Contents -- Preface -- Subriemannian geometry and subelliptic partial differential equations (by Der-Chen Chang, Peter C. Greiner and Jingzhi Tie) -- 1. Euclidean Laplacian and elliptic operators -- 2. The Heisenberg group and the sub-Laplacian -- 3. The Hamilton-Jacobi equation and the heat kernel -- 4. SubRiemann geometry associated to step 3 sub-Laplacian -- Acknowledgments -- References -- Defective values of double Meissel's formula and reduction of space-time requirement of Meissel-Lehmer-Lagarias-Odlyzko's algorithm an experimental program to find T(1021) (by Chen Guangxiao) -- 1. Preface -- 2. Re-explanation of ideas of M-L-L-0's Algorithm -- 3. Estimation of time-space in computing defective sum -- 4. Some Remarks -- References -- Hardy space of holomorphic functions in infinite complex variables (by Zeqian Chen) -- 1. Introduction -- 2. Hardy space of infinite complex variables -- 3. Multipliers and the N-shift -- 4. von Neumann's inequality -- References -- The law of the iterated logarithm for pluriharmonic functions in the unit ball of Cn (by Zeqian Chen, Caiheng Ouyang) -- 1. Introduction -- 2. Preliminaries -- 3. Proof of Theorem -- Acknowledgement -- References -- Proper holomorphic mappings between some generalized Hartogs triangles (by Zhihua Chen) -- 1. Introduction -- 2. Preliminary -- 3. The proof of theorem 1.2 -- References -- Semigroups of holomorphic mappings with boundary fixed points and spirallike mappings (by Mark Elin and David Shoikhet) -- 0. Introduction -- 1. General description of cones G(B) and G[T] -- 2. Generators of one-dimensional type -- 3. Differential equations for starlike and spirallike mappings in H = Cn -- Acknowledgments -- References -- Invariant mappings in geometric function theory (by Carl H. FitzGerald) -- 1. One Variable Invariant Functions -- 2. Several Variable Invariant Mappings.

3. Conclusion -- 4. Acknowledgement -- The Cauchy Theorem for domains of arbitrary connectivity in Ftiemann surfaces (by P. M. Gauthier) -- 1. Introduction -- 2. Green-Goursat -- 3. Approximation -- 4. Natural domains -- 5. Nonrectifiable boundary -- 6. Discontinuous boundary functions -- References -- The distortion theorems for convex mappings in several complex variables (by Sheng Gong) -- 1. Introduction -- 2. The estimate of Jf(z)Jf(z) '. -- 3. Distortion theorem for linear invariant family -- 4. Distortion theorem for bounded symmetric domains -- References -- Anti-holomorphiclly reversible holomorphic maps that are not holomorphically reversible (by Xianghong Gong) -- 1. Introduction and results -- 2. Estimates for periodic points of a special family of holomorphic maps -- 3. Formally linearizable maps -- References -- Basic properties of Loewner chains in several complex variables (by Ian Graham, Gabriela and Mirela Kohr) -- 1. Introduction and preliminaries -- 2. The generalization of the Caratheodory class -- 3. Loewner chains and the Loewner differential equation -- 4. Lipschitz continuity and its consequences -- 5. The Roper-Suffridge extension operator -- Acknowledgments -- References -- The Euler-Lagrange cohomology on symplectic manifolds (by Han-Ying Guo, Jianzhong Pan, Ke Wu and Bin Zhou) -- 1. Introduction -- 2. The Euler-Lagrange Cohomology Group of Degree 1 -- 2.1. The Euler-Lagrange 1-Forms in the Lagrange Mechanics -- 2.2. The Euler-Lagmnge I-Form on a Symplectic Manifold -- 2.3. The Euler-Lagrange Cohomology Group of Degree 1 -- 3. The Euler-Lagrange Cohomology Groups on Symplectic Manifolds -- 3.1. The Euler-Lagrangian Cohomology Group of Degree 2k - 1 -- 3.2. Some Operators -- 3.3. The Spaces Xik-1 ( M , w ) and H (M, u) -- 3.4. The Other Euler-Lagrange Cohomology Groups.

