harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{d f}{d \kappa}= - 2 \frac{M_{c}}{I_{c}}\left[ 1 + \frac{\delta M}{M c}- \frac{\delta I}{I_{c}}\right] = - \frac{2}{3}\; - \; \frac{1}{3}\varepsilon \ln ( \varepsilon ) \; - \; \varepsilon \; .
\frac{B}{A}= \frac{\Gamma ( 1 - | \alpha | ) \left( | \alpha | - \frac{m \bar{\lambda}_{\theta}}{2 \pi}\right)}{\Gamma ( 1 + | \alpha | ) \left( | \alpha | + \frac{m \bar{\lambda}_{\theta}}{2 \pi}\right)}\left( \frac{k r_{0}}{2}\right)^{2 | \alpha |}.
\div_*\bigl(U_0^\epsilon f_0\bigr) \,=\, U_0^\epsilon \cdot \nabla f_0 + \div_*(U_0^\epsilon) f_0 \,=\, (U_0^\epsilon - U_0^0) \cdot \nabla f_0 - \frac{\epsilon U_{0,r}^\epsilon}{1+\epsilon R}f_0\,,
\int_0^T \bar{H}( \eta^{\nu}(t)) \beta(t)dt = \int_0^T \left( \bar{H}( \eta^{\nu}(t)) - \bar{H}_{}( \eta^{\nu}(t)) \right) \beta(t)dt & + \int_0^T \bar{H}_{}( \eta^{\nu}(t)) \beta(t)dt .
L_{\Sigma +}= \epsilon^{\mu \nu \rho \sigma}\bigg [ \dots + \bigg ( C -{\frac{B^{2}}{4 A}}\bigg ) \Sigma_{\mu \nu}^{+ a b}\Sigma_{\rho \sigma a b}^{+}-{\frac{B E}{2 A}}G_{\mu \nu}^{+ a b}\Sigma_{\rho \sigma a b}^{+}+ \dots \bigg ] ,
\|y\xi\|^2 =\langle y \xi,y \xi\rangle =\langle y(\sum_{n}\alpha_n e_n+\phi),y (\sum_{k}\alpha_ke_k+\phi)\rangle =\langle y(\sum_{n}\alpha_n e_n),y (\sum_{k}\alpha_ke_k)\rangle =\sum_n \alpha_n^2 &\leq \sum_n \alpha_n^2+\langle \phi,\phi\rangle=\langle\sum_{n}\alpha_n e_n+\phi,\sum_{k}\alpha_k e_k+\phi\rangle=\|\xi\|^2.
a_{k^1,k^2}(t^1,t^2,\cdot)=\sum_{k^3 \in \mathcal{K}_3}(\mathcal{J}^{u,\aleph_{t^2_{|k^2|}},G}_{k^3}a_{k^1,k^2,k^3}(t^1,t^2,\cdot))\circ \mathcal{G}^{t^2_{|k^2|}}.
\widetilde{V}_{\rm SUSY}^{(0^{2})}\big(x;\omega ,N,- \Vert \psi_{0}\Vert^{2}\big) = V ( x;\omega ,N+1 ) +E_{1}( \omega ){}-2\Big[ \log \big({-}2\omega \psi_{1}(x;\omega,N) \Psi_{0}^{(0)}\big(x;\omega ,N,- \Vert \psi_{0}\Vert^{2}\big)-W(\psi_{0},\psi_{1}\,|\, x)\big) \Big]^{\prime \prime},
||-\frac{1}{\omega}\sum_{j=1}^{\infty}\int_{0}^{\frac{2}{\sqrt{\omega}\lambda(t)}}r^{2j}\omega^{j}\lambda(t)^{2j}\phi_{j}(r^{2}) F_{4}(t,r\lambda(t)) dr||_{L^{2}(\rho(\omega \lambda(t)^{2}) d\omega)}\leq \frac{C (\log(\log(t)))^{2}}{t^{2}\log^{b+1-2b\alpha}(t)}
\mathcal{R^+}= \frac{1}{2}\sum_{m \geq 1}\sum_{n \geq 1}\frac{A(n, m)}{(m^2 n)^{\frac{1}{2}}}g\Bigl( \frac{m^2 n}{N}\Bigr) \sum_{c > 0}\frac{S(n, p; c)}{c}H_{m, n}^+ \Bigl (\frac{4 \pi \sqrt{n p}}{c}\Bigr )
\left. \begin{array}{l l l}{a_{1}}{=}{( 0 . 7 7 , 1 . 9 5 ) , \quad a_{2}= ( 3 . 2 5 , 1 . 9 5 ) ;}{b_{1}}{=}{( 1 . 3 2 , 1 . 9 5 ) , \quad b_{2}= ( 2 . 7 0 , 1 . 9 5 ) .}\end{array}\right.
