arXiv subjects

David Schmitz

Publications and source records attributed to David Schmitz.

18 recordsLinked to original sources

arXiv2025

Continuous Inverse Ambiguous Functions on Lie Groups

In a previous study, the first author defines an inverse ambiguous function on a group $G$ to be a bijective function $f : G \to G$ satisfying the functional equation $f^{-1}(x) = f(x^{-1})$ for all $x \in G$. In this paper, we investigate the existence of continuous inverse ambiguous functions on classical Lie groups. In particular, we look at tori, elliptic curves over various fields, vector spaces, additive matrix groups, and multiplicative matrix groups.

math.GR↗

arXiv2023

A proof of the corrected Sister Beiter cyclotomic coefficient conjecture inspired by Zhao and Zhang

The largest coefficient (in absolute value) of a cyclotomic polynomial $Φ_n$ is called its height $A(n)$. In case $p$ is a fixed prime it turns out that as $q$ and $r$ range over all primes satisfying $p<q<r$, the height $A(pqr)$ assumes a maximum $M(p)$. In 1968, Sister Marion Beiter conjectured that $M(p)\leq (p+1)/2$. In 2009, this was disproved for every $p\ge 11$ by Yves Gallot and Pieter Moree. They proposed a Corrected Beiter Conjecture, namely $M(p)\leq 2p/3$. In 2009, Jia Zhao and Xianke Zhang posted on the arXiv what they thought to be a proof of this conjecture. Their work was never accepted for publication in a journal. However, in retrospect it turns out to be essentially correct, but rather sketchy at some points. Here we supply a lot more details. \par The bound $M(p)\le 2p/3$ allows us to improve some bounds of Bzdęga from 2010 for ternary cyclotomic coefficients. It also makes it possible to determine $M(p)$ exactly for three new primes $p$ and study the fine structure of $A(pqr)$ for them in greater detail.

math.NT↗

arXiv2022

SNOWMASS Neutrino Frontier NF10 Topical Group Report: Neutrino Detectors

We discuss here future neutrino detectors with physics goals ranging from the eV to the EeV scale. The focus is on future enabling technologies for such detectors, rather than existing detectors or those under construction. The report includes methodologies across the broad spectrum of neutrino physics: liquid noble and other cryogenic detectors, includin LAr and LXe TPCs; photon-based detectors including technologies enabling hybrid Cherenkov/scintillation detectors; low-threshold detectors which use a wide variety of technologies to probe physics like coherent neutrino-nucleus scattering or detection of cosmic background neutrinos; and ultra-high energy detectors including optical and radio detectors, as well as tracking detectors for use at the forward physics facility of the LHC

hep-ex↗

arXiv2020

Rationality of Seshadri constants on general blow ups of $\mathbb{P}^2$

Let $X$ be a projective surface and let $L$ be an ample line bundle on $X$. The global Seshadri constant $\varepsilon(L)$ of $L$ is defined as the infimum of Seshadri constants $\varepsilon(L,x)$ as $x\in X$ varies. It is an interesting question to ask if $\varepsilon(L)$ is a rational number for any pair $(X, L)$. We study this question when $X$ is a blow up of $\mathbb{P}^2$ at $r \ge 0$ very general points and $L$ is an ample line bundle on $X$. For each $r$ we define a $\textit{submaximality threshold}$ which governs the rationality or irrationality of $\varepsilon(L)$. We state a conjecture which strengthens the SHGH Conjecture and assuming that this conjecture is true we determine the submaximality threshold.

math.AG↗

arXiv2018

On exterior powers of the tangent bundle on toric varieties

We study the positivity of exterior powers of the tangent sheaf on toric varieties in order to generalize results by Campana and Peternell about 3-folds with nef second exterior power of the tangent bundle. Using the theory of equivariant vector bundles and the toric MMP, we establish in the smooth case a criterion for the positivity of $Λ^m\mathcal T_X$ in terms of wall relations. As an application, we classify smooth toric varieties of arbitrary dimension $n\ge3$ with $Λ^2\mathcal T_X$ nef and those of dimension $n\ge 4$ with $Λ^3\mathcal T_X$ ample.

math.AG↗

arXiv2017

On the Mori theory and Newton-Okounkov bodies of Bott-Samelson varieties

We prove that on a Bott-Samelson variety $X$ every movable divisor is nef. This enables us to consider Zariski decompositions of effective divisors, which in turn yields a description of the Mori chamber decomposition of the effective cone. This amounts to information on all possible birational morphisms from $X$. Applying this result, we prove the rational polyhedrality of the global Newton-Okounkov body of a Bott-Samelson variety with respect to the so called `horizontal' flag. In fact, we prove the stronger property of the finite generation of the corresponding global value semigroup.

math.AG↗

arXiv2017

Revisiting Open eXchange Points with Software Defined Networking

The introduction of SDN in Service Providers' networks like GEANT is a challenging task. Prototypes of new generation services have to exhibit "carrier grade" characteristics, meet the high level expectations and specialized needs of GEANT customers. In this demonstration, we present the SDN based prototype of GEANT Open service, used by GEANT customers and approved commercial partners to interconnect using Layer 2 circuits. Currently, this service is delivered through a set of Open eXchange Points leveraging on legacy solutions. The SDN based prototype has been realized on top of ONOS and leverages on hardware switches for the data plane. During the demo, which runs inside the GEANT Testbed Service, we show how operators can deploy services and manage the SDN based infrastructure.

cs.NI↗

arXiv2017

On the postulation of lines and a fat line

In this note we show that the union of $r$ general lines and one fat line in ${\mathbb P}^3$ imposes independent conditions on forms of sufficiently high degree $d$, where the bound on $d$ is independent of the number of lines. This extends former results of Hartshorne and Hirschowitz on unions of general lines, and of Aladpoosh on unions of general lines and one double line.

