Quantum stochastic convolution cocycles II

Schuermann's theory of quantum Levy processes, and more generally the theory of quantum stochastic convolution cocycles, is extended to the topological context of compact quantum groups and operator space coalgebras. Quantum stochastic convolution cocycles on a C*-hyperbialgebra, which are Markov-regular, completely positive and contractive, are shown to satisfy coalgebraic quantum stochastic differential equations with completely bounded coefficients, and the structure of their stochastic generators is obtained. Automatic complete boundedness of a class of derivations is established, leading to a characterisation of the stochastic generators of *-homomorphic convolution cocycles on a C*-bialgebra. Two tentative definitions of quantum Levy process on a compact quantum group are given and, with respect to both of these, it is shown that an equivalent process on Fock space may be reconstructed from the generator of the quantum Levy process. In the examples presented, connection to the algebraic theory is emphasised by a focus on full compact quantum groups.Comment: 32 pages, expanded introduction and updated references. The revised version will appear in Communications in Mathematical Physic

Quantum stochastic convolution cocycles III

Every Markov-regular quantum Levy process on a multiplier C*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C*-bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Levy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C*-bialgebra, to locally compact quantum groups and multiplier C*-bialgebras. Strict extension results obtained by Kustermans, together with automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic convolution cocycles.Comment: 20 pages; v2 corrects some typos and no longer contains a section on quantum random walk approximations, which will now appear as a separate submission. The article will appear in the Mathematische Annale

Improving CMB non-Gaussianity estimators using tracers of local structure

Local non-Gaussianity causes correlations between large scale perturbation modes and the small scale power. The large-scale CMB signal has contributions from the integrated Sachs Wolfe (ISW) effect, which does not correlate with the small scale power. If this ISW contribution can be removed, the sensitivity to local non-Gaussianity is improved. Gravitational lensing and galaxy counts can be used to trace the ISW contribution; in particular we show that the CMB lensing potential is highly correlated with the ISW signal. We construct a nearly-optimal estimator for the local non-Gaussianity parameter \fnl and investigate to what extent we can use this to decrease the variance on {\fnl}. We show that the variance can be decreased by up to $20\%$ at Planck sensitivity using galaxy counts. CMB lensing is a good bias-independent ISW tracer for future more sensitive observations, though the fractional decrease in variance is small if good polarization data is also available.Comment: 8 pages, 3 figures. Comments welcom

Quantum Feynman-Kac perturbations

We develop fully noncommutative Feynman-Kac formulae by employing quantum stochastic processes. To this end we establish some theory for perturbing quantum stochastic flows on von Neumann algebras by multiplier cocycles. Multiplier cocycles are constructed via quantum stochastic differential equations whose coefficients are driven by the flow. The resulting class of cocycles is characterised under alternative assumptions of separability or Markov regularity. Our results generalise those obtained using classical Brownian motion on the one hand, and results for unitarily implemented flows on the other.Comment: 27 pages. Minor corrections to version 2. To appear in the Journal of the London Mathematical Societ

Warnock 40 Years on: The Development of Special Educational Needs Since the Warnock Report and Implications for the Future

The report of the Warnock Committee “Special Educational Needs,” published in 1978, provided the first comprehensive review of special educational needs (SEN) in England and the basis for subsequent legislation, from the Education Act 1981 to the recent Children and Families Act 2014. The Warnock Report has been highly influential with respect to the development of both national and local policy and practice for SEN, primarily in England but also in the UK generally and internationally. We conceived the “Warnock 40 Years On” research topic published in this journal as an opportunity to examine developments for children and young people with SEN or a disability (SEND) since the Report's publication. As editors of this research topic, in this paper we provide an overview of major elements within SEND, drawing on both the individual constituent papers and the broader policy and research literature. We examine the origins and precursors of the Warnock Review, including then current progressive conceptual development and practice, leading to an examination of the Review. We explore the legislative structure arising from the Report and implementation of the SEND system—from the Education Act 1981 onwards. We critically examine a number of key aspects of the Warnock Report and how these developments of the SEND system have taken these forward, including: categorization of SEN; provision made to meet the special needs of children and young people with SEN or a disability, including inclusion; early identification and intervention; the important rights and role of parents; prevalence of different types of SEN and the complex interaction between SEN and a range of factors, including social disadvantage and ethnicity; and a range of SEND research. Finally we consider the implications for future development of the SEND system, in particular whether its limitations can be addressed better by “patching up” the system or by “system change.” Although focused on England, this paper has relevance—as did the Warnock Report—to SEND internationally