harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{i=1}^3 (\log f_i)'(\log f_i)''' - (\frac{V'}{V})'\overline{\mathcal{T}}V^2 & \leq -\sum_{i=1}^3 \Big((\log f_i)''\Big)^2 + \frac{2V'}{V}\sum_{i=1}^3 (\log f_i)'(\log f_i)'' & = -\sum_{i=1}^3 \Big((\log f_i)''\Big)^2 + \frac{2}{\overline{\mathcal{T}}V^2}\Big\{\sum_{i=1}^3 (\log f_i)'(\log f_i)''\Big\}^2
V_{2}( \phi , \sigma ) = \beta_{1}\frac{( \phi^{\dag}\phi )^{2}\sigma}{M_{\mathrm{p l}}}+ \beta_{2}\frac{( \phi^{\dag}\phi ) \sigma^{2}\sigma^{\ast}}{M_{\mathrm{p l}}}+ \beta_{3}\frac{( \phi^{\dag}\phi ) \sigma^{3}}{M_{\mathrm{p l}}}+ \mathrm{{h . c .}.}
\begin{aligned}\textrm{(P1a)}\min_{T_{\textrm{a}}\in \mathbb{N}, \; \mathbf{d}_{t}\in \mathcal{D}}T_{\textrm{a}}\textrm{s.t.}\;\, \textrm{C14:}B^R_{\boldsymbol{\rho},T_{\textrm{a}}}= V_{n},\;\forall\boldsymbol{\rho},\end{aligned}
\begin{array}{ll}C_{31}:=\min \left[\{2^{p+q-1}pAC(5N)^{p+q}N^{-1}\}^{-1/(p+q)},\right. &\quad\{2^{p+q-1}qAC(5N)^{p+q}N^{-1}\}^{-1/(p+q)}, &\quad\left.\{2^{r-1}rBC(5N)^rN^{-1}\}^{-1/(r+1)}\right],\end{array}
f(p|X_{1}=x_{1},...,X_{n}=x_{n})=\frac{L(p)f(p)}{\int_{0}^{1}L(r)f(r)dr}=\frac{p^{s}(1-p)^{n-s}}{\int_{0}^{1}r^{s}(1-r)^{n-s}dr}
\tilde{\Theta}^{\pm; L}_{\lambda, T}:= \tilde{\Theta}^{n, L}_{\lambda, T, \epsilon_k^\pm}= \Theta^{\epsilon^\pm;L_n}_{\lambda_n}\ \prod_{p = 1}^{n}\tilde{\Phi}^{\pm[\frac{k}{2}| \gamma_{p}|^2; (\lambda| \gamma_{p})]}\ (\tau, -\beta_{p}+ \gamma^\vee_{p}+ \sum_{i=1}^{p -1}(\gamma^\vee_{p}| \gamma_i) \beta_i),
\ln \frac{f(x)}{f(x+1)}+\ln\frac{\Phi_0(x)+\frac{\kappa_0}{x}}{\Phi_0(x+1)+\frac{\kappa_0}{x+1}}=2\ln \frac{x+1}{x+\frac 12}+\ln\frac{x+\frac 14+\frac{\kappa_0}{x}}{x+\frac 54+\frac{\kappa_0}{x+1}}= \frac{-1/16 + 2\kappa_0}{x^3}+O\left(\frac{1}{x^4}\right).
f := \displaystyle\prod_{\chi}f_\chi = \displaystyle\prod_{\chi}{\rm char}(C_\infty(\chi)) &h := \displaystyle\prod_{\chi}h_\chi = \displaystyle\prod_{\chi}{\rm char}\left(E_\infty(\chi)/(\overline{\mathcal{E}_\infty}(\chi))\right).
D ( x , x^{\prime}) = \int{\frac{d^{4}k}{( 2 \pi )^{4}}}\int{\frac{d^{2}{\vec{q}}}{( 2 \pi )^{2}}}{\frac{e^{i k_{\mu}( x^{\mu}- x^{\mu}) + i{\vec{q}}\cdot ({\vec{x}}-{\vec{x}^{\prime}})}}{k_{4}^{2}+ q^{2}}},
\phi_{1}(p)=\mathcal{F}\left\{\psi_{1}(x)\right\}=-A\pi\left(\frac{\hslash^{2}}{a}\right) \frac{\cos\left( ap/\hslash\right)}{p^{2}-\left( \pi\hslash/2a\right)^{2}},\left\vert x\right\vert <a,
\textup{(1)}ha''g = b''; \textup{(2)}gb''h = a''; \textup{(3)}a'' g = g b''; \textup{(4)}b'' h = h a''; \textup{(5)}ha'g = b'; \textup{(6)}gb'h = a'; \textup{(7)}a' g = g b'; \textup{(8)}b' h = h a'\,.
\begin{gathered}\cfrac{u_{m+1,n+1}\,u_{m,n}-u_{m+1,n}\,u_{m,n+1}}{u_{m,n}^3\, \Delta x_m\, \delta y_n}=1, \Delta x_{m,n}=h, \delta x_{m,n}=0,\Delta y_{m,n}=0,\delta y_{m,n}=k\end{gathered}
\Omega_{D}^{k}({\cal A}) = \left( \begin{array}{c c}{{\Lambda^{k}( X )}}&{{0}}\\{{0}}&{{\Lambda^{k}( X )}}\\ \end{array}\right) + \left( \begin{array}{c c}{{0}}&{{\Lambda^{k - 1}( X )}}\\{{\Lambda^{k - 1}( X )}}&{{0}}\\ \end{array}\right) \, .
