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We give an in-depth analysis of a 1-parameter family of electrified droplets first described in D. Khavinson et. al. (2005). We also investigate a technique for searching for new solutions to the droplet equation, and rederive via this technique a 1-parameter family of physical droplets, which were first discovered by D. Crowdy (1999). We speculate on extensions of these solutions, in particular to the case of a droplet with multiple connected components.
This is an auxiliary note to [12]. To be precise, here we have gathered the proofs of all the statements in [12, Section 5] that happen to have points of contact with techniques recently developed in Chousionis-Pratt [5] and Chunaev [6].
We obtain a dimensional reduction result for the law of a class of stochastic differential equations using a supersymmetric representation first introduced by Parisi and Sourlas.
We study the Regge and hard scattering limits of the one-loop amplitude for massless open string states in the type I theory. For hard scattering we find the exact coefficient multiplying the known exponential falloff in terms of the scattering angle
We prove the existence of a roof function for arclength null quadrature domains having finitely many boundary components. This bridges a gap toward classification of arclength null quadrature domains by removing an a priori assumption from previous classification results.
Expressions for the summation of a new series involving the Laguerre polynomials are obtained in terms of generalized hypergeometric functions. These results provide alternative, and in some cases simpler, expressions to those recently obtained in the literature.