Some Uniqueness Theorems with Truncated Multiplicities of Meromorphic Mappings

In this article, some uniqueness theorems of meromorphic mappings in several complex variables sharing hyperplanes in general position are proved with truncated multiplicities.     

On Best Approximations from RS-sets in Complex Banach Spaces

The concept of an RS-set in a complex Banach space is introduced and the problem of best approximation from an RS-set in a complex space is investigated. Results consisting of characterizations, uniqueness and strong uniqueness are established,     

A UNICITY THEOREM FOR ENTIRE FUNCTIONS OF SEVERAL COMPLEX VARIABLES

A unicity theorem concerning the total derivative for entire functions of several complex variables is proved.     

A RESULT OF SYSTEMS OF NONLINEAR COMPLEX ALGEBRAIC DIFFERENTIAL EQUATIONS

In this article, we mainly investigate the behavior of systems of complex differential equations when we add some condition to the quality of the solutions, and obtain an interesting result, which extends Gaekstatter and Laine＇s result concerning complex differential equations to the systems of algebraic differential equations.     

共享三个值的亚纯函数的唯一性问题

This paper investigate the uniqueness problems for meromorphic functions that share three values CM and proves a uniqueness theorem on this topic which can be used to improve some previous related results.     

Equivalence of three kinds of well-posed-ness of discontinuous Riemann-Hilbert problem for elliptic complex equation in multiply connected domains

In this article, we first introduce the general linear elliptic complex equation of first order with certain conditions, and then propose discontinuous Riemann-Hilbert problem and some kinds of modified well-posed-ness for the complex equation. Then we verify the equivalence of three kinds of well-posed-ness. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.     

Uniqueness Theorems for Meromorphic Mappings in Several Complex Variables into PN（c） with Two Families of Moving Targets

The authors prove some uniqueness theorems for meromorphic mappings in several complex variables into the complex projective space PN(C)with two families of moving targets,and the results obtained improve some earlier work.     

Admissible solutions of systems of complex partial difference equations on Cn

In a recent papers,some authors applied Nevanlinna theory to prove some results on complex difference equations reminiscent of the classical Malmquist theorem in complex differential equations. We will mainly investigate Malmquist theorem of a type of systems of complex partial difference equations on Cn,improvements and extensions of such results are presented in this paper.     

SLANT IMMERSIONS OF COMPLEX SPACE FORMS AND CHEN＇S INEQUALITY

A submanifold in a complex space form is called slant it it has constant Wirtinger angles. B, Y, Chen and Y. Tazawa proved that there do not exist minimal proper slant surfaces in CP^2 and CH^2. So it seems that the slant immersion has some interesting properties. The authors have great interest to consider slant immersions satisfying some additional conditions, such as unfull first normal bundles or Chen‘s equality holding. They prove that there do not exist n-dimensional Kaehlerian slant immersion sin CP^n and CH^n with unfull first normal bundles. Next, it is seen that every Kaehlerian slant submanifold satisfying an equality of Chen is minimal which is similar to that of Lagrangian immersions. But in contrast, it is shown that a large class of slant immersions do not exist thoroughly. Finally, they give an application of Chen‘s inequality to general slant immersions in a complex projective space, which generalizes a result of Chen.     

SOME PROPERTIES OF HOLOMORPHIC CLIFFORDIAN FUNCTIONS IN COMPLEX CLIFFORD ANALYSIS

In this article, we mainly develop the foundation of a new function theory of several complex variables with values in a complex Clifford algebra defined on some subdomains of Cn+1, so-called complex holomorphic Cliffordian functions. We define the complex holomorphic Cliffordian functions, study polynomial and singular solutions of the equation D△mf=0, obtain the integral representation formula for the complex holomorphic Cliffordian functions with values in a complex Clifford algebra defined on some submanifolds of Cn+1, deduce the Taylor expansion and the Laurent expansion for them and prove an invariance under an action of Lie group for them.     

Uniform approximation by polynomials on compacta of special form

A lower estimate of the least deviations is obtained and polynomials of the best uniform approximation are found for some functions given on compact sets of the complex plane containing complete preimages Q ^?1(v _j ) of several points v _j ?? ? for some polynomial Q(z) of complex variable.     

