Herein, we use hybrid resampling to address (a) the long-standing problem of inference on change times and changed parameters in change-point ARX-GARCH models, and (b) the challenging problem of valid confidence intervals, after variable selection under sparsity assumptions, for the parameters in linear regression models with high-dimensional stochastic regressors and asymptotically stationary noise. For the latter problem, we introduce consistent estimators of the selected parameters and a resampling approach to overcome the inherent difficulties of post-selection confidence intervals. For the former problem, we use a sequential Monte Carlo for the latent states (representing the change times and changed parameters) of a hidden Markov model. Asymptotic efficiency theory and simulation and empirical studies demonstrate the advantages of the proposed methods.

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Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact K\"ahler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called {\em real bisectional curvature} for Hermitian manifolds. When the metric is K\"ahler, this is just the holomorphic sectional curvature $H$, and when the metric is non-K\"ahler, it is slightly stronger than $H$. We classify compact Hermitian manifolds with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case. The underlying reason for the extension is that the Schwarz lemma of Wu-Yau works the same when the target metric is only Hermitian but has nonpositive real bisectional curvature.

This paper presents a method of obtaining geometric registrations between high-genus (g≥1) surfaces. Surface registration between simple surfaces, such as simply connected open surfaces, has been well studied. However, very few works have been carried out for the registration of high-genus surfaces. The high-genus topology of the surface poses a great challenge for surface registration. A possible approach is to partition surfaces into simply connected patches and registration can be done in a patch-by-patch manner. Consistent cuts are required, which are usually difficult to obtain and prone to error. In this work, we propose an effective way to obtain geometric registration between high-genus surfaces without introducing consistent cuts. The key idea is to conformally parameterize the surface into its universal covering space, which is either the Euclidean plane or the hyperbolic disk embedded in R2. Registration can then be done on the universal covering space by iteratively minimizing a shape mismatching energy measuring the geometric dissimilarity between the two surfaces. The Beltrami coefficient of the mapping is considered and adjusted in order to control the bijectivity of the mappings in each iteration. Our proposed algorithm effectively computes a smooth registration between high-genus surfaces that matches geometric information as much as possible. The algorithm can also be applied to find a smooth registration minimizing any general energy functionals. Numerical experiments on high-genus surface data show that our proposed method is effective for registering high-genus surfaces with geometric matching. We also applied the method to register anatomical structures for medical imaging, which demonstrates the usefulness of the proposed algorithm.

By using certain idea developed in minimal submanifold theory we study rigidity problem for self-shrinkers in the present paper. We prove rigidity results for the squared norm of the second fundamental form of self-shrinkers, either under point-wise conditions or under integral conditions.