harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Delta=\frac{1}{r^2}\frac{\partial}{\partial r}r^2\frac{\partial}{\partial r}+\frac{1}{r^2\sin{\varphi}}\frac{\partial}{\partial \varphi}\sin\varphi\frac{\partial}{\partial \varphi}+\frac{1}{r^2\sin^2{\varphi}}\frac{\partial^2}{\partial \theta^2}
0= ({\cal M}^2)^2 - \frac1{4 k_q}\tilde{\gamma}\tilde g^2 (q_0^2+\bar{q}_0^2){\cal M}^2 +\frac1{4 k_T k_q}(\tilde{\gamma}\tilde g^2 -\frac1{k_T}) q_0^2 \bar{q}_0
\sqrt{T}(\hat{r}_{\bar{\tau},ij}-r_{\bar{\tau},ij}) =\mathbb{V}_{T,ij}+\nabla_{1}G_{ij}^{\top}\sqrt{T}\{\hat{\beta}_{i}(\tau_{i})-\beta_{i}(\tau_{i})\}& \ \ \ +\nabla_{2}G_{ij}^{\top}\sqrt{T}\{\hat{\beta}_{j}(\tau_{j})-\beta_{j}(\tau_{j})\}+o_{p}(1),
\cos \frac{2 \pi (k+j)}{n-1}- \frac{1}{n}\cos \frac{2 \pi (k+j)n}{n-1}=\cos \frac{2 \pi (k'+j)}{n-1}- \frac{1}{n}\cos \frac{2\pi (k'+j) n}{n-1}=\gamma^K
a ( \lambda ) = \rho_{g}^{2}\frac{\left( e^{\lambda}+ 1 \right)^{4}}{e^{2 \lambda}}\qquad \mathrm{a n d}\qquad b ( \lambda ) = \frac{\left( e^{\lambda}- 1 \right)^{2}}{\left( e^{\lambda}+ 1 \right)^{2}},
\begin{aligned}A_{a}(x,\xi)=\frac{1}{2}|\nabla_{x}\varphi_{a}(x,\xi)|^2+I^L_{a}(x_a,x^a)-\frac{1}{2}|\xi|^2-\frac{1}{2}i\Delta_{x}\varphi_{a}(x,\xi)\end{aligned}
\delta_t \rho^1 + \sum\limits_{q=1}^{2}\widetilde{Q}_{1}^{(q)}(\mathbf{B}_q\rho) = r_3^{1}+ r_4^1 + r_5^1, \delta_t^{(2)}\rho^n + \sum\limits_{q=1}^{2}\widetilde{Q}_{n}^{(q)}(\mathbf{B}_q\rho) = r_3^{n}+ r_4^{n}+ r_5^{n}, n\geq 2, \rho^0=0,
+\frac{a^{2l+2}}{b^{2l+2}-a^{2l+2}}\Big(\frac{r^l}{r'^{l+2}}+\frac{r'^l}{r^{l+2}}+\frac{l}{l+2}\frac{b^{2l+2}}{r^{l+2}r'^{l+2}}\Big)\Bigg)\frac{\sin((l+1)\theta)}{\sin\theta}.
\mathcal{H}:=\mathcal{H}_1-N^{-1}\mathcal{V}\quad\mathrm{where,}&\mathcal{H}_1=\int\Delta_x\delta(x-y)a_{x}^{\ast}a_y\,\mathrm{d}x\,\mathrm{d}y, & \mathcal{V}:=\frac{1}{2}\int v_N(x-y)a_x^{\ast}a_y^{\ast}a_{x}a_{y}\,\mathrm{d}x\,\mathrm{d}y &\mathrm{with}\,\,\,v_N(x):=N^{3\beta}v(N^\beta x)
(1 + 2c^{- 2}U - c^{- 2}\sum_{j = 1}^{3}\left( \frac{dx^{j}}{dt}\right)^{2})^{1/2}\frac{d}{dt}\left( (1 + 2c^{- 2}U - c^{- 2}\sum_{j = 1}^{3}\left( \frac{dx^{j}}{dt}\right)^{2})^{- 1/2}\frac{dx^{i}}{dt}\right) + \frac{\partial U}{\partial x^{i}}= 0
f_{\mu \nu}(x):=\frac{4}{R_{P}^{4}g^{2}}\frac{\left< \mathrm{Tr}\mathbf{W}[\partial S]\mathrm{Tr}\mathbf{P}_{\mu \nu}(x)\right> -\left< \mathrm{Tr}\mathbf{W}[\partial S]\right> \left< \mathrm{Tr}\mathbf{P}_{\mu \nu}(x)\right>}{\left< \mathrm{Tr}\mathbf{W}[\partial S]\right>}
\frac{1}{k^{2}}\rightarrow \lim_{\alpha \rightarrow 1 + i \epsilon}\frac{1}{\alpha^{2}k_{0}^{2}- \vec{k}^{2}}= \frac{1}{k^{2}+ 2 i \epsilon k_{0}^{2}}, \epsilon \rightarrow 0^{+}.
