harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g\left(\omega = \frac{2\pi \left[n-1\right]}{N_{\mathrm{F}}^{\mathrm{h}}}\right) = \left(\mathbf{f}_{N_{\mathrm{F}}^{\mathrm{h}}}^n\right)^{\mathrm{T}}\boldsymbol{\rho}_{k_l}^\mathrm{h}= \mathcal{F}\left\lbrace \boldsymbol{\rho}_{k_l}^\mathrm{h}\right\rbrace^{N_\mathrm{F}^{\mathrm{h}}}_n,
2\pi i f(z)=-\int_{p-ie^{i\alpha}[0,+\infty)}\left(\int_{e^{-i\alpha}[0,+\infty)}f(\zeta)e^{\omega \zeta}d\zeta\right)e^{-\omega z}d\omega+ +\int_{p+ie^{-i\alpha}[0,+\infty)}\left(\int_{e^{i\alpha}[0,+\infty)}f(\zeta)e^{\omega \zeta}d\zeta\right)e^{-\omega z}d\omega.
v_1+ \dots+v_m+v'_1+\dots+v'_m begin{align*}1ex] =\ell_0\delta-\delta+n-m-\ell_0\delta-\delta(2m-1)-\ell_{1,1}\delta-\delta-(m-1) begin{align*}1ex] =n-2m+1-(2m+1)\delta-\ell_{1,1}\delta begin{align*}1ex] \ge n-2m+\tfrac12-(2m+1)\delta>n-2m
(f^{'}\cos \theta \cos 2 \theta + \frac{2}{r}f \sin \theta \sin2 \theta) \frac{\partial \gamma^2}{\partial x^2}+(f^{'}\sin\theta \sin 2 \theta + \frac{2}{r}f \cos \theta \cos 2 \theta)\frac{\partial \gamma^1}{\partial x^1}
E \sim \int_{0}d r ~ r^{2}~ \left( \kappa_{1}( f^{\prime})^{2}+ \kappa_{2}\frac{1}{r^{2}}( \sin f )^{2}+ \kappa_{3}\frac{1}{r^{2}}( f^{\prime})^{2}( \sin f )^{2}+ \kappa_{4}\frac{1}{r^{4}}( \sin f )^{4}\right) .
\mathbb{P}_{\!x}\{X_{\tau_{ab}}\in \mathrm{d}z\}/\mathrm{d}z=\sum_{k=0}^{N-1}(-1)^kH_k^-(x)\,\delta_a^{(k)}(z)+\sum_{k=0}^{N-1}(-1)^kH_k^+(x)\,\delta_b^{(k)}(z)
\Theta = \frac{\kappa}{2 \pi}\int_{D}d^2 x ~\epsilon^{ij}\left< \partial_i g^{-1}\partial_j g g^{-1}d g \right> + \frac{\kappa}{2 \pi}\oint_{\partial{D}}d \varphi \left< g^{-1}\partial_{\varphi}g g^{-1}d g \right>,
SO_{\alpha}(G^*)-SO_{\alpha}(G^{*'}) = \left(d_{G^*}^2(v_{i-1})+9\right)^{\alpha}-\left(d_{G*}^2(v_{i-1})+4\right)^{\alpha}+2\cdot13^{\alpha}+2\cdot 8^{\alpha}-3\cdot20^{\alpha}-17^{\alpha}&\leq 10^{\alpha}-5^{\alpha}+2\cdot13^{\alpha}+2\cdot 8^{\alpha}-3\cdot20^{\alpha}-17^{\alpha}<0,
\sigma_k^2 U_k(\sigma_k t) &= e^{- \sigma_k N t}D^2 F(\bar u_k,\bar v_k)(t) - \sigma_k^2 \beta_k^{2}\left (\begin{array}{cc}W_1(\sigma_k t) 0 0 W_2(\sigma_k t)\end{array}\right) &\longrightarrow\; D^2 F(\bar u,\bar v)(t)
\int_{2B}\omega{(B_{xu})}^{-\frac{n-1}{n}}\omega(u)du\leq C \sum_{k\geq 0}\int_{u\in 2B, \omega_{(B_{xu})}\approx 2^{-k}\omega(2B)}2^{\frac{k(n-1)}{n}}\omega(2B)^{-\frac{n-1}{n}}\omega(u)du \leq C \omega(B)^{-\frac{n-1}{n}}\sum_{k\geq 0}2^{\frac{k(n-1)}{n}}2^{-k}\omega(B) = C \omega(B)^{\frac{1}{n}}.
\lvert u_n\rvert_S =\lvert(\phi v_0)_n\rvert_S-(n-i) &=\lvert(\phi v_0)_n\rvert_S-n+i &\geq \lvert(\phi v_0)_n\rvert_S-n+(n-2\lvert I\rvert+1) &=\lvert(\phi v_0)_n\rvert_S-2\lvert I\rvert+1
\mathcal{A}(y,p^1,p^2,f):= (\mathcal{A}_1(y,p^1,p^2,f),\mathcal{A}_2 (y,p^1,p^2,f),\mathcal{A}_3(y,p^1,p^2,f),\mathcal{A}_4(y,p^1,p^2,f))
\begin{array}{|c|c|c|c|}\hline$Group$ $Order$ $Generators$ $Remarks$ \hline \hline$VII$* 120\times 4 ($I$),F'' Lift(S_5) $VIII$* 120\times 4 ($II$),F' Lift(S_5) $IX$* 720\times 4 ($III$),F'' Lift(S_6) \hline\end{array}
| (\int_0^\tau A^{1/2}e^{-(\tau -s) A}g(s) ds, x) | &= |\int_0^\tau (g(s) ,{A^*}^{1/2}e^{-(\tau -s) A^*}x) | &\le (\int_0^\tau \| g(s) \|^2 ds)^{1/2}( \int_0^\tau \|{A^*}^{1/2}e^{-(\tau -s) A^*}x \|^2 ds)^{1/2}&\le C \| x \| (\int_0^\tau \| g(s) \|^2 ds)^{1/2}.
