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Norbert Hungerbühler

Publications and source records attributed to Norbert Hungerbühler.

27 records · Page 2Linked to original sources
arXiv2018

Communicating harmonic pencils of lines

Suppose there are $n$ harmonic pencils of lines given in the plane. We are interested in the question whether certain triples of these lines are concurrent or if triples of intersection points of these lines are collinear, provided that we impose suitable conditions on the initial harmonic pencils. Such conditions can be that certain of the given lines coincide, are concurrent or that certain intersection points are collinear. The study of these questions for $n=2, 3, 4$ sheds light on some well known affine configurations and provides new results in the projective setting. As applications, we will formulate generalizations or stronger versions of the theorems of Pappus, Desargues, Ceva and Menelaos. Notably, the generalized theorems of Ceva and Menelaos suggest a new way to generalize the terms collinearity and concurrency.

math.HO↗
arXiv2018

Resistors in dual networks

Let $G$ be a finite plane multigraph and $G'$ its dual. Each edge $e$ of $G$ is interpreted as a resistor of resistance $R_e$, and the dual edge $e'$ is assigned the dual resistance $R_{e'}:=1/R_e$. Then the equivalent resistance $r_e$ over $e$ and the equivalent resistance $r_{e'}$ over $e'$ satisfy $r_e/R_e+r_{e'}/R_{e'}=1$. We provide a graph theoretic proof of this relation by expressing the resistances in terms of sums of weights of spanning trees in $G$ and $G'$ respectively.

math.CO↗
arXiv2018

A Theorem of Fermat on Congruent Number Curves

A positive integer $A$ is called a congruent number if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a congruent number if and only if the congruent number curve $y^2=x^3-A^2x$ has a rational point $(x,y)\in\mathbb Q^2$ with $y\neq 0$. Using a theorem of Fermat, we give an elementary proof for the fact that congruent number curves do not contain rational points of finite order.

math.NT↗
arXiv2016

The One-Sided Isometric Extension Problem

Let $Σ$ be a codimension one submanifold of an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We give a necessary condition for an isometric immersion of $Σ$ into $\mathbb R^q$ equipped with the standard Euclidean metric, $q\geqslant n+1$, to be locally isometrically $C^1$-extendable to $M$. Even if this condition is not met, "one-sided" isometric $C^1$-extensions may exist and turn out to satisfy a $C^0$-dense parametric $h$-principle in the sense of Gromov.

math.DG↗
arXiv2016

Poncelet's Theorem in the four nonisomorphic finite projective planes of order 9

We study Poncelet's Theorem in the four non-isomorphic finite projective planes of order 9. Among these planes, only the Desarguesian plane turns out to be a Poncelet plane, while the other three planes which are constructed over the miniquaternion near-field of order 9, are not. This gives a complete discussion of Poncelet's Theorem in finite projective planes of order 9.

math.CO↗
arXiv2015

Steiner's Porism in finite Miquelian Möbius planes

We investigate Steiner's Porism in finite Miquelian Möbius planes constructed over the pair of finite fields $GF(p^m)$ and $GF(p^{2m})$, for $p$ an odd prime and $m \geq 1$. Properties of common tangent circles for two given concentric circles are discussed and with that, a finite version of Steiner's Porism for concentric circles is stated and proved. We formulate conditions on the length of a Steiner chain by using the quadratic residue theorem in $GF(p^m)$. These results are then generalized to an arbitrary pair of non-intersecting circles by introducing the notion of capacitance, which turns out to be invariant under Möbius transformations. Finally, the results are compared with the situation in the classical Euclidean plane.

math.CO↗
arXiv2014

A Poncelet Criterion for special pairs of conics in $PG(2,p)$

We study Poncelet's Theorem in finite projective coordinate planes over the field $GF(p)$ and concentrate on a particular pencil of conics. For pairs of such conics we investigate whether we can find polygons with $n$ sides, which are inscribed in one conic and circumscribed about the other, so-called Poncelet Polygons. By using suitable elements of the dihedral group for these pairs, we prove that the length $n$ of such Poncelet Polygons is independent of the starting point. In this sense Poncelet's Porism is valid. By using Euler's divisor sum formula for the totient function, we can make a statement about the number of different conic pairs, which carry Poncelet Polygons of length $n$. Moreover, we will introduce polynomials whose zeros in $GF(p)$ yield information about the relation of a given pair of conics. In particular, we can decide for a given integer $n$, whether and how we can find Poncelet Polygons for pairs of conics in the given coordinate plane. We will see that this condition is closely connected with the theory of quadratic residues.

math.CO↗
arXiv2011

Soliton solutions of the mean curvature flow and minimal hypersurfaces

Let (M,g) be an oriented Riemannian manifold of dimension at least 3 and X a vector field on M. We show that the Monge-Ampère differential system (M.A.S.) for X-pseudosoliton hypersurfaces on (M,g) is equivalent to the minimal hypersurface M.A.S. on (M,g') for some Riemannian metric g', if and only if X is the gradient of a function u, in which case g'=exp(-2u)g. Counterexamples to this equivalence for surfaces are also given.

math.DG↗