{\cal P}_{n}^{\pm}( \beta ) = N_{\pm}^{2}( \beta ) \left[ | T_{n}( \beta ) |^{2}+ | T_{n}( - \beta ) |^{2}{\pm}\left( T_{n}^{*}( \beta ) T_{n}( - \beta ) + T_{n}( \beta ) T_{n}^{*}( - \beta ) \right) \right] .
\psi_{\lambda}^{( N_{m})}(x)=\widehat{A}\big( w_{\nu_{m}}^{( N_{m-1})}\big) \psi_{\lambda}^{( N_{m-1})}(x)=\widehat{A}\big( w_{\nu_{m}}^{( N_{m-1})}\big) \cdots \widehat{A}( w_{\nu_{1}}) \psi_{\lambda}(x).
J_{ij}=\frac{\partial\sigma_{j}}{\partial\lambda_{i}}|_{\lambda=0}=2[\hat{x}_{j\alpha}-\hat{x}_{j\beta}][\frac{\partial\sigma_{i}}{\partial x_{j\alpha}}m_{j\alpha}^{-1}-\frac{\partial\sigma_{i}}{\partial x_{j\beta}}m_{j\beta}^{-1}].
\frac{\tilde{\phi}^{n+1}-2\phi^n+\phi^{n-1}}{\tau^2}+\eta^n\frac{\tilde{\phi}^{n+1}-\phi^{n-1}}{2 \tau}= -G(\phi^n)+\lambda^n\phi^n,\mathbf{x}\in\mathbb{R}^d, & \phi^{n+1}= \tilde{\phi}^{n+1}/\|\tilde{\phi}^{n+1}\|,n\geq 1.
\mathcal P_a(t',I):=\ \Big\{P \ :\P\in{[2,n]\choose k-1},\I\subset P,\ [2,t']\cap P=\emptyset\Big\}, \mathcal P_b(t',I):=\ \Big\{P\ :\P\in{[2,n]\choose k},\ [2,t']\subset P,\I\cap P=\emptyset\Big\},
\frac{1}{\pi}\liminf_{r\rightarrow1-0}\frac{\int_{\{z\in D:2\psi\ge \log r\}}|F_0|^2\tilde\rho}{1-r}= \frac{M(Z_0,J,\tilde\rho)}{\pi\int_{0}^{+\infty}c(t)e^{-t}dt}.
\mathbb{E}\left[\|f_1-(\Pi_Vf_1)_1\|_n^2\right] =\|f_1-(\Pi_Vf_1)_1\|^2 &\leq\frac{1}{(1-\rho_0^2)}\|f_1-\Pi_Vf_1\|^2 &\leq\frac{1}{(1-\rho_0^2)}\|f_1-\Pi_{V_1}f_1\|^2.
\sum_{j = 0}^k{\sum_{s = 0}^{k - j}{\binom kj\binom{k-j}s\left({\frac{{f_3}}{{f_1}}}\right)^j \left({\frac{{f_2}}{{f_1}}}\right)^s X_{m - ak - (c - a)j - (b - a)s}}}= \frac{{X_m}}{{f_1^k}}\,,
\log\L=\frac{1}{2}\Bigg[\log\Big( 1+ \frac{u}{p}\Big) + \log\Big( 1+ \frac{v}{p}\Big)+\log(uv)+\log\Big( 1 + \frac{p}{u}\Big) + \log\Big( 1+ \frac{p}{v}\Big) \Bigg]\,.
