According to the Ringel-Green theorem, the generic composition algebra of the Hall algebra provides a realization of the positive part of the quantum group. Furthermore, its Drinfeld double can be identified with the whole quantum group, in which the BGP-reflection functors coincide with Lusztig’s symmetries. It is first asserted that the elements corresponding to exceptional modules lie in the integral generic composition algebra, hence in the integral form of the quantum group. Then it is proved that these elements lie in the crystal basis up to a sign. Eventually, it is shown that the sign can be removed by the geometric method. The results hold for any type of Cartan datum.

Property A and uniform embeddability are notions of metric geometry which imply the coarse Baum-Connes conjecture and the Novikov conjecture. In this paper, the authors prove the permanence properties of property A and uniform embeddability of metric spaces under large scale decompositions of finite depth.

The authors prove the global exact boundary controllability for the cubic semi-linear wave equation in three space dimensions, subject to Dirichlet, Neumann, or any other kind of boundary controls which result in the well-posedness of the corresponding initial-boundary value problem. The exponential decay of energy is first established for the cubic semi-linear wave equation with some boundary condition by the multiplier method, which reduces the global exact boundary controllability problem to a local one. The proof is carried out in line with [2, 15]. Then a constructive method that has been developed in [13] is used to study the local problem. Especially when the region is star-complemented, it is obtained that the control function only need to be applied on a relatively open subset of the boundary. For the cubic Klein-Gordon equation, similar results of the global exact boundary controllability are proved by such an idea.

Decay of the energy for the Cauchy problem of the wave equation of variable coefficients with a dissipation is considered. It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coefficients. In particular, some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given.

For a Riemann surface X of conformally finite type (g, n), let d _T, d _L and $d_{P_i } $ (i = 1, 2) be the Teichmüller metric, the length spectrum metric and Thurston’s pseudometrics on the Teichmüller space T(X), respectively. The authors get a description of the Teichmüller distance in terms of the Jenkins-Strebel differential lengths of simple closed curves. Using this result, by relatively short arguments, some comparisons between d _T and d _L, $d_{P_i } $ (i = 1, 2) on T _ɛ(X) and T(X) are obtained, respectively. These comparisons improve a corresponding result of Li a little. As applications, the authors first get an alternative proof of the topological equivalence of d _T to any one of d _L, $d_{P_1 } $ and $d_{P_2 } $ on T(X). Second, a new proof of the completeness of the length spectrum metric from the viewpoint of Finsler geometry is given. Third, a simple proof of the following result of Liu-Papadopoulos is given: a sequence goes to infinity in T(X) with respect to d _T if and only if it goes to infinity with respect to d _L (as well as $d_{P_i } $ (i = 1, 2)).

Let $\mathbb{D}_n $ be the generalized unit disk of degree n. In this paper, Riemannian metrics on the Siegel-Jacobi disk $\mathbb{D}_n $ × ℂ^(m,n) which are invariant under the natural action of the Jacobi group are found explicitly and the Laplacians of these invariant metrics are computed explicitly. These are expressed in terms of the trace form.

The authors are concerned with a class of one-dimensional stochastic Anderson models with double-parameter fractional noises, whose differential operators are fractional. A unique solution for the model in some appropriate Hilbert space is constructed. Moreover, the Lyapunov exponent of the solution is estimated, and its Hölder continuity is studied. On the other hand, the absolute continuity of the solution is also discussed.

The authors define the Gauss map of surfaces in the three-dimensional Heisenberg group and give a representation formula for surfaces of prescribed mean curvature. Furthermore, a second order partial differential equation for the Gauss map is obtained, and it is shown that this equation is the complete integrability condition of the representation.

The concept of Koszul differential graded (DG for short) algebra is introduced in [8]. Let A be a Koszul DG algebra. If the Ext-algebra of A is finite-dimensional, i.e., the trivial module _A k is a compact object in the derived category of DG A-modules, then it is shown in [8] that A has many nice properties. However, if the Ext-algebra is infinite-dimensional, little is known about A. As shown in [15] (see also Proposition 2.2), _A k is not compact if H(A) is finite-dimensional. In this paper, it is proved that the Koszul duality theorem also holds when H(A) is finite-dimensional by using Foxby duality. A DG version of the BGG correspondence is deduced from the Koszul duality theorem.