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This paper is devoted to the study of complex and unexpected phenomena that are observed in twodimensional mappings (or in three-dimensional flows) with homoclinic tangencies. In particular, we show that, in the C-topology, for an arbitrary finite r, in any neighborhood of a system with a quadratic homoclinic tangency there are nonrough systems with homoclinic tangencies of arbitrarily high orders, i.e., systems of an arbitrarily high codimension. Such phenomena were not observed in bifurcation theory previously. The study of systems with homoclinic tangencies was initiated in [4]. First and foremost, three classes of such systems were distinguished in that paper. Namely, let L be a saddle periodic motion, and let Γ be a homoclinic trajectory along which the stable and unstable invariant manifolds of L are quadratically tangent to each other (see Fig. 1). Let λ and γ be multipliers of L, |λ| 1. Assume that |λγ| = 1; moreover, without loss of generality, we can assume that |λγ| < 1. Let U be a small neighborhood of the closure Γ ∪ L of the homoclinic trajectory, and let N be the set of all trajectories that lie entirely in U . Depending on the signs of multipliers and on the signs of certain coefficients that characterize the way in which the stable and unstable manifolds adjoin to Γ, the systems with homoclinic tangencies fall into one of the following three classes: (1) for systems of the first class, the set N is trivial: N = {L,Γ}; (2) for systems of the second class, N is a nontrivial, nonuniformly hyperbolic set that admits a complete description in the language of symbolic dynamics (via some quotient system of the topological Bernoulli scheme consisting of three symbols); (3) for systems of the third class, N still contains nontrivial, hyperbolic subsets, but, generally speaking, the set N is not exhausted by them; moreover, the everywhere dense nonroughness takes place on bifurcation films of systems of the third class. (In [10, 11], a similar classification was carried out for the multidimensional case, including the case of systems with homoclinic tangencies of an arbitrary finite order.) To be more specific, according to [4], systems that have nonrough periodic motions are dense in any one-parameter family of systems with homoclinic tangencies of the third class in which the quantity

During the past several years, there has been an increase in the development of data assimilation methods and their application. The first data assimilation applications were in meteorology, and now the data assimilation technique is an important component of numerical weather forecasting. The data revolution in oceanography is bringing the daily practice of physical oceanography closer to that of dynamic meteorology. There are now large observational data sets of temperature and salinity in the Atlantic from global projects such as WOCE, SECTIONS, TOGA, COARE, PIRATA, and others. Improved assimilation techniques are needed in order to have ocean models fully exploit the new observational facilities. Data assimilation improves the estimate of the ocean and atmospheric physical state by extracting as much information as possible from both the measurements and the dynamic model, combining them in an optimal way. Assimilation may be used to improve initial and/or boundary conditions and to evaluate poorly known model parameters. Data assimilation in ocean models has been ongoing in the scientific literature for approximately 30 years (for a review, see [8]). It is necessary to understand which processes can be realistically described by the ocean models and which is the role of measurements. Two extreme opinions exist. One considers that the ocean is like a quasi-stationary medium and that it is sufficient to make only a few measurements in order to have a good description of the ocean state. From this point of view, the numerical models are useless (or simply wrong) and data assimilation techniques are restricted to a statistical correction process. The second opinion is that the ocean is a highly turbulent fluid that does not have a memory about its previous states. In this case, information generated by measurements becomes meaningless in a short time interval, and it is necessary to continuously monitor the ocean. Thus, only well-designed resolution models would be able to describe real processes in the ocean and data assimilation would be useful only to obtain correct initial (and possible boundary) conditions. Reality, however, lies somewhere in between these two extreme points of view. Thus, the scientific question is how to represent this situation numerically. The answer depends on both the model and the data assimilation techniques as well as on the local dynamic. There are two general concepts which have been discussed for data assimilation. The first is the “variational/adjoint” method, which has been the most popular scheme (see, e.g., [22, 24, 28]). An example of this technique is to assume that the model initial (and/or boundary) conditions are unknown parameters and that there is observation data distributed over some time interval. The technique seeks the optimal initial conditions by comparing the model trajectory with the measurements with respect to some criteria. This can be formulated as a constrained minimization problem. The present paper does not deal with this class of methods so there is no need to go into the details of this problem. The other class of methods is the “sequential data assimilation.” Starting from some initial condition the model solution is sequentially updated whenever measurements are available. The model solution will approach the observed state under certain conditions. This group of methods requires an updating scheme, which combines the model solution and the measurements to find the “best” state estimate. The Kalman-filter method belongs to this class of data assimilation technique. The Kalman filter is derived in a number of books on control theory (e.g., [1, 11]). In oceanography, the Kalmanfilter has been used in [7, 16]. The main idea of this method is to solve dynamic equations using the error covariance matrix equation, where an error is the difference between the model and observation values. These applications have been presented in [9, 16] and are commonly used. In this paper, another method for the definition of the error covariance matrix is presented. It continues the studies published in our paper by K. Belyaev, S. Meyers, and J. J. O’Brien.

