An Erratum to this paper has been published: https://doi.org/10.1134/S0001434620110188

Abstract For a projectively invariant subgroup \(C\) of a reduced \(p\)-group \(G\), a nondecreasing sequence of ordinals and the symbol \(\infty\) is constructed in which the \(k\)th position, \(k=0,1,2,\dots\), is occupied by the minimum of heights in \(G\) of all nonzero elements of the subgroup \(p^kC[p]\). It is proved that if all elements of this sequence are integers, then the subgroup \(C\)

Abstract We prove the existence in rational spaces of distance graphs without short odd cycles whose chromatic number increases exponentially with the dimension of the space.

Abstract The Banach Contraction Mapping Principle has many generalizations and extensions in different directions. Here we define \((\varepsilon,\psi)\)-Uniformly Local Weak Contractions and show that these mappings admit unique fixed points. We obtain a generalization of two existing results. We construct an example which illustrates of our main result in which the mapping is neither a \(\psi\)-weak

Abstract It is known that, among all the differentiable homeomorphic changes of variable, only the functions \(\varphi_1 (x)=x\) and \(\varphi_2 (x)=1-x\), \(x\in[0,1]\), preserve the absolute convergence of Fourier–Haar series everywhere. It is established that the class of all differentiable homeomorphic changes of variable that preserve absolute convergence everywhere will not become wider if we

Abstract In this paper, we investigate the global structure of positive solutions of $$\begin{cases} u''''(x)=\lambda h(x)f(u(x)), & 0 0\) is a parameter, \(h\in C[0,1]\), \(f\in C[0,\infty)\) and \(f(s)>0\) for \(s>0\). We show that the problem has three positive solutions suggesting suitable conditions on the nonlinearity. Furthermore, we also establish the existence of infinitely many positive solutions

Abstract A complete description of multipliers of Fourier–Haar series in Lorentz spaces is given. Necessary and sufficient conditions for sequences of complex numbers to belong to the class \(m(L_{p,r}\to L_{q,s})\) of multipliers of Fourier–Haar series are obtained.

Abstract For the perturbed Hamiltonian of a multifrequency resonance harmonic oscillator, a new approach to calculating the coefficients in the procedure of quantum averaging is proposed. The procedure of quantum averaging is transferred to the space of the graded algebra of symbols by using twisted product introduced in the paper. As a result, the averaged Hamiltonian is represented as a function

Abstract The sub-Laplacian plays a key role in CR geometry. In this paper, we investigate eigenvalues of the sub-Laplacian on bounded domains of strictly pseudoconvex CR manifolds, strictly pseudoconvex CR manifolds submersed in Riemannian manifolds. We establish some Levitin–Parnovski-type inequalities and Cheng–Huang–Wei-type inequalities for their eigenvalues. As their applications, we derive some

Abstract The density of smooth functions in weighted Sobolev spaces that are anisotropic with respect to the order of the derivatives and the weight functions is established.

Abstract We provide estimates for the eigenvalues of non-self-adjoint Sturm–Liouville operators with periodic and antiperiodic boundary conditions for special potentials that are trigonometric polynomials. Moreover, we give error estimations and, finally, we present an example.

Abstract The paper deals with the study of dynamical systems admitting compact attractors, which describe the behavior of trajectories of the system at infinity and play an important role in the qualitative theory of stability of motion. The properties of trajectories of a dynamical system in the attraction domain of the attractors and on the boundary are established by using concepts such as elliptic

Abstract Properties of homogeneous spaces having an invariant \(\mathcal E\)-structure (a generalization of the structure of a homogeneous dual manifold) are studied. Simply connected homogeneous spaces of dimension \(<5\) with such a structure are described up to a diffeomorphism.

Abstract Unitary transformations and canonical representatives of a family of real-valued harmonic fourth-degree polynomials in three complex variables are studied. The subject relates to the study of Moser normal equations for real hypersurfaces of four-dimensional complex spaces and isotropy groups (holomorphic stabilizers) of such surfaces. The dimension of the stabilizer for a particular strictly

Abstract Potential functions associated with eigenfunctions of the discrete Sturm–Liouville operator are studied in a loaded space. On the basis of representations of kernels with respect to the corresponding orthogonal polynomials, an estimate for the growth of potential functions is obtained.

