In this paper, we study the geometry of non-Archimedean Gromov-Hausdorff metric. This is the first part of our series work, which we try to establish some facts about the counterpart of Gromov-Hausdorff metric in the non-Archimedean spaces. One of the motivation of this work is to find some implied relations between this geometry and number theory via p-adic analysis, so that we can use the for...

We prove an analog of Schmid’s SL2-orbit theorem for a class of variations of mixed Hodge structure which includes logarithmic deformations, degenerations of 1-motives and archimedean heights. In particular, as consequence this theorem, we obtain a simple formula for the asymptotic behavior of the archimedean height of a flat family of algebraic cycles which depends only on the weight filtratio...

The aim of this paper is to prove a Calabi theorem for metrized line bundles over non-archimedean analytic spaces, and apply it to endomorphisms with the same polarization and the same set of preperiodic points over a complex projective variety. The proof uses Arakelov theory on Berkovich’s non-archimedean analytic spaces even though the results on dynamical systems can be purely stated over co...

In this paper, we investigate the Hyers-Ulam stability for the system of additive, quadratic, cubicand quartic functional equations with constants coecients in the sense of dectic mappings in non-Archimedean normed spaces.

For more than two millennia, ever since Euclid’s geometry, the so called Archimedean Axiom has been accepted without sufficiently explicit awareness of that fact. The effect has been a severe restriction of our views of space-time, a restriction which above all affects Physics. Here it is argued that, ever since the invention of Calculus by Newton, we may actually have empirical evidence that t...

In this paper, we introduce the concept of best proximal contraction theorems in non-Archimedean fuzzy metric space for two mappings and prove some proximal theorems. As a consequence, it provides the existence of an optimal approximate solution to some equations which contains no solution. The obtained results extend further the recently development proximal contractions in non-Archimedean fuz...

Archimedean classes and convex subgroups play important roles in the study of ordered groups. In this paper, we show that ACA0 is equivalent to the existence of a set of representatives for the Archimedean classes of an ordered abelian group. Hahn’s Theorem is the strongest known tool for classifying orders on abelian groups. It states that every ordered abelian group can be embedded into produ...

Two concepts of being Archimedean are defined for arbitrary categories. 1. The case of usual semigroups For convenience, let us recall two versions of concepts of being Archimedean in the usual case of algebraic structures. In this regard, a sufficiently general setup is as follows. Let (E,+,≤) be a partially ordered semigroup, thus we have satisfied (1.1) x, y ∈ E+ =⇒ x+ y ∈ E+ where E+ = {x ∈...

The following is a proof which is independent of this characterisation. First assume that ‖ ‖ is non-archimedean. Let x, y ∈ K. Using that ‖ ‖ extends | | we then obtain |x + y| = ‖x + y‖ ≤ max{‖x‖, ‖y‖} = max{|x|, |y|} which shows that | | is non-archimedean. Now assume that | | is non-archimedean. Let x, y ∈ K̂. Let ε > 0. Since K is dense in K̂ there exist u, v ∈ K such that ‖x − u‖ < ε and ‖y...

We define the universal 1-adic thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process. We show that the transposition of the perfection process at the real archimedean place is identical to the “dequantization” process and yields Viro’s tropical real hyperfield T R. Then we pr...