F(\bar{y},z) = \sum_{s}\oint\frac{\bar{du}}{2 \pi i}\frac{1}{(\bar{y}-\bar{u})^2}\oint \frac{dv}{2 \pi i}\frac{1}{(v - z)}\frac{\xi_s'(v)}{\xi_s(v)(1 - \bar{\xi}_s(u)\xi_s(v))}
\dot C_n = \dot M_n - \sum_{j=1}^{n-1}\binom{n-1}{j}\big(\dot M_j C_{n-j}+ M_j \dot C_{n-j}\big) &\,\,\, = \dot M_n - G(M_1) M_{n-1}- 2 \sigma (n-1) M_{n-2}- \sum_{j=1}^{n-1}\binom{n-1}{j}\big(\dot M_j - (n-j) M_j \big) C_{n-j}\,,
r_{i}\leq \frac{1}{x_{i}}\left( (d_{i}-t+1)d'_{t}+\sum_{h=1}^{t-1}d'_{h}\right)+ \frac{1}{x_{i}}\sum_{k=1}^{s-1}(x_{k}-1)d_{i}= \phi^{2}_{s,t}
\displaystyle \prod_{n = 1 \atop n \equiv \pm ( j + 1 ) ( \mathrm{m o d}5 )}^{\infty}( 1 - q^{n})^{- 1}= \displaystyle \sum_{r = 0}^{\infty}\frac{q^{r^{2}+ j r}}{( q )_{r}}, ~ j = 0 , 1 ,
\prod_{i=1}^n \left( uX_i + vX_i^{-1}+ w \right) \prod_{1 \leq p < q \leq n}\left( u E_{k_p}+ v E_{k_q}^{-1}+ w E_{k_p}E_{k_q}^{-1}\right) \tilde{s}_{(k_n,k_{n-1},\ldots,k_1)}(X_1,X_2,\ldots,X_n),
\beta_\varepsilon = \frac{(-1)^{i+1}k_d^{-1}}{\sqrt{1+\varepsilon^2|\bar{\tau}_\varepsilon^i|^2}}\left\{\varepsilon\,\overline{\partial^\circ g_i}+\varepsilon^2\bar{g}_i\bar{\tau}_\varepsilon^i\cdot\Bigl(\nabla\overline{V_\Gamma}+\overline{\nabla_\Gamma V_\Gamma}\Bigr)\right\}\quad\Gamma_\varepsilon^i(t)
\mathrm{d}\varepsilon_{y,z}= \frac{1}{128 \pi^4}J_{x,y}(\omega_1,\omega_2)^2 \, \mathrm{d}_{2\kappa}\times \mathrm{d}_{2\kappa}& + \frac{1}{64 \kappa^2 \pi^4}J_{x,y}(\omega_1,\omega_2)^2 \sum_{j=1}^{k-1}\mathrm{d}^{(j/2\kappa,(\kappa-j)/2\kappa)},
S = S_{0}- \frac{n + 1}{2 ( n - 1 )}\ln ( S_{0}) - \Gamma_{n}( \Xi ) + \frac{l^{2}\Xi^{2}[ ( n - 4 ) \Xi^{2}+ 2 ]}{1 6 [ ( n + 2 ) \Xi^{2}- n - 1 ]}\frac{q^{2}}{S_{0}^{2}}+ . . .
[\bar \epsilon Q \ , \bar Q \epsilon' ]^* = \bar \epsilon ( /{\hskip - 0.27 cm{\bf P}}+\Gamma^{11}Z){/{\hskip - 0.27 cm{\bf P}}\over 2{\bf P}^2}( /{\hskip - 0.27 cm{\bf P}}+\Gamma^{11}Z) \epsilon'=
\|\mu_\epsilon\|_{C^k_{\epsilon,\gamma+2,\delta}(\{r\leq r_\epsilon\})}=\epsilon^{\gamma+2}\|(p_\epsilon^{-1})^*\mu_\epsilon\|_{C^k_{\gamma+2}(\{R\leq \epsilon^{-1}r_\epsilon\})}&=\epsilon^{\gamma+4}\|\mu^-\|_{C^k_{\gamma+2}(\{R\leq \epsilon^{-1}r_\epsilon\})}&\leq Cr_\epsilon^{4+\gamma}
|\hat h (\hat y_n^*\hat x)| = |\hat h ((\hat y_n-\hat \psi)^*\hat x)|\leq \|(\hat y_n-\hat \psi)^*\hat x\|_1= \|\mathcal F((\iota\otimes (y_nh-\psi)^*)\Delta(x))\|_1 & \leq K\|(\iota\otimes (y_nh-\psi)^*)\Delta(x)\|_\infty \leq K\|y_nh-\psi\|_1\|x\|_\infty \to 0,
(\Omega \Omega^* )_N (\vec{x})\ =\e^{2e_o \sigma_N (\vec{x})}\ +\ \frac{2e^2_o}{N}e^{2e_o \sigma_N (\vec{x})}(X_i \sigma_N (\vec{x}))^2 \ +\o(e^2_o N^{-2})\ ,
A^{T}= \Gamma_{t}^{T}\cdots \Gamma_{1}^{T}= ( - \eta )^{t}{\cal C}\Gamma_{t}\cdots \Gamma_{1}= \eta^{t}{\cal C}A^{- 1}{\cal C}^{- 1}= ( - )^{t ( t + 1 ) / 2}\eta^{t}{\cal C}A{\cal C}^{- 1}\, .
\begin{aligned}\Theta^s_k(M)=\Big\{\theta=(\theta_{ij})\in\mathbb{R}^{n\times n}:\theta_{ii}=0,\, \theta_{ij}=\theta_{ji}=Q_{z(i)z(j)}, z\in[k]^{n}, & ~~~ Q = Q^T\in[-M,M]^{k\times k}\Big\}.\end{aligned}
\int_{T_n(s)}G_{s-t_n}(x-x_n) \prod_{j=1}^{n-1}G_{t_{j+1}-t_j}(x_{j+1}- x_j) d\pmb{t}= \int_{T_n(t)}G_{s-t_n}(x-x_n) \prod_{j=1}^{n-1}G_{t_{j+1}-t_j}(x_{j+1}- x_j) d\pmb{t}.
