arXiv subjects

Alexey Ukhalov

Publications and source records attributed to Alexey Ukhalov.

5 recordsLinked to original sources

arXiv2023

On a Geometric Approach to the Estimation of Interpolation Projectors

Suppose $\Omega$ is a closed bounded subset of ${\mathbb R}^n,$ $S$ is an $n$-dimensional non-degenerate simplex, $\xi(\Omega;S):=\min \left\{\sigma\geq 1: \, \Omega\subset \sigma S\right\}$. Here $\sigma S$ is the result of homothety of $S$ with respect to the center of gravity with coefficient $\sigma$. Let $d\geq n+1,$ $\varphi_1(x),\ldots,\varphi_d(x)$ be linearly independent monomials in $n$ variables, $\varphi_1(x)\equiv 1,$ $\varphi_2(x)=x_1,\ \ldots, \ \varphi_{n+1}(x)=x_n.$ Put $\Pi:={\rm lin}(\varphi_1,\ldots,\varphi_d).$ The interpolation projector $P: C(\Omega)\to \Pi$ with a set of nodes $x^{(1)},\ldots, x^{(d)}$ $ \in \Omega$ is defined by equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ Denote by $\|P\|_{\Omega}$ the norm of $P$ as an operator from $C(\Omega)$ to $C(\Omega)$. Consider the mapping $T:{\mathbb R}^n\to {\mathbb R}^{d-1}$ of the form $T(x):=(\varphi_2(x),\ldots,\varphi_d(x)). $ We have the following inequalities: $ \frac{1}{2}\left(1+\frac{1}{d-1}\right)\left(\|P\|_{\Omega}-1\right)+1$ $ \leq \xi(T(\Omega);S)\leq \frac{d}{2}\left(\|P\|_{\Omega}-1\right)+1. $ Here $S$ is the $(d-1)$-dimensional simplex with vertices $T\left(x^{(j)}\right).$ We discuss this and other relations for polynomial interpolation of functions continuous on a segment. The results of numerical analysis are presented.

math.MG↗

arXiv2021

Estimates for Interpolation Projectors and Related Problems in Computational Geometry

This paper contains a survey of results obtained by the authors mostly during the past few years and published by 2021. In particular, we present the best of known estimates of numerical characteristics related to the research theme. Sections: 1. Introduction. 2. The case when $n+1$ is an Hadamard number. 3. Estimates for the minimal absorption index of a cube by a simplex. 4. Estimates for the minimal norm of a projector in linear interpolation on a cube in ${\mathbb R}^n$. 5. Estimates of numbers $ξ_n^\prime$ and $θ_n^\prime$. 6. Simplices satisfying the inclusions $S\subset Q_n\subset nS$. 7. Perfect simplices. 8. Equisecting simplices. 9. Properties of $(0,1)$-matrices of order $n$ having maximal determinant. 10. Problems for a simplex and a Euclidean ball. 11. Linear interpolation on a Euclidean ball. Bibliography: 56 titles. Keywords: simplex, cube, Euclidean ball, homothety, axial diameter, absorption index, Hadamard number, interpolation, projector, norm, estimate.

math.MG↗

arXiv2019

Interpolation by Linear Functions on an $n$-Dimensional Ball

By $B=B(x^{(0)};R)$ we denote the Euclidean ball in ${\mathbb R}^n$ given by the inequality $\|x-x^{(0)}\|\leq R$. Here $x^{(0)}\in{\mathbb R}^n, R>0$, $\|x\|:=\left(\sum_{i=1}^n x_i^2\right)^{1/2}$. We mean by $C(B)$ the space of continuous functions $f:B\to{\mathbb R}$ with the norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|$ and by $Π_1\left({\mathbb R}^n\right)$ the set of polynomials in $n$ variables of degree $\leq 1$, i.e., linear functions on ${\mathbb R}^n$. Let $x^{(1)}, \ldots, x^{(n+1)}$ be the vertices of $n$-dimensional nondegenerate simplex $S\subset B$. The interpolation projector $P:C(B)\to Π_1({\mathbb R}^n)$ corresponding to $S$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ We obtain the formula to compute the norm of $P$ as an operator from $C(B)$ into $C(B)$ via $x^{(0)}$, $R$ and coefficients of basic Lagrange polynomials of $S$. In more details we study the case when $S$ is a regular simplex inscribed into $B_n=B(0,1)$.

math.MG↗

arXiv2018

Properties of 0/1-Matrices of Order n Having Maximum Determinant

We give some necessary conditions for maximality of $0/1$-determinant. Let ${\bf M}$ be a nondegenerate $0/1$-matrix of order $n$. Denote by $\bf A$ the matrix of order $n+1$ which appears from ${\bf M}$ after adding the $(n+1)$th row $(0,0,\ldots,0,1)$ and the $(n+1)$th column consisting of $1$'s. Suppose ${\bf A}^{-1}=(l_{ij}),$ then for all $i=1,\ldots,n$ we have $\sum_{j=1}^{n+1} |l_{ij}|\geq 2.$ Moreover, if $|\det({\bf M})|$ is equal to the maximum value of a $0/1$-determinant of order $n$, then $\sum_{j=1}^{n+1} |l_{ij}|= 2$ for all $i=1,\ldots,n$. Keywords: maximum 0/1-deteminant, simplex, cube, axial diameter

math.MG↗

arXiv2017

Five-dimensional Perfect Simplices

Let $Q_n=[0,1]^n$ be the unit cube in ${\mathbb R}^n$, $n \in {\mathbb N}$. For a nondegenerate simplex $S\subset{\mathbb R}^n$, consider the value $ξ(S)=\min \{σ>0: Q_n\subset σS\}$. Here $σS$ is a homothetic image of $S$ with homothety center at the center of gravity of $S$ and coefficient of homothety $σ$. Let us introduce the value $ξ_n=\min \{ξ(S): S\subset Q_n\}$. We call $S$ a perfect simplex if $S\subset Q_n$ and $Q_n$ is inscribed into the simplex $ξ_n S$. It is known that such simplices exist for $n=1$ and $n=3$. The exact values of $ξ_n$ are known for $n=2$ and in the case when there exist an Hadamard matrix of order $n+1$, in the latter situation $ξ_n=n$. In this paper we show that $ξ_5=5$ and $ξ_9=9$. We also describe infinite families of simplices $S\subset Q_n$ such that $ξ(S)=ξ_n$ for $n=5,7,9$. The main result of the paper is the existence of perfect simplices in ${\mathbb R}^5$. Keywords: simplex, cube, homothety, axial diameter, Hadamard matrix

math.MG↗