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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.

In the present paper, we give a description of the derived category of coherent sheaves on a Fano threefold of index 1 and degree 12 (the variety V 12). It can easily be shown that if X is a V 12 variety, then its derived category contains an exceptional pair of vector bundles ( $$\mathcal{U},\mathcal{O}_X$$ ), where $$\mathcal{O}_X$$ is the trivial bundle, and $$\mathcal{U}$$ is the Mukai bundle of rank 5 (which induces the embedding X → Gr(5, 10)). The orthogonal subcategory $$\mathcal{A}_X = ^ \bot \left\langle {\mathcal{U},\mathcal{O}} \right\rangle \subset \mathcal{D}^b (X)$$ can be treated as the nontrivial part of the derived category of X. The main result of the present paper is the construction of the category equivalence $$\mathcal{A}_X \cong \mathcal{D}^b (C^ \vee )$$ , where C ∨ is the curve of genus 7 which can be canonically associated to X according to the results due to Iliev and Markushevich. In the construction of the equivalence, we make use of the geometric results due to Iliev and Markushevich, as well as the Bondal and Orlov results about derived categories. As an application, we prove that the Fano surface of X (which is the surface parametrizing conics on X) is isomorphicto S 2 C ∨, the symmetric square of the corresponding curve of genus 7.

—We prove the following main theorem of the theory of ( r,q )-polycycles. Supposea nonseparable plane graph satisfies the following two conditions:(1) each internal face is an r -gon, where r ≥ 3;(2) the degree of each internal vertex is q ,where q ≥ 3, and the degree of each boundaryvertex is at most q and at least 2.Then it also possesses the following third property:(3) the vertices, the edges, and the internal faces form a cell complex.Simple examples show that conditions (1) and (2) are independent even provided condition (3)is satisfied. These are the defining conditions for an ( r,q )-polycycle. Key words : polycycle, cell complex, barycentric subdivision, plane graph. This paper completes the series of papers [1–10] concerning the so-called ( r,q )-polycycles. Thedefinition of a polycycle in this series includes three conditions. One of these three conditions turnsout to be superfluous. The two others are independent. These are the defining conditions of an( r,q )-polycycle. This is the main theorem of the present paper.Now, let us give our initial definition of an (

and proved that, for q = 2, 3, . . . , the relation A(1/q) = 1/q holds, the extremal function is φ1/q(x) , and hence derived the asymptotics A(h) = h+O(h) , h → 0 . A. Yu. Popov conjectured that, as h → 0 , the stronger asymptotics A(h) = h+O(h) is valid. Further progress in the Turan problem was made in the papers of Gorbachev and Manoshina [2– 6]. They proved Popov’s conjecture and reduced the Turan problem for rational h to a finitedimensional extremal problem, and evaluated A(h) for h = 2/q, 3/q, 4/q, p/(2p+ 1) . In this paper, the Turan problem is solved for all rational h . We need some results from [6]. Suppose that p, q ∈ N , (p, q) = 1, p ≤ q/2 . Consider the following linear programming problem.

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.

