harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\langle \bar{W}^{4}| W^{4}| W^{4}| \bar{W}^{4}\rangle = \frac{1}{x_{1 3}^{8}x_{4 2}^{8}}\left[ a_{2}+ \frac{b_{4}}{s}+ \frac{c_{1}}{s^{2}}+ \frac{b_{1}}{s^{3}}+ \frac{a_{1}}{s^{4}}\right] \; ,
\lim_{n\rightarrow \infty}E_\gamma(g_n-\phi) = \lim_{n\rightarrow \infty}\frac{1}{\lambda_n^2}E_\gamma(\lambda_n (g_n-\phi)) + \frac{2(\lambda_n^{p-1}-1)}{p+1}\|g_n-\phi\|^{p+1}_{L^{p+1}}\geq \frac{c-\|\phi\|^2_{L^2}}{c}I_\gamma(c).
\frac{1}{p+1}\bigg|\int\big(|R_1+\sigma R_2|^{p+1}-R_1^{p+1}-R_2^{p+1}-(p+1)\sigma R_1R_2^p-(p+1)\sigma R_2R_1^p\big)\bigg| &\lesssim \int R_1^2R_2^{p-1}+R_2^2R_1^{p-1}=O\bigg(\frac{1}{|x_1-x_2|^3}\bigg).
u_T(x) - \sigma_i(x_0) v_{iT}(x, a(x_0)) = \int_{0}^{\infty}\int_{\mathbb{R}}g_{x_0}^T(x_1- y, x_2-s) G(a(x_0), y, s) \, \textrm{d}y \, \textrm{d}s.
\P(C'_{k_1,j_1}\cap B') + \P( C'_{k_1,j_2}\cap B') = \P(C'_{k_1,j_1}) + \P( C'_{k_1,j_2})= \P(C_{k_1,j_1})-p + \P( C_{k_1,j_2})+p &= \P(C_{k_1,j_1}) + \P( C_{k_1,j_2})=\P(C_{k_1,j_1}\cap B) + \P( C_{k_1,j_2}\cap B),
A(k,\theta) := \sum_{j=0}^{\alpha-1}\sum_{m=0}^{2g}m^k \frac{\sin (2 \pi \theta(2m-j))}{2m-j}+\sum_{j=0}^{\alpha-1}\sum_{m=1}^{2g}m^k \frac{\sin (2 \pi \theta(2m+j))}{2m+j}=
\widehat{h}= \displaystyle \frac{\widehat{H}}{{\cal M}}= \displaystyle \frac{2 \pi}{l}\left( \widehat{L_{0}}+ \widehat{\overline{{L}}_{0}}- \displaystyle \frac{c}{1 2}\widehat{I}+ \kappa \displaystyle \frac{l^{2 - h}}{\left( 2 \pi \right)^{1 - h}}\widehat{B}\right)
\Delta_{n}(x)=\sqrt{n}\left| f_{n}(x)-{\bf E\,}f_{n}(x)\right| =\frac{1}{h_{n}\sqrt{n}}\left| \,\sum_{i=1}^{n}\left\{K\left( \frac{x-X_{i}}{h_{n}}\right) -{\bf E}\,K\left( \frac{x-X}{h_{n}}\right) \right\}\right|{\bf .}
\delta=\frac{P_{\rm max}-P_{\rm min}}{\kappa_{\rm t}-\kappa_{\rm b}}, \,\,{\rm and}\,\, P_{\rm bias}=\frac{P_{\rm min}\kappa_{\rm t}-P_{\rm max}\kappa_{\rm b}}{\kappa_{\rm t}-\kappa_{\rm b}}.
=\frac{L^{\prime}u}{2c^{2}}+\frac{L/2}{u}\sqrt{...}=\frac{Lu}{2c^{2}\sqrt{...}}+\frac{L\sqrt{...}}{2u}=\frac{Lu}{2c^{2}\sqrt{...}}\cdot\frac{u}{u}+\frac{L\sqrt{...}}{2u}\cdot\frac{c^{2}\sqrt{...}}{c^{2}\sqrt{...}}
\int_t^\infty e^{-b \tau + \eta |\tau - s |}d\tau = \begin{dcases}\frac{e^{-bt}}{b- \eta}e^{\eta(t-s)}, t \geq s \frac{e^{-bt}}{b+\eta}e^{\eta (s-t)}- \frac{e^{-bs}}{b+\eta}+ \frac{e^{-bs}}{b-\eta}, t \leq s. \end{dcases}
\begin{array}{l l l}{X_{3 7_{1}}= \gamma_{\omega_{2}}^{\frac{1}{2 n}},}&{X_{3 7_{2}}= \gamma_{\omega_{1}}^{- \frac{1}{2 n}}\gamma_{\omega_{2}}^{- \frac{1}{2 n}},}&{X_{3 7_{3}}= \gamma_{\omega_{1}}^{\frac{1}{2 n}}.}\\ \end{array}
V(0) ={1 \over{4\alpha}}[ n_1 n_1'{\rm Tr}(Y^2) \mu_0 + n_3 n_3'{\rm Tr}(\lambda_i \lambda_j) \mu_3 + n_8 n_8'{\rm Tr}(T_a T_b) \mu_8 ] \ < \ 0 \ .
(2h) = -\ \frac{i}{6(2\pi)^4}\tr \int d^dk\ \frac{\stackrel{\displaystyle (Sk)}{\scriptscriptstyle (-p)\phantom{+}}\stackrel{\displaystyle (S(k+q))}{\scriptscriptstyle (-q)\phantom{+qqqqqqq}}\stackrel{\displaystyle (S(k-p))}{\scriptscriptstyle (p+q)\phantom{+qqqqqqq}}}{k^{2l}(k-p)^{2l}(k+q)^{2l}}
I\left(U;Y\mid S=s\right)\geq I\left(U;Z\mid S=s\right) ~~ ;\forall s\in \mathcal{S}\mathrm{and}~\forall p\left(u,x,y,z\mid s\right)=p\left(u\mid s\right)p\left(x\mid u,s\right) p\left(y,z\mid x,s\right).
