A partition relation for partially ordered sets

We introduce a partition relation which is an alternate for measuring how badly the ordinary partition relation fails, we develop its corresponding partition calculus and we determine its status for various typical partially ordered sets.     

Synthesis,characterization, and photoresponsive behavior of new azobenzene‐containing polyethers

Two new polyethers, bearing azobenzene moiety in the side chain, were synthesized in excellent yields by means of anionic polymerization of 4‐glycidyloxyazobenzene and 4‐cyano‐4′‐glycidyloxyazobenzene (leading to azo‐P1 and azo‐P2 polymers, respectively) with the system polyiminophosphazene base t‐Bu‐P₄/3,5‐di‐tert‐butylphenol as initiator. The polymers were characterized with respect to their molecular weights, structure, and calorimetric features. The polyether bearing cyanoazobenzene group in the side chain was found to exhibit nematic phase up to 200 °C. E–Z isomerization of both polymers in tetrahydrofuran solution, after irradiating with UV light at 364 nm for 15 min, was investigated by means of UV–visible absorption spectroscopy. In the case of glycidylic monomers as well as the resulting polymers, E–Z isomerization was also investigated by means of ¹H NMR, by direct irradiation in the NMR probe in deuterated 1,1,2,2‐tetrachloroethane solution. By this technique, in the case of 4‐cyano‐4′‐glycidyloxyazobenzene, it was found that irradiation led to a photostationary state corresponding to an amount of Z isomer equal to 25%. For azo‐P1 polymer, Z–E or “reverse” isomerization was investigated at 60, 70, or 80 °C directly in the NMR probe; as expected, the process followed a first‐order rate law. The kinetic constants as well as the activation energy for the process were determined in this temperature range. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 5426–5436, 2009     

A Population-Based Approach to the Resource-Constrained Project Scheduling Problem

We present a population-based approach to the RCPSP. The procedure has two phases. The first phase handles the initial construction of a population of schedules and these are then evolved until high quality solutions are obtained. The evolution of the population is driven by the alternative application of an efficient improving procedure for locally improving the use of resources, and a mechanism for combining schedules that blends scatter search and path relinking characteristics. The objective of the second phase is to explore in depth those vicinities near the high quality schedules. Computational experiments on the standard j120 set, generated using ProGen, show that our algorithm produces higher quality solutions than state-of-the-art heuristics for the RCPSP in an average time of less than five seconds.     

Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations

Summary. A nonlinear Galerkin method using mixed finite elements is presented for the two-dimensional incompressible Navier-Stokes equations. The scheme is based on two finite element spaces and for the approximation of the velocity, defined respectively on one coarse grid with grid size and one fine grid with grid size and one finite element space for the approximation of the pressure. Nonlinearity and time dependence are both treated on the coarse space. We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin solution is of the order of $H^2$, both in velocity ( and pressure norm). We also discuss a penalized version of our algorithm which enjoys similar properties. Received October 5, 1993 / Revised version received November 29, 1993     

Stability of nonautonomous differential equations in Hilbert spaces

We introduce a large class of nonautonomous linear differential equations v^′=A(t)v in Hilbert spaces, for which the asymptotic stability of the zero solution, with all Lyapunov exponents of the linear equation negative, persists in v^′=A(t)v+f(t,v) under sufficiently small perturbations f. This class of equations, which we call Lyapunov regular, is introduced here inspired in the classical regularity theory of Lyapunov developed for finite-dimensional spaces, that is nowadays apparently overlooked in the theory of differential equations. Our study is based on a detailed analysis of the Lyapunov exponents. Essentially, the equation v^′=A(t)v is Lyapunov regular if for every k the limit of Γ(t)^1/t as t→∞ exists, where Γ(t) is any k-volume defined by solutions v₁(t),…,v_k(t). We note that the class of Lyapunov regular linear equations is much larger than the class of uniformly asymptotically stable equations.     

Cyclometalated Iridium(III) Complexes Containing Semicarbazone Ligands: Synthesis,Characterization, Photophysical and Biological Studies

The synthesis, crystal structure, photophysical properties, and biological activity of the novel bis‐cyclometalated complexes [Ir(ptpy)₂(vnsc)] ( 2 ) and [Ir(ptpy)₂(acsc)] ( 3 ) [ptpy = 2‐(p‐tolyl)pyridinato, vnsc = vanillin semicarbazone, acsc = acetone semicarbazone] are described. The new compounds were prepared by the reaction of [{Ir(μ‐Cl)(ptpy)₂}₂] ( 1 ) with the corresponding semicarbazone ligands under basic conditions. The molecular structure of compound 3 was confirmed by a single‐crystal X‐ray diffraction study. The complex crystallized from chloroform as a mono‐ solvate in the orthorhombic space group Pcab with eight molecules in the unit cell.