In this paper, we shall survey results about the group-theoretic properties of stable groups. These can be classified into three main categories, according to the strength of the assumptions needed: chain conditions, generic types, and some form of rank. Each category has its typical application: Chain conditions often allow us to deduce global properties from local ones, generic properties are used to get definable groups from undefinable ones, and rank is necessary to interpret fields in certain group actions. While originally (...) the main input came from algebraic group theory, founded on the similarities between rank and dimension, increasingly methods borrowed from finite group theory have come to play a rôle, emphasizing the study of characteristic subgroups and characteristic families of subgroups. (shrink)

A minimal field of non-zero characteristic is algebraically closed.

The following theorems are proved about the Frattini-free componentG Φ of a soluble stable ℜ-group: a) If it has a normal subgroupN with nilpotent quotientG Φ/N, then there is a nilpotent subgroupH ofG Φ withG Φ=NH. b) It has Carter subgroups; if the group is small, they are all conjugate. c) Nilpotency modulo a suitable Frattini-subgroup (to be defined) implies nilpotency. The last result makes use of a new structure theorem for the centre of the derivative of the Frattini-free component (...) of a centreless soluble ℜ-group. (shrink)

We define an $\mathfrak{R}$-group to be a stable group with the property that a generic element can only be algebraic over a generic. We then derive some corollaries for $\mathfrak{R}$-groups and fields, and prove a decomposition theorem and a field theorem. As a nonsuperstable example, we prove that small stable groups are $\mathfrak{R}$-groups.

We define a generalized version of CM-triviality, and show that in the presence of enough regular types, or solubility, a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial stable group is abelian and connected. The first result makes use of a generalized version of the analysis of bad groups.

We define an R-group to be a stable group with the property that a generic element (for any definable transitive group action) can only be algebraic over a generic. We then derive some corollaries for R-groups and fields, and prove a decomposition theorem and a field theorem. As a nonsuperstable example, we prove that small stable groups are R-groups.

We show that a stable groupG satisfying certain commutator conditions is nilpotent. Furthermore, a soluble stable group with generically splitting automorphism of prime order is nilpotent-by-finite. In particular, a soluble stable group with a generic element of prime order is nilpotent-by-finite.

We prove that a stable solvable group G which satisfies xn = 1 generically is of finite exponent dividing some power of n. Furthermore, G is nilpotent-by-finite. A second result is that in a stable group of finite exponent, involutions either have big centralisers, or invert a subgroup of finite index (which hence has to be abelian).

We define a reasonably well-behaved class of ultraimaginaries, i.e. classes modulo [Formula: see text]-invariant equivalence relations, called tame, and establish some basic simplicity-theoretic facts. We also show feeble elimination of supersimple ultraimaginaries: If [Formula: see text] is an ultraimaginary definable over a tuple [Formula: see text] with [Formula: see text], then [Formula: see text] is eliminable up to rank [Formula: see text]. Finally, we prove some uniform versions of the weak canonical base property.

An infinite field with only countably many pure types is algebraically closed.

We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimension. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that a pseudofinite $\widetilde {\mathfrak M}_c$ -group of finite positive dimension contains a finite-by-abelian subgroup of positive dimension, and a pseudofinite group of dimension 2 contains a soluble subgroup of dimension 2.

We generalise various properties of quasiendomorphisms from groups with regular generic to small abelian groups. In particular, for a small abelian group such that no infinite definable quotient is connected-by-finite, the ring of quasi-endomorphisms is locally finite. Under some additional assumptions, it decomposes modulo some nil ideal into a sum of finitely many matrix rings.

If * is a binary partial function which happens to be a group law on some infinite subset of some model of a stable theory, then this subset can be embedded into a definable group such that * becomes the group operation.

We prove that a stable solvable group $G$ which satisfies $x^n = 1$ generically is of finite exponent dividing some power of $n$. Furthermore, $G$ is nilpotent-by-finite. A second result is that in a stable group of finite exponent, involutions either have big centralisers, or invert a subgroup of finite index (which hence has to be abelian).

