Forking in simple theories and CM-triviality

[cat] Aquesta tesi té tres objectius. En primer lloc, estudiem generalitzacions de la jerarquia no ample relatives a una família de tipus parcials. Aquestes jerarquies en permeten classificar la complexitat del “forking” respecte a una família de tipus parcials. Si considerem la família de tipus algebraics, aquestes generalitzacions corresponen a la jerarquia ordinària, on el primer i el segon nivell corresponen a one-basedness i a CM-trivialitat, respectivament. Fixada la família de tipus regulars “no one-based”, el primer nivell d'una d'aquestes possibles jerarquies no ample ens diu que el tipus de la base canònica sobre una realització és analitzable en la família. Demostrem que tota teoria simple amb suficients tipus regulars pertany al primer nivell de la jerarquia dèbil relativa a la família de tipus regulars no one-based. Aquest resultat generalitza una versió dèbil de la “Canonical Base Property” estudiada per Chatzidakis i Pillay. En segon lloc, discutim problemes d'eliminació de hiperimaginaris assumint que la teoria és CM-trivial, en tal cas la independència del “forking” té un bon comportament. Més concretament, demostrem que tota teoria simple CM-trivial elimina els hiperimaginaris si elimina els hiperimaginaris finitaris. En particular, tota teoria petita simple CM-trivial elimina els hiperimaginaris. Cal remarcar que totes les teories omega-categòriques simples que es coneixen són CM-trivials; en particular, aquelles teories obtingudes mitjançant una construcció de Hrushovski. Finalment, tractem problemes de classificació en les teories simples. Estudiem la classe de les teories simples baixes; classe que inclou les teories estables i les teories supersimples de D-rang finit. Demostrem que les teories simples amb pes finit acotat també pertanyen a aquesta classe. A més, provem que tota teoria omega-categòrica simple CM-trivial és baixa. Aquest darrer fet resol parcialment una pregunta formulada per Casanovas i Wagner.[eng] The development of first-order stable theories required two crucial abstract notions: forking independence, and the related notion of canonical base. Forking independence generalizes the linear independence in vector spaces and the algebraic independence in algebraically closed fields. On the other hand, the concept of canonical base generalizes the field of definition of an algebraic variety. The general theory of independence adapted to simple theories, a class of first-order theories which includes all stable theories and other interesting examples such as algebraically closed fields with an automorphism and the random graph. Nevertheless, in order to obtain canonical bases for simple theories, the model-theoretic development of hyperimaginaries --equivalence classes of arbitrary tuple modulo a type-definable (without parameters) equivalence relation-- was required. In the present thesis we deal with topics around the geometry of forking in simple theories. Our first goal is to study generalizations of the non ample hierarchy which will code the complexity of forking with respect to a family of partial types. We introduce two hierarchies: the non (weak) ample hierarchy with respect to a fixed family of partial types. If we work with respect to the family of bounded types, these generalizations correspond to the ordinary non ample hierarchy. Recall that in the ordinary non ample hierarchy the first and the second level correspond to one-basedness and CM-triviality, respectively. The first level of the non weak ample hierarchy with respect to some fixed family of partial types states that the type of the canonical base over a realization is analysable in the family. Considering the family of regular non one-based types, the first level of the non weak ample hierarchy corresponds to the weak version of the Canonical Base Property studied by Chatzidakis and Pillay. We generalize Chatzidakis' result showing that in any simple theory with enough regular types, the canonical base of a type over a realization is analysable in the family of regular non one-based types. We hope that this result can be useful for the applications; for instance, the Canonical Base Property plays an essential role in the proof of Mordell-Lang for function fields in characteristic zero and Manin-Mumford due to Hrushovski. Our second aim is to use combinatorial properties of forking independence to solve elimination of hyperimaginaries problems. For this we assume the theory to be simple and CM-trivial. This implies that the forking independence is well-behaved. Our goal is to prove that any simple CM-trivial theory which eliminates finitary hyperimaginaries --hyperimaginaries which are definable over a finite tuple-- eliminates all hyperimaginaries. Using a result due to Kim, small simple CM-trivial theories eliminate hyperimaginaries. It is worth mentioning that all currently known omega-categorical simple theories are CM-trivial, even those obtained by an ab initio Hrushovski construction. To conclude, we study a classification problem inside simple theories. We study the class of simple low theories, which includes all stable theories and supersimple theories of finite D-rank. In addition, we prove that it also includes the class of simple theories of bounded finite weight. Moreover, we partially solve a question posed by Casanovas and Wagner: Are all omega-categorical simple theories low? We solve affirmatively this question under the assumption of CM-triviality. In fact, our proof exemplifies that the geometry of forking independence in a possible counterexample cannot come from finite sets

Small- scale structure of infaunal polychaete communities in an estuarine environment: Methodological approach

