harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Big \langle J_{\mu_{1}}^{( 1 )}( x_{1}) J_{\mu_{2}}^{( 1 )}( x_{2}) . . . . . J_{\mu_{n}}^{( 1 )}( x_{n}) J_{\nu_{1}}^{( I_{1})}( y_{1}) J_{\nu_{2}}^{( I_{2})}( y_{2}) . . . . . J_{\nu_{k}}^{( I_{k})}( y_{k}) \Big \rangle_{0}^{\theta}
\frac{d}{dt}(\begin{matrix}M_{x}\\M_{y}\\M_{z}\end{matrix})=(\begin{matrix}-\frac{1}{T_{2}}&\gamma B_{z}&-\gamma B_{y}\\ -\gamma B_{z}&-\frac{1}{T_{2}}&\gamma B_{x}\\ \gamma B_{y}&-\gamma B_{x}&-\frac{1}{T_{1}}\end{matrix})(\begin{matrix}M_{x}\\M_{y}\\M_{z}\end{matrix})+(\begin{matrix}0\\ 0\\ \frac{M_{0}}{T_{1}}\end{matrix})
\gamma_{g , 9}: | D_{\mathrm{i n v}}\rangle \rightarrow{\frac{1}{\sqrt{D}}}\sum_{\alpha}\sum_{\beta , \gamma}c_{\alpha \beta}{\overline{{c}}}_{\gamma \alpha}| d_{\beta}\otimes{\overline{{d}}}_{\gamma}\rangle ={\frac{1}{\sqrt{D}}}\sum_{\alpha}| d_{\alpha}\otimes{\overline{{d}}}_{\alpha}\rangle = | D_{\mathrm{i n v}}\rangle ~ ,
\begin{array}{l}4\cos^2\left(\frac{2\pi r}{p+1}\right)-1 = 3 - \frac{16\pi^2r^2}{p^2}+O\left(\frac{r^2}{p^3}\right), 1-4\cos \left(\frac{2(p-1)\pi s}{p^2+1}\right) \cos\left( \frac{2(p+1)\pi s}{p^2+1}\right) = -3 + \frac{16\pi^2s^2}{p^2}+O\left(\frac{s^2}{p^3}\right).\end{array}
\mathrm{PD}^{-1}([f]) =\bigg( \int_{\overline{\alpha}_y}\mathrm{PD}^{-1}([f]) \bigg)\mathrm{PD}^{-1}([\overline{\alpha}_x])- \bigg( \int_{\overline{\alpha}_x}\mathrm{PD}^{-1}([f]) \bigg)\mathrm{PD}^{-1}([\overline{\alpha}_y]) &- \sum_{i = 1}^{n-1}\bigg( \int_{\partial \overline{D}_i}\mathrm{PD}^{-1}([f]) \bigg)\mathrm{PD}^{-1}([\overline{\alpha}_i]),
F ( T ) = \frac{\Delta n^{2}}{\pi a}\left( \frac{2 3}{3 8 4}- \frac{\zeta ( 3 )}{8 \pi^{2}}t^{3}+ \frac{7}{1 0 8 0}t^{4}- \frac{2 2}{1 4 1 7 5}t^{6}+ \frac{4 6}{5 5 1 2 5}t^{8}+{\cal O}( t^{1 0}) \right)
(- 1)^{N-M-1}d_{j-l}=(2N+1)b_j^{(N-M-1)}(-\frac n2-M-1)-a_{j}^{(N-M-1)}(-\frac n2-M-1) = 2N\frac{(N-M-1)!(-4)^{N-M-j+l-1}}{(j-l)!(2N-2M-2j+2l-1)!}(j-l-2N+M+\frac 32)_{N-M-j+l-1}.
\left.\begin{aligned}\boldsymbol{y}_t^{*}:=(y_t^{*,i})_{1\leq i\leq N}:=(P_t x_t^{*,i}+\varphi_t^{*,i})_{1\leq i\leq N}, &\boldsymbol{z}_t^{*}:=(z_t^{*,i,j})_{1\leq i,j\leq N}:=(P_t\sigma+\Lambda_t^{*,i,j})_{1\leq i,j\leq N},\boldsymbol{z}_t^{0,*}:=(z_t^{0,*,i})_{1\leq i\leq N}:=(P_t\sigma_0+\Lambda_t^{0,*,i})_{1\leq i\leq N}.\end{aligned}\right.
\displaystyle\sum\limits_{e=1}^{l_{r-1}}\sum\limits_{f=1}^{l_n}b^{t}_{ef}p^{r-1}_{\tilde{e}e}p^{n}_{\tilde{f}f}=\sum\limits_{f=1}^{l_n}b^{f}_{\tilde{e}\tilde{f}}p^{r}_{11}p^{n}_{ft}.
d Q_{n}( \zeta ) \left[ 1 - \sqrt{1 -{\frac{c_{1}^{( n )}}{d \zeta}}-{\frac{c_{2}^{( n )}}{d \zeta^{2}}}}\right] \equiv \sum_{i = 1}^{n}p_{i}\zeta^{i}+ p_{0}+ \sum_{j = 1}^{\infty}p_{- j}\zeta^{- j},
\Psi_{h}( z_{1}, \dots , z_{N}) = U^{- 1}( z_{1}, \dots , z_{N}) \exp \left( -{\frac{1}{2 \pi \kappa}}\sum_{\alpha}\lim_{z \rightarrow z_{\alpha}}\int^{z}{\frac{\hat{Q}_{\alpha}^{2}}{z - z_{\alpha}}}d z \right)
\left\{Q_{2^{j_{0}}\rho_{z_{i}}}(z_{i})\right\}_{i\in\mathbb{N}}D_{\kappa\lambda}\subset\bigcup_{z_{0}\in D_{\kappa\lambda}}Q_{\rho_{z_{0}}}(z_{0})\subset\bigcup\limits_{i=1}^{\infty}Q_{5\times 2^{j_{0}}\rho_{z_{i}}}(z_{i}).