Properties of the idempotently convex hull of a two-point set in a free semimodule over the idempotent semiring Rmax min and in a free semimodule over a linearly ordered idempotent semifield are studied. Construction algorithms for this hull are proposed. Some asymptotic physical problems (such as quasiclassical approximation in quantum mechan- ics (1)), as well as many problems of optimization theory, mathematical economics, etc., admit a natural and simple formulation in terms of algebraic structures involving the operations of mini- mization or maximization (2, 3). Such algebraic structures are the object of the actively developing field of idempotent mathematics (3-5). Interesting, important and useful constructions and re- sults of traditional mathematics over number fields and similar structures have counterparts over idempotent semifields and semirings formulated in the spirit of Bohr's correspondence principle in quantum theory (6, 7). This correspondence can be far from obvious, though. In this paper, we consider the simplest problem of idempotent convex geometry (developed, in particular, in (8) and (7)), the construction of the convex hull of a two-point set in an idempotent semimodule, and prove that the algorithmic complexity of its solution increases with the growth of the semimodule's dimension. We consider the number line with the operations ⊕ = max and � = + and the additional element −∞, which plays the part of 0, i.e., is assumed to have the properties −∞ ⊕ a = a, −∞ � a = −∞. The ⊕ operation is commutative, associative, and idempotent ( a ⊕ a = a), the � operation is commutative and distributive with respect to ⊕. In idempotent analysis, the above- mentioned properties are considered to be the axioms of an idempotent semiring. The structure defined above has also the property of invertibility of theoperation and, for that reason, is called the idempotent semifield Rmax +. The idempotent semiring Rmax min is an important example of a semiring which is not a semi- field. It includes the entire number line with the additional elements −∞ and +∞ and has two operations ⊕ = max and � = min. The elements −∞ and +∞ are considered to be the elements 0 and 1 of the semiring, i.e., are assumed to have the properties −∞ ⊕ a = a, a � (+∞ )= a. In any idempotent semiring, the ⊕ operation induces a partial order: ab if and only if a ⊕ b = b; a ≺ b if and only if ab and ab. In both semirings considered above, this order is linear, because for any elements a and b ,w e haveab or ba; therefore, for any elements a and b of these semirings the operation a ∧ b of taking the lower bound is also defined. In the paper, we consider semimodules S n of column vectors of the form (a 1 , . .. ,a n ), a i ∈ S, with coordinatewise operations of generalized addition and multiplication by a scalar from the

In this paper, we suggest a new functional approach to the study of group classes which enables us to describe all formations and Fitting classes of finite groups in the language of functions. The notions of \(\omega\)-fibered formation and of \(\omega\)-fibered Fitting class with direction ϕ are introduced. A direction ϕ is defined as a mapping of the set \(\mathbb{P}\) of all primes into the set of all nonempty Fitting formations. The existence of infinitely many mappings of this kind makes it possible to construct new forms of formations and Fitting classes for a given nonempty set \(\omega\). In particular, an \(\omega\)-local formation is an \(\omega\)-fibered formation with a direction ϕ such that ϕ\((p) = \mathfrak{B}_{p'} \mathfrak{N}_p \) for any prime p. In the paper we study some basic properties of \(\omega\)-fibered formations and of \(\omega\)-fibered Fitting classes with direction ϕ and obtain the structure of their minimal satellites for a given ϕ.

In the paper, on the basis of the notion of saturation introduced in [1], we give a characterization of locally finite simple groups \(Re(P),Sz(Q),{\text{ and }}L_2 (P)\) with an Abelian Sylow 2-subgroup in the classes of mixed and periodic groups. A part of the main results was announced in [2]. Theorem 2 of the paper generalizes results of [3, 4].

In this paper, we study the problem of the variation (if any) of the sets of convergence and divergence everywhere or almost everywhere of a multiple Fourier series (integral) of a function \(f \in L_p \), \(p \geqslant 1\), \(f(x) = 0\), on a set of positive measure \(\mathfrak{A} \subset \mathbb{T}^N = [ - \pi ,\pi )^N \), \(N \geqslant 2\), depending on the rotation of the coordinate system, i.e., depending on the element \(\tau \in \mathcal{F}\), where \(\mathcal{F}\) is the rotation group about the origin in \(\mathbb{R}^N \). This problem has been reduced to the study of the change in the geometry of the sets \(\tau ^{ - 1} (\mathfrak{A}) \cap \mathbb{T}^N \) (where \(\tau ^{ - 1} \in \mathcal{F}\) satisfies \(\tau ^{ - 1} \cdot \tau = 1\)) and \(\mathbb{T}^N \backslash {\text{supp}}(f \circ \tau )\) depending on the “rotation,” i.e., on \(\tau \in \mathcal{F}\). In the present paper, we consider two settings of this problem (depending on the sense in which the Fourier series of the function \(f \circ \tau \) is understood) and give (for both cases) possible solutions of the problem in the class \(L_1 (\mathbb{T}^N )\), \(N \geqslant 2\).

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.

The present paper is devoted to the well-known problem of determining the maximum number of elementsτ m (s) of a sphericals-code (−1-2; to be exact, here we consider the Delsarte functionw m (s) related toτ m (s) via the inequalityτ m (s) ≤w m (s). In this paper, the solution of the equationw m (s)=N is obtained form=4 andN=24,25. As a consequence, we obtain the assertion that among any 25 (24) points on the unit sphere in the space ℝ4 there always exist two points with angular distance between them strictly less than 60.5° (61.41°).

