Liquid-phase chlorination ofC-chlorovinylsilanes

Liquid-phase chlorination of a number of chloro(chlorovinyl)methylsilanes was investigated. A number of novelC-chlorosilanes were characterized by IR and¹H NMR spectra. Some regularities of these reactions were determined; correlations between the structure of chloro(chlorovinyl)methylsilanes and their reactivities were identified.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 11, pp. 2256–2260, November, 1995.     

Singularities and noncommutative Frobenius manifolds

We prove that the quaternionic miniversal deformations of an A _n singularity have the structure of a noncommutative Frobenius manifold in the sense of the extended cohomological field theory.     

A note on 4-dimensional locally conformally flat Walker manifolds

A 4-dimensional Walker metric on a semi Riemanian manifold M, for the canonical metric with c = 0, have been investigated by M. Chaichi, E. García—Río and Y. Matsushita. The paper generalizes these notions to the case of constant c ≠ 0. Specially the form of defining functions of this metric in locally conformally flat 4-dimensional Walker manifolds is found.     

On the bidirectional vortex and other similarity solutions in spherical coordinates

The bidirectional vortex refers to the bipolar, coaxial swirling motion that can be triggered, for example, in cyclone separators and some liquid rocket engines with tangential aft-end injectors. In this study, we present an exact solution to describe the corresponding bulk motion in spherical coordinates. To do so, we examine both linear and nonlinear solutions of the momentum and vorticity transport equations in spherical coordinates. The assumption will be that of steady, incompressible, inviscid, rotational, and axisymmetric flow. We further relate the vorticity to some power of the stream function. At the outset, three possible types of similarity solutions are shown to fulfill the momentum equation. While the first type is incapable of satisfying the conditions for the bidirectional vortex, it can be used to accommodate other physical settings such as Hill’s vortex. This case is illustrated in the context of inviscid flow over a sphere. The second leads to a closed-form analytical expression that satisfies the boundary conditions for the bidirectional vortex in a straight cylinder. The third type is more general and provides multiple solutions. The spherical bidirectional vortex is derived using separation of variables and the method of variation of parameters. The three-pronged analysis presented here increases our repertoire of general mean flow solutions that rarely appear in spherical geometry. It is hoped that these special forms will permit extending the current approach to other complex fluid motions that are easier to capture using spherical coordinates.     

Integrable models for vicious and friendly walkers

We consider random walks of two essentially different classes of random walkers, namely, of vicious and friendly ones, on one-dimensional lattices with periodic boundary conditions. The walkers are called vicious since, arriving at a lattice site, they annihilate not only one another but all the remaining walkers as well. On the contrary, an arbitrary number of friendly walkers can share the same lattice sites. It is shown that a natural model describing the behavior of friendly walkers is an integrable model of the boson type. A representation of the generating function for the number of the lattice paths performed by a fixed number of friendly walkers for a certain number of steps is obtained. Bibliography: 22 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 59–74.     

Essentially normal Hilbert modules and Khomology Ⅲ: Homogenous quotient modules of Hardy modules on the bidisk

In this paper, we study the homogenous quotient modules of the Hardy module on the bidisk. The essential normality of the homogenous quotient modules is completely characterized. We also describe the essential spectrum for a general quotient module. The paper also considers K-homology invariant defined in the case of the homogenous quotient modules on the bidisk. This work is partially supported by the National Natural Science Foundation of China (Grant No. 10525106), the Young Teacher Fund, the National Key Basic Research Project of China (Grant No. 2006CB805905) and the Specialized Research for the Doctoral Program     

On the limit cycles of a quintic planar vector field

This paper concerns the number and distributions of limit cycles in a Z_2-equivariant quintic planar vector field.25 limit cycles are found in this special planar polynomial system and four different configurations of these limit cycles are also given by using the methods of the bifurcation theory and the qualitative analysis of the differential equation.It can be concluded that H(5)≥25=5~2, where H(5)is the Hilbert number for quintic polynomial systems.The results obtained are useful to study the weakened 16th Hilbert problem.