harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{M_m(Z_1)}{M_m(Z_2)}= \begin{cases}1 \textrm{if}Z_1,Z_2<R_1, \left(\prod_{j=1}^\nu\hat R_j/\hat{Z_1}\right)(\hat{Z_1}/\hat{Z_2})^{\nu}\textrm{if}Z_1<R_1\leq Z_2, \left(\prod_{j=\mu+1}^\nu\hat R_j/\hat{Z_1}\right)(\hat{Z_1}/\hat{Z_2})^{\nu}\textrm{if}R_1\leq Z_1\leq Z_2 ,\end{cases}
\aligned X_t =X_0+\int_0^t\sigma(s,\omega,X_s)dB_s+\int_0^tb(s,\omega,X_s)ds &\quad+\int_0^{t+}\int_Uf_1(s,\omega,X_{s-},u)\tilde{N}_k(ds,du)+\int_0^{t+}\int_Uf_2(s,\omega,X_{s-},u)N_k(ds,du) \endaligned
\displaystyle \sum_{n=0}^{+\infty}H(n)\, H'(-n) = \displaystyle \sum_{n=0}^{+\infty}H^{q}(n)H'^{p}(-n)\left[H^{p}(n)H'^{q}(-n)\right] \leq \mathbb{E}\left[\xi_1^p\right]^p\,\mathbb{E}[(-{\xi_1'})^q]^q \displaystyle \sum_{n=0}^{+\infty}\left[(n+1)^{-q^2}H^{q}(n) \, (n+1)^{-p^2}H'^{p}(-n) \right].
-\sum_{n=0}^N \frac{1}{2\pi}\int_{-\infty}^\infty \frac{2(2n+a+\frac12)}{(2n+a+\frac12)^2+(t+b)^2}\hat{g}(\frac12+it)dt+(\log N)f(1) + O\big(\frac{1}{N}\big).
A=0, B=-\tfrac{\eta}{\sqrt{1-q+\eta\widetilde{\theta}}},C=-\tfrac{\widetilde{\theta}+\tfrac{\eta\tau}{1-q}-\sqrt{\theta^2-4\tau}}{2\sqrt{1-q+\eta\widetilde{\theta}}}D=-\tfrac{\widetilde{\theta}+\tfrac{\eta\tau}{1-q}+\sqrt{\theta^2-4\tau}}{2\sqrt{1-q+\eta\widetilde{\theta}}}
\begin{aligned}J^4_{\lambda}(u,v) = \int g_{[<\lambda]}^{x_0}\left(M_\lambda(u) E_\lambda(v) - 2 P_\lambda(u) P_\lambda(v)\right) \ + g_{[<\lambda]}\left(M_\lambda(v) E_\lambda(u)- 2 P_\lambda(u) P_\lambda(v)\right)\, dx.\end{aligned}
\frac{4}{\sqrt{- \triangle}}\frac{1}{\sqrt{4 m^{2}- \triangle}}{\mathrm a r c t a n h}\left( \frac{\sqrt{- \triangle}}{\sqrt{4 m^{2}- \triangle}}\right) = \int_{m}^{\infty}\!{\mathrm d}\tilde{m}\, \frac{1}{\sqrt{{\tilde{m}}^{2}- m^{2}}}\, \frac{1}{4{\tilde{m}}^{2}- \triangle}.
F_{Coriolis}=-2(\omega\times p)=-2(\omega\times)p=[\begin{matrix}0&-2\omega_{3}&2\omega_{2}\\ 2\omega_{3}&0&-2\omega_{1}\\ -2\omega_{2}&2\omega_{1}&0\end{matrix}][\begin{matrix}p_{1}\\p_{2}\\p_{3}\end{matrix}]
\mathbf{K}^{\varepsilon,n}_{e,\rho}(z')=\frac{\varepsilon^{-d+1}}{\rho^\alpha}\int_{\mathbb{R}^d}\mathbf{K}_{n}(z)\varphi_{\zeta\rho}(z'-z)\frac{\langle\phi^{e,\varepsilon}_\rho(z'-z),\eta^{\kappa}_{y_{\tau}}\rangle}{|z'-z|^{d-\alpha}}dz-\mathbf{c}(\varepsilon,\kappa,\tau,\zeta)\varphi_{\rho}(z')\mathbf{K}_{n}(z')~~\forall~~z'\in \mathbb{R}^d,
m ={\frac{2^{\frac{1}{3}}\pi^{4 / 3}\Gamma ( 5 / 9 )}{\Gamma ( 2 / 3 ) \Gamma ( 8 / 9 )}}\left({\frac{2 \Gamma ( 1 1 / 1 6 )}{\Gamma ( 5 / 1 6 ) \Gamma ( 3 / 4 )}}\right)^{8 / 9}g^{8 / 9}\approx 4 . 9 2 7 7 9 1 2 2 4 4 \ldots g^{8 / 9}.
