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This paper studies conditions for local (in time) solvability of a qualitatively new singularllimit problem, the free (unknown) boundary problem appearing recently. In fact, there are not so many different free boundary problems, which corresponds to not so large a variety of principally different phase transitions of the first and second kinds. Therefore, the appearance of principally new problems elicits interest. This paper studies structural features of a certain problem on the basis of a certain method developed previously, precisely, the localization method [1, 3, 9].

The purpose of this paper is to determine the improve system reliability by introducing the redundancy of components. This paper considers one warm stand-by unit with one main unit. We study the reliability of such systems using probability models in terms of Sedyakin’s model. A parametric family of Birnbaum-Saunders distributions is considered as the distribution of failure times both of main and stand-by units. Parametric point and interval estimation is obtained from censored data. Bibliography: 17 titles.

The present paper continues a cycle 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 algebra is described (in terms of quivers with relations) for algebras of semidihedral type, namely, of the family SD(3\(\mathcal{B}\))1. Bibliography: 16 titles.

Recently Liebeck, Nikolov, and Shalev noticed that finite Chevalley groups admit fundamental SL2-factorizations of length 5N, where N is the number of positive roots. From a recent paper by Smolensky, Sury, and Vavilov, it follows that the elementary Chevalley groups over rings of stable rank 1 admit such factorizations of length 4N. In the present paper, we establish two further improvements of these results. Over any field the bound here can be improved to 3N. On the other hand, for SL(n, R), over a Bezout ring R, we further improve the bound to 2N = n 2-n. Bibliography: 25 titles.

The problem of characterization of integrals as linear functionals is considered in the paper. It starts from the familiar results of F. Riesz (1909) and J. Radon (1913) on integral representation of bounded linear functionals by Riemann–Stieltjes integrals on a segment and by Lebesgue integrals on a compact in \( {\mathbb{R}^n} \), respectively. After works of J. Radon, M. Frechet, and F. Hausdorff the problem of characterization of integrals as linear functionals took the particular form of the problem of extension of Radon’s theorem from \( {\mathbb{R}^n} \) to more general topological spaces with Radon measures. This problem has turned out difficult and its solution has a long and rich history. Therefore, it may be naturally called the Riesz–Radon–Frechet problem of characterization of integrals. The important stages of its solution are connected with such mathematicians as S. Banach, S. Saks, S. Kakutani, P. Halmos, E. Hewitt, R. E. Edwards, N. Bourbaki, V. K. Zakharov, A. V. Mikhalev, et al. In this paper, the Riesz–Radon–Fr´echet problem is solved for the general case of arbitrary Radon measures on Hausdorff spaces. The solution is given in the form of a general parametric theorem in terms of a new notion of the boundedness index of a functional. The theorem implies as particular cases well-known results of the indicated authors characterizing Radon integrals for various classes of Radon measures and topological spaces.

In the present paper, norm series for Honda formal groups are studied. The Steinberg relation for the classical Hilbert symbol is generalized to Honda formal groups. Necessary and sufficient conditions for a series to satisfy the generalized Steinberg relation are obtained. In the paper, such series are called norm series. Bibliography: 8 titles.

Lately, the following problem attracted a lot of attention in various contexts: find the shortest factorization G = UU - UU - …U ± of a Chevalley group G = G(Φ, R) in terms of the unipotent radical U = U(Φ, R) of the standard Borel subgroup B = B(Φ, R) and the unipotent radical U - = U -(Φ, R) of the opposite Borel subgroup B - = B - (Φ, R). So far, the record over a finite field was established in a 2010 paper by Babai, Nikolov, and Pyber, where they prove that a group of Lie type admits the unitriangular factorization G = UU - UU - U of length 5. Their proof invokes deep analytic and combinatorial tools. In the present paper, we notice that from the work of Bass and Tavgen one immediately gets a much more general results, asserting that over any ring of stable rank 1 one has the unitriangular factorization G = UU - UU - of length 4. Moreover, we give a detailed survey of traingular factorizations, prove some related results, discuss prospects of generalization to other classes of rings, and state several unsolved problems. Another main result of the present paper asserts that, in the assumption of the Generalized Riemann’s Hypothesis, Chevalley groups over the ring $$ \mathbb{Z}\left[ {\frac{1}{p}} \right] $$ admit the unitriangular factorization G = UU - UU - UU - of length 6. Otherwise, the best length estimate for Hasse domains with infinte multiplicative groups that follows from the work of Cooke and Weinberger, gives 9 factors. Bibliography: 67 titles.