\left| \sum_{n = 1}^W e\left(\overline{\xi}_{x + nw}\right)\right|^2 \leq \frac{W(W+1)}{2}+ \frac{W+1}{2}\left| \sum_{n = 1}^{W-1}e\left(\overline{\xi}_{x + (n+1)w}- \overline{\xi}_{x + nw}\right)\right|.
\widetilde{\mathcal{I}}_{[m,j]}:=\{(i_{1},\ldots,i_{m})\in\mathbb{Z}^{m}:-p\leq i_{1}\leq (m-1)p+j\,\,\mbox{and}\,\,i_{k-1}-p\leq i_{k}\leq (m-k)p+j,\,\,2\leq k\leq m\}.
K:=\min\Big\{\frac{|\chi_2\lambda_2\mu_2-\chi_1\lambda_1\mu_1|+\chi_1\mu_1|\lambda_1-\lambda_2|}{\lambda_2},\frac{\chi_2\mu_2|\lambda_1-\lambda_2| + |\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1|}{\lambda_{1}}\Big\}.
g_{1}(\nabla^{^{M_1}}_{U}V,X) =g_{1}(\phi^{2}V,\nabla^{^{M_1}}_{U}X)+g_{1}(\omega\phi V,\nabla^{^{M_1}}_{U}X)-g_{1}(\omega V,\mathcal{T}_{U}\mathcal{B}X) &-g_{1}(\omega V,\mathcal{H}\nabla^{^{M_1}}_{U}\mathcal{C}X)-g_{1}(V,\phi U)\eta (X).
{\cal M}^{(x,0)}={\cal M}^{(x,0)}_{(K,\widehat{K})}:={\cal M}({\cal S}(\widehat{\pi}^{(x,0)}_r, \widehat{\pi}^{(x,0)}_l)) \widehat{{\cal M}}^{(x,0)}= \widehat{{\cal M}}^{(x,0)}_{(K,\widehat{K})}:={\cal M}(\widehat{{\cal S}}(\widehat{\pi}^{(x,0)}_r, \widehat{\pi}^{(x,0)}_l)).
Q^{(m)}_\nu(k) = Q^{(m-1)}_\nu(k)\frac{q^{k+1}-1}{q-1}- \frac{q^{m-2}-1}{q-1}\sum_{a=1}^k q^aQ^{(m-1)}_\nu(a-1) &= Q^{(m-1)}_\nu(k)\frac{q^{k+1}-1}{q-1}- Q^{(m)}_\nu(k-1)\frac{q^{m-1}-q}{q-1}&= Q^{(m-1)}_\nu(k)\frac{q^{k+1}-1}{q-1}- \left(Q^{(m)}_\nu(k) - q^{k}Q^{(m-1)}_\nu(k)\right)\frac{q^{m-1}-q}{q-1}.
\frac{d}{dt}| \varphi_t(\omega, x^\prime) - \varphi_t(\omega,x)|^2 = 2 \left\langle b(\varphi_t(\omega, x^\prime)) - b(\varphi_t(\omega, x)) ,\varphi_t(\omega, x^\prime) - \varphi_t(\omega, x) \right\rangle &\leq 2 |\varphi_t(\omega, x^\prime) - \varphi_t(\omega, x)|^2
e^{- 2 \pi \big ( 1 - \frac{g}{\pi + g N}\big ) \sum_{i = 1}^{n_{a}}\sum_{j = 1}^{n_{a}}\Big [ \big ( 1 \! - \! \delta_{i j}\big ) \big ( C^{0 , S}( x_{i}^{( a )}, x_{j}^{( a )}) + C^{0 , S}( y_{i}^{( a )}, y_{j}^{( a )}) \big ) - 2 C^{0 , S}( x_{i}^{( a )}, y_{j}^{( a )}) \Big ]}
\lim_{t\to \infty}\frac{D_{1-x/t}-K_{d}}{A_{d}(b_{d}(t))}= \lim_{t\to \infty}\left[ \frac{{\bf{I}}(x,t)}{A_{d}(b_{d}(t))}- \frac{{\bf{II}}(x,t)}{A_{d}(b_{d}(t))}\right]= \frac{c_{d}K_{d}}{\alpha \rho}(x^{-\rho/\alpha}-1).
\frac{\partial}{\partial{{z}^{i}}}=\frac{1}{2}\left( \frac{\partial}{\partial{{x}^{i}}}-\sqrt{-1}\frac{\partial}{\partial{{y}^{i}}}\right) \frac{\partial}{\partial{{\overline{z}}^{i}}}=\frac{1}{2}\left( \frac{\partial}{\partial{{x}^{i}}}+\sqrt{-1}\frac{\partial}{\partial{{y}^{i}}}\right)
\omega_{\mu}{}^{ab}(e)= -e^{\nu[a}\left( \partial_{\mu}e_{\nu}{}^{b]}-\partial_{\nu}e_\mu{}^{b]}\right) - e^{\rho[a}e^{\sigma b]}\left( \partial_{\sigma}e_{c\rho}\right) e_{\mu}{}^{c}\ .
