We define and investigate a uniformly locally o-minimal structure of the second kind in this paper. All uniformly locally o-minimal structures of the second kind have local monotonicity, which is a local version of monotonicity theorem of o-minimal structures. We also demonstrate a local definable cell decomposition theorem for definably complete uniformly locally o-minimal structures of the second kind. We define dimension of a definable set and investigate its basic (...) properties when the given structure is a locally o-minimal structure which admits local definable cell decomposition. (shrink)

Consider a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group. Let $f:X \rightarrow R^n$ be a definable map, where X is a definable set and R is the universe of the structure. We demonstrate the inequality $\dim ) \leq \dim $ in this paper. As a corollary, we get that the set of the points at which f is discontinuous is of dimension smaller than $\dim $. We (...) also show that the structure is definably Baire in the course of the proof of the inequality. (shrink)

We consider locally o-minimal structures possessing tame topological properties shared by models of DCTC and uniformly locally o-minimal expansions of the second kind of densely linearly ordered abelian groups. We derive basic properties of dimension of a set definable in the structures including the addition property, which is the dimension equality for definable maps whose fibers are equi-dimensional. A decomposition theorem into quasi-special submanifolds is also demonstrated.

In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and prove that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably simply connected, open definable subsets and, definable paths and definable homotopies on such locally definable manifolds can be lifted to locally definable covering maps. These properties allow us to obtain the main properties of the general o-minimal fundamental group, including: invariance (...) and comparison results; existence of universal locally definable covering maps; monodromy equivalence for locally constant o-minimal sheaves – from which one obtains, as in algebraic topology, classification results for locally definable covering maps, o-minimal Hurewicz and Seifert–van Kampen theorems. (shrink)

We study first-order expansions of ordered fields that are definably complete, and moreover either are locally o-minimal, or have a locally o-minimal open core. We give a characterisation of structures with locally o-minimal open core, and we show that dense elementary pairs of locally o-minimal structures have locally o-minimal open core.

This paper discusses sufficient conditions for a definably complete expansion of a densely linearly ordered abelian group to have uniformly locally o‐minimal open cores of the first/second kind and strongly locally o‐minimal open core, respectively.

Let M=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}=}$$\end{document} be a weakly o-minimal expansion of a dense linear order without endpoints. Some tame properties of sets and functions definable in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} which hold in o-minimal structures, are examined. One of them is the intermediate value property, say IVP. It is shown that strongly continuous definable functions in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} (...) \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} satisfy an extended version of IVP. After introducing a weak version of definable connectedness in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document}, we prove that strong cells in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} are weakly definably connected, so every set definable in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} is a finite union of its weakly definably connected components, provided that M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} has the strong cell decomposition property. Then, we consider a local continuity property for definable functions in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} and conclude some results on cell decomposition regarding that property. Finally, we extend the notion of having no dense graph which was examined for definable functions in and related to uniform finiteness, definable completeness, and others. We show that every weakly o-minimal structure M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} having cell decomposition, satisfies NDG, i.e. every definable function in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} has no dense graph. (shrink)

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o‐minimal structure. This fact together with the results in a previous paper implies a tame dimension theory and a decomposition theorem into good‐shaped definable subsets called quasi‐special submanifolds. Using this fact, we investigate definably complete locally o‐minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly (...) to o‐minimal expansions of ordered fields, Łojasiewicz's inequality, Tietze's extension theorem and affiness of pseudo‐definable spaces hold true for such structures under the extra assumption that the domains of definition and the pseudo‐definable spaces are definably compact. Here, a pseudo‐definable space is a topological space having finite definable atlases. We also demonstrate Michael's selection theorem for definable set‐valued functions with definably compact domains of definition. (shrink)

We define a discrete closure operator for definably complete locally o‐minimal structures. The pair of the underlying set of and the discrete closure operator forms a pregeometry. We define the rank of a definable set over a set of parameters using this fact and call it ‐dimension. A definable set X is of dimension equal to the ‐dimension of X. The structure is simultaneously a first‐order topological structure. The dimension rank of a set definable in the (...) first‐order topological structure also coincides with its dimension. (shrink)

We describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a lattice. We propose a definition of a lattice in an arbitrary first-order structure. We then use it to describe, uniformly, various structure theorems for o-minimal groups, each time recovering a lattice that captures some significant invariant of the group at hand. The analysis first goes through a local level, where a (...) pertinent notion of pregeometry and generic elements is each time introduced. (shrink)

We prove that a function definable with parameters in an o-minimal structure is bounded away from ∞ as its argument goes to ∞ by a function definable without parameters, and that this new function can be chosen independently of the parameters in the original function. This generalizes a result in [1]. Moreover, this remains true if the argument is taken to approach any element of the structure , and the function has limit any element of the (...) class='Hi'>structure. (shrink)

We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o-minimal structures on $(\mathbb {R},<)$ have the property, as do all expansions of $(\mathbb {R},+,\cdot,\mathbb {N})$. Our main analytic-geometric result is that any such expansion of $(\mathbb {R},<,+)$ by Boolean combinations of open sets (of any arities) either is o-minimal or defines an isomorph of $(\mathbb N,+,\cdot )$. We also show that any given expansion (...) of $(\mathbb {R}, <, +,\mathbb {N})$ by subsets of $\mathbb {N}^n$ (n allowed to vary) has the property if and only if it defines all arithmetic sets. Variations arise by considering connected components or quasicomponents instead of path components. (shrink)

In this article, we develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous “multivalued functions.” This generalizes known statements about weakly o-minimal, C-minimal, and P-minimal theories.

Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures.

L’objet de ce texte est de montrer que des fonctions qui appartiennent à une famille vérifiant une propriété de finitude a priori non uniforme sont en fait définissables dans une structure o-minimale.

For a signature L with at least one constant symbol, an L‐structure is called minimal if it has no proper substructures. Let be the set of isomorphism types of minimal L‐structures. The elements of can be identified with ultrafilters of the Boolean algebra of quantifier‐free L‐sentences, and therefore one can define a Stone topology on. This topology on generalizes the topology of the space of n‐marked groups. We introduce a natural ultrametric on, and show that the Stone (...) topology on coincides with the topology of the ultrametric space iff the ultrametric space is compact iff L is locally finite (that is, L contains finitely many n‐ary symbols for any ). As one of the applications of compactness of the Stone topology on, we prove compactness of certain classes of metric spaces in the Gromov‐Hausdorff topology. This slightly refines the known result based on Gromov's ideas that any uniformly totally bounded class of compact metric spaces is precompact. (shrink)

Suppose G is a definably connected, definable group in an o-minimal expansion of an ordered group. We show that the o-minimal universal covering homomorphism equation image: equation image→ G is a locally definable covering homomorphism and π1 is isomorphic to the o-minimal fundamental group π of G defined using locally definable covering homomorphisms.

A structure (M, $ ,...) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any (...) quasi-o-minimal ordered group is abelian; any quasi-o-minimal ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U-rank 1. (shrink)

Wilkie 5 397) proved a “theorem of the complement” which implies that in order to establish the o-minimality of an expansion of with C∞ functions it suffices to obtain uniform bounds on the number of connected components of quantifier free definable sets. He deduced that any expansion of with a family of Pfaffian functions is o-minimal. We prove an effective version of Wilkie's theorem of the complement, so in particular given an expansion of the ordered field with finitely many (...) C∞ functions, if there are uniform and computable upper bounds on the number of connected components of quantifier free definable sets, then there are uniform and computable bounds for all definable sets. In such a case the theory of the structure is effectively o-minimal: there is a recursively axiomatized subtheory such that each of its models is o-minimal. This implies the effective o-minimality of any expansion of with Pfaffian functions. We apply our results to the open problem of the decidability of the theory of the real field with the exponential function. We show that the decidability is implied by a positive answer to the following problem ): given a language L expanding the language of ordered rings, if an L-sentence is true in every L-structure expanding the ordered field of real numbers, then it is true in every o-minimal L-structure expanding any real closed field. (shrink)

