正电荷分布诱导对数电位的平衡点。I.广义de Bruijn-Springer关系
Equilibrium points of logarithmic potentials induced by positive charge distributions. I. Generalized de Bruijn-Springer relations
摘要：
本文定义了加权多元优序的概念，它是由原子概率测度的Choquet序所诱导的欧氏空间中向量序列的预序。我们用随机阵和凸函数刻画了这个预序，并用它来描述离散平面电荷组态产生的对数势的平衡点的分布.在$n$正电荷的情况下，我们证明了平衡点满足$\binom{n}{2}$加权优控制关系，并由$n-1$这种关系唯一确定.进一步证明了平衡点和带电粒子的Hausdorff几何是由后者的加权标准差控制的。利用紧正规Hilbert空间算子的有限秩扰动，我们建立了无穷电荷分布的类似关系。我们还讨论了一个层次的加权de Bruijn-Springer关系和惯性定律，存在的零Borel级数与积极的$l^1$ -系数，和经营者版本的Clunie-Eremenko-Rossi猜想。
英文摘要：
A notion of weighted multivariate majorization is defined as a preorder on sequences of vectors in Euclidean space induced by the Choquet ordering for atomic probability measures. We characterize this preorder both in terms of stochastic matrices and convex functions and use it to describe the distribution of equilibrium points of logarithmic potentials generated by discrete planar charge configurations. In the case of $n$ positive charges we prove that the equilibrium points satisfy $\binom{n}{2}$ weighted majorization relations and are uniquely determined by $n-1$ such relations. It is further shown that the Hausdorff geometry of the equilibrium points and the charged particles is controlled by the weighted standard deviation of the latter. By using finite-rank perturbations of compact normal Hilbert space operators we establish similar relations for infinite charge distributions. We also discuss a hierarchy of weighted de Bruijn-Springer relations and inertia laws, the existence of zeros of Borel series with positive $l^1$-coefficients, and an operator version of the Clunie-Eremenko-Rossi conjecture.
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