Das Königsberger Brückenproblem und der Algorithmus von Hierholzer

Article for general public (2023)

Das mathematische Teilgebiet der Graphentheorie eignet sich hervorragend, um im Rahmen des Schulunterrichts behandelt zu werden. Es ist leicht zugänglich, kann auf anschauliche Art erläutert werden und ... [more ▼]

Das mathematische Teilgebiet der Graphentheorie eignet sich hervorragend, um im Rahmen des Schulunterrichts behandelt zu werden. Es ist leicht zugänglich, kann auf anschauliche Art erläutert werden und hat zahlreiche konkrete Alltagsanwendungen, etwa im Bereich der Logistik und Routenplanung. Dieser Artikel gibt einen kurzen Einblick in die Graphentheorie, indem er Teile eines Kurses zu diesem Thema vorstellt, der am Scienteens Lab, dem Schülerlabor der Universität Luxemburg, angeboten wird. [less ▲]

Universal radial limits of meromorphic functions in the unit disk

in Comptes Rendus. Mathématique (2022), 360

We consider the space of meromorphic functions in the unit disk $\D$ and show that there exists a dense $G_{\delta}$-subset of functions having universal radial limits. Our results complement known ... [more ▼]

We consider the space of meromorphic functions in the unit disk $\D$ and show that there exists a dense $G_{\delta}$-subset of functions having universal radial limits. Our results complement known statements about holomorphic functions and further imply the existence of meromorphic functions having maximal cluster sets along certain subsets of $\D$. [less ▲]

Non-normality, topological transitivity and expanding families

in Mathematical Proceedings of the Cambridge Philosophical Society (2022), 173(3), 511-523

We investigate the behaviour of families of meromorphic functions in the neighbourhood of points of non-normality and prove certain covering properties that complement Montel’s Theorem. In particular, we ... [more ▼]

We investigate the behaviour of families of meromorphic functions in the neighbourhood of points of non-normality and prove certain covering properties that complement Montel’s Theorem. In particular, we also obtain characterisations of non-normality in terms of such properties. [less ▲]

Seasonal low-degree changes in terrestrial water mass load from global GNSS measurements

in Journal of Geodesy (2017), 91(11), 1329-1350

Large-scale mass redistribution in the terrestrial water storage (TWS) leads to changes in the low-degree spherical harmonic coefficients of the Earth's surface mass density field. Studying these low ... [more ▼]

Large-scale mass redistribution in the terrestrial water storage (TWS) leads to changes in the low-degree spherical harmonic coefficients of the Earth's surface mass density field. Studying these low-degree fluctuations is an important task that contributes to our understanding of continental hydrology. In this study, we use global GNSS measurements of vertical and horizontal crustal displacements that we correct for atmospheric and oceanic effects, and use a set of modified basis functions similar to Clarke et al. (2007) to perform an inversion of the corrected measurements in order to recover changes in the coefficients of degree-0 (hydrological mass change), degree-1 (center of mass shift) and degree-2 (flattening of the Earth) caused by variations in the TWS over the period January 2003 - January 2015. We infer from the GNSS-derived degree-0 estimate an annual variation in total continental water mass with an amplitude of $(3.49 \pm 0.19) \times 10^{3}$ Gt and a phase of $70 \pm 3^{\circ}$ (implying a peak in early March), in excellent agreement with corresponding values derived from the Global Land Data Assimilation System (GLDAS) water storage model that amount to $(3.39 \pm 0.10) \times 10^{3}$ Gt and $71 \pm 2^{\circ}$, respectively. The degree-1 coefficients we recover from GNSS predict annual geocentre motion (i.e. the offset change between the center of common mass and the center of figure) caused by changes in TWS with amplitudes of $0.69 \pm 0.07$ mm for GX, $1.31 \pm 0.08$ mm for GY and $2.60 \pm 0.13$ mm for GZ. These values agree with GLDAS and estimates obtained from the combination of GRACE and the output of an ocean model using the approach of Swenson et al. (2008) at the level of about 0.5, 0.3 and 0.9 mm for GX, GY and GZ, respectively. Corresponding degree-1 coefficients from SLR, however, generally show higher variability and predict larger amplitudes for GX and GZ. The results we obtain for the degree-2 coefficients from GNSS are slightly mixed, and the level of agreement with the other sources heavily depends on the individual coefficient being investigated. The best agreement is observed for $T_{20}^C$ and $T_{22}^S$, which contain the most prominent annual signals among the degree-2 coefficients, with amplitudes amounting to $(5.47 \pm 0.44) \times 10^{-3}$ and $(4.52 \pm 0.31) \times 10^{-3}$ m of equivalent water height (EWH), respectively, as inferred from GNSS. Corresponding agreement with values from SLR and GRACE is at the level of or better than $0.4 \times 10^{-3}$ and $0.9 \times 10^{-3}$ m of EWH for $T_{20}^C$ and $T_{22}^S$, respectively, while for both coefficients, GLDAS predicts smaller amplitudes. Somewhat lower agreement is obtained for the order-1 coefficients, $T_{21}^C$ and $T_{21}^S$, while our GNSS inversion seems unable to reliably recover $T_{22}^C$. For all the coefficients we consider, the GNSS-derived estimates from the modified inversion approach are more consistent with the solutions from the other sources than corresponding estimates obtained from an unconstrained standard inversion. [less ▲]

A comparison of interannual hydrological polar motion excitation from GRACE and geodetic observations

in Journal of Geodynamics (2016), 99

Continental hydrology has a large influence on the excitation of polar motion (PM). However, these effects are far from being completely understood. Current global water storage models differ ... [more ▼]

