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In the present paper, necessary and sufficient conditions are given for the equality of the power rezidue symbols $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ and $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ in the cyclotomic field ℚ(ζ n), 2 ∤ n, for a ∈ ℤ, ( a, n) = 1. This result is a generalization of the classical Eisenstein reciprocity law and its continuation in a Hasse’s paper. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

It is shown that an approach earlier used by the author for classification of closed simply connected 6-manifolds (reduction to the problem of calculating certain bordism groups) can also be applied for easily obtaining the results by Barden (1965) on classification of closed simply connected 5-manifolds. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

The Euler problem on the stability of a thin elastic rod under compressing forces is widely known. In a number of papers this problem is generalized to finding the shape of such a rod under the simultaneous action of a compressing force and a torsional moment. The shape is determined by solving a complex nonlinear boundary value problem by numerical methods. In this paper, an approach providing a full analytical solution of the problem for a broad range of external conditions is discussed. [ABSTRACT FROM AUTHOR]

In the problem of gliding flight, i.e., aircraft longitudinal movement with its engines switched off, the aircraft was traditionally assumed to be an absolutely rigid body, the Earth was not rotating and had a plane surface, and the atmosphere was quiet. Further, the attack angle and the air density were assumed constant. The aim of this paper is to discuss the applicability limits of these assumptions. [ABSTRACT FROM AUTHOR]

In the present paper, a finite-dimensional phenomenological model of unsteady interaction of a rigid plate with a flow is proposed. It is assumed that the plate performs translational motion across the flow. The internal dynamics of the flow is modeled by the attached second order dynamical system. It is shown that the model allows satisfactory agreement with experimental data. With the developed model an inverse problem of dynamics is examined for the situation where the plate performing uniform translational motion at some moment begins uniform deceleration and finally stops. It is shown that for sufficiently large values of the plate acceleration for some time range the flow does not resist the motion of the plate but “accelerates” it. It is shown also that the equations of motion in the context of the proposed model can be reduced to the integro-differential form, and comparison with the known model of S. M. Belotserkovsky is performed. The structural resemblance of the motion equations for a body in flow in both models is noted. The domain of applicability of the quasi-stationary model is examined. [ABSTRACT FROM AUTHOR]

In airborne gravimetry the data received from an airborne survey system is used to obtain the specific gravity along the aircraft path and to construct maps of the gravity anomaly. The quality of the maps depends largely on both the instrument noise and modeling errors. The paper discusses the errors in the estimated gravity caused by imperfect calibration of the gravimeter and how to calibrate the instrument using the survey flights data and the gravity maps. [ABSTRACT FROM AUTHOR]

The Steward platform is employed in dynamic simulation of a number of controlled mechanical systems, in control of the active surface of a telescope mirror, and other constructions. Position and orientation of this platform is regulated by changing the lengths of its supporting bars, and corresponding change of their angles relative to the base. Viable motions of the platform depend on how the bars are hinged to the base and to the platform. This paper studies dynamics and stabilization of a platform leaning on three bars of controlled length under wind loads. This problem is of interest for design of active mirror surfaces of radio telescopes. [ABSTRACT FROM AUTHOR]

This paper investigates algorithms of motion planning for a six-legged walking machine over complex terrain. Application of the Rodrigues-Hamilton parameters for describing the orientation of the body enables one to develop algorithms of motion planning for the body and legs in an absolute coordinate frame with automatic adaptation to the surface. Experiments with a laboratory scale walking machine have demonstrated the efficiency of proposed algorithms. [ABSTRACT FROM AUTHOR]

This is an attachment to a letter of the author to Alexander Zvonkin (dated November 1, 1995). It is reproduced here with the kind permission of the author. [ABSTRACT FROM AUTHOR]

In this paper a retrial queuing system is considered with the help of two-way communication where the server is subject to random breakdowns. This is a M/M/1//N type of system so the population of the source is finite. The server becoming idle enables calls the customers in the orbit (outgoing call or secondary customers). The service time of the primary and secondary customers follows exponential distribution with different rates μ1 and μ2 respectively. All the random variables included in the model construction are assumed to be totally independent of each other. The novelty of this paper is to show the effect of the different distributions of failure time on the main performance measures such as the mean waiting time of an arbitrary customer or the utilization of the service unit. In order to achieve a valid comparison a fitting process is done; thus, in case of every distribution the mean value and dispersion is the same. Graphical illustrations are given with the help of the self-developed simulation program. [ABSTRACT FROM AUTHOR]

