First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints

In this paper we consider a mathematical program with semidefinite cone complementarity constraints (SDCMPCC). Such a problem is a matrix analogue of the mathematical program with (vector) complementarity constraints (MPCC) and includes MPCC as a special case. We first derive explicit formulas for the proximal and limiting normal cone of the graph of the normal cone to the positive semidefinite cone. Using these formulas and classical nonsmooth first order necessary optimality conditions we derive explicit expressions for the strong-, Mordukhovich- and Clarke- (S-, M- and C-)stationary conditions. Moreover we give constraint qualifications under which a local solution of SDCMPCC is a S-, M- and C-stationary point. Moreover we show that applying these results to MPCC produces new and weaker necessary optimality conditions.     

13C NMR quantification of mono and diacylglycerols obtained through the solvent‐free lipase‐catalyzed esterification of saturated fatty acids

In the present investigation, we studied the enzymatic synthesis of monoacylglycerols (MAG) and diacylglycerols (DAG) via the esterification of saturated fatty acids (stearic, palmitic and an industrial residue containing 87% palmitic acid) and glycerol in a solvent‐free system. Three immobilized lipases (Lipozyme RM IM, Lipozyme TL IM and Novozym 435) and different reaction conditions were evaluated. Under the optimal reaction conditions, esterifications catalyzed by Lipozyme RM IM resulted in a mixture of MAG and DAG at high conversion rates for all of the substrates. In addition, except for the reaction of industrial residue at atmospheric pressure, all of these products met the World Health Organization and European Union directives for acylglycerol mixtures for use in food applications. The products were quantified by ¹³C NMR, with the aid of an external reference signal which was generated from a sealed coaxial tube filled with acetonitrile‐d3. After calibrating the area of this signal using the classical external reference method, the same coaxial tube was used repeatedly to quantify the reaction products. Copyright © 2012 John Wiley & Sons, Ltd.     

An investigation of preservice teachers’ use of guess and check in solving a semi open-ended mathematics problem

Open-ended problems have been regarded as powerful tools for teaching mathematics. This study examined the problem solving of eight mathematics/science middle-school teachers. A semi-structured interview was conducted with (PTs) after completing an open-ended triangle task with four unique solutions. Of particular emphasis was how the PTs used a specific heuristic strategy. The results showed that the primary strategy PTs employed in attempting to solve the triangle problem task was guess and check; however, from the PTs’ reflections, we found there existed misapplications of guess and check as a systematic problem-solving strategy. In order to prepare prospective teachers to effectively teach, teacher educators should pay more attention to the mathematical proficiency of PTs, particularly their abilities to systematically and efficiently use guess and check while solving problems and explain their solutions and reasoning to middle-school students.     

A class of solvable coupled nonlinear oscillators with amplitude independent frequencies

Existence of amplitude independent frequencies of oscillation is an unusual property for a nonlinear oscillator. We find that a class of N coupled nonlinear Liénard type oscillators exhibit this interesting property. We show that a specific subset can be explicitly solved from which we demonstrate the existence of periodic and quasiperiodic solutions. Another set of N coupled nonlinear oscillators, possessing the amplitude independent nature of frequencies, is almost integrable in the sense that the system can be reduced to a single nonautonomous first order scalar differential equation which can be easily integrated numerically.