The Cauchy problem for the barotropic compressible Navier--Stokes equations on the whole two-dimensional space with vacuum as far field density is considered. When the shear viscosity is a positive constant and the bulk one is a power function of density with the power bigger than four-thirds, the global existence and uniqueness of strong and classical solutions is established. It should be remarked that there are no restrictions on the size of the data.

Finding the maximum eigenvalue of a symmetric tensor is an important topic in tensor computation and numerical multilinear algebra. In this paper, we introduce a new class of structured tensors called W-tensors, which not only extends thewell-studied nonnegative tensors by allowing negative entries but also covers several important tensors arising naturally from spectral hypergraph theory.We then show that finding the maximum H-eigenvalue of an even-order symmetric W-tensor is equivalent to solving a structured semidefinite program and hence can be validated in polynomial time. This yields a highly efficient semidefinite program algorithm for computing themaximumH-eigenvalue of W-tensors and is based on a new structured sums-of-squares decomposition result for a nonnegative polynomial induced by W-tensors. Numerical experiments illustrate that the proposed algorithm can successfully find the maximum H-eigenvalue of W-tensors with dimension up to 10,000, subject to machine precision. As applications, we provide a polynomial time algorithm for computing the maximum H-eigenvalues of large-size Laplacian tensors of hyperstars and hypertrees, where the algorithm can be up to 13 times faster than the state-of-the-art numerical method introduced by Ng, Qi, and Zhou in 2009. Finally, we also show that the proposed algorithm can be used to test the copositivity of a multivariate form associated with symmetric extended Z-tensors,whose order may be even or odd.

We introduce and analyze lower ($Ricci$) curvature bounds $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$for metric measure spaces $ {\left( {M,d,m} \right)} $ . Our definition is based on convexity properties of the relative entropy $ Ent{\left( { \cdot \left| m \right.} \right)} $ regarded as a function on the$L$_{2}-Wasserstein space of probability measures on the metric space $ {\left( {M,d} \right)} $ . Among others, we show that $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$implies estimates for the volume growth of concentric balls. For Riemannian manifolds, $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$if and only if $ Ric_{M} {\left( {\xi ,\xi } \right)} $ ⩾$K$ $ {\left| \xi \right|}^{2} $ for all $ \xi \in TM $ .

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Multivariate cryptography is one of the main candidates to guarantee the security of communication in the presence of quantum computers. While there exist a large number of secure and efficient multivariate signature schemes, the number of practical multivariate encryption schemes is somewhat limited. In this paper we present our results on creating a new multivariate encryption scheme, which is an extension of the original SimpleMatrix encryption scheme of PQCrypto 2013. Our scheme allows fast en- and decryption and resists all known attacks against multivariate cryptosystems. Furthermore, we present a new idea to solve the decryption failure problem of the original SimpleMatrix encryption scheme.