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\begin{align} \CofP &:= \int \tfrac12 x q(x)^2\,dx \qtq{generates} \tfrac{d}{dt} q = (xq)' = xq' + q = \tfrac{d q_\lambda}{d\lambda} \bigr|_{\lambda=1}. \end{align}

\begin{equation} \beta(\lambda \kappa; q_\lambda) = \beta(\kappa; q) \quad\text{for any $\lambda >0$.} \end{equation}

\begin{equation} \beta(\kappa; q+c) + {\textstyle\int} (q+c)\,dx = \tfrac{\kappa^2}{(\kappa-c)^2} \Bigl[\beta(\kappa -c;q) + {\textstyle\int} q\,dx\Bigr] + \tfrac{c \kappa}{\kappa-c}. \end{equation}

\begin{equation} [ \mc U_\lambda f](x) = \sqrt\lambda\,f(\lambda x). \end{equation}

\begin{align*} \mc{U}_\lambda (\mc L(q) + \kappa )^{-1} = \lambda (\mc L(q_\lambda) + \lambda\kappa)^{-1} \;\!\mc U_\lambda . \end{align*}

\begin{align*} \beta(\lambda\kappa; q_\lambda) = \lambda \langle \mc U_\lambda q_+, (\mc L(q_\lambda) + \lambda\kappa)^{-1} \;\!\mc U_\lambda q_+\rangle = \langle \mc U_\lambda q_+, \;\!\mc U_\lambda (\mc L(q) + \kappa)^{-1} q_+\rangle = \beta(\kappa;q). \end{align*}

\begin{align} \{ \beta(\kappa), \CofP \} &= -\kappa \tfrac{\partial\beta}{\partial\kappa}. \end{align}

\begin{align} \CofE := \int \tfrac12 x q(x) \cdot \h\partial q(x) - \tfrac13 x q(x)^3\,dx . \end{align}

\begin{align} \partial_x \bigl(\tfrac{\delta\ }{\delta q} \CofE\bigr) = \bigl[ x\h q' +\tfrac12 \h q\bigr]' - [ x q^2]' = x (\h q'' -2qq') - q^2 +\tfrac32 \h q'. \end{align}

\begin{align} \{ \beta(\kappa), \CofE \} &= \kappa^2 \tfrac{\partial\beta}{\partial\kappa} + \kappa \beta(\kappa), \end{align}

\begin{align} \tfrac{d}{dt} q = \begin{cases} - \kappa^2 \big( m + \ol m + |m|^2 \big)' + \kappa q' & \text{ on $\R$,}\\ - \kappa^2 \big( m + \ol m + |m|^2 \big)' + \bigl[ \kappa +\tint \bigr] q' & \text{ on $\T$.} \end{cases} \end{align}

\begin{equation} \tfrac{d}{dt} \mc{L} = [\mc{P}_\kappa,\mc{L}] \end{equation}

\begin{equation} \mc{P}_\kappa:= i\kappa^3(\mc L +\kappa)^{-1} - i\kappa^2 (m+1)C_+(\ol{m}+1) + \kappa\partial, \end{equation}

\begin{equation} \mc{P}_\kappa:=i\kappa^2\bigl[\kappa+\beta(\kappa)+\tint\bigr](\mc L +\kappa)^{-1} - i\kappa^2 (m+1)C_+(\ol{m}+1) + \bigl[\kappa +\tint \bigr]\partial . \end{equation}

\begin{align} \tfrac{d}{dt} q_+ = \mc P_\kappa q_+. \end{align}

\begin{equation} \tfrac{d}{dt} q = - \kappa^2 \big( \tfrac{\del\beta}{\del q} \big)' + \bigl[\kappa+\tint\bigr] \big(\tfrac{\del P}{\del q} \big)' = - \kappa^2 \big( m + \ol m + |m|^2 \big)' + \bigl[\kappa+\tint\bigr] q'. \end{equation}

