We prove linear and polynomial growth properties of sets and functions that are existentially definable in the ordered group of integers with divisibility. We determine the laws of addition with order and divisibility.

We prove linear and polynomial growth properties of sets and functions that are existentially definable in the ordered group of integers with divisibility. We determine the laws of addition with order and divisibility.

We prove linear and polynomial growth properties of sets and functions that are existentially definable in the ordered group of integers with divisibility. We determine the laws of addition with order and divisibility.

Van den Dries, L. and J. Holly, Quantifier elimination for modules with scalar variables, Annals of Pure and Applied Logic 57 161–179. We consider modules as two-sorted structures with scalar variables ranging over the ring. We show that each formula in which all scalar variables are free is equivalent to a formula of a very simple form, uniformly and effectively for all torsion-free modules over gcd domains . For the case of Presburger arithmetic with scalar variables the result takes (...) a still simpler form, and we derive in this way the polynomial-time decidability of the sets defined by such formulas. (shrink)

I solve here some problems left open in “T-convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular,Twill denote a completeo-minimal theory extending RCF, the theory of real closed fields. Let (,V) ⊨Tconvex, let=V/m(V)be the residue field, with residue class mapx↦:V↦, and let υ:→ Γ be the associated valuation. “Definable” will mean “definable with parameters”.The main goal of this article is to determine the structure induced by(,V)on its residue fieldand on its (...) value group Γ. In [9] we expanded the ordered fieldto a model ofTas follows. Take a tame elementary substructure′ ofsuch thatR′ ⊆VandR′maps bijectively ontounder the residue class map, and make this bijection into an isomorphism′ ≌ofT-models. (We showed such′ exists, and that this gives an expansion ofto aT-model that is independent of the choice of′.). (shrink)

We extend the field of Laurent series over the reals in a canonical way to an ordered differential field of “logarithmic-exponential series” , which is equipped with a well behaved exponentiation. We show that the LE-series with derivative 0 are exactly the real constants, and we invert operators to show that each LE-series has a formal integral. We give evidence for the conjecture that the field of LE-series is a universal domain for ordered differential algebra in Hardy fields. We define (...) composition of LE-series and establish its basic properties, including the existence of compositional inverses. Various interesting subfields of the field of LE-series are also considered. (shrink)

For any group satisfying a suitable chain condition, we construct a finitely additive measure on it that is invariant under certain actions.

In lieu of an abstract, here is a brief excerpt of the content:BreathingLuk Van den Dries (bio)This text, "Breathing," was conceived for the book From Act to Acting: Fabre's Guidelines for the Performer of the 21st Century (2021). The book was conceived and designed by Jan Fabre, author, theatre artist, and visual artist, active since the 1970s. The book was written by Luk Van den Dries, dramaturg and theatre researcher of the University of Antwerp, in tight collaboration with (...) Jan Fabre and the three most important performers/teachers of the Belgian theatre company, Troubleyn (Annabelle Chambon, Cédric Charron, and Ivana Jozic). The book sheds new light on the physical, mental, and vocal training of the performer and furthermore gives insight into Fabre's revolutionary thoughts on contemporary theatre.1Breathing is the first of the ten performative principles that together create the basic grammar in which the language of the performers is grounded.Three-phased breathing (one part in, one part out, one part nothing) is a basic condition in the series of exercises Jan Fabre developed throughout his career. This respiration technique returns in all exercises. Also, it is the first principle that the performers are taught: by dividing breathing into three equal parts, energy is built up and stress or fatigue is canalized. It is a way to rebalance, to find a place of rest after a strenuous effort, or it can also be used as a springboard to a heightened performance. Very regularly during individual exercises, Fabre will refer to this respiration technique, especially in specific situations of high adrenaline, but of course the performers should learn to use this respiration technique autonomously. They should become so familiar with the rhythm of this breathing that they automatically fall back on it when necessary. Because this respiration technique is so crucial to Fabre's acting pedagogy, it is described in detail as the very first exercise in his book. Its basic principles are explained below as well, as the first performative principle.Breathing is a sign of life, and on this primal and essential level it needs to be understood that, through breathing, the performer gives life. He breathes life into the performance. Fabre himself has described this basic condition for the birth of theatre through breathing very precisely: [End Page 30]The most important weapon of the warrior of beauty is his respiration. When the performance starts, life stops. Life itself dies. The warrior of beauty is alive, but life itself is dead. The warrior of beauty remains with just his own breath. When he enters the stage, he needs to breathe life into deceased life. Like a god gives his breath to create life. In the same way he will create life with his breath. How will he achieve this? By breathing correctly: one-third inhaling through the nostrils, one-third exhaling through pursed lips, one-third without breathing. And then the miracle happens. The breath becomes imagination and the imagination breathes. The warrior of beauty is the master of breath and life.2Fabre's abdominal breathing differs from regular forms of breathing, such as techniques derived from Asian yoga techniques, Chi-kung or Tai-chi, as it stresses different parts. At the same time, it is closely related to it, because these practices also adopt the notion that breathing in the first place rouses vitality and spirit and that the right kind of breathing thus aims at controlling life currents and bringing about a clear mind. In yoga it is called pranayama: prana meaning spirit or force of life, ayama meaning 'control' or 'grasp.' Yoga knows many variants in directing the breath, depending on the asanas (yoga postures) in which they are bedded. Sometimes focusing mainly on the diaphragm and the flanks, sometimes on abdominal breathing. Also, suspending your breath (kumbhaka) is a yoga technique, applied at the end of inhalation as well as after exhalation and during taking breath.In Fabre's case, abdominal breathing is stressed: first, air is inhaled through the nose, subsequently sent in the direction of the peritoneal cavity and the flanks, and then again exhaled via a very precise blowing technique (a dosed flow of air through an imaginary straw). During the cycle the diaphragm remains... (shrink)

