On Equilibrium Points of Logarithmic and Newtonian Potentials

Clunie, J. G.; Erëmenko, Alexandre; Rossi, John

doi:10.1112/jlms/s2-47.2.309

Cited by 40 publications

(39 citation statements)

References 7 publications

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“…As a second application of Theorem 1 we give a unified proof of the following results recently obtained by J. Clunie, J. Langley, J. Rossi, and the second author [6,8].…”

Section: Corollarymentioning

confidence: 99%

On the singularities of the inverse to a meromorphic function of finite order

Bergweiler¹,

Erëmenko²

1995

Rev. Mat. Iberoam.

328

257

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Our main result implies the following theorem: Let f be a transcendental meromorphic function in the complex plane. If f has finite order ρ, then every asymptotic value of f , except at most 2ρ of them, is a limit point of critical values of f .We give several applications of this theorem. For example we prove that if f is a transcendental meromorphic function then f f n with n ≥ 1 takes every finite non-zero value infinitely often. This proves a conjecture of Hayman. The proof makes use of the iteration theory of meromorphic functions.

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“…As a second application of Theorem 1 we give a unified proof of the following results recently obtained by J. Clunie, J. Langley, J. Rossi, and the second author [6,8].…”

Section: Corollarymentioning

confidence: 99%

On the singularities of the inverse to a meromorphic function of finite order

Bergweiler¹,

Erëmenko²

1995

Rev. Mat. Iberoam.

328

257

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“…Various results concerning the existence of equilibrium points for Newtonian and logarithmic potentials have been obtained in, e.g., [8], [14] and [24]. It should be emphasized though that these deal almost exclusively with unbounded discrete charge configurations.…”

Section: Operator Versions Of the Clunie-eremenko-rossi Conjecturementioning

confidence: 99%

“…It should be emphasized though that these deal almost exclusively with unbounded discrete charge configurations. Note that Conjecture 2 may actually be viewed as a natural analog for bounded discrete charge configurations of the following well-known conjecture of Clunie-Eremenko-Rossi; see, e.g., [8 A catchy albeit somewhat loose rephrasing of this conjecture is as follows: every flat universe has infinitely many resting points.…”

Section: Operator Versions Of the Clunie-eremenko-rossi Conjecturementioning

confidence: 99%

“…To begin with, in this case the fundamental question dealing with the existence of equilibrium points is already quite delicate and, as a matter of fact, still an open problem (cf. [8,14,24]). …”

Section: Theorem 4 In the Above Notation One Has H(w Z) ≤ H(w E Zmentioning

confidence: 99%

“…In this way we arrive at a great many related questions on the interplay between value distribution of meromorphic functions and variations of discrete spectra of compact normal Hilbert space operators under finite-rank perturbations. In §6-7 we give an operator theoretic approach to the well-known Clunie-Eremenko-Rossi conjecture [8] dealing with the existence of zeros of certain Borel series and we consider possible extensions of this conjecture as well as their potential implications (Conjectures 4-6).…”

Section: Introductionmentioning

confidence: 99%

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

Borcea

2007

Trans. Amer. Math. Soc.

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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 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 finiterank 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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Localization of Zeros in Cauchy–de Branges Spaces

Abakumov¹,

Baranov

Belov

2019

Trends in Mathematics

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We study the class of discrete measures in the complex plain with the following property: up to a finite number, all zeros of any Cauchy transform of the measure (with ℓ 2 -data) are localized near the support of the measure. We find several equivalent forms of this property and prove that the parts of the support attracting zeros of Cauchy transforms are ordered by inclusion modulo finite sets.

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On Equilibrium Points of Logarithmic and Newtonian Potentials

Cited by 40 publications

References 7 publications

On the singularities of the inverse to a meromorphic function of finite order

On the singularities of the inverse to a meromorphic function of finite order

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

Localization of Zeros in Cauchy–de Branges Spaces

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