In this short note, we give a proof of our partial C ⁰-estimate for Kähler–Einstein metrics. Our proof uses a compactness theorem of Cheeger–Colding–Tian and L ²-estimate for $\bar{\partial}$-operator.

We propose the single-index hazards model for censored survival data. As an extension of the Cox model and many transformation models, this model allows nonparametric modeling of covariate effects in a parsimonious way via a single index. In addition, the relative importance of covariates can be assessed via this model. We consider two commonly used profile likelihood methods for parameter estimation: the local profile likelihood method and the stratified profile likelihood method. It is shown that both methods may give consistent estimators under certain restrictive conditions, but in general they can yield biased estimation. Simulation studies are also conducted to demonstrate these bias phenomena. The existence and nature of the failures of these two commonly used approaches is somewhat surprising.

In this paper, we obtain the Hölder continuity of the solutions of SPDEs with reflection, which have singular drifts (random measures).

By the standard theory, the stable Q _k+1,k−Q _k,k+1/$Q_{k}^{dc'}$ divergence-free element converges with the optimal order of approximation for the Stokes equations, but only order k for the velocity in H ¹-norm and the pressure in L ²-norm. This is due to one polynomial degree less in y direction for the first component of velocity, which is a Q _k+1,k polynomial of x and y. In this manuscript, we will show by supercloseness of the divergence free element that the order of convergence is truly k+1, for both velocity and pressure. For special solutions (if the interpolation is also divergence-free), a two-order supercloseness is shown to exist. Numerical tests are provided confirming the accuracy of the theory.

In this article, we consider the problem of lifting the GW theory of a symplectic divisor to that of the ambient manifold in the context of symplectic birational geometry. In particular, we generalize Maulik–Pandharipande’s relative/absolute correspondence to relative-divisor/absolute correspondence. Then, we use it to lift a minimal uniruled invariant of a divisor to that of the ambient manifold.

We undertake a local analysis of combinatorial independence as it connects to topological entropy within the framework of actions of sofic groups.