Symmetric (co)homologies of Lie algebras

Cohomologies of Lie algebras are usually calculated using the Chevalley-Eilenberg cochain complex of skew-symmetric forms. We consider two cochain complexes consisting of forms with some symmetric properties. First, cochains C^*(L) are symmetric in the last 2 arguments, skew-symmetric in the others and satify moreover some kind of Jacobi condition in the last 3 arguments. In characteristic 0, its cohomologies are isomorphic to the cohomologies of the factor-complex C^*(L,L’)/C^*+1(L,K). Second, a symmetric version C_λ^*(A) is defined for an associative algebra A. It is a subcomplex of the cyclic cochain complex. These symmetric cochain complexes are used for the calculation of 3-cohomologies of Cartan Type Lie algebras with trivial coefficients.     

On Meromorphically Normal Families of Meromorphic Mappings of Several Complex Variables into P(C)

We shall prove some meromorphic normality criteria for families of meromorphic mappings of several complex variables into P^N(C), the complex N-dimensional projective space, related to Nochka's Picard type theorems. Some related results (e.g., extension theorems, improved normality criteria, and quasi-normality criteria) will be obtained also. The technique in this paper mainly depends on Stoll's normality criteria for families of non-negative divisors on a domain of Cⁿ.     

Real hypersurfaces in complex two-plane grassmannians with parallel structure Jacobi operator

We give some non-existence theorems for Hopf real hypersurfaces in complex two-plane Grassmannians G ₂(? ^m+2) with parallel structure Jacobi operator R _ξ.     

Integration of complex polynomial differential equations in higher dimension,a first step: Lie transverse structure

We prove that a one-dimensional holomorphic foliationswith generic singularities on a complex projective space ?P ^m+1, m ≥ 2, exhibiting a Lie group transverse structure in some Zariski open subset, is logarithmic. That is, it is given by a system of m closed rational one-forms with simple poles. The foliation is given by a linear vector field in some affine space ? ^m+1 ? ?P ^m+1 if, and only if, it exhibits only one singularity in this affine space. An application to foliations invariant under Lie group transverse actions is given.     

Spectral geometry for almost isospectral hermitian manifolds

The following question was posed by M. Berger: Is it possible to determine from the spectrum of the real Laplacian whether or not a manifold is Kähler? The Kähler condition for Hermitian manifolds is found out from the invariants of the spectrum of some differential operators acting on forms of type (p, q). P. Gilkey and H. Donnelly proved the Berger conjecture for the complex Laplacian and the reduced complex Laplacian respectively. In this paper we consider the Berger conjecture of almost isospectral Hermitian manifolds about the complex Laplacian acting on forms of type (p, q). Then we can show that a closed complexm(≥ 3)-dimensional Hermitian manifold which is strongly (?2/m)-isospectral to the complex projective space CP ^m with the Fubini-Study metric is holomorphically isometric to CP ^m .     

L P estimates for ∂-equation on generalized complex ellipsoids

The estimate of a holomorphic supporting function for the generalized complex ellipsoid in ?ⁿ is given, This domain is not decoupled. By using this estimate, the best possibleL ^p estimates for the ?-equation and some results of function theory on generalized complex ellipsoids are proved.     

The Cauchy problem for critical and subcritical semilinear parabolic equations in Lr (II), initial data in critical Sobolev spaces Ḣ−s,rs

By presenting some time–space L^p−L^r estimates, we will establish the local and global existence and uniqueness of solutions for semilinear parabolic equations with the Cauchy data in critical Sobolev spaces of negative indices. Our results contain the complex (derivative) Ginzburg–Landau equation and the Cahn–Hilliard equation as special cases.     

Koszul duality in deformation quantization and Tamarkin's approach to Kontsevich formality

Let α be a quadratic Poisson bivector on a vector space V. Then one can also consider α as a quadratic Poisson bivector on the vector space V^∗[1]. Fixed a universal deformation quantization (prediction of some complex weights to all Kontsevich graphs [12]), we have deformation quantization of the both algebras S(V^∗) and Λ(V). These are graded quadratic algebras, and therefore Koszul algebras. We prove that for some universal deformation quantization, independent on α, these two algebras are Koszul dual. We characterize some deformation quantizations for which this theorem is true in the framework of the Tamarkin's theory [19].