In (Arch. Math. 57 (1991), pp. 446–455), R. Farré proved a positivstellensatz for real-series closed fields. Here we consider p-valued fields 〈K, vp〉 with a non-trivial valuation v which satisfies a compatibility condition between vp and v. We use this notion to establish the p-adic analogue of real-series closed fields; these fields are called henselian residually p-adically closed fields. Fir...

We study model theoretic properties of valued fields (equipped with a real-valued multiplicative valuation), viewed as metric structures in continuous first order logic. For technical reasons we prefer to consider not the valued field (K, |·|) directly, but rather the associated projective spaces KP, as bounded metric structures. We show that the class of (projective spaces over) metric valued ...

We develop the notion of deformations using a valuation ring as ring of coefficients. This permits to consider in particular the classical Gerstenhaber deformations of associative or Lie algebras as infinitesimal deformations and to solve the equation of deformations in a polynomial frame. We consider also the deformations of the enveloping algebra of a rigid Lie algebra and we define valued de...

a necessary and sufficient condition in terms of lower cut sets are given for the insertion of a $gamma-$continuous function between two comparable real-valued functions.

In this paper, we first introduce some function spaces, with certain locally convex topologies, closely  related to the space of real-valued continuous functions on $X$,  where $X$ is a $C$-distinguished topological space. Then, we show that their dual spaces can be identified in a natural way with certain spaces of Radon measures.

in this paper we study the relationships existing between total measurability in variation and gould type fuzzy integrability (introduced and studied in [21]), giving a special interest on their behaviour on atoms and on finite unions of disjoint atoms. we also establish that any continuous real valued function defined on a compact metric space is totally measurable in the variation of a regula...

We show that every polynomial overring of the ring Int(Z) of polynomials which are integer-valued over Z may be considered as the ring of polynomials which are integer-valued over some subset of Ẑ, the profinite completion of Z with respect to the fundamental system of neighbourhoods of 0 consisting of all non-zero ideals of Z.

We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact  (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...

The concepts of tropical semiring and tropical hypersurface, are extended to the case of an arbitrary ordered group. Then, we define the tropicalization of a polynomial with coefficients in a Krull-valued field. After a close study of the properties of the operator “tropicalization” we conclude with an extension of Kapranov’s theorem to algebraically closed fields together with a valuation over...