Renorming spaces with greedy bases

We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given \vare>0, so that the basis becomes (1+\vare)-democratic, and hence (2+\vare)-greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is (1+\vare)-greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applications include the Haar systems in $L_p[0,1]$, $1<p<\infty$, and in dyadic Hardy space $H_1$, as well as the unit vector basis of Tsirelson space

Involutions on Banach Spaces and Reflexivity

spaces. E is said to be finitely representable in F if, given e&gt; 0 and a finite dimensional subspace E0 of E, there exists a subspace F0 of F such that d(E0, F0) _&lt; 1 + e, where d(E0, F0) = inf { T T-1 ß T is an isomorphism from E0 onto F0} denotes the Banach-Mazur distance coefficient. E is said to be super-reflexive if every Banach space which is finitely representable in E is re-flexive. Super-reflexivity has been characterized in terms of the notion of J-convexity: suppose that n&gt; _ 1 and that e&gt; 0; E is said to be J(n, e)-convex if, for all Xl,..., x, • in the unit ball of E, we have inf Xl q-&apos;&apos; &apos; q- Xk-- Xk+l..... Xn I-- • 7 /-- e. l&lt;k&lt;n--1 The &quot;if &quot; part of the following theorem was proved in [12] and [5], and the &quot;only if &quot; part was proved in [10]. TnEOaEM A. E is super-ret•exive //&apos;and only iœE is J(n, e)-convex for some n&gt; 1 ande&gt; 0

Coefficient Quantization for Frames in Banach Spaces

Let $(e_i)$ be a fundamental system of a Banach space. We consider the problem of approximating linear combinations of elements of this system by linear combinations using quantized coefficients. We will concentrate on systems which are possibly redundant. Our model for this situation will be frames in Banach spaces.Comment: 33 page

Selection of Australian Root Nodule Bacteria for Broad-Scale Inoculation of Native Legumes

The unique and diverse native Australian perennial legumes are under current investigation for use as pastures in Australian agriculture. Identification of root nodule bacteria (RNB) that can fix nitrogen effectively for the plant is a critical factor for the success of a legume species in agriculture (Howieson et al., 2000). Some legumes under investigation are relatively promiscuous (Lange, 1961). This trait may allow the development of a single, broad-scale inoculant that could allow inoculation of multiple species of agricultural importance, whilst more effective, specific RNB are developed in time. Aimed to identify strains that can form effective symbioses with several native legume species of potential interest to agriculture, this experiment screened putative indigenous RNB on 5 native legumes

Property A and CAT(0) cube complexes

Property A is a non-equivariant analogue of amenability defined for metric spaces. Euclidean spaces and trees are examples of spaces with Property A. Simultaneously generalising these facts, we show that finite-dimensional CAT(0) cube complexes have Property A. We do not assume that the complex is locally finite. We also prove that given a discrete group acting properly on a finite-dimensional CAT(0) cube complex the stabilisers of vertices at infinity are amenable