\nabla_\mu\nabla^\mu T+\nabla_\mu T\nabla^\mu\Phi-\frac{e^{-2\Phi}\left(\nabla_\mu\nabla_\nu T-\nabla_\mu\Phi\nabla_\nu T\right)\nabla^\mu T\nabla^\nu T}{1+e^{-2\Phi}\nabla_\alpha T\nabla^\alpha T}-\frac{e^{2\Phi}}{V}\frac{dV}{dT}=0,
{}^tSA + TQ + C_1= \begin{pmatrix}3s_{21}- 2s_{11}- t_{21}+ c_{11}s_{11}- 2t_{12}- t_{22}+ c_{12}3s_{22}- 2t_{21}- t_{11}+ c_{21}s_{12}- t_{12}- 2t_{22}+ c_{22}\end{pmatrix},
EI_{T}^{k}(\lambda){}=\frac{1}{(2\pi H_{2,T}(0))^{k}}\sum_{\substack{\nu=(\nu_{1},\ldots,\nu_{1}) (1,\ldots,2k)}}\prod_{l=1}^{p}\mathit{cum}\bigl(d_{T}(\lambda_{i}),i\in\nu_{l}\bigr) &\quad{}\times\prod_{i=1}^{k}\delta(\lambda_{2i-1}-\lambda) \prod_{i=1}^{k}\delta(\lambda_{2i}+\lambda).
q_2 = \frac{t_d}{t_{(\ell-1)d}}\frac{1}{\tan \left( \theta^+_{\chi}\right)}\left( \frac{1}{\frac{1}{\ln \left( -\tan \left( \theta^+_{\chi}\right) \right)}+\frac{1}{\tan \left( \theta^+_{\chi}\right) + 1}}- 1 \right) q_1 \;,
|v(g) - 1 | = \frac{1}{\mu(F)}\left| \int 1_{F\cap g^{-1}F}(x) u(\alpha(g,x)) - 1_{F\cap g^{-1}F}(x) - 1_{F\setminus g^{-1}F}(x) \d\mu(x) \right| &\leq \frac{1}{\mu(F)}\int 1_{F\cap g^{-1}F}(x) | u(\alpha(g,x)) - 1 | + 1_{F\setminus g^{-1}F}(x) \d\mu(x) &\leq \epsilon + \frac{\mu(F\setminus g^{-1}F)}{\mu(F)}&\leq 2\epsilon.
\sum_{t=1}^T\sum_{i=1}^n\frac{1}{2}\Big(\frac{q_{i,t+1}^\top g_{i,t}(x_{i,t})}{\gamma_t}+\frac{\sigma\|\epsilon^z_{i,t}\|^2}{2\gamma_0}\Big)\le h_T(y)+\tilde{h}_T(y)+\hat{h}_T(y),~\forall y\in\mathcal{X}_T,
\|\chi_{H\setminus S_{-\sigma,0}(\nu)}(i\partial_t+\Delta)u\|_{L_{t,x}^2(|V|^{-1})}\leq\|Vu\|_{L_{t,x}^2(|V|^{-1})}&=\|u\|_{L_{t,x}^2(|V|)}&\leq C\|V\|_{\mathcal{L}^{2,p}}\|\nabla u\|_{L_{t,x}^2}<\infty.
\left<W(C)\right>=\exp\left(\sum\limits_{k=1}^{\infty}\frac{(ig)^k}{k!}\int\limits_\Sigma^{}d\sigma_{\mu_1\nu_1}(x_1)\cdots\int\limits_\Sigma^{}d\sigma_{\mu_k\nu_k}(x_k)\left<\left<F_{\mu_1\nu_1}(x_1)\cdots F_{\mu_k\nu_k}(x_k)\right>\right>\right).
\sum_{k=N}^M a_k \bigg(\sum_{i=k}^M a_i \bigg)^s b_k \approx \sum_{k=N+1}^M (b_k - b_{k-1}) \bigg(\sum_{i= k}^M a_i\bigg)^{s+1}+ \bigg(\sum_{k=N}^M a_k\bigg)^{s+1}b_N.
{\cal D}\bar{\cal M}\equiv{\cal D}\bar{B}_{\mu\nu}{\cal D}\bar{B}_{\mu5}{\cal D}\bar{B}_{55}{\cal D}\bar{\zeta}_{\mu}{\cal D}\bar{\zeta}_{4}{\cal D}\bar{\zeta}_{\mu,\theta}{\cal D}\bar{\zeta}_{4,\theta}{\cal D}\bar{\eta}{\cal D}\bar{\eta}_{,\theta}\,,
J(b, s)= \left [ \hat{\theta}- \hat{\sigma}\, b \left(\frac{\hat{\tau}}{\hat{\sigma}}\right)- \hat{\sigma}\, s \left(\frac{\hat{\tau}}{\hat{\sigma}}\right), \,\hat{\theta}- \hat{\sigma}\, b \left(\frac{\hat{\tau}}{\hat{\sigma}}\right)+ \hat{\sigma}\, s \left(\frac{\hat{\tau}}{\hat{\sigma}}\right) \right ].
\begin{aligned}F^{ij}w_{ij}= 2 \sum_k F^{ij}v_{ki}v_{kj}+ 2 \sum_k F^{ij}v_k v_{kij}& \geq \frac{3}{2}(1-\phi_u d)^2 F^{ii}U_{ii}^2 - \epsilon F^{ii}U_{ii}^2 - \epsilon C_1 w^2 \sum_i F^{ii}- Cw^{3/2}\sum_i F^{ii}& \geq (1-\phi_u d)^2 F^{ii}U_{ii}^2 - \epsilon C_1 w^2 \sum_i F^{ii}- Cw^{3/2}\sum_i F^{ii},\end{aligned}
{}\int_{\Omega}{\vert\nabla{z_{n}}\vert}^{p}\phi^{p}=-p\int_{\Omega}\phi^{p-1}z_{n}{\vert\nabla{z_{n}}\vert}^{p-2}\nabla{\phi}.\nabla{z_{n}}+\int_{\Omega}\frac{\lambda_{0}z_{n}\phi^{p}}{(z_{n}+\frac{1}{n})^{\delta}}+\int_{\Omega}z_{n}^{q+1}\phi^{p}.
