A well known result of Beurling asserts that if f is a function which is analytic in the unit disc $\Delta =z\in ℂ:\left|z\right|<1$ and if either f is univalent or f has a finite Dirichlet integral then the set of points $e^{i\theta }$ for which the radial variation $V(f,e^{i\theta })={\int }_0^1\left|f^{\text{'}}\left(r{e}^{i\theta }\right)\right|dr$ is infinite is a set of logarithmic capacity zero. In this paper we prove that this result is sharp in a very strong sense. Also, we prove that if f is as above then the set of points $e^{i\theta }$ such that $(1-r)|f^{\text{'}}\left(r{e}^{i\theta }\right)|\ne o\left(1\right)$ as r → 1 is a set of logarithmic capacity zero. In particular, our results give...

Let f(z) be a conformal mapping of an annulus A(R) = 1 < |z| < R and let f(A(R)) be a ring domain bounded by a circle and a k-circle. If R(φ) = w : arg w = φ, and l(φ) - 1 is the linear measure of f(A(R)) ∩ R(φ), then we determine the sharp lower bound of $l\left({\phi }_1\right)+l\left({\phi }_2\right)$ for fixed ${\phi }_1$ and ${\phi }_2$$(0\le ...$

We study the ramification of the Gauss map of complete minimal surfaces in ${ℝ}^m$ on annular ends. This is a continuation of previous work of Dethloff-Ha (2014), which we extend here to targets of higher dimension.

Let a sequence $\left\{{\lambda }_n\right\}\subset ℝ$ be given such that the exponential system $\left\{\exp \right(i{\lambda }_{n}x\left)\right\}$ forms a Riesz basis in $L^2(-\pi ,\pi )$ and $\left\{{\xi }_n\right\}$ be a sequence of independent real-valued random variables. We study the properties of the system $\{exp(i({\lambda }_n+{\xi }_n)x\left)\right\}$ as well as related problems on estimation of entire functions with random zeroes and also problems on reconstruction of bandlimited signals with bandwidth $2\pi$ via their samples at the random points $\{{\lambda }_n+{\xi }_n\}$.

For certain ensembles of random polynomials we give the expected value of the zero distribution (in one variable) and the expected value of the distribution of common zeros of m polynomials (in m variables).

Spatially homogeneous random walks in ${\left({\mathbb{Z}}_{+}\right)}^2$ with non-zero jump probabilities at distance at most $1$, with non-zero drift in the interior of the quadrant and absorbed when reaching the axes are studied. Absorption probabilities generating functions are obtained and the asymptotic of absorption probabilities along the axes is made explicit. The asymptotic of the Green functions is computed along all different infinite paths of states, in particular along those approaching the axes.