K_1 =\int_{\Gamma_\gamma}|F(z)|\left|\frac{dz}{z}\right|=2\int_0^\infty |te^{-i\gamma}e^{-t(\cos(\gamma)-i\sin(\gamma))}|\frac{dt}{t}&=2\int_0^\infty e^{-t\cos(\gamma)}dt\lesssim \frac{1}{\cos(\gamma)},
\Phi_{R}= \left( \begin{array}{c c c c c}{- m}&{0}&{0}&{0}&{0 \nonumber}\\{0}&{- m}&{0}&{0}&{0 \nonumber}\\{0}&{0}&{- m}&{0}&{0 \nonumber}\\{.}&{.}&{.}&{.}&{. \nonumber}\\{0}&{0}&{0}&{0}&{( k - 1 ) m}\\ \end{array}\right) ,
\frac{1}{3}-2cr^{\frac{1}{2}}\leq c\sum_{j=0}^{\infty}(2^{-j}r)^{1-\frac{n+\alpha}{p}}\log^{-\frac{\lambda}{p}}\left(e+(2^{-j}r)^{-\frac{1}{2}}\right)\left(\int_{B^n(x,2^{-j}r)^b}\Psi(\vert\nabla u\vert)\,d\mu_{\alpha}\right)^{\frac{1}{p}},
F^{(u,v)}_{p}K^{-l}:= \bigoplus_{0 \leq i_1 < \ldots < i_l \leq l}F^{(u,v)}_{p -\sum_{k=1}^l c_{i_k}}(D_{V}/ D_V J_A) e_{i_1\ldots i_{i_l}}\, ,
(T_4)_{ij}x^ix^j = -\frac{n-6}{12(n-1)}|W(p)|^2r^2+\frac{32}{9(n-4)(n-2)}\sum_{k,l,s}[(W_{ikls}(p)+W_{ilks}(p))x^i]^2 -\frac{4}{3(n-4)(n-2)(n-1)}|W(p)|^2r^2+\frac{16(7n-8)}{(n-4)(n-2)}\sigma_1(A)_{,ij}(p)x^ix^j+O(r^3),
\mathrm d\rho_{\infty}Y_+ v_{m,\ell}= v_{m+1,\ell}, \mathrm d\rho_{\infty}X_+ v_{m,\ell}= -\frac{1}{2\pi N}v_{m+2,\ell}, \mathrm d\rho_{\infty}Y_- v_{m,\ell}= -2\pi N m v_{m-1,\ell}, \mathrm d\rho_{\infty}X_- v_{m,\ell}= \pi N m(m-1) v_{m-2,\ell}- \frac{\ell}{4}(2k+\ell-1)v_{m,\ell-2}.
\begin{array}{c}(\tau_{zz}-\tau \frac{s}{2})Y + 2 \tau_z Y_z + \tau (Y_{zz}+\frac{s}{2}Y) = (e^{-2u}(\xi_2 - \xi_1))_z f_{\bar{z}}+ e^{-2u}(\xi_2 - \xi_1) (-e^{2u}f + e^{2u}h X)+ \sigma_z X + \sigma \{e^{-2u}(\xi_2 - \xi_1) f_{\bar{z}}+ \sigma X \}.\end{array}
Q_{0}= - \sum_{\kappa = - \infty}^{\infty}\vert \kappa \vert \left\{N_{\kappa}^{+}- N_{\kappa}^{--}\left[ \eta_{m \kappa}\left( 0 \right) - \eta_{- m \kappa}\left( 0 \right) \right] \frac{1}{\pi}+ \left[ \eta_{\infty \kappa}\left( \infty \right) - \eta_{- \infty \kappa}\left( \infty \right) \right] \frac{1}{\pi}\right\}\, .
\omega_S^{(ij)}= \omega_S, \omega_{S\cup\{i\}}^{(ij)}= \omega_{S\cup\{j\}}, \omega_{S\cup\{j\}}^{(ij)}= \omega_{S\cup\{i\}}, \omega_{S\cup\{i,j\}}^{(ij)}= \omega_{S\cup\{i,j\}}.
(g\circ f)^\alpha=\bigcup_{\nu\in N}(g\circ f)^{\alpha_\nu}\subset\bigcup_{\nu\in N}(g^{\alpha_\nu}\odot f^{\alpha_\nu})\subset\bigcup_{\nu\in N}(g^{\alpha}\odot f^{\alpha})=(g^{\alpha}\odot f^{\alpha}).
i{\cal D}^{- 1}( p^{2}) = Z_{\mu}^{( 0 )}\mu^{2}+ Z_{4}^{( 0 )}\frac{\beta^{4}\sigma^{2}}{2}+ N \beta^{2}\Sigma ( p^{2}) - Z_{\phi}^{( 0 )}p^{2}\, ,
E[|B^{n,4}_\tau|^2] = n^{2\alpha+ 1}\int_{[(\delta+\frac 1n)\wedge \tau,\tau]^2}(t-s_1)^{\alpha}(t-s_2)^{\alpha}E\left[ \sigma'(X_{\eta_n(s_1)-\delta}) M_{s_1}^{n,3}\sigma'(X_{\eta_n(s_2)-\delta}) M_{s_2}^{n,3}\right] ds_1ds_2,
\int_{P}\Phi^{q}\d\sigma = vol(P,d\sigma)\frac{d!q!}{(q+d)!}\sum_{k\in \mathbb{N}^{d+1}, |k|=q}\Phi(s_{1})^{k_{1}}\cdot\cdot\cdot \Phi(s_{d+1})^{k_{d+1}},
\begin{aligned}T^n((\eta, +1),(\eta + 2 n^{\gamma -1}, +1)) = p^n_+(\eta), T^n((\eta, -1), (\eta - 2 n^{\gamma -1},-1)) = p^n_-(\eta), T^n((\eta, +1),(\eta, -1)) = \max\left(0,p^n_-(\eta) - p^n_+(\eta) \right), T^n((\eta, -1),(\eta, +1)) = \max\left(0,p^n_+(\eta) - p^n_-(\eta) \right),\end{aligned}
\mathrm{Tr}(\Omega(c)|D|^{2-p}e^{-s^2D^2}) = -\mathrm{Tr}(a_0\left(\prod_{k=1}^p\partial(a_k)\right)|D|^{2-p}e^{-s^2D^2}\Gamma) = -\mathrm{Tr}(\Gamma a_0\left(\prod_{k=1}^p\partial(a_k)\right)|D|^{2-p}e^{-s^2D^2}).
