Calcul stochastique non adapté pour des processus à deux paramètres: formules de changement de variables de type Stratonovitch et de type Skorohod

Résumé Nous établissons une formule de changement de variables de type Stratonovitch pour des processus non adaptés indéxés par [0, 1]² qui est analogue à celle existant en calcul déterministe. Pour cela nous définissons une intégrale de Stratonovitch simple (resp. double): Nous en déduisons une formule de changement de variables de type Skorohod ne contenant pas d'intégrale curviligne. Nous utilisons des régularisations du processus de Wiener obtenues par convolution avec une approximation de l'unité.

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Summary We prove a Stratonovitch-type change of variable formula for anticipative processes on [0, 1]². The formula is the same as the existing one from deterministic calculus. In order to do so we define simple and double Stratonovitch integrals. We deduce a Skorohod-type change of variable formula which does not contain any line integral. Our method consists in using regularization of the Wiener process obtained by convolution.

     

Hamiltonian Monodromy via Picard-Lefschetz Theory

 In this paper, we investigate the ``Hamiltonian' monodromy of the fibration in Liouville tori of certain integrable systems via (real) algebraic geometry. Using Picard-Lefschetz theory in a relative Prym variety, we determine the Hamiltonian monodromy of the ``geodesic flow on SO(4)'. Using a relative generalized Jacobian, we prove that the Hamiltonian monodromy of the spherical pendulum can also be obtained by the Picard-Lefschetz formula. Received: 28 September 2001 / Accepted: 12 April 2002 Published online: 12 August 2002     

Undecidability of automorphism groups

We give a simple (and easy to apply) technique that gives the undecidability of the theory of many automorphism groups: Let G be a group of automorphisms of a structure. Suppose that is not the identity and has no non-singleton finite orbits. If the centraliser of g is transitive on the support of g and satisfies a further technical condition, then the subgroup generated by g is equal to the double centraliser of g. Thus if G contains such an element g that is conjugate to all its positive powers, then one can interpret addition and multiplication of natural numbers in the theory of G using the parameter g; consequently, G has undecidable theory. Received: 9 October 2000 / in final form: 2 October 2001 / Published online: 29 April 2002     

A characterization of reciprocal processes via an integration by parts formula on the path space

 We characterize in this paper the class of reciprocal processes associated to a Brownian diffusion (therefore not necessarily Gaussian) as the set of Probability measures under which a certain integration by parts formula holds on the path space . This functional equation can be interpreted as a perturbed duality equation between Malliavin derivative operator and stochastic integration. An application to periodic Ornstein-Uhlenbeck process is presented. We also deduce from our integration by parts formula the existence of Nelson derivatives for general reciprocal processes. Received: 25 October 2000 / Revised version: 7 September 2001 / Published online: 13 May 2002     

Anion Cryptates: Synthesis,Crystal Structures,and Complexation Constants of Fluoride and Chloride Inclusion Complexes of Polyammonium Macrobicyclic Ligands

Three macrobicyclic octamines 1–3 and the macrotricyclic hexadecamine 14 have been synthesized. The octamines 1–3 bind anionic substrates when protonated. The stability constants of the complexes between the protonated forms of the macrobicyclic polyamines and halide anions have been determined by pH-metric measurements. The stability constants in H₂O are very high; 1 in its hexaprotonated form binds F⁻ with high selectivity (selectivity F⁻/Cl⁻ > 10⁸), while 3 exhibits strong stability constants for both F⁻ and Cl⁻. Three X-ray structures have been obtained, one where F⁻ is held inside the cavity of 1 · 6H⁺, one where Cl⁻ is included in 3 · 6H⁺, and 3 · 6H⁺ where the cavity is empty.     

Marine natural products

This review covers the literature published in 2006 for marine natural products, with 758 citations (534 for the period January to December 2006) referring to compounds isolated from marine microorganisms and phytoplankton, green algae, brown algae, red algae, sponges, cnidaria, bryozoans, molluscs, tunicates and echinoderms. The emphasis is on new compounds (779 for 2006), together with their relevant biological activities, source organisms and country of origin. Biosynthetic studies, first syntheses, and syntheses that lead to the revision of structures or stereochemistries, have been included.     

Activity coefficients at infinite dilution of organic compounds in 1-(meth)acryloyloxyalkyl-3-methylimidazolium bromide using inverse gas chromatography

Activity coefficients at infinite dilution, gammainfinity, of organic compounds in two new room-temperature ionic liquids (n-methacryloyloxyhexyl-N-methylimidazolium bromide (C10H17O2MIM)(Br) at 313.15 and 323.15 K and n-acryloyloxypropyl-N-methylimidazolium bromide(C6H11O2MIM)(Br)) were determined using inverse gas chromatography. Phase loading studies of the net retention volume per gram of packing as a function of the percent phase loading were used to estimate the influence of concurrent retention mechanisms on the accuracy of activity coefficients at infinite dilution of solutes in both ionic liquids. It was found that most of the solutes were retained largely by partition with a small contribution from adsorption and that n-alkanes were retained predominantly by interfacial adsorption on ionic liquids studied in this work. The solvation characteristics of the two ionic liquids were evaluated using the Abraham solvation parameter model.     

Functional [6]pericyclynes: synthesis through [14+4] and [8+10] cyclization strategies

Critical analysis of possible strategies for the synthesis of novel carbo-benzene derivatives suggests several [(18-n)+n] routes for the preparation of hexaoxy[6]pericyclyne precursors. Beyond the previously attempted [9+9] symmetrical scheme (n=9), the a priori most selective strategies are those for which n=1, 4, 7, 10, 13, and 16. They involve a cyclizing double-propargylation of a C(18-n) omega-bis-terminal-skipped oligoyne containing (19-n)/3 triple bonds with a C(n) omega-dicarbonyl-skipped oligoyne containing (n-1)/3 triple bonds. To complement the previously used [11+7] strategy, the [14+4] and [8+10] strategies were thus explored. They proved to be efficient, affording seven novel hexaoxy[6]pericyclynes corresponding to six different substitution patterns. These compounds were obtained in 7-15 steps as mixtures of stereoisomers. Thus, by using dibenzoylacetylene as the C(4) electrophile, a [14+4] strategy allowed the synthesis of two hexaphenyl representatives with two or four free carbinol vertices. This approach also afforded tetraphenyl representatives in which the two remaining carbinoxy vertices were substituted with either two alkynyl or one 4-anisyl and one 4-pyridyl groups. By using the hexacarbonyldicobalt complex of butynedial as the C(4) electrophile, a [14+4] strategy also allowed the isolation of a tetraphenylhexaoxy[6]pericyclyne with two adjacent unsubstituted carbinol vertices. A regioisomer with two opposite unsubstituted carbinol vertices was obtained through an [8+10] strategy and its oxidation afforded the corresponding pericyclynedione. Several attempts at synthesizing diphenylhexaoxy[6]pericyclynes with four unsubstituted carbinoxy vertices are described. Only an [8+10] strategy allowed the generation of a fragile diphenylhexaoxy[6]pericyclyne with four adjacent unsubstituted carbinoxy vertices, which could be partly characterized. These results show that the synthesis of the nonsubstituted hexahydroxy[6]pericyclyne, the ring carbo-mer of [6]cyclitol, is a difficult challenge.