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The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a point, and make it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields.

This study characterizes extremely low frequency (ELF) magnetic field (MF) levels in 10 car models. Extensive measurements were conducted in three diesel, four gasoline, and three hybrid cars, under similar controlled conditions and negligible background fields. Averaged over all four seats under various driving scenarios the fields were lowest in diesel cars (0.02 μT), higher for gasoline (0.04–0.05 μT) and highest in hybrids (0.06–0.09 μT), but all were in-line with daily exposures from other sources. Hybrid cars had the highest mean and 95th percentile MF levels, and an especially large percentage of measurements above 0.2 μT. These parameters were also higher for moving conditions compared to standing while idling or revving at 2500 RPM and higher still at 80 km/h compared to 40 km/h. Fields in non-hybrid cars were higher at the front seats, while in hybrid cars they were higher at the back seats, particularly the back right seat where 16%–69% of measurements were greater than 0.2 μT. As our results do not include low frequency fields (below 30 Hz) that might be generated by tire rotation, we suggest that net currents flowing through the cars’ metallic chassis may be a possible source of MF. Larger surveys in standardized and well-described settings should be conducted with different types of vehicles and with spectral analysis of fields including lower frequencies due to magnetization of tires.

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Arithmetic motivic integration re-interprets the classical measure on p-adic fields, and p-adic manifolds, in a geometric way. The main benefit of such an interpretation is that it allows one to isolate the dependence on p, so that one can perform integration in a field-independent way, and then "plug in" p at the very end. Even though this is not the only achievement of the theory, it will be our main focus in these notes. Hence, we begin with a brief review of the properties of the field of p-adic numbers, and integration on p-adic manifolds.2.1. The p-adic numbers. Let p be a fixed prime. Throughout these notes our main example of a local field will be the field Q p of p-adic numbers, which is the completion of Q with respect to the p-adic metric.2.1.1. Analytic definition of the field Q p . Every non-zero rational number x ∈ Q can be written in the form x = a b p n , where n ∈ Z, and a, b are integers relatively prime to p. The power n is called the valuation of x and denoted ord(x). Using the valuation map, we can define a norm on Q: |x| p = p −ord(x) if x is non-zero and |0| p = 0. This norm induces a metric on Q, which satisfies a stronger triangle inequality than the standard metric:This property of the metric is referred to as the ultrametric property.The set Q p , as a metric space, is the completion of Q with respect to this metric. The operations of addition and multiplication extend by continuity from Q to Q p and make it a field. The set {x ∈ Q p | ord(x) ≥ 0} is denoted Z p and called the ring of p-adic integers. Algebraic definition of the fieldThere is a way to define Q p without invoking analysis. Consider the rings Z/p n Z. They form a projective system with natural mapsThe projective limit is called Z p , the ring of p-adic integers. The field Q p is then defined to be its field of fractions.2.1.3. Basic facts about Q p .• The two definitions of Q p agree, and Q p is a field extension of Q.• Topology on Q p : if we use the analytic definition, then Q p comes equipped with a metric topology. It follows from the strong triangle inequality that Q p is totally disconnected in this topology. It is easy to prove that the sets p n Z p , as n ranges over Z, form a basis of neighbourhoods of 0. If one uses the algebraic definition of Q p , then the topology for Q is defined by declaring that these sets form a basis of neighbourhoods of 0, and the basis of neighbourhoods at any other point is obtained by translating them. • The set Z p ⊂ Q p is open and compact in this topology. It follows that each p n Z p is also a compact set, which, in turn, implies that Q p is locally compact. Note that Z p (in the analytic definition) has the description Z p = {x ∈ Q p | |x| p ≤ 1}, so it is the closed unit ball in our metric space (somewhat counter-intuitively).