\begin{array}{l}r=r_1^{e_1}\cdots r_h^{e_h}{r'_1}^{e_1'}\cdots{r'_{h'}}^{e'_{h'}}; h,h'\ge 0;~~\mbox{all $e_i,e_i'$ are positive;}begin{align*}6pt]n=r_1^{u_1}\cdots r_h^{u_h}{p_1}^{v_1}\cdots{p_{\ell}}^{v_{\ell}}, \ell\ge 1,~~\mbox{all $u_i,v_i$ are positive.}\end{array}
\mathcal{P}(\mathbf{u}) = \frac{1}{2}\int_{G}\sum_{|\beta| \leq k}c_{\beta}\Big(\big(\nabla^{\alpha}\mathbf{u}\big)_{|\alpha| \leq s}\Big)^{\beta}\mathrm{d}\mathbf{x}\mathcal{P}(\mathbf{u}) = \mathrm{TV}(\mathbf{u}),
\begin{array}{rcl}a^\pm(\zeta)+d^\pm(\zeta) = \rho(\zeta^{\pm2})^{\pm 1}\displaystyle\frac{\bar{\alpha}(\zeta^{-1})}{\bar{\alpha}(\zeta)}~, b^\pm(\zeta)+c^\pm(\zeta) = \rho(\zeta^{\pm 2})^{\pm 1}q^{\pm 1}\displaystyle\frac{1+(q^{-1}\zeta)^{\pm 1}}{1+(q\zeta)^{\pm 1}}\displaystyle\frac{\bar{\beta}(\zeta^{-1})}{\bar{\beta}(\zeta)}~.\end{array}
\bigg\|\dfrac{u^k_1-u^k_0}{t^k_1-t^k_0}\bigg\|\le\Tilde\mu+1\;\;\mbox{and}\;\;\sum^{k-2}_{j=0}\bigg\|\dfrac{u^k_{j+2}-2u^k_{j+1}+u^k_j}{h_k}\bigg\|\le\Tilde\mu+1,
{\cal D}_{\Gamma}( w ) = \sqrt{\mathrm{B e r}\; \Omega_{0}\vert_{\Gamma}}\equiv \sqrt{\mathrm{B e r}\frac{\partial_{r}z^{A}}{\partial w^{\mu}}\Omega_{( 0 ) A B}\frac{\partial_{l}z^{B}}{\partial w^{\nu}}},
\Pr_{\rho\sim P_\epsilon}[\rho\notin D_k(H\rho^\intercal)] \leq 2e^{-\frac{\sqrt{N}}{3\epsilon}}+\frac{N}{k^*}\mathop{\max}_{\substack{S\subseteq \{\epsilon N\pm N^{3/4}\}1\leq|S|\leq 2}}\Big\{\frac{1}{\binom{N}{S}}\sum_{j=0}^N \Pr_{v\sim\mu_{t}}\big[|v H|=j\big] K_S(j)^2 -1 \Big\}.
\mathbb{E}[M_{\tau}] =\sum_{k=1}^{n}\mathbb{P}[M_{\tau}\ge k]=\frac{n+1}{3}, \,\,~\mathbb{P}[M_{\tau}\ge k] =\frac{\binom{n-2}{k-1}}{\binom{n}{k-1}}=\frac{(n-k)(n-k+1)}{n(n-1)},
\mu^{2}\frac{\partial}{\partial \mu^{2}}\Gamma_{m}( 0 | \phi , \phi_{a}^{*}, \bar{\phi}, \eta ) = \bigr ( \beta_{g}g \frac{\partial}{\partial g}+ \gamma_{Q}\mathbf{\tilde{N}}_{Q}+ \gamma_{B}\mathbf{\tilde{N}}_{B}\bigr ) \Gamma_{m}( 0 | \phi , \phi_{a}^{*}, \bar{\phi}, \eta ) ,
s_{\theta}(x|t)=\frac{\det [(x_{j}|t)^{\theta_{i}+n-i}]_{1\leq i,j\leq n}}{\det [(x_{j}|t)^{n-i}]_{1\leq i,j\leq n}}~,(x_{j}|t)^{r}\overset{\beta =0}{=}\prod_{i=1}^{r}(x_{j}+t_{i})\;.