Related Works: Original Paper: Lou Van Den Dries. $T$-Convexity and Tame Extensions II. J. Symbolic Logic, Volume 62, Issue 1 , 14--34. Project Euclid: euclid.jsl/1183745182.

The Euclidean algorithm on the natural numbers ℕ = {0,1,…} can be specified succinctly by the recursive programwhere rem is the remainder in the division of a by b, the unique natural number r such that for some natural number q,It is an algorithm from the remainder function rem, meaning that in computing its time complexity function cε, we assume that the values rem are provided on demand by some “oracle” in one “time unit”. It is easy to prove thatMuch (...) more is known about cε, but this simple-to-prove upper bound suggests the proper formulation of the Euclidean's optimality among its peers—algorithms from rem:Conjecture. If an algorithm α computes gcd from rem with time complexity cα, then there is a rational number r > 0 such that for infinitely many pairs a > b > 1, cα > r log2a. (shrink)

The Euclidean algorithm on the natural numbers ℕ = {0,1,…} can be specified succinctly by the recursive programwhere rem is the remainder in the division of a by b, the unique natural number r such that for some natural number q,It is an algorithm from the remainder function rem, meaning that in computing its time complexity function cε, we assume that the values rem are provided on demand by some “oracle” in one “time unit”. It is easy to prove thatMuch (...) more is known about cε, but this simple-to-prove upper bound suggests the proper formulation of the Euclidean's optimality among its peers—algorithms from rem:Conjecture. If an algorithm α computes gcd from rem with time complexity cα, then there is a rational number r > 0 such that for infinitely many pairs a > b > 1, cα > r log2a. (shrink)

The differential field of transseries extends the field of real Laurent series and occurs in various contexts: asymptotic expansions, analytic vector fields, and o-minimal structures, to name a few. We give an overview of the algebraic and model-theoretic aspects of this differential field and report on our efforts to understand its elementary theory.

L. van den Dries proved that the theory of n-valued rings has a model companion. We show here that this result is still true when the valuation rings are required to satisfy given inclusion relations (we restrict ourselves to the case of residual characteristic zero).

Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T. We give a quantifier elimination relative to T for the theory of pairs (R, V) where $\mathscr{R} \models T$ and V ≠ R is the convex hull of an elementary substructure of R. We deduce (...) that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to T for the theory of pairs (R, N) with R a model of T and N a proper elementary substructure that is Dedekind complete in R. We deduce that the theory of such "tame" pairs is complete. (shrink)

I solve here some problems left open in “T-convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular,Twill denote a completeo-minimal theory extending RCF, the theory of real closed fields. Let (,V) ⊨Tconvex, let=V/m(V)be the residue field, with residue class mapx↦:V↦, and let υ:→ Γ be the associated valuation. “Definable” will mean “definable with parameters”.The main goal of this article is to determine the structure induced by(,V)on its residue fieldand on its (...) value group Γ. In [9] we expanded the ordered fieldto a model ofTas follows. Take a tame elementary substructure′ ofsuch thatR′ ⊆VandR′maps bijectively ontounder the residue class map, and make this bijection into an isomorphism′ ≌ofT-models. (We showed such′ exists, and that this gives an expansion ofto aT-model that is independent of the choice of′.). (shrink)

Vector spaces over fields are pseudofinite, and this remains true for vector spaces over division rings that are finite-dimensional over their center. We also construct a division ring such that the nontrivial vector spaces over it are not pseudofinite, using Richard Thompson's group F. The idea behind the construction comes from a first-order axiomatization of the class of division rings all whose nontrivial vector spaces are pseudofinite.

