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Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

Zoltán M. Balogh, Jeremy T. Tyson, Kevin Wildrick (2013)

Analysis and Geometry in Metric Spaces

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the...

Dimers and cluster integrable systems

Alexander B. Goncharov, Richard Kenyon (2013)

Annales scientifiques de l'École Normale Supérieure

We show that the dimer model on a bipartite graph Γ on a torus gives rise to a quantum integrable system of special type, which we call acluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space Ł Γ of line bundles with connections on the graph Γ . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs Γ 1 and Γ 2 areequivalentif the Newton polygons of the corresponding partition functions...

Diophantine approximation on Veech surfaces

Pascal Hubert, Thomas A. Schmidt (2012)

Bulletin de la Société Mathématique de France

We show that Y. Cheung’s general Z -continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates in an appropriate sense. The saddle connection continued fractions then allow one to recognize certain transcendental...

Discrete complex analysis – the medial graph approach

Alexander I. Bobenko, Felix Günther (2013)

Actes des rencontres du CIRM

We discuss a new formulation of the linear theory of discrete complex analysis on planar quad-graphs based on their medial graphs. It generalizes the theory on rhombic quad-graphs developed by Duffin, Mercat, Kenyon, Chelkak and Smirnov and follows the approach on general quad-graphs proposed by Mercat. We provide discrete counterparts of the most fundamental objects in complex analysis such as holomorphic functions, differential forms, derivatives, and the Laplacian. Also, we discuss discrete versions...

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...

Disjoint hypercyclic operators

Luis Bernal-González (2007)

Studia Mathematica

We introduce the concept of disjoint hypercyclic operators. These are operators performing the approximation of any given vectors with a common subsequence of iterates applied on a common vector. The notion is extended to sequences of operators, and applied to composition operators and differential operators on spaces of analytic functions.

Currently displaying 1301 – 1320 of 6220