Trajectory classification

The approach outlined, considering the trajectories in the i-rj plane instead of f(/) and T)(t) individually, has numerous applications. For an introduction to the theory of trajectory classification see D. A. Sanchez, Ordinary Differential Equations and Stability Theory (San Francisco Freeman, 1968). 

The different theoretical models for analyzing particle deposition kinetics from suspensions can be classified as either deterministic or stochastic. The deterministic methods are based on the formulation and solution of the equations arising from the application of Newton s second law to a particle whose trajectory is followed in time, until it makes contact with the collector or leaves the system. In the stochastic methods, forces are freed of their classic duty of determining directly the motion of particles and instead the probability of finding a particle in a certain place at a certain time is determined. A more detailed classification scheme can be found in an overview article [72]. 

The Matsuzaka Elbow-Jet classifier (Fig. 11) is based on a transverse flow principle (26). The stream of feed particles are accelerated to minimize the effect of gravity, and introduced into an air jet at right angles. The particles are fanned out in the classification zone with the trajectories for particles of the same hydrodynamic behavior, ie, size and shape, being the same. Classification is achieved by mounting one or more cutters in the classification zone, thus dividing the feed into two or more fractions. A stream of fine particles of less than 5 Jm can be produced in this manner. 

Classification, Missiles are commonly classified by their launch and target environments, as well as by popular names. Other methods of classifying guided missiles are by trajectory, speed (subsonic, sonic or supersonic), propulsion (air breathers, usually jets or rockets), guidance (command, Inertial or homing), payload (such as nuclear, high explosive, or electronic jammer), and purpose (strategic or tactical, offensive or defensive)... 

Fig. 2 Positional detection and mean-square displacement (MSD). (a) The x, y-coordinates of a particle at a certain time point are derived from its diffraction limited spot by fitting a 2D-Gaussian function to its intensity profile. In this way, a positional accuracy far below the optical resolution is obtained, (b) The figure depicts a simplified scheme for the analysis of a trajectory and the corresponding plot of the time dependency of the MSD. The average of all steps within the trajectory for each time-lag At, with At = z, At = 2z,... (where z = acquisition time interval from frame to frame) gives a point in the plot of MSD = f(t). (c) The time dependence of the MSD allows the classification of several modes of motion by evaluating the best fit of the MSD plot to one of the four formulas. A linear plot indicates normal diffusion and can be described by = ADAt (D = diffusion coefficient). A quadratic dependence of on At indicates directed motion and can be fitted by = v2At2 + ADAt (v = mean velocity). An asymptotic behavior for larger At with = [1 - exp (—AA2DAt/)] indicates confined diffusion. Anomalous diffusion is indicated by a best fit with = ADAf and a < 1 (sub-diffusive)...

Separation in these devices known as winnowing machines [3], is achieved due to the difference between trajectories of coarse and fine particles in the separation zone (Fig. lb). Their operation and efficiency are strongly affected by the stochastic factors of the process, in particular by uncertainties in feeding and particles aerodynamic interactions. In most cases coarse particles prevent proper classification of fines. Separation efficiency of these devices is usually low. They are normally used for separation of solid particles according to densities (e.g. grain from peel), rather than by size. Sometimes crossflow separation in horizontal streams is used in combination with other separation principles. 

The results calculated for the flight trajectories of particles with various diameters and moisture contents are shown in Fig. 6.22. The figure indicates that the radial gas flow exhibits certain classification effect for particles with various diameters. However, the moisture content of the particle has almost no effect on the flying distance. These theoretical results illustrate that the arrangement of the upper overflow discharging port is totally unfeasible. 

The above classification may be considered to be exhaustive for the most wide-spread simple (from the viewpoint of their dynamics) multicomponent systems where all of the whole trajectories i(p) and (p) start and finish at unstable and... 

Using the expression (5.22) together with Tables 5.5 and 5.6 on the base of the general principles reported in Sect. 5.2 one can carry out an exhaustive classification of the four-component systems as it has been already done for terpolymerization in Sect. 5.3. However, when the forth monomer is added, the number of the system types increases from 7 (see Fig. 6) to 41 (see Fig. 9) and that is a reason why the results of the complete theoretical analysis cannot be represented in the framework of this review. Without appealing to the classification and using only the algorithm described in Sect. 5.2 one may present a phase portrait of any concrete four-component system and hence predict the qualitative character of its dynamic behavior before the computer calculations of trajectories x(p) are performed. 

All the mentioned types of the nontrivial dynamic behavior are excluded for the systems where the reactivity ratios ry can be described by the expressions of the well-known Alfrey-Price Q-e scheme [20], and as a result they are to follow the simplified terminal model (see Sect. 4.6). In these systems, due to the relations Bj(X)/Bj(x) = ajj/ajj which holds for all i and j, the functions 7e,-(2) according to relations (4.10) are the ratios of the homogeneous polynomials of degree 2. Besides, for the calculations of the coefficients ak of Eq. (5.11) one can use the simple formulae presented in terms of determinants Dj and D [6, p. 265]. The theoretical analysis [202] leads to the conclusion that in such systems even the limited cycles are not possible and all azeotropes are certainly unstable. Hence any trajectory H(p) and X(p) when p -> 1 inevitably approaches the SP corresponding to the homopolymer the number of which can be from 1 to m. The set of systems obtained due to the classification within the framework of the simplified model essentially impoverishes in comparison with the general case of the terminal copolymerization model since some types of systems cannot be principally realized under the restrictions which the Q-e scheme puts on the reactivity ratios r. ... 

In a cross-flow classifier, the feed material enters the flow medium at one point in the classification chamber, at an angle to the direction of fluid flow with a component of velocity transverse to the flow and is fanned out under the action of field, inertia and drag forces. Particles of different sizes describe different trajectories and so can be separated according to size. 

As we ve seen, in one-dimensional phase spaces the flow is extremely confined— all trajectories are forced to move monotonically or remain constant. In higherdimensional phase spaces, trajectories have much more room to maneuver, and so a wider range of dynamical behavior becomes possible. Rather than attack all this complexity at once, we begin with the simplest class of higher-dimensional systems, namely linear systems in two dimensions. These systems are interesting in their own right, and, as we ll see later, they also play an important role in the classification of fixed points of nonlinear systems. We begin with some definitions and examples. 

Classification of back trajectories by sources of pollution (Chen et al., 2004)... 

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