Search

We correct an error in the paper of Z. Shen, S. Li, and J. Zhang published in [4]. In addition, we give an answer to a question posed by the authors.

The present paper is devoted to traversing a maze by a collective of automata. This part of automata theory gave rise to a fairly wide range of diverse problems ([1], [2]), including those related to problems of the theory of computational complexity and probability theory. It turns out that the consideration of complicated algebraic objects, such as Burnside groups, can be interesting in this context. In the paper, we show that the Cayley graph a finitely generated group cannot be traversed by a collective of automata if and only if the group is infinite and its every element is periodic.

In the present paper, necessary conditions for the metric and topological projectivity of closed ideals of Banach algebras are given. In the case of commutative Banach algebras, a criterion for the metric and topological projectivity of ideals admitting a bounded approximate identity is obtained. The main result of the paper is as follows: a closed ideal of an arbitrary C*-algebra is metrically or topologically projective if and only if it admits a self-adjoint right identity.

It is well known that many operators in analysis, for example, the Hilbert operator or the conjugate function operator are bounded in the space Lp for p ∈ (1,∞), but they are unbounded in L1 or L∞. In Yano’s paper [1] (also see [2]), the extrapolation theorems were first used to define the “extremal” space in the limit case, which led to the appearance of many papers [3]–[5] generalizing the Yano and Zygmund theorems. In the present paper, we consider another approach to the proof of extrapolation theorems for quasilinear operators in Marcinkiewicz spaces. In this approach, the geometric properties of these spaces are taken into account. A similar approach was first demonstrated for the Lorentz spaces in [6]. Comparing our results with the results of the above papers and, in particular, with [3], we note that, first, we consider operators in the scale of Marcinkiewicz spaces rather than in the scale of Lebesgue spaces, and second, which is more important, from the character of growth of the operator norm as the exponent tends to the critical exponent, we write out the exact Marcinkiewicz space, which is the extrapolation space. The theorems given below include both the classical Zygmund theorem (see, e.g., Example 1 at the end of the paper) and recent results about extrapolation of operators near L∞ (see [7]). We assume that (Ω,Σ, μ) is a space equipped with a σ-finite measure, S(μ) is the set of measurable functions on Ω, and χ(D) is the characteristic function of the set D; moreover, we assume that the measure is continuous. For f : Ω → R, we let f∗ denote its permutation in nonascending order. We assume that X is a symmetric space [8] and ψ(X, s) = {‖χ(D)|X‖ : μ(D) = s}

The calculation of the exact value of the rth order nonlinearity of a Boolean function (the power of the distance between the function and the set of functions is at most r) or the derivation of a lower bound for it is a complicated problem (especially for r > 1). Lower bounds for nonlinearities of different orders in terms of the value of algebraic immunity were obtained in a number of papers. These estimates turn out to be sufficiently strong if the value of algebraic immunity is maximum or close to maximum. In the present paper, we prove a statement that allows us to obtain fairly strong lower bounds for nonlinearities of different orders and for many functions with low algebraic immunity.

The paper [1] contained the proof of the theorem on the existence of a gigantic component in a random distance graph for the case in which the edge probability is equal to p = c/N , where c > 1. There it was also stated that “all the other vertices are contained in components of size o(N)”. In the present paper, we succeed in showing that “all the other vertices are contained in components of size O(lnN)”. This result is a significant step forward, because it is truly an analog of the Erdős–Renyi theorem for the classical model (see [2]). In the present paper, we consider the problem of the threshold probability of the existence of a gigantic component for a series of random distance graphs of special form. Set n = 4k, k ∈ N, and N = C n 1 and consider a complete distance graph GN = (VN , EN ) such that

In previous papers of the author, the cascade search principle was proposed, which makes it possible to construct a set-valued self-map of a metric spaceX from a set-valued functional or a collection of set-valued maps of X so that the new map generates a multicascade, i.e., a set-valued discrete dynamical system whose limit set coincides with the zero set of the given functional, with the coincidence set of the given collection, or with the common preimage of a closed subspace under the maps from this collection. Stability issues of cascade search were studied. This paper is devoted to a generalization and local modifications of the cascade search principle and their applications to problems concerning local search and approximation of common preimages of subspaces and coincidence sets for finite collections of set-valued maps of metric spaces.

