In this paper we consider some problems in the Arakelov geometry of Mg, the moduli space of smooth projective curves of genus g over C. Specifically, we are interested in naturally metrized line bundles over Mg and their extensions to Mg, the Deligne-Mumford compactification of Mg. These line bundles typically occur when computing the archimedean height of a curve. Here they arise in computatio...

We consider the class of polynomial optimization problems inf{f(x) : x ∈ K} for which the quadratic module generated by the polynomials that define K and the polynomial c − f (for some scalar c) is Archimedean. For such problems, the optimal value can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically. Moreover, the Ar...

We prove that if X,X are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly unbounded) connected component C of Trop(X) ∩ Trop(X) lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a s...

‎In 1897‎, ‎Hensel introduced a normed space which does‎ ‎not have the Archimedean property‎. ‎During the last three decades‎ ‎theory of non--Archimedean spaces has gained the interest of‎ ‎physicists for their research in particular in problems coming‎ ‎from quantum physics‎, ‎p--adic strings and superstrings‎. ‎In this paper‎, ‎we prove‎ ‎the generalized Hyers--Ulam--Rassias stability for a‎ ...

In this paper, the concept of semiprimary fuzzy ideals of an ordered semigroup S is introduced. Some characterizations for an ordered semigroup S to be a semilattice of archimedean (rarchimedean, t-archimedean) ordered subsemigroups are given by some binary relations on S and the fuzzy radical of fuzzy ideals (fuzzy right ideals, fuzzy bi-ideals) of S. Furthermore, some characterizations for an...

In this paper, we solve the additive ρ-functional inequalities ‖f(x+ y)− f(x)− f(y)‖ ≤ ∥∥∥∥ρ(2f (x+ y 2 ) − f(x)− f(y) )∥∥∥∥ , (1) ∥∥∥∥2f (x+ y 2 ) − f(x)− f(y) ∥∥∥∥ ≤ ‖ρ (f(x+ y)− f(x)− f(y))‖ , (2) where ρ is a fixed non-Archimedean number with |ρ| < 1 or ρ is a fixed complex number with |ρ| < 1. Using the direct method, we prove the Hyers-Ulam stability of the additive ρ-functional inequalit...

There is a natural analytification functor from the category of locally separated algebraic spaces locally of finite type over C to the category of complex-analytic spaces [Kn, Ch. I, 5.17ff]. (Recall that a map of algebraic spaces X → S is locally separated if the diagonal ∆X/S : X → X ×S X is an immersion. We require algebraic spaces to have quasi-compact diagonal over SpecZ.) It is natural t...

Several researchers have recently established that for every Turing degree c, the real closed field of all c-computable real numbers has spectrum {d : d′ ≥ c′′}. We investigate the spectra of real closed fields further, focusing first on subfields of the field R0 of computable real numbers, then on archimedean real closed fields more generally, and finally on nonarchimedean real closed fields. ...

An Archimedean graph in R 2 is any finite subgraph of the vertices and edges of one of the 11 Archimedean tilings of R E shown in Figures 1-3. Archimedean filings are commonly characterized (as in Figures 1-3) by the order in which regular polygons (with unit edges) occur around each vertex of the tiling; see Grtinbaum and Shephard [3]. We use the same description for Archimedean subgraphs. Whe...

Let K be a non-archimedean field with residue field k, and suppose that k is not an algebraic extension of a finite field. We prove two results concerning wandering domains of rational functions φ ∈ K(z) and Rivera-Letelier’s notion of nontrivial reduction. First, if φ has nontrivial reduction, then assuming some simple hypotheses, we show that the Fatou set of φ has wandering components by any...