# New PDF release: A biplot method for multivariate normal populations with

By Calvo M., Villarroya A., Oller J.M.

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Extra info for A biplot method for multivariate normal populations with unequal covariance matrices

Example text

1 we have uα Lr¯ (Bα ) ≥ C −1 µ2α log 1 µα 2/3 . Inserting into (64), we obtain αµ4α log 1 µα 4/3 ≤ C µ4α log 1 µα 4/3 + αµ4α log 1 . µα Once again we obtain α ≤ C, a contradiction. 1 is thus established in the remaining limit case n = 6. 2. We adapt some ideas from [6]. Let (M, g) be a smooth compact Riemannian manifold without boundary, n ≥ 3. 1. Suppose that there exist ε¯ > 0 and Aε¯ > 0 such that (65) u 2 L2∗ (M,g) ≤ K2 {|∇g u|2 + c(n)Rg u2 } dvg + Aε¯ u 2 L1 (M,g) , M for all u ∈ H 1 (M ) such that diamg (supp) u < ε¯.

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