In this paper, the following system of substitutions in a 3-letter alphabet $$\sum { = \left\langle {\left. {a,b,c} \right|a^2 \to bc,b^2 \to ac,c^2 \to ab} \right\rangle }$$ is considered. A detailed proof of results that were described briefly in the author’s paper [1] is presented. They give an answer to the specific question on the possibility of giving a polynomial upper bound for the lengths of derivations from a given word in the system Σ stated in the literature. The maximal possible number of steps in derivation sequences starting from a given word W is denoted by D(W). The maximum of D(W) for all words of length |W| = l is denoted by D(l). It is proved that the function D(W) on wordsW of given length |W| = m+2 reaches its maximum only on words of the form W = c2bm and W = bma2. For these words, the following precise estimate is established: Open image in new window where ⌌3m2/2⌍ for odd |m| is the round-up of 3m2/2 to the nearest integer.

This paper is devoted to the study of so-called finitely bi-Lipschitz mappings, which are a far-reaching generalization of isometries and quasi-isometries. We obtain several criteria for the homeomorphic extension to the boundary of finitely bi-Lipschitz homeomorphisms f between domains in ℝn, n ≥ 2, whose outer dilatations KO(x, f) satisfy the integral constraints $$\int {\Phi (K_O^{n - 1} (x,f))dm(x) < \infty } $$ with an increasing convex function Φ: [0,∞] → [0,∞]. Note that the integral conditions on the function Φ (obtained in the paper) are not only sufficient, but also necessary for the continuous extension of f to the boundary.

A new operation of product of groups, the n-periodic product of groups for odd exponent n ≥ 665, was proposed by the author in 1976 in the paper [1]. This operation is described on the basis of the Novikov-Adyan theory introduced in the monograph [2] of the author. It differs from the classic operations of direct and free products of groups, but has all of the natural properties of these operations, including the so-called hereditary property for subgroups. Thus, the well-known problem of A. I. Mal’tsev on the existence of such new operations was solved. Unfortunately, in the paper [1], the case where the initial groups contain involutions, was not analyzed in detail. It is shown that, in the case where the initial groups contain involutions, this small gap is easily removed by an additional restriction on the choice of defining relations for the periodic product. It suffices to simply exclude products of two involutions of previous ranks from the inductive process of defining new relations for any given rank α. It is suggested that the adequacy of the given restriction follows easily from the proof of the key Lemma II.5.21 in the monograph [2]. We also mention that, with this additional restriction, all the properties of the periodic product given in [1] remain true with obvious corrections of their formulation. Moreover, under this restriction, one can consider n-periodic products for any period n ≥ 665, including even periods.

for semiflows {Tt} with invariant measure μ was begun in [1], [2], where it was established that, for absolutely continuous probability measures ν on the semiaxis and bounded functions f , the quantities Ftf(x) as t → +∞ have, for μ-almost all x, the same limit as in the case of the classical uniform averaging in the Birkhoff–Khinchin theorem. The study of the averaging Ftf(x) was continued in [3] (see also [4, Chap. 10]), where it was found that, for unbounded functions f , it may happen that the nonuniform averaging does not converge. In that paper, additional conditions ensuring such a convergence were obtained. Similar questions for stochastic equations were considered in [5]. The case of averaging with an operator semigroup was studied in [6]. In the present paper, we study the convergence of the means Ftf(x) in Lp(μ), as well as some applications of the results obtained. In particular, we are referring to an analog of the ergodic Wiener–Wintner theorem. In contrast to [3], [5], [6], a weak mixing condition is imposed on the semigroup in a number of assertions in present paper. Suppose that (X,A , μ) is a probability space. Let us recall that the semigroup {Tt}t≥0 of transformations of the spaceX preserving the measure μ is said to be weakly mixing (see [7, p. 28]) if, for any of two functions f, g ∈ L2(μ), the following relation holds

In the paper, it is proved that, for any odd n ≥ 1039, there are words u(x, y) and υ(x, y) over the group alphabet {x, y} such that, if a and b are any two noncommuting elements of the free Burnside group B(m, n), then, for some k, the elements u(ak, b) and υ(ak, b) freely generate a free Burnside subgroup of the group B(m, n). In particular, the facts proved in the paper imply the uniform nonamenability of the group B(m, n) for odd n, n ≥ 1039.

