Estimating Invariant Probability Densities for Dynamical Systems

Knowing a probability density (ideally, an invariant density) for the trajectories of a dynamical system allows many significant estimates to be made, from the well-known dynamical invariants such as Lyapunov exponents and mutual information to conditional probabilities which are potentially more suitable for prediction than the single number produced by most predictors. Densities on typical attractors have properties, such as singularity with respect to Lebesgue measure, which make standard density estimators less useful than one would hope. In this paper we present a new method of estimating densities which can smooth in a way that tends to preserve fractal structure down to some level, and that also maintains invariance. We demonstrate with applications to real and artificial data.     

Geometric and Combinatorial Realizations of Crystal Graphs

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For type A_n⁽¹⁾, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids. Presented by P. Littleman Mathematics Subject Classifications (2000) Primary 16G10, 17B37. Alistair Savage: This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and was partially conducted by the author for the Clay Mathematics Institute.     

Influence of adsorbed anions on the dehydroxylation of synthetic goethite

The influence of surface structure on the dehydroxylation of synthetic goethite has been studied using heat-flux differential scanning calorimetry. Adsorption of various anions by goethite causes a progressive change in both the amount and environment of surface hydroxyls and this is reflected by changes in the DSC curves. These alterations indicate the importance of surface features in determining the course of dehydroxylation and further suggest that, in studies of this type of reaction, the use of synthetic products known to be free of surface contaminants could prove to be of great value in understanding dehydroxylation mechanisms.

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Zusammenfassung Der Einfluß von Oberflächenstrukturen auf die Dehydroxylierung synthetischer Goethite wurde unter Anwendung der Wärmestrom-Differential-Abtastkalorimetrie untersucht. Die Adsorption verschiedener Anione durch Goethit verursacht eine progressive Änderung sowohl der Menge als auch der Umgebung der Oberflächenhydroxyle und dies wird durch die Änderungen der DSC-Kurven wiedergespiegelt. Diese Änderungen zeigen die Bedeutung der Oberflächenbeschaffenheiten bei der Bestimmung des Verlaufs der Dehydroxylierung und weisen ferner darauf hin, daß bei Studien dieses Reaktionstyps der Einsatz synthetischer Produkte, die bekanntlich frei von Oberflächenverunreinigungen sind, sich hinsichtlich der Klärung der Dehydroxylierungsmechanismen als sehr wertvoll erweisen könnte.

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Résumé On a étudié par analyse calorimétrique différentielle (DSC) l'influence de la structure de surface sur la déshydroxylation de goethites synthétiques. L'adsorption d'anions divers par la goethite entraîne une variation progressive à la fois de la quantité et de l'environnement des hydroxyles de surface, ce qui se reflète dans les variation des courbes DSC. Ces altérations indiquent l'importance des propriétés de surface quand il s'agit de déterminer le cours de la déshydroxylation et suggèrent en outre que l'emploi de produits synthétiques connus comme étant exempts de contaminants de surface, pourrait être de grande valeur dans des études de ce type de réactions, pour comprendre les mécanismes de déshydroxylation.

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Total synthesis of (+)-discodermolide: an improved endgame exploiting a Still-Gennari-type olefination with a C1-C8 beta-ketophosphonate fragment

[structure: see text] An improved, third-generation, total synthesis of (+)-discodermolide, a potent microtubule-stabilizing anticancer agent of marine sponge origin, is achieved in 11.1% yield over 21 steps. Key steps include a Still-Gennari HWE olefination, performed using NaH as the base, between C1-C8 beta-ketophosphonate 7 and C9-C24 aldehyde 8, introducing the (8Z)-alkene with 10:1 selectivity, and K-Selectride reduction of the derived enone 16, installing the (7S)-configuration.     

Stereoselective reductase-catalysed deoxygenation of sulfoxides in aerobic and anaerobic bacteria

Direct and indirect evidence, of unexpected stereoselective reductase-catalysed deoxygenations of sulfoxides, was found. The deoxygenations proceeded simultaneously, with the expected dioxygenase-catalysed asymmetric sulfoxidation of sulfides, during some biotransformations with the aerobic bacterium Pseudomonas putida UV4. Stereoselective reductase-catalysed asymmetric deoxygenation of racemic alkylaryl, dialkyl and phenolic sulfoxides was observed, without evidence of the reverse sulfoxidation reaction, using anaerobic bacterial strains. A purified dimethyl sulfoxide reductase, obtained from the intact cells of the anaerobic bacterium Citrobacter braakii DMSO 11, yielded, from the corresponding racemates, enantiopure alkylaryl sulfoxide and thiosulfinate samples.     

Primary amido substituted diborane4 compounds and imidodiborate4 anions

Treatment of the diborane(4) compound B(2)(NMe(2))(4) with aniline or 2,6-dimethylaniline results in the primary amido compounds B(2)(NHR)(4)(R = Ph, 2,6-Me(2)C(6)H(3)); subsequent treatment with n-BuLi in toluene in each case affords the first examples of anionic imidodiborates namely Li(4)(thf)(6)B(2)(NPh)(4) and Li(4)(thf)(4)B(2)(N-2,6-Me(2)C(6)H(3))(4); all complexes have been characterised crystallographically.     

Selective, controllable, and reversible aggregation of polystyrene latex microspheres via DNA hybridization

The directed three-dimensional self-assembly of microstructures and nanostructures through the selective hybridization of DNA is the focus of great interest toward the fabrication of new materials. Single-stranded DNA is covalently attached to polystyrene latex microspheres. Single-stranded DNA can function as a sequence-selective Velcro by only bonding to another strand of DNA that has a complementary sequence. The attachment of the DNA increases the charge stabilization of the microspheres and allows controllable aggregation of microspheres by hybridization of complementary DNA sequences. In a mixture of microspheres derivatized with different sequences of DNA, microspheres with complementary DNA form aggregates, while microspheres with noncomplementary sequences remain suspended. The process is reversible by heating, with a characteristic "aggregate dissociation temperature" that is predictably dependent on salt concentration, and the evolution of aggregate dissociation with temperature is observed with optical microscopy.     

Mixed-metal cluster chemistry. 26 . Proclivity for “all-terminal” or “plane-of-bridging-carbonyls” ligand disposition in tungsten-triiridium clusters

Reaction of WH(CO)₃(η-C₅Me₅) with IrCl(CO)₂(4-H₂NC₆H₄Me) affords WIr₃(μ-CO)₃(CO)₈(η-C₅Me₅) in low yield. A structural study reveals a WIr₂-centred plane of bridging carbonyls, in contrast to the crystal structure of WIr₃(CO)₁₁(η-C₅H₅) (all-terminal carbonyl distribution). DFT calculations reveal an increasing proclivity to adopt an all-terminal CO disposition for clusters MIr₃(CO)₁₁(η-C₅H₅) in the gas phase on proceeding from M=Cr to Mo and then W, consistent with structural studies in the solid state for which the tungsten-containing cluster is the only all-terminal example. Increasing electron donation from the ligands in the tungsten system (either from phosphine substitution or cyclopentadienyl permethylation) suffices to impose a plane of bridging carbonyls in the ground state structure. ¹³C NMR fluxionality studies reveal that CO exchange mechanism(s) for WIr₃(CO)₁₁(η-C₅H₅) and the related tetrahedral cluster W₂Ir₂(CO)₁₀(η-C₅H₅)₂ are very fast and involve all carbonyls on the clusters. DFT calculations on MIr₃(CO)₁₁(η-C₅H₅) (M=Cr, Mo) substantiate a ‘merry-go-round’ mechanism for carbonyl scrambling in these systems, a result which is consistent with the scrambling behaviour seen in the NMR fluxionality studies on the W-containing congener.     

Actin-binding marine macrolides: total synthesis and biological importance

Marine organisms produce a fascinating range of structurally diverse secondary metabolites, which often possess unusual and sometimes unexpected biological activities. This structural diversity makes these marine natural products excellent molecular probes for the investigation of biochemical pathways. Recently, a number of novel and stereochemically complex macrolides, having a large macrolactone (22- to 44-membered) ring, that interact with the actin cycloskeleton have been isolated from different marine sources. Actin, like tubulin, is a major component of the cytoskeleton and has important cellular functions. Although the details of these interactions are still under investigation, these marine macrolides are becoming increasingly important as novel molecular probes to help elucidate the cellular functions of actin. Owing to their potent antitumor activities, these compounds, for example the aplyronines, also have potential for preclinical development in cancer chemotherapy. Their appealing molecular structures, with an abundance of stereochemistry, and biological significance, coupled with the extremely limited availability from the marine sources, have stimulated enormous interest in the synthesis of these compounds. This review summarizes the biological properties of these unusual marine natural products and features the recently completed total syntheses of swinholide A, scytophycin C, aplyronine A, mycalolide A--all of these being potent cytotoxic agents that target actin--and a diastereoisomer of ulapualide A. Rather than detailing each individual step of these multistep total syntheses, the different synthetic strategies, key reactions, and methods adopted for controlling the stereochemistry are compared.     

Total synthesis of the microtubule-stabilizing agent (-)-laulimalide

[structure: see text] The total synthesis of the potent microtubule-stabilizing anticancer agent (-)-laulimalide has been achieved in 27 steps and 2.9% overall yield. Notable features are the use of Jacobsen HDA chemistry for the enantioselective construction of the side chain dihydropyran, a diastereoselective aldol coupling using chiral boron enolate methodology, a Mitsunobu macrolactonization, and a Sharpless AE to introduce the epoxide onto des-epoxy-laulimalide.