It is known that certain one parameter families of unimodal maps of the interval have a topological universality with regard to their dynamic behavior [ 1, 2]. As a parameter is smoothly increased, a fascinating varie...It is known that certain one parameter families of unimodal maps of the interval have a topological universality with regard to their dynamic behavior [ 1, 2]. As a parameter is smoothly increased, a fascinating variety of dynamic behaviors are produced. For some families the behaviors are monotonic in the parameter, while in others they are not [3]. The question is what sort of conditions on a one parameter family will ensure this monotonicity of the behavior with the parameter？ The answer is unknown and will not be given here. What we do instead is to investigate certain geometric-dynamic-combinatorial consequences of assuming that the family has this monotonicity. Specifically, using tools of symbolic dynamics, state space is ＂course grained＂ with a finite alphabet. We decompose a non-invertible map into nonlinear but invertible pieces. From these invertible pieces, we form inverse maps via composition along words. Equations of motion are developed for both forward and inverse orbits （in both the variables of state space and the parameter）, and an equation relating forward and inverse motions at fix-points is exhibited. Finally, we deduce a list of conditions, each of which is equivalent to monotone behavior. One of these conditions states that simple parity characteristics of words correspond to definite dynamics near fixed-points and vice versa.展开更多
A synthetic method for a new unsymmetrical Schiff base and its Ln(Ⅲ)complexes including multi > C = N — groups is reported. The complexes are characterized byelemental analysis, IR spectra, ~1H and ^(13)C NMR, es...A synthetic method for a new unsymmetrical Schiff base and its Ln(Ⅲ)complexes including multi > C = N — groups is reported. The complexes are characterized byelemental analysis, IR spectra, ~1H and ^(13)C NMR, especially 2D-COSY ~1H, ~1H NMR spectra. Thegeneral formula of the obtained complexes is [Ln_3(TBLY)(NO_3)_3] · nH_2O (Ln = La, n = 3; Ln = Nd,n=5; Ln = Gd, Dy, Yb, Y, n = 7), where TBLY = tetraglycol aldehyde-2,4-dihydroxy benzaldehydebis-lysine Schiff base. In addition, the evidence for existence of > C = CH — NH — group issupported by the AM1 method. The complexes obtained may be used as a catalyst. Conversion rate of80% with the viscosity-average molecular weight 220000 for the polymerization of methyl methacrylate(MMA) without addition of any cocatalyst has been obtained.展开更多
文摘It is known that certain one parameter families of unimodal maps of the interval have a topological universality with regard to their dynamic behavior [ 1, 2]. As a parameter is smoothly increased, a fascinating variety of dynamic behaviors are produced. For some families the behaviors are monotonic in the parameter, while in others they are not [3]. The question is what sort of conditions on a one parameter family will ensure this monotonicity of the behavior with the parameter？ The answer is unknown and will not be given here. What we do instead is to investigate certain geometric-dynamic-combinatorial consequences of assuming that the family has this monotonicity. Specifically, using tools of symbolic dynamics, state space is ＂course grained＂ with a finite alphabet. We decompose a non-invertible map into nonlinear but invertible pieces. From these invertible pieces, we form inverse maps via composition along words. Equations of motion are developed for both forward and inverse orbits （in both the variables of state space and the parameter）, and an equation relating forward and inverse motions at fix-points is exhibited. Finally, we deduce a list of conditions, each of which is equivalent to monotone behavior. One of these conditions states that simple parity characteristics of words correspond to definite dynamics near fixed-points and vice versa.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 29671026) Natural Science Foundation of Zhejiang Province (Grant No. 296062) and the Laboratory of MRAMP (Grant No. 971502). References
文摘A synthetic method for a new unsymmetrical Schiff base and its Ln(Ⅲ)complexes including multi > C = N — groups is reported. The complexes are characterized byelemental analysis, IR spectra, ~1H and ^(13)C NMR, especially 2D-COSY ~1H, ~1H NMR spectra. Thegeneral formula of the obtained complexes is [Ln_3(TBLY)(NO_3)_3] · nH_2O (Ln = La, n = 3; Ln = Nd,n=5; Ln = Gd, Dy, Yb, Y, n = 7), where TBLY = tetraglycol aldehyde-2,4-dihydroxy benzaldehydebis-lysine Schiff base. In addition, the evidence for existence of > C = CH — NH — group issupported by the AM1 method. The complexes obtained may be used as a catalyst. Conversion rate of80% with the viscosity-average molecular weight 220000 for the polymerization of methyl methacrylate(MMA) without addition of any cocatalyst has been obtained.
