Sentence examples for multiplicity of lambda from inspiring English sources

If the multiplicity of (lambda omega_{0})) is 1, then we see that there exists an (Msubsetmathbf{W}), where M is a neighborhood of (omega_{0}), such that the multiplicity of (lambda =lambda omega)) is 1 for each (omegain M).

(a) If the multiplicity of (lambda (omega_{0}) ) is 1, then, by Lemma 4, there exists a neighborhood N of (omega_{0} ) such that the multiplicity of (lambda (omega) ) is 1 for any (omega in N ).

(b) If the multiplicity of (lambda (omega) ) is l ((l=2, ldots,2n )) for all ω in some neighborhood N of (omega_{0} ) in Ω.

Using Lemma 3.2, we see that there exists a neighborhood M of (omega_{0}) such that the multiplicity of (lambda (omega)) is 1 for all (omegainmathbf{M}).

Let (lambda =lambda ( omega) ) be an eigenvalue of the operator L. If the multiplicity of (lambda (omega_{0}) ) is 1, then there exists a neighborhood N of (omega_{0} ) belonging to Ω such that the multiplicity of (lambda (omega) ) is 1 for every ω in N. If (lambda (omega_{0}) ) is simple, then (Delta '(lambda (omega _{0}))neq 0 ).

If ( lambda _0 ) is a scattering resonance then, in the notation of (2.16) with ( A( lambda ) = R_0 ( lambda ) ), the multiplicity of ( lambda _0 ) is defined as begin{aligned} m ( lambda _0 ) = dim, {mathrm{span}}, left{ A_1 left( L^2_{mathrm{comp},} right), ldots, A_J left( L^2_{mathrm{comp},} right) right}.

If the multiplicity of eigenvalue (lambda (omega) ) is l ((l=1,2, ldots,2n )) for all (omega in N ), and (Nin Omega ) is a neighborhood of (omega_{0}).

The function (D z)={mathcal D}(lambda (z))) is analytic in ({mathbb D}) whose zeros are given by begin{aligned} z_j=z(lambda _j), quad j=1,2,dots, N, end{aligned}where (lambda _j) are zeros (counting with multiplicity) of ({mathcal D}(lambda )) in (Lambda ={mathbb C}{setminus }[-d,d]), i.e., eigenvalues of H (counted with algebraic multiplicity).

end{aligned}Then begin{aligned} F s)sin (pi s -G s cos (pi s -G s cos }{2s}sum _{j=1}^Nr_j|lambda _j|^s, end{aligned} (3.1)where (r_j) is the multipiicity of the eigenvalue (lambda _j).

end{aligned} (1.6 If (zeta =0) is a resonance of multiplicity one, we also assume that begin{aligned} int _{e_j}(1+x^2)|v_j x)|,dxmultiplicity of the eigenvalue (lambda _j), we have begin{aligned} sum _{j=1}^Nr_j|lambda _j|^{1/2}-pi ^{-1}int _0^infty log a k),dk=frac{L_1}{4}equiv -frac{1}{4}sum _{j=1}^nint _0^infty v_j x),dx.

If there is an eigenvalue of constant multiplicity near (lambda (0)) for s small, it follows from (2.1) that P s) and (lambda (s)) are (C^2).