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Congruences mod p for a prime p and partial products of the numbers 1,…, p − 1 are obtained. Bibliography: 2 titles.

It is shown that an approach earlier used by the author for classification of closed simply connected 6-manifolds (reduction to the problem of calculating certain bordism groups) can also be applied for easily obtaining the results by Barden (1965) on classification of closed simply connected 5-manifolds. Bibliography: 11 titles.

This paper is a survey of results on countable families with natural degree spectra. These results were obtained by a modification of the methodology proposed by Wechner, who first found a family of sets with the spectrum consisting precisely of nonzero Turing degrees. Based on this method, many researchers obtained examples of families with other natural spectra. In addition, in this paper we extend these results and present new examples of natural spectra. In particular, we construct a family of finite sets with the spectrum consisting of exactly non-K-trivial degrees and also we find new sufficient conditions on $$ {\Delta}_2^0 $$ -degree a, which guarantees that the class {x : x ≰ a} is the degree spectrum of some family. Finally, we give a survey of our recent results on the degree spectra of α-families, where α is an arbitrary computable ordinal.

A dynamical system $$ {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla 1\mathrm{n}\;\rho \cdot \nabla u=0& in\kern0.6em {\mathrm{\mathbb{R}}}_{+}^3\times \left(0,T\right),\\ {}{\left.u\right|}_{t=0}={\left.{u}_t\right|}_{t=0}=0& in\kern0.6em \overline{{\mathrm{\mathbb{R}}}_{+}^3},\\ {}{\left.{u}_z\right|}_{z=0}=f& for\kern0.36em 0\le t\le T,\end{array}} $$ is under consideration, where ρ = ρ(x, y, z) is a smooth positive function; f = f(x, y, t) is a boundary control; u = uf (x, y, z, t) is a solution. With the system one associates a response operator R : f ↦ uf|z = 0. The inverse problem is to recover the function ρ via the response operator. A short representation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way to use them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.

In this paper, the structure of relative disposition of 3-vertex cutsets in a 3-connected graph is studied. All such cutsets are divided into structural units – complexes of flowers, of cuts, of single cutsets, and trivial complexes. The decomposition of the graph by a complex of each type is described in detail. It is proved that for any two complexes C1 and C2 of a 3-connected graph G there is a unique part of the decomposition of G by C1 that contains C2. The relative disposition of complexes is described with the help of a hypertree T (G) – a hypergraph any cycle of which is a subset of a certain hyperedge. It is also proved that each nonempty part of the decomposition of G by the set of all of its 3-vertex cutsets is either a part of the decomposition of G by one of the complexes or corresponds to a hyperedge of T (G). This paper can be considered as a continuation of studies begun in the joint paper by D. V. Karpov and A. V. Pastor “On the structure of a 3-connected graph,” published in 2011. Bibliography: 10 titles.

The word maps $$ \tilde{w}:\kern0.5em {\mathrm{GL}}_m{(D)}^{2k}\to {\mathrm{GL}}_n(D) $$ and $$ \tilde{w}:\kern0.5em {D}^{\ast 2k}\to {D}^{\ast } $$ for a word $$ w=\prod \limits_{i=1}^k\left[{x}_i,{y}_i\right], $$ where D is a division algebra over a field K, are considered. It is proved that if $$ \tilde{w}\left({D}^{\ast 2k}\right)=\left[{D}^{\ast },{D}^{\ast}\right], $$ then $$ \tilde{w}\left({\mathrm{GL}}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right), $$ where En(D) is the subgroup of GLn(D), generated by transvections, and Z(En(D)) is its center. Furthermore if, in addition, n > 2, then $$ \tilde{w}\left({E}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right). $$ The proof of the result is based on an analog of the “Gauss decomposition with prescribed semisimple part” (introduced and studied in two papers of the second author with collaborators) in the case of the group GLn(D), which is also considered in the present paper.