In the present paper, analogs of the Eilenberg-Watts theorem are proved for categories of finitely generated modules over finite-dimensional algebras for right exact and left exact functors. Furthermore, for left exact functors the corresponding bimodules are described explicitly. The main aim of this paper is to present how, with these versions of the Eilenberg-Watts theorem, we obtain some new descriptions of the Nakayama functor and the inverse Nakayama functor in the case of self-injective algebras. Bibliography: 4 titles.

Let μ be a Borel charge (i.e., a real Borel measure) on ℝ, and let $ {P_{(y)}}(t) = \frac{y}{\pi \left( {{y^2} + {t^2}} \right)},y > 0 $ , t ∈ ℝ, denote the Poisson kernel. Bourgain proved that for a nonnegative μ and for many points t ∈ ℝ, the variation of the function $ y \mapsto \left( \mu * {P_{ {(y)}}} \right)(x) $ on (0, 1] is finite. This is true, in particular, for so-called B-points x introduced in a previous author’s paper, In the present paper, we give new descriptions of B-points which are adjusted to some applications of this notion. Bibliography: 5 titles.

The Q-index of a simple graph is the largest eigenvalue of its signless Laplacian, or Q-matrix. In our previous paper [1] we gave three lower and three upper bounds for the Q-index of nested split graphs, also known as threshold graphs. In this paper, we give another two upper bounds, which are expressed as solutions of cubic equations (in contrast to quadratics from [1]). Some computational results are also included.

In 1998, Kleinbock & Margulis established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent $w_{0}(\vv x) = 1/n$ for almost every point $\vv x$ on a non-degenerate submanifold $\cM$ of $\R^n$. In this paper the simultaneous inhomogeneous analogue of Sprindzuk's conjecture is established. More precisely, for any `inhomogeneous' vector $\bm\theta\in\R^n$ we prove that the simultaneous inhomogeneous Diophantine exponent $w_{0}(\vv x, \bm\theta)= 1/n$ for almost every point $\vv x$ on $M$. The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent $w_0(\vv x)=1/n$ for almost all $\vv x\in \cM$ if and only if for any $\bm\theta\in\R^n$ the inhomogeneous exponent $w_0(\vv x,\bm\theta)=1/n$ for almost all $\vv x\in \cM$. The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered in \cite{Beresnevich-Velani-new-inhom}. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas of \cite{Beresnevich-Velani-new-inhom} while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory., Comment: Dedicated to A.O. Gelfond on what would have been his 100th birthday 13 pages

In this paper, a simple shear flow in a half-space, which has interesting properties from the point of view of boundary regularity, is described. It is a solution with a bounded velocity field to both the homogeneous Stokes system and the Navier–Stokes equation, and satisfies the homogeneous initial and boundary conditions. The gradient of the solution may become unbounded near the boundary. The example significantly simplifies an earlier construction by K. Kang, and shows that the boundary estimates obtained in a recent paper by the first author are sharp. Bibliography: 4 titles.

In this paper, we present the main results of the study of multidimensional three-websW(p, q, r) obtained by the method of external forms and moving Cartan frame. The method was developed by the Russian mathematicians S. P. Finikov, G. F. Laptev, and A. M. Vasiliev, while fundamentals of differential-geometric (p, q, r)-webs theory were described by M. A. Akivis and V. V. Goldberg. Investigation of (p, q, r)-webs, including algebraic and geometric theory aspects, has been continued in our papers, in particular, we found the structure equations of a three-web W(p, q, r), where p = λl, q = λm, and r = λ(l + m − 1). For such webs, we define the notion of a generalized Reidemeister configuration and proved that a three-web W(λl, λm, λ(l + m − 1)), on which all sufficiently small generalized Reidemeister configurations are closed, is generated by a λ-dimensional Lie group G. The structure equations of the web are connected with the Maurer–Cartan equations of the group G. We define generalized Reidemeister and Bol configurations for three-webs W(p, q, q). It is proved that a web W(p, q, q) on which generalized Reidemeister or Bol configurations are closed is generated, respectively, by the action of a local smooth q-parametric Lie group or a Bol quasigroup on a smooth p-dimensional manifold. For such webs, the structure equations are found and their differential-geometric properties are studied.

