Dilations for systems of imprimitivity acting on Banach spaces

Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-known result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space. This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces.     

A study of shear-stress relaxation anomalies in binary mixtures of monodisperse polystyrenes

We report measurements of the nonlinear relaxation moduli after a step-shear strain of polystyrene solutions with nearly monodisperse and with bidisperse distributions of molecular weight. We find, as have others, that for monodisperse solutions with M/M_e > 60, there are anomalies, such as an unusually low nonlinear modulus and a kink in a plot of shear stress versus time after the step strain. Here M is the polymer molecular weight and M_e is the entanglement molecular weight. We find that in the bidisperse solutions the anomalies persist as long as M_w/M_e > 60, where M_w is the weight-averaged molecular weight of the bidisperse solution. The persistence of the anomalies in bidisperse solutions disagrees with a theory of Marrucci and Grizzuti that attributes the anomalies to strain inhomogeneities similar to shear banding. The Marrucci-Grizzuti theory predicts that as little as 10% short chains in the bidisperse mix should eliminate the anomalies, whereas in the experiments reported here at least 30% is required. Nevertheless the way in which the anomalies disappear at high strains when one increases the fraction of low-molecular-weight component is qualitatively similar to the theoretical predictions and supports the notion that strain inhomogeneities occur in these systems. © 1992 John Wiley & Sons, Inc.     

Step planar extension of polymer melts using a lubricated channel

A technique to do step planar extension on polymer melts has been developed using a rectangular channel with lubricated walls and the linear motor of the Rheometrics System Four rheometer. Using this method we probe the stress relaxation of two polymer melts, a linear low density polyethylene (LLDPE) and a highly branched low density polyethylene (IUPAC X), and compare the step planar extensional data to step shear data. Since a step planar deformation is theoretically equivalent to a step shear in a rotating frame of reference, we expect that the nonlinear modulus for step planar extension should be equivalent to that for step shear. Although we find the time dependence of the stress relaxation modulus to be the same in both shear and planar extension, the strain dependence is surprisingly different for the two experiments.     

The BKZ as an alternative to the Wagner model for fitting shear and elongational flow data of an LDPE melt

For the low density polyethylene Melt I, which is the melt for which the most complete set of shear and elongational data exists, the semi-empirical single integral Wagner model gives an excellent data-fit, but suffers the drawback of having no entropic constitutive equation, that is a relationship between strain history and elastic free energy from which viscous heating and cohesive failure can be predicted. We show here that the BKZ model, which does possess an entropic constitutive equation, gives as good a fit as does the Wagner model to both the shear and elongational data.     

Characterization of Glycan Structures of Chondroitin Sulfate-Glycopeptides Facilitated by Sodium Ion-Pairing and Positive Mode LC-MS/MS

Purification and liquid chromatography-tandem mass spectrometry (LC-MS/MS) characterization of glycopeptides, originating from protease digests of glycoproteins, enables site-specific analysis of protein N- and O-glycosylations. We have described a protocol to enrich, hydrolyze by chondroitinase ABC, and characterize chondroitin sulfate-containing glycopeptides (CS-glycopeptides) using positive mode LC-MS/MS. The CS-glycopeptides, originating from the Bikunin proteoglycan of human urine samples, had ΔHexAGalNAcGlcAGalGalXyl-O-Ser hexasaccharide structure and were further substituted with 0-3 sulfate and 0-1 phosphate groups. However, it was not possible to exactly pinpoint sulfate attachment residues, for protonated precursors, due to extensive fragmentation of sulfate groups using high-energy collision induced dissociation (HCD). To circumvent the well-recognized sulfate instability, we now introduced Na⁺ ions to form sodiated precursors, which protected sulfate groups from decomposition and facilitated the assignment of sulfate modifications. Sulfate groups were pinpointed to both Gal residues and to the GalNAc of the hexasaccharide structure. The intensities of protonated and sodiated saccharide oxonium ions were very prominent in the HCD-MS2 spectra, which provided complementary structural analysis of sulfate substituents of CS-glycopeptides. We have demonstrated a considerable heterogeneity of the bikunin CS linkage region. The realization of these structural variants should be beneficial in studies aimed at investigating the importance of the CS linkage region with regards to the biosynthesis of CS and potential interactions to CS binding proteins. Also, the combined use of protonated and sodiated precursors for positive mode HCD fragmentation analysis will likely become useful for additional classes of sulfated glycopeptides.

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Catalytic asymmetric aza-Darzens reaction with a vaulted biphenanthrol magnesium phosphate salt

Conditions for a catalytic asymmetric aza-Darzens aziridine synthesis mediated by a vaulted biphenanthrol (VAPOL) magnesium phosphate salt is described. Using simple substrates, this methodology explores the scope and reactivity of a new magnesium catalyst for an aziridination reaction capable of building chirality and complexity simultaneously.     

A posteriori error estimates for mixed finite element approximations of parabolic problems

We derive residual based a posteriori error estimates for parabolic problems on mixed form solved using Raviart–Thomas–Nedelec finite elements in space and backward Euler in time. The error norm considered is the flux part of the energy, i.e. weighted L ²(Ω) norm integrated over time. In order to get an optimal order bound, an elementwise computable post-processed approximation of the scalar variable needs to be used. This is a common technique used for elliptic problems. The final bound consists of terms, capturing the spatial discretization error and the time discretization error and can be used to drive an adaptive algorithm.