Dual approach to minimization on the set of pareto-optimal solutions

LetX ^* be the set of Pareto-optimal solutions of a multicriteria programming problem. We are interested in finding a vectorxX ^* which minimizes another criterion. SinceX ^* is a nonconvex set, our problem is that of minimization over a nonconvex set. By exploiting the fact that the number of criteria is often very small compared with the number of variables, we use a dual approach to obtain a practical algorithm. We report preliminary numerical results on problems with up to 100 variables and 5 criteria.The authors express their special thanks to Professor H. P. Benson and the referees for many valuable comments and suggestions and for pointing out valuable references. They also thank Dr. Y. Ishizuka for the references of Theorem 2.2.     

Copolymerization reactivity of poly(methyl methacrylate) macromer

Poly(methyl mehtacrylate)(PMMA) macromers with several vinyl groups at both chain ends were synthesized by the mechanical scission reaction of the main chain in the presence of p-divinylbenzene(p-DVB). The radical copolymerization of this macromer with styrene(St) or MMA was carried out in benzene at 60°C and the reactivity ratio of both monomers (r₂) was calculated from a kinetic scheme of copolymerization. As a result, the effect of molecular weight and concentration of macromers was not observed in both copolymerization systems. The value of r₂, however, decreased as the number of end vinyl groups in a macromer (N) increased. These results are discussed in some detail as we describe the construction of the kinetic model of copolymerization.     

Aqueous solution behavior of alkali-soluble polyethylene periodic polyelectrolyte, poly(ethylene- per-ethylene-per-acrylic acid)

A sequence-ordered, periodic copolymer of ethylene, ethylene, and acrylic acid, poly (ethylene-per-ethylene-per-acrylic acid) (PEEA), with M _w=1.44×10⁵ has been synthesized by alternating copolymerization of 1,3-butadiene and methyl acrylate, followed by hydrogenation and hydrolysis. Aqueous solution and dissociation properties of the alkali-soluble PEEA were explored by potentiometric titration and intrinsic viscosity at 25 °C. The pH values of PEEA were almost constant (pH = 6.48 ∼ 6.55) with an increasing degree of dissociation (α) from 0.3 to 0.8 at C _s=50 mN NaCl. Correspondingly, the plots of negative logarithm of apparent dissociation constant (pK _a) against α showed a reversed S-shape curve over the whole α, indicating an extensive precipitation and subsequent tran-sition from compact to coiled conformation. The intrinsic viscosity steeply increased with α above 0.4 up to 9.97 dl/g at α = 1.0. Good agreement between the observed electrostatic potential and that calculated from the rod model with a smeared charge density was observed in the region of α higher than 0.9. The dissociation and dissolution processes of PEEA with neutralization in water were described. Received: 14 April 1998 Accepted: 3 June 1998     

In situ STM study of potential-dependent height change of a tetrathiafulvalene derivative embedded in alkanethiol self-assembled monolayers on Au(111)

Electroactive tetrathiafulvalene thiol islands embedded in an n-alkanethiol SAM matrix were studied under potential control using in situ scanning tunneling microscopy. Unlike previously studied stochastic switching, the apparent height of the island could be intentionally controlled in the present system by choosing the appropriate potential and the island size. The dependence of the height change on the potential and the size is explained as structural change of the island induced by the charging effect of the electroactive moiety.     

Synthesis of end-functionalized (1 → 6)-2,5-anhydro-D-glucitols via anionic cyclopolymerization of 1,2:5,6-dianhydro-3,4-di-O-methyl-D-mannitol for preparing graft copolymers

End-functionalized (1→6)-2,5-anhydro-3,4-di-O-methyl-D-glucitols ( 3a–c ) were synthesized by the anionic cyclopolymerization of 1,2:5,6-dianhydro-3,4-di-O-methyl-D-mannitol ( 1 ), followed by treatment with a terminating agent such as 4-vinylbenzyl ( 2a ), oxetanyl ( 2b ), and methacryloyl group ( 2c ). The end-functionalization proceeded in a high efficiency at 73–98%. The radical copolymerization of styrene with 3a yielded a polymer ( 5a ) whose GPC trace exhibited a unimodal peak. 5a was polystyrene with (1→6)-2,5-anhydro-3,4-di-O-methyl-D -glucitol as pendant groups whose structure was confirmed by the ¹H NMR spectrum.