MIXED NONLINEAR OSCILLATION OF SECOND ORDER FORCED DYNAMIC EQUATIONS

By using a technique similar to the one introduced by Kong [J Math Anal Appl 229 (1999) 258-270] and employing an arithmetic-geometric mean inequality, we establish oscillation criteria for second-order forced dynamic equations on time scales containing mixed nonlinearities of the for

Enzymatic ring-opening (co)polymerization of lactide stereoisomers catalyzed by lipases. Toward the in situ synthesis of organic/inorganic nanohybrids

© 2015 Elsevier B.V.Lipase-based catalysts were tested for the ring-opening polymerization of d-, l- and d,l-lactide isomers, highlighting the different specificity of the enzyme toward these isomers. Free form of Candida antarctica lipase B (CALB) and its clay- and acrylic resin- immobilized forms were compared. For l- and d,l-lactide monomers only short oligomers were obtained. The acrylic resin immobilized form of CALB (NOVO-435) led to a complete conversion of d-lactide to PDLA with a Mn of 2600 g/mol, whereas the clay-immobilized and free forms of CALB exhibited slower kinetics and produced chains of lower Mn. Copolymerization reactions between ε-caprolactone and lactide isomers were performed using NOVO-435 as bio-catalyst. Random copolyesters were successfully synthesized by copolymerizing d-lactide with ε-caprolactone. Better results were obtained with a two-step reaction, starting from presynthesized polycaprolactone chains, compared with the one-pot copolymerization. Conducting this two-step copolymerization in the presence of organo-modified montmorillonite allowed the successful synthesis of copolymer/clay nanohybrids

Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions

In this article we discuss some of the qualitative properties of fractional difference operators. We especially focus on the connections between the fractional difference operator and the monotonicity and convexity of functions. In the integer-order setting, these connections are elementary and well known. However, in the fractional-order setting the connections are very complicated and muddled. We survey some of the known results and suggest avenues for future research. In addition, we discuss the asymptotic behavior of solutions to fractional difference equations and how the nonlocal structure of the fractional difference can be used to deduce these asymptotic properties