Let f(z) be a holomorphic cusp form of weight κ with respect to the full modular group SL2(Z). Let L(s, f) be the automorphic L-function associated with f(z) and χ be a Dirichlet character modulo q. In this paper, t...Let f(z) be a holomorphic cusp form of weight κ with respect to the full modular group SL2(Z). Let L(s, f) be the automorphic L-function associated with f(z) and χ be a Dirichlet character modulo q. In this paper, the authors prove that unconditionally for k =1/n with n ∈ N,and the result also holds for any real number 0 < k < 1 under the GRH for L(s, f ■χ).The authors also prove that under the GRH for L(s, f ■χ),for any real number k > 0 and any large prime q.展开更多
We consider the extended Doimodel for nematic liquid crystalline polymers in-planar shear flow,which is inhomogeneous in shear direction.We study the formation of microstructure and the dynamics of defects.We discreti...We consider the extended Doimodel for nematic liquid crystalline polymers in-planar shear flow,which is inhomogeneous in shear direction.We study the formation of microstructure and the dynamics of defects.We discretize the Fokker-Plank equation using the spherical harmonic spectralmethod.Five in-plane flow modes and eight out-of-plane flow modes are replicated in our simulations.In order to demonstrate the validity of our method in simulating liquid crystal dynamics,we replicated weak shear limit results and detected defects.We also demonstrate numerically that the Bingham closure model,which maintains energy dissipation,is a reliable closure model.展开更多
A general nonhomogeneous extension of the Doi’s kinetic theory with trans-lational diffusion and nonlocal potential is proposed to describe the microstructuresand defect dynamics of Liquid Crystal Polymer (LCP) solut...A general nonhomogeneous extension of the Doi’s kinetic theory with trans-lational diffusion and nonlocal potential is proposed to describe the microstructuresand defect dynamics of Liquid Crystal Polymer (LCP) solutions. The long-range elas-ticity of polymer molecules is depicted by a kernel type potential, from which onecan derive the well-known Marrucci-Greco potential with weak spatial distortion as-sumption. Applying quasi-equilibrium closure approximation, we get a second-ordermoment model for isotropic long-range elasticity, and this reduced moment modelmaintains the energy dissipation. Implemented by the invariant-based fitting method,the moment model is a decent tool for numerical simulations of defect dynamics andtexture evolution in LCP solutions. The numerical results of in-plane rotational caseshow that the reduced second-order moment model qualitatively predicts complicatednonhomogeneous director dynamics under moderate nematic potential strength, andthe translational diffusion plays an important role in defect dynamics.展开更多
基金supported by the National Natural Science Foundation of China(No.11301299)the Natural Science Foundation of Shandong Province(No.ZR2012AQ001)the Specialized Research Fund for the Doctoral Program of Higher Education(New Teachers)(Nos.20110131120001,20120131120075)
文摘Let f(z) be a holomorphic cusp form of weight κ with respect to the full modular group SL2(Z). Let L(s, f) be the automorphic L-function associated with f(z) and χ be a Dirichlet character modulo q. In this paper, the authors prove that unconditionally for k =1/n with n ∈ N,and the result also holds for any real number 0 < k < 1 under the GRH for L(s, f ■χ).The authors also prove that under the GRH for L(s, f ■χ),for any real number k > 0 and any large prime q.
基金The authors would like to thank Prof.Sharon Murrel for her help in revising the En-glish.Guanghua Ji is partially supported by the National Science Foundation of China 10726015Pingwen Zhang is partially supported by the special funds for Major State Research Projects 2005CB321704National Science Foundation of China for Distin-guished Young Scholars 10225103 and 20490222.
文摘We consider the extended Doimodel for nematic liquid crystalline polymers in-planar shear flow,which is inhomogeneous in shear direction.We study the formation of microstructure and the dynamics of defects.We discretize the Fokker-Plank equation using the spherical harmonic spectralmethod.Five in-plane flow modes and eight out-of-plane flow modes are replicated in our simulations.In order to demonstrate the validity of our method in simulating liquid crystal dynamics,we replicated weak shear limit results and detected defects.We also demonstrate numerically that the Bingham closure model,which maintains energy dissipation,is a reliable closure model.
基金The authors would like to thank Prof.Sharon Murrel for her help in revising English.The research of Pingwen Zhang is partially supported by the State Key Basic Research Project of China 2005CB321704the National Science Foundation of China for Distinguished Young Scholars 10225103The research of Guanghua Ji is partially supported by National Science Foundation of China 10801014.
文摘A general nonhomogeneous extension of the Doi’s kinetic theory with trans-lational diffusion and nonlocal potential is proposed to describe the microstructuresand defect dynamics of Liquid Crystal Polymer (LCP) solutions. The long-range elas-ticity of polymer molecules is depicted by a kernel type potential, from which onecan derive the well-known Marrucci-Greco potential with weak spatial distortion as-sumption. Applying quasi-equilibrium closure approximation, we get a second-ordermoment model for isotropic long-range elasticity, and this reduced moment modelmaintains the energy dissipation. Implemented by the invariant-based fitting method,the moment model is a decent tool for numerical simulations of defect dynamics andtexture evolution in LCP solutions. The numerical results of in-plane rotational caseshow that the reduced second-order moment model qualitatively predicts complicatednonhomogeneous director dynamics under moderate nematic potential strength, andthe translational diffusion plays an important role in defect dynamics.
