We observe the heat flux exchanged by the hot tip of a scanning thermal microscope, which is an instrument based on the atomic force microscope. We first vary the pressure in order to analyze the impact on the hot tip temperature. Then the distance between the tip and a cold sample is varied down to few nanometers, in order to reach the ballistic regime. We observe the cooling of the tip due to the tip-sample heat flux and compare it to the current models in the literature.

Scanning Thermal Microscopy measurements with a resistive microprobe electrically heated were performed for different probe temperatures, for probe free in air and in contact with various specimens. The measured relative difference of Joule power dissipated in the probe when tip is in contact with a sample and when it is free in air is studied for different magnitude of the electrical current that heats the probe. A variation of this signal, never outlined before, is observed. A predictive modeling is used to explain these results and identify from the experimental data the global thermal conductance of the probe-sample thermal exchange for experiments performed in ambient conditions.

In 1965, during the first All-Union Symposium on Group Theory, Kargapolov presented the following two problems: (a) describe the universal theory of free nilpotent groups of class m; (b) describe the universal theory of free groups (see [18, 1.28 and 1.27]). The first of these problems is still open and it is known [25] that a positive solution of this problem for an m ≤ 2 should imply the decidability of the universal theory of the field of the rationals (this last problem is equivalent to Hilbert's tenth problem for the field of the rationals which is a difficult open problem; see [17] and [20] for discussions on this problem). Regarding the second problem, Makanin proved in 1985 that a free group has a decidable universal theory (see [15] for stronger results), however, the problem of deriving an explicit description of the universal theory of free groups is open. To try to solve this problem Remeslennikov gave different characterization of finitely generated groups with the same universal theory as a noncyclic free group (see [21] and [22] and also [11]). Recently, the author proved in [8] that a free metabelian group has a decidable universal theory, but the proof of [8] does not give an explicit description of the universal theory of free metabelian groups.

The aim of this paper is to describe (without proofs) an analogue of the theory of nontrivial torsion-free divisible abelian groups for metabelian groups. We obtain illustrations for “old-fashioned” model theoretic algebra and “new” examples in the theory of stable groups. We begin this paper with general considerations about model theory. In the second section we present our results and we give the structure of the rest of the paper. Most parts of this paper use only basic concepts from model theory and group theory (see [14] and especially Chapters IV, V, VI and VIII for model theory, and see for example [23] and especially Chapters II and V for group theory). However, in Section 5, we need some somewhat elaborate notions from stability theory. One can find the beginnings of this theory in [14], and we refer the reader to [16] or [21] for stability theory and to [22] for stable groups.

§1. Some model theoretic considerations. Denote by the theory of torsion-free abelian groups in the language of groups ℒgp . A finitely generated group G satisfies iff G is isomorphic to a finite direct power of ℤ. It follows that axiomatizes the universal theory of free abelian groups and that the theory of nontrivial torsion-free abelian groups is complete for the universal sentences. Denote by the theory of nontrivial divisible torsion-free abelian groups.