This document presents the Bonn PRINTEGER Consensus Statement: Working with Research Integrity—Guidance for research performing organisations. The aim of the statement is to complement existing instruments by focusing specifically on institutional responsibilities for strengthening integrity. It takes into account the daily challenges and organisational contexts of most researchers. The statement intends to make research integrity challenges recognisable from the work-floor perspective, providing concrete advice on organisational measures to strengthen integrity. The statement, which was concluded February 7th 2018, provides guidance on (...) the following key issues: § 1.Providing information about research integrity § 2.Providing education, training and mentoring § 3.Strengthening a research integrity culture § 4.Facilitating open dialogue § 5.Wise incentive management § 6.Implementing quality assurance procedures § 7.Improving the work environment and work satisfaction § 8.Increasing transparency of misconduct cases § 9.Opening up research § 10.Implementing safe and effective whistle-blowing channels § 11.Protecting the alleged perpetrators § 12.Establishing a research integrity committee and appointing an ombudsperson § 13.Making explicit the applicable standards for research integrity. (shrink)

We study local strengthenings of the simplicity condition. In particular, we define and study a local Lascar rank, as well as short, low, supershort and superlow theories. An example of a low, non supershort theory is given.

A review of fraud in medical research in Britain, Europe, the USA and Australia. It includes a history of known cases of fraud since 1974 and discusses ways for detecting and dealing with fraud that have been devised by government agencies, pharmaceutical companies, academic institutions and scientific publications (especially medical journals).

America frequently talks about impeaching a president, but the impeachment provisions of the American constitution are widely misunderstood. In High Crimes and Misdemeanors, constitutional scholar Frank O. Bowman, III offers unprecedented clarity to the question of impeachment, tracing its roots to medieval England through its adoption in the Constitution and 250 years of American experience. By examining the human and political history of those who have faced impeachment, Bowman demonstrates that the Framers intended impeachment to be a flexible tool, (...) adaptable to the needs of any age. Written in a lively, engaging style, the book combines a deep historical and constitutional analysis of the impeachment clauses, a coherent theory of when impeachment should be used to protect constitutional order against presidential misconduct, and a comprehensive presentation of the case for and against impeachment of President Trump. It is an indispensable work for the present moment. (shrink)

An ω-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU-rank. Every definable subgroup is commensurable with an acl( $\emptyset$ )-definable subgroup. Every finitely based regular type in a CM-trivial ω-categorical simple theory is non-orthogonal to a type of SU-rank 1. In particular, a supersimple ω-categorical CM-trivial theory has finite SU-rank.

We reconstruct the group action in the group configuration theorem. We apply it to show that in an ω-categorical theory a finitely based pseudolinear regular type is locally modular, and the geometry associated to a finitely based locally modular regular type is projective geometry over a finite field.

A totally ordered group G is essentially periodic if for every definable non-trivial convex subgroup H of G every definable subset of G is equal to a finite union of cosets of subgroups of G on some interval containing an end segment of H; it is coset-minimal if all definable subsets are equal to a finite union of cosets, intersected with intervals. We study definable sets and functions in such groups, and relate them to the quasi-o-minimal groups introduced in Belegradek (...) et al. . Main results: An essentially periodic group G is abelian; if G is discrete, then definable functions in one variable are ultimately piecewise linear. A group such that every model elementarily equivalent to it is coset-minimal is quasi-o-minimal , and its definable functions in one variable are piecewise linear. (shrink)

Non-$n$-ampleness as defined by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of $\Sigma$-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.

If ${\mathfrak {F}}$ is a type-definable family of commensurable subsets, subgroups or subvector spaces in a metric structure, then there is an invariant subset, subgroup or subvector space commensurable with ${\mathfrak {F}}$. This in particular applies to type-definable or hyper-definable objects in a classical first-order structure.

If G is an omega-stable group with a normal definable subgroup H, then the Sylow-2-subgroups of G/H are the images of the Sylow-2-subgroups of G. /// Sei G eine omega-stabile Gruppe und H ein definierbarer Normalteiler von G. Dann sind die Sylow-2-Untergruppen von G/H Bilder der Sylow-2-Untergruppen von G.

We show that if p is a real type which is almost internal in a formula φ in a simple theory, then there is a type p' interalgebraic with a finite tuple of realizations of p, which is generated over φ. Moreover, the group of elementary permutations of p' over all realizations of φ is type-definable.

