654 research outputs found
The Liouville-type theorem for integrable Hamiltonian systems with incomplete flows
For integrable Hamiltonian systems with two degrees of freedom whose
Hamiltonian vector fields have incomplete flows, an analogue of the Liouville
theorem is established. A canonical Liouville fibration is defined by means of
an "exact" 2-parameter family of flat polygons equipped with certain pairing of
sides. For the integrable Hamiltonian systems given by the vector field
on where
is a complex polynomial in 2 variables, geometric properties of
Liouville fibrations are described.Comment: 6 page
Green Economy: Regional Priorities
The article is dedicated to transforming the economy of Russian regions to a green economy, which is an essential factor for the sustainable development. This is important not only for Russia but the whole world because our country has the great natural capital and provides important environmental services that support the planet biosphere. Based on the analysis of economic, social and ecological statistical data and Human Development Index (HDI) we have shown that the development of Russian Federal Districts is very unbalanced and each Russian region has its own way to new economic model. For instance, it is necessary to increase the well-being in the North Caucasus Federal District, it is important to reach higher life expectancy at birth in the Siberian and the Far Eastern Districts. It is necessary to move from the «brown» economy to a green one by using the human capital (building a knowledge economy), by applying Best Available Technologies (Techniques), by investing in efficiency of use of natural resources and by increasing energy efficiency. The transition to a green economy will help to achieve social equity and the development of human potential; it helps to move from the exploitation of non-renewable natural capital to renewable human capital. All these socio-economic measures should give decoupling effect, make risks lower, reduce the exploitation of natural capital, stop the environmental degradation and prevent the ecological crisis. Transition to the green economic model has to be accompanied by new economic development indicators, which take into account social and environmental factors.The research was supported by the grant of the Russian Foundation for Basic Research No. 14-06-00075
Deformations of circle-valued Morse functions on surfaces
Let be a smooth connected orientable compact surface. Denote by
the space of all Morse functions having no critical
points on the boundary of and such that for every boundary component of
the restriction is either a constant map or a covering
map. Endow with the -topology. In this note the
connected components of are classified. This result extends the
results of S. V. Matveev, V. V. Sharko, and the author for the case of Morse
functions being locally constant on the boundary of .Comment: 8 pages, 4 figure
Connected components of spaces of Morse functions with fixed critical points
Let be a smooth closed orientable surface and be the space
of Morse functions on having exactly critical points of local minima,
saddle critical points, and critical points of local maxima,
moreover all the points are fixed. Let be the connected component of a
function in . By means of the winding number introduced by Reinhart
(1960), a surjection is constructed. In
particular, , and the Dehn twist about the boundary of any
disk containing exactly two critical points, exactly one of which is a saddle
point, does not preserve . Let be the group of orientation
preserving diffeomorphisms of leaving fixed the critical points, be the connected component of in , and
the set of diffeomorphisms preserving
. Let be the subgroup of generated by
and all diffeomorphisms which preserve some
functions , and let be its subgroup
generated and the Dehn twists about the components of level
curves of functions . We prove that if , and construct an epimorphism
, by means of
the winding number. A finite polyhedral complex associated to the
space is defined. An epimorphism and finite generating sets for the groups
and in terms of the 2-skeleton of the complex
are constructed.Comment: 12 pages with 2 figures, in Russian, to be published in Vestnik
Moskov. Univ., a typo in theorem 1 is correcte
Special framed Morse functions on surfaces
Let be a smooth closed orientable surface. Let be the space of Morse
functions on , and the space of framed Morse functions, both
endowed with -topology. The space of special framed
Morse functions is defined. We prove that the inclusion mapping
is a homotopy equivalence. In the
case when at least critical points of each function of are
labeled, homotopy equivalences
and are
proved, where is the complex of framed Morse functions,
is the universal moduli
space of framed Morse functions, is the group of
self-diffeomorphisms of homotopic to the identity.Comment: 8 pages, in Russia
Topology of the spaces of Morse functions on surfaces
Let be a smooth closed orientable surface, and let be the space of
Morse functions on such that at least critical points of each
function of are labeled by different labels (enumerated). Endow the space
with -topology. We prove the homotopy equivalence where is one of the manifolds , and the point in dependence on the sign of ,
and is the universal moduli space of framed Morse
functions, which is a smooth stratified manifold. Morse inequalities for the
Betti numbers of the space are obtained.Comment: 15 pages, in Russia
ON THE CONVERGENCE OF MAPPINGS WITH k-FINITE DISTORTION.
We prove that a locally uniform limit of a sequence of homeomorphisms with finite k-distortion is also a mapping with finite k-distortion. We obtain also an estimation for the distortion coefficient of the limit mapping
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