In [1], a class of quadratic stochastic operators mapping the finite-dimensional simplex into itself was singled out, and these operators were called bistochastic quadratic operators. They are closely related to the notion of majorization and are applied not only to problems in population genetics [1], [2], but also to the problems in economics [3]. In mathematical economics, the bistochastic quadratic operator is called thewelfare operator. Bistochastic quadratic operators were first studied in [1], where a necessary and sufficient condition for the bistochasticity of operators was obtained. This theorem will be presented below and used throughout the paper. In the present paper, it is shown that the set of bistochastic quadratic operators is a convex polyhedron. Therefore, an analog of Birkhoff’s theorem on the extreme points of the set of bistochastic matrices [4] is of interest. In this paper, the problem is partially solved and, more precisely, we obtain a sufficient condition for points to be extreme and, for the two-dimensional simplex, a necessary and sufficient condition as well. Moreover, we find the number of extreme points of the set of bistochastic quadratic operators in the two-dimensional simplex. Let us now pass to some necessary notions.

In the present paper, we consider estimates for the orders of zeros of polynomials in functions satisfying a system of algebraic differential equations and possessing a special D-property defined in the paper. The main result obtained in the paper consists of two theorems for the two cases in which these estimates are given. These estimates are improved versions of a similar estimate proved earlier in the case of algebraically independent functions and a single point. They are derived from a more general theorem concerning the estimates of absolute values of ideals in the ring of polynomials, and the proof of this theorem occupies the main part of the present paper. The proof is based on the theory of ideals in rings of polynomials. Such estimates may be used to prove the algebraic independence of the values of functions at algebraic points.

In this paper, we introduce the notion of Laplacian spectrum of an infinite countable graph in a different way than in the papers by B. Mohar. We prove some basic properties of this type of spectrum. The approach used is in line with our approach to the limiting spectrum of an infinite graph. The technique of the Laplacian spectrum of finite graphs is essential in this approach.

It is well known that, for cooperative games with transferable utility (and with crisp payoffs), the set of reasonable imputations is nonempty. It is also known for what values of $$\varepsilon$$ the set of reasonable imputations belongs to the $$\varepsilon$$ -core. Then the $$\varepsilon$$ -core is also nonempty. This result is of considerable interest, because the 0-core of a cooperative game can be empty, but if the $$\varepsilon$$ -core is nonempty in this case for some small $$\varepsilon>0$$ , then there exist imputations such that the difference in the properties between them and the imputations from the 0-core is small. In this paper, these results are generalized to the case of games with fuzzy payoffs.

In the present paper, we prove a fixed-point theorem for completely continuous multivalued mappings defined on a bounded convex closed subset X of the Hilbert space H which satisfies the tangential condition \(F(x) \cap (x + T_X (x)) \ne \emptyset\), where TX(x) is the cone tangent to the set X at a point x. The proof of this theorem is based on the method of single-valued approximations to multivalued mappings. In this paper, we consider a simple approach for constructing single-valued approximations to multivalued mappings. This approach allows us not only to simplify the proofs of already-known theorems, but also to obtain new statements needed to prove the main theorem in this paper.

