harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{-\pi/2}^{+\pi/2}~\frac{d\xi^1}{\pi}~ O_{M}^{cl}(\{\sigma_{+}+ \xi^1 \}; \{\sigma_{-}- \xi^1 \}) \equiv \int_{-\pi/2}^{+\pi/2}~ \frac{d\xi^1}{\pi}~ O_{M}^{cl}(\{\sigma^{0}\};\{\sigma^1 + \xi^1 \})
\chi(g_iw_jw_\ell) = S(\Delta(g_i)\Delta(w_j), \Delta(w_\ell)) = S(\sum_k \lambda^i_{jk}\Delta(w_k), \Delta(w_\ell)) = \sum_{k = 1}^{n^2}\lambda^i_{jk}\chi(w_k\cdot w_\ell),
\sum_{q\in\mathcal{Q}}\;\sideset{}{^\star}\sum_{\chi\bmod{q}}L\left(\tfrac{1}{2},\pi\otimes\chi\right)(1+\chi(-1))\bar\chi(\ell)=\frac{\lambda_\pi(\ell,1)}{\sqrt{\ell}}Y+O\left(\ell^2\:Q^{2-1/2013+\varepsilon}\right),
\P(X_1>s/t)-t^2\P(X_1>s) = t^2\left(\frac{1}{\sqrt{t^2+s^2}(\sqrt{t^2+s^2}+s)}-\frac{1}{\sqrt{1+s^2}(\sqrt{1+s^2}+s)}\right) &=t^2(1-t^2)\frac{1+\frac{s}{\sqrt{t^2+s^2}+\sqrt{1+s^2}}}{(t^2+s^2+s\sqrt{t^2+s^2})(1+s^2+s\sqrt{1+s^2})}.
f(x_{1},x_{2},...,x_{m}|p_{1},p_{2},...,p_{m})=\frac{n!}{\prod x_{i}!}\prod p_{i}^{x_{i}}=(\begin{matrix}n\\x_{1},x_{2},...,x_{m}\end{matrix})p_{1}^{x_{1}}p_{2}^{x_{2}}\cdot\cdot\cdot p_{m}^{x_{m}}
\lim_{\bar{\gamma}\rightarrow \infty}F_{\gamma}(\gamma, \bar{\gamma}) = \lim_{\bar{\gamma}\rightarrow \infty}\int_0^{\gamma}f_{\gamma}(t, \bar{\gamma}) t & = \int_0^{\gamma}\frac{t^{N-1}}{(\bar{\gamma}/N_T)^N (N-1)!}t &=\frac{1}{(\bar{\gamma}/N_T)^N N!}\gamma^{N}&=G(\bar{\gamma}) \gamma^{N},
\langle \psi^{a_1}_1\gamma_1,\dots,\psi^{a_k}_k\gamma_k,1\rangle_{g,k+1,\beta}=\sum_{i=1}^{k}\langle \psi^{a_1}_1\gamma_1,\dots,\psi^{a_{i-1}}_{i-1}\gamma_{i-1},\psi^{a_i-1}_i\gamma_{a_i},\psi^{a_{i+1}}_{i+1}\gamma_{i+1},\dots, \psi^{a_k}_k\gamma_k\rangle_{g,k,\beta},
K\big( \eta^{(3)}\big) =\sigma K\big( \eta^{(2)}\big) \sigma \hphantom{K\big( \eta^{(3)}\big)}{}=\sum_{n,m=0}^{\infty}(-1)^{m+n}\alpha_{n}(x(u_{2},0))^{\ast}H_{1}\rho\big(\vert z_{2}\vert^{-2\kappa},\vert z_{2}\vert^{2\kappa}\big) \alpha_{m}(x( u_{2},0)) \overline{z_{2}}^{m}z_{2}^{n},
\sup\limits_{0<t<T}\int_\Omega\rho_x^2dx+\iint_{Q_T}\big(\rho_t^2+\theta\rho_x^2\big) dxdt\leq C, begin{align*}1mm] \left|\rho(x,t)-\rho(y,s)\right|\leq C\left(|x-y|^{1/2}+|s-t|^{1/4}\right),\quad\forall (x, t), (y, s) \in \overline Q_T.
\int (\theta(x)-x)^{2}\bar{\mu}_{Q,t}(dx) = \frac{1}{2}\iint ((\theta(x)-x)-(\theta(y)-y))^{2}\bar{\mu}_{Q,t}(dx)\bar{\mu}_{Q,t}(dy) \int (x-\theta(x))H\mu(\theta(x))\bar{\mu}_{Q,t}(dx) = \frac{1}{2}\iint ((x-\theta(x))-(y-\theta(y)))\cot((\theta(x)-\theta(y))/2)\bar{\mu}_{Q,t}(dx)\bar{\mu}_{Q,t}(dy)
\kappa_1 =E\{T(\psi)\}=\eta^{1/2}(\xi^{rst}\lambda_{rs,t}+\xi^{rs}\lambda_{rs})+O(n^{-1}), \kappa_2 ={\rm var}\{T(\psi)\}=1+O(n^{-1}), \kappa_3 ={\rm skew}\{T(\psi)\}=\eta^{3/2}(\lambda^{1r}\lambda^{1s}\lambda^{1t}\lambda_{rst}+3\lambda^{1r}\lambda^{1s}\lambda^{1t}\lambda_{rs,t}-6\xi^{rs1}\lambda^{1t}\lambda_{rs,t}-6\xi^{11})+O(n^{-1}),
0= -\partial_x^2V_n+DW(V_n)=-\partial_x^2U_0-\partial_x^2\Phi_n+DW(U_0+\Phi_n)= -\partial_x^2U_0+DW(U_0) -\partial_x^2\Phi_n + D^2W(U_0)\Phi_n + \frac12 \frac{d^2}{ds^2}|_{s=s_n}DW(U_0+s\Phi_n) = L\Phi_n + \frac12 D^3 W(U_0+s_n\Phi_n)[\Phi_n,\Phi_n].
