We study the first integrals polynomial and rational in momenta of the geodesic flows (including those in a magnetic field) on two-dimensional surfaces.

The asymptotic formula is obtained for the capacity of a generalized condenser when parts of its plates contract to prescribed points. In contrast to the previous studies of capacity, we consider condensers with variable potential levels and establish the third term of the asymptotics.

Using the Pontryagin maximum principle for the time-optimal problem in coordinates of the first kind, we find the extremals of an arbitrary left-invariant sub-Finsler metric on the Engel group defined by a distribution of rank 2.

Under consideration are the bundle of paracomplex structures and the related problems of the existence of a paracomplex structure on a manifold. We obtain some explicit descriptions for the bundle of paracomplex structures for spheres of dimensions 2, 4, and 6. The existence is proved of a nonitegrable almost paracomplex structure on the six-dimensional sphere.

We study the generalized Leibniz brackets on the coordinate algebra of the \( n \)-dimensional sphere. In the case of the one-dimensional sphere, we show that each of these is a bracket of vector type. Each Jordan bracket on the coordinate algebra of the two-dimensional sphere is a generalized Poisson bracket. We equip the coordinate algebra of a sphere of odd dimension with a Jordan bracket whose

We obtain some analogs of the Liouville property for the function that is harmonic on the exterior of a Jordan domain \( G\subset{\mathbb{C}} \) and has constant boundary values of the function itself and its normal derivative. We show that these conditions cannot be relaxed in general.

The study of the Cauchy problem for solutions of the heat equation in a cylindrical domain with data on the lateral surface by the Fourier method raises the problem of calculating the inverse Laplace transform of the entire function \( \cos\sqrt{z} \). This problem has no solution in the standard theory of the Laplace transform. We give an explicit formula for the inverse Laplace transform of \( \cos\sqrt{z}

We prove some exponential Chebyshev inequality and derive the large deviation principle and the law of large numbers for the graphons constructed from a sequence of Erdős–Rényi random graphs with weights. Also, we obtain a new version of the large deviation principle for the number of triangles included in an Erdős–Rényi graph.

We prove the area formula for the classes of graph mappings on the Carnot groups and nilpotent graded groups with sub-Lorentzian structure which are of arbitrary dimension and depth. The number of spacelike and timelike directions is arbitrary as well.

We construct an example of a quasi-Kähler structure of cohomogeneity 1 on \( S^{2}\times S^{4} \).

Studying the behavior of solutions to the Cauchy problem for a family of nonlinear functional differential equations with delay which arise in the living system models, we establish the conditions that provide some exponential decay estimates for components of solutions. We find the parameters of simultaneous exponential estimates as solutions to nonlinear inequalities built from the majorants of the

Under study are the boundary value problems that describe the equilibria of two-dimensional elastic bodies with thin weakly curved inclusions in the presence of delamination, which means that there is a crack between the inclusions and an elastic body. Some inequality-type nonlinear boundary conditions are imposed on the crack faces that exclude mutual penetration. This puts the problems into the class

Considering compact pseudodifferential operators with symbols whose smoothness in \( x \) vanishes on a prescribed set, we obtain some validity conditions for the Weyl spectral asymptotics of singular values. These results are applied to the symbols whose decay order as \( |\xi|\to\infty \) is a nonsmooth function of \( x \).

We describe the statements of boundary value problems for multidimensional higher order equations of mixed type and propose a method for constructing coercive boundary value problems. Also, we prove the generalized and regular solvability of these boundary value problems in Sobolev spaces.

We obtain a characterization of the multidimensional bilinear Hardy inequality in weighted Lebesgue spaces.

A sequence of integers is called a T-sequence if there exists a Hausdorff group topology on the integers such that the sequence converges to 0. Given a finite set S of primes, we construct some Hausdorff group topology on the integers such that every increasing sequence with terms divisible only by primes from S converges to 0. Also we answer in the affirmative the question on T-sequences which was

Basing on the notion of algebraic function in real function theory, we study the question of the syntactic definition of the functions on universal algebras whose graphs are algebraic or piecewise-algebraic sets.