\Pi \tilde L_a f(\xi)= (2\pi)^{-d}\sum_{m\in\mathbb{Z}^d}\int_{\mathbb{R}^d}\int_{\mathbb{T}^d}e^{i(\varphi_a(x,\xi+2\pi m)-\varphi_a(x,\eta))}f(\eta) d\eta dx = (2\pi)^{-d}\sum_{m\in\mathbb{Z}^d}\int_{\mathbb{R}^d}\int_{\mathbb{T}^d}e^{i(\varphi_a(x,\xi)+2\pi x\cdot m-\varphi_a(x,\eta))}f(\eta) d\eta dx .
2 \mathrm{Re}\langle u_{\alpha}\otimes \Phi_y\,|\, H_y R_y \rangle = -2 \mathrm{Re}(\overline{\Phi}^{(0)}_y\kappa)\|\Phi_{\#}^y\|^2_{\#}-C \alpha(\varepsilon_7 + \alpha ) \|R_y^{\#}\|^2 & -C \alpha (\varepsilon_8 + \alpha) \|R_*\|^2 - C\alpha \varepsilon_7 \|P R^{\#}_y\|^2+ O(\alpha^4).
(S_0,S_1) +(S_0,S_2)+\frac{1}{2}(S_1,S_1)+(S_1,S_2)+(S_1,S_3)+\frac{1}{2}(S_2,S_2) &= \frac{i}{\hbar}\left(\Omega_0(S_3) + \Omega_0(S_4) + \Omega_0(S_5) + \Omega_1(S_0)+\Omega_1(S_2)\right)+\frac{1}{2}\left(\frac{i}{\hbar}\right)^2\Omega_2(S_0^2 + 2S_0S_2 + 2S_0S_3).
\mathcal{W}_{\varepsilon_{j}}(v_{j},\Sigma_{\rho})\leq\sum_{i=1}^{k}\mathcal{W}_{\varepsilon_{j}}(v_{j},R_{\rho}^{i})+M_{W}c_{\Sigma}\hat{\sigma}\mathcal{H}^{n-2}(\Sigma^{n-2})~.
u_{0 , l}( r ) = \left( \begin{array}{l}{{\phi_{0 , l , 1}( r )}}\\{{0}}\\ \end{array}\right) , \mathrm{\;}{}l \geq 1 ; u_{0}^{I I}( r ) = \left( \begin{array}{l}{{\phi_{0 , 1}^{i r}( r )}}\\{{0}}\\ \end{array}\right) , \;{}l = 0 .
B_{1}= \left( \begin{array}{c c}{0}&{0}\\{0}&{v t I}\\ \end{array}\right) , B_{5}= \left( \begin{array}{c c}{0}&{0}\\{0}&{P_{2}}\\ \end{array}\right) , B_{9}= \left( \begin{array}{c c}{0}&{0}\\{0}&{Q_{2}}\\ \end{array}\right) .