3.5. Euler-Lagrange Cohomology and Harmonic Cohomology -- 3.6. The Relative Euler-Lagrange Cohomology -- 4. The General Volume-Preserving Hamiltonian Equations -- 4.1. The Derivation of the Equations -- 4.2. On The Canonical Hamiltonian Equations, The l'race of 2-Forms and The Poisson Bracket -- 4.3. One Possible Application -- 5. Discussions and Conclusions -- Acknowledgement -- References -- A new inequality and its applications (by Hu Ke) -- 1. A new inequality -- 2. Some applications of the Theorem 1 -- References -- Extended Cesbro operators on the Bloch space in the unit ball of Cn (by Hu, Zhangjian) -- 1. Introduction -- 2. Main Theorems -- References -- On the criteria for Schatten von Neumann class of composition operators on Hardy and Bergman spaces in domains in Cn (by Song-Ying Li) -- 1. Introduction and main theorems -- 2. The Proof of Theorem 1.3 -- 3. The proof of Theorem 1.1 -- 4. The Proof of Theorem 1.4 -- 5. Proof of Proposition 1.2 -- References -- The higher order linear partial differential integral equations on closed smooth manifolds in Cn (by Liangyu Lin, Chunhui Qiu and Yusheng Huang) -- 1. Introduction -- 2. Definitions and main theorems -- Acknowledgments -- References -- The new characteristics for spirallike mappings of type a on bounded balanced pseudoconvex domains (by Hao Liu) -- 1. Introduction -- 2. Preliminaries -- 3. Main Theorems and Proof -- References -- The growth and 1/2-covering theorems for quasi-convex mappings (by Taishun Liu and Wenjun Zhang) -- 1. Quasi-convex mapping of type A and Quasi-convex mappings in Complex Banach Space -- 2. Several Lemmas -- 3. The Growth and Covering Theorems of Quasi-Convex Mappings -- References -- Intermediate value theorem for functions of classes of Riemann surfaces (by Makoto Masumoto) -- 1. Introduction -- 2. Main theorem -- 3. Peano curve method -- 4. Examples.

References -- Integral formula for differential forms of type (P, Q) on complex Finsler manifolds (by Chunhui Qui and Tongde Zhong) -- 1. Introduction -- 2. Complex Finsler manifolds and invariant integral kernel -- 3. Invariant integral kernel in local coordinates -- 4. The Koppelman formula for differential forms of type (P, 4) -- Acknowledgments -- References -- Holomorphic mappings of domains in Cn onto convex domains (by Ted J. Suffridge) -- 1. Fundamental Concepts -- 2. Extension to the Boundary -- 3. Convex Domains that Contain A Line -- 4. Other Circular Domains in Cn -- 5. Open Problems -- References -- Rigidity of proper holomorphic mappings between bounded symmetric domains (by Zhen-Han Tu) -- Acknowledgments -- References -- A Hadamard theorem on algebraic curves (by Shi-Kun Wang and Hui-Ping Zhang) -- 1. Preliminary -- 2. Hardamard Three-Circle Theorem on Riemann Surface -- Acknowledgement -- References -- Hodge-Laplace operator on complex Finsler manifolds (by Chunping Zhong and Tongde Zhong) -- 1. Introduction -- 2. Complex Finsler metric on complex manifold -- 2.1. Case I: If ( M, F ) is a compact Hermitian manifold -- 2.2. Case II: If ( M , F ) is a strongly pseudoconvex compact complex Finsler manifold -- 3. Hermitian product on projectivized tangent bundle PTM -- 4. Global Hermitian inner product and Hodge-Laplace operator on compact complex Finsler manifolds -- Acknowledgments -- References -- Weighted composition operators on the Lipschitz space in polydiscs (by Zehua Zhou) -- 1. Introduction -- 2. Some Lemmas -- 3. The Proof of Theorem 1 -- 4. The Proof of Theorem 2 -- References.