\begin{array}{l}\displaystyle{\inf}\; \mathbb{E}_{\xi_2,\ldots,\xi_{T_{\max}}}\Big[ \displaystyle{\sum_{t=1}^{T_{\max}}}\; f_{t}(x_{t}, x_{t-1}, \xi_t)\Big] x_t \in X_t( x_{t-1}, \xi_t )\;\mbox{a.s.}, x_{t}\;\mathcal{F}_t\mbox{-measurable,}t=1,\ldots,T_{\max},\end{array}
\begin{cases}\lambda_3\lambda_2 = \frac{(a-2) \pm \sqrt{a^2 - 4a}}{2}\lambda_2\lambda_1 = \frac{(b-2) \pm \sqrt{b^2 - 4b}}{2}\lambda_1\lambda_3 = \frac{(c-2) \pm \sqrt{c^2 - 4c}}{2}\end{cases}
\frac{K}{2}\int_{R}\sqrt{- g}g^{r r}\phi ( - k , r ) \partial_{r}\phi ( k , r ) \frac{d^{4}k}{( 2 \pi )^{4}}- \frac{K}{2}\int_{L}\sqrt{- g}g^{r r}\phi ( - k , r ) \partial_{r}\phi ( k , r ) \frac{d^{4}k}{( 2 \pi )^{4}}\ ,
\frac{1}{d^4}\geq 1\,, & p=d+ \frac{y_1^2}{|y|}(\alpha-c_1)|k'_\eta(|y|)| \geq d \geq c_1\,, & q^2 = (\alpha -c_1)^2 \frac{y_1^2 y_2^2}{|y|^2}(k'_\eta(|y|))^2 \leq 9 (1-c_1)^2 \,, & |m| \leq \frac{42 c_2 (1-c_1^2)}{c_1}\eta + \frac{c_2^2 (1-c_1^2)}{c_1^2}\eta^2\,.
+\frac{1}{\sqrt{2\pi}}\sum_{n=1}^{\infty}\sum\limits_{k=1}^{N-1}\left(\frac{(-q_{k,n})\int\limits_0^tu_n(s)e^{-q_{k,n}(t-s)}ds}{l^{(1)}_n(-q_{k,n})}\right)\psi_n(x).
\psi^2+\psi{b}=\frac{s^2x^2}{(p+1)^2}\left(\frac{-\log(s)}{p}+\frac{1}{p+1}\right)\left(\frac{-\log(s)}{p}+\frac{1}{p+1}+\frac{2(p+1)\log(s)}{p(s+1)}\right).
0\le \widetilde \eta\le 1, \widetilde \eta\equiv 1 \mbox{on}B_{1/4}, \widetilde \eta\equiv 0 \mbox{on}\mathbb{R}^N\setminus B_{5/16}, |\nabla \widetilde\eta|\le c_N' \mbox{and}|D^2\widetilde\eta|\leq c_N'.
\det\left( - \frac{(\delta - |w|)}{\delta}df_{z}(w)\right) ~= \det \left(Id - \frac{1}{\delta - |w|}(z-w) \left(\frac{w}{|w|}\right)^{T}\right) \crcr ~ = 1 - \frac{1}{(\delta - |w|)|w|}\langle z - w, w \rangle.
L_{v}(f)\; =\;L_{v\chi}(f)\;=\;\lim_{r\downarrow 0}W_r(\overline{v\chi},f)\;=\;-\omega(v\chi,f) =\;\langle v\chi,S^*f\rangle-\langle S^*(v\chi),f\rangle\;=\;\langle v\chi,\overline{S}f\rangle-\langle v\chi,\overline{S}f\rangle\;=\;0\,,
t_{1-\alpha,D}(X):=\varepsilon^2 \sum_{j=1}^{D \wedge M}X_j^{-2}+ C(\alpha) \varepsilon^2 \sqrt{\sum_{j=1}^{D \wedge M}X_j^{-4}}+(1+\sqrt{x_{\alpha/2}}) \left[\sigma^2 \ln^{3/2}(1/\sigma) \vee a_{D \wedge M}^{-2}\right],
\|u^*(x)\|_{L^2(B^n)}\lesssim\sum_{k=0}^{\infty}\sum_{\ell=1}^{d(k)}\Big\|\int_0^\infty \tilde{J}_{\nu(k)}(r\rho)e^{it(r)\rho^2}\rho^{-\frac14}g(\rho)\mathrm{d}\rho\Big\|_{L^2_{r}([0,1])}.
f(G_{1}', G_{2}', s_{1}, s_{2}) = |E(G_{1}')| + |E(G_{2}')| + s_{2}|G_{1}'| + s_{1}|G_{2}'| &=\big( |E(G_{1})| - d_{G_{1}}(u_{1}) \big) + \big( |E(G_{2})| + d_{G_{2}}(u_{1}) \big) + s_{2}( |G_{1}| - 1) + s_{1}(|G_{2}| + 1) &= f(G_{1}, G_{2}, s_{1}, s_{2}) + \big( d_{G_{2}}(u_{1}) - d_{G_{1}}(u_{1}) \big) - (s_{2}- s_{1}).
\nabla_{2k+2}=\sum_{s=0}^{k}(-1)^{s-1}\nabla_s\Delta_{2k+2-s}+\sum_{\substack{r=k+2 r}}^{2k+1}\nabla_r\Delta_{2k+2-r}-\sum_{\substack{t=k+1 t}}^{2k}\nabla_t\Delta_{2k+2-t}=\sum_{s=0}^{2k+1}(-1)^{s-1}\nabla_s\Delta_{2k+2-s}.