Much clinical and ethnographic evidence suggests that humans, like many other organisms, are selected to avoid close inbreeding because of the fitness costs of inbreeding depression. The proximate mechanism of human inbreeding avoidance seems to be precultural, and to involve the interaction of genetic predispositions and environmental conditions. As first suggested by E. Westermarck, and supported by evidence from Israeli kibbutzim, Chinese sim-pua marriage, and much convergent ethnographic and clinical evidence, humans negatively imprint on intimate associates during a critical period (...) of early childhood.There is also much evidence that, like other social animals, humans do not seek to maximize outbreeding, but rather to maintain an optimal balance between outbreeding and inbreeding.Closeinbreeding reduces fitness through inbreeding depression, butsomeinbreeding brings the benefits of nepotism. For simple, stateless, horticultural societies, the optimal balance seems to be achieved by a combination of precultural inbreeding avoidance of relatives with anr≤·25 and cultural rules of preferential marriage with kin withr≥·25. Adjustment of the coefficient of inbreeding to other ecological settings seems to be largely cultural. An interactive model of “culture in nature” is presented, in which culture is seen as coevolving with genes to produce the maxiniization of individual inclusive fitness. (shrink)

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].

Background and aim With the increasing interest in lifestyle, health and consequences of unhealthy lifestyles for the healthcare system, a new kind of solidarity is gaining importance: lifestyle solidarity. While it might not seem fair to let other people pay for the costs arising from an unhealthy lifestyle, it does not seem fair either to punish people for their lifestyle. However, it is not clear how solidarity is assessed by people, when considering disease risks or lifestyle risks. The aim of (...) this study was to investigate the degree of solidarity with lifestyle as well as with other factors that are related to health outcomes—for example, old age—and the relation between this degree of solidarity and various characteristics. Methods This cross-sectional study is part of the Dutch longitudinal SMILE study. Data on the degree of solidarity with different lifestyles and old age, and the relation between the degree of solidarity and various demographic and other variables were obtained in a questionnaire survey. Results Solidarity with smokers and overweight people was moderate, as was solidarity with older people. Respondents were ambivalent about athletes. Respondents who were younger, male and highly educated, and those with a healthy lifestyle, a small social network, high quality of life and an internal locus of control, showed low solidarity. Conclusions Solidarity with an unhealthy lifestyle and old age is moderate and the degree of solidarity varies among the different subgroups. (shrink)

Aim:This study explores experiences of patients suffering from Korsakoff’s syndrome. It contributes to improved reflection on the value of patient knowledge.Background:An ethics of care perspective states the importance of moving to patients in their vulnerable state of being, and to figure out patients’ individual needs necessary to provide good care. The information given by patients suffering from Korsakoff’s syndrome might be mistaken, invented and even not true. The value of these patients’ experiences and knowledge had not been researched to date.Method:Data (...) from six in-depth interviews were analysed concurrently through the constant comparative method.Ethical considerations:The principles of voluntariness, confidentiality and anonymity were respected during the research process.Findings:Four important themes within patients’ knowledge emerged: health condition, involuntary admission into the nursing home, the daily routine and interpretation of the caring abilities of healthcare professionals. The findings show that our respondents were able to provide a window into their inner perspective.Conclusion:This study deepens our understanding of the value of patient knowledge in situations where patients see the world in a different manner as do healthcare professionals. The conversations revealed personal subjective knowledge which our respondents create and live by. Healthcare professionals are encouraged to discover and affirm the understandings of these patients, and not be prepossessed due to the diagnosis or general nursing home rules. Understanding patients’ unique knowledge through appreciation of the complexity and richness of different views on the situation can assist to give a more humane response to individual needs and pain. Health staff in healthcare institutions are encouraged to evaluate the – unintended – effect of rigidly applied nursing home rules, and the government is challenged to rethink its policy about care, and provide more resources permitting relational care that nurtures the dignity of patients suffering from Korsakoff’s syndrome. (shrink)