Continental hydrology has a large influence on the excitation of polar motion (PM). However, these effects are far from being completely understood. Current global water storage models differ significantly from one another and are unable to completely represent the complex hydrological cycle, particularly at interannual scales. A promising alternative to study hydrological effects on PM is given by the GRACE satellite mission. In this study, we assess the ability of GRACE to investigate interannual hydrological PM excitations. For this purpose, we use the latest GRACE Release-05 data from three different processing centers (CSR, GFZ, JPL) that we convert into estimates of hydrological PM excitation, $\chi_1^H$ and $\chi_2^H$. In addition to these gravimetric excitations, we also consider geodetic hydrological excitations, which we calculate by removing modelled atmospheric and oceanic effects from precise observations of full PM excitations. We remove signals with frequencies $\geq 1$ cpy from the series and compare the resulting estimates of interannual hydrological excitations for the period 2004.5 - 2014.5. The comparison between geodetic and gravimetric excitations reveals some discrepancies for $\chi_1^H$, likely to be related to inadequately modelled atmospheric and oceanic effects. On the other hand, good agreement is observed for $\chi_2^H$. For both components, the best agreement between geodetic and gravimetric excitations is obtained for the estimate from CSR. Very good agreement is obtained between GRACE-derived excitations from different processing centers, in particular for CSR and JPL. Both the comparisons between geodetic and gravimetric, and the comparisons between the different gravimetric excitations give substantially better results for $\chi_2^H$ than for $\chi_1^H$, leading to the conclusion that geodetic and gravimetric $\chi_2^H$ can be more reliably determined than $\chi_1^H$. Although there are still some discrepancies between geodetic and gravimetric interannual hydrological excitations, we conclude that GRACE and potential follow-on missions are valuable tools to study the interannual effects of continental hydrology on the excitation of PM. [less ▲]

Seasonal Variations of Low-degree Spherical Harmonics Derived from GPS Data and Loading Models

Scientific Conference (2014, September 30)

An assessment of degree-2 Stokes coefficients from Earth rotation data

in Geophysical Journal International (2013), 195((1)), 249-259

Variations in the degree-2 Stokes coefficients C20, C21 and S21 can be used to understand long and short-term climate forcing. Here, we derive changes in these coefficients for the period 2003 ... [more ▼]

Variations in the degree-2 Stokes coefficients C20, C21 and S21 can be used to understand long and short-term climate forcing. Here, we derive changes in these coefficients for the period 2003 January–2012 April using Earth rotation data. Earth rotation data contain contributions from motion terms (the effects of winds and currents) and contributions from the effects of mass redistribution. We remove the effects of tides, atmospheric winds and oceanic currents from our data. We compare two different models of atmospheric and oceanic angular momentum for removing the effects of winds and currents: (1) using products from the National Centers for Environmental Prediction and (2) using data from the European Centre for Medium-range Weather Forecasts (ECMWF). We assess the quality of these motion models by comparing the two resulting sets of degree-2 Stokes coefficients to independent degree-2 estimates from satellite laser ranging (SLR), GRACE and a geophysical loading model. We find a good agreement between the coefficients from Earth rotation and the coefficients from other sources. In general, the agreement is better for the coefficients we obtain by removing winds and currents effects using the ECMWF model. In this case, we find higher correlations with the independent models and smaller scatters in differences. This fact holds in particular for C20 and C21, whereas we cannot observe a significant difference for S21. At the annual and semiannual periods, our Earth rotation derived coefficients agree well with the estimates from the other sources, particularly for C21 and S21. The slight discrepancies we obtain for C20 can probably be explained by errors in the atmospheric models and are most likely the result of an over-/underestimation of the annual and semiannual contributions of atmospheric winds to the length-of-day excitation. [less ▲]

On two classes of universal meromorphic functions

in Complex Variables and Elliptic Equations (2013), 58(10), 1343-1354

We consider two classes of meromorphic functions, which have universal approximation properties with respect to translations, and prove that both are residual subsets of the space of all meromorphic ... [more ▼]

We consider two classes of meromorphic functions, which have universal approximation properties with respect to translations, and prove that both are residual subsets of the space of all meromorphic functions. Furthermore, we show that the two classes do not coincide. [less ▲]

Universal distribution of limit points

in Acta Mathematica Hungarica (2011), 133(3), 288-303

We consider sequences of functions that have in some sense a universal distribution of limit points of zeros in the complex plane. In particular, we prove that functions having universal approximation ... [more ▼]

We consider sequences of functions that have in some sense a universal distribution of limit points of zeros in the complex plane. In particular, we prove that functions having universal approximation properties on compact sets with connected complement automatically have such a universal distribution of limit points. Moreover, in the case of sequences of derivatives, we show connections between this kind of universality and some rather old results of Edrei/MacLane and Pólya. Finally, we show the lineability of the set of what we call Jentzsch-universal power series. [less ▲]

Universal rational expansions of meromorphic functions

in Computational Methods and Function Theory (2011), 11(1), 317-324

Motivated by known results about universal Taylor series, we show that every function meromorphic on a domain $G$ can be expanded into a series of rational functions, whose partial sums have universal ... [more ▼]

Motivated by known results about universal Taylor series, we show that every function meromorphic on a domain $G$ can be expanded into a series of rational functions, whose partial sums have universal approximation properties on arbitrary compact sets $K \subset G^c$. [less ▲]