Gas phase oxidation of alkoxo ligands in bis(peroxo)[MO(O2)2(OR)]- and trisoxo [MO3(OR)]- anions (M = Cr, Mo, W)

The anions [M(VI)O(O(2))(2)(OR)](-) and [M(VI)O(3)(OR)](-)(M = Cr, Mo, W; R = H, Me, Et, (n)Pr, (i)Pr) were transferred to the gas phase by the electrospray process. Their decomposition was examined by multistage mass spectrometry and collisional activation experiments. The molybdate and tungstate anions [M(VI)O(O(2))(2)(OR)](-) underwent parallel elimination of aldehyde (ketone) and dioxygen while the equivalent chromate underwent loss of dioxygen only. The peroxo ligands were the source of oxidising equivalents in both reactions. For each alkoxo ligand, the total yield of aldehyde for the tungstate system exceeded that for the molybdate system. Collisional activation of [M(VI)O(3)(OMe)](-) led to clean elimination of formaldehyde with the metal centre supplying the oxidising equivalents. For larger alkoxo ligands, only the chromate centre eliminated aldehyde, while the molybdate and tungstate centres underwent clean loss of alkene. Threshold activation voltages indicated that the peroxo ligands of [W(VI)O(O(2))(2)(OMe)](-) are more oxidising than the tungstate centre of [W(VI)O(3)(OMe)](-). (2)H and (18)O isotope tracing experiments were consistent with a formal hydride transfer mechanism operating for oxidation of alkoxo ligand in each system. In the solid state, anions [M(VI)O(O(2))(2)(OR)](-) are typically pentagonal pyramidal (oxo in apical site) while [M(VI)O(3)(OR)](-) are tetrahedral. The data indicate that an equatorial ligand position is the site of alkoxo oxidation in [M(VI)O(O(2))(2)(OR)](-) anions. Comparisons of the gas phase data with those for a solution phase system are made.     

Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space

Let (W,,H) be an abstract Wiener space assume two _i ,i=1,2 probabilities on (W,(W)). We give some conditions for the Wasserstein distance between ₁ and ₂ with respect to the Cameron-Martin space to be finite, where the infimum is taken on the set of probability measures on W×W whose first and second marginals are ₁ and ₂. In this case we prove the existence of a unique (cyclically monotone) map T=I _W +, with :WH, such that T maps ₁ to ₂. Moreover, if ₂, then T is stochastically invertible, i.e., there exists S:WW such that ST=I _{W 1} a.s. and TS=I _{W 2} a.s. If, in addition, ₁=, then there exists a 1-convex function in the Gaussian Sobolev space such that =. These results imply that the quasi-invariant transformations of the Wiener space with finite Wasserstein distance from can be written as the composition of a transport map T and a rotation, i.e., a measure preserving map. We give also 1-convex sub-solutions and Ito-type solutions of the Monge-Ampère equation on W. Mathematics Subject Classification (2000)60H07, 60H05,60H25, 60G15, 60G30, 60G35, 46G12, 47H05, 47H1, 35J60, 35B65,35A30, 46N10, 49Q20, 58E12, 26A16, 28C20cf. Theorem 6.1 for the precise hypothesis about ₁ and ₂.In fact this hypothesis is too strong, cf. Theorem 6.1. AcknowledgementThe authors are grateful to Françoise Combelles for all the bibliographical help that she has supplied for the realization of this research. We thank also the anonymous referee for his particular attention and valuable remarks.     

Gas-phase oxidation of isomeric butenes and small alkanes by vanadium-oxide and -hydroxide cluster cations

Bare vanadium-oxide and -hydroxide cluster cations (V(m)O(n)H(o)+, m = 2-4, n = 1-10, o = 0, 1) were generated by electrospray ionization in order to examine their intrinsic reactivity toward isomeric butenes and small alkanes using mass spectrometric techniques. Two of the major reactions described here concern the activation of C-H bonds of the alkene/alkane substrates resulting in the transfer of two hydrogen atoms and/or attachment of the dehydrogenated hydrocarbon to the cluster cations; these processes are classified as oxidative dehydrogenation (ODH) and dehydrogenation, respectively. For the dehydrogenation of butene, it evolved as a general trend that high-valent clusters prefer ODH resulting in the addition of two hydrogen atoms to the cluster concomitant with elimination of neutral butadiene, whereas low-valent clusters tend to add the diene with parallel loss of molecular hydrogen. Deuterium labeling experiments suggest the operation of a different reaction mechanism for V2O2(+) and V4O10(+) compared to the other cluster cations investigated, and these two cluster cations also are the only ones of the vanadium-oxide ions examined here that are able to dehydrogenate small alkanes. The kinetic isotope effects observed experimentally imply an electron transfer mechanism for the ion-molecule reactions of the alkanes with V4O10(+).     

On Fractional Brownian Processes

We use Liouville spaces in order to prove the existence of some different fractional -Brownian motion ( 0 < 1 ), or fractional ( , )-Brownian sheets. There are also applications to the Wiener stochastic integral with respect to these -Brownian.     

Opérateurs linéaires gaussiens

We extend operators from the Cameron-Martin space to Gaussian Lusinian locally convex space. We then are allowed to give sense to the Mehler formula for every such bounded operator. An application is made to Hilbert-Schmidt operators. Next we show that capacities asociated to second quantizations of operators are tight on compact sets, and this is a general result even if the underlying space is not a Banach space.

Le deuxième auteur a exposé ce travail au colloque d'Amersfoort, août 1991.