Volume 58, Nº 1 (2017)
- Ano: 2017
- Artigos: 22
- URL: https://journals.rcsi.science/0037-4466/issue/view/10403
Article
Centralizers of generalized skew derivations on multilinear polynomials
Resumo
Let R be a prime ring of characteristic different from 2, let Q be the right Martindale quotient ring of R, and let C be the extended centroid of R. Suppose that G is a nonzero generalized skew derivation of R and f(x1,..., xn) is a noncentral multilinear polynomial over C with n noncommuting variables. Let f(R) = {f(r1,..., rn): ri ∈ R} be the set of all evaluations of f(x1,..., xn) in R, while A = {[G (f(r1,..., rn)), f(r1,..., rn)]: ri ∈ R}, and let CR(A) be the centralizer of A in R; i.e., CR(A) = {a ∈ R: [a, x] = 0, ∀x ∈ A }. We prove that if A ≠ (0), then CR(A) = Z(R).
A monotone path-connected set with outer radially lower continuous metric projection is a strict sun
Resumo
A monotone path-connected set is known to be a sun in a finite-dimensional Banach space. We show that a B-sun (a set whose intersection with each closed ball is a sun or empty) is a sun. We prove that in this event a B-sun with ORL-continuous (outer radially lower continuous) metric projection is a strict sun. This partially converses one well-known result of Brosowski and Deutsch. We also show that a B-solar LG-set (a global minimizer) is a B-connected strict sun.
Sub-Riemannian distance on the Lie group SL(2)
Resumo
We find the distances between arbitrary elements of the Lie group SL(2) for the left invariant sub-Riemannian metric also invariant with respect to the right shifts by elements of the Lie subgroup SO(2) ⊂ SL(2), in other words, the invariant sub-Riemannian metric on the weakly symmetric space (SL(2) × SO(2))/ SO(2) of Selberg.
On DP-coloring of graphs and multigraphs
Resumo
While solving a question on the list coloring of planar graphs, Dvořák and Postle introduced the new notion of DP-coloring (they called it correspondence coloring). A DP-coloring of a graph G reduces the problem of finding a coloring of G from a given list L to the problem of finding a “large” independent set in the auxiliary graph H(G,L) with vertex set {(v, c): v ∈ V (G) and c ∈ L(v)}. It is similar to the old reduction by Plesnevič and Vizing of the k-coloring problem to the problem of finding an independent set of size |V(G)| in the Cartesian product G□Kk, but DP-coloring seems more promising and useful than the Plesnevič–Vizing reduction. Some properties of the DP-chromatic number χDP (G) resemble the properties of the list chromatic number χl(G) but some differ quite a lot. It is always the case that χDP (G) ≥ χl(G). The goal of this note is to introduce DP-colorings for multigraphs and to prove for them an analog of the result of Borodin and Erdős–Rubin–Taylor characterizing the multigraphs that do not admit DP-colorings from some DP-degree-lists. This characterization yields an analog of Gallai’s Theorem on the minimum number of edges in n-vertex graphs critical with respect to DP-coloring.
The height of faces of 3-polytopes
Resumo
The height of a face in a 3-polytope is the maximum degree of the incident vertices of the face, and the height of a 3-polytope, h, is the minimum height of its faces. A face is pyramidal if it is either a 4-face incident with three 3-vertices, or a 3-face incident with two vertices of degree at most 4. If pyramidal faces are allowed, then h can be arbitrarily large; so we assume the absence of pyramidal faces. In 1940, Lebesgue proved that every quadrangulated 3-polytope has h ≤ 11. In 1995, this bound was lowered by Avgustinovich and Borodin to 10. Recently, we improved it to the sharp bound 8. For plane triangulation without 4-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that h ≤ 20 which bound is sharp. Later, Borodin (1998) proved that h ≤ 20 for all triangulated 3-polytopes. Recently, we obtained the sharp bound 10 for triangle-free 3-polytopes. In 1996, Horňák and Jendrol’ proved for arbitrarily 3-polytopes that h ≤ 23. In this paper we improve this bound to the sharp bound 20.
On quasivarieties of axiomatic rank 3 of torsion-free nilpotent groups
Resumo
We study the lattice of quasivarieties of axiomatic rank at most 3 of torsion-free nilpotent groups of class at most 3. We prove that this lattice has cardinality of the continuum and includes a sublattice that is order isomorphic to the set of real numbers. Also we establish that the lattice of quasivarieties of axiomatic rank at most 2 of these groups is a 5-element chain.
The Fω-normalizers of finite groups
Resumo
Given a nonempty set ω of primes and a nonempty formation F of finite groups, we define the Fω-normalizer in a finite group and study their properties (existence, invariance under certain homomorphisms, conjugacy, embedding, and so on) in the case that F is an ω-local formation. We so develop the results of Carter, Hawkes, and Shemetkov on the F-normalizers in groups.
