In the long-time scale, we consider the fluid dynamical limits for the kinetic equations when the fluctuation is decomposed into even and odd parts with respect to the microscopic velocity with different scalings. It is shown that when the background state is an absolute Maxwellian, the limit fluid dynamical equations are the incompressible Navier-Stokes equations with viscous heating. This is different from the case when the even and odd parts of the fluctuation have the same scaling where the standard incompressible Navier-Stokes equations without viscous heating are obtained. On the other hand, when the background is a local Maxwellian, it is shown that the above even-odd decomposition leads to a non-classical fluid dynamical system without viscous heating which has been used to describe the ghost effect in the kinetic theory. In addition, the above even-odd decomposition is justified rigorously for the Boltzmann equation for the former case when the background is an absolute Maxwellian.

In the presence of vacuum, the physical entropy for polytropic gases behaves singularly and it is thus a challenge to study its dynamics. It is shown in this paper that the boundedness of the entropy can be propagated up to any finite time provided that the initial vacuum presents only at far fields with sufficiently slow decay of the initial density. More precisely, for the Cauchy problem of the onedimensional heat conductive compressible Navier--Stokes equations, the global well-posedness of strong solutions and uniform boundedness of the corresponding entropy are established, as long as the initial density vanishes only at far fields with a rate no more than $O(\frac{1}{x^2})$. The main tools of proving the uniform boundedness of the entropy are some singularly weighted energy estimates carefully designed for the heat conductive compressible Navier--Stokes equations and an elaborate De Giorgi type iteration technique for some classes of degenerate parabolic equations. The De Giorgi type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy.

In this article our main goal is to establish an L2 version of the twisted invariance of plurigenera. We equally simplify the original “two towers” approach of Yum-Tong Siu.

In this paper, a homotopy continuation method for the computation of nonnegative Z-/H-eigenpairs of a nonnegative tensor is presented. We show that the homotopy continuation method is guaranteed to compute a nonnegative eigenpair. Additionally, using degree analysis, we show that the number of positive Z-eigenpairs of an irreducible nonnegative tensor is odd. A novel homotopy continuation method is proposed to compute an odd number of positive Z-eigenpairs, and some numerical results are presented.

We give a generalizations of lower Ricci curvature bound in the framework of graphs. We prove that the Ricci curvature in the sense of Bakry and Emery is bounded below by 1 on locally finite graphs. The Ricci flat graph in the sense of Chung and Yau is proved to be a graph with Ricci curvature bounded below by zero. We also get an estimate for the eigenvalue of Laplace operator on finite graphs: 1 dD (exp (dD+ 1) 1)