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## On Pareto Optimality Conditions in the Case of Two-dimension Non-convex Utility Space

The paper suggests a new --- to the best of the author's knowledge --- characterization of Pareto-optimal decisions for the case of two-dimensional utility space which is not supposed to be convex. The main idea is to use the angle distances between the bisector of the first quadrant and points of utility space. A necessary and sufficient condition for Pareto optimality in the form of an equation is derived. The first-order necessary condition for optimality in the form of a pair of equations is also obtained.

This paper contains the research of neuroeconomics results such as formulation and analysis of Ultimatum game (see Alan G. Sanfey, 2003) and neuromarketing (see Patrick Renvoisé, 2005). As a result the rational behavior of consumer during the decision-making of consume object prejudiced. In particular the axiom of reflexiveness of the rational utility theory was disproved. That axiom maintains that the fixed set of goods is not worse that itself. A conclusion that consumer choice based on the utility criterion depends not only on the set of goods but on the consume environment was made. The hypothesis of irrational behavior allowed to formalize floating utility criterion and correlation between the basket of products utility and consume environment during the consumer decision-making. Based on floating utility criterion the problem of optimal consumer’s budget distribution in conditions of integral utility maximization on limited time interval and consideration of the predicted environment factors value posed. Then the problem of intertemporal consumer choise for floating criterion was posed. The solution analysis of that problems had allowed to draw a conclusion of a significant influence of the predicted environment factors value exactness on an optimal solution and a dependence of that exactness on a consumer satisfaction.

The decision two-criteria task of simultaneously reached maximum through the free point’s method modification is constructed using the principle of guaranteed result.

Proceedings include extended abstracts of reports presented at the III International Conference on Optimization Methods and Applications “Optimization and application” (OPTIMA-2012) held in Costa da Caparica, Portugal, September 23—30, 2012.

The concept of economic equilibrium under uncertainty is applied to a model of insurance market where, in distinction to the classic Borch's model of a reinsurance market, risk exchanges are allowed between the insurer and each insured only, not among insureds themselves. Conditions characterizing an equilibrium are found. A variant of the conditions, based on the Pareto optimality notion and involving risk aversion functions of the agents, is derived. An existence theorem is proved. Computation of the market premiums and optimal indemnities is illustrated by an example with exponential utility functions.

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.