Distortion of the triangle isoperimetricity coefficient under quasiconformal mapping

When solving problems of mathematical modeling on triangular and terahedral computational grids, it becomes necessary to estimate the error of the obtained solution, which depends on the degree of non-degeneracy of triangulation triangles. Therefore, long and narrow (“splinter”) triangles are avoided. We introduce the ratio (Δ) = |𝜕Δ| 𝑛-1 |Δ| , called isoperimetricity coefficient of an 𝑛-dimensional simplex Δ. The value (Δ) characterizes the deviation of an arbitrary simplex Δ from the regular simplex, since the minimum value is reached on the regular simplex based on isoperimetric inequality. Let the mapping : → (𝐷,𝐷 ⊂ R𝑛) is homeomorphic and differentiable almost everywhere. Denoted by ,Λ are the smallest and largest eigenvalues of the operator (𝑑𝑥0𝑓)𝑇 (𝑑𝑥0𝑓), respectively. For some interior point 𝑥0 ∈ Δ at which the mapping is differentiable, we denote = 𝐵(𝑥0, 𝑓, Δ) = max 𝑘=0,𝑛 |𝐻𝑘| |𝑃𝑘 - 𝑥0| , where = 𝐻(𝑥0, 𝑃𝑘) = 𝑓(𝑃𝑘) - 𝑓(𝑥0) - 𝑑𝑥0𝑓(𝑃𝑘 - 𝑥0). For a pair of simplex vertices and , we introduce the notation = |𝑃𝑖 - 𝑥0| + |𝑃𝑗 - 𝑥0|, 0 6 > and the area of the triangle is 𝑆, and (·)𝑧0 is the incenter of Δ, : → 𝐷′ is a differentiable quasiconformal mapping with the coefficient = ‖𝑑𝑧0𝑓‖2 · √ 1 + + √ 2𝐵 ‖𝑑𝑧0𝑓‖2 · √ 1 - - √ 2𝐵 . Then if function show_eabstract() { $('#eabstract1').hide(); $('#eabstract2').show(); $('#eabstract_expand').hide(); }