\begin{equation} \widetilde{\mc P}_\kappa := - i\kappa^2 (m+1)C_+(\ol{m}+1) + \bigl[\kappa +\tint \bigr]\partial . \end{equation}

\begin{equation} \tfrac{d}{dt} \mc{L} = \kappa^2 C_+ m'(\ol m +1) + \kappa^2 C_+ (m+1) \ol m' - \bigl[\kappa +\tint \bigr] C_+ q' \end{equation}

\begin{align} [\widetilde{\mc P}_\kappa,\mc{L}] &= \kappa^2 \bigl\{ m' C_+(\ol{m}+1) + (m+1)C_+ \ol{m}' \big\} - \bigl[\kappa +\tint \bigr] C_+q' \\ &- i\kappa^2 C_+ q(m+1)C_+(\ol{m}+1) + i\kappa^2 (m+1)C_+(\ol{m}+1)C_+ q \notag\\ &=\text{RHS}\eqref{5:42} -\kappa^2 C_+ m' [1 - C_+] (\ol{m}+1) -\kappa^2 C_+ (m+1) [1 - C_+] \ol m' \notag\\ &- i\kappa^2 C_+ q(m+1)C_+(\ol{m}+1) + i\kappa^2 (m+1)C_+(\ol{m}+1)C_+ q \notag \end{align}

\begin{align*} C_+ m' & [1 - C_+] (\ol{m}+1) f + C_+ (m+1) [1 - C_+] \ol m' f \\ &= -i\kappa C_+ m [1 - C_+] (\ol{m}+1)f + i\kappa C_+ (m+1) [1 - C_+] \ol m f \\ &+ i C_+ [q(m+1)]_+ [1 - C_+] (\ol{m}+1) f - i C_+ (m+1) [1 - C_+] [(\ol m+1)q]_- f\\ &= i C_+ q(m+1) [1 - C_+] (\ol{m}+1) f - i C_+ (m+1) [1 - C_+] (\ol m+1) q f \\ &= - i C_+ q(m+1) C_+ (\ol{m}+1) f + i C_+ (m+1) C_+ (\ol m+1) qf \\ &= - i C_+ q(m+1) C_+ (\ol{m}+1) f + i C_+ (m+1) C_+ (\ol m+1) C_+ qf . \end{align*}

\begin{align*}C_+(\ol m +1)q_+ &= C_+(\ol m +1)q = [1-C_-](\ol m +1)q + \bigl[ \beta(\kappa) + \tint \bigr] \\ &= (\ol m +1)q - i\ol m' - \kappa \ol m + \bigl[ \beta(\kappa) + \tint \bigr]. \end{align*}

\begin{align*}C_+(m+1)(\ol m +1)q &= C_+(\ol m +1)C_+ (m+1)q + C_+(\ol m +1)[1-C_+] (m+1)q \\ &= C_+(\ol m +1)(-im' +\kappa m), \end{align*}

\begin{align*} i\kappa^2 C_+ & (m+1) C_+(\ol m +1)q_+ \\ &= i\kappa^2C_+ \Bigl[ (m+1)(\ol m +1)q + (m+1)\bigl[ - i\ol m' - \kappa \ol m + \beta(\kappa) + \tint \bigr] \Bigr]\\ &= i\kappa^2C_+ \Bigl[ (\ol m +1)(-im' +\kappa m) + (m+1)\bigl[ - i\ol m' - \kappa \ol m + \beta(\kappa) + \tint \bigr] \Bigr] \\ &= \kappa^2 C_+\bigl[ m+\ol m + |m|^2\bigr]' + i \kappa^3C_+(m-\ol m) + i\kappa^2\bigl[ \beta(\kappa) + \tint \bigr] (m+1). \end{align*}

\begin{align*} i \kappa^3C_+ \ol m = i \kappa^3 {\textstyle\int} \ol m = i\kappa^2 \bigl[ \beta(\kappa) + \tint \bigr]. \end{align*}

