Degenerate, definition

SmA phases, and SmA and SmC phases, meet tlie line of discontinuous transitions between tire N and SmC phase. The latter transition is first order due to fluctuations of SmC order, which are continuously degenerate, being concentrated on two rings in reciprocal space ratlier tlian two points in tire case of tire N-SmA transition [18,19 and 20], Because tire NAC point corresponds to the meeting of lines of continuous and discontinuous transitions it is an example of a Lifshitz point (a precise definition of tliis critical point is provided in [18,19 and 20]). The NAC point and associated transitions between tire tliree phases are described by tire Chen-Lubensky model [97], which is able to account for tire topology of tire experimental phase diagram. In tire vicinity of tire NAC point, universal behaviour is predicted and observed experimentally [20]. 

Since aj always is positive or zero, and Ei — Eq always is positive or zero ( o is by definition the lowest energy), this completes the proof. Furthermore, in order for the equal sign to hold, all =0 since , o - q is non-zero (neglecting degenerate ground states). This in turns means that ao = ], owing to the nonnalization of, and consequently the wave function is the exact solution. 

Figure 23 shows sections of with J fixed, C2 = 0 and various values of L/N, where V = 27 is the total polyad action. There is an obvious conical point, X, at K, L) = (7,0). In addition, the concentric contours degenerate to points Y and Z at K, L) = (—7, V). It is important for what follows to explore the character of the point X. As a preliminary, the Hamiltonian may be reduced from the above seven-parameter form, to one involving two essential parameters, both of which vary with the total action. The appropriate definitions [14, 28, 29], modified to conform with the present (7, K) notation, are... 

For a degenerate energy eigenvalue, the several corresponding eigenfunctions of H may not initially have a definite parity. However, each eigenfunction may be written as the sum of an even part V e(q) and an odd part V o(q)... 

Resorting to the definition of the Fourier transform, Eq. (2.9), we notice that for the redundant coordinates the harmonic kernel degenerates and becomes exp (0) = 1. Thus for the redundant coordinates the Fourier transform turns into a simple integration with respect to the respective reciprocal coordinate20 - a projection . 

We shall assume that the electron and hole gases on the surface of semiconductor are not degenerate. Then, by definition, ... 

The framework in Figure 9.18 illustrates a relationship between a Trader who makes Orders from a Distributor. In the framework, we don t care how the Trader gets rid of stock, nor how the Distributor acquires it. We have shown this as a degenerate collaboration, as it will next be refined as a unit. (Notice that we ve made all the types substitutable except Date. So we would likely have Date defined as an actual type somewhere in the package in which this framework definition appears or in the packages it imports.)... 

Intrinsic Nucleophilicity vs. Leaving Group Ability. As pointed out earlier, our definition of intrinsic nucleophilicity in terms of barriers to degenerate exchange reactions necessarily implies that nucleophilicity and leaving group ability are equivalent. This can be seen in either of two ways. First,... 

This shows that the Hessian of / is positive definite (resp. negative definite) on (resp. N ). Therefore / is non-degenerate in the sense of Bott, i.e. the set of critical points is a disjoint union of submanifolds of X, and the Hessian is non-degenerate in the normal direction at any critical point. We put = dim N = 2 dime N which is the index of / at the critical manifold Cj,. Note that the index is always even in this case. 

BIOMINERALIZATION DEBYE-HOCKEL TREATMENT DEFINITE INTEGRAL Degenerate rearrangement,... 

For degenerate states a problem arises with the definition of cumulants. We consider here only spin degeneracy. Spatial degeneracy can be discussed on similar lines. For S 0 there are (2S + 1) different Afs-values for one S. The n-particle density matrix p Ms) = of a single one of these states does not... 

If one tries to apply the definition (39) naively to degenerate states, one is faced with the problem that, for example, and have a different transformation behavior with respect to the spin rotation group SU2, hence would have no acceptable transformation behavior at all. The way out of this dilemma is to define the irreducible tensor components of in terms of those of yPf and y. ... 

The definitions given in this section are valid both for degenerate and nondegenerate states. 

Following the concepts on page 53 and 57 we find out a third concept for experimental strategies to change order parameters in the A/S sense regarding structure and reactivity (for definition of A/S see )). This concept fits for degenerated subsystems on the level... 

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