\langle \tilde{\Theta}_\pm \tilde{\Theta}_\pm D_a\varphi,\tilde{\Theta}_\pm B\varphi\rangle =\pm \langle D_a\varphi,\tilde{\Theta}_\pm B\varphi\rangle &=\mp \langle D_a\varphi,B\tilde{\Theta}_\pm \varphi\rangle &=\pm \langle B\varphi, D_a\tilde{\Theta}_\pm \varphi\rangle=\langle B\varphi, \tilde{\Theta}_\pm D_a\varphi\rangle
\begin{array}{lll}\Lambda^2_7 V_7^\ast = \{\alpha^2 + \ast(\alpha^2 \wedge \varphi) \mid \alpha^2 \in \Lambda^2 V_7^\ast\}, \mbox{and}begin{align*}2mm]\Lambda^2_{14}V_7^\ast = \{2 \alpha^2 - \ast(\alpha^2 \wedge \varphi) \mid \alpha^2 \in \Lambda^2 V_7^\ast\}.\end{array}
E\left(\frac{a}{s^r}\alpha_{ij}\right)E\left(s^dx\beta_{il}\right)E\left(\frac{a}{s^r}\alpha_{ij}\right)^{-1}=E_{\frac{a}{s^r}\alpha_{ij}}E_{s^dx\beta_{il}}^*E_{\frac{a}{s^r}\alpha_{ij}}^{-1}.
U_{j}=-\frac{1}{\alpha}\left[\frac{a-1}{a-b}B^{a-2}-\frac{b-1}{a-b}B^{b-2}\right] &V_{j}=-\frac{1}{\alpha}\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]p^{\alpha}_{j}q^{-\alpha}_{j}
L^{+ 4}( x , \theta^{+}, u ) = \mathrm{T r}\; ( A^{+ +}\partial^{+ \alpha}A_{\alpha}^{+}-{\frac{1}{2}}A^{+ \alpha}D^{+ +}A_{\alpha}^{+}+ A^{+ +}A^{+ \alpha}A_{\alpha}^{+}) \ .
\nabla_{(1-q) \partial_{t_i}}\left.\right._\circ \left.\right. S^{K\textnormal{th}}= S^{K\textnormal{th}}\left.\right._\circ \left.\right. (1-q) \partial_{t_i}T^{K\textnormal{th}}\left.\right._\circ \left.\right. \nabla_{(1-q) \partial_{t_i}}= (1-q) \partial_{t_i}\left.\right._\circ \left.\right. T^{K\textnormal{th}}
\beta C_{n,1}+ \sum_{k=2}^{K}\frac{2\mu_{k}\left(C_{n,k}- (n-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k}\left| \mathbb{P}_{n}\right.\stackrel{d}{\to}W + \sum_{k=1}^{K-1}\frac{2\mu_{k+1}Z_{k}-\mu_{k+1}^2}{4(k+1)}.
{4M^{2}}\,\left[ D\epsilon^{\beta \gamma \kappa \lambda}\partial_{\nu}{\partial_{\beta}}T_{(\kappa \lambda )}{}^{\nu}+2a\partial_{\nu}{\partial_{\beta}}T^{(\beta \gamma )\nu}\right] =e^2\,\partial_{\nu}\left({\partial}^{\gamma}\lambda^{\nu}-{\partial}^{\nu}\lambda^{\gamma}\right).
\Omega_{\beta_1,\bar{\beta}_2}=\frac{1}{2}\tanh(\beta_2(a)-\beta_1(a))(\frac{1}{\sinh(2\beta_1(a)}-\frac{1}{\sinh(2\beta_2(a)})\chi([\theta(e_{\beta_2}),e_{\beta_1}]) & + \frac{1}{2}\tanh(\beta_1+\beta_2)(\frac{1}{\sinh(2\beta_1(a)}+\frac{1}{\sinh(2\beta_2(a)})\chi([\theta(e_{\beta_2}),\sigma(e_{\beta_1})])
\frac{[x^{n-\Sigma_{\mathrm{k}}}]S(x)}{[x^n]S(x)}= w_n^{\Sigma_{\mathrm{k}}}\bigg( \exp\bigg\{-\frac{\big(A_{1,1}(w_n)-(n-\Sigma_{\mathrm{k}})\big)^2}{2A_{2,2}(w_n)}\bigg\}+o(1) \bigg)
\log{\cal D}\mbox{et}(\bar{P}^{\dag}\bar{P}) = - \frac{d}{ds}Z(0) = - \frac{d}{ds}\left[ \frac{1}{\Gamma (s)}\int_{0}^{\infty}\! dt \:t^{s -1}\mbox{Tr}(e^{-t \bar{P}^{\dag}\bar{P}}) \right]
\mathbb{P}\left(n_{k+1}=1\right)=\sum_{N\geq n_{1}\geq\cdots\geq n_{k}\geq1}\frac{1}{Nn_{1}\cdots n_{k}}=\frac{1}{N}S\left(\underset{k}{\underbrace{H_{1},\ldots,H_{1}}};N,1\right).
z_{tt}(0, \cdot) = \big(A^{-1}+ \gamma\big)^{-1}\Big(f(z^{0}, \nabla z^{0}) - a(z^{0}) A z^{0}+ \alpha A \theta^{0}\Big), \theta_{t}(0, \cdot) = -\tfrac{1}{\beta}\big(\eta A \theta^{0}+ \sigma \theta^{0}+ \alpha z^{1}\big).
\beta F = - \frac{N}{\beta^{3}}\int d \theta d \phi \int_{r_{H}+ h}^{L}d r \frac{\sqrt{g_{4}}}{( - g_{t t}^{'})^{2}}= - N \int_{0}^{\beta}d \tau \int d \theta d \phi \int_{r_{H}+ h}^{L}d r \sqrt{g_{4}}\frac{1}{\beta_{l o c a l}^{4}},
\left\Vert y_{t}-y_{1}^{s,t}\right\Vert \leq C_{p}\omega \left( s,t\right)^{\theta}\forall \left[ s,t\right] =\left[ t_{j}^{l},t_{j+1}^{l}\right] \subseteq \left[ s_{0},t_{0}\right] l=0,1,\dots ,n
e^{- 1}\mathcal{L}= \frac{1}{2}R -{\frac{1}{4}}G_{I J}F_{\mu \nu}{}^{I}F^{\mu \nu J}- \frac{1}{2}\mathcal{G}_{i j}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+{\frac{e^{- 1}}{4 8}}\epsilon^{\mu \nu \rho \sigma \lambda}C_{I J K}F_{\mu \nu}^{I}F_{\rho \sigma}^{J}A_{\lambda}^{K}.