\delta^2 =\delta^2(\cos^2{\psi}+\sin^2{\psi}) &=\tfrac{1}{4}(1-(\gamma\beta)^2)^2+\gamma^2(\beta^2-\sigma^2) &=\tfrac{1}{4}(1-2(\gamma\beta)^2+(\gamma\beta)^4)+\gamma^2(\beta^2-\sigma^2) &=\tfrac{1}{4}(1-2\gamma\sigma+(\gamma\beta)^2)(1+2\gamma\sigma+(\gamma\beta)^2)
\frac{1}{m}\left( \sum_{i =1}^{k_m}\sum_{\stackrel{j=1}{j\neq i}}^{k_m}\frac{\lambda_i \lambda_j}{(\lambda_i-\lambda_j)^2}+ \sum_{i =1}^{k_m}(n-k_m)\frac{\lambda_i\sigma^2}{(\sigma^2-\lambda_i)^2}\right) \leq \frac{(\sqrt{\delta+1}-1)^2}{1+\varepsilon},
\int\limits_{-\infty}^\infty\bigl|\widehat u\!\left(\xi',\xi_n-i\eta\right)\bigr|^2\,\mathrm d\xi_n =\int\limits_{-\infty}^\infty\left|\frac{\widehat f\!\left(\xi',\xi_n-i\eta\right)}{q\!\left(\xi_n-i\eta\right)-\lambda}\right|^2\,\mathrm d\xi_n\ll\int\limits_{-\infty}^\infty\bigl|\widehat f\!\left(\xi',\xi_n\right)\bigr|^2\,\mathrm d\xi_n,
\partial_t\omega = \frac{\partial_x^2 h}{(1+h_x^2)^{3/2}}+ \Big( \partial_t \phi - \frac{\partial_x^2 h}{(1+h_x^2)^{3/2}}\Big) - T_{\partial_tB}P_{\geq 1}h - T_B\partial_tP_{\geq 1}h &= \frac{\partial_x^2 h}{(1+h_x^2)^{3/2}}+ I + II + III,
|\widetilde{G}_{\omega,[a,b],E,q}(k_{1},k_{2})| =\begin{cases}\frac{|P_{\omega,q,[a,k_{1}-1]}(E)P_{\omega,q,[k_{2}+1,b]}(E)|}{|P_{\omega,q,[a,b]}(E)|}, k_{1}\leq k_{2}\frac{|P_{\omega,q,[a,k_{2}-1]}(E)P_{\omega,q,[k_{1}+1,b]}(E)|}{|P_{\omega,q,[a,b]}(E)|}, k_{2}\leq k_{1}\end{cases}.
g(r)=r^{\mu_{1}}\ \ \mbox{with}\ \mu_{1}>0; & g(r)=\big(\ln(1+r)\big)^{\mu_{2}}\ \ \mbox{with}\ \mu_{2}>1; & g(r)=r^{\mu_{3}}\big(\ln(1+r)\big)^{\mu_{4}}\ \ \mbox{with}\ \mu_{3}>0,\, \mu_{4}\geq0; & g(r)=\ln(1+r)\Big(\ln\big(1+\ln(1+r)\big)\Big)^{\mu_{5}}\ \ \mbox{with}\ \mu_{5}>1.
0= \langle \tau_{H}, 2\alpha_{2}\rangle\{(2m_{1}+m_{2}+2m_{3})(1-(\cot \langle 2\alpha_{2}, H\rangle )^{2}) &+(n_{2}+2n_{3})(1-(\tan \langle 2\alpha_{2}, H\rangle )^{2})-2m_{1}\}(\alpha_{1}+2\alpha_{2}).
\langle \:{\hat{\phi}}_{\bar{t}}\prod_{i \in S}\phi_{i}\: \rangle_{g , \:{\Delta_{1}}}= \frac{1}{2}\: B_{\bar{t}}^{' \alpha \beta}\langle \: \phi_{\alpha}\phi_{\beta}\prod_{i \in S}\phi_{i}\: \rangle_{g - 1}\,
e^{-\int_{\xi_j^-}^{t \wedge{\bar \rho}_j}G_j(v) \: dv}X^{[\xi_j^-, \xi_j]}_j(t \wedge{\bar \rho}_j) = \int_{\xi_j^-}^{t \wedge \xi_j \wedge{\bar \rho}_j}\mu X_{j-1}(u) e^{-\int_{\xi_j^-}^u G_j(v) \: dv}\: du + Z_j^{[\xi_j^-, \xi_j]}(t \wedge{\bar \rho}_j),
\delta S = \frac{1}{4 \pi}\, \mathrm{I m}\, \, \left\{- \frac{i}{2 4}\int \! \! d^{1 2}z \D^{\alpha{\mathbf j}}L_{{\mathbf i}{\mathbf j}}D_{\alpha}^{\mathbf i}{\cal F}- 3 \int \! \! d^{8}z_{+}\sigma{\cal A}{\cal A}_{D}- \frac{i}{4 8}\int \! \! d^{1 2}z \ ( H -{\bar{H}}){\bar{\cal A}}{\cal A}_{D}\right\}.
\delta(t_{k+1}) \leq\delta(t_{0})+0.5\beta_{\max}\mathsf{T}_{\max}\sum^{k}_{l=0}L\beta^{l}_{\max}\|e(t_{0})\| &\leq\delta(t_{0})+0.5\|e(t_{0})\|\mathsf{T}_{\max}L\sum^{k}_{l=0}\beta^{l+1}_{\max}&\leq\delta(t_{0})+\frac{L\beta_{\max}\mathsf{T}_{\max}\|e(t_{0})\|}{2(1-\beta_{\max})}.