Moral principles or the rules of conduct are based in the society. If the purpose of ethics in research is to take into consideration the needs and the rights of the experimental subject, his social milieu must then largely determine the ethical considerations of a projected study. The inability to comprehend such rights may often be due to ignorance, disease and his societal values. Blood letting, biopsy and post-mortem examinations may so conflict with local beliefs that so called 'consent' to (...) these is much more than a surrender of the rights of the individual. It is difficult to conform with the highest principles of ethics in research in any uniform society. It is more so when the many variables are further complicated by cultural demands and social taboos. The best custodians of ethical standards must relate intimately with the norms of the local population. (shrink)

The main result of this paper is Theorem 3.1 which is a criterion for weak o-minimality of a linearly ordered structure in terms of realizations of 1-types. Here we also prove some other properties of weakly o-minimal structures. In particular, we characterize all weakly o-minimal linear orderings in the signature $\{ . Moreover, we present a criterion for density of isolated types of a weakly o-minimal theory. Lastly, at the end of the paper we present some (...) remarks on the Exchange Principle for algebraic closure in a weakly o-minimal structure. (shrink)

In this note I construct a proper o-minimal expansion of the ordered additive group of rationals.

_The purpose_ of the article is to clarify the philosophical principle of the local in the context of modern political governance. _The theoretical basis_ of the research embraces scenario analysis, dialectical and existential approaches, as well as philosophical anthropology and philosophy of communication. Local communities are a specific reflection of the connection between a person and a place. The specifics of the formation of a special mode of being, which forms and reproduces relations of loyalty, mutual understanding, and a common (...) feeling in the locus, are established. _Originality._ The article for the first time focuses on the anthropological content of local existence, which is the matrix of civil culture and the prospects of its integration into the global cultural space. _Conclusions._ Local identity as an object of purposeful formation in the conditions of nation-building requires considering the philosophical provisions that were introduced in the anthropological discourse. At the current stage, Ukrainian civil culture strives to invent specific local meanings of political identity in the conditions of external aggression. Its main components will be consensus, mutual respect, integrity, and the will to unite in order to protect statehood. Interdisciplinary meanings of locality form subject-object relations within which modern political management should be carried out. Taking into account the ethnic, cultural, and socio-functional, and role aspects of locality promises to ensure effective decision-making and increase the well-being of modern local communities. The political context of locality is formed based on the interaction of local political actors and institutions. Their examination within the framework of the case studies approach allows for empirically substantiating the local priorities of political actions. It also indicates the specific functionality of local political and administrative structures. at the local level, the discussed source of locality is the question of the origin or authenticity of the local population. For political governance, this is one of the meaningful contexts for justifying local autonomy or even sovereignty. (shrink)

We introduce and study some local versions of o-minimality, requiring that every definable set decomposes as the union of finitely many isolated points and intervals in a suitable neighbourhood of every point. Motivating examples are the expansions of the ordered reals by sine, cosine and other periodic functions.

We prove, by explicit construction, that not all sets definable in polynomially bounded o-minimal structures have mild parameterization. Our methods do not depend on the bounds particular to the definition of mildness and therefore our construction is also valid for a generalized form of parameterization, which we call G-mild. Moreover, we present a cell decomposition result for certain o-minimal structures which may be of independent interest. This allows us to show how our construction can produce polynomially bounded, model (...) complete expansions of the real ordered field which, in addition to lacking G-mild parameterization, nonetheless still have analytic cell decomposition. (shrink)

Let $\scr{R}$ be an o-minimal structure over a real closed field R. Given a simplicial complex K and some definable subsets S₁,...,S l of its realization $|K|$ in R we prove that there exist a subdivision K' of K and a definable triangulation $\phi ^{\prime}\colon |K^{\prime}|\rightarrow |K|$ of $|K|$ partitioning S₁,...,S l with $\phi ^{\prime}$ definably homotopic to $id_{|K|}$ . As an application of this result we obtain the semialgebraic Hauptvermutung.

