An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials

In this paper, a new adaption of homotopy analysis method is presented to handle nonlinear problems. The proposed approach is capable of reducing the size of calculations and easily overcome the difficulty arising in calculating complicated integrals. Furthermore, the homotopy polynomials that decompose the nonlinear term of the problem as a series of polynomials are introduced. Then, an algorithm of calculating such polynomials, which makes the solution procedure more straightforward and more effective, is constructed. Numerical examples are examined to highlight the significant features of the developed techniques. The algorithms described in this paper are expected to be further employed to solve nonlinear problems in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd.     

Diels–Alder reactions and transformations of 2-cyclopenten-1-one with a chiral anthracene template

Excellent regio- and diastereoselectivity were achieved in the microwave assisted Diels–Alder reaction between (S)-9-(1-methoxyethyl) anthracene and 2-cyclopenten-1-one. The addition of Grignard reagents to the ketone cycloadduct gave poor levels of diastereoselectivity, however, the reduction was significantly more stereoselective. Flash vacuum pyrolysis of the reduced material furnished the corresponding allylic alcohol in good yield and ee.     

Glass transition (Tg) and dielectric loss in Na3PO4–Pb3(PO4)2–BiPO4 phosphate glasses

Thermal and dielectric loss properties of Na₃PO₄-Pb₃(PO₄BiPO₄ (Na₂O-PbO-Bi₂O₃-P₂O₅) phosphate glasses, have been studied by the differential scanning calorimetry (DSC) and electrical factor loss (tgδ) measurements. Experiments have been carried out from ambient temperature to 500°C and show a strong influence of sodium ions on T_g and tgδ.     

Limit distributions of generalized order statistics in a stationary Gaussian sequence

In this paper we study the limit distributions of extreme, intermediate and central m-generalized order statistics, as well as m-dual generalized order statistics, of a stationary Gaussian sequence under equi-correlated set up. Moreover, the result of extremes is extended to a wide subclass of generalized order statistics, as well as dual generalized order statistics, when the parameters γ_1,n, γ_2,n,?…?, γ_n,n are assumed to be pairwise different.     

In vivo lipid diffusion coefficient measurements in rat bone marrow

The diffusion coefficient of lipids, D_l, within bone marrow, fat deposits and metabolically active intracellular lipids in vivo will depend on several factors including the precise chemical composition of the lipid distribution (chain lengths, degree of unsaturation, etc.) as well as the temperature. As such, D_l may ultimately prove of value in assessing abnormal fatty acid distributions linked to diseases such as cystic fibrosis, diabetes and coronary heart disease. A sensitive temperature dependence of D_l may also prove of value for MR-guided thermal therapies for bone tumors or disease within other fatty tissues like the breast. Measuring diffusion coefficients of high molecular weight lipids in vivo is, however, technically difficult for a number of reasons. For instance, due to the much lower diffusion coefficients compared to water, much higher b factors than those used for central nervous system applications are needed. In addition, the pulse sequence design must incorporate, as much as possible, immunity to motion, susceptibility and chemical shift effects present whenever body imaging is performed. In this work, high b-factor line scan diffusion imaging sequences were designed, implemented and tested for D_l measurement using a 4.7-T horizontal bore animal scanner. The gradient set available allowed for b factors as high as 0.03 μs/nm² (30,000 s/mm²) at echo times as short as 42 ms. The methods were used to measure lipid diffusion coefficients within the marrow of rat paws in vivo, yielding lipid diffusion coefficients approximately two orders of magnitude smaller than typical tissue water diffusion coefficients. Phantom experiments that demonstrate the sensitivity of lipid diffusion coefficients to chain length and temperature were also performed.     

On Legendre polynomial approximation with the VIM or HAM for numerical treatment of nonlinear fractional differential equations

The variational iteration method and the homotopy analysis method, as alternative methods, have been widely used to handle linear and nonlinear models. The main property of the methods is their flexibility and ability to solve nonlinear equations accurately and conveniently. This paper deals with the numerical solutions of nonlinear fractional differential equations, where the fractional derivatives are considered in Caputo sense. The main aim is to introduce efficient algorithms of variational iteration and homotopy analysis methods that can be simply used to deal with nonlinear fractional differential equations. In these algorithms, Legendre polynomials are effectively implemented to achieve better approximation for the nonhomogeneous and nonlinear terms that leads to facilitate the computational work. The proposed algorithms are capable of reducing the size of calculations, improving the accuracy and easily overcome the difficulty arising in calculating complicated integrals. Numerical examples are examined to show the efficiency of the algorithms.     

Self‐Assembled Heteroleptic Zn(II) Dithiocarbamate‐Based 2D‐Interwoven Supramolecular Giant Macrocycles and Their Redox Properties

This article presents a new heteroleptic Zn(II) dithiocarbamate ( 1 ) having molecules linked together via H···H and π···π interactions to form a giant hexanuclear macrocycle. These macrocycles are connected together through S···H interactions to offer a unique two‐dimensional cloth‐like interwoven architecture in the solid state. The redox behavior of the complex deposited on the surface of a glassy carbon electrode was investigated in different pH media. The analyte was found to oxidize and reduce in a pH‐dependent irreversible manner. On the basis of the electrochemical findings, redox mechanistic pathways were proposed.