Representations of Free Groups on Rational Line Q

A.M.W. Glass and S.H.McCleary have given the 2 transitive representation of the countable free l group F η(1<η≤ω 0 ).In this paper we shall give the highly ordered transitive representation of countable free groups on the rational line Q, which generalizes their results. As applications,we obtain the highly ordered transitive representation for the direct product of countable free groups,and the inverse limit of countable free groups must be an action on the set Q.

Comparison of anatomical and functional outcomes of different surgical techniques in myopic macular hole without retinal detachment

AIM:To define the anatomic and functional outcomes of pars plana vitrectomy(PPV)with internal limiting membrane(ILM)peeling,inverted ILM flap and free ILM patch graft technique for the treatment of myopic macular hole(MH)without retinal detachment.METHODS:Sixty-four eyes of 64 patients who underwent PPV for myopic MH were included.Group 1 consists of patients underwent ILM peeling(n=26),and Groups 2 and 3 consists of patient underwent free ILM patch graft(n=20)and inverted ILM flap procedure(n=18)respectively.Outcomes following surgery were MH closure and best corrected visual acuity(BCVA)in logMAR at 6mo.RESULTS:Closure of MH was obtained in 20 eyes(76.9%)of the Group 1,in 16 eyes(80%)of the Group 2 and in 16 eyes(88.9%)of the Group 3.The mean preoperative and postoperative BCVA was 1.60±0.53 logMAR and 1.27±0.58 logMAR,respectively(P<0.05).There was no significant difference in the postoperative BCVA and anatomical closure rates in the three groups.Although the anatomical closure rate did not differ significantly in the groups,closure of MH tended to be better in the inverted ILM flap technique group at 6mo.CONCLUSION:Different surgical techniques may provide favorable visual and anatomical results for myopic MH surgery.ILM flap techniques offer higher closure rates compared to ILM peeling technique.However,in terms of visual outcomes,the study reveals no difference in three surgical techniques.

Implicit Asymptotic Preserving Method for Linear Transport Equations

The computation of the radiative transfer equation is expensive mainly due to two stiff terms:the transport term and the collision operator.The stiffness in the former comes from the fact that particles(such as photons)travel at the speed of light,while that in the latter is due to the strong scattering in the optically thick region.We study the fully implicit scheme for this equation to account for the stiffness.The main challenge in the implicit treatment is the coupling between the spacial and angular coordinates that requires the large size of the to-be-inverted matrix,which is also ill-conditioned and not necessarily symmetric.Our main idea is to utilize the spectral structure of the ill-conditioned matrix to construct a pre-conditioner,which,along with an exquisite split of the spatial and angular dependence,significantly improve the condition number and allows a matrix-free treatment.We also design a fast solver to compute this pre-conditioner explicitly in advance.Our method is shown to be efficient in both diffusive and free streaming limit,and the computational cost is comparable to the state-of-the-art method.Various examples including anisotropic scattering and two-dimensional problems are provided to validate the effectiveness of our method.