Polymerization of D-α-methylbenzyl methacrylate by n-butyllithium and the tacticity and optical rotation of the polymer

D -α-Methylbenzyl methacrylate, [α] = +51.3° (neat), was polymerized by n-butyllithium in toluene–tetrahydrofuran mixtures of various solvent ratios at ?78°C. The polymers obtained were converted into poly(methyl methacrylate)s, which were analyzed for tacticity by high resolution NMR spectroscopy. A linear relationship was obtained between the optical rotation and the isotacticity of poly(D -α-methylbenzyl methacrylate). The extrapolation of the data gave +120° and +99° for [α] of the fully isotactic and syndiotactic polymers, respectively. The copolymerization of the D - and L -isomers in toluene gave copolymers which were less isotactic than the homopolymer of the D -isomer. The optical rotation of the copolymer was proportional to the excess of one isomer in the polymer.     

Chromatic numbers of quadrangulations on closed surfaces

It has been shown that every quadrangulation on any nonspherical orientable closed surface with a sufficiently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface N_k has chromatic number at least 4 if G has a cycle of odd length which cuts open N_k into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface N_k admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 100–114, 2001     

Estimation of the laser beam enabling rectangular processing by the inverse diffraction theory

A small rectangular laser spot has been widely studied for laser processing and laser repair technology. It is usually made by converging the input beam with the lens. Faithful reproduction of the laser spot is dependent on NA of the imaging lens. The small rectangular spot can be obtained by high NA lens, which is limited by many factors, such as high energy loss due to the reflection on the surface, large mass and volume, and strong sensitivity to aberrations and misalignment. On the other hand, the beam cannot be faithfully reproduced because of the diffraction with the low NA lens which has no such the limitations. One of the alternative ways to produce small rectangular profiles by using the low NA lens system is to estimate the input beam profile leading to the output profile of the sharp rectangular shape.     

Vertex‐disjoint cycles containing specified vertices in a bipartite graph

A minimum degree condition is given for a bipartite graph to contain a 2‐factor each component of which contains a previously specified vertex. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 145–166, 2004     

SH-wave in a cylindrically anisotropic solid

Propagation of a transient SH-wave in a cylindrically anisotropic elastic solid is considered and an exact closed form solution for a special class of anisotropy is obtained. This is a case where a square root of the shear rigidity ratio is an integer. In this case we find an interesting wave pattern. However, the singularity at the wave front is of the same order as the one in the isotropic solid.     

Biodegradations of block copolymers composed of L- or D,L-lactide and six-membered cyclic carbonates prepared with organolanthanide initiators

Diblock copolymerizations of L - or D,L -lactide (LA) with trimethylene carbonate (TMC) or 2,2-dimethyltrimethylene carbonate (2,2-DTMC) with SmMe(C₅Me₅)₂-(tetrahydrofuran) as an initiator and triblock copolymerizations of L - or D,L -LA/cyclic carbonates/L - or D,L -LA with [Sm(C₅Me₅)₂]₂(PhCCCCPh) as an initiator generated the desired block copolymers. This article describes the comparison of biodegradabilities by proteinase K and a compost and mechanical properties between the resulting di- or triblock copolymers and random copolymers composed of L - or D,L -LA and cyclic carbonates. The scanning electron microscopic profiles of resulting polymers were measured to understand the morphological change during biodegradation. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 3572–3588, 2003