Symmetric behavior

Taking into account the symmetrical behavior of both the reduced volume fraction axes, a rheological equation expressing the mid-point of the phase inversion region may be written ... 

The asymmetry in the effects of variations of X/Cp and pD 2 large jS arises through the influence on the reactant concentration in the reaction zone. A symmetric behavior could be anticipated if the reaction extended throughout the entire thickness of the deflagration. 

In both of the examples above, we have assumed that pA B is independent of composition and that the dependence of pA on the temperature is linear. Clearly if the temperature dependence is nonlinear, but still monotonically increasing or decreasing functions of T, we shall obtain UCST and LCST, respectively. If, on the other hand, pAAB is also a function of xA, then we lose the symmetrical behavior of the T(x) diagram, but still the general phenomenon is the same. 

From Table C.6, we see that what we eall irreducible representations represent distinct rhythms of pluses and minuses, which after making the square, give the fully symmetric behavior. 

The process of mixture and reaction is controlled by large vortices. The inflow is symmetric and, after the entrance, the boundary effects break down the symmetric behavior. The evolution of the mixing layer and the breakdown of the bigger vortices is verified. 

A simple combinational logic circuit with two NOR gates is shown in Fig. 1.56(a). This circuit with a feedback path providing a closed loop as shown in Fig. 1.56(b) has two stable states and, hence, can be used to store one binary digit of information. Since the gates function symmetrically it is convenient to draw the circuit in symmetric form as shown in Fig. 1.56(c), to convey the perception of symmetric behavior. 

Figure 1 The canonical cusp catastrophe function, Az = -f + 6x, at different values of the parameter a. The left panel illustrates the symmetric behavior for h = 0 the right panel for 6 = 1 illustrates the asymmetric behavior which occurs whenever b is different from zero. In the horizontal plane (a, x) we have drawn the locus of maxima as dotted lines, of minima as dashed lines and of inflections as dashed-dotted lines. The triply degenerate (or catastrophe) point occurs at a = 0 and 6 = 0 and bifurcation between single and double well modes occurs there for symmetric systems. For asymmetric systems, the bifurcation occurs on the fold line (at a = —1.9 for 6 = 1) where the cusp function has a doubly degenerate horizontal inflection point at x = 0.8.

Fig. 1. Angular dependence (a) of the resonance field and (b) of the linewidth Af/ in La,923Sro(,5Cu04 at 70K by rotating the c-axis with respect to the magnetic field. A purely axial symmetric behavior is indicated by the solid lines. The inset shows the absorption derivative indicating a Lorentzian lineshape of the EPR spectrum for the crystal orientation c H. From Kochdaev et al. (1997).

This symmetrical behavior is similar to that which we encountered for the special case (4.3.88) and (4.3.89) and similarly for the pair distribution function. 

Of the five parts of trans d can always be considered symmetric as far as the Pauli principle is concerned (no matter which type of particle—a fermion or a boson—the nucleus is). The electronic wavefunction eiect almost always symmetric. For homonuclear diatomic molecules, there is usually a + superscript on the term symbol of the ground electronic state that implies symmetric behavior however, some diatomic molecules—O2 is the noteworthy one—have a superscript minus (—) in their term symbol, indicating that the groimd electronic state is actually antisymmetric Ignoring these rare exceptions (but see the end-of-chapter exercises), ultimately the and the partition functions combine to determine the overall symmetry of Q for the molecule. 

Titanium discharges very quickly, relying almost completely on double-layer charge injection. Unlike platinum or iridium, titanium does not exhibit symmetrical behavior for cathodic and anodic pulses. As a result, the anodic charge injection is almost completely consumed in the formation of oxide. Repeating the experiment produces nearly identical results, despite the fact that the electrode potential has increased. 

The velocity profiles and the axial distribution of volume fraction of solid particles are shown in Fig 4.12. The solid velocity pattern did show a non-axial symmetric behavior. The axial distribution of the solid phase was close to uniform. Figure 4.13 displays the outlet fractions of hydrogen, methane, and CO2 in the gas phase as achieved for the SMR and SE-SMR processes operated in a BFB reactor. In the SMR results, the outlet molar fraction of H2 is only 75 %, thus a considerable amount of CO2 and CH4 are emitted out of the reactor. In the SE-SMR process results, both the conversion of methane and the adsorption of CO2 are larger than 99 %. The simulation results show that the integration of CO2 sorption in the SMR process can enhance the methane conversion to hydrogen to near the equilibrium composition in the BFB reactor. 

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