Collective Tunnelling of Strongly Interacting Cold Atoms in a Double-Well Potential

It is known that under resonance conditions, a group of strongly interacting bosonic atoms, trapped in a double-well potential, mimics a single particle, performing Rabi oscillations between the wells. By implication, all atoms need to tunnel at roughly the same time, even though the Bose–Hubbard Hamiltonian accounts only for one-atom-at-a-time transfers. The mechanism of this collective behavior is analyzed, the Rabi frequencies in the process are evaluated, and the limitation of this simple picture is discussed. In particular, it is shown that the small rapid oscillations superimposed on the slow Rabi cycle result from splitting the transferred cluster at the sudden onset of tunnelling, and disappear if tunnelling is turned on gradually.     

On the approximation of convex functions by Bernstein-type operators

As a consequence of Jensen's inequality, centered operators of probabilistic type (also called Bernstein-type operators) approximate convex functions from above. Starting from this fact, we consider several pairs of classical operators and determine, in each case, which one is better to approximate convex functions. In almost all the discussed examples, the conclusion follows from a simple argument concerning composition of operators. However, when comparing Szász-Mirakyan operators with Bernstein operators over the positive semi-axis, the result is derived from the convex ordering of the involved probability distributions. Analogous results for non-centered operators are also considered.     

Unexpected differences in the alpha-halogenation and related reactivity of sulfones with perhaloalkanes in KOH-t-BuOH

Most alkyl phenyl sulfones are readily alpha-chlorinated with CCl(4) and alpha-brominated with CBrCl3 in KOH-t-BuOH via radical-anion radical pair (RARP) reactions. While isopropyl mesityl sulfone (4) is easily alpha-chlorinated with CCl(4), it was completely recovered when treated with the more reactive CBrCl3. Subsequent investigations showed the latter result to be due to the poor acidity of 4 together with the rapid depletion of CBrCl3 and KOH by their reaction with each other, and led to a variety of other important results. 4-Hydroxyphenyl isopropyl sulfone (6) is unreactive with either CCl4 or CBrCl3 in KOH-t-BuOH, its phenoxide anion strongly reducing the electronegativity of the sulfonyl group, thereby inhibiting alpha-anion formation. This effect is reversed by the electron-withdrawing influence of two alpha-phenyls, so that benzhydryl 4-hydroxyphenyl sulfone (8) is readily alpha-halogenated in KOH-t-BuOH with CCl4 or CBrCl3. On further contact with KOH-t-BuOH the alpha-halogenated sulfones from 8 are decomposed into benzophenone and phenol. While the alpha-halogenated derivatives of 4-methoxyphenyl benzhydryl sulfone (9) are stable to base, they are decomposed even under mildly acidic conditions into 4-methoxyphenyl 4-methoxybenzenethiolsulfonate (9c), phenol, and benzophenone. Mono-alpha-halogenation of benzyl phenyl sulfone (10) enhances the rate of the subsequent halogenation, so that alpha,alpha-dihalogenation is attained while much substrate is still present and the mono-alpha-halogenated product is not detected. The ease of reductive debromination of alpha-bromo sulfones with Cl3C- was correlated with the stability of the formed alpha-anions, explaining the success with alpha-bromobenzylic sulfones but failure with alpha-bromoalkyl sulfones. In the presence of air and the absence of competing halogenation, formation of the alpha-anions of alkyl aryl sulfones is quickly accompanied by oxidative cleavage by atmospheric O2, leading to the formation of arenesulfonyl alcohols, arenesulfonyl halides, and haloarenes.     

Copper bronze catalyzed Heck reaction in ionic liquids

Heck reaction of aryl iodides and activated aryl bromides catalyzed by copper bronze in tetrabutylammonium bromide as solvent and tetrabutylammonium acetate as base was developed. The effective catalysts are Cu nanoparticles deriving from the reaction of iodobenzene with copper bronze. These nanoparticles are very stable in tetraalkylammonium salts, are easily recycled, and can be stored for months without a loss of catalytic efficiency. [reaction: see text]     

Determination of decabromodiphenyl ether in sediments using selective pressurized liquid extraction followed by GC–NCI-MS

A method based on selective pressurized liquid extraction (SPLE) followed by gas chromatography–negative ion chemical ionization-mass spectrometry (GC–NCI-MS) has been evaluated for analysis of decabromodiphenyl ether (PBDE-209) in sediment samples. Instrumental operating conditions such as source temperature and system pressure were optimized in the NCI-MS system, giving an instrumental detection limit of 2 pg. The limit of determination of the entire SPLE–GC–NCI-MS procedure was around 50 pg g^–1 dry weight (dw), with repeatability of replicates between 4 and 21% relative standard deviation. Application of the method to 13 different river and marine sediment samples collected in Spain revealed that levels of decabromodiphenyl ether ranged between 2 and 132 ng g^–1 dry weight.     

Pd nanoparticles as efficient catalysts for Suzuki and Stille coupling reactions of aryl halides in ionic liquids

Pd-catalyzed Suzuki and Stille cross-couplings of aryl bromides and chlorides were carried out in quaternary ammonium salts as solvents under mild conditions and with the recycling of the catalyst.     

Model-based clustering of probability density functions

Complex data such as those where each statistical unit under study is described not by a single observation (or vector variable), but by a unit-specific sample of several or even many observations, are becoming more and more popular. Reducing these sample data by summary statistics, like the average or the median, implies that most inherent information (about variability, skewness or multi-modality) gets lost. Full information is preserved only if each unit is described by a whole distribution. This new kind of data, a.k.a. “distribution-valued data”, require the development of adequate statistical methods. This paper presents a method to group a set of probability density functions (pdfs) into homogeneous clusters, provided that the pdfs have to be estimated nonparametrically from the unit-specific data. Since elements belonging to the same cluster are naturally thought of as samples from the same probability model, the idea is to tackle the clustering problem by defining and estimating a proper mixture model on the space of pdfs. The issue of model building is challenging here because of the infinite-dimensionality and the non-Euclidean geometry of the domain space. By adopting a wavelet-based representation for the elements in the space, the task is accomplished by using mixture models for hyper-spherical data. The proposed solution is illustrated through a simulation experiment and on two real data sets.     

A Convenient Synthesis of ω-[(Arylphosphonomethyl)Amino] Alkylphosphonic and Carboxylic Acids via in Situ–Generated Arylideneaminoalkyl- Phosphonic or Carboxylic Acids

ω-[(Arylphosphonomethyl)amino]alkylphosphonic and carboxylic acids are prepared in an easy work-up procedure, by addition of diethyl phosphite to in situ–generated arylideneaminoalkylphosphonic or carboxylic acids followed by hydrolysis with hydrochloric acid