[en] We describe homogeneous canonical transformations of the cotangent bundle of a manifold with conical singular points and compute the index of an elliptic Fourier integral operator obtained by the quantization of such a transformation. The answer involves the index of an elliptic Fourier integral operator on a smooth manifold and the residues of the conormal symbol

[en] We consider a control system described by a non-linear first-order evolution equation on an evolution triple of Banach spaces (a 'Gelfand triple') with a mixed multivalued control constraint whose values are non-convex closed sets in the control space. Besides the original system, we consider systems with the following control constraints: the constraint whose values are the closed convex hulls of the values of the original constraint, and the constraint whose values are the extreme points of the convexified constraint that belong to the original one. We study topological properties of the sets of admissible 'state-control' pairs for the same system with various constraints and consider the relations between them. An example of a non-linear parabolic control system is worked out in detail

[en] We consider the Hamiltonian H of a system of n pseudo-relativistic electrons in a Coulomb field of n₀ fixed nuclei. Under the assumption that the total charge of electrons and nuclei is non-negative, it is proved that the discrete spectrum of H is infinite, and a spectral asymptotic formula is derived (without taking the Pauli exclusion principle into account). The results are extended to systems of the same type with long-range potentials more general than Coulomb potentials. It is also proved that the discrete spectrum is finite in the short-range case

[en] If the Hodge conjecture (respectively the Tate conjecture or the Mumford-Tate conjecture) holds for a smooth projective variety X over a field k of characteristic zero, then it holds for a generic member X_t of a k-rational Lefschetz pencil of hypersurface sections of X of sufficiently high degree. The Mumford-Tate conjecture is true for the Hodge Q-structure associated with vanishing cycles on X_t. If the transcendental part of the second cohomology of a K3 surface S over a number field is an absolutely irreducible module under the action of the Hodge group Hg(S), then the punctual Hilbert scheme Hilb²(S) is a hyperkaehler fourfold satisfying the conjectures of Hodge, Tate and Mumford-Tate

[en] We prove the local representability and local finite basis property of varieties of associative rings and algebras over an arbitrary associative-commutative Noetherian ring Φ.

[en] We consider a problem on a half-line with a Neumann-type boundary condition. We prove the global existence of solutions and find the leading term of their large-time asymptotic expansion. The case of large initial data and critical non-linearity is considered in the case when the non-linear term of the equation decays in time as rapidly as the linear terms.

[en] We prove that for any odd number n≥1003, every non-cyclic subgroup of the 2-generator free Burnside group of exponent n contains a subgroup isomorphic to the free Burnside group of exponent n and infinite rank. Various families of relatively free n-periodic subgroups are constructed in free periodic groups of odd exponent n≥665. For the same groups, we describe a monomorphism τ such that a word A is an elementary period of rank α if and only if its image τ(A) is an elementary period of rank α+1.

[en] We consider the factorization I-K=(I-U⁺)(I-U^-), where I is the identity operator, K is an integral operator acting on some Banach space of functions summable with respect to a measure μ on (a,b) subset of (-∞,+∞) continuous relative to the Lebesgue measure, (Kf)(x)=∫_a^bk(x,t)f(t)μ(dt), x element of (a,b), and U^± are the desired Volterra operators. A necessary and sufficient condition is found for the existence of this factorization for a rather wide class of operators K with positive kernels and for Hilbert-Schmidt operators.

[en] We study open equivariant projective embeddings of homogeneous spaces such that the complement of the open orbit has codimension at least 2. We establish a criterion for the existence of such an embedding, prove that the set of isomorphism classes of such embeddings is finite, and give a construction of the embeddings in terms of Geometric Invariant Theory. A generalization of Cox's construction and the theory of bunched rings enable us to describe in combinatorial terms the basic geometric properties of embeddings with small boundary

[en] We establish interconnections between the conditions of weak convexity in the sense of Vial, weak convexity in the sense of Efimov-Stechkin, and proximal smoothness of sets in Banach spaces. We prove a theorem on the separation by a sphere of two disjoint sets, one of which is weakly convex in the sense of Vial and the other is strongly convex. We also prove that weakly convex and proximally smooth sets are locally connected, and study questions related to the preservation of the conditions of weak convexity and proximal smoothness under passage to the limit