When a differential field K having n commuting derivations is given together with two finitely generated differential extensions L and M of K, an important problem in differential algebra is to exhibit a common differ...When a differential field K having n commuting derivations is given together with two finitely generated differential extensions L and M of K, an important problem in differential algebra is to exhibit a common differential extension N in order to define the new differential extensions L<span style="white-space:nowrap;"><span style="white-space:nowrap;">∩M and the smallest differential field <span style="white-space:nowrap;">(L,M ) <span style="white-space:nowrap;"><span style="white-space:nowrap;">⊂ N containing both L and M. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules L and M over the non-commutative ring <span style="white-space:nowrap;">D = K [d<sub>1</sub>,...,d<sub>n</sub>] = K [d] of differential operators with coefficients in K, we may similarly look for a differential module N containing both L and M in order to define <span style="white-space:nowrap;">L∩M and <span style="white-space:nowrap;">L+M. This is exactly the situation met in linear or non-linear OD or PD control theory by selecting the inputs and the outputs among the control variables. However, in many recent books and papers, we have shown that controllability was a built-in property of a control system, not depending on the choice of inputs and outputs. The purpose of this paper is thus to revisit control theory by showing the specific importance of the two previous problems and the part plaid by N in both cases for the parametrization of the control system. An important tool will be the study of differential correspondences, a modern name for what was called B<span style="white-space:nowrap;"><span style="white-space:nowrap;">äcklund problem during the last century, namely the elimination theory for groups of variables among systems of linear or nonlinear OD or PD equations. The main difficulty is to revisit differential homological algebra by using noncommutative localization as a way to generalize the symbolic calculus in the style of Heaviside and Mikusinski. Finally, when M is a D-module, this paper is using for the first time the fact that the system <span style="white-space:nowrap;">R = hom<sub>K</sub> (M,K) is a D-module for the Spencer operator acting on sections, avoiding thus behaviours, trajectories and signal spaces in a purely formal way, contrary to a few recent works on this difficult subject.展开更多
The extensional model of the South China Sea(SCS)has been widely studied,but remains under debate.Based on the latest high-quality multi-channel seismic data,bathymetric data,and other obtained seismic profiles,the as...The extensional model of the South China Sea(SCS)has been widely studied,but remains under debate.Based on the latest high-quality multi-channel seismic data,bathymetric data,and other obtained seismic profiles,the asymmetric characteristics between the conjugate margins of the SCS are revealed and extensional model of the SCS margin is discussed further.Spatial variation of morphology,basement structure,and marginal faults are discovered among the SCS margin profiles.As for the NS-trending variation,the basement of northern margin displays in the shape of step downwards to the sea,while the basement of southern margin is composed of wide rotated and tilted blocks,without any obvious bathymetric change.The variation also exists in the development of marginal faults between the conjugate margins,and detachment fault system is identified on the southern margin.Along the southern margin from east to west,the Eastern and Southwestern Basins developed different structural units.Based on the tectonic contrast of the conjugate margins,differential extensional model is proposed to explain the spatial variation of the SCS structure,which introduces detachment faults controlling the evolution of the SCS.The upper crust above the detachment fault was deformed by simple shear,while the lower crust and upper mantle below the detachment fault was deformed by pure shear.Because of the different lateral transfer between the upper brittle faulting and the lower ductile extensional regions,there developed marginal plateau(Liyue basin)and outer rise(Zhenghe massif)on the lower plate margin of the Eastern Basin and the Southwestern Basin,respectively.The evolution of the present SCS may be influenced by the diachronous close of the paleo-SCS.展开更多
文摘When a differential field K having n commuting derivations is given together with two finitely generated differential extensions L and M of K, an important problem in differential algebra is to exhibit a common differential extension N in order to define the new differential extensions L<span style="white-space:nowrap;"><span style="white-space:nowrap;">∩M and the smallest differential field <span style="white-space:nowrap;">(L,M ) <span style="white-space:nowrap;"><span style="white-space:nowrap;">⊂ N containing both L and M. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules L and M over the non-commutative ring <span style="white-space:nowrap;">D = K [d<sub>1</sub>,...,d<sub>n</sub>] = K [d] of differential operators with coefficients in K, we may similarly look for a differential module N containing both L and M in order to define <span style="white-space:nowrap;">L∩M and <span style="white-space:nowrap;">L+M. This is exactly the situation met in linear or non-linear OD or PD control theory by selecting the inputs and the outputs among the control variables. However, in many recent books and papers, we have shown that controllability was a built-in property of a control system, not depending on the choice of inputs and outputs. The purpose of this paper is thus to revisit control theory by showing the specific importance of the two previous problems and the part plaid by N in both cases for the parametrization of the control system. An important tool will be the study of differential correspondences, a modern name for what was called B<span style="white-space:nowrap;"><span style="white-space:nowrap;">äcklund problem during the last century, namely the elimination theory for groups of variables among systems of linear or nonlinear OD or PD equations. The main difficulty is to revisit differential homological algebra by using noncommutative localization as a way to generalize the symbolic calculus in the style of Heaviside and Mikusinski. Finally, when M is a D-module, this paper is using for the first time the fact that the system <span style="white-space:nowrap;">R = hom<sub>K</sub> (M,K) is a D-module for the Spencer operator acting on sections, avoiding thus behaviours, trajectories and signal spaces in a purely formal way, contrary to a few recent works on this difficult subject.
基金supported by the National Natural Science Foundation of China(Grant No.41006031)Strategic Priority Research Program of the Chinese Academy of Sciences(Grant No.XDA11030102)the National High Technology Research and Development Program of China(Grant No.2009AA093401)
文摘The extensional model of the South China Sea(SCS)has been widely studied,but remains under debate.Based on the latest high-quality multi-channel seismic data,bathymetric data,and other obtained seismic profiles,the asymmetric characteristics between the conjugate margins of the SCS are revealed and extensional model of the SCS margin is discussed further.Spatial variation of morphology,basement structure,and marginal faults are discovered among the SCS margin profiles.As for the NS-trending variation,the basement of northern margin displays in the shape of step downwards to the sea,while the basement of southern margin is composed of wide rotated and tilted blocks,without any obvious bathymetric change.The variation also exists in the development of marginal faults between the conjugate margins,and detachment fault system is identified on the southern margin.Along the southern margin from east to west,the Eastern and Southwestern Basins developed different structural units.Based on the tectonic contrast of the conjugate margins,differential extensional model is proposed to explain the spatial variation of the SCS structure,which introduces detachment faults controlling the evolution of the SCS.The upper crust above the detachment fault was deformed by simple shear,while the lower crust and upper mantle below the detachment fault was deformed by pure shear.Because of the different lateral transfer between the upper brittle faulting and the lower ductile extensional regions,there developed marginal plateau(Liyue basin)and outer rise(Zhenghe massif)on the lower plate margin of the Eastern Basin and the Southwestern Basin,respectively.The evolution of the present SCS may be influenced by the diachronous close of the paleo-SCS.
