An improved Hardy-Sobolev inequality and its application

For , a bounded domain, and for , we improve the Hardy-Sobolev inequality by adding a term with a singular weight of the type . We show that this weight function is optimal in the sense that the inequality fails for any other weight function more singular than this one. Moreover, we show that a series of finite terms can be added to improve the Hardy-Sobolev inequality, which answers a question of Brezis-Vazquez. Finally, we use this result to analyze the behaviour of the first eigenvalue of the operator as increases to for .

     

Lagrangian finite element method for the analysis of two-dimensional sloshing problems

A new version of a numerical algorithm for the Lagrangian treatment of incompressible fluid flows with free surfaces is developed. The novel features of the present method are the adoptions of the Lagrangian finite element method and the velocity correction technique. The use of the velocity correction approach makes the computational scheme extremely simple in algorithmic structure. Hence, the present method is particularly attractive for large-scale problems. The techniques discussed here are applied to some two-dimensional sloshing problems, which may indicate the versatility and effectiveness of the present method.     

Rapid hydrocarbon type separation of vacuum residues

Summary A simple and fast column chromatographic procedure for the separation of vacuum residues into saturates, mono-, di-, polyaromatics and polars has been standardized using silica-alumina-bauxite adsorbents and gradient elution with n-pentane, benzene, and methanol. The cut points have been monitored employing ultraviolet and infrared spectroscopy. The method is applicable for preparative separation of short residues from processed products.

---

Rasche Trennung der Kohlenwasserstofftypen von Vakuum-RückstÄnden

     

Database diversity assessment: New ideas, concepts, and tools

We present some new ideas for characterizing and comparing largechemical databases. The comparison of the contents of large databases is nottrivial since it implies pairwise comparison of hundreds of thousands ofcompounds. We have developed methods for categorizing compounds into groupsor series based on their ring-system content, using precalculatedstructure-based hashcodes. Two large databases can then be compared bysimply comparing their hashcode tables. Furthermore, the number of distinctring-system combinations can be used as an indicator of database diversity.We also present an indepen- dent technique for diversity assessment calledthe saturation diversity approach. This method is based on picking as manymutually dissimilar compounds as possible from a database or a subsetthereof. We show that both methods yield similar results. Since the twomethods measure very different properties, this probably says more about theproperties of the databases studied than about the methods.     

Cobalt and manganese nets via their wires: facile transformation in metal-diorganophosphates

The manganese and cobalt complexes [M(dtbp)2]n (M=Mn, Co; dtbp=di-tert-butyl phosphate), which exist as one-dimensional molecular wires, transform to [M(dtbp)2(bpy)2.2H2O]n by the addition of 4,4-bipyridine (bpy) at room temperature; the latter compounds form noninterpenetrating rectangular grid structures.     

New substrates in Belousov-Zhabotinsky reaction employing mixed media

Several substrates have been employed in different oscillatory systems. A serious limitation is the low solubility in water. This has been overcome by employing aqueous-organic mixed media in the iodate-system as well as the uncatalyzed and catalyzed [Ce(III) or Ferroin] systems. The present paper deals with the study of fifteen new substrates in the bromate-Mn(II) oscillatory system employing aqueous-organic mixed media.

---

. . , - , (Ce(III) ), . -Mn(II) - .

     

Non-interpenetrating transition metal diorganophosphate 2-dimensional rectangular grids from their 1-dimensional wires: structural transformations under mild conditions

The manganese, cobalt, and cadmium complexes [M(dtbp)2]n (M = Mn (1) and Co 2) and [Cd(dtbp)2(H2O)]n (3) (dtbp = di-tert-butyl phosphate), which exist as one-dimensional molecular wires, transform to non-interpenetrating rectangular grids [M(dtbp)2(bpy)2.2H2O]n (M = Mn (4), Co (5), and Cd (6)) by the addition of 4,4-bipyridine (bpy) at room temperature. Products 4-6 have also been prepared by a room-temperature reaction or by solvothermal synthesis in methanol through a direct reaction between the metal acetate, di-tert-butyl phosphate, and 4,4'-bipyridine (bpy) in a 1:2:2 molar ratio. Single-crystal X-ray structure determination of 4-6 shows that these compounds are composed of octahedral transition metal ions woven into a two-dimensional grid structure with the help of bpy spacer ligands. The axial coordination sites at the metal are occupied by bulky unidentate dtbp ligands, which prevent any interpenetration of the individual grids. The change of reaction conditions from solvothermal to hydrothermal, for the attempted synthesis of a magnesium grid structure, however leads to the isolation of an organic phosphate [(H2bpy)(H2PO4)2] (7) and an inorganic phosphate [Mg(HPO4)(OH2)3] (8). Compound 7 can also be prepared quantitatively from a direct reaction between bpy and H3PO4. The new organic phosphate 7 is a unique example of a phosphate material with alternating layers of [H2bpy]2+ cations and [H2PO4]- anions that are held together by hydrogen bonds. Solid-state thermal decomposition of 4-6 produced the respective metaphosphate materials [M(PO3)2] (M = Mn (9), Co (10), and Cd (11)). All new metal-organic phosphates have been characterized by elemental analysis, thermal analysis (TGA, DTA, DSC), and IR and NMR spectroscopy. The metaphosphate ceramic materials were characterized by IR spectral and powder X-ray diffraction studies.