K-space trajectory

Fig. 2 (a) A RARE pulse sequence, and (b) its k-space trajectory. The order of the phase... 

Fig. 18. (a) RARE pulse sequence, and (b) its associated k-space trajectory. The order of the phase encoding is shown by the numbers to the left of the raster. After each line in k-space, the spins are returned to the same point on the fcxqead) axis prior to the application of the refocusing pulse shown by the dashed line and arrow. 

Two studies have examined the role of detection methods in the reliability of detection of brain metabolites in humans. A study has been carried out on the reproducibility of H NMR data obtained using spectroscopic imaging in the human brain. A PRESS-based pulse sequence with dual-band RF pulses was used to excite the resonances of Cho, creatine plus phosphocreatine (tCr) and NAA while simultaneously suppressing water and lipids. Spiral k-space trajectories were used for fast spectral and spatial encoding of 1 cm voxels throughout the brain. Automated quantification of the spectrum of each... 

Fig. 5.4.2 [Ljul] Trajectories in k space, (a) Line-scan method, (b) Back-projection method, (c) Modified back-projection method, (d) Fourier imaging, (e) Echo-planar imaging.

Fig. 1.21. Phase space trajectories for different noise intensities. We see successive period doubling of the small amplitude oscillations until chaos emerges. Soon after, spiking starts. Note the small parameter range (from a to e) Tjog = 0.00157,0.00158,0.0015826,0.001585,0.001586 K = 0.1. Subplot e is an enlargement of f. [41]...

Fig. 6.21. Mixing 5% periodic cells with 95% chaotic cells results in periodic cAMP oscillations in the final suspension. Shown are the phase space trajectories for the chaotic population 1 (upper left), the homogeneous, periodic population 2 (upper right), and the final mixed suspension, which oscillates on a limit cycle (lower part of figure) differing from that of population 2. The curves are obtained by integrations of eqns (6.3) for the pure population 1 or 2, and of eqns (6.9) for the mixed suspension. For the chaotic population 1, the parameter values correspond to point C in fig. 6.2 (k i - 2.625 min" ), while for the periodic population 2, = 2.4257 min" in both cases, v = v/p, -1.407 min (Li et...

The Levinthal s paradox emerges from the (semi-)classical picture of the molecules conformations defined in the f-space of the one-dimensional model of Figure 9.4. Within this strategy, the particle bears a definite position h in every instant of time. Thus every conformational change can be represented by a trajectory (path) in K-space, following the shape of V(k). 

Fig. 5 shows a number of kinetic energy and longitudinal stress profiles at r=100 for a three-dimensional filament (Fig. 1) with the periodic blocks again made up of 5x5x5 bcc unit cells. The interaction potential was B of Fig. 2a, also for a-iron. The lattice was initially at an equilibrium temperature of 38 K. Up expressed in its longitudinal Mach number was 0.1. The different cases, A to F, corresponded to different initial conditions characterized by two parameters (TqjTi), with Tq denoting the initial position of the phase space trajectory of the... 

We now examine how a next-amplitude-map was obtained from tire attractor shown in figure C3.6.4(a) [171. Consider tire plane in tliis space whose projection is tire dashed curve i.e. a plane ortliogonal to tire (X (tj + t)) plane. Then, for tire /ctli intersection of tire (continuous) trajectory witli tliis plane, tliere will be a data point X (ti + r), X (ti + 2r))on tire attractor tliat lies closest to tire intersection of tire continuous trajectory. A second discretization produces tire set Xt- = k = 1,2,., I This set is used in tire constmction... 

It has been shown that there is a two-dimensional cut of the PES such that the MEP lies completely within it. The coordinates in this cut are 4, and a linear combination of qs-q-j. This cut is presented in fig. 64, along with the MEP. Motion along the reaction path is adiabatic with respect to the fast coordinates q -q and nonadiabatic in the space of the slow coordinates q -qi-Nevertheless, since the MEP has a small curvature, the deviation of the extremal trajectory from it is small. This small curvature approximation has been intensively used earlier [Skodje et al. 1981 Truhlar et al. 1982], in particular for calculating tunneling splittings in (HF)2- The rate constant of reaction (6.45a) found in this way is characterized by the values T<. = 20-25 K, = 10 -10 s , = 1-4 kcal/mol above T, which compare well with the experiment. 

Pagels show that thermodynamic depth is proportional to the difference between the state s thermodynamic entropy (i.e. its coarse grained entropy) and its finegrained entropy, given by fcex volume of points in phase space corresponding to the system s trajectory, where k], is Boltzman s constant. 

Figure 14. Classical trajectories for the H + H2(v = l,j = 0) reaction representing a 1-TS (a-d) and a 2-TS reaction path (e-h). Both trajectories lead to H2(v = 2,/ = 5,k = 0) products and the same scattering angle, 0 = 50°. (a-c) 1-TS trajectory in Cartesian coordinates. The positions of the atoms (Ha, solid circles Hb, open circles He, dotted circles) are plotted at constant time intervals of 4.1 fs on top of snapshots of the potential energy surface in a space-fixed frame centered at the reactant HbHc molecule. The location of the conical intersection is indicated by crosses (x). (d) 1-TS trajectory in hyperspherical coordinates (cf. Fig. 1) showing the different H - - H2 arrangements (open diamonds) at the same time intervals as panels (a-c) the potential energy contours are for a fixed hyperradius of p = 4.0 a.u. (e-h) As above for the 2-TS trajectory. Note that the 1-TS trajectory is deflected to the nearside (deflection angle 0 = +50°), whereas the 2-TS trajectory proceeds via an insertion mechanism and is deflected to the farside (0 = —50°).

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