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Schlesinger (S) transformations can be combined with a direct rational (R) pull-back of a hypergeometric 2 × 2 system of ODEs to obtain $$ {RS}_4^2 $$ -pullback transformations to isomonodromic 2 × 2 Fuchsian systems with 4 singularities. The corresponding Painleve VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. The paper demonstrates direct computations (involving polynomial syzygies) of Schlesinger transformations that affect several singular points at once, and presents an algebraic procedure of computing algebraic Painleve VI solutions without deriving full RS-pullback transformations.

For complex interpolation nodes, we study the problem of interpolation by series of exponential functions whose exponents form a set, which is condensed at infinity in a certain direction. We obtain a criterion for all sets of nodes from a special class. For arbitrary sets of nodes, we obtain a necessary condition for the solvability of a more general problem of interpolation by functions that can be represented as Radon integrals of an exponential function over a set of exponents. The paper also contains well-known results on interpolation, which, in particular, allow studying the multipoint holomorphic Vallee Poussin problem for convolution operators.

We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semidiscrete 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.

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.

In this paper, we consider the problem on the existence of the upper (lower) envelope of a convex cone or, more generally, a convex set for functions on the projective limit of vector lattices with values in the completion of the Kantorovich space or on the extended real line. We propose vectorial, ordinal, and topological dual interpretations of the existence conditions for such envelopes and the method of constructing it. Applications to the problem on the existence of a nontrivial (pluri)subharmonic and/or (pluri)harmonic minorant for functions in domains of a finite-dimensional real or complex space are considered. We also propose general approaches to problems on the nontriviality of weight classes of holomorphic functions, describing zero (sub)sets for such classes of holomorphic functions, and to the problem of representing a meromorphic function as a ratio of holomorphic function from a given weight class.

In this paper, we present results of the existence of a linear continuous right inverse operator for the operator of the representation of analytic functions in a bounded convex domain of the complex plane by series of quasi-polynomials and exponentials. We also present closely related results on the A. F. Leontiev interpolating function and, more generally, on the interpolating functional and the corresponding Pommiez operator. We examine cyclic vectors and closed invariant subspaces of the Pommiez operator in weighted spaces of entire functions.

This paper is devoted to representing exponential systems in various subspaces of the space H(D) of functions that are analytic in a bounded convex domain D. We consider two kinds of such subspaces: uniformly weighted spaces H(D,𝜑) and spaces of the type of Carleman classes H(D,ℳ).

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.

In this paper, we find the dependence of the reaction force N(y) of a curved trough of arbitrary shape described by a function y(x). Based on the extremum condition dN/dx valid for any point of the abscissa axis, we examine the equation N(y, y', y'') = const whose solutions determine the desired class of curves y(x). We obtain an analytic solution of this equations and perform numerical simulations for various values of parameters. Examples of functions y(x) for which N = const are presented.

This paper is a review of results concerning the construction of asymptotics of solutions to degenerate linear differential equations with holomorphic coefficients. We consider both cases of ordinary and partial differential equations.

In this paper, we consider a certain class of anisotropic second-order elliptic equations of divergent type with variable indices of nonlinearities. We examine conditions of the solvability in the whole space ℝn, n ≥ 2. We prove the existence of solutions without restrictions to the growth rate as |x| → ∞.

This paper is devoted to the study of the motion of a platform maintained on an aircraft in a predetermined position by a system of gyroscopes, which does follows oscillations of the aircraft. Such systems are used for determining the angular position of the aircraft. We briefly review key results on this issue without detailed proofs.

In this paper, we develop a method for solving planar nonstationary problems on the influence of surface pressure on an elastic-porous half-space. Using surface influence functions, we obtain an integral relation, which expresses the desired displacements with the surface pressure. A numericalanalytical algorithm is constructed. For calculating singular integrals, we derive special quadrature formulas based on the canonical regularization of integrands.

In this paper, we consider problems of discrete approximation of special integral operators with the Calderon–Zygmund kernel. We introduce discrete spaces and bounded discrete operators acting in these spaces; then we use these operators for the search for approximate solutions of the corresponding equations. We state theorems on the solvability of equations with discrete operators, compare integral operators with their discrete analogs, and obtain estimates of errors of approximate solutions.

