The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties oflk,singular values of rea...The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties oflk,singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest lk,S-singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest lk,s- singular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.展开更多
We consider the problem of supersonic flow of a Chaplygin gas past a delta wing with a shock or a rarefaction wave attached to the leading edges.The flow under study is described by the three-dimensional steady Euler ...We consider the problem of supersonic flow of a Chaplygin gas past a delta wing with a shock or a rarefaction wave attached to the leading edges.The flow under study is described by the three-dimensional steady Euler system.In conical coordinates,this problem can be reformulated as a boundary value problem for a nonlinear equation of mixed type.The type of this equation depends fully on the solutions of the problem itself,and thus it cannot be determined in advance.We overcome the difficulty by establishing a crucial Lipschitz estimate,and finally prove the unique existence of the solution via the method of continuity.展开更多
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant Nos. 11371109, 11426075）, the Natural Science Foundation of Heilongjiang Province （No. QC2014C001）, and the Fundamental Research Funds for the Central Universities.
文摘The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties oflk,singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest lk,S-singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest lk,s- singular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.
文摘We consider the problem of supersonic flow of a Chaplygin gas past a delta wing with a shock or a rarefaction wave attached to the leading edges.The flow under study is described by the three-dimensional steady Euler system.In conical coordinates,this problem can be reformulated as a boundary value problem for a nonlinear equation of mixed type.The type of this equation depends fully on the solutions of the problem itself,and thus it cannot be determined in advance.We overcome the difficulty by establishing a crucial Lipschitz estimate,and finally prove the unique existence of the solution via the method of continuity.
