We study the problem of the stability of the extremals of the potential energy functional. By the stability of an extremal surface we mean the sign-definiteness of its second variation. For estimating the second variation of the functional, we use the properties of the eigenvalues of symmetric matrices. Also, we prove an analog of Alexandrov’s Theorem on the variational property of a sphere.

Considering a stationary stochastic process with independent increments (Lévy process), we study the probability of the first exit from a strip through its upper boundary. We find the two-sided inequalities for this probability under various conditions on the process.

The Gersten group \( G \) is the split extension \( F_{3}\rtimes_{\varphi}{𝕑} \) of the free group \( F_{3} \) with basis \( \{a,b,c\} \) by the automorphism \( \varphi:a\mapsto a,b\mapsto ba,c\mapsto ca^{2} \). We describe the generators and structure of the group \( \operatorname{Out}(G) \) and prove that \( \operatorname{Out}(G)\cong(F_{3}\times{𝕑}^{3})\rtimes({𝕑}_{2}\times{𝕑}_{2}) \).

We consider a second kind representation for solutions to a first order general uniformly elliptic linear system in a simply connected plane domain \( G \) with the \( W^{k-\frac{1}{p}}_{p} \)-boundary. We prove that the operator of the system is an isomorphism of Sobolev’s space \( W^{k}_{p}(\overline{G}) \), \( k\geq 1 \), \( p>2 \), under appropriate assumptions about coefficients and the boundary

We describe retracts in free metabelian groups and obtain some structural results for verbally closed subgroups in free polynilpotent and free metabelian groups.

We consider the solvability of boundary value problems and some inverse problems for operator-differential equations of odd order which can be referred to as mixed type equations since the coefficient of the higher order time derivative can change sign. Studying the general classes of boundary value problems with nonlocal boundary conditions, we establish the existence and uniqueness theorems of regular

Suppose that \( n \) is an odd integer, \( n\geq 5 \). We prove that a periodic group \( G \), saturated with finite simple orthogonal groups \( O_{n}(q) \) of odd dimension over fields of odd characteristic, is isomorphic to \( O_{n}(F) \) for some locally finite field \( F \) of odd characteristic. In particular, \( G \) is locally finite and countable.

There is a series of publications about the Riemann boundary value problem on spiral-like arcs. Some constraints are imposed on the shape of the turns which derive the latter close to approximating concentric circles in form. This article solves the problem for the spirals that may lose concentricity. We show that this effect can entail changes in the solvability of the boundary value problem.

On the space of continuous functions from a line segment to a reflexive Banach space, we consider some operator whose values are closed convex subsets of the space. If the values are singletons, the operator becomes a well-known single-valued history-dependent operator. We study the properties of the operator, prove a fixed-point theorem analogous to the fixed-point theorem for single-valued history-dependent

Under study are the branching time temporal logics with temporal accessibility relations for the agents different in length and content. We find an algorithm for solving the problems of satisfiability and decidability of the logic through describing finite satisfiable models of computable size (through the size of the input formulas). We remove the constraints on the size of all temporal accessibility

An abelian group \( A \) is quotient divisible if \( A \) has no torsion divisible subgroups but possesses a free subgroup \( F \) of finite rank such that \( A/F \) is a torsion divisible group. Quotient divisible groups were introduced by Beaumont and Pierce in the class of torsion-free groups in 1961, and by Wickless and Fomin, in the general case in 1998. This paper deals with the abelian groups

We find the abnormal extremals on four-dimensional connected Lie groups with left-invariant sub-Finsler quasimetric defined by a seminorm on a two-dimensional subspace of the Lie algebra generating the algebra. In terms of the structure constants of a Lie algebra and the Minkowski support function of the unit ball of the seminorm on the two-dimensional subspace of a Lie algebra which defines a quasimetric

We introduce the class of split Malcev Poisson algebras as the natural extension of split (noncommutative) Poisson algebras. We show that if \( P \) is a split Malcev Poisson algebra then \( P=\oplus_{j\in J}I_{j} \) with \( I_{j} \) a nonzero ideal of \( P \) such that \( \{I_{j_{1}},I_{j_{2}}\}=I_{j_{1}}I_{j_{2}}=0 \) for \( j_{1}\neq j_{2} \). Under some conditions, the above decomposition of \(

The Radon transform \( R \) maps a function \( f \) on \( {𝕉}^{n} \) to the family of the integrals of \( f \) over all hyperplanes. The classical Reshetnyak formula (also called the Plancherel formula for the Radon transform) states that \( \|f\|_{L^{2}({𝕉}^{n})}=\|Rf\|_{H^{(n-1)/2}_{(n-1)/2}({𝕊}^{n-1}\times{𝕉})} \), where \( \|\cdot\|_{H^{(n-1)/2}_{(n-1)/2}({𝕊}^{n-1}\times{𝕉})} \) is some special

