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A solution of a differentiated equation that has solutions only for particular values of some parameters. Compare: eigenvalue.
[Ger. eigen, particular, peculiar to]
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Wakayama, Correspondence between eigenfunctions of non-commutative harmonic oscillators and holomorphic solutions of Heun's differential equations.
Denote by [[lambda].sub.1] > 0 and [[phi].sub.1] (x) the first eigenvalue and the corresponding eigenfunction of the following eigenvalue problem
Convergence of reflection coefficients [S.sup.TE(TE).sub.11], defined by error coefficient [[delta].sub.S] versus eigenfunctions numbers P and L for structure 1 from Fig.
If I is a partition, then each part of I is a singleton Lyndon factor, which is why the corresponding eigenfunctions have much simpler expressions.
The sought-for eigenfunction of the perturbed operator can be decomposed in terms of the unperturbed operator as
Corresponding to the n-th eigenvalue [[lambda].sub.n], the eigenfunction [u.sub.n] admits at most n weak sign graphs; in particular, for n = 2, the eigenfunction u2 divides F into precisely two weak sign graphs (see [section] 9.3).
In terms of the Sturm-Liouville theory, the eigenfunctions [C.sub.0]([k.sub.m][rho]) are orthogonal each other and constitute a complete basis set for region (I).
Another sampling formula for f(t), which was invented and studied in [30, 31], is based on the eigenfunctions [[psi].sub.l]([OMEGA], x) = [[psi].sub.l] (x) of the Finite (Truncated)
To find the first eigenvalues and eigenfunctions of the problem (3.3) with sparse matrices A and B, one can use various numerical methods [31].
Therefore, for the permanent environmental effect, three eigenvalues and corresponding eigenfunctions needed to account for more than 97.37 percent of variation.
This can be recast in terms of fourier analysis on the hypercube: A function is noise sensitive iff most of its fourier weight is on "high energy" eigenfunctions of the random walk operator.we propose to use noise sensitivity ideas in three main directions:(a) address some outstanding questions in the classical case of iid inputs: Universality in critical planar percolation; The friedgut-kalai conjecture on fourier entropy vs influence; Noise in first passage percolation.(b) in statistical physics, A key example is the critical planar fk-ising model, With noise being glauber dynamics.
[28], 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.