We begin with a note of general character. Most readers are familiar with the theory of algebraic varieties over a closed field and have become accustomed to identifying a variety with the set of its geometric points. This gives only a superficial representation of the heart of problems in the nonclosed case. Therefore, in the studies of algebraic groups defined over a nonclosed field or ring, there is a strict necessity to consider these groups as group objects in the category of schemes. Although the theory of group schemes is explicitly presented in proceedings of the seminar of Grothendieck and Demazure [58], there exists a compromise way. Let k be an arbitrary field, ks be the separable closure of k, G = Gal(ks/k) be the Galois group of the extension ks/k, and X be an algebraic variety over the field k. Then we have an object containing abundant information about the variety X. This is the ks-variety X = X ⊗k ks, considered together with the action of the Galois group G on X by the second factor. Therefore, we obtain an abundant family of associated G-sets and Galois G-modules: points X(ks), differential forms, modules Pic X, Br X, and so on. Therefore, we can decompose the study of the variety X into two steps: the geometric step (the study of the variety X over a closed field ks), and the algebraic step (the study of Galois modules corresponding to the scheme X considered over the field k). It is impossible to include both of these steps in one work, since such a work would be too large. In this paper, we present a detailed review of the second, algebraic aspect of studying varieties; the geometric part is assumed to be known. Of course, the decomposition of the study into two steps is too rough. Even from the formal standpoint, the variety X is only one fiber of a k-scheme X. For example, in the case of number fields, there exist p-adic varieties X⊗kkp with their own families of Galois modules. Further, arithmetical questions demand that one define integral structures on a variety X. The following problems immediately arise: the classification of these integral structures, the calculation of reduction of the integral model, and so on. Tamagawa measures or the Siegel formula appeared as tools in this field. We recall some recent events. In his well-known report at the International Mathematical Congress (1962), A. Borel [3] presented many achievements and newly arisen problems in the arithmetic of linear algebraic groups, which was a new topic at that time. In particular, he pointed to the dramatic situation in the study of such important characteristics of connected algebraic k-groups G as the Tamagawa numbers τ(G) or the sets Ш(G) of principal homogeneous spaces of G, which are trivial in all completions of the number field k. Counter-examples to all hypotheses on the structure of the numbers τ(G) or [Ш(G)] were soon constructed. Now we see that, at that time, the arithmetic of algebraic groups did not have enough additional deep information on the birational essence of the variety of the group G. A very important source of additional information was the Hironaka paper [59] on the reduction of singularities. In their earlier studies in surface theory, Yu. I. Manin and I. R. Shafarevich called attention to the fact that the group H1(k,Pic X) is a birational invariant in the class of smooth projective surfaces X defined over a field k and successfully used this fact (Manin [30, 31]). The structure of the module Pic X gives information about the arithmetic of the surface X; this fact is noted by Segre [70]. Even the first application of birational tools to study the properties of linear groups gave nontrivial results (Voskresenskii [6]). The birational geometry gave new possibilities for the study of cause–effect relations in the vast world which is called the theory of linear algebraic groups and their homogeneous spaces. At present, the birational geometry of algebraic groups has its own individuality and contains a number of