\delta\left(\left[\begin{array}{cc}0 M 0 0 \end{array}\right]\left[\begin{array}{cc}I_{\mathcal{A}}0 0 0 \end{array}\right]\right)=\delta\left(\left[\begin{array}{cc}I_{\mathcal{A}}0 0 0 \end{array}\right]\left[\begin{array}{cc}0 M 0 0 \end{array}\right]\right),
(\begin{matrix}a_{1}\\a_{2}\\a_{3}\\a_{4}\\a_{5}\end{matrix})=(\begin{matrix}1&1&1&1&1\\ -3&-2&-1&0&1\\ 9&4&1&0&1\\ -27&-8&-1&0&1\\ 81&16&1&0&1\\ \end{matrix})^{-1}(\begin{matrix}0\\ 0\\ 0\\ 0\\ 24\end{matrix})=(\begin{matrix}1\\ -4\\ 6\\ -4\\ 1\end{matrix}).
\|\partial_tv\|_{\dot{B}_{p,1}^{\frac3p}}= \|\Delta v-\mathbf{P}(v\cdot\nabla v)+\mathbf{P}({h}\cdot\nabla{h})\|_{\dot{B}_{p,1}^{\frac3p}}\le \|v\|_{\dot{B}_{p,1}^{2+\frac3p}}+\|v\|_{\dot{B}_{p,1}^{\frac3p}}\|v\|_{\dot{B}_{p,1}^{1+\frac{3}{p}}}+\|{h}\|_{\dot{B}_{p,1}^{\frac3p}}\|{h}\|_{\dot{B}_{p,1}^{1+\frac{3}{p}}},
2\pi_*(h^k e f) = \pi_*( e\cdot (h^{n+k}+ c_1(V/F^\perp) h^{n+k-1}+ \cdots + c_n(V/F^\perp) h^k ) ) = \rho_*\left( h^{n+k}+ c_1(V/F^\perp) h^{n+k-1}+ \cdots + c_n(V/F^\perp) h^k \right) = c_{k+1}(V-E-F^\perp),
v_i = F^1_i, \frac{1}{\rho}\left( A_{iJKj}u_{j,K}- \beta_{Ji}\theta -(C_{iJKLIj}u_{j,IL}+ M_{iJKL}\tau_{,L}- B_{iJKj}v_{j})_{,K}\right)_{,J}= F^2_i, \theta = F^3, \frac{1}{a}\left( -\beta_{Ki}v_{i,K}+ (m_{IJ}\theta_{,J}+M_{jLKI}u_{j,LK}+ K_{IJ}\tau_{,J})_{,I}\right) = F^4.
\textrm{Re}\big(Q_t^w(x,D_x)i(\textrm{Im}q)^w(x,D_x)e^{-\frac{t}{2}q^w}u,e^{-\frac{t}{2}q^w}u\big)_{L^2}= \textrm{Re}\big(i(\textrm{Im}q)^w(x,D_x)e^{-\frac{t}{2}q^w}u,Q_t^w(x,D_x)e^{-\frac{t}{2}q^w}u\big)_{L^2}= \frac{1}{2}\big([Q_t^w(x,D_x),i(\textrm{Im}q)^w(x,D_x)]e^{-\frac{t}{2}q^w}u,e^{-\frac{t}{2}q^w}u\big)_{L^2},
Q_M (\alpha ,\beta)=1-\mathrm{exp}\left(-\frac{a^2}{2}\right)\sum_{k=0}^{\infty}(-1)^n\frac{L_k^{(M-1)}\left(\frac{a^2}{2}\right)}{\Gamma(M+k+1)}\left(\frac{b^2}{2}\right)^{k+M}
f(x) = x^7 - x^5 ( z_1 + z_2 + z_3 + z_4 + z_5 + z_6) \qquad\; + x^3 ( z_1 z_3 + z_1 z_4 + z_1 z_5 + z_1 z_6 + z_2 z_4 + z_2 z_5 + z_2 z_6 + z_3 z_5 + z_3 z_6 + z_4 z_6) \qquad\; - x (z_1 z_3 z_5 + z_1 z_3 z_6 + z_1 z_4 z_6 + z_2 z_4 z_6).
D_{i_{01}}^2\times\cdots\times D_{i_{0p_0}}^2\times\partial D^{d(w)}\cup\left(\left(\bigcup\limits_{k = 1}^p D_{i_{01}}^2\times\cdots\times D_{i_{0(k-1)}}^2\times S_{i_{0k}}^1\times D_{i_{0(k+1)}}^2 \times\cdots\times D_{i_{0p_0}}^2\right)\times D_{w}^{d(w)}\right).