The paper analyzes the product of m regular permutation groups G1· . . . · Gm, where m ≥ 2 is a natural number. Each of the regular permutation groups is a subgroup of the symmetric permutation group S(Ω) of degree |Ω| for the set Ω. M. M. Glukhov proved that for k = 2 and m = 2, 2-transitivity of the product G1· G2 is equivalent to the absence of zeros in the corresponding square matrix with the number of rows and columns equal to |Ω| − 1. Also M. M. Glukhov has given necessary conditions of 2-transitivity of such a product of regular permutation groups. In this paper, we consider the general case for any natural m and k such that m ≥ 2 and k ≥ 2. It is proved that k-transitivity of the product of regular permutation groups G1· . . . · Gm is equivalent to the absence of zeros in the square matrix with the number of rows and columns equal to (|Ω| − 1)!/(|Ω| − k)!. We obtain correlation between the number of arcs corresponding to this matrix and a natural number l such that the product (PsQt)l is 2-transitive, where P,Q ⊆ S(Ω) are some regular permutation groups and the permutation st is an (|Ω| − 1)-cycle. We provide an example of the building of AES ciphers such that their round transformations are k-transitive on a number of rounds.

A 3-connected graph G is said to be critical if for any vertex υ ∈ V (G) the graph G − υ is not 3-connected. Entringer and Slater proved that any critical 3-connected graph contains at least two vertices of degree 3. In this paper, a classification of critical 3-connected graphs with two vertices of degree 3 is given in the case where these vertices are adjacent. The case of nonadjacent vertices of degree 3 will be studied in the second part of the paper, which will be published later.

Let rk(n) denote the number of lattice points on a k-dimensional sphere of radius $$ \sqrt{n} $$ . The generating function $$ {\zeta}_k(s)=\sum \limits_{n=1}^{\infty }{r}_k(n){n}^{-s},\kern0.5em k\ge 2, $$ is Epstein’s zeta function. The paper considers the Riesz mean of the type $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{1}{\Gamma \left(\rho +1\right)}\sum \limits_{n\le x}{\left(x-n\right)}^{\rho }{r}_3(n), $$ where ρ > 0; the error term Δρ(x; ζ3) is defined by $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{\uppi^{3/2}{x}^{\rho +3/2}}{\Gamma \left(\rho +5/2\right)}+\frac{x^{\rho }}{\Gamma \left(\rho +1\right)}{\zeta}_3(0)+{\Delta}_{\rho}\left(x;{\zeta}_3\right). $$ K. Chandrasekharan and R. Narasimhan (1962, MR25#3911) proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\Big({x}^{1/2+\rho /2\Big)}& \left(\rho >1\right),\\ {}{\Omega}_{\pm}\left({x}^{1/2+\rho /2}\right)& \left(\rho \ge 0\right).\end{array}} $$ In the present paper, it is proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\left(x\log x\right)& \left(\rho =1\right),\\ {}O\left({x}^{2/3+\rho /3+\varepsilon}\right)& \left(1/2

In a joint work with S. Bianchini [8] (see also [6, 7]), we proved a quadratic interaction estimate for the system of conservation laws $$ \left\{\begin{array}{l}{u}_t+f{(u)}_x=0,\\ {}u\left(t=0\right)={u}_0(x),\end{array}\right. $$ where u : [0, ∞) × ℝ → ℝn, f : ℝn → ℝn is strictly hyperbolic, and Tot.Var.(u0) ≪ 1. For a wavefront solution in which only two wavefronts at a time interact, such an estimate can be written in the form $$ \sum \limits_{t_j\;\mathrm{interaction}\ \mathrm{time}}\frac{\left|\sigma \left({\alpha}_j\right)-\sigma \left({\alpha}_j^{\prime}\right)\right|\left|{\alpha}_j\right|\left|{\alpha}_j^{\prime}\right|}{\left|{\alpha}_j\right|+\left|{\alpha}_j^{\prime}\right|}\le C(f)\mathrm{Tot}.\mathrm{Var}.{\left({u}_0\right)}^2, $$ where αj and $$ {\alpha}_j^{\prime } $$ are the wavefronts interacting at the interaction time tj, σ(·) is the speed, |·| denotes the strength, and C(f) is a constant depending only on f (see [8, Theorem 1.1] or Theorem 3.1 in the present paper for a more general form). The aim of this paper is to provide the reader with a proof for such a quadratic estimate in a simplified setting, in which: • all the main ideas of the construction are presented; • all the technicalities of the proof in the general setting [8] are avoided.

