On function field Mordell-Lang: the semiabelian case and the socle theorem

We here aim to complete our model-theoretic account of the function field Mordell-Lang conjecture, avoiding appeal to dichotomy theorems for Zariski geometries, where we now consider the general case of semiabelian varieties. The main result is a reduction, using model-theoretic tools, of the semiabelian case to the abelian case.Comment: 43 pages. Some minor corrections and clarifications were made following a referee's repor

On function field Mordell-Lang and Manin-Mumford

We present a reduction of the function field Mordell-Lang conjecture to the function field Manin-Mumford conjecture, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski structures. In this version 2, the quantifier elimination result in positive characteristic is extended from simple abelian varieties to all abelian varieties, completing the main theorem in the positive characteristic case. In version 3, some corrections are made to the proof of quantifier elimination in positive characteristic, and the paper is substantially reorganized.Comment: 21 page

Dimension, matroids, and dense pairs of first-order structures

A structure M is pregeometric if the algebraic closure is a pregeometry in all M' elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power of omega and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding a field, while not pregeometric in general, do have an unique existential matroid. Generalising previous results by van den Dries, we define dense elementary pairs of structures expanding a field and with an existential matroid, and we show that the corresponding theories have natural completions, whose models also have a unique existential matroid. We extend the above result to dense tuples of structures.Comment: Version 2.8. 61 page

Some Definability Results in Abstract Kummer Theory

Let $S$ be a semiabelian variety over an algebraically closed field, and let $X$ be an irreducible subvariety not contained in a coset of a proper algebraic subgroup of $S$. We show that the number of irreducible components of $[n]^{-1}(X)$ is bounded uniformly in $n$, and moreover that the bound is uniform in families $X_t$. We prove this by purely Galois-theoretic methods. This proof applies in the more general context of divisible abelian groups of finite Morley rank. In this latter context, we deduce a definability result under the assumption of the Definable Multiplicity Property (DMP). We give sufficient conditions for finite Morley rank groups to have the DMP, and hence give examples where our definability result holds.Comment: 21 pages; minor notational fixe

Semiabelian varieties over separably closed fields, maximal divisible subgroups, and exact sequences

Given a separably closed field K of positive characteristic and finite degree of imperfection we study the # functor which takes a semiabelian variety G over K to the maximal divisible subgroup #G of G(K). We show that the # functor need not preserve exact sequences. The main result is an example where #G does not have "relative Morley rank", yielding a counterexample to a claim of Hrushovski. The methods involve studying preservation of exact sequences by the # functor as well as issues of descent. We also develop the notion of an iterative D-structure on a group scheme over an iterative Hasse field, as well as giving characteristic 0 versions of our results.Comment: 55 pages In this version 3, some corrections and clarifications are made: in section 2.3 on relative Morley rank. Also in section 5.2 where more explanation is given of D-structures in positive characteristic. In an appendix we give a proof of the exactness of the functor taking a semiabelian variety to its universal vectorial extensio

Beyond aspect: will be -ing and shall be -ing

This article discusses the synchronic status and diachronic development of will be -ing and shall be -ing (as in I’ll be leaving at noon).2 Although available since at least Middle English, the constructions did not establish a significant foothold in standard English until the twentieth century. Both types are also more prevalent in British English (BrE) than American English (AmE). We argue that in present-day usage will/shall be -ing are aspectually underspecified: instances that clearly construe a situation as future-in-progress are in the minority. Similarly, although volition-neutrality has been identified as a key feature of will/shall be -ing, it is important to take account of other, generally richer meanings and associations, notably ‘future-as-matter-of-course’ (Leech 2004), ‘already-decided future’ (Huddleston & Pullum et al. 2002) and non-agentivity. Like volition-neutrality, these characteristics appear to be relevant not only in contemporary use, but also in their historical expansion. We show that the construction has evolved from progressive aspect towards more subjectivised evidential meaning

Hydration and cooling in elite athletes: relationship with performance, body mass loss and body temperatures during the Doha 2019 IAAF World Athletics Championships