of the productX. It is easy to see that the Σ-product is an everywhere dense subspace of the product. It is natural to consider Σ-products that do not coincide with the product of spaces. Such Σ-products are said to be natural. Obviously, a natural Σ-product cannot be a separable space and cannot be a paracompact set. In 1959, Corson [1] proved that a Σ-product of complete metric spaces is a normal space. In 1977, Gul’ko [3] and Rudin (see [4]) independently proved that a Σ-product of metric spaces is a normal space. Then the author of the present paper [5] sharpened this result by proving that a Σ-product of paracompact p-spaces is normal if and only if this Σ-product has countable tightness. Paracompact p-spaces can be defined as the preimages of metric spaces under closed compact mappings [6], and Lashnev spaces are the images of metric spaces under closedmappings [7]. It is well known that Lashnev spaces are Frechet–Urysohn paracompact sets (in particular, of countable tightness) (see, e.g., [8]) but need not be p-spaces. In this connection, Kodama posed the following problem, which is widely known at present (see, e.g., [9]): Is aΣ-product of Lashnev spaces a normal space? In [9], it was proved that a positive answer to this question is impossible, because there is a nonnormal Σ-product of Lashnev spaces in a set-theoretical model. The problem of the existence of a “naive” example of such kind remains open. The main result in the present paper is the following assertion.

In this paper, we obtain inequalities for trigonometric and algebraic polynomials supplementing and strengthening the classical results going back to papers of S. N. Bernstein and I. I. Privalov. The method of proof is based on the construction of the conformal and univalent mapping from a given trigonometric polynomial and on the application of results of the geometric theory of functions of a complex variable to this mapping.

It is shown in this paper that \({h\left( r \right)}\)is a smoothly varying function of order ς if and only if the function \(\rho \left( r \right) = \left( {\ln h\left( r \right)} \right)/\ln r\) is a perfect proximate order, i.e., an infinitely differentiable (in a neighborhood of >+∞) function for which the conditions \(\lim _{r \to + \infty } \rho \left( r \right) = \rho\), \(\rho \in \mathbb{R}\), and \(\lim _{r \to + \infty } r^n \ln \rho ^{\left( n \right)} \left( r \right) = 0\) for all \(n \in \mathbb{N}\) are satisfied. Consequences of the result indicated above are also obtained in this paper.

The first part of this paper is concerned with the proof of a discrete analog of the Poisson summation formula. In the second part, we describe an elementary proof of a functional equation for the function $$\theta (t)$$ , based on the summation formula derived in the paper.

The paper is devoted to the study of properties of a class of subgroups H in Lie groups G that was recently introduced by the author. A closed subgroup H in a Lie group G is said to be plesio-uniform if there is a closed subgroup P of G that contains H and for which P is uniform in G and H is quasi-uniform in P. In the paper we give answers to several natural questions concerning plesio-uniform subgroups. It is proved that one obtains the same notion of plesio-uniformity when transposing the conditions of uniformity and quasi-uniformity in the definition of plesio-uniformity of a subgroup. If a closed subgroup H of G contains a plesio-uniform subgroup, then H is also plesio-uniform. Other properties of plesio-uniform subgroups are also considered.

A “topological digression”, which is a consequence of the Brouwer fixed-point theorem, appears in Sec. 4.1 in the book of Lee and Markus [1]. Under some restrictions on the continuous mapping, this consequence states that the interior of the image of the unit ball in Rn is nonempty under such a mapping. This consequence was widely used by these authors (e.g., in the proof of Theorem 3 in Sec. 4.1 [1]). This consequence was also used by the author of this paper in [2], [3]. In the present paper, we prove a generalization of this result to the case of Banach spaces.

The paper is devoted to the Jacobian Conjecture: a polynomial mappingf∶ℂ2→ℂ2 with a constant nonzero Jacobian is polynomially invertible. The main result of the paper is as follows. There is no four-sheeted polynomial mapping whose Jacobian is a nonzero constant such that after the resolution of the indeterminacy points at infinity there is only one added curve whose image is not a point and does not belong to infinity.

The present paper deals with estimates of the chromatic number of a space with forbidden one-color triangles. New lower bounds for the quantity under study, which are sharper than all bounds obtained so far, are presented.

We study the complexity of the realization of Boolean functions by circuits in infinite complete bases containing all monotone functions with zero weight (cost of use) and finitely many nonmonotone functions with unit weight. The complexity of the realization of Boolean functions in the case where the only nonmonotone element of the basis is negation was completely described by A. A. Markov: the minimum number of negations sufficient for the realization of an arbitrary Boolean function f (the inversion complexity of the function f) is equal to ⌈log2(d(f) + 1)⌉, where d(f) is the maximum (over all increasing chains of sets of values of the variables) number of changes of the function value from 1 to 0. In the present paper, this result is generalized to the case of the computation of Boolean functions over an arbitrary basis B of prescribed form. It is shown that the minimum number of nonmonotone functions sufficient for computing an arbitrary Boolean function f is equal to ⌈log2(d(f)/D(B) +1)⌉, where D(B) = max d(ω); the maximum is taken over all nonmonotone functions ω of the basis B.

