Communications in Contemporary Mathematics

In this paper, all irreducible weight modules with finite dimensional weight spaces over the twisted Heisenberg–Virasoro algebra are determined. There are two different classes of them. One class is formed by simple modules of intermediate series, whose nonzero weight spaces are all 1-dimensional; the other class consists of the irreducible highest weight modules and lowest weight modules.

The structure and embedding of subgroups permuting with the system normalizers of a finite soluble group are studied in the paper. It is also proved that the class of all finite soluble groups in which every subnormal subgroup permutes with the Sylow subgroups is properly contained in the class of all soluble groups whose subnormal subgroups permute with the system normalizers while this latter is properly contained in the class of all supersoluble groups.

We show that on any closed contact manifold of dimension greater than 1 a contact structure with vanishing contact homology can be constructed. The basic idea for the construction comes from Giroux. We use a special open book decomposition for spheres. The page is the cotangent bundle of a sphere and the monodromy is given by a left-handed Dehn twist. In the resulting contact manifold, we exhibit a closed Reeb orbit that bounds a single finite energy plane like in the computation for the overtwisted case. As a result, the unit element of the contact homology algebra is exact and so the contact homology vanishes. This result can be extended to other contact manifolds by using connected sums. The latter is related to the plumbing or 2-Murasugi sum of contact open books. We shall give a possible description of this construction and some conjectures about the plumbing operation.

In this paper, we study the extension and restriction theorems of the anisotropic Besov spaces in a Lipschitz hyper-surface in space-time domain. We hope that such theorems will be useful in solving a parabolic type equations in a time-varying domain.

In this paper, we establish the Picone inequalities for p-sub-Laplacian on the Heisenberg group. Then, we prove Hardy inequalities with weight functions involving partial variables and a comparison principle. As an immediate consequence, we obtain the uniqueness for p-sub-Laplace equation involving singular indefinite weights. Also we prove the existence of positive solutions.

In this paper, we introduce the notion of weakly symmetric Finsler spaces and study some geometrical properties of such spaces. In particular, we prove that each maximal geodesic in a weakly symmetric Finsler space is the orbit of a one-parameter subgroup of the full isometric group. This implies that each weakly symmetric Finsler space has vanishing S-curvature. As an application of these results, we prove that there exist reversible non-Berwaldian Finsler metrics on the 3-dimensional sphere with vanishing S-curvature. This solves an open problem raised by Z. Shen.

We study the decay of Fourier modes of solutions to the two-dimensional Navier–Stokes System on a flat cylinder and the unit square with slip boundary conditions. Under some suitable assumptions on the initial velocity, we obtain quantitative decay estimates of the Fourier modes.