\phi(r,t)_{\mathrm{in}}\approx{{\cal N}_{\mathrm{in}}\over\sqrt{2\omega}\; r_H}\exp\left[\mp i \omega \left\{t +{(r-r_H)[ \omega^2 - c^2 k_\perp^2]\over 2 \;c_H\; \omega}\right\}\right].
h^{\beta}( a^{2}) ={\frac{1}{2 \pi{\hat{\beta}}^{2}}}\left\{\frac{\pi}{2 a}+{\frac{1}{2}}\ln a +{\frac{1}{2}}( \gamma - \ln 4 \pi ) - \frac{\zeta ( 3 )}{1 6 \pi^{2}}a^{2}+ \frac{3 \, \zeta ( 5 )}{2 5 6 \, \pi^{4}}a^{4}\right\}+{\cal O}( a^{6}) \; .
\mbox{Re}\langle (\tilde{H}- z - i \epsilon \tilde{S}) f , f \rangle_{{\mathcal G}^{-1},{\mathcal G}^1}= Q_H (f,f) - \mbox{Re}(z) ||f||_{L^2}^2 - \mbox{Im}\langle (\tilde{H}- z - i \epsilon \tilde{S}) f , f \rangle_{{\mathcal G}^{-1},{\mathcal G}^1}= \mbox{Im}(z)||f||_{L^2}^2 + \epsilon Q_{[H,iA]}(f,f) \ \geq \ \mbox{Im}(z)||f||_{L^2}^2 .
2\sum_{n=0}^{\infty}\textup{PD}_\textup{t}(6n+3)q^n \equiv\left(\dfrac{1}{f_2^2}-\dfrac{a(q)}{f_2^2}\right)\dfrac{f_{6}^3}{f_2}&\equiv \dfrac{f_{6}^3}{f_2^3}-\dfrac{f_{6}^3}{f_2^3}\left(a(q^4)+6q\dfrac{f_4^2f_{12}^2}{f_2f_6}\right)~(\textup{mod}~4),
\sum_{\sigma\subset \mu_\ell}c_{\ell}\frac{r_\ell}{v( \mu_\ell )}+ c_{\kappa}\frac{1}{v( \kappa)}=0, \sigma \subset \kappa, \sum_{\sigma\subset \mu_\ell}c_{\ell}\frac{r_\ell}{v( \mu_\ell )}=0, \sigma \not \subset \kappa,
\left\vert K\left( k,j\right) \right\vert =\left\vert \varphi_{z_{j}}\left(z_{k}\right) -\varphi_{z_{j}}\left( z_{j}\right) \right\vert \leq\left(\log\frac{1}{1-\left\vert z_{j}\right\vert^{2}}\right)^{-1}
\delta x^\mu=\imath\psi^\mu \epsilon\;,\;\;\delta e=\imath\chi\epsilon\;\;,\delta\psi^\mu=\frac{z^\mu}{2e}\epsilon\;\;,\delta\psi^{2n+1}=\frac{m}{2}\epsilon,\;\;\delta\chi=\dot\epsilon\;,\;\;\delta\kappa=0\;,
K_t = M_t^{-1}= \exp \Big( \int_0^t h(X(s,Z)) dw(s) +\frac{1}{2}\int_0^t h^2(X(s,Z))ds \Big) &=\exp \Big( \int_0^t h(X(s,Z)) dR(s) -\frac{1}{2}\int_0^t h^2(X(s,Z))ds \Big).
\begin{aligned}\min_{u\in U(x^{\bar{u}}(t))}\lambda^{\bar{u}}(t)^T f(x^{\bar{u}}(t),u) =\min_{u\in U}\left[ \left(\lambda^{\bar{u}}(t) \right)^T f(x^{\bar{u}}(t),u) + \sum_{i=1}^{p}\mu_{i}^{\bar{u}}(t)g_{i}(x^{\bar{u}}(t),u)\right] &= \lambda^{\bar{u}}(t)^T f(x^{\bar{u}}(t),\bar{u}(t)) = 0\end{aligned}
\left( \begin{array}{c c}{2 A_{[{\tt i}}{}^{[{\tt I}}\delta_{{\tt j}]}{}^{{\tt J}]}}{\Sigma_{\tt i j K L}}{\Sigma^{\tt k l I J}}{2 A_{[{\tt k}}{}^{[{\tt K}}\delta_{{\tt l}]}{}^{{\tt L}]}}\end{array}\right)
\pi_{21}=1, \pi_{13}=\pi_{15}=\pi_{17}=\pi_{18}=0, \pi_{14}=1, \pi_{16}=-1, \pi_{23}=\pi_{24}=\pi_{26}=\pi_{28}=0, \pi_{25}=1, \pi_{27}=-1, X_1 = \left[ \frac{e_2 e_3}{e_4 e_5 e_7}\right], X_2 = \left[ \frac{e_4 e_6 e_7^2}{e_1^2 e_8}\right].
L = - \frac{1}{4}F_{\mu \nu}^{a}F^{a \, \mu \nu}+ \frac{1}{2}\sum_{i = 1}^{2}\left( D_{\mu}A_{i}\right)^{a}\left( D^{\mu}A_{i}\right)^{a}- \frac{e^{2}}{2}f^{a b c}A_{4}^{b}A_{5}^{c}f^{a d e}A_{4}^{d}A_{5}^{e}+
N_{\widetilde{\varphi_\ell^{H_1}}}(t) =\sum_{0<j<\ell,j\equiv 0\pmod{4}}\frac{(-1)^{\frac{\ell}{4}}}{(-1)^{\ell/4}+2^{(\ell-4)/2}}t^{j - 4}+t^{\ell- 4}&\equiv \frac{(-1)^{\ell/4}}{(-1)^{\ell/4}+2^{(\ell-4)/2}}\frac{1}{1-t^4}+\left(1-\frac{(-1)^{\ell/4}}{(-1)^{\ell/4}+2^{(\ell-4)/2}}\right)t^{\ell- 4}\pmod{t^{\ell-3}}.