J(X_\tau,\alpha,\beta)=\int_{X_\tau}d^{m+n}z \prod_{i=1}^{m}z_i^\alpha(1-z_i)^\beta \prod_{\hat{i}=\hat{1}}^{\hat{n}}z_{\hat{i}}^{\alpha'}(1-z_{\hat{i}})^{\beta'}\left[\prod z_{ij}^{2\rho}\prod z_{\hat{i}\hat{j}}^{2\rho'}\prod z_{i\hat{j}}^{2\sigma}\right]_\tau.
\phi_0^{(-1)}={1\over{J^{-1}_H}'},\qquad\; \phi_1^{(-1)}={J^{-1}_H\over{J^{-1}_H}'},\qquad \; \phi_2^{(-1)}={\left(J^{-1}_H\right)^2\over{J^{-1}_H}'}.
\jmath_{U_1^n}\left(u_1^n, d\right) = \sum_{i = 1}^n \frac{\min(\theta_n, \sigma_{n, i}^2)}{2\theta_n}\left(\frac{x_i^2}{\sigma_{n,i}^2}- 1\right) + &\frac{1}{2}\sum_{i = 1}^n \log\frac{\max(\theta_n, \sigma_{n, i}^2)}{\theta_n},
R_{k}( x_{1}, \dots , x_{k}) = \left( \frac 2 \pi \right)^{k}\left( \prod_{p = 1}^{k}x_{p}\right) \lim_{\{a_{f}\}\to \infty}\frac{\det [ C_{N}( z_{p}, \zeta_{q}) ]_{p , q = 1 , \dots , k + N_{f}}}{\det [ C_{N}( a_{f}, i m_{g}) ]_{f , g = 1 , \dots , N_{f}}}
\mathbb{E}[\overline{V}_t ] \le \frac{c_2}{c_1}=\frac{\eta^2}{2c_1(2a\lambda_2-c_1)}+\frac{b | Tr(L) |}{2c_1 N}+\frac{m\tau^2\tilde{\gamma}(N-1)}{2c_1 N}.
\mu^{1/\alpha}\circledast\mathbf{b}_{\alpha,\rho}=\mu^{\boxtimes 1/\alpha}\boxtimes\mathbf{b}_{\alpha,\rho}, \alpha\in(0,1] , \rho \in \{0,1/2,1\}; & \mu^{1/\alpha}\circledast\mathbf{b}_{\alpha,\rho}=\mu \circlearrowright \mathbf{b}_{\alpha,\rho}, (\alpha,\rho)\in\mathfrak{A}.
P_{1,2N+1}(i,j,q) = P_{1,2N}(i,j,q) + \chi(i>1) q^{3N+2}P_{2,2N}(i-1,j,q), P_{1,2N+2}(i,j,q) = P_{1,2N+1}(i,j,q) + q^{3(N+1)}P_{2,2N+1}(i,j,q), P_{2,2N+1}(i,j,q) = P_{2,2N}(i,j,q) + \chi(j>1) q^{3N+1}P_{1,2N}(i,j-1,q), \intertext{and}P_{2,2N+2}(i,j,q) = P_{2,2N+1}+ q^{3(N+1)}P_{1,2N+1}(i,j,q),
\delta_{t}(e_C)^{k}_{ij}-\mu\Delta_{h}(e_C)^{k}_{ij}- \mathfrak{w}_{1,1}\Delta_{h}(e_C)^{k}_{ij}=\frac{1}{2}\left[g(u^k_{ij})-g((U_C)^k_{ij})\right] +\frac{1}{2}\left[g(u^0_{ij})-g((U_C)^0_{ij})\right]+ (R)^{k}_{ij},(x_i,y_j)\in\Omega_{h},
d\beta^{n}=e^{-2n\rho}(-2\,d\rho\wedge d\theta-2\,d\rho\wedge\alpha_{n-1})\wedge(d\alpha_{n-1})^{n-1}&=-2e^{-2n\rho}\,d\rho\wedge d\theta\wedge(d\alpha_{n-1})^{n-1}
\begin{pmatrix}\nu_0,\nu_1,\dots,\nu_i\end{pmatrix}A_{i+1}= \begin{pmatrix}1,-1,\dots,(-1)^i\end{pmatrix}\quad\textup{where}A_{i+1}=\begin{pmatrix}\binom{s+r-2-2i}{s-1-i+j-k}\end{pmatrix}_{0\leq j,k\leq i}.
\|(f\ast g)\tau^{k}\|_{1}\leq \int\int |f(y)\|g(y^{-1}x)\|\tau^{k}(x)|dydx &=\int\int |f(y)\|g(y^{-1}x)\|\tau^{k}(x)|dxdy &=\int\int |f(y)\|g(x)\|\tau^{k}(yx)|dxdy &\leq\|f\tau^{k}\|_{1}\|g\tau^{k}\|_{1}.
