This work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity. We also introduce the free boundary problem for the compressible primitive equations.

In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Khler Calabi-Yau surfaces or Hopf surfaces.

Estimation of longitudinal data covariance structure poses significant challenges because the data usually are collected at irregular time points. A viable semiparametric model for covariance matrixes has been proposed that allows one to estimate the variance function nonparametrically and to estimate the correlation function parametrically by aggregating information from irregular and sparse data points within each subject. But the asymptotic properties of the quasi-maximum likelihood estimator (QMLE) of parameters in the covariance model are largely unknown. We address this problem in the context of more general models for the conditional mean function, including parametric, nonparametric, or semiparametric. We also consider the possibility of rough mean regression function and introduce the difference-based method to reduce biases in the context of varying-coefficient partially linear mean regression

<div>Without Abstract

Continuing the program of [DS] and [U1], we introduce refinements of the Donaldson-Smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components. In cases where a corresponding analogue of the Gromov-Taubes invariant is easy to define, our invariants agree with those analogues. We also prove a vanishing result for some of the invariants that count nodal curves.

All continuous GL(n) covariant valuations that map convex bodies to convex bodies are completely classified. This establishes a characterization of moment and projection operators and shows that the Holmes-Thompson area is the unique Minkowski area that is also a bivaluation.

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

We investigate, by numerical simulation, the shear layer instability associated with the outer layer of a spiral vortex formed behind an impulsively started thin ellipse. The unstable free shear layer undergoes a secondary instability. We connect this instability with the dynamics of corner vortices adjacent to the tip of the ellipse by observing that the typical turnover time of the corner vortex matches the period of the unstable mode in the shear layer. We suggest that the corner vortex acts as a signal generator, and produces periodic perturbation which triggers the instability.