Let f∈C^1(R^2, R^2), f(0)=0. The Jacobian Conjecture states that if for any x∈R^2, the eigenvalues of the Jacobian matrix Df(x) have negative real parts, then the zero solution of the differential equation x=f(x) is...Let f∈C^1(R^2, R^2), f(0)=0. The Jacobian Conjecture states that if for any x∈R^2, the eigenvalues of the Jacobian matrix Df(x) have negative real parts, then the zero solution of the differential equation x=f(x) is globally asymptotically stable. In this paper we prove that the conjecture is true.展开更多
We propose a novel method to compute globally injective parameterizations with arbitrary positional constraints on disk topology meshes.Central to this method is the use of a scaffold mesh that reduces the globally in...We propose a novel method to compute globally injective parameterizations with arbitrary positional constraints on disk topology meshes.Central to this method is the use of a scaffold mesh that reduces the globally injective constraint to a locally flipfree condition.Hence,given an initial parameterized mesh containing flipped triangles and satisfying the positional constraints,we only need to remove the flips of a overall mesh consisting of the parameterized mesh and the scaffold mesh while always meeting positional constraints.To successfully apply this idea,we develop two key techniques.Firstly,an initialization method is used to generate a valid scaffold mesh and mitigate difficulties in eliminating flips.Secondly,edgebased remeshing is used to optimize the regularity of the scaffold mesh containing flips,thereby improving practical robustness.Compared to state-of-the-art methods,our method is much more robust.We demonstrate the capability and feasibility of our method on a large number of complex meshes.展开更多
基金supported by the National Natural Science Foundation of China
文摘Let f∈C^1(R^2, R^2), f(0)=0. The Jacobian Conjecture states that if for any x∈R^2, the eigenvalues of the Jacobian matrix Df(x) have negative real parts, then the zero solution of the differential equation x=f(x) is globally asymptotically stable. In this paper we prove that the conjecture is true.
基金supported by the National Natural Science Foundation of China(61802359,62025207)USTC Research Funds of the Double First-Class Initiative(YD0010002003).
文摘We propose a novel method to compute globally injective parameterizations with arbitrary positional constraints on disk topology meshes.Central to this method is the use of a scaffold mesh that reduces the globally injective constraint to a locally flipfree condition.Hence,given an initial parameterized mesh containing flipped triangles and satisfying the positional constraints,we only need to remove the flips of a overall mesh consisting of the parameterized mesh and the scaffold mesh while always meeting positional constraints.To successfully apply this idea,we develop two key techniques.Firstly,an initialization method is used to generate a valid scaffold mesh and mitigate difficulties in eliminating flips.Secondly,edgebased remeshing is used to optimize the regularity of the scaffold mesh containing flips,thereby improving practical robustness.Compared to state-of-the-art methods,our method is much more robust.We demonstrate the capability and feasibility of our method on a large number of complex meshes.