\Pi_{\mathcal{K}}(g(x_t)+\lambda_t) =\Pi_{\mathcal{K}}\big(g(\overline{x})+tg'(\overline{x})\xi^*+\overline{\lambda}+tv^*+o(t)\big) =\Pi_{\mathcal{K}}(g(\overline{x})+\overline{\lambda}) +t\Pi_{\mathcal{K}}'(g(\overline{x})\!+\!\overline{\lambda};g'(\overline{x})\xi^*+v^*)+o(t) =g(\overline{x})+tg'(\overline{x})\xi^*+o(t),
H ( x_{1}, x_{2}) = \frac{1}{x_{1}^{2}- x_{2}^{2}}\log \left( \frac{\mu^{2}- x_{1}^{2}}{\mu^{2}- x_{2}^{2}}\right) + \frac{\mu^{2}- x_{1}x_{2}}{( \mu^{2}- x_{1}^{2}) ( \mu^{2}- x_{2}^{2})}.
b_{\frac{n+2}{2}}= x_{\frac{n+2}{2}}\left(x_{\frac{n}{2}}x_{\frac{n+2}{2}}\right) = \left(x_{\frac{n+2}{2}}x_{\frac{n}{2}}\right)x_{\frac{n+2}{2}}= (-1)^{\frac{n+2}{2}}zx_{\frac{n+2}{2}}= (-1)^{\frac{n+2}{2}}c_{\frac{n+2}{2}},
\chi = \left( 1 2 8 \pi^{2}\right)^{- 1}\int d^{4}x g \varepsilon_{\mu \nu}^{\tau \omega}R_{\tau \omega \alpha \beta}\varepsilon_{\rho \sigma}^{\alpha \beta}R^{\mu \nu \rho \sigma}+ b o u n d a r y t e r m s
\real \left( \frac{zw_{{\nu}}^{\prime}(z)}{w_{{\nu}}(z)}\right) \geq1-\sum_{n\geq1}\frac{\left\vert z\right\vert}{j_{{\nu},n}^{4}-\left\vert z\right\vert}=\frac{\left\vert z\right\vert w_{{\nu}}^{\prime}(\left\vert z\right\vert )}{w_{{\nu}}(\left\vert z\right\vert )},
w_{\lambda_0}(x)=C\int_{\Sigma_{\lambda_0}}(\frac{1}{|x-y|^{n-\alpha}}-\frac{1}{|x-y^{\lambda_0}|^{n-\alpha}})(\frac{f(|y^{\lambda_0}|^{n-\alpha}v(y^{\lambda_0}))}{|y^{\lambda_0}|^{n+\alpha}}-\frac{f(|y|^{n-\alpha}v(y))}{|y|^{n+\alpha}})dy.
\begin{array}{lll}|\tilde{\Lambda}_n(s,x) - \tilde{\Lambda}(s,x)| = |\Lambda_n(s,x,u_n(s,x)) - \Lambda(s,x,u(s,x))| \leq |\Lambda_n(s,x,u_n(s,x)) - \Lambda(s,x,u_n(s,x))| + |\Lambda(s,x,u_n(s,x)) - \Lambda(s,x,u(s,x))| := I_1(n,s,x) + I_2(n,s,x) \ . \end{array}
K^{a,i,\tau}_{k}(T):=\sum_{l\geq k}\|Q|_{U^{(k)}_i}(k,l+1)^{-1}\|\|Z(l+1)\|\langle\|Q(l)\|\rangle^{a}\|Q(l+1)\|^\tau, &C^{a,i,\tau}_{k}(T):= \sum_{0 \leq l < k}\|Q|_{E^{(l+1)}_i}(l+1,k)\|\|Z(l+1)\|\langle \|Q(l)\|\rangle^{a}\|Q(l+1)\|^\tau.
A(q)=\sum_{n\ge0}\frac{q^{(n+1)^{2}}(-q;q^{2})_{n}}{(q;q^{2})_{n+1}^{2}}=\sum_{n\ge0}\frac{q^{n+1}(-q^{2};q^{2})_{n}}{(q;q^{2})_{n+1}}
\left[ E^{2}- P_{z}^{2}- \frac{m^{2}}{4}- 4 \omega \left( \frac{E}{m}+ ( N + \left| m \right| + \frac{1}{2}) \right) \right] + i \omega P_{z}b_{-}+ i \omega \left( N + \left| m \right| + \frac{1}{2}\right) P_{z}b_{+}= 0
- \partial_{q}W = \frac{4 \pi \hbar}{T_{B H}}\int_{- \infty}d z r^{2}F \biggl [ \frac{\partial_{q}F}{F}\langle T_{\tau}^{\tau}+ T_{z}^{z}\rangle + 2 \frac{\partial_{q}r}{r}\langle T_{\theta}^{\theta}+ T_{\phi}^{\phi}\rangle \biggr ] + \cdots
h_{0}^{( 1 )}( k ) = - k^{2}- m^{2}- \int{\frac{d^{4}l}{( 2 \pi )^{4}}}{\frac{R_{0 0}( k , l ) \, ( l^{2}+ m^{2})}{( l^{2}+ m^{2})^{2}+ [ h_{1}^{( 0 )}( l ) ]^{2}+ [ h_{2}^{( 0 )}( l ) ]^{2}+ [ h_{3}^{( 0 )}( l ) ]^{2}}}\, .
\delta P^{A}_{\mu(1)}=\frac{1}{2\pi\alpha'}\sqrt{-\bar{G}}\left( (\bar{D}_{ij}\;^{A}\Phi^{j}_{(1)})\bar{n}^{i}_\mu+(\bar{\Omega}_{i}\;^{AB}-\bar{G}^{AB}\bar{\Omega}_{iC}\;^{C})\Phi^{i}_{(1)}\bar{x}_{\mu,B}\right)
\left. \begin{array}{l l}\frac{\partial V(\textbf{x})}{\partial x_E}\!\!\!\! = \frac{\nu_E x_E - \frac{1}{2}[ (y_2-y_3) R_1^2 + (y_3-y_1) R_2^2 + (y_1-y_2) R_3^2]}{\Lambda}&~~ - \mu_E \frac{\nu_E R_E^2 - \nu_1 R_1^2 -\nu_2 R_2^2 -\nu_3 R_3^2}{2\Lambda^2}.\end{array}\right.