The interconnection between the maximum principle [13] and dynamic programming [3], the two main methods in the theory of optimal control and differential games, is now understood much better due to the theory of viscosity solutions of firstand second-order nonlinear PDEs [2, 8, 14]. The value functions in optimal control and differential games have been proved to be generalized (viscosity) solutions of the corresponding Hamilton–Jacobi–Bellman–Isaacs (HJBI) equations. The classical method of characteristics (MC), which reduces the solution of a PDE to the integration of an ODE (characteristic) system, is one of the attractive and powerful tools for solving nonlinear first-order PDEs arising in control theory and mathematical physics [5]. Characteristics are of special interest in control theory, since they represent optimal paths [10]. This is clear in regarding the regular characteristics. The nonsmoothness of the generalized (e.g., viscosity) solution and/or of the Hamiltonian (left-hand-side function of the PDE), i.e., the presence of singular surfaces, is often referred to as an obstacle to the implementation of the MC. On the other hand, the construction of singular surfaces and lines is an essential and interesting part of the solution of a problem in control theory. The method of singular characteristics (MSC) shows that one can overcome this obstacle by using the same notion of characteristics suitably modified [11,12]. In this paper, a new notion of singular characteristics (SC) is suggested, which, together with the classical (regular) ones, form generalized characteristics. SC are effective for the construction of singular lines, surfaces, and manifolds of a nonsmooth solution (to firstor second-order PDEs), which carry essential information on the corresponding control or physical problem or phenomenon. Singular characteristics were found owing to investigation of singular paths in differential games and optimal control [4,9,10]. Regular paths in these domains are known to be governed by a Hamiltonian ODE system, the characteristic system for the HJBI-equation. In many cases, singular paths are described by similar equations by using the so-called singular controls. The attempt to eliminate singular controls from these equations has led to the discovery of singular characteristics, which appear to be inherent not only to game or control problems but also to the general nonlinear first-order PDEs. Owing to the notion of SC, many singular lines and surfaces known in control theory have received their invariant description in terms of general (abstract) PDEs. The possible interconnection between singular paths and characteristics (in some generalized sense) was mentioned in [7]. This general mathematical insight into the nature of a singularity, as a rule, simplifies the solution procedure of a game or control problem. Correspondingly, the experience accumulated during the construction of solutions to the latter problems is highly useful for the understanding the structure of singularities of the viscosity solutions to general PDEs. The considerations of the present paper are given in general mathematical terms rather than in terms specific for control theory. The ODE systems of singular characteristics are derived for several types of well-known singular surfaces in optimal control and differential games. The solutions of particular problems are presented.

The objective of the paper is to find simple but rather general conditions giving the opportunity to extend the Kolmogorov theorem on the law of the iterated logarithm to the case of unbounded summands with finite variances. The results obtained in the paper include both the Kolmogorov and the Hartman-Wintner theorems. Bibliography: 11 titles.

This paper deals with a software tool to evaluate the main characteristics of a nonhomogeneous finite-source queueing model to describe the performance of a multi-terminal system subject to random breakdowns under FIFO, priority processor sharing, and polling service disciplines. The model studied here is actually a closed queueing network network with three nonindependent service stations (CPU, terminals, and repairman), and a finite number of customers (jobs), which have different service rates at the service stations. The aim of this paper is to introduce the FQM (finite-source queueing model) program package, which was developed at the Institute of Mathematics and Informatics of Lajos Kossuth University in Debrecen, Hungary, and to investigate the performance of the above-mentioned finite-source queueing models. At the end we give a sample result to illustrate the problem in question.

This paper is aimed at finding asymptotic formulas for solutions to the mixed boundary problem for the Poisson equation in a domain obtained by joining singularly degenerating domains. In this paper, which is the second part of the work (the first part was published in No. 18), the main attention is given to three-dimensional problems in which a thin plate or a periodic family of thin rods is joined to a massive body (the distances between the rods are comparable with the diameters of their cross-sections). The distinctive feature of such problems is that an integral equation arises as one of the limit problems. Bibliography: 48 titles.

