Exact and analytical solutions for a nonlinear sigma model

We consider wave solutions to nonlinear sigma models in n dimensions. First, we reduce the system of governing PDEs into a system of ODEs through a traveling wave assumption. Under a new transform, we then reduce this system into a single nonlinear ODE. Making use of the method of homotopy analysis, we are able to construct approximate analytical solutions to this nonlinear ODE. We apply two distinct auxiliary linear operators and show that one of these permits solutions with lower residual error than the other. This demonstrates the effectiveness of properly selecting the auxiliary linear operator when performing homotopy analysis of a nonlinear problem. From here, we then obtain residual error‐minimizing values of the convergence control parameter. We find that properly selecting the convergence control parameter makes a drastic difference in the magnitude of the residual error. Together, appropriate selection of the auxiliary linear operator and of the convergence control parameter is shown to allow approximate solutions that quickly converge to the true solution, which means that few terms are needed in the construction of such solution. This, in turn, greatly improves computational efficiency. Copyright © 2013 John Wiley & Sons, Ltd.     

On the choice of auxiliary linear operator in the optimal homotopy analysis of the Cahn-Hilliard initial value problem

Analytical solutions for the Cahn-Hilliard initial value problem are obtained through an application of the homotopy analysis method. While there exist numerical results in the literature for the Cahn-Hilliard equation, a nonlinear partial differential equation, the present results are completely analytical. In order to obtain accurate approximate analytical solutions, we consider multiple auxiliary linear operators, in order to find the best operator which permits accuracy after relatively few terms are calculated. We also select the convergence control parameter optimally, through the construction of an optimal control problem for the minimization of the accumulated L ²-norm of the residual errors. In this way, we obtain optimal homotopy analysis solutions for this complicated nonlinear initial value problem. A variety of initial conditions are selected, in order to fully demonstrate the range of solutions possible.     

A new synthesis of pyrrolidines by way of an enantioselective Mannich/diastereoselective hydroamination reaction sequence

A new two-step synthesis of highly substituted pyrrolidines has been developed. Chiral silane Lewis acid promoted enantioselective Mannich reactions of silyl ketene imines with acylhydrazones may be used to access bishomoallylic benzoic hydrazides that in turn may be cyclized to pyrrolidines by way of the thermal hydroamination reaction reported recently by Beauchemin. Importantly, excellent diastereoselectivity may be realized in the hydroamination reactions.     

Isotopic analyses by ICP-MS in clinical samples

This critical review focuses on inductively coupled plasma mass spectrometry (ICP-MS) based applications for isotope abundance ratio measurements in various clinical samples relevant to monitoring occupational or environmental exposure, human provenancing and reconstruction of migration pathways as well as metabolic research. It starts with a brief overview of recent advances in ICP-MS instrumentation, followed by selected examples that cover the fields of accurate analyte quantification using isotope dilution, tracer studies in nutrition and toxicology, and areas relying upon natural or man-made variations in isotope abundance ratios (Pb, Sr, actinides and stable heavy elements). Finally, some suggestions on future developments in the field are provided.     

Conformational analysis of 1,1,2,2-tetraaryl-disilanes

A study of the static stereochemistry of 1,1,2,2-tetraphenyldisilane (1) and 1,1,2,2-tetramesityldisilane (2) by empirical force field calculations, X-ray diffraction, and ¹H NMR reveals that the preference for the anti conformation, exhibited by the unclamped 1,1,2,2-tetraarylethanes, is sustained in the analogous disilanes, although in somewhat attenuated form. This anti preference stands in contrast to the gauche preference of 1,1,2,2-tetraalkyldisilanes. Examination of ³J_HH coupling constants for R₂HSiSiHR₂; (R = phenyl, mesityl, 2,6-dimethyl-phenyl, t-butyl, cyclohexyl) suggests the existence of a Karplus relation for H-Si-Si-H systems.     

