Persistent eigenvalues

Even in the approximate treatment using the bivariational principle and a truncated finite basis, it is usually easily checked that the eigenfunctions f a corresppnding to the persistent eigenvalues belong to the domain D(u 1) of... 

In a similar way, one can plot the eigenvalues of the matrix H = ri 2T + ri 1 for any choice of basis 0 as functions of the parameter t] = pelx. Since the eigenvalues / are in general going to be complex, it is evidently necessary to carefully distinguish between the approximations to real persistent eigenvalues, to continuous eigenvalues in the complex plane, and to true physical resonances. 

Thus a wavepacket initiated in well A passes to well B by a curve crossing. Prof. Fleming showed an interesting case of persistent coherence in such a situation, despite the erratic pattern of the eigenvalue separations. An alternative, possibly more revealing approach, is to employ Stuckelberg-Landau-Zener theory, which relates the interference (i.e., coherence) in the two different wells via the area shown in Fig. 2. A variety of applications to time-independent problems may be found in the literature [1]. 

It is clear that if F bejongs to the domain D(U), then the function F = t/ P also belongs to L2 and E = E is a persistent discrete eigenvalue. On the other hand, if P belongs to the complement C(U), the function F = U V is outside L2 and it cannot be an eigenfunction to H hence E becomes a lost eigenvalue, one that does not necessarily belong to the spectrum of H. 

It is evident that the spectra of H and W are essentially complex conjugate. For a persistent real eigenvalue E = E, one has... 

It is hence evident that this expectation value is going to be complex, even if is a persistent real eigenvalue. Let us now plot the function... 

Even if the conditions for persistent, lost, and new eigenvalues are completely clear for the exact eigenvalue problems to the operators T and Tt, it is considerably more difficult to translate them to the approximate eigenvalue problems associated with the application of the bi-variational principle for the operators T and Tt to truncated basis sets. In this connection, the relations (A. 1.40-1.49) may turn out to be useful in formulating the problem. Some of the computational aspects, particularly the choice of the dual basis sets, are further discussed in reference A. 

Proof. If El and E2 are unstable, then dissipativeness and uniform persistence (previous proposition) yield the existence of an interior rest point for -k(x, t) (Theorem D.3). If E exists then it is unique and has all eigenvalues negative (Lemma 5.2). Suppose that exists and that Ei is asymptotically stable. Then, since asymptotically stable, Theorem E.l contradicts the uniqueness of E. A similar argument applies if E2 is asymptotically stable. Note that the computations leading up to Lemma 5.1 explicitly determine the signs of the eigenvalues for linearization about Ei and E2. ... 

A Matrices and Their Eigenvalues B Differential Inequalities C Monotone Systems D Persistence... 

By far, singular value decomposition (SVD) is the most popular algorithm to estimate the rank of the data matrix D. As a drawback of SVD, the threshold that separates significant contributions from noise is difficult to settle. Other eigenvalue-based and error functions can be utilized in a similar way, but the arbitrariness in the selection of the significant factors still persists. For this reason, additional assays may be required, especially in the case of complex data sets. 

Once the limiting distribution is reached, it will persist this result is called the steady-state condition. If there exists one eigenvalue of unity, there exists only one limiting distribution, which is independent of the initial distribution. It can be shown that a unique limiting distribution exists if any state of the system can be reached from any other state with a nonzero multistep transition probability. In this case the system (or Markov chain) is ergodic. 

The nonnegative heteroclinic orbit will persist as v decreases, as long as no bifurcation occurs in the vector field of (5.64a) and (5.64c) and (1/2, 1 /2) remains a saddle poinf and (0, 0) a sfable node, whose eigenvectors lie strictly within the positive quadrant. For w < y, the eigenvalues for the fixed point (0, 0) are... 

Consider a family of systems of differential equations which is C -smooth (r > 3) with respect to the variables x R, y G R (m > 0) and parameters e MP p > 1). Let the system have, at = 0, an equilibrium state O with a pair of purely imaginary characteristic exponents the rest are assumed to lie to the left of the imaginary axis. Since the equilibrium state has no zero eigenvalues, it persists in a small neighborhood of s = 0. Without loss of generality we may suppose that it resides at the origin for all small e. Let us assume that this is a pair of characteristic exponents closest to the imaginary axis... 

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