\frac{1}{C\sqrt{\delta}}\leq \|\nabla u\|_{L^{\infty}}\leq \frac{C}{\sqrt{\delta}}\mbox{for}d=2,~\qquad\frac{\log\delta^{-1}}{C~\delta}\leq \|\nabla u\|_{L^{\infty}}\leq \frac{C \log\delta^{-1}}{\delta}\mbox{for}d=3,
\sum_{a\leq x}\!{\vphantom{\sum}}^\ast A_k(a)=\frac{1}{2\pi i}\int_{k+\epsilon-i\infty}^{k+\epsilon+i\infty}f_k(s)\frac{x^s}{s}ds=\frac{1}{2\pi i}\int_{k+\epsilon-i\infty}^{k+\epsilon+i\infty}f_0(s-k)\frac{x^s}{s}ds,
f ({\bf r}) = \frac{1}{2 \pi}e^{i \theta}\left( \frac{\partial}{\partial r}+ \frac{i}{r}\frac{\partial}{\partial \theta}\right) \int_{D}d^{2}r^{\prime}\ln{|{\bf r - r^{\prime}}|}e^{2 \phi ({\bf r^{\prime}})}b ( r^{\prime}e^{i \theta^{\prime}}) + C ,
\begin{aligned}\phi''(z^*) +\left(\frac{1}{z^*-s}-\frac{1}{z^*-s+u(s)}\right)\phi'(z^*) &+\left(\frac{\frac{\left(u'(s)\right)^2}{4u(s)}-\frac{u'(s)}{2u(s)}-su(s)+u^2(s)}{z^*-s}+\frac{\frac{u'(s)}{2u(s)}}{z^*-s+u(s)}-z^*\right)\phi(z^*)=0,\end{aligned}
J_a\propto\left| \begin{array}{ccccc}m_e m_d m_c m_b m_a\cr\epsilon_e-\epsilon_a v_{de}v_{ce}0 0\cr0 \epsilon_d-\epsilon_a 0 0 v_{ad}\cr0 0 \epsilon_c-\epsilon_a v_{bc}v_{ac}\cr0 0 0 \epsilon_c-\epsilon_a v_{ab}\end{array}\right| \, .
\frac{1}{m}+ \frac{4}{n}= \frac{1}{3}( m + n ) ( \frac{1}{m}+ \frac{4}{n}) = \frac{1}{3}[ 5 + \frac{n}{m}+ \frac{4 n}{n}] \geq \frac{1}{3}( 5 + 2 \sqrt{\frac{n}{m}\frac{4 m}{n}})
\left\langle T_{\ell}^{i j}( p ) T_{\ell^{\prime}}^{k m}( - p ) \right\rangle = -{\frac{3}{4 \kappa_{1 0}^{2}{\alpha^{\prime}}^{3}}}F_{\ell}^{i j ; k m}( p ) \delta_{\ell \ell^{\prime}},
O_i = \{f \in B_i \mid \exists j \in [i+1,t] : f \notin U(\sigma_{j+1}) \wedge \forall \ell \in [i+1,j]: f \ne w_{\ell}\}N_i = \{f \in B_i \mid \forall j \in [i+1, t] : f \in U(\sigma_{j+1}) \wedge \forall \ell \in [i+1,t]: f \ne w_{\ell}\}B_i^* = B_i \setminus \{O_i \cup N_i \}\enspace .
\begin{array}{l}\displaystyle \# \{\sum_{\substack{j\in J\setminus J_{k}}}\Phi(B_{t_{j}},t_{j})(B_{t_{j+1}}-B_{t_{j}}) | \forall J_{k}=\{i_1,\cdots,i_k\}\subset J, 1\leq k\leq K(n)\}=C_{n}^{K(n)}(2^{K(n)}-1).\end{array}
K_{\lambda,1}(z,w) =\sum_{n=0}^{\infty}\frac{(n+\lambda+2)(n+\lambda+1)}{\lambda+1}\phi_{n}(z)\overline{\phi_{n}(w)}, K_{\lambda,2}(z,w) =\frac{1}{2\lambda+2}\sum_{n=0}^{\infty}\frac{\Gamma(n+\lambda+4)}{\Gamma(n+\lambda+1)}\phi_{n}(z)\overline{\phi_{n}(w)}.
I(\omega) \leq \beta(\omega) \left( C^{2}(1 + | r_{\xi}(\omega) |)^{-2r}+ C^{2}(1 + | r_{\xi^{\prime}}(\omega) |)^{-2r}\right) & = C^{2}\beta(\omega) \left( (1 + | r_{\xi}(\omega) |)^{-2r}+ (1 + | r_{\xi^{\prime}}(\omega) |)^{-2r}\right).
L(f, 3) = \frac{4\pi^{3}}{3}\int_{0}^{\infty}b^{2}(e^{-2\pi u}) c(e^{-6\pi u}) u^{2}du &= \frac{4\pi^{3}}{3\sqrt{3}}\int_{0}^{\infty}b (e^{-2\pi u}) c(e^{-6\pi u}) \cdot c(e^{- \frac{2\pi}{3u}}) u du.
\frac{\psi_{x,y-1,z}(a,b)}{\psi_{x,y-1,z-1}(a,b)}\frac{\psi_{x,y,z-1}(a,b)}{\psi_{x,y,z}(a,b)}+\frac{\psi_{x+1,y-1,z-1}(a,b)}{\psi_{x,y-1,z-1}(a,b)}\frac{\psi_{x-1,y,z}(a,b)}{\psi_{x,y,z}(a,b)}=1.