Under study is the integral inequality that has as kernel a nonnegative polynomial in the powers of the difference of arguments and a large parameter N. We establish some inequality whose form agrees with the celebrated Gronwall-Bellman inequality in which the argument of the exponent depends linearly on N.

Considering the partially commutative groups of the varieties that include the variety of all length 2 nilpotent groups, we study the questions of factor cancellation in direct products, coincidence of elementary theories, characterization of the group by its elementary theory, and the possibility of direct product decomposition.

We prove completeness theorems for the set of all d-closed d-bounded sets in a (q1, q2)-quasimetric space (X, d) equipped with suitable analogs of the Hausdorff distance.

Under study are the notions of transitivity, regularity, and σ-regularity for Boolean-valued algebraic systems of set-theoretic signature. The notion of a universe over an arbitrary extensional Boolean-valued system is introduced. Some description is proposed for the structure of the universe by means of various hierarchies. The results are used for proving the uniqueness of a Boolean-valued universe

Studying the properties of almost ω-categorical quite o-minimal theories, we prove that the arbitrary families of pairwise weakly orthogonal nonalgebraic 1-types in these theories are orthogonal. We also establish the binarity of these theories.

Studying Boolean algebras with distinguished subalgebras, we establish the existence of continuum many simple superatomic Boolean algebras with distinguished subalgebra whose elementary theories are distinct and each of them lacks countably saturated models. The subalgebras coincide with the Boolean algebras modulo their Fréchet ideals.

We characterize the topological spaces that are homeomorphic to the spectra of posets with certain properties.

We continue the study of multivalued mappings with the BAD (bounded angular distortion) property which was initiated in the author’s works in 2018. Using the technique of ultrafilters, we obtain a full description for the mutual adherence domains of the images of points under mappings of BAD class. We exhibit the example that illustrates all possible cases in the general formula for adherence domains

Given a root class \(\mathscr{C}\) of groups and a tree product P in which each amalgamated subgroup lies in the centers of the corresponding vertex groups, we point out certain sufficient conditions for the \(\mathscr{C}\)-residuality of P. In particular, we show that the tree product of solvable groups of bounded derived length with central amalgamated subgroups is residually solvable.

We prove the theorem that fully describes relatively intrinsically computable relations on Boolean algebras with a distinguished set of atoms.

We describe the (n + s)-dimensional solvable Leibniz algebras whose nilradical has characteristic sequence (m1, …, ms), where m1 +⋯+ ms = n. The completeness and cohomological rigidity of this algebra are proved.

We show the spectral universality of the class of structures that are linear orders with an additional binary relation and hence with an n-ary relation for each n ≥ 2. To this end, we study the category of the structures. We then obtain some other algorithmic properties of the category by using the notion of a computable functor which was studied in recent papers by other authors.

We prove that, for every 0 < ϵ < 1, there exists a measurable set E ⊂ T = [0, 1]2 with measure ∣E∣ > 1 − ϵ such that, for all f ∈ L1(T) and 0 < η < 1, we can find \(\tilde f \in {L^1}(T)\) with \(\int\!\!\!\int_T {{\rm{|}}f(x,y) - \tilde f(x,y){\rm{|}}dxdy \le \eta } \) coinciding with f(x, y) on E whose double Fourier-Franklin series converges absolutely to f almost everywhere on T.

We construct the example of an admissible set \(\mathbb{A}\) such that there exists a positive computable \(\mathbb{A}\)-numbering of the family of all \(\mathbb{A}\)-c.e. sets, whereas any negative computable \(\mathbb{A}\)-numberings are absent.

We introduce the concept of KΣ-structure and prove the existence of a universal Σ-function in the hereditarily finite superstructure over this structure. We exhibit some examples of families of KΣ-structures of the theory of trees.