\frac{d}{ds}\Big\lvert_{s=0}g_H(T_*N,N^\perp) =g_H(\nabla_{N^\perp}T_*N,N^\perp)+g_H(T_*N,\nabla_{N^\perp}N^\perp)\lvert_{s=0}&=g_H(T_*N,N)^2\kappa_1\lvert_{s=0}-g_H(TN,N)\kappa_\eta\lvert_{s=0}&=\kappa_1-\kappa_\eta<0.
\sigma^1 \,=\,\left( \begin{array}{cc}0 1 1 0 \end{array}\right) \;\; , \;\;\sigma^2 \,=\,\left( \begin{array}{cc}0 -i i 0 \end{array}\right) \;\; , \;\;\sigma^3 \,=\,\left( \begin{array}{cc}1 0 0 - 1 \end{array}\right) \;.
\sum_{k=0}^a\frac{(-a)_{k}(\beta)_{k}}{(1)_{k}^2}\cdot z^k={}_2F_1\bigg[\begin{matrix}-a \beta &1\end{matrix}\bigg|\,z\bigg]=(1-z)^a{}_2F_1\bigg[\begin{matrix}-a 1-\beta &1\end{matrix}\bigg|\,\frac{z}{z-1}\bigg] = (1-z)^a\sum_{k=0}^a\frac{(-a)_{k}(1-\beta)_{k}}{(1)_{k}^2}\cdot\frac{z^k}{(z-1)^k}.
\frac{d \phi^{2}}{d \phi^{1}}= - \frac{\partial W^{1}}{\partial \phi^{1}}\left( \frac{\partial W^{1}}{\partial \phi^{2}}\right)^{- 1}\hspace{0 . 5 c m}\equiv \hspace{0 . 5 c m}\frac{\partial W^{1}}{\partial \phi^{1}}d \phi^{1}+ \frac{\partial W^{1}}{\partial \phi^{2}}d \phi^{2}= d W^{1}= 0
S^m(f_1,\dots,f_m) (x) \ge C_m |x|^{1-m}\int_{B^{m-1}(0,\frac1{300m})}|\tilde{u}|^{-\frac1p}\left(-\log(|\tilde{u}|)\right)^{-\frac{2}{p}}d\tilde{u}&\gtrsim \begin{cases}|x|^{1-m}\frac1p = m-1, \infty \frac1p >m-1. \end{cases}
w(\xi,s)=(1+t)^{\frac{d}{2}}r^{-A}u(r,t)=(1+t)^{\frac{d}{2}}r^{-A}U(r)u_*(r,t)=(1+t)^{\frac{d}{2}}U_d(r)u_*(r,t) \mbox{with $\xi=(1+t)^{-\frac{1}{2}}r$ and $s=\log(1+t)$},
Z \left[ J^{\mu}, \overline{{{\eta}}}, \eta \right] = \int \left[ d A_{\mu}\right] \Im \left( A_{\mu}\right) \exp \left( - \left\langle \frac 1 4 F_{\mu \nu}F^{\mu \nu}\right\rangle + \left\langle J^{\mu}A_{\mu}\right\rangle + \left\langle \overline{{{\eta}}}^{\prime}\left( i \gamma^{\mu}\partial_{\mu}\right)^{- 1}\eta^{\prime}\right\rangle \right) ,
I =c_n\int^\pi_{-\pi}z^{-n}\frac{\phi_{2n+1}(z)}{z-1}\, x^j(1-\cos\theta)f_3(\theta)d\theta &= -\frac{c_n}{2}\int^\pi_{-\pi}z^{-n}\phi_{2n+1}(z)x^j(1-1/z)f_3(\theta)d\theta.
\sum_{r_1\geq\dots\geq r_n\geq 0}\frac{q^{r_1^2+\cdots+r_n^2+r_i+\cdots+r_n}}{(q)_{r_1-r_2}\cdots(q)_{r_{n-1}-r_n}(q^2;q^2)_{r_n}}=\frac{(q^{2n+2};q^{2n+2})_{\infty}}{(q)_{\infty}}\, \cdot \theta(q^i;q^{2n+2}).
g \simeq \tilde{\Lambda}^{2}\, ( - A^{2}\eta^{2}d s^{2}+ d \eta^{2}+ d \rho^{2}) + \tilde{\Lambda}^{- 2}\rho^{2}d \phi^{2}, \qquad \tilde{\Lambda}\equiv 1 + \frac{B^{2}\rho^{2}}{4},
V_A(\beta)^{n_1}V_A(2\beta)^{n_2}V_A(0)^{2k-n_1-n_2}=e^{-\i N \frac{\beta}{2}n_1}e^{-\i N \beta n_2}\mathcal{Z}_A(0)^k \overline{\mathcal{Z}_A(0)}^{k-n_1-n_2}\overline{\mathcal{Z}_A(\beta)}^{n_1}\overline{\mathcal{Z}_A(2\beta)}^{n_2}.
\psi = \psi \left( \varepsilon^{a b}\right) = \left( 1 , 0 , \sqrt{{\frac{1}{2}}}\varepsilon^{a b}, 0 , \sqrt{{\frac{3 . 1}{4 . 2}}}\varepsilon^{\left( a b \right.}\varepsilon^{\left. c d \right)}, 0 , . . . \right) \; .
\epsilon^{i_{1}. . . i_{N}j_{1}. . . j_{M}}\epsilon_{i_{1}. . . i_{N}l_{1}. . . l_{M}}= N ! \ \delta_{[ l_{1}. . . l_{M}]}^{j_{1}. . . j_{M}}: = N ! \ \delta_{[ l_{1}}^{j_{1}}. . . \delta_{l_{M}]}^{j_{M}}\
d-r =2\cdot 3^{k-1}+2\cdot 3^{k-2}+\cdots+2\cdot 3^2+0\cdot 3^1+1\cdot 3^0, r-1-h =2\cdot 3^{k-1}+2\cdot 3^{k-2}+\cdots+2\cdot 3^2+2\cdot 3^1+1\cdot 3^0-h.
