We present a simple and direct method for solving the Beltrami equation ∂ f /∂ z = μ ∂ f /∂z for a quasiconformal self-mapping of planar disk without approximating singular integrals or infinite series. Given a triangulation of the disk, it is a simple matter to discretize the Beltrami equation as a sistem of linear equations. However, natural attempts to require the boundary to be a circle (so that the disk is mapped to a disk) normally produce nonlinear conditions which would require iterative methods for the solution. We show how to eliminate all nonlinear conditions, and prove that the Least-Squares solution to the linear system is a good approximation for the solution of the Beltrami equation in the sense that as the triangular mesh is refined appropriately, the discrete solution approaches the true μ-conformal mapping.
An effective algorithm is presented for solving the Beltrami equation ∂f /∂z = µ ∂f /∂z in a planar disk. The disk is triangulated in a simple way and f is approximated by piecewise linear mappings; the images of the vertices of the triangles are defined by an overdetermined system of linear equations. (Certain apparently nonlinear conditions on the boundary are eliminated by means of a symmetry construction.) The linear system is sparse and its solution is obtained by standard least-squares, so the algorithm involves no evaluation of singular integrals nor any iterative procedure for obtaining a single approximation of f . Numerical examples are provided, including a deformation in a Teichmüller space of a Fuchsian group.
Herein, we present a novel topic variation detection method that combines a topic extraction method and a change-point detection method. It extracts topics from time-series text data as the feature of each time and detects change points from the changing patterns of the extracted topics. We applied this method to analyze the valuable, albeit underutilized, text dataset containing the Japanese Prime Minister’s (PM’s) detailed daily activities for over 32 years. The proposed method and data provide novel insights into the empirical analyses of political business cycles, which is a classical issue in economics and political science. For instance, as our approach enables us to directly observe and analyze the PM’s actions, it can overcome the empirical challenges encountered by previous research owing to the unobservability of the PM’s behavior. Our empirical observations are primarily consistent with recent theoretical developments regarding this topic. Despite limitations, by employing a completely novel method and dataset, our approach enhances our understanding and provides new insights into this classic issue.
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