arXiv subjects

H. -M. Teichert

Publications and source records attributed to H. -M. Teichert.

3 recordsLinked to original sources

arXiv2017

Extremal properties and morphisms of finite ultrametric spaces and their representing trees

We study extremal properties of finite ultrametric spaces $X$ and related properties of representing trees $T_X$. The notion of weak similarity for such spaces is introduced and related morphisms of labeled rooted trees are found. It is shown that the finite rooted trees are isomorphic to the rooted trees of nonsingular balls of special finite ultrametric spaces. We also found conditions under which the isomorphism of representing trees $T_X$ and $T_Y$ implies the isometricity of ultrametric spaces $X$ and $Y$.

math.MG↗

arXiv2015

How rigid the finite ultrametric spaces can be?

A metric space $X$ is rigid if the isometry group of $X$ is trivial. The finite ultrametric spaces $X$ with $|X| \geq 2$ are not rigid since for every such $X$ there is a self-isometry having exactly $|X|-2$ fixed points. Using the representing trees we characterize the finite ultrametric spaces $X$ for which every self-isometry has at least $|X|-2$ fixed points. Some other extremal properties of such spaces and related graph theoretical characterizations are also obtained.

math.MG↗

arXiv2014

On spaces extremal for the Gomory-Hu inequality

Let $(X,d)$ be a finite ultrametric space. In 1961 E.C. Gomory and T.C. Hu proved the inequality $|Sp(X)|\leqslant |X|$ where $Sp(X)=\{d(x,y)\colon x,y \in X\}$. Using weighted Hamiltonian cycles and weighted Hamiltonian paths we give new necessary and sufficient conditions under which the Gomory-Hu inequality becomes an equality. We find the number of non-isometric $(X,d)$ satisfying the equality $|Sp(X)|=|X|$ for given $Sp(X)$. Moreover it is shown that every finite semimetric space $Z$ is an image under a composition of mappings $f\colon X\to Y$ and $g\colon Y\to Z$ such that $X$ and $Y$ are finite ultrametric space, $X$ satisfies the above equality, $f$ is an $\varepsilon$-isometry with an arbitrary $\varepsilon>0$, and $g$ is a ball-preserving map.

math.MG↗