\frac{d}{d \phi^2}\left ( v(\phi^2) \, (\exp (2 \sinh^{-1}\phi^2)-2)^{\lambda/2 - 3/2}\right ) = \frac{1}{2}\frac{k \; (\exp(2\sinh^{-1}\phi^2) -2)^{\lambda/2 - 3/2}}{\phi^2 (2 + 2 V')^{3/2}-1}
d^2\!f(\overline{x}|\overline{v})(w) =\inf_{z\in\mathbb{R}^n}\big\{d^2\vartheta(F(\overline{x})) (dF(\overline{x})(w))\,|\,d^2F(\overline{x})(w|z))-\langle\overline{v},z\rangle\}=d^2\vartheta(F(\overline{x}))(dF(\overline{x})(w))+\langle w,\nabla^2\!F(\overline{x})w\rangle\sup_{\xi\in \Lambda_{\overline{x},\overline{v}}}\xi
F(f) = \mathbb E \sum_{n=1}^{\infty}\langle f_n, \tilde g_n \rangle = \mathbb E \sum_{n=1}^{\infty}\mathbb E_{n}\langle f_n, \tilde g_n \rangle = \mathbb E \sum_{n=1}^{\infty}\langle f_n,\mathbb E_{n}\tilde g_n \rangle = \mathbb E \sum_{n=1}^{\infty}\langle f_n, g_n \rangle.
\begin{aligned}s_b(N_k^{n}) =\frac{n+1}{2}(b-1)(2^{k}-1) &+(b-C_{n}^{n-1})+(b-C_{n}^{n-3})+\cdots + (b-C_{n}^{0}) &+(C_{n}^n-1)+(C_{n}^{n-2}-1)+\cdots +(C_{n}^{1}-1),\end{aligned}
J_{p}^{a}\overline{{{| n , m \rangle}}}_{\jmath}= \left( t n + m ( 2 \jmath + 1 - m ) \right) \mu^{a}( \jmath - \jmath_{0}, \jmath_{1}, m ) | n - p , m - a \rangle_{\jmath}\; \; \forall J_{p}^{a}\in{\cal E}_{+}
D_{\mu \nu}^{+}( x , x^{\prime}) \equiv \frac{1}{4}\left( 4 F^{\alpha}{}_{( \mu}( x ) \; F_{\nu ) \alpha}( x^{\prime}) + \eta_{\mu \nu}\; F_{\lambda \beta}( x ) \; F^{\lambda \beta}( x^{\prime}) \right) ;
\frac{d e}{d \omega}= \frac{\gamma^{3}}{2 \pi^{3}R_{0}^{3}}\; \frac{\omega^{2}+ ( \gamma v \omega_{0})^{2}}{\omega^{2}}\; \frac{v^{3}w^{2}}{w^{2}+ ( 2 \gamma v )^{2}}\; h_{\gamma}( w ) ,
I(P,L) \le akb^j \left( (m/k)^{1/2}n^{3/4}+ (m/k) \right)+ kB \left( (m/k)^{2/3}n^{1/3}s^{1/3}+ n \right ) & = ak^{1/2}b^j m^{1/2}n^{3/4}+ ab^j m + k^{1/3}B m^{2/3}n^{1/3}s^{1/3}+ kB n ,
A = \{x \in \textstyle\prod X_{n, k}\colon (\exists m)(\forall n >m)(\exists j)(\forall k >j)(x(n, k) \in A_{n,k})\}, B =\{x \in \textstyle\prod X_{n, k}\colon (\exists m)(\forall n >m)(\exists j)(\forall k >j)(x(n , k) \in B_{n,k})\}.
\left\{0 < b_{1}+ b_{2}< 1 , b_{1}b_{2}\leq 0 \right\}\cup \left\{b_{1}+ b_{2}\geq 1 , b_{1 , 2}> 0 \right\}\cup \left\{b_{1}+ b_{2}< 0 , b_{1}b_{2}< 0 \right\}.
T_{m}(f_{1},f_{2})(x):=\int_{\mathbb{R}^{4}}m(\xi,\eta)\hat{f_{1}}(\xi_{1},\eta_{1})\hat{f_{2}}(\xi_{2},\eta_{2})e^{2\pi ix\cdot((\xi_{1},\eta_{1})+(\xi_{2},\eta_{2}))}d\xi d\eta,
A_{x_{\perp}}^{2 1}= \int_{R^{2}}d^{2}z \, \psi_{- \frac{1}{2}}^{\dagger}( z + \omega ) \frac{\partial}{\partial x_{\perp}}\psi_{\frac{1}{2}}( z - \omega ) = \frac{\bar{x}_{\perp}}{2 \pi | x_{\perp}|}e^{2 i x \cdot \omega}( I + J ) .
