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.     

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.     

Detection of VX contamination in soil through solid-phase microextraction sampling and gas chromatography/mass spectrometry of the VX degradation product bis(diisopropylaminoethyl)disulfide

A solid-phase microextraction (SPME) and gas chromatography-mass spectrometry (GC-MS) sampling and analysis method was developed for bis(diisopropylaminoethyl)disulfide (a degradation product of the nerve agent VX) in soil. A 30-min sampling time with a polydimethylsiloxane-coated fiber and high temperature alkaline hydrolysis allowed detection with 1.0 microg of VX spiked per g of agricultural soil. The method was successfully used in the field with portable GC-MS instrumentation. This method is relatively rapid (less than 1 h), avoids the use of complex preparation steps, and enhances analyst safety through limited use of solvents and decontamination of the soil before sampling.     

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.     

Synthesis of trans‐disubstituted cyclam ligands appended with two 6‐hydroxymethylpyridin‐2‐ylmethyl sidearms: Crystal structures of the 1,8‐dimethyl‐4,ll‐di(6‐hydroxymethylpyridin‐2‐ylmethyl)cylam ligand and its Co(II) and Ni(II) complexes

Two new trans‐disubstituted cyclam ligands; 1,8‐di(6‐hydroxymethylpyridin‐2‐ylmethyl)‐1,4,8,11‐tetra‐azacyclotetradecane ( 5 ) and 1,8‐dimethyl‐4, 11‐di(6‐hydroxymethylpyridin‐2‐ylmethyl)‐1,4,8,11 ‐tetraaza‐cyclotetradecane ( 6 ); have been synthesized and characterized. The crystal structures of ligand 6 and its Ni(II) and Co(II) complexes have been determined. Crystal data are given for 6 , space group, P2₁/c, a = 11.095 (6) Å, b = 9.467 (5) Å, c = 13.283 (8) Å; β = 106.95 (5)°, Z = 2, R = 0.0715; for [Ni 6 ](C10₄)₂, space group P2₁/c, a = 9.4848 (14) Å, b = 33.941(6) Å, c = 9.793(2) A, β = 95.264(14)°, Z = 4, R = 0.0567; for [Co 6 ](C10₄)₂, space group, P2₁/c, a = 9.440 (6) Å, b = 33.848 (13) Å, c = 9.820 (3) Å, β = 95.16(3)°, Z = 4, R = 0.0718. In both complexes, the metal atoms are six‐coordinate with only one of the pendants interacting with the central metal atom and the other pendant remaining uncoordinated.