\frac{1}{2}\sum_{n \neq 0}\left\{\frac{\omega_n}{2}\chi_n^2 +\frac{1}{2\omega_n}[(\partial_{\,z}+F^\prime)\chi_n]^2-\frac{\omega_n}{2}\chi_n^2 - \frac{1}{2\omega_n}[(\partial_{\,z}+F^\prime)\chi_n]^2\right\}\equiv 0\, ,
\int_\Omega|u|^{2p/(p+1)}\,dx =\int_\Omega|u|^{2p/(p+1)}\frac{V^{p/(p+1)}}{V^{p/(p+1)}}\,dx\le\left(\int_\Omega Vu^2\,dx\right)^{p/(p+1)}\left(\int_\Omega\frac{1}{V^p}\,dx\right)^{1/(p+1)},
(\lambda-\partial_t-\Delta)^{-\frac{\alpha}{2}}h(t,x):=\int_{t}^\infty\int_{\mathbb R^d}e^{-\lambda(s-t)}\frac{1}{(4\pi (s-t))^{\frac{d}{2}}}\frac{1}{(s-t)^{\frac{2-\alpha}{2}}}e^{-\frac{|x-y|^2}{4(s-t)}}h(s,y)dsdy,
\frac{s^{2}}{s^{2}-\kappa^{2}\nu^{2}}\frac{\Gamma_{\frac{\kappa}{s}}(\nu+1)}{\nu s^{\nu}}=\frac{s}{s+|\kappa|\nu}\frac{\Gamma(\nu)}{|2\kappa|^{\nu}}\frac{\Gamma(\frac{s}{|2\kappa|}-\frac{\nu}{2})}{\Gamma(\frac{s}{|2\kappa|}+\frac{\nu}{2})}
\left(\frac{t}{e^{t}-1}\right)^{r}e^{xt}=\sum_{n=0}^{\infty}B_{n}^{\left(r\right)}\left(x\right)\frac{t^{n}}{n!},\quad\left(\mbox{see \cite{key-11,key-12,key-13,key-14,key-15,key-16,key-17,key-18,key-19,key-20,key-21}}\right).
\times \left\langle \prod_{b = 1}^{N}\prod_{j = 1}^{n_{b}}\left[ : e^{+ i 2 \sqrt{\pi}\varphi^{( b )}( x_{j}^{( b )})}:_{Q^{0 , S}}: : e^{- i 2 \sqrt{\pi}\varphi^{( b )}( y_{j}^{( b )})}:_{Q^{0 , S}}\right] \right\rangle_{Q^{0 , S}}\leq
\sum_{k=0}^D \sum_{i\in \mathcal{C}_k^{\{1,\dots,D\}}}\sum_{h=0}^{D+1-k}\lambda^{D+1-(h+k)}\Biggl[ \binom{D+1-k}{h}- \Bigl(p_0+\sum_{j\not \in i}p_j\Bigr)\binom{D-k}{h}\Biggr] \frac{\partial^{h+k}p}{\partial t^{h}\partial x_{i_1}\cdots \partial x_{i_k}}= 0
(\mathcal{J}Q)_{\pi_{\overline{\theta}^{*}}}(\mathrm{S}_t, \mathrm{A}_t) = R_t(\mathrm{S}_t, \mathrm{A}_t)+\gamma\sum_{\mathrm{S}^{'}\subseteq \overline{\mathbf{S}}}p(\mathrm{S}_t, \mathrm{A}_t, \mathrm{S}^{'}) (\mathcal{H}Q)(\mathrm{S}^{'}, \mathrm{A}^{'}),
\sum_{n=1}^{N}\frac{(q^2;q^2)_N}{(q^2;q^2)_{N-n}}\frac{(-1)^{n-1}q^{n^2}}{(1-q^{2n})(q;q^2)_n}=\sum_{m=1}^{\infty}\left( \sum_{n=1}^{N}\sum_{\substack{\vec{\pi}\in S_3 |\vec{\pi}|=m}}w(\vec{\pi})\right)q^m.