The paper deals with the inverse problem of reconstructing a Riemannian manifold from its boundary data. This problem has been solved by the boundary control method, and at the moment there are several variants of solving it. In the paper, one more version of the procedure, which recovers the manifold from scalar spectral or dynamical data, is proposed. This version is the simplest one in regard to the devices in use: geometrical optics, polar representation of operators, etc. are not employed and only a controllability property of a relevant dynamical system is applied. Without substantial changes, this version is applicable to a more complicated (vector) problem of electrodynamics for the Maxwell system. The simplicity of the procedure proposed provides additional chances for its numerical realizability. At the end of the paper, a plan of numerical experiment is discussed. To draw attention to such new options is one of the main aims of the paper. Bibliography: 9 titles.

In this paper, we prove some uniqueness theorems of entire functions and their derivatives that share an entire of function whose order is less than their order. The results in this paper improve those given by G. G. Gundersen and L. Z. Yang, J. P. Wang, J. M. Chang, and Y. Z. Zhu, X. M. Li and H. X. Yi, and others. Some examples are provided to show that the results in this paper are the best possible.

The paper deals with the structure of intermediate subgroups of the general linear group GL(n, k) of degree n over a field k of odd characteristic that contain a nonsplit maximal torus related to a radical extension of degree n of the ground field k. The structure of ideal nets over a ring that determine the structure of intermediate subgroups containinga transvection is given. Let $$ K = k\left( {\sqrt[n]{d}} \right) $$ be a radical degree-n extension of a field k of odd characteristic, and let T =(d) be a nonsplit maximal torus, which is the image of the multiplicative group of the field K under the regular embedding in G =GL(n, k). In the paper, the structure of intermediate subgroups H, T ≤ H ≤ G, that contain a transvection is studied. The elements of the matrices in the torus T = T (d) generate a subring R(d) in the field k.Let R be an intermediate subring, R(d) ⊆ R ⊆ k, d ∈ R. Let σR denote the net in which the ideal dR stands on the principal diagonal and above it and all entries of which beneath the principal diagonal are equal to R. Let σR denote the net in which all positions on the principal diagonal and beneath it are occupied by R and all entries above the principal diagonal are equal to dR. Let E(σR) be the subgroup generated by all transvections from the net group G(σR). In the paper it is proved that the product TE(σR) is a group (and thus an intermediate subgroup). If the net σ associated with an intermediate subgroup H coincides with σR,then TE(σR) ≤ H ≤ N(σR),where N(σR) is the normalizer of the elementary net group E(σR) in G. For the normalizer N(σR),the formula N(σR)= TG(σR) holds. In particular, this result enables one to describe the maximal intermediate subgroups. Bibliography: 13 titles.

The paper deals with some additional details concerning the parametrization of the highest Weyl orbit of equations on the highest weight orbit in the adjoint representations of Chevalley groups of types E 7 and E 8 , as given in the author’s paper “Numerology of square equations.” Bibliography: 25 titles.

The aim of this paper is to study the correlation between the properties of rings and the structure of their additive groups. In this paper, all multiplications on reduced algebraically compact groups and on divisible torsion-free Abelian groups are described. Absolute Jacobson radicals and absolute nil-radicals in some classes of Abelian groups are found. Using these results, we will give a description of semisimple groups and also of groups on which every ring is a nilpotent ring (a nil-ring, a radical ring) in some concerned classes of Abelian groups.

In this paper, properties of solutions of the convolution-type integral equation $$ \left( {1 + w(x)} \right)P(x) = \left( {m*P} \right)(x) + Cm(x) $$ on the real axis are studied. The main concern is to find conditions for the function w(x) and the kernel m(x) sufficient for the existence of an admissible solution P(x), i.e., a solution which has a nonzero limit at infinity. The main results of the paper are the uniqueness theorem for the admissible solution for rapidly decreasing kernels m and the existence theorem for one-sided compactly supported kernels m.