In one of his papers, Dragalin proposed a very general approach to the construction of models for nonstandard logics of uniform-algebra type and, in particular, for intuitionistic logic (see [1]). The presentation (clear enough, but without the obvious details) is accompanied by a number of arithmetical examples (see also [1]). In the last of these examples, Kleene’s slash realizability is considered (see [2, column C]), and the author gives a “ . . . model corresponding to Kleene’s slash realizability . . . ” [1, p. 194]. However, the connection between this model and Kleene’s slash realizability is as follows: “ . . . ‖φ‖ = T ⇔ ((|φ)∧HA φ)”; (see also [1, p. 195]; cf. [3]). Of course, by means of the above-mentioned model (which corresponds exactly to the formula realizability from [3]) one can prove the properties of disjunctiveness and existentiality for arithmetic (this is just the result the author seeks to obtain, using a suitable uniform algebra). However, Kleene’s slash realizability does not coincide with deducibility in the intuitionistic HA arithmetic. As mentioned above, the paper [1] contains a number of examples of functional pseudoboolean algebras corresponding to various HA models (including, in particular, realizability models). Exact formulation of functional pseudoboolean algebra is given in [1], and here we shall only briefly mention facts needed in what follows. If F is the set of forms of a functional pseudoboolean algebra, then for any f and g from F (where f and g are forms of the same arity), there exist the forms f ∧ g , f ∨ g , f ⊃ g in F of equal arity. After that, a functional algebraic model for logic-mathematical language is defined, and for a given functional algebraic model, we define the value of any formula of our language in it. The main distinction from usual algebraic models is in the fact that the value ‖φ‖ of the formula φ is a certain form fφ from a functional pseudoboolean algebra. Now we can fix the language of arithmetic, the object area, and the function Ĉnst . After that, each model is defined by specifying the following set: B (a pseudoboolean algebra), F , and Pr-valuation of predicates. Let a given functional pseudoboolean algebra be a model for Kleene’s slash realizability, i.e., suppose formulas of the language can be mapped to the set of forms so that any formula φ is |realizable if and only if fφ ∈ 1 (the unit of algebra B). Consider two distinct statements, φ and ψ , undecidable in HA, i.e., HA φ, HA ψ, HA ¬φ, HA ¬ψ. Since φ and ψ cannot be deduced in HA, the formulas ¬φ and ¬ψ are nondeducible |-realizable formulas of the language of arithmetic. If the forms corresponding to them in the functional algebraic model are F¬φ and F¬ψ , respectively, then these forms belong to algebra’s 1 , and then the form F¬φ ∨F¬ψ = F¬φ∨¬ψ (which corresponds in the functional algebraic model to the formula ¬φ∨¬ψ , see [1, p. 187]), also belongs to our algebra’s unit, and hence, the formula ¬φ∨¬ψ is |realizable. But this implies that HA φ or HA ψ , which is impossible by the choice of φ and ψ . Thus, we have proved the following theorem

The main subject of this paper is the study of a general linear boundary-value problem with Drazin or right Drazin (respectively, left Drazin) invertible operators corresponding to initial boundary operators. The obtained results are then employed to solve a Schro¨ dinger equation.

In 1955, M. A. Krasnosel’skii proved a fixed-point theorem for a single-valued map which is a completely continuous contraction (a hybrid theorem). Subsequently, his work was continued in various directions. In particular, it has stimulated the development of the theory of condensing maps (both single-valued and set-valued); the images of such maps are always compact. Various versions of hybrid theorems for set-valued maps with noncompact images have also been proved. The set-valued contraction in these versions was assumed to have closed images and the completely continuous perturbation, to be lower semicontinuous (in a certain sense). In this paper, a new hybrid fixed-point theorem is proved for any set-valued map which is the sum of a set-valued contraction and a compact set-valued map in the case where the compact set-valued perturbation is upper semicontinuous and pseudoacyclic. In conclusion, this hybrid theorem is used to study the solvability of operator inclusions for a new class of operators containing all surjective operators. The obtained result is applied to solve the solvability problem for a certain class of control systems determined by a singular differential equation with feedback.