\nabla_{\phi p}R_0{}^{\beta}{}_{l \bar{q}m}=\nabla_{0p}R_0{}^{\beta}{}_{l\bar{q}m}+U_{ps}^{\beta}R_0{}^s{}_{l \bar{q}m}-U_{pl}^s R_0{}^{\beta}{}_{s \bar{q}m}-U_{pm}^s R_0{}^{\beta}{}_{l \bar{q}s}.
T(z_{1},z_{2})=\frac{Y(z_{1},z_{2})}{X(z_{1},z_{2})}=\frac{\sum_{p,q=0,0}^{m,n}b_{p,q}z_{1}^{-p}z_{2}^{-q}}{\sum_{p,q=0,0}^{m,n}a_{p,q}z_{1}^{-p}z_{2}^{-q}}
\partial_{\mu}\left( \sqrt{- k}( k^{- 1})^{[ \mu \nu ]}e^{- \phi}- \frac{2 T_{p}}{T_{p}^{\prime}}\frac{1}{( n - 2 ) !}\epsilon^{\mu \nu \gamma_{1}\ldots \gamma_{n - 2}}( C^{( n - 2 )})_{\gamma_{1}\ldots \gamma_{n - 2}}\right) = 0 .
\left\{- \hbar^{2}\frac{\partial^{2}}{\partial x^{2}}+ \hbar^{2}\frac{\partial^{2}}{\partial y^{2}}+{( 8 \pi \chi_{0}^{3})}^{2}\left[ x \left( x - a_{0}- \frac{\alpha}{4 \pi \chi_{0}^{3}}\right) - y^{2}\right] \right\}\Psi_{1}( x , y ) = 0 ~ .
\nabla_x(\varrho\, m_0)=\frac{1}{D}\frac{x}{\left<x\right>}\varrho\, m_0,(\partial_i \sigma^{ij})\partial_j(\varrho\, m_0)=-\lambda_1(\xi)\xi_j \varrho \partial_j m_0,\partial_{ij}(\varrho\, m_0)=\varrho \partial_{ij}m_0,
\frac{1}{2}\frac{d}{dt}\|u\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2}=\int_{\mathbb{R}^{3}}(v-w)u\cdot\nabla\Psi dx, \int_{\mathbb{R}^{3}}\big(\partial_{t}v\Psi+\nabla\cdot(v\nabla\Psi)\Psi-\Delta v\Psi+(u\cdot\nabla) v\Psi\big)dx=0, \int_{\mathbb{R}^{3}}\big(\partial_{t}w\Psi-\nabla\cdot(w\nabla\Psi)\Psi-\Delta w\Psi+(u\cdot\nabla) w\Psi\big)dx=0.
\sigma_\infty = \sum_{s \ge 0}\sum_{p_1,\ldots,p_s}(-1)^{\binom{s+1}{2}}\frac{1}{s!}[ d_{Y \otimes X}, \partial_{t_{p_1}}] \cdots [ d_{Y \otimes X}, \partial_{t_{p_s}}] \theta_{p_1}\cdots \theta_{p_s}+ (t )\,.
P(\boldsymbol{Y}=\boldsymbol{y}) =\frac{P(\vert\boldsymbol{Y}\vert=\vert\boldsymbol{y}\vert)}{\binom{\vert\boldsymbol{y}\vert +\vert \boldsymbol{\alpha}\vert -1}{\vert\boldsymbol{y}\vert}}\prod_{j=1}^J \binom{y_j+\alpha_j-1}{y_j}.
0=\lim_{i\rightarrow\infty}\frac{dA}{dt}\left( t_{i}\right) =-\int_{0}^{2\pi}w^{1-\frac{1}{\alpha}}\left( \theta\right) d\theta+\dfrac{{L\!}\left( 0\right)}{2\pi}\int_{0}^{2\pi}w\left( \theta\right)d\theta,\ \ \ 1-\frac{1}{\alpha}>0.
P_{c}\left(V_{c},\Pi\right)=\sum_{j\in J}e^{-\hat{r}T_{j}}\left(\sum_{i\in I_{j}}-\Pi\left(t_{i},T_{j}\right)V_{c}\left(t_{i},T_{j}\right)+s_{c}p_{c}D\left(T_{j}\right)\right),
\rho(h_1(1)+\mathfrak{a}(B,D_\flat,E_\flat)1+\mathfrak{e}(\rho;B,D_\flat,E_\flat)\rho (h_1(1)+1)+1) = \rho h_1(1)+\rho\mathfrak{a}(B,D_\flat,E_\flat)+\rho\mathfrak{e}(\rho;B,D_\flat,E_\flat)\rho (h_1(1)+1)+\rho = \rho\mathfrak{a}(B,D_\flat,E_\flat)+\rho^2\mathfrak{e}(\rho;B,D_\flat,E_\flat)+\rho+(\rho +\rho^2\mathfrak{e}(\rho;B,D_\flat,E_\flat))h_1(1).
Z({\cal A},G,q) ={\rm e}^{F({\cal A},G,q)}=\sum_{N=1}^{\infty}Z_{N}({\cal A},G)q^N =\prod_{n=-\infty}^{+\infty}Z^{(\rm cov)}(G,{\rm e}^{-{\cal A}v(n)}q)~.
|x_{N+k}| = |x_N|e^{\sum_{i=0}^{k-1}\ln\left(1-h_{N+i}x_{N+i}^2 \right)}< |x_N| e^{\sum_{i=0}^{k-1}\left(-h_{N+i}x_{N+i}^2 \right)}&< |x_N| e^{-L^2\sum_{i=0}^{k-1}h_{N+i}}.