The present paper is devoted to the study of normal subgroups of the general linear group over a ring and the centrality of the extension St (n, R) → E(n,R). The notions of the standard commutator formula and the standard normal structure of GL(n, R), E(n, R), and St (n,R) and their relationships are discussed. In particular, it is shown that the normality of E(n, R) in GL (n,R) and the standard distribution of subgroups normalized by E (n, R) follow from some conditions of linear dependence in R. Also, it is proved that the standardness of the normal structure of GL (n,R) and the centrality of K2(n, R) in St (n,R) follow from the same conditions over a quotient ring R/I, provided that si I≤n}-1. Under certain additional assumptions (for example, I is contained in the Jacobson radical of R), the converse is also true. The standard technique due to H. Bass, Z. I. Borevich, N. A. Vavilov, L. N. Vaserstein, W. van der Kallen, A. A. Suslin, M. S. Tulenbaev, and others is used and developed in this paper. Bibliography: 21 titles.

In the previous paper by the authors, it is shown that a sense-preserving homeomorphism, absolutely continuous on lines (ACL), under certain integral conditions on the complex dilation is conformal at a point. This result includes the Teichmuller-Wittich-Belinski theorem. In the present paper, it is shown that under weakened conditions the local behavior of such a homeomorphism is similar to that of a quasiconformal mapping. Bibliography: 2 titles.

The paper continues the author's previous paper and deals with the case where the existence of the exponential moment of the distribution under consideration is not assumed. Bibliography: 11 titles.

In this paper, we consider a stochastic queueing model for the performance evaluaton of a real-life computer system consisting of n terminals connected with a CPU. A user at terminal i has thinking and processing time depending on the index i. Let us suppose that the operational system is subject to random breakdowns, which may be software or hardware ones, stopping the service both at the terminals and at the CPU. The failure-free operation times of the system and the restoration times are random variables. Busy terminals are also subject to random breakdowns not affecting the system operation. The failure-free operation times and the repair times of a busy terminal i are random variables with distribution function depending on index i. The breakdowns are serviced by a single repairman providing preemptive priority to the system's failure, while the restoration at the terminals are carried out according to the FIFO rule. We assume that each user generates only one job at a time, and he waits at the CPU before he starts thinking again, that is, the terminal is inactive while waiting at the CPU, and it cannot break down. All random variables involved in the model construction are assumed to be exponentially distributed and independent of each other. The aim of this paper is to investigate the effect of different service disciplines, such as FIFO, processor sharing, priority processor sharing, and polling, on the main performance measures, such as utilizations, response times, throughput, and mean queue length.

This paper deals with a nonhomogeneous finite-source queueing model to describe the performance of a multiterminal system subject to random breakdowns under the polling service discipline. The model studied here is a closed queueing network which has three service stations. a CPU (single server), terminals (infinite server), a repairman (single server), and a finite number of customers (jobs) that have distinct service rates at the service stations. The CPU's repair has preemptive priority over the terminal repairs, and failure of the CPU stops the service of the other stations, thus the nodes are not independent. It can be viewed as a continuation of papers by the authors (see references), which discussed a FIFO (first-in, first-out) and a PPS (priority processor sharing) serviced queueing model subject to random breakdowns. All random variables are assumed to be independent and exponentially distributed. The system behavior can be described by a Markov chain, but the number of states is very large. The purpose of this paper is to give a recursive computational approach to solve steady-state equations and to illustrate the problem in question using some numerical results.

sp-Groups form an interesting and informative class of Abelian mixed groups. In this paper, we systematically study self-small sp-groups of finite rank and their endomorphism rings.