In this paper, we consider the Dirichlet problem for harmonic functions on a two-dimensional complex of a special type. We prove that this problem is a Fredholm problem in the Holder class and its index is zero.

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.

This paper is the third in a series of papers, the aim of which is to construct algebraic geometry over metabelian Lie algebras., Comment: 14 pages

with μ = Lnσ and θ = α−1. Thus, estimation of scale and shape in the model PARσ,α is equivalent to estimation of location and scale in Ex(θ, μ). Robust estimation of θ is treated in a very important paper [3]. In their paper, the authors introduced new estimators of “generalized quantile” type and compared them with several well-established estimators (corresponding to methods of trimming, least squares, and quantiles). The authors examined asymptotic relative efficiency with respect to the (efficient but nonrobust) maximum likelihood estimator and breakdown point. Consider the original statistical model M1 = (R+,B+, {Pθ,μ,1, θ > 0,−∞ < μ < +∞}) where R+ is the real positive half-line, B+ is the family of the Borel subsets of R+, and Pθ,μ,1 is the exponential distribution Ex(θ, μ) with p.d.f. fθ,μ(x), given in (2), and its extension

The present paper is a revised Russian translation of the paper “A new approach to representation theory of symmetric groups,” Selecta Math., New Series, 2, No. 4, 581–605 (1996). Numerous modifications to the text were made by the first author for this publication. Bibliography: 35 titles.

We introduce interpolation analogs of the Schur Q-functions — the multiparameter Schur Q-functions. We obtain for them several results: a combinatorial formula, generating functions for one-row and two-row functions, vanishing and characterization properties, a Pieri-type formula, a Nimmo-type formula (a quotient of two Pfaffians), a Giambelli-Schur-type Pfaffian formula, a determinantal formula for the transition coefficients between multiparameter Schur Q-functions with different parameters. We write an explicit Pfaffian expression for the dimension of a skew shifted Young diagram. This paper is a continuation of the author's paper math. CO/0303169 and a partial projective analog of the paper q-alg/9605042 by A. Okounkov and G. Olshanski and the paper math. CO/0110077 by G. Olshanski, A. Regev, and A. Vershik. Bibliography: 36 titles.

In this part of the paper, we investigate the structure of an arbitrary measure μ supported by a polyhedral cone C in R d in the case where the decumulative distribution function gμ of the measure μ satisfies certain continuity conditions. If a face Γ of the cone C satisfies appropriate conditions, the restriction μ|Γint of the measure μ to the interior part of Γ is proved to be absolutely continuous with respect to the Lebesgue measure λΓ on the face Γ. Besides, the density of the measure μ|Γint is expressed as the derivative of the function gμ multipied by a constant. This result was used in the first part of the paper to find the finite-dimensional distributions of a monotone random field on a poset. Bibliography: 6 titles.

We continue a series of papers in which the Yoneda algebra is computed for algebras of dihedral and semidihedral types. In this paper, the Yoneda algebra is computed for one more family of algebras, namely, for the family D(3\(\mathcal{L}\)) (in the classification of K. Erdmann). In order to find the minimal resolutions of simple modules, we used for the first time a C++ program implemented by the second author. Bibliography: 10 titles.

The present paper continues a series of papers of the author (some of them are written in collaboration), in which the Yoneda algebras are calculated for several families of algebras of dihedral and semidihedral type (in K. Erdmann’s classification). In the paper, the Yoneda algebras are described (in terms of quivers with relations) for the algebras of semidihedral type that form the family SD(3\(\mathcal{K}\)). Bibliography: 10 titles.

This paper supplements the previous paper of the author under the same title. An analog of the Schwarz boundary lemma is proved for non-univalent regular mappings of subsets of the unit disk onto a disk. Based on this result, certain strengthened inequalities of Bernstein type for algebraic polynomials are obtained. The generalized Mendeleev problem is discussed. Two-sided bounds for the module of the derivative of a polynomial with critical points on an interval are established. Bounds for the coefficients of polynomials under certain constraints are provided. Bibliography: 16 titles.

In the paper we introduce formal calculi which are a generalization of propositional modal logics. These calculi are called fuzzy modal logics. We introduce the concept of a fuzzy Kripke model and consider a semantics of these calculi in the class of fuzzy Kripke models. The main result of the paper is the completeness theorem of a minimal fuzzy modal logic in the class of fuzzy Kripke models.

