Fuzzy trajectory

This energy difference may be interpreted in terms of two elliptical trajectories separated by Ae and with a phase lag between the leading and following edges of the element, ApAq that moves along the fuzzy trajectory. The two edges remain separated in time by a fixed amount A and define the elements A and B at -q0 and p respectively. 

From a structural point of view, mechanism in a single crystal can be much closer to a set of identical atomic trajectories than to the kind of fuzzy statistical average with which one must be content in solution. It is not surprising that with this kind of structural uniformity the site problems that plague kinetic studies in rigid glasses disappear. Adherence to first-order rate laws can be as close in single crystals as it is in fluids, and equally valid activation parameters can be obtained for thermal unimolecular reactions of reaction intermediates [12]. 

Bandemer considered the role of fuzzy set theory in analytical chemistry. The applications they described focused on pattern recognition problems, the calibration of analytical methods,quality control, and component identification and mixture evaluation. Gordon and Somorjai applied a fuzzy clustering technique to the detection of similarities among protein substructures. A molecular dynamics trajectory of a protein fragment was analyzed. In the following subsections, some applications based on the hierarchical fuzzy clustering techniques presented in this chapter are reviewed. 

The phenomenon of diffraction introduces a certain fuzziness or uncertainty into the trajectories followed by the diffracting particles. Figure 3.3 shows the situation at the moment of collision of a diffracting particle (electron, light atom or molecule) with a surface. The first uncertainty arises in the position of impact of the particle with the surface. Diffraction can occur only if the interaction with the surface extends over more than one lattice spacing, which means that the uncertainty in... 

Amodel-based closed-loop controller was designed for maximization of polymer production under safe process condition in emulsion copolymerization processes, while keeping the copolymer composition constant [195]. The interesting feature of the proposed controller was the use of a fuzzy model for design of the optimum reference trajectories. 

Further publications [16-18] have applied bio-inspired techniques to solve the tracking problem for the dynamic model of a unicycle mobile robot, using a fuzzy logic controller that provides the required torques to reach the desired velocity and trajectory inputs (Fig. 5.1). 

Once we have found optimal values for the gain constants, the next step is to find the optimal values for the input/output membership functions of the fuzzy controller. Our goal is that in the simulations, the linear and angular velocities reach zero and the mobile robot follow the reference trajectory represented in Eq. (6.4) and showed in Fig. 6.12. 

When applied the CRA to the fuzzy tracking controller simulations confirmed that the controller found was able to reach the reference trajectory. The algorithm was compared with the results obtained using GAs and again, the result errors were smaller. 

The error found by the CRA applied to the type-2 fuzzy tracking controller was larger than the errors obtained for the type-1 FLC. Despite this, the trajectory obtained was acceptable. 

As resulting of these experiments, the type-2 fuzzy logic controller performed better than the type-1 FLC, for a disturbance of e < 40, the mobile robot was able to return to the reference trajectory. In a previous woik [1], these tests were made applying a disturbance of e = 0.03 maximum. 

Hence if some physical measurement is recorded of the electron s position or velocity, the outcome will always be a fuzzy or blurred picture. One of the important implications of Heisenberg s Uncertainty Principle shown by this calculation is that it rules out the existence of definite paths (trajectories) of electrons. 

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