We propose a characterization of the adjacency of vertices in the class of permutation polytopes generated by arbitrary subsets of symmetric groups. In particular, this class contains polytopes for the well-known classical problems, such as the assignment problem, 2-and 3-combinations, the traveling salesman problem and their various modifications. Up to now, the problem of vertex adjacency has been studied for a single polytope only. In the present paper, we obtain, for general permutation polytopes, necessary and sufficient conditions that guarantee that two given vertices are adjacent (or not) to each other. The conditions are formulated in terms of permutations and of the solvability of certain special systems of linear equations. The presently known adjacency criteria for vertices of polytopes for the assignment problem are simple corollaries of our conditions. The latter allow us to develop a general algorithmic scheme for recognizing vertex adjacency of a general permutation polytope and estimate its complexity.

In this paper, we study the integrability of optimal mappings T taking a probability measure μ to another measure g · μ. We assume that T minimizes the cost function c and μ satisfies some special inequalities related to c (the infimum-convolution inequality or the logarithmic c-Sobolev inequality). The results obtained are applied to the analysis of measures of the form exp(−|x|α).

We pose the problem of minimization for the relative preimage problem and prove a minimization theorem. The relative common preimage problem for a finite system of maps, whose particular cases are the relative coincidence and common root problems for finitely many maps, is reduced to the relative preimage problem. As corollaries, results concerning the relative coincidence problems for two maps, problems of fixed points and roots are obtained; these results, with certain distinctions, can be found in known papers by other authors.

In this paper, we establish a lower bound for the dimension of the vector spaces spanned over ℚ by 1 and the sums of the values of the Riemann zeta function at even and odd points. As a consequence, we obtain numerical results on the irrationality and linear independence of the sums of zeta values at even and odd points from a given interval of the positive integers.

In this paper, an example of a variety of nonassociative algebras is constructed in which the system of Capelli identities of small rank is satisfied, whereas the colength function has exponential growth. As is well known, in the associative case and in the Lie case, the validity of Capelli identities implies the polynomial growth of the colength.

In this paper, it is proved that the intersection of the radicals of nilpotent residues for the generalized pure braid group corresponding to an irreducible finite Coxeter group or an irreducible imprimitive finite complex reflection group is always trivial. The proof uses the solvability of the Riemann—Hilbert problem for analytic families of faithful linear representations by the Lappo-Danilevskii method. Generalized Burau representations are defined for the generalized braid groups corresponding to finite complex reflection groups whose Dynkin—Cohen graphs are trees. The Fuchsian connections for which the monodromy representations are equivalent to the restrictions of generalized Burau representations to pure braid groups are described. The question about the faithfulness of generalized Burau representations and their restrictions to pure braid groups is posed.

We consider finite groups only. Let ω be a nonempty subset of the set P of all primes, and let f : ω ∪ {ω′} → {formations of groups} and δ : P → {nonempty Fitting formations of groups} be some functions. The formation consisting of all groups G such that $${G \mathord{\left/ {\vphantom {G O}} \right. \kern-\nulldelimiterspace} O}_\omega (G) \in f(\omega ')$$ and $${G \mathord{\left/ {\vphantom {G G}} \right. \kern-\nulldelimiterspace} G}_{\delta (p)} \in f(p)$$ for any p ∈ ω ∩ π (G) is referred to as an ω-fan formation with direction δ. Let ℌ be some class of groups; an ω-fan formation $$\mathfrak{F}$$ with direction δ is said to be an ℌωδ-critical formation if $$\mathfrak{F}\not \subseteq \mathfrak{H}$$ and any proper ω-fan subformation with direction δ in $$\mathfrak{F}$$ is contained in the class ℌ. In the paper, a description of the structure of the ℌωδ-critical formations is presented.

In 1894, Pincherle proved a theorem relating the existence of a minimal solution of three-term recursion relations to the convergence of a continued fraction. The present paper deals with solutions of an infinite system $$q_n = \sum\limits_{j - 1}^{k - 1} {_{Pk - j,n} } q_{n - j} ,\quad p_{1,n} \ne 0,\quad n = 0,1, \ldots ,$$ of k-term recursion relations with coefficients in a field F. We study the connection between such relations and multidimensional ((k − 2)-dimensional) continued fractions. A multidimensional analog of Pincherle's theorem is established.

