Symmetric functions of two noncommuting variables

We prove a noncommutative analogue of the fact that every symmetric analytic function of $(z,w)$ in the bidisc $\mathbb{D}^2$ can be expressed as an analytic function of the variables $z+w$ and $zw$. We construct an analytic nc-map $S$ from the biball to an infinite-dimensional nc-domain $\Omega$ with the property that, for every bounded symmetric function $\varphi$ of two noncommuting variables that is analytic on the biball, there exists a bounded analytic nc-function $\Phi$ on $\Omega$ such that $\varphi=\Phi\circ S$. We also establish a realization formula for $\Phi$, and hence for $\varphi$, in terms of operators on Hilbert space

Geodesics, retracts, and the norm-preserving extension property in the symmetrized bidisc

A set in a domain in has the norm-preserving extension property if every bounded holomorphic function on has a holomorphic extension to with the same supremum norm. We prove that an algebraic subset of the symmetrized bidischas the norm-preserving extension property if and only if it is either a singleton, itself, a complex geodesic of , or the union of the set and a complex geodesic of degree in . We also prove that the complex geodesics in coincide with the nontrivial holomorphic retracts in . Thus, in contrast to the case of the ball or the bidisc, there are sets in which have the norm-preserving extension property but are not holomorphic retracts of . In the course of the proof we obtain a detailed classification of the complex geodesics in modulo automorphisms of . We give applications to von Neumann-type inequalities for -contractions (that is, commuting pairs of operators for which the closure of is a spectral set) and for symmetric functions of commuting pairs of contractive operators. We find three other domains that contain sets with the norm-preserving extension property which are not retracts: they are the spectral ball of matrices, the tetrablock and the pentablock. We also identify the subsets of the bidisc which have the norm-preserving extension property for symmetric functions

A geometric characterization of the symmetrized bidisc

The symmetrized bidisc G def = {(z + w, zw) : |z| < 1, |w| < 1} has interesting geometric properties. While it has a plentiful supply of complex geodesics and of automorphisms, there is nevertheless a unique complex geodesic in G that is invariant under all automorphisms of G. Moreover, G is foliated by those complex geodesics that meet in one point and have nontrivial stabilizer. We prove that these properties, together with two further geometric hypotheses on the action of the automorphism group of G, characterize the symmetrized bidisc in the class of complex manifolds

Poly(propylene glycol) and urethane dimethacrylates improve conversion of dental composites and reveal complexity of cytocompatibility testing.

OBJECTIVES: To determine the effects of various monomers on conversion and cytocompatibility of dental composites and to improve these properties without detrimentally affecting mechanical properties, depth of cure and shrinkage. METHODS: Composites containing urethane dimethacrylate (UDMA) or bisphenol A glycidyl methacrylate (Bis-GMA) with poly(propylene glycol) dimethacrylate (PPGDMA) or triethylene glycol dimethacrylate (TEGDMA) were characterized using the following techniques: conversion (FTIR at 1 and 4mm depths), depth of cure (BS EN ISO 4049:2009 and FTIR), shrinkage (BS EN ISO 17304:2013 and FTIR), strength and modulus (biaxial flexural test) and water sorption. Cytocompatibility of composites and their liquid phase components was assessed using three assays (resazurin, WST-8 and MTS). RESULTS: UDMA significantly improved conversion, BFS and depth of cure compared to Bis-GMA, without increasing shrinkage. UDMA was cytotoxic at lower concentrations than Bis-GMA, but extracts of Bis-GMA-containing composites were less cytocompatible than of those containing UDMA. PPGDMA improved conversion and depth of cure compared to TEGDMA, without detrimentally affecting shrinkage. TEGDMA was shown by all assays to be highly toxic. Resazurin, but not WST-8 and MTS, suggested that PPGDMA exhibited improved cytocompatibility compared to TEGDMA. SIGNIFICANCE: The use of UDMA and PPGDMA results in composites with excellent conversion, depth of cure and mechanical properties, without increasing shrinkage. Composites containing UDMA appear to be slightly more cytocompatible than those containing Bis-GMA. These monomers may therefore improve the material properties of dental restorations, particularly bulk fill materials. The effect of diluent monomer on cytocompatibility requires further investigation

