Stationary states classification

States of the system may be divided into three groups (a) nonstationary state, VxF 0 (b) nondegenerate stationary state, VxF=0, det(F,) 0 (Vij = d2V/dxidxJ) (c) degenerate stationary state Vx F = 0, det(fy) = 0 analogously to the classification used in elementary catastrophe theory. 

The above observation may be employed for finding the remaining sensitive states of the system (5.2) on the basis of knowledge of its stability matrix a j and the classification of stationary states given in Section 5.1. The sensitive states are those corresponding to such values of the control parameters e that the eigenvalues of the stability matrix ay(e) are of the following form ... 

Classification of catastrophes associated with a loss of stability by a spatially homogeneous stationary state (wave phenomena)... 

The stationary state (x2, y2, z2) will be stable when all the roots of equation (6.106) have negative real parts. We will investigate the conditions under which this stationary state loses stability, that is under which at least one solution with a positive real part appears. Next, in the region of control parameters corresponding to instability of the state (x2, y2, z2) we shall examine possible catastrophes of codimension 2. It follows from the classification given in Section 5.5 that the bifurcations of codimension one and two of a sensitive state corresponding to the requirement = 0 are theoretically possible the Hopf bifurcation for which a sensitive state is of... 

Reaction conditions (i) and (ii) resemble the Semenov classifications of stable and unstable behaviour. For (iii), the reaction conditions are called parametrically sensitive. With absolute control of system parameters, any degree of self-heating can be produced and a complete range of maximum temperature excesses attained up to the adiabatic flame temperature. Physically such exact control is impossible and althou in the laboratory we should expect to see occasional temperature rises of the order of 100 K, repeatable non-explosive temperatures will be practically bounded by the steady-state limits. For simple systems, therefore, stationary-state treatments are still of great value. First, they impose a stability bound, inasmuch as conditions stable under stationary-state theory always remain stable under... 

When spin terms are omitted, the operators that commute with h, and thus permit a full classification of each stationary state in terms of corresponding constants of the motion, are... 

In chromatography, one phase is held immobile or stationary, and the other one is passed over it (the mobile phase). The designations GC and LC refer to the physical state of the mobile phase. Further classifications can be made by naming the mobile and stationary phases thus we have gas-solid (GS), gas-liquid (GL), liquid-liquid (LL), and liquid-solid (LS) chromatography. More recently, supercritical fluids have been used as mobile phases, and these techniques have been named supercritical fluid chromatography (SFC) irrespective of the state of the stationary phase. Other names have also become popular, and Table 1 shows a complete classification scheme. Included in the classification scheme are not only the states of the two phases but also the configuration of the chromato-... 

In order to form a bridge between the laboratory (chemical) experiments and the theoretical (mathematical) models we refer to Table I. In a traditional approach, experimental chemists are concerned with Column I of Table I. As this table implies there are various types of research areas thus research interests. Chemists interested in the characteristics of reactants and products resemble mathematicians who are interested in characteristics of variables, e.g. number theorists, real and complex variables theorists, etc. Chemists who. are interested in reaction mechanism thus in chemical kinetics may be compared to mathematicians interested in dynamics. Finally, chemists interested in findings resulting from the study of reactions are like mathematicians interested in critical solutions and their classifications. In chemical reactions, the equilibrium state which corresponds to the stable steady states is the expected result. However, it is recently that all interesting solutions both stationary and oscillatory, have been recognized as worthwhile to consider. 

Separation of the components, or solutes, of a sample results from differences in their rates of adsorption, solution, or reaction with the mobile and stationary phases. In the light of these observations distinguishing the numerous chromatographic techniques only on the basis of specification of the physical states of the stationary and mobile phases is inadequate, and a more adequate classification of these techniques must additionally also take into account (i) the nature of the separation e.g. adsorption, and (ii) the configuration of the system e.g. columnar. Table 4.4 gives a system of classification which incorporates these considerations. 

Properties of excited electronic states of polyatomic molecules may be conveniently classified as stationary or nonstationary. A comparison of the present knowledge about properties in these two classifications is interesting. Theoreticians and spectroscopists have been rather successful in developing good descriptions of energy levels and geometries of excited states, and hence the stationary type is in a comparatively advanced state of development. For example, the spectroscopic work of... 

The criterion of classification may be either the physico-chemical features of the systems and processes (the state of mobile and stationary phases, the shape of distribution isotherms, the nature of sorption which determines the distribution, etc.) or the technical aspects (the method of sample injection, the arrangement of the stationary phase, the procedure, etc.). 

However, the most frequently employed classification of chromatographic methods, which is independent of the separation mechanism, is based on the state of the mobile and stationary phases (Table 1.1.). 

Table 1.1. Classification of chromatographic methods according to the states of mobile and stationary phases...

A convenient classification of chromatographic techniques can be made in terms of the physical state of the phases employed in the separation process (Fig. 3) If the mobile phase is a gas, the separation techniques are known as gas-liquid chromatography (GLC) or gas-solid chromatography (GSC) when the stationary phase is a liquid or solid, respectively. GLC is the more popular separation mode and is often simply referred to as GC. If the mobile phase is a supercritical fluid, the separation technique is known as supercritical fluid chromatography (SFC) whether the stationary phase is an... 

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