S_{S U S Y}\left( \left. \begin{array}{c c}{{a}}&{{d}}\\{{b}}&{{c}}\\ \end{array}\right| \theta_{1}- \theta_{2}\right) \times S_{S G}\left( \theta_{1}- \theta_{2}\right)_{\epsilon_{1}\epsilon_{2}}^{\epsilon_{1}^{\prime}\epsilon_{2}^{\prime}}
\partial_{x_j}( I_{h,T}\mathbf{u}(X) - \mathbf{u}(X) )=\sum_{i\in\mathcal{I}^{s'}}\partial_{x_j}\Phi_{i,T}(X)(\mathbf{E}_i^s(X) + \mathbf{F}_i^s(X)) + \sum_{i\in\mathcal{I}}\partial_{x_j}\Phi_{i,T}(X)\mathbf{R}_i^s(X), ~s = \pm,
\frac{1}{[(k+p)^{2}-m^{2}]}=\sum_{j=0}^{N}\frac{\left( -1\right)^{j}\left(p^{2}+2p\cdot k\right)^{j}}{\left( k^{2}-m^{2}\right)^{j+1}}+\frac{\left(-1\right)^{N+1}\left( p^{2}+2p\cdot k\right)^{N+1}}{\left(k^{2}-m^{2}\right)^{N+1}[\left( k+p\right)^{2}-m^{2}]}\cdot
M_n=n!B_n =\sum_{m=0}^n\frac{(-1)^{n-m}n!(n+1-m)}{m!}v^m\frac{d^m}{dv^m}=\sum_{l=0}^n\frac{(-1)^{l}n!(l+1)}{(n-l)!}v^{n-l}\frac{d^{n-l}}{dv^{n-l}}.
2\lambda \int_{\O}u\phi ~dx = \int_{u>t_p}(2\lambda u)\phi ~dx + 2\lambda \int_{u\leq t_p}u\phi ~dx \leq \int_{\Omega}u^p \phi ~dx + 2\lambda t_p \int_{\O}\phi ~dx = \lambda \int_{\O}u\phi ~dx + 2\lambda t_p,
R(\varPsi) = \sup_{\mu\in\mathcal{D}_{\rho}}\bigg\{\int_0^1 \varPsi(p)\;\mu(dp) - R^*(\mu)\bigg\}, \intertext{with}R^*(\mu) =\sup_{\varPsi\in \mathbb{Q}_{}}\bigg\{\int_0^1 \varPsi(p)\;\mu(dp) - R(\varPsi) \bigg\}.
\widetilde s-\widetilde J =(s+n\widehat\tau)-\left[J_\tau+\left(n\widehat\tau\wedge\frac{d}{p}\right)\right] =s-J_\tau-n+\left(n\widehat\tau-\frac{d}{p}\right)_+, \left(n\widehat\tau-\frac{d}{p}\right)_+ =\left(\left[n\tau-\frac{n}{p}+\frac{d}{p}\right]_+-\frac{d}{p}\right)_+ =n\left(\tau-\frac{1}{p}\right)_+,
N\left( bH \Big( |\tilde{A}|^2-\frac{1}{2}\tilde{H}^2 \Big) \right) \le bC(S)H\Big( |\tilde{A}|^2-\frac{1}{2}\tilde{H}^2 \Big) +bC(S)H^2\sqrt{\Big( |\tilde{A}|^2-\frac{1}{2}\tilde{H}^2 \Big)}\le bC(S,\Sigma_0)H^{3-\sigma}\le \frac{1}{4}H^3 +C(S,\Sigma_0,b).
D_{\alpha}^{\mathbf i}= \partial_{\alpha}^{\mathbf i}+ i{\bar{\theta}}^{\dot{\alpha}{\mathbf i}}\partial_{\alpha \dot{\alpha}}\ , \qquad \quad{\bar{D}}_{\dot{\alpha}{\mathbf i}}= -{\bar{\partial}}_{\dot{\alpha}{\mathbf i}}- i{\theta}_{\mathbf i}^{\alpha}\partial_{\alpha \dot{\alpha}}.
{\cal L}_{F F}= \bar{\Psi}[ - i \frac{2}{3}L_{\mu}( \partial_{\mu}+ i a_{\mu}+ i C_{\mu}) + M ] \Psi - \frac{i}{8 \pi ( \alpha - 1 )}\epsilon_{\mu \nu \lambda}a_{\mu}\partial_{\nu}a_{\lambda}\; ,
Z_{k}= i e^{i\pi \mbox{\boldmath$\hat{\scriptstyle\psi}\scriptstyle_{\perp}$}\cdot\mbox{\boldmath$\scriptstyle\tau$}/2}e^{i(2k+1)\pi\mbox{\boldmath$\hat{\scriptstyle\psi}$}\cdot\mbox{\boldmath$\scriptstyle\tau $}x^{3}/L}\quad\mbox{with}\quad\mbox{\boldmath$\hat{\psi}$}\cdot\mbox{\boldmath$\hat{\psi}_{\perp}$}=0
\sup_{z\in\mathbb{B}_n}\beta(\varphi_{kj}(z),\rho(z)) \leq \sup_{z\in \mathbb{B}_n}k_{V^{-1}\mathbb{B}_n}(V^{-1}\varphi_{kj}(z),V^{-1}\rho (z)) &= \sup_{z\in V^{-1}\mathbb{B}_n}k_{V^{-1}\mathbb{B}_n}(V^{-1}\varphi_{kj}V(z),V^{-1}\rho V(z)).
-2a_1 (\bar{\Lambda}_1^0)^2(D_\zeta\sigma_1)+ 2 (D_{\zeta}\sigma_1) \, \mathcal Poly_4(0,\zeta^0,\bar\Lambda_0) &= -a_1 (\bar \Lambda_1^0)^2 (D_\zeta\sigma_1) &=-a_1 (\bar \Lambda_1^0)^2 D^2_{\zeta\zeta}\sigma_1(0,\zeta^0)(\zeta-\zeta^0)-a_1 (\bar \Lambda_1^0)^2\bigl((D_\zeta\sigma_1) - D^2_{\zeta\zeta}\sigma_1(0,\zeta^0)(\zeta-\zeta^0)\bigr) .
