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A rough curvature-dimension condition for metric measure spaces

Anca-Iuliana Bonciocat (2014)

Open Mathematics

We introduce and study a rough (approximate) curvature-dimension condition for metric measure spaces, applicable especially in the framework of discrete spaces and graphs. This condition extends the one introduced by Karl-Theodor Sturm, in his 2006 article On the geometry of metric measure spaces II, to a larger class of (possibly non-geodesic) metric measure spaces. The rough curvature-dimension condition is stable under an appropriate notion of convergence, and stable under discretizations as...

Cauchy problems for discrete affine minimal surfaces

Marcos Craizer, Thomas Lewiner, Ralph Teixeira (2012)

Archivum Mathematicum

In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes,...

Cheeger inequalities for unbounded graph Laplacians

Frank Bauer, Matthias Keller, Radosław K. Wojciechowski (2015)

Journal of the European Mathematical Society

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.

Convex bodies associated to linear series

Robert Lazarsfeld, Mircea Mustață (2009)

Annales scientifiques de l'École Normale Supérieure

In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens...

Curvature on a graph via its geometric spectrum

Paul Baird (2013)

Actes des rencontres du CIRM

We approach the problem of defining curvature on a graph by attempting to attach a ‘best-fit polytope’ to each vertex, or more precisely what we refer to as a configured star. How this should be done depends upon the global structure of the graph which is reflected in its geometric spectrum. Mean curvature is the most natural curvature that arises in this context and corresponds to local liftings of the graph into a suitable Euclidean space. We discuss some examples.

Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

David Cimasoni (2012)

Journal of the European Mathematical Society

Let be a flat surface of genus g with cone type singularities. Given a bipartite graph Γ isoradially embedded in , we define discrete analogs of the 2 2 g Dirac operators on . These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair Γ for these discrete Dirac operators to be Kasteleyn matrices of the graph Γ . As a consequence, if these conditions are met, the partition function of the dimer...

Dreieckszerlegungen

Werner Raffke (1991)

Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry

Generalized Staircases: Recurrence and Symmetry

W. Patrick Hooper, Barak Weiss (2012)

Annales de l’institut Fourier

We study infinite translation surfaces which are -covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition for recurrence of their straight-line flows. Extending results of Hubert and Schmithüsen, we provide examples of infinite non-arithmetic lattice surfaces, as well as surfaces with infinitely generated Veech groups.

Motion of spirals by crystalline curvature

Hitoshi Imai, Naoyuki Ishimura, TaKeo Ushijima (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Modern physics theories claim that the dynamics of interfaces between the two-phase is described by the evolution equations involving the curvature and various kinematic energies. We consider the motion of spiral-shaped polygonal curves by its crystalline curvature, which deserves a mathematical model of real crystals. Exploiting the comparison principle, we show the local existence and uniqueness of the solution.

On stabbing triangles by lines in 3-space

Boris Aronov, Jiří Matoušek (1995)

Commentationes Mathematicae Universitatis Carolinae

We give an example of a set P of 3 n points in 3 such that, for any partition of P into triples, there exists a line stabbing Ω ( n ) of the triangles determined by the triples.

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