12 páginas, 3 figuras, 2 tablasThis study compares different methods for the estimation of minimal areas (viz. species/area curves, diversity/area curves, similarity/area curves, variance/mean ratio vs. area curves) as community structure descriptions. The comparisons are based upon two polychaete taxocoenoses from muddy and sandy habitats, located in a semienclosed shallow-water Mediterranean bay (Alfacs Bay, Ebro Delta, NW Mediterranean).The mud community appeared to be very homogeneous, with very low diversity. This community displayed high structural simplicity (related to various stress factors), and therefore, qualifies as a physically controlled community). The diversity index was stabilized for areas of 37 cm2, quantitative similarity (Kulcznski index) was higher than 0·7 for areas of 90 cm2, and density of individuals was stabilized for areas of 120 cm2. Therefore, and area of 120 cm2 is suggested as being representative of the community structure. However, it was impossible to define a qualitatively adequate sampling area ( more than 300cm2). The sand community displayed hig structural complexity, with hig species richness and high diversity. This community was characterized by high environmental stability and high variability of microhabitats, as is frequent in biologically accomodated communities. Tne number of individuals became homogeneous for areas of 600-1000 cm2, diversity was stabilized around 300 cm2 and a Kulczynski similarity index of 0·7 was alredy attained at areas of 1000cm2. Thus, a quantitatively representative sampling area of between 700 and 1000 cm2 was suggested. Moreover, the more general pattern of species distribution (with an important set of common species) was directly related to the relatively low qualitative minimal area (400 cm2)Peer reviewe

Higher COVID-19 pneumonia risk associated with anti-IFN-α than with anti-IFN-ω auto-Abs in children

We found that 19 (10.4%) of 183 unvaccinated children hospitalized for COVID-19 pneumonia had autoantibodies (auto-Abs) neutralizing type I IFNs (IFN-alpha 2 in 10 patients: IFN-alpha 2 only in three, IFN-alpha 2 plus IFN-omega in five, and IFN-alpha 2, IFN-omega plus IFN-beta in two; IFN-omega only in nine patients). Seven children (3.8%) had Abs neutralizing at least 10 ng/ml of one IFN, whereas the other 12 (6.6%) had Abs neutralizing only 100 pg/ml. The auto-Abs neutralized both unglycosylated and glycosylated IFNs. We also detected auto-Abs neutralizing 100 pg/ml IFN-alpha 2 in 4 of 2,267 uninfected children (0.2%) and auto-Abs neutralizing IFN-omega in 45 children (2%). The odds ratios (ORs) for life-threatening COVID-19 pneumonia were, therefore, higher for auto-Abs neutralizing IFN-alpha 2 only (OR [95% CI] = 67.6 [5.7-9,196.6]) than for auto-Abs neutralizing IFN-. only (OR [95% CI] = 2.6 [1.2-5.3]). ORs were also higher for auto-Abs neutralizing high concentrations (OR [95% CI] = 12.9 [4.6-35.9]) than for those neutralizing low concentrations (OR [95% CI] = 5.5 [3.1-9.6]) of IFN-omega and/or IFN-alpha 2

Finite groups contain large centralizers

Every finite non-abelian group of order n has a non-central element whose centralizer has order exceeding n^{1/3}. The proof does not rely on the classification of finite simple groups, yet it uses the Feit-Thompson theorem.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu

Macroinfaunal Polychaetes of a Mediterranean, shallow- water bay: A hyerarchical approach

Peer reviewe

Distribución del macrobentos en el Delta del Ebro

VII Simposio Ibérico de Estudios del Bentos Marino, October 1991, Murcia, SpainPeer reviewe

Distribución del meiobentos en el Delta del Ebro

VII Simposio Ibérico de Estudios del Bentos Marino, October 1991, Murcia, SpainPeer reviewe

On the class of flat stable theories

A new notion of independence relation is given and associated to it, the class of flat theories, a subclass of strong stable theories including the superstable ones is introduced. More precisely, after introducing this independence relation, flat theories are defined as an appropriate version of superstability. It is shown that in a flat theory every type has finite weight and therefore flat theories are strong. Furthermore, it is shown that under reasonable conditions any type is non-orthogonal to a regular one. Concerning groups in flat theories, it is shown that type-definable groups behave like superstable ones, since they satisfy the same chain condition on definable subgroups and also admit a normal series of definable subgroup with semi-regular quotients.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu

The dp-rank of abelian groups

An equation to compute the dp-rank of any abelian group is given. It is also shown that its dp-rank, or more generally that of any one-based group, agrees with its Vapnik–Chervonenkis density. Furthermore, strong abelian groups are characterised to be precisely those abelian groups A such that there are only finitely many primes p such that the group A/pA is infinite and for every prime p, there are only finitely many natural numbers n such that (p^n A)[p]/(p^{n + 1} A)[p] is infinite. Finally, it is shown that an infinite stable field of finite dp-rank is algebraically closed.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu