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In this paper, numerical characteristics of identities of non-associative algebras are studied. A short survey of the results of the last 10 years in this area is presented., 1. Introduction In the present paper, we study numerical characteristics of polynomial identities of associative and nonassociative algebras. Numerical methods are used in PI-theory since the 1970s. Results of research [...]

This paper belongs to our series of works on algebraic geometry over arbitrary algebraic structures. In this one, there will be investigated seven equivalences (namely: geometric, universal geometric, quasi-equational, universal, elementary, and combinations thereof) in specific classes of algebraic structures (equationally Noetherian, [q.sub.[omega]]-compact, [u.sub.[omega]]-compact, equational domains, equational co-domains, etc.). The main questions are the following: (1) Which equivalences coincide inside a given class K, which do not? (2) With respect to which equivalences a given class K is invariant, with respect to which it is not?, 1. Introduction In the series of papers on universal algebraic geometry [2-10], we consecutively develop the technique for investigations in algebraic geometry over an arbitrary algebraic structure [??] in a [...]

This paper defines Artin groups with m-gon structure and proves that for m > 3 in this class the problems of equality and conjugacy of words are solvable., The results of this paper were announced in [9]. The authors introduce a new class of Artin groups whose graphs consist of n-gons for n [greater than or equal to] [...]

The paper is devoted to the construction of an 'integral' on an infinite-dimensional space, combining the approaches proposed previously and at the same time the simplest. A new definition of the construction and study its properties on a special class of functionals is given. An introduction of a quasi-scalar product, an orthonormal system, and applications in physics (path integral, loop space, functional derivative) are proposed. In addition, the paper contains a discussion of generalized functionals. Bibliography: 15 titles., 1. INTRODUCTION The term 'functional integral' has a long history and for the first time was proposed by Daniell [1] in 1919. Since then it has been playing an important [...]

We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semi-discrete variant). First, we linearize the equation around its arbitrary solution u. Then we construct a differential (respectively, difference) equation compatible with the linearized equation for any choice of u. This equation defines a surface called a generalized invariant manifold. In a sense, the manifold generalizes the symmetry, which is also a solution to the linearized equation. In this paper, we concentrate on continuous and discrete models of hyperbolic type. It is known that such kinds of equations have two hierarchies of symmetries, corresponding to the characteristic directions. We have shown that a properly chosen generalized invariant manifold allows one to construct recursion operators that generate these symmetries. It is surprising that both recursion operators are related to different parametrizations of the same invariant manifold. Therefore, knowing one of the recursion operators for the hyperbolic type integrable equation (having no pseudo-constants) we can immediately find the second one. Keywords and phrases: integrability, Lax pair, invariant manifold, recursion operator, quad equation. AMS Subject Classification: 35[L.sup.1]0, 39A14, 1. Introduction. In our papers [3-5], we proposed a direct method for searching Lax pairs and recursion operators for integrable models. The method is based on the construction of invariant [...]

In this paper, we study the generalized interpolation problem in the class of entire functions of exponential type defined by a certain majorant from the convergence class. Keywords and phrases: interpolation sequence, entire function, convergence class, [bar.[partial derivative]]-problem. AMS Subject Classification: 30E05, 1. Introduction. In this paper, we consider the generalized interpolation problem in the class of entire functions of exponential type, which is determined by a certain majorant from the convergence [...]

In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude. Keywords and phrases: Bethe--Sommerfeld hypothesis, Laplacian, band, oscillating boundary. AMS Subject Classification: 35P05, 47A10, 35B27, 1. Introduction. The classical Bethe--Sommerfeld conjecture is formulated as follows: the spectrum of a multidimensional periodic differential operator contains a finite number of internal lacunas. This hypothesis was confirmed for [...]

