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## Kac–Moody groups and cosheaves on Davis building

We investigate smooth representations of complete Kac-Moody groups. We approach representation theory via geometry, in particular, the group action on the Davis realisation of its Bruhat-Tits building. Our results include an estimate on projective dimension, localisation theorem, unimodularity and homological duality.

Options for developing an effective insulation shell for extreme temperature conditions are considered. From the point of view of resistance to low and variable temperatures, stress strength and water absorption, polyethylene foam insulation shells are quite applicable to insulation systems. We study the experience of using seamless polyethylene shells in stationary and mobile residential modules operated in the Arctic region, as well as in insulation systems based on the principle of a "thermal blanket" to preserve snow used in ski resorts, when storing snow, it is covered with rolled polyethylene foam with a mechanical connection of sheets followed by a tent cover

In the present paper the game theory is applied to an important open question in economics: providing microfoundations for often-used types of production function. Simple differential games of bargaining are proposed to model a behavior of workers and capital-owners in processes of formation of a set of admissible factor prices or participants’ weights (moral-ethical assessments). These games result, correspondingly, in a factor price curve and a weight curve – structures dual to production function. Ultimately, under constant bargaining powers of the participants, the Cobb-Douglas production function is received.

In Soviet period absence of market prices led to extremely inefficient land use and spatial development of cities. Centralized planning system was not flexible and responsive to changing demand, preoccupied with minimization of construction costs and characterized by very low density of land use. In 20 years after the beginning of market economic reforms and mass privatization of real property the situation in land use and spatial development of Russian cities didn’t change much. Main reasons of this are: unclear, non-specified and often not registered property rights; quasi-monopoly of the state on urban lands; absence of clear distinction between federal, regional and municipal lands; high transaction costs and administrative barriers for developers; still very much administrative approach to planning and land use regulation, absence of real dialog with community development groups and NGOs. In this legal and institutional environment regional and/or local authorities often act in interests of big and influential investors and developers, scarifying interests of community as well as of small private owners and tenants. As a result we can see a further worsening of the urban environment, decreasing of green areas, disappearance of historical character of whole parts of city centers, sprawl developments in suburbia etc.

To measure transaction costs and administrative risks in urban development and construction, a survey of developers, builders and real estate agents was undertaken in St Petersburg and Leningrad region, the results of which are presented in the paper.

The results of studies of the properties and features of the use of rolled polyethylene foam with or without a metallized coating are presented. The tensile strength of the product for products with a metallized coating is 80-92 kPa, without a metallized coating-80-87 kPa, and for the weld - 29-32 kPa. Insulation systems have been developed (which have found wide practical application) and a full-scale heat engineering assessment of these systems and the condition of the wooden frame in the building has been carried out. It is established that the thermal resistance of the structure is 2.96 m2 K / W, the heat transfer resistance is 3.12 m2 KK / V. The humidity of the frame wood is 7.7-7.8%.

The Bill on amendments to Part II of the Civil Code of the Russian Federation consists of legislation provisions on a new limited interest in land - a right to development. The principal difference between the new model of development and the right to development concept in Russian and foreign treaties and the Concept of civil legislation development is pointed out in the article. Hence, a comparative analysis of the notion, provisions on the right to development entitlement and the correspondence between the right to development and the right to building is conducted by the author. The problems which can rise during the practical realization of the aforementioned model of the right to development are highlighted in the article.

We consider a monopolistic firm that sells seasonal goods. The firm seeks the minimum of the total advertising expenditure during the selling period, given that some previously defined levels of goodwill and sales have to be reached at the end of the period. The only control allowed is on advertising while goodwill and sales levels are considered as state variables. More precisely we consider a linear optimal control problem for which the general position condition does not hold so that the application of Pontryagin's Maximum Principle may not be useful to determine a solution. Therefore the dual of the problem is studied and solved. Moreover, a necessary and sufficient condition for the feasibility of the primal problem is determined.

Two factors of energy efficiency of the technical insulation system are considered, namely, the operational stability of thermal insulation products and the reduction of energy costs for their production. A special feature of the cylinder technology is that a thin section, wound on a rolling pin, is pre-formed from a mineral wool carpet made of mineral wool and after winding is subjected to heat treatment. When using a binder, the curing of which takes place in the temperature range of 130-140 ° C, it becomes possible to optimize the parameters of heat treatment, aimed at increasing the uniformity of the curing of the binder in the product and directly reducing energy costs. Based on the results of the research, an algorithm was developed, a computer program was compiled and tested to select the parameters of heat treatment, taking into account the properties of mineral fibers, binder and mineral wool carpet.

A new approach is proposed revealing duality relations between a physical side of economy (resources and technologies) and its institutional side (institutional relationsd between social groups). Production function is modeled not as a primal object but rather as a secondary one defined in a dual way by the institutional side. Differential games of bargaining are proposed to model a behavior of workers and capitalists in process of prices or weights formation. These games result, correspondingly, in a price curve and in a weight curve - structures dual to a production function. Ultimately, under constant bargaining powers of the participants, the Cobb-Douglas production function is generated.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.