On subspace lattices I. Closedness type properties and tensor products

We study properties of subspace lattices related to the continuity of the map Lat and the notion of reflexivity. We characterize various “closedness” properties in different ways and give the hierarchy between them. We investigate several properties related to tensor products of subspace lattices and show that the tensor product of the projection lattices of two von Neumann algebras, one of which is injective, is reflexive.

Whitney's type theorems for infinite dimensional spaces

It is proved that for any $f$ is an element of $C^k(L,R)$, where k is a natural number and L is a closed linear subspace of a nuclear Frechet space $X$, the function $f$ can be extended to a function of class $C^{k-1}$ defined on the entire space $X$. It is also proved that for any $f$ is an element of $C^k(L, R)$, where $k$ is a natural number of infinity and L is a closed linear subspace of a dual $X$ of a nuclear Frechet space, the function $f$ can be extended to a function of class $C^k$ defined on the entire space $X$. In addition, it is proved that under these conditions, the existence of a linear extension operator is equivalent to the complementability of the subspace.

Flow-mediated dilatation of the brachial artery is a poorly reproducible indicator of microvascular function in Type I diabetes mellitus

Aim: Two Type I diabetes and control group comparator studies were conducted to assess the reproducibility of FMD and to analyse blood flow data normally discarded during FMD measurement.

Design: The studies were sequential and differed only with regard to operator and ultrasound machine. Seventy-two subjects with diabetes and 71 controls were studied in total.

Methods: Subjects had FMD measured conventionally. Blood velocity waveforms were averaged over 10 pulses post forearm ischaemia and their component frequencies analysed using the wavelet transform, a mathematical tool for waveform analysis. The component frequencies were grouped into 11 bands to facilitate analysis.

Results: Subjects were well-matched between studies. In Study 1, FMD was significantly impaired in subjects with Type I diabetes vs. controls (median 4.35%, interquartile range 3.10-4.80 vs. 6.50, 4.79-9.42, P < 0.001). No differences were detected between groups in Study 2, however. However, analysis of blood velocity waveforms yielded significant differences between groups in two frequency bands in each study.

Conclusions: This report highlights concerns over the reproducibility of FMD measures. Further work is required to fully elucidate the role of analysing velocity waveforms after forearm ischaemia.

Monte Carlo radiation hydrodynamics:Methods, tests and application to Type Ia supernova ejecta

In astrophysical systems, radiation-matter interactions are important in transferring energy and momentum between the radiation field and the surrounding material. This coupling often makes it necessary to consider the role of radiation when modelling the dynamics of astrophysical fluids. During the last few years, there have been rapid developments in the use of Monte Carlo methods for numerical radiative transfer simulations. Here, we present an approach to radiation hydrodynamics that is based on coupling Monte Carlo radiative transfer techniques with finite-volume hydrodynamical methods in an operator-split manner. In particular, we adopt an indivisible packet formalism to discretize the radiation field into an ensemble of Monte Carlo packets and employ volume-based estimators to reconstruct the radiation field characteristics. In this paper the numerical tools of this method are presented and their accuracy is verified in a series of test calculations. Finally, as a practical example, we use our approach to study the influence of the radiation-matter coupling on the homologous expansion phase and the bolometric light curve of Type Ia supernova explosions. © 2012 The Authors Monthly Notices of the Royal Astronomical Society © 2012 RAS.

Local operator multipliers and positivity

We establish an unbounded version of Stinespring's Theorem and a lifting result for Stinespring representations of completely positive modular maps defined on the space of all compact operators. We apply these results to study positivity for Schur multipliers. We characterise positive local Schur multipliers, and provide a description of positive local Schur multipliers of Toeplitz type. We introduce local operator multipliers as a non-commutative analogue of local Schur multipliers, and characterise them extending both the characterisation of operator multipliers from [16] and that of local Schur multipliers from [27]. We provide a description of the positive local operator multipliers in terms of approximation by elements of canonical positive cones.