L = L(d,\beta) \leq 11 + \big(1 + \lceil \log_2 \beta \rceil \big) \cdot \Big(11 + \frac{\beta}{d-1}\Big) \leq 11 + \big(1 + \lceil \log_2 \beta \rceil \big) \cdot \Big(11 + \frac{2\beta}{d}\Big)
\theta \left( \begin{array}{c}{\psi_{+}}\\{\psi_{-}}\\ \end{array}\right) = D^{( s )}( i \sigma_{2}) \left( \begin{array}{c}{\psi_{-}^{*}}\\{\psi_{+}^{*}}\\ \end{array}\right) , \pm \mathrm{= ( a n t i ) p a r t i c l e \thinspace \thinspace d o u b l i n g}
\delta\Psi\bigg(\phi^A,\,\phi^*_A;\, \frac{\delta_l}{\delta\phi^A},\,\frac{\delta}{\delta\phi^*_A}\bigg)= \delta\Psi\bigg(\varphi^i,\,\psi^\alpha,\,\phi^*_A;\, \frac{\delta_l}{\delta\varphi^i},\,\frac{\delta_l}{\delta\psi^\alpha},\, \frac{\delta}{\delta\phi^*_A}\bigg)
\begin{array}{l l}{{( a ) ~ \mathrm{F o r ~ e a c h ~ ( t h r e e ~ p o i n t ) ~ v e r t e x}}}&{{: - i ( 2 \pi )^{2}c^{a b c},}}\\{{( b ) ~ \mathrm{F o r ~ e a c h ~ p r o p a g a t o r}}}&{{: \frac{i}{( 2 \pi )^{2}( p_{a}^{2}- m_{a}^{2})},}}\\{{( c ) ~ \mathrm{L o o p ~ i n t e g r a t i o n}}}&{{: \int d^{2}l .}}\\ \end{array}
{\frac{1}{- ( \gamma \cdot D )^{2}+ M^{2}}}={\frac{1}{-{\bf D}^{2}+ \Phi^{2}+ M^{2}}}+{\frac{1}{-{\bf D}^{2}+ \Phi^{2}+ M^{2}}}\left({\frac{i}{2}}\gamma_{\mu}\gamma_{\nu}G_{\mu \nu}\right){\frac{1}{-{\bf D}^{2}+ \Phi^{2}+ M^{2}}}+ \cdots
(P\psi)(y)=\sum_{a\in\alpha}\zeta(y_a)\psi(y_a), (L \psi)(y, \ell) = \begin{cases}\sum_{a \in \alpha}\zeta(y_a) \psi(y_a, \tau(y_a)-1), \ell=0 \psi(y, \ell-1), \ell \ge 1 \end{cases}.
\sum_{{\lambda}= 1}^{2}{\epsilon}_{\mu}^{({\lambda})}( k ){\epsilon}_{\nu}^{({\lambda})}( k ) = - g_{{\mu}{\nu}}+ \frac{n_{\mu}k_{\nu}+ n_{\nu}k_{\mu}}{k_{-}}- n^{2}\frac{k_{\mu}k_{\nu}}{k_{-}^{2}}.
- \beta F^{\mathrm{{\tiny S t r i n g}}}( \beta ) = - \frac{N^{2}}{8}\int_{0}^{\infty}\frac{d t}{2}t^{- 1}\left[ \theta_{3}\left( 0 , \frac{2 \pi i t}{\beta^{2}}\right) - \theta_{2}\left( 0 , \frac{2 \pi i t}{\beta^{2}}\right) \right] f ( t )
L_k(\mathbf{a}_{k, 3}) + \sum_{j=1}^t V_k(\mathbf{a}_{k, 3j},\mathbf{a}_{k, 3j+3}) = \sum_{j=1}^tV^\circ_k(\mathbf{a}_{k, 3j}) + L_k(\mathbf{a}_{k, 3+3t})
\begin{array}{ccccccc}\nabla Z_{AB}= P_{ABCD}\overline{Z}^{CD}\longrightarrow \nabla \widehat{Z}_{AB}={\cal P}_A^{A^\prime}\,{\cal P}_B^{B^\prime}P_{A^\prime B^\prime CD}\, \overline{Z}^{CD}\ , \end{array}
W_{e}( 0 ) = \oint p_{\tilde{\eta}}( \tilde{a}) d \tilde{a}= 2 \int_{0}^{\lambda^{- 1 / 2}}z ( 1 - \lambda z^{2})^{1 / 2}d z ={\frac{2}{3 \lambda}}={\frac{\pi \tau^{2}}{G}}\ .
\mathcal{I}_1^*(h,k) = \sum_{\substack{\alpha\in A \beta\in B}}\frac{1}{2(2\pi i )^2}\int_{(\epsilon)}\int_{(\epsilon)}\mathcal{J}\,ds_2\,ds_1 + O\big( X^{\varepsilon}Q^{\frac{3}{2}+\varepsilon}(hk)^{\varepsilon}\big),
\mu_n(x) = \sum_{x=g_1\ldots g_n}\mu(g_1) \cdot \ldots \cdot \mu(g_n) &= \sum_{x^{-1}=g^{-1}_n \ldots g^{-1}_1}\check{\mu}(g_1^{-1})\cdot \ldots \cdot \check{\mu}(g_n^{-1}) &= \check\mu^n(x^{-1}).
\begin{aligned}p_{A_2}(N,\ell;2)= \frac{1}{\ell}+\frac{1}{\ell}\binom{N}{\ell}^{-1}&\times\sum_{r=1}^{N-1}(-1)^{r}\cdot\binom{N-1}{r}^{-1}\left[\binom{N-1}{\ell-1}-\binom{r-1}{\ell-1}\right].\end{aligned}
\begin{aligned}{\frac{8(e^{-\lambda_{n,1}^{(1)}}-e^{-\lambda_{n,1}^{(2)}})}{h^2(q_1)e^{G_1^*(q_1)}\pi m}\Big( D(\mathbf{q}) +O(\delta^\sigma)\Big)}{= O(\sum_{i=1}^2(\lambda_{n,1}^{(i)})^2e^{-\frac{3}{2}\lambda_{n,1}^{(i)}})+O(\sum_{i=1}^2( e^{-(1+\frac{\sigma}{2})\lambda_{n,1}^{(i)}}),}\end{aligned}
\sum_{i_1=0}^{r-1}\sum_{i_2=0}^{s-1}\binom{i_1+i_2}{i_1}^2\sum_{m_1+m_2<p}\binom{m_1+m_2}{m_1}^2\cdot\left(1+2p((i_1+i_2)H_{m_1+m_2}-i_1H_{m_1}-i_2H_{m_2})\right).
