Abstract. The linear complexity L2(G ) of a finite group G is the minimal number of additions, subtractions and multiplications by complex constants of absolute value <2 sufficient to evaluate a suitable Fourier transform of ~G. Combining and modifying several classical FFT-algorithms, we show that Lz(G ) =< 81G I log21GI for any finite metabelian group G.
Biological invasions by alien species have substantial economic impacts and are a major driver of the ongoing decline and loss of biodiversity. Through humans, the North American pond slider (Trachemys scripta) has acquired a global distribution over the last decades and is currently listed among the worst invasive reptile species. However, in more recent times, other freshwater chelonian species have increasingly been recorded far outside their native distribution ranges as well, not only on the same continent but also on others. Despite that, the impact of alien chelonians on their respective new ecosystems remains unclear. The long-term effects and severity of impacts of alien populations mostly depend on whether they ultimately succeed in establishing themselves. This is not entirely resolved for chelonians in Central Europe. To answer that, we investigated wild populations of three non-native chelonian species from North America in Germany (Pseudemys concinna, Graptemys pseudogeographica and Trachemys scripta) applying population genetic approaches. We revealed the successful reproduction of all three species in Germany and provide the very first record for the reproduction of P. concinna and G. pseudogeographica in a temperate continental climate zone outside their native distribution. Based on our unambiguous evidence of natural reproduction, we call for dedicated studies to verify how widespread established populations are and to investigate the existing and potential impacts of all three species in a range of ecosystems along a climatic gradient. Such data is urgently needed to revise the current risk assessments of non-native chelonians, especially in Central European countries.
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