\mathbf{E}\left(F\left((S^{\varepsilon,L,K}_n)_{n=0}^{\lfloor L/\varepsilon^2\rfloor}\right)\right)=\mathbf{E}\left(F\left((S^{\varepsilon}_n)_{n=0}^{\lfloor L/\varepsilon^2\rfloor}\right)\:\vline\:\mathcal{A}_{\varepsilon,L,K},\:S^\varepsilon_{\lfloor L/\varepsilon^2\rfloor}>0\right),
0 = (a_0,a_1,\ldots, a_{n-1}) \Sigma E^{\mu+1}&= (a_0,a_1,\ldots, a_{n-1}) \left(\begin{array}{c|c}0 I_{n-1}\hline1 0 \end{array}\right) \left(\begin{array}{c|c}0 1 \hline I_{n-1}0\end{array}\right) \Sigma E^{\mu+1}&= (a_{n-1},a_0,\ldots, a_{n-2}) \left(\begin{array}{c|c}0 1 \hline I_{n-1}0\end{array}\right) \Sigma E^{\mu+1}.
\int_{S^2}a^i \tilde X^i ( 2W_0 \tilde \Delta X_0^{(3)}+ 2 \tilde \nabla W_0 \tilde \nabla X_0^{(3)}) dS^2 = -2 \frac{a^i a^j}{a^0}\int_{S^2}W_0 \tilde \nabla \tilde X^i \tilde \nabla P_j dS^2 = - 8 \frac{a^i a^j}{a^0}\int_{S^2}W_0 \tilde X^i P_j dS^2.
E(\sup_{s,t\in[0,T],|t-s|\leq \epsilon}| X^n_{t\wedge \tau_N^n}-X^n_{s\wedge \tau_N^n}|^{2l}) \leq C(l)\sum_{j=1}^{k}E(\sup_{t\in[(j-1)\epsilon,j\epsilon \wedge T]}| X^n_{t\wedge \tau_N^n}-X^n_{(j-1)\epsilon\wedge \tau_N^n}|^{2l}) \leq C(N,l)(T+\epsilon)\epsilon^{l-1}.
\overset{I \otimes J}{T}\!\!\!_{12}= \overset{I}{T_1}\overset{J}{T_2}, \eta(1) = \overset{\mathbb{C}}{T}, \Delta(\overset{I}{T^{a}_{b}}) = \overset{I}{T^{a}_{i}}\otimes \overset{I}{T^{i}_{b}}, \varepsilon(\overset{I}{T}) = \mathbb{I}_{\dim(I)}, S(\overset{I}{T}) = \overset{I}{T}{^{-1}},
\frac{\partial}{\partial x^{\nu}}\left( e^{\mu\lambda\sigma\tau}x_{\sigma}T_{\tau}{}^{\nu}\right) +e^{\mu\lambda\sigma\tau}T_{\sigma\tau}& \ +e^{\mu\lambda\sigma\tau}x_{\sigma}X_{\tau}{}+\frac{1}{c}e^{\mu\lambda\sigma\tau}x_{\sigma}F_{\tau\nu}j^{\nu}=0^{\mu\lambda}.
a_{nn}\!=\!\Vert f_n\Vert^2+ \Vert g_n^c\Vert^2\!=\!\Vert (T_{kn}(2)-t_{kn}(2)){\bf 1}\Vert^2\!+\!t_{kn}^2(2)\Vert x_{1k}{\bf 1}\Vert^2\Vert(T_{\alpha_{1n}}-c_{1n}){\bf 1}\Vert^2.
\frac{\prod_{k = 1}^{n_{f}}{\mathcal{Z}}\left( J , m_{f}^{( k )}\right)}{\prod_{l = 1}^{n_{b}}{\mathcal{Z}}\left( J , m_{b}^{( l )}\right)}= \mathrm{c o n s t}\; \int{\mathcal{D}}{\mathcal{F}}{\mathcal{D}}{\mathcal{B}}\; \exp \left\{-{\mathcal{A}}_{\mathcal{F}}( J ) -{\mathcal{A}}_{\mathcal{B}}( J ) \right\}
f=\frac{1}{2\pi}\int_{[0,2\pi)}\left({\textstyle\sum\limits_{k\in\mathbb{S}}}a_{k}(t)\Psi_{k,t}(x)\right) dt+\frac{1}{2\pi}\sum_{k\in\mathbb{Z}\backslash\mathbb{S}}\int_{[0,2\pi)}a_{k}(t)\Psi_{k,t}dt,
M_{\ell,n}A :=\{(a,a_1,\dots,a_\ell)\mid n\cdot a+a_1+\dots+a_\ell=0\}\subset A\times A^\ell,, N_{n+\ell-1}:=\{(b_1,\dots,b_{n+\ell})\mid b_1+\dots+b_{n+\ell}=0\}\subset A^{n+\ell}\,.