w(E_\lambda)=\int_0^{|E_\lambda|}w^* \leq \Big(\frac{|E_\lambda|}{|E_{\lambda_o}|}\Big)^\frac{1}{\delta}\int_0^{|E_{\lambda_o}|}w^*= \Big(\frac{|E_\lambda|}{|E_{\lambda_o}|}\Big)^\frac{1}{\delta}w(E_{\lambda_o}).
a_D=\partial_a{\cal F}={\sqrt 2\over \pi}\int_{\Lambda^2}^u{dx \sqrt{x-u}\over \sqrt{x^2-{\Lambda^4}}},\qquad a={\sqrt 2\over\pi}\int_{-{\Lambda^2}}^{\Lambda^2}{dx \sqrt{x-u}\over \sqrt{x^2-{\Lambda^4}}},
\mathfrak{X}_{[m]\times P}(L_d^{n_0}, L_{i_1}^{n_1}, \cdots, L_{i_s}^{n_s})=(L_{i_1+1}^{n_0+1}, L_{i_2+1}^{n_1}, \cdots, L_{i_s+1}^{n_{s-1}}, L_0^{n_s-1}),
[ \hat{E}, \hat{x}] = - i \frac{\hbar}{m}\hat{p}, \, \, \, \, \, \, [ \hat{E}, \hat{p}] = i m \omega^{2}\hbar \hat{x}, \, \, \, \, \, \, [ \hat{x}, \hat{p}] = i \hbar ( 1 + \frac{1}{m c^{2}}\hat{E}) \, ,
y_{\pm}\simeq \frac{a}{t}\, \sum_{n = - \infty}^{\infty}\left[ ( n \pm b )^{2}+ c^{2}\right]^{1 / 2}, \ \b = \frac{\beta t}{4 \pi}\sqrt{\rho ( \lambda^{1 1}- \lambda^{2 2})}, \ \c^{2}= \frac{\rho \beta^{2}t^{2}( \lambda^{1 1}+ \lambda^{2 2})}{1 6 \pi^{2}}\simeq \frac{\alpha_{0}^{2}}{1 6 \pi^{2}}.
\begin{aligned}D_{|N_{\kappa,\gamma}|}^*(P_{N,\kappa,\gamma}) \:\:\: = \:\:\: \sup_{J}\frac{\mathcal{D}_{N_{\kappa,\gamma}}(J)}{|N_{\kappa,\gamma}|}\leq \:\:\: \frac{d}{N}+\frac{1}{|N_{\kappa,\gamma}|}\prod_{i=1}^d\frac{i+1}{i}\left(\frac{p_i-1}{2\log(p_i)}\log(N)+\frac{p_i+1}{2}\right). \end{aligned}
\left| T_2(s_1,s_2) \right| \lesssim \varepsilon^{3/2}\Delta^{1/2}, \left| \partial_{s_1}T_2(s_1,s_2) \right| \lesssim \varepsilon^{3/2}\Delta^{-1/2}, \left| \partial_{s_2}T_2(s_1,s_2) \right| \lesssim \varepsilon^{3/2}\Delta^{-1/2}.
perm(\sum )=\sum_{\pi \in S_{2\left\vert \alpha^{(l)}\right\vert}}\prod_{i=1}^{2\left\vert \alpha^{(l)}\right\vert}a_{i\pi(i)}\leq (2\left\vert \alpha^{(l)}\right\vert)!\prod_{i=1}^{2\left\vert \alpha^{(l)}\right\vert}a_{ii}.
