On a non-abelian invariant on complex surfaces of general type Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this paper,we give certain homotopy and diffeomorphism versions as a generalization to an earlier result due to W.S.Cheung,Bun Wong and Stephen S.T.Yau concerning a local rigidity problem of the tangent bundle over compact surfaces of general type.

Some lower bounds for the canonical volume of 3-folds of general type

Let V be a smooth projective 3-fold of general type. We show that the canonical volume K3 of V has a lower bound 1/10 provided q（V） 〉 0 and Ka ≥ 3 provided P4（V） ≥ 5. We also give some clean and better form for Noether inequalities for minimal 3-folds of general type.

The 4-canonical Maps of 3-folds of General Type

The aim of this paper is to show that the 4-canonical maps of 3-folds of general type is birational when pg≥7 under some given conditions.

A characterization of Burniat surfaces with K^2 = 4 and of non nodal type

Let S be a minimal surface of general type with pg(S) = 0 and K_S^2= 4. Assume the bicanonical map ψ of S is a morphism of degree 4 such that the image of ψ is smooth. Then we prove that the surface S is a Burniat surface with K^2= 4 and of non nodal type.

Oscillatory Behavior of Solutions of General Third Order Mixed Neutral Difference Equations

The objective of this paper is to study the oscillatory and asymptotic properties of the general mixed type third order neutral difference equation of the form △（aαn△2（xn＋ bnxn-τ1＋ cnxn＋τ2）） ＋ qnx^a n＋1-σ＋ pnx^βn＋1＋=σ2 0,where {an}, {bn}, {cn}, {qn} and {pn} are positive real sequences, both α and β are ratios of odd positive integers, τ1, τ2, σ1 and σ2 are positive integers. We establish some sufficient conditions which ensure all solutions are either oscillatory or converge to zero.

A FURTHER GENERALIZATION OF THE MODIFIED MBIUS INVERSION FORMULA

Ⅰ. INTRODUCTIONThe Mbius inversion formula in number theory has been modified recently by Chen to account for the continuous variables such

Some birationality criteria on 3-folds with p_g>1 Dedicated to the memory of Professor XIAO Gang

We give some birationality criteria for m(m = 4,5,6,7) on general type 3-folds with pg 2 by means of an intensive classification. In particular,we show that 7 is not birational if and only if pg(X) = 2and X admits a genus 2 curve family of canonical degree23. When the canonical volume is large,we also characterize the birationality of 4,5 and 6.

On the canonical map of surfaces with q≥6

We carry out an analysis of the canonical system of a minimal complex surface S of general type with irregularity q > 0.Using this analysis,we are able to sharpen in the case q > 0 the well-known Castelnuovo inequality KS2≥3pg(S) + q(S)-7.Then we turn to the study of surfaces with pg=2q-3 and no fibration onto a curve of genus > 1.We prove that for q≥6 the canonical map is birational.Combining this result with the analysis of the canonical system,we also prove the inequality:KS2≥7χ(S) + 2.This improves an earlier result of Mendes Lopes and Pardini (2010).