Critical value analysis

Consequently, when D /Dj exceeds the critical value, close to the bifurcation one expects to see the appearance of chemical patterns with characteristic lengtli i= In / k. Beyond the bifurcation point a band of wave numbers is unstable and the nature of the pattern selected (spots, stripes, etc.) depends on the nonlinearity and requires a more detailed analysis. Chemical Turing patterns were observed in the chlorite-iodide-malonic acid (CIMA) system in a gel reactor [M, 59 and 60]. Figure C3.6.12(a) shows an experimental CIMA Turing spot pattern [59]. 

Samples were tested on in a melt of salts (75% Na SO, 25% NaCl) at 950°C in an air atmosphere for 24 hours. Micro X-rays spectrum by the analysis found that the chemical composition of carbides of an alloy of the ZMI-3C and test alloys differs noticeably. In the monocarbide of phase composition of an alloy of the ZMI-3C there increased concentration of titanium and tungsten is observed in comparison with test alloys containing chemical composition tantalum. The concentration of more than 2% of tantalum in test alloys has allowed mostly to deduce tungsten from a mono carbide phase (MC) into solid solution. Thus resistance of test alloys LCD has been increased essentially, as carbide phase is mostly sensitive aggressive environments influence. The critical value of total molybdenum and tungsten concentration in MC should not exceed 15%. 

To further clarify the role of magnetic effects on compressibility, a shock compression experiment was performed on an fee 28.5-at. % Ni sample whose initial temperature was raised to 130°C. As is shown in Table 5.1, the compressibility was found to decrease to a value consistent with the nonmagnetic compressibility. Thus, the sharp change in compressibility, the critical values for the transition, and the magnitudes of the compressibility under the various conditions give a clear demonstration that a second-order magnetic transition has been observed, and we will proceed with a quantitative analysis of the transition. 

At the critical value a = oi = 1, however, becomes unstable and the a-dependent fixed point becomes stable. This exchange of stability between two fixed points of a map is known as a transcritical bifurcation. By using the same linear-stability analysis as above, we see that remains stable if — 1 < a(l — Xjjj) < 1, or for all a such that 1 < a < 3. Something more interesting happens at a — 3. 

From the previous analysis it follows that the harmonic elastic capacitor collapses when approaches its critical value... 

Analysis of the dynamics of SCR catalysts is also very important. It has been shown that surface heterogeneity must be considered to describe transient kinetics of NH3 adsorption-desorption and that the rate of NO conversion does not depend on the ammonia surface coverage above a critical value [79], There is probably a reservoir of adsorbed species which may migrate during the catalytic reaction to the active vanadium sites. It was also noted in these studies that ammonia desorption is a much slower process than ammonia adsorption, the rate of the latter being comparable to that of the surface reaction. In the S02 oxidation on the same catalysts, it was also noted in transient experiments [80] that the build up/depletion of sulphates at the catalyst surface is rate controlling in S02 oxidation. 

The reliability of multispecies analysis has to be validated according to the usual criteria selectivity, accuracy (trueness) and precision, confidence and prediction intervals and, calculated from these, multivariate critical values and limits of detection. In multivariate calibration collinearities of variables caused by correlated concentrations in calibration samples should be avoided. Therefore, the composition of the calibration mixtures should not be varied randomly but by principles of experimental design (Deming and Morgan [1993] Morgan [1991]). 

As follows from the previous analysis for quasi and ordinary particles gases there exists a critical value of parameters a and b for which the least value of the distribution function for observable frequencies is observed. From the physical point of view this is in agreement with the absolute minimal realization of the most probable state. As in any equilibrium distribution, there is an unique most probable state which the system tends to achieve. In consequence we conclude that the observable temperature of the relic radiation corresponds to this state. Or, what is the same, the temperature of such radiation correspond to the temperature originated in the primary microwave cosmic background and the primitive quantum magnetic flow. 

Abstract. Classical regular and chaotic dynamics of the particle bound in the Coulomb plus linear potential under the influence of time-periodical perturbations is treated using resonace analysis. Critical value of the external field at which chaotization will occur is evaluated analytically based on the Chirikov criterion of stochasticity. 

In this paper we consider the QCD counterpart of this problem. Namely, we address the problem of regular and chaotic motion in periodically driven quarkonium. Using resonance analysis based on the Chirikov criterion of stochasticity we estimate critical values of the external field strength at which quarkonium motion enters into chaotic regime. 

Having established that the standard deviations of two sets of data agree at a reasonable confidence level it is possible to proceed to a comparison of the mean results derived from the two sets, using the t-test in one of its forms. As in the previous case, the factor is calculated from the experimental set of results and compared with the table of critical values (Table 2.3). If /jX ) exceeds the critical value for the appropriate number of degrees of freedom, the difference between the means is said to be significant. When there is an accepted value for the result based on extensive previous analysis t is computed from equation (2.9)... 

A stability analysis made by Ryan and Johnson (1959) suggests that the transition from laminar to turbulent flow for inelastic non-Newtonian fluids occurs at a critical value of the generalized Reynolds number that depends on the value of The results of this analysis are shown in Figure 3.7. This relationship has been tested for shear thinning and for Bingham... 

In Chapter 3, the conditions for a chain branching explosion were developed on the basis of a steady-state analysis. It was shown that when the chain branching factor a at a given temperature and pressure was greater than some critical value acrit, the reacting system exploded. Obviously, in that development no induction period or critical chain ignition time rc evolved. 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

Fig. 32 a-d. Phase transition a from the extended coil b to a globule state as found by scaling analysis [83] c the transition is caused by lowering the surface pressure below a certain critical value TTc at which the fraction of adsorbed monomers 3=N2d/N undergo discrete changes d hereby, the depends critically on the side chain length... 

This crude analysis is based on the behavior postulated by the Born equation. However, ion-pair formation equilibrium constants have been observed to deviate ma edly from that behavior (22/ -222)1 Oakenful, and Fenwick (222) found a maximum in the ion-pair formation constants of several alkylamines with carboxylic acids when determined at various methanol-water solvent compositions as shown by their data in Fig. 54. The results demonstrate that in this system the stability constant decreases with increasing organic solvent concentration above a.critical value which yields maximum stability. The authors suggested that this was due to a weakening of hydrophobic interactions between the ion-pair forming species by increased alcohol concentrations. In practice the effect of added organic solvent has been either to decrease the retention factor or to have virtually no effect. 

Assuming normally distributed sampling and analysis errors (and no bias), the NIOSH accuracy standard is met if the true coefficient of variation of the total error, denoted by CVp, is no greater than 0.128. However, estimates of CVp (denoted by CVp), which were obtained in the laboratory validations, are themselves subject to appreciable random errors of estimation. Therefore, a "critical value" for the CVp was needed (i.e. the value not to be exceeded by an experimental CVp if the method is to be judged acceptable). 

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