Kolmogorov Width of Classes of Smooth Functions on the Sphere

Let ^d−1{(x₁,…,x_d) ^d:x²₁+···+x²_d=1} be the unit sphere of the d-dimensional Euclidean space ^d. For r>0, we denote by B^r_p (1p∞) the class of functions f on ^d−1 representable in the formwhere dσ(y) denotes the usual Lebesgue measure on ^d−1, and P^λ_k(t) is the ultraspherical polynomial.For 1p,q∞, the Kolmogorov N-width of B^r_p in L^q( ^d−1) is given bythe left-most infimum being taken over all N-dimensional subspaces X_N of L^q( ^d−1).The main result in this paper is that for r2(d−1)²,where A_NB_N means that there exists a positive constant C, independent of N, such that C⁻¹A_NB_NCA_N.This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions.     

OR and the challenge to improve the NHS: modelling for insight and improvement in in-patient flows

This paper considers efforts to improve in-patient flows, a particularly urgent issue in the National Health Service (NHS). The context is described and related to reasons why OR has been making relatively little contribution. The paper argues that large complex models may often be unnecessary and even get in the way of providing clear insight and guidance for problem owners. The importance of understanding the generic working of systems to lead to improvement, and the limitations of simply describing them, is stressed. It is demonstrated that some very simple models can be of significant practical value in understanding and managing complex systems, changing mindsets and driving collection and use of operationally valuable data. Recommendations for more effective engagement with the NHS are offered.     

Predicting the Mechanical Properties of Binary Blends of Immiscible Polymers

We couple a morphological study of an immiscible binary AB mixture with a micromechanical simulation to determine how the spatial distribution of the A and B domains and the interfacial region (interphase) affects the mechanical behavior of the blend. The morphological studies are conducted through a three-dimensional Cahn-Hilliard (CH) simulation. Through the CH calculations, we obtain the size and structure of the domains for different blend compositions. The output of the CH model serves as the input to the Lattice Spring Model (LSM), which consists of a three-dimensional network of springs. In particular, the location of the different phases is mapped onto the LSM lattice and the appropriate force constants are assigned to the LSM sites. A stress is applied to the LSM lattice and we calculate the elastic response of the material. We find that the local stress and strain fields are highly dependent on the morphology of the system. By integrating the morphological and mechanical models, we can isolate how modifications in the composition of the mixture affect the macroscopic behavior. Thus, we can establish how choices made in the components affect the ultimate performance of the material.     

Results of radius scaling experiments and analysis of neon K-shellradiation data from an inductively driven Z-pinch

The K-shell radiated energy (yield) from neon Z-pinch implosions with annular, gas-puff nozzle radii of 1, 1.75, and 2.5 cm was measured for implosion times from 50 to 300 ns while systematically keeping the implosion kinetic energy nearly constant. The implosions were driven by the Hawk inductive-storage generator at the 0.65-MA level. Initial neutral-neon density distributions from the nozzles were determined with laser interferometry. Measured yields are compared with predictions from zero-dimensional (0-D) scaling models of ideal. One-dimensional (1-D) pinch behavior to both benchmark the scaling models, and to determine their utility for predicting K-shell yields for argon implosions of 200 to >300 ns driven by corresponding currents of 4 to 9 MA, such as envisioned for the DECADE QUAD. For all three nozzles, the 0-D models correctly predict the Z-pinch mass for maximum yield. For the 1and 1.75-cm radius nozzles, the scaling models accurately match the measured yields if the ratio of initial to final radius (compression ratio) is assumed to be 8:1. For the 2.5-cm radius nozzle, the measured yields are only one-third of the predictions. Analysis of K-shell spectral measurements suggest that as much as 70% (50%) of the imploded mass is radiating in the K-shell for the 1-cm (1.75-cm) radius nozzle. That fraction is only 10% for the 2.5-cm radius nozzle. The 0-D scaling models are useful for predicting 1-D-like K-shell radiation yields (better than a factor-of-two accuracy) when a nominal (≈10:1) compression ratio is assumed. However, the compression ratio assumed in the models is only an “effective” quantity, so that further interpretations based on the 0-D analysis require additional justification. The lower-than-predicted yield for the 2.5-cm radius nozzle is associated with larger radius and not with longer implosion time, and is probably a result of two-dimensional effects     

A microcalorimetric study of the association of hexacyanoferrate(II) ions with calcium and magnesium ions in aqueous solution

Using flow microcalorimetry, the ion association reaction M²⁺(aq)+Fe(CN) ₆ ^4– (aq)=MFe(CN) ₆ ^2– (aq) (M=Ca, Mg) has been studied at 25°C over the ionic strength range 0.02 to 0.08 mol-dm^–3. Analyses of the data to obtain H^o, the enthalpy change at infinite dilution, are described. The value obtained for H^o is sensitive to the kind of functions used to correct for non-ideal behavior.     

Energy migration in novel pH-triggered self-assembled beta-sheet ribbons

Energy migration between tryptophan residues has been experimentally demonstrated in self-assembled peptide tapes. Each peptide contains 11 amino acids with a Trp at position 6. The peptide self-assembly is pH-sensitive and forms amphiphilic tapes, which further stack in ribbons (double tapes) and fibrils in water depending on the concentration. Fluorescence spectra, quenching, and anisotropy experiments showed that when the pH is lowered from 9 to 2, the peptide self-assembly buries the tryptophan in a hydrophobic and restricted environment in the interior of stable ribbons as expected on the basis of the peptide design. These fluorescence data support directly and for the first time the presence of such ribbons which are characterized by a highly packed and stable hydrophobic interior. In common with Trp in many proteins, fluorescence lifetimes are nonexponential, but the average lifetime is shorter at low pH, possibly due to quenching with neighboring Phe residues. Unexpectedly, time-resolved fluorescence anisotropy does not change significantly with self-assembly when in water. In highly viscous sucrose-water mixtures, the anisotropy decay at low pH was largely unchanged compared to that in water, whereas at high pH, the anisotropy decay increased significantly. We concluded that depolarization at low pH was not due to rotational diffusion but mainly due to energy migration between adjacent tryptophan residues. This was supported by a master equation kinetic model of Trp-Trp energy migration, which showed that the simulated and experimental results are in good agreement, although on average only three Trp residues were visited before emission.     

The chemical synthesis of beta-(1-->4)-linked D-mannobiose and D-mannotriose

A D-cellobiose derivative was converted to D-mannobiose via simultaneous epimerization at C-2 and C-2'. Subsequent beta-D-glucosylation and epimerization at C-2" gave D-mannotriose.