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.

Alcohol-Related Problems in Eastern Europe : A Comparative Perspective

This thesis investigates the association between alcohol consumption and alcohol-related harm in Eastern Europe. The main aim was to estimate to what extent changes in per capita alcohol consumption have an impact on different forms of alcohol-related mortality, and to put the results in an international comparative perspective. The thesis includes four papers; the first two papers use aggregate time-series analysis to assess how changes in per capita consumption affect rates in suicide mortality and fatal non-intentional injuries in several Eastern European countries, respectively. The third paper applies the same methodological approach to analyse the population-level relationship between alcohol and homicide in Russia and the U.S.. The fourth paper employs survey data to assess how the risk of experiencing alcohol-related problems in relation to volume of consumption in the Baltic countries compares to Sweden and Italy. The results of the first three papers suggests: (i) that changes in per capita consumption are significantly related to changes in mortality rates of suicide, non-intentional injuries and homicide in the countries under study; (ii) that the relationship is stronger for men than for women, and (iii) that the relationship tends to be stronger in the countries with more detrimental drinking patterns, e.g. Russia. The results of the fourth paper suggest that the risk of experiencing alcohol-related problems in relation to level of drinking in the Baltic countries is similar to the corresponding risk in Sweden, but considerably stronger than in Italy. In conclusion, the findings support the significance of a public health approach to alcohol-related problems in Eastern Europe, i.e., policy measures directed towards total alcohol consumption. In addition, strategies aimed at reducing the occurrence of binge drinking seem to have great potential for reducing alcohol-related harm and mortality in Eastern European countries.