Some gaps in two of my publications dated 1958 and 1959 years are indicated, and the way of closing them is given. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

Congruences mod p for a prime p and partial products of the numbers 1,..., p − 1 are obtained. Bibliography: 2 titles. [ABSTRACT FROM AUTHOR]

This paper discusses different approaches to constructing approximate mathematical models. [ABSTRACT FROM AUTHOR]

In this paper, we present a technique for constructing generalized solutions of the Cauchy problem for abstract integro-differential equations with degeneration in Banach spaces. A generalized solution is constructed as the convolution of the fundamental operator function (fundamental solution, influence function) of the integro-differential operator of the equation with a generalized function of a special form, which involves all input data of the original problem. Based on the analysis of the representation for the generalized solution, we obtain sufficient solvability conditions for the original Cauchy problem in the class of functions of finite smoothness. Under these sufficient conditions, the generalized solution constructed turns out to be a classical solution with the required smoothness. The abstract results obtained in the paper are applied to the study of applied initial-boundary-value problems from the theory of oscillations in viscoelastic media. [ABSTRACT FROM AUTHOR]

In this paper, we consider problems for partial differential equations that contain a small parameter in the principal part. We construct exact solutions of the original singularly perturbed problems and, passing to the limit, we state a correspondence between these solutions and solutions of the limit problems obtained from the original problems if the small parameter is set equal to zero. Examples given in this paper show that among singularly perturbed problems for partial differential equations, there are problems that possess characteristic properties of regularly perturbed problems and allow constructing asymptotic solutions by methods of the theory of regular perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we examine the optimal control problem for a hyperbolic system with differential constraints on the boundary, taking into account the delay. Controls are selected from the class of smooth functions that satisfy pointwise constraints. Problems of this type arise, in particular, in modeling the processes of population dynamics. The approach proposed in this paper is based on the use of "internal variations" of the control, which preserves the smoothness of the control function and ensures the fulfillment of pointwise constraints. We obtain an estimate of the state increment, prove a necessary optimality condition, and develop a scheme of an iterative method. [ABSTRACT FROM AUTHOR]

The drop of the smoothness of an analytic function compared to the smoothness of its modulus is discussed for the unit ball of ℂn. The paper is devoted to local aspects of the problem. [ABSTRACT FROM AUTHOR]

In this paper, we suggest a new type of tasks for control theory for stochastic dynamical systems — programmed control with Probability 1 (PCP1). PCP1 is an application of an invariant theory. We use the PCP1 concept for dynamical processes described by a system of Itô differential equations with jump-diffusion (GSDES). The considered equations include the drift, the diffusion, and the jumps, together or not. Features of our approach are both a wide set of dynamical systems and investigation of such systems for their unique trajectories. Our method is based on the concept of a stochastic first integral (SFI) for GSDES and its equations which author studied before. The purpose of the present paper is to construct a differential equation system (both stochastic and deterministic) using a known set of FIs for the investigating process. Several examples are given. [ABSTRACT FROM AUTHOR]

In this paper, we study a two-state Hard-Core (HC) model with activity λ > 0 on a Cayley tree of order k ≥ 2. It is known that there are λcr, λ cr 0 , and λ cr ′ such that • for λ ≤ λcr this model has a unique Gibbs measure μ*, which is translation invariant. The measure μ* is extreme for λ < λ cr 0 and not extreme for λ > λ cr ′ ; • for λ > λcr there exist exactly three 2-periodic Gibbs measures, one of which is μ*, the other two are not translation invariant and are always extreme. The extremity of these periodic measures was proved using the maximality and minimality of the corresponding solutions of some equation, which ensures the consistency of these measures. In this paper, we give a brief overview of the known Gibbs measures for the HC-model and an alternative proof of the extremity of 2-periodic measures for k = 2, 3. Our proof is based on the tree reconstruction method. [ABSTRACT FROM AUTHOR]