\ln \epsilon \int \prod_{i=1}^n d^Dy_i^\mu \left\{\left({\partial\over \partial y_1^\mu}S^{(n)}\right) \left({\partial \over\partial y_{1\mu}}S^{(n)}\right) +{\partial^2\over \partial y_1^2}S^{(n)}\right\}e^{S^{(n)}}= 0.
\left( \frac{1}{2}+\frac{1}{2}\cos^{q_{1}}x\right)^{2/\left( 3q_{1}\right)}<\frac{\sin x}{x}<\left( \frac{1}{2}+\frac{1}{2}\cos^{q_{2}}x\right)^{2/\left( 3q_{2}\right)}
\left|\mathbb{E}(\widehat{\alpha}_{t,\varepsilon}(x)-\widehat{\alpha}_{t,\varepsilon}(y))^n\right|\leq c\,|x-y|^\lambda\sum_{S_2}\int_{E^{n}}\int_{\mathbb{R}^{n}}\prod_{i=1}^{2n-1}|u_{i}|^{\frac{\lambda}{2}\alpha_i}\exp\prod_{j=1}^\infty(1+4\lambda_{j}^2)^{-\frac14}dpd\ell.
\frac{1}{N^m}= \exp\left(-m\log N\right) < \exp\left(-\tau\cdot \frac{\log r}{\log(a\varepsilon)}\cdot \log N\right) = \exp\left(s \log r\right) \left( s = \frac{\tau \log N}{\log\left(\frac{1}{a\varepsilon}\right)}\right) = r^s.
\Gamma_{A = 0}= 2 V \int_{0}^{\infty}\frac{d t}{2 t}\left[ \mathrm{T r}_{\mathrm{N S}}\left( \frac{1 + ( - 1 )^{\cal F}}{2}\, q^{2 L_{0}- 1}\right) - \mathrm{T r}_{\mathrm{R}}\left( \frac{1 + ( - 1 )^{\cal F}}{2}\, q^{2 L_{0}}\right) \right]
\leq \;\int \prod_{a=1}^{N-1}\prod_{i=1}^{n_a}d^Dz_i^{(a)}\;\prod_{b<c}^{N-1}\bigg| \; f^{(b,c)}\bigg| \;\prod_{d=1}^{N-1}\Big|\!\Big|\!\Big| f^{(d,N)}\Big|\!\Big|\!\Big|_N
s_a^1(x,\xi) =\sum_{z\in\mathbb{Z}^d}f\left[z\right]\left(e^{i(\varphi_a(x-z,\xi)-\varphi_a(x,\xi))}-e^{-iz\cdot\nabla_x\varphi_a(x,\xi)}\right) &=\sum_{z\in\mathbb{Z}^d}f\left[z\right]e^{-iz\cdot\nabla_x\varphi_a(x,\xi)}\left(e^{i\Phi_a(x,\xi,z)}-1\right) ,
A_{p}\left( \sum_{n=1}^{N}\left\vert a_{n}\right\vert^{2}\right)^{\frac{1}{2}}\leq \left( \int_{0}^{1}\left\vert \sum_{n=1}^{N}a_{n}r_{n}\left(t\right) \right\vert^{p}dt\right)^{\frac{1}{p}}\leq B_{p}\left(\sum_{n=1}^{N}\left\vert a_{n}\right\vert^{2}\right)^{\frac{1}{2}}
L_{1 1}^{( \pm )}L_{1 2}^{( + )}= q^{\mp 1}L_{1 2}^{( + )}L_{1 1}^{( \pm )}, \; \; L_{2 2}^{( \pm )}L_{1 2}^{( + )}= - q^{\mp 1}L_{1 2}^{( + )}L_{2 2}^{( \pm )}
2=I(\alpha_{k+1},\alpha_{k})= I(\hat{\alpha}\cup{(\gamma,M)},\hat{\alpha}\cup{(\delta,1)}\cup{(\gamma,M-p_{i})}) = c_{1}(\xi|_{Z},\tau)+Q_{\tau}(Z,Z)+2(M-p_{i})Q_{\tau}(Z,Z_{\gamma}) +\sum_{k=M-p_{i}+1}^{M}(2\lfloor k\theta \rfloor+1)-\mu_{\tau}(\delta).
