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.     

Molecular dynamics averaging of Xe chemical shifts in liquids

The Xe nuclear magnetic resonance chemical shift differences that afford the discrimination between various biological environments are of current interest for biosensor applications and medical diagnostic purposes. In many such environments the Xe signal appears close to that in water. We calculate average Xe chemical shifts (relative to the free Xe atom) in solution in eleven liquids: water, isobutane, perfluoro-isobutane, n-butane, n-pentane, neopentane, perfluoroneopentane, n-hexane, n-octane, n-perfluorooctane, and perfluorooctyl bromide. The latter is a liquid used for intravenous Xe delivery. We calculate quantum mechanically the Xe shielding response in Xe-molecule van der Waals complexes, from which calculations we develop Xe (atomic site) interpolating functions that reproduce the ab initio Xe shielding response in the complex. By assuming additivity, these Xe-site shielding functions can be used to calculate the shielding for any configuration of such molecules around Xe. The averaging over configurations is done via molecular dynamics (MD). The simulations were carried out using a MD technique that one of us had developed previously for the simulation of Henry's constants of gases dissolved in liquids. It is based on separating a gaseous compartment in the MD system from the solvent using a semipermeable membrane that is permeable only to the gas molecules. We reproduce the experimental trends in the Xe chemical shifts in n-alkanes with increasing number of carbons and the large chemical shift difference between Xe in water and in perfluorooctyl bromide. We also reproduce the trend for a given solvent of decreasing Xe chemical shift with increasing temperature. We predict chemical shift differences between Xe in alkanes vs their perfluoro counterparts.     

Probabilistic approach to homoclinic chaos

Three-dimensional systems possessing a homoclinic orbit associated to a saddle focus with eigenvalues ±i, – and giving rise to homoclinic chaos when the Shil'nikov condition < is satisfied are studied. The 2D Poincaré map and its 1D contractions capturing the essential features of the flow are given. At homoclinicity, these 1D maps are found to be piecewise linear. This property allows one to reduce the Frobenius—Perron equation to a master equation whose solution is analytically known. The probabilistic properties such as the time autocorrelation function of the state variablex are explicitly derived.     

Optimized Quantitative Method for Determining Isoflavones and Equol in Bovine Digestive Fluids and Feces

Many studies have been conducted on the impact of animal feed on isoflavones and their metabolite concentrations in bovine milk, but few studies have focused on the development and validation of analytical protocols for quantifying these compounds in biological matrices other than milk and plants. The purpose of this study was to develop a method that would enable four isoflavones and equol in cows’ feces and digestive fluids to be quantified simultaneously. The method is based on aglycones released by methanolic ultrasound-assisted extraction, followed by enzymatic hydrolysis and high-performance liquid chromatography tandem–mass spectrometry analysis. The sample preparation was optimized using the Box–Behnken design. The selected extraction conditions were 80°C, 10?min, and 50% methanol for digestive fluids and 70°C, 35?min, and 60% methanol for feces. For hydrolysis, the selected conditions were 37°C, 1?h, and a pH of 6 for both matrices. The analytical method showed a good linear regression model ranging from 5 to 125?ng?mL^?1. Both inter- and intraday accuracy (≤8.5 and ≤12.3%) and precision (≤11.1 and ≤15.2%) were suitable. No matrix effects were found. There was good repeatability and extract stability for at least 4 days of storage at???20 and 6°C. All recoveries were in the acceptable range of 70–120% for both matrices, except for biochanin A in feces, where the value was approximately 43%. This sensitive and reliable method will be useful for monitoring the passage of isoflavones and equol in the digestive system of ruminants.     

Nickel-catalyzed intramolecular C–H arylation using aryl pivalates as electrophiles

This paper describes a method for nickel catalyzed intramolecular C–H arylation using aryl pivalates as electrophiles. The transformation is efficient for the synthesis of diverse electronically and sterically differentiated dibenzofurans. Additionally, the method could be expanded toward the synthesis of carbazoles. Preliminary mechanistic studies of the transformation are also described.     

Constructions for Terraces and R‐Sequencings,Including a Proof That Bailey's Conjecture Holds for Abelian Groups

Every abelian group of even order with a noncyclic Sylow 2‐subgroup is known to be R‐sequenceable except possibly when the Sylow 2‐subgroup has order 8. We construct an R‐sequencing for many groups with elementary abelian Sylow 2‐subgroups of order 8 and use this to show that all such groups of order other than 8 also have terraces. This completes the proof of Bailey's Conjecture in the abelian case: all abelian groups other than the noncyclic elementary abelian 2‐groups have terraces. For odd orders it is known that abelian groups are R‐sequenceable except possibly those with noncyclic Sylow 3‐subgroups. We show how the theory of narcissistic terraces can be exploited to find R‐sequencings for many such groups, including infinitely many groups with each possible of Sylow 3‐subgroup type of exponent at most 3¹² and all groups whose Sylow 3‐subgroups are of the form or .     

Nuclear magnetic shielding and chirality IV. The odd and even character of the shielding response to a chiral potential

We investigate the odd and even character of the shielding response in a chiral molecule (modeled by a Ne8 helix) when subjected to a chiral potential. We establish that the diastereomeric splittings are a measure of odd powers of Vodd. Implications for diastereomeric, splittings of Xe in handed cages with handed tethers are discussed.