Let X be a compact toric Kaehler manifold with $-K_X$ nef. Let $L\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on X. Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of X as virtual counts of holomorphic discs with Lagrangian boundary condition L. We prove a formula which equates such open GW invariants with closed GW invariants of certain X-bundles over $\mathbb{P}^1$ used to construct the Seidel representations for $X$. We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields a formula for the disc potential of X, an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya-Oh-Ohta-Ono.

We study a robust and efficient eigensolver for computing the positive dense spectrum of the two-dimensional transmission eigenvalue problem (TEP) which is derived from the Maxwell’s equation with complex media in pseudo-chiral model and the transverse magnetic mode. The discretized governing equations by the N ́ed ́elec edge element result in a large-scale quadratic eigenvalue problem (QEP). We estimate half of the positive eigenvalues of the QEP are on some interval which forms a dense spectrum of the QEP. The quadratic Jacobi-Davidson method with a so-called non-equivalence deflation technique is proposed to compute the dense spectrum of the QEP. Intensive numerical experiments show that our proposed method makes the convergence efficiently and robustly even it needs to compute more than 5000 desired eigenpairs. Numerical results also illustrate that the computed eigenvalue curves can be approximated by the non- linear functions which can be applied to estimate the density of the eigenvalues for the TEP.

In this paper, we present an efficient \GQR\ algorithm for solving the linear response eigenvalue problem $\scrH\bx ={\lambda}\bx $, where $\scrH$ is $\bPi^-$-symmetric with respect to $\Gamma_0 = \diag(I_n,-I_n)$. Based on newly introduced $\Gamma$-orthogonal transformations, the \GQR\ algorithm preserves the $\bPi^-$-symmetric structure of $\scrH$ throughout the whole process, which guarantees the computed eigenvalues to appear pairwise $(\lambda,-\lambda)$ as they should. With the help of a newly established implicit $\Gamma$-orthogonality theorem, we incorporate the implicit multi-shift technique to accelerate the convergence of the \GQR\ algorithm. Numerical experiments are given to show the effectiveness of the algorithm.

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We consider localized deformation for initial data sets of the Einstein field equations with the dominant energy condition. Deformation results with the weak inequality need to be handled delicately. We introduce a modified constraint operator to absorb the first order change of the metric in the dominant energy condition. By establishing the local surjectivity theorem, we can promote the dominant energy condition to the strict inequality by compactly supported variations and obtain new gluing results with the dominant energy condition. The proof of local surjectivity is a modification of the earlier work for the usual constraint map by the first named author and R. Schoen and by P. Chru\'sciel and E. Delay, with some refined analysis.

We propose an inverse iterative method for computing the Perron pair of an irreducible nonnegative third order tensor. The method involves the selection of a parameter $\theta_k$ in the $k$th iteration. For every positive starting vector, the method converges quadratically and is positivity preserving in the sense that the vectors approximating the Perron vector are strictly positive in each iteration. It is also shown that $\theta_k=1$ near convergence. The computational work for each iteration of the proposed method is less than four times (three times if the tensor is symmetric in modes two and three, and twice if we also take the parameter to be $1$ directly) that for each iteration of the Ng--Qi--Zhou algorithm, which is linearly convergent for essentially positive tensors.

We construct SYZ mirrors of the local Calabi--Yau manifolds of type $\widetilde{A}$ by developing an equivariant SYZ theory for the toric Calabi--Yau manifolds of infinite-type. The equations for the SYZ mirrors involve the Riemann theta functions and generating functions of the open Gromov--Witten invariants. We obtain explicit formulae for the generating functions which are open analogs of the Yau--Zaslow formula in dimension $2$, and show that they have nice modular properties. We also relate the SYZ mirror pairs with mirror symmetry for the abelian varieties and hypersurfaces therein.

We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results in terms of mirror symmetry and symplectic geometry. As an interesting application, we construct an explicit global deformation quantization of an affine del Pezzo surface as a noncommutative mirror to an elliptic orbifold.

This paper gives a new way of constructing Landau-Ginzburg mirrors using deformation theory of Lagrangian immersions motivated by the works of Seidel, Strominger-Yau-Zaslow and Fukaya-Oh-Ohta-Ono. Moreover we construct a canonical functor from the Fukaya category to the mirror category of matrix factorizations. This functor derives homological mirror symmetry under some explicit assumptions. As an application, the construction is applied to spheres with three orbifold points to produce their quantum-corrected mirrors and derive homological mirror symmetry. Furthermore we discover an enumerative meaning of the (inverse) mirror map for elliptic curve quotients.

Morphing is the process of changing a geometric model or an image into another. The process generally involves rigid body motions and non-rigid deformations. It is well known that there exists a unique conformal mapping from a simply connected surface into a unit disk by the Riemann mapping theorem. On the other hand, a 3D surface deformable model can be built via various approaches such as mutual parameterization from direct interpolation or surface matching using landmarks. In this paper, a numerical methods of 3D surface morphing based on deformable model and conformal mapping is demonstrated. We take the advantage of the unique representation of 3D surfaces by the mean curvatures $H$ and the conformal factors $\lambda$ associated with the Riemann mapping and build up the deformation model by consistently registering the landmarks on the conformal parametric domains. As a result, the correspondence of the $(H, \lambda)$ between two surfaces can be defined and a 3D deformation field can be reconstructed. Furthermore, by composition of the M\"obius transformation and the 3D deformation field, a smooth morphing sequence can be generated over a consistent mesh structure via the cubic spline homotopy. Several numerical experiments on the face morphing are presented to demonstrate the robustness of our approach.

We present a novel solution to automatic semantic modeling of indoor scenes from a sparse set of low-quality RGB-D images. Such data presents challenges due to noise, low resolution, occlusion and missing depth information. We exploit the knowledge in a scene database containing 100s of indoor scenes with over 10,000 manually segmented and labeled mesh models of objects. In seconds, we output a visually plausible 3D scene, adapting these models and their parts to fit the input scans. Contextual relationships learned from the database are used to constrain reconstruction, ensuring semantic compatibility between both object models and parts. Small objects and objects with incomplete depth information which are difficult to recover reliably are processed with a two-stage approach. Major objects are recognized first, providing a known scene structure. 2D contour-based model retrieval is then used to recover smaller objects. Evaluations using our own data and two public datasets show that our approach can model typical real-world indoor scenes efficiently and robustly.

3D scene modeling has long been a fundamental problem in computer graphics and computer vision. With the popularity of consumer-level RGB-D cameras, there is a growing interest in digitizing real-world indoor 3D scenes. However, modeling indoor 3D scenes remains a challenging problem because of the complex structure of interior objects and poor quality of RGB-D data acquired by consumer-level sensors. Various methods have been proposed to tackle these challenges. In this survey, we provide an overview of recent advances in indoor scene modeling techniques, as well as public datasets and code libraries which can facilitate experiments and evaluation.

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