\xi_{l,n,i}^{(\alpha )}= 2{( - 1)^i}\sqrt{{4^\alpha}{l^{- 2(1 + \alpha )}}\left({l - x_{l,n,i}^{(\alpha )}}\right)\,x_{l,n,i}^{(\alpha )}\,\varpi_{l,n,i}^{(\alpha )}}\,\forall l, n; i = 0, \ldots, n.
\chi_{n}= \sum b_{\vec{n}, r , s , t}\lambda^{n_{1}}( \lambda y )^{n_{2}}( \lambda y_{c})^{n_{3}}( \lambda / \Delta_{-})^{n_{4}}e^{r \Delta_{+}( y - y_{c})}e^{s \Delta_{-}y}e^{t \Delta_{-}y_{c}},
\begin{aligned}0 \leq \log d_L + G_1(\alpha; |t|) \cdot r_1 + G_2(\alpha; |t|) \cdot 2r_2 + \Re\Big\{\frac{1}{\alpha+it}+ \frac{1}{\alpha+1+it}\Big\}& - \sum_{\rho}\Re\Big\{\frac{1}{\alpha+1-\rho}+ \frac{1}{\alpha+1+it-\rho}\Big\}+ \frac{1}{\alpha}+ \frac{1}{\alpha+1}.\end{aligned}
X_{k,s}' =-\sum_{l=0}^{k-1}(-1)^{k-l}X_{l,s}U(d_{k}-d_{l}-1) \ \ +\begin{cases}X_{k,s-1}+\sum_{l=0}^{k-1}(-1)^{k-l}X_{l,s-1}U(d_{k}-d_{l}) s>1 \delta_{k,0}s=1,\end{cases}
K^{3, 3}= \left(\begin{array}{cccc}0 0 \alpha begin{align*}6pt] 0 0 \beta begin{align*}6pt] 0 0 0\end{array}\right) \mbox{and}K(x)= K^{k+2, k+2}:= \left(\begin{array}{cccc}0_{k-1, k-1}0_{k-1, 3}begin{align*}6pt]0_{3, k - 1}K^{3,3}\end{array}\right) \mbox{for}k \ge 1.
\widetilde{Au}=\begin{pmatrix}(Au,f_0) (Au,f_1) (Au,f_2) (Au,f_3)\end{pmatrix}=\begin{pmatrix}\frac{P}{G}\frac{E-\frac{EP}{G^2}}{(e-\frac{E^2}{G^2})^{1/2}}0 \left(P-\frac{P^2}{G^2}-(\frac{(E-\frac{EP}{G^2})^2}{e-\frac{E^2}{G^2}})\right)^{1/2}\end{pmatrix},
Z_{0}=\frac{1}{N!h^{3N}}\int\prod_{i}d\vec{p}_{i}d\vec{r}_{i}exp\{-\beta H_{0}(\{r_{i},p_{i}\})\}=\frac{V^{N}}{N!h^{3N}}(\frac{2\pi m}{\beta})^{\frac{3N}{2}}.
\theta_1 = \frac{1}{(1-2|w|^2)^2}\left(2|\phi|^2 -(w\phi+\bar w\bar\phi)^2\right),\theta_2 = \frac{1}{(1-2|w|^2)^2}\left(2|dw|^2+ (wd\bar w-\bar w dw)^2\right).
\underline{e}= e, \underline{s}= e + s, \underline{t}= e + t, \underline{st}= e + s + t + st, \underline{ts}= e + s + t + ts, \underline{sts}= e + s + t + st + ts + sts, \underline{tst}= e + s + t + st + ts + tst, \underline{stst}= e + s + t + st + ts + sts + tst + stst.
J_\mu^{(1)}=\partial_\mu \phi +{\textstyle \frac 12}\varepsilon_{\mu \nu}[\partial^\nu \phi ,\phi ]-{\textstyle \frac 12}[J_\mu ,\phi (-\infty)]=Z_\mu -{\textstyle \frac 12}[J_\mu ,\phi (-\infty )].
N_5(2a,2b,2c) =N_1(a,b,c)^2, N_5(2a+1,2b,2c) =N_1(a,b,c)N_1(a+1,b,c), N_5(2a+1,2b+1,2c) =N_1(a+1,b,c)N_1(a,b+1,c).
I_G\, := \, \langle\,x_{i,j}^2-1, x_{1,j}-1, x_{i,1}-1\colon\, i,j\in G\setminus\{1\}\,\rangle\, + \langle\,p_{i,j,k}(\text X)\colon\, i,j,k\in G\,\rangle\, + \ \langle\,\sum_{j\in G}x_{i,j}\colon\, i\in G\setminus\{1\}\,\rangle\, \subset \mathbb{Q}[X_G].
\left(v\alpha_{1}^{a_{1}}\alpha_{2}^{a_{2}}\alpha_{3}^{a_{3}}\right)^{r}=\left(\prod_{j=0}^{r-1}\rho^{k_{j}}v\tau^{a_{2}v_{2}j}\right)\left(\alpha_{1}^{a_{1}}\alpha_{2}^{a_{2}}\alpha_{3}^{a_{3}}\right)^{r}&=\rho^{l_{1}}v^{r}\tau^{l_{2}a_{2}v_{2}}\left(\alpha_{1}^{a_{1}}\alpha_{2}^{a_{2}}\alpha_{3}^{a_{3}}\right)^{r},
+\gamma^{4}[\partial_{1}a(\partial_{2}b_{4}+\partial_{1}a_{4})-\partial_{2}a(\partial_{1}b_{4}-\partial_{2}a_{4})+4c(-\partial_{1}a_{2}+\partial_{2}b_{2})+4d(\partial_{2}a_{2}+\partial_{1}b_{2})
\mathbb{E}\left[\omega_{t_{i}}\left|\omega_{t_{0}},\omega_{t_{1}},\cdots\omega_{t_{i-1}},\omega_{t_{i+1}},\omega_{t_{n}}\vphantom{\frac{t_{i+1}}{t_{i+1}}}\right.\right]=\frac{t_{i+1}-t_{i}}{t_{i+1}-t_{i-1}}\omega_{t_{i-1}}+\frac{t_{i}-t_{i-1}}{t_{i+1}-t_{i-1}}\omega_{t_{i+1}}.