A 3-path \( uvw \) in a 3-polytope is an \( (i,j,k) \)-path if \( d(u)\leq i \), \( d(v)\leq j \), and \( d(w)\leq k \), where \( d(x) \) is the degree of a vertex \( x \). It is well known that each 3-polytope has a vertex of degree at most 5 called minor. A description of 3-paths in a 3-polytope is minor or major if the central item of its every triplet is at least 6. Back in 1922, Franklin proved

Under consideration is the class of nonlinear systems of nonautonomous differential equations of neutral type with a variable delay that can be unbounded. Using a Lyapunov–Krasovskii functional, we establish some estimates of solutions that allow us to conclude whether the solutions are stable. In the case of exponential and asymptotic stability, we estimate the attraction domains and the rate of stabilization

We obtain approximation theorems for analytic functions by the shifts \( F(\zeta(s+i\tau)) \) with \( \tau\in{𝕉} \), where \( \zeta(s) \) is the Riemann \( \zeta \)-function, while \( F \) is some operator on the space of analytic functions, on short intervals \( [T,T+H] \) with \( T^{1/3}(\log T)^{26/15}\leq H\leq T \) as \( T\to\infty \).

Let \( \alpha \) be an algebraic number of degree \( d\geq 2 \). We consider the set \( E(\alpha) \) of positive integers \( n \) such that the primitive \( n \)th root of unity \( e^{2\pi i/n} \) is expressible as a quotient of two conjugates of \( \alpha \) over \( {𝕈} \). In particular, our results imply that \( E(\alpha) \) is small. We prove that \( |E(\alpha)|

The notion of weakly injective \( S \)-act can be regarded as a generalization of the notion of injective \( S \)-act. This article describes the finite monoids over which each weakly injective \( S \)-act has a primitively normal theory. Moreover, we show that the primitive normality of the class of all principally weakly injective \( S \)-acts is equivalent to \( S \) being totally ordered.

We consider a problem for a first-order differential equation in a Hilbert space. We replace the initial data with some condition that includes the integral of the solution over the entire time interval on which we solve the problem. It is proved that the problem has a unique solution on an arbitrary time interval under the condition that some of the data are positive.

We establish sufficient conditions for the almost compactness of partially integral operators in \( L_{p} \).

Under study is the multidimensional inverse problem of determining the convolutional kernel of the integral term in an integro-differential wave equation. The direct problem is represented by a generalized initial-boundary value problem for this equation with zero initial data and the Neumann boundary condition in the form of the Dirac delta-function. For solving the inverse problem, the traces of

Given a prime \( p \) and a partition \( \sigma=\{\{p\},\{p\}^{\prime}\} \) of the set of all primes, we describe the structure of the nonnilpotent finite groups whose every Schmidt subgroup is \( \sigma \)-subnormal.

We consider some system of delay differential equations describing the interaction between predator and prey populations and accounting for the age structure of the predator population. Under study are the asymptotic properties of solutions to this system. We establish the estimates that characterize the solution stabilization rate at infinity as well as the estimates for the attraction domains of

We consider a waveguide composed of two not necessity equal semi-infinite strips (trunks) and a rectangle (resonator) connected by narrow openings in the shared walls. As their diameter vanishes, we construct asymptotics for the scattering coefficients, justifying them by the technique of weighted spaces with detached asymptotics. We establish a criterion for the possibility of observing, at a given

Providing some examples of Lagrangian cycles that arise as a generalization of Mironov’s construction to the case of Grassmann manifolds \( \operatorname{Gr}_{{𝔺}}(k,n+1) \), we show that these manifolds enjoy all data necessary for this generalization, the natural real structure, and an incomplete toric action. We also provide new concrete examples.

Introducing some classes of the functions defined by Poisson integrals on the segment \( [-1,1] \) and studying approximations by Fourier–Chebyshev rational integral operators on the classes, we establish integral expressions for approximations and upper bounds for uniform approximations. In the case of boundary functions with a power singularity on \( [-1,1] \), we find the upper bounds for pointwise

We consider the subgroups \( H \) in a symplectic or orthogonal group over a finite field of odd characteristic such that \( O_{r}(H)\neq 1 \) for some odd prime \( r \). We obtain a refinement of the well-known Aschbacher Theorem on subgroups of classical groups for this case.

We establish the coarea formula as an expression for the measure of a subset of a Carnot group in terms of the sub-Lorentzian measure of the intersections of the subset with the level sets of a vector function. We describe the conditions for the level sets of vector functions to be spacelike and find the metric characteristics of these surfaces. Also, we address a series of relevant questions, in particular

We obtain some new estimates that show the extremality of the Rademacher system in the set of sequences of independent functions considered in rearrangement invariant spaces.

