Presentations of subgroups of the braid group generated by powers of band generators

According to the Tits conjecture proved by Crisp and Paris (2001) [4], the subgroups of the braid group generated by proper powers of the Artin elements σi are presented by the commutators of generators which are powers of commuting elements. Hence they are naturally presented as right-angled Artin groups.The case of subgroups generated by powers of the band generators aij is more involved. We show that the groups are right-angled Artin groups again, if all generators are proper powers with exponent at least 3. We also give a presentation in cases at the other extreme, when all generators occur with exponent 1 or 2. Such presentations are distinctively more complicated than those of right-angled Artin groups.     

Measurability of automorphisms of topological groups

Relations between the measurability and continuity of algebraic automorphisms of topological groups depending on the types of groups are examined. Various cases are considered and theorems on the continuity of measurable automorphisms are proved; for instance, such theorems are proved for separable locally compact groups and automorphisms measurable with respect to nonnegative Haar measures. On the other hand, examples of nonmetrizable nonseparable compact groups with Haar measures and of non-locally-compact separable metrizable groups with measures μ quasi-invariant with respect to dense subgroups admittings μ-measurable discontinous automorphisms are given. Translated fromMatenmaticheskie Zametki, Vol. 68, No. 1, pp. 105–112, July, 2000. An erratum to this article can be found online at .     

Properties of topologies of information retrieval systems

This paper studies topological properties of different topologies that are possible on the space of documents as they are induced by queries in a query space together with a similarity function between queries and documents. The main topologies studied here are the retrieval topology (introduced by Everett and Cater) and the similarity topology (introduced by Egghe and Rousseau).The studied properties are the separation properties T₀, T₁, and T₂ (Hausdorff), proximity and connectedness. Full characterizations are given for the diverse topologies to be T₀, T₁, or T₂. It is shown that the retrieval topology is not necessarily a proximity space, while the similarity topology and the pseudo-metric topology always are proximity spaces. A characterization of connectedness in terms of the Boolean NOT-operator is given, hereby showing the intimate relationship between IR and topology.     

On Unimodality of Hilbert Functions of Gorenstein Artin Algebras of Embedding Dimension Four

We prove that the Hilbert functions of Gorenstein Artin algebras R/I of embedding dimension four are unimodal provided I has a minimal generator in degree less than five. It is still an open question as to whether all Gorenstein Hilbert functions in codimension four are SI-sequences. It is not even known if they are all unimodal. In this article, we prove that Hilbert functions of all Gorenstein Artin algebras starting with (1, 4, 10, 20, h ₄,…), where h ₄ = 34 are unimodal. Combining this with previously known results, we obtain that all Gorenstein Hilbert functions (1, 4, h ₂, h ₃, h ₄,…4, 1) are unimodal if h ₄ ≤ 34.     

On Two Families of Hopf Algebras of Dimension 2 m

Abstract

In this paper we construct two families of semisimple Hopf algebras of dimension 2 ⁿ⁺¹, n ≥ 3. They are all constructed as Radford's biproducts. For these examples and their duals we compute their grouplike elements, centers, character algebras and Grothendieck rings. Comparing these facts we are able to show that depending on the dimension, representatives of one of the families are selfdual. We also prove that Hopf algebras from these families are neither triangular nor cotriangular and that their cocycle deformations are trivial.     

Optimization of multiple covering of a bounded set with circles

Numerical algorithms for the optimization of multiple covering of a bounded set G in the plane P with equal circles are proposed. The variants in which G is a connected bounded set in P or a finite set in P are considered. The circles may be centered at arbitrary points of G or at points belonging to a given set. Minimization of the radius of the given number of circles and minimization of the number of circles of a given radius are considered. Models and solution algorithms are described, and estimates of the solutions provided by most variants are given. Numerical results are presented.     

Normal families of functions and groups of pseudoconformal diffeomorphisms of quaternion and octonion variables

This paper is devoted to the specific class of pseudoconformal mappings of quaternion and octonion variables. Normal families of such functions are defined and investigated. Four criteria of a family to be normal are proved. Then groups of pseudoconformal diffeomorphisms of quaternion and octonion manifolds are investigated. It is proved that they are finite-dimensional Lie groups for compact manifolds. Their examples are given. Many characteristic features are found in comparison with commutatiive geometry over R or C. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 18, Functional Analysis, 2006.     

On cross products of orthonormal systems of functions

The continual analog of an orthonormal system of functions is an orthonormal kernel. In this article the concept of cross product of orthonormal systems of functions is introduced, and it is shown that the cross product of any two orthonormal systems which are complete in L₂ is a complete orthonormal kernel with respect to Lebesgue measure on half-axes. The properties of the cross product of two orthonormal systems which are complete in L₂, each of which is uniformly bounded, are studied, as are the properties of the cross product of a Haar system on an orthonormal system of functions, complete in L₂, which are uniformly bounded.Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 469–480, March, 1973.     