Purpose: To characterise hydration, cooling, body mass loss, and core (Tcore) and skin (Tsk) temperatures during World Athletics Championships in hot-humid conditions. Methods: Marathon and race-walk (20 km and 50 km) athletes (n=83, 36 women) completed a pre-race questionnaire. Pre-race and post-race body weight (n=74), Tcore (n=56) and Tsk (n=49; thermography) were measured. Results: Most athletes (93%) had a pre-planned drinking strategy (electrolytes (83%), carbohydrates (81%)) while ice slurry was less common (11%; p<0.001). More men than women relied on electrolytes and carbohydrates (91%–93% vs 67%–72%, p≤0.029). Drinking strategies were based on personal experience (91%) rather than external sources (p<0.001). Most athletes (80%) planned pre-cooling (ice vests (53%), cold towels (45%), neck collars (21%) and ice slurry (21%)) and/or midcooling (93%; head/face dousing (65%) and cold water ingestion (52%)). Menthol usage was negligible (1%–2%). Pre-race Tcore was lower in athletes using ice vests (37.5°C±0.4°C vs 37.8°C±0.3°C, p=0.024). Tcore (pre-race 37.7°C±0.3°C, post-race 39.6°C±0.6°C) was independent of event, ranking or performance (p≥0.225). Pre-race Tsk was correlated with faster race completion (r=0.32, p=0.046) and was higher in non-finishers (did not finish (DNF); 33.8°C±0.9°C vs 32.6°C±1.4°C, p=0.017). Body mass loss was higher in men than women (−2.8±1.5% vs −1.3±1.6%, p<0.001), although not associated with performance. Conclusion: Most athletes’ hydration strategies were pre-planned based on personal experience. Ice vests were the most adopted pre-cooling strategy and the only one minimising Tcore, suggesting that event organisers should be cognisant of logistics (ie, freezers). Dehydration was moderate and unrelated to performance. Pre-race Tsk was related to performance and DNF, suggesting that Tsk modulation should be incorporated into pre-race strategies

Hydration and cooling in elite athletes: relationship with performance, body mass loss and body temperatures during the Doha 2019 IAAF World Athletics Championships.

PURPOSE: To characterise hydration, cooling, body mass loss, and core (Tcore) and skin (Tsk) temperatures during World Athletics Championships in hot-humid conditions. METHODS: Marathon and race-walk (20 km and 50 km) athletes (n=83, 36 women) completed a pre-race questionnaire. Pre-race and post-race body weight (n=74), Tcore (n=56) and Tsk (n=49; thermography) were measured. RESULTS: Most athletes (93%) had a pre-planned drinking strategy (electrolytes (83%), carbohydrates (81%)) while ice slurry was less common (11%; p<0.001). More men than women relied on electrolytes and carbohydrates (91%-93% vs 67%-72%, p≤0.029). Drinking strategies were based on personal experience (91%) rather than external sources (p<0.001). Most athletes (80%) planned pre-cooling (ice vests (53%), cold towels (45%), neck collars (21%) and ice slurry (21%)) and/or mid-cooling (93%; head/face dousing (65%) and cold water ingestion (52%)). Menthol usage was negligible (1%-2%). Pre-race Tcore was lower in athletes using ice vests (37.5°C±0.4°C vs 37.8°C±0.3°C, p=0.024). Tcore (pre-race 37.7°C±0.3°C, post-race 39.6°C±0.6°C) was independent of event, ranking or performance (p≥0.225). Pre-race Tsk was correlated with faster race completion (r=0.32, p=0.046) and was higher in non-finishers (did not finish (DNF); 33.8°C±0.9°C vs 32.6°C±1.4°C, p=0.017). Body mass loss was higher in men than women (-2.8±1.5% vs -1.3±1.6%, p<0.001), although not associated with performance. CONCLUSION: Most athletes' hydration strategies were pre-planned based on personal experience. Ice vests were the most adopted pre-cooling strategy and the only one minimising Tcore, suggesting that event organisers should be cognisant of logistics (ie, freezers). Dehydration was moderate and unrelated to performance. Pre-race Tsk was related to performance and DNF, suggesting that Tsk modulation should be incorporated into pre-race strategies