To find eigenvalues and eigenfunctions
, we can use various numerical methods .
Eigenvalues illustrate the quantity of variation described by the corresponding eigenfunction
(Kirkpatrick et al., 1990).
, where adaptively chosen eigenfunctions
of certain local eigenvalue problems extended by zero to the rest of the domain, are added to the standard average Schwarz coarse space.
Since Theorem 4 is only true in the case of sufficiently small and regular potentials of compact support, this shows that even in the case of randomised initial data the observability constants (eigenfunction
observability constants) can be very close for strong conditions relating V and u>.
By Lemma 7, we have that the spectral radius [mathematical expression not reproducible] has a positive eigenfunction
After finding the eigenvalue of matrix A, [[lambda].sup.(c)], the corresponding eigenvector d can be used to construct the conditional eigenfunction
of the conditional covariance
Suppose that u is a nonnegative eigenfunction
corresponding to the eigenvalue [lambda].
The method described herein is more efficient than equivalent eigenfunction
expansions and can also provide field solutions with greater resolution and accuracy.
The total Hilbert space of the eigenfunctions
is split into two subspaces and the symmetry of the motion of the electron around the magnetic flux makes Pauli's criterion inapplicable [9-12]: that means the condition of admissibility of the wave function in the region of AB potential.
For example, the eigenfunctions
of a non-Hermitian operator H are biorthogonal with nontrivial consequences for their normalization.
 discussed the excess pore-air and pore-water pressure dissipations in the plane-strain consolidation of unsaturated soft soils using eigenfunction
expansion and Laplace transform techniques.
(1) "though the form in which the degeneration of the free electron is now treated in the paper would seem to be unambiguous" (2) "The difficulty was the eigenfunction
in the presence of a magnetic field, a point which I think is not completely cleared up yet" (3) "Therefore the eigenfunctions
you have used in your paper are not so simply as we thought connected with the "spin" " (4) "but this has no influence on your results, only the sentences where spin comes in could not have quite the definite form you gave them" The second letter,(11,23) dated 2nd Dec.