In this paper, we formulate mathematical foundations of applications of boundary integral equations with strongly singular integrals understood in the sense of finite Hadamard value to numerical solution of certain boundary-value problems. We describe numerical schemes for solving boundary strongly singular equations based on quadrature formulas and the collocation method. Also, we make references to known results on the mathematical justification of the numerical methods described in the paper. Keywords and phrases: hypersingular integral, boundary integral equation, boundary-value problem, numerical methods. AMS Subject Classification: 30E25, 45E99, 65N38, 1. Introduction. The method of boundary integral equations is based on the representation of a solution of a boundary-value problem as an integral over the boundary of a domain and [...]

In this paper, we study the problem of boundary control of string oscillations, which is carried out over a period of time less than the critical time. The control is performed by a displacement of one end of the string, whereas at the other end a uniform boundary condition with oblique derivative is given, and the direction of this derivative does not coincide with characteristics. The classical statement of the problem is considered. Necessary and sufficient conditions for the existence of a unique control are found, and the control itself is obtained in an explicit analytical form. Keywords and phrases: wave equation, displacement control, time less than critical, boundary conditions with oblique derivative. AMS Subject Classification: 35L05, 1. Statement of the problem. In this paper, we consider boundary control of string oscillations described by the one-dimensional wave equation. The problem consists of transferring the string by a [...]

The study of the Dirichlet problem in the unit disk D with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua [48] has been devoted to boundary-value problems (only with Holder continuous data) for the generalized analytic functions, i.e., continuous complex valued functions h(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form [[partial derivative].sub.z]h + ah + bh = c, where it was assumed that the complex valued functions a, b and c belong to the class [L.sup.p] with some p > 2 in smooth enough domains D in C. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincare and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called A-harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions h : D [right arrow] C with the sources g : [[partial derivative].sub.z]h = g [member of] [L.sup.p], p > 2, and to generalized harmonic functions U with sources G : [DELTA]U = G [member of] [L.sup.p],p>2. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincare problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations [DELTA] U = G with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media. Keywords. Logarithmic potential, generalized analytic and harmonic functions; Poisson equations; Riemann, Hilbert, Dirichlet, Neumann and Poincare boundary-value problems., 1. Introduction The well-known monograph of Vekua [48] has been devoted to the theory of the generalized analytic functions, i.e., continuous complex valued functions h(z) of the complex variable z [...]

The main goal of this paper is to study algorithmic properties of countable linear orders by constructing effective presentations of these structures on the set of natural numbers. In 1991, C. Jockusch and R. Soare constructed a low linear order without computable presentations. Earlier, in 1989, R. Downey and M. Moses showed that each low discrete linear order has a computable copy. It is natural to ask for which order types of low presentations the existence of a computable presentation is sufficient. This question (namely, research program) was stated by R. Downey in 1998: Describe the order property P such that, for any low linear order L, P(L) implies the existence of a computable presentation of L. In this paper, we give a detailed review of the main results in this direction. These results are mostly obtained by the author or in co-authorship. Keywords and phrases: countable linear orders, computable structures, computable presentability, low degrees. AMS Subject Classification: 03D45, 03C57, CONTENTS Introduction 199 1. Low Linear Orders 200 2. [Low.sub.2] Linear Orders 219 3. Counterexamples 227 References 231 Introduction The main goal of this paper is to study algorithmic properties [...]

This paper is a survey of results on countable families with natural degree spectra. These results were obtained by a modification of the methodology proposed by Wechner, who first found a family of sets with the spectrum consisting precisely of nonzero Turing degrees. Based on this method, many researchers obtained examples of families with other natural spectra. In addition, in this paper we extend these results and present new examples of natural spectra. In particular, we construct a family of finite sets with the spectrum consisting of exactly non-K-trivial degrees and also we find new sufficient conditions on [[DELTA].sup.0.sub.2]-degree a, which guarantees that the class {x : x [??] a} is the degree spectrum of some family. Finally, we give a survey of our recent results on the degree spectra of [alpha]-families, where [alpha] is an arbitrary computable ordinal. Keywords and phrases: degree spectra, countable family, enumeration of family, algebraic structure, [alpha]-family. AMS Subject Classification: 03D20, 03D45, 1. Introduction. The problem of description of Turing degrees enumerating a given family of sets appeared in the context of the so-called Lempp problem (1997, [2] 9.5): Is there a [...]