\sum_{\mathclap{j\in \{k|d_k=d_i,k=1,\ldots,n\}}}(v_j^{\Re}v_j^{\Re\,T}+v_j^{\Im}v_j^{\Im\,T}) ~~~= \Re \sum_{\mathclap{j\in \{k|d_k=d_i,k=1,\ldots,n\}}}(v_j^{\Re}+iv_j^{\Im})(v_j^{\Re}+iv_j^{\Im})^{*}= \Re(P_M(d_i))
r(t)=[(\gamma_{LG}\frac{96\lambda V^{4}}{\pi^{2}\eta}(t+t_{0}))^{\frac{1}{2}}+(\frac{\lambda(t+t_{0})}{\eta})^{\frac{2}{3}}\frac{24\rho gV^{\frac{3}{8}}}{7\cdot96^{\frac{1}{3}}\pi^{\frac{4}{3}}\gamma_{LG}^{\frac{1}{3}}}]^{\frac{1}{6}}
\lim_{n\to\infty}\int_{t_0}^t \Big\langle \partial_t f_n(\tau),\Phi'(f_n(\tau))\Big\rangle_{(H^1)',H^1}\, \mathrm{d}\tau = \int_{t_0}^t \Big\langle \partial_t f(\tau),\Phi(f(\tau)) \Big\rangle_{(H^1)',H^1}\, \mathrm{d}\tau = 0\,.
\int \! d X \int \! d P \; e^{i ( - \frac{2}{3}\, X^{3}- 2 ( x + p^{2}- P ) X )}\star \delta ( P - E ) = \, 2^{2 / 3}\pi \, \, A i ( 2^{2 / 3}( x + p^{2}- E ) ) \; .
H T = \pi + 2 \bar{k}K ( \bar{k}) \sqrt{\frac{1 - 2 \bar{k}^{2}}{1 - \bar{k}^{2}}}- 2 K ( \bar{k}) E ( \bar{\phi}, \bar{k^{\prime}}) + 2 [ K ( \bar{k}) - E ( \bar{k}) ] F ( \bar{\phi}, \bar{k^{\prime}}) ,
\inf_{\varphi\in C^{\infty}_{cs}((0,\infty))}\frac{\int_{0}^{\infty}(\partial_r\varphi)^2\sinh^ndr}{\int_{0}^{\infty}\varphi^2\cosh^2\sinh^{n-2}dr}\leq \inf_{\varphi\in C^{\infty}_{cs}((0,\infty))}\frac{\int_{0}^{\infty}(\partial_r\varphi)^2\sinh^ndr}{\int_{0}^{\infty}\varphi^2\sinh^{n-2}dr}=\frac{(n-1)^2}{4}.
e_n H^{(1)\prime}_{n-1+{D\over 2}}(|\omega| r')+f_nH^{(2)\prime}_{n-1+{D\over 2}}(|\omega| r')-d_n H^{(1)\prime}_{n-1+{D\over 2}}(|\omega|r')={(2n+D-2)\Gamma\left ({D-2\over 2}\right )\over 4 (\pi r')^{D\over 2}|\omega|}.
\min_{\overline{\mathbf{\Phi}}}~~ \mathsf{Tr}(\overline{\mathbf{\Phi}}\mathbf{Y}\overline{\mathbf{\Phi}}^H\mathbf{Z}) - 2\Re\{\mathsf{Tr}(\overline{\mathbf{\Phi}}\mathbf{X})\}\mathrm{s.t.}~~ \overline{\mathbf{\Phi}}^H\overline{\mathbf{\Phi}}= \mathbf{I}_M,
\mathbb{E}[ d({\bf S}_i, \hat{\bf S}) ] & = \sum_{{\bf s}_i}P_{\bf S}({\bf s}_i) \left( - \frac{1}{M}\sum_{j=1}^M s_{i,j}\log P( \hat{s}_{i,j}= s_{i,j}) \right) & = \sum_{{\bf s}_i}P_{\bf S}({\bf s}_i) \left( - \frac{1}{M}\sum_{j=1}^M s_{i,j}\log( \sum_{y^i}P( \hat{s}_{i,j}= s_{i,j}| y^i) P_{Y^i}(y^i) ) \right).
