Weak Convergence and Banach Space-Valued Functions: Improving the Stability Theory of Feynman’s Operational Calculi

In this paper we investigate the relation between weak convergence of a sequence \(\left\{ \mu_{n}\right\} \) of probability measures on a Polish space S converging weakly to the probability measure μ and continuous, norm-bounded functions into a Banach space X. We show that, given a norm-bounded continuous function f:S→X, it follows that \(\lim_{n\to\infty}\int_{S}f\, d\mu_{n}=\int_{S}f\, d\mu\)—the limit one has for bounded and continuous real (or complex)—valued functions on S. This result is then applied to the stability theory of Feynman’s operational calculus where it is shown that the theory can be significantly improved over previous results.     

Assembly,characterization, and photocatalytic activities of TiO<Subscript>2</Subscript> nanotubes/CdS quantum dots nanocomposites

The semiconductor quantum dots (QDs) can be very efficient to tune the response of photocatalyst of TiO₂ to visible light. In this study, CdS QDs formed in situ with about 8 nm have been successfully deposited onto the surfaces of TiO₂ nanotubes (TNTs) to form TNTs/CdS QDs nanocomposites by use of a simple bifunctional organic linker, thiolactic acid. The diffuse reflectance spectroscopy (DRS) spectra of as prepared samples showed that the absorption edge of the TNTs/CdS composite is extended to visible range, with absorption edge at 530 nm. The photocatalytic activity and stability of TNTs/CdS were also evaluated for the photodegradation of rhodamine B. The results showed that when TNTs/CdS QDs was used, photocatalytic degradation of RhB under visible light irradiation reached 91.6%, higher than 45.4 and 30.5% for P25 and TNTs, respectively. This study indicated that the TNTs/CdS QDs nanocomposites were superior catalysts for photodegradation under visible light irradiation compared with TNTs and P25 samples, which may find wide application as a powerful photocatalyst in environmental field.     

Improvement of <Emphasis Type="Italic">n</Emphasis>-ZnO/<Emphasis Type="Italic">p</Emphasis>-Si photodiodes by embedding of silver nanoparticles

The photo-current of n-ZnO/p-Si heterojunction photodiodes was improved by embedding Ag nanoparticles in the interface (ZnO/nano-P_Ag/p-Si), and the ratio between photo- and dark-current increased by about three orders more than that of a n-ZnO/p-Si specimen. The improvement in the photo-current resulted from the light scattering of embedded Ag nanoparticles. The I–V curve of n-ZnO/p-Si degraded after thermal treatment (A-ZnO/p-Si) because the silicon robbed the oxygen from ZnO to form amorphous silicon dioxide and left an oxygen vacancy. Notably, the properties of ZnO/nano-P_Ag/p-Si were better in the time-dependent photoresponse under 10 V bias. Ag nanoparticles (15–20 nm) scattered the UV light randomly and increased the probability for the absorption of ZnO to enhance the properties of the photodiode.     

Pulsed spark-discharge assisted synthesis of colloidal gold nanoparticles in ethanol

A green method, using pulsed spark-discharge (PSD) to synthesize gold nanoparticles (AuNPs) in ethanol, is studied in this article. Unlike conventional methods for metal nanoparticles synthesis, the PSD method does not require the addition of chemical surfactants and stabilizers. The size of PSD–AuNPs is examined by transmission electron microscopy, with a range 5–50 nm. The chemical compounds, crystal structure, and surface plasmon resonance of PSD–AuNPs are studied using energy dispersive X-ray spectroscopy, X-ray diffraction, and UV–Visible spectroscopy, respectively. Zeta potential analysis shows that a negative charge (−40 mV) on the surface of the PSD–AuNPs may be contributing to the stability of the suspension. During the gold electrodes discharge in the ethanol, under an intensive electric field and thermal energy, bulk metallic gold and ethanol may produce AuNPs and varieties of chemical derivatives, which are also studied by GC/MS and FTIR to investigate the suspension mechanism. The analysis results show that there is an oxidation reaction of ethanol occurring during the PSD process to produce ethanol derivatives, such as acetaldehyde, acetic acid, and ethyl acetate, which may modify the surface of AuNPs by coordination of oxygen atoms. However, only acetic acid can form a negative charge by the deprotonation of the carboxylic group of surface in ethanol, resulting in the creation of a repulsion force between the particles to form the stable colloid system. The experimental results indicate that PSD is an alternative green process to synthesize gold nanoparticles suspension in ethanol. Moreover, with a gold rod consumption rate of 15 mg/L, concentrations of gold nanoparticles ~9 ppm have been observed; therefore, the net production rate is around 60%.     