Abstract We consider the equation \(G(x,x)=y\), where \(G\colon X\times X\to Y\) and \(X\) and \(Y\) are metric spaces. This operator equation is compared with the “model” equation \(g(t,t)=0\), where the function \(g\colon \mathbb{R}_+\times \mathbb{R}_+ \to\mathbb{R}\) is continuous, nondecreasing in the first argument, and nonincreasing in the second argument. Conditions are obtained under which

Abstract Let a weakly convex function (in the general case, nonconvex and nonsmooth) satisfy the quadratic growth condition. It is proved that the gradient projection method for minimizing such a function on a set converges with linear rate on a proximally smooth (nonconvex) set of special form (for example, on a smooth manifold), provided that the constant of weak convexity of the function is less

Abstract A discrete dynamical system generated by a homeomorphism of a compact manifold is considered. A sequence \(\omega_n\) of periodic \(\varepsilon_n\)-trajectories converges in the mean as \(\varepsilon_n\to 0\) if, for any continuous function \(\varphi\), the mean values on the period \(\overline\varphi(\omega_n)\) converge as \(n\to\infty\). It is shown that \(\omega_n\) converges in the mean

Abstract An algorithm is presented which determines in a finite number of steps whether an initial finite binary automaton is spherically transitive. Since the class of deterministic functions coincides with the class of functions satisfying the Lipschitz condition with constant 1 on the ring of \(p\)-adic integers, the algorithm is based on an ergodicity criterion for a deterministic function given

Abstract Isotropic Nikol’skii–Besov spaces with norms whose definition, instead of the modulus of continuity of certain order of partial derivatives of functions of fixed order, involves the “\(L_p\)-averaged” modulus of continuity of functions of the corresponding order are studied. We construct continuous linear mappings of such spaces of functions given on bounded domains of \((1,\dots,1)\)-type

Abstract Boundedness conditions for spherical and spatial variable-order Riesz potential-type operators with integrable density in variable-exponent Hölder spaces are proved.

Abstract The present paper is devoted to traversing a maze by a collective of automata. This part of automata theory gave rise to a fairly wide range of diverse problems ([1], [2]), including those related to problems of the theory of computational complexity and probability theory. It turns out that the consideration of complicated algebraic objects, such as Burnside groups, can be interesting in

We provide examples of finite non-abelian groups acting on non-trivial Severi–Brauer surfaces. A Severi–Brauer surface over a field K is a surface that becomes isomorphic to P over the algebraic closure of K. Concerning birational automorphisms of non-trivial Severi– Brauer surfaces (i.e. those that are not isomorphic to P) the following is known. Theorem 1 ([Sh20, Theorem 1.2(ii)]). Let S be a non-trivial

We study the Weyl-type solutions of the differential system with a singularity $y'-x^{-1}Ay-q(x)y=\rho By$ in the case of integrable potential $q(\cdot)$.

Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 - b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_{\!q}$ such that $\sqrt[3]{b} \notin \mathbb{F}_{\!q}$. This article tries to resolve the problem of constructing a rational $\mathbb{F}_{\!q}$-curve on the Kummer surface of the direct product $E_b \!\times\! E_b^\prime$, where $E_b^\prime$ is the quadratic $\mathbb{F}_{\!q}$-twist of

In this paper, we are interested to analyze a nonlocal nonlinear parabolic equation with fractional Laplacian. We show that there are no nontrivial global weak solutions using the test function method.

The Dirichlet problem in the half-space for elliptic differential-difference equations with operators that are compositions of differential and difference operators is considered. For this problem, classical solvability or solvability almost everywhere (depending on the constraints imposed on the boundary data) is proved, an integral representation of the found solution in terms of a Poisson-type formula

We consider 3-subgroups in groups of birational automorphisms of rationally connected threefolds and show that any 3-subgroup can be generated by at most five elements. Moreover, we study groups of regular automorphisms of terminal Fano threefolds and prove that in all cases which are not among several explicitly described exceptions any 3-subgroup of such group can be generated by at most four elements

Abstract We refine the classical boundedness criterion for sums of sine series with monotone coefficients \(b_k\): the sum of a series is bounded on \(\mathbb R\) if and only if the sequence \({\{kb_k\}}\) is bounded. We derive a two-sided estimate of the Chebyshev norm of the sum of a series via a special norm of the sequence \(\{kb_k\}\). The resulting upper bound is sharp, and the constant in the

Abstract In the present paper, a complete description of the inverse subsemigroups of the bicyclic semigroup is given. Isomorphism conditions for two inverse subsemigroups are found. From the inverse subsemigroups under consideration, a category is constructed, and the existence of a functor from this category to the category of finite ordered sets of numbers is proved.

