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Vol 210, No 2 (2019)
- Year: 2019
- Articles: 5
- URL: https://journals.rcsi.science/0368-8666/issue/view/7450
Hadamard's theorem for mappings with relaxed smoothness conditions
Abstract
The paper puts forward sufficient conditions for a mapping from $\mathbb R^n$ to $\mathbb R^n$ to be a global homeomorphism. As an application, the Hadamard theorem for differentiable mappings and conditions for the existence and uniqueness of a coincidence point of a covering mapping and a Lipschitz mapping on $\mathbb R^n$ are derived. Covering mappings of metric spaces and mappings covering at a point are studied. Bibliography: 23 titles.
Matematicheskii Sbornik. 2019;210(2):3-23
3-23
A self-symmetric cycle in a system of two diffusely connected Hutchinson's equations
Abstract
The so-called bi-local model is considered for Hutchinson's equation. This is a system of two identical nonlinear delay differential equations connected by means of linear diffusion terms. The question of the existence, asymptotic behaviour and stability of a particular periodic solution of this system, such that a certain phase shift takes the coordinates of this solution back to this solution, are investigated. Bibliography: 19 titles.
Matematicheskii Sbornik. 2019;210(2):24-74
24-74
On the measure of inclusion in relatively free algebras with the identity of Lie nilpotency of degree 3 or 4
Abstract
This work is concerned with the concept of a graded subspace of the polylinear part of a relatively free algebra and with the measure of inclusion of such a subspace. Other asymptotic characteristics are also considered. In the case of relatively free algebras with the identity of Lie nilpotency of degree 3 and 4, the measure of inclusion is computed for many subspaces; in particular, for the centre and the $T$-space generated by the commutator this measure is $1/2$. Bibliography: 17 titles.
Matematicheskii Sbornik. 2019;210(2):75-86
75-86
Dissections of trapezoids into trapezoids homothetical to trapezoids in a given family
Abstract
We prove several theorems related to the dissection of trapezoids into trapezoids that are homothetical to given ones. We prove that, using homotheties of a trapezoid with rational ratio of bases, we can tile any trapezoid with rational ratio of bases and the same angles, and no other trapezoid can be tiled. We also consider trapezoids whose ratio of bases is a quadratic irrationality. For certain pairs of trapezoids we prove that their homotheties can tile any trapezoid with the same angles and the ratio of bases belonging to the same quadratic field. For some other class of trapezoids with a quadratic-irrational ratio of bases, we present a necessary condition on trapezoids that can be tiled with given ones. This condition is remarkable because it contains a transcendental function. This is the first occurrence of a transcendental function in problems of tiling polygons with similar polygons. Bibliography: 8 titles.
Matematicheskii Sbornik. 2019;210(2):87-114
87-114
Tauberian class estimates for vector-valued distributions
Abstract
We study Tauberian properties of regularizing transforms of vector-valued tempered distributions. The transforms have the form $M^\mathbf f_\varphi(x,y)=(\mathbf f\ast\varphi_y)(x)$, where the kernel $\varphi$ is a test function and $\varphi_y(\cdot)=y^{-n}\varphi(\cdot/y)$. We investigate conditions which ensure that a distribution that a priori takes values in a locally convex space actually takes values in a narrower Banach space. Our goal is to characterize spaces of Banach-space-valued tempered distributions in terms of so-called class estimates for the transform $M^\mathbf f_\varphi(x,y)$. Our results generalize and improve earlier Tauberian theorems due to Drozhzhinov and Zav'yalov. Special attention is paid to finding the optimal class of kernels $\varphi$ for which these Tauberian results hold. Bibliography: 24 titles.
Matematicheskii Sbornik. 2019;210(2):115-142
115-142