When D:ξ→η is a linear ordinary differential (OD) or partial differential (PD) operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1:</su...When D:ξ→η is a linear ordinary differential (OD) or partial differential (PD) operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1:</sub>η→ξ such that Dξ = η implies D<sub>1</sub>η = 0. When D is involutive, the procedure provides successive first-order involutive operators D<sub>1</sub>,...,D<sub>n </sub>when the ground manifold has dimension n. Conversely, when D<sub>1</sub> is given, a much more difficult “inverse problem” is to look for an operator D:ξ→η having the generating CC D<sub>1</sub>η = 0. If this is possible, that is when the differential module defined by D<sub>1</sub> is “torsion-free”, that is when there does not exist any observable quantity which is a sum of derivatives of η that could be a solution of an autonomous OD or PD equation for itself, one shall say that the operator D<sub>1</sub> is parametrized by D. The parametrization is said to be “minimum” if the differential module defined by D does not contain a free differential submodule. The systematic use of the adjoint of a differential operator provides a constructive test with five steps using double differential duality. We prove and illustrate through many explicit examples the fact that a control system is controllable if and only if it can be parametrized. Accordingly, the controllability of any OD or PD control system is a “built in” property not depending on the choice of the input and output variables among the system variables. In the OD case and when D<sub>1</sub> is formally surjective, controllability just amounts to the formal injectivity of ad(D<sub>1</sub>), even in the variable coefficients case, a result still not acknowledged by the control community. Among other applications, the parametrization of the Cauchy stress operator in arbitrary dimension n has attracted many famous scientists (G. B. Airy in 1863 for n = 2, J. C. Maxwell in 1870, E. Beltrami in 1892 for n = 3, and A. Einstein in 1915 for n = 4). We prove that all these works are already explicitly using the self-adjoint Einstein operator, which cannot be parametrized and the comparison needs no comment. As a byproduct, they are all based on a confusion between the so-called div operator D<sub>2</sub> induced from the Bianchi operator and the Cauchy operator, adjoint of the Killing operator D which is parametrizing the Riemann operator D<sub>1</sub> for an arbitrary n. This purely mathematical result deeply questions the origin and existence of gravitational waves, both with the mathematical foundations of general relativity. As a matter of fact, this new framework provides a totally open domain of applications for computer algebra as the quoted test can be studied by means of Pommaret bases and related recent packages.展开更多
Sequence operation theory (SOT) is a powerful tool for solving complex probabilistic problems in power system. However, the basic single dimension SOT cannot satisfy the requirement of multi-state; multi-attribute ana...Sequence operation theory (SOT) is a powerful tool for solving complex probabilistic problems in power system. However, the basic single dimension SOT cannot satisfy the requirement of multi-state; multi-attribute analysis, which is often the case in actual power system practice. To address this problem, multidimensional sequence operation theory (MSOT) is developed. On the basis of previous research, this paper first categorizes the situations by the number of state variables; the number of attribute values,; defines the multidimensional sequence for single state variable; multiple attribute values, as well as the multidimensional sequence for multiple state variables; multiple attribute values. Corresponding to those definitions, four types of operations between two discrete multidimensional sequences are derived respectively. Therefore, the sequence is extended from single dimensional to multidimensional, establishing an integrated theory of multidimensional sequence operation. In particular, the basic single dimension SOT can be viewed as a special case of MSOT with only one state variable; one attribute value. Finally, the paper demonstrates the effectiveness of MSOT through an example of risk evaluation in power system generation scheduling.展开更多
The progress of grey system models is reviewed, and the general grey numbers, the grey sequence op- erators and several most commonly used grey system models are introduced, such as the absolute degree of grey inciden...The progress of grey system models is reviewed, and the general grey numbers, the grey sequence op- erators and several most commonly used grey system models are introduced, such as the absolute degree of grey incidence model, the grey cluster model based on endpoint triangular whitenization functions, the grey cluster model based on center-point triangular whitenization functions, the grey prediction model of the model GM （ 1,1）, and the weighted multi-attribute grey target decision model.展开更多