\frac{1}{2\pi i}\int_{\delta^{0}}res\left(\frac{\omega_\beta}{f^k}\right)=\frac{B\left(\frac{1}{2},\frac{\beta_2+1}{d}\right)B\left(A_{\beta'},A_{\beta'}\right)}{2\pi i}\sum_{j}C_j(\lambda)F\left(a_j,b_j,c_j;\frac{1}{\lambda}\right),
V^{(1)}(\Phi) = \frac{\lambda}{\alpha^2}\Phi^4 +\frac{1}{2}\left[ \frac{\gamma^2 (4 \pi )^2}{2 \theta^4}-\frac{8\lambda}{\theta^2}\right] \Phi^4 \left( \ln\frac{\Phi^2}{\mu^2}-\frac{25}{6}\right).
\int_0^\infty \sum_{i=1}^\infty e^{-\mu_i t}(\phi_i, g)(\phi_i, f)\, dt = \sum_{i=1}^\infty (\phi_i,g)(\phi_i, f)\int_0^\infty e^{-\mu_i t}\, dt &= \sum_{i=1}^\infty \frac{(\phi_i, g)(\phi_i, f)}{\mu_i}= 2\,(g, T_{\Phi, \Gamma}f)\,.
\lvert\gamma(x)\rvert\;\omega_2(\phi(x)) =\lim_{\alpha}\frac{\lvert\gamma(x)\rvert\;\| l_{\phi(x)}(\chi_{{U_\alpha}})\|_{p,\omega_2}}{\lambda_2(U_\alpha)^{\frac{1}{p}}}&\le\lim_\alpha\frac{\|T\|\;\|T^{-1}\|\;\omega_1(x)\;\|\chi_{U_\alpha}\|_{p,\omega_2}}{\lambda_2(U_\alpha)^{\frac{1}{p}}}\leq\|T\|\;\|T^{-1}\|\omega_1(x).
\parallel W\parallel_{D(s)\times\Pi}=\sup_{\xi\in\Pi,\textbf{j}\in\mathbb{Z}^d}{\sum_{k\in\mathbb{Z}^n}\left(|\widehat{W}(k;\xi)|+|\partial_{\xi_{\textbf{j}}}\widehat{W}(k;\xi)|\right)e^{|k|s}}.
\frac{\sin \left(\beta -\frac{\pi}{n}\right)}{\sin \frac{2\pi}{n}}= -\frac{1}{2}\Rightarrow \beta = \beta_0(n) = \frac{\pi}{n}-\arcsin \left(\frac{1}{2}\sin \frac{2\pi}{n}\right) = \frac{\pi^3}{2n^3}+ \frac{\pi^5}{8n^5}+ O\left(\frac{1}{n^7}\right).
\mathbf{E}^s_i(X)= \left( (A_i - \widetilde{Y}_i)^T\otimes I_2 \right)\left( M^s(\widetilde{Y}_i)-\overline{M}^s \right)\emph{Vec}(\nabla\mathbf{u}^s(X)), &\mathbf{F}^s_i(X)= -\left( (\widetilde{Y}_i-\overline{X}_i)^T\otimes I_2 \right)\left(\overline{M}^s - I_4 \right)\emph{Vec}(\nabla\mathbf{u}^s(X)).
T^{(1)}_1(E)={\rm e}^{\pm \frac{\pi}{2}i}\frac{Q^{\mp}(E q^{\frac{5}{4}})}{Q^{\mp}(E q^{\frac{1}{4}})}+ \frac{Q^{\mp}(E q^{\frac{3}{4}})Q^{\mp}(E q^{-\frac{3}{4}})}{Q^{\mp}(E q^{\frac{1}{4}}) Q^{\mp}(E q^{-\frac{1}{4}})}+{\rm e}^{\mp \frac{\pi}{2}i}\frac{Q^{\mp}(E q^{-\frac{5}{4}})}{Q^{\mp}(E q^{-\frac{1}{4}})}
\rho = - \frac{1}{2}\log \left[ \left( \frac{\pi T e^{2}}{\sqrt{2 p_{+}p_{-}}}\right)^{2}k_{( + )}k_{( - )}\frac{\partial_{+}b \partial_{-}\overline{{b}}}{\left( 1 + b \overline{{b}}\right)^{2}}\right]
\mathcal{A}_{SG}=\int_{-\infty}^{0}dx \int_{-\infty}^{\infty}dt~\frac{1}{4\pi}(\partial_{\mu}\varphi)^2+2\mu \cos (2\beta \varphi)+2\mu_{B}\int_{-\infty}^{\infty}dt \cos \beta(\varphi(0,t)-\varphi_{0})\,,
\begin{aligned}F(\theta,\phi,\pi) =-F(-\theta,-\phi,0)=F(\theta,\pi-\phi,0), G(\theta,\phi,\pi) =G(-\theta,-\phi,0)=-G(\theta,\pi-\phi,0), H(\theta,\phi,\pi) =-H(-\theta,-\phi,0)=-H(\theta,\pi-\phi,0).\end{aligned}
\int_0^{\infty}y^{2}\nu_1(x,dy) =\int_0^{\infty}y^{2}|\nu_1(x,dy)-\nu_2(x,dy)+\nu_2(x,dy)| \leq\int_0^{\infty}y^{2}\nu_2(x,dy)+\int_0^{\infty}y^{2}|\nu_1(x,dy)-\nu_2(x,dy)|
\begin{aligned}\sum_{i=1}^{n}c_{p,r}(C_{i,n})\leq \left[\sum_{l=0}^{r}\left(N^{lp}C_{H}^{lp/2}\left(\frac{M_{r}}{c}\right)^{lp}\right)\right]^{1/p}\cdot\begin{aligned}\sum_{i=1}^{n}\left[P\left(\bigcap_{j=1}^{N}\{-\alpha_{j}-c\leq X_{j}\leq\alpha_{j}+c\}\right)\right]^{1/p},\end{aligned}\end{aligned}
