Schematic trajectory

A root velocity can be defined as the rate of advance of a given value of an H function root. For any given composition, roots with lower index numbers have lower velocities. An arbitrary initial noncoherent boundary involving variations of all roots thus is resolved, upon undisturbed development, into separate variations of the roots. This is shown by schematic trajectories of root values in a distance-time diagram in Figure 6. After resolution, each trajectory bundle involves variation of... 

Figure 3.3 Dynamics of the system (3.56)-(3.57). Left panels show the phase space, the two nullclines F(x,z) = 0 and G(x,z) = 0 and schematic trajectories. The arrows indicate the direction of motion. Right panels show the time evolution of x (solid line) and z (dashed line). In all panels q = 0.01, and the square labelled by S indicates the fixed point. First row / = 0.4, ei = 0.04. The fixed point is stable and dynamics, starting from P, finally leads to this steady state. Second row / = 0.4, ei = 0.04. The fixed point is unstable and concentrations oscillate in a limit cycle. Third row / = 4, ei = 0.004. The system lies on the stable fixed point S until time=l. Then its position is slightly displaced in the direction of P (jump not visible in the scale of the right panel), so that a excitation pulse occurs, after which the system returns to the steady state S.

FIGURE 3.43 Schematic trajectories of ions converting inside FAIMS from precursor X to product Y with Ec values differing by more than the instrumental resolution solid and dashed fines are for the species in and not in equilibrium, respectively. The transition does or would occur in the end (a), the middle (b, d), and the beginning (c) of separation FAIMS is set for Ec of X in (a, b) and Y in (c, d). 

Fig. 29.1 (a) A schematic view of a liquid jet injected perpendicular to a uniform incoming gas stream (b) a sample cross-sectional element (c) schematic trajectory of the droplets formed at the CBL... 

Fig. 3. Phase plane for the local dynamics of the model. The axes are the variables u and V. Shown are the system nullclines the v-nullcline, g u,v) = 0, is the line v = u, and the u-nullcline, /(u, v) = 0, has a backward N shape consisting of three lines u — 0,u = 1, and u = uih(w) = (w + b)/a. An excitable fixed point sits at the origin where the u and v nullclines intersect. th is the excitability threshold for the fixed point. Schematic trajectories for two initial conditions are shown. The initial condition to the left of the threshold decays directly to the fixed point. The initial condition to the right of the threshold undergoes a large excursion before returning to the fixed point.

Figure 6.5 Schematic trajectories illustrating the re-crossing of a transition state. Trajectory 1 corresponds to the situation idealised hy TST.

Figure 6 shows a two-dimensional schematic view of an individual ion s path in the ion implantation process as it comes to rest in a material. The ion does not travel in a straight path to its final position due to elastic collisions with target atoms. The actual integrated distance traveled by the ion is called the range, R The ion s net penetration into the material, measured along the vector of the ion s incident trajectory, which is perpendicular to the... 

Eig. 4. Schematic diagram of a Lagrangian trajectory model where Lf(/) represents the air column height in both Eulerian and Lagrangian (, Tj, ... 

Schematic DRD shown in Fig. 13-59 are particularly useful in determining the imphcations of possibly unknown ternary saddle azeotropes by postulating position 7 at interior positions in the temperature profile. It should also be noted that some combinations of binary azeotropes require the existence of a ternaiy saddle azeotrope. As an example, consider the system acetone (56.4°C), chloroform (61.2°C), and methanol (64.7°C). Methanol forms minimum-boiling azeotropes with both acetone (54.6°C) and chloroform (53.5°C), and acetone-chloroform forms a maximum-boiling azeotrope (64.5°C). Experimentally there are no data for maximum or minimum-boiling ternaiy azeotropes. The temperature profile for this system is 461325, which from Table 13-16 is consistent with DRD 040 and DRD 042. However, Table 13-16 also indicates that the pure component and binary azeotrope data are consistent with three temperature profiles involving a ternaiy saddle azeotrope, namely 4671325, 4617325, and 4613725. All three of these temperature profiles correspond to DRD 107. Experimental residue cui ve trajectories for the acetone-...

Fig. 3.48. Schematic diagram of particle trajectories undergoing scattering at the surface and channeling within the crystal. The depth scale is compressed relative to the width of the channel, to display the trajectories [3.120].

Fig. 3.61. Schematic illustration of projectile trajectories, showing focusing collisions when the projectiles impinge under a critical angle (5 c [3.150].

Fig. 13.5. Schematic representation of the potential energy surfaces of the ground state (S ,) and the excited state (.5,) of a nonadiabatic photoreaction of reactant R. Depending on the way the classical trajectories enter the conical intersection region, different ground-state valleys, which lead to products P and can be reached. Reproduced from Angew. Chem. Int. Ed. Engl. 34 549 (1995) by permission of Wiley-VCH.

The analytical method of jet trajectory study developed by Shepelev allows the derivation of several other useful features and is worth describing. On the schematic of a nonisothermal jet supplied at some angle to the horizon (Fig. 7.25), 5 is the jet s axis, X is the horizontal axis, and Z is the vertical axis. The ordinate of the trajectory of this jet can be described as z = xtga a- Az, where Az is the jet s rise due to buoyancy forces. To evaluate Az, the elementary volume dW with a mass equal to dm dV on the jet s trajectory was considered. The buoyancy force influencing this volume can be described as dP — g(p -Pj). Vertical acceleration of the volume under the consideration is j — dP / dm — -p,)/ g T,-T / T. Vertical... 

