Hopf algebra of multi-decorated rooted forests, free matching Rota-Baxter algebras and Gr\"obner-Shirshov bases

Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study of algebraic structures with replicating operations. To understand their algebraic and combinatorial nature, we first use rooted forests with multiple decoration sets to construct free Hopf algebras with multiple Hochschild 1-cocycle conditions. Applying the universal property of the underlying operated algebras and the method of Gr\"obner-Shirshov bases, we then construct free objects in the category of matching Rota-Baxter algebras which is a generalization of Rota-Baxter algebras to allow multiple Rota-Baxter operators. Finally the free matching Rota-Baxter algebras are equipped with a cocycle Hopf algebra structure.Comment: 27 page

Effects of ultrasound on electrochemical oxidation mechanisms of p-substituted phenols at BDD and PbO2 anodes

The effects of low-frequency (40 kHz) ultrasound are investigated with regard to the effectiveness and mechanisms of electrochemical oxidation of p-substituted phenols (p-nitrophenol, p-hydroxybenzaldehyde, phenol, p-cresol, and p-methoxyphenol) at BDD (boron-doped diamond) and PbO2 anodes. Although ultrasound improved the disappearance rates of p-substituted phenols at both the BDD and PbO2 anodes, the degree of enhancement varied according to the type of p-substituted phenol and type of anode under consideration. At the BDD anode, the % Increase values were in the range 73-83% for p-substituted phenol disappearance and in the range 60-70% for COD removal. However, at the PbO2 anode, the corresponding %Increase values were in the range 50-70% for disappearance of p-substituted phenols and only 5-25% for COD removal, much lower values than obtained at the BDD anode. Further investigations on the influence of ultrasound on the electrochemical oxidation mechanisms at BDD and PbO2 anodes revealed that the different increase extent were due to the specialized electrochemical oxidation mechanisms at these two anodes. The hydroxyl radicals were mainly free at the BDD electrodes with a larger reaction zone, but adsorbed at the PbO2 electrodes with a smaller reaction zone. Therefore, the enhancement due to ultrasound was greater at the BDD anode than at the PbO2 anode. (C) 2010 Elsevier Ltd. All rights reserved.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000280422800007&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=8e1609b174ce4e31116a60747a720701ElectrochemistrySCI(E)EI21ARTICLE205569-55755