Using Hartogs' fundamental theorem for analytic functions in several complex variables, we establish a multiple $q$-exponential differential operational identity for the analytic functions in several variables, which can be regarded as a multiple $q$-translation formula. This multiple $q$-translation formula is a fundamental result and play a pivotal role in $q$-mathematics. Using this $q$-translation formula, we can easily recover many classical conclusions in $q$-mathematics and derive some new $q$-formulas. Our work reveals some profound connections between the theory of complex functions in several variables and $q$-mathematics.

In this paper we consider the Monge-Ampère type equations on compact almost Hermitian manifolds. We derive C^∞ a priori estimates under the existence of an admissible C-subsolution. Finally, we obtain an existence result if there exists an admissible supersolution.

Based on the action of the mapping class group on the space of measured foliations, we construct a new boundary of the mapping class group and study the structure of this boundary. As an application, for any point in Teichmüller space, we consider the orbit of this point under the action of the mapping class group and describe the closure of this orbit in the Thurston compactification and the Gardiner-Masur compactification of Teichmüller space. We also construct some new points in the Gardiner-Masur boundary of Teichmüller space.

In this note, we present an L² extension approach to the effectiveness result of strong openness property of multiplier ideal sheaves.