An iterative starting method to control parasitism for the Leapfrog method

The Leapfrog method is a time-symmetric multistep method, widely used to solve the Euler equations and other Hamiltonian systems, due to its low cost and geometric properties. A drawback with Leapfrog is that it suffers from parasitism. This paper describes an iterative starting method, which may be used to reduce to machine precision the size of the parasitic components in the numerical solution at the start of the computation. The severity of parasitic growth is also a function of the differential equation, the main method and the time-step. When the tendency to parasitic growth is relatively mild, computational results indicate that using this iterative starting method may significantly increase the time-scale over which parasitic effects remain acceptably small. Using an iterative starting method, Leapfrog is applied to the cubic Schrödinger equation. The computational results show that the Hamiltonian and soliton behaviour are well-preserved over long time-scales.     

Nonlinear dynamics of imperfectly-supported pipes conveying fluid

In this paper, the nonlinear dynamics of a pipe imperfectly supported at the upstream end and free at the other and conveying fluid is investigated. The imperfect support is modelled via cubic translational and rotational springs. The equation of motion is obtained via Hamilton’s principle for an open system, and the Galerkin method is used for discretizing the resulting partial differential equation. The dynamics of a system with either strong rotational or strong translational stiffness is examined in details. Numerical results show that similarly to a cantilevered pipe, the system undergoes a supercritical Hopf bifurcation leading to period-1 limit cycle oscillations. The Hopf bifurcation may, however, occur at a much lower flow velocity compared to the perfect system. At higher flow velocities, quasi-periodic and chaotic-like motions may be observed. The amplitude of transverse displacement is generally much higher than that for a cantilevered pipe, mainly due to large-amplitude rigid-body motion. In addition, effects of the mass ratio, internal dissipation, hardening- or softening-type nonlinearity, as well as concentrated- or distributed-type nonlinearity on the dynamics of the system are examined.     

Facile Synthesis of Ag/Ag3PO4 Composites with Highly Efficient and Stable Photocatalytic Performance under Visible Light

The Ag/Ag₃PO₄ composites with various shapes (spheres, polyhedral, and microcubes) were synthesized by a facile precipitation method and a subsequent light‐reduction route at room temperature. The as‐prepared Ag/Ag₃PO₄ composites were characterized in detail by X‐ray diffraction, Fourier transform infrared spectra, X‐ray photoelectron spectroscopy, scanning electron microscopy, UV–vis diffuse reflection, and photoluminescence spectroscopy. The growth processes of different morphologies Ag/Ag₃PO₄ composites are also discussed. The decomposition test of rhodamine B (RhB) indicated that the Ag/Ag₃PO₄ composites enhanced the photocatalytic performance compared with pure Ag₃PO₄, which was attributed to the surface plasmon resonance (SPR) of Ag nanoparticles and the stability of the photocatalysts. Moreover, uniform cubes of Ag/Ag₃PO₄ showed the highest photocatalytic activity and could completely degrade RhB in 7 min, which could be primarily ascribed to the cubic structure of Ag/Ag₃PO₄ with strong visible‐light absorption and efficient separation of the photo‐generated electrons and holes. Furthermore, the possible photocatalytic mechanism is also discussed.     

Mixed convection boundary layer flow of non-Newtonian fluids along vertical wavy plates

Mixed convection boundary layer flows of non-Newtonian fluids over the wavy surfaces are studied by the coordinate transformation and the cubic spline collocation numerical method. The effects of the wavy geometry, the buoyancy parameter and the generalized Prandtl number for pseudoplastic fluids, Newtonian fluids and dilatant fluids on the skin-friction coefficient, local and mean Nusselt numbers have been graphically studied. Results show that both higher generalized Prandtl numbers and buoyancy parameters are seen to enhance the influence of wavy surfaces on the local Nusselt number, irrespective of whether the fluids are Newtonian fluids or non-Newtonian fluids. Moreover, the irregular surfaces have higher total heat flux than that of corresponding flats plate for any fluid.     

pH-Dependent Release from Monoolein Cubic Phase Containing Hydrophobically Modified Chitosan

Monoolein (MO) cubic phase incorporating hydrophobically modified chosan (Hm chitosan) was prepared to obtain a pH-dependent release. Following calorimetric study, Hm chitosan had little effect on the crystal structure of MO cubic phase under acidic condition where Hm chitosan is readily soluble. At a higher pH (e.g., pH 9.0), however, the crystal structure of MO cubic phase was disturbed, possibly due to the insolubilization of Hm chitosan at the alkali condition. Whether the dye included in the cubic phase is anionic (amaranth) or cationic (methylene blue), the release from the cubic phase was suppressed as the pH of release medium increased. The structural change of cubic phase caused by the insolubilization of Hm chitosan, or the blockage of the water channel of the cubic phase by precipitated Hm chitosan would be responsible for the suppressed released.     

On constrained Volterra cubic stochastic operators

We consider constrained Volterra cubic stochastic operators and construct several Lyapunov functions for the constrained Volterra cubic stochastic operators. We prove that such kind operators do not have periodic trajectories. Finally, we show that the set of all constrained Volterra cubic stochastic operators is a convex compact set and find the extreme points of this set.     

A categorical invariant for cubic threefolds

We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class.     

Infrared and Raman spectra of magnesium ammonium phosphate hexahydrate (struvite) and its isomorphous analogues. IX: Spectra of protiated and partially deuterated cubic magnesium caesium phosphate hexahydrate

Infrared and Raman spectra of cubic magnesium caesium phosphate hexahydrate, MgCsPO₄·6H₂O (cF100), and its partially deuterated analogues were analyzed and compared to the previously studied spectra of the hexagonal analogue, MgCsPO₄·6H₂O (hP50). The vibrational spectra of the cubic and hexagonal dimorphic analogues are similar, especially in the regions of HOH stretching and bending vibrations. In the difference IR spectrum of the slightly deuterated analogue (<5% D), one distinctive band appears at 2260 cm⁻¹ with a small shoulder at around 2170 cm⁻¹, but only one band is expected in the region of the OD stretchings of isotopically isolated HDO molecules. The small weak band could possibly result from second-order transitions (a combination of HDO bending and some libration of the same species) rather than statistical disorder of the water molecules. By comparing the IR spectra in the region of external vibrations of water molecules of the protiated compound recorded at RT (room temperature) and at LNT (liquid nitrogen temperature) and those in the series of the partially deuterated analogues, it can be stated with certainty that the bands at 924 and 817 cm⁻¹ result from librations of water molecules, rocking and wagging respectively. And the band at 429 cm⁻¹ can be safely attributed to a stretching Mg–O_w mode. In the ν₃(PO₄) and ν₄(PO₄) region in the infrared spectra, one band in each is observed, at 995 and 559 cm⁻¹, respectively. In the region of the ν₁ modes, in the Raman spectrum of the protiated compound, one very intense band was observed at 930 cm⁻¹ which is only insignificantly shifted to 929 cm⁻¹ in the spectrum of the perdeuterated compound. The band at 379 cm⁻¹ in the Raman spectrum could be assigned to the ν₂(PO₄) modes. With respect to the phosphate ion vibrations, the comparison between the two polymorphic forms of MgCsPO₄·6H₂O and their deuterated compounds shows that ν₁(PO₄) and ν₃(PO₄) appear at lower wavenumbers in the cubic phase than in the hexagonal phase. These data are in full agreement with the lower repulsion potential at the cubic lattice sites compared with that for the hexagonal lattice sites.