Eigenvalue relation

This means that the v(i), when viewed as eolumn veetors, obey the eigenvalue identity M v(i)> = mi v(i)>. These same veetors, when viewed as row veetors (and thus eomplex eonjugated), also obey the eigenvalue relation, but in the "turn over" form [Pg.542]

Taking the logarithm of this operator—which is an algebraic function of the kind just mentioned—we have the corresponding eigenvalue relation ... 

Case Eigenvalue relations Signs el e2 Types of contours Geometric interpretation center of contours Figure... 

The stability matrix carries the necessary information related to the vicinity of the trajectory and provides an efficient numerical procedure for computing the response function. It plays an important role in the field of classical chaos the sign of its eigenvalues (related to the Lyapunov exponents) controls the chaotic nature of the system. Interference effects in classical response functions have a different origin than their quantum counterparts. For each initial phase-space point we need to launch two trajectories with very close initial conditions. [For 5(n) we need n trajectories.] The nonlinear response is obtained by adding the contributions of these trajectories and letting them interfere. 

To define K, M, and X, see inequality Eq. (42), we need to define the following procedure. First the plot, Figure 2.8 below, displays the dashed test circle C with the center etest marked with a cross. In addition we have so-called probe circles Cp(e, Me)) defined by the eigenvalue relation... 

Taking the complex conjugate of the eigenvalue relation H V = E V, one obtains H V = P, which means that, without loss of generality, the eigenfunctions F may be assumed to be real. If one further assumes that the operator U is restricted to fulfill the condition Eq. (1.57) or... 

In the eigenvalue relation TC = AC, where Tf = T, the eigenfunction C is said to belong to the continuum if C does not belong to L2 but C is still finite at infinity. An alternative formulation is that C is a derivative of a function H(A), which itself belongs to L2 ... 

Non-trivial solutions of Eqns. (2.3.13) - (2.3.16) subject to homogenous boundary conditions (2.3.17) and (2.3.18) will exist only for particular combinations of the parameters a, f3, w, and Re. This then produces the dispersion or eigenvalue relation as... 

It should be observed that, even if the superoperator M is not self-adjoint, the pair of adjoint superoperators—M and ft—may have rather interesting properties.5 Let us assume that M and ft have eigenvalue relations of the type ... 

The determination of the equation for the free surface shape and the eigenvalue relation between the amplification factor (3, the wave number k, and the integer n is carried out just as in the evaluation of the dispersion relation for plane waves. In particular, from the kinematic boundary condition Eq. 

The excess pressure is then found from the dynamic boundary condition Eq. (10.4.26). Using the unsteady Bernoulli equation + d(f)ldt - 0, we can eliminate the unknown constant A to give the eigenvalue relation... 

These equations are typical eigenvalue relations the constant ImE appearing on the right is an eigenvalue of the momentum operator. 

The last equality sign follows from the eigenvalue relation Eq. (6.102). Since... 

This equation is to be solved simultaneously with the eigenvalue relation (7.254). The joint solution yields the criticality requirements for the slab. It is convenient to write this pair of equations in the form... 

This condition is to be used in conjunction with the eigenvalue relation (7.254). As in the case of the set (7.262), (7.270) together with (7.254) yield either the size or the fuel concentration, given the other. 

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