This paper presents the working notes by S. V. Kerov (1946–2000) written in 1993. The author introduces a multidimensional analog of the classical hypergeometric distribution. This is a probability measure Mn on the set of Young diagrams contained in the rectangle with n rows and m columns. The fact that the expression for Mn defines a probability measure is a nontrivial combinatorial identity, which is proved in various ways. Another combinatorial identity analyzed in the paper expresses a certain coherence between the measures Mn and Mn+1. A connection with Selberg-type integrals is also pointed out. The work is motivated by harmonic analysis on the infinite-dimensional unitary group. Bibliography: 25 titles.

This paper is an English translation (made by V. Gelfreich) of V. F. Lazutkin’s work that was published in 1984 by VINITI and thus was not easily available for readers. In the paper, a formula for an exponentially small angle of separatrix splitting of the Chirikov standard map was obtained for the first time. Bibliography: 16 titles and 17 titles added by the translator.

In the first paper of the series, we proved the standardness of a subgroup H containing a split maximal torus in the split spinor group Spin(n,R) over a field K of characteristic different from 2 containing at least 7 elements under one of the following additional assumptions: (1) H is reducible, (2) H is imprimitive, (3) H contains a nontrivial root element. In the present paper, we complete the proof of a result announced by the author in 1990 and prove the standardness of all intermediate subgroups, provided that n=2l and \(\left| K \right| \geqslant 9\). For an algebraically closed K, this follows from a classical result of Borel and Tits, and for a finite K this was proved by Seitz. Similar results for subgroups of the orthogonal groups SO(n,R) were previously obtained by the author not only for fields, but for any commutative semilocal rings R with residue fields large enough. Bibliography: 52 titles.

In this paper, we present a general introduction to the K-theory of C *-algebras and survey of our previous papers, where the functor N 0 from the category of von Neumann algebras to the category of Abelian groups was defined. We investigate the properties of this functor (in particular, its interrelation with the functor K 0) and point out some applications of the functor N 0 in noncommutative geometry. In addition, we recall facts of theory of C *-algebras, von Neumann algebras, and Hilbert C *-modules.

Loop groups are very important in differential geometry, algebraic topology, and theoretical physics [7, 9, 16, 37, 47], but nothing was known about Gaussian quasi-invariant differentiable measures on them. Only the simplest possible representations associated with path integrals were constructed for loop groups of the circle, i.e., for the manifold M = S1 and (real) Riemannian manifolds N [37]. On the other hand, quasi-invariant measures can be used for construction of regular unitary representations [30–32,35]. Moreover, quasi-invariant measures are helpful for investigation of a group itself. In the previous papers of the author [33, 34], loop groups of Riemannian manifolds M and N were investigated, where either M = S is an n-dimensional real sphere, n = 1, 2, . . . , or M = S∞ is the unit sphere in a real separable Hilbert space l2(R). This was progress in comparison with previous works of others authors, which considered only loop groups for the simplest case M = S1. This paper treats arbitrary complex separable connected metrizable manifolds M and N . For example, products of odd-dimensional real spheres S2n−1×S2m−1 can be endonved with structures of complex manifolds in different ways [29]. Other numerous examples of complex manifolds can be found in [25] and references therein; they are domains in Cn, the complex torus Cn/D, where D is a discrete additive subgroup of Cn generated by a basis τ1, . . . , τ2n of C n over R; the quotient space G/D of a complex Lie group G by a discrete subgroup D, submanifolds of the complex Grassmann manifold Gp,q(C), and also their different products and submanifolds. In general, there are complex compact manifolds which are not Kahler manifolds [28,36]. For construction of loop groups here, we use manifolds M with some mild additional conditions. When M is finite-dimensional over C, we assume that it is compact. This condition is not very restrictive, since each locally compact space admits the Alexandrov (one-point) compactification (see [13, Theorem 3.5.11]). When M is infinite-dimensional over C, it is assumed that M is embedded as a closed bounded subset in the corresponding Banach space XM over C. This is necessary for defining a group structure on the quotient space of a free loop space. The free loop space is considered as consisting of continuous functions f : M → N which are holomorphic on M \ M ′ and preserving distinguished points f(s0) = y0, where M ′ is a closed real submanifold depending on f of codimension codimRM ′ = 1, s0 ∈ M , and y0 ∈ N are distinguished points. There are two reasons to consider such a class of mappings. The first is the need to correctly define compositions of elements in the loop group (see below). The second is the isoperimetric inequality for holomorphic loops, which can impose the condition that the loop be constant on a sufficiently small neighborhood of s0 in M if this loop is in some small neighborhood of w0, where w0(M) := {y0} is a constant loop (see [23, Remark 3.2]). In this paper, loop groups of different classes are considered. Classes analogous to Gevrey classes of f : M \M ′ → N are considered for construction of dense loop subgroups and quasi-invariant measures. Henceforth, we consider only orientable manifolds M and N , since for a nonorientable manifold, there always exists its orientable double covering manifold (see [1, 6.5]). Loop commutative monoids with cancellation property are quotients of families of mappings f from M into a manifold N with f(s0) = y0

This review paper is devoted to the numerical analysis of abstract differential equations in Banach spaces. Most of the finite difference, finite element, and projection methods can be considered from the point of view of general approximation schemes (see, e.g., [207,210,211] for such a representation). Results obtained for general approximation schemes make the formulation of concrete numerical methods easier and give an overview of methods which are suitable for different classes of problems. The qualitative theory of differential equations in Banach spaces is presented in many brilliant papers and books. We can refer to the bibliography [218], which contains about 3000 references. Unfortunately, no books or reviews on general approximation theory appear for differential equations in abstract spaces during last 20 years. Any information on the subject can be found in the original papers only. It seems that such a review is the first step towards describing a complete picture of discretization methods for abstract differential equations in Banach spaces. In Sec. 2 we describe the general approximation scheme, different types of convergence of operators, and the relation between the convergence and the approximation of spectra. Also, such a convergence analysis can be used if one considers elliptic problems, i.e., the problems which do not depend on time. Section 3 contains a complete picture of the theory of discretization of semigroups on Banach spaces. It summarizes Trotter–Kato and Lax–Richtmyer theorems from the general and common point of view and related problems. The approximation of ill-posed problems is considered in Sec. 4, which is based on the theory of approximation of local C-semigroups. Since the backward Cauchy problem is very important in applications and admits a stochastic noise, we also consider approximation using a stochastic regularization. Such an approach was never considered in the literature before to the best of our knowledge. In Sec. 5, we present discrete coercive inequalities for abstract parabolic equations in Cτn([0, T ];En), C τn([0, T ];En), L p τn([0, T ];En), and Bτn([0, T ];C (Ωh)) spaces.

In this paper we investigate a single server retrial queue with a finite number of heterogeneous sources of calls. It is assumed when a given source is idle it will generate a primary call after an exponentially distributed time. If the server is free at the time of the request’ s arrival then the call starts to be served. The service time is also exponentially distributed. During the service time the source cannot generate a new primary call. After service the source moves into free state and can generate a new call again. If the server is busy at the time of the arrival of a primary call, then the source starts generating so called repeated calls with exponentially distributed times until it finds the server free. As before, after service the source becomes free and can generate a new primary call again. We assume that the primary calls, repeated attempts and service times are mutually independent. This queueing system and its variants could be used to model magnetic disk memory systems, local area networks with CSMA/CD protocols and collision avoidance local area networks. The novelty of this model is the heterogeneity of the calls, which means that each call is characterized by its own arrival, repeated and service rates. The aim of the paper is to give the usual steady-state performance measures of the system. To do so, an efficient software tool, MOSEL ( Modeling, Specification and Evaluation Language ) developed at the University of Erlangen, Germany, is used to formulate and solve the problem. Several sample numerical results illustrate the power of the tool showing the effect of different parameters on the system measures.

This paper is a review of certain results related to the holomorphic continuation of functions given on the boundary of a bounded domain D ⊂ Cn, n > 1, to this domain. The subject is far from new, but here we speak of functions with a one-dimensional holomorphic continuation property along complex lines and curves and also of boundary higher-dimensional variants of the Morera theorem. On the complex plane C, the results on functions with the one-dimensional holomorphic continuation property are trivial, and there are no boundary Morera theorems. Therefore, most of the results of the paper are essentially higher-dimensional. We note that ordinary (nonboundary) Morera theorems in domains of the space Cn are well known (see, e.g., [10, 26]). The first result pertaining to our topic was obtained in [3] by M. L. Agranovskii and R. E. Val’skii, who studied functions with the one-dimensional holomorphic continuation property in a ball. The proof was based on the properties of the automorphism group of the ball. E. L. Stout [44] has used the complex Radon transform and extended the Agranovskii–Val’skii theorem to arbitrary bounded domains with smooth boundaries. An alternative proof of the Stout theorem was obtained in [16] by A. M. Kytmanov, who has applied the Bochner–Martinelli integral. The idea of using integral representations (of Bochner–Martinelli, Cauchy–Fantappie, and logarithmic residue) turns out to be useful in studying functions with the one-dimensional holomorphic continuation property along complex curves [17,18]. The so-called Morera property is weaker than the one-dimensional holomorphic continuation property. It consists of the vanishing of integrals of a given function over the intersection of the boundary of the domain considered with complex lines (complex planes). E. Grinberg [37] has studied functions with the Morera property in a ball (in fact, this result is contained in the work [3] of M. L. Agranovskii and R. E. Val’skii). J. Globevnik and E. L. Stout [33] have obtained the boundary Morera theorem for an arbitrary bounded domain. A local variant of the Morera theorem was considered by J. Globevnik in [32] and D. Govekar in [36]. In [20], functions with the Morera properties along complex curves were considered. In Sec. 1 of the review, we consider global boundary analogs of the Morera theorem along complex and real planes. In the second section, we present various results related to the Morera property along complex lines. In Sec. 3, we consider different variants of the Morera theorem in a ball. In Sec. 4, we present global analogs of the boundary Morera theorems along complex curves. In Sec. 5, we present local variants of the Morera theorems. Section 6 contains results on functions with the Morera property in classical domains. Section 7 is devoted to the Morera theorems in unbounded domains. In the last section, we formulate certain unsolved problems.

In the preceding paper, the authors published an existence theorem for basic submodules of right modules over right Noetherian, serial rings. The aim of the present paper is to prove a uniqueness theorem for basic submodules over such rings. Bibliography: 13 titles.

Discrete analogs of extrema of curvature and generalizations of the four-vertex theorem to the case of polygons and polyhedra are suggested and developed. For smooth curves and polygonal lines in the plane, a formula relating the number of extrema of curvature to the winding numbers of the curves (polygonal lines) and their evolutes is obtained. Also are considered higher-dimensional analogs of the four-vertex theorem for regular and shellable triangulations., Several changes in the last section. In the original version of this paper we claimed that any regular triangulation of a convex d-polytope has at least d ears. For a proof we used the same arguments as in Schatteman's paper [22]. Since this paper has certain gaps (see our paper [1]), the d -ears problem of a regular triangulation is still open

A projective transformation of a pseudo-Riemannian manifoldM is an automorphism of the induced Riemannian connection of the projective structure that takes geodesics in M to geodesics again. For the first time, the problem of determining Riemannian spaces admitting continuous transformation groups preserving geodesics was considered by Sophus Lie for the case of two-dimensional surfaces (see [306]). However, as Fubini wrote in the preface to [262], “the famous mathematician did not succeed in solving this problem” (which Fubini called the “Lie problem”). Having criticized Lie’s method, which, to Fubini’s opinion, could not be used for the general solution of the problem, Fubini developed his own approach based on the infinitesimal calculus, later named the “Lie differentiation.” The subsequent development of the theory of projective transformations on linear connection spaces is connected with the names of E. Cartan, Eisenhart, Thomas, Knebelman, Schouten, Yano, Egorov, Vranceanu, Kobayashi, and others. It is known that in the spaces of constant curvature S, when considered in the small, the complete projective group coincides with the projective group of pseudo-Euclidean space, i.e., with the group of bilinear substitutions, and depends on n(n+ 2) parameters. In the spaces V n of nonconstant curvature, the order of the complete projective group does not exceed n(n−2)−5 (Egorov [91]), and, moreover, in the majority of cases, this group consists of similarity transformations (homotheties) or isometries. In 1903, in “Turin Academy Notes,” Fubini’s paper “Groups of geodesic transformations” was published [262], which as mentioned above, laid the foundations for a systematic definition and study of Riemannian spaces admitting infinitesimal projective transformations. Later on, Solodovnikov [156–158] continued Fubini’s research and completely solved the problem posed; the works of Fubini and Solodovnikov contain a classification of the Riemannian spaces V , n ≥ 3, in terms of (local) groups of projective transformations which are larger than homothety groups. The conclusions of Fubini and Solodovnikov are based on the assumption that the metrics considered are positive definite. Taking a given signature as a condition considerably complicates the problem and requires a basically new approach for its solution. It was proposed in the papers of the author [5– 11], where the problem of defining all pseudo-Riemannian manifolds with Lorentz signature (+ − . . .−) (Lorentz manifolds) of dimension n ≥ 3 admitting nonhomothetic infinitesimal projective and affine transformations was solved, and for each of them, the maximal projective and affine Lie algebras, together with the homothetic and isometric subalgebras, were defined. This paper includes a survey of the results in the theory of projective transformations of pseudoRiemannian manifolds, in particular, the solution of the classical geometrical problem of determining the pseudo-Riemannian metrics with corresponding geodesics (Sec. 5) and the Lie problem (Sec. 6). By using the technique of skew-normal frames developed by the author in [31] (see Sec. 1), all two-dimensional

The theory of lower semicontinuous linear mappings for dioid-valued functions and the abstract (axiomatic) theory of Fenchel-Moreau conjugations have developed parallelly and independently. In this paper we show that these theories are closely related. We introduce the concept of conjugations for functions with values in the top enlargement of a boundedly complete idempotent semifield. Main result: There exists an isomorphism between the complete semimodules of all lower semicontinuous linear mappings and all conjugations. We show that the above framework and the framework of functions with values in the canonical enlargement of a complete lattice ordered group, used in our previous papers to introduce and study abstract conjugations, are essentially equivalent.

The paper is a continuation of the authors' previous paper on the topology of higher local fields. A useful class of systems of topological generators for the additive and multiplicative groups is described. Bibliography: 3 titles.

The theory of topological vector spaces (TVS), being a foundation of modern functional analysis, is now considered as a completely mature, or, to be more specific, dead mathematical discipline. This pessimistic view is based on a picture, where, on the one hand, a well-known system of facts is stated, facts that have been considered classical since the times of Mackey and Grothendieck, and, on the other hand, an opinion exists explicitly or implicitly, that any deviation from this system inevitably tends to a disappointment. “Do not seek any other patterns or connections but those that we have described here, because any of your assumptions will find a counterexample in Nature”; this phrase should have been cited as an afterward to the three textbooks on topological vector spaces translated into Russian [12, 41, 43], which represent the face of this science from the sixties until now. Indeed, after the famous Enflo counterexample [22], the development of the theory of topological vector spaces (which was planned originally as an extension of a clear and simple discipline, linear algebra) was transformed into a sequence of reports on the oddities one can encounter in modern mathematics. The result of this process is a general skepticism about and a loss of interest in this field. The number of papers on the theory of TVS is declining, while the attitude of other mathematicians now consists of polite perplexity or lenient irony. And add to that the fact that the vast majority of specialists in functional analysis do not concern themselves with topological vector spaces but consciously concentrate on Banach theory. (According to Compumath, for example, in 1996 only 7 papers were devoted to locally convex and topological vector spaces, while 612 were devoted to Banach spaces. In the theory of topological algebras, the ratio is 6:90.) The dominant position of Banach theory in functional analysis justifies a natural question: what are the advantages of the Banach case over the more general topological situation? At the conceptual level, the difference becomes most apparent in the theory of topological algebras. Let us cast a critical look. The first impression we will get when comparing Banach and non-Banach theories of topological algebras is as follows. While the concept of Banach algebra emerges naturally from intuitive expectations, faciliated by an acquaintance with general algebra and the theory of Banach spaces (and leads to the profound theory of Banach algebras), the situation with general topological algebras looks completely different. We find here unexpectedly that the very attempts to define a topological algebra (and topological module) lead to results that contradict intuition. Indeed, intuitively it is clear that a “good” definition of topological algebra (and topological module) must fulfill at least the following minimal list of requirements

