Spatial design matrices and associated quadratic forms: structure and properties

The paper provides significant simplifications and extensions of results obtained by Gorsich, Genton, and Strang (J. Multivariate Anal. 80 (2002) 138) on the structure of spatial design matrices. These are the matrices implicitly defined by quadratic forms that arise naturally in modelling intrinsically stationary and isotropic spatial processes. We give concise structural formulae for these matrices, and simple generating functions for them. The generating functions provide formulae for the cumulants of the quadratic forms of interest when the process is Gaussian, second-order stationary and isotropic. We use these to study the statistical properties of the associated quadratic forms, in particular those of the classical variogram estimator, under several assumptions about the actual variogram.     

Some remarks on Zienkiewicz-Zhu estimator

In this article, Zienkiewicz-Zhu estimator is analyzed for the piecewise linear finite element approximate solution of an elliptic problem. The estimator is proved to be equivalent to the error for the Poisson equation with a homogeneous Dirichlet boundary condition for any triangular regular mesh. No assumptions are needed about the regularity of the solution (i.e., solutions with corner singularities are not excluded). The estimator is also proved to be asymptotically exact on subdomains where the solution is smooth when parallel meshes are used. Therefore, its behavior is similar to that of other well-known estimators. © 1994 John Wiley & Sons, Inc.     

Critical percolation exploration path and SLE 6: a proof of convergence

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE ₆. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE ₆ that we prove suffices for the Smirnov–Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops. Research of Charles M.Newman was partially supported by the US NSF under grants DMS-01-04278 and DMS-06-06696.     

Flow through a porous medium with mass removal and diffusion

A one-dimensional model of filtration through a saturated porous medium with a mutual action between the solid matrix and the flow is investigated. The substances removed from the porous soil are both particles transported by the liquid flow and substances which diffuse in the fluid. Received June 30, 1997     

The extended loop group: An infinite dimensional manifold associated with the loop space

A set of coordinates in the non-parametric loop-space is introduced. We show that these coordinates transform under infinite dimensional linear representations of the diffeomorphism group. An extension of the group of loops in terms of these objects is proposed. The enlarged group behaves locally as an infinite dimensional Lie group. Ordinary loops form a subgroup of this group. The algebraic properties of this new mathematical structure are analyzed in detail. Applications of the formalism to field theory, quantum gravity and knot theory are considered.     

The effect of the anion on the physical properties of trihalide-based N,N-dialkylimidazolium ionic liquids

Low viscosity, high density trihalide-based 1-n-butyl-3-methylimidazolium ionic liquids have been prepared and characterised. Key physical properties (density, conductivity, melting point, refractive index, surface tension and diffusion coefficient) of the ionic liquids have been determined and are compared with those of other 1,3-dialkylimidazolium molten salts. The relationship between anion identity and the physical properties of the ionic liquids under investigation is discussed.     

Evidence against the hopping mechanism as an important electron transfer pathway for conformationally constrained oligopeptides

The rate constant of intramolecular electron transfer through oligopeptides based on the alpha-aminoisobutyric acid residue was determined as a function of the peptide length and found to depend weakly on the donor-acceptor separation. By measuring the electron-transfer activation energy and estimating the energy gap between donor and bridge, we were able to discard the electron hopping mechanism.     

Thermal lens spectrometric determination of hexavalent chromium

The applicability of thermal lens spectrometry (TLS) for quantification and routine determination of hexavalent chromium was investigated by using a collinear dual beam thermal lens spectrometer. In aqueous solutions the LOD of 0.1 μg l⁻¹ was obtained for Cr(VI) by using 160 mW laser power. The performance of the technique was verified by the determination of hexavalent chromium in standard reference water samples (NIST SRM 1643a and NIST SRM 1643c) and comparing the results for Cr(VI) in CCA (Cr, Cu, As)-treated timber extracts to concentrations obtained by atomic absorption spectrometry (AAS). Good agreement between the TLS results and reported values for Cr(VI) in SRMs as well as AAS results for Cr(VI) in CCA-treated timbers confirmed that TLS is a reliable and accurate analytical technique applicable for the determination of Cr(VI) in aqueous solutions at concentration levels 0.5–100 μg l⁻¹.