As three control points are fixed and the fourth control point varies, the planar cubic C-curve may take on a loop, a cusp, or zero to two inflection points, depending on the position of the moving point. The plane ca...As three control points are fixed and the fourth control point varies, the planar cubic C-curve may take on a loop, a cusp, or zero to two inflection points, depending on the position of the moving point. The plane can, therefore, be partitioned into regions labelled according to the characterization of the curve when the fourth point is in each region. This partitioned plane is called a "characterization diagram". By moving one of the control points but fixing the rest, one can induce different characterization diagrams. In this paper, we investigate the relation among all different characterization diagrams of cubic C-curves based on the singularity conditions proposed by Yang and Wang (2004). We conclude that, no matter what the C-curve type is or which control point varies, the characterization diagrams can be obtained by cutting a common 3D characterization space with a corresponding plane.展开更多
A k-colouring （not necessarily proper） of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We co...A k-colouring （not necessarily proper） of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclie k-eolourings such that each colour class induces a graph with a given （hereditary） property. In particular, we consider aeyclic k-eolourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyelic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree 4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with A（G） ≤ 4 can be acyelically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.展开更多
We carry out an analysis of the canonical system of a minimal complex surface S of general type with irregularity q > 0.Using this analysis,we are able to sharpen in the case q > 0 the well-known Castelnuovo ine...We carry out an analysis of the canonical system of a minimal complex surface S of general type with irregularity q > 0.Using this analysis,we are able to sharpen in the case q > 0 the well-known Castelnuovo inequality KS2≥3pg(S) + q(S)-7.Then we turn to the study of surfaces with pg=2q-3 and no fibration onto a curve of genus > 1.We prove that for q≥6 the canonical map is birational.Combining this result with the analysis of the canonical system,we also prove the inequality:KS2≥7χ(S) + 2.This improves an earlier result of Mendes Lopes and Pardini (2010).展开更多
Given any (2,4)-elliptic surface with nine smooth rational curves,eight (2)-curves and one (3)-curve,forming a Dynkin diagram of type [2,2][2,2][2,2][2,2,3],we show that a fake projective plane can be constructed from...Given any (2,4)-elliptic surface with nine smooth rational curves,eight (2)-curves and one (3)-curve,forming a Dynkin diagram of type [2,2][2,2][2,2][2,2,3],we show that a fake projective plane can be constructed from it by taking a degree 3 cover and then a degree 7 cover.We also determine the types of singular fibres of such a (2,4)-elliptic surface.展开更多
Abstract We identify R^7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S^6. It is known that a cone over a surface M in S^6 ...Abstract We identify R^7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S^6. It is known that a cone over a surface M in S^6 is an associative submanifold of R^7 if and only if M is almost complex in S^6. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S^6 are the equation for primitive maps associated to the 6-symmetric space G2/T^2, and use this to explain some of the known results. Moreover, the equation for S^1-symmetric almost complex curves in S^6 is the periodic Toda lattice, and a discussion of periodic solutions is given.展开更多
Let G be a finite group. The degree(vertex) graph Γ(G) attached to G is a character degree graph.Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adja...Let G be a finite group. The degree(vertex) graph Γ(G) attached to G is a character degree graph.Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor(m, n) > 1. In this paper, we classify all graphs with four vertices that occur as Γ(G) for nonsolvable groups G.展开更多
基金Project supported by the National Natural Science Foundation of China (No. 60473130)the National Basic Research Program(973) of China (No. 2004CB318000)
文摘As three control points are fixed and the fourth control point varies, the planar cubic C-curve may take on a loop, a cusp, or zero to two inflection points, depending on the position of the moving point. The plane can, therefore, be partitioned into regions labelled according to the characterization of the curve when the fourth point is in each region. This partitioned plane is called a "characterization diagram". By moving one of the control points but fixing the rest, one can induce different characterization diagrams. In this paper, we investigate the relation among all different characterization diagrams of cubic C-curves based on the singularity conditions proposed by Yang and Wang (2004). We conclude that, no matter what the C-curve type is or which control point varies, the characterization diagrams can be obtained by cutting a common 3D characterization space with a corresponding plane.
基金supported by the Minister of Science and Higher Education of Poland(Grant No.JP2010009070)
文摘A k-colouring （not necessarily proper） of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclie k-eolourings such that each colour class induces a graph with a given （hereditary） property. In particular, we consider aeyclic k-eolourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyelic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree 4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with A（G） ≤ 4 can be acyelically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.
基金supported by FCT (Portugal) through program POCTI/FEDER and Project PTDC/MAT/099275/2008by MIUR (Italy) through project PRIN 2007 "Spazi di moduli e teorie di Lie"
文摘We carry out an analysis of the canonical system of a minimal complex surface S of general type with irregularity q > 0.Using this analysis,we are able to sharpen in the case q > 0 the well-known Castelnuovo inequality KS2≥3pg(S) + q(S)-7.Then we turn to the study of surfaces with pg=2q-3 and no fibration onto a curve of genus > 1.We prove that for q≥6 the canonical map is birational.Combining this result with the analysis of the canonical system,we also prove the inequality:KS2≥7χ(S) + 2.This improves an earlier result of Mendes Lopes and Pardini (2010).
基金supported by the National Research Foundation of Korea funded by the Ministry of Education,Science and Technology (Grant No. NRF-2007-2-C00002)
文摘Given any (2,4)-elliptic surface with nine smooth rational curves,eight (2)-curves and one (3)-curve,forming a Dynkin diagram of type [2,2][2,2][2,2][2,2,3],we show that a fake projective plane can be constructed from it by taking a degree 3 cover and then a degree 7 cover.We also determine the types of singular fibres of such a (2,4)-elliptic surface.
文摘Abstract We identify R^7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S^6. It is known that a cone over a surface M in S^6 is an associative submanifold of R^7 if and only if M is almost complex in S^6. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S^6 are the equation for primitive maps associated to the 6-symmetric space G2/T^2, and use this to explain some of the known results. Moreover, the equation for S^1-symmetric almost complex curves in S^6 is the periodic Toda lattice, and a discussion of periodic solutions is given.
基金supported by National Natural Science Foundation of China(Grant No.10871032)
文摘Let G be a finite group. The degree(vertex) graph Γ(G) attached to G is a character degree graph.Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor(m, n) > 1. In this paper, we classify all graphs with four vertices that occur as Γ(G) for nonsolvable groups G.
