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The universal Teichmu¨ller space is an infinitely dimensional generalization of the classical Teichmu¨ller space of Riemann surfaces. It carries a natural Hilbert structure, on which one can define a natural Riemannian metric, the Weil-Petersson metric. In this paper we investigate the Weil-Petersson Riemannian curvature operator ˜ Q of the universal Teichmu¨ller space with the Hilbert structure, and prove the following: (i) ˜ Q is non-positive definite. (ii) ˜ Q is a bounded operator. (iii) ˜ Q is not compact; the set of the spectra of ˜ Q is not discrete. As an application, we show that neither the Quaternionic hyperbolic space nor the Cayley plane can be totally geodesically immersed in the universal Teichmu¨ller space endowed with the Weil-Petersson metric.

Sure screening technique has been considered as a powerful tool to handle the ultrahigh dimensional variable selection problems, where the dimensionality p and the sample size n can satisfy the NP dimensionality logp =O(na) for some a>0[J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 (2008) 849–911]. The current paper aims to simultaneously tackle the “universality” and “effectiveness” of sure screening procedures. For the “universality,” we develop a general and unified framework for nonparametric screening methods from a loss function perspective. Consider a loss function to measure the divergence of the response variable and the underlying nonparametric function of covariates. We newly propose a class of loss functions called conditional strictly convex loss, which contains, but is not limited to, negative log likelihood loss from one-parameter exponential families, exponential loss for binary classification and quantile regression loss. The sure screening property and model selection size control will be established within this class of loss functions. For the “effectiveness,” we focus on a goodness-of-fit nonparametric screening (Goffins) method under conditional strictly convex loss. Interestingly, we can achieve a better convergence probability of containing the true model compared with related literature. The superior performance of our proposed method has been further demonstrated by extensive simulation studies and some real scientific data example.

We show that complex hypercontractivity gives better constants than real hypercontractivity in comparison inequalities for (low) moments of Rademacher chaoses (homogeneous polynomials on the discrete cube).

In this paper we establish the local-in-time existence and uniqueness of strong solutions to the free boundary problem of the full compressible Navier-Stokes equations in three-dimensional space. The vanishing density and temperature condition is imposed on the free boundary, which captures the motions of the non-isentropic viscous gas surrounded by vacuum with bounded entropy. We also assume some proper decay rates of the density towards the boundary and singularities of derivatives of the temperature across the boundary on the initial data, which coincides with the physical vacuum condition for the isentropic flows. This extends the previous result of Liu [arXiv: 1612.07936] by removing the spherically symmetric assumption and considering more general initial density and temperature profiles.