Positive solutions and dynamics of a finite difference reaction–diffusion system

We investigate the dynamics and methods of computation for some nonlinear finite difference systems that are the discretized equations of a time-dependent and a steady-state reaction–diffusion problem. The formulation of the discrete equations for the time-dependent problem is based on the implicit method for parabolic equations, and the computational algorithm is based on the method of monotone iterations using upper and lower solutions as the initial iterations. The monotone iterative method yields improved upper and lower bounds of the solution in each iteration, and the sequence of iterations converges monotonically to a solution for both the time-dependent and the steady-state problems. An important consequence of this method is that it leads to a bifurcation point that determines the dynamic behavior of the time-dependent problem in relation to the corresponding steady-state problem. This bifurcation point also determines whether the steady-state problem has one or two non-negative solutions, and is explicitly given in terms of the physical parameters of the system and the type of boundary conditions. Numerical results are presented for both the time-dependent and the steady-state problems under various boundary conditions, including a test problem with known analytical solution. These numerical results exhibit the predicted dynamic behavior of the time-dependent solution given by the theoretical analysis. Also discussed are the numerical stability of the computational algorithm and the convergence of the finite difference solution to the corresponding continuous solution of the reaction–diffusion problem. © 1993 John Wiley & Sons, Inc.     

Sulfonated poly(aryl ether ether ketone ketone)s containing fluorinated moieties as proton exchange membrane materials

A series of sulfonated poly(aryl ether ether ketone ketone)s statistical copolymers with high molecular weights were synthesized via an aromatic nucleophilic substitution polymerization. The sulfonation content (SC), defined as the number of sulfonic acid groups contained in an average repeat unit, could be controlled by the feed ratios of monomers. Flexible and strong membranes in sodium sulfonate form could be prepared by the solution casting method, and readily transformed to their proton forms by treating them in 2 N sulfuric acid. The polymers showed high T_gs, which increased with an increase in SC. Membranes prepared from the present sulfonated poly(ether ether ketone ketone) copolymers containing the hexafluoroisopropylidene moiety (SPEEKK‐6F) and copolymers containing the pendant 3,5‐ditrifluoromethylphenyl moiety (SPEEKK‐6FP) had lower water uptakes and lower swelling ratios in comparison with previously prepared copolymers containing 6F units. All of the polymers possessed proton conductivities higher than 1 × 10^?2 S/cm at room temperature, and proton conductivity values of several polymers were comparable to that of Nafion at high relative humidity. Their thermal stability, oxidative stability, and mechanical properties were also evaluated. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2299–2310, 2006     

Cytotoxic Constituents from the Stem Bark of Zanthoxylum Pistaciiflorum

Two new esters, methyl 4‐(prenyloxy)dihydrocinnamate and methyl 4‐(geranyloxy)dihydrocinnamate, together with fourteen known compounds have been isolated from the stem bark of Zanthoxylum pistaciiflorum. The structures of two new compounds were determined through spectral analyses. Among the isolates, four compounds exhibited effective cytotoxicities against P‐388 and HT‐29 cell lines in vitro.     

Synthesis and Mesogenic Properties of Liquid Crystals Containing Terminal Cyclohexylmethanol,Cyclopentylmethanol, Cyclobutylmethanol and Cyclopropylmethanol Moiety

Four series of compounds 11?50 containing terminal alicyclic rings such as cyclohexylmethyl, cyclopentylmethyl, cyclobutylmethyl, and cyclopropylmethyl rings were synthesized and their liquid crystal behavior studied. The ring size and the length of flexible alkoxy chain influence the phase formation in different ways. While the smaller ring and the shorter alkoxy chain tend to favor the formation of the N phase, the larger ring and the longer alkoxy chain tend to favor the formation of the SmC phase. All the compounds except 11 and 21 exhibit SmA phases. The widest temperature range of the N, SmA, and SmC phases are found in the compounds 41 , 46 , and 20 , respectively, which are 75 °C for 41 , 115 °C for 46 , and 100 °C for 20 .     

Synthesis and Second‐Order Nonlinearities of Chiral Prolinol‐Substituted Chromophores

A new series of thiophene‐ and furan‐containing chromophores with a chiral prolinol donor and a sulfone acceptor has been synthesized. The UV‐vis absorptions, second‐order nonlinear optical properties, and X‐ray crystal structures are described.     

Radiation patterns of electric dipole antenna in the presence of a biisotropic sphere

Analytic expressions of the dyadic Green's function of an arbitrary electric current source embedded in a general biisotropic sphere are presented at first, and the radiation characteristics of an electric dipole antenna placed on the surface of a biisotropic sphere lens are examined. The effects of the electric-and the magnetic-coefficients ( _e and _m ) of biisotropic medium on the radiation patterns of electric dipole antenna are investigated in detail.     

Vacuum ultraviolet 178.7 nm emission in sodium vapour under two-photon resonance excitation

We have observed a fixed wavelength emission at 178.7 nm in sodium vapour under 578.7 nm two-photon resonance excitation. The proposed non-linear wave mixing scheme is described by _{178.7 nm} = 2_L + _{465.7 nm}; where _178.7 is the 178.7 nm photon frequency, _L is the laser-photon frequency, and _465.7 is the 465.7 nm photon frequency. This 465.7 nm emission comes from another six-wave mixing process involving two hyper-electronic Raman scattering photons. The excitation spectrum of the 178.7 nm emission has a typical multiwave mixing pattern with a competing effect appearing at higher temperatures under two-photon resonance excitation. Numerical analysis indicates that this vacuum ultraviolet emission has a poor phase-match condition that will depress the emission intensity to a certain extent. This makes the observation more difficult compared with other reported four-wave mixing generated emissions. Fortunately, on the one hand, it is enhanced by quasi-auto-ionization resonance when the 3s–5s transition is coupled to the sodium continuum by a 330.2 nm photon. On the other hand, its wavelength sits so close to the sodium Cooper minimum that weak absorption will not suppress this vacuum ultraviolet emission further.