In this paper, Yau’s conjecture on harmonic functions in Riemannian manifolds is generalized to Alexandrov spaces. It is proved that the space of harmonic functions with polynomial growth of a fixed rate is finite dimensional and strong Liouville theorem holds in Alexandrov spaces with nonnegative curvature.

The author first constructs a Lie algebra $\mathfrak{L}: = \mathfrak{L}(q,w_d )$ from rank 3 quantum torus, which is isomorphic to the core of EALAs of type A _d−1 with coordinates in quantum torus C _q ^d, and then gives the necessary and sufficient conditions for the highest weight modules to be quasifinite. Finally the irreducible ℤ-graded quasifinite $\mathfrak{L}$-modules with nonzero central charges are classified.

The notion of an ideal family of weighted subspaces of a discrete metric space X with bounded geometry is introduced. It is shown that, if X has Yu’s property A, the ideal structure of the Roe algebra of X with coefficients in $\mathcal{B}$(H) is completely characterized by the ideal families of weighted subspaces of X, where $\mathcal{B}$(H) denotes the C*-algebra of bounded linear operators on a separable Hilbert space H.

The authors study the continuity of barrier function B _c(x) with respect to the parameter. A sufficient condition which makes B _c(x) be continuous with respect to c is obtained, and an example of discontinuity when the condition is not satisfied is also constructed.

The BCL system, a kind of equations governing the motion of the free surface of water waves in ℝ³, is studied. Some results on the global existence, uniqueness and regularity of solutions to such system with small initial data are obtained.

In this paper, the Kähler conditions of the Chern-Finsler connection in complex Finsler geometry are studied, and it is proved that Kähler Finsler metrics are actually strongly Kähler.

The concept of locally strong compactness on domains is generalized to general topological spaces. It is proved that for each distributive hypercontinuous lattice L, the space SpecL of nonunit prime elements endowed with the hull-kernel topology is locally strongly compact, and for each locally strongly compact space X, the complete lattice of all open sets $\mathcal{O}$(X) is distributive hypercontinuous. For the case of distributive hyperalgebraic lattices, the similar result is given. For a sober space X, it is shown that there is an order reversing isomorphism between the set of upper-open filters of the lattice $\mathcal{O}$(X) of open subsets of X and the set of strongly compact saturated subsets of X, which is analogous to the well-known Hofmann-Mislove Theorem.

The authors study a diffusive prey-predator model subject to the homogeneous Neumann boundary condition and give some qualitative descriptions of solutions to this reaction-diffusion system and its corresponding steady-state problem. The local and global stability of the positive constant steady-state are discussed, and then some results for non-existence of positive non-constant steady-states are derived.

The authors investigate the global behavior of the solutions of the difference equation $x_{n + 1} = \frac{{ax_{n - l} x_{n - k} }}{{bx_{n - p} + cx_{n - q} }}, n = 0, 1, \cdots ,$ where the initial conditions x _−r, x _−r+1, x_−r+2, …, x ₀ are arbitrary positive real numbers, r = max{l, k, p, q} is a nonnegative integer and a, b, c are positive constants. Some special cases of this equation are also studied in this paper.

The authors prove that the crossed product of an infinite dimensional simple separable unital C*-algebra with stable rank one by an action of a finite group with the tracial Rokhlin property has again stable rank one. It is also proved that the crossed product of an infinite dimensional simple separable unital C*-algebra with real rank zero by an action of a finite group with the tracial Rokhlin property has again real rank zero.