\rho(\tilde{\gamma}) = (\rho_0\circ\eta^{-1}\circ\eta'\circ\eta'^{-1})(\tilde{\gamma}) &= (\rho_0\circ\eta^{-1}\circ\eta')(x,\gamma) &= \rho_0(\varphi(\gamma)x,\gamma) &= \rho_0(\varphi(\gamma),1)\rho_0(x,\gamma).
m\ge{[\frac{2\pi}{100\sqrt{G}}\frac{C^{2}}{3^{2/3}t_{*}}a^{3/2}M^{5/6}]}^{3/4}={(\frac{2\pi}{100\sqrt{G}})}^{3/4}\frac{C^{3/2}}{\sqrt{3}{t_{*}}^{3/4}}a^{9/8}M^{5/8}
\begin{aligned}T := \left\{\small{\begin{pmatrix}e^{i\theta_1}0 0 0 e^{i\theta_2}0 0 0 e^{i\theta_3}\end{pmatrix}}: \theta_1+\theta_2+\theta_3 = 0\right\}\simeq \ S\big(U(1)\times U(1)\times U(1)\big) \simeq U(1)\times U(1) \ .\end{aligned}
u_{r}(r,\theta)=\frac{2}{3}(\frac{R^{3}}{r^{3}}-1)B_{1}P_{1}(cos\theta)+\sum_{n=2}^{\infty}(\frac{R^{n+2}}{r^{n+2}}-\frac{R^{n}}{r^{n}})B_{n}P_{n}(cos\theta),
\begin{array}{c}{u^{\alpha}= u_{0}^{\alpha}+ \xi^{\alpha}s - \frac 1 2 \left( \stackrel{s}{\Gamma}_{\beta \gamma}^{\alpha}\right)_{0}\xi^{\beta}\xi^{\gamma}s^{2}-}{\frac 1{3 !}\left( \Gamma^{\alpha}{}_{\beta \gamma \delta}\right)_{0}\xi^{\beta}\xi^{\gamma}\xi^{\delta}s^{3}- \cdots}\end{array}
\frac{e^{L \kappa_L}}{q\left( \kappa_L \right)}\leq\frac{1}{f\left( \cos{g_L\left( \kappa_L \right)}\right)}=\frac{1}{2 - \cos{g_L\left( \kappa_L \right)}- \sqrt{\left( 2 - \cos{g_L\left( \kappa_L \right)}\right)^2 - 1}}=2 - \cos{g_L\left( \kappa_L \right)}+ \sqrt{\left( 2 - \cos{g_L\left( \kappa_L \right)}\right)^2 - 1}.
\mu_{(\boldsymbol{x},\boldsymbol{\phi}(t,\boldsymbol{x}))}^{N,i}+\frac{1}{N-1}\underset{j\neq i}{\sum}h\Big(\mu_{(\boldsymbol{x},\boldsymbol{\phi}(t,\boldsymbol{x}))}^{N,j}\Big)=-R^{-1}B\Big[P_t\nu^{N,i}_{\boldsymbol{x}}+\frac{1}{N-1}\underset{j\neq i}{\sum}\phi^{N,j}(t,\boldsymbol{x})\Big].
\sum_{n=0}^{\infty}\textup{PD}_\textup{t}(48n+24)q^n \equiv \dfrac{f_2f_3f_4f_{12}}{f_1^3f_6}-2q\dfrac{f_6^2f_{12}^4}{f_3^8}&\equiv \dfrac{f_2f_4f_{12}}{f_6}\cdot\dfrac{f_3}{f_1^3}-2qf_{12}^3 &\equiv \dfrac{f_2f_4f_{12}}{f_6}\left(\dfrac{f_4^6f_6^3}{f_2^9f_{12}^2}+3q\dfrac{f_4^2f_6f_{12}^2}{f_2^7}\right)-2qf_{12}^3~(\textup{mod}~4),
S_r=\frac{1}{\det G\cdot W^r}\sum_{1\leq i_1<\ldots<i_r\leq n}^n\ddot{f}_{i_1}\ldots \ddot{f}_{i_r}\det\ [ e_1+f'_1\nabla F,\ldots,e_{i_1},\ldots, e_{i_r},\ldots, e_n+f'_n\nabla F].
\begin{bmatrix}\begin{array}{c|cc}0 0 1 \hline1 0 0 0 1 0\end{array}\end{bmatrix},\begin{bmatrix}\begin{array}{c|cc}0 1 0 \hline0 0 1 1 0 0\end{array}\end{bmatrix}:\begin{bmatrix}\mathbb C \mathbb C^2\end{bmatrix}\to \begin{bmatrix}\mathbb C \mathbb C^2\end{bmatrix}
\begin{array}{l}g_{\tau\tau}= \partial_{\tau}X^M \partial_{\tau}X^N G_{MN}= G_{00}=-e^{2V}g_{\sigma\sigma}= \partial_{\sigma}\rho \partial_{\sigma}\rho\,G_{\rho\rho}+ G_{\theta\theta}= (\rho^{\prime})^2 e^{-2V}\Delta^{-1}+\rho^2 \Delta\end{array}
R(\theta) = \frac{\sin \left( \frac{2 \pi}{5}- \frac{3 i \theta}{5}\right)\sin \left( \frac{\pi}{5}- \frac{3 i \theta}{5}\right)}{\sin \left( \frac{2 \pi}{5}+ \frac{3 i \theta}{5}\right) \sin \left( \frac{\pi}{5}+\frac{3 i \theta}{5}\right)}.
\begin{aligned}\sigma =(\log k_0)^{-N_2}, \alpha =1+(\log k_0)^{-2N_2}, \varepsilon_0 =(\log k_0)^{-N_3}; \eta_0 =\frac{2k_0^2\alpha}{\pi\varepsilon_0}, T_0 =\frac{2\eta_0}{2k_0+1}.\end{aligned}
\Delta \subset \bigcap_{i\in\mathcal{D}_1\cup\cdots\cup\mathrm{D}_{M+1}}\{x\in\mathbb{R}^n |\langle x,v_i\rangle\leq\lambda_i=t_{i}\}& \subset\bigcap_{i\in\mathcal{D}_1\cup\cdots\cup\mathrm{D}_{M+1}}\{x\in\mathbb{R}^n |\langle x,v_i\rangle\leq t_{M+1}\}=\widetilde{\Delta}^n_0.
