477 research outputs found
Measurements of Riemannian two-disks and two-spheres
We prove that any Riemannian two-sphere with area at most 1 can be
continuously mapped onto a tree in a such a way that the topology of fibers is
controlled and their length is less than 7.6. This result improves previous
estimates and relies on a similar statement for Riemannian two-disks
Stable norms of non-orientable surfaces
We study the stable norm on the first homology of a closed, non-orientable
surface equipped with a Riemannian metric. We prove that in every conformal
class there exists a metric whose stable norm is polyhedral. Furthermore the
stable norm is never strictly convex if the first Betti number of the surface
is greater than two
Systolic geometry and simplicial complexity for groups
Twenty years ago Gromov asked about how large is the set of isomorphism
classes of groups whose systolic area is bounded from above. This article
introduces a new combinatorial invariant for finitely presentable groups called
{\it simplicial complexity} that allows to obtain a quite satisfactory answer
to his question. Using this new complexity, we also derive new results on
systolic area for groups that specify its topological behaviour.Comment: 35 pages, 9 figure
Minimal length product over homology bases of manifolds
Minkowski's second theorem can be stated as an inequality for -dimensional
flat Finsler tori relating the volume and the minimal product of the lengths of
closed geodesics which form a homology basis. In this paper we show how this
fundamental result can be promoted to a principle holding for a larger class of
Finsler manifolds. This includes manifolds for which first Betti number and
dimension do no necessarily coincide, a prime example being the case of
surfaces. This class of manifolds is described by a non-vanishing condition for
the hyperdeterminant reduced modulo of the multilinear map induced by the
fundamental class of the manifold on its first -cohomology group
using the cup product.Comment: 24 page
Marco, registro y concepción. Notas sobre las relaciones entre tres conceptos claves en didáctica
Las palabras «marco», «registro» y «medio» designan conceptos de amplia utilización por los investigadores en didáctica de las matemáticas, con el fin de modelizar situaciones o de analizar las actividades de los alumnos. Sin embargo, y a pesar de una literatura importante, su uso plantea problemas recurrentes para distinguirlas y relacionarlas (o vincularlas). Proponemos una solución a tales problemas, analizando las relaciones que mantienen entre ellos estos tres conceptos clave de la didáctica de las matemáticas. Mostraremos que la distinción entre «marco» (o «encuadre») y «concepción» debe ser buscada en el anclaje problemático de cada uno de estos conceptos; para los «marcos (o «encuadres»), se sitúa en primer plano el análisis matemático, para las «concepciones», tal lugar corresponde al análisis de las situaciones generadoras de obligaciones (o limitaciones) para el sujeto que aprende. En lo que a los «registros» se refiere, mostramos que son una herramienta indispensable para el funcionamiento de los «marcos» (o encuadres), como concepciones, y de sus relaciones; entre dos marcos, o dos concepciones, los registros tienen una función de mediación semiótica
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