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We present the first computational tool to help ordinary users create transforming pop-up books. In each transforming pop-up, when the user pulls a tab, an initial flat 2D pattern, i.e. a 2D shape with a superimposed picture, such as an airplane, turns into a new 2D pattern, such as a robot, standing up from the page. Given the two 2D patterns, our approach automatically computes a 3D pop-up mechanism that transforms one pattern into the other; it also outputs a design blueprint, allowing the user to easily make the final model. We also present a theoretical analysis of basic transformation mechanisms; combining these basic mechanisms allows more flexibility of final designs. Using our approach, inexperienced users can create models in a short time; previously, even experienced artists often took weeks to manually create them. We demonstrate our method on a variety of real world examples.

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For the three-dimensional full compressible Navier–Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid, we establish the global existence and uniqueness of classical solutions with smooth initial data which are of small energy but possibly large oscillations where the initial density is allowed to vanish. Moreover, for the initial data, which may be discontinuous and contain vacuum states, we also obtain the global existence of weak solutions. These results generalize previous ones on classical and weak solutions for initial density being strictly away from a vacuum, and are the first for global classical and weak solutions which may have large oscillations and can contain vacuum states.

In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-K\"ahler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds $\S^{2n-1}\times \S^1$. We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifold such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and nonnegative, but their Riemannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian manifolds with nonnegative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature.