For a commutative ring R, we define the notions of deformed Picard algebroids and deformed twisted differential operators on a smooth, separated, locally of finite type R-scheme and prove these are in a natural bijection. We then define the pullback of a sheaf of twisted differential operators that reduces to the classical definition when R = ℂ. Finally, for modules over twisted differential operators, we prove a theorem for the descent under a locally trivial torsor.

According to the written historical sources, the Gepids were a Germanic tribe that settled in the Carpathian Basin during the Migration Period. They were allies of the Huns, and an independent Gepid Kingdom arose after the collapse of the Hun Empire. In this period, the Carpathian Basin was characterized by so-called row-grave cemeteries. Due to the scarcity of historical and archaeological data, we have a poor knowledge of the origin and composition of these barbarian populations, and this is still a subject of debate. To better understand the genetic legacy of migration period societies, we obtained 46 full mitogenome sequences from three Gepid cemeteries located in Transylvania, Romania. The studied samples represent the Classical Gepidic period and illustrate the genetic make-up of this group from the late 5th and early 6th centuries AD, which is characterized by cultural markers associated with the Gepid culture in Transylvania. The genetic structure of the Gepid people is explored for the first time, providing new insights into the genetic makeup of this archaic group. The retrieved genetic data showed mainly the presence of Northwestern European mitochondrial ancient lineages in the Gepid group and all population genetic analyses reiterated the same genetic structure, showing that early ancient mitogenomes from Europe were the major contributors to the Gepid maternal genetic pool.

The geographic area broadly corresponding to the Upper Basin of Tisa delineates the north-eastern extremity of the Carpathian Basin, which has given the evolutions taking place there over time specific characteristics. Regardless of the historical period, this area has been a connecting space between the regions north of the Carpathian Mountains and territories situated in the direction of the Superior Danube, but mostly the entire Tisa Plain and the Transylvanian Basin towards the south-east. There are many settlements that can be dated roughly to the second half of the 6 th century and the first half of the 7 th century, alongside some funerary discoveries. However, there are few sites that were investigated extensively, at least according to current publication records. The inventories of the dwellings and of the few reported graves are lacking in diversity as handmade pottery is the norm. The current examination offers indirect proof of the agricultural activities and the domestic crafts that were undertaken there at the time, which were potentially connected to a certain degree of specialization in tool and iron utensil production, and the manufacturing of the raw matter this required. A simple, autarchic economic model can be reconstructed from the data as there are few indications of external contacts -thus, a model similar to the one commonly attributed to the Slavs of that period.

There are few flat cremation graves in the geographic area of the upper basin of the Tisa River which can be dated between the second half of the 6th century and the first half of the 7th century, and none of them are recent findings. The current article re-examines these graves and makes new observations in relation to their location, the internal topography of the graveyards (or smaller clusters of graves), the appearance of the graves and the scarce inventories, and, last but not least, their chronological placement.

We consider the problem of describing Gröbner-Shirshov bases for free associative algebras in finite terms. To this end we consider parametrised elements of an algebra and give methods for working with them which under favourable conditions lead to a basis given by finitely many patterns. On the negative side we show that in general there can be no algorithm. We relate our study to the problem of verifying that a given set of words in certain groups yields Bokut' normal forms (or groups with a standard basis).