\Pi^{\mu \nu}( k ) = ( P_{( 1 )}^{\mu \nu}+ P_{( 2 )}^{\mu \nu}) \Pi^{( 1 )}( k^{2}) + m \sqrt{k^{2}}( P_{( 1 )}^{\mu \nu}- P_{( 2 )}^{\mu \nu}) \Pi^{( 2 )}( k^{2}) .
\partial_i = (1+s_i)\frac{1}{x_i-x_{i+1}}, C_{s_i}=(1+s_i)\frac{t^{-\frac12}-t^{\frac12}x_i^{-1}x_{i+1}}{1-x_i^{-1}x_{i+1}}, D_{i, i+1}= (1+s_i)\frac{1}{1-x_ix^{-1}_{i+1}}, D_{i+1,i}= (1+s_i)\frac{1}{1-x^{-1}_ix_{i+1}}.
Z_{\gamma_1,\gamma_2}(t)\Rightarrow Y_\rho(t) =t \cdot \left[\frac{(\rho+1)^{-1}+(2\sqrt{\rho})^{-1}}{\sqrt{2(\rho+1)^{-2}+(2\rho)^{-1}}}\cdot X_1+ \frac{(\rho+1)^{-1}-(2\sqrt{\rho})^{-1}}{\sqrt{2(\rho+1)^{-2}+(2\rho)^{-1}}}\cdot X_2\right],
\overline{\chi}_{(\lambda_1,\lambda_2)}^\pm(s_1s_2)=\begin{cases}\overline{\chi}^+_{\lambda_1}(s_1)\overline{\chi}^\pm_{\lambda_2}(s_2)+\overline{\chi}^-_{\lambda_1}(s_1)\overline{\chi}^\mp_{\lambda_2}(s_2) s_1\in\tilde{A}_{n_1}s_2\in\tilde{A}_{n_2}, 0 \end{cases}
\frac{\partial}{\partial s}K_L(x,t,\cdot,\cdot)=-L_sK_L(x,t,\cdot,\cdot)+R(g_0(s))K_L(x,t,\cdot,\cdot), \frac{\partial}{\partial s}g=2Ric(g_0(\tau)), \lim_{\tau \to 0}K_L(x,t,\cdot,\tau)=\delta_x.
X_{\gamma}(J_{\alpha \beta}^i) = X_{\gamma}g\big{(}J_{Z_i}X_{\alpha}, X_{\beta}\big{)}= X_{\gamma}g\big{(}Z_i, T(X_{\alpha}, X_{\beta}) \big{)}&= g \big{(}\nabla_{X_{\gamma}}Z_i, T(X_{\alpha}, X_{\beta}) \big{)}+ g\big{(}Z_i, \nabla_{X_{\gamma}}(T(X_{\alpha}, X_{\beta})) \big{)}.
\aligned\sum_{k_1+k_2=k+1}\sum_{\beta_1+\beta_2=\beta}\sum_{i=1}^{k-k_2+1}&(-1)^*{\frak m}^P_{k_1,\beta_1}(h_1,\ldots,{\frak m}^P_{k_2,\beta_2}(h_i,\ldots,h_{i+k_2-1}),\ldots,h_{k}) = 0,\endaligned
\gamma_\xi^{(\beta)}(X) \simeq \frac{\pi^2}{4 5}b_\xi (X){\cal T}_\xi^4 (X) - \frac{1}{1 2}c_\xi (X){\cal M}_\xi^2 (X){\cal T}_\xi^2 (X) + \frac{1}{6 \pi}{\cal M}_\xi^3 (X) \,{\cal T}^{(\xi)}(X) .
\begin{array}{c}{L ={\tilde{\sigma}}\left[ 2 c \epsilon + c{\bf e^{\prime}}_{1}^{2}+{\bf p}({\bf x^{\prime}}-{\bf e}_{+}) + \right.}{\left. +{\bf p}_{+}({\bf e^{\prime}}_{+}-{\bf e}_{1}) - \sum_{i , j}d_{i j}({\bf e}_{i}{\bf e}_{j}- \eta_{i j}) \right] ,}\end{array}
\langle O_{1}(x_{1})O_{2}(x_{2})O_{3}(x_{3})\rangle=\frac{C_{123}}{|x_{12}|^{\Delta_{1}+\Delta_{2}-\Delta_{3}}|x_{13}|^{\Delta_{1}+\Delta_{3}-\Delta_{2}}|x_{23}|^{\Delta_{2}+\Delta_{3}-\Delta_{1}}},
\delta{\cal P}_{i}^{\left( 1 \right) a}= - \epsilon_{0 i j k}\pi^{j k a}, \; \delta{\cal P}_{i}^{\left( 2 \right) a}= - \frac{1}{2}\epsilon_{0 i j k}F^{j k a}, \; \delta{\cal P}^{a}= - \partial^{i}{\cal P}_{i}^{\left( 2 \right) a},
\binom{2p}{p}(T\xi^{p}v_1,T\xi^{p}v_2) = -\sum_{r=0}^{p-1}\binom{2p}{r}(T\xi^r v_1, T\xi^{2p-r}v_2)-\sum_{r=p+1}^{2p}\binom{2p}{r}(T\xi^{r}v_1, T\xi^{2p-r}v_2) = -\sum_{r=0}^{p-1}\binom{2p}{r}(T\xi^r v_1, T\xi^{2p-r}v_2)-\sum_{r=0}^{p-1}\binom{2p}{r}(T\xi^{2p-r}v_1, T\xi^{r}v_2).
