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Homological Projective duality (HP-duality) theory, introduced by Kuznetsov \cite{Kuz07HPD}, is one of the most powerful frameworks in the homological study of algebraic geometry. The main result (HP-duality theorem) of the theory gives complete descriptions of bounded derived categories of coherent sheaves of (dual) linear sections of HP-dual varieties. We show the theorem also holds for more general intersections beyond linear sections. More explicitly, for a given HP-dual pair $(X,Y)$, then analogue of HP-duality theorem holds for their intersections with another HP-dual pair $(S,T)$, provided that they intersect properly. We also prove a relative version of our main result. Taking $(S,T)$ to be dual linear subspaces (resp. subbundles), our method provides a more direct proof of the original (relative) HP-duality theorem.

This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magneto-hydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies Serrin’s condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanishing or vacuum appearing in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well known Serrin’s blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid.

In this paper, high order central finite difference schemes in a finite interval are analyzed for the diffusion equation. Boundary conditions of the initial-boundary value problem (IBVP) are treated by the simplified inverse Lax-Wendroff (SILW) procedure. For the fully discrete case, a third order explicit Runge-Kutta method is used as an example for the analysis. Stability is analyzed by both the GKS (Gustafsson, Kreiss and Sundstr\"om) theory and the eigenvalue visualization method on both semi-discrete and fully discrete schemes. The two different analysis techniques yield consistent results. Numerical tests are performed to demonstrate and validate the analysis results.

In this paper, we construct a new suboptimal filter by deriving the Ito's stochastic differential equations of the estimation of higher order central moments satisfy and imposing some conditions to form a closed system. The essentially infinite-dimensional cubic sensor problem has been investigated in detail numerically to illustrate the reasonableness of the imposed conditions, and the numerical experiments support our discussion. A 2-dimensional polynomial filtering problem has also been experimented.