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## Construction of Strongly Time-Consistent Subcores in Differential Games with Prescribed Duration

A new strongly time-consistent (dynamically stable) optimality principle is proposed in a cooperative differential game. This is done by constructing a special subset of the core of the game. It is proposed to consider this subset as a new optimality principle. The construction is based on the introduction of a function V^ that dominates the values of the classical characteristic function in coalitions. Suppose that V (S, x¯ (τ), T −τ) is the value of the classical characteristic function computed in the subgame with initial conditions x¯ (τ), T −τ on the cooperative trajectory. Define V^(S;X0,T−t0)=maxt0≤τ≤TV(S;x∗(τ),T−τ)V(N;X∗(τ),T−τ)V(N;x0,T−t0) Using this function, we construct an analog of the classical core. It is proved that the constructed core is a subset of the classical core; thus, we can consider it as a new optimality principle. It is also proved that the newly constructed optimality principle is strongly time-consistent.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

A game with {\em{restricted cooperation}} is a triple (N,v,\Omega), where N is a finite set of players, \Omega is a non-empty collection oft feasible coalitions , and v a characteristic function defined on \Omega. U.Faigle (1989) obtained necessary and sufficient conditions for the non-emptiness of the core for games with restricted cooperation. Unlike the classical TU games the cores for games with restricted cooperation may be unbounded. Recently Grabisch and Sudh\"olter (2012) studied the core for games whose collections of feasible coalitions has a hierarchical structure generated by a partial order relation of players.For this class of games they proposed a new concept -- the bounded core -- whose definition can be extended to the general class of games with restricted cooperation as the union of all bounded faces of the core. For this class of games the bounded core can be empty even the core is not empty. An axiomatization of the bounded core for the whole class of games with restricted cooperation is given with the help of axioms efficiency, boundedness, bilateral consistency, a weakening of converse consistency, and ordinality. Another axiomatization of the core is given for the subclass of games with non-empty cores that are bounded. The characterizing axioms are non-emptiness, covariance, boundedness, consistency, the reconfirmation property, superadditivity, and continuity.

The paper describes a recent study aimed at investigating the most efficient data imputation algorithm for several methods of data analysis such as regression modeling, factor analysis, descriptive statistics, and correlation analysis. The lack of recommendations when choosing the data imputation algorithm poses the problem of choice ambiguity in each situation.

The authors consider that the data imputation algorithm should be selected according to the method employed after data improvement. In other words, it is believed that for each data analysis method the efficiency of the same data imputation algorithm is different. The statistical experiment was used to evaluate the efficiency of several data imputation algorithms for each method of data analysis.

The core idea of statistical experiment was to compare the results of each method application used in the etalon data set (without missing values) with the results obtained on a large number of artificial subsamples generated from the original data set where missing values were filed with comparable data imputation algorithms.

Generation of subsamples was carried out via the bootstrap procedure, which allowed to undertake

statistical evaluation and to build confidence intervals for each parameter before and after the data imputation.

Through this experiment the authors managed to evaluate the efficiency of such data imputation algorithms as imputation with the average trend measures, the EM algorithm, the imputation via regression model and Hot Deck algorithm for the mentioned methods of data analysis.

This article is the foreword to «Pure Theory of Law» by Hans Kelsen reedited in 2008. The author of this article, M. Jestaedt, mentions the most common mistakes and misunderstandings suffered by the legal theory of Kelsen. Jestaedt also stresses ambiguity in reception of Kelsenian ideas. He describes the context of creation and publication of «Pure Theory of Law», and accentuates the revolutionary character of this book for its époque. Jestaedt underlines the role the adherents and neophytes of Hans Kelsen played in development of his legal doctrine. Among the most important ideas of this doctrine Jestaedt cites those about self-referential character of law and about purity. Kelsen requires such purity from the methodology of legal studies and not from law which is the object of these studies.

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.