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Andreas Schweizer

Publications and source records attributed to Andreas Schweizer.

At least 19 recordsLinked to original sources
arXiv2026

Splittability, non-splittability, and automatic splittability of metacyclic $p$-groups

Let $P$ be a finite metacyclic $p$-group where $p$ is an odd prime. We refine the notions split metacyclic and non-split metacyclic group by distinguishing whether a metacyclic structure (i.e. a cyclic normal subgroup $K$ of $P$ such that $P/K$ is also cyclic) is non-splittable or splittable or automatically split. We show that these different types can be characterised in terms of $|K|$. We also determine which of these types can coexist on the same group $P$.

math.GR↗
arXiv2025

Rational functions that share finite values with their first derivative

We treat shared value problems for rational functions $R(z)$ and their derivative $R'(z)$ in the plane and on the sphere. We also consider shared values for the pair $R(w)$ and $\partial_{z} R = \lambda w \cdot R'(w)$ on ${\mathbb C} \setminus \{ 0 \}$ and $\widehat{\mathbb C}$, again with rational functions $R$. In ${\mathbb C} \setminus \{ 0 \}$ this is related to shared values of meromorphic functions $f : {\mathbb C} \to \widehat{\mathbb C}$ and $f'$ through $f(z)=R(w)$ with $w=\exp(\lambda z)$, while on $\widehat{\mathbb C}$ this is connected to shared limit values in a similar fashion.

math.CV↗
arXiv2025

Meromorphic functions that partially share values with their first derivative

We consider uniqueness results for meromorphic functions $f:{\mathbb C} \to \widehat{\mathbb C}$ such that for certain values $a\in {\mathbb C}$ the implication $f(z)=a \Rightarrow f'(z)=a$ holds, i.e. that $f$ and $f'$ share values {\it partially}. In particular, we give a result for four partially shared values.

math.CV↗
arXiv2024

On the automorphisms of the Drinfeld modular groups

Let $A$ be the ring of elements in an algebraic function field $K$ over $\mathbb{F}_q$ which are integral outside a fixed place $\infty$. In contrast to the classical modular group $SL_2(\mathbb{Z})$ and the Bianchi groups, the {\it Drinfeld modular group} $G=GL_2(A)$ is not finitely generated and its automorphism group $\mathrm{Aut}(G)$ is uncountable. Except for the simplest case $A=\mathbb{F}_q[t]$ not much is known about the generators of $\mathrm{Aut}(G)$ or even its structure. We find a set of generators of $\mathrm{Aut}(G)$ for a new case. \par On the way, we show that {\it every} automorphism of $G$ acts on both, the {\it cusps} and the {\it elliptic points} of $G$. Generalizing a result of Reiner for $A=\mathbb{F}_q[t]$ we describe for each cusp an uncountable subgroup of $\mathrm{Aut}(G)$ whose action on $G$ is essentially defined on the stabilizer of that cusp. In the case where $\delta$ (the degree of $\infty$) is $1$, the elliptic points are related to the isolated vertices of the quotient graph $G\setminus\mathcal{T}$ of the Bruhat-Tits tree. We construct an infinite group of automorphisms of $G$ which fully permutes the isolated vertices with cyclic stabilizer.

math.NT↗
arXiv2022

A uniqueness problem concerning entire functions and their derivatives

We determine all entire functions $f$ such that for nonzero complex values $a\neq b$ the implications $f=a \Rightarrow f' =a$ and $f' =b \Rightarrow f=b$ hold. This solves an open problem in uniqueness theory. In this context we give a normality criterion, which might be interesting in its own right.

math.CV↗
arXiv2022

Bielliptic quotient modular curves of $X_0(N)$

Let $N\geq 1$ be a non-square free integer and let $W_N$ be a non-trivial subgroup of the group of the Atkin-Lehner involutions of $X_0(N)$ such that the modular curve $X_0(N)/W_N$ has genus at least two. We determine all pairs $(N,W_N)$ such that $X_0(N)/W_N$ is a bielliptic curve and the pairs $(N,W_N)$ such that $X_0(N)/W_N$ has an infinite number of quadratic points over $\mathbb{Q}$.