Let P be the ω-orbit of a point under a unary function definable in an o-minimal expansion ℜ of a densely ordered group. If P is monotonically cofinal in the group, and the compositional iterates of the function are cofinal at +\infty in the unary functions definable in ℜ, then the expansion (ℜ, P) has a number of good properties, in particular, every unary set definable in any elementarily equivalent structure is a disjoint union of open intervals and (...) finitely many discrete sets. (shrink)

We consider the extent to which certain properties of definably complete structures may persist in structures which are not definably complete, particularly in the weakly o-minimal structures.

A structure is locally finite if every finitely generated substructure is finite; local sentences are universal sentences all models of which are locally finite. The stretching theorem for local sentences expresses a remarkable reflection phenomenon between the finite and the infinite models of local sentences. This result in part requires strong axioms to be proved; it was studied by the second named author, in a paper of this Journal, volume 53. Here we correct and extend this paper; (...) in particular we show that the stretching theorem implies the existence of inaccessible cardinals, and has precisely the consistency strength of Mahlo cardinals of finite order. And we present a sequel due to the first named author: (i) decidability of the spectrum $\operatorname{Sp}(\varphi)$ of a local sentence φ, below ω ω ; where $\operatorname{Sp}(\varphi)$ is the set of ordinals α such that φ has a model of order type α (ii) proof that $\operatorname{beth}_\omega = \sup\{\operatorname{Sp}(\varphi): \varphi \text{local sentence with a bounded spectrum}\}$ (iii) existence of a local sentence φ such that $\operatorname{Sp}(\varphi)$ contains all infinite ordinals except the inaccessible cardinals. (shrink)

By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.

0-categorical o-minimal structures were completely described by Pillay and Steinhorn 565–592), and are essentially built up from copies of the rationals as an ordered set by ‘cutting and copying’. Here we investigate the possible structures which an 0-categorical weakly o-minimal set may carry, and find that there are some rather more interesting examples. We show that even here the possibilities are limited. We subdivide our study into the following principal cases: the structure is 1-indiscernible, in which case (...) all possibilities are classified up to binary structure; the structure is 2-indiscernible, classified up to ternary structure; the structure is 3-indiscernible, in which case we show that it is k-indiscernible for every finite k. We also make some remarks about the possible structures of higher arities which an 0-categorical weakly o-minimal structure may carry. (shrink)

_Purpose._ This article aims at outlining the consider Ukrainian identity in the context of European collective action through the prism of value orientations/approaches. _Theoretical basis._ The following methods were used in order to cover the problem as objectively as possible: historical, analytical, comparative, socio-geographical, behavioral, and dialectical. The use of these methods contributed to tracing the peculiarities of identity and collective action in the dynamics of the historical process and social development. _Originality._ The role of identity in collective action formation (...) is considered. It was established that identity determines the socio-species structure and projects social development. After all, identification for a person in the formation process is a basic need that determines a person’s belonging, not identity. During the formation of "I" there is a combination of "I1" + "I2" + "Other" = "WE". This process is complex and long-term, but it is precisely the process that gives you the opportunity to clearly understand your "I" and know yourself. It was found that identity as a social phenomenon, which is related to the formation of the "I", is determined by certain circumstances, which can be conditionally divided into objective and subjective. After analyzing the scientific literature, a generalization was made that identity is not a new phenomenon, but it has certain features at each stage of social development. This became most noticeable in the conditions of globalization, when countries that are actually different perceive the "Other", unite around certain problems, and then this union forms a system of mutual relations. It is thanks to this approach that alliances, multicultural associations, etc. are formed, and an example of this is the heterogeneous European identity. Special attention is drawn to the role of values, which are the landmarks in the process of identity and the formation of collective action. Based on political, social, economic, historical processes and ontological features, it was established that identity is a dynamic process, since values during a person’s life, the existence of the state, etc. are variables. This is confirmed by the value orientation and various socio-political processes aimed at the formation of Homo in countries that have the status of "between". _Conclusions._ The article illustrates and justifies the existence of several forms of identity according to which the types of identity are formed. The fact proves that identity is not a constant process and varies due to multiple factors: social, political, economic, and religious as well as due to formations – regional, local, civilizational, etc. As practice shows, the problem of identity mostly has situational nature and arises due to the need for unification. The socio-historical process manifests the significance of the values in the formation of a collective European heterogeneous identity. These processes are influenced by many factors: geographical, historical, mythological, etc. (shrink)