An associative ring R is called an E-ring if the canonical homomorphism R ∼= E(R+) is an isomorphism. Additive groups of E-rings are called E-groups. In other words, an Abelian group A is an E-group if and only if A ∼= End A and the endomorphism ring E(A) is commutative. In this paper, we give a survey of the main results on E-groups and E-rings and also consider some of their generalizations: E-closed groups, T -rings, A-rings, the groups admitting only commutative multiplications, etc.

The paper contains both known and new results on automorphism-invariant modules, automorphism-coinvariant modules and modules which are invariant or coinvariant with respect to idempotent endomorphisms of its hull and its cover, respectively. The main results are given with proofs.

This paper is a survey of recent works on invariants of actions of Hopf algebras. Its highlights are results on integrality of H-module PI algebras over subrings of invariant elements obtained by P. Etingof and M. Eryashkin. Older results are also reviewed.

In this paper, we present results on the homological classification of semirings for the last ten years and certain new results.

In this paper, we describe the technique of extremely monotonic functions in the theory of computable linear orders. The basic definitions of extremely monotonic functions and their generalizations are given, and a number of their basic properties and applications are presented.

The categoricity spectrum of a computable structure S is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable presentations of S. The degree of categoricity of S is the least degree in the categoricity spectrum of S. This paper is a survey of results on categoricity spectra and degrees of categoricity for computable structures. We focus on the results about degrees of categoricity for linear orders and Boolean algebras. We construct a new series of examples of degrees of categoricity for linear orders.

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.

In this paper, the length of the group algebra of a dihedral group in the modular case is computed under the assumption that the order of the group is a power of two. Various methods for studying the length of a group algebra in the modular case are considered. It is proved that the length of the group algebra of a dihedral group of order 2k+1 over an arbitrary field of characteristic 2 is equal to 2k.

The paper presents a matrix representation of filter banks corresponding to spline wavelets with shifted supports. The matrix form of the decomposition and reconstruction filters simplifies the representation of nonuniform nonstationary wavelet transforms built on nonuniform grids on a finite interval. Such a representation of filters is used, for example, in constructing error-correcting codes.

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.

A finite computational process using arithmetic operations only is called a rational algorithm. Presently, no rational algorithm for checking the congruence of arbitrary complex matrices A and B is known. The situation can be different if both A and B belong to a special matrix class. For instance, there exist rational algorithms for the cases where both matrices are Hermitian, unitary, or accretive. This paper proposes a rational algorithm for checking whether two involutive matrices A and B are congruent.

The story of the friendship and collaboration between Vladimir Abramovich Rokhlin and the Nizhny Novgorod mathematician Dmitry Andreevich Gudkov during the last period of Rokhlin’s mathematical biography, when he worked in the topology of real algebraic varieties, in which he obtained remarkable results. The paper is based on the correspondence of 1971–1982 preserved in Gudkov’s archive containing 15 letters by V. A.Rokhlin and 8 letters by D. A. Gudkov.

We study the Gnedin–Kingman graph, which corresponds to Pieri’s rule for the monomial basis {Mλ} in the algebra QSym of quasisymmetric functions. The paper contains a detailed announcement of results concerning the classification of indecomposable semifinite harmonic functions on the Gnedin–Kingman graph. For these functions, we also establish a multiplicativity property, which is an analog of the Vershik–Kerov ring theorem.

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.

The following general variant of deterministic “Hats” game is analyzed. Several sages wearing colored hats occupy the vertices of a graph, the kth sage can have hats of one of h(k) colors. Each sage tries to guess the color of his own hat merely on the basis of observing the hats of his neighbors without exchanging any information. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colors. For complete graphs and cycles, the problem of describing the function h(k) for which the sages win is solved in the present paper. A “theory of constructors,” i.e., a collection of theorems demonstrating how one can construct new graphs for which the sages win is developed. A new game “Rook check ” equivalent to the Hats game on a 4-cycle is introduced and completely analyzed.

The goal of the paper is to study vertices of degree 6 of minimal and contraction critical 6-connected graph, i.e., a 6-connected graph that looses 6-connectivity both upon removal and upon contraction of any edge. It is proved that if x and z are adjacent vertices of degree 6, then x and z have at least 4 common neighbors. In addition, a detailed description of the neighborhood of the set {x, z} is given. An infinite series of examples of minimal and contraction critical 6-connected graphs for which the fraction of vertices of degree 6 equals $$ \frac{11}{17} $$ is constructed.