math.AG↗

arXiv2016

On numerical Newton-Okounkov bodies and the existence of Minkowski bases

Towards the boundary of the big cone, Newton-Okounkov bodies do not vary continuously and in fact the body of a boundary class is not well defined. Using the global Okounkov body one can nonetheless define a numerical invariant, the numerical Newton-Okounkov body. We show that if a normal projective variety has a rational polyhedral global Okounkov body, it admits a Minkowski basis provided one includes numerical Newton-Okounkov bodies above non-big classes. Under the same assumption, we also show that the dimension of the numerical Newton-Okounkov body is the numerical Kodaira dimension.

math.AG↗

arXiv2016

Newton-Okounkov bodies and complexity functions

We show that quite universally the holonomicity of the complexity function of a big divisor on a projective variety does not predict the polyhedrality of the Newton-Okounkov body associated to every flag.

math.AG↗

arXiv2015

On the boundedness of the denominators in the Zariski decomposition on surfaces

Zariski decompositions play an important role in the theory of algebraic surfaces. For making geometric use of the decomposition of a given divisor, one needs to pass to a multiple of the divisor in order to clear denominators. It is therefore an intriguing question whether the surface has a 'universal denominator' that can be used to simultaneously clear denominators in all Zariski decompositions on the surface. We prove in this paper that, somewhat surprisingly, this condition of bounded Zariski denominators is equivalent to the bounded negativity of curves that is addressed in the Bounded Negativity Conjecture. Furthermore, we provide explicit bounds for Zariski denominators and negativity of curves in terms of each other.

math.AG↗

arXiv2015

Global Okounkov bodies for Bott-Samelson varieties

We use the theory of Mori dream spaces to prove that the global Okounkov body of a Bott-Samelson variety with respect to a natural flag of subvarieties is rational polyhedral. In fact, we prove more generally that this holds for any Mori dream space which admits a flag of Mori dream spaces satisfying a certain regularity condition. As a corollary, Okounkov bodies of effective line bundles over Schubert varieties are shown to be rational polyhedral. In particular, it follows that the global Okounkov body of a flag variety $G/B$ is rational polyhedral. As an application we show that the asymptotic behaviour of dimensions of weight spaces in section spaces of line bundles is given by the counting of lattice points in polytopes.

math.AG↗

arXiv2015

Minkowski decomposition and generators of the moving cone for toric varieties

We prove that for smooth projective toric varieties, the Okounkov body of a $T$-invariant pseudo-effective divisor with respect to a $T$-invariant flag decomposes as a finite Minkowski sum of indecomposable polytopes. We prove that these indecomposable polytopes form a Minkowski base and that they correspond to the rays in the secondary fan. Moreover, we present an algorithm to find the Minkowski base.

math.AG↗

arXiv2014

On the polyhedrality of global Okounkov bodies

We prove that the existence of a finite Minkowski base for Okounkov bodies on a smooth projective variety with respect to an admissible flag implies rational polyhedrality of the global Okounkov body. As an application of this general result, we deduce that the global Okounkov body of a surface with finitely generated pseudo-effective cone with respect to a general flag is rational polyhedral. We give an alternative proof for this fact which recovers the generators more explicitly. We also prove the rational polyhedrality of global Okounkov bodies in the case of certain homogeneous 3-folds using inductive methods.

math.AG↗

arXiv2013

Minkowski decomposition of Okounkov bodies on surfaces

We prove that the Okounkov body of a big divisor with respect to a general flag on a smooth projective surface whose pseudo-effective cone is rational polyhedral decomposes as the Minkowski sum of finitely many simplices and line segments arising as Okounkov bodies of nef divisors.

math.AG↗

arXiv2012

Volumes of Zariski chambers

Zariski chambers are natural pieces into which the big cone of an algebraic surface decomposes. They have so far been studied both from a geometric and from a combinatorial perspective. In the present paper we complement the picture with a metric point of view by studying a suitable notion of chamber sizes. Our first result gives a precise condition for the nef cone volume to be finite and provides a method for computing it inductively. Our second result determines the volumes of arbitrary Zariski chambers from nef cone volumes of blow-downs. We illustrate the applicability of this method by explicitly determining the chamber volumes on Del Pezzo and other anti-canonical surfaces.

math.AG↗

arXiv2012

Zariski chambers on surfaces of high Picard number

We present an improved algorithm for the computation of Zariski chambers on algebraic surfaces. The new algorithm significantly outperforms the so far available method and allows therefore to treat surfaces of high Picard number, where huge chamber numbers occur. As an application, we efficiently compute the number of chambers supported by the lines on the Segre-Schur quartic.

math.AG↗

arXiv2011

Integrated monitoring of multi-domain backbone connections -- Operational experience in the LHC optical private network

Novel large scale research projects often require cooperation between various different project partners that are spread among the entire world. They do not only need huge computing resources, but also a reliable network to operate on. The Large Hadron Collider (LHC) at CERN is a representative example for such a project. Its experiments result in a vast amount of data, which is interesting for researchers around the world. For transporting the data from CERN to 11 data processing and storage sites, an optical private network (OPN) has been constructed. As the experiment data is highly valuable, LHC defines very high requirements to the underlying network infrastructure. In order to fulfil those requirements, the connections have to be managed and monitored permanently. In this paper, we present the integrated monitoring solution developed for the LHCOPN. We first outline the requirements and show how they are met on the single network layers. After that, we describe, how those single measurements can be combined into an integrated view. We cover design concepts as well as tool implementation highlights.

cs.NI↗