Academic literature on the topic 'Polynomial functions : equivalence relations : symmetries'

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.

AbstractA three-dimensional Direct Numerical Simulation (DNS) database of statistically planar $$H_{2} -$$ H 2 - air turbulent premixed flames with an equivalence ratio of 0.7 spanning a large range of Karlovitz number has been utilised to assess the performances of the extrapolation relations, which approximate the stretch rate and curvature dependences of density-weighted displacement speed $$S_{d}^{*}$$ S d ∗ . It has been found that the correlation between $$S_{d}^{*}$$ S d ∗ and curvature remains negative and a significantly non-linear interrelation between $$S_{d}^{*}$$ S d ∗ and stretch rate has been observed for all cases considered here. Thus, an extrapolation relation, which assumes a linear stretch rate dependence of density-weighted displacement speed has been found to be inadequate. However, an alternative extrapolation relation, which assumes a linear curvature dependence of $$S_{d}^{*}$$ S d ∗ but allows for a non-linear stretch rate dependence of $$S_{d}^{*}$$ S d ∗ , has been found to be more successful in capturing local behaviour of the density-weighted displacement speed. The extrapolation relations, which express $$S_{d}^{*}$$ S d ∗ as non-linear functions of either curvature or stretch rate, have been found to capture qualitatively the non-linear curvature and stretch rate dependences of $$S_{d}^{*}$$ S d ∗ more satisfactorily than the linear extrapolation relations. However, the improvement comes at the cost of additional tuning parameter. The Markstein lengths LM for all the extrapolation relations show dependence on the choice of reaction progress variable definition and for some extrapolation relations LM also varies with the value of reaction progress variable. The predictions of an extrapolation relation which involve solving a non-linear equation in terms of stretch rate have been found to be sensitive to the initial guess value, whereas a high order polynomial-based extrapolation relation may lead to overshoots and undershoots. Thus, a recently proposed extrapolation relation based on the analysis of simple chemistry DNS data, which explicitly accounts for the non-linear curvature dependence of the combined reaction and normal diffusion components of $$S_{d}^{*}$$ S d ∗ , has been shown to exhibit promising predictions of $$S_{d}^{*}$$ S d ∗ for all cases considered here.