Stability definitions

Many other definitions for stability are known and they do not always look alike. Each of them characterizes a required property of the solution under study. Though the Lyapunov definition of stability seems to be the most natural, in many cases it cannot be used. No stability definition can fit every real case. Some other versions of this concept are given in ref. 20 in which further references can be found. 

The Consequences of Different Stability Definitions How Stable Are Ethyl and Fluoromethyl Radicals ... 

Once life-threatening injuries are stabilized, definitive treatment is initiated. This treatment can include... 

Another possibility is to replace the stability definition by its robust version ... 

Process Validation (Performance) qnalification Characterization Robnstness Consistency Variability Reprodncibility Optimization Stability Definition Understanding Capability... 

Following the stability definition, Xs and Ts describe a stable steady state if the deviations AX and AT approach zero with increasing time. But this can only be the case if the real terms of pi and p2 are less than zero. Based on this logic it follows for the stability condition ... 

The formal analysis of the closed-loop dynamics can be done with an extension of the nonlocal closed-loop stability analysis of a continuous reactor with temperature cascade controller presented before [22] in conjunction with the stability definitions (Eq. 7) given in section 2. Here it suffices to mention that, the closed-loop motion is stable if the filter and estimator gains are chosen not faster than the characteristic frequency Oj of the jacket hydraulic dynamics [26], and the secondary and primary control gains are chosen so that there is adequate dynamic separation between coj, coj and 0). This is,(Eq. 30),... 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

The traditional view of emulsion stability (1,2) was concerned with systems of two isotropic, Newtonian Hquids of which one is dispersed in the other in the form of spherical droplets. The stabilization of such a system was achieved by adsorbed amphiphiles, which modify interfacial properties and to some extent the colloidal forces across a thin Hquid film, after the hydrodynamic conditions of the latter had been taken into consideration. However, a large number of emulsions, in fact, contain more than two phases. The importance of the third phase was recognized early (3) and the lUPAC definition of an emulsion included a third phase (4). With this relation in mind, this article deals with two-phase emulsions as an introduction. These systems are useful in discussing the details of formation and destabilization, because of their relative simplicity. The subsequent treatment focuses on three-phase emulsions, outlining three special cases. The presence of the third phase is shown in order to monitor the properties of the emulsion in a significant manner. 

A modified definition of resonance energy has been introduced by Dewar (66T(S8)75, 69JA6321) in which the reference point is the corresponding open-chain polyene. In principle this overcomes the difficulties inherent in comparing observed stability with that of an idealized molecule with pure single and double bonds, as thermochemical data for the reference acyclic polyenes are capable of direct experimental determination. In practice, as the required data were not available, recourse was made to theoretical calculations using a semiempirical SCF-MO method. The pertinent Dewar Resonance Energies are listed in Table 30. 

NMR data for 4-methyloxazole have been compared with those of 4-methylthiazole the data clearly show that the ring protons in each are shielded. In a comprehensive study of a range of oxazoles. Brown and Ghosh also reported NMR data but based a discussion of resonance stabilization on pK and UV spectral data (69JCS(B)270). The weak basicity of oxazole (pX a 0.8) relative to 1-methylimidazole (pK 7.44) and thiazole (pK 2.44) demonstrates that delocalization of the oxygen lone pair, which would have a base-strengthening effect on the nitrogen atom, is not extensive. It must be concluded that not only the experimental measurement but also the very definition of aromaticity in the azole series is as yet poorly quantified. Nevertheless, its importance in the interpretation of reactivity is enormous. 

Rejection Rejection is defined in Background and Definitions. The highest-rejection membranes are those designed for single-pass production of potable water from the sea. The generally accepted criterion is 99.4 percent rejection of NaCl. Some membranes, notably cellulose triacetate fibers are rated even higher. A whole range of membranes is available as rejection requirements ease, and membranes with excellent chlorine resistance and hydrolytic stability can be made with salt rejection over 90 percent. 

Full eomplexation with Pt(II) is reaehed at the presenee of an aseorbie aeid (optimum 4 mg/em ) during 10-15 min at 65 5°C. At these eonditions eomplexes of Pd lose stability and within a day there is their full destmetion aeeompanying with deeolouration of solutions. That makes possible definition of a Pd and Pt at joint presenee, ineluding at H SO media. 

Good heat transfer on the outside of the reactor tube is essential but not sufficient because the heat transfer is limited at low flow rates at the inside film coefficient in the reacting stream. The same holds between catalyst particles and the streaming fluid, as in the case between the fluid and inside tube wall. This is why these reactors frequently exhibit ignition-extinction phenomena and non-reproducibility of results. Laboratory research workers untrained in the field of reactor thermal stability usually observe that the rate is not a continuous function of the temperature, as the Arrhenius relationship predicts, but that a definite minimum temperature is required to start the reaction. This is not a property of the reaction but a characteristic of the given system consisting of a reaction and a particular reactor. 

Aris (1969) pointed out that the mathematical definition of the CSTR stability problem and the catalyst particle problem cooled by the feed flow were essentially identical. 

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