In the present paper, we calculate the Mandel parameter QM for an oscillator-like system generated by generalized Chebyshev polynomials. The sign of the Mandel parameter QM characterizes the deviation of the excitation statistics from the Poisson one. This paper is a continuation of our works (2005). [ABSTRACT FROM AUTHOR]

The present paper is a continuation of the three-body one-dimensional scattering problem in the presence of bound states. Full description and justification of the generalized eigenfunction asymptotics in the case of repulsive potentials have a simple geometric description. In the case of the presence of the bound states, additional terms appear in the eigenfunction asymptotics. In previous papers, during the analysis of Faddeev's equations in coordinate representation an operator of special form appeared which was connected to scattering amplitude and which did not have a simple description. In the current paper, some properties of that operator are described. In particular, we derive the solvability of Faddeev's equations. [ABSTRACT FROM AUTHOR]

In this paper, the structural properties of a function are characterized by moduli of continuity. The classical Hardy–Littlewood theorem describes a relation between the smoothness of the analytic function boundary values at the boundary of its analyticity and the growth rate of the modulus of its higher-order derivatives. An analogue of the Hardy–Littlewood theorem has been obtained for functions from the class Hp and higher-order moduli of continuity. [ABSTRACT FROM AUTHOR]

Let K be a field with at least 19 elements. It is proved that any subgroup of GL(4, K) generated by a pair of 2-tori contains unipotent elements of rank 1 or 2. Taking into account previous papers of N. A. Vavilov and the author, this result is valid for any general linear group. It is one of the first steps in studying subgroups generated by a pair of microweight tori in Chevalley groups. [ABSTRACT FROM AUTHOR]

Several comments on the papers by L. V. Kantorovich reproduced in this volume. Bibliography: 16 titles. [ABSTRACT FROM AUTHOR]

Let 풜 be an algebra over a field 픽 generated by a set of matrices 풮. The paper considers algorithmic aspects of checking whether 풜 coincides with the full matrix algebra. Laffey has shown that for 픽 = ℂ, under the assumption that 풮 contains a Jordan matrix from a certain class, there is a fast method for checking whether 풜 possesses nontrivial invariant subspaces and, consequently, coincides with the full algebra by Burnside's theorem. This paper extends the class to the largest subclass of Jordan matrices on which the algorithm works correctly. Examples demonstrating different types of behavior of other matrix systems are provided. [ABSTRACT FROM AUTHOR]

The paper proves that an indecomposable nonnegative matrix is chainable if and only if it satisfies Kolmogorov's condition, under which the Markov chain determined by a stochastic matrix obeys the multidimensional local limit theorem. [ABSTRACT FROM AUTHOR]

This paper presents a formula expressing Macaulay constants of a numerical polynomial through its minimizing coefficients. From this, we obtain that Macaulay constants of Kolchin dimension polynomials do not decrease. For the minimal differential dimension polynomial ω G / F (this concept was introduced by W. Sitt) we will prove a criterion for Macaulay constants to be equal. In this case, as our example shows, there are no bounds from above to the Macaulay constants of the polynomial ω ξ / F for G = F ξ. [ABSTRACT FROM AUTHOR]

In this paper, Jordan–Kronecker invariants are calculated for all nilpotent 6- and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on the lambda parameter on a Lie coalgebra, i.e., on the linear space dual to a Lie algebra. For some space 픤 proposed in the paper, two skew-symmetric matrices are defined for all points x on this linear space. To understand the behaviour of the matrix pencil (A − λB)(x), we consider Jordan–Kronecker invariants for this pencil and how they change with x (the latter is done for 6-dimensional Lie algebras). [ABSTRACT FROM AUTHOR]

In this paper, we introduce a certain topology on the group DiffF (M) of all Cr-diffeomorphisms of the foliated manifold (M; F), where r ≥ 0. This topology depends on the foliation and is called the F-compact-open topology. It coincides with the compact-open topology when F is an n-dimensional foliation. If the codimension of the foliation is n, then the convergence in this topology coincides with the pointwise convergence, where n = dim M. We prove that some subgroups of the group DiffF (M) are topological groups with the F-compact-open topology. Throughout this paper, we use smoothness of the class C∞. [ABSTRACT FROM AUTHOR]