A (m)= \frac{\alpha}{2m(2m-1)}- \frac{\alpha}{2 ( 2m-1)(2m+\alpha-1)}+ \sum_{k=2}^{\infty}\frac{(-1)^{k+1}\alpha^k}{k (2m-1)^k}+ \sum_{k=1}^{\infty}\frac{B_{2k}}{2k}\left( - \frac{1}{(2m+\alpha-1)^{2k}}+ \frac{1}{(2m-1)^{2k}}\right).
{\rm ev}_{x}^{\mathfrak{m}}(e_{a_{1}}\cdots e_{a_{k}})=\widetilde{{\rm ev}_{x}^{\mathfrak{m}}}\left(\mathbb{I}_{{\rm dch}}(\overrightarrow{1}_{z};a_{1},\dots,a_{k};-\overrightarrow{1}_{1-z})\right),
\partial_t \psi = \frac{1}{2}\partial_x^2 \psi + \left( \frac{4 \zeta^2 - \frac{1}{4}}{2 \cosh^2 x}- \frac{4(\xi + \sigma)^2 - \frac{1}{4}}{2 \sinh^2 x}- \frac{e^{4t}}{16}\cosh 4x + \left( \sigma - \frac{1}{2}\right) e^{2t}\cosh 2x \right) \psi ,
(v\otimes a)_{(0)}\otimes p((v \otimes a)_{(1)}) = v_{(0)}\otimes a_{(2)}\otimes p(S(a_{(1)}) v_{(1)}a_{(3)}) & = v \otimes a_{(2)}\otimes p(S(a_{(1)}) a_{(3)}) & = v \otimes a \otimes 1
\langle T_\eta^\eta \rangle^C=\frac 1{2C}\phi_c^{^{\prime}}\phi_c^{^{\prime}}+2\xi \frac{C^{^{\prime}}}{C^2}\phi_c\phi_c^{^{\prime}}+\frac{3\xi}C(\frac{C^{^{\prime}}}C+k)\phi_c^2+\frac{m^2}2\phi_c^2-2\kappa \phi_c^4+\frac 32\lambda^2\phi_c^4
\frac{\delta S}{\delta \theta^{\alpha\beta}}= \frac{\beta}{2}\int d^2x{\mathrm{Tr}}\left[-\frac i4 \{F_{\alpha\beta}, \Phi\}\star F_{01}-\frac i4 \Phi \star \{F_{0\alpha}, F_{\beta 1}\}+ \frac i4 \Phi \star\{F_{0\beta}, F_{\alpha 1}\}\right]\,.
\begin{array}{r c l}|\psi_i> = |q_i>\otimes|p_\lambda=0>\otimes|-+>=\sum_{n=0}^\infty|n>\otimes|p_\lambda=0>\otimes|-+><n|q_i>\ \ \ , |\psi_f> = |q_f>\otimes|p_\lambda=0>\otimes|-+>=\sum_{n=0}^\infty|n>\otimes|p_\lambda=0>\otimes|-+><n|q_f>\ ,\end{array}
\mathbb{H}_{\Gamma}:=\frac{\mathbb{C}[t_0,t_1,t_2,t_3,t_4,t_5,t_6,t_7,t_8]}{(t_0+2t_1+3t_2+4t_3+5t_4+3t_5+2t_6+4t_7+6t_8)}
{\frac{c_{A}}{2 4}}\sum_{r}\left[ H_{1}, A_{i}^{c}( r ) \right] \left[ H_{1}, A_{i}^{c}( - r ) \right] \sum_{k}{\frac{1}{\omega_{k}^{3}}}= -{\frac{c_{A}}{2 4}}{\frac{1}{2 \pi^{2}}}l o g \left({\frac{\Lambda}{\mu}}\right) \int d^{3}x E^{2}
\Omega_{a\nu}(az,bz)=\frac{\bar{K}_{\nu}^{(b)}(bz)/\bar{K}_{\nu}^{(a)}(az)}{\bar{K}_{\nu}^{(a)}(az)\bar{I}_{\nu}^{(b)}(bz)-\bar{K}_{\nu}^{(b)}(bz)\bar{I}_{\nu}^{(a)}(az)}.
