Vol 95, No 1 (2017)

A generalized equilibrium for conflict problems with partially overlapping game sets of participants is proposed; this equilibrium turns out to be nonempty even in those cases where all known equilibria are empty.

New lower bounds are found for the minimum number of colors needed to color all points of a Euclidean space in such a way that any two points at a distance of 1 have different colors.

In this paper we prove that for a linearly ordered skewfield the groups of quitients of the semigroup G_n(D) coincides with the group GL_n(D) for n ≥ 3.

For the general integrability case of M. Adler and P. van Moerbeke, invariant relations are obtained in which the rank of the momentum map is 1. Thereby, special periodic solutions generating the edges of the bifurcation diagram are defined. All phase variables are expressed in terms of a set of constants and one auxiliary variable, for which a differential equation integrable in elliptic functions time is given. An explicit expression for the characteristic exponent determining the type of special periodic solutions is presented, which makes it possible to study the character of stability of the obtained solution.

A quantum anomaly is the breaking of symmetry with respect to some transformations after the quantization of a classical Hamiltonian or Lagrangian system. It is shown that both the Noether theorems (including their infinite-dimensional versions) and the explanation of the origin of quantum anomalies can be obtained by using similar formulas for derivatives of functions whose values are measure in the former case and pseudomeasures in the latter.

Let α be an automorphism of a finite group G. For a positive integer n, let E_G,n(α) be the subgroup generated by all commutators [...[[x,α],α],…,α] in the semidirect product G 〈α〉 over x ∈ G, where α is repeated n times. By Baer’s theorem, if E_G,n(α)=1, then the commutator subgroup [G,α] is nilpotent. We generalize this theorem in terms of certain length parameters of E_G,n(α). For soluble G we prove that if, for some n, the Fitting height of E_G,n(α) is equal to k, then the Fitting height of [G,α] is at most k + 1. For nonsoluble G the results are in terms of the nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F_h* (H) = H, where F₀* (H) = 1, and F_i+1* (H) is the inverse image of the generalized Fitting subgroup F*(H/F_i*(H)). Let m be the number of prime factors of the order |α| counting multiplicities. It is proved that if, for some n, the generalized Fitting height E_G,n(α) of is equal to k, then the generalized Fitting height of [G,α] is bounded in terms of k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λE_G,n(α)= k, then the nonsoluble length of [G,α] is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.

We consider a natural generalization to the class of all GLP-algebras of the so-called reduction property for the polymodal provability algebras in arithmetic. An analogue of this property is established for the free GLP-algebras and for some topological GLP-algebras (GLP-spaces).

A random graph G(n, p) is said to obey the (monadic) zero–one k-law if, for any monadic formula of quantifier depth k, the probability that it is true for the random graph tends to either zero or one. In this paper, following J. Spencer and S. Shelah, we consider the case p = n^−α. It is proved that the least k for which there are infinitely many α such that a random graph does not obey the zero–one k-law is equal to 4.

Strategic games in which the profit of each participant is the sum of local profits gained by means certain “objects” and shared by all players using these objects are considered. If each object satisfies a regularity condition, then the game has an exact potential. Both universal classes of potential games considered previously in the literature, overflow games and games with structured utility functions, are contained in the class under consideration and satisfy the regularity condition.

A constructive method is proposed for finding finite-dimensional submanifolds in the space of smooth functions that are invariant with respect to flows defined by evolutionary partial differential equations. Conditions for the stability of these submanifolds are obtained. Such submanifolds are constructed for generalized Rapoport–Leas equations that arise in the theory of porous media flows.

Given a closed weakly regular d-thick subset S of ℝⁿ, we prove the existence of a bounded linear extension operator Ext: Tr_|SW_p¹ (ℝⁿ, γ) → W_p¹(ℝⁿ, γ) for p ∈ (1, ∞), 0 ≤ d ≤ n, r ∈ (max{1, n − d}, p), l ∈ ℕ, and \(\gamma \in {A_{\frac{p}{r}}}\)(ℝⁿ). In particular, we prove that a linear bounded trace space exists in the case where S is the closure of an arbitrary domain in ℝⁿ, γ ≡ 1, and p > n − 1. The obtained results supplement those of previous studies, in which a similar problem was considered either in the case of p ∈ (n, ∞) without constraints on the set S or in the case of p ∈ (1, ∞) under stronger constraints on the set S.

The Hudson–Parthasarathy equation and the Itô–Schrödinger equation (known also as the Belavkin equation) describe a Markov approximation of the dynamics of open quantum systems. The former is a stochastic version of the classical Heisenberg equation, while the latter is a stochastic version of the classical Schrödinger equation (but this analogy is not complete). Two versions of stochastic Heisenberg equations are considered, one of which uses a white noise operator constructed from (non–self-adjoint) birth and death operators and the other uses a white noise operator constructed from (self-adjoint) coordinate and momentum operators.

A numerical method is proposed for constructing an external polyhedral estimate for the trajectory tube of a nonlinear dynamic system described by a differential inclusion. The method is based on the approximation of cross sections of the trajectory tube (reachable sets) for an auxiliary system described by the convex hull of the graph of the differential inclusion. It produces polyhedral estimates suitable for the direct study of tubes via computer visualization and for the solution of more general problems.

A variational method for the optimal control of moving sources governed by a parabolic equation with nonlocal integral conditions is considered. For this problem, an existence and uniqueness theorem is proved, necessary optimality conditions in the form of pointwise and integral maximum principles are obtained, sufficient conditions for the Fréchet differentiability of the cost functional are found, and an expression for its gradient is derived.