In this paper, we examine the unique solvability of the Gevrey problem for a mixed parabolic equation with singular coefficients in a band. We prove the existence and uniqueness of a solution to the problem stated. The uniqueness of a solution is proved by the method of energy integrals and the existence by methods of the theory of Volterra, Fredholm, and singular integral equations.

In this paper, we examine the solvability of a new class of nonclassical direct problems for multidimensional loaded parabolic equations with Cauchy data. We obtain sufficient conditions for the solvability of the problem; the proof is based on the method of weak approximation. By an example, we demonstrate the application of the theorem proved to the study of inverse problems for multidimensional parabolic equations with Cauchy data.

In this paper, we consider a loaded parabolic equation of a special form in an unbounded domain with Cauchy data. The equation is one-dimensional and its right-hand side depends on the unknown function u(t, x) and traces of this function and its derivatives by the spatial variable at a finite number of different points of space. Such equations appear after the reduction of some identification problems for coefficients of one-dimensional parabolic equations with Cauchy data to auxiliary direct problems. We obtain sufficient conditions of the global solvability and sufficient conditions of the solvability of the problem considered in a small time interval. We search for solutions in the class of sufficiently smooth bounded functions. We examine the uniqueness of the classical solution found and prove the corresponding sufficient conditions. We also obtain an a priori estimate of a solution that guarantees the continuous dependence of the solution on the right-hand side of the equation and the initial conditions.

The paper continues the author's studies of the question on the existence of quadratic differentials Q(z)dz2 having given structure of trajectories and poles of high orders. It is shown that such differentials can be considered as the limits of sequences of quadratic differentials that have poles of second order with trajectories asymptotically similar to logarithmic spirals and realize extremal configurations in suitable families of nonoverlapping domains. It is established that there exist differentials Q(z)dz2 of indicated form having given initial terms of the Laurent expansions in the vicinities of the poles of Q(z)dz2 of order not smaller than three. Some discrepancies in an earlier paper are corrected. Bibliography: 9 titles.

In this paper, we consider the problem of diffraction of an acoustic wave by a cone of arbitrary shape. The problem is to calculate in a high frequency approximation the wave scattered by the cone vertex. The calculation is based on a formula of Smyshlyaev. The acoustic wave satisfies the Newmann boundary conditions. The same problem for the Dirichlet boundary conditions was considered by Babich, Dement'ev, and Samokish. In this paper, we apply a similar method. We calculate the direction diagram for the wave scattered by the cone vertex, in the so-called Smyshlyaev oasis. Calculations show that for the case when the exact solution can be written explicitly (axisymmetric wave on a circular cone), the approximate solution is very close to the exact one. Bibliography: 8 titles.

This paper contiues the series of papers devoted to designing algorithms for computing the second and subsequent terms of the ray series for the vector of longitudinal displacements in isotropic nonhomogeneous elastic media. The method proposed in this paper essentially differs from the previous ones. It is based on expanding the amplitudes, the eikonal, and the given problem parameters into power series with respect to the coordinates transversal to the direction of propagation of the waves considered. The methods for computing the correction term of the ray expansion for the vector of longitudinal displacements are compared. Bibliography: 10 titles.

This paper is an extension of our studies of the computational aspects of spectral problems for rational matrices pursued in previous papers. Methods of solution of spectral problems for both one-parameter and two-parameter matrices are considered. Ways of constructing irreducible factorizations (including minimal factorizations with respect to the degree and size of multipliers) are suggested. These methods allow us to reduce the spectral problems for rational matrices to the same problems for polynomial matrices. A relation is established between the irreducible factorization of a one-parameter rational matrix and its irreducible realization used in system theory. These results are extended to the case of two-parameter rational matrices. Bibliography: 15 titles.

