arXiv subjects

Henning Wunderlich

Publications and source records attributed to Henning Wunderlich.

4 recordsLinked to original sources

arXiv2023

A geometric view on planar graphs and its application to coloring

While planar graphs are flat from a topological viewpoint, we observe that they are not from a geometric one. We prove that every planar graph can be embedded into a surface consisting of spheres, glued together in a tree-like fashion. As a technical ingredient we prove a statement implying an inverse of the Jordan curve theorem. This statement helps to identify cycles in the planar graph, corresponding to circles of latitude on the spheres. The tree-like embedding then allows for an inductive construction of a four-coloring of the planar graph. Hence, this yields a simple proof of the Four-Color Theorem.

math.GM↗

arXiv2020

Characterization of Fréchet Spaces and Application to Hausdorff MNC

In this short note, we give a characterization of Fréchet spaces via properties of their metric. This allows us to prove that the Hausdorff measure of noncompactness (MNC), defined over Fréchet spaces, is indeed an MNC. As first applications, we lift well-known fixed-point theorems for contractive and condensing operators to the setting of Fréchet spaces.

math.FA↗

arXiv2019

On a characterization of spaces satisfying open mapping and equivalent theorems

For classes of topological vector spaces, we analyze under which conditions open-mapping, continuous-inverse, and closed-graph properties are equivalent. Here, closure under quotients with closed subspaces and closure under closed graphs are sufficient. We show that the class of barreled Ptak spaces is exactly the largest class of locally-convex topological vector spaces, which contains all Banach spaces, is closed under quotients with closed subspaces, is closed under closed graphs, is closed under continuous images, and for which an open-mapping theorem, a continuous-inverse theorem, and a closed-graph theorem holds. An analogous, weaker result also holds for the strictly larger class of barreled infra-Ptak spaces.

math.FA↗

arXiv2012

A note on a problem in communication complexity

In this note, we prove a version of Tarui's Theorem in communication complexity, namely $PH^{cc} \subseteq BP\cdot PP^{cc}$. Consequently, every measure for $PP^{cc}$ leads to a measure for $PH^{cc}$, subsuming a result of Linial and Shraibman that problems with high mc-rigidity lie outside the polynomial hierarchy. By slightly changing the definition of mc-rigidity (arbitrary instead of uniform distribution), it is then evident that the class $M^{cc}$ of problems with low mc-rigidity equals $BP\cdot PP^{cc}$. As $BP\cdot PP^{cc} \subseteq PSPACE^{cc}$, this rules out the possibility, that had been left open, that even polynomial space is contained in $M^{cc}$.

cs.CC↗