\begin{array}{r c l}{}&{}&{{\cal V}= \{f \in{\cal L}( V , W ) \vert f_{| W}\mathrm{i s a s c a l a r o p e r a t o r}\},}\\{}&{}&{{\cal W}= \{f \in{\cal L}( V , W ) \vert f_{| W}= 0 \}.}\\ \end{array}
\begin{aligned}P_Y=-\frac{1}{2}\left(P\Omega_{[1]}\right)_X, &(\Omega_{[i]})_{XXX}-(\Omega_{[i]})_X=-P\left(P\Omega_{[i+1]}\right)_X, & P_T=\frac{(\Omega_{[n]})_{XXX}-(\Omega_{[n]})_X}{2P}=\Delta_X.\end{aligned}i=1,\dots, n-1,
\|w-\begin{pmatrix}x^2 0 0 y^2\end{pmatrix}\| = \| \begin{pmatrix}x^{\frac{1}{2}}yx^{\frac{1}{2}}x^{\frac{3}{2}}y^{\frac{1}{2}}+x^{\frac{1}{2}}y^{\frac{3}{2}}y^{\frac{1}{2}}x^{\frac{3}{2}}+y^{\frac{3}{2}}x^{\frac{1}{2}}y^{\frac{1}{2}}xy^{\frac{1}{2}}\end{pmatrix}\|<\delta,
\sum_{i\in I}v_{i}^{2}\langle \pi_{Z_{i}}C^{\prime}f,\pi_{Z_{i}}Cf \rangle =\Vert v_{i}(C^{*}\pi_{Z_{i}}C^{\prime})^{\frac{1}{2}}f \Vert^{2}\geq(A-\epsilon^{2})\Vert f\Vert^{2}
\mathcal{Z}_{m,n}^\gamma (z,\bar z) : = (\gamma+m+1)_n(1 - |z|^2)^{-\nu +m}\nabla^{\nu}_{m}\left( z^n (1 - |z|^2)^{\nu-m}\right) &=(-1)^{m+n}\left(1-\mid z\mid^{2}\right)^{-\gamma}\dfrac{\partial^{m+n}}{\partial z^{m}\partial \overline{z}^{n}}\left(\left(1-|z|^{2}\right)^{\gamma+m+n}\right),
\left( E^{C_2}_{*+|n\sigma| - n\sigma}\overset{u}{\longrightarrow}E^{C_2}_{*+|n\sigma| - (n+1)\sigma}\right) = \left( E^{C_2}_* \{1, \dots , u^n \}\overset{u}{\longrightarrow}E^{C_2}_{*+|n\sigma| - (n+1)\sigma}\right) & = \left( E^{C_2}_{*}\{u , \dots , u^{n+1}\}\to E^{C_2}_{*+|n\sigma| - (n+1)\sigma}\right),
\begin{bmatrix}0 2\xi_1- 1 1-2\xi_1 0\end{bmatrix}\begin{bmatrix}x_1 x_2\end{bmatrix}+\begin{bmatrix}0 -1\end{bmatrix}+\begin{bmatrix}2\lambda x_1 -\lambda\end{bmatrix}=\begin{bmatrix}0 0\end{bmatrix}.
M(s)=\sum_{m\sim M}\frac{a_m}{m^s},Z^\flat(s)=\sum_{n\asymp x/M}\frac{h(n)\chi(n)}{n^s}-\delta_\chi \frac{\varphi(q)}{q}H(1-s),\delta_\chi=\begin{cases}1, , 0, \end{cases}
\{I_1,I_2\}= \left(\frac{\partial I_1}{\partial q_1}\frac{\partial I_2}{\partial p_1}-\frac{\partial I_1}{\partial p_1}\frac{\partial I_2}{\partial q_1}\right)+\left(\frac{\partial I_1}{\partial q_2}\frac{\partial I_2}{\partial p_2}-\frac{\partial I_1}{\partial p_2}\frac{\partial I_2}{\partial q_2}\right)=0.
{\cal L}_{e f f}= I m \: [ \: \int d^{2}\theta \: d^{2}\bar{\theta}\frac{1}{2}K ( \Phi^{a}, \bar{\Phi}^{a}, V^{a}) + \int d^{2}\theta f_{a b}( \Phi ) \, W^{a}\, W^{b}+ c \: c \: ] ,
DG(a,b) =\left[\begin{array}{cc}\frac{\partial_{xx}\xi(a,b)}{\gamma}-\frac{1+a}{(1-a)^3}\frac{\partial_{xy}\xi(a,b)}{\gamma}\frac{\partial_{xy}\xi(a,b)}{1-\gamma}\frac{\partial_{yy}\xi(a,b)}{1-\gamma}-\frac{1+b}{(1-b)^3}\end{array}\right]
\tilde{\gamma}_{i}(\gamma_{5},\sigma_3,\sigma_2,\gamma_1,Z_{a}h,Z_{b}h,V_a) =O(1)|Z_a-Z_b|h+\tilde{\gamma}_{i}(\gamma_{5},\sigma_3,\sigma_2,\gamma_1,Z_{a}h,Z_{a}h,V_a) &=O(1)|Z_a-Z_b|h+O(1)Z_{a}h+\tilde{\gamma}_{i}(\gamma_{5},\sigma_3,\sigma_2,\gamma_1,0,0,V_a) &=\gamma_i+O(1)Z_{a}h+O(1)|\gamma_4|h.