I_2(\alpha^{(1)},\alpha^{(2)},\dots,\alpha^{(N)})=\sum\limits_{k_1=1}^{n_1}\sum\limits_{k_2=1}^{n_2}\dots\sum\limits_{k_{N}=1}^{n_N}B(k_1,k_2,\dots,k_{N})\alpha_{k_1}^{(1)}\alpha_{k_2}^{(2)}\dots\alpha_{k_{N}}^{(N)}.
v^n_++v^n_-+(-2+\partial_{\beta}W^{\mu_0}_{v^0}+\sigma^0)v^n+v^0\sigma^n+\lambda^n=R_n &-c_+^0c^n+(-c^0+2-\partial_{\beta}W_{v^0}^{\mu^0}-\sigma^0)c^n_+-c_+^0\sigma^n-\partial_{\beta}\partial_{\beta}W_{v^0}^{\mu_0}c_+^0v^n=\tilde{R_n}.
\widetilde m_{\alpha,\lambda}(s) = \frac{\lambda}{\lambda+s^\alpha}= \frac{\lambda/s^\alpha}{1+\lambda/s^\alpha}= -\sum_{k=1}^\infty \left(-\frac{\lambda}{s^\alpha}\right)^{k}\implies m_{\alpha,\lambda}(x) = -\sum_{k=1}^\infty (-\lambda)^k \frac{x^{\alpha k-1}}{\Gamma(\alpha k)}
H_{-}\chi(r,\theta)= \left(\begin{array}{cc}m ie^{i\theta}\left(\partial_{r}+\frac{i}{r}\partial_{\theta}+\frac{eB}{2}r\right) ie^{-i\theta}\left(\partial_{r}-\frac{i}{r}\partial_{\theta}-\frac{eB}{2}r\right) -m\end{array}\right)\chi(r,\theta)=E\chi(r,\theta),
g_{\mu \nu}= v_{0}h_{\mu \nu}+ \kappa v_{0}( \varphi h_{\mu \nu}+ \varphi_{\mu \nu}) \; , \qquad \mathrm{a n d}\qquad I_{\mu}^{\ \nu}= \left[ \delta_{\mu}^{\ \nu}+ \kappa \frac{\varphi_{\mu \alpha}h^{\alpha \nu}}{( 1 + \kappa \varphi )}\right] \; .
\begin{array}{r c l}{{\hat{M}_{a - 1}}}&{{=}}&{{- R_{5}P_{a}\, , \ \ \ ( a = 0 , . . . , 4 )}}\\{{\tilde{M}_{6 a}}}&{{=}}&{{R_{5}P_{a}\, , \ \ \ ( a = 5 , . . . , 9 ) \, .}}\\ \end{array}
r^{*,\pi}_t(x_t|y^{t-1}_{t-M},y_t)=\Big(\frac{q_t(y_t|y^{t-1}_{t-M},x_t)}{\nu^{\pi}_{t}(y_t|y^{t-1}_{t-J})}\Big){\pi}_{t}(x_t|y^{t-1}_{t-J}),~t\in\mathbb{N}_0^{n-1}
n_{x^{(l)}}\!\left(y^{(k)}\right)= \; n_{{(x-1)}^{(l)}}\!\left((y-1)^{(k)}\right)+n_{{(x+1)}^{(l)}}\!\left((y-1)^{(k)}\right)+n_{{(x-1)}^{(l)}}\!\left( (y+1)^{(k)}\right) &+n_{{(x+1)}^{(l)}}\!\left( (y+1)^{(k)}\right)=n_{{(x-1)}^{(l)}}\!\left(y^{(k)}\right)+n_{{(x+1)}^{(l)}}\!\left(y^{(k)}\right),
\left. \left( S + S_{\mathrm{g f}}+ S_{\mathrm{g h}}\right) \right|_{\mathrm{b i l i n e a r}}= \int d^{4}x \sqrt{- g}\left({\Phi}\hat{H}{\Phi}^{T}+ c_{\mu}\hat{H}_{\mathrm{g h}c^{\mu}}\right) ,
f_d(a) = \begin{cases}1 + |a| a \leq 0 \left(\sum_{i = 0}^{d - 1}a^i\right)^{-1}0 \leq a \leq 1 1 + 1/d - a 1 \leq a \leq 1 + 1/d 0 a \geq 1 + 1/d\end{cases}.
= 1 - \frac{1 1}{3 6}\times 3 + 3 - \frac{1 1}{3 6}\times 5 + 5 - \frac{1 1}{3 6}\times 7 + 7 - \frac{1 1}{3 6}\times 9 + 9 + \frac{1 1}{3 6}\times 1 1 + 1 1 - \frac{1 1}{3 6}\times 1 3 7
\delta_{n,m}=\sum_{\mu}\mathcal{C}_{n}^{\left( +\right)}\left( E_{\mu}\right) \mathcal{C}_{m}^{\left( +\right)}\left( E_{\mu}\right)+\int\limits_{\Omega_{c}^{\left( +\right)}\left( E\right)}\mathcal{C}_{n}^{\left( +\right)}\left( E\right) \mathcal{C}_{m}^{\left( +\right)}\left( E\right) dE
\begin{array}{c}\langle \widehat{q}_1^2 f,f \rangle_{L^2 (\mathbb{R}^2)}=E^2 (x_1^{(0)})^4 + 2 \lambda E (x_1^{(0)})^3 + \left(\lambda^2 + \frac{3}{2}a E^2 \right) (x_1^{(0)})^2 + \frac{3}{2}\lambda a E x_1^{(0)}+ +\frac{\lambda^2 a}{4}+ \frac{\theta^2}{4 \lambda^2 b}+ \frac{3 a^2 E^2}{16}\end{array}
\int \prod_{i>j}[dc^{ij}][d\bar{c}^{ij}][dA_{\mu}^{ij}] e^{-S_{\rm off-diag}}= \exp \left[{1 \over 8 \pi}\int_{\Sigma_g}R \sum_{i>j}\log (\lambda_i -\lambda_j) \right]~.