In the paper, the set of exponents of exponential growth (growth exponents) for a finitely generated group with respect to all possible generators of this group is studied. It is proved that the greatest lower bound of this set is attained for the free products of a cyclic group of prime order and a free group of finite rank.

In this paper, we prove Brennan's conjecture for conformal mappings f of the disk {z : | z| < 1} assuming that the Taylor coefficients of the function log(zf′(z)/f(z)) at zero are nonnegative. We also obtain inequalities for the integral means over the circle |z| = r of the squared modulus of the function zf′(z)/f(z).

The results of Kasparov, Connes, Higson, and Loring imply the coincidence of the functors [[qℂ ⊗ K, B ⊗ K]] = [[C 0(ℝ2) ⊗ K, B ⊗ K]] for any C*-algebra B; here[[A, B]] denotes the set of homotopy classes of asymptotic homomorphisms from A to B. Inthe paper, this assertion is strengthened; namely, it is shown that the algebras qℂ ⊗ K and C 0(ℝ2) ⊗ K are equivalent in the category whose objects are C*-algebras and morphisms are classes of homotopic asymptotic homomorphisms. Some geometric properties of the obtained equivalence are studied. Namely, the algebras qℂ ⊗ K and C 0(ℝ2) ⊗ K are represented as fields of C*-algebras; it is proved that the equivalence is not fiber-preserving, i.e., is does not take fibers to fibers. It is also proved that the algebras under consideration are not homotopy equivalent.

In the present paper, we consider products of lengths of algebraic numbers whose sum or product is a chosen algebraic number. These products are used to construct a new height function for algebraic numbers. With the help of this function, a metric on the set of all algebraic numbers, which induces the discrete topology, is introduced.

In the present paper, we prove an assertion allowing us to extend results related to the presence or absence of real zeros of functions of Mittag-Leffler type $$E_{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \left( {z;\mu } \right) = \sum\limits_{k = 0}^\infty {\frac{{z^k }}{{\Gamma \left( {\alpha k + \mu } \right)}}} $$ for certain values of α and µ to more extensive ranges of these parameters. We give a geometric description of the sets of pairs (α, µ) for which the function E 1/α(z ; µ) has and does not have real zeros.

Davenport and Swinnerton-Dyer found the first 19 extremal ternary cubic forms g i, which have the same meaning as the well-known Markov forms in the binary quadratic case. Bryuno and Parusnikov recently computed the Klein polyhedra for the forms g 1 – g 4. They also computed the “convergents” for various matrix generalizations of the continued fractions algorithm for multiple root vectors and studied their position with respect to the Klein polyhedra. In the present paper, we compute the Klein polyhedra for the forms g 5, – g 7 and the adjoint form g 7 * . Their periods and fundamental domains are found and the expansions of the multiple root vectors of these forms by means of the matrix algorithms due to Euler, Jacobi, Poincare, Brun, Parusnikov, and Bryuno, are computed. The position of the “convergents of the continued fractions” with respect to the Klein polyhedron is used as a measure of quality of the algorithms. Euler’s and Poincare’s algorithms proved to be the worst ones from this point of view, and the Bryuno one is the best. However, none of the algorithms generalizes all the properties of continued fractions.

In this paper, we use Hermite-Pade approximations of the second kind to obtain a lower bound for the absolute value of a linear form, with integer coefficients, in values of polylogarithmic functions at a rational point. This estimate takes into account the growth of all coefficients of the linear form.

In this paper, using the Steklov function, we introduce the modulus of continuity and define the classes of functions \(W_{2,\varphi }^{r,k} \) and \(W_\varphi ^{r,k} \) in the spaces L2 and C. For the class \(W_{2,\varphi }^{r,k} \), we calculate the order of the Kolmogorov width and, for the class \(W_\varphi ^{r,k} \), we obtain an estimate of the error of a quadrature formula.