The paper characterizes linear operators converting column majorization into weak, directional, and strong majorizations. An example of a linear converter from weak, directional, and strong majorizations to column majorization preserving none of these majorizations is provided. Bibliography: 27 titles., 1. INTRODUCTION Let [M.sub.n,m] be the space of all real n x m matrices; in the case where m = n, we write [M.sub.n]. Throughout the paper, the following matrices [...]

The paper considers parallel preconditioned iterative methods in Krylov subspaces for solving systems of linear algebraic equations with large sparse symmetric positive-definite matrices resulting from grid approximations of multidimensional problems. For preconditioning, generalized block algorithms of symmetric successive over-relaxation or incomplete factorization type with matching row sums are used. Preconditioners are based on variable-triangular matrix factors with multiple alternations in triangular structure. For three-dimensional grid algebraic systems, methods are based on nested factorizations, as well as on two-level iterative processes. Successive approximations in Krylov subspaces are computed by applying a family of conjugate direction algorithms with various orthogonality and variational properties, including preconditioned conjugate gradient, conjugate residual, and minimal error methods. Bibliography: 23 titles., 1. INTRODUCTION The aim of this paper is to design and study parallel iterative methods in Krylov subspaces for solving systems of linear algebraic equations (SLAEs) Au = f, A [...]

The paper considers a Riemann surface (in a broad sense of the term in the Hurwitz-Courant terminology) and an open set with a compact closure on this surface. It is proved that, following Aikawa-Ohtsuka, with a condenser on a given open set one can associate a family of vector measures, whose modules are computed directly with the aid of the weighted capacity (with Muchenhoupt weight) of the condenser. [ABSTRACT FROM AUTHOR]

The inverse factorial series expansion for the ratio of products of gamma functions whose arguments are linear functions of the variable is found. A recurrence relation for the coefficients in terms of the Nørlund-Bernoulli polynomials is provided, and the half-plane of convergence is determined. The results obtained naturally supplement a number of previous investigations of the gamma ratios, which began in the 1930-ies. The expansion obtained in this paper plays a crucial role in the study of the behavior of the delta-neutral Fox’s H-function in the neighborhood of its finite singular point. A particular case of the inverse factorial series expansion is used in deriving a possibly new identity for the Nørlund-Bernoulli polynomials. [ABSTRACT FROM AUTHOR]

The paper suggests a general approach to deriving upper bounds for the spectral radii of weighted digraphs. The approach is based on the generalized Wielandt lemma (GWL), which reduces the problem of bounding the spectral radius of a given block matrix to bounding the Perron root of the matrix composed of the norms of the blocks. In the case of the adjacency matrix of weighted graphs and digraphs where all the blocks are square positive (semi)definite matrices of the same order, the GWL takes an especially nice simple form. The second component of the approach consists in applying known upper bounds for the Perron root of a nonnegative matrix. It is shown that the approach suggested covers, in particular, the known upper bounds of the spectral radius and allows one to describe the equality cases. [ABSTRACT FROM AUTHOR]

The concept of the minimal polyanalytic polynomial was introduced by M. Huhtanen in connection with the generalized Lanczos process as applied to a normal matrix. The paper discusses the possibility of finding an equivalent of the characteristic polynomial in the set of polyanalytic polynomials. [ABSTRACT FROM AUTHOR]

Let matrices A,C ∈ Mn have eigenvalues α1,..., αn and γ1,..., γn, respectively. The set of complex numbers DC(A) = {det(A−UCU*) : U ∈ Mn, U*U = In} is called the C-determinantal range of A. The paper studies various conditions under which the relation DC(R S) = DC(S R) holds for some matrices R and S. [ABSTRACT FROM AUTHOR]

The Correction Notice in the Journal of Mathematical Sciences addresses an error in the article "Smoothness of Solutions to the Damping Problem for Nonstationary Control System with Delay." The correction pertains to the title of the paper, which should read as "Smoothness of Solutions to the Damping Problem for Nonstationary Control System with Delay." The original article was published in Vol. 278, No. 4, in 2024, and the correction was made in Vol. 285, No. 3, in October 2024. [Extracted from the article]

The classification problem for finite and infinite algebraic lattices has also been extensively addressed. A wild classification problem contains the classification problem of pairs of matrices up to simultaneous similarity. In this paper, we prove that the classification problem for graphs is wild by reducing the classification problem for finite 2-nilpotent p-groups to the classification problem for graphs. [ABSTRACT FROM AUTHOR]

In this paper, we consider diffeomorphic mappings of a nondegenerate parallelepiped onto another such parallelepiped. We prove that the image of any face of the first parallelepiped under such a mapping is a certain face of the second parallelepiped. Moreover, we also obtain several auxiliary results on the transformation of boundary points of a nondegenerate parallelepiped under the action of a diffeomorphism. Keywords and phrases: homeomorphism, diffeomorphism, boundary of a set, parallelepiped. AMS Subject Classification: 26B05, Introduction. This paper is based on lectures on the theory of differential forms given by one of the authors (A. V. Bad'in) at the Faculty of Physics of the Lomonosov [...]

In this paper, we consider boundary-value problems for a new class of hybrid systems of differential equations whose coefficients contain the Dirac delta-function. Hybrid systems are systems that contain both ordinary and partial differential equations; such systems appear, for example, when equations of motion of mechanical systems of rigid bodies attached to a beam by elastic bonds are derived from the Hamilton-Ostrogradsky variational principle. We present examples that lead to such systems and introduce the notions of generalized solutions and eigenvalues of a boundary-value problem. We also compare results of numerical simulations based on methods proposed in this paper with results obtained by previously known methods and show that our approach is reliable and universal. Keywords and phrases: eigenvalue, boundary-value problem, hybrid system, differential equation, frequency equation. AMS Subject Classification: 39A20, CONTENTS 1. Introduction 625 2. Systems Leading to Hybrid Systems of Differential Equations 627 3. Hybrid System of Differential Equations 633 4. General Approach to Constructing the Frequency Equation 640 [...]

In this paper, we present results of the development of certain parallel numerical methods for solving three-dimensional evolutionary and stationary problems of diffusion and heat transfer. We present a detailed description of a special, explicit iteration scheme for parabolic equations and discuss a multigrid technology used for solving elliptic equations and implicit schemes for parabolic equations. Keywords and phrases: parabolic equations, elliptic equations, multigrid method, Chebyshev parameters, adaptation. AMS Subject Classification: 65M06, 65M22, 65M55, 1. Introduction. In this paper, we present some algorithms for numerical solution of three-dimensional differential equations of diffusion and heat transfer. The main emphasis in the development of algorithms is [...]

An embedded system of algebras of vertex subsets of weighted digraph is constructed. The properties of minimal spanning forests restricted to atoms of the corresponding algebras are investigated. Bibliography: 9 titles., 1. NOTATION AND DEFINITIONS The present paper is a continuation of paper [1]; the definitions and notation correspond to those adopted in it. The vertex set of a digraph G [...]

Earlier, a model of a time series with heavy tails constructed from a Gaussian time series was developed. In the present paper, the reverse problem is considered: an estimator of the copula function is built; the copula function is a nonlinear function that maps Gaussian variables to the variables from the Frechet maximum domain of attraction. The statistical properties of this estimator are considered for a stationary time series with a low rate of covariance decay., 1. Introduction This paper deals with the problem concerning the building of a model by data with potentially heavy tails. This problem is quite common for financial and economical data, [...]

This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p- and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain. Keywords. p-Laplacian, p(x)-Laplacian, Dirichlet problem, Robin problem, transmission problem, degenerate oblique derivative problem, angular and conical points., 1. Introduction In this paper, I present a survey of the last main results about solutions to the Dirichlet [2,3] and Robin [9-11, 17] boundary problems, the Robin transmission problem [...]

