Based on the theory of Fermat reals we introduce new topologies on spaces of Colombeau generalized points and derive some of their fundamental properties. In particular, we obtain metric topologies on the space of near-standard generalized points that induce the standard Euclidean topology on the reals. We also give a new description of the sharp topology in terms of the natural extension of th...

In this work, we aim to prove algebra properties for generalized Sobolev spaces W ∩L on a Riemannian manifold, whereW s,p is of Bessel-typeW s,p := (1+L)(L) with an operator L generating a heat semigroup satisfying off-diagonal decays. We don’t require any assumption on the gradient of the semigroup. To do that, we propose two different approaches (one by a new kind of paraproducts and another ...

Let K be a eld that is complete with respect to a nonarchimedean absolute value such that K has a countable dense subset. We prove that the Berkovich analyti cation V an of any d-dimensional quasi-projective scheme V over K embeds in R. If, moreover, the value group of K is dense in R>0 and V is a curve, then we describe the homeomorphism type of V an by using the theory of local dendrites.

We study the Erdős distance problem over finite Euclidean and non-Euclidean spaces. Our main tools are graphs associated to finite Euclidean and non-Euclidean spaces that are considered in Bannai-Shimabukuro-Tanaka (2004, 2007). These graphs are shown to be asymptotically Ramanujan graphs. The advantage of using these graphs is twofold. First, we can derive new lower bounds on the Erdős distanc...

There are many problems and configurations in Euclidean geometry that were never extended to the framework of (normed or) finite dimensional real Banach spaces, although their original versions are inspiring for this type of generalization, and the analogous definitions for normed spaces represent a promising topic. An example is the geometry of simplices in non-Euclidean normed spaces. We pres...

Compact metric spaces that admit intrinsic isometries to the Euclidean d-space are considered. Roughly, the main result states that the class of such spaces coincides with the class of inverse limits of Euclidean d-polyhedra. §

Consider an n-point metric space M = (V, δ), and a transmission range assignment r : V → R that maps each point v ∈ V to the disk of radius r(v) around it. The symmetric disk graph (henceforth, SDG) that corresponds to M and r is the undirected graph over V whose edge set includes an edge (u, v) if both r(u) and r(v) are no smaller than δ(u, v). SDGs are often used to model wireless communicati...