class becomes clear for most combinatorial optimization problems, it remains an open question whether Feedback Vertex Set can be approximated within a constant ratio in directed graph case. In this paper we present an approximation algorithm with performance bound L max −1, where L max is the largest length of essential cycles in the graph G(V,E). The worst case bound is \(\left\lfloor {\sqrt {{{\left| V \right|}^2} - \left| V \right| - \left| E \right| + 1} } \right\rfloor \) which, in general, is inferior to Seymour’s recent result [14], but becomes a small constant for some graphs. Furthermore, we prove the so-called pseudo ∈-approximate property, i.e. FVS can be divided into a class of disjoint NP -complete subproblems, and our heuristic becomes ∈-approximate for each one of these subproblems.Keywordsapproximationboundfeedback vertex set NP -complete