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OALib Journal期刊

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Deformation of K?hler manifolds
Junyan Cao
Mathematics , 2012,
Abstract: It has been shown by Claire Voisin in 2003 that one cannot always deform a compact K\"ahler manifold into a projective algebraic manifold, thereby answering negatively a question raised by Kodaira. In this article, we prove that under an additional semipositivity or seminegativity condition on the canonical bundle, the answer becomes positive, namely such a compact K\"ahler manifold can be approximated by deformations of projective manifolds.
Numerical dimension and a Kawamata-Viehweg-Nadel type vanishing theorem on compact K?hler manifolds
Junyan Cao
Mathematics , 2012, DOI: 10.1112/S0010437X14007398
Abstract: Let $X$ be a compact K\"ahler manifold and let $(L, \varphi)$ be a pseudo-effective line bundle on $X$. We first define a notion of numerical dimension of pseudo-effective line bundles with singular metrics, and then discuss the properties of this type numerical dimension. We finally prove a very general Kawamata-Viehweg-Nadel type vanishing theorem on an arbitrary compact K\"ahler manifold.
Ohsawa-Takegoshi extension theorem for compact K?hler manifolds and applications
Junyan Cao
Mathematics , 2014,
Abstract: Our main goal in this article is to prove a new extension theorem for sections of the canonical bundle of a weakly pseudoconvex K\"ahler manifold with values in a line bundle endowed with a possibly singular metric. We also give some applications of our result.
Kodaira dimension of algebraic fiber spaces over surfaces
Junyan Cao
Mathematics , 2015,
Abstract: In this short note we prove the Iitaka C_nm conjecture for algebraic fiber spaces over surfaces
Hole probabilities of SU(m+1) Gaussian random polynomials
Junyan Zhu
Mathematics , 2014,
Abstract: In this paper, we study hole probabilities $P_{0,m}(r,N)$ of SU(m+1) Gaussian random polynomials of degree $N$ over a polydisc $(D(0,r))^m$. When $r\geq1$, we find asymptotic formulas and decay rate of $\log{P_{0,m}(r,N)}$. In dimension one, we also consider hole probabilities over some general open sets and compute asymptotic formulas for the generalized hole probabilities $P_{k,1}(r,N)$ over a disc $D(0,r)$.
Survey on fusible numbers
Junyan Xu
Mathematics , 2012,
Abstract: We point out that the recursive formula that appears in Erickson's presentation "Fusible Numbers" is incorrect, and pose an alternate conjecture about the structure of fusible numbers. Although we are unable to solve the conjecture, we succeed in establishing some basic properties of fusible numbers. We suggest some possible approaches to the conjecture, and list further problems in the final chapter.
A remark on compact K?hler manifolds with nef anticanonical bundles and its applications
Junyan Cao
Mathematics , 2013,
Abstract: Let $(X, \omega_X)$ be a compact K\"ahler manifold such that the anticanonical bundle $-K_X$ is nef. We prove that the slopes of the Harder-Narasimhan filtration of the tangent bundle with respect to a polarization of the form $\omega_X^{n-1}$ are semi-positive. As an application, we give a characterization of rationally connected compact K\"ahler manifolds with nef anticanonical bundles. As another application, we give a simple proof of the surjectivity of the Albanese map.
The nilpotent variety and invariant polynomial functions in the Hamiltonian algebra
Junyan Wei
Mathematics , 2014,
Abstract: Premet has conjectured that the nilpotent variety of any finite-dimensional restricted Lie algebra is an irreducible variety. In this paper, we prove this conjecture in the case of Hamiltonian Lie algebra. and show that its nilpotent variety is normal and a complete intersection. In addition, we generalize the Chevalley Restriction theorem to Hamiltonian Lie algebra. Accordingly, we give the generators of the invariant polynomial ring.
Additivity of the approximation functional of currents induced by Bergman kernels
Junyan Cao
Mathematics , 2014,
Abstract: In this note, we give a positive answer to a question raised by Jean-Pierre Demailly in 2013, and show the additivity of the approximation functional of closed positive $(1,1)$-currents induced by Bergman kernels.
Natural Language Processing for Foreign Languages Learning as Computer-based Learning Tools
Ying Zhang,Junyan Liu
Modern Applied Science , 2009, DOI: 10.5539/mas.v3n1p125
Abstract: This paper outlines a framework to use computer and natural language techniques for various levels of learners to learn foreign languages in Computer-based Learning environment. We propose some ideas for using the computer as a practical tool for learning foreign language where the most of courseware is generated automatically. We then describe how to build Computer Based Learning tools, discuss its effectiveness, and conclude with some possibilities using on-line resources.
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