∇ $^{2}$G$_{·}$$_{,k}$ =      ∂ $^{2}$G ∂ω$_{1}$∂ω$_{k}$ ... ∂ $^{2}$G ∂ω 2 k ... ∂ $^{2}$G ∂ω$_{N}$ ∂ω$_{k}$ ∂ $^{2}$G ∂µ∂ω$_{k}$     $^{=}$       − θ$_{1}$θ$_{k}$e$^{ω}$ k µ e$^{ω$_{1}$}$ µ µ { ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ } 2 ... θ$_{k}$e$^{ω}$ k µ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ − θ$_{k}$e$^{ω}$ k µ µ { ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ } 2 ... − θ$_{N}$ θ$_{k}$e$^{ω}$ k µ e$^{ω}$ N µ µ { ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ } 2 θ$_{k}$e$^{ω}$ k µ N i $_{=1}$θ$_{i}$ ( ω$_{i}$ − ω$_{k}$ ) e$^{ω}$ i µ µ 2 { ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ } 2       = θ$_{k}$e$^{ω}$ k µ µ { ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ } 2      − θ$_{1}$e$^{ω$_{1}$}$ µ ... ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ − θ$_{k}$e$^{ω}$ k µ ... − θ$_{N}$ e$^{ω}$ N µ 1 µ ∑ N i $_{=1}$θ$_{i}$ ( ω$_{i}$ − ω$_{k}$ ) e$^{ω}$ i µ      (82) 47 Thus, ∇ $^{2}$G$_{·}$$_{,k}$ in (82) shows the k th column of ∇ $^{2}$G . ∇ $^{2}$G$_{·}$$_{,}$$_{1}$ denotes the first column, ∇ $^{2}$G$_{·}$$_{,}$$_{2}$ denotes the second column, and so forth. Provided that j, k ∈ { 1 , ..., N } and j = k , ∂ $^{2}$G ∂ω 2 k = ∂ ∂ω$_{k}$ θ$_{k}$e$^{ω}$ k µ ∑ N i $_{=1}$¯ θ$_{i}$e$^{ω}$ i µ = ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ · θ$_{k}$ µ e$^{ω}$ k µ − θ$_{k}$e$^{ω}$ k µ · θ$_{k}$ µ e$^{ω}$ k µ [ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ] 2 = θ$_{k}$e$^{ω}$ k µ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ − θ$_{k}$e$^{ω}$ k µ µ [ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ] 2 ∂ $^{2}$G ∂ω$_{j}$∂ω$_{k}$ = ∂ ∂ω$_{j}$ θ$_{k}$e$^{ω}$ k µ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ = ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ · 0 − θ$_{k}$e$^{ω}$ k µ 1 $_{µ}$θ$_{j}$ e$^{ω}$ j µ [ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ] 2 = − 1 µ θ$_{k}$θ$_{j}$ [ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ] $_{2}$e$^{ω}$ k µ e$^{ω}$ j µ