A brief review of the emergence and development of the nonlocal approach to the problem of turbulent diffusion with a discussion of the physical reasons of the nonlocality is given. The main attention is paid to fractional differential operators. In concluding the paper, the author's original results on applications to the diffusion of cosmic rays in the interstellar galactic medium are presented. Keywords and phrases: Brownian motion, anomalous diffusion, fractional Laplacian, Levy-Feldheim distribution, self-similarity AMS Subject Classification: 91B70, 1. Introduction. In this paper, we will focus on the nonlocal method of describing the transport process in turbulent media, here referred to as 'turbulent diffusion' for brevity. It must [...]

The paper proposes a new mathematical FitzHugh--Nagumo model with memory, which describes the propagation of a nerve impulse in a membrane. The model consists of an integro-differential equation with initial conditions (the Cauchy problem). The difference kernel (memory function) of the model equation was chosen as a power function so that it can be rewritten in terms of the fractional derivative. For the Cauchy problem, an explicit finite-difference scheme is constructed, and an investigation of its stability and convergence was performed in computer experiments. The finite-difference scheme was implemented in the Maple computer program, with the help of which the simulation results were visualized, and oscillograms and phase trajectories were obtained. Keywords and phrases: hereditarity, FitzHugh--Nagumo model, fractional derivative, finite-difference scheme. AMS Subject Classification: 37M05, 34A08, 1. Introduction. In this paper, a mathematical model of the FitzHugh-Nagumo dynamical system was studied taking into account the hereditarity effect. The model was proposed by J. Nagumo, S. Arimoto, [...]

In this paper, we suggest a method of constructing logical neural networks on the basis of variable-valued logical functions. We prove the theorem on a possibility of representation of any logical function as a logical neural network. The proof given in the paper contains an algorithm of the construction of the logical neural network. We point out the possibility of the generalization of the result obtained to the case of fuzzy logic. Keywords and phrases: variable-valued predicate, data mining, variable-valued logical function, training sample, logical neural network, fuzzy logic. AMS Subject Classification: 68T27, 1. Introduction. The actuality of solving the problem of intellectualization of informational and control systems follows from the modern tendency of increasing quantity and quality of different data flows and [...]

This paper is concerned with small UAV (unmanned aerial vehicle) control algorithm development during landing. The UAV landing problem for small UAVs (less than 20 kg) remains actual despite the great amount of books, papers, and monographs devoted to this topic. In this paper, a model with V-shaped keel is under consideration. Mechanization of such a vehicle consists of flaperons and ailerons. To describe a flight, a system of nonlinear differential equations is developed. The automatic control both for longitudinal and lateral motion as well as for thrust is designed. The stability analysis of the controlled system is conducted by plotting stability regions in the space of the feedback coefficients. The results of a numerical modeling of a flight with external disturbances are given., 1. Introduction The problem of UAV design remains actual in Russian aerial industry. The main use of a plane type UAVs is observation, reconnaissance and aerial photography of objects on [...]

This paper shows that the stability or nonstability of switched systems does not depend on the eigenvalues of the matrices. The result is obtained by giving examples of no stable (resp. no unstable) switched linear systems consisting of stable (resp. unstable) matrices. Moreover, for the first time all kinds of eigenvalues are considered, as well as two general results for establishing the stability or nonstability of switched linear systems are presented. [ABSTRACT FROM AUTHOR]

In this paper, we considered a power of a non-Hermitian random matrix and prove that the empirical distribution of its singular values converges to the Fuss-Catalan distribution almost surely. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive consequences of a result of Götze and Zaitsev (). We show that the i.i.d. case of this result implies a multidimensional version of some results of Sakhanenko (). We establish bounds for the rate of strong Gaussian approximation of sums of i.i.d. R d-valued random vectors ξ j having finite moments E IIξ jIIγ, γ>2. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

We apply a probabilistic approach developed in our previous papers, which allows us to solve the Cauchy problem for nonlinear parabolic equations and systems and to construct solutions of problems arising in financial mathematics in search for arbitrage-free option prices on nonideal markets. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