On a system of entire functions of class A which is biorthogonal to a lacunar power system on the ray
Resumo
We construct a system of entire functions of exponential type of class A biorthogonal with weight to some power system on the ray. The indicator diagram of such a function is a segment of the imaginary axis. Functions analytic in a circular lacuna are represented by biorthogonal series.
Sufficient conditions for the existence of 0’-limitwise monotonic functions for computable η-like linear orders
Resumo
We find new sufficient conditions for the existence of a 0’-limitwise monotonic function defining the order for a computable η-like linear order L, i.e., of a function G such that L ∑q∈ℚG(q). Namely, we define the notions of left local maximal block and right local maximal block and prove that if the sizes of these blocks in a computable η-like linear order L are bounded then there is a 0’-limitwise monotonic function G with L = ∑q∈ℚG(q).
Graph surfaces on five-dimensional sub-Lorentzian structures
Resumo
Studying the space-like graph surfaces of codimension 2 on the five-dimensional sub-Lorentzian structures with two negative directions of distinct degrees, we determine the differential properties of graph mappings and prove the area formula for the corresponding image surfaces.
k-invariant nets over an algebraic extension of a field k
Resumo
Let K be an algebraic extension of a field k, let σ = (σij) be an irreducible full (elementary) net of order n ≥ 2 (respectively, n ≥ 3) over K, while the additive subgroups σij are k-subspaces of K. We prove that all σij coincide with an intermediate subfield P, k ⊆ P ⊆ K, up to conjugation by a diagonal matrix.
Existence of weak solutions to the three-dimensional problem of steady barotropic motions of mixtures of viscous compressible fluids
Resumo
We consider the boundary value problem describing the steady barotropic motion of a multicomponent mixture of viscous compressible fluids in a bounded three-dimensional domain. We assume that the material derivative operator is common to all components and is defined by the average velocity of the motion, but keep separate velocities of the components in other terms. Pressure is common and depends on the total density. Beyond that we make no simplifying assumptions, including those on the structure of the viscosity matrix; i.e., we keep all terms in the equations, which naturally generalize the Navier–Stokes model of the motion of one-component media. We establish the existence of weak solutions to the boundary value problem.
On pronormality and strong pronormality of Hall subgroups
Resumo
We study several well-known questions on pronormality and strong pronormality of Hall subgroups. In particular, we exhibit the examples of finite groups (a) having a Hall subgroup not pronormal in its normal closure (this solves Problem 18.32 of The Kourovka Notebook in the negative); (b) having a Hall subgroup pronormal but not strongly pronormal; and (c) that are simple, having a Hall subgroup, and not strongly pronormal (this solves Problem 17.45(b) of The Kourovka Notebook in the negative).
Narrow orthogonally additive operators in lattice-normed spaces
Resumo
We consider a new class of narrow orthogonally additive operators in lattice-normed spaces and prove the narrowness of every C-compact norm-laterally-continuous orthogonally additive operator from a Banach–Kantorovich space V into a Banach space Y. Furthermore, every dominated Urysohn operator from V into a Banach sequence lattice Y is also narrow. We establish that the order narrowness of a dominated Urysohn operator from a Banach–Kantorovich space V into a Banach space with mixed norm W implies the order narrowness of the least dominant of the operator.
Solvability of the inhomogeneous Cauchy–Riemann equation in projective weighted spaces
Resumo
We establish an analog of Hörmander’s Theorem on solvability of the inhomogeneous Cauchy–Riemann equation for a space of measurable functions satisfying a system of uniform estimates. The result is formulated in terms of the weight sequence defining the space. The same conditions guarantee the weak reducibility of the corresponding space of entire functions. Basing on these results, we solve the problem of describing the multipliers in weighted spaces of entire functions with the projective and inductive-projective topological structure. Applications are obtained to convolution operators in the spaces of ultradifferentiable functions of Roumieu type.
Representation of the Green’s function of the exterior Neumann problem for the Laplace operator
Resumo
We represent the Green’s function of the classical Neumann problem for the exterior of the unit ball of arbitrary dimension. We show that the Green’s function can be expressed through elementary functions. The explicit form of the function is written out.
Separability of the subgroups of residually nilpotent groups in the class of finite π-groups
Resumo
Given a nonempty set π of primes, call a nilpotent group π-bounded whenever it has a central series whose every factor F is such that: In every quotient group of F all primary components of the torsion subgroup corresponding to the numbers in π are finite. We establish that if G is a residually π-bounded torsion-free nilpotent group, while a subgroup H of G has finite Hirsh–Zaitsev rank then H is π’-isolated in G if and only if H is separable in G in the class of all finite nilpotent π-groups. By way of example, we apply the results to study the root-class residuality of the free product of two groups with amalgamation.
Gröbner–Shirshov bases for some Lie algebras
Resumo
We give Gröbner–Shirshov bases for the Drinfeld–Kohno Lie algebra Ln in [1] and the Kukin Lie algebra AP in [2], where P is a semigroup. By way of application, we show that Ln is free as a ℤ-module and exhibit a ℤ-basis for Ln. We give another proof of the Kukin Theorem: If P has the undecidable word problem then so isAP.