\mathfrak{X}(L_i, L_i) =(L_{i+1}, I_0), \mathfrak{X}^{2n-i-2}(L_{i+1}, L_0) =(L_{2n-1}, L_{2n-i-2}), \mathfrak{X}(L_{2n-1}, L_{2n-i-2}) =(L_{2n-i-1}, L_{2n-i-1}), \mathfrak{X}(L_{2n-i-1}, L_{2n-i-1}) =(L_{2n-i}, L_{0}), \mathfrak{X}^{i-1}(L_{2n-i}, L_{0}) =(L_{2n-1}, L_{i-1}), \mathfrak{X}(L_{2n-1}, L_{i-1}) =(L_{i}, L_{i}).
dx_t = \beta_t(y_t, x_t)dt +\alpha_t(y_t, x_t)dw_t + \int_{Z}\lambda_t(y_{t},x_{t},z)\tilde N(dt,dz), \,\,\, t \in [0,T], x_t = \xi_t, \,\,\, t \in [-H, 0 ],
\pi\cdot Kf\left( \frac12 \right) = \lim_{\epsilon \to 0}\int_0^{1/2-\epsilon}\frac{f(y)}{1/2 - y}\, dy + \int_{1/2}^\infty \left[ \frac{1}{1/2 - y}\chi_{\{y> 1/2 +\epsilon\}}+ \frac{1}{y}\right] \, dy & = \int_0^{x_0}\frac{f(y)}{1/2-y}\, dy + \int_{x_0}^{1/2}\frac{f(y)-f(1/2)}{1/2-y}\, dy + \log \left( \frac12 - x_0 \right).
|v_n| = 2 \left| \sin \frac{\pi}{2}(-\varphi)^{n}(1 - 2\varepsilon(-\varphi)^{-n})\right| &= \pi \varphi^{n}\, |1 - 2\varepsilon(-\varphi)^{-n}| \, (1+\mathcal{O}(\varphi^{2n})) &\leq \pi \varphi^n\sqrt{5}\left( 1+\mathcal{O}(\varphi^{2n})\right) .
\vline\quad\begin{aligned}2\sum_{j\in E_{\mathrm{ss}}[i]}\alpha_{ij}+ \sum_{k\in E_{\mathrm{sa}}[i]}\alpha_{ik}& \frac{1}{2}\sum_{j\in E_{\mathrm{ss}}[i]}\alpha_{ij}^2 + \frac{1}{2}\sum_{k\in E_{\mathrm{sa}}[i]}\alpha_{ik}^2=f(U^{(0)},V^{(0)}).\end{aligned}
\frac{C^q_i}{\beta_{i,q}}=\prod_{k=1}^{m^+}\left(\frac{\eta_k-\beta_{i,q}}{\eta_k}\right)^{m_k}\prod_{k=1,k \neq i}^{M}\frac{\beta_{k,q}}{\beta_{k,q}-\beta_{i,q}},\ \for \ \ 1\leq i \leq M.
\frac12 C_1 \alpha(C_1) = C_1^2 \left( 1- e^{-\frac{1}{2C_1}}\right) \geq C_1^2 \left( 1- \frac{1}{1+\frac{1}{2C_1}}\right) = \frac{C_1}{2+\frac{1}{C_1}}\geq \frac{C_1}{2+\frac18}= \frac{8}{(1-\lambda)^2}.
dq^\prime_t= (\tilde\gamma^T)^{-1}(t^*,q^\prime_t)\left(-\partial_t\psi(t^*,q^\prime_t)-\nabla_q V(t^*,q^\prime_t)+\tilde F(t^*,q^\prime_t,\psi(t^*,q^\prime_t))\right) dt & +\tilde S^\prime(t^*,q^\prime_t)dt+(\tilde\gamma^T)^{-1}(t^*,q^\prime_t)\sigma(t^*,q^\prime_t)\circ dW_t.
\mathcal{D}_{\theta_{0}}^{- w}\mathcal{H}(\theta) = \mathcal{J}^{w}_{\theta_{0}}\mathcal{H}(\theta) = \frac{1}{\Gamma(w)}\int_{\theta_{0}}^{\theta}(\theta - x)^{w - 1}\mathcal{H}(x)\nabla x.
| I \rangle_{M}: = \mu_{I}\exp \left( -{\frac{1}{2}}a^{\dagger}C a^{\dagger}\right) | 0 \rangle , \ \ \ \mu_{I}\mu_{3}^{M}= \det{}^{\frac{d}{2}}\left( 1 - M_{0}\right) = \det{}^{\frac{d}{2}}\left({\frac{( 1 - T )^{2}}{1 - T + T^{2}}}\right) ,
s=\left(\begin{array}{ccc}0 0 1 1 0 0 0 1 0\end{array}\right),\,t=\left(\begin{array}{ccc}0 -1 0 1 0 0 0 0 1\end{array}\right),\, u=\left(\begin{array}{ccc}i 0 0 0 1 0 0 0 1\end{array}\right)
\tilde{F}( k ) ={\frac{1}{4 \pi \eta}}\int_{- \infty}^{+ \infty}d q \tilde{F}( q ) e^{i{\frac{k - q}{2}}}{\frac{\sin{\frac{k - q}{2}}}{\frac{k - q}{2}}}( \mathrm{s i g n}( q ) + \eta ) ( 1 - e^{- 2 \pi q \eta \lambda}) ,
c_0(n)F(n) + c_1(n)F(n + 1) + \cdots + c_k(n)F(n + k) = \sum_{j = 0}^k \left[c_j(n)\sum_i f(i, n + j)\right] = \sum_i \sum_{j = 0}^k c_j(n)f(i, n + j) = \sum_i [g(i + 1, n) - g(i, n)] = 0.
e^{\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}}e^{\beta\hat{a}^{\dagger}-\beta^{*}\hat{a}}=e^{(\alpha+\beta)\hat{a}^{\dagger}-(\beta^{*}+\alpha^{*})\hat{a}}e^{(\alpha\beta^{*}-\alpha^{*}\beta)/2}.