\displaystyle D^G(x')^{1/2}\theta(x')=\sum_{y\in \mathcal{X}^M(x)}\sum_{N^G_M(y)}D^M(g^{-1}g_y^{-1}x'g_yg)^{1/2}\theta^M(g^{-1}g_y^{-1}x'g_yg)
\begin{array}{c}F_{1}=x_1+x_2,\F_{2}=2(x_0+x_1+x_2) begin{align*}1.5pt]G_{1}=x_0^{2}+x_0x_2+x_1x_2+x_2^{2},\G_{2}=x_0^{2}+x_0x_1+2x_1^{2}+x_2^{2}begin{align*}1.5pt]F=2x_0^{2}x_1+2x_0x_1^{2}+2x_1^{3}+2x_0x_1x_2+2x_1^{2}x_2+x_2^{3}.\end{array}
\mathbb{E}\left( ( \mathbb{P}_{n}^{(N)}(f)-P(f) )^{2}\right)-\frac{1}{n}\mathbb{V}\left( \mathbb{G}^{(N)}(f)\right) & =\left\vert \mathbb{E}\left( \mathbb{P}_{n}^{(N)}(f)\right)-P(f)\right\vert^{2}+\frac{1}{n}\mathbb{V}\left( \mathbb{R}_{n}^{(N)}(f)\right)+\frac{2}{n}\mathrm{Cov}\left( \mathbb{G}_{n}^{(N)}(f),\mathbb{R}_{n}^{(N)}(f)\right).
N =\frac{1}{2 \pi\alpha}\left[\frac{1}{2}\ln \left(\frac{k^2(t) r^{2\alpha}}{\Lambda(t)^{2\alpha}}\right) -1+\frac{1}{r^{2\alpha}/\Lambda(t)^{2\alpha}-1}\ln\left(\frac{k^2(t) r^{2\alpha}}{\Lambda(t)^{2\alpha}}\right )\right]
|D^{(k)}_t-D^{(\ell)}_s|^2 =|D^{(k)}_t|^2+|D^{(\ell)}_s|^2-2|D^{(k)}_t|\cdot|D^{(\ell)}_s|\cos\theta &=\left(|D^{(k)}_t|-|D^{(\ell)}_s|\right)^2 +2|D^{(k)}_t|\cdot|D^{(\ell)}_s|\left[1-\cos\theta\right].
\mathcal{D}_{1}=\sup_{R > 0}\left(\int_{|z| \geq R}|z^+|^{\frac{\sigma-l-1}{2}q}\,e^{-q\langle \rho,\, z^+ \rangle}e^{q|\rho|\frac{|z^+|}{2}}e^{-\frac{\xi}{2}|z^+|q}|z|^{-\beta q}dx\right)^{\frac{1}{q}}\left(\int_{|z| \leq R}|z|^{\alpha p(1-p')}dz\right)^{\frac{1}{p'}}<\infty.
\times \,\exp \left\{- \sum_{-\infty}^{+\infty}[{\bar\chi}_n^{(1)}(\not \! d + \omega_n + e \varphi)\chi_n^{(1)}+{\bar\chi}_n^{(2)}(\not \! d - \omega_n - e \varphi)\chi_n^{(2)}]\right\}\;.
R_0 =\{(s,t;m(e))=(t,s;m(e))\mid e=\{s,t\}\in E\}&\mathrel{\hphantom{=}}\cup\{\tilde\alpha_i v = (\tilde\alpha_i)_\#(v)\tilde\alpha_i\mid 0\le i\le 4, v\in V\}&\mathrel{\hphantom{=}}\cup\{\varepsilon_i^2 v = (\varepsilon_i)_\#^2(v)\varepsilon_i^2\mid 1\le i\le 4, v\in V\}&\mathrel{\hphantom{=}}\cup\{\iota v = v^{-1}\iota\mid v\in V\}.
Z = \beta^{N / 2}\Gamma ( ( N - 1 ) / 2 )^{- 1}{\LARGE \int_{0}^{\infty}}d \sigma{\LARGE \int_{- \infty}^{\infty}}d \sigma_{1}( 4 \beta^{2}\sigma^{2}+ ( 2 \beta \sigma_{1}+ v )^{2}+ m^{2})^{M}\sigma^{N - 2}e x p [ - \beta ( \sigma^{2}+ \sigma_{1}^{2}) ]
e_1=\begin{pmatrix}1 0 0\end{pmatrix},e_2=\begin{pmatrix}0 1 0\end{pmatrix},e_3=\begin{pmatrix}0 0 1\end{pmatrix}, R(\varphi)=\begin{pmatrix}1 0 0 0 \cos(\varphi) -\sin(\varphi) 0 \sin(\varphi) \cos(\varphi)\end{pmatrix}, E=\begin{pmatrix}1 0 0 0 1 0 0 0 -1 \end{pmatrix}.
\vartheta^{\Gamma}(g_{1}\delta_{q}g_{2}) = \vartheta^{\Gamma}(g_{1}\sigma_{q}(g_{2}) \delta_{q}) & = \vartheta(g_{1}) \ \vartheta \circ \sigma_{q}\circ \vartheta^{-1}(\vartheta(g_{2})) \ \vartheta^{\Gamma}(\delta_{q}) & = \vartheta^{\Gamma}(g_{1}\delta_{q}) \ \vartheta^{\Gamma}(g_{2}).
\left\vert \frac{\nu^{n}}{\lambda-\nu}\sum_{k=1}^{n}\tau_{k-1}\nu^{-k}\right\vert \leq\frac{M\left( 1-\left\vert \nu\right\vert^{n}\right)}{\left\vert \lambda-\nu\right\vert \left( 1-\left\vert \nu\right\vert\right)}=O(1),
Y\bigl(e^{z(1-zz_0)L_{1}}(1-zz_0)^{-2\deg}\omega,z_0/(1-zz_0)\bigr) =(1-zz_0)^{-4}Y\bigl(\omega,z_0/(1-zz_0)\bigr) =(1-zz_0)^{-4}L(z_0/(1-zz_0)).
