Approximation of conformal mappings using conformally equivalent triangular lattices

Consider discrete conformal maps defined on the basis of two conformally equivalent triangle meshes, that is edge lengths are related by scale factors associated to the vertices. Given a smooth conformal map $f$, we show that it can be approximated by such discrete conformal maps $f^\epsilon$. In particular, let $T$ be an infinite regular triangulation of the plane with congruent triangles and only acute angles (i.e.\ $<\pi/2$). We scale this tiling by $\epsilon>0$ and approximate a compact subset of the domain of $f$ with a portion of it. For $\epsilon$ small enough we prove that there exists a conformally equivalent triangle mesh whose scale factors are given by $\log|f'|$ on the boundary. Furthermore we show that the corresponding discrete conformal maps $f^\epsilon$ converge to $f$ uniformly in $C^1$ with error of order $\epsilon$.Comment: 14 pages, 3 figures; v2 typos corrected, revised introduction, some proofs extende

Constructing solutions to the Bj\"orling problem for isothermic surfaces by structure preserving discretization

In this article, we study an analog of the Bj\"orling problem for isothermic surfaces (that are more general than minimal surfaces): given a real analytic curve $\gamma$ in ${\mathbb R}^3$, and two analytic non-vanishing orthogonal vector fields $v$ and $w$ along $\gamma$, find an isothermic surface that is tangent to $\gamma$ and that has $v$ and $w$ as principal directions of curvature. We prove that solutions to that problem can be obtained by constructing a family of discrete isothermic surfaces (in the sense of Bobenko and Pinkall) from data that is sampled along $\gamma$, and passing to the limit of vanishing mesh size. The proof relies on a rephrasing of the Gauss-Codazzi-system as analytic Cauchy problem and an in-depth-analysis of its discretization which is induced from the geometry of discrete isothermic surfaces. The discrete-to-continuous limit is carried out for the Christoffel and the Darboux transformations as well.Comment: 29 pages, some figure

Discrete complex analysis on planar quad-graphs

We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph yields more instructive proofs of discrete analogs of several classical theorems and even new results. We provide discrete counterparts of fundamental concepts in complex analysis such as holomorphic functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss discrete versions of important basic theorems such as Green's identities and Cauchy's integral formulae. For the first time, we discretize Green's first identity and Cauchy's integral formula for the derivative of a holomorphic function. In this paper, we focus on planar quad-graphs, but we would like to mention that many notions and theorems can be adapted to discrete Riemann surfaces in a straightforward way. In the case of planar parallelogram-graphs with bounded interior angles and bounded ratio of side lengths, we construct a discrete Green's function and discrete Cauchy's kernels with asymptotics comparable to the smooth case. Further restricting to the integer lattice of a two-dimensional skew coordinate system yields appropriate discrete Cauchy's integral formulae for higher order derivatives.Comment: 49 pages, 8 figure

The influence of the ectomycorrhizal fungus Rhizopogon subareolatus on growth and nutrient element localisation in two varieties of Douglas fir (Pseudotsuga menziesii var. menziesii and var. glauca) in response to manganese stress

Acidification of forest ecosystems leads to increased plant availability of the micronutrient manganese (Mn), which is toxic when taken up in excess. To investigate whether ectomycorrhizas protect against excessive Mn by improving plant growth and nutrition or by retention of excess Mn in the hyphal mantle, seedlings of two populations of Douglas fir (Pseudotsuga menziesii), two varieties, one being menziesii (DFM) and the other being glauca (DFG), were inoculated with the ectomycorrhizal fungus Rhizopogon subareolatus in sand cultures. Five months after inoculation, half of the inoculated and non-inoculated seedlings were exposed to excess Mn in the nutrient solution for further 5 months. At the end of this period, plant productivity, nutrient concentrations, Mn uptake and subcellular compartmentalisation were evaluated. Non-inoculated, non-stressed DFM plants produced about 2.5 times more biomass than similarly treated DFG. Excess Mn in the nutrient solution led to high accumulation of Mn in needles and roots but only to marginal loss in biomass. Colonisation with R. subareolatus slightly suppressed DFM growth but strongly reduced that of DFG (−50%) despite positive effects of mycorrhizas on plant phosphorus nutrition. Growth reductions of inoculated Douglas fir seedlings were unexpected since the degree of mycorrhization was not high, i.e. ca. 30% in DFM and 8% in DFG. Accumulation of high Mn was not prevented in inoculated seedlings. The hyphal mantle of mycorrhizal root tips accumulated divalent cations such as Ca, but not Mn, thus not providing a barrier against excessive Mn uptake into the plants associated with R. subareolatus

Mycorrhization of fagaceae forests within mediterranean ecosystems

Mediterranean Fagaceae forests are valuable due to their ecological and socioeconomic aspects. Some profitable plant species, such as Castanea (timber and chestnut), Quercus (timber and cork), and Fagus (timber), encounter in this habitat the excellent edaphoclimatic conditions to develop. All Fagaceae plants are commonly associated to ECM fungal species, which are found in these forests in quite stable communities, mainly enriched in Russulaceae and Telephoraceae species. Currently, the Mediterranean Basin is considered as one of the global biodiversity hotspots, since many of their endemic plant species are not found elsewhere and are now under threat. Due to climate changing and introduction of disease agents, Fagaceae forests are facing an adaptation challenge to both biotic and abiotic threats. Although ECM communities are highly disturbed by climate factors and tree disease incidence, they could play an important role in increasing water availability to the plant and also improving plant tree defense against pathogens. Recent advances, namely, on genomics and transcriptomics, are providing tools for increasing the understanding of Fagaceae mycorrhization process and stress responses to biotic and abiotic stresses. Such studies can provide new information for the implementation of the most adequate management policies for protecting threaten Mediterranean forests.info:eu-repo/semantics/publishedVersio