This paper is devoted to the geometry of symmetric Killing tensors and Codazzi tensors of ranks p ≥ 2 and includes, in addition to the new results obtained in this paper, a survey on this topic from earlier works. [ABSTRACT FROM AUTHOR]

In this paper, we examine the differential-geometric aspect of constant-rank mappings of smooth manifolds based on the concept of a graph as a smooth submanifold in the space of the direct product of the original manifolds. The nonmaximality of the rank provides the fibered nature of the graph. A Riemannian structure on manifolds enriches the geometry of the graph, which now essentially depends on the induced field of the metric tensor; we characterize relatively affine, projective, and g-umbilical mappings. The final part of the paper is devoted to mappings of Euclidean spaces of the types described earlier in terms of V. T. Bazylev's constructive graph. [ABSTRACT FROM AUTHOR]

In this paper, we examine topological partitions of topological spaces that arise in connection with continuous mappings of topological spaces. The content of the paper is closely related to such classical fundamental concepts of general topology as compactness, homogeneity, and Čech completeness. New facts related to these concepts are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider recurrent behavior of bounded solutions for a functional integro-differential equation arising from heat conduction in materials with memory. Prior to the main results, we give a new version of composite theorem on measure pseudo almost automorphic functions involved in delay. Based on recently obtained results on the uniform exponential stability as well as contraction mapping principle, we prove some existence and uniqueness theorems on the recurrence of bounded mild solutions for the addressed equations with infinite delay. Finally, we finish this paper with an example on partial integro-differential equation which frequently comes to light in the study of heat conduction. [ABSTRACT FROM AUTHOR]

In this paper, we consider an optimal control problem for a system described by a parabolic linear partial differential equation with the Neumann boundary condition. We impose some constraints on the control. The performance functional has the integral form. The control time T is fixed. The initial condition is not given by a known function but belongs to a certain set (incomplete information about the initial state). To obtain optimality conditions for the Neumann problem, a generalization of the Dubovitskii–Milyutin theorem was applied. The problem formulated in this paper describes the process of optimal heating for which we do not have exact information about the initial temperature of the heating object. We also present an example in which admissible controls and one of the initial conditions are given by means of the norm constraints. [ABSTRACT FROM AUTHOR]

Abstract The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio–Ryazanov–Srebro–Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk–Petkov–Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries. We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Carathéodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Carathéodory prime ends are obtained. [ABSTRACT FROM AUTHOR]

This review paper is a continuation of two previous review papers devoted to properties of modules over discrete valuation domains. The first part of this work was published in the volumes 121 (Part I, Chaps. 1–4, Secs. 1–22) and 122 (Part II, Chaps. 5–8, Secs. 23–39) of the series Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory (English translation: Journal of Mathematical Sciences (New York), vol. 145, No. 4, pp. 4997–5117 (2007), and vol. 151, No. 5, pp. 3225–3371 (2008)). In this review paper, we preserve the numeration of chapters and sections of parts I and II. The first part consists of Chapter 9 "Appendix," Secs. 40–42. In Sec. 40, we consider p-adic torsion-free modules with isomorphic automorphism groups. Section 41 is devoted to torsion-free modules over a complete discrete valuation domain with isomorphic radicals of their endomorphism rings. The volume of the paper does not allow us to provide the proofs of all the results that appeared after publication of the previous parts and directly related to the issues under consideration in it. In the final Sec. 42, we describe some of these new results. [ABSTRACT FROM AUTHOR]

The present paper generalizes the results of our paper "On the finite presentability of the group F/[M, N]" to an arbitrary family of normal subgroups {Ni | i ∈ I} in a free group F. We obtain conditions for finite presentability of the quotient group F/ ∏ [Ni, Nj]. In both papers, the proof of the main result is based on the properties of verbal wreath products introduced by Alfred Lvovich Shmelkin. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper provides a brief overview of various definitions of multivariate regularly varying functions and some of their applications in probability theory and related fields. [ABSTRACT FROM AUTHOR]

The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin's paper in the journal "Russian Mathematical Surveys" in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ ∈ (1; 2:57] and generalized this result to the case of multidimensional complex space. [ABSTRACT FROM AUTHOR]