math.NT↗
arXiv2021

Quasi-inner automorphisms of Drinfeld modular groups

Let $A$ be the set of elements in an algebraic function field $K$ over ${\mathbb F}_q$ which are integral outside a fixed place $\infty$. Let $G=GL_2(A)$ be a {\it Drinfeld modular group}. The normalizer of $G$ in $GL_2(K)$, where $K$ is the quotient field of $A$, gives rise to automorphisms of $G$, which we refer to as {\it quasi-inner}. Modulo the inner automorphisms of $G$ they form a group $Quinn(G)$ which is isomorphic to ${\mathrm Cl}(A)_2$, the $2$-torsion in the ideal class group ${\mathrm Cl}(A)$. The group $Quinn(G)$ acts on all kinds of objects associated with $G$. For example, it acts freely on the cusps and elliptic points of $G$. If ${\mathcal T}$ is the associated Bruhat-Tits tree the elements of $Quinn(G)$ induce non-trivial automorphisms of the quotient graph $G\setminus{\mathcal T}$, generalizing an earlier result of Serre. It is known that the ends of $G\setminus{\mathcal T}$ are in one-one correspondence with the cusps of $G$. Consequently $Quinn(G)$ acts freely on the ends. In addition $Quinn(G)$ acts transitively on those ends which are in one-one correspondence with the vertices of $G\setminus{\mathcal T}$ whose stabilizers are isomorphic to $GL_2({\mathbb F}_q)$.

math.NT↗
arXiv2019

Torsion of rational elliptic curves over different types of cubic fields

Let $E$ be an elliptic curve defined over $\Q$, and let $G$ be the torsion group $E(K)_{tors}$ for some cubic field $K$ which does not occur over $\Q$. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) $G$ can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves $E/\Q$ together with cubic fields $K$ so that $G= E(K)_{tors}$.

math.NT↗
arXiv2019

An Automatic Genetic Algorithm Framework for the Optimization of Three-dimensional Surgical Plans of Forearm Corrective Osteotomies

3D computer-assisted corrective osteotomy has become the state-of-the-art for surgical treatment of complex bone deformities. Despite available technologies, the automatic generation of clinically acceptable, ready-to-use preoperative planning solutions is currently not possible for such pathologies. Multiple contradicting and mutually dependent objectives have to be considered, as well as clinical and technical constraints, generally requiring iterative manual adjustments. This leads to unnecessary efforts and unbearable clinical costs, hindering also the quality of patient treatment. In this paper, we propose an optimization framework for the generation of ready-to-use preoperative planning solutions in a fully automatic fashion. An automatic diagnostic assessment using patient-specific 3D models is performed for 3D malunion quantification and definition of the optimization parameters. Afterward, clinical objectives are translated into the optimization module, and controlled through tailored fitness functions based on a weighted and multi-staged optimization approach. The optimization is based on a genetic algorithm capable of solving multi-objective optimization problems with non-linear constraints. The framework outputs a complete preoperative planning solution including position and orientation of the osteotomy plane, transformation to achieve the bone reduction, and position and orientation of the fixation plate and screws. A qualitative validation was performed on 36 consecutive cases of radius osteotomy where solutions generated by the optimization algorithm (OA) were compared against the gold standard (GS) solutions generated by experienced surgeons. Solutions were blinded and presented to 6 readers, who voted OA solutions to be better in 55% of the time. The quantitative evaluation was based on different error measurements, showing average improvements with respect to the GS.

physics.med-ph↗
arXiv2019

Bielliptic intermediate modular curves

We determine which of the modular curves $X_Δ(N)$, that is, curves lying between $X_0(N)$ and $X_1(N)$, are bielliptic. Somewhat surprisingly, we find that one of these curves has exceptional automorphisms. Finally we find all $X_Δ(N)$ that have infinitely many quadratic points over $\mathbb{Q}$.

math.NT↗
arXiv2017

Several types of solvable groups as automorphism groups of compact Riemann surfaces

