A graded subring of an inverse limit of polynomial rings

We study the power series ring R= K[[x1,x2,x3,...]]on countably infinitely many variables, over a field K, and two particular K-subalgebras of it: the ring S, which is isomorphic to an inverse limit of the polynomial rings in finitely many variables over K, and the ring R', which is the largest graded subalgebra of R. Of particular interest are the homogeneous, finitely generated ideals in R', among them the generic ideals. The definition of S as an inverse limit yields a set of truncation homomorphisms from S to K[x1,...,xn] which restrict to R'. We have that the truncation of a generic I in R' is a generic ideal in K[x1,...,xn]. It is shown in Initial ideals of Truncated Homogeneous Ideals that the initial ideal of such an ideal converge to the initial ideal of the corresponding ideal in R'. This initial ideal need no longer be finitely generated, but it is always locally finitely generated: this is proved in Gröbner Bases in R'. We show in Reverse lexicographic initial ideals of generic ideals are finitely generated that the initial ideal of a generic ideal in R' is finitely generated. This contrast to the lexicographic term order. If I in R' is a homogeneous, locally finitely generated ideal, and if we write the Hilbert series of the truncated algebras K[x1,...,xn] module the truncation of I as qn(t)/(1-t)n, then we show in Generalized Hilbert Numerators that the qn's converge to a power series in t which we call the generalized Hilbert numerator of the algebra R'/I. In Gröbner bases for non-homogeneous ideals in R' we show that the calculations of Gröbner bases and initial ideals in R' can be done also for some non-homogeneous ideals, namely those which have an associated homogeneous ideal which is locally finitely generated. The fact that S is an inverse limit of polynomial rings, which are naturally endowed with the discrete topology, provides S with a topology which makes it into a complete Hausdorff topological ring. The ring R', with the subspace topology, is dense in R, and the latter ring is the Cauchy completion of the former. In Topological properties of R' we show that with respect to this topology, locally finitely generated ideals in R'are closed.

Palaeoenvironment in North-Western Romania during the last 15 000 years

The objectives of this thesis are to establish a chronological framework for environmental changes during the last 15,000 years in northwest Romania, to reconstruct the vegetation development, and to evaluate the underlying processes for forest dynamics. Furthermore, an overview of earlier and ongoing pollenstratigraphic work in Romania is provided. Sediments from two former crater lakes, Preluca Tiganului and Steregoiu, situated in the Gutaiului Mountains, on the western extremity of the Eastern Carpathians at 730 m and 790 m a.s.l., respectively were obtained and analysed for high-resolution pollen, macrofossils, charcoal, mineral magnetic parameters and organic matter. The chronostratigraphic framework was provided by dense AMS 14C measurements. Cold and dry climatic conditions are indicated by the occurrence of open vegetation with shrubs and herbs, and cold lake water prior to 14,700 cal. yr BP. The climatic improvement at the beginning of the Lateglacial interstadial (around 14,700 cal. yr BP) is seen by the development of open forests. These were dominated by Pinus and Betula, but contained also new arriving tree taxa, such as Populus, Alnus and Prunus. The gradual establishment of forests may have led to a stabilization of the soils in the catchment. Between ca. 14,100 and 13,800 cal. yr BP the forest density became reduced to stands of Pinus, Betula, Alnus, Larix and Populus trees and grassland expanded, suggesting colder climatic conditions. Picea arrived as a new taxon at around 13,800 cal. yr BP, and between 13,800 and 12,900 cal. yr BP, the surroundings of the sites were predominantly covered by Picea forest. This forest included Betula, Pinus, Alnus, Larix and Populus and, from 13,200 cal. yr BP onwards also Ulmus. At ca. 12,900 cal. yr BP, the forest became significantly reduced and at 12,600 cal. yr BP, a recurrence of open vegetation with stands of Larix, Pinus, Betula, Salix and Alnus is documented, lasting until 11,500 cal. yr BP. This distinct change in vegetation may by taken as a strong decline in temperature and moisture availability. At the transition to the Holocene, at ca. 11,500 cal. yr BP, Pinus, Betula and Larix quickly expanded (from small local stands) and formed open forests, probably as a response to warmer and more humid climatic conditions. At 11,250 cal. yr BP Ulmus and Picea expanded and the landscape became completely forested. The rapid increase of Ulmus and Picea after 11,500 cal. yr BP may suggest the existence of small residual populations close to the study sites during the preceding cold interval. Ulmus was the first and most prominent deciduous taxa in the early Holocene in the Gutaiului Mountains. From ca. 10,750 cal. yr BP onwards Quercus, Tilia, Fraxinus and Acer expanded and Corylus arrived. A highly diverse, predominantly deciduous forest with Ulmus, Quercus, Tilia, Fraxinus, Acer, Corylus and Picea developed between 10,700 and 8200 cal. yr BP, which possibly signifies more continental climatic conditions. The development of a Picea-Corylus dominated forest between 8200 and 5700 cal. yr BP is likely connected to a more humid and cooler climate. The establishment of Carpinus and Fagus was dated to 5750 cal. yr BP and 5200 cal. yr BP, respectively. The dominance of Fagus during the late Holocene, from 4000 cal. yr BP onwards, may have been related to cooler and more humid climatic conditions. First signs of human activities are recorded around 2300 cal. yr BP, but only during the last 300 years did local human impact become significant. The vegetation development recorded in the Gutaiului Mountains during the Lateglacial is very similar to reconstructions based on lowland sites, whereas higher elevation sites seem not to have always experienced visible vegetation changes. The time of tree arrival and expansion during the past 11,500 cal. yr BP seems to have occurred almost synchronously across Romania. The composition of the forests during the Holocene in the Gutaiului Mountains is consistent with that reconstructed at mid-elevation sites, but differs from the forest composition at higher elevations. Important differences between the Gutaiului Mountains and other studied sites in Romania are a low representation of Carpinus and a late and weak human impact. The available data sets for Romania give evidence for the presence of coniferous and cold-tolerant deciduous trees before 14,700 cal. yr BP. Glacial refugia for Ulmus may have occurred in different parts of Romania, whereas the existence of Quercus, Tilia, Corylus and Fraxinus has not been corroborated.