This paper is devoted to exploiting the restrictions of Riesz–Morrey potentials on either unbounded or bounded domains in Euclidean spaces.

We propose a simple finite difference scheme for Navier--Stokes equations in primitive formulation on curvilinear domains. With proper boundary treatment and interplay between covariant and contravariant components, the spatial discretization admits exact Hodge decomposition and energy identity. As a result, the pressure can be decoupled from the momentum equation with explicit time stepping. No artificial pressure boundary condition is needed. In addition, it can be shown that this spatially compatible discretization leads to uniform inf-sup condition, which plays a crucial role in the pressure approximation of both dynamic and steady state calculations. Numerical experiments demonstrate the robustness and efficiency of our scheme.

In the present paper, we prove the 2-part of Birch and Swinnerton-Dyer conjecture for an explicit infinite family of rank 0 quadratic twists of the modular elliptic curve χ_0(14), using an explicit form of the Waldspurger formula. We also give an explicit infinite family of rank 1 quadratic twists of χ_0(14) whose Tate–Shafarevich group is of odd cardinality.

Using the degeneration formula for Donaldson-Thomas invariants [L-W,MNOP2], we proved formulae for blowing up a point, simple flops, and extremal transitions.

We prove a formula which relates Euler characteristic of moduli spaces of stable pairs on local K3 surfaces to counting invariants of semistable sheaves on them. Our formula generalizes KawaiYoshioka’s formula for stable pairs with irreducible curve classes to arbitrary curve classes. We also propose a conjectural multiple cover formula of sheaf counting invariants which, combined with our main result, leads to an Euler characteristic version of KatzKlemm-Vafa conjecture for stable pairs.