A stability condition is provided for a class of vorticity boundary formulas used with the second order finite-difference numerical scheme for the vorticity-stream function formulation of the unsteady incompressible Navier-Stokes equations. These local vorticity boundary formulas are derived using the no-slip boundary condition for the velocity. A new form of these long-stencil formulas is needed to classify the stability property, in which local terms are controlled by global quantities via discrete elliptic regularity for the stream functions.

Analogous of Bessel and Flett potentials are defined and studied for the Dunkl transform associated with a family of weighted functions that are invariant under a reflection group. We show that the Dunkl-Bessel potentials, of positive order, can be represented by an integral involving the k-heat transform and we give some applications of this result. Also, we obtain an explicit inversion formula for the Dunkl-Flett potentials, which are interpreted as negative fractional powers of a certain operator expressed in terms of the Dunkl-Laplacian.

This article studies the structure and its evolution of incompressible flows with the anti-symmetry using a combination of rigorous analysis and numerical simulations, with an application to an example of oceanic flow. In particular, necessary and sufficient conditions for 2D divergence-free vector fields with anti-symmetry are obtained, and a detailed numerical simulation for a simplified model of Marsigli oceanic flow is provided to explore and verify the structure and its transitions of the flow. It is expected that the study will lead to useful insights to the understanding of the flow dynamics from both the mathematical and physical points of view.

Variational method is a useful mathematical tool in various areas of research, espe- cially in image processing. Recently, solving image processing problems on general manifolds using variational techniques has become an important research topic. In this paper, we solve several image processing problems on general Riemann surfaces using variational models defined on surfaces. We propose an explicit method to solve variational problems on general Riemann surfaces, using the conformal parameterization and covariant derivatives defined on the surface. To simplify the com- putation, the surface is firstly mapped conformally to the two dimensional rectangular domains, by computing the holomorphic 1-form on the surface. It is well known that the Jacobian of a conformal map is simply the scalar multiplication of the conformal factor. Therefore with the conformal para- meterization, the covariant derivatives on the parameter domain are similar to the usual Euclidean differential operators, except for the scalar multiplication. As a result, any variational problem on the surface can be formulated to a 2D problem with a simple formula and efficiently solved by well developed numerical scheme on the 2D domain. With the proposed method, we solve various image processing problems on surfaces using different variational models, which include image segmen- tation, surface denoising, surface inpainting, texture extraction and automatic landmark tracking. Experimental results show that our method can effectively solve variational problems and tackle image processing tasks on general Riemann surfaces.

All the surfaces in real life are Riemann surfaces, therefore with conformal structures. Two surfaces share the same conformal structure, if there exists a conformal (angle-preserving) mapping between them. Conformal modules are the complete invariants of conformal structures, which can be treated as shape descriptors for shape analysis applications.

This work focuses on the computational methods of conformal modules for genus zero surfaces with boundaries, including topological quadrilaterals, annuli, multiply connected annuli. The algo- rithms are based on both holomorphic 1-forms and discrete curvature flows, which are rigorous and practical. The conformal module shape descriptors are applied for shape classification and compari- son. Experiments on surfaces acquired from real world demonstrate the efficiency and efficacy of the conformal module method.