An evolutionary problem of small motions of an ideal barotropic liquid filling a rotating isotropic elastic body is studied in the paper. Moreover, the corresponding spectral problem arising in the study of normal motions of the mentioned system is considered. First, we state the evolutionary problem, then we pass to a second-ordered differential equation in some Hilbert space. Based on this equation, we prove the uniqueness theorem for the strong solvability of the corresponding mixed problem. The spectral problem is studied in the second part of the paper. A quadratic spectral sheaf corresponding to the spectral problem was derived and studied. Problems of localization, discreteness, and asymptotic form of the spectrum are considered for this sheaf. The statement of double completeness with a defect for a system of eigenelements and adjoint elements and the statement of essential spectrum of the problem are proved.

This paper deals with an extremal problem concerning hypergraph colorings. Let k be an integer. The problem is to find the value mk(n) equal to the minimum number of edges in an n-uniform hypergraph not admitting two-colorings of the vertex set such that every edge of the hypergraph contains k vertices of each color. In this paper, we obtain the exact values of m2(5) and m2(4), and the upper bounds for m3(7) and m4(9).

The paper proposes a mathematical formalism for describing artificial sense-of-touch systems. Mathematical models for obtaining, processing, and interpreting tactile information are provided. Problems of encoding and reproducing tactile information are formulated and algorithms for solving these problems are proposed. The problem of interpreting tactile information is considered, and the corresponding simple mathematical model is studied. Within the framework of this simple model, an exact solution of the interpretation problem is obtained for the case of finite deformations, and the insufficiency of the linear method (Hooke’s law) for describing problems of interpreting tactile data is shown. The presentation of mathematical models for the theory of artificial sense-of-touch systems in this paper is the first such detailed presentation in the Russian literature. These models are of interest to mathematicians, mechanicians, physicians, and engineers who construct or use artificial sense-of-touch systems.

We study the problem of realization of a given generalized oscillator by a system of N generalized oscillators of a different type. We consider a generalized oscillator related to a fixed system of orthogonal polynomials that are determined by three-term recurrent relations and the corresponding three-diagonal Jacobi matrix J. The case N =2 was considered in a previous paper. It was shown that in this case the orthogonality measure is symmetric with respect to rotation at angle π. In this paper, we consider the case N =3. We prove that such a problem has a solution only in two cases. In the first case, the Jacobi matrix related to the given “composite” generalized oscillator has block-diagonal form and consists of similar 3×3 blocks. In the second (more interesting) possible case, the Jacobi matrix is not block-diagonal. For this matrix, we construct the corresponding system of orthogonal polynomials. This system decomposes into three series which are related to Chebyshev polynomials of the second kind. The main result of the paper is a solution of the moment problem for the corresponding Jacobi matrix. In this case, the constructed measure is symmetric with respect to rotation at angle 2π/3. Bibliography: 6 titles.

For every hyperbolic toral automorphism T, the present author has defined in his previous paper some unbounded T-invariant second-order difference operators related to the so-called homoclinic group of T. These operators were considered in the space L2 with respect to the Haar measure. It is shown in the present paper that such operators give rise to transition semigroups in the space of continuous functions on the torus and generate dynamically invariant Markov processes. This leads almost immediately to a family of invariant measures for the automorphism T.Along with a short discussion, some open questions about properties of these measures are posed. Bibliography: 9 titles.

The paper deals with the so-called M-transform, which maps divergence-free vector fields in Ω T := {x ∈ Ω| dist(x, ∂Ω) < T}, Ω ⊂⊂ $$ \mathbb{R} $$ 3, to the space of transversal fields. The latter space consists of vector fields in Ω T tangential to the equidistant surfaces of the boundary ∂Ω. In papers devoted to the dynamical inverse problem for the Maxwell system, in the framework of the BC-method, the operator M T was defined for T < T ω, where T ω depends on the geometry of Ω. This paper provides a generalization for arbitrary T. It is proved that M T is partially isometric, and its intertwining properties are established. Bibliography: 6 titles.

In this paper, we consider structured automata without output signals whose state sets are endowed with an algebraic structure of hypergraphs. The main result of the paper is a theorem where we obtain necessary and sufficient conditions for the possibility of defining on the state set of some automaton A a structure of a hypergraph H such that the automaton A will be the universal hypergraphic automaton.

In the present paper we study connections in the fiberings associated with the Grassmann manifold and the space of the centered planes. The work is related to the studies in differential geometry. In the paper, we use the method of continuations and scopes of G. F. Laptev which generalizes the moving frame method and the exterior forms method of Cartan; the method depends on calculation of exterior differential forms. In the paper, we develop a new method of research in Grassman manifolds and some generalization of the method which includes the theory of the induced connections of the spaces of planes and centered planes in the n-dimensional projective space.

A point q is embraced by a set of points S if q is interior to the convex hull of S [8]. In some illumination applications where points of S are lights and q is a point to be illuminated, the embracing concept is related to a good illumination [4, 6], also known as the ∆-guarding [12] and the well-covering [10]. In this paper, we are not only interested in convex dependency (which is actually the embracing notion) but also in proximity. Suppose that the sites of S are lights or antennas with limited range; due to their limited power, they cover a disk of a given radius r centered at the sites of S. Only the points lying in such disks are illuminated. If we want to embrace the point q with the minimum range r, we need to know which is the closest light sq to q such that q lies in the convex hull formed by sq and the lights of S closer to q than sq. This subset of S related to point q is called the closest embracing set for q in relation to S and its cardinality is the closest embracing number of q. By assigning every point q in the convex hull of S to its closest embracing site sq, we obtain a partition of the convex hull that we call the embracing Voronoi diagram of S. This paper proves some properties of the embracing Voronoi diagrams and describes algorithms to compute such diagrams, as well as the levels in which the convex hull is decomposed regarding the closest embracing number.

The present paper continues a cycle of papers, in which the Yoneda algebras were calculated for several families of algebras of dihedral and semidihedral type in the classification by K. Erdmann. Using the technique of a previous paper, a description of the Yoneda algebras for both families of local algebras occurring in this classification is given. Namely, a conjecture about the structure of the minimal free resolution of a (unique) simple module is stated, which is based on some empirical observations, and after establishing this conjecture, “cohomology information" is derived from the resolution discovered, and, as a result, this allows us to describe the Yoneda algebras of the algebras under consideration, It is noted that a similar technique was applied in computation of the Hochschild cohomology algebra for some finite-dimensional algebras. Bibliography: 23 titles.

In a previous paper, the order of the stable Grothendieck group of a symmetric special biserial algebra was computed. The present paper is the final part of that paper. The precise structure of K 0(stmod (Λ)) is determined. Bibliography: 2 titles.

In this paper, the construction of CLFs for nonlinear systems and a new inverse optimal control law are presented. The construction of a CLF for an affine nonlinear system is reduced to the construction of a CLF for a simpler system, and a new LgV type control law with respect to a CLF is provided. This control law is a generalization of Sontag’s formula and contains a design parameter. Tuning this parameter gives many suboptimal solutions for the optimization problem. Also, the gain margin and sector margin of the control law are calculated. Examples are provided to illustrate the main theoretical results of the paper.

In [3], the inversion of an integral operator of potential type with constant characteristic generated by the many-dimensional generalized shift was obtained. In this paper, the author obtains a generalization of the results from [3] to the case of a shift of mixed type, i.e., on a part of the variable generalized shifts of integral nature adopted to deal with the Bessel singular differential operator act, whereas on the other part, the ordinary shift act. Also, it should be noted that in contrast to [3], the integral of B-potential type with homogeneous characteristic is considered in this paper. This generalization is attained by introducing general hypersingular integrals of the general form [8].

The aim of the present (mostly expository) paper is to show the relationship of a generalization of Kazhdan’s property (T) for C*-algebras introduced in our recent paper to that of B. Bekka. It is shown that our definition coincides with Bekka’s definition for group C*-algebras of locally compact groups, whereas, in general, these definitions are distinct. Criteria for a C*-algebra to possess our property (T) are given. A number of examples of C*-algebras with and without property (T) are considered. Relations to K-theory are studied.

This paper is a follow-up to a previous paper where the logical characterization of n-ary synchronous relations due to Eilenbeig, Elgot, and Shepherdson was investigated in the case where the alphabet has infinitely many letters. Here we show that modifying one of the predicates leads to a completely different picture for infinite alphabets, though it does not change the expressive power for finite alphabets. Indeed, roughly speaking, being able to express the fact that two words end with the same symbol leads to an undecidable theory, already for the Σ2 fragment. Finally, we show that the existential fragment is decidable. Bibliography: 19 titles.

In this paper, the author performs a global qualitative study of plane polynomial dynamical systems and suggests a new geometric approach to solving the sixteenth Hilbert problem on the maximum number and mutual location of their limit cycles in two special cases of such systems. First of all, using the geometric properties of four parameters rotating the vector field of a new canonical system constructed in the paper, the author proposes the proof of his early conjecture, which asserts that the maximum number of limit cycles of an arbitrary quadratic system is equal to 4, and their location (3: 1) is uniquely possible [4]. Then using the same geometric approach, the author solves the primary problem for the polynomial Lienard system (in this special case, it is considered as the thirteenth Smale problem), and generalizing the obtained results, the author formulates the theorem on the maximum number of limit cycles enclosing one singular point in the case of a polynomial system. Moreover, applying the Wintner-Perko termination principle for multiple limit cycles, the author develops an alternative approach to solving the sixteenth Hilbert problem, and using this approach, the author completes the global qualitative study of a general cubic Lienard system having three singular points in the finite part of the plane. In conclusion, the author discusses one more known approach to solving the sixteenth Hilbert problem.

In this paper, we describe the range of the Lp-norm of a function under fixed Lp-norms with two other different exponents p and under a natural multiplicative restriction of the type of the Muckenhoupt condition. Particular cases of such results are simple inequalities as the interpolation inequality between two Lp-norms as well as such nontrivial inequalities as the Gehring inequality or the reverse Holder inequality for Mackenhoupt weights. The basic method of our paper is the search for the exact Bellman function of the corresponding extremal problem. Bibliography: 5

The paper is devoted to Koosis' approach to the Carleson theorem. We show that this approach also works for other related questions. The main emphasis in this paper is on the method. Bibliography: 10 titles.

This review paper is devoted to some questions related to investigations of bases in PI-algebras. The central point is generalization and refinement of the Shirshov height theorem, of the Amitsur–Shestakov hypothesis, and of the independence theorem. The paper is mainly inspired by the fact that these topics shed some light on the analogy between structure theory and constructive combinatorial reasoning related to the “microlevel,” to relations in algebras and straightforward calculations. Together with the representation theory of monomial algebras, height and independence theorems are closely connected with combinatorics of words and of normal forms, as well as with properties of primary algebras and with combinatorics of matrix units. Another aim of this paper is an attempt to create a kind of symbolic calculus of operators defined on records of transformations.

This is the first paper of a conceived series under the common title “The boundary control method in inverse problems.” The aim of the series is to expound systematically an approach to inverse problems based upon its relationship with control theory. The 1d-variant of the method is shown with the example of the classical problem of recovering the density of an inhomogeneous string, and both dynamical and spectral statements of the problem are considered. The paper is written in such a way as to serve as an introduction to the multidimensional BC-method: the basic tools and constructions are amenable to further generalization to multidimensional problems. Bibliography: 31 titles.

This paper is a sequel to the author's papers [1–8] devoted to lower multiplicative bounds on L 1 norms and their applications. Now we give estimates for L 1 norms of exponential sums and prove the result announced in [8].

Let A be a class of Abelian groups, A ∈ A, and End(A) be the additive endomorphism group of the group A. The group A is said to be defined by its endomorphism group in the class {ie208-01} if for every group B ∈ B such that End(B) ≅ End(A) the isomorphism B ≅ A holds. The paper considers the problem of definability of a periodic Abelian group A such that End-End(A) ≅ End(A). The classes of periodical Abelian groups, of divisible Abelian groups, of reduced Abelian groups, of nonreduced Abelian groups, and of all Abelian groups are investigated in this paper.

