Let$D$⊂$C$be a simply connected domain that contains 0 and does not contain any disk of radius larger than 1. For$R$>0, let$ω$_{$D$}($R$) denote the harmonic measure at 0 of the set {$z$:|$z$|≽$R$}⋔∂$D$. Then it is shown that$there exist$β>0$and C$>0$such that for each such D$,$ω$_{$D$}($R$)≤$Ce$^{−$βR$},$for every R$>0. Thus a natural question is: What is the supremum of all β′s , call it β_{0}, for which the above inequality holds for every such$D$? Another formulation of the problem involves hyperbolic metric instead of harmonic measure. Using this formulation a lower bound for β_{0}is found. Upper bounds for β_{0}can be obtained by constructing examples of domains$D$. It is shown that a certain domain whose boundary consists of an infinite number of vertical half-lines, i.e. a comb domain, gives a good upper bound. This bound disproves a conjecture of C. Bishop which asserted that the strips of width 2 are extremal domains. Harmonic measures on comb domains are also studied.

Consider an age-dependent, single-species branching process defined by a progeny number distribution, and a lifetime distribution associated with each independent particle. In this paper, we focus on the associated inverse problem where one wishes to formally solve for the progeny number distribution or the lifetime distribution that defines the Bellman-Harris branching process. We derive results for the existence and uniqueness (the identifiability) of these two distributions given one of two types of information: the extinction time probability of the entire process (extinction time distribution), or the distribution of the total number of particles at one fixed time. We demonstrate that perfect knowledge of the distribution of extinction times allows us to formally determine either the progeny number distribution or the lifetime distribution. Furthermore, we show that these constructions are unique. We then consider “data” consisting of a perfectly known total number distribution given at one specific time. For a process with known progeny number distribution and exponentially distributed lifetimes, we show that the rate parameter is identifiable. For general lifetime distributions, we also show that the progeny distribution is globally unique. Our results are presented through four theorems, each describing the constructions in the four distinct cases.

In connection with the recent proposal for possible singularity formation at the boundary for solutions of three-dimensional axisymmetric incompressible Euler’s equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite-time blowup from smooth data.

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