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Equilibrium points of logarithmic potentials induced by positive charge distributions. I. Generalized de Bruijn-Springer relations

Author: Julius Borcea
Journal: Trans. Amer. Math. Soc. 359 (2007), 3209-3237
MSC (2000): Primary 31A15; Secondary 30C15, 47A55, 60E15
Published electronically: January 19, 2007
MathSciNet review: 2299452
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Abstract: 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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Additional Information

Julius Borcea
Affiliation: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden

Keywords: Logarithmic potentials, electrostatic equilibrium, Hausdorff geometry, multivariate majorization, compressions of normal operators
Received by editor(s): April 29, 2005
Published electronically: January 19, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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