The paper deals with methods of computation of distributions of location for maxima and minima for diffusions with jumps. As an example, we obtain explicit formulas for distributions of location for the maximum of the process which is equal to the sum of a Brownian motion and the compound Poisson process. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

The paper presents a method of adaptive estimation for a class of probability density functions. This method is a continual analog of some known methods. Bibiligraphy: 10 titles. [ABSTRACT FROM AUTHOR]

The paper studies the region of values of the system { f( z1), f( z2), c2},where z j, j=1, 2, are arbitrary fixed points of the disk | z|<1; f ∈ T, and the class T consists of all functions f( z) = z + c2 z2 + ··· regular in the disk | z| < 1 and satisfying the condition Im f( z)·Im z>0 for Im z > 0 for Im z ≠ 0. The region of values of f( z1) in the subclass of functions f ( z) ∈ T with prescribed values c2 and f( z2) is determined. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

The paper considers issues related to constructing asymptotic formulas for deviations of Riesz sums. The error estimates in the spaces C and L p in terms of the best approximations are obtained. Bibliography: 9 titles. [ABSTRACT FROM AUTHOR]

The paper considers the capacity of a polycondenser and the module of a family of measures associated with the class of curves connecting the polycondenser’s plates. It is proved that these quantities are equal. Also a relation between the polycondenser’s capacity and the module of a family of vector measures associated with the class of functions admissible for the polycondenser’s capacity is established. The results obtained generalize the earlier results by M. Ohtsuka and H. Aikawa. [ABSTRACT FROM AUTHOR]

The paper presents sharp inequalities for the moduli of rational functions under certain constraints on the modulus of the independent variable. These inequalities supplement some results of Govil, Mohapatra, and Dubinin. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if two generalized incidence rings I( P1 ,R1) and I( P2 ,R2) are elementarily equivalent, then the corresponding ordered sets ( P1 ,R1) and ( P2 ,R2) are elementarily equivalent. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a survey of the first advances in the study of diverse interconnections between epigroups and their subepigroup lattices. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that every variety of commutative unary algebras of a finite signature has a finite basis of identities. [ABSTRACT FROM AUTHOR]

In this paper, we discuss measures of divergence and focus on a recently introduced measure of divergence, the socalled BHHJ measure. A general class of such measures is introduced, and goodness of fit tests for multinomial populations are presented. Simulations are performed to check the appropriateness of the proposed test statistics. Bibliography: 20 titles. [ABSTRACT FROM AUTHOR]

Known well is the problem of finding configurations of points on the Euclidean sphere S n that can be put into one level surface of any continuous function on S n by a rotation of S n. The paper is devoted to various ways of transferring this problem to the ease of a normed space. Here is one of the results. Let f and g be two even continuous functions on an n-dimensional normed space E, and let f(0) < f( x) for all nonzero x ∈ E. Then E contains n unit vectors e1,..., e n such that for any 1 ≤ i ≤ j ≤ n we have f( e i + e j): f( e i − e j) and g( e i + e j): g( e i − e j). Bibliography: 16 titles. [ABSTRACT FROM AUTHOR]

In the paper, a special class of suitable weak solutions to the three-dimensional nonstationary Navier–Stokes equations is considered, and a reverse Hölder inequality for them is proved. An interesting feature of this class is that it contains solutions having majorants invariant to the Navier-Stokes scaling. Bibliography: titles. [ABSTRACT FROM AUTHOR]

The paper presents an error bound of the Ritz method for the problem of minimizing the functional in the space $$ {\mathop W\limits^{\text{o}} }^{1}_{2} {\left( {0,1} \right)} $$ in the case where the standard assumption on the continuity of q(t) is replaced by the condition q2(t)t(1-t) ∈ L(0,1). In the case where q(t) is continuous, the new bound is sharper than the known one. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]

The paper preserits same exact geometric characteristics of removable sets for two classes of weighted capacities. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]

An estimate for the Riemann localization principle for trigonometric series is established. At the same time, there is an error in treating this problem in the paper [3] which is now being corrected. [ABSTRACT FROM AUTHOR]