Shift map

Despite bearing no direct relation to any physical dynamical system, the onedimensional discrete-time piecewise linear Bernoulli Shift map nonetheless displays many of the key mechanisms leading to deterministic chaos. The map is defined by (see figure 4.2) ... 

Next, we introduce the shift mapping, 7, which truncates the first orbit symbol and shifts all other symbols one place to the left ... 

The dynamical system F/, 7, consisting of the set of admissible sequences under / and the shift map, 7, is the n -order symbolic dynamical system induced by f The power of the method resides in the following - provided that the... 

Fig. 2.9.12 Phase shift maps measured as the difference in the signal phases with and without current pulses. The rf, field gradient and current pulse program are shown in Figure 2.9.2. The phase shift maps must be unwrapped as the original evaluation yields phase angles only in the principal range between —7t/2 and +7r/2 as a consequence of the 7r periodicity of the tangent function formed as the quotient of the imaginary and real signal...

Fig. 2 Chemical shift perturbation and chemical shift mapping, (a) Portions of the [15N, 1H]-HSQC spectra of Bcf-xL recorded in absence (black) and in presence of each of the four molecules (in colors). Resonance assignments for amino acid residues that exhibit large shifts are reported, (b) Structure of Bc1-Xl in complex with the BH3 peptide from Bak (PDB code 1BXL) showing the chemical shift changes in Bcl-xL upon ligand binding (blue, large shits yellow, no shifts the Bak peptide is reported in cyan). Adapted from [48]...

This kind of shift mapping has been used extensively in NMR studies of protein-ligand interactions. Shuker et al.1/2 used this method to advantage in a lead generation approach that effectively covalently links two... 

Fig.2 The calculated 13C chemical shift map of the Cp carbon in N-acetyl-N -methyl-L-alanine amide obtaining using the FPT INDO method. The chemical shifts were calculated at 15° intervals for the dihedral angles(< >,i /).

Figure 8.2.9 CSI experiment using four samples stacked vertically. The resulting contour plot is a spatial/chemical-shift map, which can be analysed by taking slices along the frequency axis to yield individual spectra for each sample position. Reprinted with permission from Hou, T., Smith, J., MacNamara, E., Macnaughtan, M. and Raftery, D., Anal. Chem., 73, 2541-2546 (2001). Copyright (2001) American Chemical Society...

Having studied the properties of the logistic mapping, a mapping with practical applications, we will now encounter a more abstract mapping, the shift map. It is very useful for fundamental investigations in chaos theory. It is defined by... 

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