On weak and strong Peano theorems

We say that the Peano theorem holds for a topological vector space $E$ if, for any continuous mapping $f : {\Bbb R}\times E \to E$ and any $(t(0), x(0))$ is an element of ${\Bbb R}\times E$, the Cauchy problem $\dot x(t) = f(t,x(t))$, $x(t(0)) = x(0)$, has a solution in some neighborhood of $t(0)$. We say that the weak version of Peano theorem holds for $E$ if, for any continuous map $f : {\Bbb R}\times E \to E$, the equation $\dot x(t) = f (t, x(t))$ has a solution on some interval. We construct an example (answering a question posed by S. G. Lobanov) of a Hausdorff locally convex topological vector space E for which the weak version of Peano theorem holds and the Peano theorem fails to hold. We also construct a Hausdorff locally convex topological vector space E for which the Peano theorem holds and any barrel in E is neither compact nor sequentially compact.

Unusual voltammetry of the reduction of O-2 in [C(4)dmim][N(Tf)(2)] reveals a strong interaction of O-2(center dot-) with the [C(4)dmim](+) cation

Voltammetric studies of the reduction of oxygen in the room temperature ionic liquid [C(4)dmim][N(Tf)(2)] have revealed a significant positive shift in the back peak potential, relative to that expected for a simple electron transfer. This shift is thought to be due to the strong association of the electrogenerated superoxide anion with the solvent cation. In this work we quantitatively simulate the microdisc electrode voltammetry using a model based upon a one-electron reduction followed by a reversible chemical step, involving the formation of the [C(4)dmim](+)center dot center dot center dot O-2(center dot-) ion-pair, and in doing so we extract a set of parameters completely describing the system. We have simulated the voltammetry in the absence of a following chemical step and have shown that it is impossible to simultaneously fit both the forward and reverse peaks. To further support the parameters extracted from fitting the experimental voltammetry, we have used these parameters to independently simulate the double step chronoamperometric response and found excellent agreement. The parameters used to describe the association of the O-2(center dot-) with the [C(4)dmim](+) were k(f) = 1.4 x 10(3) s(-1) for the first-order rate constant and K-eq = 25 for the equilibrium constant.

Electroreduction of Chlorine Gas at Platinum Electrodes in Several Room Temperature Ionic Liquids: Evidence of Strong Adsorption on the Electrode Surface Revealed by Unusual Voltammetry in Which Currents Decrease with Increasing Voltage Scan Rates

Voltammetry is reported for chlorine, Cl-2, dissolved in various room temperature ionic liquids using platinum microdisk electrodes. A single reductive voltammetric wave is seen and attributed to the two-electron reduction of chlorine to chloride. Studies of the effect of voltage scan rate reveal uniquely unusual behavior in which the magnitude of the currents decrease with increasing scan rates. A model for this is proposed and shown to indicate the presence of strongly adsorbed species in the electrode reaction mechanism, most likely chlorine atoms, Cl*((ads)).

Strong erbium luminescence in the near-infrared telecommunication window

A new type of near-infrared emitting rare-earth complex has been synthesised, consisting of three bis(perfluoroalkylsulfonyl)imide ligands and one 1,10-phenanthroline molecule. The chelate rings formed by the rare-earth ion and the bidentate ligands do not contain any carbon atoms and can hence be considered as 'inorganic' chelate rings. The absence of C-H stretching vibration modes in the first coordination sphere of the rare-earth ion and the presence of a light-harvesting moiety (1,10-phenanthroline) bound to the rare-earth ion result in a complex that can be efficiently excited and exhibits intense near-infrared luminescence. (C) 2004 Elsevier B.V. All rights reserved.