A new case configuration in R^3, the conjugate-nest consisted of one regular tetrahedron and one regular octahedron is discussed. If the configuration is a central configuration, then all masses of outside layer are e...A new case configuration in R^3, the conjugate-nest consisted of one regular tetrahedron and one regular octahedron is discussed. If the configuration is a central configuration, then all masses of outside layer are equivalent, the masses of inside layer are also equivalent. At the same time the following relation between ρ（r =√3/3ρ is the radius ratio of the sizes） and mass ratio τ=~↑m/m must be satisfied τ=~↑m/m=ρ（ρ＋3）（3＋2ρ＋ρ^2）^-3/2＋ρ（-ρ＋3）（3-2ρ＋ρ^2）^-3/2-4.2^-3/2ρ^-2-^-1ρ^-2/2（1＋ρ）（3＋2ρ＋ρ^2）^-3/2＋2（ρ-1）（3-2ρ＋ρ^2）^-3/2-4（2√2）^-3ρ, and for any mass ratio τ, when mass ratio r is in the open interval （0, 0.03871633950 ... ）, there exist three central configuration solutions（the initial configuration conditions who imply hamagraphic solutions） corresponding radius ratios are r1, r2, and r3, two of them in the interval （2.639300779… , ＋∞） and one is in the interval （0.7379549890…, 1.490942703… ）. when mass ratio τ is in the open interval （130.8164950… , ＋∞）, in the same way there have three corresponding radius ratios, two of them in the interval （0, 0.4211584789... ） and one is in the interval （0.7379549890…, 1.490942703…）. When mass ratio τ is in the open interval （0.03871633950…, 130.8164950…）, there has only one solution r in the interval （0.7379549890…, 1.490942703… ）.展开更多
The purpose of this paper is to study relations among equivariant operations on 3-dimensional small covers. The author gets three formulas for these operations. As an application, the Nishimura＇s theorem on the const...The purpose of this paper is to study relations among equivariant operations on 3-dimensional small covers. The author gets three formulas for these operations. As an application, the Nishimura＇s theorem on the construction of oriented 3-dimensional small covers and the Lu-Yu＇s theorem on the construction of all 3-dimensional small covers are improved. Moreover, for a construction of 3-dimensional 2-torus manifolds, it is shown that all operations can be obtained by using the equivariant surgeries.展开更多
基金NSF of China(10231010)NSF of Chongqing EducationCommittee(071105)NSF of SXXYYB(070X)
文摘A new case configuration in R^3, the conjugate-nest consisted of one regular tetrahedron and one regular octahedron is discussed. If the configuration is a central configuration, then all masses of outside layer are equivalent, the masses of inside layer are also equivalent. At the same time the following relation between ρ（r =√3/3ρ is the radius ratio of the sizes） and mass ratio τ=~↑m/m must be satisfied τ=~↑m/m=ρ（ρ＋3）（3＋2ρ＋ρ^2）^-3/2＋ρ（-ρ＋3）（3-2ρ＋ρ^2）^-3/2-4.2^-3/2ρ^-2-^-1ρ^-2/2（1＋ρ）（3＋2ρ＋ρ^2）^-3/2＋2（ρ-1）（3-2ρ＋ρ^2）^-3/2-4（2√2）^-3ρ, and for any mass ratio τ, when mass ratio r is in the open interval （0, 0.03871633950 ... ）, there exist three central configuration solutions（the initial configuration conditions who imply hamagraphic solutions） corresponding radius ratios are r1, r2, and r3, two of them in the interval （2.639300779… , ＋∞） and one is in the interval （0.7379549890…, 1.490942703… ）. when mass ratio τ is in the open interval （130.8164950… , ＋∞）, in the same way there have three corresponding radius ratios, two of them in the interval （0, 0.4211584789... ） and one is in the interval （0.7379549890…, 1.490942703…）. When mass ratio τ is in the open interval （0.03871633950…, 130.8164950…）, there has only one solution r in the interval （0.7379549890…, 1.490942703… ）.
基金Project supported by Fudan Universitythe Fujyukai Foundation and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2009-0063179)
文摘The purpose of this paper is to study relations among equivariant operations on 3-dimensional small covers. The author gets three formulas for these operations. As an application, the Nishimura＇s theorem on the construction of oriented 3-dimensional small covers and the Lu-Yu＇s theorem on the construction of all 3-dimensional small covers are improved. Moreover, for a construction of 3-dimensional 2-torus manifolds, it is shown that all operations can be obtained by using the equivariant surgeries.
