We introduce and study the notions of a PAC-substructure of a stable structure, and a bounded substructure of an arbitrary substructure, generalizing [10]. We give precise definitions and equivalences, saying what it means for properties such as PAC to be first order, study some examples (such as differentially closed fields) in detail, relate the material to generic automorphisms, and generalize a "descent theorem" for pseudo-algebraically closed fields to the stable context. We also point out that the elementary invariants of pseudo-algebraically (...) closed fields from [6] are also valid for pseudo-differentially closed fields. (shrink)

Given a lovely pair P ≺ M of models of a simple theory T, we study the structure whose universe is P and whose relations are the traces on P of definable (in ℒ with parameters from M) sets in M. We give a necessary and sufficient condition on T (which we call weak lowness) for this structure to have quantifier-elimination. We give an example of a non-weakly-low simple theory.

Care of the critically ill newborn includes support for the birth mother/parents with regular updates around the clinical condition of the baby, and involvement in discussions around complex decision-making issues. Discussions around continuation or discontinuation of life-sustaining are challenging even in the most straightforward of cases, but what happens when the birth mother is critically unwell? Such cases can lead to uncertainty around who should assume the parental role for these difficult discussions. In this round table discussion, we explore the (...) ethical, moral and legal uncertainties raised by coincident severe maternal and neonatal illness in the context of surrogacy. (shrink)

For G a group definable in some structure M, we define notions of “definable” compactification of G and “definable” action of G on a compact space X , where the latter is under a definability of types assumption on M. We describe the universal definable compactification of G as View the MathML source and the universal definable G-ambit as the type space SG. We also point out the existence and uniqueness of “universal minimal definable G-flows”, and discuss issues of amenability (...) and extreme amenability in this definable category, with a characterization of the latter. For the sake of completeness we also describe the universal compactification and universal G-ambit in model-theoretic terms, when G is a topological group. (shrink)

We introduce the notion of a lovely pair of models of a simple theory T, generalizing Poizat's “belles paires” of models of a stable theory and the third author's “generic pairs” of models of an SU-rank 1 theory. We characterize when a saturated model of the theory TP of lovely pairs is a lovely pair , finding an analog of the nonfinite cover property for simple theories. We show that, under these hypotheses, TP is also simple, and we study forking (...) and canonical bases in TP. We also prove that assuming only that T is low, the existentially universal models of the universal part of a natural expansion TP+ of TP, are lovely pairs, and “simple Robinson universal domains”. (shrink)

We develop a basic theory of rosy groups and we study groups of small Uþ-rank satisfying NIP and having finitely satisfiable generics: Uþ-rank 1 implies that the group is abelian-by-finite, Uþ-rank 2 implies that the group is solvable-by-finite, Uþ-rank 2, and not being nilpotent-by-finite implies the existence of an interpretable algebraically closed field.

Adequate measurement is an essential component of the assessment of mental health disorders and symptoms such as depression and anxiety. The present study investigated sex-specific differences in the Patient Health Questionnaire-9 and Generalized Anxiety Disorder-7. This comprehensive cross-sectional design study pursued four objectives: measurement invariance of PHQ-9 and GAD-7 between male and female; depression and anxiety prevalence differences; cross-sex differences in the relationship between depression and anxiety; and a comparison of symptom heterogeneity. A sample of 1966 students from South Africa (...) completed the PHQ-9 and the GAD-7. Data analyses for measurement invariance, latent class analysis, inter-variable correlations and group comparisons were conducted in Mplus. The two-dimensional PHQ-9 achieved scalar invariance, while the GAD-7 yielded metric invariance. The somatic and non-somatic latent dimensions of depression were compared and showed no significant difference between male and female groups. The positive relationship between depression and anxiety was also not significantly different between the two groups. While the PHQ-9 symptoms formed three classes in the male group, and four classes in the female group, the GAD-7 had the same number of classes and a similar pattern between the two groups. These findings hold implications for the measurement, assessment and understanding of symptom manifestation and distribution, as well as the treatment of depression and anxiety in South Africa. (shrink)

We study structures equipped with generic predicates and/or automorphisms, and show that in many cases we obtain simple theories. We also show that a bounded PAC field is simple. 1998 Published by Elsevier Science B.V. All rights reserved.

We prove here:Theorem. LetT be a countable complete superstable non ω-stable theory with fewer than continuum many countable models. Then there is a definable groupG with locally modular regular generics, such thatG is not connected-by-finite and any type inG eq orthogonal to the generics has Morley rank.Corollary. LetT be a countable complete superstable theory in which no infinite group is definable. ThenT has either at most countably many, or exactly continuum many countable models, up to isomorphism.

