scholarly journals On Some Problems for a Simplex and a Ball in Rn

2018 ◽  
Vol 25 (6) ◽  
pp. 680-691
Author(s):  
Mikhail V. Nevskii
Keyword(s):  
Convex Body ◽  
Geometric Interpretation ◽  
Unit Cube ◽  
Linear Functions ◽  
Barycentric Coordinates ◽  
Regular Simplex ◽  
Euclidean Ball ◽  
Lagrange Polynomials ◽  
Dimensional Unit ◽  
Unit Euclidean Ball

Let \(C\) be a convex body and let \(S\) be a nondegenerate simplex in \({\mathbb R}^n\). Denote by \(\tau S\) the image of \(S\) under homothety with a center of homothety in the center of gravity of \(S\) and the ratio \(\tau\). We mean by \(\xi(C;S)\) the minimal \(\tau>0\) such that \(C\) is a subset of the simplex \(\tau S\). Define \(\alpha(C;S)\) as the minimal \(\tau>0\) such that \(C\) is contained in a translate of \(\tau S\). Earlier the author has proved the equalities \(\xi(C;S)=(n+1)\max\limits_{1\leq j\leq n+1}\max\limits_{x\in C}(-\lambda_j(x))+1\)  (if \(C\not\subset S\)), \(\alpha(C;S)=\sum\limits_{j=1}^{n+1} \max\limits_{x\in C} (-\lambda_j(x))+1.\)Here \(\lambda_j\) are the linear functions that are called the basic Lagrange polynomials corresponding to \(S\). The numbers \(\lambda_j(x),\ldots, \lambda_{n+1}(x)\) are the barycentric coordinates of a point \(x\in{\mathbb R}^n\). In his previous papers, the author investigated these formulae in the case when \(C\) is the \(n\)-dimensional unit cube \(Q_n=[0,1]^n\). The present paper is related to the case when \(C\) coincides with the unit Euclidean ball \(B_n=\{x: \|x\|\leq 1\},\) where \(\|x\|=\left(\sum\limits_{i=1}^n x_i^2 \right)^{1/2}.\) We establish various relations for \(\xi(B_n;S)\) and \(\alpha(B_n;S)\), as well as we give their geometric interpretation. For example, if \(\lambda_j(x)=l_{1j}x_1+\ldots+l_{nj}x_n+l_{n+1,j},\) then \(\alpha(B_n;S)=\sum\limits_{j=1}^{n+1}\left(\sum\limits_{i=1}^n l_{ij}^2\right)^{1/2}\). The minimal possible value of each characteristics \(\xi(B_n;S)\) and \(\alpha(B_n;S)\) for \(S\subset B_n\) is equal to \(n\). This value corresponds to a regular simplex inscribed into \(B_n\). Also we compare our results with those obtained in the case \(C=Q_n\).

scholarly journals Linear Interpolation on a Euclidean Ball in Rⁿ

2019 ◽  
Vol 26 (2) ◽  
pp. 279-296
Author(s):  
Mikhail V. Nevskii ◽  
Alexey Yu. Ukhalov
Keyword(s):  
Linear Interpolation ◽  
Continuous Functions ◽  
Linear Functions ◽  
Regular Simplex ◽  
Euclidean Ball ◽  
Integer Number ◽  
Space Of Continuous Functions ◽  
Lagrange Polynomials ◽  
Computational Data ◽  
Theoretical Results

For \(x^{(0)}\in{\mathbb R}^n, R>0\), by \(B=B(x^{(0)};R)\) we denote a Euclidean ball in \({\mathbb R}^n\) given by~the inequality \(\|x-x^{(0)}\|\leq R\), \(\|x\|:=\left(\sum_{i=1}^n x_i^2\right)^{1/2}\). Put \(B_n:=B(0,1)\). 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 \(\Pi_1\left({\mathbb R}^n\right)\) 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 \(P:C(B)\to \Pi_1({\mathbb R}^n)\) corresponding to \(S\) is defined by the equalities \(Pf\left(x^{(j)}\right)=%f_j:=f\left(x^{(j)}\right).\) Denote by \(\|P\|_B\) the norm of \(P\) as an operator from \(C(B)\) into \(C(B)\). Let us define \(\theta_n(B)\) as minimal value of \(\|P\|\) under the condition \(x^{(j)}\in B\). In the paper, we obtain the formula to compute \(\|P\|_B\) making use of \(x^{(0)}\), \(R\), and coefficients of basic Lagrange polynomials of \(S\). In more details we study the case when \(S\) is a regular simplex inscribed into \(B_n\). In this situation, we prove that \(\|P\|_{B_n}=\max\{\psi(a),\psi(a+1)\},\) where \(\psi(t)=\frac{2\sqrt{n}}{n+1}\bigl(t(n+1-t)\bigr)^{1/2}+\bigl|1-\frac{2t}{n+1}\bigr|\) \((0\leq t\leq n+1)\) and integer \(a\) has the form \(a=\bigl\lfloor\frac{n+1}{2}-\frac{\sqrt{n+1}}{2}\bigr\rfloor.\) For this projector, \(\sqrt{n}\leq\|P\|_{B_n}\leq\sqrt{n+1}\). The equality \(\|P\|_{B_n}=\sqrt{n+1}\) takes place if and only if \(\sqrt{n+1}\) is an integer number. We give the precise values of \(\theta_n(B_n)\) for \(1\leq n\leq 4\). To supplement theoretical results we present computational data. We also discuss some other questions concerning interpolation on a Euclidean ball.

scholarly journals On Properties of a Regular Simplex Inscribed into a Ball

2021 ◽  
Vol 28 (2) ◽  
pp. 186-197
Author(s):  
Mikhail Viktorovich Nevskii
Keyword(s):  
Uniform Norm ◽  
Linear Functions ◽  
Regular Simplex ◽  
Euclidean Ball ◽  
Minimal Norm ◽  
Lagrange Polynomials

Let  $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of continuos functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon ${\mathbb R}^n$. The interpolation projector  $P:C(B)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$,  $j=1,\ldots, n+1$.The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\|P\|_B=\max_{x\in B}\sum |\lambda_j(x)|.$ Here $\lambda_j$ are the basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having the nodes in the vertices of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum |\lambda_j(y)|$. Also we formulate some geometric conjecture which implies that $\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$.  We prove that this conjecture holds true at least for $n=1,2,3,4$. 

scholarly journals Geometric Estimates in Interpolation on an n-Dimensional Ball

2019 ◽  
Vol 26 (3) ◽  
pp. 441-449
Author(s):  
Mikhail V. Nevskii
Keyword(s):  
Legendre Polynomial ◽  
Polynomial Interpolation ◽  
Linear Interpolation ◽  
Continuous Functions ◽  
Unit Cube ◽  
Linear Functions ◽  
Dimensional Unit ◽  
Euclidean Unit Ball ◽  
Dimensional Ball ◽  
Geometric Estimates

Suppose \(n\in {\mathbb N}\). Let \(B_n\) be a Euclidean unit ball in \({\mathbb R}^n\) given by the inequality \(\|x\|\leq 1\), \(\|x\|:=\left(\sum\limits_{i=1}^n x_i^2\right)^{\frac{1}{2}}\). By \(C(B_n)\) we mean a set of continuous functions \(f:B_n\to{\mathbb R}\) with the norm \(\|f\|_{C(B_n)}:=\max\limits_{x\in B_n}|f(x)|\). The symbol \(\Pi_1\left({\mathbb R}^n\right)\) denotes a set of polynomials in \(n\) variables of degree \(\leq 1\), i.e. linear functions upon \({\mathbb R}^n\). Assume that \(x^{(1)}, \ldots, x^{(n+1)}\) are vertices of an \(n\)-dimensional nondegenerate simplex \(S\subset B_n\). The interpolation projector \(P:C(B_n)\to \Pi_1({\mathbb R}^n)\) corresponding to \(S\) is defined by the equalities \(Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).\) Denote by \(\|P\|_{B_n}\) the norm of \(P\) as an operator from \(C(B_n)\) on to \(C(B_n)\). Let us define \(\theta_n(B_n)\) as the minimal value of \(\|P\|_{B_n}\) under the condition \(x^{(j)}\in B_n\). We describe the approach in which the norm of the projector can be estimated from the bottom through the volume of the simplex. Let \(\chi_n(t):=\frac{1}{2^nn!}\left[ (t^2-1)^n \right] ^{(n)}\) be the standardized Legendre polynomial of degree \(n\). We prove that \(\|P\|_{B_n}\geq\chi_n^{-1}\left(\frac{vol(B_n)}{vol(S)}\right).\) From this, we obtain the equivalence \(\theta_n(B_n)\) \(\asymp\) \(\sqrt{n}\). Also we estimate the constants from such inequalities and give the comparison with the similar relations for linear interpolation upon the \(n\)-dimensional unit cube. These results have applications in polynomial interpolation and computational geometry.

scholarly journals On Optimal Interpolation by Linear Functions on an n-Dimensional Cube

2018 ◽  
Vol 25 (3) ◽  
pp. 291-311
Author(s):  
Mikhail V. Nevskii ◽  
Alexey Yu. Ukhalov
Keyword(s):  
Hadamard Matrix ◽  
Continuous Functions ◽  
Unit Cube ◽  
Linear Functions ◽  
Optimal Interpolation ◽  
Maximum Volume ◽  
Space Of Continuous Functions ◽  
Lagrange Polynomials ◽  
Geometric Character ◽  
Dimensional Cube

