Approximation and Extension in Normed Spaces of Infinitely Differentiable Functions

We consider questions of rational and polynomial approximation,and related extension questions, for various normed spaces ofinfinitely differentiable functions on perfect compact subsetsof the complex plane C and the real line R. We obtain an approximationtheorem for compact planar sets which are, in a precise sense,locally radially self-absorbing. All smoothly-bounded compactsets are of this type. We give a variety of results and counterexampleson extension, mainly in the one-dimensional case. We also provepolynomial approximation theorems for totally-disconnected sets,linear sets, and some others.     

On the Role of Charge Transfer in Many-Body Non-Covalent Interactions**

Charge transfer is one of the mechanisms involved in non-covalent interactions. In molecular dimers, its contribution to pairwise interaction energies has been studied extensively using a variety of interaction energy decomposition schemes. In polar interactions such as hydrogen bonds, it can contribute ten or several tens of percent of the interaction energy. Less is known about its importance in higher-order interactions in many-body systems, mainly because of the lack of methods applicable to this problem. In this work, we extend our method for the quantification of the charge-transfer energy based on constrained DFT to many-body cases and apply it to model trimers extracted from molecular crystals. Our calculations show that charge transfer can account for a large fraction of the total three-body interaction energy. This also has implications for DFT calculations of many-body interactions in general as it is known that many DFT functionals struggle to describe charge-transfer effects correctly.     

纳米 Ce1MgxZr1-xO2 固体多相催化剂的制备、表征及催化合成四氢苯并[b]吡喃衍生物的性能

 A series of Ce1MgxZr1-xO2 mixed metal oxides with different molar ratios were prepared by simple co-precipitation and were characterized by X-ray diffraction, infrared spectroscopy, scanning electron microscopy, energy dispersive spectroscopy, temperature-pro- grammed desorption of CO2, and N2 adsorption techniques. The prepared materials were tested for catalytic activity by the synthesis of tet-rahydrobenzo[b]pyran derivatives using a three component reaction (aromatic aldehydes, malononitrile, and dimedone) in an ethanol me-dium. The best catalytic activity was obtained with Ce1Mg0.6Zr0.4O2. The particle size or crystallite size was estimated using the De-bye-Scherrer equation. The addition of magnesium oxide into the ceria-zirconia lattice resulted in the formation of nanosized particles rang-ing from 5.41 to 9.78 nm. This work describes the catalytic behavior of magnesium oxide in mixed metal oxide systems.