Weyl formula for multi-quasi-elliptic operators. of Schrödinger type

In this work we consider a general class of Schr?dinger type operators, associated with multi-quasi-elliptic symbols. We give a precise estimate of the remainder of the so-called Weyl asymptotic formula for the eigenvalues of these operators. In order to reach our aim, we use the Weyl–H?rmander calculus, with locally temperate metrics and weights, and interpolation techniques. Received: February 14, 2000; in final form: October 29, 2000?Published online: July 13, 2001     

Lingshuine,an Unexpected Passerini Product from the Hainan Sponge Axinyssa variabilis

Lingshuine ( 1 ), a novel sesquiterpene amide, along with three known N‐containing compounds, 2 – 4 , have been isolated from the Hainan sponge Axinyssa variabilis. The structure of 1 was elucidated by detailed analysis of the spectroscopic data and by chemical methods. Lingshuine represents the first example of a Passerini product formed during the isolation process.     

On the jump behavior of distributions and logarithmic averages

The jump behavior and symmetric jump behavior of distributions are studied. We give several formulas for the jump of distributions in terms of logarithmic averages, this is done in terms of Cesàro-logarithmic means of decompositions of the Fourier transform and in terms of logarithmic radial and angular local asymptotic behaviors of harmonic conjugate functions. Application to Fourier series are analyzed. In particular, we give formulas for jumps of periodic distributions in terms of Cesàro–Riesz logarithmic means and Abel–Poisson logarithmic means of conjugate Fourier series.     

Thermo gravimetric analysis of supramolecular complexes of p-tert-butylcalix[6]arene and ammonium cations: crystal structure of diethylammonium complex

Thermo gravimetric analysis of ammonium-p-tert-butylcalix[6]arene salts were carried out to determine the stoichiometry of the inclusion compounds and the temperature of leaving of the ammonium cation, as amine, from the solid phase by a reverse proton transfer reaction to calixarene. The chain length of the amine molecules and their relative basicity order are the main factors which determine the decomplexation temperatures for the ammonium salts of piperidinium, diethylammonium, ethylenediammonium, morpholinium, s-butylammonium, butylammonium, hexylammonium and triethylammonium salts of p-tert-butylcalix[6]arene anion. The solid state structure of bis-diethylammonium-p-tert-butylcalix[6]arene dianion is reported. The calix moiety adopts a 1,2,3 alternate conformation, with one ammonium as exo-calix and other as endo-calix, with an intricate array of hydrogen bonds between phenol, phenolate and ammonium cations.     

Structure of premixed flames of propylene oxide: Molecular beam mass spectrometric study and numerical simulation

The knowledge of the combustion chemistry of oxygenated fuels is essential for the development of detailed kinetic mechanisms suitable for the combustion processes involving biofuels. Moreover, epoxidized olefins, are increasingly used as chemical intermediates or as bulk chemicals. Nevertheless, a dearth of data for their reactivity in the oxidative environment can be observed in the current literature. This study reports the experimental and the model characterization of the flame structure of propylene oxide at stoichiometric and fuel-rich conditions at atmospheric pressure. To this aim, the species mole fractions in three premixed flames stabilized on a flat-flame burner have been quantitatively measured by using the flame sampling molecular beam mass spectrometry. Three chemical kinetic mechanisms retrieved from the current literature involving propylene oxide chemistry have been validated against the novel experimental data. In general, the predictions appeared to be in satisfactory agreement with measurements except for acetaldehyde and ketene. The rate of production analysis in the flame has shown that the discrepancies observed for these species are related basically to the incorrect ratio between the rates of primary reaction pathways of propylene oxide destruction.     

The ABC Matrix

The atom-bond connectivity index—abbreviated ABC—has proven to describe the heat of formation of alkanes with an accuracy comparable to that of high-level ab initio and DFT methods. Its physical interpretation in terms of quantum-theory, its generalization, and many of its mathematical properties have been reported. Here, a probabilistic interpretation of the generalized ABC index is provided. It indicates that the terms defining this index represent the probability of visiting a nearest neighbor edge from one side or the other of a given edge in a graph. This interpretation is general enough as to cover the case of polarizing bonds in a molecular context. We then introduce a matrix representation of these probabilities in the form of generalized ABC matrices. Using this representation we study some indices related to the eigenvalues of the ABC matrices, such as the ABC energy and the ABC Estrada index. We provide some bounds for these parameters in general graphs.     

Polymesomorphism and orientation in liquid crystalline poly(triethylene glycol p,p′-bibenzoate)

The phase transitions and the orientational behavior of liquid crystalline poly(triethylene glycol p,p′-bibenzoate) have been studied. The real-time synchrotron diffraction results indicate that, on cooling from the isotropic melt, an orthogonal SmA mesophase is formed first, and later it is transformed into a tilted SmC mesophase. However, the SmA mesophase is stable in a rather wide temperature interval, and the transformation into the SmC phase occurs at temperatures close to the glass transition, so that not very high tilting angles are attained. The uniaxial deformation of the SmC mesophase indicates that usual parallel orientation of the molecular axes in relation to the stretching direction is obtained at high strain rates, while anomalous perpendicular orientation occurs at low deformation rates, with the smectic layers aligned with the stretching direction and the molecular axes almost perpendicular. A mixture of the two types of orientation is observed at intermediate rates, with rather interesting features.     

Synthesis,structural characterization and growth features of some (Ph)Se–Cd cluster compounds (Ph = phenyl)

Cd(SePh)₂, CdBr₂·4H₂O/CdI₂ and PCh₃ (Ph = phenyl; PCh₃ = tricyclohexylphosphine) react to give [(SePh)₂Cd₂Br₂(PCh₃)₂] (1), [(SePh)₂Cd₂I₂(PCh₃)₂] (2) and [(SePh)₇Cd₄Br(PCh₃)]_n (3).     

Metal-ion responsive redox polyelectrolyte multilayers

We present polyelectrolyte multilayer modified electrodes exhibiting novel chemically responsive redox behaviour due to the combination of both redox and metal-ion-ligand functionalities on the same sites.