Vol 58, No 6 (2017)
- Year: 2017
- Articles: 19
- URL: https://journals.rcsi.science/0037-4466/issue/view/10450
Article
The Generalized Davies Problem for Polyharmonic Operators
Abstract
The Davies problem is connected with the maximal constants in Hardy-type inequalities. We study the generalizations of this problem to the Rellich-type inequalities for polyharmonic operators in domains of the Euclidean space. The estimates are obtained solving the generalized problem under an additional minimal condition on the boundary of the domain. Namely, for a given domain we assume the existence of two balls with sufficiently small radii and the following property: the balls have only a sole common point; one ball lies inside the domain and the other is disjoint from the domain.
On a Certain Sub-Riemannian Geodesic Flow on the Heisenberg Group
Abstract
Under study is an integrable geodesic flow of a left-invariant sub-Riemannian metric for a right-invariant distribution on the Heisenberg group. We obtain the classification of the trajectories of this flow. There are a few examples of trajectories in the paper which correspond to various values of the first integrals. These trajectories are obtained by numerical integration of the Hamiltonian equations. It is shown that for some values of the first integrals we can obtain explicit formulae for geodesics by inverting the corresponding Legendre elliptic integrals.
Computability of Distributive Lattices
Abstract
The class of (not necessarily distributive) countable lattices is HKSS-universal, and it is also known that the class of countable linear orders is not universal with respect to degree spectra neither to computable categoricity. We investigate the intermediate class of distributive lattices and construct a distributive lattice with degree spectrum {d: d ≠ 0}. It is not known whether a linear order with this property exists. We show that there is a computably categorical distributive lattice that is not relatively Δ20-categorical. It is well known that no linear order can have this property. The question of the universality of countable distributive lattices remains open.
On 2-Closedness of the Rational Numbers in Quasivarieties of Nilpotent Groups
Abstract
The dominion of a subgroup H of a group G in a class M is the set of all elements a ∈ G that have equal images under every pair of homomorphisms from G to a group of M coinciding on H. A group H is said to be n-closed in M if for every group G = gr(H, a1,..., an) of M that contains H and is generated modulo H by some n elements, the dominion of H in G (in M) is equal to H. We prove that the additive group of the rational numbers is 2-closed in every quasivariety M of torsion-free nilpotent groups of class at most 3 whenever every 2-generated group of M is relatively free.
Lattices with Defining Relations Close to Distributivity
Abstract
We consider the 3-generated lattices whose generators enjoy the defining relations of the type a∨(b∧c) = (a∨b)∧(a∨c). Moreover, if the lattice is finite then we obtain its diagram; otherwise, we prove that the corresponding lattice is infinite.
Evolution of the Yamabe Constant Under Bernhard List’s Flow
Abstract
Let g(t) be a solution of Bernhard List’s flow on a closed manifold. We obtain a pointwise control on the volume of g(t). Then under an essential assumption, we achieve a formula for the evolution of the Yamabe constant Y(g(t)) when g(t) is evolving by Bernhard List’s flow.
Negative Dense Linear Orders
Abstract
Considering dense linear orders, we establish their negative representability over every infinite negative equivalence, as well as uniformly computable separability by computable gaps and the productivity of the set of computable sections of their negative representations. We construct an infinite decreasing chain of negative representability degrees of linear orders and prove the computability of locally computable enumerations of the field of rational numbers.
On the Chief Factors of Parabolic Maximal Subgroups of Special Finite Simple Groups of Exceptional Lie Type
Abstract
Considering the finite simple groups F4(2n) and G2(pn), where p ≤ 3, we give a description of the chief factors of a parabolic maximal subgroup involved in its unipotent radical. For every parabolic maximal subgroup of the groups F4(2n), G2(2n), and G2(3n), we give the fragment of its chief series that involves in the unipotent radical of this parabolic subgroup. The generators of the corresponding chief factors are presented in three tables.
Slices and Levels of Extensions of the Minimal Logic
Abstract
We consider two classifications of extensions of Johansson’s minimal logic J. Logics and then calculi are divided into levels and slices with numbers from 0 to ω. We prove that the first classification is strongly decidable over J, i.e., from any finite list Rul of axiom schemes and inference rules, we can effectively compute the level number of the calculus (J + Rul). We prove the strong decidability of each slice with finite number: for each n and arbitrary finite Rul, we can effectively check whether the calculus (J + Rul) belongs to the nth slice.
Stochastic Equations with an Unbounded Operator Coefficient and Multiplicative Noise
Abstract
Under study is a stochastic operator-differential equation with multiplicative noise in the space of Hilbert-valued generalized random variables. Existence and uniqueness of solutions to the Cauchy problem are proved for the case of an unbounded operator coefficient at the white noise. The equation of population dynamics with a stochastically perturbed multiplication operator is presented as an example.
Simple 5-Dimensional Right Alternative Superalgebras with Trivial Even Part
Abstract
We study the simple right alternative superalgebras whose even part is trivial; i.e., the even part has zero product. A simple right alternative superalgebra with the trivial even part is singular. The first example of a singular superalgebra was given in [1]. The least dimension of a singular superalgebra is 5. We prove that the singular 5-dimensional superalgebras are isomorphic if and only if suitable quadratic forms are equivalent. In particular, there exists a unique singular 5-dimensional superalgebra up to isomorphism over an algebraically closed field.
On the Inhomogeneous Conservative Wiener–Hopf Equation
Abstract
We prove the existence of a solution to the inhomogeneous Wiener–Hopf equation whose kernel is a probability distribution generating a random walk drifting to −∞. Asymptotic properties of a solution are found depending on the corresponding properties of the free term and the kernel of the equation.
The Rogers Semilattices of Generalized Computable Enumerations
Abstract
We study the cardinality and structural properties of the Rogers semilattice of generalized computable enumerations with arbitrary noncomputable oracles and oracles of hyperimmune Turing degree. We show the infinity of the Rogers semilattice of generalized computable enumerations of an arbitrary nontrivial family with a noncomputable oracle. In the case of oracles of hyperimmune degree we prove that the Rogers semilattice of an arbitrary infinite family includes an ideal without minimal elements and establish that the top, if present, is a limit element under the condition that the family contains the inclusion-least set.