Under consideration are the construction and properties of some special class of second other tangent sets on using the technique of nonstandard analysis.

Under study is the spectral asymptotics of the Sturm–Liouville problem with a singular self-conformal weight measure. We assume that the conformal iterated function system generating the weight measure satisfies a stronger version of the bounded distortion property. The power exponent of the main term of the eigenvalue counting function asymptotics is obtained under the assumption. This generalizes

We prove that (1): the characteristic function of each independent set in each regular graph attaining the Delsarte–Hoffman bound is a perfect coloring; (2): each transversal in a uniform regular hypergraph is an independent set in the vertex adjacency multigraph of a hypergraph attaining the Delsarte–Hoffman bound for this multigraph; and (3): the combinatorial designs with parameters \( t \)-\( (v

We prove that a simple finite-dimensional unital right alternative superalgebra over an algebraically closed field of characteristic 0 has semisimple even part if its even part is strongly alternative.

We consider linear evolutionary systems of partial differential equations with constant coefficients of general form. We suppose that the matrix of operators at the higher time derivative of the sought vector-function is degenerate. These systems are called partial differential-algebraic equations (DAEs). The index is the most important characteristic defining the structure complexity of these equations

We consider the so-called simplest formula for local approximation by polynomial splines of order \( n \) (Schoenberg splines). The spline itself and all derivatives except that of the highest order, approximate a given function and its corresponding derivatives with the second order. We show that the jump of the highest derivative of order \( n-1 \); i. e., the value of discontinuity, divided by the

We prove that studying the universal theories of generalized rigid metabelian groups reduces to those of the pairs \( (A,R) \), where \( R \) is a commutative integral domain and \( A \) is a nontrivial torsion-free subgroup of the multiplicative group \( R^{\ast} \) generating \( R \).

Considering periodic \( \zeta \)-functions and periodic Hurwitz \( \zeta \)-functions, we obtain joint universality for approximation to a collection of analytic functions by generalized shifts of the \( \zeta \)-functions.

Under study are the graph mappings constructed from the contact mappings of arbitrary Carnot groups. We establish the well-posedness conditions for the problem of maximal surfaces, introduce a suitable notion of the increment of the (sub-Lorentzian) area functional, and prove that this functional is differentiable. The necessary maximality conditions for graph surfaces are described in terms of the

We describe the nonabelian simple finite groups whose every nonsolvable local maximal subgroup is a Hall subgroup, and the nonsolvable finite groups whose all nonsolvable superlocals are Hall subgroups.

The goal of this article is to present some method of optimal extension of positive order continuous and \( \sigma \)-order continuous operators on quasi-Banach function spaces with values in Dedekind complete quasi-Banach lattices. The optimal extension of such an operator is the smallest extension of the Bartle–Dunford–Schwartz type integral. It is also shown that if a positive operator sends order

We introduce the notion of endomorph \( E({\mathcal{A}}) \) of a \( ( \)super\( ) \)algebra \( {\mathcal{A}} \) and prove that \( E({\mathcal{A}}) \) is a simple \( ( \)super\( ) \)algebra if \( {\mathcal{A}} \) is not an algebra of scalar multiplication. If \( {\mathcal{A}} \) is a right-symmetric (super)algebra then \( E({\mathcal{A}}) \) is right-symmetric as well. Thus, we construct a wide class

We consider a one-parametric family of conformal mappings from the upper half-plane onto the family of the polygons obtained by an angle-preserving shift of one or more vertices of some initial polygon. We obtain some differential equation for the family of mappings and some system of ordinary second-order differential equations for the preimages of the vertices of the family of polygons with Cauchy

We describe classes of local coordinates on the Carnot–Carathéodory spaces of lower smoothness which permit the homogeneous approximation of quasimetrics and basis vector fields. We establish the minimal smoothness that is required for these classes to coincide with the class of the already-described privileged coordinates in the infinite smoothness case. Moreover, we apply these results to prove the

Under study is the structure of subsemigroup lattices of semigroups of elementary types. We establish that the subsemigroup lattices of semigroups of elementary types are lattice-universal. Also, we show that, for a series of classes \( {\mathbf{K}} \) of algebraic structures, each subsemigroup lattice of the semigroup of elementary types of the structures from \( {\mathbf{K}} \) contains the ideal

We prove that the set of extreme points of the set of dilation–invariant Banach limits is unclosed in the weak\( {}^{*} \) topology.

In an article published forty years ago, Kutateladze proposed a machinery for studying the extremal structure of convex sets of linear operators which was based on the theory of Kantorovich spaces. The purpose of this article is to extend a portion of the so-arising theory to the convex sets of positive multilinear operators from the Cartesian product of vector lattices to a Kantorovich space. The

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.