It follows from results of A. Yudin, V. Yudin, E. Belinskii, and I. Liflyand that if $$m \geqslant 2$$ and a $$2\pi $$ -periodic (in each variable) function $$f(x) \in C(T^m )$$ belongs to the Nikol'skii class $$h_\infty ^{(m - 1)/2} (T^m )$$ , then its multiple Fourier series is uniformly convergent over hyperbolic crosses. In this paper, we establish the finality of this result. More precisely, there exists a function in the class $$h_\infty ^{(m - 1)/2} (T^m )$$ whose Fourier series is divergent over hyperbolic crosses at some point.

In this paper, the following sharp estimate is proved: $$\int_{0}^{2{\pi }} {\left| {F\prime \left( {e^{i\theta } } \right)} \right|^p d\theta \leqslant \sqrt {\pi } 2^{1 + p} \frac{{\gamma \left( {{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + {p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}} {{\gamma \left( {1 + {p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}} ,\quad p > - 1,$$ where F is the conformal mapping of the domain \(D^ - = \left\{ {\zeta :\left| \zeta \right| > 1} \right\}\) onto the exterior of a convex curve, with \(F\prime \left( \infty \right) = 1\). For p=1, this result is due to Polya and Shiffer. We also obtain several generalizations of this estimate under other geometric assumptions about the structure of the domain F(D-).

This paper is devoted to the study of the rate of convergence of pure greedy algorithms in Hilbert space. We obtain upper bounds for the rate of convergence of pure greedy algorithms for functions from the class Aα, β(D).

The class of branched coverings over \(\mathbb{C}^2\) traditionally called exotic arouses interest because of its connection with the Jacobian conjecture. In this paper, we construct a series of examples of such coverings; in particular, methods of construction of coverings with arbitrarily many sheets, as well as with unknotted branch curves, are described. In addition, some topological characteristics of these coverings are computed, which allows us to answer some questions about a possible counterexample to the Jacobian conjecture.

Let C[-1,1] be the Banach space of continuous complex functions \(f\) on the interval [-1,1] equipped with the standard maximum norm \(\left\| f \right\|\); let \(\omega \left( \cdot \right) = \omega \left( { \cdot ,f} \right)\) be the modulus of continuity of \(f\); and let \(R_n = R_n \left( f \right)\) be the best uniform approximation of \(f\) by rational functions (r.f.) whose degrees do not exceed \(n = 1, 2, \ldots \). The space C[-1,1] is also regarded as a pre-Hilbert space with respect to the inner product given by \(\left( {f,g} \right) = \left( {1/\pi } \right)\int_{ - 1}^1 {f\left( x \right)g\left( x \right)} \left( {1 - x^2 } \right)^{ - 1/2} dx\). Let \(z_n = \{ z_1 , z_2 , \ldots z_n \} \) be a set of points located outside the interval [-1,1]. By \(F\left( { \cdot ,f,z_n } \right)\) we denote an orthoprojection operator acting from the pre-Hilbert space C[-1,1] onto its (\({n + 1}\))-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set \(z_n \). In this paper, we show that if \(f\) is not a rational function of degree \( \leqslant n\), then we can find a set of points \(z_n = z_n \left( f \right)\) such that \(\left\| {f\left( \cdot \right) - F\left( { \cdot ,f,z_n } \right)} \right\| \leqslant 12R_n ln\frac{3}{{\omega ^{ - 1} \left( {R_n /3} \right)}}.\)

ACKNOWLEDGMENTSThis paper was written during a visit to the Laboratory of Numerical Analysis of Universityof Pierre and Marie Curie (Paris 6) in the framework of the program CNRS “Jumelage Franco-Russe.”TheauthorisgratefultotheUniversityforhospitalityandtoCNRSforsupportingthevisit.ThisresearchwassupportedbytheRussianFoundationforBasicResearchundergrantsnos.02-01-00248and03-01-06300andbytheprogram“LeadingScientificSchools,”grantno.304.2003.1.REFERENCES

The problem of reducing polynomial matrices to canonical form by using semiscalar equivalent transformations is studied. This problem is wild as a whole. However, it is tame in some special cases. In the paper, classes of polynomial matrices are singled out for which canonical forms with respect to semiscalar equivalence are indicated. We use this tool to construct a canonical form for the families of coefficients corresponding to the polynomial matrices. This form enables one to solve the classification problem for families of numerical matrices up to similarity.

