arXiv subjects
Publications and source records attributed to Marcelina Mocanu.
We prove several formulas for the Hilbert metric in the unit disk and apply these results to study quasiregular mappings of the unit disk $\mathbb{B}^2$ onto a bounded convex domain $D$. The main result deals with the H\"older continuity of these mappings with respect to Hilbert metrics of $\mathbb{B}^2$ and $D$. Also several open problems are formulated.
We study in the setting of a metric space $\left( X,d\right) $ some generalizations of four hyperbolic-type metrics defined on open sets $G$ with nonempty boundary in the $n-$dimensional Euclidean space, namely Gehring-Osgood metric, Dovgoshey- Hariri-Vuorinen metric, Nikolov-Andreev metric and Ibragimov metric. In the definitions of these generalizations, the boundary $\partial G$ of $G$ and the distance from a point $x$ of $G$ to $\partial G$ are replaced by a nonempty proper closed subset $M$ of $X$ and by a $1-$Lipschitz function positive on $X\setminus M$, respectively. For each generalization $\rho $ of the hyperbolic-type metrics mentioned above we prove that $\left( X\setminus M,\rho \right) $ is a Gromov hyperbolic space and that the identity map between $\left( X\setminus M,d\right) $ and $% \left( X\setminus M,\rho \right) $ is quasiconformal. For the Gehring-Osgood metric and the Nikolov-Andreev metric we improve the Gromov constants known from the literature. For Ibragimov metric the Gromov hyperbolicity is obtained even if we replace the distance from a point $x$ to $\partial G$ by any positive function on $X\setminus M$
We discuss the problem of the reflection of light on spherical and quadric surface mirrors. In the case of spherical mirrors, this problem is known as the Alhazen problem. For the spherical mirror problem, we focus on the reflection property of an ellipse, and show that the catacaustic curve of the unit circle follows naturally from the equation obtained from the reflection property of an ellipse. Moreover, we provide an algebraic equation that solves Alhazen's problem for quadric surface mirrors.
We introduce a new intrinsic metric in subdomains of a metric space and give upper and lower bounds for it in terms of well-known metrics. We also prove distortion results for this metric under quasiregular maps.
Answering a question about triangle inequality suggested by R. Li, A. Barrlund introduced a distance function which is a metric on a subdomain of ${\mathbb R}^n\,.$ We study this Barrlund metric and give sharp bounds for it in terms of other metrics of current interest. We also prove sharp distortion results for the Barrlund metric under quasiconformal maps.
An ancient optics problem of Ptolemy, studied later by Alhazen, is discussed. This problem deals with reflection of light in spherical mirrors. Mathematically this reduces to the solution of a quartic equation, which we solve and analyze using a symbolic computation software. Similar problems have been recently studied in connection with ray-tracing, catadioptric optics, scattering of electromagnetic waves, and mathematical billiards, but we were led to this problem in our study of the so-called triangular ratio metric.