Homomorphisms from functional equations: the Goldie equation

The theory of regular variation, in its Karamata and Bojani´c-Karamata/de Haan forms, is long established and makes essential use of the Cauchy functional equation. Both forms are subsumed within the recent theory of Beurling regular variation, developed elsewhere. Various generalizations of the Cauchy equation, including the Gołab–Schinzel functional equation (GS) and Goldie's equation (GBE) below, are prominent there. Here we unify their treatment by algebraicization: extensive use of group structures introduced by Popa and Javor in the 1960s turn all the various (known) solutions into homomorphisms, in fact identifying them 'en passant', and show that (GS) is present everywhere, even if in a thick disguise

Back to the Eneolithic: Exploring the Rudki-type ornaments from Poland

For a long time, the Eneolithic attribution of the Rudki-type double spiral ornaments was contested by a wide academic audience, and therefore, this new and extraordinary category of the copper metalwork seemed to have fallen into scientific oblivion. In this paper, we contribute to the debate about cultural attribution of the Rudki-type double spiral ornaments considering their chemical and isotope characteristics (using ED XRF and MC-ICP-MS) and the manufacturing technology (OM, X-ray, CT). Noticeably, this study represents the first documented implementation of the lead isotope analysis (LIA) for the Eneolithic metalwork from Poland. The new scientific analyses give ground to the hypothesis that the Rudki-type double spiral ornaments were produced by the Baden culture metalworker(s) who practiced somewhere in the Carpathian Basin and who have used copper ore mined in the Slovak Ore Mountains (Spania Dolina-Banska Bystrica-Kremnica mine complex). These ornaments were redistributed towards the northern ecumene of the Baden culture complex. The new owners, the Funnel Beaker (TRB) culture communities from the region of modern Poland, deposited the ornaments in hoards (Kaldus, Przeuszyn and Rudki) during the mid-4th millennium BC. The results, furthermore, indicate that the so-called Baden spiral metalwork package must be now complemented by the Rudki-type double spiral ornaments. Remarkably, this package also found an echo in pottery decoration, as documented by a narrative scene incised on an amphora from Kaldus, which could be also interpreted as one of the earliest known proofs for the wagon transport in Europe, alongside the famous ones reported from Bronocice or Flintbek.Material Culture Studie

Orthogonalities and functional equations

In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations

Analysis of medieval mtDNA from Napole cemetery provides new insights into the early history of Polish state

Contemporary historical anthropology and classical archaeology are concerned not only with such fundamental issues as the origins of ancient human populations and migration routes, but also with the formation and development of inter-population relations and the mixing of gene pools as a result of inter-breeding between individuals representing different cultural units. The contribution of immigrants to the analysed autochthonous population and their effect on the gene pool of that population has proven difficult to evaluate with classical morphological methods. The burial of one individual in the studied Napole cemetery located in central Poland had the form of a chamber grave, which is typical of Scandinavian culture from that period. However, this fact cannot be interpreted as absolute proof that the individual (in the biological sense) was allochtonous. This gives rise to the question as to who was actually buried in that cemetery. The ancient DNA results indicate that one of the individuals had an mtDNA haplotype typical of Iron Age northern Europe, which suggests that he could have arrived from that area at a later period. This seems to indirectly confirm the claims of many anthropologists that the development of the early medieval Polish state was significantly and directly influenced by the Scandinavians

Quartz luminescence applied in palaeoenvironmental reconstruction of a dune

Optically stimulated luminescence (OSL) and thermoluminescence (TL) dating were applied for studying the evolution of a dune at the archaeological site of Kałdus (Lower Vistula Valley, Poland), where excavation revealed a settlement sequence. The dating results are supported by investigations of optical bleaching characteristics of the dune quartz. The luminescence studies presented here are a part of a wider interdisciplinary project studying the dynamics of the local geomorphology and its relationship to human activity at the site from prehistoric times until early medieval ages