For each, algebraic multiplicity of is equal to 1, and the corresponding eigenfunction has exactly simple zeros in.
Therefore, we could sample the observed intensity values for each reflection multiple times (multiplicity value = 34), and we made the assumption that the strongest intensity roughly approximated the complete intensity.
(v) For each, the algebraic multiplicity of is 1.
For each the algebraic multiplicity of the characteristic value of is equal to 1. Let under the product topology.
For (u,ω,k)= 300,100,20) we found that c=297 yields the best iterated Guruswami-Sudan strategy (for a single multiplicity) where each step consumes z GS (c,k) time (a fraction of a second on a standard workstation, see Section 4.3) with success probability p GS (u,ω,k,c ≈0.96 %.
The closed testing procedure was applied for multiplicity considerations of multiple primary endpoints.
Then the imbedding E ≲ L q (ℝ N ) is continuous if q ∈ [2, 2*] and compact if q ∈ [2, 2*[. Hence, the eigenvalue problem - Δ u + V ( x ) u = λ u, x ∈ ℝ N. possesses a sequence of positive eigenvalue 0 < λ 1 < λ 3 < ⋯ < λ k < ⋯ → ∞. with finite multiplicity for each λ k.
For each subsystem (mathsf{R}_) and multiplicity function (kappa_{alpha}) we have the Dunkl operators T_{i}f(x)=frac{partial f(x)}{partial{x_{i}}}+sum_{alphainmathsf{R}_ kappa_{alpha} frac{f(x -f( sigma_{alpha}x -flangle x, alpharangle}alpha_{i}, quad i=1, ldotsigma_{alpha}xC^{1}{langle{R}^{m})).
Statistical analyses of differences between tumor multiplicity for each mouse strain were evaluated using the Mann–Whitney U-test included in the Prism 5 software package (GraphPad Software, La Jolla, CA).
The median numbers of tumors per tumor-bearing rat (multiplicity) for each group were compared using the nonparametric Mann–Whitney test [ 27].
