Constructive Description Research Articles

Homogeneous monounary algebras

Homogeneous algebras have been investigated by Marczewski [4], Ganter, P lonka and Werner [3] and Csakany [1]. Some related problems for graphs were studied by Droste, Giraudet and Macpherson [2]. In this paper we deal with homogeneous monounary algebras. We show that a connected monounary algebra is homogeneous if and only if either (i) (A, f) is a cycle, or (ii) card f−1(x) = card f−1(y) for each x, y ∈ A. Further, we prove that for each cardinal α > 0 there is, up to isomorphism, a unique connected monounary algebra (Bα, f) possessing no cycle and having the property that card f−1(x) = α for each x ∈ Bα; we give a constructive description of (Bα, f). The case of non-connected monounary algebras can be easily reduced to the connected case. Next, we find necessary and sufficient conditions under which a monounary algebra can be embedded into a homogeneous monounary algebra.

Optimality Conditions for Irregular Inequality-Constrained Problems

We consider feasible sets given by conic constraints, where the cone defining the constraints is convex with nonempty interior. We study the case where the feasible set is not assumed to be regular in the classical sense of Robinson and obtain a constructive description of the tangent cone under a certain new second-order regularity condition. This condition contains classical regularity as a special case, while being weaker when constraints are twice differentiable. Assuming that the cone defining the constraints is finitely generated, we also derive a special form of primal-dual optimality conditions for the corresponding constrained optimization problem. Our results subsume optimality conditions for both the classical regular and second-order regular cases, while still being meaningful in the more general setting in the sense that the multiplier associated with the objective function is nonzero.

On Abnormally Factorizable Finite Solvable Groups

We study hereditary formations closed with respect to the operation of taking products of abnormal subgroups of finite solvable groups. We obtain a constructive description of solvable local hereditary formations of finite groups with the property indicated.

The Kegel--Shemetkov Problem on Lattices of Generalized Subnormal Subgroups of Finite Groups

We study into formations \(\mathfrak{F}\) for which the set of all \(\mathfrak{F}\)-subnormal (\(\mathfrak{F}\)-accessible) subgroups form a sublattice of the lattice of all subgroups in every finite group. A constructive description of all soluble hereditary formations of finite groups with given properties is furnished.

Monge-Ampére equations and characteristic connection functors

We investigate contact equivalence of Monge-Ampére equations. We define a category of Monge-Ampére equations and introduce the notion of a characteristic connection functor. This functor maps the category of Monge-Ampére equations to the category of affine connections. We give a constructive description of the characteristic connection functors corresponding to three subcategories, which include a large class of Monge-Ampére equations of elliptic and hyperbolic type. This essentially reduces the contact equivalence problem for Monge-Ampére equations in the cases under study to the equivalence problem for affine connections. Using E. Cartan's familiar theory, we are thus able to state and prove several criteria of contact equivalence for a large class of elliptic and hyperbolic Monge-Ampére equations.

SCTL-MUS: A Formal Methodology for Software Development of Distributed Systems. A Case Study

Abstract. This paper introduces an iterative model for the software development process of distributed systems. It is based on dealing with the system evolution and maintenance activities as similar stages of the system development. In order to formalise this model, a multi-valued causal temporal logic, referred to as Simple Causal Temporal Logic (SCTL), is defined for the acquisition and specification of the functional requirements. A Model of Unspecified States (MUS) is also defined with a double goal: firstly, to show the fundamental aspects of system behaviour, which has been specified through a set of SCTL requirements; and, secondly, to verify the consistency and completeness of the specified requirements. The combination of SCTL and MUS allows obtaining the specification of the initial architecture of the system formally. Besides, the design decisions are stored with the goal of making the evolution and maintenance tasks easier. The translation between MUS and a constructive formal description technique (LOTOS) is automatic from the definition of architectural operators.

Structure of certain classes of groups with locally cyclic Abelian subgroups

We investigate groups with locally cyclic Abelian subgroups. We give a constructive description of locally solvable groups of this type that contain a nonidentity periodic part; we also describe locally solvable groups all Abelian subgroups of which are cyclic.

Isochronicity and commutation of polynomial vector fields

We study a connection between the isochronicity of a center of a polynomial vector field and the existence of a polynomial commuting system. We demonstrate an isochronous system of degree 4 which does not commute with any polynomial system. We prove that among the Newton polynomial systems only the Lienard and Abel systems may commute with transversal polynomial fields. We give a full and constructive description of centralizers of the Abel polynomial systems. We give new examples of isochronous systems.

On the stability of global non-radial pulsations of neutron stars

A neutron star is the cosmic nuclear object in which the energy of gravitational pull is brought to equilibrium by elastic energy stored in the neutron Fermi-continuum. Evidence for the viscoelastic behaviour of a stellar nuclear matter provides a seismological model of pulsar glitches interpreted as a sudden release of the elastic energy. In laboratory nuclear physics, the signatures of viscoelasticity of nuclear matter are found in the current investigations on the collective nuclear dynamics, in which a heavy nucleus is modelled by a spherical piece of viscoelastic Fermi-continuum compressed to the normal nuclear density. It is plausible to expect, therefore, that the motions of self-gravitating nuclear matter constituting the interior of neutron stars should be governed by the equations of an elastic solid, rather than by hydrodynamic equations describing the behaviour of gaseous plasma inside the main sequence stars. In this paper, we present arguments that elastodynamic equations, originally introduced in the context of nuclear collective dynamics, can provide a proper account of elasticity in the large scale motions of neutron matter under its own gravity. Emphasis is placed on mathematical physics underlying the constructive description of the continuum mechanics and the rheology of macroscopic nuclear matter. The capability of the elastodynamic approach is examined by analysis of oscillatory dynamics of a neutron star in the standard homogenous model, operating with a spherical mass of self-gravitating degenerate neutron matter whose viscoelastic behaviour is described in terms of the spheroidal and torsional gravitational-elastic eigenmodes, inherenly related to viscoelasticy. The energy variational principle is utilized to compute the frequencies of viscoelastic gravitational pulsations and their relaxation time. The method is demonstrated for both the idealized homogeneous model and the neutron star models constructed on realistic equations of state. Finally, we derive analytic conditions for the stability of a neutron star to linear elastic deformations accompanying the non-radial pulsations, and discuss the fingerprints of these pulsations in the electromagnetic activity of radiopulsars.

On one smoothing problem in uniform metric

We obtain the constructive description of functions that realize the solution of smoothing problem in uniform metric.

On the Phase Diagram of the Random Field Ising Model on the Bethe Lattice

The ferromagnetic Ising model on the Bethe lattice of degree k is considered in the presence of a dichotomous external random field ξ x = ±α and the temperature T≥0. We give a description of a part of the phase diagram of this model in the T−α plane, where we are able to construct limiting Gibbs states and ground states. By comparison with the model with a constant external field we show that for all realizations ξ = {ξ x = ±α} of the external random field: (i) the Gibbs state is unique for T > T c (k ≥ 2 and any α) or for α > 3 (k = 2 and any T); (ii) the ±-phases coexist in the domain {T 0; and (iv) the Gibbs state is unique for 3≥α≥2 at any T. We show that the residual entropy at T = 0 is positive for 3≥α≥2, and we give a constructive description of ground states, by so-called dipole configurations.

A Constructive Description of SAGBI Bases for Polynomial Invariants of Permutation Groups

LetRbe a commutative ring with 1, letR[X1,…,Xn] be the polynomial ring inX1,…,XnoverR, and letGbe an arbitrary group of permutations of {X1,…,Xn}. This note presents a detailed analysis and a constructive combinatorial description of SAGBI bases for theR-algebra ofG-invariant polynomials. Our main result is a ground ring independent characterization of all rings of polynomial invariants of permutation groupsGhaving a finite SAGBI basis.

Structure of one class of groups with conditions of denseness of normality for subgroups

We give a constructive description of locally graded groups G satisfying the following condition: For any pair of subgroups A and B such that A<B, there exists a normal subgroup N that belongs to G and is such that A≦N≦B.

Structure of locally graded nonnilpotent CDN[]-groups

We prove a theorem that gives a constructive description of locally graded nonnilpotent CDN []-groups.

Construction of nonnilpotent biprimary dispersible groups with metacyclic subgroups of nonprimary index

We give a constructive description of nonnilpotent biprimary dispersible groups with metacyclic subgroups of nonprimary index and classify them into 21 types.

Localized classical waves created by defects

We study acoustic and electromagnetic waves in a periodic medium (or any other background medium with a spectral gap) disturbed by a single defect, i.e., a local disturbance analogous to a well potential in solid state physics. We show that defects do not change the essential spectrum of the associated nonnegative operators and can only create isolated eigenvalues of finite multiplicity in a gap of the periodic medium, with the eigenmodes decaying exponentially. We give a constructive and simple description of defects in acoustic and dielectric media, including a simple condition on the parameters of the medium and of the defect, which ensures the rise of a localized eigenmode with the corresponding eigenvalue in a specified subinterval of the given gap of the periodic medium.

One-dimensional random-field ising model: Gibbs states and structure of ground states

We consider the random Gibbs field formalism for the ferromagnetic ID dichotomous random-field Ising model as the simplest example of a quenched disordered system. We prove that for nonzero temperatures the Gibb state is unique for any realization of the external field. Then we prove that asT→0, the Gibbs state converges to a limit, a ground state, for almost all realizations of the external field. The ground state turns out to be a probability measure concentrated on an infinite set of configurations, and we give a constructive description of this measure.

The weighted BMO condition and a constructive description of classes of analytic functions satisfying this condition

The problem of local polynomial approximation of analytic functions prescribed in finite domains with a quasiconformal boundary is investigated in weighted plane integral metrics; a constructive description of the class of analytic functions satisfying a weak version of the known BMO condition in obtained.

Continuity of functions operating on characteristic functions

AbstractFor a locally compact abelian group G let P0(G) denote the set of all characteristic functions on G (i.e., continuous positive definite functions φ with φ(0) = 1). A complex-valued function f is said to operate on P0(G) if f(φ(·)) ∈ P0(G) whenever φ ∈ P0(G). The natural domain for functions operating on P0(G) is the set D(G) = {z ∈ C: z = φ(g), g ∈ G, φ ∈ P0(G)}. It is known that every function operating on P0(G) is continuous on the interior of D(G) for each infinite group G. On the other hand, functions discontinuous on D(G) operate on P0(G) for any discrete G. We show that, for any non-discrete group G, every function operating on P0(G) is continuous on D(G). Together with some earlier results, this statement allows us to obtain a constructive description of operating functions on the whole set D(G).

The weighted BMO condition and a constructive description of classes of analytic functions satisfying this condition

The weighted BMO condition and a constructive description of classes of analytic functions satisfying this condition