Elliptic difference equations and interior regularity

Summary Letu be a solution of an elliptic differential equation and letu _h be a solution of a corresponding elliptic difference equation. It is proved that ifu _h converges tou at a certain rate as the mesh-widthh tends to zero, then the appropriate difference quotient ofu h converges to the corresponding derivative ofu at the same rate whenh tends to zero.     

Separation of calcium and aluminum from iron by an ion exchange method

An ion exchange method has been worked out for the séparation of calcium and aluminum from iron. In the first step iron and aluminum are taken up in a column filled with an anion exchanger in the citrate form, whereas calcium passes into the effluent. Subsequently, aluminum is cluted by means of concentrated hydrochloric acid. The method has been adapted to the analysis of lime.     

Barriers to internal rotation in 1,3,5-trineopentylbenzenes: V—conformational studies by carbon-13 magnetic resonance

The temperature dependence of the carbon-13 NMR signals in three completely substituted 1,3,5-trineopentylbenzenes has been studied by the Fourier transform technique. The assignment of all peaks was based on studies of 1,3,5-trineopentylbenzene and some of its mono- and disubstituted derivatives. Nearly all carbon signals showed a splitting into several peaks at low temperature, which gave information about the existing rotamers in the completely substituted compounds. Spectra for one compound were studied in a narrow temperature interval in order to test the reliability of rate constants obtained from ¹³C FT spectra. Good agreement was found between previously reported values obtained from ¹H NMR and those obtained in this investigation.     

Determination of the homogeneity range and refinement of the crystal structure of Fe2P

The homogeneity range of Fe₂P has been determined using the annealing and quenching technique in combination with X-ray powder diffraction methods. The iron-rich limit is invariant at Fe_2.00P for temperatures up to 1150°C. The phosphorus-rich limit varies from Fe_2.00P at low temperatures to Fe_1.94P at 1100°C. The crystal structure of Fe_2.00P has been refined from X-ray diffraction data obtained by single-crystal diffractometry. The thermal vibrations differ appreciably in magnitude for the two non-equivalent types of iron atom.     

Combined Sea and Land Transportation

An important problem today in the field of transportation is the standardization of the cargo, e.g. by using containers, and the design of the handling and transportation equipment for the specific cargo to be transported.The paper presents a method for determining the transportation system with emphasis on sea transport. Thus the cargo is to be transported by sea from the factory to customers spread over a large region, e.g. Europe. The problem is to select the ports of call, the quantities to be delivered at the ports, as well as the size and type of vessel.This problem resembles the warehouse location problem (the location of ports) but requires in addition the determination of ship size, type of ship and whether one or more ports should be called at on each journey with a single ship. A discussion is also presented as to the possibility of considering randomness in the system with respect to customer demand and weather conditions.The method used resembles that suggested by Baumol and Wolfe for the ware-house location problem. A concave function of the quantities delivered at each port is derived and this is then shown to converge to a local optimum.An example is solved to illustrate the method.     

Microstructure and dynamics in lyotropic liquid crystals. Principles and applications of nuclear spin relaxation

Abstract

Nuclear spin relaxation of quadrupolar nuclei provides access to a wide range of properties of lyotropic liquid crystals, ranging from the molecular ordering and dynamics at the interface to the macroscopic viscoelastic behaviour. We emphasize here the unique capability of the spin relaxation method to provide detailed geometric and dynamic information relating to the microstructure of lyotropic liquid crystals, i.e. the metric, curvature, and fluctuations of the dividing interface that separates polar and non-polar regions. This information is conveyed to the spin system via the translational diffusion of surfactants or counterions over the interface. The general principles of the spin relaxation method, as applied to lyotropic liquid crystals, are described, with emphasis on the model-independent information content of the relaxation observables and on the relation to microstructure. Specific results for lamellar, hexagonal, cubic, and nematic phases are also described.     

Uncertainty of standard addition experiments: a novel approach to include the uncertainty associated with the standard in the model equation

A new model equation for determining the measurement result in standard addition experiments was derived and successfully applied to the quantitative determination of rhodium in automotive catalysts. Existing equations for standard addition experiments with gravimetric preparation were changed in order to integrate the novel idea of including the uncertainty associated with the standard into the model equation. Using this novel equation combined with the ordinary least squares algorithm for the regression line also yielded a new formula for the associated measurement uncertainty. This uncertainty accounts for the first time for the uncertainty associated with the standard. The derivation for the model equation and the resulting associated measurement uncertainty is shown for gravimetric standard addition experiments both with and without an internal standard.     

Antiferroelectric siloxane liquid crystal dimers with large molecular tilt under an applied electric field

The scope of the present study is the response of a series of antiferroelectric dimeric or bi‐mesogenic siloxanes to an applied electric field with focus on their pretransitional behaviour and the field‐induced antiferro–ferroelectric (AF–F) transition. Most of these compounds possess a molecular tilt close to 45° and spontaneous polarization in the field‐induced ferroelectric (F) phase in the range of 250–300 nC cm^?2. In the dimers with a spacer length exceeding five carbons, a transformation from first to second order of the field‐induced AF–F transition is found with temperature. Several different indications for this transformation are identified and their characteristics are discussed in the framework of the existing theoretical models. A large field‐induced in‐plane deviation of the sample optic axis was observed in the pretransitional region of several of the siloxane dimers and is likely due to the flexibility of the linking chains. The potential of the antiferroelectric bimesogenic siloxanes for displaying high contrast images and grey scale capability is shortly discussed. The large molecular tilt close to 45° in combination with the field‐induced AF–F transition of second order seems to be the most attractive features of these materials.     

Estimation of optimal backward perturbation bounds for the linear least squares problem

In this paper a method of estimating the optimal backward perturbation bound for the linear least squares problem is presented. In contrast with the optimal bound, which requires a singular value decomposition, this method is better suited for practical use on large problems since it requiresO(mn) operations. The method presented involves the computation of a strict lower bound for the spectral norm and a strict upper bound for the Frobenius norm which gives a gap in which the optimal bounds for the spectral and the Frobenius norm must be. Numerical tests are performed showing that this method produces an efficient estimate of the optimal backward perturbation bound.     

Solvent extraction procedures combined with back-extraction for trace metal determinations by atomic absorption spectrometry

Metal dithiocarbamate extractions with three different organic solvents (chloroform, freon and 4-methylpentan-2-one) followed by back-extraction with nitric acid are described. The back-extraction is studied in detail and simple rapid procedures are described for each solvent. The dilute nitric acid solution obtained is suitable for use with both flame and electrothermal atomic absorption spectrometry. Quantitative recoveries were obtained for Cd, Cu, Fe, Ni, Pb and Zn. Cobalt could be rapidly back-extracted only from chloroform and freon.