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.     

An Asymptotic Ratio in the Complete Binary Tree

Let T _n be the complete binary tree of height n considered as the Hasse-diagram of a poset with its root 1 _n as the maximum element. For a rooted tree T, define two functions counting the embeddings of T into T _n as follows A(n;T)=|{S T _n  : 1 _n ∈S, S≅T}|, and B(n;T)=|{S T _n :1 _n ∉S, S≅T}|. In this paper we investigate the asymptotic behavior of the ratio A(n;T)/B(n;T), and we show that lim  _n→∞[A(n;T)/B(n;T)]=2^ℓ;−1−1, for any tree T with ℓ leaves. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.