论文标题
某些常规球形构型电势的绝对最小值
Absolute Minima of Potentials of Certain Regular Spherical Configurations
论文作者
论文摘要
我们使用近似理论的方法在球形$ $（2M-3）$的潜力上找到绝对的最小值 - 设计具有非平地指数$ 200M $的设计，其中包含在$ M $平行的超平面型，$ M \ geq 2 $中，其位置满足某些其他假设。点之间的相互作用由DOT产品的函数描述，该函数的订单为200万美元，200万美元$和200万美元的订单衍生品。这包括经典的库仑，瑞斯和对数电势以及距离平方的完全单调电位。我们通过表明在单位球上的一组顶点的潜力的绝对最低限度来说明了这一结果。 $ e_8 $ $ e_8 $的$ e_8 $ root lattice的配置潜力的绝对最低限度是在$ \ mathbb r^8 $中固定在单位球上的$ s^7 $，以$ 2160 $的$ s^7 $上的$ 2160 $获得。
We use methods of approximation theory to find the absolute minima on the sphere of the potential of spherical $(2m-3)$-designs with a non-trivial index $2m$ that are contained in a union of $m$ parallel hyperplanes, $m\geq 2$, whose locations satisfy certain additional assumptions. The interaction between points is described by a function of the dot product, which has positive derivatives of orders $2m-2$, $2m-1$, and $2m$. This includes the case of the classical Coulomb, Riesz, and logarithmic potentials as well as a completely monotone potential of the distance squared. We illustrate this result by showing that the absolute minimum of the potential of the set of vertices of the icosahedron on the unit sphere $S^2$ in $\mathbb R^3$ is attained at the vertices of the dual dodecahedron and the one for the set of vertices of the dodecahedron is attained at the vertices of the dual icosahedron. The absolute minimum of the potential of the configuration of $240$ minimal vectors of $E_8$ root lattice normalized to lie on the unit sphere $S^7$ in $\mathbb R^8$ is attained at a set of $2160$ points on $S^7$ which we describe.