\begin{align*} i\kappa^2 C_+ & (m+1) C_+(\ol m +1)q_+ = \kappa^2 C_+\bigl( m+\ol m+|m|^2\bigr)' + i\kappa^2 \bigl[\kappa + \beta(\kappa) + \tint \bigr] m, \end{align*}

\begin{equation} \tfrac{d}{dt} \beta(\vk;q(t))= 0 \qtq{for any} \vk\geq \kappa_0. \end{equation}

\begin{equation} Q_* = \big\{ e^{J\nabla (t_1\hbo + t_2\hk)}(q) : q\in Q,\ t_1,t_2\in\R,\ \kappa\geq\kappa_0(A) \big\}. \end{equation}

\begin{align} \tfrac{d}{dt} n &= [\mc P_\kappa,R(\vk)]q_+ + R(\vk) \mc P_\kappa q_+ = \mc P_\kappa R(\vk) q_+ = \mc P_\kappa n. \end{align}

\begin{align} \tfrac{d}{dt} n = \mc P n = - in'' - 2C_+([q-q_+] n)' - 2q_+n' \end{align}

\begin{equation} %\end{equation}

\begin{align} \tfrac{d}{dt} n= \bigl\{ \mc{L}n - C_+(q_- n) \bigr\}' - iq_+\mc{L}n + q'_+n - iq_+C_+(q n), \end{align}

\begin{align} \tfrac{d}{dt}n ={}&\bigl\{ \kappa R(\kappa) \mc{L} n - \kappa^2 C_+\big[\ol{m} R(\kappa)n \big] \bigr\}' - i \kappa q_+ R(\kappa) \mc{L}n + \kappa m'n\notag\\ & - i\kappa m C_+( [q-q_-+\kappa\ol{m}] n ) - i \kappa C_+(q_+\ol{m}) \cdot \mc{L} R(\kappa) n \\ & + \kappa C_+\big( |m|^2 \big)' \cdot n - i\kappa [1-C_-](q_+\ol{m})\cdot mn . \notag \end{align}

\begin{align*} \tfrac{d}{dt} n&= \textup{RHS\eqref{m dot BO 3}} + [\tint ] n' \qtq{under \eqref{BO},} \\ \tfrac{d}{dt} n&= \textup{RHS\eqref{m dot Hk 2}} + [\tint ] n' \qtq{under the $H_\kappa$ flow.} \end{align*}

\begin{align*} \tfrac{d}{dt} n &= - in'' - 2C_+([q -q_+]n)' - 2q_+n' \\ &= \bigl\{ - in' - C_+(qn) - C_+([q-q_+]n) \bigr\}' + (q_+n)' - 2q_+n' \\ &=\bigl\{ \mc{L}n - C_+([q-q_+]n) \bigr\}' + q'_+n - iq_+\mc{L}n - iq_+C_+(qn)\\ &=\textup{RHS\eqref{m dot BO 3}} + [\tint ] n'. \end{align*}

\begin{align*} i\kappa^3 R(\kappa,q)n + \kappa\partial n &= i\kappa^2n + \kappa\partial n - i\kappa\mc L n + i \kappa \mc L R(\kappa) \mc L n\\ &= i\kappa^2n + i\kappa C_+ (qn) + i \kappa\mc L R(\kappa) \mc L n,\\ i\kappa^2 \btint R(\kappa)n &= i\kappa C_+\bigl(\btint n\bigr) - i\kappa \btint R(\kappa) \mc L n,\\ - i\kappa^2 (m+1)C_+\bigl([\ol{m}+1]n\bigr) &= -i\kappa^2 n - i\kappa^2 C_+([m+\ol m]n) - i\kappa^2 m C_+ (\ol m n). \end{align*}