In this paper, we construct two polyconvolutions (generalized convolutions) with weight \(\gamma = x^{ - \nu }\) generated by the Hankel transform possessing the factorization relations \({\text{H}}_\nu [h_1 ](x) = x^{ - \nu } {\text{H}}_\mu {\text{[}}f](x){\text{H}}_\mu {\text{[}}g](x),{\text{ H}}_\mu [h_1 ](x) = x^{ - \nu } {\text{H}}_\nu {\text{[}}f](x){\text{H}}_\mu {\text{[}}g](x).\) Here Hμ is the Hankel transform operator of order μ. Conditions for the existence of the constructed polyconvolutions are found. On their basis, using the differential properties of the Hankel transform, we obtain two more polyconvolutions. The derived constructions allow us to solve new classes of integral and integro-differential equations and systems of equations.

In this paper, maximal non-Hamiltonian graphs ( MNH graphs), i.e., non-Hamiltonian graphs such that the addition of any new edge violates their property of being non-Hamiltonian are studied. It is shown that the study of MNH graphs can be reduced to the study of the so-called simplified MNH graphs. Restrictions on the structure of maximal cliques of simplified MNH graphs are obtained, the orders and the number of such graphs are estimated.

A polyhedron P admits a Coxeter decomposition if P can be tiled by finitely many Coxeter polyhedra such that any two tiles having a common face are symmetric with respect to this face. In this paper, we classify Coxeter decompositions of compact convex pyramids and triangular prisms in the hyperbolic space $$\mathbb{H}^3 $$ .

A Lambert cube Q(α,β,γ) is a combinatorial cube with dihedral angles α, β, and γ assigned to the three mutually noncomplanar edges and right angles at the remaining edges. In this paper, we classify the Lambert cubes in S3, \(\mathbb{E}^3 \), and \(\mathbb{H}^3 \) such that the group GQ generated by the reflections with respect to the faces of a cube Q is discrete.

In the paper, the convergence properties of the Rogers--Ramanujan continued fraction $$1 + \frac{qz}{1 + \tfrac{q^2 z}{1 + \cdots}}$$ are studied for q = exp (2 π i τ), where τ is a rational number. It is shown that the function H q to which the fraction converges is a counterexample to the Stahl conjecture (the hyperelliptic version of the well-known Baker--Gammel--Wills conjecture). It is also shown that, for any rational τ, the number of spurious poles of the diagonal Pade approximants of the hyperelliptic function H q does not exceed one half of its genus.

A connected subset of \({\mathbb{R}}^2\) consisting of unit squares with integral vertices is called a convex polyomino or is simply said to be xy-convex if it intersects any horizontal or vertical line exactly in one closed interval. In this paper, a geometric representation for xy-convex sets is described, allowing us to obtain, by elementary combinatorial methods, known formulas for the number of convex polyominoes contained in a rectangle of given size.

Let $$\mathfrak{B}$$ $$\user2{(}\mathfrak{D}\user2{)}$$ be the variety of associative (special Jordan, respectively) algebras over an infinite field of characteristic 2 defined by the identity ((((x 1,x 2),x 3), ((x 4,x 5),x 6)), (x 7,x 8)) = 0 (((x 1 x 2 · x 3)(x 4 x 5 · x 6))(x 7 x 8) = 0, respectively). In this paper, we construct infinite independent systems of identities in the variety $$\mathfrak{B}$$ ( $$\mathfrak{D}$$ , respectively). This implies that the set of distinct nonfinitely based subvarieties of the variety $$\mathfrak{B}$$ $$\user2{(}\mathfrak{D}\user2{)}$$ has the cardinality of the continuum and that there are algebras in $$\mathfrak{B}$$ $$\user2{(}\mathfrak{D}\user2{)}$$ with undecidable word problem.

A closed set is called a cut between two disjoint sets if any continuum intersecting both these sets intersects the cut. The main result of this paper is that, for any compact space, the dimension defined by induction on the basis of the notion of cut does not exceed the covering dimension.