\textbf{R}_{r}=\left(\boldsymbol{\gamma}_{r}\right)+\sigma^2 \mathbf{I}, \textbf{R}_{d}^b =\left(\boldsymbol{\gamma}_{d}^b \right)+\sigma^2 \mathbf{I}, \textbf{R}_{d}^m =\left(\boldsymbol{\gamma}_{d}^m \right)+\sigma^2 \mathbf{I},
f(f(p^{k}))=f(k^{p})=f(\prod_{i=1}^{m}q_{i}^{pn_{i}})=\prod_{i=1}^{m}(pn_{i})^{q_{i}}=p^{\sum_{i=1}^{n}q_{i}}\prod_{i=1}^{m}n_{i}^{q_{i}}=p^{\sum_{i=1}^{n}q_{i}}f(k)
\begin{array}{l}H_{1}=\dfrac{1}{(1+3h+2\alpha_{1}+2\alpha_{3})t_{1}}\{q_{2}^2p_{2}^2+\dfrac{1}{2}q_{1}q_{2}p_{1}+\dfrac{1}{2}q_{2}^2p_{2}-\eta t_{1}2q_{1}p_{2}begin{align*}10pt]\phantom{H_{1}=}+(1+h+2\alpha_{1}-\alpha_{2}+2\alpha_{3})q_{2}p_{2}+\dfrac{1}{2}\alpha_{1}q_{2}-\eta t_{1}(p_{1}-t_{2}p_{2})\}.\end{array}
v_\ell(q^n - 1) - v_\ell(q - 1) = \sum_{k=1}^{n-1}v_\ell(q - \zeta_n^k) = \sum_{k=1}^{n-1}\min(v_\ell(q - 1), v_\ell(1 - \zeta_n^k)) = \sum_{k=1}^{n-1}v_\ell(1 - \zeta_n^k),
u_1=\frac{x_1-\frac{a}{x_1}}{x_1x_2-\frac{ab}{x_1x_2}}=\frac{x_2(x_1^2-a)}{x_1^2x_2^2-ab},u_2=\frac{x_2-\frac{b}{x_2}}{x_1x_2-\frac{ab}{x_1x_2}}=\frac{x_1(x_2^2-b)}{x_1^2x_2^2-ab}.
\left( \left(p^{-n}\right)'(z) \right)^s =\left( \big( n c ( cz+d ) + 1\big)^{-2}\right)^s = e^{-2si\cdot \varphi(n,z)}\left(\left( c (cz+d) \right)^{-2}\right)^s \left(\left( n + \frac{1}{c (cz+d)}\right)^{-2}\right)^s
\|A\|_{A^2(D)}= \|\left[\Phi^{\mathrm{H}}_{\tau, \, 0}(\psi^{n}u_\pi)\right](\pi-\cdot)\|_{A^2(D)}&= \|\Phi^{\mathrm{H}}_{\tau, \, 0}(\psi^{n}u_\pi)\|_{A^2(D)}&\lesssim \|\psi^{n}u_\pi \|_{L^2(0, \tau)}&\leq \|u_\pi \|_{L^2(0, \tau)}\|\psi\|_{L^\infty(0, \infty)}^{n}
\hat P_n(x) =Q\begin{bmatrix}\frac{t_{n+1}(x)-\frac{t_{n+1}(-1)}{t_n(-1)}t_n(x)}{x+1}0 0 \frac{t_{n+1}(x)-\frac{t_{n+1}(1)}{t_n(1)}t_n(x)}{x-1}\end{bmatrix}Q^\top, Q := \frac{1}{\sqrt 2}\begin{bmatrix}1 1 1 - 1\end{bmatrix},
{\widehat \Sigma}= \left( \begin{array}{c c}{\Sigma_{R R},}&{\Sigma_{R A}}\\{\Sigma_{A R},}&{\Sigma_{A A}}\\ \end{array}\right) , \qquad{\bar{\Sigma}}= \left( \begin{array}{c c}{\Gamma_{a a}^{( 2 )},}&{\Gamma_{a r}^{( 2 )}}\\{\Gamma_{r a}^{( 2 )},}&{\Gamma_{r r}^{( 2 )}}\\ \end{array}\right) .
\begin{array}{lll}dim(\iota_{D}^{*}W^{-}_{i}(\nu^j))_{p}= dim(\rho'_{S^{k}Y}!(\mathcal{L}_{1 ,j,1}(-\Delta_{Y})))_{p}-dim(\rho'_{S^{k}Y}!(\mathcal{L}_{1 ,j,2}(-2\Delta_{Y})))_{p}&= \chi (Y,\lambda_{1 ,j,1}(-p))-\chi (Y,\lambda_{1 ,j,2}(-2p)). \end{array}
t\delta+\left(\!\begin{array}{c}|\mu| 2\end{array}\!\right)+\mathrm{ct}(\mu)-\frac{n(n-1)}{2}-\mathrm{ct}(\lambda) = t\delta-t|\mu|-\mathrm{ct}(\lambda)+\mathrm{ct}(\mu)-\frac{t(t-1)}{2}.