We study properties of sets having the minimum length (one-dimensional Hausdorff measure) in the class of closed connected sets Σ ⊂ ℝ2 satisfying the inequality max yϵM dist (y, Σ) ≤ r for a given compact set M ⊂ ℝ2 and given r > 0. Such sets play the role of the shortest possible pipelines arriving at a distance at most r to every point of M where M is the set of customers of the pipeline. In this paper, it is announced that every maximum distance minimizer is a union of finitely many curves having one-sided tangent lines at every point. This shows that a maximum distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact set M, which distinguishes it from a solution of the Steiner problem (an example of a Steiner tree with infinitely many branching points can be found in a paper by Paolini, Stepanov, and Teplitskaya). Moreover, the angle between these lines at each point of a maximum distance minimizer is at least 2π/3. Also, we classify the behavior of a minimizer Σ in a neighborhood of any point of Σ. In fact, all the results are proved for a more general class of local minimizers.

The paper considers the category ( $$ \mathcal{T} $$ , S, X) consisting of mappings S : $$ \mathcal{T} $$ −→ $$ \mathcal{T} $$ of spaces $$ \mathcal{T} $$ with distinguished subsets X ⊂ $$ \mathcal{T} $$ . Let rX (i, x0) be the distribution function of points of an S-orbit x0, x1 = S(x0), . . . , xi−1 = Si−1(x0) getting into X, and let δX (i, x0) be the deviation defined by the equation rX (i, x0) = aX i + δX (i, x0), where aX i is the average value. If δX (i, x0) = O(1), then such sets X are called bounded remainder sets. In the paper, bounded remainder sets X are constructed in the following cases: (1) the space $$ \mathcal{T} $$ is the circle, torus, or the Klein bottle; (2) the map S is a rotation of the circle, a shift or an exchange mapping of the torus; (3) X is a fixed subset X ⊂ $$ \mathcal{T} $$ or a sequence of subsets depending on the iteration number i = 0, 1, 2, . . .. Bibliography: 27 titles.

It is known that the monoid wreath product of any two semigroup varieties that are atoms in the lattice of all semigroup varieties may have a finite as well as an infinite lattice of subvarieties. If this lattice is finite, then as a rule it has at most eleven elements. This was proved in a paper of the author in 2007. The exclusion is the monoid wreath product Sl w N 2 of the variety of semilattices and the variety of semigroups with zero multiplication. The number of elements of the lattice L(Sl w N 2) of subvarieties of Sl w N 2 is still unknown. In our paper, we show that the lattice L(Sl w N 2) contains no less than 33 elements. In addition, we give some exponential upper bound of the cardinality of this lattice.

This paper is devoted to the study of groups from the category M of R-power groups. We examine problems on the commutation of the tensor completion with basic group operations and on the exactness of the tensor completion. Moreover, we introduce the notion of a variety and obtain a description of abelian varieties and some results on nilpotent varieties of A-groups. We prove the hypothesis on irreducible coordinate groups of algebraic sets for the nilpotent R-groups of nilpotency class 2, where R is a Euclidean ring. We state that the analog to the Lyndon result for the free groups (see [10]) holds in this case, whereas the analog to the Myasnikov–Kharlampovich result fails.The paper is dedicated to partial R-power groups which are embeddable to their A-tensor completions. The free R-groups and free R-products are described with usual group-theoretical free constructions.

The present paper is devoted to graph-links with many components and consists of two parts. In the first part of the paper we classify vertices of a labeled graph according to the component they belong to. Using this classification, we construct an invariant of graph-links. This invariant shows that the labeled second Bouchet graph generates a nonrealizable graph-link. In the second part of the work we introduce the notion of an oriented graph-link. We define a writhe number for the oriented graph-link and we get an invariant of oriented graph-links, the Jones polynomial, by normalizing the Kauffman bracket with the writhe number.

This paper deals with an extremal problem concerning hypergraph colorings. Let k be an integer. The problem is to find the value m k (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 at least k vertices of each color. In this paper, we obtain upper bounds of m k (n) for small k and n, the exact value of m 4(8), and a lower bound for m 3(7).