v_n x v_n^{-1}-x =\sum\limits_{k=n+1}^\infty \left(v_n y_k v_n^{-1}- y_k\right)+\sum\limits_{k=1}^n \left(v_n y_k v_n^{-1}- y_k\right) \stackrel{\eqref{invv}}{=}\sum\limits_{k=1}^{n}\left(v_n y_k v_n^{-1}- y_k\right)\stackrel{\eqref{vvvv}}{=}-2y_n + \sum\limits_{k=1}^{n-1}\left(v_n y_k v_n^{-1}- y_k\right)
t(1+t^{2})\frac{\partial u}{\partial t}=\frac{u^{2}\Delta_{g(t)}u}{2}-\frac{u^{3}}{4}(R_{g(t)}-t^{2}\overline{R}) &+u\left(\frac{1+3t^{2}}{2}+\frac{t^{2}(1+t^{2})|M|_{g(t)}^{2}}{4}\right).
\mathcal{H}\left( P,Q\right) =\frac{1}{2}\left[ \left( P-\theta Q\right)^{T}\alpha^{-1}\left( P-\theta Q\right) +Q^{T}\eta Q\right] =\frac{1}{2}\dot{Q}^{T}\alpha\dot{Q}+\frac{1}{2}Q^{T}\eta Q.
F_{r}\left(w_{1},\dots,w_{r}\right)=\frac{1}{w_{r}}\left[F_{r-1}\left(w_{1},\dots,w_{r-1}\right)- F_{B}\left(-w_{r}\right)F_{r-1}\left(w_{1},\dots,w_{r-2},w_{r-1}+w_{r}\right) \right]
\mathrm{SNDR}=\frac{E^2[\gamma g(\gamma)]}{E[g^2(\gamma)]-E^2[\gamma g(\gamma)]-E^2[g(\gamma)] + \sigma_v^2/A^2}&=\frac{E^2[\gamma g(\gamma)]}{var[g(\gamma)]-E^2[\gamma g(\gamma)] + \sigma_v^2/A^2}
\hat{Q}(z)=\hat{Q}_{1}(z)+\hat{Q}_{2}(z);\hat{Q}_{1}\in N_{\hat{\kappa}_{1}}(H),\hat{Q}_{2}\in N_{\hat{\kappa}_{2}}\left( H \right);\hat{\kappa}_{1}+\hat{\kappa}_{2}=\kappa .
\left( \eta_{a}^{\prime}( \sigma ) + i 4 \pi G \epsilon_{a}^{b}\eta_{b}( \sigma ){\frac{\delta}{\delta \eta_{2}( \sigma )}}+ i 4 \pi G \eta_{3}( \sigma ) \epsilon_{a b}{\frac{\delta}{\delta \eta_{b}( \sigma )}}\right) \Psi ( \eta , q , X ) \hspace *{\fill}
\langle \Psi_0^{*-1}\left(\theta(T)\right), \phi\rangle =\big\langle \theta(T), \big(\sum_{i=1}^\infty a_{\alpha,i}\otimes x^*_{\alpha,i}\big)_\mathcal{U}+W_0\big\rangle=\lim_\mathcal{U}\sum_{i=1}^\infty x^*_{\alpha,i}(T(a_{\alpha,i}))=\langle T, \phi\rangle.
H^{fer}_D = \frac{1}{\sqrt{2}}\int d^3x \xi^{\dagger}(\vec{x})\left(\frac{1}{i \partial_{-}- e V_{-}}\ast \xi\right)(\vec{x}) + \frac 1 2 \frac{e^2}{2L}\int d^2x_\perp \Gamma^i (x_\perp) \left({\cal G}_{(\perp)}[{\cal M}^2] \ast\Gamma^i \right) (x_\perp)
I(X^n\rightarrow Y^n) \triangleq \sum_{t=0}^n I(X^t;Y_t|Y^{t-1})= \sum_{t=0}^n{\bf E}\Big\{\log \Big( \frac{d{\bf P}_{Y_t|Y^{t-1}, X^t}(\cdot|Y^{t-1}, X^t)}{d{\bf P}_{Y_t|Y^{t-1}}(\cdot|Y^{t-1})}(Y_t)\Big)\Big\}.