\begin{equation} \begin{aligned} \tfrac{d}{dt} n ={}& i \kappa \mc L R(\kappa) \mc L n - i\kappa C_+\bigl(\bigl[\kappa m +\kappa \ol m - q - \tint\bigr] n\bigr) - i\kappa^2 m C_+ (\ol m n) \\ & - i\kappa \btint R(\kappa) \mc L n + i\kappa^2\beta(\kappa) R(\kappa)n + \btint n'. \end{aligned} \end{equation}

\begin{align*} \bigl[\kappa m +\kappa \ol m - q - \tint \bigr] = (\kappa m - q_+) + (\kappa \ol m - q_-) = - \mc L m - \ol{\mc L m}. \end{align*}

\begin{align*} i \kappa \mc L R(\kappa) \mc L n - i\kappa \btint R(\kappa) \mc L n = \bigl( \kappa R(\kappa) \mc L n \bigr)' - i\kappa C_+\bigl([q_+ + q_-] R(\kappa) \mc L n\bigr). \end{align*}

\begin{equation} \begin{aligned} \tfrac{d}{dt} n ={}& \big( \kappa R(\kappa)\mc{L} n \big)' - i\kappa q_+ R(\kappa)\mc{L}n - i\kappa^2 mC_+(\ol{m}n) + i\kappa (\mc{L}m) n \\ & + i\kappa C_+\big[ (\ol{\mc{L}m}) n \big] - i\kappa C_+\big[ q_-\mc{L}R(\kappa)n \big] + i\kappa^2\beta(\kappa) R(\kappa)n +\btint n'. \end{aligned} \end{equation}

\begin{align*} i\kappa C_+\Bigl[ ( \ol{\mc{L}m}) n - q_- \mc{L}R(\kappa)n \Bigr] &= i\kappa C_+\Bigl[ \ol{\mc{L}m}\cdot (\mc L +\kappa) R(\kappa) n - \ol{(\mc{L}+\kappa)m} \cdot \mc{L}R(\kappa)n \Bigr] \\ &= i\kappa^2 C_+\Bigl[ \ol{\mc{L}m} \cdot R(\kappa) n - \ol{m} \cdot \mc{L}R(\kappa)n \Bigr], \end{align*}

\begin{align*} i\kappa C_+\Bigl[ ( \ol{\mc{L}m}) n - q_- \mc{L}R(\kappa)n \Bigr] &= - \kappa^2 C_+\big[ \ol{m}\,R(\kappa)n \big]' + i\kappa^2 [1-C_-](q_+\ol{m})\cdot R(\kappa)n. \end{align*}

\begin{align*} i\kappa^2[1-C_-](q_+\ol{m})\cdot R(\kappa)n + i\kappa^2\beta(\kappa) R(\kappa)n &= i\kappa^2 C_+( q_+\ol{m})\cdot R(\kappa)n \\ &= i\kappa C_+( q_+\ol{m})\cdot [ n - R(\kappa)\mc L n ]. \end{align*}

\begin{align} \tfrac{d}{dt} n ={}& \big( \kappa R(\kappa)\mc{L} n \big)' - i\kappa q_+ R(\kappa)\mc{L}n - i\kappa^2 mC_+(\ol{m}n) + i\kappa (\mc{L}m) n \notag \\ & - \kappa^2 C_+\big[ \ol{m}\,R(\kappa)n \big]' - i\kappa C_+( q_+\ol{m}) \cdot R(\kappa)\mc L n + i\kappa C_+( q_+\ol{m})\cdot n + \btint n'. \end{align}

