We prove, by using the main inequality of Reich and Strebel, that any n K-quasiconformal germs defined on n disjoint domains in the Riemann sphere can be glued by one (K + ε)-quasiconformal homeomorphism, where ε is a positive number which can go to zero as the domains of germs shrinking to n points. This generalizes a result in [8] where only the case K = 1 has been considered.

We study Fano manifolds X admitting an unsplit dominating family of rational curves and we prove that the Generalized Mukai Conjecture holds if X has pseudoindex i_X = (dim X)/3 or dimension dim X = 6. We also show that this conjecture is true for all Fano manifolds with i_X > (dim X)/3.

By using Dziok-Srivastava operator a new subclass of analytic functions generalized k-parabolic starlike functions, denoted by k–SP_l,m (α₁;γ), is introduced. For this class the Fekete-Szegö problem is completely solved. Various known or new special cases of our results are also point out.

In this article, we prove a finiteness theorem for meromorphic mappings of C^m into Pⁿ(C) sharing 2n + 2 hyperplanes without counting multiplicity.

The purpose of this work is twofold. The first is to solve a uniqueness problem of meromorphic mappings posed by T. Cao and H. Yi in [1]. The second is to generalize several previous uniqueness theorems of meromorphic mappings "partially" sharing a few moving targets, which were given by Z. Chen and M. Ru [2], Z. Chen and Q. Yan [3], D. Thai and S. Quang [13].

In this paper, we investigate two hyperbolic flows obtained by adding forcing terms in direction of the position vector to the hyperbolic mean curvature flows in [5, 8]. For the first hyperbolic flow, as in [8], by using support function, we reduce it to a hyperbolic Monge-Ampère equation successfully, leading to the short-time existence of the flow by the standard theory of hyperbolic partial differential equation. If the initial velocity is non-negative and the coefficient function of the forcing term is non-positive, we also show that there exists a class of initial velocities such that the solution of the flow exists only on a finite time interval [0,T_max), and the solution converges to a point or shocks and other propagating discontinuities are generated when t → T_max. These generalize the corresponding results in [8]. For the second hyperbolic flow, as in [5], we can prove the system of partial differential equations related to the flow is strictly hyperbolic, which leads to the short-time existence of the smooth solution of the flow, and also the uniqueness. We also derive nonlinear wave equations satisfied by some intrinsic geometric quantities of the evolving hypersurface under this hyperbolic flow. These generalize the corresponding results in [5].

A surface Σ₀ is a graph in R⁴ if there is a unit constant 2-form w in R⁴ such that ‹e₁ ∧ e₂, w› ≥ v₀ > 0, where {e₁, e₂} is an orthonormal frame on Σ₀. In this paper, we investigate a 2-dimensional surface Σ evolving along a mean curvature flow with a forcing term in direction of the position vector. If v₀ ≥ ${1 \over \sqrt {2}}$ holds on the initial graph Σ₀ which is the immersion of the surface Σ, and the coefficient function of the forcing vector is nonnegative, then the forced mean curvature flow has a global solution, which generalizes part of the results of Chen-Li-Tian in [2].

Two types of elementary and direct proofs of the classification theorem for CR automorphisms of generalized ellipsoids by [MM10] are given. The first proof is to use an invariant decomposition of holomorphic tangent bundle, and the second is to use infinitesimal CR automorphisms. The advantages of our proofs are: (1) The conditions on the number n_j of the variables and the exponents m_j in the defining equation are weakened, (2) the proofs are more direct than [MM10], (3) our proofs may be applicable to wider class of hypersurfaces.

We prove quenched large deviation principles governing the position of the random walk on a supercritical site percolation on the integer lattice. A feature of this model is non-ellipticity of transition probabilities. Our analysis is based on the consideration of so-called Lyapunov exponents for the Laplace transform of the first passage time. The rate function is given by the Legendre transform of the Lyapunov exponents.

Let (R,$\mathfrak{m}$) be a commutative Noetherian local ring. A finitely generated R-module M is called sequentially generalized Cohen-Macaulay module if there is a filtration M₀ $\subseteq$ M₁ $\subseteq$ ··· $\subseteq$ M_t = M of submodules of M such that 0 = dim M₀ < dim M₁ < ··· < dim M_t and each M_i/M_i–1 is a generalized Cohen-Macaulay module. In this paper we study the asymptotic behavior of good systems of parameters, introduced in [N. T. Cuong, D. T. Cuong, On sequentially Cohen-Macaulay modules, Kodai Math. J. 30 (2007), 409-428], of sequentially generalized Cohen-Macaulay modules.

We consider a positive solution of the Emden equation with the critical Sobolev exponent on a geodesic ball in S³. In the case of the Dirichlet boundary condition, Bandle and Peletier [2] proved the precise result on the existence of a positive radial solution. We investigate the same equation with the third kind boundary condition and obtain a more general result. Namely we prove that the existence and the nonexistence of solutions depend on the geodesic radius and the boundary condition. Moreover the set of solutions consists of a unique radial classical solution and a continuum of singular solutions.

A limit theorem for the maximum processes of a class of null recurrent linear diffusions is proved. The limiting distribution is a mixture of the Mittag-Leffler distribution.

Given k ≥ 2 let α₁, ..., α_k be transcendental numbers such that α₁, ..., α_k–1 are algebraically independent over Q and α_k $\in$ Q(α₁, ..., α_k–1), but α_k, ≠ (aα_i + c)/b for some i $\in$ {1, ..., k – 1} and some a, b $\in$ N, c $\in$ Z satisfying gcd(a,b) = 1. We prove that then there exists a nonnegative integer q such that the set of so-called Dirichlet exponents log(n + α_j, where n runs through the set of all nonnegative integers for j = 1, ..., k – 1 and n = q, q + 1, q + 2, ... for j = k, is linearly independent over Q. As an application we obtain a joint universality theorem for corresponding Hurwitz zeta functions ζ(s, α₁), ..., ζ(s, α_k) in the strip {s $\in$ C: 1/2 < $\Re$(s) < 1}. In our approach we follow a recent result of Mishou who analyzed the case k = 2.