On Numerical Characteristics of а Simplex and their Estimates

Abstract

Let \(n\in {\mathbb N}\), and let \(Q_n=[0,1]^n\) be the \(n\)-dimensional unit cube. For a nondegenerate simplex \(S\subset {\mathbb R}^n\), by \(\sigma S\) we denote the homothetic image of \(S\) with the center of homothety in the center of gravity of S and the ratio of homothety \(\sigma\). We apply the following numerical characteristics of the simplex. Denote by \(\xi(S)\) the minimal \(\sigma>0\) with the property \(Q_n\subset \sigma S\). By \(\alpha(S)\) we denote the minimal \(\sigma>0\) such that \(Q_n\) is contained in a translate of a simplex \(\sigma S\). By \(d_i(S)\) we mean the \(i\)th axial diameter of \(S\), i.\,e. the maximum length of a segment contained in \(S\) and parallel to the \(i\)th coordinate axis. We apply the computational formulae for \(\xi(S)\), \(\alpha(S)\), \(d_i(S)\) which have been proved by the first author. In the paper we discuss the case \(S\subset Q_n\). Let \(\xi_n=\min\{ \xi(S): S\subset Q_n\}. \) Earlier the first author formulated the conjecture: {\it if \(\xi(S)=\xi_n\), then \(\alpha(S)=\xi(S)\).} He proved this statement for \(n=2\) and the case when \(n+1\) is an Hadamard number, i.\,e. there exists an Hadamard matrix of order \(n+1\). The following conjecture is a stronger proposition: {\it for each \(n\), there exist \(\gamma\geq 1\), not depending on \(S\subset Q_n\), such that \(\xi(S)-\alpha(S)\leq \gamma (\xi(S)-\xi_n).\)} By \(\varkappa_n\) we denote the minimal \(\gamma\) with such a property. If \(n+1\) is an Hadamard number, then the precise value of \(\varkappa_n\) is 1. The existence of \(\varkappa_n\) for other \(n\) was unclear. In this paper with the use of computer methods we obtain an equality $$\varkappa_2 = \frac{5+2\sqrt{5}}{3}=3.1573\ldots $$ Also we prove a new estimate $$\xi_4\leq \frac{19+5\sqrt{13}}{9}=4.1141\ldots,$$ which improves the earlier result \(\xi_4\leq \frac{13}{3}=4.33\ldots\) Our conjecture is that \(\xi_4\) is precisely \(\frac{19+5\sqrt{13}}{9}\). Applying this value in numerical computations we achive the value $$\varkappa_4 = \frac{4+\sqrt{13}}{5}=1.5211\ldots$$ Denote by \(\theta_n\) the minimal norm of interpolation projection on the space of linear functions of \(n\) variables as an operator from \(C(Q_n)\) in \(C(Q_n)\). It is known that, for each \(n\), $$\xi_n\leq \frac{n+1}{2}\left(\theta_n-1\right)+1,$$ and for \(n=1,2,3,7\) here we have an equality. Using computer methods we obtain the result \(\theta_4=\frac{7}{3}\). Hence, the minimal \(n\) such that the above inequality has a strong form is equal to 4. %, a principal architecture of common purpose CPU and its main components are discussed, CPUs evolution is considered and drawbacks that prevent future CPU development are mentioned. Further, solutions proposed so far are addressed and new CPU architecture is introduced. The proposed architecture is based on wireless cache access that enables reliable interaction between cores in multicore CPUs using terahertz band, 0.1-10THz. The presented architecture addresses the scalability problem of existing processors and may potentially allow to scale them to tens of cores. As in-depth analysis of the applicability of suggested architecture requires accurate prediction of traffic in current and next generations of processors we then consider a set of approaches for traffic estimation in modern CPUs discussing their benefits and drawbacks. The authors identify traffic measurements using existing software tools as the most promising approach for traffic estimation, and use Intel Performance Counter Monitor for this purpose. Three types of CPU loads are considered including two artificial tests and background system load. For each load type the amount of data transmitted through the L2-L3 interface is reported for various input parameters including the number of active cores and their dependences on number of cores and operational frequency.