A proper elementary extension of a model is called small if it realizes no new types over any finite set in the base model. We answer a question of Marker, and show that it is possible to have an o-minimal structure with a maximal small extension. Our construction yields such a structure for any cardinality. We show that in some cases, notably when the base structure is countable, the maximal small extension has maximal possible cardinality.

This chapter re-examines the evidence for local ecclesiastical and other burial sites in pre-Viking Ireland. It compares local churches and cemetery settlements in pre-Viking Ireland with those found in England and Wales. The chapter describes the density of the pre-Viking ecclesiastical sites in Ireland, church density and social structure in Anglo-Saxon England, and the local ecclesiastical sites in Cornwall and Wales.

It is shown that if is an o-minimal structure such that is a dense total order and ≾ is a parameter-definable partial order on M, then ≾ has an extension to a definable total order.

Let $\scr{M}=\langle M,+,<,0,S\rangle $ be a linear o-minimal expansion of an ordered group, and $G=\langle G,\oplus ,e_{G}\rangle $ an n-dimensional group definable in $\scr{M}$ . We show that if G is definably connected with respect to the t-topology, then it is definably isomorphic to a definable quotient group U/L, for some convex ${\ssf V}\text{-definable}$ subgroup U of $\langle M^{n},+\rangle $ and a lattice L of rank equal to the dimension of the 'compact part' of G.

[John Dupré] This paper attacks some prominent contemporary attempts to provide reductive accounts of ever wider areas of human behaviour. In particular, I shall address the claims of sociobiology (or evolutionary psychology) to provide a universal account of human nature, and attempts to subsume ever wider domains of behaviour within the scope of economics. I shall also consider some recent suggestions as to how these approaches might be integrated. Having rejected the imperialistic ambitions of these approaches, I shall briefly advocate (...) a more pluralistic approach to the understanding of human behaviour, and one which leaves some space for the possibility of genuine human autonomy. /// [John O'Neill] One response to Dupré's criticism of rational choice theory's unifying aspirations is that it is aimed at over-ambitious versions of the theory. Immodesty about the scope of rational choice theory may look more plausible given suitable modesty in assumptions about the rational agent. The paper examines problems with one immodest version of the theory-public choice theory-and show how these shed light on problems in modest versions employing minimal assumptions about the preference structure of rational agents. However, while rational choice theory may fail in its unifying ambitions, I argue those aspirations are defensible. (shrink)

[John Dupré] This paper attacks some prominent contemporary attempts to provide reductive accounts of ever wider areas of human behaviour. In particular, I shall address the claims of sociobiology to provide a universal account of human nature, and attempts to subsume ever wider domains of behaviour within the scope of economics. I shall also consider some recent suggestions as to how these approaches might be integrated. Having rejected the imperialistic ambitions of these approaches, I shall briefly advocate a more pluralistic (...) approach to the understanding of human behaviour, and one which leaves some space for the possibility of genuine human autonomy. /// [John O'Neill] One response to Dupré's criticism of rational choice theory's unifying aspirations is that it is aimed at over-ambitious versions of the theory. Immodesty about the scope of rational choice theory may look more plausible given suitable modesty in assumptions about the rational agent. The paper examines problems with one immodest version of the theory-public choice theory-and show how these shed light on problems in modest versions employing minimal assumptions about the preference structure of rational agents. However, while rational choice theory may fail in its unifying ambitions, I argue those aspirations are defensible. (shrink)

We use a result due to Rolin, Speissegger, and Wilkie to show that definable sets in certain o-minimal structures admit definable parameterizations by mild maps. We then use this parameterization to prove a result on the density of rational points on curves defined by restricted Pfaffian functions.

