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.     

Compensated Compactness, Paracommutators, and Hardy Spaces

LetB₁: ⁿ× ^N₁→ ^m₁,B₂: ⁿ× ^N₂→ ^m₂andQ: ^m₂→ ^m₁be bilinear forms which are related as follows: ifμandνsatisfyB₁(ξ, μ)=0 andB₂(ξ, ν)=0 for someξ≠0, thenμ^τQν=0. Supposep⁻¹+q⁻¹=1. Coifman, Lions, Meyer and Semmes proved that, ifuL^p( ⁿ) andvL^q( ⁿ), and the first order systemsB₁(D, u)=0,B₂(D, v)=0 hold, thenu^τQvbelongs to the Hardy spaceH¹( ⁿ), provided that both (i)p=q=2, and (ii) the ranks of the linear mapsB_j(ξ, ·) : ^N_j→ ^m₁are constant. We apply the theory of paracommutators to show that this result remains valid when only one of the hypotheses (i), (ii) is postulated. The removal of the constant-rank condition whenp=q=2 involves the use of a deep result of Lojasiewicz from singularity theory.     

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.     

Synthesis and characterization of carbohydrate-based phospholipids

Novel carbohydrate-based phospholipids containing two saturated C(12) (dilauroyl ribo-phosphocholine) (DLRPC), C(14) (dimyristoyl ribo-phosphocholine) (DMRPC), and C(20) (diarachadonyl ribo-phosphocholine) (DARPC) carboxylic acid chains were synthesized. The physical properties of the supramolecular structures formed by these compounds were compared to those formed by their direct glycerol analogues dilauroyl phosphocholine (DLPC), dimyristoyl phosphocholine (DMPC), and diarachadonyl phosphocholine (DAPC). Modulated differential scanning calorimetry (MDSC) and X-ray diffraction data indicated that with chain lengths < or =14 carbons, the carbohydrate backbone increased the thermal stability of the bilayer below the phase-transition temperature (T(m)) as compared to the glycerol-based lipids. With longer chains (C(20)), the bilayer structure was destabilized as compared to glycerol-based lipids. NMR studies of a DMRPC vesicle dispersion reveal split choline headgroup signals and distinct magnetization transfer effects arising from the "inner" and "outer" surfaces of the bilayer vesicle. Modulated differential scanning calorimetry also demonstrated that glycerol- and carbohydrate-based lipids mix, as evidenced by a single intermediate T(m). In addition, carbohydrate-based lipid/cholesterol mixtures exhibited a decrease in enthalpy with an increase in cholesterol concentration. Unlike glycerol phospholipids, carbohydrate lipids were resistant to enzymatic degradation by phospholipase A(2) (PLA(2)).     

Synthesis, structure, and kinetics and stereochemistry of base-catalyzed hydrolysis of meso- and rac-[Co2(tmpdtne)Cl2]4+, bis(pentaamine) complexes devoid of deprotonatable NH centers

The bis(pentadentate) ligand tmpdtne binds two Co(II) centers, and the entity is readily oxidized to the dicobalt(III) derivative [Co(2)(tmpdtne)Cl(2)](4+) which has been separated into two isomeric forms. NMR studies establish these as meso and rac isomers arising from the different or same absolute configurations for the asym configuration about each Co(III) center. Each dinuclear ion base hydrolyses to the dihydroxo derivative [Co(2)(tmpdtne)(OH)(2)](4+) with retained asym configurations about each metal ion and also retained rac or meso configurations. The kinetics for the stepwise loss of the two Cl(-) ligands is uniphasic, and data are presented to show that the loss of the first chloride is rate determining and is followed by very rapid intramolecular and loss of the second Cl(-) via a hydroxo-bridged species to yield the observed dihydroxo derivative. Meso and rac forms of the latter have been crystallized. The X-ray crystal structure of the rac-dihydroxo complex is reported, and it establishes the configurations of all the complexes reported. The (1)H NMR spectra for the hydroxo ions show very high field Co-OH resonances (ca. delta-0.5 ppm) not observed previously for such ions, and this result is discussed in the context of published (1)H NMR data for bridged Co-OH-Co species. The base hydrolysis kinetics for the dichloro ions are first order in [OH(-)], and deprotonation at an alpha-CH(2) center (alpha to a pyridyl) is identified as the source of the catalysis, since there is no NH center available for deprotonation on the ligand. These data further support the new pseudoaminate base hydrolysis mechanism first reported in 2003. The values of k(OH) for the second-order base-catalyzed reaction are ca. 4.0 M(-1) s(-1) for both the rac and meso isomers, and these results are discussed in terms of the increased acidities of these 4+ cations compared to their 2+ ion counterparts.     

Cycloaldol approach to the isobenzofuran core of eunicellin diterpenes

[reaction: see text] A novel cycloaldol approach to the isobenzofuran core common to many of the eunicellin diterpenes is described. The cycloaldol precursor was prepared by aldol addition of (S)-(+)-carvone and methacrolein followed by etherification to a glycolate ester. Chemoselective enolization of the glycolate ester led to the cycloaldol adduct in high yield and diastereoselectivity. An oxidative rearrangement-allylic diazene rearrangement sequence established the requisite cis ring fusion.