This paper is devoted to the applicability of the matrix method for solving the heat equation for multilayer media in the case where a phase transition is possible in some layer. We consider only stationary processes in the absence of internal heat sources. We propose a general method for layer systems with translation, axial, or central symmetry based on the technique of generalized Bers powers. Using this method, we perform calculations for one substance, when after a phase transition, the system becomes a two-layer system. We consider the dependence of the coordinate of the phase-transition point on the external temperature and compare results obtained for media with types of symmetry indicated above. A temperature field is constructed for multilayer media with various types of symmetry when a phase transition has occurred in a certain layer. [ABSTRACT FROM AUTHOR]

The paper deals with three numerical approaches that allow one to construct computational technologies for solving nonconvex optimization problems. We propose to use the developed algorithms based on modifications of the tunnel search algorithm, the Luus–Yaakola method, and the expert algorithm. The presented techniques are implemented within the framework of the software package and are used for solving nonconvex optimization problems of various classes, in particular, the minimization of the potential function for the Sutton–Chen atomic-molecular cluster, the problem of parametric identification of nonlinear dynamic systems, and the nonconvex optimal control problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the relations between parameters of the m-component mixture distributions which imply the stochastic and failure rate comparisons. Then we apply the failure rate and stochastic ordering techniques to construct the upper and lower bounds for the steady-state performance indexes in a multiserver queueing system with multicomponent exponential-Pareto mixture service time distribution. The uniform distance between multicomponent mixture distribution and its parent distribution is discussed. The obtained theoretical results are then illustrated by a few numerical examples based on the regenerative simulation multiserver queueing systems with mixture service time distributions. [ABSTRACT FROM AUTHOR]

We consider the problem of approximating an M/G/1 queueing system by an M/PH/1 system. Namely, the unknown general service time distribution G is approximated by a phase-type distribution PH. The approximation as well as the estimation of the parameters by means of statistical methods results in perturbations of the system that may affect its performance measures. In this work, we provide by means of the strong stability method, the mathematical justification of the approximation method by phase-type distributions that is already used in several works. We prove the robustness of the underlying Markov chain in each case and estimate an upper bound of the deviation of the stationary vector, resulting from the perturbation of the service-time distribution. We provide numerical examples and compare the perturbation bounds obtained in this paper with the estimates of the real deviation of the stationary vector obtained by simulation. [ABSTRACT FROM AUTHOR]

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 Hölder 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 𝜕zh + ah + b h ¯ = c ; where it was assumed that the complex valued functions a; b and c belong to the class Lp 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, Poincar´e 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 → ℂ with the sources g : 𝜕zh = g ∈ Lp, p > 2 , and to generalized harmonic functions U with sources G : △U = G ∈ Lp, 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 Poincar´e problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the stability of Markov chains with general state space. We present new conditions for the strong stability of Markov chains after a small perturbation of their transition kernels. Also, we obtain perturbation bounds with respect to different quantities. [ABSTRACT FROM AUTHOR]

The paper tells about the main features and events of the ergodic seminar organized and headed by V. A. Rokhlin at the Leningrad State University. The seminar was active in 1960–1970. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The present paper considers a discrete analog of the parking problem. Let n be an integer. If n > 1, then we randomly locate an interval (t, t + 1) with integer endpoints on a segment [0, n]. Thus, the original segment is divided into two: [0, t] and [t + 1, n], and each of them is further considered separately likewise the original one. The phrase "randomly" in this problem means that t is a uniformly distributed on a set {1,... , n − 1} random variable. The process of location of the intervals finishes when the lengths of all the remaining intervals are less than 2. Define as Xn the total amount of the located intervals. In the present paper, the expectations {Xn} are calculated. The process described above can be interpreted as a parking process of cars with handlebars on the left. Hence, the driver is able to leave his car only if the place on his left is free. This is exactly the case when the driver cannot take the left end place of any free segment. In this case, Xn stands for the amount of the parked cars. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can fail near the boundary of these subdomains even for an infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary ∂Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one-dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. Also there was obtained the smoothness (in Sobolev spaces W 2 k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding ε-neighborhoods of certain points. However, the smoothness (in Hölder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Hölder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Hölder space. [ABSTRACT FROM AUTHOR]