M_N=Nm_0\,\frac{\sqrt{2}}{3}\left(\,\sqrt{1+\sqrt{1-\left(\frac{g^2N}{4\pi m_0^2\alpha_0}\right)^2}}+\,\frac{4\pi m_0^2\alpha_0}{g^2N}\,\sqrt{1-\sqrt{1-\left(\frac{g^2N}{4\pi m_0^2\alpha_0}\right)^2}}\;\right)
\times\Bigg[ \int_{\Lambda}\prod_{c=1}^N \left[ \prod_{j=1}^{n_c}d^2x_{j}^{(c)}d^2y_{j}^{(c)}\right] \prod_{d<e}^N\Big| B(d,e) \Big|^{\frac{g}{\pi+gN}q}\Bigg]^{\frac{1}{q}}
[\Delta_\perp]^{-1}(x_\perp) \rightarrow[\Delta_\perp - m^2]^{-1}(x_\perp) = - \int \frac{d^2k_\perp}{(2\pi)^2}\frac{e^{i k_\perp \cdot x_\perp}}{k_\perp^2+m^2}= - \frac{1}{2\pi}K_{0}(m\sqrt{x_\perp^2}),
A_{\mu}^{( N )}d x^{\mu}= \frac{1}{2 e}\left( \begin{array}{c c c c c}{{( N - 1 ) \cos \theta d \phi}}&{{\omega_{1}\Theta}}&{{0}}&{{\ldots}}&{{0}}\\{{\omega_{1}\bar{\Theta}}}&{{( N - 3 ) \cos \theta d \phi}}&{{\omega_{2}\Theta}}&{{\ldots}}&{{0}}\\{{\vdots}}&{{}}&{{\ddots}}&{{}}&{{\vdots}}\\{{0}}&{{\ldots}}&{{0}}&{{\omega_{N - 1}\bar{\Theta}}}&{{( 1 - N ) \cos \theta d \phi}}\\ \end{array}\right)
\partial^\alpha (f \circ g)(x) = \sum_{n=1}^{|\alpha|}\left[ f^{(n)}(g(x)) \cdot \sum_{\gamma \in \Gamma_{\alpha,n}}\bigg( C_{\gamma}\cdot \prod_{j=1}^n (\partial^{\gamma_j}g) (x) \bigg) \right] \forall \, x \in U \, ,
\hat{h}{\blacktriangleright}\big(\hat{f}\hat{g}\big)=m\big({\bigtriangleup}\big(\hat{h}\big){\blacktriangleright}\big(\hat{f}\otimes\hat{g}\big)\big)=\big(\hat{h}_{(1)}{\blacktriangleright}\hat{f}\big)\big(\hat{h}_{(2)}{\blacktriangleright}\hat{g}\big)
\varepsilon_{j}( \theta ) =\delta_{j,0}\, mR \, \cosh \theta + \sum_{k=0}^{p-2}\int_{-\infty}^\infty \Phi_{j,k}(\theta-\theta') \log \big( 1+ e^{-\varepsilon_k( \theta')}\, \big)\,\, d \theta', \qquad j=0,1,\ldots,p-2.
\delta \Gamma^{\sigma}= \frac{i 3 6 g^{2}}{M^{2}}\int \frac{d^{4}k}{( 2 \pi )^{4}}\int_{0}^{1}d x \; d y \; \delta ( x + y - 1 ) \Biggl \{\frac{N_{1}^{\sigma}+ \epsilon N_{2}^{\sigma}}{\left[ k^{2}- \Delta_{1}\right]^{2}}- \frac{N_{3}^{\sigma}}{\left[ k^{2}- \Delta_{2}\right]^{2}}\Biggr \},
f(\mu ,s,n) = \sum_{\nu = 1}^\infty{\frac{{H_{\nu ,n}}}{{\nu^s}}}- \sum_{\nu = 1}^\infty{\frac{{H_{\nu ,n}}}{{(\nu + \mu )^s}}}- \sum_{\nu = 1}^\mu{\frac{{H_{\nu ,n}}}{{\nu^s}}}- \zeta (n + s)+H_{\mu,n+s}\,.
T_1 := \sum_{j> j_0}\sum_{m\ge j}\overline\kappa_{m,j}\overline \nu_{m,j}, T_2 := \sum_{j\le j_0}\sum_{m > j_0}\overline \kappa_{m,j}\overline \nu_{m,j}, T_3 := \sum_{j\le j_0}\sum_{m=j}^{j_0}\overline \kappa_{m,j}\overline \nu_{m,j}.
\gamma_n(\alpha) =\frac i{2\pi}d\bar\alpha_n\wedge \bigwedge_{k<n}\frac i{2\pi}d \alpha_k\wedge d\bar \alpha_k= \sqrt{1-|w|^2}e^{-i\psi}\; \frac{d\psi}{2\pi}\wedge d\beta_{n-1}(w)
\left( \left( \mathbf{I}+\bar{K}(\nu,0)\right) r_{0},r_{0}\right) =\left( \left( \mathbf{I}+\bar{K}(\nu,0)\right) r_{0},r_{0}\right) & =1-F\left( i\lambda\right) \backsim a_{0}\lambda^{2}=a_{0}\nu\text{,}
\sum_{i = 1}^{N}\delta \alpha_{i}\, \lambda_{i}= \sum_{i = 1}^{N}( \alpha_{i}- 1 ) \delta \left[ \sum_{j \neq i}2 \pi ( \alpha_{j}- 1 ) G ( \omega_{i}- \omega_{j}| \tau ) \right] + \frac{1}{2}\delta \left[ \sum_{i = 1}^{N}( \alpha_{i}- 1 )^{2}\log | 2 \pi \eta^{2}| \right]
\xi^{(n)}_1= \sum_{\nu\in\{0,1\}^n}\tilde\xi_{\nu}, &\tilde\xi_{\nu_n,\ldots,\nu_1}=\prod_{i=2}^n Y_{i,q^{2\sum_{j=i+1}^n\nu_j+\nu_i-\nu_{i-1}+i}s_1}^{-\nu_i+\nu_{i-1}}\cdot Y_{1,q^{2\sum_{j=1}^n\nu_j}s_1^{2\nu_1}}Y_{\bar 1,q^{2\sum_{j=1}^n\nu_j}s_1}^{-1}\,.
\overline{{{( \Delta N )^{2}}}}= \frac{\partial^{2}\, \psi_{d}\, ( \beta , \gamma )}{\partial \, \beta^{2}}\ , \ \overline{{{( \Delta{\cal R})^{2}}}}= \frac{\partial^{2}\, \psi_{d}\, ( \beta , \gamma )}{\partial \, \gamma^{2}}\, .
