The Weinstein Conjecture for Hamiltonian Fibrations

In this note we extend to non trivial Hamiltonian fibrations over symplectically uniruled manifolds a result of Lu's, \cite{Lu}, stating that any trivial symplectic product of two closed symplectic manifolds with one of them being symplectically uniruled verifies the Weinstein Conjecture for closed separating hypersurfaces of contact type, under certain technical conditions. The proof is based on the product formula for Gromov-Witten invariants ($GW$-invariant) of Hamiltonian fibrations derived in \cite{H}.Comment: 15 page

Indecomposable representations and oscillator realizations of the exceptional Lie algebra G_2

In this paper various representations of the exceptional Lie algebra G_2 are investigated in a purely algebraic manner, and multi-boson/multi-fermion realizations are obtained. Matrix elements of the master representation, which is defined on the space of the universal enveloping algebra of G_2, are explicitly determined. From this master representation, different indecomposable representations defined on invariant subspaces or quotient spaces with respect to these invariant subspaces are discussed. Especially, the elementary representations of G_2 are investigated in detail, and the corresponding six-boson realization is given. After obtaining explicit forms of all twelve extremal vectors of the elementary representation with the highest weight {\Lambda}, all representations with their respective highest weights related to {\Lambda} are systematically discussed. For one of these representations the corresponding five-boson realization is constructed. Moreover, a new three-fermion realization from the fundamental representation (0,1) of G_2 is constructed also.Comment: 29 pages, 4 figure

Language Teaching from the View of Formative Assessment

As a two-way activity, formative assessment collects feedbacks from students to help teachers better the teaching methods. At the same time, it makes students have a clear understanding about their performance on the in-class and after-class participation. Traditional ways of evaluation prevent students from active and creative learning. Even though there are so many problems existing, formative assessment makes remarkable contributions to collect teaching feedbacks, increase teaching efficiency and improve learners’ learning performance. For college English teaching, the task is to break the traditional test grading assessment pattern, and set up a completely new system to give effective evaluation on both the learning process and result