\int_{\mathbb{R}^n}\rho_{y,s}(x,t) \, d\mu_s (y) \leq \int_{B_R (0)}\rho_{y,s}(x,t) \, d\mu_s (y) + 2^{n-1}e^{-\frac{3}{8}\frac{(R/2)^2}{2(s-t)}}D_1,
\langle \mathrm{d}\Phi_X , H_p v \rangle = \bigl\langle \mathrm{d}\Phi_X , T \mu (v) - V_p ( \nabla^\ast_v \mu ) \bigr\rangle = v \bigl[ \langle \mu, X \rangle \bigr] - \langle \nabla_v^\ast \mu, X \rangle = \langle \mu, \nabla_v X \rangle ,
\frac{{\rm d}}{{\rm d}s}\left(\frac{\alpha(s)-P_S(s)}{R_S(s)}\right)=\frac{1}{R_S}\left[\mathbf{t}+\frac{\rho\rho'}{\tau R_S^2}\varsigma\,\mathbf{n}+\left(\frac{\rho'\,^2}{\tau^2R_S^2}-1\right)\varsigma\,\mathbf{b}\right].
||S_{\theta_i}\Delta_3||_{s,\ast,T}\le C\theta_i^{s-6}||\Delta_3||_{6,\ast,T}\le C\theta_i^{s-6}\cdot\delta\theta_i^{8-\alpha}=C\delta\theta_i^{s+2-\alpha}, \quad\mbox{for}\,\,\,s\ge 6,
L_{eff}= - \Biggl( \frac{1}{2}\gamma^a_{~b \mu}F_{a~~m}^{~b \mu~ n \nu}\gamma^m_{~n \nu}+\frac{1}{2}\lambda^b_{~\nu}D_{b~a}^{~\nu~\mu}\lambda^a_{~\mu}+ \lambda^m_{~n}\biggl( G^{~~n j ~b c}_{m ~~a}\nabla_j +T^{~n~b c}_{m~ a}\biggr)\gamma^a_{~bc}\Biggr) e
E_i\left\{\boldsymbol{\eta}(j)\boldsymbol{\eta}(l)\right\}= \begin{cases}\mu_{\eta,i}^2\boldsymbol{1}\boldsymbol{1}^\top , j\neq l \sigma^2_{\eta,i}\boldsymbol{I}+ \mu_{\eta,i}^2\boldsymbol{1}\boldsymbol{1}^\top , j=l\end{cases},
\langle \bar{q}q \rangle =\langle \bar{q}q \rangle_v \cos ( 2\pi (x-x_0)/L)\ ,\quad\langle \bar{q}\,{\rm i}\gamma^5 q \rangle = -\langle \bar{q}q \rangle_v\sin (2 \pi (x-x_0)/L)\ ,
\int L^{-1}(\tau,\tau^{\prime})j(\tau^{\prime})d\tau \; d\tau^{\prime}= \frac{e^{\frac{ge_{0}F}{2}}}{\cosh\frac{ge_{0}F}{2}}\int_{0}^1 e^{-ge_{0}F\tau^{\prime}}j(\tau^{\prime})d\tau^{\prime}.
\|\nabla_p s_f\|_{p-Sl}^2 = \frac{1}{3p-1}(2s_{f}(p)^2+2(1-p)s_{f}(p)\delta+\delta^2(1-p)p) +\min (\|a(t)\|^2+\|b(t)\|^2+\|c(t)\|^2)
V \left( t , \bar{t}, ( E , M ) \right) \sim | Z^{\mathrm{l o c a l}}( t , \bar{t}, ( q , p ) ) |^{2}= \left| e^{\frac{K ( t , \bar{t})}{2}}[ X^{\Lambda}( t ) q_{\Lambda}- F_{\Lambda}( t ) \, p^{\Lambda}] \right|^{2}\ ,
\begin{aligned}\sum_{m=1}^{k+l}(U_2^nKU_1^{-n}\, U_1^{k-m}e, U_2^{l-m+1}\bar e)=\sum_{m=1}^{k+l}(K U_1^{k-m-n}e, U_2^{l-m-n+1}\bar e) = \sum_{m=1}^{k+l}\eta_{m+n-1}=\sum_{m=n}^{n+k+l-1}\eta_m.\end{aligned}
|T'| =2^{v-m-2\nu+2}\sum_{j\,{\rm odd}}\binom{\nu}{j}3^{\nu-j}&=2^{v-m-2\nu+2}\cdot\frac{1}{2}\big((3+1)^\nu-(3-1)^\nu\big) &=2^{t+2}-2^{t+2-\nu}~~~~\hbox{(as $t=v-m-1$).}