The Einstein-Kähler metric on the third Cartan-Hartogs domain

In this paper we discuss the Einstein-Kahler metric on the third Cartan-Hartogs domain Y111（n, q; K）. Firstly we get the complete Einstein Kahler metric with explicit form on Y111（n, q; K） in the case of K=q/2 ＋ 1/q-1. Secondly we obtain the holomorphic sectional curvature under this metric and get the sharp estimate for this holomorphic curvature. Finally we prove that the complete Einstein-Kahler metric is equivalent to the Bergman metric on Y111（n, q; K） in case of K=q/2＋1/q-1.     

Regularization for unconstrained vector optimization of convex functionals in Banach spaces

An operator regularization method is considered for ill-posed vector optimization of weakly lower semicontinuous essentially convex functionals on reflexive Banach spaces. The regularization parameter is chosen by a modified generalized discrepancy principle. A condition for the estimation of the convergence rate of regularized solutions is derived. This article was submitted by the author in English.     

3D anisotropy measurement methodology for surface microstructures

We propose a simple method able to quantify three‐dimensional (3D) anisotropy in topographic microstructure measurements. The anisotropy quantification of 3D surface data is based on a horizontal‐cut analysis yielding level sets. For a specific level set, we study xy anisotropy with the use of a directional chord length analysis. In the height profile direction, the z direction, anisotropy is quantified using the framework of mean Euler characteristic and more generally that of Minkowski functionals. We exemplify the method with the use of 3D heights profiles from atomic force microscopy measurements. Copyright © 2012 John Wiley & Sons, Ltd.     

Dixmier traces of operators on Banach and Hilbert spaces

We present a simple approach to the theory of traces on the Marcinkiewicz operator ideal Particular attention is paid to the Dixmier traces, which may be obtained from both dilation‐invariant functionals and shift‐invariant functionals on $\mathfrak l_\infty$. Since the concept of positivity does not make sense for traces defined on operator ideals over Banach spaces, traces are not assumed to be positive in the Hilbert space setting, as well. This standpoint raises some new problems. The present studies will be continued in 35 , 36 .     

L P boundedness of area function associated with operators on product spaces

Let L be the infinitesimal generator of an analytic semigroup on L2 (Rn) with suitable upper bounds on its heat kernels. Assume that L has a bounded holomorphic functional calculus on L2(Rn). In this paper,we define the Littlewood- Paley g function associated with L on Rn × Rn, denoted by GL(f)(x1, x2), and decomposition, we prove that ‖SL(f)‖p ≈‖GL(f)‖p ≈‖f‖p for 1 ＜ p ＜∞.     

Characterization of quadratic mappings through a functional inequality

We provide conditions under which every solution (f,?) of the functional inequality

     

Normality concerning shared values

Let be a family of meromorphic functions in a plane domain D, and a and b be finite non-zero complex values such that . If for and , then is normal. We also construct a non-normal family of meromorphic functions in the unit disk Δ={|z|<1} such that for every and in Δ, where m is a given positive integer. This answers Problem 5.1 in the works of Gu, Pang and Fang. This work was supported by National Natural Science Foundation of China (Grant Nos. 10671093, 10871094) and the Natural Science Foundation of Universities of Jiangsu Province of China (Grant No. 08KJB110001), the Qing Lan Project of Jiangsu, China and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry     

Regularization of a three-element functional equation

In this paper we study the three-element functional equation

$(V\Phi )(z) \equiv \Phi (iz) + \Phi ( - iz) + G(z)\Phi \left( {\frac{1}{z}} \right) = g(z), z \in R,$

, subject to

$R: = \{ z:\left| z \right| < 1, \left| {\arg z} \right| < \frac{\pi }{4}\} .$

We assume that the coefficients G(z) and g(z) are holomorphic in R and their boundary values G ⁺(t) and g ⁺(t) belong to H(Γ), G(t)G(t ^?1) = 1. We seek for solutions Φ(z) in the class of functions holomorphic outside of \(\bar R\) such that they vanish at infinity and their boundary values Φ^?(t) also belong to H(Γ). Using the method of equivalent regularization, we reduce the problem to the 2nd kind integral Fredholm equation.

     

A criterion for the Generalized Riemann hypothesis

In this paper, we study the automorphic L-functions attached to the classical automorphic forms on GL（2）, i.e. holomorphic cusp form. And we also give a criterion for the Generalized Riemann Hypothesis （GRH） for the above L-functions.