Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and turbulence.Several decompositions of the velocity gradient tensor,su...Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and turbulence.Several decompositions of the velocity gradient tensor,such as the triple decomposition of motion(TDM)and normal-nilpotent decomposition(NND),have been proposed to analyze the local motions of fluid elements.However,due to the existence of different types and non-uniqueness of Schur forms,as well as various possible definitions of NNDs,confusion has spread widely and is harming the research.This work aims to clean up this confusion.To this end,the complex and real Schur forms are derived constructively from the very basics,with special consideration for their non-uniqueness.Conditions of uniqueness are proposed.After a general discussion of normality and nilpotency,a complex NND and several real NNDs as well as normal-nonnormal decompositions are constructed,with a brief comparison of complex and real decompositions.Based on that,several confusing points are clarified,such as the distinction between NND and TDM,and the intrinsic gap between complex and real NNDs.Besides,the author proposes to extend the real block Schur form and its corresponding NNDs for the complex eigenvalue case to the real eigenvalue case.But their justification is left to further investigations.展开更多
By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial....By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial. The definition of the regularity of square tensors is generalized to the rectangular tensors, and in the regular case, a root of the Echaracteristic polynomial of a special rectangular tensor is an E-singular value of the rectangular tensor. Moreover, the best rank-one approximation of a real partially symmetric rectangular tensor is investigated.展开更多
We successfully employ an automatic centroid moment tensor(CMT) inversion system to infer the CMT solutions of the February 12,2014 MS7.3 Yutian,Xinjiang earthquake using near-field seismic waveforms(4° < △ &...We successfully employ an automatic centroid moment tensor(CMT) inversion system to infer the CMT solutions of the February 12,2014 MS7.3 Yutian,Xinjiang earthquake using near-field seismic waveforms(4° < △ < 12°) observed by the virtual China seismic networks,which have been recently set up.The results indicate that this event occurred on a rupture plane(strike 243°,dip 70°,and rake-18°),showing left-lateral strike-slip faulting with a minor normal-faulting component.The centroid in the horizontal direction is located nearly 13 km east of the epicenter(36.123° N,82.499° E),and the best-fitting centroid depth is around 10 km.The total scalar moment,M0,is retrieved with an average value of 3.05 × 1019N·m,corresponding to moment magnitude MW6.92.Most of the energy is released within about 14 s.Moreover,we discuss about the potential application of this system in earthquake disaster decision.展开更多
We study the concept of module twistor for a module over an algebra. This concept provides a unifying framework for various deformed constructions of modules over algebras, such as module R-matrices,(n-factor iterated...We study the concept of module twistor for a module over an algebra. This concept provides a unifying framework for various deformed constructions of modules over algebras, such as module R-matrices,(n-factor iterated) twisted tensor products and L-R-twisted tensor products of algebras. Among the main results,we find the relations among these constructions. Furthermore, we study some properties of module twistors.展开更多
文摘Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and turbulence.Several decompositions of the velocity gradient tensor,such as the triple decomposition of motion(TDM)and normal-nilpotent decomposition(NND),have been proposed to analyze the local motions of fluid elements.However,due to the existence of different types and non-uniqueness of Schur forms,as well as various possible definitions of NNDs,confusion has spread widely and is harming the research.This work aims to clean up this confusion.To this end,the complex and real Schur forms are derived constructively from the very basics,with special consideration for their non-uniqueness.Conditions of uniqueness are proposed.After a general discussion of normality and nilpotency,a complex NND and several real NNDs as well as normal-nonnormal decompositions are constructed,with a brief comparison of complex and real decompositions.Based on that,several confusing points are clarified,such as the distinction between NND and TDM,and the intrinsic gap between complex and real NNDs.Besides,the author proposes to extend the real block Schur form and its corresponding NNDs for the complex eigenvalue case to the real eigenvalue case.But their justification is left to further investigations.
文摘By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial. The definition of the regularity of square tensors is generalized to the rectangular tensors, and in the regular case, a root of the Echaracteristic polynomial of a special rectangular tensor is an E-singular value of the rectangular tensor. Moreover, the best rank-one approximation of a real partially symmetric rectangular tensor is investigated.
基金funded by Special Oceanic Scientific Research Program(201405026)Science for Earthquake Resilience Program(XH12060Y)Special Seismological Industry Research Program(201208003)
文摘We successfully employ an automatic centroid moment tensor(CMT) inversion system to infer the CMT solutions of the February 12,2014 MS7.3 Yutian,Xinjiang earthquake using near-field seismic waveforms(4° < △ < 12°) observed by the virtual China seismic networks,which have been recently set up.The results indicate that this event occurred on a rupture plane(strike 243°,dip 70°,and rake-18°),showing left-lateral strike-slip faulting with a minor normal-faulting component.The centroid in the horizontal direction is located nearly 13 km east of the epicenter(36.123° N,82.499° E),and the best-fitting centroid depth is around 10 km.The total scalar moment,M0,is retrieved with an average value of 3.05 × 1019N·m,corresponding to moment magnitude MW6.92.Most of the energy is released within about 14 s.Moreover,we discuss about the potential application of this system in earthquake disaster decision.
基金supported by National Natural Science Foundation of China (Grant Nos. 11201285 and 11371238)the First-class Discipline of Universities in Shanghai
文摘We study the concept of module twistor for a module over an algebra. This concept provides a unifying framework for various deformed constructions of modules over algebras, such as module R-matrices,(n-factor iterated) twisted tensor products and L-R-twisted tensor products of algebras. Among the main results,we find the relations among these constructions. Furthermore, we study some properties of module twistors.
