Closable multipliers

Let $(X,\mu)$ and $(Y,\nu)$ be standard measure spaces. A function $\nph\in L^\infty(X\times Y,\mu\times\nu)$ is called a (measurable) Schur multiplier if the map $S_\nph$, defined on the space of Hilbert-Schmidt operators from $L_2(X,\mu)$ to $L_2(Y,\nu)$ by multiplying their integral kernels by $\nph$, is bound-ed in the operator norm. The paper studies measurable functions $\nph$ for which $S_\nph$ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if $\nph$ is of Toeplitz type, that is, if $\nph(x,y)=f(x-y)$, $x,y\in G$, where $G$ is a locally compact abelian group, then the closability of $\nph$ is related to the local inclusion of $f$ in the Fourier algebra $A(G)$ of $G$. If $\nph$ is a divided difference, that is, a function of the form $(f(x)-f(y))/(x-y)$, then its closability is related to the ``operator smoothness'' of the function $f$. A number of examples of non closable, norm closable and w*-closable multipliers are presented.

Research Ethics in Divided and Violent Societies: Seizing the Ethical Opportunity

This chapter explores how to conduct social research in divided and violent societies by developing the concept of the ‘ethical opportunity’. The ‘ethical opportunity’ is situated in a brief discussion of ‘action’ and feminist approaches to research. It argues that seizing the ethical opportunity requires researchers to: plan for their personal safety, plan for participants’ personal safety and plan how they will communicate and disseminate their results. It draws on the author’s personal experience researching in South Africa, Zimbabwe and Northern Ireland, concluding that it is in the communication and dissemination phase that researchers’ hopes for ‘making a difference’ may be realised or dashed. It cautions would-be researchers to manage their own – and research participants’ – expectations about what social research can achieve. Its effects may not often be as transforming and liberating as idealistic researchers hope for, but that should not dissuade them from striving towards those ends.

Joining together or pushing apart? Building relations and exploring difference through shared education in Northern Ireland

In divided societies, the promotion of cross-cultural contact through the education system has been central to efforts to improve intergroup relations. This approach is informed by an understanding of the contact hypothesis, which suggests that positive contact with a member of a different group should contribute to improvements in attitudes towards the group as a whole. While a substantial body of research provides support for contact theory, critics have argued that its emphasis on harmonious encounters can result in the neglect of group differences and associated issues of conflict and discrimination during contact. The research discussed in this article explores this tension with reference to two shared education projects in Northern Ireland. Research data, gathered primarily through interviews with pupils, confirms that divisive issues are rarely addressed during contact and explores several influences on this: the nature of pupils’ relationships, the programme structure, and the prevailing social norms of avoidance.

Recognising difference while promoting cohesion: the role of collaborative networks in education

The chapter outlines various approaches taken to the challenge of divided societies through education systems and links this to debates on political approaches to conflict resolution in Northern Ireland. The education debate has structural and affective components, while the political debate revolves around the privileging of identity or commonality. Applying the principle of recognition and the importance of tolerance is seen to provide a range of policy options, but the evidence of education interventions in Northern Ireland does not highlight any one policy option as the most effective, nor did the end of the violent conflict produce a new consensus on a way forward. The chapter argues that a key additional element to effective practice might be provided through the promotion of shared education through collaborative networks of schools which allows for the continued operation of separate schools while promoting an integrated experience for pupils and teachers.

Difference equation approach to two-thermocouple sensor characterisation in constant velocity flow environments

Thermocouples are one of the most popular devices for temperature measurement due to their robustness, ease of manufacture and installation, and low cost. However, when used in certain harsh environments, for example, in combustion systems and engine exhausts, large wire diameters are required, and consequently the measurement bandwidth is reduced. This article discusses a software compensation technique to address the loss of high frequency fluctuations based on measurements from two thermocouples. In particular, a difference equation sDEd approach is proposed and compared with existing methods both in simulation and on experimental test rig data with constant flow velocity. It is found that the DE algorithm, combined with the use of generalized total least squares for parameter identification, provides better performance in terms of time constant estimation without any a priori assumption on the time constant ratios of the thermocouples.

Exploiting a priori time constant ratio information in difference equation two-thermocouple sensor characterisation

The characterization of thermocouple sensors for temperature measurement in varying-flow environments is a challenging problem. Recently, the authors introduced novel difference-equation-based algorithms that allow in situ characterization of temperature measurement probes consisting of two-thermocouple sensors with differing time constants. In particular, a linear least squares (LS) lambda formulation of the characterization problem, which yields unbiased estimates when identified using generalized total LS, was introduced. These algorithms assume that time constants do not change during operation and are, therefore, appropriate for temperature measurement in homogenous constant-velocity liquid or gas flows. This paper develops an alternative ß-formulation of the characterization problem that has the major advantage of allowing exploitation of a priori knowledge of the ratio of the sensor time constants, thereby facilitating the implementation of computationally efficient algorithms that are less sensitive to measurement noise. A number of variants of the ß-formulation are developed, and appropriate unbiased estimators are identified. Monte Carlo simulation results are used to support the analysis.