Sign-Changing Solutions for Discrete Dirichlet Boundary Value Problem

Using invariant sets of descending flow and variational methods, we establish some sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions for second-order nonlinear difference equations with Dirichlet boundary value problem. Some results in the literature are improved.

A Facile Synthesis of 9,10-Dimethoxybenzo[6,7]- ox-epino[3,4-<i>b</i>]quinolin-13(6<i>H</i>)-one and Its Derivatives

A concise and efficient method for the synthesis of novel 9,10-imethoxybenzo[6,7]oxepino[3,4-b]quinolin13(6H)-one and its derivatives 7a-p has been developed via the intramolecular Friedel-Crafts acylation reactions of 6,7-dimethoxy-2-(phenoxymethyl)quinoline-3-carboxylic acids 6a-p with polyphosphoric acid (PPA) as catalyst and solvent under mild conditions. The key intermediates 6a-p were prepared through the in situ formation of ethyl 6,7-dimethoxy-2-(phenoxymethyl)quinoline-3-carboxylates 5a-p followed by hydrolysis with aqueous ethanolic sodium hydroxide solution. The novel synthetic method has the advantages of good yields, easy work-up, and environmentally friendly character, which may provide a novel highly efficient process for making quinoline and related azaheterocycle libraries.

Global asymptotic stability in a pseudorabies virus model with age structure

Pseudorabies is a highly contagious disease caused by pseudorabies virus(PRV)or suid herpesvirus 1(SuHV1),causing significant economic losses to the swine industry in countries where the disease exists.In this paper,we formulate an age structure model of pseudorabies virus that takes into account disease-related mortality and vertical trans-mission.We find a threshold to determine the stability and existence of the disease.We show that there is always a globally asymptotically stable boundary equilibrium if and only if R_(02)<1+q,which means that the disease always exists in piglets and will die out in adult pigs.When R_(02)>1+q,the boundary equilibrium is unstable and there exists a unique disease-endemic equilibrium,which is globally asymptotically stable.We give detailed proofs of our theoretical results and numerical examples.Brief concluding re-marks are also provided.

Platinum-Catalyzed Asymmetric Ring-opening Reaction of Oxabenzonorbornadiene with Terminal Alkynes

A novel platinum-catalyzed asymmetric ring-opening reaction of oxabenzonorbornadiene with terminal alkynes is described.The reaction affords optically active cis-2-alkynyl-1,2-dihydronaphthalen-1-ols in moderate yields with good enantioselectivity in the presence of catalytic amounts of Pt(COD)Cl_(2)/(S)-BINAP and an excess of zinc powder.The products were obtained exclusively with the relative cis-configuration of the ring substituents and the prevalent(1R,2S)-configuration of the stereocenters,as determined by single crystal X-ray diffraction analysis.