Robust fixed points

There are two reasons for having many points with which to fix a temperature scale. One is that, as mentioned before, few materials affected by heat change length equally or linearly. Having many points allows scales to be calibrated in short ranges, where nonlinearity is less likely to have a pronounced effect. The second is that few, if any, thermometers can read all temperatures. Most thermometers are calibrated to read a small range of temperatures. Many fixing points allows for a robust system of calibration. Unfortunately, most of these points require expensive equipment, and even then they are not easy to obtain and/or verify. 

The solutions of (3.68)-(3.69) for positive parameters and generic initial conditions are oscillations, of amplitude fixed by the initial conditions, around the fixed point Z — a /a2, P = b2/b. The equation of this family of closed trajectories is ai In Z + b2 In P — biP — a2Z = constant. The oscillations are suggestive of the population oscillations observed in some real predator-prey systems, but they suffer from an important drawback the existence of the continuous family of oscillating trajectories is structurally unstable systems similar to (3.68)-(3.69) but with small additional terms either lack the oscillations, or a single limit cycle is selected out of the continuum. Thus, the model (3.68)-(3.69) can not be considered a robust model of biological interactions, which are never known with enough... 

This fixed-point iteration provides a highly efficient scheme for the purification of density matrices [8]. As shown in Section 10.7.7, it converges quadratically in the local region and has robust global convergence characteristics. 

For the reasons described, no specific test will be advanced here as being superior, but Huber s model and the classical one for z = 2 and z = 3 are incorporated into program HUBER the authors are of the opinion that the best recourse is to openly declare all values and do the analysis twice, once with the presumed outliers included, and once excluded from the statistical analysis in the latter case the excluded points should nonetheless be included in tables (in parentheses) and in graphs (different symbol). Outliers should not be labeled as such solely on the basis of a fixed (statistical) rule the decision should primarily reflect scientific experience. The justification must be explicitly stated in any report cf. Sections 4.18 and 4.19. If the circumstances demand that a mle be set down, it is best to use a robust model such as Huber s its sensitivity for the problem at hand, and the typical rate for false positives, should be investigated by, for example, a Monte... 

From the previous analysis, we conclude that a robust plantwide control structure will fix the combined isobutane + recycle (Fj) and fresh butene flow (F0), as illustrated in Figure 9.4. The desired production rate and selectivity could be achieved in a 3-m3 reactor, operated at 268 K. The operating point shows low sensitivity to errors in the manipulated variable Fj (Figure 9.5). This design seems to ensure feasible operation even if the temperature decreases to 260 K (Figure 9.6) or the catalyst activity becomes 40% of the initial value (Figure 9.7), irrespective of the purity of the butene feed stream (Figure 9.8). 

Traditional control design approaches, consider a fixed operating point in the hope that the resulting controller is robust enough to stabilize the system for different operating conditions. On the other hand, adaptive control incorporates the time-varying dynamical properties of the model (a new value of gas composition) and considers the disturbances acting at the plant (load power variation). 

Fuzzy connectedness is a very robust segmentation technique that effectively handles inhomogeneity (coming from both tissue properties and the imaging procedure). It is also robust against the selection of seed points (if boundaries are avoided). Since the only user input (apart from the VOI selection) is the seed specification, and all other parameters are fixed, our segmentation inherits the reproducibility and efficiency of the fuzzy connectedness method. 

The usual practice was to erect a few switch fuses, distribution boards, socket outlets and lighting points on an ad hoc basis, connect them with unprotected PVC-insulated conductors draped over the structure and operate the installation at mains voltage. Non-weatherproof apparatus, of the type used for indoor fixed wiring installations, was employed - usually with indifferent quality to save cost and often with insufficient robustness for site use. 

---