\begin{align*} i\kappa \bigl[ (\mc{L}m)& +C_+( q_+\ol{m} )\bigr] n\\ &= \kappa m'n - i\kappa [q - q_-] mn - i\kappa nC_+(q_- m) + i\kappa nC_+( q_+\ol{m} )\\ &= \kappa m'n - i\kappa [q - q_-] mn - i\kappa nC_+ \big\{ \ol{q_+} m - \ol{m}q_+ \big\} \\ &= \kappa m'n - i\kappa [q - q_-] mn - i\kappa nC_+ \big\{ \ol{(\mc{L}+\kappa)m} \cdot m - \ol{m}\cdot (\mc{L}+\kappa)m \big\} \\ &= \kappa m'n - i\kappa [q - q_-] mn - i\kappa nC_+ \big\{ \ol{\mc{L}m} \cdot m - \ol{m}\cdot\mc{L}m \big\} \\ &= \kappa m'n - i\kappa m C_+\bigl( [q - q_-] n \bigr) + \kappa nC_+\big( |m|^2 \big)' - i\kappa mn [1-C_-](q_+\ol{m}). \end{align*}

\begin{equation} \begin{aligned} \sup_{|t|\leq T} \norm{ q_j(t) - q_\ell(t) }_{H^s} \leq{}& \sup_{|t|\leq T} \bnorm{ e^{tJ\nabla\hk} (q_j^0) - e^{tJ\nabla\hk} (q_\ell^0) }_{H^s} \\ &+ 2 \sup_{q\in Q_*}\,\sup_{|t|\leq T} \bnorm{ e^{tJ\nabla(\hbo-\hk)} (q) - q }_{H^s} . \end{aligned} \end{equation}

\begin{equation} \lim_{\kappa\to\infty}\, \sup_{q\in Q_*}\,\sup_{|t|\leq T} \bnorm{ e^{tJ\nabla(\hbo-\hk)} (q) - q }_{H^s} = 0 . \end{equation}

\begin{equation} \lim_{\kappa\to\infty}\,\sup_{q\in Q_*}\,\sup_{|t|\leq T} \norm{ n(t) - n(0) }_{H^{s+1}} = 0. \end{equation}

\begin{equation} \lim_{\kappa\to\infty}\,\sup_{q\in Q_*}\,\sup_{|t|\leq T} \norm{ n(t) - n(0) }_{H^{-2}} = 0. \end{equation}

\begin{equation} \lim_{\kappa\to\infty}\,\sup_{q\in Q_*}\,\sup_{|t|\leq T} \norm{ \tfrac{dn}{dt} }_{H^{-2}} = 0, \end{equation}

\begin{align} \tfrac{d}{dt} n ={}& \big\{ \mc{L} R(\kappa)\mc{L} n - C_+\big[ \big( q_- - \kappa^2\ol{m} R(\kappa) \big) n \big] \big\}' - i q_+\mc{L} R(\kappa)\mc{L}n \notag\\ &+ \big(q'_+ - \kappa m'\big) n - i(q_+-\kappa m) C_+(qn) - i\kappa m C_+([q_- - \kappa \ol m]n) \\ &+ i \kappa C_+(q_+\ol{m}) \cdot \mc{L} R(\kappa) n - \kappa C_+\big( |m|^2 \big)' \cdot n + i \kappa [1-C_-](q_+\ol{m}) \cdot mn.\!\!\!\! \notag \end{align}

\begin{equation} \norm{ \kappa m - q_+ }_{H^s} = \norm{\mc{L}m}_{H^s} \lesssim \norm{m}_{H^{s+1}} \to 0 \quad\text{as }\kappa\to\infty \end{equation}

\begin{equation} \tfrac{d}{dt} q = \big(m + \ol{m} + |m|^2\big)', \end{equation}

\begin{equation} \mcPbk:= -i\kappa(\mc L +\kappa)^{-1} + i (m+1)C_+(\ol{m}+1) . \end{equation}

\begin{equation} \tfrac{d}{dt} q_+(t) = \mcPbk q_+(t) \qtq{and} \tfrac{d}{dt} n(x;q(t)) = \mcPbk n(x;q(t)) \end{equation}

\begin{equation} q_+(t,z) = \tfrac{1}{2\pi i} I_+ \Big( \big( X - t\kappa R(\kappa;q^0)^2 - z \big)^{-1} q^0_+\Big) \end{equation}