\Phi_{1}^{m}\equiv \frac{\sigma_{1}+ i \sigma_{2}}{2}\otimes \psi_{- 1 / 2}^{m}| 0 , k \rangle ~ ~ , ~ ~ \Phi_{2}^{m}\equiv \frac{\sigma_{1}- i \sigma_{2}}{2}\otimes \psi_{- 1 / 2}^{m}| 0 , k \rangle
\begin{cases}u_t + uu_{x}= -\partial_xp*(\frac{\alpha^2}{2}u_x^2+u^2 + c_0u+\frac{1}{2}\tilde{\rho}^2+\tilde{\rho}) , t > 0, x\in \mathbb{R}\tilde{\rho_t}+ u\tilde{\rho}_x = -u_x\tilde{\rho}- u_x , t > 0, x\in \mathbb{R}u(x,0) = u_0(x), x\in \mathbb{R}\tilde{\rho}(0,x) = \tilde{\rho_0}(x) x\in \mathbb{R}\end{cases}
f'(X^*;D) =(\mathcal{A}^*(\mathcal{A}(X^*)-b))\bullet D &\le\|\mathcal{A}^*(\mathcal{A}(X^*)-b)\|_2 &\le\|\mathcal{A}^*\|_{2,2}\|\mathcal{A}(X^*)-b\|_2 &\le\|\mathcal{A}^*\|_{2,2}\|b\|_2,
g (v^{\Omega_{\tau,\sigma,J,x}})_\tau = (v^{\Omega_{\tau,\sigma,J,x}}\omega_\tau^{\Omega_{\tau,\sigma,J,x}}(g))_\tau = \omega_{\sigma,J,x}(g) (v^{\Omega_{\tau,\sigma,J,x}})_\tau,
T_2T_1(z_1^{m_1}z_2^{m_2}\eta)=M_{z_2}^{\alpha_2}D[U](\alpha_1 z_1^{m_1+1}z_2^{m_2}\eta)=\alpha_1 \alpha_2 z_1^{m_1+1}z_2^{m_2+1}U^{m_1+1}\eta,
[\langle\Phi^2(t)\rangle-\langle\Phi^2(0)\rangle] =\int \frac{d^3k}{(2\pi)^3}\frac{1}{2\omega_{<}(k)}[{\cal{U}}^+_k(t){\cal{U}}^-_k(t)-1] \coth[\beta_i\omega_{<}(k)/2]
Z_{Q}(\lambda )=\sum_{a=0}^{2(\mathsf{N}+1)}z_{a}\cosh^{a}2\lambda ,\text{with}z_{2(\mathsf{N}+1)}=\frac{2\kappa_{+}\kappa_{-}\cosh (\alpha_{+}+\alpha_{-}-\beta_{+}+\beta_{-}-(\mathsf{N}+1)\eta )}{\sinh \zeta_{+}\sinh \zeta_{-}}.
\int_{H}\left((u')^2 - (\tilde{w}')^2 \right) = \int_{H}\big((u')^2 - (\tilde{l}')^2 \big)+ \int_{H}\big((\tilde{l}')^2 - (\tilde{w}')^2 \big) & \geq \int_{H}\big( (u')^2 - (\tilde{l}')^2 \big) - \int_{H}|(\tilde{l}')^2 - (\tilde{w}')^2|,
\left\langle T_{ii}(x)\right\rangle =\frac{1}{4}\sum_{n}\omega_{n}\left|\phi_{n}\right|^{2}-\frac{1}{4}\sum_{n}\frac{1}{\omega_{n}}\left| \vec{\nabla}\phi_{n}\right|^{2}+\frac{1}{2}\sum_{n}\frac{1}{\omega_{n}}\left|\partial_{i}\phi_{n}\right|^{2}
\omega(z;q):=\sum_{n=0}^{\infty}\frac{z^n q^{2n^2+2n}}{(q;q^2)_{n+1}(zq;q^2)_{n+1}}, \nu(z; q):=\sum_{n=0}^{\infty}\frac{q^{n^2+n}}{(-zq;q^2)_{n+1}}.
\int_{\mathbb{R}^n}f(y)\,dy =\sum^{\infty}_{j=1}M_j \int_{\mathbb{R}^n}\psi_j (y)\,dy=I\sum^{\infty}_{j=1}M_j r^{n}_{j}=I\sum^{\infty}_{j=1}\varepsilon_j <\infty.
\begin{array}{c}{( q^{-{\frac{\sigma^{3}}{2}}}\otimes \sigma^{+}+ \sigma^{+}\otimes q^{\frac{\sigma^{3}}{2}}) \downarrow \otimes \downarrow =}\\{= q^{1 / 2}\downarrow \otimes \uparrow + q^{- 1 / 2}\uparrow \otimes \downarrow}\\ \end{array}
|\nabla^l_g u|_g := \begin{cases}|\nabla^l_g u|, \ \ \ \ \ \ \ \ \ \ \ \mbox{if $l$ is even}, \langle \nabla^l_g u, \nabla^l_g u \rangle^{\frac{1}{2}}_g ,\ \ \mbox{if $l$ is odd}. \end{cases}
\sum_{n=1}^{\infty}\frac{(-1)^{n}}{(n+2)^{a}}\sum_{n=1}^{\infty}\frac{\overline{H}_{n}^{(c)}(-1)^{(n+1)}}{(n+1)^{b}}=\zeta(\overline{a},\overline{b},\overline{c})
\begin{cases}\frac{1}{K}\nabla_{A, s}\psi + D^{g_3}_{A(s)}\psi =0 \frac{1}{K}E_A + *_3 F_{A(s)}+ \frac{1}{2}*_3 F_{A_{K^{-1}}}-r(q_3(\psi) -iKdt) + \frac{i}{2}*_3\wp_3=0. \end{cases}
\mu \left( \left( \bigcap_{\gamma \in F_p}\bigsqcup_{\substack{\sigma \in k^{F_p}\sigma(e) = \jmath(\gamma)}}D_{\gamma \cdot \sigma}\right) \triangle \left( \bigcap_{\gamma \in F_p}\bigsqcup_{\substack{\sigma \in k^{F_p}\sigma(e) = \jmath_p(\gamma)}}\gamma^b D_\sigma \right) \right) < (|F_p| k^{|F_p|}) ( 4 \delta k^{2 |F|}).
a_{V,W,U}\circ \gamma_{U,V\otimes W}\circ a_{U,V,W}=(V\otimes \gamma_{U,W})\circ a_{V,U,W}\circ (\gamma_{U,V}\otimes W), a_{W,U,V}^{-1}\circ \gamma_{U\otimes V,W}\circ a_{U,V,W}^{-1}=(\gamma_{U,W}\otimes V)\circ a_{U,W,V}^{-1}\circ (U\otimes \gamma_{V,W}).