The paper deals with the uniformly elliptic equation (aij(x)uxi)xj=f(x) in an unbounded domain Ω⊂ℝn and its solution u(x) that satisfies the homogeneous Neumann condition. The function f has a compact support. The domain Ω has the following structure: assume that {rm} is an increasing sequence of positive numbers, hm=rm+1−rm, and the ratio hm+1/hm lies between positive constants C1 and C2. The intersection of Ω with the spherical layer between the spheres of radius rm and rm+1 with center at the origin satisfies a certain inequality of isoperimetric type. It is shown in this paper that the set of solutions splits up into three classes: (i) the solutions for which\(\mathop {\overline {\lim } }\limits_{\left| x \right| \to \infty } u(x) = \infty \) and\(\mathop {\underline {\lim } }\limits_{\left| x \right| \to \infty } U(x) = \infty \); moreover, it is shown that these limits are attained with nearly the same speed (if C1/C2=1, then the speed is not less than the exponential one); (ii) the solutions for each of which a constant C exists such that\(\mathop {\lim }\limits_{\left| x \right| \to \infty } u(x) = C\) and u(x)−C changes its sign for large |x|; here, the convergence to C is rapid (for C1/C2=1 this convergence is not slower than the exponential one); (iii) the solutions that do not change their sign for large |x| and increase or decrease to +∞ or −∞, respectively, with low speed (for C1/C2=1 with a linear speed) (one exception is possible here: a slow convergence to a constant). Bibliography: 7 titles.

The transfer matrix of the 6-vertex model of two-dimensional statistical physics commutes with many (more complicated) transfer matrices, but these latter, generally, do not commute between each other. The studying of their action in the eigenspaces of the 6-vertex model transfer matrix becomes possible due to a ``multiplicative property'' of the {\em vacuum curves} of $\cal L$-operators from which transfer matrices are built. This approach allowed, in particular, to discover for the first time the fact that the dimensions of abovementioned eigenspaces must be multiples of (big enough) degrees of the number 2., An English version of the author's article which recently appeared in Zapiski Nauvhnyh Seminarov POMI (S-Petersburg) and is based on two older papers (of 1987) mentioned in the bibliography. The author's Russian paper of 1986 from the bibliography will soon appear in English, too

This paper is the first in a series of papers that lay the foundations of enumeration theory for finite groups including the classical inversion calculus on segments of the natural series and on lattices of subsets of finite sets. Since it became possible to calculate the Mobius function on all subgroups of finite nilpotent groups, the Mobius inversion on these groups began to play a decisive role. The efficiency of the inversion method as a regular technique suitable for solution of enumeration problems of group theory is illustrated with a number of concrete and very important enumerations. Bibliography: 13 titles.

An algorithm for computing the second term of the series of the ray method in the case of elastic inhomogeneous isotropic media is proposed. The main idea of the approach to the problem can be formulated as follows. Let a central (or support) ray of a ray tube be known. If we inroduce the ray-centered coordinates s, q1, q2 in the vicinity of the central ray, then the rays of the ray tube can be described by functions qi=qi(s, γ1, γ2), i=1,2, where s is the are length of the central ray and γj, j=1, 2, are the ray parameters. On the one hand, we show that the integrand of the second term of the series of the ray method can be expressed via the derivatives of the functions qi with respect to γj of the first, second, and third orders. On the other hand, additional differential equations for the derivatives as functions of s can be obtained from Euler's equations for the rays. This paper also contains initial conditions for the derivatives in the case of a point source. Thus, we obtain an algorithm involving additional differential equations for the derivatives\(\frac{{\partial q_i }}{{\partial \gamma _j }},\frac{{\partial ^2 q_i }}{{\partial \gamma _j \gamma _k }},\frac{{\partial ^3 q_i }}{{\partial \gamma _j \gamma _k \gamma _1 }}\), and the initial conditions for them at the source. The algorithm for calculating the mixed components of a vector of displacement is elaborated in detail. Unfortunately, the Russian version of this paper contains some errors, which have been corrected in the English translation of the paper.