\begin{equation} \chi_y(x)=\tfrac{iy}{x+iy} \qtq{satisfies} \lim_{y\to\infty} \mcPbk \chi_y = 0 \end{equation}

\begin{align*} [\kappa R(\kappa) - 1]\chi_y &= R(\kappa) C_+ q \kappa R_0(\kappa) \chi_y + [\kappa R_0(\kappa) - 1]\chi_y \\ &= R (\kappa)q_+ - R(\kappa) C_+(q-q\chi_y) + [R(\kappa) C_+ q + 1][\kappa R_0(\kappa) - 1]\chi_y. \end{align*}

\begin{equation} [X,\mcPbk] = - \kappa R(\kappa,q)^2 \end{equation}

\begin{align} -i\kappa [X, R] = i\kappa R [X,\mc L] R = - \kappa R^2 - i\kappa R [ X, C_+ q] R . \end{align}

\begin{align} [X,i(m+1)C_+(\ol{m}+1)]f &= i[X,(m+1)] C_+ (\ol{m}+1) f + i(m+1) [X, C_+(\ol{m}+1)]f \notag\\ &= i [X, m]C_+(1+\ol{m}) f \notag \\ &= - \tfrac{1}{2\pi} m \cdot I_+\bigl( f + C_+(\ol{m} f) \bigr) \notag\\ & = - \tfrac{1}{2\pi} R q_+ \cdot I_+\bigl( f + C_+(\ol{m} f) \bigr). \end{align}

\begin{align*} [X,i(m+1)C_+(\ol{m}+1)]f &= i\kappa R [ X, C_+ q] R f. \end{align*}

\begin{align} \bigl( |\partial| + \kappa \bigr) (m+\ol m) = q + C_+(qm)+C_-(q\ol m). \end{align}

\begin{equation} \begin{aligned} \bigl\| \langle x\rangle \bigl[ m + \ol m + |m|^2 \bigr]' \bigr\|_{L^2} &\lesssim \bigl[1 + \| m \|_{L^\infty} \bigr] \| \langle x\rangle m' \|_{L^2} \\\ &\lesssim \bigl[1 + \| m \|_{H^{s+1}} \bigr]^2 \| \langle x\rangle q \|_{L^2} . \end{aligned} \end{equation}

\begin{equation} U(t)^* \:\!\! R(\kappa;q(t)) U(t) = R(\kappa;q^0) \qtq{and} q_+(t) = U(t) q^0_+ \qtq{for all} t\in \R. \end{equation}

\begin{equation} Y_1(t) := \big( X - t\kappa R(\kappa;q^0)^2 - z \big)^{-1} \qtq{and} Y_2(t):=U(t)^* (X-z)^{-1} U(t) . \end{equation}

\begin{equation} \tfrac{d}{dt} Y(t)= \kappa Y(t) R(\kappa;q^0)^2 Y(t) \qtq{with} Y(0)=(X-z)^{-1} . \end{equation}

\begin{align*} \tfrac{d}{dt} Y_2(t)=U(t)^* [(X-z)^{-1},\mcPbk] U(t) &= - U(t)^* (X-z)^{-1} [X,\mcPbk] (X-z)^{-1} U(t) \\ &=\kappa Y_2(t) U(t)^* R(\kappa;q(t))^2 U(t) Y_2(t) \\ &= \kappa Y_2(t) R(\kappa;q^0)^2 Y_2(t) . \end{align*}

\begin{equation} \big( X - t\kappa R(\kappa;q^0)^2 - z \big)^{-1} q^0_+ = U(t)^* (X-z)^{-1} U(t) q^0_+ \end{equation}

\begin{align*}q_+(t,z) = \lim_{y\to\infty} \tfrac{1}{2\pi i} \bigl\langle U(t)^* \chi_y, \big( X - t\kappa R(\kappa;q^0)^2 - z \big)^{-1} q^0_+ \bigr\rangle. \end{align*}

