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.

Wolfgangi Bolyai de Bolya Tentamen iuventutem studiosam in elementa matheseos purae elementaris ac sublimioris methodo intuitiva evidentiaque huic propria introducendi, cum appendice triplici.

With reproduction of original title-pages.

Competition and fixation of cohorts of adaptive mutations under Fisher geometrical model

One of the simplest models of adaptation to a new environment is Fisher's Geometric Model (FGM), in which populations move on a multidimensional landscape defined by the traits under selection. The predictions of this model have been found to be consistent with current observations of patterns of fitness increase in experimentally evolved populations. Recent studies investigated the dynamics of allele frequency change along adaptation of microbes to simple laboratory conditions and unveiled a dramatic pattern of competition between cohorts of mutations, i.e., multiple mutations simultaneously segregating and ultimately reaching fixation. Here, using simulations, we study the dynamics of phenotypic and genetic change as asexual populations under clonal interference climb a Fisherian landscape, and ask about the conditions under which FGM can display the simultaneous increase and fixation of multiple mutations-mutation cohorts-along the adaptive walk. We find that FGM under clonal interference, and with varying levels of pleiotropy, can reproduce the experimentally observed competition between different cohorts of mutations, some of which have a high probability of fixation along the adaptive walk. Overall, our results show that the surprising dynamics of mutation cohorts recently observed during experimental adaptation of microbial populations can be expected under one of the oldest and simplest theoretical models of adaptation-FGM.

Synteza i właściwości oksiranowych pochodnych estrów kwasów fosfonowych

Wydział Chemii

Geometriundervisning enligt Van Hieles utvecklingsnivåer : En kvalitativ studie om överensstämmelsen mellan lärares planering, genomförande och uppfattning av geometriundervisning i årskurs 3

Denna studie har undersökt överensstämmelsen mellan lärares planering, genomförande och uppfattning av geometriundervisning i årskurs 3. Bakgrunden till undersökningen är att elever i årskurs 4 uppvisar låga resultat i geometri. Bakgrunden är även att lärare som fokuserar på kommunikation, samarbete och utmaningar i geometriundervisningen kan se till att elever får arbeta med annat än enskilt räknande av rutinuppgifter, något som studier efterfrågar. Utifrån detta behövs en studie som undersöker lärares geometriundervisning närmare, gärna ur flera perspektiv. I denna studie samlades data om två lärares geometriundervisning in genom observationer, intervjuer och innehållsanalyser. Dataanalysen har utgått från Van Hieles utvecklingsnivåer. Resultatet visar att lärarna föredrar en balans mellan enskilt arbete med rutinuppgifter och alternativ till detta men har ett omedvetet fokus på olika geometriska aspekter vid varje stadie som tillhör undervisningen (planering, genomförande, uppfattning). Konsekvenserna blir exempelvis att en Van Hiele-nivå som ska bearbetas enligt planering inte alls berörs i genomförandet av lektionen, samt att nivåer hoppas över. Det finns även utmaningar i att skapa balans mellan matematik och aktivitet samt en avsaknad av geometriskt innehåll i en del läroböcker. Elever i årskurs 4 kanske har låga resultat i geometri för att nivåer av geometrilärande hoppas över i undervisningen?