Lie groups of bi-M<span style="white-space:nowrap;">öbius transformations are known and their actions on non orientable n-dimensional complex manifolds have been studied. In this paper, m-M<...Lie groups of bi-M<span style="white-space:nowrap;">öbius transformations are known and their actions on non orientable n-dimensional complex manifolds have been studied. In this paper, m-M<span style="white-space:nowrap;">öbius transformations are introduced and similar problems as those related to bi-M<span style="white-space:nowrap;">öbius transformations are tackled. In particular, it is shown that the subgroup generated by every m-M<span style="white-space:nowrap;">öbius transformation is a discrete group. Cyclic subgroups are exhibited. Vector valued m-M<span style="white-space:nowrap;">öbius transformations are also studied.展开更多
Analytic atlases on <img src="Edit_948e45b7-cbef-425e-bb79-28648b859994.png" width="23" height="22" alt="" /> can be easily defined making it an n-dimensional complex mani...Analytic atlases on <img src="Edit_948e45b7-cbef-425e-bb79-28648b859994.png" width="23" height="22" alt="" /> can be easily defined making it an n-dimensional complex manifold. Then with the help of bi-M<span style="white-space:nowrap;"><span style="white-space:nowrap;">öbius transformations in complex coordinates Abelian groups are constructed making this manifold a Lie group. Actions of Lie groups on differentiable manifolds are well known and serve different purposes. We have introduced in previous works actions of Lie groups on non orientable Klein surfaces. The purpose of this work is to extend those studies to non orientable n-dimensional complex manifolds. Such manifolds are obtained by factorizing <img src="Edit_7e5721ee-bb7f-4224-bd52-7d4641ac1c73.png" width="23" height="22" alt="" /> with the two elements group of a fixed point free antianalytic involution of <img src="Edit_ddfdac64-b296-48c5-9bb2-932eb3d76008.png" width="23" height="22" alt="" />. Involutions h(z) of this kind are obtained linearly by composing special M<span style="white-space:nowrap;"><span style="white-space:nowrap;">öbius transformations of the planes with the mapping <img src="Edit_4cda269a-9399-41ae-a5da-0c9d18a419ab.png" width="89" height="24" alt="" /><img src="Edit_4cda269a-9399-41ae-a5da-0c9d18a419ab.png" width="85" height="22" alt="" />. A convenient partition of <img src="Edit_9e899708-41b0-4351-a12b-cc6efb5b1581.png" width="23" height="22" alt="" /> is performed which helps defining an internal operation on <img src="Edit_7cd42987-68f8-4e4c-9382-cbc68b56377e.png" width="49" height="26" alt="" /> and finally actions of the previously defined Lie groups on the non orientable manifold <img src="Edit_5740b48c-f9ea-438d-a87d-8cdc1f83662b.png" width="49" height="25" alt="" /> are displayed.展开更多
Let be a Clifford matrix in dimension n, c≠0, g（x）=（ax+b）（cx+d）<sup>-1</sup>, （x） be a Poincare extension in R<sup>n+1</sup>of g（x）. In 1985, L. Ahlfors established three conjugate c...Let be a Clifford matrix in dimension n, c≠0, g（x）=（ax+b）（cx+d）<sup>-1</sup>, （x） be a Poincare extension in R<sup>n+1</sup>of g（x）. In 1985, L. Ahlfors established three conjugate classes of M（R<sup>n</sup>）which are hyperbolic, elliptic and parabolic and their identifications. Here,展开更多
The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F.Defining primes in a non-abelian monoid is highly non-trivial,which relies on a concept calle...The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F.Defining primes in a non-abelian monoid is highly non-trivial,which relies on a concept called"castling".Three types of castlings are essential to grasp the arithmetics.The divisor functionτon Thompson's monoid S satisfiesτ(uv)≤τ(u)τ(v)for any u,v∈S.Then the limitτ_(0)(u)=lim_(n→∞)(τ(u~n))^(1/n)exists.The quantityC(S)=sup_(1≠u∈S)τ_(0)(u)/τ(u)describes the complexity for castlings in S.We show thatC(S)=1.Moreover,the MCbius function on S is calculated.And the Liouville functionCon S is studied.展开更多
文摘Lie groups of bi-M<span style="white-space:nowrap;">öbius transformations are known and their actions on non orientable n-dimensional complex manifolds have been studied. In this paper, m-M<span style="white-space:nowrap;">öbius transformations are introduced and similar problems as those related to bi-M<span style="white-space:nowrap;">öbius transformations are tackled. In particular, it is shown that the subgroup generated by every m-M<span style="white-space:nowrap;">öbius transformation is a discrete group. Cyclic subgroups are exhibited. Vector valued m-M<span style="white-space:nowrap;">öbius transformations are also studied.
文摘Analytic atlases on <img src="Edit_948e45b7-cbef-425e-bb79-28648b859994.png" width="23" height="22" alt="" /> can be easily defined making it an n-dimensional complex manifold. Then with the help of bi-M<span style="white-space:nowrap;"><span style="white-space:nowrap;">öbius transformations in complex coordinates Abelian groups are constructed making this manifold a Lie group. Actions of Lie groups on differentiable manifolds are well known and serve different purposes. We have introduced in previous works actions of Lie groups on non orientable Klein surfaces. The purpose of this work is to extend those studies to non orientable n-dimensional complex manifolds. Such manifolds are obtained by factorizing <img src="Edit_7e5721ee-bb7f-4224-bd52-7d4641ac1c73.png" width="23" height="22" alt="" /> with the two elements group of a fixed point free antianalytic involution of <img src="Edit_ddfdac64-b296-48c5-9bb2-932eb3d76008.png" width="23" height="22" alt="" />. Involutions h(z) of this kind are obtained linearly by composing special M<span style="white-space:nowrap;"><span style="white-space:nowrap;">öbius transformations of the planes with the mapping <img src="Edit_4cda269a-9399-41ae-a5da-0c9d18a419ab.png" width="89" height="24" alt="" /><img src="Edit_4cda269a-9399-41ae-a5da-0c9d18a419ab.png" width="85" height="22" alt="" />. A convenient partition of <img src="Edit_9e899708-41b0-4351-a12b-cc6efb5b1581.png" width="23" height="22" alt="" /> is performed which helps defining an internal operation on <img src="Edit_7cd42987-68f8-4e4c-9382-cbc68b56377e.png" width="49" height="26" alt="" /> and finally actions of the previously defined Lie groups on the non orientable manifold <img src="Edit_5740b48c-f9ea-438d-a87d-8cdc1f83662b.png" width="49" height="25" alt="" /> are displayed.
文摘Let be a Clifford matrix in dimension n, c≠0, g（x）=（ax+b）（cx+d）<sup>-1</sup>, （x） be a Poincare extension in R<sup>n+1</sup>of g（x）. In 1985, L. Ahlfors established three conjugate classes of M（R<sup>n</sup>）which are hyperbolic, elliptic and parabolic and their identifications. Here,
基金Supported by National Natural Science Foundation of China(Grant No.11701549)。
文摘The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F.Defining primes in a non-abelian monoid is highly non-trivial,which relies on a concept called"castling".Three types of castlings are essential to grasp the arithmetics.The divisor functionτon Thompson's monoid S satisfiesτ(uv)≤τ(u)τ(v)for any u,v∈S.Then the limitτ_(0)(u)=lim_(n→∞)(τ(u~n))^(1/n)exists.The quantityC(S)=sup_(1≠u∈S)τ_(0)(u)/τ(u)describes the complexity for castlings in S.We show thatC(S)=1.Moreover,the MCbius function on S is calculated.And the Liouville functionCon S is studied.