\begin{equation} \sum c_j \beta(\kappa_j) = \langle q_+, \phi(\mc L)q_+\rangle \qtq{where} \phi(E) = \sum c_j (E+\kappa_j)^{-1} \end{equation}

\begin{equation} q_+(t,z) = \tfrac{1}{2\pi i} I_+ \Big( \big( X - t \psi(\mc L_{q_0}) - z \big)^{-1} q^0_+\Big) \qtq{with} \psi(E) = \phi(E) + E\phi'(E).\end{equation}

\begin{equation} q_+(t,z) = \tfrac{1}{2\pi i} I_+ \big\{ \big( X - 2t\mc{L}_{q_0} - z \big)^{-1} q_+^0 \big\} \end{equation}

\begin{equation} \CofB(\vk) := \tfrac12 \int x q [n + \ol n] \,dx \end{equation}

\begin{equation} \bigl\{ \CofB(\vk) , \beta(\kappa) \bigr\} = -\kappa \langle q_+, R(\kappa)R(\vk)R(\kappa) q_+\rangle = -\kappa \tfrac{\partial}{\partial\kappa} \ \tfrac{\beta(\kappa)-\beta(\vk)}{\kappa-\vk}. \end{equation}

\begin{equation} \langle g_+, X f_+\rangle + \langle X f_+, g_+ \rangle = \int_{-\infty}^\infty \ol{\widehat g(\xi)} \cdot i \widehat f'(\xi) \, d\xi = \int x f(x) g(x) \,dx. \end{equation}

\begin{equation} \CofB(\vk) = \tfrac12 \langle n, X q_+ \rangle + \tfrac12 \langle X q_+, n \rangle . \end{equation}

\begin{align*} \bigl\{ \CofB(\vk) , \beta(\kappa) \bigr\} &= \tfrac12 \langle n, [X,\mcPbk] q_+ \rangle + \tfrac12 \langle [X,\mcPbk] q_+, n \rangle. \end{align*}

\begin{align} \CofB(\vk) = \vk^{-1} \CofP - \vk^{-2} \CofE + O(\vk^{-3}\bigr). \end{align}

\begin{align*} \Bigl\{ {\int} \tfrac12 x^2 q^2 \,dx ,\ \beta(\kappa) \Bigr\} = 2\kappa \frac{d}{d\kappa} \CofB(\kappa) \end{align*}

\begin{align*} \VofP\bigl(q(t)\bigr)= - t^2\bigl( \kappa \tfrac{d^2\beta}{d\kappa^2} + \kappa^2\tfrac{d^3\beta}{d\kappa^3} \bigr)(\kappa;q(0)) + 2t\kappa \tfrac{d}{d\kappa} \CofB(\kappa;q(0)) + \VofP\bigl(q(0)\bigr). \end{align*}

\begin{equation} \Krk(b', a'C) < \infty \text{and} \Krk(b'', a'C) = \infty \end{equation}

\begin{equation} \Krk(b' ,a''C) = \infty \text{and} \Krk(b'', a''C) < \infty. \end{equation}

\begin{align*} f(g_v \cdot y) & \in \Stab_P(B) g^{-1} \cdot f(\chi(\sigma)h g_v \cdot y)\\ & \subseteq \Stab_P(B) \pi^{-1} \Stab_P(A) \cdot f(\chi(\sigma)h \cdot y) \\ & \subseteq \Stab_P(B) g^{-1} \Stab_P(A) g \Stab_P(B) \cdot f(y) \\ & = \Stab_P(B) \cdot f(y), \end{align*}

\begin{equation} \begin{gathered} \begin{xymatrix}{ K_0(C^*_{\mathbb C}(D))\ar[r]^-{(j_\mathbb C)_*}\ar[d]^-{(i_D)_*}&K_0(C^*_{\mathbb C}(X))\ar[d]^-{(i_X)_*}\\ K_0(C^*_H(D))\ar[r]^-{(j_H)_*}&K_0(C^*_H(X)). } \end{xymatrix} \end{gathered} \end{equation}

\begin{align} \langle d_jf,g\rangle_{X^{j+1}}=&\frac12 \sum_{s\in X^{j+1}}m(s)df(s)\overline{g(s)}\nonumber \\ =&\frac12 \sum_{s\in X^{j+1}}m(s)\overline{g(s)}\sum_{r\subset s}f(r) \\ =&\frac12 \sum_{r\in X^{j}}m(r)f(r) \overline{\sum_{r\subset s}\frac{m(s)}{m(r)}g(s)}\ . \end{align}

\begin{equation} d_j^*g(r)=\sum_{r\subset s}\frac{m(s)}{m(r)}g(s) \end{equation}

\begin{equation} \Delta_j(X)=d_j^*d_j+d_{j-1}d_{j-1}^*\ . \end{equation}

\begin{equation} d_j^*d_jf=-d_{j-1}d_{j-1}^*f\ . \end{equation}

\begin{equation} \tau f=\begin{cases}f &\text{ if }f\in\ell^2(X)_{\text{even}}\\ -f &\text{ if }f\in\ell^2(X)_{\text{odd}}\\ \end{cases} \end{equation}

\begin{equation} P^{j,n}:=\{I:\{1,\dots,j\}\to\{1,\dots,n\}; I \,\,{\rm strictly \,\, monotone}\} \end{equation}

\begin{equation} \partial (s) := \cup_{i=1}^j \{(-1)^{j-i} (\lfloor s \rfloor;\prescript{}{i}{\hat{s}}) \}\bigcup\cup_{i=1}^j\{(-1)^{i} (\lceil s \rceil;(\prescript{}{i}{\hat{s}})^*) \}\ . \end{equation}

\begin{align*} % \end{align*}

\begin{equation} (d_hf)(s)=\sum_{r \subset s}f(r); \quad( d_h^*f)(s)=\frac1{h^2} \sum_{s \subset r}f(r)\ . \end{equation}

\begin{equation} \langle dx^I;dx^{I'}\rangle_{\bigwedge\nolimits^j(\Z^n)}:=\begin{cases} 1 &\text{ if } I=I'\\ 0&\text{ if else } \end{cases}\ . \end{equation}

\begin{equation} \omega(\mu)=\sum_{I\in P^{j,n}_+} \omega_{I}(\mu) dx^{I}, \end{equation}

\begin{equation} \langle \omega , \eta\rangle_{\Omega_c^j(\Z^n)}=\sum_{\mu\in\Z^n}\sum_{I\in P^{j,n}_+} \omega_{I}(\mu)\overline{\eta_{I}(\mu)}\ . \end{equation}

\begin{equation} \tilde d_{0} \omega=\sum_{l=1}^n(\mathcal{D}_{l} \omega) d x_l. \end{equation}

\begin{equation} ({U}_jf)(\mu):=\sum_{I\in P^{j,i}_+} f(\mu;\delta_{I(1)},\hdots, \delta_{I(j)}) \, dx^I\ . \end{equation}

\begin{align} [(U_1\circ d_0)f](\mu)=\sum_{i=1}^j (d_0f)(\mu;\delta_j)dx^{i}=\sum_{i=1}^j (d_0f)(\mu,\mu+\delta_i)dx^{i}=&\sum_{i=1}^j (f(\mu+\delta_i)-f(\mu))dx^{i}\nonumber \\ =&\sum_{i=1}^j \mathcal{D}_if(\mu)dx^{i} \end{align}

\begin{equation} \tilde d_j (\sum_{I\in P^{j,n}_+}\omega_I dx^I)=\sum_{I\in P^{j,n}_+}(\tilde d_0 \omega_I)\wedge dx^I. \end{equation}