Let G be a group definable in an o-minimal structure M. In this paper we show: Theorem. If G is a two-dimensional definably connected nonabelian group, then G is centerless and G is isomorphic to R+R*>0, for some real closed field R. Theorem. If G is a three-dimensional nonsolvable, centerless, definably connected group, then either G SO3 or G PSL2, for some real closed field R.

We study the groups Gal L and Gal KP, and the associated equivalence relations EL and EKP, attached to a first order theory T. An example is given where EL≠ EKP. It is proved that EKP is the composition of EL and the closure of EL. Other examples are given showing this is best possible.

A lay movement known as the Poor of Christ, incorrectly referred to as the Waldensians, started in the Roman Church in 1176 and rapidly spread through Europe despite severe persecution by the church. Through their values and methods, they impacted on the communities where they were present. This article aims to show their role and contributions indicating its impact on the 16th century Reformation and relevance for the church today. It does this by examining selected themes from the Poor of (...) Christ: their life-transforming faith, the strengthening of laity, the equal role of women in ministry, the importance of teaching and living the faith, a focused rather than fenced theology and loyalty to the one Church of Christ. The article also draws reference to the COVID-19 pandemic and explores what we can learn from the Poor of Christ in addressing this situation.Contribution: This article puts the spotlight on an overlooked lay movement that contributed significantly in preparing the climate for the 16th Century Reformation. Their example shows that the real strength of church is through members empowered to holistically live their faith. The article shows the current relevance of their values and methods. (shrink)

This article assesses the shortcomings and the disconnectedness of the current academic theological education in South Africa. It offers a brief history to provide a guiding principle for academic theology. It then proceeds to show the current disconnect and challenges between academic theology and the church, with its primary focus on academic theology. Drawing on original research and reflection on these responses, commodification, euro-centricity and rankings are seen as three traps of modern academics. These three areas have distorted the true (...) content of theological reflection. This article thus clearly highlights the current problem and motivates the need for academic theology and the local church to reconnect with each other. With this article focussing on academic institutions, it calls for the academy to change not in nature but in content, and to draw its content for the local context. (shrink)

We prove (Proposition 2.1) that if $\mu$ is a generically stable measure in an NIP (no independence property) theory, and $\mu(\phi(x,b))=0$ for all $b$ , then for some $n$ , $\mu^{(n)}(\exists y(\phi(x_{1},y)\wedge \cdots \wedge\phi(x_{n},y)))=0$ . As a consequence we show (Proposition 3.2) that if $G$ is a definable group with fsg (finitely satisfiable generics) in an NIP theory, and $X$ is a definable subset of $G$ , then $X$ is generic if and only if every translate of $X$ does not (...) fork over $\emptyset$ , precisely as in stable groups, answering positively an earlier problem posed by the first two authors. (shrink)

In this paper we construct a non-CM-trivial stable theory in which no infinite field is interpretable. In fact our theory will also be trivial and ω-stable, but of infinite Morley rank. A long term aim would be to find a nonCM-trivial theory which has finite Morley rank (or is even strongly minimal) and does not interpret a field. The construction in this paper is direct, and is a “3-dimensional” version of the free pseudoplane. In a sense we are cheating: the (...) original point of the notion ofCM-triviality was to describe the geometry of a strongly minimal set, or even of a regular type. In our example, non-CM-triviality will come from the behaviour of three orthogonal regular types.A stable theory is said to beCM-trivial if wheneverA⊆Band acl(Ac) ∩ acl(B) = acl(A) inTeq, then Cb(stp(c/A)) ⊆ Cb(stp(c/B)). ( An infinite stable field will not beCM-trivial.) The notion is due to Hrushovski [3], where he gave several equivalent definitions, as well as showing that his new strongly minimal sets constructed “ab ovo” wereCM-trivial. The notion was studied further in [6] where it was shown thatCM-trivial groups of finite Morley rank are nilpotent-by-finite. These results were generalized in various ways to the superstable case in [8]. (shrink)

A theory T is called almost [MATHEMATICAL SCRIPT CAPITAL N]0-categorical if for any pure types p1,…,pn there are only finitely many pure types which extend p1 ∪…∪pn. It is shown that if T is an almost [MATHEMATICAL SCRIPT CAPITAL N]0-categorical theory with I = 3, then a dense linear ordering is interpretable in T.