Vol 59, No 6 (2018)

We calculate asymptotics for the Kolmogorov ε-entropy of the compact set of infinitely differentiable aperiodic functions embedded continuously into the space of continuous functions on a closed finite interval.

We define two scales of the mappings that depend on two real parameters p and q, with n−1 ≤ q ≤ p < ∞, as well as a weight function θ. The case q = p = n and θ ≡ 1 yields the well-known mappings with bounded distortion. The mappings of a two-index scale are applied to solve a series of problems of global analysis and applications. The main result of the article is the a.e. differentiability of mappings of two-index scales.

For constructing an enrichment of a language with restricted quantifiers, we extend the notion of term by the construction of the terms conditional and recursive with respect to lists. We show that the so-obtained extension of the language of formulas with restricted quantifiers over structures with hereditary finite lists is a conservative enrichment and the new terms are Δ-definable in the basic language.

Under study is the Cauchy problem for the fractional diffusion equation with a Caputo derivative. The existence and uniqueness theorems for a smooth solution are proven in a weighted H¨older space.

The recent articles of Arutyunov and Greshnov extend the Banach and Hadler Fixed-Point Theorems and the Arutyunov Coincidence-Point Theorem to the mappings of (q₁, q₂)-quasimetric spaces. This article addresses similar questions for f-quasimetric spaces.

Given a function f: R ₊² → R₊ with f(r₁, r₂) → 0 as (r₁, r₂) → (0, 0), an f-quasimetric space is a nonempty set X with a possibly asymmetric distance function ρ: X² → R+ satisfying the f-triangle inequality: ρ(x, z) ≤ f(ρ(x, y), ρ(y, z)) for x, y, z ∈ X. We extend the Banach Contraction Mapping Principle, as well as Krasnoselskii’s and Browder’s Theorems on generalized contractions, to mappings of f-quasimetric spaces.

We discuss the problem of existence of a smooth endomorphism of a closed n-dimensional manifold carrying a differential n-form into a prescribed volume form. Of course, we assume that the integrals of these forms over the whole manifold are equal. The solution of this problem for the n-dimensional torus reduces to the problem of small denominators well known in analysis.

Using the higher-order derivatives of Lyapunov functions, we obtain sufficient tests of various types for the uniform-ultimate boundedness in the sense of Poisson of solutions and various types of equiultimate boundedness of solutions in the sense of Poisson.

We introduce the notion of homological multiplicity for an oriented link in a thickened orientable closed surface. Using the notion, we establish some lower bounds for the crossing number of a link in thickened surfaces.