T \equiv \beta_{H}^{- 1}= \left. \frac{\tilde{A}^{\prime}}{4 \pi \sqrt{\tilde{A}\tilde{B}}}\right|_{r_{+}}\nonumber \, = \frac{\tilde{d}}{4 \pi r_{+}}\left[ 1 - \left( \frac{r_{-}}{r_{+}}\right)^{\tilde{d}}\right]^{( \tilde{d}- 2 ) / 2 \tilde{d}},
\lambda_1^\star = \left(\frac{z_2}{b}\right)\frac{l_2^{-1}z_2^{-l_2}-l_1^{-1}z_2^{-l_1}}{l_1^{-1}z_2^{-l_1}+l_2^{-1}z_2^{-l_2}}-1 \kappa_1^\star =z\left[l_1^{-1}+l_2^{-1}-1\right]-z_2\frac{l_1^{-1}z_2^{-l_1}-l_2^{-1}z_2^{-l_2}}{l_1^{-1}z_2^{-l_1}+l_2^{-1}z_2^{-l_2}}\left[l_1^{-1}-l_2^{-1}+\ln{\hat{x}_2}-\ln{x^\star_2}\right],
{\rm det\,}\left((b_{j,\ell})^{n-d}_{j,\ell=0}\right)\neq0, &b_{0,\ell}=B_{G,M_1}(p,a_{\ell}),\ \ \ell=0,\ldots,n-d, &b_{j,\ell}=(Z_{j,x}B_{G,M})(p,a_\ell),\ \ \ell=0,\ldots,n-d, j=1,\ldots,n-d.
\Psi^{(p,g)}(\Phi_{(p,g)}(a)) = \Psi^{(p,g)}(\Theta_{x\otimes g, y\otimes g}) = \Psi_{(p,g)}(x\otimes g)\Psi_{(p,g)}(y\otimes g)^* = \psi_p(x) U_g (\psi_p(y) U_g)^* = \psi_p(x) U_g U_g^* \psi_p(y)^* = \psi_p(x) \psi_p(y)^* = \psi^{(p)}(\theta_{x,y}) = \psi^{(p)}(\phi_p(a)) = \psi_{e}(a)=\Psi_{(e,e)}(a).
\alpha=\int_{\partial\Omega}y^\perp\cdot\left\{\big(T(u,p)+a\,u\otimes y^\perp\big)\,\nu\right\}\,d\sigma_y+\int_\Omega y^\perp\cdot f\,dy, &\beta=\int_{\partial\Omega}\nu\cdot u\,d\sigma.
E(\Sigma, X, T_0)=\frac{1}{8\pi}\int_\Sigma \left[f (1+|\nabla\tau|^2)+\Delta \tau \sinh^{-1}(\frac{f \Delta\tau}{|H_0| |H|})-\alpha_{H_0}(\nabla \tau)+\alpha_H(\nabla \tau)\right] d\Sigma,
\frac{C(\alpha(s),\beta(s);z)}{C(\alpha(1-s),\beta(1-s);z)}=\pi^{-2g(s-\frac{1}{2})}\prod_{j=1}^g \left((-1)^{\epsilon_j (\alpha_j-\beta_j)}\cdot\frac{\Gamma(1-\beta_j)}{\Gamma(\alpha_j)}\right).
i \ln W_{q \overline{{{q}}}}^{\mathrm{S R}}= \int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}d t \, \left\{- \frac{4}{3}\frac{\alpha_{s}}{r}+ \frac{1}{2}\frac{4}{3}\frac{\alpha_{s}}{r}( \delta^{h k}+ \hat{r}^{h}\hat{r}^{k}) \dot{z}_{1}^{h}\dot{z}_{2}^{k}\right\}.
L^* L = W^* A^* \overline{W}W^T A W = W^* A^* A W = \Lambda^2, L L^* = W^T A W W^* A^* \overline{W}= W^T A \overline{A}\overline{W}= \overline{W^* A^* A W}= \overline{\Lambda^2}= \Lambda^2,
\Gamma_a^{112}(x,y) =\frac{|x|^2(x\otimes y)}{64\pi}\int_0^\infty \int_t^\infty e^{-|x|^2/4s}\,\frac{ds}{s^4}\, dt+M^{112}_a(x,y) &=\frac{x\otimes y}{4\pi|x|^2}+M^{112}_a(x,y)
\sqrt{(1-\frac{2GM}{rc^{2}})dt^{2}-\frac{1}{c^{2}}(1-\frac{2GM}{rc^{2}})^{-1}dr^{2}-\frac{r^{2}}{c^{2}}d\theta^{2}-\frac{r^{2}}{c^{2}}sin^{2}\theta d\phi^{2}}
F_r(\phi,\psi) =F_r(\phi)+F_{r-1}(\psi)=\sum_{k=0}^rE_k(\phi)+\sum_{k=0}^{r-1}E_k(\psi)+\|\psi\|_{L^2}^2 &=s^2\|\partial_t\phi\|^2_{H^r}+\|D\phi\|^2_{H^r}+s^2\|\nabla_t\psi\|^2_{H^{r-1}}+\|D\psi\|^2_{H^{r-1}}+\|\psi\|_{L^2}^2.
z_N(s) = \frac{4}{\sqrt{\pi}}\int_{0}^{1}\frac{1}{\sqrt{1-\rho}}\left[ \frac{1}{2}+ \sum_{n=1}^{N}\exp\left[-n^2\pi \left(1-\rho\right)^{\frac{1}{s}}\right] \right] d\rho.
