### Article

## Построение сильно-динамически устойчивых подъядер в дифференциальных играх с предписанной продолжительностью

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* ^ V^ that dominates the values of the classical characteristic function in coalitions. Suppose that *V*(*S*,*x* ¯ (*τ*),*T*−*τ*) V(S,x¯(τ),T−τ) is the value of the classical characteristic function computed in the subgame with initial conditions *x* ¯ (*τ*) x¯(τ) , *T*−*τ* T−τ on the cooperative trajectory. Define

*V* ^ (*S*;*x* 0 ,*T*−*t* 0 )=max *t* 0 ≤*τ*≤*T* *V*(*S*;*x* ∗ (*τ*),*T*−*τ*)*V*(*N*;*x* ∗ (*τ*),*T*−*τ*) *V*(*N*;*x* 0 ,*T*−*t* 0 ). 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 proved also that the newly constructed optimality principle is strongly time-consistent.

Various Condorcet consistent social choice functions based on majority rule (tournament solutions) are considered in the general case, when ties are allowed: the core, the weak and strong top cycle sets, versions of the uncovered and minimal weakly stable sets, the uncaptured set, the untrapped set, classes of k-stable alternatives and k-stable sets. The main focus of the paper is to construct a unified matrix-vector representation of a tournament solution in order to get a convenient algorithm for its calculation. New versions of some solutions are also proposed.

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.

A cooperative finite-stage dynamic *n*-person prisoner's dilemma is considered. The time-consistent subset of the core is proposed. The Shapley value for the stochastic model of the *n*-person prisoner's dilemma is calculated in explicit form.

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.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

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.