\sum_{G(\mathcal{O}_{\mathbf{h}})\backslash\mathfrak{X}/G(\mathcal{O}_{\mathbf{h}})}\lambda_{\mathbf{f}}(\xi)\cdot\chi_{\mathbf{h}}(\det(r))|\det(r)|_{\mathbf{h}}^{-s}\mathbf{f}(g)=D(s,\mathbf{f},\chi)\mathbf{f}(g).
|c(s(\xi-i\eta))| \asymp\prod_{\alpha\in\Sigma_0^+}\left(1+|\langle \xi,s^{-1}\alpha_0\rangle|\right)^{-\frac{m_\alpha+m_{2\alpha}}2}&=\prod_{\alpha\in\Sigma_0^+}\left(1+|\langle \xi,\alpha_0\rangle|\right)^{-\frac{m_\alpha+m_{2\alpha}}2},
\partial_{q_{N_r}}\ln Z_L^{(n)}(\{q_{N_s}\})=\sum_{p=a,b}\int\limits_{C_p^{(r)}}dtW_{p,N_r}^{(o)}(t)+\sum_{p=a,b}\int\limits_{C_p^{(r)}}dt \tilde W_{p,N_r}(t)+\int\limits_{C_r}dt \hat W_{N_r}(t)+I_{(an)}
\begin{array}{ccc}1 = sf begin{align*}.5em]\left(Au\right)_{1}= u_{+}s\theta_{1}\frac{\partial f}{\partial\theta_{1}}begin{align*}.5em]\left(Au\right)_{2}= u_{+}s\theta_{2}\frac{\partial f}{\partial\theta_{2}}\vdots \left(Au\right)_{d-1}= u_{+}s\theta_{d-1}\frac{\partial f}{\partial\theta_{d-1}}\end{array}
\tilde{F}_{i}=[\begin{matrix}\vdots&\vdots&&\vdots\\ \tilde{f}_{i1}&\tilde{f}_{i2}&\cdot\cdot\cdot&\tilde{f}_{i|J_{i}|}\\ \vdots&\vdots&&\vdots\\ \end{matrix}]_{N\times|J_{i}|},
g_{1}(x_{1})[v_{\delta^{1}}+u_{\delta^{1,n}}g_{n}(x_{n})]=[g_{1}(x_{1}-x_{n})-g_{1}(-x_{n})]\sum_{\substack{\delta\in\{0,1\}^{n-1}, \delta_{1}=1}}v_{\delta}f_{n}^{\delta_{n}}(x_{n})\prod_{i=2}^{n-1}g_{i}^{\delta_{i}}(-x_{n})
\mu^{\prime \rho}= - \left( \partial_{\sigma}\eta^{* \alpha ( \rho )}\right) B_{\alpha}^{( \sigma )}+ \eta^{* \alpha ( \sigma )}\partial_{\sigma}B_{\alpha}^{( \rho )}+ \varepsilon^{\alpha \beta \gamma \delta}A_{\alpha \beta}^{* ( \sigma )}\partial_{\sigma}A_{\gamma \delta}^{* ( \rho )},
(\widehat{T}_{i}f)(x) = \alpha\delta f(x)+(1+\beta\gamma)f(s_{i}x)+\alpha\gamma\sum_{j=1}^{a_{i}(x)}f(s_{i}x+ja^{\vee}_{i}+v_{i+1}) +(\alpha\delta+\beta\gamma)\sum_{j=1}^{a_{i}(x)-1}f(s_{i}x+ja^{\vee}_{i})+\beta\delta\sum_{j=0}^{a_{i}(x)-1}f(s_{i}x+ja^{\vee}_{i}-v_{i+1}).
{\delta'}_{Q+S}x^{\alpha \beta}= 0\,, \qquad{\delta'}_{Q+S}\lambda^{4\alpha}= (x^{\alpha \beta}+ 2i A_{44}\lambda^{4 \alpha}\lambda^{4 \beta})\xi^4_\beta \,, \qquad{\delta'}_{Q+S}(1^I_i,2^I_i,3^I_i) =0 \, .
Z_{N, j}:= [C_{1, N}^{j}] - [C_{2, N}^{j}] - \Sigma_{i=j}^{N-1}[C_{3,i}^{j}] - \Sigma_{i=1}^{j-1}[C_{4,i}^{j}] + \Sigma_{i=1}^{j-1}[C_{5,i}^{j}] - \Sigma_{i=j}^{N-1}[C_{5,i}^{j}]
C_{i_1,\ldots,i_k,i'_1,\ldots,i'_{k'}}(1) = C_{i_1,\ldots,i_k}(1)C_{i'_1,\ldots,i'_{k'}}+ C_{i'_1,\ldots,i'_{k'}}(1)C_{i_1,\ldots,i_k}&+ \sum_{r=1}^k\sum_{s=1}^{k'}\widetilde{C}_{i_r,i'_s}(1)C_{i_1,\ldots,\widehat{i_r},\ldots,i_k}C_{i'_1,\ldots,\widehat{i'_s},\ldots,i'_{k'}}
\frac{r(|W_k|+q_k -2,{\mathbf x})}{|W_k|+q_k-2}\le \frac{|W_k| + q_k + q_{k-1}-2}{|W_k|+q_k -2}&= 1 + \frac{\eta_k}{t_k +1-\varepsilon_k}< 1 + \frac{\eta_k}{t_k +1}+ \varepsilon_k.
S_{L}[{\hat{\Phi}}] \ = \{\frac{4 \pi}{L + 1}}\mathrm{T r}[{\frac{1}{2}}({\hat{\cal L}}_{i}{\hat{\Phi}})^{2}+{\frac{r}{2}}{\hat{\Phi}}^{2}+{\frac{\lambda}{4 !}}{\hat{\Phi}}^{4}] \ .
