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In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degenerate. As application we obtain the Bergman kernel asymptotics for adjoint semi-positive line bundles over complete K\"ahler manifolds, on the set where the curvature is positive. We also prove the asymptotics for big line bundles endowed with singular Hermitian metrics with strictly positive curvature current. In this case the full asymptotics holds outside the singular locus of the metric.

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Let $q=p^{e}$ be a prime power and $\ell$ be an integer with $0\leq \ell\leq e-1$. The $\ell$-Galois hull of classical linear codes is a generalization of the Euclidean hull and Hermitian hull. We provide a necessary and sufficient condition under which a codeword of a GRS code or an extended GRS code belongs to its $\ell$-Galois dual code, generalizing both the Euclidean case and Hermitian case in the literature. By using four different tools: 1) the norm mapping from $\mathbb{F}_{q}^{\ast}$ to $\mathbb{F}_{p^{\ell}}^{\ast}$; 2) the direct product of two cyclic subgroups; 3) the coset decomposition of a cyclic group; 4) an additive subgroup of $\mathbb{F}_{q}$ and its cosets, we construct eleven families of $q$-ary MDS codes with $\ell$-Galois hulls of arbitrary dimensions, and give the related eleven families of $[[n,k,d;c]]_{q}$ entanglement-assisted quantum error-correcting codes (EAQECCs) with relatively large minimum distance in the sense that $2d=n-k+2+c$. We show that developing the theory of $\ell$-Galois hulls of $q$-ary MDS codes in this paper enables us to obtain new $q$-ary EAQECCs with different kinds of length sets via different $\ell$, where $2\ell\mid e$.

We consider a L 2-contraction (a L 2-type stability) of large viscous shock waves for the multi-dimensional scalar viscous conservation laws, up to a suitable shift by using the relative entropy methods. Quite different from the previous results, we find a new way to determine the shift function, which depends both on the time and space variables and solves a viscous Hamilton-Jacobi type equation with source terms. Moreover, we do not impose any conditions on the anti-derivative variables of the perturbation around the shock profile. More precisely, it is proved that if the initial perturbation around the viscous shock wave is suitably small in L 2-norm, then the L 2-contraction holds true for the viscous shock wave up to a suitable shift function. Note that BV-norm or the L-norm of the initial perturbation and the shock wave strength can be arbitrarily large. Furthermore, as the time t tends to infinity, the L 2-contraction holds