In this paper we outline some recent results concerning the existence of steady solutions to the Euler equation in ${ℝ}^3$ with a prescribed set of (possibly knotted and linked) thin vortex tubes.

Dati due elementi $f$ e $g$ in un'algebra uniforme $A$, sia $G=f(M_A/f({\partial }_A)$. Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione $\lambda \to \log \max g(f^{-1}(\lambda ))$ è subarmonica su $G$ e che l’applicazione $\lambda \to g(f^{-1}(\lambda ))$ è analitica nel senso di Oka.

Let a < 0, Ω = ℂ -(-∞, a] and U = z: |z| < 1. We consider the class $S_H(U,\Omega )$ of functions f which are univalent, harmonic and sense preserving with f(U) = Ω and satisfy f(0) = 0, $f_z\left(0\right)>0$ and $f_{z̅}\left(0\right)=0$. We describe the closure $\overline{S_H(U,\Omega )}$ of $S_H(U,\Omega )$ and determine the extreme points of $\overline{S_H(U,\Omega )}$.

Let a < 0 < b and Ω(a,b) = ℂ - ((-∞, a] ∪ [b,+∞)) and U= z: |z| < 1. We consider the class $S_H(U,\Omega (a,b))$ of functions f which are univalent, harmonic and sense-preserving with f(U) = Ω and satisfying f(0) = 0, $f_z\left(0\right)>0$ and $f_z̅\left(0\right)=0$.

A holomorphic function $f$ on a simply connected domain $\Omega$ is said to possess a universal Taylor series about a point in $\Omega$ if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta $K$ outside $\Omega$ (provided only that $K$ has connected complement). This paper shows that this property is not conformally invariant, and, in the case where $\Omega$ is the unit disc, that such functions have extreme angular boundary behaviour.

We present a number of Wiener’s type necessary and sufficient conditions (in terms of divergence of integrals or series involving a condenser capacity) for a compact set E ⊂ ℂ to be regular with respect to the Dirichlet problem. The same capacity is used to give a simple proof of the following known theorem [2, 6]: If E is a compact subset of ℂ such that $d\left(t^{-1}E\cap |z-a|\le 1\right)\ge const>0$ for 0 < t ≤ 1 and a ∈ E, where d(F) is the logarithmic capacity of F, then the Green function of ℂ E with pole at infinity is Hölder continuous....