A ring K is a unique addition ring (a UA-ring) if its multiplicative semigroup (K, · ) can be equipped with a unique binary operation + transforming this semigroup to a ring (K, ·, +). An Abelian group is called an End-UA-group if its endomorphism ring is a UA-ring. In the paper, we find End-UA-groups in the class of nonreduced Abelian groups.

In this paper, we investigate the homological properties of the functor categories (mod−R, Ab) and ((mod−R)op, Ab). Some new homological dimensions in these functor categories such as FP-projecive dimensions and cotorsion dimensions for functors and functor categories are introduced and studied. We also characterize functor categories of homological dimensions zero and explore the connections among some different homological dimensions.

The aim of this paper is to establish the boundedness of certain sublinear operators with rough kernel generated by Calderon–Zygmund operators and their commutators on generalized Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. The Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.

In the present paper,we describe conditions on a vector group C which are necessary and sufficient for the class of completely decomposable torsion-free Abelian groups to be a C EH-class.

In the paper, it is proved that the distribution of a measurable polynomial on an infinite-dimensional space with log-concave measure is absolutely continuous if the polynomial is not equal to a constant almost everywhere. A similar assertion is proved for analytic functions and for some other classes of functions. Properties of distributions of norms of polynomial mappings are also studied. For the space of measurable polynomial mappings of a chosen degree, it is proved that the L1-norm with respect to a log-concave measure is equivalent to the L1-norm with respect to the restriction of the measure to an arbitrarily chosen set of positive measure.

In the paper, the semigroup of weak limits of the powers of an infinite transformation of rank one of Chacon type is completely described.

In the paper, the additive complexity of matrices formed by positive integer powers of greatest common divisors and least common multiples of the indices of the rows and columns is considered. It is proved that the complexity of the n × n matrix formed by the numbers GCD r (i, k) over the basis {x + y} is asymptotically equal to rn log2 n as n→∞, and the complexity of the n × n matrix formed by the numbers LCM r (i, k) over the basis {x + y,−x} is asymptotically equal to 2rn log2 n as n→∞.

In this paper, the following problem is studied. For what multipliers {λ k,n } do the linear means of the Fourier series of functions f ∈ L 1[−π, π], $$\begin{array}{*{20}c} {\sum\limits_{k = - \infty }^\infty {\lambda _{k,n} \widehat{f_k }e^{ikx} ,} } & {where \widehat{f_k } is the kth Fourier coefficient, } \\ \end{array} $$ , converge as n→∞ at all points at which the derivative of the function ∫ 0 f exists? In the case λ k,n = (1 − |k|/(n + 1)), a criterion of the convergence of the (C, 1)-means and, in the general case λ k,n = ϕ(k/(n + 1)), a sufficient condition of the convergence at all such points (i.e., almost everywhere) are obtained. In the general case, the answer is given in terms of whether ϕ(x) and xϕ′(x) belong to the Wiener algebra of absolutely convergent Fourier integrals. New examples are given.

In this paper, we establish upper bounds for the moduli of zeros of Hermite–Pade approximations of type I for a system of exponential functions \(\left\{ {{e^{{\lambda _{{p^z}}}}}} \right\}_{p = 0}^k\), where \(\left\{ {{\lambda _p}} \right\}_{p = 0}^k\) are various arbitrary complex numbers. The proved statements supplement and generalize well-known results due to Saff and Varga, as well as those due to Stahl and Wielonsky, on the behavior of zeros of Hermite–Pade approximations for a set of exponential functions \(\left\{ {{e^{pz}}} \right\}_{p = 0}^k\).