\frac{a^3_{7,\ell}(\phi)}{(1-4 \alpha_2^2/\lambda_{\ell}^2)^{1/2}(1-4 \alpha_1^2/\lambda_{\ell}^2)^{1/2}}= 8^3 \frac{\gamma_4^3}{\ell^{4 \cdot 3}}+O(\frac{1}{\ell^3 \phi^3})
PL_{mk}= \begin{cases}-L-35\log_{10}(d_{mk}), d_{mk}>d_1 -L-10\log_{10}(d_1^{1.5}d_{mk}^2), d_0<d_{mk}\leq d_1 -L-10\log_{10}(d_1^{1.5}d_0^2), d_{mk}\leq d_0\end{cases},
\left\| \sum_{k_1 + k_2 + k_3 = k}^* \frac{(k_1 + k_2)u_{k_1}v_{k_2}w_{k_3}}{(k_1 + k_2 - c_1k)(k_1+k_2 - c_2 k )}\right\|_{X^{s_1,b-1}_1}\lesssim \|u\|_{X^{s,1/2}_1}\|v\|_{X^{s,1/2}_\alpha}\|w\|_{X^{s,1/2}_\alpha}.
\int_X|Ric(\omega(t))+\omega(t)|^2_{\omega(t)}\omega(t)^n &=\int_X(\partial_tS(t))\omega(t)^n+\int_X(S(t)+n)\omega(t)^n &=\partial_t\left(\int_XS(t)\omega(t)^n\right)+\int_X(S(t)+1)(S(t)+n)\omega(t)^n,
g_1(A) =t_{J_k(v)-1,q}-a &=(a-r-1)b+v a(J_k(v)-J_{k-1}(v)-1)-a &+\frac{v a(a-r)J_{k-1}(v)}{J_k(v)}+(v a J_{k-1}(v)+b J_k(v))\,.
f_{a(k-1)}(x) - f_{a(k)}(x) =\sum_{a(k-1) \leq \alpha < a(k)}\left( xq^{\alpha}f_{v(\alpha)}\left(xq^{Nw(\alpha)}\right) + dxq^{\alpha}f_{v(\alpha)}\left(xq^{N(w(\alpha)-1)}\right)\right).
\begin{array}{r c l}{v_{\mu}A_{\nu}^{a}= i \, \mathrm{s g n}( k ){\epsilon}_{\nu \mu \rho}{\partial}^{\rho}c^{a}}{,}{v_{\mu}c^{a}= 0 \, ,}{v_{\mu}{\bar{c}^{a}}= A_{\mu}^{a}}{,}{v_{\mu}B^{a}= - D_{\mu}c^{a}\, ,}\end{array}
|\mathcal{T}_i-\mathcal{T}_j|=\frac{|\mathcal{R}_i-\mathcal{R}_j|}{g_{ij}}[1+o(1)], \mbox{for}~ \delta\ll 1, \mbox{and}~ i\in \{1,\ldots,N\}, ~j\in \mathcal{N}_i.
\prod_{i}\frac{1-x_itu}{1-x_iu}=1+(1-t)\sum_{r=1}^\infty\, \sum_{i}\frac{x_i(1-x_i^{r})}{(1-x_i)}\prod_{i\ne j}\frac{x-tx_j}{x_i-x_j}\, (u^r-u^{r+1}),
\sum_{n =1}^{\infty}\frac{(b/a)_n a^n}{(1- c q^n) (b)_n}=\frac{( b/c )_{\infty}}{(b)_{\infty}}\sum_{n=0}^{\infty}\frac{(c)_n ( b/c)^n}{(q)_n}\sum_{m=1}^{\infty}\frac{a^m - b^m}{1- c q^{m+n}}.
\left( T^{- 1}\right)^{2}= v_{0^{\prime}}+ \vec{\gamma}\cdot \vec{v}= \gamma^{m}v_{m}, \quad \mathrm{{w i t h \, \, \, \,}}v_{0^{\prime}}^{2}- \vec{v}^{2}= 1 . \mathrm{{\,}}
F(x)=\mathop{\rm lim}_{\varepsilon\nearrow\frac{1}{M}}(f^{\varepsilon})_{\varepsilon}(x)=\mathop{\rm lim}_{\varepsilon\nearrow\frac{1}{M}}\mathop{\rm inf}_{z\in\mathcal{H}}\mathop{\rm sup}_{y\in\mathcal{H}}\{f(y)-\frac{|y-z|^{2}}{2\varepsilon}+\frac{|z-x|^{2}}{2\varepsilon}\}
T_{\alpha}\Phi_{\alpha}(z) \begin{pmatrix}0 0 1 \end{pmatrix}= \frac{2\pi}{\sqrt{3}}z^{-\gamma + \frac{1}{3}}\left( \mathbb{I}+ \mathcal O(z^{-1})\right)L_{\alpha}(z) \begin{pmatrix}0 0 1 \end{pmatrix}e^{-3 z^{\frac{1}{3}}}
D = e^{- \gamma_{5}\phi ( r ) \}i \not \! \partial \e^{- \gamma_{5}\phi ( r )}= \left( \begin{array}{c c}{{0}}&{{\sigma^{- 1}( \partial_{r}+ B )}}\\{{\sigma \ ( - \partial_{r}+ B )}}&{{0}}\\ \end{array}\right) ,
\begin{cases}|\xi_2 + \frac{N}{2}- \alpha| \leq 2^{30}A^{-1}M_0^{\frac{1}{2}}N, |\eta_2 - (\sqrt{2}+1)^{\frac{2}{3}}\xi_2+ C'(\alpha, M_0,N) | \leq 2^{55}A^{-1}N,\end{cases}