1. We show that if p is a real type which is internal in a set $\sigma$ of partial types in a simple theory, then there is a type p' interbounded with p, which is finitely generated over $\sigma$ , and possesses a fundamental system of solutions relative to $\sigma$ . 2. If p is a possibly hyperimaginary Lascar strong type, almost \sigma-internal$ , but almost orthogonal to $\sigma^{\omega}$ , then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts (...) on p while fixing $\sigma$ generically. In case p is $\sigma-internal$ and T is stable, this is the binding group of p over \sigma$. (shrink)

This paper completes the proof of the group configuration theorem for simple theories started in [1]. We introduce the notion of an almost hyperdefinable structure, and show that it has a reasonable model theory. We then construct an almost hyperdefinable group from a polygroup chunk.

In recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or in the $\omega$-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity. The proof necessitated an extension of the model-theoretic framework to include almost hyperimaginaries, and (...) the study of polygroups. (shrink)

A totally ordered group G is called coset-minimal if every definable subset of G is a finite union of cosets of definable subgroups intersected with intervals with endpoints in G{±∞}. Continuing work in Belegradek et al. 1115) and Point and Wagner 261), we study coset-minimality, as well as two weak versions of the notion: eventual and ultimate coset-minimality. These groups are abelian; an eventually coset-minimal group, as a pure ordered group, is an ordered abelian group of finite regular rank. Any (...) pure ordered abelian group of finite regular rank is ultimately coset-minimal and has the exchange property; moreover, every definable function in such a group is piecewise linear. Pure coset-minimal and eventually coset-minimal groups are classified. In a discrete coset-minimal group every definable unary function is piece-wise linear 261), where coset-minimality of the theory of the group was required). A dense coset-minimal group has the exchange property ); moreover, any definable unary function is piecewise linear, except possibly for finitely many cosets of the smallest definable convex nonzero subgroup. Finally, we give some examples and open questions. (shrink)

The integration of ethics into the day-to-day work of research and innovation is an important but difficult challenge. However, with the Aachen method for identification, classification and risk analysis of innovation-based problems an approach from an engineering perspective is presented that enables the integration of ethical, legal and social implications into the day-to-day work of R&I practitioners. AMICAI appears in particular capable of providing a procedural guidance for R&I practitioners based on a method established in engineering science, breaking down the (...) object of consideration into partial aspects and prioritizing the innovation-based problems in dependence of potential risk. This enables the user to apply AMICAI continuously during all stages of the research and development process and to analyze and choose between certain sociotechnical alternatives. In this way, problems that affect ethical, legal, and social aspects can be understood, reflected and considered in the mostly technically focused R&D process. The paper gives a general guidance about AMICAI by describing principles and assumptions, providing the steps of analysis and application aids, giving an example application, explaining the necessary adjustments of AMICAI compared to the methodical basis of failure mode, effects, and criticality analysis and discussing the advantages and limits. AMICAI’s simple applications can stimulate interdisciplinary cooperation in the R&D process and be a starting point for the development of an “open RRI risk analysis platform” allowing society to evaluate innovation-based problems. (shrink)

Introduction Joseph Dunne, Attracta Ingram, Frank Litton This volume of essays has two main objectives: first, to pay tribute to Fergal O'Connor, ...

O'Loughlin, Frank Over recent years there has been a great deal of discussion and disagreement about the nature of marriage and family both in the public sphere and within the church. Almost every aspect of human relationships and sexuality has come up for discussion. The two synods on marriage and the family that took place in Rome in 2014 and 2015 have been a watershed for Catholics in these discussions. Similar discussions will need to continue for some time yet (...) if we are to come to greater clarity and depth in our understanding of the many issues involved. In this article I am seeking to put forward elements of the Christian tradition concerning marriage that, I hope, may be helpful background in these current discussions. (shrink)

Questo libro ha come scopo aiutare gli studenti di filosofia a formare un proprio giudizio filosofico rispetto a Kant. Tutto il testo è strutturato in una serie organicamente connessa di domande la cui risposta deve essere trovata con la ragione. Frank O'Farrell irlandese è stato ordinario per più di trent'anni per la cattedra di metafisica presso la Pontificia Università Gregoriana.