H_{2,1}(x_{2,0}^{-})^3x_{1,0}^{-}v^{+}&=[H_{2,1},(x_{2,0}^{-})^3]x_{1,0}^{-}v^{+}+(x_{2,0}^{-})^3 H_{2,1}x_{1,0}^{-}v^{+}&=-2\big(x_{2,1}^{-}(x_{2,0}^{-})^2+x_{2,0}^{-}x_{2,1}^{-}x_{2,0}^{-}+(x_{2,0}^{-})^2 x_{2,1}^{-}\big)x_{1,0}^{-}v^{+}&+\frac{1}{2}\big((a+\frac{3}{2})^2-(a-\frac{3}{2})^2\big)(x_{2,0}^{-})^3 x_{1,0}^{-}v^{+}.
(0,1,2,0,1,2,5,{\bf 2},{3},{\bf 2},{3},8,8,{\bf 4}) \rightarrow (0,1,2,0,1,2,5,{2},{3},{3},{\bf 4},8,8,{\bf 4}) &\rightarrow (0,1,2,0,1,2,5,{2},{3},{3},{\bf 4},4,{\bf 4}) &\rightarrow (0,1,2,0,1,2,5,{2},{5},{3},{\bf 5},5,{\bf 4})=f_{5,4}(s)=f_5^*(s).
Z ( \eta ) \approx \int{\cal D}a{\cal D}A_{\alpha}^{a}\ \exp \left[ i \int{\frac{1}{2}}( \partial_{\mu}a - i \eta_{\mu})^{2}+ \Lambda_{Q C D}^{4}\cos{\frac{a}{f_{0}}}+{\cal L}_{Y M}\right] \ .
- \omega \mathrm{~}D_{{\bf{k}}}^{*}( -{\bf{q}}) = \omega_{{\bf{k}}}( -{\bf{q}}) D_{{\bf{k}}}^{*}( -{\bf{q}}) + ( \frac{v_{{\bf{q}}}}{V}) \Lambda_{{\bf{k}}}({\bf{q}}) \sum_{{\bf{k}}^{'}}[ \Lambda_{{\bf{k}}^{'}}({\bf{q}}) D_{{\bf{k}}^{'}}^{*}( -{\bf{q}}) + \Lambda_{{\bf{k}}^{'}}( -{\bf{q}}) C_{{\bf{k}}^{'}}({\bf{q}}) ] + \Lambda_{{\bf{k}}}({\bf{q}})
\sum_{i=c}^{e}\tbinom{(p-w)-b}{2(p-w)-2b+i-r}\tbinom{a-(p-w)}{2a-2(p-w)-i-1}= \sum_{v=A+c}^{A+e}\tbinom{(p-w)-b}{v}\tbinom{a-(p-w)}{(A + B) - v}.
B_{\lambda}[w_1,w_2]=\int_{\mathbb{T}^d}[ m_{\lambda}D\psi_1\cdot D^2_{pp}H \cdot D\psi_2+f_1D_pH_{\lambda}D\psi_2-f_2D_pH_{\lambda}D\psi_1 &+g'(m_{\lambda})f_1f_2+D\psi_1Df_2-Df_1D\psi_2+h_1f_2 -h_2f_1].
\Big\Vert \mathcal{K}_{}\Big[\Phi^{\textbf{e}}(\xi,t;z,\tau)\Big]\Big\Vert_{\mathrm{H}^{1,\frac{1}{2}}\Big(\partial \Omega\times \mathbb{R}\Big)}^2 = \frac{1}{4}\Big\Vert\Phi^{\textbf{e}}(\xi,t;z,\tau)\Big\Vert_{\mathrm{H}^{1,\frac{1}{2}}\Big(\partial \Omega\times \mathbb{R}\Big)}^2.
D_2 = B(v_m) + \sum_{i=1}^n \Big[ \big( D(w_i) + A\big)(w_i) + \sum_{v_j \prec w_i}\sum_{p\in (w_i,v_j)^{\circ}}\,D(p) \,(p)\Big] &+ \sum_{i=1}^{m-1}\Big[D(v_i) (v_i) + \sum_{w_j \prec v_i}\sum_{p \in (v_i,w_j)^{\circ}}\,D(p) \,(p)\Big].
