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Common Universal Meromorphic Functions for Translation and Dilation Mappings Meyrath, Thierry in Computational Methods and Function Theory (in press) We consider translation and dilation mappings acting on the spaces of meromorphic functions on the complex plane and the punctured complex plane, respectively. In both cases, we show that there is a dense ... [more ▼] We consider translation and dilation mappings acting on the spaces of meromorphic functions on the complex plane and the punctured complex plane, respectively. In both cases, we show that there is a dense $G_{\delta}$-subset of meromorphic functions that are common universal for certain uncountable families of these mappings. While a corresponding result for translations exists for entire functions, our result for dilations has no holomorphic counterpart. We further obtain an analogue of Ansari’s Theorem for the mappings we consider, which is used as a key tool in the proofs of our main results. [less ▲] Detailed reference viewed: 11 (0 UL)Non-normality, topological transitivity and expanding families Meyrath, Thierry ; in Mathematical Proceedings of the Cambridge Philosophical Society (in press) 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 ▲] Detailed reference viewed: 34 (1 UL)On unavoidable families of meromorphic functions Meyrath, Thierry in Canadian Mathematical Bulletin (in press) We prove several results on unavoidable families of meromorphic functions. For instance, we give new examples of families of cardinality three that are unavoidable with respect to the set of meromorphic ... [more ▼] We prove several results on unavoidable families of meromorphic functions. For instance, we give new examples of families of cardinality three that are unavoidable with respect to the set of meromorphic functions on $\C$. We further obtain families consisting of less than three functions that are unavoidable with respect to certain subsets of meromorphic functions. In the other direction, we show that for every meromorphic function $f$, there exists an entire function that avoids $f$ on $\C$. [less ▲] Detailed reference viewed: 55 (9 UL)Compositionally universal meromorphic functions Meyrath, Thierry in Complex Variables and Elliptic Equations (2019), 64(9), 1534-1545 For a sequence of holomorphic maps $(\vp_n)$ from a domain $\Omega_2$ to a domain $\Omega_1$, we consider meromorphic functions $f$ on $\Omega_1$ for which the sequence of compositions $(f \circ \vp_n ... [more ▼] For a sequence of holomorphic maps $(\vp_n)$ from a domain $\Omega_2$ to a domain $\Omega_1$, we consider meromorphic functions $f$ on $\Omega_1$ for which the sequence of compositions $(f \circ \vp_n)$ is dense in the space of all meromorphic functions on $\Omega_2$, endowed with the topology of spherically uniform convergence on compact subsets. We generalize and unify several known results about universal meromorphic functions and provide new examples of sequences of holomorphic maps, for which there exist universal meromorphic functions. We also consider meromorphic functions that have in some sense a maximally erratic boundary behavior in general domains $\Omega \subset \C, \Omega \neq \C$. As a corollary, we obtain that meromorphic functions on general domains are generically non-extendable. [less ▲] Detailed reference viewed: 128 (10 UL)GRACE era variability in the Earth’s oblateness: A comparison of estimates from six different sources Meyrath, Thierry ; ; van Dam, Tonie in Geophysical Journal International (2017), 208(2), 1126-1138 We study fluctuations in the degree-2 zonal spherical harmonic coefficient of the Earth's gravity potential, $C_{20}$, over the period 2003-2015. This coefficient is related to the Earth's oblateness and ... [more ▼] We study fluctuations in the degree-2 zonal spherical harmonic coefficient of the Earth's gravity potential, $C_{20}$, over the period 2003-2015. This coefficient is related to the Earth's oblateness and studying its temporal variations, $\Delta C_{20}$, can be used to monitor large-scale mass movements between high and low latitude regions. We examine $\Delta C_{20}$ inferred from six different sources, including satellite laser ranging (SLR), GRACE and global geophysical fluids models. We further include estimates that we derive from measured variations in the length-of-day (LOD), from the inversion of global crustal displacements as measured by GPS, as well as from the combination of GRACE and the output of an ocean model as described by \cite{sunetal2016}. We apply a sequence of trend- and seasonal moving average filters to the different time series in order to decompose them into an interannual, a seasonal and an intraseasonal component. We then perform a comparison analysis for each component, and we further estimate the noise level contained in the different series using an extended version of the three-cornered-hat method. For the seasonal component, we generally obtain a very good agreement between the different sources, and except for the LOD-derived series, we find that over 90\% of the variance in the seasonal components can be explained by the sum of an annual and semiannual oscillation of constant amplitudes and phases, indicating that the seasonal pattern is stable over the considered time period. High consistency between the different estimates is also observed for the intraseasonal component, except for the solution from GRACE, which is known to be affected by a strong tide-like alias with a period of about 161 days. Estimated interannual components from the different sources are generally in agreement with each other, although estimates from GRACE and LOD present some discrepancies. Slight deviations are further observed for the estimate from the geophysical models, likely to be related to the omission of polar ice and groundwater changes in the model combination we use. On the other hand, these processes do not seem to play an important role at seasonal and shorter time scales, as the sum of modelled atmospheric, oceanic and hydrological effects effectively explains the observed $C_{20}$ variations at those scales. We generally obtain very good results for the solution from SLR, and we confirm that this well-established technique accurately tracks changes in $C_{20}$. Good agreement is further observed for the estimate from the GPS inversion, showing that this indirect method is successful in capturing fluctuations in $C_{20}$ on scales ranging from intra- to interannual. Obtaining accurate estimates from LOD, however, remains a challenging task and more reliable models of atmospheric wind fields are needed in order to obtain