UDC 512.5 In this paper, the normalizer N (σ) of the elementary net group E(σ) associated with a nonsplit maximal torus T (d) in the general linear group GL(n, k) over a field k of odd characteristic is computed. The nonsplit maximal torus T = T (d) is determined by the radical extension k( n √ d) of degree n of the ground field k (minisotropic torus). Bibliography: 18 titles. The problem of describing the overgroups of a split maximal torus in linear groups and in Chevalley groups is virtually solved. The fundamental contribution to the solution of this problem was made by the Leningrad–Petersburg algebraic school (Z. I. Borevich, N. A. Vavilov, and their pupils; for example, see [1, 2, 4–6]). The description of overgroups of a nonsplit torus is a significantly less investigated area. At the present time, a complete description of overgroups of a nonsplit torus has been obtained only for some special fields such as finite or local fields. For finite fields, this was carried out in papers by W. Kantor and G. Seitz (see [15, 17, 18]). Important results on overgroups of a nonsplit torus for local and global fields have been obtained by V. P. Platonov [16]. The case of the field of real numbers has been considered in [14]. For arbitrary fields, the study of overgroups of a nonsplit torus has been conducted in papers [3, 7–13]. The present paper is devoted to an investigation of intermediate subgroups of the general linear group that contain the nonsplit maximal torus related to a radical extension K = k( n √ d) of the ground field k, d ∈ k. The elementary net groups E(σ) associated with the torus T = T (d) and their normalizer N (σ) (which is an overgroup of the nonsplit torus) in the general linear group G =G L( n, k) are of primary importance in studies of the intermediate subgroups mentioned above (see [7–11]). In the present paper, we calculate the normalizer N (σ) of the elementary net group E(σ) associated with a nonsplit maximal torus T (d) in the general linear group GL(n, k )o ver afi eld k of odd characteristic. Moreover, the nonsplit maximal torus T = T (d) is determined by the radical degree-n extension k( n √ d) of the ground field k (minisotropic torus).

The paper considers upper bounds for the infinity norm of the inverse for matrices in two subclasses of the class of (nonsingular) H-matrices, both of which contain the class of Nekrasov matrices. The first one has been introduced recently and consists of the so-called S-Nekrasov matrices. For S-Nekrasov matrices, the known bounds are improved. The second subclass consists of the socalled QN- (quasi-Nekrasov) matrices, which are defined in the present paper. For QN-matrices, an upper bound on the infinity norm of the inverses is established. It is shown that in application to Nekrasov matrices the new bounds are generally better than the known ones. Bibliography: 15 titles.

Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. The paper deals with the question about the behavior of the concentration function of the random variable $\sum\limits_{k=1}^{n}X_k a_k$ according to the arithmetic structure of vectors $a_k$. Recently, the interest to this question has increased significantly due to the study of distributions of eigenvalues of random matrices. In this paper we formulate and prove multidimensional generalizations of the results Eliseeva and Zaitsev (2012). They are also the refinements of the results of Friedland and Sodin (2007) and Rudelson and Vershynin (2009)., Comment: 13 pages

Comultiplication modules over not necessarily commutative rings are studied. All rings are assumed to be associative and with nonzero identity element; all modules are assumed to be unitary. Expressions of the form “A is an invariant ring” mean that “AA and AA are right and left invariant rings.” If A is a ring and M is a right A-module, then for any subset B in A, we denote by lM (B) the left annihilator {m ∈ M | mB = 0} of B in M . According to [2], a module MA is called a comultiplication module if for any submodule X of M , there exists an ideal B of the ring A such that X = lM (B). Comultiplication modules are studied in many papers (see, e.g., [1–6]), where the focus is mainly on comultiplication modules over commutative rings. Theorem 1 ([1]). Let A be a commutative ring and let M be a comultiplication A-module. (1) M is an essential extension of a direct sum of pairwise nonisomorphic simple modules, and any submodule of the module M that is a finite direct sum of cyclic modules is a cyclic module. (2) If the module M is nonsingular, then M is a projective semisimple module. In connection with Theorem 1, we prove Theorem 2, which is the main result of the present paper. Theorem 2. Let A be a ring and let M be a comultiplication right A-module. (1) Any submodule of the module M that is a finite direct sum of cyclic modules is a cyclic module. (2) If A is an invariant ring with commutative multiplication of ideals, then M is an essential extension of a direct sum of pairwise nonisomorphic simple modules; in addition, if the module M is nonsingular, then M is a projective semisimple module. In connection with Theorem 2, we note that the class of all invariant rings with commutative multiplication of ideals contains all commutative rings, all rings of formal power series in one variable over division rings, all factor rings of any direct products of division rings, and all strongly regular rings. (A ring A is said to be strongly regular if every principal one-sided ideal of A is generated by a central idempotent.) The proof of Theorem 2 is decomposed into a series of assertions; some of the assertions are of independent interest. We present the necessary notation and definitions. A ring A is said to be right invariant if all right ideals of A are ideals. A ring A is said to be dual if B = rA(lA)(B) for any right ideal B of A and C = lA(rA)(C) for any left ideal C of A. We denote by J(M) the Jacobson radical of the module M , i.e., J(M) is the intersection of all maximal submodules in M ; we have J(M) = M if M does not have maximal submodules. We denote by Soc(M) the socle of the module M , i.e., Soc(M) is the sum of all minimal submodules in M , and Soc(M) = 0 if M does not have minimal submodules. A ring A is said to be semiperfect if A/J(A) is an Artinian ring and all idempotents of A/J(A) are lifted to idempotents of the ring A. A cyclic module M is said to be local if the module M/J(M) is simple. A submodule X of the module M is said to be essential if X has the nonzero intersection with every nonzero submodule in M . In this case, we say that M is an essential extension of the module X. A module MA is said to be nonsingular if M does not have nonzero elements whose annihilators are essential right ideals of the ring A. A module M is said to be cocyclic if M is an essential extension of a simple module. A module M is said to be finite-dimensional if M does not contain an infinite direct sum of nonzero submodules. A module M is said to be quotient finite-dimensional if all factor modules Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 4, pp. 217–224, 2011/12. 1072–3374/13/1915–0743 c © 2013 Springer Science+Business Media New York 743 of the module M are finite-dimensional. For a module M , a submodule X in M is said to be superfluous (in M) if X + Y = M for each proper submodule Y in M . Any submodule of a factor module of M is called a subfactor of M . A module MA is said to be faithful if rA(M) = 0. The following two assertions are known; they can be directly verified. Lemma 3 ([1, Lemma 1.2]). Let A be a ring, B be an ideal in A, and let M be a right A-module with MB = 0. Then M is a comultiplication A-module if and only if M is a comultiplication A/B-module. Lemma 4 ([2, Theorem 3.17, Lemma 3.7]). Let A be a ring and let M be a right A-module. The following conditions are equivalent : (1) M is a comultiplication module; (2) X = lM ( rA(X) ) for every submodule X of the module M ; (3) every submodule of the module M is a comultiplication module. Lemma 5. Let A be a ring, M be a comultiplication right A-module, and let X be a nonzero submodule in M . (1) X is a comultiplication A/rA(X)-module and f(X) ⊆ X for any homomorphism f : X → M . In particular, M does not contain a direct sum of two isomorphic nonzero modules. (2) If X = X1⊕· · ·⊕Xn is a submodule in M and there exists a module N such that every module Xi is a homomorphic image of the module N , then X is a homomorphic image of the module N . (3) If X = X1 ⊕ · · · ⊕ Xn is a submodule in M and every module Xi is cyclic, then X is a cyclic module. (4) In M , every finitely generated semisimple submodule is cyclic. (5) If there exists an element m ∈ M with rA(m) = 0, then A is a comultiplication right A-module. (6) If the ring A is right invariant and M contains a faithful cyclic submodule mA, then A is a comultiplication right A-module. (7) If the module M is nonsingular and the multiplication of right ideals of the ring A is commutative, then M is a projective semisimple module. Proof. (1) By Lemma 4(3) and Lemma 3, X is a comultiplication A/rA(X)-module. Since M is a comultiplication module, X = rM (B) for some ideal B in A. Then f(X)B ⊆ f(XB) = f(0) = 0, whence f(X) ⊆ rM (B) = X. (2) By assumption, there exist epimorphisms hi : N → Xi, i = 1, . . . , n. We denote by h the homomorphism h1 + · · · + hn from N into X = X1 ⊕ · · · ⊕Xn. Let πi be the natural projection from the module X onto the module Xi, i = 1, . . . , n. By (1), πi ( h(N) ) ⊆ h(N) for all i. Therefore, h(N) = π1 ( h(N) ⊕ · · · ⊕ πn ( h(N) ) = X1 ⊕ · · · ⊕Xn = X, and h is the required epimorphism from the module N onto the module X. (3) Since every module Xi is cyclic, there exist epimorphisms hi : AA → Xi, i = 1, . . . , n. By (2), there exists an epimorphism from the module AA onto the module X. Therefore, X is a cyclic module. (4) Since every finitely generated semisimple module is a finite direct sum of simple modules, the assertion follows from (3) and the property that every simple module is cyclic. (5) By Lemma 4, mAA is a comultiplication module. In addition, mAA is a free cyclic module with free generator m. Now it is directly verified that AA is a comultiplication module. (6) Since the ring A is right invariant, rA(mA) = rA(m). Therefore, (6) follows from (5). (7) First, we prove that M is a semisimple module. Let Y be a submodule in M . By Zorn’s lemma, there exists a submodule Z in M such that Y ∩Z = 0 and Y ⊕Z is an essential submodule in M . We set N = Y ⊕ Z. It is sufficient to prove that M = N . We assume the contrary. Then there exists a nonzero element m ∈ M \ N . Since M is a comultiplication module, there exists an ideal B of the ring A with N = lM (B). Then mB = 0. Since N is an essential submodule in M , we have that mC ⊆ N for some

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.

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.

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

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.

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).

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.

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.

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.

Let C be an Abelian group. A class X of Abelian groups is called a CE• H-class if for any groups A, B ∈ X, it follows from the existence of isomorphisms E• (A) ≅ E• (B) and Hom(C,A) ≅ Hom(C,B) that there is an isomorphism A ≅ B. In this paper, conditions are studied under which the class $$ {\mathfrak{I}}_{\mathrm{cd}}^{\mathrm{ad}} $$ of completely decomposable almost divisible Abelian groups and class $$ {\mathfrak{I}}_{\mathrm{cd}}^{\ast } $$ of completely decomposable torsion-free Abelian groups A where Ω(A) contains only incomparable types are CE• H-classes, where C is a completely decomposable torsion-free Abelian group.

Among the finitely generated groups of diffeomorphisms of the line and the circle, groups that act freely on the orbit of almost every point of the line (circle) are allocated. The paper is devoted to the study of the structure of the set of finitely generated groups of orientation-preserving diffeomorphisms of the line and the circle of C(1) smoothness with a given number of generators and the property noted above. It is shown that such a set contains a massive subset (contains a countable intersection of open everywhere dense subsets). Such a result for finitely generated groups of orientation-preserving diffeomorphisms of the circle, in the case of C(2) smoothness, was obtained by the author earlier.

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.

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/ ( ∞ ⊕

Let X be an irreducible projective variety over an algebraically closed field of characteristic zero. For ≥ 3, if every (r−2)-plane $$\overline {x_1 , \ldots ,x_{r - 1} } $$ , where the x i are generic points, also meets X in a point x r different from x 1,..., x r−1, then X is contained in a linear subspace L such that codim L X ≥ r − 2. In this paper, our purpose is to present another derivation of this result for r = 3 and then to introduce a generalization to nonequidimensional varieties. For the sake of clarity, we shall reformulate our problem as follows. Let Z be an equidimensional variety (maybe singular and/or reducible) of dimension n, other than a linear space, embedded into ℙr, where r ≥ n + 1. The variety of trisecant lines of Z, say V 1,3(Z), has dimension strictly less than 2n, unless Z is included in an (n + 1)-dimensional linear space and has degree at least 3, in which case dim V 1,3(Z) = 2n. This also implies that if dim V 1,3(Z) = 2n, then Z can be embedded in ℙ n + 1. Then we inquire the more general case, where Z is not required to be equidimensional. In that case, let Z be a possibly singular variety of dimension n, which may be neither irreducible nor equidimensional, embedded into ℙr, where r ≥ n + 1, and let Y be a proper subvariety of dimension k ≥ 1. Consider now S being a component of maximal dimension of the closure of $$\{ l \in \mathbb{G}(1,r)|\exists p \in Y, q_1 , q_2 \in Z\backslash Y, q_1 , q_2 ,p \in l\} $$ . We show that S has dimension strictly less than n + k, unless the union of lines in S has dimension n + 1, in which case dim S = n + k. In the latter case, if the dimension of the space is strictly greater than n + 1, then the union of lines in S cannot cover the whole space. This is the main result of our paper. We also introduce some examples showing that our bound is strict.

