This paper is a brief survey of our recent study on the connection between two parts of Hilbert′s 16th problem and equivariant bifurcation problem. We hope to understand the following questions: can we use the period...This paper is a brief survey of our recent study on the connection between two parts of Hilbert′s 16th problem and equivariant bifurcation problem. We hope to understand the following questions: can we use the periodic solution family of ( m-1) degree planar Hamiltonian systems with Z q equivariant (or D q equivariant) symmetry to realize some schemes of ovals for planar algebraic curves? On the contrary, if an algebraic curve of degree m has maximal number of branches of ovals (it is called M -curve), can we make his all ovals become limit cycles of a planar polynomial system? What schemes of distribution of limit cycles can be realized by polynomial system.展开更多
Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure...Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.展开更多
文摘This paper is a brief survey of our recent study on the connection between two parts of Hilbert′s 16th problem and equivariant bifurcation problem. We hope to understand the following questions: can we use the periodic solution family of ( m-1) degree planar Hamiltonian systems with Z q equivariant (or D q equivariant) symmetry to realize some schemes of ovals for planar algebraic curves? On the contrary, if an algebraic curve of degree m has maximal number of branches of ovals (it is called M -curve), can we make his all ovals become limit cycles of a planar polynomial system? What schemes of distribution of limit cycles can be realized by polynomial system.
基金Supported by the National Natural Science Foundation of China (Grant Nos.10771028 60533060)+1 种基金the programof New Century Excellent Fellowship of NECCfunded by a DoD fund (Grant No.DAAD19-03-1-0375)
文摘Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.