It is well-known that a linear stochastic operator (a Markov operator) associated with a square stochastic matrix is a surjection of the simplex if and only if it is bijective. The similar problem was open for nonlinear stochastic operators (nonlinear Markov operators) associated with stochastic hyper-matrices (higher dimensional matrices). In this paper, we solved this problem for quadratic stochastic operators acting on the simplex. Namely, we showed that a quadratic stochastic operator associated with a cubic stochastic matrix is a surjection of the simplex if and only if it is bijective. Moreover, we also described all surjective quadratic stochastic operators of the simplex.

In the paper, it is proved that, if f(x1,..., xn)g(y1,..., ym) is a multilinear central polynomial for a verbally prime T-ideal Γ over a field of arbitrary characteristic, then both polynomials f(x1,..., xn) and g(y1,..., ym) are central for Γ.

In the paper, quadratic mappings acting from one finite-dimensional space to another are studied. Sufficient conditions for the stable surjectivity of a quadratic surjective mapping (i.e., for the condition that every quadratic mapping sufficiently close to a given one is also surjective) are obtained. The existence problem for nontrivial zeros of a surjective quadratic mapping acting from Rn to Rn is studied. For n = 3, the absence of these zeros is proved.

In this paper, we obtain order estimates for the Kolmogorov, Gelfand and linear widths of some discrete function classes on trees, which are generated by a two-weighted summation operator.

An associative ring R is called a unique addition ring (UA-ring) if its multiplicative semigroup (R, · ) can be equipped with a unique binary operation+ transforming the triple (R, ·, +) to a ring. An R-module A is said to be an End-UA-module if the endomorphism ring EndR(A) of A is a UA-ring. In the paper, the torsion-free End-UA-modules over commutative Dedekind domains are studied. In some classes of Abelian torsion-free groups, the Abelian groups having UA-endomorphism rings are found.

Let H be a weak Hopf algebra, A a right weak H-comodule algebra, and B the subalgebra of the H-coinvariant elements of A. Let A/B be a right weak H-Galois extension. In this paper, a spectral sequence for Ext which yields an estimate for the global dimension of A in terms of the corresponding data for H and B is constructed. Next, the relationship between the finitely presented dimensions of A and its subalgebra B are given. Further, the case in which A is an n-Gorenstein algebra is studied.

Let R be a commutative ring with nonzero identity and J(R) be the Jacobson radical of R. The Jacobson graph of R, denoted by JR, is a graph with vertex-set RJ(R), such that two distinct vertices a and b in RJ(R) are adjacent if and only if 1 − ab is not a unit of R. Also, the line graph of the Jacobson graph is denoted by L(JR). In this paper, we characterize all finite commutative rings R such that the graphs L(JR) are planar, toroidal or projective.

Zlobin proved an integral identity relating two different methods for constructing Diophantine approximations to the values of the Riemann zeta function at integer points. In the present paper, several variants of generalizing this integral identity are proved.

The present paper deals with a generalization of a well-known theorem for a set A ⊆ G, where G is an arbitrary Abelian group. According to this classical result, it follows from |A + A| < (3/2)|A| or |A − A| < (3/2)|A| that A ⊆ H, where H is a coset with respect to some subgroup of G and |H| ≤ (3/2)|A|. Consider the sets A2 ± Δ(A) ⊆ G2 (two-dimensional sum and difference). Here A2 = A × A is the set of pairs of elements from A and Δ(A) is the diagonal set Δ(A) = {(a, a) ∈ G × G | a ∈ A}. The main result involves the given sets and is as follows. If |A2 ± Δ(A)| < 7/4|A|2, then A ⊆ H + xfor somex ∈ Gand subgroupH ⊆ G, where |H| < 3/2|A

The paper describes a method for determining integer solutions of a homogeneous polynomial system with integer coefficients which has finitely many solutions in the projective space over the field of complex numbers under the assumption that these solutions have a certain property.

In this paper, we prove direct and inverse theorems on the approximation of functions by Fourier–Laplace sums in the spaces S(p,q)(σm−1), m ≥ 3, in terms of best approximations and moduli of continuity and consider the constructive characteristics of function classes defined by the moduli of continuity of their elements. The given statements generalize the results of the author’s work carried out in 2007.