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## Estimating the index of increase via balancing deterministic and random data

We introduce and explore an empirical index of increase that works in both deterministic and random environments, thus allowing to assess monotonicity of functions that are prone to random measurement errors. We prove consistency of the index and show how its rate of convergence is influenced by deterministic and random parts of the data. In particular, the obtained results suggest a frequency at which observations should be taken in order to reach any pre-specified level of estimation precision.We illustrate the index using data arising from purely deterministic and error-contaminated functions, which may or may not be monotonic.

The paper describes the necessary metaphysical grounds and central points of J. Searle’s general theory of social reality. It shows how in a world of physical particles and fields of force, the diversity of social life is constructed with the help of one kind of logical and linguistic operations, i.e. declarations of status functions.

The Realist interpretation of 'War and Peace' - articulated by Martin Wight and Stanley Hoffmann - is based on Tolstoy's understanding of history as it is elaborated in his account of the Napoleonic invasion in the second epilogue of the book. There Tolstoy puts forward a mechanistic view of international relations which are assumed to be governed by inexorable laws of history determining human behaviour and limiting man's exercise of free will. However, Tolstoy's subjection of man to the workings of impenetrable laws of history in the second epilogue is at variance with a multiplicity of conscious moral choices that his three main characters - Nikolay Rostov, Andrey Bolkonsky and Pierre Bezukhov - make throughout the book. It is argued that the different treatment of the freedom vs. necessity problem in the fictional and historical narrative can only be understood contextually, i.e. from within Tolstoy' rejection of the Enlightenment tradition of scientific and moral inquiry.

The paper raises the question as how we can include the category of subject within physicalist ontology without postulating metaphysical freedom of will. Significance of the issue is justified through the analysis of the notion of subject in the everyday moral discourse. Suggested answer can be described as compatibilistic. Author claims that category of the subject might be highlighted in the physical world, if we could find its causally effective feature, and such feature is intentionality.

*This article discusses the alternative possibilities condition in libertarian accounts of free will. It examines G.E. Moore’s conditional analysis and libertarian critics of this approach, than goes to libertarian view on the ability to do otherwise and its ontological conditions. Finally it shows that the ability to otherwise doesn’t suffice for freedom of will in libertarian sense and what is still needed remains mystery.*

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.