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.

Deictic Demonstratives in Japanese, Finnish and Swedish : First and Third Language Perspectives

The goal of this thesis is twofold. Firstly, it investigates the actual, native use of spatial-deictic demonstratives in Japanese, Finnish and Swedish. Secondly, it investigates and elucidates the interlanguage of Finnish-speaking and Swedish-speaking learners of Japanese regarding their use of Japanese spatial-deictic demonstratives in the light of respective native use and, in comparison to the descriptions of demonstratives in the teaching materials used. Thus, the present study deals with analyses of two sets of empirical data: data produced by native-speaking informants (L1 data) and data produced by language learners (L2 data). These were elicited by Discourse Completion Tasks (DCTs) designed, collected and analyzed using both quantitative and qualitative methods by the author. The results showed that the actual use of demonstratives by the native informants was not always in accordance with the way described in grammars. The typological similarities between Japanese and Finnish were in this study not reflected in the native use of demonstratives, and some uses were not solely based on the spatial relations between the referent, the speaker and the addressee, but rather on social-interactional factors. The main findings regarding the learner data revealed some differences in the usage rate of the demonstratives between the two Finnish-speaking groups and the one Swedish-speaking learner group studied. There were, however, no particular differences found between them regarding the type of demonstrative used. It is suggested that these differences are first and foremost connected both with the teaching materials used and the more or less heterogeneous linguistic environment in which the learners reside, and only thereafter with the typological similarities or differences between their respective native languages, Finnish and Swedish, and the target language, Japanese. It is further argued that the learners’ use of the different Japanese demonstratives, that is the type of demonstrative used, could be explained in terms of familiarity with the grammar. That is, when the situations used in the DCTs were exemplified in teaching materials and were familiar to them, the learners seemed to use Japanese demonstratives as they are described in the teaching materials and as the native Japanese speakers use them. When the situations used in the DCTs were not exemplified in the teaching materials, the learners seem to rely more on their native language. The results, thus, suggest that the learners’ interlanguage is influenced by the grammar of the target language known to the learners, but also by the number of languages (or varieties) that the learners have contact with at the time of learning. The results of the present study have implications for the teaching of Japanese in at least two ways. Firstly, the importance of grammar instruction must be emphasized since its effect on the learners’ language is apparent. Secondly, the contents of teaching materials should be revised on the basis of the native speakers’ actual use of the grammar.