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We construct positive loops of Legendrian submanifolds in several instances. In particular, we partially recover G. Liu’s result stating that any loose Legendrian admits a positive loop, under some mild topological assumptions on the Legendrian. Moreover, we show contractibility of the constructed loops under an extra topological assumption.

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 $2\ell\mid e$.

For the three-dimensional full compressible Navier–Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid, we establish the global existence and uniqueness of classical solutions with smooth initial data which are of small energy but possibly large oscillations where the initial density is allowed to vanish. Moreover, for the initial data, which may be discontinuous and contain vacuum states, we also obtain the global existence of weak solutions. These results generalize previous ones on classical and weak solutions for initial density being strictly away from a vacuum, and are the first for global classical and weak solutions which may have large oscillations and can contain vacuum states.

The Gauss-Bonnet theorem and the Cohn-Vossen inequality show that the only complete surface with positive curvature is either the sphere, RP2, or the plane. In higher dimension, the curvature tensor is far more complicated. There are several commonly used partial components of the curvature tensor that had been studied for the whole century. The simplest one is the scalar curvature which is the average of all curvatures at one point. It appeared in the Hilbert action for general relativity. This was studied extensively in connection with general relativity.