This paper is devoted to the calculation of superfluid current in the A-phase of helium-3. This calculation is connected with the solution of a nonhomogeneous ordinary differential equation arising in the low-temperature limit from the Gor'kov equation for normal and anomalous Green functions. The new solution is obtained in the form of the expansion in the eigenfunctions of the problem of an electron in a uniform magnetic field. In the formal high-temperature limit the result obtained in the paper reproduces the known result. Bibliography: 18 titles.

This paper continues previous papers devoted to the realistic (interval) statistics. It is assumed that the input data are distorted by measurement errors, and the greatest possible errors are known. This paper deals with the influence of measurement errors on regression, cluster, and discriminant analysis.

In the paper [Ya. P. Lumel'skii and Ya. M. L'vovskii,Dokl. Akad. Nauk SSSR,246, No. 3 (1979)], asymptotic formulas for multidimensional Polya distributions, including both polynomial and multidimensional hypergeometric distributions, were deduced. The derivation of those formulas was rather bulky and constituted the paper [Ya. P. Lumel'skii and Ya. M. L'vovskii,Dep. VINITI, No. 3155-79, Moscow (1979)]. The aim of the present paper is to realize another approach to constructing asymptotic expansions for multidimensional distributions of more general structure generated by plans of Polya random walks [Ya. P. Lumel'skii,Dokl. Akad. Nauk SSSR,209, No. 6 (1973)]. This allows us not only to prove the results of [Ya. P. Lumel'skii and Ya. M. L'vovskii,Dep. VINITI, No. 3155-79, Moscow (1979)] but to derive new asymptotic formulas as well.

The paper continues the author's previous papers. The aim of it is to study the ellipsoids\(\Phi _{N,\Delta } \subset l_p \), which arises in problems of simple hypothesis testing. Bibliography: 11 titles.

The paper deals with an alphabet-code model and with an associative algebra over a field generated by the alphabet for which the code words are the determinant relations. The restrictions on the overlapping of the code words are closely connected to the restrictions on the homology groups of the algebra. The paper presents the study of this connection quantitatively expressed via the Poincare series of the algebra. Bibliography: 6 titles.

The paper deals with convolution type integral equations of the first kind. For exact solutions it is typical that the solutions are discontinuous or unbounded. The aim of the paper is to construct approximate solutions which may also be discontinuous or unbounded. A priori information about the solution and estimates for the approximate solutions, as well as for the differences between the approximate solution and the exact one, are given in terms of Nikol'skii-Besov spaces with differentiation of fractional orders. Bibliography: 8 titles.

The paper deals with the uniqueness and stability of generalized solutions to the mixed boundary-value problem for the elasticity theory system in an unbounded domain, coinciding with a cone in a neighborhood of infinity. It is assumed that the boundary of the domain consists of two parts: the Dirichlet condition is prescribed on one of them, denoted by Γ1, and the Neumann condition is prescribed on the other. The paper contains sufficient conditions (in terms of metric properties of Γ1) for the validity of the Korn and Hardy inequalities, which imply the uniqueness and stability of the solution to the considered problem in appropriate function spaces. Bibliography: 8 titles.

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.

In this paper, a review of mathematical methods of randomized machine learning is presented.

When creating new technological processes based on the use of high-intensity energy flows, it is necessary to take into account the finite speed of heat transfer. This can be done using the hyperbolic heat transfer equation obtained by A. V. Lykov within the framework of nonequilibrium phenomenological thermodynamics as a consequence of a generalization of the Fourier law for flows and the equation of thermal balance. In previous works, V. N. Khankhasaev modeled the process of switching off an electric arc in a gas flow using this equation. In this paper, we present a mathematical model of this process including the period of stable burning of an electric arc until the shutdown moment, which consists of the replacement of the strongly hyperbolic heat transfer equation by a hyperbolicparabolic equation. For the mixed heat transfer equation obtained, we state certain boundary-value problems, solve them by numerical algorithms, and obtain temperature fields that are consistent with the available experimental data.