Approximations of relations by continuous functions

Let X be a Tychonoff space, C(X) be the space of all continuous real-valued functions defined on X and CL(X×R) be the hyperspace of all nonempty closed subsets of X×R. We prove the following result. Let X be a countably paracompact normal space. The following are equivalent: (a) dimX=0; (b) the closure of C(X) in CL(X×R) with the Vietoris topology consists of all F∈CL(X×R) such that F(x)≠∅ for every x∈X and F maps isolated points into singletons; (c) each usco map which maps isolated points into singletons can be approximated by continuous functions in CL(X×R) with the locally finite topology. From the mentioned result we can also obtain the answer to Problem 5.5 in [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] and to Question 5.5 in [R.A. McCoy, Comparison of hyperspace and function space topologies, Quad. Mat. 3 (1998) 243-258] in the realm of normal, countably paracompact, strongly zero-dimensional spaces. Generalizations of some results from [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] are also given.     

Analytic Calculation of B 4 for Hard Spheres in Even Dimensions

We exactly calculate the fourth virial coefficient for hard spheres in even dimensions for D = 4, 6, 8, 10, and 12.     

New results for virial coefficients of hard spheres inD dimensions

We present new results for the virial coefficientsB _k for κ<- 10 for hard spheres in dimensionsD = 2,..., 8.     

Femtosecond study of Cu(H(2)O) dynamics

The short-time nuclear dynamics of Cu(H(2)O) is investigated using femtosecond photodetachment-photoionization spectroscopy and time-dependent quantum wave packet calculations. The Cu(H(2)O) dynamics is initiated in the electronic ground state of the complex by electron photodetachment from the Cu(-)(H(2)O) complex, where hydrogen atoms are oriented toward Cu. Several time-resolved resonant multiphoton ionization schemes are used to probe the ensuing reorientation and dissociation. Immediately following photodetachment, the neutral complex is far from its minimum energy geometry and possesses an internal energy comparable to the Cu-H(2)O dissociation energy and undergoes both large-amplitude H(2)O motion and dissociation. Dissociation is observed to occur on three distinct time scales: 0.6, 8, and 100 ps. These results are compared to the results of time-dependent J=0 wave packet calculations, propagating the initial anion vibrational wave functions on the ground-state potential of the neutral complex. An excellent agreement is obtained between the experimental results and the ionization signals derived from the calculated probability amplitudes. Related experiments and calculations are carried out on the Cu(D(2)O) complex, with results very similar to those of Cu(H(2)O).     

Numerical and similarity solutions for reversible population balance equations with size-dependent rates

Population balance equations (PBEs) for reversible aggregation-fragmentation processes are important to particle agglomeration and dissolution, polymerization and degradation, liquid droplet coalescence and breakup, and floc coagulation and disintegration. Moment solutions provide convenient solutions to the PBEs, including steady state and similarity solutions, but may not be feasible for complex forms of size-dependent rate coefficients and stoichiometric kernels. Numeric solutions are thus necessary not only for applications, but also for the study of the mathematics of PBEs. Here we propose a numerical method to solve PBEs and compare the results to moment solutions. The numeric results are consistent with known steady state and asymptotic long-time similarity solutions and show how processes can be approximated by self-similar formulations.     

Temperature effects for isothermal polymer crystallization kinetics

We adopt the cluster size distribution model to investigate the effect of temperature on homogeneous nucleation and crystal growth for isothermal polymer crystallization. The model includes the temperature effects of interfacial energy, nucleation rate, growth and dissociation rate coefficients, and equilibrium solubility. The time dependencies of polymer concentration, number and size of crystals, and crystallinity (in Avrami plots) are presented for different temperatures. The denucleation (Ostwald ripening effect) is also investigated by comparing moment and numerical solutions of the population balance equations. Agreement between the model results and temperature-sensitive experimental measurements for different polymer systems required strong temperature dependence for the crystal-melt interfacial energy.     

Experimentally determined oscillator strengths and linewidths for the Schumann-Runge band system of molecular oxygen—II. The (2-0) to (5-0) bands

Experimental oscillator strengths and predissociation linewidths have been measured at room temperature for the (2-0) to (5-0) Schumann-Runge bands of molecular oxygen using the Adelaide 6 m vacuum ultraviolet monochromator operated at a resolution of about 0.06 Å. Photoelectric detectors were used to measure the ultraviolet absorption at normally two different gas pressures for 37 groups of rotational lines, and the resulting data were interpreted using an equivalent width type of analysis. The variation with N″ of the oscillator strengths within each band was found to be smaller than that measured earlier for the higher vibrational bands. Measured oscillator strengths agree well with previously accepted values except for the (3-0) band, while the linewidths of the present work agree well with recent theoretical values for the (3-0) and (4-0) bands but are larger for the (2-0) and (5-0) bands. A modification was performed on the parameters of a recent theoretical predissociation model to produce excellent overall agreement between experimental and theoretical linewidths from ν' = 2 to 14. The present work provides results which can be used to construct accurate synthetic absorption profiles for studies of the atmospheric absorption of ultraviolet radiation by the Schumann-Runge bands.     

Densely Continuous Forms in Vietoris Hyperspaces

For countably paracompact normal spaces X and locally compact separable metric spaces Y, a characterization is given for the closure of the set of densely continuous forms from X to Y in the hyperspace of nonempty closed subsets of X × Y under the Vietoris topology. This shows that for such X having no isolated points, every closed subset of X × R that is dense over X can be Vietoris approximated by a semicontinuous function on X.     

Unique pitch evolution in the smectic-C+alpha phase

Employing resonant x-ray diffraction, we observed unique pitch evolutions in the smectic-C*(alpha) phase in mixtures of two antiferroelectric liquid crystals. Our results show that the pitch in this phase continuously evolves across 4 layers, contradicting a theoretical model that predicts that the smectic-C*(FI2) phase intervenes in the smectic-C*(alpha) phase. The phase sequences we found can be explained by another model that includes one type of long-range interaction among smectic layers.     

Swellable polymer films containing Au nanoparticles for point-of-care therapeutic drug monitoring using surface-enhanced Raman spectroscopy

Large (10 × 10 cm) sheets of surface-enhanced Raman spectroscopy (SERS) active polymer have been prepared by stabilising metal nanoparticle aggregates within dry hydroxyethylcellulose (HEC) films. In these films the aggregates are protected by the polymer matrix during storage but in use they are released when aqueous analyte droplets cause the films to swell to their gel form. The fact that these “Poly-SERS” films can be prepared in bulk but then cut to size and stored in air before use means that they provide a cost effective and convenient method for routine SERS analysis. Here we have tested both Ag and Au Poly-SERS films for use in point-of-care monitoring of therapeutic drugs, using phenytoin as the test compound. Phenytoin in water could readily be detected using Ag Poly-SERS films but dissolving the compound in phosphate buffered saline (PBS) to mimic body fluid samples caused loss of the drug signal due to competition for metal surface sites from Cl⁻ ions in the buffer solution. However, with Au Poly-SERS films there was no detectable interference from Cl⁻ and these materials allowed phenytoin to be detected at 1.8 mg L⁻¹, even in PBS. The target range of detection of phenytoin in therapeutic drug monitoring is 10–20 mg L⁻¹. With the Au Poly-SERS films, the absolute signal generated by a given concentration of phenytoin was lower for the films than for the parent colloid but the SERS signals were still high enough to be used for therapeutic monitoring, so the cost in sensitivity for moving from simple aqueous colloids to films is not so large that it outweighs the advantages which the films bring for practical applications, in particular their ease of use and long shelf life.