The continuity of superposition operators on some sequence spaces defined by moduli

Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = (ϕ k) let λ(Φ) = {x = (x _k ): (ϕk(|x _k |)) ∈ λ}. Given another sequence of modulus functions Ψ = (ψ_k), we characterize the continuity of the superposition operators P _f from λ(Φ) into μ (Ψ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕk (k ∈ ℕ) are unbounded and the topologies on the sequence spaces λ(Φ) and μ(Ψ) are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type. This research was supported by Estonian Science Foundation Grant 5376.     

α-Trifluoroacetyl-δ-valerolactone: synthesis, acyl-lactone rearrangement and unexpected easy decarboxylation of the rearranged product

Claisen condensation of δ-valerolactone and ethyl trifluoroacetate leads to α-trifluoroacetyl-δ-valerolactone, existing completely in CDCl₃ solution as an enol. Upon treatment of this cyclic 1,3-ketoester with aqueous HCl, an acyl-lactone rearrangement takes place. The structure of the resulting pyran was confirmed by an X-ray analysis. The ability of the rearrangement product to lose its carboxylic group under unexpected mild conditions was used for the synthesis of the corresponding 3-unsubstituted cyclic hemi-ketal.     

Effect of group II metal cations on catecholate oxidation.

The unexpected effects of Ca(2+) on the free-radical chain reactions of dopamine, norepinephrine, isoproterenol, and pyrocatechol oxidation are studied using oxygen consumption measurements, EPR-spectroscopy, UV/VIS spectrophotometry, and by potentiometric titration. It is found that the formation of Ca(2+)-catecholate complexes is accompanied by an increase in the dissociation constants (K(ai) ) of their phenolic hydroxyls. At pH>pK(ai) and in the presence of alkaline-earth metal cations, the rate of catecholate oxidation increases (Ca(2+), Mg(2+)> Sr(2+), Ba(2+)), whereas on addition of Zn ions the rate decreases. The effects of Group II metal cations on catecholate autoxidation are concomitant with a transient increase of the EPR signal for metal-semiquinonate complexes. Therefore, the effects of Ca(2+) and other alkaline-earth metal cations on catecholate autoxidation can be defined as 1) additional deprotonation of catechol OH-groups involved in the formation of M(2+)-catecholate complexes, the latter exceeding catechols in the susceptibility to dioxygen-induced oxidation and 2) formation of relatively stable free-radical intermediates responsible for chain propagation.     

Finite size effects in tightly meshed polymer networks

Molecular dynamics computer simulations on regular, tightly meshed model networks exhibit variations of the network density with system size. We show that these variations are due mainly to network elasticity. A theoretical expression derived on the basis of the self-consistent-field approach yields finite size scaling behavior in good accord with the simulation for a wide range of thermodynamic conditions.     

Stabilising small clusters: synthesis and characterisation of thermolabile [Gd4F7(15-crown-5)4][AsF6]5.6 SO2

Addition of 15-crown-5 to [GdF(AsF6)2], both dissolved in liquid SO2, and crystallisation at -30 degrees C has led to the isolation of the tetranuclear ionic complex [Gd4F7(15-crown-5)4][AsF6]5.6 SO2 which is stable up to--10 degrees C where SO2 loss leads to loss of crystallinity.     

Pentafluorethylschwefeltrifluorid: Synthese und Reaktionen

Pentafluorethyl Sulfurtrifluoride: Synthesis and Reactions By oxidation of (C₂F₅S?)₂ ( 1 ) with AgF₂ at 0°C a mixture of C₂F₅SF₃ ( 2 ) and C₂F₅SF₅ ( 3 ) besides C₂F₅S(O)F ( 4 ) is formed. With elemental fluorine only 3 is isolated, an intermediate in this reaction is (C₂F₅SF₄?)₂ ( 5 ). At ?40 to ?30°C the mixture of 2, 3 and 4 was reacted with TASF and AsF₅, to give TAS⁺ C₂F₅SF₄^? ( 6 ), TAS⁺ C₂F₅S(O)F₂^? ( 7 ) and C₂F₅SF₂⁺AsF₆^? ( 8 ), respectively. While 6 and 7 decompose rapidly in solution even at low temperatures, of thermally stable 8 the solid state structure was determined by x-ray diffraction.     

Bootstrap,wild bootstrap,and asymptotic normality

Summary We show for an i.i.d. sample that bootstrap estimates consistently the distribution of a linear statistic if and only if the normal approximation with estimated variance works. An asymptotic approach is used where everything may depend onn. The result is extended to the case of independent, but not necessarily identically distributed random variables. Furthermore it is shown that wild bootstrap works under the same conditions as bootstrap.This work has been supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 123 Stochastische Mathematische Modelle