Ukrainian Mathematical Journal最新文献

We apply the technique of matching functions in the setting of contact Anosov flows satisfying a bunching assumption. This allows us to generalize the 3-dimensional rigidity result of Feldman and Ornstein [Ergodic Theory Dynam. Syst., 7, No. 1, 49–72 (1987)]. Namely, we show that if two Anosov flow of this kind are C⁰ conjugate, then they are C^r conjugate for some r ∈ [1, 2) or even C^∞ conjugate under certain additional assumptions. This, e.g., applies to geodesic flows on compact Riemannian manifolds of 1/4-pinched negative sectional curvature. We can also use our result to recover Hamendstädt’s marked length spectrum rigidity result for real hyperbolic manifolds.

We develop new Hermite–Hadamard-type integral inequalities for p-convex functions in the context of q-calculus by using the concept of recently defined T_q-integrals. Then the obtained Hermite–Hadamard inequality for p-convex functions is used to get a new Hermite–Hadamard inequality for coordinated p-convex functions. Furthermore, we present some examples to demonstrate the validity of our main results. We hope that the proposed ideas and techniques may stimulate further research in this field.

We provide a cohomology of n-Hom–Lie color algebras, in particular, a cohomology governing oneparameter formal deformations. Then we also study formal deformations of the n-Hom–Lie color algebras and introduce the notion of Nijenhuis operator on an n-Hom–Lie color algebra, which may give rise to infinitesimally trivial (n − 1)th-order deformations. Furthermore, in connection with Nijenhuis operators, we introduce and discuss the notion of product structure on n-Hom–Lie color algebras.

We revisit Schmidt’s theorem connecting the Schmidt rank of a tensor with the codimension of a certain variety and adapt the proof to the case of arbitrary characteristic. We also establish a sharper result for this kind for homogeneous polynomials, assuming that the characteristic does not divide the degree. Further, we use this to relate the Schmidt rank of a homogeneous polynomial (resp., a collection of homogeneous polynomials of the same degree) with the codimension of the singular locus of the corresponding hypersurface (resp., intersection of hypersurfaces). This gives an effective version of Ananyan–Hochster’s theorem [J. Amer. Math. Soc., 33, No. 1, 291–309 (2020), Theorem A].

We establish necessary and sufficient conditions for the convergence of the Baum–Katz series for the sums of elements of linear mth order autoregressive sequences of random variables.

We consider the problem of characterization of topological spaces embedded into countably compact Hausdorff topological spaces. We study the separation axioms for subspaces of Hausdorff countably compact topological spaces and construct an example of a regular separable scattered topological space that cannot be embedded into an Urysohn countably compact topological space.

Let 𝕂 be an algebraically closed field of characteristic zero, let 𝕂[x₁,…,x_n] be the polynomial algebra, and let W_n(𝕂) be the Lie algebra of all 𝕂-derivations on 𝕂[x₁,…,x_n]. For any derivation D with linear components, we describe the centralizer of D in W_n(𝕂) and propose an algorithm for finding the generators of this centralizer regarded as a module over the ring of constants of the derivation D in the case where D is a basic Weitzenböck derivation. In a more general case where a finitely generated integral domain A over the field 𝕂 is considered instead of the polynomial algebra 𝕂[x₁,…,x_n] and D is a locally nilpotent derivation on A, we prove that the centralizer C_DerA(D) of D in the Lie algebra DerA of all 𝕂-derivations on A is a “large” subalgebra of Der A. Specifically, the rank of C_DerA(D) over A is equal to the transcendence degree of the field of fractions Frac(A) over the field 𝕂.

Based on the Steklov operator, we consider a modulus of smoothness for functions in some Banach function spaces, which can be not translation invariant, and establish its main properties. A constructive characterization of the Lipschitz class is obtained with the help of the Jackson-type direct theorem and the inverse theorem on trigonometric approximation. As an application, we present several examples of related (weighted) function spaces.

For a nonlinear autonomous boundary-value problem posed for an ordinary differential equation in the critical case, we establish constructive conditions for its solvability and propose a scheme for the construction of solutions based on the use of Adomian’s decomposition method.

We extend the concept of generalized weakly demicompact and relatively weakly demicompact operators on linear relations and present some outstanding results. Moreover, we address the theory of Fredholm and upper semi-Fredholm relations and make an attempt to establish connections with these operators.