Eigenvalue positive dominant

Theorem 3. The time-independent eigenvalue problem with no up-scattering Equation (9), has a positive dominant eigenvalue Ao (i.e., Ao > Aa for all Aa Ao) which is simple corresponding to an eigenfunction O = ( J, ... 

If some of the reactions of (A3.4.138) are neglected in (A3.4.139). the system is called open. This generally complicates the solution of (A3.4.141). In particular, the system no longer has a well defined equilibrium. However, as long as the eigenvalues of K remain positive, the kinetics at long times will be dominated by the smallest eigenvalue. This corresponds to a stationary state solution. 

For a given set of parameters (/ , n, k, and y) we may ask which mode has the largest positive eigenvalue and postulate that this will be dominant. Treating n as a continuous variable for the moment, we can maximize Agiving... 

Boundaries between regions of dominance by different modes correspond to parameter values at which the positive eigenvalues for two adjacent wave numbers n and (n + 1) become equal (and larger than those for all the other modes). 

Here Dt is a positive proportionality constant ( diffusion constant for Et), Jfz is z-ward flow induced by the gradient, and superscript e denotes eigenmodt character of the associated force or flow. The proportionality (13.25) corresponds to Fick s first law of diffusion when Et is dominated by mass transport or to Fourier s heat theorem when Et is dominated by heat transport, but it applies here more deeply to the metric eigenvalues that control all transport phenomena. In the near-equilibrium limit (13.25), the local entropy production rate (13.24) is evaluated as... 

The existence of complex eigenvalues of the value matrix IV implies that the coefficients in Eqn. (11.15) are complex and rules out the existence of a real-valued potential function. Transient oscillations in the concentrations may occur, but in the limit of long times the system nevertheless converges toward the dominant eigenvector. The corresponding largest eigenvalue is real and positive, and hence all oscillations in concentrations have to fade out inevitably. 

The eigenvalue with the least positive real part, which dominates near the corner (r < 1), obviously occurs for n = 1. Numerical values of 4i and /q for 2a < 146° were tabulated... 

Theorem 2 shows that /x2, and hence is a continuous function of a for a > 0. Let O be the quasi-positive eigenfunction corresponding to the dominant eigenvalue of Equation (14) and be the positive adjoint... 

As a approaches zero in Equation (14), /xj approaches the dominant (positive) eigenvalue of the problem for U = 0. Thus is bounded from below /x2 > 8 > 0. Thus for a sufficiently small, /x > a. 

Ao is the eigenvalue whose existence is established by means of the preceding lemmas. Its simplicity can be established by applying Theorem 1 to Equation (14) with a = Ao. Moreover, by construction, the corresponding eigenfunction is quasi-positive and the adjoint eigenfunction is positive. The dominance of Ao can be shown by use of the same technique as applied in proving Theorem 1. 

In eqs. (18) and (19), the expressions for /i(0,(p) and (0,(p) are the same as those given by eqs. (11) and (12), respectively. The superscripts in (17)-(19) represent the order of the contribution in perturbation to the eigenvalues of the spin-Hamiltonian. Further, it is noted that for the central sextet (M= 1/2), AS (M,/w), is zero, while AB M,m) is positive regardless of the absolute sign of the hyperfine-interaction constant (A). This means that the forbidden line position, 5+(M,/w), is at a higher magnetic field value than the forbidden line position, BJ M,m), unless the value of D is sufficiently large to render the dominant contribution. 

The dynamic behaviour of the system tends to be dominated by the motion associated with either the positive eigenvalues or the smallest negative ones, since those with large negative eigenvalues tend to relax to their local equilibria very quickly and therefore do not influence the slower modes. 

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