Small time convergence of subordinators with regularly or slowly varying canonical measure

We consider subordinators $X_{\alpha }={\left(X_{\alpha }\left(t\right)\right)}_{t\ge 0}$ in the domain of attraction at 0 of a stable subordinator ${\left(S_{\alpha }\left(t\right)\right)}_{t\ge 0}$ (where $\alpha \in (0,1)$); thus, with the property that ${\overline{\Pi }}_{\alpha }$, the tail function of the canonical measure of $X_{\alpha }$, is regularly varying of index $?\alpha \in (?1,0)$ as $x\downarrow 0$. We also analyse the boundary case, $\alpha =0$, when ${\overline{\Pi }}_{\alpha }$ is slowly varying at 0. When $\alpha \in (0,1)$, we show that ${\left(t{\overline{\Pi }}_{\alpha }\left(X_{\alpha }\left(t\right)\right)\right)}^{?1}$ converges in distribution, as $t\downarrow 0$, to the random variable ${\left(S_{\alpha }\left(1\right)\right)}^{\alpha }$. This latter random variable, as a function of $\alpha$, converges in distribution as $\alpha \downarrow 0$ to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in $\mathbb{D}[0,1]$), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from the process. The $\alpha =0$ case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe.     

Coordination chemistry of [Cp*Fe(η5-P3C2tBu2)] (Cp* = η5-C5Me5) with copper(I) halides – Formation of oligomeric and polymeric compounds

The reaction of the 1,2,4-triphosphaferrocene [Cp*Fe(η⁵-P₃C₂tBu₂)] (1) with CuX (X = Cl, Br, I) in a 1:1 stoichiometric ratio leads to the formation of the oligomeric compounds [{Cu(μ-X)}₆(μ₆-X)Cu(MeCN)₃{μ,η²-(Cp*Fe(η⁵-P₃C₂tBu₂))}₂{μ₃,η³-(Cp*Fe(η⁵-P₃C₂tBu₂))}] (X = Cl (2), Br (3)) and [{Cu(μ-I)}₃{Cu(μ₃-I)}₃Cu(μ₆-I){μ,η²-(Cp*Fe(η⁵-P₃C₂tBu₂))}₃{η¹-(Cp*Fe(η⁵-P₃C₂tBu₂))}] (4) revealing Cu(I) halide cages surrounded by 1,2,4-triphosphaferrocene moieties. The reaction of [Cp*Fe(η⁵-P₃C₂tBu₂)] with CuI in a 1:4 stoichiometry leads to the formation of the two-dimensional polymer [{Cu(μ-I)}₄{Cu(μ₃-I)(MeCN)}₂{μ₃,η³-(Cp*Fe(P₃C₂tBu₂))}]_n (5). The oligomeric compounds show dynamic behavior in solution monitored by ³¹P NMR spectroscopy. All compounds are additionally characterized by single crystal X-ray diffraction.     

Zur Thermodynamik der Metallcarbonate. 4. Mitteilung. Potentiometrische Untersuchungen am System Mn2+?CO2?H2O

The solubility and the complex equilibria in the system Mn²⁺-CO₂-H₂O have been investigated at 25°C in solutions of the constant ionic strength 3 M (Na)ClO₄. From experimental data the following values for equilibrium constants and Gibbs free energies of formation are deduced: A predominance area diagram for the system Mn²⁺-H₂-CO_2(g)-H₂S_(g) including MnCO_3(ppt.), α-MnS and Mn²⁺ is given.     

Na-Li-[V3O8] insertion electrodes: Structures and diffusion pathways

The potential insertion-electrode compounds Na_1.2[V₃O₈] (NaV) and Na_0.7Li_0.7[V₃O₈] (NaLiV) were synthesized from mixtures of Na₂CO₃, Li₂CO₃ and V₂O₅, which were melted at 750° and subsequently cooled to room temperature. The structures of NaV and LiV contain sheets of polymerized (VO_n) polyhedra, which are topologically identical to the sheet of polymerized polyhedra in Li_1.2[V₃O₈] (LiV). Vanadium occurs in three different coordination environments: [2+3] V(1), [2+2+2] V(2) and [1+4+1] V(3). Calculated bond-valence sums indicate that V⁴⁺ occurs preferentially at the V(3) site, which agrees with the general observation that [6]-coordinated V⁴⁺ prefers [1+4+1]-rather than [2+2+2]-coordination. The M-cations Na and Li occur at three distinct sites, M(1), M(2) and M(3) between the vanadate sheets. The M(1)-site is fully occupied and has octahedral coordination. The M(2) sites are partly occupied in NaV and NaLiV, in which they occur in [4]- and [6]-coordination, respectively. Li partly occupies the M(3) site in NaLiV, in which it occurs in [3]-coordination. The M(2) and M(3) sites in NaLiV occur closer to the vanadate sheets than the M(2) sites in NaV and LiV. The shift in these cation positions is a result of the larger distance between the vanadate sheets in NaLiV than in LiV, which forces interstitial Li to move toward one of the vanadate sheets to satisfy its coordination requirements. Bond-valence maps for the interstitial cations Na and Li are presented for NaV, NaLiV and LiV. These maps are used to determine other potential cation positions in the interlayer and to map the regions of the structure where the Na and Li have their bond-valence requirements satisfied. These regions are potential pathways for Na and Li diffusion in these structures, and are used to explain chemical diffusion properties of Na and Li in the Na-Li-[V₃O₈] compounds.