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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].
Let $X, Y$ be two independent identically distributed (i.i.d.) random variables taking values from a separable Banach space $(mathcal{X}, |cdot|)$. Given two measurable subsets $F, Ksubseteqcal{X}$, we established distribution free comparison inequal
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-
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.
We prove that for any given upper semicontinuous function $varphi$ on an open subset $E$ of $mathbb C^nsetminus{0}$, such that the complex cone generated by $E$ minus the origin is connected, the homogeneous Siciak-Zaharyuta function with the weight