We study Dirichlet Laplacian in a screw-shaped region, i.e. a straight twisted tube of a non-circular cross section. It is shown that a local perturbation which consists of slowing down the twisting in the mean gives rise to a non-empty discrete spectrum

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We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such equations arise from the theory of semiconductors and are composed of two continuity equations coupled with a Poisson equation. In the case that the continuity equations are non degenerate, we prove the convergence of the scheme and then the existence of solutions to the problem. The key point of the proof relies on the construction of an approximate gradient of the electric potential which allows us to deal with coupled terms in the continuity equations. Finally, a numerical example is given to show the efficiency of the scheme.

A counterpart of the modular double for quantum superalgebra Uq(osp(1\2)) is constructed by means of supersymmetric quantum mechanics. We also construct the R-matrix operator acting in the corresponding representations, which is expressed via quantum dilogarithm.

This is a continuation of our first paper in [WY16]. There are two purposes of this paper: One is to give a proof of the main result in [WY16] without going through the argument depending on numerical effectiveness. The other one is to provide a proof of our conjecture, mentioned in [TY], where the assumption of negative holomorphic sectional curvature is dropped to quasi-negative. We should note that a solution to our conjecture is also provided by Diverio-Trapani [DT]. Both proofs depend on our argument in [WY16]. But our argument here makes use of the argument given by the second author and Cheng in [CY75].