In [12], P. Scowcroft and L. van den Dries proved a cell decomposition theorem for p-adically closed fields. We work here with the notion of P-minimal fields defined by D. Haskell and D. Macpherson in [6]. We prove that a P-minimal field K admits cell decomposition if and only if K has definable selection. A preprint version in French of this result appeared as a prepublication [8].

In this paper we develop a differential analogue of o-minimal cell decomposition for the theory CODF of closed ordered differential fields. Thanks to this differential cell decomposition we define a well-behaving dimension function on the class of definable sets in CODF. We conclude this paper by proving that this dimension is closely related to both the usual differential transcendence degree and the topological dimension associated, in this case, with a natural differential topology on ordered differential fields.

We prove that forp-optimal fields a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strongp-minimality. Then we turn to stronglyp-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to ap-adic one. For such fieldsK, we prove that every definable subset ofK×Kdwhose fibers overKare inverse images by the valuation of subsets of the value group is (...) semialgebraic. Combining the two we get a preparation theorem for definable functions onp-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are in definable bijection iff they have the same dimension. (shrink)

C-minimality is a variant of o-minimality in which structures carry, instead of a linear ordering, a ternary relation interpretable in a natural way on set of maximal chains of a tree. This notion is discussed, a cell-decomposition theorem for C-minimal structures is proved, and a notion of dimension is introduced. It is shown that C-minimal fields are precisely valued algebraically closed fields. It is also shown that, if certain specific ‘bad’ functions are not definable, then algebraic closure has (...) the exchange property, and for definable sets dimension coincides with the rank obtained from algebraic closure. (shrink)

Strong cell decomposition property has been proved in non-valuational weakly o-minimal expansions of ordered groups. In this note, we show that all o-minimal traces have strong cell decomposition property. Also after introducing the notion of irrational nonvaluational cut in arbitrary o-minimal structures, we show that every expansion of o-minimal structures by irrational nonvaluational cuts is an o-minimal trace.

We develop a notion of cell decomposition suitable for studying weak p-adic structures definable). As an example, we consider a structure with restricted addition.

Continuous extension cell decomposition in o-minimal structures was introduced by Simon Andrews to establish the open cell property in those structures. Here, we define strong $\mathrm{CE}$-cells in weakly o-minimal structures, and prove that every weakly o-minimal structure with strong cell decomposition has $\mathrm{SCE}$-cell decomposition if and only if its canonical o-minimal extension has $\mathrm{CE}$-cell decomposition. Then, we show that every weakly o-minimal structure with $\mathrm{SCE}$-cell decomposition satisfies $\mathrm{OCP}$. Our last (...) result implies that every o-minimal structure in which every definable open set is a union of finitely many open $\mathrm{CE}$-cells, has $\mathrm{CE}$-cell decomposition. (shrink)

We study a reduct ${\mathcal{L}_*}$ of the ring language where multiplication is restricted to a neighbourhood of zero. The language is chosen such that for p-adically closed fields K, the ${\mathcal{L}_*}$ -definable subsets of K coincide with the semi-algebraic subsets of K. Hence structures (K, ${\mathcal{L}_*}$ ) can be seen as the p-adic counterpart of the o-minimal structure of semibounded sets. We show that in this language, p-adically closed fields admit cell decomposition, using cells similar to p-adic semi-algebraic (...) cells. From this we can derive quantifier-elimination, and give a characterization of definable functions. In particular, we conclude that multiplication can only be defined on bounded sets, and we consider the existence of definable Skolem functions. (shrink)

We obtain a p-adic semilinear cell decomposition theorem using methods developed by Denef in [Journal fur die Reine und Angewandte Mathematik, vol. 369 (1986), pp. 154-166]. We also prove that any set definable with quantifiers in (0, 1, +, =, λq, Pn){n∈N,q∈Qp} may be defined without quantifiers, where λq is scalar multiplication by q and Pn is a unary predicate which denotes the nonzero nth powers in the p-adic field Qp. Such a set is called a p-adic semilinear (...) set in this paper. Some further considerations are discussed in the last section. (shrink)

In [T. Brihaye, C. Michaux, C. Rivière, Cell decomposition and dimension function in the theory of closed ordered differential fields, Ann. Pure Appl. Logic .] the authors proved a cell decomposition theorem for the theory of closed ordered differential fields which generalizes the usual Cell Decomposition Theorem for o-minimal structures. As a consequence of this result, a well-behaving dimension function on definable sets in CODF was introduced. Here we continue the study of this (...) class='Hi'>cell decomposition in CODF by proving three additional results. We first discuss the relation between the δ-cells introduced in the above-mentioned reference and the notion of Kolchin polynomial in differential algebra. We then prove two generalizations of classical decomposition theorems in o-minimal structures. More exactly we give a theorem of decomposition into definably d-connected components and a differential cell decomposition theorem for a particular class of definable functions in CODF. (shrink)

