In this paper, we first establish an $L^2$-type Dolbeault isomorphism for logarithmic differential forms by H\"{o}rmander's $L^2$-estimates. By using this isomorphism and the construction of smooth Hermitian metrics, we obtain a number of new %Akizuki-Kodaira-Nakano type vanishing theorems for sheaves of logarithmic differential forms on compact K\"ahler manifolds with simple normal crossing divisors, which generalize several classical vanishing theorems, including Norimatsu's vanishing theorem, Gibrau's vanishing theorem, Le Potier's vanishing theorem and a version of the Kawamata-Viehweg vanishing theorem.

We present a discontinuous Galerkin (DG) scheme with suitable quadrature rules \cite{Tchen2017entropy} for ideal compressible magnetohydrodynamic (MHD) equations on structural meshes. The semi-discrete scheme is analyzed to be entropy stable by using the symmetrizable version of the equations as introduced by Godunov \cite{godunov1972symmetric}, the entropy stable DG framework with suitable quadrature rules \cite{Tchen2017entropy}, the entropy conservative flux in \cite{chandrashekar2016entropy} inside each cell and the entropy dissipative approximate Godunov type numerical flux at cell interfaces to make the scheme entropy stable. {\color{blue}{ The main difficulty in the generalization of the results in \cite{Tchen2017entropy} is the appearance of the non-conservative ``source terms'' added in the modified MHD model introduced by Godunov \cite{godunov1972symmetric}, which do not exist in the general hyperbolic system studied in \cite{Tchen2017entropy}. Special care must be taken to discretize these ``source terms'' adequately so that the resulting DG scheme satisfies entropy stability. }} Total variation diminishing / bounded (TVD/TVB) limiters and bound-preserving limiters are applied to control spurious oscillations. We demonstrate the accuracy and robustness of this new scheme on standard MHD examples.

We derive an asymptotic formula which counts the number of abelian extensions of prime degrees over rational function fields. Specifically, let l be a rational prime and K a rational function field F_q(t) with l\nmid q. Let Discf (F/K) denote the finite discriminant of F over K. Denote the number of abelian l-extensions F/K with deg (Discf (F/K)) = (l − 1)αn by a_l(n), where α = α(q, l) is the order of q in the multiplicative group (Z/lZ)^\times. We give a explicit asymptotic formula for a_l(n). In the case of cubic extensions with q ≡ 2 (mod 3), our formula gives an exact analogue of Cohn’s classical formula.

No abstract uploaded!

No abstract uploaded!