A System of Poisson Equations for a Nonconstant Varadhan Functional on a Finite State Space

Given a discrete-time Markov chain with finite state space and a stationary transition matrix, a system of "local" Poisson equations characterizing the (exponential) Varadhan's functional J(·) is given. The main results, which are derived for an arbitrary transition structure so that J(·) may be nonconstant, are as follows: (i) Any solution to the local Poisson equations immediately renders Varadhan's functional, and (ii) a solution of the system always exist. The proof of this latter result is constructive and suggests a method to solve the local Poisson equations.     

Structure and conformational motion of seven-coordinate diorganotin(IV) complexes derived from salen and salan type ligands

Reaction of R₂SnCl₂ (R = Me, nBu, Ph) and the potassium salts of salenN₃H₃ (N,N′-bis(salicylidene)diethylenetriamine) and saleanN₃H₅ (N,N′-bis(o-hydroxybenzyl)diethylenetriamine) provided diorganotin(IV) complexes of the composition [Me₂Sn(salenN₃H)]·solvate (solvate = 2.5H₂O, MeOH or DMSO), [nBu₂Sn(salenN₃H)]·H₂O, [Ph₂Sn(salenN₃H)]·2EtOH and [Me₂Sn(saleanN₃H₃)]·2.5H₂O. In all compounds the tin atoms are seven-coordinate and have pentagonal-bipyramidal coordination environments, in which the organic substituents attached to the tin atoms occupy the axial positions. This occurs both in solution and the solid state; however, in solution the molecules are involved in conformational equilibria that require the presence of intermediates, in which the N → Sn bonds are dissociated. Although the [saleanN₃H₃]²⁻ ligand is more flexible and basic, a very similar complexing behavior to that of [salenN₃H]²⁻ has been found, and there is evidence that it is even a weaker ligand. Both ligands show the tendency to adopt a curved conformation within the complex, thus indicating that the dynamic process resembles the flapping of butterfly wings. However, the folding is reduced with increasing steric bulk of the organic substitutents attached to the tin atoms. The six-membered heterocyclic rings in the [R₂Sn(salenN₃H)] derivatives have envelope conformation, while those in [Me₂Sn(saleanN₃H₃)] have distorted boat-conformation. Thus, small changes in the hybridization and basicity of the nitrogen atoms cause significant differences of the stability and the dynamic behavior of the resulting molecules.     

FTIR-ATR Monitoring and SEC/RI/MALLS Characterization of ATRP Synthesized Hyperbranched Polyacrylates

This work reports the synthesis at 1 L scale of hyperbranched polyacrylates based upon acrylate/diacrylate monomers such as n-butyl acrylate (nBA)/1,6-hexanediol diacrylate (HDDA) and using atom transfer radical polymerization (ATRP). A FTIR-ATR immersion probe was used to monitor the polymerization reaction. The dynamics of the build-up of polymer structure was studied by off-line analysis of samples at different reaction times by size exclusion chromatography (SEC) with detection of refractive index (RI) and multi-angle laser light scattering (MALLS) signals, leading to molecular weight distribution and z-average radius of gyration. Kinetic measurements and observed parameters of the molecular architecture are compared with theoretical predictions which can be used to design new synthesis strategies to improve the homogeneity of hyperbranched polymers. Another goal of this study was elucidating the impact on polymerization of secondary reactions such as intramolecular cyclizations. For comparison purposes, FRP (conventional radical polymerization) of the same monomers is also considered.     

Nonlinear diffusion with a bounded stationary level surface

We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class C²$C^2$. Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a proper C²$C^2$-subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the whole RN${\mathbb{R}}^N$ of some substance whose density is initially a characteristic function of the complement of a domain with bounded C²$C^2$ boundary, and obtain similar results. These results are also extended to the heat flow in the sphere SN${\mathbb{S}}^N$ and the hyperbolic space HN${\mathbb{H}}^N$.     

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Kinetic measurements of diethylfumarate (DEF) electroreduction to diethylsuccinate in aqueous solutions of pH ranging from 1 to 10 were carried out by the polarographic technique under different experimental conditions, and DEF adsorption was measured by the chronocoulometric technique. The rate changes produced by changes in pH, in buffer concentration, in the potential _d at the outer Helmholtz plane or in the charge density σ_i at the inner Helmholtz plane due to specifically adsorbed iodide ions, were thoroughly examined. The effects of these changes are amenable to a straightforward interpretation if the overall electrode process is assumed to be controlled by an enol-keto transformation of the enol form of diethylsuccinate for 1$R^{?}\stackrel{+H^{+}}{\to }RH$

Theoretical study of species formed by IA and IB metals or ions and a CN ligand

Multireference second-order perturbation calculations have been carried out for copper and sodium complexes involving one CN ligand. Neutral compounds are strongly linked, cyanides M-CN being more stable than isocyanides M-NC (M = Cu, Na) as in the case of the pair HCN/HNC. Cations are much less strongly linked, Cu-NC⁻ being predicted to be the only system having a fairly bound structure, at variance with the dominance of the cyanide form in the HCN⁺/HNC⁺ pair.     

Evolving Systems of Stochastic Differential Equations

Journal of Theoretical Probability - We introduce Evolving Systems of Stochastic Differential Equations. This model generalizes the well-known stochastic differential equations with Markovian...     

Finite-state approximations for denumerable state discounted markov decision processes

A finite-state iterative scheme introduced by White [9] to approximate the optimal value function of denumerable-state Markov decision processes with bounded rewards, is extended to the case of unbounded rewards. Convergence theorems that, when applied to the case of bounded rewards, give stronger results than those in [9] are proved. Moreover, bounds on the rates of convergence under several assumptions are given and the extended scheme is used to obtain policies with asymptotic optimality properties.This research was supported in part by the Consejo Nacional de Ciencia y Tecnología under Grant PCCBBNA 020630, and in part by the Universidad Autónoma Agraria Antonio Narro.     

Solutions of elliptic equations with a level surface parallel to the boundary: Stability of the radial configuration

A positive solution of a homogeneous Dirichlet boundary value problem or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of its level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. We show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls \({B_{{r_e}}}\) and \({B_{{r_i}}}\), with the difference r_e-r_i (linearly) controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and elaborates arguments developed by Aftalion, Busca, and Reichel.