Let \(\mathfrak{M}\) be a set of finite groups. Given a group G, denote by \(\mathfrak{M}(G)\) the set of all subgroups of G isomorphic to the elements of \(\mathfrak{M}\). A group G is said to be saturated with groups from \(\mathfrak{M}\) (saturated with \(\mathfrak{M}\), for brevity) if each finite subgroup of G lies in an element of \(\mathfrak{M}(G)\). We prove that a periodic group G saturated

We consider a class of diffeomorphisms G = (G1, G2) : M → N ⊂ R1 × R2 between Riemannian manifolds, where N is equipped with the product metric, which allows us to investigate the pairs of foliations on M consisting of level sets of coordinates G1 and G2, respectively. We give some lower and upper estimates for the product of conjugate modules of these pairs of foliations that depend on the properties

Let p be a prime and let σ = {{p}, {p}′} be a partition of the set ℙ of all primes. We prove the following conjecture by Skiba: If each complete Hall set of type σ in a finite group G is reducible to some subgroup H of G then H is σ-subnormal in G.

The author had earlier defined the concept of an r-group, generalizing the concept of a rigid (solvable) group. This article proves that every metabelian r-group is equationally Noetherian; i.e., each system of equations in finitely many variables with coefficients in the group is equivalent to some finite subsystem.

We study the compressed zero-divisor graph of a finite associative ring R. In particular, we describe commutative finite rings with compressed zero-divisor graphs of order 2. Moreover, we find all graphs of order 3 that are the compressed zero-divisor graphs of some finite rings.

The Manturov (2, 3)-group G23 is the group generated by three elements a, b, and c with defining relations a2 = b2 = c2 = (abc)2 = 1. We explicitly calculate the Anick chain complex for G23 by algebraic discrete Morse theory and evaluate the Hochschild cohomology groups of the group algebra \(\mathbb{k}G_3^2\) with coefficients in all 1-dimensional bimodules over a field \(\mathbb{k}\) of characteristic

The Cauchy-Dirichlet problem for the anisotropic parabolic equation with variable exponents in the presence of a nonlinear source and gradient term is considered. We prove the existence and uniqueness of a weak solution that is Lipschitz continuous in the space variables.

A subgroup A is called seminormal in a group G if there exists a subgroup B such that G = AB and AX is a subgroup of G for every subgroup X of B. Studying a group of the form G = AB with seminormal supersoluble subgroups A and B, we prove that \(G^{\mathfrak{U}}=\left(G^{\prime}\right)^{\mathfrak{N}}\). Moreover, if the indices of the subgroups A and B of G are coprime then \(G^{\mathfrak{U}}=\lef

We prove some two- and three-point distortion theorems for rational functions that generalize some recent results on Bernstein-type inequalities for polynomials and rational functions. The rational functions under study have either majorants or restrictions on location of their zeros. The proofs are based on the new version of the Schwarz Lemma and univalence condition for regular functions which was

We find sharp asymptotics of the probability that the trajectory of a compound renewal process crosses (or does not cross) an arbitrary remote boundary. In particular, some limit theorems are obtained for the distribution of the maximum of the process in the domain of large deviations. We also give some applications to the classical ruin probability problem in insurance theory.

We show that the Jordan bracket on an associative commutative superalgebra is extendable to the superalgebra of fractions. In particular, we prove that a unital simple abelian Jordan superalgebra is embedded into a simple superalgebra of a Jordan bracket. We also study the unital simple Jordan superalgebras whose even part is a field. We demonstrate that each of these superalgebras is either a superalgebra

We consider a nonautonomous first-order linear differential equation with several delays and nonnegative coefficients. Some new sufficient conditions for the oscillation of all solutions are obtained in the form of an estimate for the lower limit of the sum of integrals of the coefficients. For the equation with one delay, the obtained oscillation conditions sharpen the classical Koplatadze-Chanturiya