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)

Let $$ {\mathcal {M}}=(M, <, \ldots ) $$ be a weakly o-minimal structure. Assume that $$ {\mathcal {D}}ef({\mathcal {M}})$$ is the collection of all definable sets of $$ {\mathcal {M}} $$ and for any $$ m\in {\mathbb {N}} $$, $$ {\mathcal {D}}ef_m({\mathcal {M}}) $$ is the collection of all definable subsets of $$ M^m $$ in $$ {\mathcal {M}} $$. We show that the structure $$ {\mathcal {M}} $$ has the strong cell decomposition property if and only if (...) there is an o-minimal structure $$ {\mathcal {N}} $$ such that $$ {\mathcal {D}}ef({\mathcal {M}})=\{Y\cap M^m: \ m\in {\mathbb {N}}, Y\in {\mathcal {D}}ef_m({\mathcal {N}})\} $$. Using this result, we prove that: (a) Every induced structure has the strong cell decomposition property. (b) The structure $$ {\mathcal {M}} $$ has the strong cell decomposition property if and only if the weakly o-minimal structure $$ {\mathcal {M}}^*_M $$ has the strong cell decomposition property. Also we examine some properties of non-valuational weakly o-minimal structures in the context of weakly o-minimal structures admitting the strong cell decomposition property. (shrink)

An analysis of two heuristic strategies for the development of mechanistic models, illustrated with historical examples from the life sciences. In Discovering Complexity, William Bechtel and Robert Richardson examine two heuristics that guided the development of mechanistic models in the life sciences: decomposition and localization. Drawing on historical cases from disciplines including cell biology, cognitive neuroscience, and genetics, they identify a number of "choice points" that life scientists confront in developing mechanistic explanations and show how different choices result (...) in divergent explanatory models. Describing decomposition as the attempt to differentiate functional and structural components of a system and localization as the assignment of responsibility for specific functions to specific structures, Bechtel and Richardson examine the usefulness of these heuristics as well as their fallibility—the sometimes false assumption underlying them that nature is significantly decomposable and hierarchically organized. When Discovering Complexity was originally published in 1993, few philosophers of science perceived the centrality of seeking mechanisms to explain phenomena in biology, relying instead on the model of nomological explanation advanced by the logical positivists (a model Bechtel and Richardson found to be utterly inapplicable to the examples from the life sciences in their study). Since then, mechanism and mechanistic explanation have become widely discussed. In a substantive new introduction to this MIT Press edition of their book, Bechtel and Richardson examine both philosophical and scientific developments in research on mechanistic models since 1993. (shrink)

Research in many fields of biology has been extremely successful in decomposing biological mechanisms to discover their parts and operations. It often remains a significant challenge for scientists to recompose these mechanisms to understand how they function as wholes and interact with the environments around them. This is true of the eukaryotic cell. Although initially identified in nineteenth-century cell theory as the fundamental unit of organisms, researchers soon learned how to decompose it into its organelles and chemical constituents (...) and have been highly successful in understanding how these carry out many operations important to life. The emphasis on decomposition is particularly evident in modern cell biology, which for the most part has viewed the cell as merely a locus of the mechanisms responsible for vital phenomena. The cell, however, is also an integrated system and for some explanatory purposes it is essential to recompose it and understand it as an organized whole. I illustrate both the virtues of decomposition (treating the cell as a locus) and recomposition (treating the cell as an object) with recent work on circadian rhythms. Circadian researchers have both identified critical intracellular operations that maintain endogenous oscillations and have also addressed the integration of cells into multicellular systems in which cells constitute units. Ó 2010 Elsevier Ltd. All rights reserved. (shrink)

We introduce CE- cell decomposition , a modified version of the usual o-minimal cell decomposition. We show that if an o-minimal structure $\mathcal{R}$ admits CE-cell decomposition then any definable open set in $\mathcal{R}$ may be expressed as a finite union of definable open cells. The dense linear ordering and linear o-minimal expansions of ordered abelian groups are examples of such structures.

