Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization

We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices). We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces. The case of quadrilateral meshes is equivalent to the cross ratio system, which provides a link to the theory of integrable systems. The extension to cyclic polygons also brings discrete conformal maps to circle domains within the scope of the theory. We provide results of numerical experiments suggesting that discrete conformal maps converge to smooth conformal maps, with convergence rates depending on the mesh quality. We consider the Fuchsian uniformization of Riemann surfaces represented in different forms: as immersed surfaces in \mathbb {R}^{3}, as hyperelliptic curves, and as \mathbb {CP}^{1} modulo a classical Schottky group, i.e., we convert Schottky to Fuchsian uniformization. Extended examples also demonstrate a geometric characterization of hyperelliptic surfaces due to Schmutz Schaller

Bond percolation on isoradial graphs: criticality and universality

In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square, triangular, and other lattices. This is achieved via the star-triangle transformation, by transporting the box-crossing property across the family of isoradial graphs. As a consequence, we obtain the universality of these models at the critical point, in the sense that the one-arm and 2j-alternating-arm critical exponents (and therefore also the connectivity and volume exponents) are constant across the family of such percolation processes. The isoradial graphs in question are those that satisfy certain weak conditions on their embedding and on their track system. This class of graphs includes, for example, isoradial embeddings of periodic graphs, and graphs derived from rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex

Finslerian 3-spinors and the generalized Duffin-Kemmer equation

The main facts of the geometry of Finslerian 3-spinors are formulated. The close connection between Finslerian 3-spinors and vectors of the 9-dimensional linear Finslerian space is established. The isometry group of this space is described. The procedure of dimensional reduction to 4-dimensional quantities is formulated. The generalized Duffin-Kemmer equation for a Finslerian 3-spinor wave function of a free particle in the momentum representation is obtained. From the viewpoint of a 4-dimensional observer, this 9-dimensional equation splits into the standard Dirac and Klein-Gordon equations.Comment: LaTeX2e, 11 pages, no figures, will be published in "Fundamental and Applied Mathematics

Finding reliable subgraphs from large probabilistic graphs

Reliable subgraphs can be used, for example, to find and rank nontrivial links between given vertices, to concisely visualize large graphs, or to reduce the size of input for computationally demanding graph algorithms. We propose two new heuristics for solving the most reliable subgraph extraction problem on large, undirected probabilistic graphs. Such a problem is specified by a probabilistic graph G subject to random edge failures, a set of terminal vertices, and an integer K. The objective is to remove K edges from G such that the probability of connecting the terminals in the remaining subgraph is maximized. We provide some technical details and a rough analysis of the proposed algorithms. The practical performance of the methods is evaluated on real probabilistic graphs from the biological domain. The results indicate that the methods scale much better to large input graphs, both computationally and in terms of the quality of the result.Reliable subgraphs can be used, for example, to find and rank nontrivial links between given vertices, to concisely visualize large graphs, or to reduce the size of input for computationally demanding graph algorithms. We propose two new heuristics for solving the most reliable subgraph extraction problem on large, undirected probabilistic graphs. Such a problem is specified by a probabilistic graph G subject to random edge failures, a set of terminal vertices, and an integer K. The objective is to remove K edges from G such that the probability of connecting the terminals in the remaining subgraph is maximized. We provide some technical details and a rough analysis of the proposed algorithms. The practical performance of the methods is evaluated on real probabilistic graphs from the biological domain. The results indicate that the methods scale much better to large input graphs, both computationally and in terms of the quality of the result.Reliable subgraphs can be used, for example, to find and rank nontrivial links between given vertices, to concisely visualize large graphs, or to reduce the size of input for computationally demanding graph algorithms. We propose two new heuristics for solving the most reliable subgraph extraction problem on large, undirected probabilistic graphs. Such a problem is specified by a probabilistic graph G subject to random edge failures, a set of terminal vertices, and an integer K. The objective is to remove K edges from G such that the probability of connecting the terminals in the remaining subgraph is maximized. We provide some technical details and a rough analysis of the proposed algorithms. The practical performance of the methods is evaluated on real probabilistic graphs from the biological domain. The results indicate that the methods scale much better to large input graphs, both computationally and in terms of the quality of the result.Peer reviewe

