An approximation algorithm for multidimensional assignment problems minimizing the sum of squared errors

Given a complete k-partite graph G=(V₁,V₂,…,V_k;E) satisfying |V₁|=|V₂|=?=|V_k|=n and weights of all  k-cliques of G, the  k-dimensional assignment problem finds a partition of vertices of G into a set of (pairwise disjoint) n k-cliques that minimizes the sum total of weights of the chosen cliques. In this paper, we consider a case in which the weight of a clique is defined by the sum of given weights of edges induced by the clique. Additionally, we assume that vertices of G are embedded in the d-dimensional space Q^d and a weight of an edge is defined by the square of the Euclidean distance between its two endpoints. We describe that these problem instances arise from a multidimensional Gaussian model of a data-association problem.We propose a second-order cone programming relaxation of the problem and a polynomial time randomized rounding procedure. We show that the expected objective value obtained by our algorithm is bounded by (5/2−3/k) times the optimal value. Our result improves the previously known bound (4−6/k) of the approximation ratio.     

Preparation and photoluminescence characteristics of polysiloxane pendant tris(2,2′-bipyridine)ruthenium (II) complex

Polysiloxanes containing pendant tris(2,2′-bipyridine)ruthenium(II) complex (Ru(bpy)3²⁺) were prepared by reaction of polysiloxane-pendant 2,2′-bipyridine (PSiO-bpy) with cis-Ru(bpy)₂Cl₂. In methanol solution, the polymer pendant Ru(bpy)3²⁺ showed absorption maximum at 456nm and emission maximum at around 609nm, both of which are shifted to longer wavelength than the monomeric Ru(bpy)₃²⁺. The lifetime τ₀ of the excited polymer complex with low Ru(bpy)3²⁺ content was almost the same as that of the monomeric one in methanol (830ns), but τ₀ of the polymer with higher complex content was shorter because of a concentration quenching. In a solid state, τ₀ was much shorter (306–503ns) than that in a methanol solution contrary to the conventional polymeric system. Higher complex content in the polymer film caused higher glass transition temperature (Tg), but shorter τ₀. These results indicate concentration quenching in the polymer film. The excited polymer pendant Ru(bpy)3²⁺ was quenched by oxygen, and the relative emission intensity followed the Stern-Volmer equation. In a methanol solution the quenching rate constant (k_q) was the same order of magnitude as the monomeric complex, and independent of the complex content in the polymer. In a film, k_q was higher for the polymer with higher complex content.     

Synthesis and surface characterization of novel perfluorooctyl-functionalized polymers with well-defined architectures

We have successfully synthesized two kinds of novel well-defined chain-end-functionalized polystyrenes with one, two, three, and four perfluorooctyl (C₈F₁₇) groups and structurally analogous block copolymers, poly[styrene-b-4-(3-perfluoroocyl)propoxystyrene], by means of living anionic polymerization, followed by the Williamson reaction with C₈F₁₇(CH₂)₃Br. The surface properties and compositions of the films prepared from both C₈F₁₇-functionalized polymers have been characterized by contact angles using water and dodecane droplets and angle-dependent XPS measurement. It was observed that C₈F₁₇ groups were segregated and preferentially enriched at their topmost surfaces and the extent of the enrichment increased with the number of C₈F₁₇ group. The dependence of extent of C₈F₁₇ group enrichment on fluorine content however appears to be different between chain-end-functionalized polymer and block copolymer.