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## Algebras of linear growth and the dynamical Mordell–Lang conjecture

Ufnarovski remarked in 1990 that it is unknown whether any finitely presented associative algebra of linear growth is automaton, that is, whether the set of normal words in the algebra form a regular language. If the algebra is graded, then the rationality of the Hilbert series of the algebra follows from the affirmative answer to Ufnarovski’s question. Assuming that the ground field has a positive characteristic, we show that the answer to Ufnarovskii’s question is positive for graded algebras if and only if the basic field is an algebraic extension of its prime subfield. Moreover, in the “only if” part we show that there exists a finitely presented graded algebra of linear growth with irrational Hilbert series. In addition, over an arbitrary infinite basic field, the set of Hilbert series of the quadratic algebras of linear growth with 5 generators is infinite. Our approach is based on a connection with the dynamical Mordell–Lang conjecture. This conjecture describes the intersection of an orbit of an algebraic variety endomorphism with a subvariety. We show that the positive answer to Ufnarovski’s question implies some known cases of the dynamical Mordell– Lang conjecture. In particular, the positive answer for a class of algebras is equivalent to the Skolem–Mahler–Lech theorem which says that the set of the zero elements of any linear recurrent sequence over a zero characteristic field is the finite union of several arithmetic progressions. In particular, the counterexamples to this theorem in the positive characteristic case give examples of algebras with irrational Hilbert series.

This chapter elaborates on entrepreneurship in developed and developing countries and focuses on the optimization of entrepreneurial activities. Various scenarios are considered: independent functioning of the market, integration in the form of reorganization (mergers and acquisitions), integration in the form of clustering, and integration in the form of innovational networks and technological parks. The optimal structure of the integration processes and best-case scenarios for its implementation to accelerate the rate and increase the quality of economic growth are substantiated. The potential for uptake of integration processes in stimulating economic growth through entrepreneurship is determined by the level of institutionalization in an economy. In developed countries, all forms of company integration are characterized by the high level of institutionalization, which allows for their effective use for economic growth. Independent companies, mergers, and acquisitions restrain economic growth and reduce its quality, while clusters, technological parks, and innovational networks accelerate the rate of economic growth and increase its quality. In developing countries, integration processes in entrepreneurship have a different influence on economic growth and require further institutionalization

In this paper as the main feature of innovation in the financial health of a company analyst view the shift to two circuits of key interests of owners of capital (financial stakeholders). Justified by differences key financial systems within the contour ownership interest and the lender three projections: liquidity, the current economic efficiency and growth. In the paper as the main feature of the innovation in the analysis of the financial health of a company is considered a transition to the two circuits of the interests of the key owners of financial capital (financial stakeholders). Justified differences of key financial indicator systems in the framework of the outline of the interests of the owner and the lender by three projections: liquidity, efficiency and the quality of growth. The examples of Russian companies have different interpretations of financial targets and indicators with a choice of activities.

This paper provides empirical analysis of macroeconomic effects of state ownership of banks. The aim is to test one of the key findings of theoretical and empirical literature of 1990s and early 2000s, namely that sizeable state ownership of commercial banks hinders financial development and economic growth. We focus on several large emerging markets including BRIC countries (Brazil, Russia, India and China) and test several specific hypotheses for the period from 1995 through 2009. Our results suggest that positive or negative sign of the government ownership impact on financial intermediation and economic growth is not constant for all times but varies depending on the type of national economy (mature market or emerging market) and, within the emerging markets category, on the level of economic development. The impact is therefore heterogeneous and not homogeneous. This finding is in contrast with the established theory but in line with the most recent empirical literature.

The article examines trends in theoretical and applied concepts about the adolescent period and in particular about how adolescents develop life goals. We discuss the blurring of the boundaries of adolescence, the postponement of life decisions, and the difficult process of separating from parents as trends in modern adolescent life. Certain features of modern society (transitivity and its consequences) complicate the ability of adolescents to plan their lives and lead them to apply tactical thinking to their lives as opposed to strategic thinking. We examine the role that the family and in particular the culture of discussing the future can play in helping adolescents plan their lives.

The conference is organized in collaboration with Polish Economic Society Branch in Toruń and Brno University of Technology (Czech Republic), BA School of Business and Finance (Latvia), Daugavpils University (Lithuania), Pereyaslav-Khmelnitsky Hryhoriy Skovoroda State Pedagogical University (Ukraine), University of Angers (France), University of Pablo de Olavide (Spain), University of Latvia (Latvia). The conference is addressed to economist from all European Union countries and Eastern Europe. It aims to bring together economists form Western, Central and Eastern Europe to discuss issues in economics, finance and business management. Main conference tracks include: 1. Macroeconomics; Microeconomics; Econometrics; International Economics 2. Financial markets; Labour markets; Institutions; 3. Business environment; Management and Marketing.

The chapter discussed the problems of the Russia’s economic competitiveness in the booming years prior to 2008 economic crisis. We estimate the competitive advantages and weaknesses, and analyze the contribution of innovations into the growth dynamics pattern.

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.