文摘When D:ξ→η is a linear ordinary differential (OD) or partial differential (PD) operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1:</sub>η→ξ such that Dξ = η implies D<sub>1</sub>η = 0. When D is involutive, the procedure provides successive first-order involutive operators D<sub>1</sub>,...,D<sub>n </sub>when the ground manifold has dimension n. Conversely, when D<sub>1</sub> is given, a much more difficult “inverse problem” is to look for an operator D:ξ→η having the generating CC D<sub>1</sub>η = 0. If this is possible, that is when the differential module defined by D<sub>1</sub> is “torsion-free”, that is when there does not exist any observable quantity which is a sum of derivatives of η that could be a solution of an autonomous OD or PD equation for itself, one shall say that the operator D<sub>1</sub> is parametrized by D. The parametrization is said to be “minimum” if the differential module defined by D does not contain a free differential submodule. The systematic use of the adjoint of a differential operator provides a constructive test with five steps using double differential duality. We prove and illustrate through many explicit examples the fact that a control system is controllable if and only if it can be parametrized. Accordingly, the controllability of any OD or PD control system is a “built in” property not depending on the choice of the input and output variables among the system variables. In the OD case and when D<sub>1</sub> is formally surjective, controllability just amounts to the formal injectivity of ad(D<sub>1</sub>), even in the variable coefficients case, a result still not acknowledged by the control community. Among other applications, the parametrization of the Cauchy stress operator in arbitrary dimension n has attracted many famous scientists (G. B. Airy in 1863 for n = 2, J. C. Maxwell in 1870, E. Beltrami in 1892 for n = 3, and A. Einstein in 1915 for n = 4). We prove that all these works are already explicitly using the self-adjoint Einstein operator, which cannot be parametrized and the comparison needs no comment. As a byproduct, they are all based on a confusion between the so-called div operator D<sub>2</sub> induced from the Bianchi operator and the Cauchy operator, adjoint of the Killing operator D which is parametrizing the Riemann operator D<sub>1</sub> for an arbitrary n. This purely mathematical result deeply questions the origin and existence of gravitational waves, both with the mathematical foundations of general relativity. As a matter of fact, this new framework provides a totally open domain of applications for computer algebra as the quoted test can be studied by means of Pommaret bases and related recent packages.
基金Supported by the National Natural Science Foundation of China (Grant No. 50777031)Program for New Century Excellent Talents in University (Grant No. NCET-07-0484)the Fok Ying-Tong Education Foundation (Grant No. 104020)
文摘Sequence operation theory (SOT) is a powerful tool for solving complex probabilistic problems in power system. However, the basic single dimension SOT cannot satisfy the requirement of multi-state; multi-attribute analysis, which is often the case in actual power system practice. To address this problem, multidimensional sequence operation theory (MSOT) is developed. On the basis of previous research, this paper first categorizes the situations by the number of state variables; the number of attribute values,; defines the multidimensional sequence for single state variable; multiple attribute values, as well as the multidimensional sequence for multiple state variables; multiple attribute values. Corresponding to those definitions, four types of operations between two discrete multidimensional sequences are derived respectively. Therefore, the sequence is extended from single dimensional to multidimensional, establishing an integrated theory of multidimensional sequence operation. In particular, the basic single dimension SOT can be viewed as a special case of MSOT with only one state variable; one attribute value. Finally, the paper demonstrates the effectiveness of MSOT through an example of risk evaluation in power system generation scheduling.
基金Supported by the Joint Research Project of Both the National Natural Science Foundation of Chinaand the Royal Society(RS)of UK(71111130211)the National Natural Science Foundation of China(90924022,70971064,70901041,71171113)+7 种基金the Major Project of Social Science Foundation of China(10ZD&014)the Key Project of Social Science Foundation of China(08AJY024)the Key Project of Soft Science Foundation of China(2008GXS5D115)the Foundation of Doctoral Programs(200802870020,200902870032)the Foundation of Humanities and Social Sciences of Chinese National Ministry of Education(08JA630039)the Science Foundation ofthe Excellent and Creative Group of Science and Technology in Jiangsu Province(Y0553-091)the Foundation of Key Research Base of Philosophy and Social Science in Colleges and Universities of Jiangsu Province(2010JDXM015)the Foundation of Outstanding Teaching Group of China(10td128)~~
文摘The progress of grey system models is reviewed, and the general grey numbers, the grey sequence op- erators and several most commonly used grey system models are introduced, such as the absolute degree of grey incidence model, the grey cluster model based on endpoint triangular whitenization functions, the grey cluster model based on center-point triangular whitenization functions, the grey prediction model of the model GM （ 1,1）, and the weighted multi-attribute grey target decision model.