\big\langle Q^{m-n}v_0,\psi^{+}(w_1)\cdots \psi^{+}(w_m)\psi^{-}(y_1)\cdots \psi^{-}(y_n)v_0\big\rangle= \frac{\prod\limits_{1\le i<j\le m}(w_i-w_j) \prod\limits_{1\le i<j\le n}(y_i-y_j)}{\prod\limits_{i=1}^{m}\prod\limits_{j=1}^{n}(w_i-y_j)}.
t^{(i_r)}_r\otimes x^{(i_s)}- x^{(i_r)}\otimes t^{(i_s)}_s = \sum_{j = 2}^{n_r}\beta_j b_j^{(i_r)}\otimes t^{(i_s)}_s - \sum_{k = 2}^{n_s}\eta_k t^{(i_r)}_r\otimes c^{(i_s)}_k.
h\lambda\sum_{k=0}^{N-1}\|\nabla v^{k+1}\|_{L^2(\Omega)}^2 +\sup_{0\leq k\leq N-1}\|v^{k+1}\|_{L^2(\Omega)}^2 \leq 8\mathcal T\|g\|_{L^2 (Q_{\mathcal T})}^2 +4 \|v_0\|_{L^2(\Omega)}^2,
ds^2 = -(dT+Jd\phi)^2 + dR^2 + \alpha^2 R^2 d\phi^2 = -(dT+Jd\phi)^2 + \alpha^2 r_0^2 (\zeta^{\alpha-1})^2 (d\zeta^2 + \zeta^2 d\phi^2),
\begin{aligned}F(X(t)) = F(X(0)) + \int_0^t \langle F_x(X(s)), \psi(s)\rangle_H\mathrm{d}s + \int_0^t\langle F_x(X(s)), \gamma(s)\rangle_H\mathrm{d}E_s + \int_0^t\langle F_x(X(s)), \phi(s)\mathrm{d}W_{E_s}\rangle_H + \frac{1}{2}\int_0^t tr(F_{xx}(X(s))(\phi(s)Q^{1/2})(\phi(s)Q^{1/2})^*) \mathrm{d}E_s \end{aligned}
K_{\Gamma}( t; z_i,b) =\frac{t^{{z_i}/{b}}\exp\{-{t}/{b}\}}{b^{{z_i}/{b}+1}\Gamma({z_i}/{b}+1)}, K_{}( t; z_i,b) =\frac{1}{\sqrt{2\pi b t^3}}\exp\Big\{-\frac{1}{2b z_i}\Big(\frac{t}{z_i}+\frac{z_i}{t}-2\Big)\Big\},
I_{\varphi}-I_{\alpha}= I -4 \ln\ln{\Lambda_{0}\over\Lambda}+ 4\ln{\Lambda_{0}^{2}\over\Lambda^{2}}-4 \ln 2+{4\over N-1}\,\sum_{i=1}^{N-1}\ln{\Lambda^{2}\over v_{i}}\raise 2pt\hbox{,}
\det(\rho_{[l]}(\tilde{\Theta})) =c_l\cdot (t-\theta)^{s_l}c_l\in \bar{K}^\times, s_l\geq 1, \sigma\bigl(\rho_{[l]}(R)\bigr) = \rho_{[l]}(R)\cdot \rho_{[l]}(\tilde{\Theta}).
p=\frac{b^{\beta\cdot x}}{1+b^{\beta\cdot x}}=\frac{b^{\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}}}{1+b^{\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}}}=\frac{1}{1+b^{-(\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2})}}
\widehat{\mathfrak{\mathbb{D}}_{\mathbf{x}}f_{\mathbf{k}}}\left(\mathbf{s}\right)=\left(\sum_{i=1}^{n}s_{i}\right)\widehat{f}_{\mathbf{k}}\left(s_{1},\ldots s_{n}\right)-\underset{x_{1}\rightarrow0^{+}}{\lim}\hat{f_{\mathbf{k}}}\left(x_{1},s_{2},...,s_{n}\right).
S_{\mathrm{F B}}= y \sum_{x}( - 1 )^{\sum_{\nu}x_{\nu}}( \bar{\psi}_{L x}^{( 1 )}\phi_{x}\psi_{R x}^{( 1 )}+ \bar{\psi}_{R x}^{( 1 )}\phi_{x}^{*}\psi_{L x}^{( 1 )}) ,
M \equiv \int_{-\infty}^{+\infty}\mathrm{d}x\, \frac{\partial \mathcal{L}}{\partial \varphi_t}\stackrel{\;\eqref{eq:lagrangian}}{=}\frac{1}{2}\int_{-\infty}^{+\infty}\mathrm{d}x\, \varphi_x = \frac{1}{2}\int_{-\infty}^{+\infty}\mathrm{d}x\, u
= \iint F_{1, h}(x) (F_{1, h}\circ T^k) (x) F_{2, h}(y)(F_{2, h}\circ T^{2k}) (y) \, d\mu(x) d\mu(y) &= \iint f_1 \otimes f_2(x, y) \,{f_1 \otimes f_2}(U^h(x, y)) \,{f_1 \otimes f_2}(U^k(x, y)) \, f_1 \otimes f_2(U^{h+k}(x, y)) \, d\mu \otimes \mu(x, y),
S_{W Z W}= \mathrm{T r}\int d^{2}z \left[ ( g^{- 1}\partial g ) ( g^{- 1}\bar{\partial}g ) - \int_{0}^{1}d t ( \hat{g}^{- 1}\partial_{t}\hat{g}) [ ( \hat{g}^{- 1}\partial \hat{g}) , ( \hat{g}^{- 1}\bar{\partial}\hat{g}) ] \right] ,
\frac{1}{\sqrt{- g}}j^{\mu}= \exp ( - 2 \lambda \phi ) g^{\mu \nu}\partial_{\nu}a + \frac{q^{2}}{1 6 \pi^{2}f_{P Q}}\epsilon^{\mu \nu \rho \sigma}( A_{\nu}^{j}\partial_{\rho}A_{\sigma j}+ \frac{q}{3}f_{i j k}A_{\nu}^{i}A_{\rho}^{j}A_{\sigma}^{k})
\|a-a\varphi_n(1_{F_n})\| \leq 2\eta + \|\varphi_n(\psi_n(a))-\varphi_n(\psi_n(a))\varphi_n(1_{F_n})\| = 2\eta + \|\varphi_n(\psi_n(a)1_{F_n})-\varphi_n(\psi_n(a))\varphi_n(1_{F_n})\| <2\eta + \sqrt{3\eta}<\varepsilon.