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Using our proof of the Poincare conjecture in dimension
three and the method of mathematical induction a short and transparent proof of
the generalized Poincare conjecture (the main theorem below) has been obtained. Main Theorem. Let Mn be a n-dimensional, connected, simply connected, compact, closed, smooth manifold and there exists a smooth
finite triangulation on Mn which is coordinated with the
smoothness structure of Mn. If Sn is the n-dimensional sphere then the manifolds Mn and Sn are homemorphic.
The epitaxial growth
of Ge on Si(111) covered with the 0.3 nm thick SiO2 film is studied
by scanning tunneling microscopy. Nanoareas of bare Si in the SiO2 film are prepared by Ge deposition at a temperature in the range of 570℃-650℃ due to the formation of
volatile SiO and GeO molecules. The surface morphology of Ge layers grown
further at 360℃-500℃ is composed of facets and
large flat areas with the Ge(111)-c(2 × 8) reconstruction which is typical of
unstrained Ge. Orientations
of the facets, which depend on the growth temperature, are identified. The growth at
250℃-300℃ produces continuous
epitaxial Ge layers on Si(111). A comparison of the surface morphology of Ge
layers grown on bare and SiO2-film covered Si(111) surfaces shows a
significantly lower Ge-Si intermixing in the latter case due to a reduction in
the lattice strain. The found approach to reduce the strain suggests the
opportunity of the thin continuous epitaxial Ge layer formation on Si(111).
In this paper we consider the stochastic systems with jumps (random impulses) generated by Erlang flow of events that lead to discontinuities in paths. These systems may be used in various applications such as a control of complex technical systems, financial mathematics, mathematical biology and medicine. We propose to use a spectral method formalism to the probabilistic analysis problem for the stochastic systems with jumps. This method allows to get a solution of the analysis problem in an explicit form.
Five habitat types have been studied in the valley forests of the Ussuri Nature Reserve. The cluster analysis was used to distinguish two types of clusters. The first one combines the anthropogenically modified forest plot, the margin, and the plot of a typical valley broad-leaved tree, while the second one combines the oak forest and the coniferous and broad-leaved valley forest. The greatest number of volant Coleoptera species were observed at the margin of the broad-leaved forest.