Under study is the inverse problem of determining the two-dimensional kernel for a system of viscoelasticity equations in a medium with slightly horizontal homogeneity in a half-space. The direct initial-boundary value problem for the displacement function contains zero initial data and the Neumann condition of a special form. The field of displacements of medium points is given for x 3 − 0 as additional

We study the multivalued mappings on a closed interval of the real line whose values are polyhedra in a separable Hilbert space. The polyhedron space is endowed with the metric of the Mosco convergence of sequences of closed convex sets. A polyhedron is defined as the intersection of finitely many closed half-spaces. The equations of the corresponding hyperplanes involve normals and reals. The normals

We consider the semigroup \({\cal P}(G)\) of primitive endomorphisms with respect to a free group G for which the group Aut( G ) coincides with the group of tame automorphisms TAut( G ). We find the necessary and sufficient conditions for a given automorphism to belong to \({\cal P}(G)\) and also the necessary and sufficient conditions for the coincidence of \({\cal P}(G)\) and Aut( G ). Considering

We consider the discrete Wiener-Hopf equation with inhomogeneous term \(g = \{ {g_j}\} _{j = 0}^\infty \in {l_\infty }\); the kernel of the equation is an arithmetic probability distribution generating a random walk drifting to +∞. We prove that the previously obtained formula for the Wiener-Hopf equation with general arithmetic kernel for g ∈ l 1 is a solution to the equation for g ∈ l ∞ and that

We classify the unital finite-dimensional simple right-alternative superalgebras with semisimple even part and prove that each of these superalgebras is either a simple associative matrix Wall algebra, or a simple alternative Shestakov superalgebra, or an asymmetric double, or an abelian superalgebra of type B n∣n , n ≥ 2, or B2∣2( v ). Furthermore, we obtain a description of right-alternative superalgebras

We classify smooth Fano weighted complete intersections of large codimension.

We study the properties of solutions in special weighted spaces to a nonhomogeneous boundary value problem in a planar angle for a singular elliptic equation of the second order with the differential Bessel operator ∂ 2/ ∂y 2 + k ∂ /( y ∂ y ), k > 0, one of the variables. Under some constraints on the weight exponents, the boundary value problem is correctly solvable. We establish a relation connecting

We describe some method that associates two chain complexes to every X and every mapping Q : X × X × X → X satisfying a few conditions motivated by Reidemeister moves. These complexes differ by boundary homomorphisms: For one complex, the boundary homomorphism is the difference of two operators; and for the other, their sum. We prove that each element of the third cohomology group of these complexes

The main result states that if E is a separable Fréchet lattice and F is a (locally solid) topological vector lattice with the σ -interpolation property then each positive linear operator T 0 from a majorizing subspace G ⊂ E into F admits extension to a continuous positive linear operator T from E into F . This fact is proved by using only the axiom of countable choice.

We consider the numerical characteristics of identities of nonassociative algebras and propose a method for constructing some algebra A ( w ) with prescribed properties of the codimension growth function. The growth of codimensions of A ( w ) is completely determined by the combinatorial complexity of the language of subwords of w .

Let tr be the canonical trace on the full matrix algebra \({{\cal M}_n}\) with unit I . We prove that if some analog of classical inequalities for the determinant and trace (or the permanent and trace) of matrices holds for a positive functional φ on \({{\cal M}_n}\) with φ ( I ) = n , then φ = tr. Also, we generalize Fischer’s inequality for determinants and establish a new inequality for the trace

We study the discrete spectra of boundary value problems for the biharmonic operator describing oscillations of a Kirchhoff plate in the form of a locally perturbed strip with rigidly clamped or simply supported edges. The two methods are applied: variational and asymptotic. The first method shows that for a narrowing plate the discrete spectrum is empty in both cases, whereas for a widening plate

Studying direct decompositions for the Fitting classes of multioperator T -groups, we establish that the direct sum of n -multiply Ω-foliated Fitting classes of T -groups with right direction is also an n -multiply Ω-foliated Fitting class with right direction, and the converse holds for some right directions: free, canonical, etc.

We provide some new integral inequalities with the Caputo fractional derivative and consider degenerate ODEs with the Caputo derivative and a fractional diffusion equation as application. Also, we prove the existence and uniqueness theorems of generalized solutions under certain conditions.

Under study are the problems of existence of the first integrals rational in momenta for a natural mechanical system on the 2-torus. We prove that a rational integral with a linear numerator and denominator reduces to a linear integral. Considering the case of a quadratic numerator and linear denominator, we obtain an analogous result under some additional assumptions on the coefficients of the first

We study the rings R whose every right ideal is a finite direct sum of automorphism-invariant right R -modules. These rings are called right Σ- a -rings. We find a representation in the form of block upper triangular rings of formal matrices for the indecomposable right Artinian right hereditary right Σ- a -rings.

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.