Abstract. We study the existential (and parts of the universal-existential) theory of equicharacteristic henselian valued fields. We prove, among other things, an existential Ax-Kochen-Ershov principle, which roughly says that the existential theory of an equicharacteristic henselian valued field (of arbitrary characteristic) is determined by the existential theory of the residue field; in particular, it is independent of the value group. As an immediate corollary, we get an unconditional proof of the decidability of the existential theory of F q ((t)).
Purpose. Female runners are known to be at greater risk from chronic running injuries than age-matched males, although the exact mechanisms are often poorly understood. The aim of the current investigation was to determine if female recreational runners exhibit distinct limb and joint stiffness characteristics in relation to their male counterparts. Methods. Fourteen male and fourteen female runners ran over a force platform at 4.0 m · s -1 . Lower limb kinematics were collected using an eight-camera optoelectric motion capture system operating at 250 Hz. Measures of limb and joint stiffness were calculated as a function of limb length and joint moments divided by the extent of limb and joint excursion. All stiffness and joint moment parameters were normalized to body mass. Sex differences in normalized limb and knee and ankle joint stiffness were examined statistically using independent samples t tests. Results. The results indicate that normalized limb (male = 0.18 ± 0.07, female = 0.37 ± 0.10 kN · kg · m -1 ) and knee stiffness (male = 5.59 ± 2.02, female = 7.34 ± 1.78 Nm · kg · rad -1 ) were significantly greater in female runners. Conclusions. On the basis that normalized knee and limb stiffness were shown to be significantly greater in female runners, the findings from the current investigation may provide further insight into the aetiology of the distinct injury patterns observed between sexes.
Abstract. In [1], Anscombe and Koenigsmann give an existential ∅-definition of the ring of formal power series F [[t]] in its quotient field in the case where F is finite. We extend their method in several directions to give general definability results for henselian valued fields with finite or pseudo-algebraically closed residue fields. §1. Introduction. The question of first order definability of valuation rings in their quotient fields has a long history. Given a valued field K, one is interested in whether there exists a first order formula ϕ in the language L = {+, −, ·, 0, 1} of rings such that the set ϕ(K) defined by ϕ in K is precisely the valuation ring, and what complexity such formula must have.Many results of this kind are known for henselian valued fields, like fields of formal power series K = F ((t)) over a field F , and their valuation ring F [[t]]. In this setting, a definition going back to Julia Robinson gives an existential definition of the valuation ring using the parameter t. Later, Ax [2] gave a definition of the valuation ring, which uses no parameters, but is not existential.Recently, Anscombe and Koenigsmann [1] succeeded to give an existential and parameter-free definition of F [[t]] in F ((t)) in the special case where F = F q is a finite field. Their proof uses the fact that F q can be defined in F q ((t)) by the quantifier-free formula x q − x = 0. In particular, their result does not apply to any infinite field F , and their formula depends heavily on q.In this note we simplify and extend their method. As a first application we get the following general definability result for henselian valued fields with finite or pseudo-algebraically closed residue fields (Theorem 2.6 and Theorem 3.5), which generalizes [1, Theorem 1.1] on F q ((t)) and [5, Theorem 6] As a further application, in Section 4, we find definitions of the valuation ring which are uniform for large (infinite) families of finite residue fields, like the following one for finite prime fields (Theorem 4.3):
Abstract. We define and investigate a family of local-global principles for fields involving both orderings and p-valuations. This family contains the PAC, PRC and PpC fields and exhausts the class of pseudo classically closed fields. We show that the fields satisfying such a local-global principle form an elementary class, admit diophantine definitions of holomorphy domains, and their orderings satisfy the strong approximation property.
Abstract. We discuss definability in the language of rings without parameters of the unique canonical henselian valuation of a field. We show that in most cases where the canonical henselian valuation is definable, it is already definable by a universal-existential or an existential-universal formula.
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