\frac{d}{dt}\left( \int_{x_1(t)}^{x_2(t)}f(t,r)\, dr \right)=f(t,x_2(t))x_2'(t)-f(t,x_1(t))x_1'(t)+\int_{x_1(t)}^{x_2(t)}{\dfrac{\partial f}{\partial t}(t,r) \, dr},
\mu_{0,T}^\mathbf{1}(A) := \mathbb P(X^1 \in A) = \mathbb P^{\gamma^1}(X^0 \in A)= \mathbb E^{\gamma^1}\mathbf{1}(X^0 \in A)=\mathbb E \gamma^1_T \mathbf{1}(X^0 \in A),
T_{\mu\nu}^{eff}=\frac{\zeta +b_{k}}{\zeta +b_{g}}\varphi_{,\mu}\varphi_{,\nu}-K\tilde{g}_{\mu\nu}+\frac{b_{g}M^{4}}{(\zeta +b_{g})^{2}}e^{-2\alpha\phi/M_{p}}\tilde{g}_{\mu\nu}+\frac{\zeta}{\zeta +b_{g}}T_{\mu\nu}^{(f,canonical)}-\frac{mF(\zeta)}{\sqrt{\zeta +b_{g}}}\overline{\Psi}^{\prime}\Psi^{\prime}\tilde{g}_{\mu\nu},
S = - \frac{1}{\lambda_{s}^{2}}\int d^{4}x \sqrt{- g}e^{- \phi}\biggl [ R + g^{\alpha \beta}\partial_{\alpha}\phi \partial_{\beta}\phi - \frac{1}{2}e^{2 \phi}g^{\alpha \beta}\partial_{\alpha}\psi \partial_{\beta}\psi \biggr ] .
m=\begin{cases}1{\rm if}\G\{\rm belongs\to\}\Phi_{16}, \Phi_{31}, \Phi_{37}, \Phi_{39}, \Phi_{43}, \Phi_{58}, \Phi_{60}\{\rm or}\ \Phi_{80}, 2{\rm if}\G\{\rm belongs\to\}\Phi_{106}\{\rm or}\ \Phi_{114}, 3{\rm if}\G\{\rm belongs\to\}\Phi_{30}.\end{cases}
\sum_{k=0}^{v}\varphi(k) =\sum_{k=1}^{v+3}\varphi(k) D(v+3-k) &-b_0c_0\varphi(2)A(v+1)-b_0c_1\varphi(1)A(v+1) &-c_0\varphi(1)\sum_{k=0}^{v+2}a_k B(v+2-k)
u_1(N) =\sum_{l=1}^{p_1}a_{l,1}E(\gamma_{l,1})+\sum_{j=1}^q c_j(N)E\left(\frac{\lambda_j}{\beta_1}\right) u_i =\sum_{l=1}^{p_i}a_{l,i}E(\gamma_{l,i}),\i=2,\dots,d
\hat{\chi}_{\mu\nu}= \Lambda^{(2)}_{\mu\nu}\, ,\hspace{1cm}\hat{\chi}_{\mu\underline{z}}={\textstyle\frac{1}{m}}\hat{\lambda}_{\mu}=\Lambda_{\mu}\, ,\hspace{1cm}\hat{\xi}^{\underline{z}}=\Lambda^{(0)}\, .
N^{( 2 l - 1 )}= \delta_{l , 1}\int d x \frac{\theta^{\prime}}{2 \pi}- \sqrt{\frac{2 m T}{\pi}}e^{- m / T}\frac{1}{( 2 l - 1 ) !}\int d x \left( \frac{\theta^{\prime}}{2 T}\right)^{2 l - 1}+ \dots
t^3_{3\overline{1}}\eta^{\overline{1}}-\overline{t}^3_{3\overline{1}}\eta^{1}=\left(-\overline{f}^2_{21}+\overline{f}^2_{12}+\overline{f}^1_{11}-\epsilon t^1_{2\overline{2}}\right)\eta^{\overline{1}}-\left(-f^2_{21}+f^2_{12}+f^1_{11}-\epsilon\overline{t}^1_{2\overline{2}}\right)\eta^1,
\chi_{-}= \nu_{-}\eta \gamma_{-}\gamma_{-}^{\prime - 1}\eta^{\prime - 1}\nu_{-}^{\prime - 1}\eta^{\prime}\gamma_{-}^{\prime}\gamma_{-}^{- 1}\eta^{- 1}, \qquad \psi_{-}= \eta \gamma_{-}\gamma_{-}^{\prime - 1}\eta^{\prime - 1}.
\Phi_\infty^{(s, u)}(z_1, z_2) : = \lim_{n \to \infty}\widetilde{\Phi}_n^{(n)}( z_1, z_2) = \frac{A_{III}^{(s, u)}( 1, z_1) B_{III}^{(s , u )}( 1, z_2) - A_{III}^{(s, u )}( 1, z_2) B_{III}^{(s, u )}( 1, z_1)}{2 ( u + z_1)(u+z_2) \cdot \big[ C_3^{(s,u)}(1) \big]^2 \cdot (z_1- z_2)}.
e_{2}\psi =\left( \begin{array}{c}-{\psi_{2}}{\psi_{1}}-{\psi_{4}^{*}}{\psi_{3}^{*}}\end{array}\right) \quad or\quad e_{3}\mathbf{)}e_{1}\psi =\left( \begin{array}{c}{\psi_{2}}{\psi_{1}}{\psi_{4}^{*}}-{\psi_{3}^{*}}\end{array}\right)
\lambda_{\mathcal{H}}(\rho^{\infty})= \min_{(\xi, \eta)\in\mathbb{R}^{n}_+}\{\frac{1}{\Delta x^2}[\xi^TA\xi+2\xi_1\eta+2\xi_{n-1}\eta+2\eta^2]\, : & \xi^T\xi+\eta^2=1, ~\eta=\sum_{i=1}^{n-1}\xi_i\}.
\int_0^{(i-1)\Delta_n}(\mathcal S(\Delta_n)-I)\Sigma_s^{\mathcal S_n}(\mathcal S(\Delta_n)-I)^* ds = (i-1)\Delta_n (e\otimes (\Delta_i\mathcal S f))(e\otimes (\Delta_i\mathcal S f))^* = (i-1)\Delta_n (\Delta_i\mathcal S f)^{\otimes 2}.
