In this paper, we present the analytical expressions for computing the minimum distance between a point and a torus, which is called the orthogonal projection point problem. If the test point is on the outside of the ...In this paper, we present the analytical expressions for computing the minimum distance between a point and a torus, which is called the orthogonal projection point problem. If the test point is on the outside of the torus and the test point is at the center axis of the torus, we present that the orthogonal projection point set is a circle perpendicular to the center axis of the torus;if not, the analytical expression for the orthogonal projection point problem is also given. Furthermore, if the test point is in the inside of the torus, we also give the corresponding analytical expression for orthogonal projection point for two cases.展开更多
In this paper, we study the intersection multiplicity of algebraic curves at a point both in R^2 and in real projective plane P^2. We introduce the fold point of curves and provide conditions for the relations between...In this paper, we study the intersection multiplicity of algebraic curves at a point both in R^2 and in real projective plane P^2. We introduce the fold point of curves and provide conditions for the relations between the intersection multiplicity of curves at a point and the folds of the point.展开更多
Consider a compact symplectic sub-orbifold groupoid S of a compact symplectic orbifold groupoid(X,ω).LetXabe the weight-a blowup of X along S,and Da=PNa be the exceptional divisor,where N is the normal bundle of S in...Consider a compact symplectic sub-orbifold groupoid S of a compact symplectic orbifold groupoid(X,ω).LetXabe the weight-a blowup of X along S,and Da=PNa be the exceptional divisor,where N is the normal bundle of S in X.In this paper we show that the absolute orbifold Gromov-Witten theory ofXacan be effectively and uniquely reconstructed from the absolute orbifold Gromov-Witten theories of X,S and Da,the natural restriction homomorphism HCR^*(X)→HCR*(S)and the first Chern class of the tautological line bundle over DQ.To achieve this we first prove similar results for the relative orbifold Gromov-Witten theories of(Xa|Da)and(Na|Da).As applications of these results,we prove an orbifold version of a conjecture of Maulik and Pandharipande(Topology,2006)on the Gromov-Witten theory of blowups along complete intersections,a conjecture on the Gromov-Witten theory of root constructions and a conjecture on the Leray-Hirsch result for the orbifold Gromov-Witten theory of Tseng and You(J Pure Appl Algebra,2016).展开更多
文摘In this paper, we present the analytical expressions for computing the minimum distance between a point and a torus, which is called the orthogonal projection point problem. If the test point is on the outside of the torus and the test point is at the center axis of the torus, we present that the orthogonal projection point set is a circle perpendicular to the center axis of the torus;if not, the analytical expression for the orthogonal projection point problem is also given. Furthermore, if the test point is in the inside of the torus, we also give the corresponding analytical expression for orthogonal projection point for two cases.
基金Teaching reform research project of Shandong vocational education(2017228)
文摘In this paper, we study the intersection multiplicity of algebraic curves at a point both in R^2 and in real projective plane P^2. We introduce the fold point of curves and provide conditions for the relations between the intersection multiplicity of curves at a point and the folds of the point.
基金supported by National Natural Science Foundation of China(Grant Nos.11890663,11821001,11826102 and 11501393)the Sichuan Science and Technology Program(Grant No.2019YJ0509)a joint research project of Laurent Mathematics Research Center of Sichuan Normal University and V.C.&V.R.Key Lab of Sichuan Province。
文摘Consider a compact symplectic sub-orbifold groupoid S of a compact symplectic orbifold groupoid(X,ω).LetXabe the weight-a blowup of X along S,and Da=PNa be the exceptional divisor,where N is the normal bundle of S in X.In this paper we show that the absolute orbifold Gromov-Witten theory ofXacan be effectively and uniquely reconstructed from the absolute orbifold Gromov-Witten theories of X,S and Da,the natural restriction homomorphism HCR^*(X)→HCR*(S)and the first Chern class of the tautological line bundle over DQ.To achieve this we first prove similar results for the relative orbifold Gromov-Witten theories of(Xa|Da)and(Na|Da).As applications of these results,we prove an orbifold version of a conjecture of Maulik and Pandharipande(Topology,2006)on the Gromov-Witten theory of blowups along complete intersections,a conjecture on the Gromov-Witten theory of root constructions and a conjecture on the Leray-Hirsch result for the orbifold Gromov-Witten theory of Tseng and You(J Pure Appl Algebra,2016).