\begin{array}{llr}\psi_k(z) + \frac{\rho}{2\pi i}\int_{L_k}\overline{n_k^2(t)}\;\overline{\psi_k(t)}E_1(t-z) \textrm{d}t = -\rho \sum_{m \neq k}\frac{1}{2\pi i}\int_{L_m}\overline{n_m^2(t)}\;\overline{\psi_m(t)}E_1(t-z) \textrm{d}t+1, \;z \in D_k\;(k=1,2,\ldots, n).\end{array}
\eta z dv-zd v =\eta z dv-z d (\eta v)+z d(\eta v)-zd v &=-\alpha z\nabla\xi\wedge(\eta v)+ zd(\eta v-v) &=\alpha z\Bigl(-\nabla\xi\wedge(\eta v)+zd({\rm I}\,+zD_{{}^\Vert})^{-1}\bigl(\nabla\xi\wedge (\eta v)-\nabla\xi\lrcorner\,(\eta v)\bigr)\Bigr).
\sum_{j=1}^r\frac{1-\alpha^j\cos(t j)}{j}\leq -\sum_{j=1}^r\frac{\alpha^j-1}{j}+\sum_{j=1}^r\frac{\alpha^j\left(1-\cos(j/r)\right)}{j}&\leq -\sum_{j=1}^r\frac{\alpha^j-1}{j}+\frac{1}{2r^2}\sum_{j=1}^rj\alpha^j &\leq -\frac{1}{2r}\sum_{j=1}^r \alpha^j+1.
\mathbb{E}\,\left[\mathcal{P}_n\,|\,\mathfrak{V}_n\wedge\mathfrak{O}_{0,n-1}\right] = \mathbf{0}= \mathbb{E}\,\left[\mathcal{P}_n\,|\,\neg\,\mathfrak{V}_n\wedge\mathfrak{O}_{0,n-1}\right]\,.
\begin{aligned}\alpha= \,\,(e^{-(q_1-q_2)}-e^{-(q_1+q_2-2q_3)})dq_1 &-(1-e^{-2(q_1-q_3)})dq_2 &+(e^{-(q_2-q_3)}-e^{-(2q_1-q_2-q_3)})dq_3,\end{aligned}
\left|1-\frac{\lambda_{s-\ell}}{\gamma_1}c_k \right|^2=1-2\frac{c_k}{\gamma_1}\cos\left(\frac{2 \left( s- \ell \right) \pi}{s}\right)+\left(\frac{c_k}{\gamma_1}\right)^2=1-2\frac{c_k}{\gamma_1}\cos\left(\frac{2 \ell \pi}{s}\right)+\left(\frac{c_k}{\gamma_1}\right)^2,
\frac{1}{\sqrt{\det \left( \hat{{\cal G}}_{\parallel}^{( 0 )}\right)}}= \frac{1}{\sqrt{\det \left( \hat{{\cal G}}_{\perp}^{( 0 )}\right)}}= \exp \left( - \beta_{T}\int_{- \infty}^{\infty}d k \rho_{0}( k ) \Phi ( k^{2}; \beta_{T}, \mu^{2}) \right)
\frac{1}{N^{h-m}}\sum\limits_{\mathbf{n}\in \mathbb{Z}^h}\Big(\prod\limits_{j=1}^d f_j(\xi_j(\mathbf{n}) + \widetilde{\mathbf{r}}_j)\Big) \frac{1}{C_{\Xi,\chi}\eta^h}\int\limits_{\mathbf{y}\in \mathbb{R}^h}F(\mathbf{y}) G(L\mathbf{y}) \boldsymbol{\chi}(\Xi(\mathbf{y}-\mathbf{n}))\, d\mathbf{y},
0 =(L_1u^f,v^g)-(u^f,L_1v^g) &=\int\limits_{Y_{3T_1}}\Big(\frac{\partial u^f}{\partial s}\overline{v^g}-u^f\frac{\overline{\partial v^g}}{\partial s}\Big)dy'ds+\int\limits_{\Gamma_3\times[T_1,T_2]}(\Lambda_1f\overline g-f\overline{\Lambda_{1}g})dy'dy_0,
\sum_{j \geq 0}L_{j}^{(m+1)}(2jt+2(m+1)t) (e^{-t}y)^{j+m+1}= \frac{1}{(1-u)^2+2tu}\frac{(ye^{-t})^{m+1}}{(1-u)^{m}}e^{2(m+1)tu/(u-1)}.
e = \frac{i}{2}\left\{d \tau + \sum_{a = 1}^{N}\frac{S_{a}}{2 i}\left({\tt \frac{1}{z - z_{a}}}d z -{\tt \frac{1}{\bar{z}- \bar{z}_{a}}}d \bar{z}\right) \right\}\left( \begin{array}{r r}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) + \frac{i}{2}\left( \begin{array}{c c}{{0}}&{{d z}}\\{{d \bar{z}}}&{{0}}\\ \end{array}\right) \, ,
u(t) = \left(\begin{array}{c c c}1 a_{1, 2}(t) a_{1, 3}(t) 0 1 a_{2, 3}(t) 0 0 1 \end{array}\right) (\, a_{1, 2}(T), a_{1, 3}(T), a_{2, 3}(T) \in k[T] \,) .
|u+v\sqrt{D}|\leqslant |u-v\sqrt{D}|+2|v|\sqrt{D}=2|v|\sqrt{D}+O_{\varepsilon,\eta}\left(\frac{1}{B}\right)\leqslant 2\sqrt{\frac{A^\prime |m|D}{\eta^\prime(2\sqrt{D}-\frac{\varepsilon^\prime}{B^2})}}B+O_{\varepsilon,\eta}\left(\frac{1}{B}\right).