Isospectral are the groups with coinciding sets of element orders. We prove that no finite group isospectral to a finite simple classical group has the exceptional groups of types \( E_{7} \) and \( E_{8} \) among its nonabelian composition factors.

Under study are the questions of closedness of an elementary net (carpet) over a field. Every complete net is closed (admissible). We construct an example of a closed elementary net over a field of characteristic 0 which cannot be supplemented to a complete net.

Each 3-polytope has obviously a face \( f \) of degree \( d(f) \) at most 5 which is called minor. The height \( h(f) \) of \( f \) is the maximum degree of the vertices incident with \( f \). A type of a face \( f \) is defined by a set of upper constraints on the degrees of vertices incident with \( f \). This follows from the double \( n \)-pyramid and semiregular \( (3,3,3,n) \)-polytope, \( h(f)

Given an arbitrary partition \( \sigma \) of the set of primes, we give some criteria for the \( \sigma \)-subnormality of a Sylow subgroup of a finite group.

The notion of \( P \)-stability is a particular case of the generalized stability of complete theories. This paper discusses some problems that are related to the \( P \)-stability of certain classes of \( S \)-acts. In particular, we describe the monoids \( S \) over which the classes of free, projective, strongly flat, divisible, regular \( S \)-acts are \( P \)-stable.

Let \( {{\mathfrak{X}}} \) be a class of finite groups containing a group of even order and closed under subgroups, homomorphic images, and extensions. Then each finite group possesses a maximal \( {{\mathfrak{X}}} \)-subgroup of odd index and the study of the subgroups can be reduced to the study of the so-called submaximal \( {{\mathfrak{X}}} \)-subgroups of odd index in simple groups. We prove a theorem

Let \( G \) be a periodic group, and let \( \omega(G)\subseteq{𝕅} \) be the spectrum of \( G \) that is the set of orders of elements in \( G \). We prove that the alternating group \( A_{7} \) is uniquely recognized by its spectrum in the class of all groups.

Let \( \pi_{x} \) be the set of primes greater than \( x \). We prove that for all \( x\in{𝕉} \) the classes of finite groups \( D_{\pi_{x}} \) and \( E_{\pi_{x}} \) coincide; i.e., a finite group \( G \) possesses a \( \pi_{x} \)-Hall subgroup if and only if \( G \) satisfies the complete analog of the Sylow Theorems for a \( \pi_{x} \)-subgroup.

We study the rings over which each square matrix is the sum of an idempotent matrix and a \( q \)-potent matrix. We also show that if \( F \) is a finite field not isomorphic to \( 𝔽_{3} \) and \( q>1 \) is odd then each square matrix over \( F \) is the sum of an idempotent matrix and a \( q \)-potent matrix if and only if \( q-1 \) is divisible by \( |F|-1 \).

Studying the properties of almost \( \omega \)-categorical weakly \( o \)-minimal theories of convexity rank 1, we prove the orthogonality of every family of pairwise weakly orthogonal nonalgebraic 1-types in the theories. Also, we establish the binarity of the theories.

We study the influence of weakly subnormal and partially subnormal subgroups on the structure of a group \( G \). In particular, we prove that a finite group \( G \) is supersoluble if and only if \( G=AB \), where \( A \) and \( B \) are supersoluble weakly subnormal subgroups in \( G \), and every Schmidt subgroup in \( G \) is partially subnormal in \( G \).

Assuming the Continuum Hypothesis (CH), we prove that there exists a perfectly normal compact topological space \( Z \) and a countable set \( E\subset Z \) such that \( Z\setminus E \) does not condense onto any compact set. The space \( Z \) enables us to answer in the negative (under CH) the following problem of Ponomarev: Is each perfectly normal compact set an \( a \)-space? We also prove that

We study the polynomial identities of alternative and Jordan algebras which imply the nilpotency of the algebras. In the case of a field of characteristic 0 we describe all these systems of identities as well as all almost nilpotent varieties of alternative and Jordan algebras. In particular, we establish a connection between the Engel identity for the Lie algebras and the standard identity for the

The paper is devoted to the normal extensions of discrete semigroups and \( * \)-homomorphisms of semigroup \( C^{*} \)-algebras. We study the normal extensions of abelian semigroups by arbitrary groups. Considering numerical semigroups, we prove that they are normal extensions of the semigroup of nonnegative integers by finite cyclic groups. On the other hand, we prove that if a semigroup is a normal

Under study are the arithmetic properties of second maximal subgroups of finite groups. Generally speaking, we investigated the problem by Monakhov [1, Problem 19.54] and developed the research of Meng and Guo [2, Theorem B] by weakening the condition of solvability.

