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## Математическое разрешение парадокса Гиббса

In this paper, we construct a new distribution corresponding to a real noble gas as well as the equation of state for it.

The actual tendency of competition between territories stimulated clasterization in high added value industries, including production sector. However cluster influence on region is still not clear. Some reasons for present circumstances could be found in the system of theoretical and methodological contradictions of modern cluster theory. Within the present investigation, basing on author’s model, we conducted cluster theory structuration and described six actual approaches to industrial clustering. Comparative analysis of the approaches mentioned showed that each of them tends to describe a particular aspect of cluster instead of drawing up a balance sheet. On this basis, we suggested a new logically correct approach, which is founded on system and agglomeration concepts combination. Further, the approach mentioned will form theoretical and methodological basis for cluster investigation.

The volume is dedicated to Boris Mirkin on the occasion of his 70th birthday. In addition to his startling PhD results in abstract automata theory, Mirkin’s ground breaking contributions in various fields of decision making and data analysis have marked the fourth quarter of the 20th century and beyond. Mirkin has done pioneering work in group choice, clustering, data mining and knowledge discovery aimed at finding and describing non-trivial or hidden structures—first of all, clusters, orderings, and hierarchies—in multivariate and/or network data.

This volume contains a collection of papers reflecting recent developments rooted in Mirkin's fundamental contribution to the state-of-the-art in group choice, ordering, clustering, data mining, and knowledge discovery. Researchers, students, and software engineers will benefit from new knowledge discovery techniques and application directions.

This Reference Module contains trusted, peer-reviewed, comprehensive content from our reference works as curated by our world-class editorial board led by Editor-in-Chief, Scott A. Elias. It is designed for faster, more relevant browsing within the subject and beyond, with topic pages for quick, clear overviews, subject hierarchies to put everything in context, and guidance to lead researchers to related knowledge.

The relationship between thermodynamics and economics has been known for a long time. The term ``thermoeconomics'' has even appeared. However, several aspects of the old thermodynamics are unacceptable in economics. For example, experts in thermodynamics believe that the diamond crystal is in the metastable state, and in due time will be transformed into graphite. However, these experts can hardly convince businessmen to part with their ancient diamonds.

The laws of economics require that the old conceptions of thermodynamicsa be mathematically scrutinized and reviewed.

The correspondence principle for quantum statistics, classical statistics and economics which associates the number of particles with the amount of money, the chemical potential with the nominal percentage, the negative pressure with debts, and the law of economic preference allowed to obtain agreement of the general theory of thermoeconomics with the latest experimental data.

The present investigation considers phenomenon of tourism cluster as a theoretical model of region tourism complex administration. Basing on authors’ approach to theoretical analysis, deep investigation of the concept “tourism cluster” was realized. Following the results of the analysis mentioned, we described a number of theoretical approaches to the category of tourism cluster. All the approaches mentioned were presented in two different ways: static and dynamic ones. The dynamic model provided a general view on historical development of the category “tourism cluster” in the scope of economic theory.

Basing on complex analysis results, we created our own “system and agglomeration” approach to the concept of “tourism cluster”. The approach mentioned was structured as follows: authors’ interpretation of the category “tourism cluster”, feature-based table, tourism cluster graphical model, authors’ methodology of tourism cluster identification and complex analysis of this type system.

New methodology was tested on the case of Perm krai.

The set of theoretical and practical innovations suggested in the present research is considered as universal one and it could be used in the practice of other regions administration. The present monograph could be recommended to the specialists working in the sphere of public administration ( specially to those of them who is responsible for tourism policy); to students of university programs related with tourism , economics, public administration and to all of those who are interested in the sphere of special development and tourism development in the Russian Federation.

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.