\left( g_{i_1}^{\alpha_1}\cdots g_{i_\ell}^{\alpha_\ell}\right)' = \left(g_{i_1}^{\alpha_1}\right)'\cdots g_{i_\ell}^{\alpha_\ell}+ \cdots + g_{i_1}^{\alpha_1}\cdots \left(g_{i_\ell}^{\alpha_\ell}\right)'.
\big\Vert\alpha(t)-\alpha(t+h)\big\Vert_{_1}=\big\Vert{F^{(N)}}^{-1}\bigl( H^{(N)}_{R_{-t}(X)}-H^{(N)}_{R_{-t-h}(X)}\bigr)\big\Vert_{_1}\Rightarrow \big\Vert\alpha(t)-\alpha(t+h)\big\Vert_{_1}\leq\big\Vert{F^{(N)}}^{-1}\big\Vert_{_1}\big\Vert H^{(N)}_{R_{-t}(X)}-H^{(N)}_{R_{-t-h}(X)}\big\Vert_{_1}.
\sigma(u) =\sigma((-1)^\delta u_1^x u_2^y)=(-1)^{\delta+x}u_1^{-x-y}u_2^x, \sigma^2(u) =\sigma((-1)^{\delta+x}u_1^{-x-y}u_2^x)=(-1)^{\delta-y}u_1^yu_2^{-x-y}.
\prod_{k = 0}^{d_j -2}q^{- (s_j +k) \dim V_j^I}\cdot \det( - \rho_j(\Phi)|V_j^I) & = \det( - \rho_j(\Phi) | V_j^I )^{d_j - 1}\cdot q^{-s_j (d_j-1)\dim V_j^I}\cdot q^{- (1 + 2 + \ldots + d_j-2) \dim V_j^I}& = \det( - \rho_j(\Phi) | V_j^I )^{d_j-1}\cdot q^{-s_j (d_j-1)\dim V_j^I}\cdot q^{- \frac{1}{2}(d_j-2)(d_j-1)\dim V_j^I}
\dot{p}_{\dot{x}}\left( \frac{{\small d\dot{x}(t)}}{{\small dt}}-\ddot{x}\right)dt+\left( \dot{p}-\frac{{\small \partial L}}{{\small \partial x}}\right) (dx- \dot{x}\,dt)-\left\langle d\dot{x}-\ddot{x}\,dt,T\gamma (\partial_t)\right\rangle dp_{\dot{x}}=0.
\Gamma\lbrack A\rbrack ={\rm tr}\int_0^{\infty}{dT\over T}\,{\rm e}^{-m^2T}\int{\cal D}x\,{\cal P}{\rm exp}\left[ - \int_0^T \!\!\! d\tau \left({1\over 4}{\dot x}^2 + igA_{\mu}\dot x^{\mu}\right) \right]
(N_1 \times N_2)\times_{X_{01}\times X_{12}}(X_{0}\times_{S}X_{1}\times_{S}X_{2})= & = (N_1 \times N_2)\times_{(X_0 \times_{S}X_1) \times(X_1 \times_{S}X_2)}(X_{0}\times_{S}X_{1}\times_{S}X_{2}) \cong (N_1 \times N_2) \times_{X_1 \times X_1}X_{1}\cong N_1\times_{X_1}N_2,
{\tilde{\langle T_{\mu}^{\mu}\rangle}}= \frac{\hbar H^{3}}{4 \pi c^{2}}( \bar{m}^{2}\gamma ) \left\lbrack ( 1 - 6 \zeta ) - \bar{m}^{2}\gamma \right\rbrack^{\frac{1}{2}}\cdot \mathrm{c t g}\: \pi \left\lbrack ( 1 - 6 \zeta ) - \bar{m}^{2}\gamma \right\rbrack^{\frac{1}{2}}
S = \int d^{3}x \left[ \frac 1 2 ( \nabla \vec{\chi})^{2}+ \frac 1 2 ( \nabla \psi )^{2}+ \frac{m_{H}^{2}}{2}\psi^{2}- 2 \zeta \mathrm{e}^{g_{m}\psi}\sum_{i = 1}^{N ( N - 1 ) / 2}\cos \left( g_{m}\vec{q}_{i}\vec{\chi}\right) \right] .
\log_2\left(\frac{|\pi_{\theta}(D_{a_1\cdots a_{n+1}})|}{2}\right)=\log_2\left(\frac{|\pi_{\theta}(D_{a_1\cdots a_n})|}{2}\right)+\log_2\left(\frac{|\pi_{\theta}(D_{a_1\cdots a_{n+1}})|}{|\pi_{\theta}(D_{a_1\cdots a_n})|}\right).
S = g(\chi^*, c,d_1) g(\chi^*, \tfrac{q_1c}{d_1},d_2) \cdots g(\chi^*, \tfrac{q_1\cdots q_{N-3}c}{d_1 \cdots d_{N-3}}, d_{N-2})g(\chi^*, \tfrac{q_1\cdots q_{N-2}c}{d_1\cdots d_{N-2}},n).
\begin{gathered}v_1 \wedge v_1 = 0, v_0 \wedge v_1 = - q^2 v_1 \wedge v_0, v_0 \wedge v_0 = - q^{-1}(q - q^{-1}) v_1 \wedge v_{-1}, v_{-1}\wedge v_1 = - v_1 \wedge v_{-1}, v_{-1}\wedge v_0 = -q^2 v_0 \wedge v_{-1}, v_{-1}\wedge v_{-1}= 0.\end{gathered}
s_{1}= \phi^{1}(\zeta) = \rho (\zeta), s_{2}= \phi^{2}(\zeta) = Im (\rho_{\zeta}\cdot (\zeta - \zeta^{\ast})), \left( t_{2k-1}, t_{2k}\right) = \left( \phi^{2k-1}(z, \zeta), \phi^{2k}(z, \zeta) \right) = \left( Re(\zeta_{k}-\zeta^{\ast}_{k}) , \, Im(\zeta_{k}- \zeta^{\ast}_{k}) \right), k = 2, \dots ,n.
