We use circle packing techniques to construct approximate solutions to the generalized Beltrami equations with simply and multiply connected regions in the plane. We show convergence of the approximate solutions. This gives a constructive proof for the existence of quasiconformal mappings with a given pair of complex dilations.

We study the solvability of the Riemann-Hilbert problem for a generalized Cauchy-Riemann system with several singularities and reveal several new phenomenon. For the number of continuous solutions we shall show that it depends not only on the index but also on the location and type of the singularities; moreover, it does not depend continuously on the coefficients of the equation.

This paper addresses the small sample size problem in linear discriminant analysis, which occurs in face recognition applications. Belhumeur, Hespanha and Kriegman (IEEE Trans. on PAMI, Vol. 19 (7) (1997), 711-720.) proposed the FisherFace method. We find out that the FisherFace method might fail since after the PCA transform the corresponding within class covariance matrix can still be singular, this phenomenon is verified with the Yale face database. Hence we propose to use an inverse Fisher criteria. Experiment results suggest that this new approach works well.

For a one-dimensional nonlinear pseudoparabolic equation we consider its initial boundary value problem on an interval. The boundary condition on the left endpoint is of Dirichlet type, but the condition on the right endpoint is replaced by a nonlocal one. Because it is given by an integral, the function involved could exhibit singularities, which distinguishes this nonlocal condition from its Dirichlet counterpart. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in a weighted Sobolev space.

Two boundary value problems are investigated for an over-determined elliptic system with several complex variables in polydisc. Necessary and sufficient conditions for the existence of finitely many linearly independent solutions and finitely many solvability conditions are derived. Moreover, the boundary value problem for any number of complex variables is treated in a unified way and the essential dierence between the case of one complex variable and that of several complex variables is revealed.