Two classes of metric spaces

The class of metric spaces (X,d) known as small-determined spaces, introduced by Garrido and Jaramillo, are properly defined by means of some type of real-valued Lipschitz functions on X. On the other hand, B-simple metric spaces introduced by Hejcman are defined in terms of some kind of bornologies of bounded subsets of X. In this note we present a common framework where both classes of metric spaces can be studied which allows us to see not only the relationships between them but also to obtain new internal characterizations of these metric properties.

On paracompactness, completeness and boundedness in uniform spaces

Recently two new types of completeness in metric spaces, called Bourbaki-completeness and cofinal Bourbaki-completeness, have been introduced in [7]. The purpose of this note is to analyze these completeness properties in the general context of uniform spaces. More precisely, we are interested in how they are related with uniform paracompactness properties, as well as with some kind of uniform boundedness.

On the uniform approximation of Cauchy continuous functions

In the context of real-valued functions defined on metric spaces, it is known that the locally Lipschitz functions are uniformly dense in the continuous functions and that the Lipschitz in the small functions - the locally Lipschitz functions where both the local Lipschitz constant and the size of the neighborhood can be chosen independent of the point - are uniformly dense in the uniformly continuous functions. Between these two basic classes of continuous functions lies the class of Cauchy continuous functions, i.e., the functions that map Cauchy sequences in the domain to Cauchy sequences in the target space. Here, we exhibit an intermediate class of Cauchy continuous locally Lipschitz functions that is uniformly dense in the real-valued Cauchy continuous functions. In fact, our result is valid when our target space is an arbitrary Banach space.

A functional representation of almost isometries

For each quasi-metric space X we consider the convex lattice SLip(1)(X) of all semi-Lipschitz functions on X with semi-Lipschitz constant not greater than 1. If X and Y are two complete quasi-metric spaces, we prove that every convex lattice isomorphism T from SLip(1)(Y) onto SLip(1)(X) can be written in the form Tf = c . (f o tau) + phi, where tau is an isometry, c > 0 and phi is an element of SLip(1)(X). As a consequence, we obtain that two complete quasi-metric spaces are almost isometric if, and only if, there exists an almost-unital convex lattice isomorphism between SLip(1)(X) and SLip(1) (Y).

La ciudad en la narrativa latinoamericana: 1950-1975

The representation of the city has always been present in the literature. A clear example of this is the famous city of Troy. The city in terms of where the actions take place, a novel in this case, despite the efforts of some works of the contemporary narrative to eradicate or reduce to its barest minimum expression, has continued to sit as a strong element of differentiation that gives the characters certain linguistic, historical, social and cultural characteristics. In the Hispanic narrative, according to historical features of the continent, the conquest, independence, and subsequently the constitution of the republics, the representation of the city acquires some unique characteristics, whose dimensions and implications, toward the second half of the twentieth century, transcend the simple notion of 'place' in which occur the facts narrated to acquire a central notion in the works, changing from being a support to become the central structure of the novel, which is able to articulate different situations, confront characters and articulate historically to the entire countries. This paper will talk mainly about the representation of the city in the published narrative between 1950 and 1975. We will try to have a transverse reading over these works through the analysis of the representation of the city that in them we can find, and that basically divided into three broad categories, each with its own specific functions: * The royal city. Corresponds to the cities that we can actually find in the American territory, and whose spaces and descriptions, historical references and territorial, it is possible to identify the reality or in any encyclopedia: streets, historical events, places, characters, etc...

Linear non-local diffusion problems in metric measure spaces

The aim of this paper is to provide a comprehensive study of some linear non-local diffusion problems in metric measure spaces. These include, for example, open subsets in ℝN, graphs, manifolds, multi-structures and some fractal sets. For this, we study regularity, compactness, positivity and the spectrum of the stationary non-local operator. We then study the solutions of linear evolution non-local diffusion problems, with emphasis on similarities and differences with the standard heat equation in smooth domains. In particular, we prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods.