Electrostatic equilibria on the unit circle via Jacobi polynomials

@article{Johnson2020ElectrostaticEO,
  title={Electrostatic equilibria on the unit circle via Jacobi polynomials},
  author={Krista E Johnson and Brian Simanek},
  journal={arXiv: Mathematical Physics},
  year={2020}
}
We use classical Jacobi polynomials to identify the equilibrium configurations of charged particles confined to the unit circle. Our main result unifies two theorems from a 1986 paper of Forrester and Rogers. 

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References

SHOWING 1-10 OF 14 REFERENCES
An electrostatics model for zeros of general orthogonal polynomials
We prove that the zeros of general orthogonal polynomials, subject to certain integrability conditions on their weight functions determine the equilibrium position of movable n unit charges in an
Electrostatics, Hyperbolic Geometry and Wandering Vectors
A family of planar discrete electrostatic systems on the unit circle with finitely atomic external fields is considered. The geometry of particles in the external field yielding a given minimum
Electrostatics and the zeros of the classical polynomials
New interpretations of the zeros of the classical polynomials as the equilibrium positions of two-dimensional electrostatic problems are given. The electrostatic problems solved include determining
A minimum energy problem and Dirichlet spaces
We analyze a minimum energy problem for a discrete electrostatic model in the complex plane and discuss some applications. A natural characteristic distinguishing the state of minimum energy from
An Electrostatic Interpretation of the Zeros of Paraorthogonal Polynomials on the Unit Circle
TLDR
It is shown that the zeros of every paraorthogonal polynomial mark the locations of a set of particles that are in electrostatic equilibrium with respect to a particular external field.
Orthogonal Polynomials
In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.
Convexity: An Analytic Viewpoint
Convexity is important in theoretical aspects of mathematics and also for economists and physicists. In this monograph the author provides a comprehensive insight into convex sets and functions
‘A’
  • P. Alam
  • Medicine
    Composites Engineering: An A–Z Guide
  • 2021
TLDR
A fluorescence-imaging-based endoscopic capsule that automates the detection process of colorectal cancer was designed and developed in the lab and offered great possibilities for future applicability in selective and specific detection of other fluorescently labelled cancers.
Sur certains polynômes
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