论文标题
超管中的可见性现象
Visibility phenomena in hypercubes
论文作者
论文摘要
我们研究了多维超振管中的一组可见晶格点。我们研究的问题将几何，概率和数字理论色调混合在一起。例如，我们证明，几乎所有具有顶点的自我可见三角形都在$ \ mathcal w = [0，n]^d $中的整数坐标的晶格中几乎等同于等级几乎等于$ \ sqrt {d} n/\ sqrt {d} n/\ sqrt {6} $，以及来自典型的典型范围。 $ \ Mathcal w $在极限上等于$ \ sqrt {7}/4 $，为$ d $和$ n/d $倾向于无限。我们还表明，存在一个有趣的数字理论常数$λ_{d，k} $，这是$ k $ - polytope在晶格$ \ mathcal w $中具有顶点的机会的限制概率。
We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic tones. For example, we prove that almost all self-visible triangles with vertices in the lattice of points with integer coordinates in $\mathcal W=[0,N]^d$ are almost equilateral having all sides almost equal to $\sqrt{d}N/\sqrt{6}$, and the sine of the typical angle between rays from the visual spectra from the origin of $\mathcal W$ is, in the limit, equal to $\sqrt{7}/4$, as $d$ and $N/d$ tend to infinity. We also show that there exists an interesting number theoretic constant $Λ_{d,K}$, which is the limit probability of the chance that a $K$-polytope with vertices in the lattice $\mathcal W$ has all vertices visible from each other.