Twistophane macrocyles with integrated 6,6'-connected-2,2'-bipyridine units: a new lead class of fluorescence sensors for metal ions

The new twistophane macrocycles 2 and 3 have been synthesised; these compounds are composed of a cyclically conjugated dehydrobenzoannulene framework that incorporates 6,6'-connected-2,2'-bipyridine moieties for the purpose of coordinating metal ions. The cyclophanes were characterised by spectroscopic techniques, and shown by molecular mechanics calculations to be helically twisted and chiral molecules that may exist in several possible ground state conformations. UV/vis spectroscopic studies revealed that 2, 3 and precursor 9 bind with different selectivities to particular members of the following small group of metal analytes: CuII, AgI, HgII, Tl1 and PdII. Significantly, 2, 3 and 9 signal the presence of CuII ions through fluorescence emission quenching output responses. Furthermore, cyclophane 3 exhibited a particularly sensitive protontriggered chromogenic fluorescence response. With respect to their unique structural features, high analyte selectivity coupled with their enhanced and characteristic fluorescence emission responses, these molecules are among the first examples representing a new lead class of chemosensory materials. Compounds 2, 3 and 9 and derivatives thereof may, therefore, be expected to find many future applications in the detection of metal-based environmental pollutants, biologically important trace elements and monitoring proton fluxes.     

Oriented crystalline monolayers and bilayers of 2 x 2 silver(I) grid architectures at the air-solution interface: their assembly and crystal structure elucidation

Oriented crystalline monolayers, approximately 14 A thick, of a 2 x 2 Ag+ grid complex, self-assembled at the air-solution interface starting from an water-insoluble ligand 3,6-bis[2-(6-phenylpyridine)]pyridazine spread on silver-ion-containing solutions, were examined by grazing-incidence X-ray diffraction and specular X-ray reflectivity using synchrotron radiation. The monolayer structure was refined, including a determination of the positions of the counter-ions, with the SHELX-97 computer program. The monolayers were transferred from the interface onto various solid supports and visualized by scanning force microscopy, and characterized by X-ray photoelectron spectroscopy in terms of molecular structure. On surface compression, the initial self-assembled monolayer undergoes a transition to a crystalline bilayer in which the two layers, almost retaining the original arrangement, are in registry. Such a phase transition is of relevance to the understanding of crystal nucleation.     

The rotating bucket of sand: Experiment and theory

The surface shape of a bucket of sand rotating about its cylindrical axis is studied experimentally and theoretically. Focusing on fast time scales on which surface shape is determined by avalanches, we identify three regimes of behavior. At intermediate and high frequencies, the surface shape is always at its critical shape determined by the Coulomb yield condition. The low frequency behavior displays an unexpected subcritical region at the center of the bucket. To understand this central region, we adapt a continuum model of surface flow developed by Bouchaud et al. and Mehta et al. The model indicates that the subcritical region is due to a nonlinear instability mechanism. (c) 1999 American Institute of Physics.     

Derivation of the order parameter of the chiral Potts model

We derive the order parameter of the chiral Potts model, using the method of Jimbo et al. The result agrees with previous conjectures.     

The Riemann Surface of the Chiral Potts Model Free Energy Function

In a recent paper we derived the free energy or partition function of the N-state chiral Potts model by using the infinite lattice inversion relation method, together with a non-obvious extra symmetry. This gave us three recursion relations for the partition function per site T _pq of the infinite lattice. Here we use these recursion relations to obtain the full Riemann surface of T _pq . In terms of the t _p ,t _q variables, it consists of an infinite number of Riemann sheets, each sheet corresponding to a point on a (2N–1)-dimensional lattice (for N>2). The function T _pq is meromorphic on this surface: we obtain the orders of all the zeros and poles. For N odd, we show that these orders are determined by the usual inversion and rotation relations (without the extra symmetry), together with a simple linearity ansatz. For N even, this method does not give the orders uniquely, but leaves only [(N+4)/4] parameters to be determined.