This paper compares sequences of independent, mean zero random variables in a rearrangement-invariant space X on [0, 1] with sequences of disjoint copies of individual terms in the corresponding rearrangement-invariant space Z X 2 on [0, ∞). The principal results of the paper show that these sequences are equivalent in X and Z X 2 , respectively, if and only if X possesses the (so-called) Kruglov property. We also apply our technique to complement well-known results concerning the isomorphism between rearrangement-invariant spaces on [0, 1] and [0, ∞). Bibliography: 20 titles.

∞n=0 ∞ pnF ∗n (x), where F ∗n (x) denotes the n-fold convolution of F (x) and where F ∗0 (x) denotes the unit mass at 0. The d.f. W (x )i s called subordinate to F (x) with subordinator {pn}. As in the univariate case in the paper, we shall assume that N satisfies condition (A): N has a generating function P (z )= E(z N ) that is analytic at z =1 . In the present paper, we discuss the relation between the asymptotic behavior of 1 − F (x) and that of 1 − F ∗n (x) and 1 − W (x). It turns out that, as in the univariate case, there are many cases in which 1 − F ∗n (x) asymptotically behaves as n(1 − F (x)) and 1 − W (x) behaves as E(N )(1 − F (x)). To specify the precise kind of asymptotic behavior, we present a form of multivariate subexponentiality. The paper is organized as follows. In Sec. 2, we briefly recall some basic properties and definitions concerning univariate subexponential d.f. In Sec. 3, we introduce and study multivariate subexponential d.f.’s. In Sec. 4, we discuss the relation with regular variation and, in Sec. 5, we provide some extensions. In our main results, we obtain first-order estimates for 1 − F ∗n (x) and 1 − W (x). In a forthcoming paper, we discuss second-order estimates. Without further comment, in the paper, we shall assume that all random vectors X, Y, Z, etc. are positive and have infinite support, i.e., the d.f. satisfies F (0+) = 0 and F (x) < 1, ∀x ∈ R d . We also use the notation F (x )=1 − F (x), and for vectors x and a ,w e setx ◦ = min(xi )a nda ∗ x =( a1x1 ,a 2x2 ,...,a dxd). 2. Univariate Subexponential Distributions In the one-dimensional case, many papers have been devoted to the tail behavior of subordinated d.f.’s. In doing so, the class of subexponential d.f.’s (notation: S) plays an important role. Extending the class S, Chover et al. [6, 7], introduced the class S(γ), where γ ≥ 0. To define these classes, let F (x) denote a d.f. in R such that F (0+) = 0 and F (x) < 1, ∀x ∈ R. Also, let f (s )= E(e −sX ) denote the generating function of X or F (x). The d.f. F (x) belongs to the subexponential class S (notation: F ∈ S) if it satisfies lim

A new method, which enables us to compute rather efficiently the Galois group of a polynomial over ℚ or over ℤ, is presented. Reductions of this polynomial with respect to different prime modules are studied, and the information obtained is used for the calculation of the Galois group of the initial polynomial. This method uses an original modification of the Chebotarev density theorem, and it is in essence a probabilistic method. The irreducibility of the polynomial under consideration is not assumed. The appendix to this paper contains tables, which enable us to find the Galois group of polynomials of degree less than or equal to 10 as a subgroup of the symmetric group. Here the final part of the paper is published. The first part is contained in a previous issue (see Vol. 134, No. 6 (2006)). Bibliography: 10 titles.

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

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

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

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

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

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

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

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

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

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

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

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