Isometric embedding of finite ultrametric spaces in Banach spaces

We prove that for any finite ultrametric space M and any infinite-dimensional Banach space B there exists an isometric embedding of M into B.

On Osgood theorem in Banach spaces

Let $X$ be a real Banach space, $\omega:[0,+\infty)\to\R$ be an increasing continuous function such that $\omega(0)=0$ and $\omega(t+s)\leq\omega(t)+\omega(s)$ for all $t,s\in[0,+\infty)$. By the Osgood theorem, if $\int_{0}^1\frac{dt}{\omega(t)}=\infty$, then for any $(t_0,x_0)\in R\times X$ and any continuous map $f: R\times X\to X$ and such that $\|f(t,x)-f(t,y)\|\leq\omega(\|x-y\|)$ for all $t\in R$, $x,y\in X$, the Cauchy problem $\dot x(t)=f(t,x(t))$, $(t_0)=x_0$ has a unique solution in a neighborhood of $t_0$ . We prove that if $X$ has a complemented subspace with an unconditional Schauder basis and $\int_{0}^1\frac{dt}{\omega(t)}

On the solvability of Hamilton's equations in Hilbert spaces

We construct a bounded function $H : l_2\times l_2 \to R$ with continuous Frechet derivative such that for any $q_0\in l_2$ the Cauchy problem $\dot p= - {\partial H\over\partial q}$, $\dot q={\partial H\over\partial p}$, $p(0) = 0$, q(0) = q_0$ has no solutions in any neighborhood of zero in R.

Structure of spectra of linear operators in Banach spaces

Descriptive characterizations of the point, the continuous, and the residual spectra of operators in Banach spaces are put forward. In particular, necessary and sufficient conditions for three disjoint subsets of the complex plane to be the point spectrum, the continuous spectrum, and the residual spectrum of a linear continuous operator in a separable Banach space are obtained.

Shared Education in contested spaces: How collaborative networks improve communities and schools

Societies which suffer from ethnic and political divisions are often characterised by patterns of social and institutional separation, and sometimes these divisions remain even after political conflict has ended. This has occurred in Northern Ireland where there is, and remains, a long-standing pattern of parallel institutions and services for the different communities. A socially significant example lies in the education system where a parallel system of Catholic and Protestant schools has been in place since the establishment of a national school system in the 1830s. During the years of political violence in Northern Ireland a variety of educational interventions were implemented to promote reconciliation, but most of them failed to create any systemic change. This paper describes a post-conflict educational initiative known as Shared Education which aims to promote social cohesion and school improvement by encouraging sustained and regular shared learning between students and broader collaboration between teachers and school leaders from different schools. The paper examines the background to work on Shared Education, describes a ‘sharing continuum’ which emerged as an evaluation and policy tool from this work and considers evidence from a case study of a Shared Education school partnership in a divided city in Northern Ireland. The paper will conclude by highlighting some of the significant social and policy impact of the Shared Education work.

European urban spaces in crisis - the mapping of affective pracitices with Living with Difference: introduction

The introduction outlines the notion of urban space and crisis in Europe while taking into account the more recent protests and riots in different cities, in and beyond Europe. It is argued that the phenomen of protest is happening alongside the economic crisis underscoring an alternative political public civic spirit expressing to a certain degree the renaissance and timely making of, what might be called in the digital age, #œuvre. Its forces and emotional properties capture a political realm that unfolds as a globalized urban transnational public space, still progressing. Further, it introduces the collection of papers for the special themed feature. Five papers look at affective practices through a Continental European lens, which places the meaning of race, migration and intersecting identity angles at the centre of debates of individual encounters in public spaces. The final and sixth paper, written by Brenda Yeoh, looks through a Singapore/East Asia lens, and comments on the common European threats as well as on the historical specificity and implications of distinctive geo-political spaces for affective practices.

Numerical and Experimental Study of the Performance Effects of Varying Vaneless Space and Vane Solidity in Radial Inflow Turbine Stators.

An extensive experimental program has been carried out on a 135?mm tip diameter radial turbine using a variety of stator designs, in order to facilitate direct performance comparisons of varying stator vane solidity and the effect of varying the vaneless space. A baseline vaned stator was designed using commercial blade design software, having 15 vanes and a vane trailing edge to rotor leading edge radius ratio (Rte/rle) of 1.13. Two additional series of stator vanes were designed and manufactured; one series having varying vane numbers of 12, 18, 24, and 30, and a further series with Rte/rle ratios of 1.05, 1.175, 1.20, and 1.25. As part of the design process a series of CFD simulations were carried out in order to guide design iterations towards achieving a matched flow capacity for each stator. In this way the variations in the measured stage efficiency could be attributed to the stator passages only, thus allowing direct comparisons to be made. Interstage measurements were taken to capture the static pressure distribution at the rotor inlet and these measurements were then used to validate subsequent numerical models. The overall losses for different stators have been quantified and the variations in the measured and computed efficiency were used to recommend optimum values of vane solidity and Rte/rle.