high-quality $\Delta C_{20}$, in particular at the seasonal scale. The combination of GRACE data and the output of an ocean model appears to be a promising approach, particularly since corresponding $\Delta C_{20}$ is not affected by tide-like aliases, and generally gives better results than the solution from GRACE, which still seems to be of rather poor quality. [less ▲] Detailed reference viewed: 314 (23 UL)Seasonal low-degree changes in terrestrial water mass load from global GNSS measurements Meyrath, Thierry ; van Dam, Tonie ; et al 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 ▲] Detailed reference viewed: 233 (15 UL)A comparison of interannual hydrological polar motion excitation from GRACE and geodetic observations Meyrath, Thierry ; van Dam, Tonie 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 ▲] Detailed reference viewed: 180 (13 UL)Mixing Taylor shifts and universal Taylor series ; Meyrath, Thierry ; in Bulletin of the London Mathematical Society (2015), 47 It is known that, generically, Taylor series of functions holomorphic in a simply connected complex domain exhibit maximal erratic behaviour outside the domain (so-called universality) and overconvergence ... [more ▼] It is known that, generically, Taylor series of functions holomorphic in a simply connected complex domain exhibit maximal erratic behaviour outside the domain (so-called universality) and overconvergence in parts of the domain. Our aim is to show how the theory of universal Taylor series can be put into the framework of linear dynamics. This leads to a unified approach to universality and overconvergence and yields new insight into the boundary behaviour of Taylor series. [less ▲] Detailed reference viewed: 94 (2 UL)Seasonal Variations of Low-degree Spherical Harmonics Derived from GPS Data and Loading Models Wei, Na ; van Dam, Tonie ; Weigelt, Matthias et al Scientific Conference (2014, September 30) Detailed reference viewed: 221 (11 UL)Limit functions of discrete dynamical systems ; Meyrath, Thierry ; in Conformal Geometry and Dynamics (2014), 18 In the theory of dynamical systems, the notion of ω-limit sets of points is classical. In this paper, the existence of limit functions on subsets of the underlying space is treated. It is shown that in ... [more ▼] In the theory of dynamical systems, the notion of ω-limit sets of points is classical. In this paper, the existence of limit functions on subsets of the underlying space is treated. It is shown that in the case of topologically mixing systems on appropriate metric spaces (X, d), the existence of at least one limit function on a compact subset A of X implies the existence of plenty of them on many supersets of A. On the other hand, such sets necessarily have to be small in various respects. The results for general discrete systems are applied in the case of Julia sets of rational functions and in particular in the case of the existence of Siegel disks. [less ▲] Detailed reference viewed: 86 (4 UL)An assessment of degree-2 Stokes coefficients from Earth rotation data Meyrath, Thierry ; van Dam, Tonie ; Weigelt, Matthias et al 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 ▲] Detailed reference viewed: 204 (14 UL)On the behaviour of the successive derivatives of meromorphic functions on the final set Meyrath, Thierry ; in Journal d'analyse mathématique (2013), 120(1), 131-149 We study the behaviour of the sequence of successive derivatives of meromorphic functions at points of the so-called final set. We prove that, whereas in many cases this sequence tends to ∞, for a special ... [more ▼] We study the behaviour of the sequence of successive derivatives of meromorphic functions at points of the so-called final set. We prove that, whereas in many cases this sequence tends to ∞, for a special class of meromorphic functions, it may have extremely wild behaviour. We also prove a connection between the derivatives of meromorphic functions from this class and so-called Dirichlet sets. [less ▲] Detailed reference viewed: 101 (0 UL)On two classes of universal meromorphic functions Meyrath, Thierry 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 ▲] Detailed reference viewed: 141 (7 UL)Universality properties of Taylor series inside the domain of holomorphy ; Meyrath, Thierry ; in Journal of Mathematical Analysis and Applications (2011), 383(1), 234-238 It is proven that the Taylor series of functions holomorphic in $\C_{\infty} \setminus \{1\}$ generically have certain universality properties on small sets outside the unit disk. Moreover, it is shown ... [more ▼] It is proven that the Taylor series of functions holomorphic in $\C_{\infty} \setminus \{1\}$ generically have certain universality properties on small sets outside the unit disk. Moreover, it is shown that such sets necessarily are polar sets. [less ▲] Detailed reference viewed: 139 (5 UL)On the universality of derived functions of the Riemann zeta-function Meyrath, Thierry in Journal of Approximation Theory (2011), 163(10), 1419-1426 We show that for functions that are universal in the sense of Voronin's theorem, some derived functions automatically share a similar universality property. In particular, this holds for the Riemann zeta ... [more ▼] We show that for functions that are universal in the sense of Voronin's theorem, some derived functions automatically share a similar universality property. In particular, this holds for the Riemann zeta-function ζ and we are thus able to state universal functions of the form F(ζ ). [less ▲] Detailed reference viewed: 119 (7 UL)Universal rational expansions of meromorphic functions Meyrath, Thierry 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 ▲] Detailed reference viewed: 105 (4 UL)Universal distribution of limit points Meyrath, Thierry ; 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 ▲] Detailed reference viewed: 95 (1 UL)Universal meromorphic approximation on Vitushkin sets ; Meyrath, Thierry ; in Journal of Contemporary Mathematical Analysis (2008), 43(6), 365-371 The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ_n} ⊂ \C, there exists a function φ, meromorphic on \C, with the following property. For every ... [more ▼] The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ_n} ⊂ \C, there exists a function φ, meromorphic on \C, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n_k} of \N such that {φ(z + λ_{n_k})} converges to f(z) uniformly on K. A similar result is obtained for arbitrary domains G \neq \C. Moreover, in case {λ_n} = {n} the function φ is frequently universal in terms of Bayart/Grivaux [3]. [less ▲] Detailed reference viewed: 99 (3 UL) |
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