Structural,electrical and optical studies on spray-deposited aluminium-doped ZnO thin films

Thin films of zinc oxide (ZnO) were deposited on cleaned glass substrates by chemical spray pyrolysis technique using Zn(CH₃COO)₂ as precursor solution. Also, aluminium-doped thin films of ZnO were prepared by using AlCl₃ as doping solution for aluminium. The dopant concentration [Al/Zn atomic percentage (at%)] was varied from 0 to 1.5 at% in thin films of ZnO prepared in different depositions. Structural characterization of the deposited films was performed with X-ray diffraction (XRD) studies. It confirmed that all the films were of zinc oxide having polycrystalline nature and possessing typical hexagonal wurtzite structure with crystallite size varying between 100.7 and 268.6 nm. The films exhibited changes in relative intensities and crystallite size with changes in the doping concentration of Al. The electrical studies established that 1 at% of Al-doping was the optimum for enhancing electrical conduction in ZnO thin films and beyond that the distortion caused in the lattice lowered the conductivity. The films also exhibited distinct changes in their optical properties at different doping concentrations, including a blue shift and slight widening of bandgap with increasing Al dopant concentration.     

Dirac particles in a gravitational field

The semiclassical approximation for the Hamiltonian of Dirac particles interacting with an arbitrary gravitational field is investigated. The time dependence of the metric leads to new contributions to the in-band energy operator in comparison to previous works in the static case. In particular we find a new coupling term between the linear momentum and the spin, as well as couplings that contribute to the breaking of the particle–antiparticle symmetry.     

Chandrasekhar: The all rounder

This paper provides a very brief introduction to three of Chandrasekhar’s famous books on Stellar Structure, Hydrodynamics and Black Holes. In particular we summarize Chandra’s treatment of the “Thermal Instability” which plays such a crucial role in the understanding of convection zones in stellar atmospheres. We also outline three important ideas in fluid dynamics which are inexplicably omitted from Chandrasekhar’s Hydrodynamic and Hydromagnetic Stability; the first is the Brunt–Väis˙alä frequency which appears in internal gravity waves and is closely related to Schwarzschild’s stability criterion; the second is the baroclinic instability which is important in atmospheric dynamics and meteorology, and the third is the conservation of potential vorticity which is central to the understanding of the planetary scale – Rossby waves.     

Isometric embeddings in cosmology and astrophysics

Recent interest in higher-dimensional cosmological models has prompted some significant work on the mathematical technicalities of how one goes about embedding spacetimes into some higher-dimensional space. We survey results in the literature (existence theorems and simple explicit embeddings); briefly outline our work on global embeddings as well as explicit results for more complex geometries; and provide some examples. These results are contextualized physically, so as to provide a foundation for a detailed commentary on several key issues in the field such as: the meaning of ‘Ricci equivalent’ embeddings; the uniqueness of local (or global) embeddings; symmetry inheritance properties; and astrophysical constraints.     

Persistence Properties and Unique Continuation of Solutions to a Two-component Camassa–Holm Equation

We will consider a two-component Camassa–Holm system which arises in shallow water theory. The present work is mainly concerned with persistence properties and unique continuation to this new kind of system, in view of the classical Camassa–Holm equation. Firstly, it is shown that there are three results about these properties of the strong solutions. Then we also investigate the infinite propagation speed in the sense that the corresponding solution does not have compact spatial support for t > 0 though the initial data belongs to C₀^￥(BbbR)C_{0}^{infty}(Bbb{R}).     

Microstructure of Polyelectrolyte Nanoaggregates Studied by Fluorescence Probe Method

The microstructure of water soluble nanoaggregates based on polyelectrolyte complex formed by the cationic comb-type copolymer poly(acrylamide -co-[3- (methacryloyl-amino)propyl] trimethylammonium chloride)-graft- polyacrylamide [P(AM-co-MAPTAC)-g-PAM] and the anionic linear polyelectrolyte sodium polyacrylate (NaPA) was investigated using the fluorescence probe technique. The fluorescence probe were 1-anilinonaphthalene-8-sulfonic acid (ANS), pyrene (Py) and 1,10-bis(1-pyrene) decane (PD). The fluorescence properties in polyelectrolyte complex solutions, which are sensitive to either micropolarity (ANS, Py) or microviscosity (PD), were related to the quantities obtained in different pure or mixed solvents. Micropolarities were quantified utilizing the polarity common index (Reichardt) E _T(30). ANS and Py showed a variation of the micropolarity with the charge ratio of the two polymers, with the lowest polarity reached at the complex neutralization. The PD probe, by its excimer-to-monomer fluorescence intensities ratio, enabled us to evidence the effect of the composition and the comb-type copolymer grafting density on the microviscosity of the interpolyelectrolytes aggregates. It has been found that the microviscosity increased with the density of the grafting PAM chains.