FIGURE 13.2 A schematic diagram of the trajectories for two different sizes of particles in a typical cyclone. 

FIGURE 13.4 A schematic diagram of the critical trajectory of a particle of diameter... 

Fig. 9.5 Schematic representation of Type-I intermittency. The plot on the LHS shows a return map for the system as it appears just below and precisely at the critical parameter value Tc. The plot on the RHS shows the return map for r > r. Note how, for r > Vc, X = Xc appears to first attract" then repel trajectories.

FIGURE 2.2. A schematic description of the evaluation of the transmission factor F. The figure describes three trajectories that reach the transition state region (in reality we will need many more trajectories for meaningful statistics). Two of our trajectories continue to the product region XP, while one trajectory crosses the line where X = X (the dashed line) but then bounces back to the reactants region XR. Thus, the transmission factor for this case is 2/3. 

Fig. 18-6 Characteristic air-mass trajectory and corresponding per mil isotopic composition of precipitation, along a transect from the subtropics to a polar ice sheet. This is a highly schematic view of the true atmospheric system.

The quadrupole ion-trap, usually referred to simply as the ion-trap, is a three-dimensional quadrupole. This type of analyser is shown schematically in Figure 3.5. It consists of a ring electrode with further electrodes, the end-cap electrodes, above and below this. In contrast to the quadrupole, described above, ions, after introduction into the ion-trap, follow a stable (but complex) trajectory, i.e. are trapped, until an RF voltage is applied to the ring electrode. Ions of a particular m/z then become unstable and are directed toward the detector. By varying the RF voltage in a systematic way, a complete mass spectrum may be obtained. 

Schematic representation of one type of mass spectrometer. An electron beam fragments gas atoms or molecules into positively charged ions. The ions are accelerated and then deflected by a magnet. Each fragment follows a trajectory that depends on its mass.

Figure 17.12 (A) Schematic presentation of deactivation and energy transfer processes in a single quantum dot placed on an Ag nanoparticle film. (B) Photoluminescence intensity trajectories of single quantum dots on a glass substrate (a) and on an Ag nanoparticle film (b). The traces in green represent background intensities. (C) Photoluminescence spectra of quantum dot solutions in the presence of...

Fig. 1. Schematic diagram of the multimass ion imaging detection system. (1) Pulsed nozzle (2) skimmers (3) molecular beam (4) photolysis laser beam (5) VUV laser beam, which is perpendicular to the plane of this figure (6) ion extraction plate floated on V0 with pulsed voltage variable from 3000 to 4600 V (7) ion extraction plate with voltage Va (8) outer concentric cylindrical electrode (9) inner concentric cylindrical electrode (10) simulation ion trajectory of m/e = 16 (11) simulation ion trajectory of rri/e = 14 (12) simulation ion trajectory of m/e = 12 (13) 30 (im diameter tungsten wire (14) 8 x 10cm metal mesh with voltage V0] (15) sstack multichannel plates and phosphor screen. In the two-dimensional detector, the V-axis is the mass axis, and V-axis (perpendicular to the plane of this figure) is the velocity axis (16) CCD camera.

To ensure the accuracy of the free energy estimate by sampling the most important set of trajectories, we choose the sequence of systems so that each successive state obeys a phase space subset relationship with the one that preceded it. This situation is illustrated schematically in Fig. 6.3. We say that a path following such a trajectory moves down the funnel [43]. 

It is instructive to simplify the above picture somewhat and consider the coalescence or sticking of two particles schematically shown in Fig. 13. One can assume that due to shear forces in the mixer, a fluidized bed in the present case, the two particles posses a relative velocity U0 which ensures collision at some point on their trajectory and possible sticking under appropriate conditions. It is essential that some binder be present at the point of contact, as depicted in the figure. From this simplified picture, allmechanisms... 

Figure 9. Schematic reprsentation of a classical trajectory moving on the Si and So energy surfaces of the H2—(CH) -NHt trans cis photoisomerization, starting near the planar Franck-Condon geometry. The geometric coordinates are (a) torsion of the C2—C3 and C3 C4 bonds and (b) asymmetric stretching coupled with pyramidalization. Both Si and So intersect at a conical intersection (Si/S0 Cl) located near the minimum of the Si surface (Min-C ) where the C2C3C4N5 torsion angle is 104°. [Reproduced with permission from [87], Copyright 2000 Amercian Chemical Society],...

It should be noted that this regime allows the possibility that the sensor is spread over several platforms and/or is comprised of several physically different sensors within each platform. It can encompass trajectory control for platforms and even control of data rates in connecting platforms to each other and to a central node. In each case the system can be viewed as consisting of many real or virtual sensors, where a virtual sensor can be a particular mode of a sensor, a position of a platform, a particular bit of a measurement made by a sensor, etc. Thus the sensor management problem may seen in all of these cases as one of choosing to switch between many different sensors, where the choice is made on the basis on knowledge of the environment. This view is schematically represented in Figure 1. 

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