We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multi-dimensional Newton's Lemma follow from our result. We give an easy proof that the latter holds in every henselian eld. Further, we apply the theorem to give a criterion for the existence of integration and solutions of certain diierential equations...

In this note we show that every n-point ultrametric embeds with constant distortion in l O(logn) p for every ∞ ≥ p ≥ 1. More precisely, we consider a special type of ultrametric with hierarchical structure called a k-hierarchically well-separated tree (k-HST). We show that any k-HST can be embedded with distortion at most 1 + O(1/k) in l O(k2 logn) p . These facts have implications to embedding...

Throughout this paper, K denotes a complete, non-trivially valued, ultrametric (or nonarchimedean) field. Sequences, infinite series and matrices have entries in K. In we record some interesting properties about the matrix classes (c0,c0) (c0,c0;P).

Let \((X, d)\) be a semimetric space and let \(G\) graph. We say that is the diametrical graph of if \(X\) vertex set adjacency vertices \(x\) \(y\) equivalent to equality \(\diam X = d(x, y)\). It shown with diameter \(d^*\) ultrametric iff d_{\varepsilon})\) \(d_{\varepsilon}(x, y) \min\{d(x, y), \varepsilon\}\) complete multipartite for every \(\varepsilon \in (0, d^*)\). A refinement last r...

Some iterative calculations can be carried out by parallel communicating processors, and yield the same results whether or not the processors are synchronized. We show that this is the case if and only if the iteration is a contraction that is strict on orbits, with respect to an ultrametric defined on the state space. The maximum number of independent processors is given by the dimension of th...

If K is a complete non-archimedean field with a discrete valuation and f ∈ K[X ] is a polynomial with non-vanishing discriminant. The first main result of this paper is about connecting the number of roots of f to the number of roots of its reduction modulo a power of the maximal ideal of the valuation ring of K. If the polynomial f is regular, we give an algorithmic method to compute the exact...

Ising spin-glass systems with long-range interactions (J(r) ∼ r) are considered. A numerical study of the critical behaviour is presented in the non-mean-field region together with an analysis of the probability distribution of the overlaps and of the ultrametric structure of the space of pure states in the frozen phase. Also in presence of diverging thermodynamical fluctuations at the critical...