Let $X$ be a compact Riemann surface of genus $g\geq 2$. Let $Aut(X)$ be its group of automorphisms and $G\subseteq Aut(X)$ a subgroup. Sharp upper bounds for $|G|$ in terms of $g$ are known if $G$ belongs to certain classes of groups, e.g. solvable, supersolvable, nilpotent, metabelian, metacyclic, abelian, cyclic. We refine these results by finding similar bounds for groups of odd order that are of these types. We also add more types of solvable groups to that long list by establishing the optimal bounds for, among others, groups of order $p^m q^n$. Moreover, we show that Zomorrodian's bound for $p$-groups $G$ with $p\geq 5$, namely $|G|\leq \frac{2p}{p-3}(g-1)$, actually holds for any group $G$ for which $p\geq 5$ is the smallest prime divisor of $|G|$.

math.CV↗
arXiv2017

What is the definition of two meromorphic functions sharing a small function?

Two meromorphic functions $f(z)$ and $g(z)$ sharing a small function $α(z)$ usually is defined in terms of vanishing of the functions $f-α$ and $g-α$. We argue that it would be better to modify this definition at the points where $α$ has poles. Related to this issue we also point out some possible gaps in proofs in the published literature.

math.CV↗
arXiv2017

A normality criterion corresponding to the defect relations

Let ${\cal F}$ be a family of meromorphic functions on a domain $D$. We present a quite general sufficient condition for ${\cal F}$ to be a normal family. This criterion contains many known results as special cases. The overall idea is that certain comparatively weak conditions on ${\cal F}$ locally lead to somewhat stronger conditions, which in turn lead to even stronger conditions on the limit function $g$ in the famous Zalcman Lemma. Ultimately, the defect relations for $g$ force normality of ${\cal F}$.

math.CV↗
arXiv2017

Metacyclic groups as automorphism groups of compact Riemann surfaces

Let $X$ be a compact Riemann surface of genus $g\geq 2$, and let $G$ be a subgroup of $Aut(X)$. We show that if the Sylow $2$-subgroups of $G$ are cyclic, then $|G|\leq 30(g-1)$. If all Sylow subgroups of $G$ are cyclic, then, with two exceptions, $|G|\leq 10(g-1)$. More generally, if $G$ is metacyclic, then, with one exception, $|G|\leq 12(g-1)$. Each of these bounds is attained for infinitely many values of $g$.

math.CV↗
arXiv2016

On the exponent of the automorphism group of a compact Riemann surface

Let $X$ be a compact Riemann surface of genus $g\geq 2$, and let $Aut(X)$ be its group of automorphims. We show that the exponent of $Aut(X)$ is bounded by $42(g-1)$. We also determine explicitly the infinitely many values of $g$ for which this bound is reached and the corresponding groups. Finally we discuss related questions for subgroups $G$ of $Aut(X)$ that are subject to additional conditions, for example being solvable.

math.CV↗
arXiv2016

Genuine non-congruence subgroups of Drinfeld modular groups

Let $A$ be the ring of elements in an algebraic function field $K$ over a finite field $F_q$ which are integral outside a fixed place $\infty$. In an earlier paper we have shown that the Drinfeld modular group $G=GL_2(A)$ has automorphisms which map congruence subgroups to non-congruence subgroups. Here we prove the existence of (uncountably many) normal genuine non-congruence subgroups, defined to be those which remain non-congruence under the action of every automorphism of $G$. In addition, for all but finitely many cases we evaluate $ngncs(G)$, the smallest index of a normal genuine non-congruence subgroup of $G$, and compare it to the minimal index of an arbitrary normal non-congruence subgroup.

math.GR↗
arXiv2015

Some remarks on bielliptic and trigonal curves

We prove some results on algebraic curves $X$ of genus $g\geq 2$ in characteristic $0$. For example: Assume that $X$ has an automorphism $σ$ of prime order $p\geq 5$. If $σ$ has no fixed points, then $X$ cannot be trigonal. On the other hand, if $σ$ has fixed points, then $X$ is bielliptic only if it belongs to one of three extremal types of curves of small genus.

math.AG↗