(\det D^2\tilde u_1)^{1/n}L(u_2-\tilde u_1+ (x_n^2-A|x'|^3)) \geq o_{|x|}(1)+(c+O(|x'|))(\det D^1\tilde u_1)^{\frac{1}{n}}\sum_{i = 1}^nu^{ii}&\geq c+o_{|x|}(1)+O(|x'|) &\geq 0
\begin{aligned}(|A u|^p)^{1/p}_{Q_{\nu r, c \nu r}(z_0)}= (\nu r)^{-1}(|A \widetilde u|^p)^{1/p}_{Q_{1, c}}, \bigg(|A u - (A u)_{Q_r (z_0)}|^p\bigg)^{1/p}_{Q_r (z_0)}= (\nu r)^{-1}\bigg(|A \widetilde u - (A\widetilde u)_{Q_{1/\nu}}|^p\bigg)_{Q_{1/\nu}}^{1/p}. \end{aligned}
\{d\widehat{w}^+=0\}=\left\{(\vec{x}, \vec{u}) \in \mathbb{P}(V^{+})_{\widehat{U}_p}: \sum_{i=1}^a x_i w_{ij}^{(1)}(\vec{u})=0\mbox{for all}1\le j\le b\right\}, &\{d\widehat{w}^-=0\}=\left\{(\vec{y}, \vec{u}) \in \mathbb{P}(V^{-})_{\widehat{U}_p}: \sum_{j=1}^b y_j w_{ij}^{(1)}(\vec{u})=0 \mbox{for all}1\le i\le a\right\}.
M_{1}(\mathcal{C}(G)) =\dfrac{q(q+1)}{2}(q^{2}-3q+2)(q^{2}-3q+1)^{2}+ \dfrac{q(q-1)}{2}(q^{2}-q)(q^{2}-q-1)^{2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ (q+1)(q^{2}-2q+1)(q^{2}-2q)^{2}= q(q-1)(q^{6}-4q^{5}+4q^{4}+2q^{3}-4q^{2}+q-1)
|V(P^*_{1,{\rm dir}}) \cap A_{i,h}| = \begin{cases}1 \textrm{for $i \in \{j, \dots, j'\}\setminus \{i_1, i_2, i_3, i_4 \}$,}m_k \textrm{for $i = i_k$ and $k \le 4$,}0 \textrm{otherwise,}\end{cases}
W = \lambda_{U}U Q_{i}\phi_{2 i}+ \lambda_{U^{\prime}}U^{\prime}Q_{i}\phi_{1 i}+ \epsilon_{i j k}( \lambda_{D}D_{i}\bar{\phi}_{2 j}Q_{k}+ \lambda_{E}E_{i}\bar{\phi}_{2 j}L_{k}+ \lambda_{D^{\prime}}\bar{\phi}_{1 i}D_{j}D_{k}^{\prime}+ \lambda_{E^{\prime}}\bar{\phi}_{1 i}E_{j}E_{k}^{\prime})
O(\lambda (D + (n + 2d\lambda)^{2n})\lambda a \log p) = O(n^2 (n + 4nda(4d+4)^n)^{2n}(4d+4)^{2n}a^3 \log p) = O(n^{2n+2}(5da(4d+4)^n)^{2n}(4d+4)^{2n}a^3 \log p) = O(2^{4n^2+9n}n^{2n+2}(d+1)^{2n^2+4n}a^{2n+3}\log p).
\delta_{\tilde{\epsilon}}x_{i}= - i{\tilde{\epsilon}}_{i}f_{i j k}x_{k}\psi_{j}\ , \quad \delta_{\tilde{\epsilon}}\psi_{i}={\tilde{\epsilon}}{\dot{x}}_{j}f_{j i k}x_{k}- i{\tilde{\epsilon}}f_{j k i}\psi_{j}\psi_{k}
\phi ( z , \bar{z}) ={\mathrm e}^{- i \pi / 4}\sum_{k = 0}^{\infty}\sum_{l = 0}^{\infty}\left[ d_{k l}z^{1 + 4 k}( z \bar{z})^{l}+ e_{k l}{\bar{z}}^{3 + 4 k}( z \bar{z})^{l}\right] .
C^{\alpha \beta}( x , y ) = \left( \begin{array}{c c}{{\{\chi^{i j}( x ) , \chi^{m n}( y ) \}}}&{{\{\chi^{i j}( x ) , \phi^{m n}( y ) \}}}\\{{\{\phi^{i j}( x ) , \chi^{m n}( y ) \}}}&{{\{\phi^{i j}( x ) , \phi^{m n}( y ) \}}}\\ \end{array}\right) ,
\mathbb{E}[|f(x,\mathbf{h},\mathbf{z})|^2]=\mathbb{E}[\mathbf{w}^{H}\mathbf{r}\mathbf{r}^{H}\mathbf{w}]=\mathbf{w}^{H}\mathbf{R}_{\mathbf{r}\mathbf{r}}\mathbf{w},
d \left[ K^{\left( 2 q + 2 \right)}\left( S \right) - C^{\left( 2 q - 1 \right)}K^{\left( 3 \right)}\left( \Gamma_{1 1}\right) \right] = d K^{\left( 2 q + 2 \right)}\left( S \right) + K^{\left( 2 q \right)}\left( S \Gamma_{1 1}\right) K^{\left( 3 \right)}\left( \Gamma_{1 1}\right) \quad ,