Generalized chordality, vertex separators and hyperbolicity on graphs

Let $G$ be a graph with the usual shortest-path metric. A graph is $\delta$-hyperbolic if for every geodesic triangle $T$, any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides. A graph is chordal if every induced cycle has at most three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. In this paper we study the relation between vertex separator sets, some chordality properties which are natural generalizations of being chordal and the hyperbolicity of the graph. We also give a characterization of being quasi-isometric to a tree in terms of chordality and prove that this condition also characterizes being hyperbolic, when restricted to triangles, and having stable geodesics, when restricted to bigons.Comment: 16 pages, 3 figure

Real valued functions and metric spaces quasi-isometric to trees

We prove that if X is a complete geodesic metric space with uniformly generated first homology group and $f: X\to R$ is metrically proper on the connected components and bornologous, then X is quasi-isometric to a tree. Using this and adapting the definition of hyperbolic approximation we obtain an intrinsic sufficent condition for a metric space to be PQ-symmetric to an ultrametric space.Comment: 12 page

Novedades en la diabetes mellitus tipo 1

Introducción: La diabetes mellitus tipo 1 se trata de una enfermedad metabólica que afecta a la secreción de insulina del organismo produciendo un desajuste en los niveles de glucosa. Esta patología muestra un cuadro agudo de hiperglucemia y la aparición de múltiples complicaciones con un riesgo vital considerable. El tratamiento se enfoca principalmente en el control de los niveles de glucosa para evitar complicaciones, para ello se centra en el uso de insulina, nutrición específica y realización de ejercicio físico. Material y Métodos: Este trabajo se ha realizado siguiendo el modelo de revisión bibliográfica sistemática, realizando una exhaustiva búsqueda bibliográfica con el objetivo de conseguir el mejor material de estudio. Resultados: El abordaje principal de la DM tipo 1 se basa en el uso de insulina para controlar los niveles de glucosa en el organismo, partiendo de esto los métodos de administración de insulina han ido evolucionando considerablemente en los últimos años creando sistemas que aseguran un gran control a la hora de afrontar esta enfermedad. Del mismo modo nuevos métodos de tratamiento basados en el trasplante o la terapia celular han ido evolucionando en los últimos años; pudiendo aportar estas vías de investigación la deseada cura de la enfermedad. Discusión: Aunque muchos de estos nuevos sistemas no presentan una eficacia del 100% siguen componiendo un gran avance en el tratamiento de la diabetes mellitus y muchos aseguran un buen control y un futuro prometedor sin complicaciones para aquellos que padecen de DM tipo 1.Grado en Enfermerí

Bushy pseudocharacters and group actions on quasi-trees

Given a group acting on a graph quasi-isometric to a tree, we give sufficent conditions for a pseudocharacter to be bushy. We relate this with the conditions studied by M. Bestvina and K. Fujiwara on their work on bounded cohomology and obtain some results on the space of pseudocharacters.Comment: 13 pages, 3 figure

Bounded distortion homeomorphisms on ultrametric spaces

It is well-known that quasi-isometries between R-trees induce power quasi-symmetric homeomorphisms between their ultrametric end spaces. This paper investigates power quasi-symmetric homeomorphisms between bounded, complete, uniformly perfect, ultrametric spaces (i.e., those ultrametric spaces arising up to similarity as the end spaces of bushy trees). A bounded distortion property is found that characterizes power quasi-symmetric homeomorphisms between such ultrametric spaces that are also pseudo-doubling. Moreover, examples are given showing the extent to which the power quasi-symmetry of homeomorphisms is not captured by the quasiconformal and bi-H\"older conditions for this class of ultrametric spaces.Comment: 20 pages, 1 figure. To appear in Ann. Acad. Sci. Fenn. Mat