Search results for "Metric geometry"
showing 10 items of 222 documents
Points massiques, cubiques Bézier rationnelles et leur points singuliers
2018
National audience; Cet articleétend l'étude des points singuliers des courbes rationnelles cubiques. Ellle porte sur les points d'inflexion, les points doubles et points de rebroussement. Les courbes cubiques rationnelleś etudiées sont décrites au moyen de la technique des points massiques. Un point massique est soit un point pondéré soit un vecteur pur. Il prend le statut de point de contrôle pour une représentation pa-ramétrique exploitable sur ordinateur dans le domaine de la géométrie de la Conception Assistée par Ordinateur. L'intérêt des points massiques est de pouvoir généraliser le tracé des courbes admettant des points doubles et de contrôler sans calcul supplémentaire l'ensemble d…
Geometry and analysis of Dirichlet forms (II)
2014
Abstract Given a regular, strongly local Dirichlet form E , under assumption that the lower bound of the Ricci curvature of Bakry–Emery, the local doubling and local Poincare inequalities are satisfied, we obtain that: (i) the intrinsic differential and distance structures of E coincide; (ii) the Cheeger energy functional Ch d E is a quadratic norm. This shows that (ii) is necessary for the Riemannian Ricci curvature defined by Ambrosio–Gigli–Savare to be bounded from below. This together with some recent results of Ambrosio–Gigli–Savare yields that the heat flow gives a gradient flow of Boltzman–Shannon entropy under the above assumptions. We also obtain an improvement on Kuwada's duality …
Evolution Problems Associated to Linear Growth Functionals: The Dirichlet Problem
2003
Let Ω be a bounded set inIR N with Lipschitz continuous boundary ∂Ω. We are interested in the problem
Capabilities of Ultrametric Automata with One, Two, and Three States
2016
Ultrametric automata use p-adic numbers to describe the random branching of the process of computation. Previous research has shown that ultrametric automata can have a significant decrease in computing complexity. In this paper we consider the languages that can be recognized by one-way ultrametric automata with one, two, and three states. We also show an example of a promise problem that can be solved by ultrametric integral automaton with three states.
On the Hierarchy Classes of Finite Ultrametric Automata
2015
This paper explores the language classes that arise with respect to the head count of a finite ultrametric automaton. First we prove that in the one-way setting there is a language that can be recognized by a one-head ultrametric finite automaton and cannot be recognized by any k-head non-deterministic finite automaton. Then we prove that in the two-way setting the class of languages recognized by ultrametric finite k-head automata is a proper subclass of the class of languages recognized by (k + 1)-head automata. Ultrametric finite automata are similar to probabilistic and quantum automata and have only just recently been introduced by Freivalds. We introduce ultrametric Turing machines an…
The branch set of a quasiregular mapping between metric manifolds
2016
Abstract In this note, we announce some new results on quantitative countable porosity of the branch set of a quasiregular mapping in very general metric spaces. As applications, we solve a recent conjecture of Fassler et al., an open problem of Heinonen–Rickman, and an open question of Heinonen–Semmes.
Finite linear spaces in which any n-gon is euclidean
1986
Abstract An n-gon of a linear space is a set S of n points no three of which are collinear. By a diagonal point of S we mean a point p off S with the property that at least two lines through p intersect S in two points. The number of diagonal points is called the type of S. For example, a 4-gon has at most three diagonal points. We call an n-gon euclidean if (roughly speaking) it contains the maximal possible number of 4-gons of type 3. In this paper, we characterize all finite linear spaces in which, for a fixed number n ⩾ 5, any n-gon is euclidean. It turns out that these structures are essentially projective spaces or punctured projective spaces.
Sobolev classes of Banach space-valued functions and quasiconformal mappings
2001
We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the “borderline case”. We show under rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree; in particular, they are absolutely continuous. This leads to an analytic characterization of quasiconformal mappings between Ahlfors regular Loewner spaces akin to the classical Euclidean situation. As a consequence, we deduce that quasisymmetric maps respect the Cheeger differentials of Lipschitz functions …
Radon–Nikodym Property and Area Formula for Banach Homogeneous Group Targets
2013
We prove a Rademacher-type theorem for Lipschitz mappings from a subset of a Carnot group to a Banach homogeneous group, equipped with a suitably weakened Radon-Nikodym property. We provide a metric area formula that applies to these mappings and more generally to all almost everywhere metrically differentiable Lipschitz mappings defined on a Carnot group. peerReviewed
A note on the distance set problem in the plane
2001
We use a simple geometric-combinatorial argument to establish a quantitative relation between the generalized Hausdorff measure of a set and its distance set, extending a result originally due to Falconer.