Space tensors in general relativity III: The structural equations

A self-consistent theory of spatial differential forms over a pair (M,Γ)is proposed. The operators d(spatial exterior differentiation), d_T (temporal Lie derivative) andL (spatial Lie derivative) are defined, and their properties are discussed. These results are then applied to the study of the torsion and curvature tensor fields determined by an arbitrary spatial tensor analysis \((\tilde \nabla ,\tilde \nabla T)\) (M,Γ). The structural equations of \((\tilde \nabla ,\tilde \nabla T)\) and the corresponding spatial Bianchi identities are discussed. The special case \((\tilde \nabla ,\tilde \nabla T) = (\tilde \nabla *,\tilde \nabla T*)\) is examined in detail. The spatial resolution of the Riemann tensor of the manifold M is finally analysed; the resultingstructure of Eintein's equations over a pair (ν₄,Γ)is established. An application to the study of the problem of motion in terms of co-moving atlases is proposed.     

Effect of SiO2 on relaxation phenomena and mechanism of ion conductivity of [Nafion/(SiO2)x] composite membranes

This report describes a study of the effect of SiO2 nanopowders on the mechanism of ionic motion and interactions taking place in hybrid inorganic-organic membranes based on Nafion. Five nanocomposite membranes of the formula [Nafion/(SiO2)x] with SiO2 ranging from 0 to 15 wt % were prepared by a solvent casting procedure. TG measurements demonstrated that the membranes are thermally stable up to 170 degrees C but with the loss water it changes the cluster environments and changes the conductivity properties. MDSC investigations in the 90-300 degrees C temperature range revealed the presence of three intense overlapping endothermal peaks indicated as I, II, and III. Peak I measures the order-disorder molecular rearrangement in hydrophilic polar clusters, II corresponds to the endothermic decomposition of -SO3 groups, and III describes the melting process in microcrystalline regions of hydrophobic fluorocarbon domains of the Nafion moiety. ESEM with EDAX measurements revealed that the membranes are homogeneous materials with smooth surfaces. DMA studies allowed us to measure two relaxation modes. The mechanical relaxation detected at ca. 100 degrees C is attributed to the motion of cluster aggregates of side chains and is diagnostic for R-SO3H...SiO2 nanocluster interactions. DMA disclosed that at SiO2/-SO3H (psi) molar ratios lower than 1.9, the oxoclusters act to restrict chain mobility of hydrophobic domains of Nafion and the dynamics inside polar cages of [Nafion/(SiO2)x] systems; at psi higher than 1.9, the oxoclusters reduce the cohesiveness of hydrophilic polar domains owing to a reduction in the density of cross-links. FT-IR and FT-Raman studies of the [Nafion/(SiO2)x] membranes indicated that the fluorocarbon chains of Nafion hydrophobic domains assume the typical helical conformation structure with a D(14pi/15) symmetry. These analyses revealed four different species of water domains embedded inside polar cages and their interconnecting channels: (a) bulk water [(H2O)n]; (b) water solvating the oxonium ions directly interacting with sulfonic acid groups [H3O+...SO3(-)-].(H2O)n; (c) water aggregates associated with H3O+ ions [H3O+.(H2O)n]; and (d) low associated water species in dimer form [(H2O)2]. The conductivity mechanism and relaxation events were investigated by broadband dielectric spectroscopy (BDS). [Nafion/(SiO2)x] nanocomposite membranes were found to possess two different molecular relaxation phenomena which are associated with the alpha-relaxation mode of PTFE-like fluorocarbon domains and the beta-relaxation mode of acid side groups of the Nafion component. Owing to their strong coupling, both these relaxation modes are diagnostic for the interactions between the polar groups of the Nafion host polymer and the (SiO2)x oxoclusters and play a determining role in the conductivity mechanism of the membranes. The studies support the proposal that long-range proton charge transfer in [Nafion/(SiO2)x] composites takes place due to a mechanism involving exchange of the proton between the four water domains. This latter proton transfer occurs owing to a subsequent combination of domain intersections resulting from the water domain fluctuations induced by the molecular relaxation events of host Nafion polymer.     

A note on scale functions and the time value of ruin for Lévy insurance risk processes

We examine discounted penalties at ruin for surplus dynamics driven by a general spectrally negative Lévy process; the natural class of stochastic processes which contains many examples of risk processes which have already been considered in the existing literature. Following from the important contributions of [Zhou, X., 2005. On a classical risk model with a constant dividend barrier. North Am. Act. J. 95-108] we provide an explicit characterization of a generalized version of the Gerber-Shiu function in terms of scale functions, streamlining and extending results available in the literature.     

Footnote to a paper by hartshorne

We prove that every curve in the biliaison class of an extremal curve in P³ can be connected to an extremal curve.     

Curves in the double plane

We study in detail locally Cohen-Macaulay curves in P⁴ which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemes H _d,g(2H) of lo-cally Cohen-Macaulay curves in 2H of degree d and arithmetic genus g, and we show that H _d,g(2H) is connected. We also discuss the Rao module of these curves and liaison and biliaison equiva-lence classes.