\Gamma [ \Phi , \bar{\Phi}] = \int d^{8}z L ( \Phi , D_{A}\Phi , D_{A}D_{B}\Phi , \ldots \bar{\Phi}, D_{A}\bar{\Phi}, D_{A}D_{B}\bar{\Phi}) + ( \int d^{6}z L_{c}( \Phi ) + h . c . )
\left( \begin{array}{c}{\partial_{\mu}\theta}\\{\sin \theta \partial_{\mu}\varphi}\\ \end{array}\right) = \left( \begin{array}{c c}{G_{\mu}}&{- H_{\mu}}\\{H_{\mu}}&{G_{\mu}}\\ \end{array}\right) \left( \begin{array}{c}{\beta}\\{- \gamma}\\ \end{array}\right) ,
V\partial_iV^{-1}=\frac{\epsilon_{iak}\tau_a\hat{r}_k}{2ir}(\cos \phi -1)-(\frac{\tau_i-\hat{r}_i\tau\cdot\hat{r}}{2ir})\sin \phi -\hat{\phi}_i(\frac{\tau\cdot\hat{r}}{2i\rho}),
u^n = -\frac{1}{2}\frac{\Delta t}{\nu+\Delta t^2}\left((w^{n}_j-w_d)+(w^{n}_i-w_d))\right)-\frac{1}{2}\frac{\Delta t^2}{\nu+\Delta t^2}\left(P_{ij}-P_{ji}\right)(w_j^n-w_i^n).
\mathrm{P}\Big(2^{-\frac{4}{3}}N^{-\frac{1}{3}}\big(\lambda_{\mathrm{max}}-4N\big) \leq x\Big)=1+ \sum_{k=1}^{N}\frac{(-1)^k}{k!}\frac{1}{(\varphi(4))^k}\int_{x}^{\infty}\cdots \int_{x}^{\infty}\mathrm{Pf}\Big[ K_{N}\Big(4N+\frac{u_i}{\varphi(4)}, 4N+\frac{u_j}{\varphi(4)}\Big) \Big]_{i,j = 1}^{k}du_1 \cdots du_k,
\begin{aligned}A \psi_k = \mu_k \psi_k, \|\psi_k\| =1, & \exists \gamma>0, \ \exists\kappa>0, \ \exists N_0> 0, \ \forall k \geq N_0, \mu_{k+1}-\mu_k \geq \kappa k^{\gamma-1};\end{aligned}
|\gamma_0^{(\beta+1)}(t)| = \tfrac{c_0}{2}|\phi^{(\beta+1)}(\pi t/\delta)| (\pi/\delta)^{(\beta+1)}\leq \tfrac{c_0}{2}C(\beta+1,\beta+1) |x_0-d|^{-\beta-1}, \;\;\;\forall t\in I,
\begin{array}{ll}\hbox{Sector}\ \ \hbox{Field content}\hbox{even}[1,1],\, [3,3],\, [5,5],\, [7,7],\, [9,9],\, [11,11],\, [13,13] \hbox{odd}[2,2],\, [4,4],\, [6,6],\, [8,8],\, [10,10],\, [12,12] \hbox{twisted}[1,13],\, [3,11],\, [5,9],\, [7,7],\, [9,5],\, [11,3],\, [13,1] \end{array}
\begin{array}{c}{{J^{+}= q^{- J^{0}+ \frac{1}{2}}\a_{1}^{+}a_{2}}}\\{{J^{-}= q^{- J^{o}- \frac{1}{2}}a_{2}^{+}a_{1}}}\\{{2 J^{0}= N_{1}- N_{2}}}\\{{\left[ J^{+}, J^{-}\right] = [ 2 J^{0}]_{q}= [ N_{1}- N_{2}]_{q}}}\\{{\left[ N , J^{\pm}\right] = [ N , J^{0}] = 0 .}}\\ \end{array}
{\frac{d}{d t}}S = \mathrm{I m}( P ) + b_{\scriptscriptstyle I}\left( \frac{1}{R}\nabla^{2}R - ( \nabla S )^{2}\right) +{\frac{b_{\scriptscriptstyle R}}{R^{2}}}\nabla \cdot ( R^{2}\nabla S ) . ~
U(X) =10X- 36X^2+ 27X^3, U(X^2) =-8X+306X^2-2160X^3+5508X^4-5832X^5+2187X^6, U(X^3) =X- 360X^2+ 10566X^3- 99144X^4+ 423549X^5- 944784X^6 &+ 1141614X^7 -708588X^8 +177147X^9.
R_{m,n}(x)=\sum_{k=0}^{min(m,n)}(\begin{matrix}m\\k\end{matrix})(\begin{matrix}n\\k\end{matrix})k!x^{k}=\sum_{k=0}^{min(m,n)}\frac{n!m!}{k!(n-k)!(m-k)!}x^{k}.
