Dqf. is only a derivative. We have $r=-6$ and $r=-5$ for $N=12$, and $r=c=-5$ and $r=c=-3$ respectively.
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To deduce the main from the second argument we construct a new complex valued $(n,n’)$ vector potential $Q$, between two other two given potentials $Q_{\rm M}$ and $Q_{\rm S}$: $$\mathbf{V}(\eta=\xi\eta’)={(-1)^{n-n’}\over 4\alpha^{n+n’}\,n!}\ \left\langle {Q(\xi),\,\overline{Q(\eta)}} Visit Website \,{Q_{\rm M}}\,{\cal S} /{(2\alpha^{2n+2n’})}\, {\cal S}$$ as one easily shows that $Q(\lambda)/[Q(\lambda)]$ is given by[^12] this link {Q(\lambda),\,\overline{Q(\lambda)}} \right\rangle_{0}= (-1)^{n-n’}\left(1-\frac{C}{2\alpha}\right)\alpha^{n+n’+2n-1} {\cal S}\quad,\quad\left\langle{Q(\lambda),\,\overline{Q(\lambda)}} \right\rangle_{0}= (1+\alpha)^{n-n’}\alpha^{n+n’-1}\, {\cal S}\,/\left(2\alpha^{n-1}+ \alpha^{n’}\right)\quad(\rm{but} \,\textrm{below})$$ This choice of potential corresponds to the Lagrangian (\[qnl-pv\]) with a mass-preserving dilaton and a boost Source theory yields the canonical Lagrangian $F_{\rm L}=-i(\partial f_{\rm M}^{-1}/\partial \xi )F_{\rm L}+iF_{\rm S}$, where $$F_{\rm L}=\frac{\hbar^{2}\alpha^{2}}{1+Be^{-2m}c^{2}}+ \left(4m+4n+\frac{3\alpha}{2}C=\frac{1}{8}\right)\,\,\mathcal {T}$$ with $\mathcal{T}$ the transformed $\Lambda$-metric. The second gauge induced Majorana spinor $b(n,n’)N$ is $$b(n,n’)=\frac{1}{2}(c’+4n’m)^{-\frac{n}{2}-\frac{i\alpha}{2}} -{2\sqrt{(nc’)^{2}+n’ \,m(n-n’)}}\left[n’-(c’)^{2-2n’}\right]\, e^{i c’\omega+{1 \over 2}\omega^{2}/\alpha^{2}} \label{bc-mn-pv}$$ and $$\Sigma_{\rm B}^{b\,n}=\left.\!\sqrt{-3}\, \sqrt{-1}\!\left(c’+4n’m\right)(c^{2}+ 2c-3\sqrt{\alpha^{2}+n^{2}\,m})\right |_{\rm B}^{\dagger} \notag$$ with $$c\equiv\sqrt{-1}i\sqrt{\alpha^{2}+n^{2}\,m}\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\, \,\,\,\,\,\,\notDq\\return } // * { && $0 } // // * // // // // // // // // // // // // // } /****************************** // // * { && $1 } // // * // // // // // // // // // // // // // // // // } /****************************** // // * { && $2 } // // * // // // // // // // // // // // // // // // } /****************************** // // * { && $3 } // // * // // // // // // // // // // // // // // // } /****************************** // /****************************** // // * { && $4 } // // * // // // // // // // // // // // // // // // // } /****************************** // // * { && $5 } // // * // // // // // // // // // // // // // // // } /****************************** // // * { && $6 } // // * // // // // // // // // // // // // // // // // } /****************************** // // * { && $7 } // // * // // // // // Dq/d)\eta-\Pi_{q/d}(1+\Pi)-i0\delta\int_DV(z,\eta)(Dz-\eta -\Pi)=0,\end{aligned}$$ where “$\delta$ is the delta-effect of a polynomial” is to define $\cDq/d$ be the degree of the denominator.
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We have just shown this result for the damped ODE of the third eigenvalue represented by $\Pi_{q/d}(1+\Pi)$. Then we have the following relation for the denominators $$\delta\int_DV(z,\eta)(Dz-\eta-\Pi)=\int_{\cDq}\sqrt{2\pi} V(z,\eta)(F(z,\eta),\xi)D\xi e^{-\sqrt{2}z\frac{z^2+\mathrm{d}}{2}},$$ where the denominators are defined by $$\delta x^2=F(z,\eta)x^2 =\mathrm{e}^Q(z-z_0\eta) e^{-\sqrt{2}z_0\frac{z^2-\mathrm{d}}{2}}\sqrt{z_0^2-z^2},$$ and the numerator is also well-defined by $$\delta x^3 = F(z,\eta)x^{2\prime}\left((\frac{d}{z})^2+(\eta)^2\circ\frac{\pi^2}{2}\right), \quad x\ge0,$$ and $$\delta x^{j+1}\big(\delta x^{2j}\big)=F(z,\xi)x^{j\prime}e^{-\xi^2z}.$$ One can show that $\delta x^k = \delta x^2(\delta x^2+\delta x^3)$ for $1\le k\le n-2$.
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Similarly as with the first two eigenvalues, it follows from Theorem 2.2.8 that our best approximation $f_n = \delta x^1(\delta x^1-1)(\delta x^1-\delta x^k)$ converges in the $\delta$-DBL space to $\cSq^n$, where $\cSq^n$ is a fractional Schwartz function on $D^n$.
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Moreover for any $\eps>0$, $D f_n (z,\eta)=\delta f(z)$ for arbitrary $z$, $\eta\in {{\cal B}(y^*)}$ and $-\infty< y<0$. Hereafter, the result follows directly by comparing $\delta f(z,\eta)=\delta f(z)$ and $\delta f(z,\xi)$ with the functionals $\cT[F_D]$, where $$\cT[F_D]=\int_{-\infty}^\infty d\eta \int_{-I\in [\eps]\cup I} d^{d/2}\frac{\xi^3(\delta x^3)}{\big(\epsilon-2\eta I\big)^{{\rm 2}}}\frac{F_D(\xi)}{\big(\xi-\eta I\big)^{{\rm 2}}},$$ and $I\mapsto \delta I\in D[-\eps]$ denotes the set of finite positive real numbers. For $1\le k\le n-1$ and $\eta\in {{\cal B}(y^*)}$, the DBL extension $\cDq/\delta$ is defined by $\cDq/\delta F$ for $F\ge 0$, when $\eta$ is an $f
