18,527 research outputs found
An embedding technique for the solution of reaction-fiffusion equations on algebraic surfaces with isolated singularities
In this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry.\ud
We create a family of smooth curves in higher dimensional space that correspond to the original curve by projection. Following this, we pose the analogous reaction-diffusion PDE on each member of this family and show that the solutions (their projection onto the original domain) approximate the solution of the original problem. Finally, we compute these approximants numerically by applying the Closest Point Method which is an embedding technique for solving PDEs on smooth surfaces of arbitrary dimension or codimension, and is thus suitable for our situation. In addition, we discuss the potential to generalize the techniques presented for higher-dimensional surfaces with multiple singularities
Codimension two Umbilic points on Surfaces Immersed in R^3
In this paper is studied the behavior of lines of curvature near umbilic
points that appear generically on surfaces depending on two parameters.Comment: 19 pages, 10 figure
Frobenius 3-Folds via Singular Flat 3-Webs
We give a geometric interpretation of weighted homogeneous solutions to the
associativity equation in terms of the web theory and construct a massive
Frobenius 3-fold germ via a singular 3-web germ satisfying the following
conditions: 1) the web germ admits at least one infinitesimal symmetry, 2) the
Chern connection form is holomorphic, 3) the curvature form vanishes
identically
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