Global regular and singular solutions for a model of gravitating particles

The existence of solutions of a nonlinear parabolic equation describing the gravitational interaction of particles is studied for the initial data in spaces of (generalized) pseudomeasures. This approach permits us to relax regularity assumptions on the initial conditions and to prove asymptotic stability results for the above problem.Mathematics Subject Classification (2000):35B40, 35K15, 82C21     

On tractability of weighted integration over bounded and unbounded regions in

We prove that for the space of functions with mixed first derivatives bounded in norm, the weighted integration problem over bounded or unbounded regions is equivalent to the corresponding classical integration problem over the unit cube, provided that the integration domain and weight have product forms. This correspondence yields tractability of the general weighted integration problem.

     

On strong tractability of weighted multivariate integration

We prove that for every dimension and every number of points, there exists a point-set whose -weighted unanchored discrepancy is bounded from above by independently of provided that the sequence has for some (even arbitrarily large) . Here is a positive number that could be chosen arbitrarily close to zero and depends on but not on or . This result yields strong tractability of the corresponding integration problems including approximation of weighted integrals over unbounded domains such as . It also supplements the results that provide an upper bound of the form when .

     

Silicon polypodands: powerful metal cation complexing agents and solid-liquid phase-transfer catalysts of new generation

Silicon polypodands 5-7 are found to be powerful complexing agents of alkali metal salts in low polarity solvents and very efficient catalysts in anion-promoted reactions under solid-liquid PTC conditions. The catalytic activity is comparable with that of the cyclic polyether PHDB18C6 8.     

A numerical model of multimode fiber PON frequency response

Frequency response of passive optical network (PON) based on multimode fibers is investigated. The network comprises fibers, connectors and splitters/couplers. It is shown that due to mode filtering at splitters, the frequency response is different for different network nodes in otherwise symmetrical network.     

Fourier analysis of scratches generated on m‐GaN substrates during polishing

Generation of scratches on surface of m‐plane GaN substrates due to polishing was studied by atomic force microscopy (AFM). For epi‐ready substrates AFM images confirm a flat surface with the atomic step roughness while a lot of scratches are visible in AFM images for partially polished GaN substrates. The Fourier analysis of AFM images show that scratches propagate easier along {c‐plane} and {a‐plane} directions on m‐plane GaN surface. This observation is an evidence of anisotropy of mechanical properties of GaN crystals in the micro‐scale. This anisotropy is directly correlated with the symmetry and atomic arrangement of m‐plane GaN.     

Studies of reaction heats and structure of complexes formed between strong N-bases and phenols

The heats of reactions between various phenols and two strong N-bases of guanidine-like character in acetonitrile, are determined. The values can be used as a measure of self-assembly abilities of the phenol molecules in the interactions with strong N-bases, playing a very important role in biological systems. In the case of TBD complexes with corresponding nitrophenols, the protonated N-base is hydrogen-bonded to the nitro group excluding the self-assembly process of the phenols. In the case of other phenols, the self-assembly abilities are dependent on pK_a values of phenols. With increasing acidity of phenols their ability to form the hydrogen-bonded chains decreases. The maximum of length of the chains is observed for 4-methylphenol, which has a similar pK_a value to that in the tyrosine residue in biological systems.