On Idomatic Partitions of Direct Products of Complete Graphs

Independent dominating sets in the direct product of four complete graphs are considered. Possible types of such sets are classified. The sets in which every pair of vertices agree in exactly one coordinate, called T ₁-sets, are explicitly described. It is proved that the direct product of four complete graphs admits an idomatic partition into T ₁-sets if and only if each factor has at least three vertices and the orders of at least two factors are divisible by 3.     

Properties of solutions of Riccati equation

The paper studies some properties of solutions of the Riccati equation

$y'(t) + a(t)y^2 (t) + b(t)y(t) + c(t) = 0$

on a semiaxis [t ₀, +∞) for different types of initial value sets. Two types of solutions are singled out: normal, that are in a sense stable, and extremal, that are non-stable in the Lyapunov sense. Relations expressing the extremal solutions by means of a given normal solution in quadratures and elementary functions are obtained and some relations between solutions the extendable to [t ₀, +∞) are derived.

     

Localization method for lines of discontinuity of an approximately defined function of two variables

A function of two variables with lines of discontinuity of the first kind is considered. It is assumed that outside the discontinuity lines the function is smooth and has a bounded partial derivative. An approximation to the function in L ₂ and a perturbation level are known. The problem in question belongs to a class of nonlinear ill-posed problems, which are solved by constructing some regularizing algorithms. We propose a simple theoretical approach to solving the problem of localizing the discontinuity lines of a function that is noisy in the space L ₂. Some conditions on the exact function are imposed ??in the small.?? Methods of averaging are constructed, and error estimates of localizing the lines (in the small) are obtained.     

Functions of a matrix and Krylov matrices

For a given nonderogatory matrix A, formulas are given for functions of A in terms of Krylov matrices of A. Relations between the coefficients of a polynomial of A and the generating vector of a Krylov matrix of A are provided. With the formulas, linear transformations between Krylov matrices and functions of A are introduced, and associated algebraic properties are derived. Hessenberg reduction forms are revisited equipped with appropriate inner products and related properties and matrix factorizations are given.     

Cardinality and nilpotency of localizations of groups andG-modules

We consider the effect of a coagumented idempotent functorJ in the the category of groups orG-modules whereG is a fixed group. We are interested in the ‘extent’ to which such functors change the structure of the objects to which they are applied. Some positive results are obtained and examples are given concerning the cardinality and structure ofJ(A) in terms of the cardinality and structure ofA, where the latter is a torsion abelian group. For non-abelian groups some partial results and examples are given connecting the nilpotency classes and the varieties of a groupG andJ(G). Similar but stronger results are obtained in the category ofG-modules.     

On algebras of operators of finite rank

Algebras of operators of finite rank are studied. Some structure theorems are presented. Results are used to study Banach representations of C¹-algebras and isolated points of the dual of C¹-algebras.     

Asymptotics of the probability of values of random Boolean expressions

The random Boolean expressions are considered that are obtained by the random and independent substitution with the probabilities p and 1 ? p of the constantly one function and constantly zero function for variables of repetition-free formulas over a given basis. The probability is studied that the expressions are equal to one. It is shown that, for each finite basis and p ? (0, 1), this probability tends to some finite limit P ₁(p) as the length of an expression grows. Explicit representation of the probability function P ₁(p) is found for all finite bases, the analytic properties of this function are studied, and its behavior is investigated in dependence on the properties of the basis.     

Quadrilateral embeddings of bipartite graphs

Current graphs and a theorem of White are used to show the existence of almost complete regular bipartite graphs with quadrilateral embeddings conjectured by Pisanski. Decompositions of K_n and K_{n, n} into graphs with quadrilateral embeddings are discussed, and some thickness results are obtained. Some new genus results are also obtained.     

Holomorphic Linearization of Commuting Germs of Holomorphic Maps

Let f ₁,…,f _h be h≥2 germs of biholomorphisms of ? ⁿ fixing the origin. We investigate the shape that a (formal) simultaneous linearization of the given germs can have, and we prove that if f ₁,…,f _h commute and their linear parts are almost simultaneously Jordanizable, then they are simultaneously formally linearizable. We next introduce a simultaneous Brjuno-type condition and prove that, in case the linear terms of the germs are diagonalizable, if the germs commute and our Brjuno-type condition holds, then they are holomorphically simultaneously linearizable. This answers a multi-dimensional version of a problem raised by Moser.     

Note on Direct Products of Certain Classes of Finite Groups

Direct products of solvable groups in which Sylow-permutability is a transitive relation are analyzed. Such groups are called 𝒫𝒮𝒯 groups, and it is well known that if the orders of two 𝒫𝒮𝒯 groups are relatively prime, then their direct product is again a 𝒫𝒮𝒯 group. Examples suggest that for solvable groups it is not necessary to have relatively prime orders to stay in the class. Some characterizations are provided, and also direct products of some other classes are discussed.     

Ultimately periodic extension of partial endomorphisms of groups

Necessary and sufficient conditions for extending a partial endomorphism μ of a groupG to a total endomorphismμ ^* of a supergroup ofG are known. In this paper conditions, which are both necessary and sufficient, forμ ^* to be ultimately periodic are obtained.