\sum_{\epsilon_{l}\leq 0}u_{\epsilon_{l}l}\left( r \right) u_{\epsilon_{l}l}\left( r^{\prime}\right) + \int_{0 +}^{\infty}\frac{\mathrm{d}k}{2 \pi}\, u_{k l}\left( r \right) u_{k l}\left( r^{\prime}\right) = \delta \left( r - r^{\prime}\right) \, .
k_{4,2}=\frac{3}{2}P e^{\Phi_{\infty}}M_1 V_1, \qquad k_{4,1}=-\frac{3}{2}V_{\Phi}P e^{\Phi_{\infty}}+\frac{51}{8}M_1 V_1 P e^{\Phi_{\infty}}-2 M_2 V_1 -\frac{3}{2}M_1 V_2 + O(M_i^4), begin{align*}2mm]
\vec{B}=curl\vec{A}=\{\frac{\partial A_{3}}{\partial x_{2}}-\frac{\partial A_{2}}{\partial x_{3}},\frac{\partial A_{1}}{\partial x_{3}}-\frac{\partial A_{3}}{\partial x_{1}},\frac{\partial A_{2}}{\partial x_{1}}-\frac{\partial A_{1}}{\partial x_{2}}\},or\Phi_{B}=dA.
\boldsymbol{\vartheta}_k \circ u(k) = v_1(k)+ \dotsb v_J(k) v_j(k) \in B_{\frac{\sigma_k}{5}}(s_j); T(k)= \boldsymbol{\vartheta}_k \circ T(k) = T_1(k)+ \dotsb + T_J(k) T_j(k) \in B_{\frac{\sigma_k}{5}}(s_j)
\mathbb{E}\big[ \mathfrak{l}^{(p,p)}\big]= \frac{1}{N}\sum_{i=1}^N \mathbb{E}\big[ \mathfrak{d}_{i,1}Q_i \mathfrak{l}^{(p-1,p)}\big]+ \frac{1}{N}\sum_{i=1}^N \mathbb{E}\big[\mathfrak{d}_{i,2}\mathcal{Q}_i \mathfrak{l}^{(p-1,p)}\big].
(I-U_{1}^{\dagger}V_{1})^{-1}A^{\dagger}\geq (I-U_{1}^{\dagger}V_{1})^{-1}U_{1}^{\dagger}(I-V_{2}U_{2}^{\dag})^{-1}=A^{\dagger}(I-V_{2}U_{2}^{\dag})^{-1}.
t^2+\left(1-\frac{4h(\Lambda_1^2,\Lambda_2^2)}{(\Lambda_1^2)^{p+1}}\left(v^2-u_1+\frac{\Lambda_1^2}{2}+O(u_1^2,u_2^2)\right)\right)t+\left(\frac{\Lambda_2^2}{\Lambda_1^2}\right)^{p+1}=0
\partial_t w = c_\mu\rho^{\alpha-1}\partial_x^2 w- (u+c_\mu\rho^{\alpha-2}\partial_x \rho )\partial_xw+ \frac{c_p}{c_\mu}\left(\gamma -2 (\alpha+1) \right)\rho^{\gamma-\alpha}w &-\frac{1}{c_\mu}(\alpha+1)\rho^{-\alpha}w^2+ \frac{c^2_p}{c_\mu}\left(\gamma - (\alpha+1) \right)\rho^{2\gamma-\alpha}.
\int_{H_{e}}\int_{\tilde{H}_{e}}\phi(t+y_1) d |D^{s}f^e_{y_1}|(t)d\mathcal{H}^{d-1}(y_1)=\int_{\mathbb{R}^d}\phi(x)\left|\langle e,\eta(x)\rangle\right|d|D^sf|(x),
\lambda\dot{a}=\frac{r}{k}(\gamma_{3}\frac{\nabla K(a)}{K(a)\lambda}+\gamma_{4}\frac{\nabla \Delta K(a)}{K(a)\lambda^{3}}) (1+o_{\frac{1}{\lambda}}(1)) & +o(\frac{1}{\lambda^{3}})+O(\frac{\vert\nabla K(a)\vert^{2}}{\lambda^{2}}+\vert \delta J(u)\vert^{2}),
P(E|S_{i})\le P(\bigcup_{i\ne i^{\prime}}A_{i^{\prime}})\le(\sum_{i\ne i^{\prime}}P(A_{i^{\prime}}))^{\rho}\le(\frac{1}{M}\sum_{i\ne i^{\prime}}(\frac{P(X_{1}^{n}(i^{\prime}))}{P(X_{1}^{n}(i))})^{s})^{\rho}.
T|T|= \begin{pmatrix}0 1 0 \ldots 0 0 0 1 0 \ldots 0 \vdots \vdots \ddots \ddots \ldots 0 \vdots \vdots . 0 \ddots 0 . . . . 0 1 0 0 . . \ldots 0 \end{pmatrix}=T
T=2\pi\sqrt{\frac{L}{g}}[\sum_{n=0}^{\infty}(\frac{(2n)!}{2^{2n}(n!)^{2}})^{2}sin^{2n}(\frac{\theta_{0}}{2})]=2\pi\sqrt{\frac{L}{g}}(1+\frac{1}{16}\theta_{0}^{2}+\frac{11}{3072}\theta_{0}^{4}+\cdot\cdot\cdot)
{\tilde{\Pi}}_{\mu \nu}( k ) \; = \; - e^{2}\, \int \frac{d^{D}p}{( 2 \pi )^{D}}\; \mathrm{t r}\left[ \gamma_{\mu}\, S_{F}( p ) \, \Gamma_{\nu}( p , p - k ) \, S_{F}( p - k ) \right]
\mathbf{K}_{1}\mathbf{K}_{1}^{\ast}\mathbf{=K}\left\vert \mathbf{K}_{2}\right\vert^{-2}\mathbf{K}^{\ast}\mathbf{=K}\left[ \boldsymbol{\mu +}\frac{1}{2}\mathbf{(\mathbf{K}^{\ast}K+I)}\right]^{-1}\mathbf{K}^{\ast}\leq \mathbf{I}_{A},
\alpha_{+}= -\frac{(i+1)z+\sqrt{2}\kappa}{(i-1)z+\sqrt{2}\kappa}, \beta_{+}= \frac{2z}{(i-1)z+\sqrt{2}\kappa}, \alpha_{-}= \frac{(i+1)z-\sqrt{2}\kappa}{(i-1)z+\sqrt{2}\kappa}, \beta_{-}= -\frac{2iz}{(i-1)z+\sqrt{2}\kappa},
\tau =\sum_{(J,K,L,M)\in\mathcal{T}}\sigma_J\sigma_K\sigma_L\sigma_M=\sum_{j=1}^s \sum_{\substack{(J,K,L,M)\in\mathcal{T}: \mathbb{F}(J,K,L,M)\in[\mathbb{F}_j]_\sim}}\sigma_J\sigma_K\sigma_L\sigma_M \leq \Bigl(d!(d-1)!\sum_{j=1}^s\gamma(\mathbb{F}_j)^{-1}\Bigr)\rho_n^2
(g-1 )^{m+n-2 k}(f_{i,m+n-k-i+1}) &=\sum_{\ell=\max(m+n-2 k-i+1,k-i+1)}^{\min(m+n-2 k,m+n-k-i)}\binom{m+n-2 k}{\ell}f_{i+\ell-m-n+2 k,m+n-k-i-\ell+1}.
