Near zero eigenvalues

Thus, with a nearly zero eigenvalue of the covariance matrix of the independent variables the estimates tend to be inflated and the results are meaningless. Therefore, in nearly singular estimation problems reducing the mean square... 

In the immediate vicinity of a saddle point there is no problem. We proceed as for minimizations either by solving a set on Unear equations to determine the Newton step Eq. (5.9), or by solving a set of eigenvalue equations to determine the near-zero eigenvalue solution Eq. (5.11). The difficulties are in the global part of the optimization. 

The short expression of these ideal structure criteria is the minimization of medium factor loadings to the benefit of near zero and high factor loadings. After rotation the descending order of the eigenvalues of factors may be revoked. 

In ID systems in vacuo the rigid rotation of the whole chain about its axis is not hindered. It follows that an additional zero eigenvalue is expected from Eq. (3.43) when applied to ID systems at k = 0. The lack of intermolecular forces for the polymer chain in vacuo carries an important consequence on the shape of the two acoustic branches near k = 0. While the three acoustic branches for 3D crystals have a discontinuity at k = 0, for 1D crystals two of the acoustical branches may approach v(0) almost asymptotically [56]. It follows that the dynamics of ID systems based on the model of an isolated chain at small k values and at very low frequencies is totally wrong and cannot be used for the interpretation of related physical properties of real systems (neutron-scattering experiments, thermodynamic properties, Debye Waller factors, mean square amplitudes, and other physical properties where thermal population is involved which is strongly affected by low-energy vibrations). 

Criterion for Adsorption—A related issue is an experimentally relevant criterion for polyelectrolyte adsorption. The scaling relations presented above for critical adsorption are determined by the condition of a zero eigenvalue 2q. This is certainly inaccessible in experiments. Often, the criterion for adsorption is chosen such that the chain is near the surface most of the time, say >90%. Such a requirement will give rise to higher critical surface charge densities and to potentially different scaling relations. 

This approach is a rather general one. Its advantage is that when the rescaling procedure has been carried out, many resonant monomials disappear. The most trivial example is a saddle-node bifurcation with a single zero eigenvalue. In this case the center manifold is one-dimensional. The Taylor expansion of the system near the equilibrium state may be written in the following form... 

C.4. 53. Below we present (following [185]) a list of asymptotic normal forms which describe the trajectory behavior of a triply-degenerate equilibrium state near a stability boundary in systems with discrete symmetry. We say there is a triple instability when a dynamical system has an equilibrium state such that the associated linearized problem has a triplet of zero eigenvalues. In such a case, the analysis is reduced to a three-dimensional system on the center manifold. Assuming that (x, y, z) are the coordinates in the three-dimensional center manifold and a bifurcating equilibrium state resides at the origin, we suppose also that our system is equivariant with respect to the transformation (x,y,z) <- (-X, -y, z). 

It is possible for H(Xn,iu) to have one or more zero eigenvalues and for Xmin stiU to be a local minimiun, but there can be no negative eigenvalues. Thus, at (and very nearly) a minimum, the Hessian is at least positive-semidefinite. Away from the near vicinity of a local minimum, however, it is quite possible for the Hessian to have negative eigenvalues, so that the search direction generated by /f (xl l)pl l = —7(xl l) is not a descent direction. 

Often, X X may be ill conditioned, i.e., it has one or more eigenvalues near zero, when the experimental design does not provide sufficient information to measure well one or more linear combinations of parameters. Therefore, QR decomposition is the favored solution method,... 

The resonances are then obtained by searching for the complex zeros of the zeta functions (4.12) in the complex surface of the energy. Assuming that the action is approximately linear, S(E,J) = T(E - Ei), while the stability eigenvalues are approximately constant near the saddle energy E, the quantization condition (4.12) gives the resonances [10]... 

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