Abstract 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

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

Abstract 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

Abstract 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

Abstract 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 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 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 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

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 described

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 \(A\).

We give more detail to our examples in [ 1 ] of K3 surfaces over \(\mathbb C\) which have an infinite automorphism group that preserves some elliptic pencil of the K3 surface.

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

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

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.

Abstract For a one-dimensional semilinear wave equation, a mixed problem with nonlinear boundary condition is considered. The uniqueness and the local and global solvability of the problem under consideration are studied depending on the type of nonlinearities in the equation and in the boundary conditions. The cases of nonexistence of a solution not only globally but even locally are considered, as

Abstract We construct a family of flat isotropic nonhomogeneous tori in \(\mathbb{H}^n\) and \(\mathbb{C}\mathrm{P}^{2n+1}\) and find necessary and sufficient conditions for their Hamiltonian minimality.

Abstract A lattice of definability subspaces in the order of rational numbers is described. It is proved that this lattice consists of five subspaces defined in the paper that are generated by the following relations: “equality,” “less,” “between,” “cycle,” and “linkage.” For each of the subspaces, its width (the minimum number of arguments of a generating relation) is found and a convenient description

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 initial boundary-value problem for systems of fourth-order partial differential equations with two independent variables is considered. By using a new unknown eigenfunction, the problem under consideration is reduced to an equivalent nonlocal problem for a system of second-order hyperbolic-type integro-differential equations with integral conditions. An algorithm for finding an approximate

Abstract A hypergraph \(H=(V,E)\) has property \(B_k\) if there exists a 2-coloring of the set \(V\) such that each edge contains at least \(k\) vertices of each color. We let \(m_{k,g}(n)\) and \(m_{k,b}(n)\), respectively, denote the least number of edges of an \(n\)-homogeneous hypergraph without property \(B_k\) which contains either no cycles of length at least \(g\) or no two edges intersecting

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 It is established that, among all continuously differentiable homeomorphic changes of variable, the absolute convergence of Fourier series in the Haar wavelet system is preserved by only those for which \(\varphi^{-1}(0)\) is binary-rational and \(\varphi'(x)=\pm 2^m\), where \(m\) is an integer and \(x\in\mathbb R\). It is also established that this condition is necessary for a continuously

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 The first example of a Banach space with the approximation property but without the bounded approximation property was given by Figiel and Johnson in 1973. We give the first example of a Banach lattice with the approximation property but without the bounded approximation property. As a consequence, we prove the existence of an integral operator (in the sense of Grothendieck) on a Banach lattice

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

Abstract An ordered topological vector space has the countable dominated extension property if any linear operator ranging in this space, defined on a subspace of a separable metrizable topological vector space, and dominated there by a continuous sublinear operator admits extension to the entire space with preservation of linearity and domination. Our main result is that the strong \(\sigma\)-interpolation

Abstract Questions related to the homogenization of the Dirichlet problem for nondivergence elliptic equations of second order with rapidly oscillating \(\varepsilon\)-periodic coefficients are studied. Error estimates of homogenization in the Sobolev spaces of order \(\sqrt\varepsilon\) are obtained. Asymptotic homogenization methods are applied.

Abstract Let \(\Lambda\) be a class of Abelian groups. A group \(A\in\Lambda\) is said to be determined by its endomorphism semigroup \(E^\star(A)\) in the class \(\Lambda\) if every isomorphism \(E^\star(A)\cong E^\star(B)\), where \(B\in\Lambda\), implies the isomorphism \(A\cong B\). The paper describes those Abelian groups in the class \(\mathscr Q\mathscr D_{\mathrm{cd}}\) of completely decomposable

Abstract Let \(f(x)\) be a function belonging to the Lebesgue class \(L^p({\mathbb R}_+)\) on the semiaxis \({\mathbb R}_+=[0,+\infty)\), \(1\le p\le 2\), and let \(\widehat{f}\) be the Fourier–Walsh transform of the function \(f\). In this paper, we give the solution of the following problem: if the function \(f\) belongs to the dyadic Dini–Lipschitz class \(\operatorname{DLip}_\oplus(\alpha,\beta

Let \(C\) be an Abelian group. A class \(X\) of Abelian groups is called a \(_CH \)-class (a \(_CEH\)-class) if, for any groups \(A\) and \(B\) in the class \(X\), the isomorphism of the groups \(\operatorname{Hom}(C,A)\) and \(\operatorname{Hom}(C,B)\) (the isomorphism of the endomorphism rings \(E(A)\) and \(E(B)\) and of the groups \(\operatorname{Hom}(C,A)\) and \(\operatorname{Hom}(C,B)\)) implies

We obtain an estimate for a Kloosterman sum with primes for an arbitrary modulus \(q\) whose length \(X\) satisfies the conditions \( q^{1/2+\varepsilon}\le X\ll q^{3/2}. \) This estimate refines the results obtained earlier by E. Fouvry, I. E. Shparlinski (2011), and the first author (2018, 2019).

In this paper, we describe recent work towards the mirror \(\mathrm P=\mathrm W\) conjecture, which relates the weight filtration on the cohomology of a log Calabi–Yau manifold to the perverse Leray filtration on the cohomology of the homological mirror dual log Calabi–Yau manifold taken with respect to the affinization map. This conjecture extends the classical relationship between Hodge numbers of

In this note, we present an unusual picture of interaction of singularities for pressureless gas dynamics.

We construct the Lie algebras of systems of \(2g\) graded heat operators \(Q_0,Q_2,\dots,Q_{4g-2}\) that determine the sigma functions \(\sigma(z,\lambda)\) of hyperelliptic curves of genera \(g=1\), \(2\), and \(3\). As a corollary, we find that the system of three operators \(Q_0\), \(Q_2\), and \(Q_4\) is already sufficient for determining the sigma functions. The operator \(Q_0\) is the Euler operator

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 been

Let \(\mathcal{A}\) be a prime \(\ast\)-algebra. In this paper, assuming that \(\Phi:\mathcal{A}\to\mathcal{A}\) satisfies \(\Phi(A\diamond B \diamond C)=\Phi(A)\diamond B \diamond C+A\diamond\Phi(B) \diamond C+A \diamond B \diamond \Phi(C)\) where \(A\diamond B = A^{*}B + B^{*}A\) for all \(A,B\in\mathcal{A}\), we prove that \(\Phi\) is additive an \(\ast\)-derivation.

It is proved that, to select a uniqueness class for the magnetic Helmholtz equation, it suffices to impose radiation conditions weaker than the Ikebe–Saito conditions. The self-adjointness of the magnetic Helmholtz operator is proved. The existence of a solution of the inhomogeneous Helmholtz equation satisfying the radiation condition is justified.

In the paper, automorphisms are studied for free groups of varieties given by a family of identities in the well-known infinite independent system of identities involving two variables that was constructed by S. I. Adian to solve the finite basis problem in group theory. It is proved that every normal automorphism (i.e., an automorphism that stabilizes any normal subgroup) of noncyclic free groups

In my paper, it is necessary to make the following corrections on p. 382. In Example 3 (line 6 from below), the equality bk = µ-k must be replaced by bk = µ-k P(Zk < 0)