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## On the Theory of Coconvex Bodies

If the complement of a closed convex set in a closed convex cone is bounded, then this complement minus the apex of the cone is called a coconvex set. Coconvex sets appear in singularity theory (they are closely related to Newton diagrams) and in commutative algebra. Such invariants of coconvex sets as volumes, mixed volumes, number of integer points, etc., play an important role. This paper aims at extending various results from the theory of convex bodies to the coconvex setting. These include the Aleksandrov–Fenchel inequality and the Ehrhart duality.

The paper analyses applicability of the market multipliers for the diagnostics of overheating of the stock market and the formation of a financial bubble. Advantages and shortcomings of employed multipliers are discussed Historical averages of multipliers are provided. Examples are given of diagnosing the overvaluation by using the multipliers. The author concludes concludes that the optimal multiplier for estimating the overvaluation of the stock market is the P/E coefficient, however other coefficients can also be used if their limitations are taken into account.

This paper aims at explaining the differences in valuation of banking firms in Russia through the impact of selected elements of corporate governance. We rely upon value-based management theory to test the hypothesis that expenses on corporate governance system create shareholder value. The price at which share stakes are acquired by strategic foreign investors is for us a criterion of market-proven value, so we use the standard valuation tool, i.e. price-to-book-value of equity (P/BV) multiple, as the dependent variable. The set of corporate governance parameters whose materiality for a would-be external investor we would like to test includes: the degree of concentration of ownership and control; maturity of corporate governing bodies; degree of Board independence; qualification of external auditors; stability of governing bodies (Management Board and Board of Directors); and availability of external credit ratings from the world’s leading rating agencies. We test our approach on a sample of acquisition deals and public offerings over the period 2004-2008 that we develop for the first time. Firstly, we find out which factors are statistically significant and relevant to a bank’s selling price. Secondly, a least squares multiple linear regression model is devised to check how each individual variable impacts the dependent variable. We discover that external investors attach value to high concentration of ownership, external credit rating coverage, stability of the Board of Directors, and involvement of well-established external auditors. Investors of a strategic nature tend to pay a higher acquisition premium. Independence of the Board of Directors might be perceived by external strategic investors as a disadvantage and might destroy shareholder value.

The differential ring of tropical fans with polynomial weights is constructed. It contains the conventional ring of tropical fans with the operation of tropical intersection, and the ring of piecewise-polynomial functions, and its differential is a generalization of the corner locus of a piecewise-linear function. As an application, the mixed volume is computed in terms of the product of the support functions of the arguments, and a new construction for the intersection theory on smooth tropical varieties is guven.

This article is the first part of the historical review of the occurrence and development of concep- tual approaches to measuring goodwill in economic science since the end of the 19 century to the 70-ies of 20 century. The problem of goodwill measurement arose in economic science at the end of the 19th century and still discussed in the academic and practitioner communities around the world. Despite numerous studies and the adoption of accounting standards issued by various pro- fessional organizations internationally, existing opinions on this issue vary and change frequently. The need to preserve the established recognition criteria, on the one hand, and the need to provide useful information, on the other, has led to a number of controversial issues in the measurement and recognition of goodwill. In the study we analyze the historical experience in the form of goodwill perceptions, identifying historical patterns suitable for improvement of modern theory and practice of measuring goodwill. Methodological basis of the study consists of the works of distinguished sci- entists in the fields of accounting, international and generally accepted standards of accounting and reporting. The authenticity of the author’s findings confirmed by a logical use of scientific methods such as historical-and-comparative, historical-and-typological and historical-and-system method. The author track back the transformation of methods of measuring goodwill in academic research and normative documents of the nineteenth and twentieth centuries. Separate section is devoted to modern concepts of goodwill measurement, which represents an alternative to the existing account- ing standards. а gradual, cumulative and cyclical process of development of methods for measuring goodwill was identified. We found that in periods of economic growth the paradigm of current value usually dominates, while in periods of recession the historical cost paradigm is rolled back.

When writing this tutorial, the contributions received by the authors with the assistance of the NTF - National Training Fund subproject "Creating a center of excellence for economics teachers ' Innovation Project Development of education and work , received diplomas Russian competitions intellectual projects " Ideas for Russia " ( 2004) and "Power" (2008 , the Public Chamber of the Russian Federation ) . The manual is intended for students and undergraduates enrolled in the direction of "Economics" and "Management" , and may also be useful to managers and professionals , both financial and non-financial corporations.

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.