In this paper, we consider equations of the form \(\user1{\ddot x}\user2{ + }B\user1{\dot x}\user2{ + }A\user1{x} = 0\), where \(\user1{x}\user2{ = }\user1{x}\left( \user1{t} \right)\) is a function with values in the Hilbert space \(\mathcal{H}\), the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in \(\mathcal{H}\). The linear operator \(\mathcal{T}\) generating the C0-semigroup in the energy space \({\mathcal{H}}_1 \times {\mathcal{H}}\) is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms.

The Steiner ratio characterizes the greatest possible deviation of the length of a minimal spanning tree from the length of the minimal Steiner tree. In this paper, estimates of the Steiner ratio on Riemannian manifolds are obtained. As a corollary, the Steiner ratio for flat tori, flat Klein bottles, and projective plane of constant positive curvature are computed.

Suppose that \(\nu\) is an arbitrary finite complex Borel measure on the interval \([0;{\text{2}}\pi {\text{, }}u{\text{(}}re^{i\varphi } )\) is its Poisson integral, \(u{\text{(}}re^{i\varphi } )\) and \(v{\text{(}}re^{i\varphi } )\) are the conjugate harmonics of \(F{\text{(}}z) = v{\text{(}}z) + iv{\text{(}}z),{\text{ }}z = re^{i\varphi }\), and \(F{\text{(}}t)\) is the nontangential limiting value of the analytic function \(F{\text{(}}z)\) as \(z \to t = e^{i\theta }\). In this paper, we consider the problem of representing the analytic function \(F{\text{(}}z)\) in terms of its boundary values \(F{\text{(}}t)\).

Let C be an Abelian group. An Abelian group A in some class \(\mathcal{X}\) of Abelian groups is said to be CH-definable in the class \(\mathcal{X}\) if, for any group B\in\(\mathcal{X}\), it follows from the existence of an isomorphism Hom(C,A) ≅ Hom(C,B) that there is an isomorphism A ≅ B. If every group in \(\mathcal{X}\) is CH-definable in \(\mathcal{X}\), then the class \(\mathcal{X}\) is called an CH-class. In the paper, conditions are studied under which a class of completely decomposable torsion-free Abelian groups is a CH-class, where C is a completely decomposable torsion-free Abelian group.

In this paper, we findall metacyclic groups (a,b:am=e,bs=e,b-1ab=ar), where m=10,14,15,20,21,22, such that the cusp forms associated with all elements of these groups by an exact representation are multiplicativeη-products. We also consider the correspondence between multiplicative η-products and elements of finite order in SL(5,C) by the adjoint representation.

Let G be a group, let A be an Abelian group, and let n be an integer such that n ≥ −1. In the paper, the sets Φn(G,A) of functions from G into A of degree not greater than n are studied. In essence, these sets were introduced by Logachev, Sal'nikov, and Yashchenko. We describe all cases in which any function from G into A is of bounded (or not necessarily bounded) finite degree. Moreover, it is shown that if G is finite, then the study of the set Φn(G,A) is reduced to that of the set Φn(G/Op(G),Ap) for primes p dividing ∣G/G′∣. Here Op(G) stands for the p-coradical of the group G, Ap for the p-component of A, and G′ for the commutator subgroup of G.

In this paper, we prove the Plancherel--Rotach asymptotic formula for the Chebyshev--Hermite functions $$( - 1)^n e^{x^2 /2} (e^{ - x^2 } )^{(n)} /\sqrt {2^n n!\sqrt \pi } $$ and their derivatives for the case in which $$ + \infty $$ belongs to the domain of definition. A method for calculating the approximation accuracy is also given.

In this paper, it is proved that for the bilinear operator defined by the operation of multiplication in an arbitrary associative algebra \(V\) with unit \(e_0 \) over the fields \(\mathbb{R}\) or \(\mathbb{C}\), the infimum of its norms with respect to all scalar products in this algebra (with \(||e_0 ||{\text{ = 1}}\)) is either infinite or at most \(\sqrt {4/3} \). Sufficient conditions for this bound to be not less than \(\sqrt {4/3} \) are obtained. The finiteness of this bound for infinite-dimensional Grassmann algebras was first proved by Kupsh and Smolyanov (this was used for constructing a functional representation for Fock superalgebras).