A perspective on non-commutative frame theory

This paper extends the fundamental results of frame theory to a non-commutative setting where the role of locales is taken over by \'etale localic categories. This involves ideas from quantale theory and from semigroup theory, specifically Ehresmann semigroups, restriction semigroups and inverse semigroups. We establish a duality between the category of complete restriction monoids and the category of \'etale localic categories. The relationship between monoids and categories is mediated by a class of quantales called restriction quantal frames. This result builds on the work of Pedro Resende on the connection between pseudogroups and \'etale localic groupoids but in the process we both generalize and simplify: for example, we do not require involutions and, in addition, we render his result functorial. We also project down to topological spaces and, as a result, extend the classical adjunction between locales and topological spaces to an adjunction between \'etale localic categories and \'etale topological categories. In fact, varying morphisms, we obtain several adjunctions. Just as in the commutative case, we restrict these adjunctions to spatial-sober and coherent-spectral equivalences. The classical equivalence between coherent frames and distributive lattices is extended to an equivalence between coherent complete restriction monoids and distributive restriction semigroups. Consequently, we deduce several dualities between distributive restriction semigroups and spectral \'etale topological categories. We also specialize these dualities for the setting where the topological categories are cancellative or are groupoids. Our approach thus links, unifies and extends the approaches taken in the work by Lawson and Lenz and by Resende.Comment: 69 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

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 $v=(-\partial f/\partial w, \partial f/\partial z)$ on ${\mathbb C}^2$ where $f=f(z,w)$ is a complex polynomial in 2 variables, geometric properties of Liouville fibrations are described.Comment: 6 page

Invariant means on Boolean inverse monoids

The classical theory of invariant means, which plays an important role in the theory of paradoxical decompositions, is based upon what are usually termed `pseudogroups'. Such pseudogroups are in fact concrete examples of the Boolean inverse monoids which give rise to etale topological groupoids under non-commutative Stone duality. We accordingly initiate the theory of invariant means on arbitrary Boolean inverse monoids. Our main theorem is a characterization of when a Boolean inverse monoid admits an invariant mean. This generalizes the classical Tarski alternative proved, for example, by de la Harpe and Skandalis, but using different methods