Analogies to machines are commonplace in the life sciences, especially in cellular and molecular biology — they shape conceptions of phenomena and expectations about how they are to be explained. This paper offers a framework for thinking about such analogies. The guiding idea is that machine-like systems are especially amenable to decompositional explanation, i.e., to analyses that tease apart underlying components and attend to their structural features and interrelations. I argue that for decomposition to succeed a system must exhibit (...) causal orderliness, which I explicate in terms of differentiation among parts and the significance of local relations. I also discuss what makes a model depict its target as machine-like, suggesting that a key issue is the degree of detail with respect to the target’s parts and their interrelations. (shrink)

Biological atomism postulates that all life is composed of elementary and indivisible vital units. The activity of a living organism is thus conceived as the result of the activities and interactions of its elementary constituents, each of which individually already exhibits all the attributes proper to life. This paper surveys some of the key episodes in the history of biological atomism, and situates cell theory within this tradition. The atomistic foundations of cell theory are subsequently dissected and discussed, (...) together with the theory’s conceptual development and eventual consolidation. This paper then examines the major criticisms that have been waged against cell theory, and argues that these too can be interpreted through the prism of biological atomism as attempts to relocate the true biological atom away from the cell to a level of organization above or below it. Overall, biological atomism provides a useful perspective through which to examine the history and philosophy of cell theory, and it also opens up a new way of thinking about the epistemic decomposition of living organisms that significantly departs from the physicochemical reductionism of mechanistic biology. (shrink)

Developing models of biological mechanisms, such as those involved in respiration in cells, often requires collaborative effort drawing upon techniques developed and information generated in different disciplines. Biochemists in the early decades of the 20th century uncovered all but the most elusive chemical operations involved in cellular respiration, but were unable to align the reaction pathways with particular structures in the cell. During the period 1940-1965 cell biology was emerging as a new discipline and made distinctive contributions to (...) understanding the role of the mitochondrion and its component parts in cellular respiration. In particular, by developing techniques for localizing enzymes or enzyme systems in specific cellular components, cell biologists provided crucial information about the organized structures in which the biochemical reactions occurred. Although the idea that biochemical operations are intimately related to and depend on cell structures was at odds with the then-dominant emphasis on systems of soluble enzymes in biochemistry, a reconceptualization of energetic processes in the 1960s and 1970s made it clear why cell structure was critical to the biochemical account. This paper examines how numerous excursions between biochemistry and cell biology contributed a new understanding of the mechanism of cellular respiration. (shrink)

Among emotions, surprise has been extensively studied in psychology. In linguistics, surprise, like other emotions, has mainly been studied through the syntactic patterns involving surprise lexemes. However, little has been done so far to correlate the reaction of surprise investigated in psychological approaches and the effects of surprise on language. This cross-disciplinary volume aims to bridge the gap between emotion, cognition and language by bringing together nine contributions on surprise from different backgrounds - psychology, human-agent interaction, linguistics. Using different methods (...) at different levels of analysis, all contributors concur in defining surprise as a cognitive operation and as a component of emotion rather than as a pure emotion. Surprise results from expectations not being met and is therefore related to epistemicity. Linguistically, there does not exist an unequivocal marker of surprise. Surprise may be either described by surprise lexemes, which are often associated with figurative language, or it may be expressed by grammatical and syntactic constructions. Originally published as a special issue of Review of Cognitive Linguistics 13:2 (2015). (shrink)

We introduce a very weak language L M on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language L M are trivial functions. We also give a definitional expansion $L\begin{array}{*{20}{c}} ' \\ M \\ \end{array} $ of L M in which K has quantifier elimination, (...) and we obtain a cell decomposition result for L M -definable sets. Our language L M can serve as a p-adic analogue of the very weak language (<) on the real numbers, to define a notion of minimality on the field of p-adic numbers and on related valued fields. These fields are not necessarily Henselian and may have positive characteristic. (shrink)

Nous étudions dans cet article les différentes formes de prédication seconde détachée en position initiale dans un corpus comparable composé de textes d’économie en anglais et en français. Ce corpus a été annoté sous le logiciel Analec. L’enjeu est de montrer en quoi un même phénomène syntaxique est exploité, sur le plan discursif, de façon divergente dans chacune des deux langues. Une étude qualitative et quantitative des prédications secondes du corpus montre que la prédication seconde prend le plus souvent la (...) forme d’une participiale en –ing en anglais, où elle se spécialise dans un rôle méta-énonciatif de balise textuelle pour le lecteur, ce qui n’est pas le cas en français. (shrink)

This study presents a sociological profile of Peruvian female industrialists and some narratives of their struggle for personal independence and entrepreneurial success within the context of global restructuring and local deindustrialization. The study adopts the classical definition of woman's economic independence as a result of women's participation in the world of paid labor.

