Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements

We derive new trace inequalities for NURBS-mapped domains. In addition to Sobolev-type inequalities, we derive discrete trace inequalities for use in NURBS-based isogeometric analysis. All dependencies on shape, size, polynomial degree, and the NURBS weighting function are precisely specified in our analysis, and explicit values are provided for all bounding constants appearing in our estimates. As hexahedral finite elements are special cases of NURBS, our results specialize to parametric hexahedral finite elements, and our analysis also generalizes to T-spline-based isogeometric analysis. We compare the bounding constants appearing in our explicit trace inequalities with numerically computed optimal bounding constants, and we discuss application of our results to a Laplace problem. We finish this paper with a brief exploration of so-called patch-wise trace inequalities for isogeometric analysis.     

Bases for rings of coinvariants

We study the multiplicative structure of rings of coinvariants for finite groups. We develop methods that give rise to natural monomial bases for such rings over their ground fields and explicitly determine precisely which monomials are zero in the ring of coinvariants. We apply our methods to the Dickson, upper triangular and symmetric coinvariants. Along the way, we recover theorems of Steinberg [17] and E. Artin [1]. Using these monomial bases we prove that the image of the transfer for a general linear group over a finite field is a principal ideal in the ring of invariants. This research is supported in part by the Natural Sciences and Engineering Research Council of Canada     

Higher order symmetrizer and application to an unusual transport equation

We study a class of stationary transport equation with nonlocal low-order tems We obtain the existence and uniqueness of a solution in sobolev spaces     

Electro-optic pulse response of ferroelectric liquid crystals

The electro-optic response of ferroelectric smectic C* liquid crystals has been studied. Anomalous switching behaviour of such materials which possess a negative dielectric anisotropy has been reported. These materials show a minimum in response time at a sufficiently high field. We present results showing the dependency of this minimum upon spontaneous polarisation and the effect of AC bias. Calculations based upon the equation of motion of the director around the cone are presented which describe this effect and its dependence on the relative magnitudes of the spontaneous polarization and dielectric anisotropy of the material. Good agreement with the experimental results is found.     

Photoemission studies of pulsed-RF plasma nitrided ultra-thin SiON dielectric layers

Results are presented of a photoemission study of the electronic structure of SiON layers formed by a pulsed-RF decoupled plasma nitration (DPN) of ultra-thin SiO₂ grown base layers approximately 1.0 nm thick. The optical thickness of these device grade nitrided dielectric layers was in the range 1.4-1.6 nm. X-ray photoelectron spectroscopy (XPS) studies indicate that the nitrogen is incorporated in a single chemical environment at concentration levels in the range 15-17%. Angle resolved XPS measurements show that the nitrogen is distributed through the layer, with the binding energy of the N 1s peak at 398.3 eV which is indicative of a Si₃N₄-like chemical species in an oxide environment. High resolution core level photoemission studies of the spin orbit stripped Si 2p⁴⁺ peak revealed full width half maximum values in the range 1.4-1.55 eV, which are significantly larger than the 1.15 eV value reported for SiO₂ layers. Synchrotron radiation photoemission studies of the valence band spectra enable the valence band off-set at the Si/SON interface to be evaluated as 2.3 eV and to infer a conduction band off-set of 2.1 eV.     

The use of charge-coupled devices in the quantitative evaluation of images, on photographic film or membranes, obtained following electrophoretic separation of DNA fragments

Charge-coupled devices can provide an excellent means for the quantification of DNA band images on membranes and photographic film. However, proper consideration must be given to important parameters such as signal-to-noise ratio and resolution, especially in applications which demand the use of large format media. When employed in the direct imaging of chemiluminescent blots, the charge-coupled device can provide equal or better sensitivity than that obtained by indirect methods using film, with the additional advantages of wide dynamic range and freedom from the vagaries of film processing.     

Teachers' funds of knowledge and the teaching and learning of mathematics in multi-ethnic primary schools: Two teachers' views of linking home and school

In this paper we explore two teachers' views on the role(s) of parents, the local community and children's home lives in the learning of mathematics in primary school. We use Moll and Greenberg's concept of ‘funds of knowledge’ and apply it to the case studies of two teachers working in the UK context. Issues of teachers' professional experience, ethnicity, class and gender emerge as significant in examining similarities and differences in the teachers' beliefs, understandings and practices in the area of linking home and school. We end with a discussion of some implications for teacher education and professional development.     

Aryl trihydroxyborates: easily isolated discrete species convenient for direct application in coupling reactions

A conceptually and practically simple alternative approach to the use of arylboron species as the organometallic component in cross-coupling processes is described whereby trihydroxyborate salts are isolated and directly employed. The protocol derives practical benefit from the ease and convenience of the isolation and subsequent use of the discrete borate salts, eliminates the need for additional base, and aids the use of correct reaction stoichiometry.