\rho^{(1)}_{\mu_1 \dots \mu_{p-1}\alpha_1}=(C \Gamma_{\nu \mu_1 \dots\mu_{p-1}})_{\beta \alpha_1}\Pi^\nu \Pi^\beta\quad, \quad \quad\rho^{(2)}_{\mu_1 \dots \mu_{p-1}\alpha_1}= (C \Gamma^\nu)_{\beta\alpha_1}\Pi_{\nu \mu_1 \dots \mu_{p-1}}\Pi^\beta\quad .
x_{u+(i_0,l)}x_{u}= \prod_{(0,0) \rightarrow v}x_{u+v}+ \prod_{v \rightarrow (0,0)}x_{u+v}y_{u+(i_0,l)}y_u = \frac{\prod_{v \rightarrow (0,0)}(1+y_{u+(i_0,l)-v})}{\prod_{(0,0) \rightarrow v}(1+y_{u+(i_0,l)-v}^{-1})}
\limsup_{h\to0}\frac{1}{h}P(N_{t-d+h}-N_{t-d}\ge 2,Z_t = j, d-h<U_t|Z_{t-d}= k, U_{t-d}= v) \le \limsup_{h\to0}\frac{1}{h}P(N_{t-d+h}-N_{t-d}\ge 2|Z_{t-d}= k, U_{t-d}= v)=0,
\hat{H}_{JC}=\hbar\omega\hat{a}^{\dagger}\hat{a}+\hbar\omega\frac{\hat{\sigma}_{z}}{2}+\frac{\hbar\Omega}{2}(\hat{a}\hat{\sigma}_{+}+\hat{a}^{\dagger}\hat{\sigma}_{-}+\hat{a}\hat{\sigma}_{-}+\hat{a}^{\dagger}\hat{\sigma}_{+}),
\frac{d}{dt}r(\boldsymbol{\zeta}^{(i)}(t)) = (\zeta_0^{(i)})'(t) + \sum_{k=1}^\infty k (\zeta_k^{(i)})'(t) \ge \sum_{k=0}^\infty (k-2) r_k(\boldsymbol{\zeta}^{(i)}(t)) - \sum_{k=1}^\infty k r_k(\boldsymbol{\zeta}^{(i)}(t)) & = -2\cdot{{1}}_{\{r(\boldsymbol{\zeta}^{(i)}(t))>0\}}\ge -2.
\begin{array}{l}{{S ({\bf X}, \sigma , \hat{g}) = S_{M}({\bf X}, \hat{g}) + \frac{1}{4 \pi}\int d^{2}z \sqrt{\hat{g}}\left( \hat{g}^{a b}\partial_{a}\sigma \partial_{b}\sigma + Q \hat{R}\sigma + \mu e^{2 \alpha \sigma}\right)}}\\{{S_{M}({\bf X}, \hat{g}) = \frac{1}{4 \pi \alpha^{\prime}}\int d^{2}z \sqrt{\hat{g}}\, \hat{g}^{a b}\partial_{a}{\bf X}\cdot \partial_{b}{\bf X}\ .}}\\ \end{array}
\begin{array}{rcr@{}l}\sigma_k(F)(x^j) = j x^{j-1}\sigma_k(H)(x^j) = (2j-k) x^j \sigma_k(E) (x^j) = (k-j) x^{j+1}\rho_k(J)(x^j) = (-1)^j x^{k-j}. \end{array}
\begin{bmatrix}B_{1}^\top \otimes A_1 - D_{1}^\top \otimes C_1 & \ddots \ddots & B_{r-1}^\top \otimes A_{k-1}- D_{r-1}^\top \otimes C_{r-1}-D_{r}^\top \otimes C_{r}B_{r}^\top \otimes A_{r}\end{bmatrix}{\cal X}={\cal E},
\frac{\operatorname*{d e t}D ( A )}{\operatorname*{d e t}D ( 0 )}\; = \; e^{- \frac{i}{2 \pi}A_{1}A_{2}}\; q^{\frac{1}{8 \pi^{2}}A_{1}^{2}}\; \frac{\vartheta ( \alpha , \tau )}{\vartheta ( 0 , \tau )}\; ,
\left(B(x,\xi) (x+p'(\xi))^2\right) \sharp \gamma(\frac{x+p'(\xi)}{\sqrt{h}}) = \left(B(x,\xi)\gamma(\frac{x+p'(\xi)}{\sqrt{h}})(x+p'(\xi)) \right)\sharp (x+p'(\xi)) + h(r_0-r'_0) \,,