\sum_{k=0}^{\infty}\left( 2k\beta+r\mid\alpha\right)_{n}\frac{\left(-1\right)^{k}\left( 2\pi\right)^{2k}}{\left( 2k\right) !}=\sum_{j=1}^{\left\lfloor n/2\right\rfloor}S\left( n,2j;\alpha,\beta,r\right)\left( -1\right)^{j}\left( 2\pi\beta\right)^{2j}.
\begin{array}[c]{c}\int_{M}A_{\alpha\beta}f^{\alpha}f^{\beta}=\int_{M}A_{\overline{\alpha}\overline{\beta}}f^{\overline{\alpha}}f^{\overline{\beta}}=0\text{\}\mathrm{and}\int_{M}Ric_{\alpha\overline{\beta}}f^{\alpha}f^{\overline{\beta}}+K\int_{M}\left\vert \nabla_{b}f\right\vert^{2}=0.\end{array}
\P(C'_{k_1,j_1}) = \P(C_{k_1,j_1}) -p, \P(C'_{k_2,j_2}) = \P(C_{k_2,j_2}) -p, \P(C'_{k_1,j_2}) = \P(C_{k_1,j_2}) +p, \P(C'_{k_2,j_1}) = \P(C_{k_2,j_1}) +p,
h(T_1',0,\eta) = \left( \prod_{j=1}^{k_0}j \hat{W}(j)\right) \int_{T_{j}\leq \tau_j \leq T_{j}'}(\eta-\tau_1 + \tau_1)(\tau_1-2\tau_2 + \tau_2) \dots \psi(\eta-\tau_1) \psi(\tau_1-2 \tau_2) \dots h_0(0, k_0, k_0 \tau_{k_0}) d\tau_1 \dots d\tau_{k}.
\sum_{m_1+\cdots+m_p=m-(n-1)p}\biggl(\prod_{i=1}^p \binom{-n+1}{m_i}\biggr)(-1)^{-m-p}L_{m_1-1}\ldots L_{m_p-1}-\delta_{p\mid n}\binom{np-m-2}{np-2}L_{-np+m}.
(1+\frac{1}{n})^{n}=1+(\begin{matrix}n\\ 1\end{matrix})\frac{1}{n}+(\begin{matrix}n\\ 2\end{matrix})\frac{1}{n^{2}}+(\begin{matrix}n\\ 3\end{matrix})\frac{1}{n^{3}}+\cdot\cdot\cdot+(\begin{matrix}n\\n\end{matrix})\frac{1}{n^{n}}.
D_u X(t) = \int_0^{u \wedge t}Z(t, s) \, A(s, X_s) \, \mbox{d}s, V(t) = \int_0^t \sum_{i=1}^m Z(t, s) \, A_i(s, X_s) \, \big\{Z(t, s) \, A_i(s, X_s) \big\}^\ast \, \mbox{d}s.
\left\|B\vec t^{(N)}\right\|_{\dot b_{p,q}^s}\gtrsim\left[\sum_{j=0}^{N-1}\left(\sum_{k\in\mathbb{Z}^n,\,|k|<2^{j-1}}2^{-jn}2^{(j-N)(F+s-\frac{n}{2})p}\right)^{\frac{q}{p}}\right]^{\frac{1}{q}}\sim\left[\sum_{j=0}^{N-1}2^{(j-N)(F+s-\frac{n}{2})q}\right]^{\frac{1}{q}}.
T_{0}( z^{\prime}) T ( z , \theta ) = \frac{\frac{c ( \theta )}{2}I ( \theta )}{( z^{\prime}- z )^{4}}+ \frac{2 + \theta}{( z^{\prime}- z )^{2}}T ( z , \theta ) + \frac{\partial_{z}T ( z , \theta )}{z^{\prime}- z}+ \cdots \: \: \: ,
\sum_{n=0}^{\infty}b(5n+4)q^{n}=35\dfrac{E_{5}^{2}E_{10}^{2}}{E_{1}^{4}E_{2}^{4}}+700q \dfrac{E_{5}^{4}E_{10}^{4}}{E_{1}^{6}E_{2}^{6}}+6875 q^2 \dfrac{E_{5}^{6}E_{10}^{6}}{E_{1}^{8}E_{2}^{8}}&+31250 q^3 \dfrac{E_{5}^{8}E_{10}^{8}}{E_{1}^{10}E_{2}^{10}}+78125 q^4 \dfrac{E_{5}^{10}E_{10}^{10}}{E_{1}^{12}E_{2}^{12}}.
{\frac{u^{2}}{l^{2}}}\sin^{2}\alpha ~ \rho^{1 2}+ \left({\frac{u^{2}}{l^{2}}}\cos^{2}\alpha -{\frac{u^{2}}{l^{2}}}\sin^{2}\alpha - 1 \right) \rho^{6}-{\frac{u^{2}}{l^{2}}}\cos^{2}\alpha = 0
D_{1 2}^{( l )}\stackrel{1}{F^{( l )}}\stackrel{2}{F^{( 1 )}}D_{1 2}^{( l ) - 1}= \left( \begin{array}{c c}{F^{( l + 1 )}}&{0}\\{0}&{- F^{( l - 1 )}}\\ \end{array}\right) .
m_\mathcal{G}(X^\ell(g_1)\cap X_{\mu,\nu}) =\begin{cases}0, \; \ell< |\nu|, N^{-\ell}-N^{-\ell-1}, \; \ell\geq |\nu|,\end{cases}\mbox{and}m_\mathcal{G}(X^\ell(g_1)) =\begin{cases}N^{-\ell}-N^{-\ell-1}, \; \kappa_\mathcal{G}(g_1)\leq \ell, N^{-\ell-1}(N-1)^2, \; 0\leq \ell< \kappa_\mathcal{G}(g_1).\end{cases}
\lim_{r\to 0}\frac{Q^*(\theta)}{r^{2j}}= \eta_j^2 Q^*(w_{1j},\ldots, w_{j-1,j}, 1, 0, \ldots, 0), \lim_{r\to 0}\frac{Q^*(\tilde{\theta})}{r^{2j+1}}= \eta_j^2 Q^*(w_{1j}, \ldots, w_{j-1,j}, 1, 0, \ldots, 0).
\frac{\partial}{\partial\rho_{ij}}H(k^{-1}\rho) =\frac1k\left(-1-\ln(\rho_{ij})\right), \frac{\partial}{\partial\rho_{ij}}E(\rho) =\frac{d}{k^2}\frac{c_\beta^2\rho_{ij}}{1-\frac{2}{k}c_\beta+\|\rho\|_2^2c_\beta/k^2}.