\tilde \phi (S^{\mu}_{\hspace{.3 true cm}\alpha \beta}+\delta^{\mu}_{\beta}S^{\lambda}_{\hspace{.3 true cm}\lambda \alpha}- \delta^{\mu}_{\alpha}S^{\lambda}_{\hspace{.3 true cm}\lambda\beta}) = 8\pi \sigma^{\mu}_{\hspace{.3 true cm}\alpha \beta}+\delta^{\mu}_{\alpha}\tilde \phi_{, \beta}- \delta^{\mu}_{\beta}\tilde \phi_{,\alpha},
\frac{1}{2}\sqrt{4\times 3}{\rm{i}}=\sqrt{3}{\rm{i}}\quad\frac{1}{2}\sqrt{4\times 3-1^2}{\rm{i}}=\frac{1+\sqrt{11}{\rm{i}}}{2}\quad\frac{1}{2}\sqrt{4\times 3-2^2}{\rm{i}}=\sqrt{2}{\rm{i}}&\frac{1}{2}\sqrt{4\times 3-3^2}{\rm{i}}=\frac{1+\sqrt{3}{\rm{i}}}{2}
b_{++}(\xi,\eta) = \frac{i}{4}\frac{|\xi| \widetilde{m}_2(\xi,\eta)}{|\xi-\eta|^{1/2}|\eta|}+ \frac{i}{4}\frac{|\xi|^{1/2}\widetilde{q}_2(\xi,\eta)}{|\xi-\eta|^{1/2}|\eta|^{1/2}},
\prod_{t=0}^{p-1}\frac{(x;\zeta)_{tq}}{(\zeta^e x;\zeta)_{tq}}= \prod_{t=0}^{p-1}\frac{(x;\zeta)_{e}}{(\zeta^{tq}x;\zeta)_{e}}= \frac{(x; \zeta)_k^e}{(x; \zeta^q)_{pe}},
\|S(f,P_1,\xi_1)-S(f,P_2,\xi_2)\| \leq \left\|S(f,P_1,\xi_1)-\int\limits_{a}^{b}f(s)ds\right\| &\qquad+\left\|S(f,P_1,\xi_1)-\int\limits_{a}^{b}f(s)ds\right\| &\leq \frac{\epsilon}{8}+\frac{\epsilon}{8}=\frac{\epsilon}{4}.
U^*( \hat{x}_3 (1-UU^*) \hat{x}_2 - \hat{x}_2 (1-UU^*) \hat{x}_3 )U = U^*( \hat{x}_1 (1-UU^*) \hat{x}_4 - \hat{x}_4 (1-UU^*) \hat{x}_1 )U.
F_{N}(a,b;t) = 1 + \frac{(1-aq)(1-q^{N+1})(1-tq^{N+1})t}{(1-tq^N)(1-bq^{N+2})(1-t)}+ \frac{(1-aq)(1-q^{N+1})^2(b-atq)tq}{(1-bq)(1-tq^{N})(1-bq^{N+2})(1-t)}F_{N}(aq,bq;tq).
\!\!\!\!\!\!\!\mathcal{O}_{\mathcal{DL}}(\alpha,\theta)=1- \int\limits_{0}^{\infty}2 \pi \lambda r \exp \left(-\pi \lambda r^2- \pi \lambda{\rm U}(\theta)r^2 -{\rm U}\left(\frac{\rho \theta r^4 |\mathcal{\tilde{C}}_{\rm{d}}(\alpha)|^2}{P_{\rm d}}\right)-\theta \frac{{\beta}\rho |\mathcal{\tilde{C}}_{\rm{d}}(\alpha)|^2 r^{8}}{P_{\rm{d}}}\right) dr,
\Phi^{(m)}_{[j_1,\cdots,j_m]}(E,{\bf g}) :=\begin{pmatrix}\phi_{j_1}(x, E,{\bf g}) \cdots \phi_{j_m}(x, E,{\bf g}) \vdots \vdots \phi^{(m-1)}_{j_1}(x, E,{\bf g}) \cdots \phi^{(m-1)}_{j_m}(x, E,{\bf g}) \end{pmatrix}.
\frac{(xz)^{n_{12}}}{\sqrt{[n_{12}]_{\mu_{12}}![j]_{\mu_2}!}}\langle0,j|0,j\rangle = \frac{(s^2 + 1)^{n_{12}/2}}{\sqrt{[j]_{\mu_2}! [n_{12}]_{\mu_{12}}!}}\langle0,j|0,j\rangle.
[h_0,h_1]=[h_0,d] = [h_1,d]=0, &h_ie_j-e_jh_i = a_{ij}e_j, h_if_j-f_jh_i=-a_{ij}f_j, &d e_j-e_j d =\delta_{0,j}e_j, d f_j-f_j d = -\delta_{0,j}f_j, &e_if_j-f_je_i = \delta_{ij}h_i, &e_je_i^3-3e_ie_je_i^2+3e_i^2e_je_i-e_i^3e_j = 0 i \neq j, &f_jf_i^3-3f_if_jf_i^2+3f_i^2f_jf_i-f_i^3f_j = 0 i \neq j.
\int_M df\wedge d^c_B f\wedge \omega_h\wedge S\wedge \eta=\frac{1}{4}\int_M (u-v)(\omega_v-\omega_u)\wedge \omega_h\wedge S\wedge \eta\leq \int_M (-h) \omega_h^2\wedge S\wedge \eta\leq 1.
\nu_i \varphi (u_1,\ldots,u_k)\nu_i^* =\left[\begin{matrix}u_1^{(a_{i,1})}0 &u_2^{(a_{i,2})}& \ddots & u_k^{(a_{i,k})}0 0^{(s_i)}\end{matrix}\right], \mbox{for}\ (u_1,\ldots,u_k)\in A
\begin{aligned}\nu \epsilon \Vert \partial_Y^2u^1_e(x,\sqrt{\epsilon}y)\Vert_{L^2}\lesssim \epsilon^{\frac{3}{4}}\Vert \partial_Y^2u^1_b(Y)\Vert_{L^2}+\epsilon^{\frac{3}{4}}\Vert \langle Y\rangle^n\partial_Y^3v^1_e\Vert_{L^2}\lesssim \epsilon^{\frac{1}{4}},\end{aligned}
\left[ \begin{array}{c c}{1}&{0}\\{0}&{1}\\ \end{array}\right] , \left[ \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right] , \left[ \begin{array}{c c}{0}&{i}\\{- i}&{0}\\ \end{array}\right] , \left[ \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right] .
b_{\pm}(\eta,z) ~=~ \bar{b}~ \exp \left[ i k ( \eta - z ) ~\pm~ i \left[ \frac{h_{r}}\eta ~+~ \frac{p^{2}}{2}\left( \frac{h_{0}\eta^2}{3}~-~ h_{r}\eta ~-~ h_{0}z^2 \right) ~+~ O(p^4) \right] ~+~ O\left(\frac 1 k \right) \right]
\tau_{\infty}=\frac{(-1)^{\mathsf{N}}\kappa_{+}^{\prime}\kappa_{-}e^{\tau_{-}-\tau_{+}^{\prime}}\prod_{b=1}^{\mathsf{N}}\alpha_{b}\beta_{b}}{\left( \zeta_{+}-1/\zeta_{+}\right) \left( \zeta_{-}-1/\zeta_{-}\right)}.
B = \frac{1}{4}\beta\left(\frac{1}{2},\frac{3}{4}\right) = \int_0^1 \frac{t^2}{\sqrt{1-t^4}}\, dt = \sum_{n=0}^{\infty}\left(\frac{1}{4}\right)^n \binom{2 n}{n}\frac{1}{4 n+3}= \frac{\sqrt{2 \pi^3}}{\Gamma^2 \left(\frac{1}{4}\right)}
\mbox{RHS of (\ref{6phi5even})}={(ax;q)_\infty\over (qx/a;q)_\infty}\sum_{n\geq 0}\sum_{j= 0}^{\lfloor{n\over 2}\rfloor}\sum_{m= 0}^{\lfloor{n-2j\over 2}\rfloor}A(n,j,m),