Under study are the right-symmetric algebras over a field \( F \) which possess a “unital” matrix subalgebra \( M_{n}(F) \). We classify all these finite-dimensional right-symmetric algebras \( {\mathcal{A}}=W\oplus M_{2}(F) \) in the case when \( W \) is an irreducible module over \( sl_{2}(F) \).

We develop the Ershov theory of C-classes for some finite families of sets in the Ershov hierarchy. We generalize the result by Muchnik on multiple \( m \)-reducibility as follows: There exists an \( m \)-universal pair of disjoint sets for each level of the Ershov hierarchy.

We prove that \( G \) is a finite \( \sigma \)-soluble group with transitive \( \sigma \)-permutability if and only if the following hold: (i) \( G \) possesses a complete Hall \( \sigma \)-set \( {\mathcal{H}}=\{H_{1},\dots,H_{t}\} \) and a normal subgroup \( N \) with \( \sigma \)-nilpotent quotient \( G/N \) such that \( H_{i}\cap N\leq Z_{\mathfrak{U}}(H_{i}) \) for all \( i \); and (ii) every

Suppose that \( n \) is an integer, \( n\geq 3 \). We prove that a periodic group saturated with a set of the finite simple groups \( O_{2n+1}(q) \), where \( q \) is congruent to \( \pm 3 \) modulo 8, is isomorphic to \( O_{2n+1}(F) \) for some locally finite field \( F \).

We establish some analogs of Ulyanov’s and Andrienko’s Theorems on the embedding of the Hölder spaces of integrable functions on zero-dimensional second countable locally compact groups into the Lebesgue spaces or other Hölder spaces. We prove that these results are unimprovable.

We prove that if a multivalued mapping \( F \) of circle to circle has the \( \eta \)-BAD property (bounded distortion of generalized angles with control function \( \eta \)) then there exist a positive integer \( N \) and a quasimöbius homeomorphism \( \varphi \) of a circle into itself such that the left inverse mapping to \( F \) is of the form \( (\varphi(z))^{N} \). Moreover, \( \varphi \) is

Some structural properties are discussed of the functions in \( P_{2}\times P_{2} \). We describe the properties of functions not belonging to the maximal subalgebras \( R_{4} \), \( R_{5} \), and \( R_{11} \) and show that the maximal cardinality of the basis in \( P_{2}\times P_{2} \) is equal to 8.

Under study is the problem of asymptotic but exponential stability for a class of linear autonomous neutral functional differential equations. We demonstrate that the asymptotic stability of an equation of the class takes place for all integrable initial functions if the roots of the characteristic equation lie on the left of and approach the imaginary axis.

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

We study the Cauchy problem in the space of continuous functions for some nonlinear differential equation of Sobolev type that simulates longitudinal waves in an infinite viscoelastic rod. Under consideration are the conditions for the existence of the global classical solution and the blow-up of the solution to the Cauchy problem on a finite time interval.

We consider a specific method for embedding a countable group that is given by generators and relations into some 2-generated group. This embedding enables us to express the images of generators of the countable group in the 2-generated group and explicitly deduce from the defining relations of the latter those of the former which inherit some special properties. The method can be used to construct

We construct some counterexamples to the statements [1, 2.3.6] claiming maximal inequalities for the spaces \( B_{p,q}^{s}(𝕉^{n}) \) and \( F_{p,q}^{s}(𝕉^{n}) \) and propose a condition for these inequalities to hold. We consider some weighted inequality on a bounded interval \( I \) of the real axis that involves \( f\in C_{0}^{\infty}(I) \) and the derivative of \( f \).

We prove that each homeomorphism \( \varphi:D\to D^{\prime} \) of Euclidean domains in \( 𝕉^{n} \), \( n\geq 2 \), belonging to the Sobolev class \( W^{1}_{p,\operatorname{loc}}(D) \), where \( p\in[1,\infty) \), and having finite distortion induces a bounded composition operator from the weighted Sobolev space \( L^{1}_{p}(D^{\prime};\omega) \) into \( L^{1}_{p}(D) \) for some weight function \(

We study the class of systems of differential equations defined by one class of matrix quasielliptic operators and establish solvability conditions for the systems and boundary value problems on \( {𝕉}^{n}_{+} \) in the special scales of weighted Sobolev spaces \( W^{l}_{p,\sigma} \). We construct the integral representations of solutions and obtain estimates for the solutions.

We study Hardy-type integral inequalities with remainder terms for smooth compactly-supported functions in convex domains of finite inner radius. New \( L_{1} \)- and \( L_{p} \)-inequalities are obtained with constants depending on the Lamb constant which is the first positive solution to the special equation for the Bessel function. In some particular cases the constants are sharp. We obtain one-dimensional