Confluence up to Garbage

No abstract available

Polynomial Growth Harmonic Functions on Finitely Generated Abelian Groups

In the present paper, we develop geometric analytic techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We develop a geometric analytic proof of the classical Heilbronn theorem and the recent Nayar theorem on polynomial growth harmonic functions on lattices \mathds{Z}^n that does not use a representation formula for harmonic functions. We also calculate the precise dimension of the space of polynomial growth harmonic functions on finitely generated abelian groups. While the Cayley graph not only depends on the abelian group, but also on the choice of a generating set, we find that this dimension depends only on the group itself.Comment: 15 pages, to appear in Ann. Global Anal. Geo

Towards a Combinatorial Proof Theory

International audienceThe main part of a classical combinatorial proof is a skew fi-bration, which precisely captures the behavior of weakening and contraction. Relaxing the presence of these two rules leads to certain substruc-tural logics and substructural proof theory. In this paper we investigate what happens if we replace the skew fibration by other kinds of graph homomorphism. This leads us to new logics and proof systems that we call combinatorial

The crucial role of particle surface reactivity in respirable quartz-induced reactive oxygen/nitrogen species formation and APE/Ref-1 induction in rat lung

Persistent inflammation and associated excessive oxidative stress have been crucially implicated in quartz-induced pulmonary diseases, including fibrosis and cancer. We have investigated the significance of the particle surface reactivity of respirable quartz dust in relation to the in vivo generation of reactive oxygen and nitrogen species (ROS/RNS) and the associated induction of oxidative stress responses in the lung. Therefore, rats were intratracheally instilled with 2 mg quartz (DQ12) or quartz whose surface was modified by either polyvinylpyridine-N-oxide (PVNO) or aluminium lactate (AL). Seven days after instillation, the bronchoalveolar lavage fluid (BALF) was analysed for markers of inflammation (total/differential cell counts), levels of pulmonary oxidants (H(2)O(2), nitrite), antioxidant status (trolox equivalent antioxidant capacity), as well as for markers of lung tissue damage, e.g. total protein, lactate dehydrogenase and alkaline phosphatase. Lung homogenates as well as sections were investigated regarding the induction of the oxidative DNA-lesion/oxidative stress marker 8-hydroxy-2'-deoxyguanosine (8-OHdG) using HPLC/ECD analysis and immunohistochemistry, respectively. Homogenates and sections were also investigated for the expression of the bifunctional apurinic/apyrimidinic endonuclease/redox factor-1 (APE/Ref-1) by Western blotting and immunohistochemistry. Significantly increased levels of H(2)O(2 )and nitrite were observed in rats treated with non-coated quartz, when compared to rats that were treated with either saline or the surface-modified quartz preparations. In the BALF, there was a strong correlation between the number of macrophages and ROS, as well as total cells and RNS. Although enhanced oxidant generation in non-coated DQ12-treated rats was paralleled with an increased total antioxidant capacity in the BALF, these animals also showed significantly enhanced lung tissue damage. Remarkably however, elevated ROS levels were not associated with an increase in 8-OHdG, whereas the lung tissue expression of APE/Ref-1 protein was clearly up-regulated. The present data provide further in vivo evidence for the crucial role of particle surface properties in quartz dust-induced ROS/RNS generation by recruited inflammatory phagocytes. Our results also demonstrate that quartz dust can fail to show steady-state enhanced oxidative DNA damage in the respiratory tract, in conditions were it elicits a marked and persistent inflammation with associated generation of ROS/RNS, and indicate that this may relate to compensatory induction of APE/Ref-1 mediated base excision repair