This paper is devoted to relations between the Kurosh problem and the Shirshov height theorem. The central point and main technical tool is the identity of algebraicity. The main result of this paper is the following. Let A be a finitely generated PI-algebra and Y be a finite subset of A. For any Noetherian associative and commutative ring {ie125-01}, let any factor of R ⊗ A such that all projections of elements from Y are algebraic over π(R) be a Noetherian R-module. Then A has bounded essential height over Y. If, furthermore, Y generates A as an algebra, then A has bounded height over Y in the Shirshov sense.

In this paper, we consider reduced semirings with supplementary conditions of annihilators, namely these are Rickart and weakly Rickart semirings. The main aim of the paper is to study functional representations of semirings. We build two sheaves of semirings and prove that a reduced Rickart semiring is presented by sections of these sheaves.

The paper is devoted to two new directions in developing the classical geometric subjects related to studying the problem of unique determination of closed convex surfaces by their intrinsic metrics. The first of these directions is the study of unique determination of domains (i.e., open connected sets) in Euclidean spaces by relative metrics of the boundaries of these domains. It appeared about 25–30 years ago and was developed owing to the efforts of Russian scientists. The first part of the paper (Secs. 3–7) contains an overview of the results referring to this direction.

In this paper, a hyperbolicity criterion for periodic solutions of nonlinear functional-differential equations is constructed in terms of zeros of the characteristic function. In the earlier papers in this area, necessary and sufficient conditions were different from each other. Moreover, it was assumed that if the period of the investigated solution is irrational, then that solution admits a rational approximation. In this paper, we obtain necessary and sufficient conditions of the hyperbolicity. It is proved (and the proof is constructive) that a rational approximation exists for any irrational period. All the results are obtained for the case of several rational delays.

The paper is concerned with a new way of deriving computable estimates for the difference between the exact solutions of elliptic variational inequalities and arbitrary functions in the corresponding energy space that satisfy the main (Dirichlet) boundary conditions. Unlike the method derived earlier, the estimates are obtained by certain transformations of variational inequalities without using duality arguments. For linear elliptic and parabolic problems, this method was suggested by the author in previous papers. The present paper deals with two different types of variational inequalities (also called variational inequalities of the first and second kind). The techniques discussed can be applied to other nonlinear problems related to variational inequalities. Bibliography: 20 titles.

We study rings over which all right modules are I0-modules. All rings are assumed to be associative and with nonzero identity element. For a module M , a submodule N of M is said to be superfluous if N +P = M for every proper submodule P of the module M . A module M is called an I0-module if every cyclic submodule of M either is superfluous in M or contains a nonzero direct summand of the module M . A ring A is called a right (left) I0-ring if AA (respectively, AA) is a right (respectively, left) I0-module. I0-modules and I0-rings were studied in [8; 11, Chap. 3; 1–3; 7; 6] and other works. In the present paper, we study rings over which all right modules are I0-modules. The main result of the present paper is Theorem 1. Theorem 1. For a ring A, the following conditions are equivalent. (1) Every right A-module is an I0-module. (2) For every right A-module M , we have that J(M) is a semisimple module and if J(M) = 0, then every nonzero submodule of the module M contains a nonzero direct summand of the module M . (3) For every right A-module M , either M has a nonzero injective direct summand or M is a semisimple module and is contained in the Jacobson radical of the injective hull of M . (4) Every cyclic right A-module either has a nonzero injective direct summand or is a semisimple module. The residue ring Z/4Z is an example of a nonsemisimple ring that satisfies the conditions of Theorem 1. In Example 11 of the present paper, we give an example of a ring A such that all right A-modules are I0-modules and A contains an infinite set of orthogonal idempotents (therefore, A is not Noetherian). It can also be proved that A is a left semi-Artinian ring and left A-modules are not necessarily I0-modules. I0-modules are close to regular modules and semiregular modules. A module M is said to be regular if every cyclic submodule of M is a direct summand of the module M . A module M is said to be semiregular if for every cyclic submodule N ofM , there exists a direct decomposition M = M1⊕M2 such thatM1 ⊆ N and N ∩ M2 is a superfluous submodule in M2. Semiregular modules were studied in [9; 10, Chap. B; 11, Chap. 4; 12, 14] and other works. It is easy to verify that every semiregular module is an I0-module, every regular module is semiregular, and every semiprimitive, semiregular module is regular. The cyclic group of order 4 is a semiregular nonregular module over the rings Z and Z/4Z. Lemma 4(4) contains an example of a semiprimitive I0-module that is not a semiregular module. The proof of Theorem 1 is decomposed into a series of assertions; some of the assertions are of independent interest. We present the necessary notation and definitions. The intersection of all maximal submodules of the module M is denoted by J(M); it is called the Jacobson radical of the module M . It is well known that J(M) coincides with the sum of all superfluous submodules of the module M (see, e.g., [13, 21.5]). A module M is said to be semiprimitive if J(M) = 0. A module M is said to be semi-Artinian if every nonzero submodule of the module M contains a simple submodule. A ring A is said to be a right V -ring if every simple right A-module is injective (this is equivalent to the property that every right A-module is semiprimitive [4, 7.32A]). A module M is said to be uniserial if any two submodules of M are comparable with respect to inclusion. A direct sum of uniserial modules is called a serial module. A module M is said to be semisimple if every submodule of M is a direct summand of Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 8, pp. 233–241, 2006. 3928 1072–3374/08/1522–3928 c © 2008 Springer Science+Business Media, Inc. the module M . A submodule N of the module M is said to be essential if, for every submodule X of the module M , the relation X ∩N = 0 implies the relation X = 0. A module M is said to be injective if for every module X and each submodule Y of the module X, any homomorphism Y → M can be extended to a homomorphism X → M . If M is an injective module and N is an essential submodule of the module M , then the module M is called the injective hull of the module N . Every module has an injective hull, which is unique up to isomorphism. Lemma 2. Let M be a nonzero right module over a ring A. (1) The module M is an I0-module if and only if every submodule of the module M either is contained in J(M) or contains a nonzero direct summand of the module M . (2) M is a semiprimitive I0-module if and only if every nonzero submodule of the module M contains a nonzero direct summand of the module M . (3) If A is a right V -ring, then M is an I0-module if and only if every nonzero submodule of the module M contains a nonzero direct summand of the module M . (4) If M is an essential extension of a semisimple module and every simple submodule of the moduleM is injective, then M is a semiprimitive I0-module. (5) If A is a right semi-Artinian right V -ring, then M is a semiprimitive I0-module. Proof. (1) The sufficiency follows from the property that J(M) contains all superfluous submodules of the module M . We prove the necessity. Let N be a submodule of the module M that is not contained in J(M). There exists a cyclic submodule X of the module N that is not contained in J(M). Since J(M) is the sum of all superfluous submodules of the module M , the module X is not a superfluous submodule of the module M . By condition (1), some nonzero direct summand Y of the module M is contained in X. Then Y ⊆ N . (2) The assertion follows from (1). (3) The assertion follows from (2) and the property that every right module over any right V -ring is semiprimitive. (4) Since M is an essential extension of a semisimple module, every nonzero submodule N of the module M contains some simple submodule S. By assumption, the module S is injective. Therefore, S is a nonzero direct summand of the module M . (5) Since A is a right semi-Artinian ring, M is an essential extension of a semisimple module. Since A is a right V -ring, every simple submodule of the module M is injective. By (4), M is a semiprimitive I0-module. Lemma 3. For a ring A, the following conditions are equivalent. (1) A is a semiprimitive right I0-ring. (2) Every nonzero right ideal of the ring A contains a nonzero idempotent. (3) Every nonzero principal right ideal of the ring A contains a nonzero idempotent. Lemma 3 follows from Lemma 2(2). Lemma 4. Let A be a ring, B be a unitary subring of the ring A, {Ai}i=1 be a countable set of copies of the ring A, D be the direct product of the rings Ai, and R be the subring in D generated by the ideal ∞ ⊕ i=1 Ai and by the subring B′ ≡ {(b, b, b, . . .) | b ∈ B}. (1) The identity elements ei of the rings Ai are central idempotents of the ring D and ei are contained in the ring R, R = {(a1, . . . , an, b, b, b, . . .) | ai ∈ A, b ∈ B}, where the positive integer n depends on the element (a1, . . . , an, b, b, b, . . .), and R has the factor ring R/ ( ∞ ⊕

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