We consider the three-dimensional mixed boundary value problem in elasticity about time harmonic oscillations of a semi-infinite anisotropic cylinder. We show that for certain position and shape of the clamping zone of the surface the elastic wave is trapped; i.e., the problem admits a nontrivial solution with exponential decay at infinity or, conversely, the absence of the trapped wave is guaranteed

Studying the elementary properties of free projective planes of finite rank, we prove that for m > n, an arbitrary ∀∃∀-formula Φ(ȳ) and a tuple ū of elements of the free projective plane \(\mathfrak{F}_{n}\) if Φ(ū) holds on the plane \(\mathfrak{F}_{m}\) then Φ(ū) holds on the plane \(\mathfrak{F}_{n}\) too. This implies the coincidence of the ∀∃-theories of free projective planes of different finite

We consider mixed problems for nonlinear equations of magnetoelasticity. Our main result in the three-dimensional case is the proof of an existence and uniqueness theorem; uniqueness is established under some extra restrictions on the smoothness of solutions. We also manage to prove the existence and uniqueness of a weak solution to the problem in the two-dimensional case; uniqueness is established

We construct an additive basis for the relatively free associative algebra F(5)(K) with the Lie nilpotency identity of degree 5 over an infinite domain K containing \({1 \over 6}\). We prove that approximately half of the elements in F(5)(K) are central. We also prove that the additive group of F(5)(ℤ) lacks the elements of simple degree ≥ 5. We find an asymptotic estimation of the codimension of T-ideal

Let An denote the alternating group of degree n. Consider a group G whose spectrum, i.e. the set of element orders, equals the spectrum of A7. Assume that G has a subgroup H isomorphic to A4 whose involutions are squares of elements of order 4. Then either O2(H) ⊆ O2(G)or G has a nonabelian finite simple subgroup.

We prove that the group of tame automorphisms of a free Lie algebra of rank 3 (as well as of a free anticommutative algebra) over an arbitrary integral domain has the structure of an amalgamated free product. We construct an example of a wild automorphism of a free Lie algebra of rank 3 (as well as of a free anticommutative algebra) over an arbitrary Euclidean ring analogous to the Anick automorphism

Studying the solvable subgroups of 2 × 2 matrix groups over Z, we find a maximal finite order primitive solvable subgroup of GL(2, Z) unique up to conjugacy in GL(2, Z). We describe the maximal primitive solvable subgroups whose maximal abelian normal divisor coincides with the group of units of a quadratic ring extension of Z. We prove that every real quadratic ring R determines h classes of conjugacy

We answer (in the negative) Questions 16.15(a) and 17.3 of The Kourovka Notebook: “Is the set of all restricted Engel elements a subgroup in every group?” and “Is it true that a group is binary nilpotent if any 4-element subset therein contains two elements generating a nilpotent subgroup?”

We prove that the Solovay set Σ is absolutely definable in a sufficiently wide sense; in particular, Σ does not depend on the choice of the ground model.

Let φ be a positive functional on a von Neumann algebra \(\mathscr{A}\) and let \(\mathscr{A}^{\rm{pr}}\) be the projection lattice in \(\mathscr{A}\). Given \(P,Q \in \mathscr{A}^{\rm{pr}}\), put ρφ(P, Q) = φ(∣P − Q∣) and dφ(P, Q) = φ(P ∨ Q − P ∧ Q). Then ρφ(P, Q) ≤ dφ(P, Q) and ρφ(P, Q) = dφ(P, Q) provided that PQ = QP. The mapping ρφ (or dφ) meets the triangle inequality if and only if φ is a tracial

We propose a general scheme for the search of a fundamental solution to the hypoelliptic diffusion equation in a “sufficiently good” sub-Riemannian manifold and the small-time asymptotics for the solution, which includes the generalized Fourier transform and the orbit method closely related to it, as well as an application of the perturbative method to the nilpotent approximation, and Trotter’s formula