Passive water control at the surface of a superhydrophobic lichen

Some lichens have a super-hydrophobic upper surface, which repels water drops, keeping the surface dry but probably preventing water uptake. Spore ejection requires water and is most efficient just after rainfall. This study was carried out to investigate how super-hydrophobic lichens manage water uptake and repellence at their fruiting bodies, or podetia. Drops of water were placed onto separate podetia of Cladonia chlorophaea and observed using optical microscopy and cryo-scanning-electron microscopy (cryo-SEM) techniques to determine the structure of podetia and to visualise their interaction with water droplets. SEM and optical microscopy studies revealed that the surface of the podetia was constructed in a three-level structural hierarchy. By cryo-SEM of water-glycerol droplets placed on the upper part of the podetium, pinning of the droplet to specific, hydrophilic spots (pycnidia/apothecia) was observed. The results suggest a mechanism for water uptake, which is highly sophisticated, using surface wettability to generate a passive response to different types of precipitation in a manner similar to the Namib Desert beetle. This mechanism is likely to be found in other organisms as it offers passive but selective water control

Urinary bisphenol A concentration and risk of future coronary artery disease in apparently healthy men and women

addresses: Epidemiology and Public Health Group, Peninsula College of Medicine and Dentistry, Barrack Road, Exeter, United Kingdom. [email protected]: Comparative Study; Journal Article; Research Support, Non-U.S. Gov'tThe endocrine-disrupting chemical bisphenol A (BPA) is widely used in food and beverage packaging. Higher urinary BPA concentrations were cross-sectionally associated with heart disease in National Health and Nutrition Examination Survey (NHANES) 2003-2004 and NHANES 2005-2006, independent of traditional risk factors.Medical Research Council UKCancer Research UKBritish Heart FoundationPeninsula Medical School, University of ExeterEuropean Regional Development FundEuropean Social Fund Convergence Programme for Cornwall and the Isles of ScillyNational Institute for Health Research Collaborations for Leadership in Applied Health Research and Car

3-extremal holomorphic maps and the symmetrised bidisc

We analyze the 3-extremal holomorphic maps from the unit disc D to the symmetrized bidisc G=def{(z+w,zw):z,w∈D} with a view to the complex geometry and function theory of G. These are the maps whose restriction to any triple of distinct points in D yields interpolation data that are only just solvable. We find a large class of such maps; they are rational of degree at most 4. It is shown that there are two qualitatively different classes of rational G-inner functions of degree at most 4, to be called aligned and caddywhompus functions; the distinction relates to the cyclic ordering of certain associated points on the unit circle. The aligned ones are 3-extremal. We describe a method for the construction of aligned rational G-inner functions; with the aid of this method we reduce the solution of a 3-point interpolation problem for aligned holomorphic maps from D to G to a collection of classical Nevanlinna–Pick problems with mixed interior and boundary interpolation nodes. Proofs depend on a form of duality for G

Calculating metalation in cells reveals CobW acquires Co(II) for vitamin B12 biosynthesis while related proteins prefer Zn(II)

Protein metal-occupancy (metalation) in vivo has been elusive. To address this challenge, the available free energies of metals have recently been determined from the responses of metal sensors. Here, we use these free energy values to develop a metalation-calculator which accounts for inter-metal competition and changing metal-availabilities inside cells. We use the calculator to understand the function and mechanism of GTPase CobW, a predicted CoII-chaperone for vitamin B12. Upon binding nucleotide (GTP) and MgII, CobW assembles a high-affinity site that can obtain CoII or ZnII from the intracellular milieu. In idealised cells with sensors at the mid-points of their responses, competition within the cytosol enables CoII to outcompete ZnII for binding CobW. Thus, CoII is the cognate metal. However, after growth in different [CoII], CoII-occupancy ranges from 10 to 97% which matches CobW-dependent B12 synthesis. The calculator also reveals that related GTPases with comparable ZnII affinities to CobW, preferentially acquire ZnII due to their relatively weaker CoII affinities. The calculator is made available here for use with other proteins