The pair (u, E) (the eigenfunction
u [member of] [V.sub.h] and the eigenvalue E) is determined from the following conditions
The first eigenfunction
suggests that milk yield is mainly controlled by genes with dissimilar effects between early and late stages of lactation.
We say that [lambda] [member of] R is an eigenvalue of the following eigenvalue problem if there exists a non-zero eigenfunction
[[phi].sub.[lambda]] [member of] V such that
This shows that the off-diagonal terms in the eigenfunction
expansion contribute less in the infinite time asymptotic regime.
Moreover, the spectral radius r([T.sub.[lambda]]) > 0 and [T.sub.[lambda]] has a positive eigenfunction
[[phi].sub.1] corresponding to its first eigenvalue [(r([T.sub.[lambda]])).sup.-1]; that is, [T.sub.[lambda]][[phi].sub.1] = r([T.sub.[lambda]])[[phi].sub.1].
where x, y is the spatial coordinate in a topological space D, the lag distance s = [absolute value of (x - y)], [[lambda].sub.n] is the eigenvalue, and [f.sub.n](x) is eigenfunction
Assume there is a nonnegative eigenfunction
corresponding to an eigenvalue [lambda] of (7).
The theoretical investigation of electromagnetic field behavior within a cylindrical inhomogeneous plasma structure is usually carried out through eigenfunction
expansions [11, 12], which consists of expanding the electromagnetic field in Bessel functions, or other eigenfunctions
appropriate to the problem's geometry, and then finding the unknown expansion coefficients by application of boundary conditions within the plasma and at the plasma container's boundaries.
The KAM is based on these commutation relations that Pauli required that the appropriate eigenfunction
be those which are square integrable and are closed under the operation of the ladder operators.
First we show a few eigenvalue trajectories together with the corresponding eigenfunction
trajectories in Section 3.1.
, on the other hand, introduced exact solutions using the eigenfunction
expansion and Laplace transform techniques.
Eigenvalues represent the amount of variation explained by the corresponding eigenfunction