Abstract [1]. In the present work, we provide bounds for Daubechies orthonormal wavelet coefficients for function spaces \(\mathcal{A}_k^p:=\{f: \|(i \omega)^k\hat{f}(\omega)\|_p< \infty\}\), \(k\in{\mathbb N}\cup\{0\}\), \(p\in(1,\infty)\).

Abstract The multicomponent wave function of the spin and vector fields is presented as a one-component function depending on the position and orientation of a zero-size moving rotating observer. It is shown that the Dirac and Maxwell equations and the fine structure constant are the result of the connection of two spaces: the Minkowski space and the space of orientations (the observer), and this relationship

Abstract Smooth solutions of the eikonal equation are studied. A relationship between the geometry of a hypersurface and the set of singular points of its metric function on both sides of this hypersurface is investigated.

Abstract Let \(\alpha > 0\) be any fixed noninteger. In the paper, we prove an analogue of the Bombieri–Vinogradov theorem for the set of primes \(p\) satisfying the condition \(\{ p^{\alpha} \} < 1/2\). This generalizes the previous result of Gritsenko and Zinchenko.

Abstract Using the restatement of the Riemann hypothesis proposed in a recent paper of Matiyasevich, we explicitly write out the system of Diophantine equations whose unsolvability is equivalent to this hypothesis.

Abstract A new method for deriving estimates of the rate of convergence of optimal methods for solving problems of smooth (strongly) convex stochastic optimization is described. The method is based on the results of stochastic optimization derived from results on the convergence of optimal methods under the conditions of inexact gradients with small noises of nonrandom nature. In contrast to earlier

Approximations on the closed interval \([-1,1]\) of functions that are combinations of classical Markov functions by partial sums of Fourier series on a system of Chebyshev–Markov rational fractions are considered. Pointwise and uniform estimates for approximations are established. For the case in which the derivative of the measure is weakly equivalent to a power function, an asymptotic expression

We improve an estimate for the additive energy of sets \(A\) with small product \(AA\). The proof uses some properties of level sets of convolutions of the indicator function of \(A\), namely, their almost invariance under multiplication by elements of \(A\).

The grand Hardy classes \(H_{p)}^{+}\) and \({}_{m}H_{p)}^{-}\), \(p>1\), of functions analytic inside and outside the unit disk, which are generated by the norms of the grand Lebesgue spaces, are defined. Riemann problems of the theory of analytic functions with piecewise continuous coefficient are considered in these spaces. For these problems in grand Hardy classes, a sufficient solvability condition

Multiplicative estimates of the \(L_p\)-norms of derivatives of a function on a domain with flexible cone condition are established.

Studying spectral properties of operator polynomials is reduced to studying the corresponding spectral properties of operators defined by operator matrices. The results are used to investigate second-order differential operators by associating them with the corresponding first-order differential operators and using their properties related to invertibility.

In the paper, the notion of an \(\mathfrak F^{\omega}\)-abnormal (and \(\mathfrak F^{\omega}\)-normal) maximal subgroup of a finite group is introduced, where \(\mathfrak F\) is a nonempty class of groups and \(\omega\) is a nonempty set of primes. The relationship between the \(\mathfrak F^{\omega}\)- abnormal maximal and normal subgroups is studied. Conditions are established under which \(\mathfrak

For the monotone symmetric threshold Boolean functions \( f^n_2(\widetilde x\mspace{2mu})=\bigvee_{1\le i it is established that a minimal contact circuit implementing \(f^n_2(\widetilde x\mspace{2mu})\) contains \(3n-4\) contacts.

A method for constructing geometric solutions of the Riemann problem for an impulsively perturbed conservation law is described. A complete classification of the possible patterns of the phase flow is given and, for each of the possible cases, the limit in the sense of Hausdorff is constructed.

We develop an approach to writing efficient short-wave asymptotics based on the representation of the Maslov canonical operator in a neighborhood of generic caustics in the form of special functions of a composite argument. A constructive method is proposed that allows expressing the canonical operator near a caustic cusp corresponding to the Lagrangian singularity of type \(A_3\) (standard cusp) in

We characterize the subsets of the space \(\ell^\infty_n\) with continuous (lower semicontinuous) metric projection. One of the characteristic properties is the strict solarity of both the set and any nonempty intersection thereof with any support coordinate hyperplane.

