A priori estimate for a family of semi-linear elliptic equations with critical nonlinearity

We consider positive solutions of $\Delta u-\mu u+Ku^{\frac{n+2}{n-2}}=0$ on $B_1$ ($n\ge 5$) where $\mu$ and $K>0$ are smooth functions on $B_1$. If $K$ is very sub-harmonic at each critical point of $K$ in $B_{2/3}$ and the maximum of $u$ in $\bar B_{1/3}$ is comparable to its maximum over $\bar B_1$, then all positive solutions are uniformly bounded on $\bar B_{1/3}$. As an application, a priori estimate for solutions of equations defined on $\mathbb S^n$ is derived.Comment: 26 page

The cohomological support locus of pluricaonical sheaves and the Iitaka fibration

Let $alb_X: X \rightarrow A$ be the Albanese map of a smooth projective variety and $f: X \rightarrow Y$ the fibration from the Stein factorization of $alb_X$. For a positive integer $m$, if $f$ and $m$ satisfy the assumptions AS(1,2), then the translates through the origin of all components of cohomological locus $V^0(\omega_X^m, alb_X)$ generates $I^*Pic^0(S)$ where $I: X \rightarrow S$ denotes the Iitaka fibration. This result applies to studying pluricanonical maps. We also considered the problem about whether a fibration is isotrivial and isogenous to a product.Comment: 16 pages. Welcome comment

Applying knowledge management in education : teaching database normalization : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Information Science at Massey University

In tertiary education, Information Science has been attracting more attention in both teaching and learning. However, along the course on the database design theory, learners always find it hard to grasp the knowledge on database normalisation and hard to apply different levels of the normal forms while designing a database. This results poor database construction and difficulties in database maintenance. In regard to this teaching and learning dilemma, academic teaching staff should, on the one hand, pay more attention to organising different teaching resources on database normalisation concepts and making the best use of the existing and newly developed resources so as to make the teaching environment more adaptive and more sharable. and on the other hand, apply different teaching methods to different students according to their knowledge levels by understanding the nature of each learner's behaviour, interests and preferences concerning the existing learning resources. However, at present there is no effective Information Technology tool to use in considering the dynamic nature of knowledge discovery, creation, transfer utilisation and reuse in this area. This provides an opportunity to examine the potentiality of applying knowledge management in education with the focus on teaching database normalisation, in terms of knowledge discovering, sharing, utilisation and reuse. This thesis contains a review of knowledge management and web mining technologies in the education environment, presents a dynamic knowledge management framework for better utilising teaching resource in the area of database normalisation and diagnoses the students' learning patterns and behaviours to assist effective teaching and learning. It is argued that knowledge management-supported education can work as a value-added process which supports the different needs of teachers and learners