A discrete string with fixed endpoints carrying a finite number of beads is determined by the masses of beads and the distances between them. The string possesses a set of simple eigenfrequencies corresponding to harmonic eigenmodes. In this paper, the following problem is studied: to find a discrete string carrying seven beads such that its eigenfrequencies coincide with the freqiencies of the notes of the first octave of the musical scale. The problem is solved in two steps. First, the spectral inverse problem is considered, i.e., we recover the string from its spectrum and a set of constants related to the normalized eigenmodes. A procedure of solving this problem is described. One of the main results of the paper is a necessary and sufficient condition for the solvability of the spectral inverse problem. The second step is numerical realization of the procedure. Bibliography: 3 titles.

The minimax signal detection problem for an admissible sets of signals forming a ball with a removed domain around its center has been considered in detail in the recent author's papers. In the present paper, we study additional possibilities arising under the assumption of positivity of the signal. Bibliography: 12 titles.

Any link in ℝ3 is isotopic to a link lying on the union T of three half-planes with common boundary line. A nontrivial theory of knots and links on T was developed by the same author in an earlier paper. In the present paper, the results obtained are interpreted in the context of M. Gusarov's theory of invariants of finite degree (cubic space theory). Bibliography: 6 titles.

The ideas of using related algebraic systems for studying properties of the original system go back to E. Galois. Later on, they were developed by R. Dedekind, G. Birkhoff, A. I. Malcev, A. Tarski, A. G. Kurosh, O. Ore, L. A. Skornjakov, G. Gratzer, and other famous algebraists. The last half of the twentieth century was marked by a special attention to lattices associated with algebras. Among them, there are, for example, lattices of subalgebras, congruences, topologies, axiomatizable classes of algebras, etc. These lattices give considerable information about the properties of the original algebras. The traditional problems in this area are the following: the description of algebras whose derived lattices satisfy a given property (e.g., modularity, distributivity), characterizations of lattices represented as a lattice associated with a given class of algebras, and so on. Up to now, substantial progress in solving these problems has been made in classes of lattices associated with unary algebras and especially with unars (a unar is an algebra with one unary operation) (see, e.g., [1–11]). Here the valuable contribution of L. A. Skornjakov to investigating unars should be pointed out ([12]). The lattices of topologies of algebras have been little studied at present. Currently, since the topological algebra is being intensively developed, one can see that this problem is becoming more and more topical ([13]). In the present paper, the problems mentioned above are solved for lattices of topologies of unars. Moreover, we obtain answers to some questions about lattices of topologies of arbitrary algebras naturally arising in this research. The attention to the class of topology lattices of unars is not by chance, since the class of topology lattices on sets coincides with the class of topology lattices of unars 〈A, f〉 of a variety defined by the identity f(x) = x. In this paper, we describe cardinalities of lattices of topologies of uncountable unars. We prove that for the topology of a unar, each of the following conditions implies the finiteness of the unar itself: finiteness of width; descending (ascending) chain condition. But the author [14] proved that for any integer n ≥ 2 and an arbitrary functional signature containing at least one symbol of arity greater than 1, there is a countably infinite algebra of this signature whose topology lattice contains exactly n elements. It is shown that there is no unar whose lattice of topologies is countably infinite. Hence, from the Lowenheim–Skolem–Tarsky theorem, it follows that the class of all lattices of topologies of unars is not axiomatizable. Further, characterizations of those unars whose lattices of topologies are modular, distributive, complemented, pseudocomplemented, Boolean, or chains, respectively, are given. We also describe the class of all modular lattices each of which is isomorphic to a lattice of topologies of some unar. Consequently, the conditions for modularity and distributivity are equivalent for lattices of topologies of unars. We also deal with the upper-subsemilattices 0(A), 1(A), and 2(A) of the lattice (A) of topologies of an arbitrary algebra A whose elements are T0, T1, and T2-topologies on the algebra A, respectively.

In papers devoted to the dynamical inverse problem for the Naxwell system, in the context of the BC method, a unitary transform M: “solenoidal field → transversal field” was introduced. In this paper, M is complemented by a transform N: “potential field → longitudinal field.” Isometry and completeness of N are established. The transform U = M ⊕ N mentioned in the title, turns out to be a unitary oprator. Bibliography: 5 titles.

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.