Ukrainian Mathematical Journal最新文献

We study the concepts of projection invariant t-extending modules and projection invariant t-Baer modules, which are generalized to the notions of π-extending and t-Baer modules, respectively. Several structural properties are obtained and some applications are developed. It is shown that the π-t-extending modules and π-t-e. Baer modules are connected with each other. Moreover, we obtain a characterization for π-t-extending modules relative to the annihilator conditions.

We establish a relationship between the subclasses of univalent functions and generalized distribution series. The main aim of our investigation is to obtain necessary and sufficient conditions for the generalized distribution series to belong to the classes 𝒯 ℱ(ρ, ϑ), 𝒯 ℋ(ρ, ϑ), 𝒯 𝒥(ρ, ϑ), and 𝒯 𝒳(ρ, ϑ) . In addition, we obtain some particular cases of our main results.

We study the exponential stability of homogeneous fractional time-varying systems and the existence of Lyapunov homogeneous function for the conformable fractional homogeneous systems. We also prove that the local and global behaviors are similar. A numerical example is given to illustrate the efficiency of the obtained results.

Let (u, v) be a pair of quasidefinite and symmetric linear functionals with {P_n}_n≥0 and {Q_n}_n≥0 as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials {R_n}_n≥0 as follows:

(begin{array}{cc}frac{{P}_{n+2}^{mathrm{^{prime}}}left(xright)}{n+2}+{b}_{n}frac{{P}_{n}^{mathrm{^{prime}}}left(xright)}{n}-{Q}_{n+1}left(xright)={d}_{n-1}left(xright),& nge 1.end{array})

We present necessary and sufficient conditions for {R_n}_n≥0 to be orthogonal with respect to a quasidefinite linear functional w. In addition, we consider the case where {P_n}_n≥0 and {Q_n}_n≥0 are monic Chebyshev polynomials of the first and second kinds, respectively, and study the relative outer asymptotics of Sobolev polynomials orthogonal with respect to the Sobolev inner product

(langle p,qrangle s=underset{-1}{overset{1}{int }}pq{left(1-{x}^{2}right)}^{-1/2}dx+{uplambda }_{1}underset{-1}{overset{1}{int }}{p}^{mathrm{^{prime}}}{q}^{mathrm{^{prime}}}{left(1-{x}^{2}right)}^{1/2}dx+{uplambda }_{2}underset{-1}{overset{1}{int }}{p}^{mathrm{^{prime}}mathrm{^{prime}}}{q}^{mathrm{^{prime}}mathrm{^{prime}}}dmu left(xright),)

where μ is a positive Borel measure associated with w and λ₁, λ₂ > 0; λ₂ is a linear polynomial of λ₁.

We establish constructive necessary and sufficient conditions of solvability and propose a scheme for the construction of solutions to a nonautonomous nonlinear periodic boundary-value problem for a Rayleightype equation unsolved with respect to the derivative. The urgency of investigation of nonautonomous boundary-value problems unsolved with respect to the derivative is explained by the fact that the analysis of traditional problems solved with respect to the derivative is sometimes significantly complicated, e.g., in the presence of nonlinearities that are not integrable in elementary functions. We consider the critical case in which the equation for generating amplitudes of a weakly nonlinear periodic boundary-value problem for a Rayleigh-type equation does not turn into the identity. The least-squares method is used to establish constructive conditions for the solvability and propose convergent iterative schemes for the construction of approximate solutions to a nonautonomous nonlinear boundary-value problem unsolved with respect to the derivative. As an example of application of the proposed iterative scheme, we find approximations to the solutions of periodic boundary-value problems unsolved with respect to the derivative in the case of periodic problem for the equation that describes the motion of a satellite on the elliptic orbit. We obtain an estimate for the range of values of a small parameter in which the iterative procedure used for the construction of solutions to a weakly nonlinear periodic boundary-value problem for a Rayleigh-type equation unsolved with respect to the derivative is convergent. To check the accuracy of the proposed approximations, we estimate the discrepancies appearing in the equation used to simulate the motion of satellites along the elliptic orbits.

We study RG-modules that do not contain nonzero G-perfect factors. In particular, it is shown that if a group G is finite and R is a Dedekind domain with some additional restrictions, then these RG-modules are G-nilpotent.

We continue the development of the theory of moduli of the families of surfaces, in particular, of strings of various dimensions m = 1, 2, . . . ,n − 1 in Euclidean spaces ({mathbb{R}}^{n}) , n ≥ 2. On the basis of the proof of the lemma on the relationships between the moduli and Lebesgue measures, we prove the corresponding analog of the Fubini theorem in terms of moduli that extends the well-known Väisälä theorem for the families of curves to the families of surfaces of arbitrary dimensions. It should be emphasized that the crucial role in the proof of the mentioned lemma is played by a proposition on measurable (Borel) hulls of sets in Euclidean spaces. In addition, we also prove a similar lemma and a proposition for the families of concentric balls.

Suppose that R is a prime ring with char(R) ≠ 2 and f(ξ₁, . . . , ξ_n) is a noncentral multilinear polynomial over C(= Z(U)), where U is the Utumi quotient ring of R. An additive mapping h : R ⟶ R is called homoderivation if h(ab) = h(a)h(b)+h(a)b+ah(b) for all a, b ∈ R. We investigate the behavior of three generalized derivations F, G, and H of R satisfying the condition

(Fleft({xi }^{2}right)=Gleft({xi }^{2}right)+Hleft(xi right)xi +xi Hleft(xi right))

for all ξ ∈ f(R) = {f(ξ₁, . . . , ξ_n) | ξ₁, . . . , ξ_n ∈ R}.

We consider graphs whose Laplacian energy is equivalent to the Laplacian energy of the complete graph of the same order, which is called an L-borderenergetic graph. First, we study the graphs with degree sequence consisting of at most three distinct integers and give new bounds for the number of vertices of these graphs to be non-L-borderenergetic. Second, by using Koolen–Moulton and McClelland inequalities, we give new bounds for the number of edges of a non-L-borderenergetic graph. Third, we use recent bounds established by Milovanovic, et al. for the Laplacian energy to get similar conditions for non-L-borderenergetic graphs. Our bounds depend only on the degree sequence of a graph, which is much easier than computing the spectrum of the graph. In other words, we develop a faster approach to exclude non-L-borderenergetic graphs.

Let p = (p_j) and q = (q_k) be real sequences of nonnegative numbers with the property that

(begin{array}{ccccccc}{P}_{m}=sum_{j=0}^{m}{p}_{j}ne 0& {text{and}}& {Q}_{m}=sum_{k=0}^{n}{q}_{k}ne 0& mathrm{for all}& m& {text{and}}& n.end{array})

Also let (P_m) and (Q_n) be regularly varying positive indices. Assume that (u_mn) is a double sequence of complex (real) numbers, which is ( (overline{N }) , p, q; α, β)-summable and has a finite limit, where (α, β) = (1, 1), (1, 0), or (0, 1). We present some conditions imposed on the weights under which (u_mn) converges in Pringsheim’s sense. These results generalize and extend the results obtained by the authors in [Comput. Math. Appl., 62, No. 6, 2609–2615 (2011)].