(2Q+\sigma_1^2)(\rho_1^2-\rho_2^2)^2 =\left| \rho_1e^{i\varphi_1}(\rho_1^2-\sigma_2^2)-\rho_2e^{i\varphi_2}(\rho_2^2-\sigma_2^2)\right|^2 &=\left| \rho_1e^{i\varphi_1}(\rho_1^2-\rho_2^2)+(\rho_1e^{i\varphi_1}-\rho_2e^{i\varphi_2})(\rho_2^2-\sigma_2^2)\right|^2.
a_i^*xa_i =[M:N]E_M(a_i^*e_N)xa_i=[M:N]E_M(a_i^*e_Nxa_i)=[M:N]E_M(a_i^*e_N\gamma_0(x)a_i) \ \ \ (\textnormal{Lemma \ref{l:me}}) &=[M:N]E_M(a_i^*e_Na_i\gamma_0(x)) \ \ \ (\gamma_0(x)\in M'\cap M_1),
S = \int d^{4}x \sqrt{- \hat{g}}e^{\psi}\left( \hat{R}- \omega ( \hat{\nabla}\phi )^{2}\right) + \sum_{i}m_{i}\int \left( - \hat{g}_{\mu \nu}d x_{i}^{\mu}d x_{i}^{\nu}\right)^{\frac{1}{2}}\; ,
M ( \Phi , \Phi^{\prime}, T ) = \left( \begin{array}{c c}{- i \Phi_{I}({\bf X}^{0}) \tilde{D}{\bf X}^{I}}&{\overline{{T}}({\bf X}^{0})}\\{T ({\bf X}^{0})}&{- i \Phi_{I}^{\prime}({\bf X}^{0}) \tilde{D}{\bf X}^{I}}\\ \end{array}\right) , \I = 1 , \dots , 9 \ .
{\tilde{\cal{E}}}( B ) = \frac{B^{2}}{2}+ \frac{1}{8 \pi^{\frac{3}{2}}}\; \int_{0}^{\infty}\frac{d s}{s^{\frac{5}{2}}}\; e^{- m^{2}s}\; \left[ ( e B s ) \coth ( e B s ) - 1 \right] \; .
\Omega_{\mathrm{n o n - p e r t}}\approx \frac{\mu^{2}}{e E}\left[ 1 + \frac{1}{2}\left( \frac{\omega}{e E}\right)^{2}k_{z}^{2}- \frac{1}{8}\left( \frac{\mu \omega}{e E}\right)^{2}+ \dots \right]
\delta_t\rho^{n-\frac{1}{2}}+ A\rho^{n-\frac{1}{2}}+ \sum\limits_{j=1}^m I^{(\alpha_j)}B_j\rho^{n-\frac{1}{2}}- \kappa \rho^{n-\frac{1}{2}}= \mathcal{R}^{n-\frac{1}{2}}
{\bf D}(x,y,z)(du,du) =\frac{1}{2}\left(\frac{1}{1+z},0,-\frac{x}{(1+z)^2}\right)\left(\begin{array}{ccc}1+x^2 xy xz xy 1+y^2 yz xz yz z^2-1\end{array}\right)\left(\begin{array}{c}\frac{1}{1+z}0 -\frac{x}{(1+z)^2}\end{array}\right) &=\frac{1}{2(1+z)^2}.
P_k(x) = \frac{1}{2\pi i}\oint_{\Sigma_k}\frac{1}{(t-k)_{k+1}}\prod_{j=1}^r \frac{\Gamma(t+1+ m_j-n)}{\Gamma(t+1+\nu_j)}\, x^t \, dt &=G_{r+1,r+1}^{0,r+1}\left( \left.{k+1,n-m_1,\ldots,n-m_r \atop 0,-\nu_1, \ldots,-\nu_r}\right|x\right) ,
w(t,\phi(t;T,x_0))= \psi(x_0) \exp \Bigl[ -\int_{t}^{T}\frac{\lambda(s,\phi(s;T,x_0))}{s}ds \Bigr] - \int_t^T \exp \Bigl[ -\int_t^{\tau}\frac{\lambda(s,\phi(s;T,x_0))}{s}ds \Bigr] \frac{g(\tau,\phi(\tau;T,x_0))}{\tau}d\tau.
(tn)_i =t_i+\sum_{k=0}^{i-1}\binom{t_1}{k}\left(-\sum_{l=0}^{i-k-1}\binom{-t_1}{l}t_{i-k-l}\right) =t_i-\sum_{r=0}^{i-1}\left(\sum_{k=0}^r\binom{t_1}{k}\binom{-t_1}{r-k}\right)t_{i-r}=t_i-\sum_{r=0}^{i-1}\binom{0}{r}t_{i-r}=t_i-t_i=0.
f_{\bf k}({\bf x},t) ={n_T^{(m-2)/2}\over\left[ 2\omega (2\pi)^m\right]^{1/2}}\left\{\Theta(-x) T_{lg}e^{-i\beta' x_1}+\Theta(x) \left[ e^{in_Tk_1x_1}+ R_{lg}e^{-in_Tk_1x_1}\right]\right\}e^{in_T(k_2x_2+\cdots +k_mx_m)}e^{-i\omega t},
\int_{1}^{\rho}\frac{( s^{2}- 1 )^{n}}{s^{2}}d s = \int_{0}^{\rho - 1}\frac{( y^{2}+ 2 y )^{n}}{( y + 1 )^{2}}d y = \frac{2^{n}( \rho - 1 )^{n + 1}}{n + 1}\, \mathrm{{A p p e l l}}F_{1}[ n + 1 , 2 , - n , n + 2 , 1 - \rho , \frac{1 - \rho}{2}]