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Let $D$ be a nonempty open set in a metric space $(X, d)$ with $\partial D \neq \emptyset$ . Define $h_{D, c} (x, y) = log (1 + c \frac{d (x, y)}{\sqrt{d_{D} (x) d_{D} (y)}}),$ where $d_{D} (x) = d (x, \partial D)$ is the distance from $x$ to the boundary of $D$ . For every $c \geq 2$ , $h_{D, c}$ is a metric. We study the sharp Lipschitz constants for the metric $h_{D, c}$ under Möbius transformations of the unit ball, the upper half space, and the punctured unit ball.

Löbell manifolds revised.

Mednykh, A.D., Vesnin, A.Yu. (2007)

Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]

$m$ -point invariants of real geometries.

Höfer, Roland (1999)

Beiträge zur Algebra und Geometrie

Metric characterizations of hyperbolic and Euclidean spaces

J. E. Valentine, S. G. Wayment (1972)

Colloquium Mathematicae

Metric properties of the Stephanos bijection.

Urban, Herbert (1995)

Beiträge zur Algebra und Geometrie

Minimal paradoxical decomposition for Mycielski's square

Glen Sherman (1991)

Fundamenta Mathematicae

Möbius gyrovector spaces in quantum information and computation

Abraham A. Ungar (2008)

Commentationes Mathematicae Universitatis Carolinae

Hyperbolic vectors, called gyrovectors, share analogies with vectors in Euclidean geometry. It is emphasized that the Bloch vector of Quantum Information and Computation (QIC) is, in fact, a gyrovector related to Möbius addition rather than a vector. The decomplexification of Möbius addition in the complex open unit disc of a complex plane into an equivalent real Möbius addition in the open unit ball $𝔹^{2}$ of a Euclidean 2-space $ℝ^{2}$ is presented. This decomplexification proves useful, enabling the resulting...

Möbius metric in sector domains

Oona Rainio, Matti Vuorinen (2023)

Czechoslovak Mathematical Journal

The Möbius metric $δ_{G}$ is studied in the cases, where its domain $G$ is an open sector of the complex plane. We introduce upper and lower bounds for this metric in terms of the hyperbolic metric and the angle of the sector, and then use these results to find bounds for the distortion of the Möbius metric under quasiregular mappings defined in sector domains. Furthermore, we numerically study the Möbius metric and its connection to the hyperbolic metric in polygon domains.

Mutation and volumes of knots in S3.

D. Ruberman (1987)

Inventiones mathematicae

O množině středů křivek konstantní křivosti hyperbolické roviny, které se dotýkají dané křivky konstantní křivosti a přímky

František Machala (1971)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica-Physica-Chemica

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