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In this paper, we will derive some twist criteria for the periodic solution of a periodic scalar Newtonian equation using the third order approximation. As an application to the forced pendulum x ̈ + ω^2 sin x = p(t), we will find an explicit bound P (ω) for the L1 norm, ∥p∥1, of the periodicforcingp(t)usingthefrequencyωasaparametersuchthattheleastamplitudeperiodic solution of the forced pendulum is of twist type when ∥p∥1 < P (ω). The bound P (ω) has the order of O(ω1/2) when ω is bounded away from resonance of orders ≤ 4 and ω → +∞.

In virtual reality/augmented reality (VR/AR) applications, especially medical image visualization, it is highly desirable to magnify region of interests. In this work, we propose a novel method for virtual magnifying glass for visualizing volumetric data based on Optimal Mass Transportation. An optimal mass transportation map deforms a volume to itself, transforms the source measure (volumetric element) to the target measure with the minimal transportation cost. Solving the optimal mass transportation problem is equivalent to a convex optimization, and can be converted to computing power Voronoi diagrams in classical computational geometry. The proposed method allows the user to accurately control the target measure, and select multiple regions of interests with irregular shapes. We demonstrate the effectiveness and efficiency of our method with several volume data sets from medical applications.

We introduce a semi-algebraic structure on the set of all isotopy classes of non-separating simple closed curves in any compact oriented surface and show that the structure is finitely generated. As a consequence, we produce a natural finite dimensional linear representation of the mapping class group of the surface. Applications to the Teichmller space, Thurston's measured lamination space, the harmonic Beltrami differentials, and the first cohomology groups of the surface are discussed.