An o-minimal structure is any relational structure in any relational type in the first order predicate calculus with equality, where one symbol is reserved to be a dense linear ordering without endpoints, satisfying the following condition: that every first order definable subset of the domain is a finite union of intervals whose endpoints are in the domain or are ±•. First order definability always allows any parameters, unless explicitly indicated otherwise.

Let ℜ be an o-minimal expansion of (ℝ, <+) and (φk)k∈ℕ be a sequence of positive real numbers such that limk→+∞f(φk)/φk+1=0 for every f:ℝ→ ℝ definable in ℜ. (Such sequences always exist under some reasonable extra assumptions on ℜ, in particular, if ℜ is exponentially bounded or if the language is countable.) Then (ℜ, (S)) is d-minimal, where S ranges over all subsets of cartesian powers of the range of φ.

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)

We introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy—Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable set can be partitioned into “basic rectifiable sets”, and that the Whitney arc property holds for basic rectifiable sets.

We propose a notion of -minimality for partially ordered structures. Then we study -minimal partially ordered structures such that is a Boolean algebra. We prove that they admit prime models over arbitrary subsets and we characterize -categoricity in their setting. Finally, we classify -minimal Boolean algebras as well as -minimal measure spaces.

The traditional image of northern Iberian mountain settlements is that they are largely egalitarian, homogeneous, and survivals of archaic forms of 'agrarian collectivism'. In this book, based both on extensive fieldwork and detailed study of local records, Brian Juan O'Neill offers a different perspective, questioning prevailing views on both empirical as well as theoretical and methodological grounds. Through a detailed examination of three major areas of social life - land tenure, cooperative labour exchanges, and marriage and inheritance practices - in (...) one particular hamlet, the author demonstrates the predominance of forms of institutionalized economic inequality and social differentiation within the peasantry. Situating the local study within a wider European and Mediterranean ethnographic and geographical framework, O'Neill offers a refreshing and challenging way of combining the research methods of anthropology with those of social and economic history. His book will appeal to anthropologists, historians, sociologists, geographers and demographers interested in the present and past social structure of European village communities, as well as to those concerned with the growing links between anthropology and history. (shrink)

Background Ensuring individual free and informed decision‐making for research participation is challenging. It is thought that preliminarily informing communities through ‘community sensitization’ procedures may improve individual decision‐making. This study set out to assess the relevance of community sensitization for individual decision‐making in research participation in rural Gambia. Methods This anthropological mixed‐methods study triangulated qualitative methods and quantitative survey methods in the context of an observational study and a clinical trial on malaria carried out by the Medical Research Council Unit Gambia. (...) Results/discussion Although 38.7% of the respondents were present during sensitization sessions, 91.1% of the respondents were inclined to participate in the trial when surveyed after the sensitization and prior to the informed consent process. This difference can be explained by the informal transmission of information within the community after the community sensitization, expectations such as the benefits of participation based on previous research experiences, and the positive reputation of the research institute. Commonly mentioned barriers to participation were blood sampling and the potential disapproval of the household head. Conclusion Community sensitization is effective in providing first‐hand, reliable information to communities as the information is cascaded to those who could not attend the sessions. However, further research is needed to assess how the informal spread of information further shapes people's expectations, how the process engages with existing social relations and hierarchies (e.g. local political power structures; permissions of heads of households) and how this influences or changes individual consent. (shrink)