Let \(n\in{\mathbb N}\), and let \(Q_n\) be the unit cube \([0,1]^n\). By \(C(Q_n)\) we denote the space of continuous functions \(f:Q_n\to{\mathbb R}\) with the norm \(\|f\|_{C(Q_n)}:=\max\limits_{x\in Q_n}|f(x)|,\) by \(\Pi_1\left({\mathbb R}^n\right)\) --- the set of polynomials of \(n\) variables of degree \(\leq 1\) (or linear functions). Let \(x^{(j)},\) \(1\leq j\leq n+1,\) be the vertices of \(n\)-dimnsional nondegenerate simplex \(S\subset Q_n\). An interpolation projector \(P:C(Q_n)\to \Pi_1({\mathbb R}^n)\) corresponding to the simplex \(S\) is defined by equalities \(Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right).\) The norm of \(P\) as an operator from \(C(Q_n)\) to \(C(Q_n)\) may be calculated by the formula \(\|P\|=\max\limits_{x\in ver(Q_n)} \sum\limits_{j=1}^{n+1} |\lambda_j(x)|.\) Here \(\lambda_j\) are the basic Lagrange polynomials with respect to \(S,\) \(ver(Q_n)\) is the set of vertices of \(Q_n\). Let us denote by \(\theta_n\) the minimal possible value of \(\|P\|.\) Earlier, the first author proved various relations and estimates for values \(\|P\|\) and \(\theta_n\), in particular, having geometric character. The equivalence \(\theta_n\asymp \sqrt{n}\) takes place. For example, the appropriate, according to dimension \(n\), inequalities may be written in the form \linebreak \(\frac{1}{4}\sqrt{n}\) \(<\theta_n\) \(<3\sqrt{n}.\) If the nodes of the projector \(P^*\) coincide with vertices of an arbitrary simplex with maximum possible volume, we have \(\|P^*\|\asymp\theta_n.\)When an Hadamard matrix of order \(n+1\) exists, holds \(\theta_n\leq\sqrt{n+1}.\) In the paper, we give more precise upper bounds of numbers \(\theta_n\) for \(21\leq n \leq 26\). These estimates were obtained with the application of maximum volume simplices in the cube. For constructing such simplices, we utilize maximum determinants containing the elements \(\pm 1.\) Also, we systematize and comment the best nowaday upper and low estimates of numbers \(\theta_n\) for a concrete \(n.\)

scholarly journals The number of lattice rules having given invariants

1992 ◽  
Vol 46 (3) ◽  
pp. 479-495 ◽  
Author(s):  
Stephen Joe ◽  
David C. Hunt
Keyword(s):  
Normal Form ◽  
Quadrature Rule ◽  
Unit Cube ◽  
Numerical Results ◽  
Smith Normal Form ◽  
Lattice Rules ◽  
Dimensional Unit ◽  
Lattice Rule ◽  
Positive Integers ◽  
Theoretical Results

A lattice rule is a quadrature rule used for the approximation of integrals over the s-dimensional unit cube. Every lattice rule may be characterised by an integer r called the rank of the rule and a set of r positive integers called the invariants. By exploiting the group-theoretic structure of lattice rules we determine the number of distinct lattice rules having given invariants. Some numerical results supporting the theoretical results are included. These numerical results are obtained by calculating the Smith normal form of certain integer matrices.

scholarly journals Bounds for the Average $L^p$-Extreme and the $L^\infty$-Extreme Discrepancy

10.37236/1951 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Michael Gnewuch
Keyword(s):  
Upper Bounds ◽  
Unit Cube ◽  
Point Sets ◽  
Dimensional Unit ◽  
Axis Parallel ◽  
Geometric Discrepancy

The extreme or unanchored discrepancy is the geometric discrepancy of point sets in the $d$-dimensional unit cube with respect to the set system of axis-parallel boxes. For $2\leq p < \infty$ we provide upper bounds for the average $L^p$-extreme discrepancy. With these bounds we are able to derive upper bounds for the inverse of the $L^\infty$-extreme discrepancy with optimal dependence on the dimension $d$ and explicitly given constants.

Hamiltonian Circuits and Paths on the n-Cube

1966 ◽  
Vol 9 (05) ◽  
pp. 557-562 ◽  
Author(s):  
H. L. Abbott
Keyword(s):  
Unit Cube ◽  
Dimensional Unit ◽  
Hamiltonian Circuits

For positive integral n let Cn denote the n-dimensional unit cube with vertices (δ1, δ2,…, δn) where δi = 0 or 1 for i=1, 2,…, n. Call two vertices of Cn adjacent if the distance between them is 1.

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