This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations.

Abstract A variable-velocity wave equation is studied on the simplest decorated graph, i.e., the topological space obtained by attaching a ray to $$\mathbb R^3$$ . The Cauchy problem with initial conditions localized on Euclidean space is considered. The leading term of an asymptotic solution of the problem under consideration as the parameter characterizing the size of the source tends to zero is

We give more details to our examples in [9] of K3 surfaces over C such that they have infinite automorphism group but it preserves some elliptic pencil of the K3

In this paper, we give a description of points at which the strong means of VilenkinFourier series converge.

Smooth dynamical systems on closed manifolds with invariant measure are considered. The evolution of the density of a nonstationary invariant measure is described by the well-known Liouville equation. For ergodic dynamical systems, the Liouville equation is expressed in Hamiltonian form. An infinite collection of quadratic invariants that are pairwise in involution with respect to the Poisson bracket

It is shown that there exist arbitrarily large natural numbers $$N$$ and distinct nonnegative integers $$n_1,\dots,n_N$$ for which the number of zeros on $$[-\pi,\pi)$$ of the trigonometric polynomial $$\sum_{j=1}^N \cos(n_j t)$$ is $$O(N^{2/3}\log^{2/3} N)$$ .

Abstract Let $$g\ne 0$$ be an entire function of exponential type in the complex plane $$\mathbb C$$ , and let $${\mathsf Z}=\{{\mathsf z}_k\}_{k=1,2,\dots}$$ be a sequence of points in $$\mathbb C$$ . We give a criterion for the existence of an entire function $$f\ne 0$$ of exponential type which vanishes on $${\mathsf Z}$$ and satisfies the constraint $$ \ln |f(iy)|\le \ln |g(iy)|+o(|y|),\qquad y\to

Abstract A logistic delay equation with diffusion, which is important in applications, is studied. It is assumed that all of its coefficients, as well as the coefficients in the boundary conditions, are rapidly oscillating functions of time. An averaged equation is constructed, and the relation between its solutions and the solutions of the original equation is studied. A result on the stability of

Abstract The paper deals with the construction of a multivalued solution of the Cauchy problem for the Hamilton–Jacobi equation with discontinuous Hamiltonian with respect to the phase variable. The constructed multivalued solution is approximated by a sequence of continuous solutions of auxiliary Cauchy problems of the Hamilton–Jacobi equation with Hamiltonian which is Lipschitz with respect to the

Abstract Properties of the approximate rotation of vector fields generated by multivalued maps of monotone type are studied. Analogs of the Hopf theorems on the extension of multivalued maps without singular points and homotopy classification of the corresponding vector fields are proved. Applications to variational inequalities and operator inclusions are outlines.

Abstract Let \(G\) be a finite group, let \(M\) be a maximal subgroup of \(G\), and let \(\mathfrak F\) be a hereditary formation consisting of solvable groups. The metanilpotency of the \(\mathfrak F\)-residual \(G^\mathfrak F\) is established under the assumption that all subgroups maximal in \(M\) are \(\mathfrak F\)-subnormal in \(G\), and the nilpotency of \(G^\mathfrak F\) is established in the

We give sufficient conditions on a matrix A ensuring the existence of a partition of this matrix into two submatrices with extremely small norm of the image of any vector. Under some weak conditions on a matrix A we obtain a partition of A with the extremely small $(1,q)$-norm of submatrices.

Given a continuous complex-valued function $$a$$ and nonnegative functions $$\rho_1$$ and $$\rho_2$$ on a two-dimensional smooth connected closed surface such that $$|a|=\rho_1\rho_2$$ and the functions $$\rho_1$$ and $$\rho_2$$ have no common zeros, it is required to find complex-valued continuous functions $$a_1$$ and $$a_2$$ satisfying the conditions $$a_1a_2=a$$ and $$|a_j|=\rho_j$$ . Necessary

We study the asymptotics of solutions to the Dirichlet problem in a domain $$\mathcal{X} \subset \mathbb{R}^3$$ whose boundary contains a singular point $$O$$ . In a small neighborhood of this point, the domain has the form $$\{ z > \sqrt{x^2 + y^4} \}$$ , i.e., the origin is a nonsymmetric conical point at the boundary. So far, the behavior of solutions to elliptic boundary-value problems has not