Surprise is treated as an affect in Aristotelian philosophy as well as in Cartesian philosophy. In experimental psychology, surprise is considered to be an emotion. In phenomenology, it is only addressed indirectly (Husserl, Heidegger, Levinas), with the important exception of Ricoeur and Maldiney; it is reduced to a break in cognition by cognitivists (Dennett). Only recently was it broached in linguistics, with a focus on lexico-syntactic categories. As for the expression of surprise, it has been studied in connection with evidentiality (...) in languages that encode surprise morphosyntactically. However, how surprise is encoded in languages that lack an evidential morphosyntactic system has been largely unexplored. This book provides new insights into the dynamics of surprise based on a heuristic hypothesis tested against the investigation of time, language and emotion. It is intended to arouse the interest of a multidisciplinary audience keen on crossing the disciplinary borders of phenomenology, cognitive sciences, and pragmatics. The theoretical approaches adopted in this collection of articles rely on experiments and corpus data. They advance knowledge by building on robust empirical results coming from psychology, microphenomenology, linguistics and physiology. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We show that every definable nested family of closed and bounded subsets of a P-minimal field K has nonempty intersection. As an application we answer a question of Darnière and Halupczok showing that P-minimal fields satisfy the “extreme value property”: for every closed and bounded subset [math] and every interpretable continuous function [math] (where [math] denotes the value group), f(U) admits a maximal value. Two further corollaries are obtained as a (...) consequence of their work. The first one shows that every interpretable subset of [math] is already interpretable in the language of rings, answering a question of Cluckers and Halupczok. This implies in particular that every P-minimal field is polynomially bounded. The second one characterizes those P-minimal fields satisfying a classical cell preparation theorem as those having definable Skolem functions, generalizing a result of Mourgues. (shrink)

We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger (...) structures within the Presburger language. (shrink)

A weakly o-minimal structure image expanding an ordered group is called nonvaluational iff for every cut left angle bracketC,Dright-pointing angle bracket of definable in image, we have that inf{y−x:xset membership, variantC,yset membership, variantD}=0. The study of nonvaluational weakly o-minimal expansions of real closed fields carried out in [D. Macpherson, D. Marker, C. Steinhorn,Weakly o-minimal structures and real closed fields, Trans. Amer. Math. Soc. 352 5435–5483. MR1781273 (2001i:03079] suggests that this class is very close to the class of o-minimal expansions of (...) real closed fields. Here we further develop this analogy. We establish an o-minimal style cell decomposition for weakly o-minimal non-valuational expansions of ordered groups. For structures enjoying such a strong cell decomposition we construct a canonical o-minimal extension. Finally, we make attempts towards generalizing the o-minimal Euler characteristic to the class of sets definable in weakly o-minimal structures with the strong cell decomposition property. (shrink)

We introduce a new notion of tame geometry for structures admitting an abstract notion of balls. The notion is named b-minimality and is based on definable families of points and balls. We develop a dimension theory and prove a cell decomposition theorem for b-minimal structures. We show that b-minimality applies to the theory of Henselian valued fields of characteristic zero, generalizing work by Denef–Pas [25, 26]. Structures which are o-minimal, v-minimal, or p-minimal and which satisfy some slight extra (...) conditions are also b-minimal, but b-minimality leaves more room for nontrivial expansions. The b-minimal setting is intended to be a natural framework for the construction of Euler characteristics and motivic or p-adic integrals. The b-minimal cell decomposition is a generalization of concepts of Cohen [11], Denef [15], and the link between cell decomposition and integration was first made by Denef [13]. (shrink)

The topology of definable sets in an o-minimal expansion of a group is not fully understood due to the lack of a triangulation theorem. Despite the general validity of the cell decomposition theorem, we do not know whether any definably compact set is a definable CW-complex. Moreover the closure of an o-minimal cell can have arbitrarily high Betti numbers. Nevertheless we prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group (...) are finitely generated and invariant under elementary extensions and expansions of the language. (shrink)

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)

Let be a weakly o‐minimal structure with the strong cell decomposition property. In this note, we show that the canonical o‐minimal extension of is the unique prime model of the full first order theory of over any set. We also show that if two weakly o‐minimal structures with the strong cell decomposition property are isomorphic then, their canonical o‐minimal extensions are isomorphic too. Finally, we show the uniqueness of the prime models in a complete weakly o‐minimal (...) theory with prime models. (shrink)

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)

O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as André–Oort conjecture. Among the many tools developed in an o-minimal setting are cohomology theories for abstract-definable continuous manifolds such as singular cohomology, sheaf cohomology and Čech cohomology, which have been used for instance to prove Pillay’s conjecture concerning definably compact groups. In the present thesis we elaborate an o-minimal de Rham cohomology theory for abstract-definable $C^{\infty (...) }$ manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function. We can specify the o-minimal cohomology groups and attain some properties such as the existence of Mayer–Vietoris sequence and the invariance under abstract-definable $C^{\infty }$ diffeomorphisms. However, in order to obtain the invariance of our o-minimal cohomology under abstract-definable homotopy we must work in a tame context that defines sufficiently many primitives and assume the validity of a statement related to Bröcker’s question.Abstract prepared by Rodrigo Figueiredo.E-mail: [email protected]: https://doi.org/10.11606/T.45.2019.tde-28042019-181150. (shrink)