One of the most striking changes in economic, thinking which comes primarily with John Maynard Keynes, is his explicit focus on time. For example, it appears early in a part of Keynes＇s writings that prediction is a ...One of the most striking changes in economic, thinking which comes primarily with John Maynard Keynes, is his explicit focus on time. For example, it appears early in a part of Keynes＇s writings that prediction is a very difficult matter. Due to the near impossibility of prediction of the future, uncertainty is a central theme relatively early of Keynes＇s economic thinking. This theme regarding prediction also implies that Keynes makes a clear distinction between short- and long-term expectations. Theory was to be concerned mainly with the present and with short period situations and the explicit attention to the concept of time, and subsequent alternative method of analysis has led to a number of innovations in economic thinking. This paper will address Keynes＇s handling of the concept of time from three angles： studying his philosophical background, his understanding of society, and his development of economic theory.展开更多
In this paper, based on the theory of rooted trees and B-series, we propose the concrete formulas of the substitution law for the trees of order = 5. With the help of the new substitution law, we derive a B-series int...In this paper, based on the theory of rooted trees and B-series, we propose the concrete formulas of the substitution law for the trees of order = 5. With the help of the new substitution law, we derive a B-series integrator extending the averaged vector field （AVF） methods for general Hamiltonian system to higher order. The new integrator turns out to be order of six and exactly preserves energy for Hamiltonian systems. Numerical exper- iments are presented to demonstrate the accuracy and the energy-preserving property of the sixth order AVF method.展开更多
Using least parameters, we expand the step-transition operator of any linear multi-step method （LMSM） up to O（τ^s＋5） with order s = 1 and rewrite the expansion of the steptransition operator for s = 2 （obtained...Using least parameters, we expand the step-transition operator of any linear multi-step method （LMSM） up to O（τ^s＋5） with order s = 1 and rewrite the expansion of the steptransition operator for s = 2 （obtained by the second author in a former paper）. We prove that in the conjugate relation G3^λτ o G1^τ =G2^τ o G3^λτ with G1 being an LMSM,（1） theorder of G2 can not be higher than that of G1; （2） if G3 is also an LMSM and G2 is a symplectic B-series, then the orders of G1, G2 and G3 must be 2, 2 and 1 respectively.展开更多
文摘One of the most striking changes in economic, thinking which comes primarily with John Maynard Keynes, is his explicit focus on time. For example, it appears early in a part of Keynes＇s writings that prediction is a very difficult matter. Due to the near impossibility of prediction of the future, uncertainty is a central theme relatively early of Keynes＇s economic thinking. This theme regarding prediction also implies that Keynes makes a clear distinction between short- and long-term expectations. Theory was to be concerned mainly with the present and with short period situations and the explicit attention to the concept of time, and subsequent alternative method of analysis has led to a number of innovations in economic thinking. This paper will address Keynes＇s handling of the concept of time from three angles： studying his philosophical background, his understanding of society, and his development of economic theory.
文摘In this paper, based on the theory of rooted trees and B-series, we propose the concrete formulas of the substitution law for the trees of order = 5. With the help of the new substitution law, we derive a B-series integrator extending the averaged vector field （AVF） methods for general Hamiltonian system to higher order. The new integrator turns out to be order of six and exactly preserves energy for Hamiltonian systems. Numerical exper- iments are presented to demonstrate the accuracy and the energy-preserving property of the sixth order AVF method.
基金This research is supported by the Informatization Construction of Knowledge Innovation Projects of the Chinese Academy of Sciences ＂Supercomputing Environment Construction and Application＂ （INF105-SCE）, and by a grant （No. 10471145） from National Natural Science Foundation of China.
文摘Using least parameters, we expand the step-transition operator of any linear multi-step method （LMSM） up to O（τ^s＋5） with order s = 1 and rewrite the expansion of the steptransition operator for s = 2 （obtained by the second author in a former paper）. We prove that in the conjugate relation G3^λτ o G1^τ =G2^τ o G3^λτ with G1 being an LMSM,（1） theorder of G2 can not be higher than that of G1; （2） if G3 is also an LMSM and G2 is a symplectic B-series, then the orders of G1, G2 and G3 must be 2, 2 and 1 respectively.
