harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int d^{3}q \int_{0}d x \, x^{\alpha^{\prime}q^{2}- \alpha^{\prime}s - 1}= \pi^{3 / 2}( \alpha^{\prime})^{- 3 / 2}\int_{0}d x x^{- 1 - \alpha^{\prime}s}[ \ln ( 1 / x ) ]^{- 3 / 2}\ ,
\alpha_{0, 0}= \frac{1}{2}\left(4 r^4+23 r^3+53 r^2+58 r+30\right), \alpha_{1, 0}=-\frac{1}{2}\left(4 r^4+23 r^3+ 59 r^2+76 r+46\right), \alpha_{0, 1}= \frac{1}{2}\left( 3 r^2 + 9 r+11\right),\alpha_{1, 1}= -\left(r^2+3 r+4\right),\alpha_{2, 0}= \frac{1}{2}(r^2+3r+4)^2.
\mathcal W_{\theta(\mathbf h,\phi_{\mathbf h})\otimes \underline{\chi},B}(g_{\infty}) = \mathcal W_{\theta(\mathbf h,\phi_{\mathbf h}),B}(g_{\infty}) = C \mathcal W_{\mathbf F_{\chi},B}(g_{\infty}) = C \mathcal W_{\mathfrak R_M\mathbf F_{\chi},B}(g_{\infty}),
G_{I J}( \phi ) = -{\frac{1}{2}}{\frac{\partial}{\partial X^{I}}}{\frac{\partial}{\partial X^{J}}}( \ln{\cal V}) |_{{\cal V}= 1}\ , \qquad g_{i j}( \phi ) = G_{I J}\partial_{i}X^{I}\partial_{j}X^{J}|_{{\cal V}= 1}\ .
W^{\rho_\sigma}_{u_i}(\lambda)* W^{\rho_\sigma}_{u_j}(u_i)(x,t)= \overline{t}W^\rho_{u_i}(\lambda)\#W^\rho_{u_j}(u_i)(x) = \overline{t}C_{i,j}W^\rho_{u_j}(\lambda)(x) = C_{i,j}W^{\rho_\sigma}_{u_j}(\lambda)(x,t)
V ( T ) = \exp \Biggl \{\int_{0}^{T}d t \rho^{\mu}( t ) \gamma_{\mu}+ \frac{1}{2}\int_{0}^{T}d t_{1}\int_{0}^{t_{1}}d t_{2}\left[ \rho_{\mu}( t_{1}) \gamma^{\mu}, \rho_{\nu}( t_{2}) \gamma^{\nu}\right] + \ldots \Biggr \}.
\delta B \vert_{\Sigma_{2}^{i}}= - 2 \epsilon_{i}{\frac{T_{3}}{T_{6}}}\chi_{2}^{1}( N_{i}) - 2 \kappa_{1 1}^{2}T_{6}\chi_{2}^{1}( N^{\prime}) \vert_{\Sigma_{2}^{i}}, \i = 1 , 2 ,
\langle f, D_{k}(g) \rangle =\lim_{M\to \infty}\lim_{J\to \infty}\sum_{j=1}^{N_J}\int_{Q_j}D_k(f)(y)g(y)d\mu(y) &=\int_{\Bbb R^n}D_k(f)(y)g(y)d\mu(y) &=\langle D_k(f), g \rangle.
H_{e-ph}=\frac{1}{\sqrt{2N}}\sum_{k,s,q,v}\gamma(\alpha,q,k,v)\omega_{qv}(c_{k,s}^{\dagger}c_{k-q,s}a_{q,v}+c_{k-q,s}^{\dagger}c_{k,s}a_{q,v}^{\dagger})
-\frac{c_0}{d+2}\int_{\partial B_1^+}\phi \le \frac{c_0^2}{4}\Theta+\Big(\frac{1}{(d+2)\Theta}\int_{\partial B_1^+}\phi\Big)^2 &\le \frac{c_0^2}{4}\Theta+\frac{|\partial B_1|}{(d+2)\Theta}\int_{\partial B_1}\phi^2= \frac{c_0^2}{4}\Theta+8d^2\int_{\partial B_1}\phi^2.
C_2 : \qquad P_m=P_m^\circ:=\left(\begin{array}{cc}0 2^{m}2^{m+1}0 \end{array}\right), G_2 : \qquad P_m=P_m^\circ:=\left(\begin{array}{cc}0 3^{m}3^{m+1}0 \end{array}\right), F_4 : P_m=P_m^\circ:=\left(\begin{array}{cccc}0 0 0 2^{m}0 0 2^m 0 0 2^{m+1}0 0 2^{m+1}0 0 0\end{array}\right).
Y_{n}^{1,\ell}=S_{\Delta t}^{n-\ell}z_\ell+\Delta t\sum_{m=\ell}^{n-1}S_{\Delta t}^{n-m}F_1'(X_m).Y_{m}^{\ell}+ \sum_{m=\ell}^{n-1}S_{\Delta t}^{n-m}e^{\tau A}\bigl(\sigma'(X_m).Y_{m}^{\ell}\bigr)\Delta W_m &+\Delta t\sum_{m=\ell}^{n-1}F_{n,1}+\sum_{m=\ell}^{n-1}S_{\Delta t}^{n-m}e^{\tau A}\sigma''(X_m).(Z_m^1,Z_m^2) \Delta W_m
g_{i_0}\, e_{i_0 i_1}\, g_{i_1}\, e_{i_1 i_2}\, g_{i_2}\, \ldots \, g_{i_{k-1}}\, e_{i_{k-1}i_k}\, g_{i_k}& = e_{i_0 i_1}\, e_{i_1 i_2}\, \ldots \, e_{i_{k-1}i_k}& = e_{ij}.
P_{j}(z) = e^{\frac{\lambda}{\xi}z}z^{-(\frac{\mu j}{\xi}- 1)}(1-z)^{-\frac{(c-j)\gamma}{\xi}}\{-\frac{(c-j+1)\gamma}{\xi}k_{1}(z) -j\left( 1+\frac{\mu}{\xi}\right) \sum_{n=0}^{j}p_{j,n}k_{2}(z) + \frac{\mu}{\xi}\sum_{n=1}^{j}n p_{j,n}k_{3}(z) -\frac{\mu}{\xi}\sum_{n=1}^{j+1}n p_{j+1,n}k_{4}(z)\}.
d_{\mathcal{P},\pm}=\left[b (\mathsf{M}^{\mathbb{R}}_{1}+ \mathsf{M}^{\mathbb{Z}}) \mp \frac{1}{2\pi}\,\partial_{1}\right]+\left[ b\, \mathsf{M}^{\mathbb{R}}_{2}\pm \frac{1}{2\pi}\,\partial_{2}\right].
(N+\beta+1-x) Q_n(x)-(N-x) Q_{n}(x+1) =\frac{(n+\alpha +1)(n+\beta+1)}{\alpha+1}\hat Q_n(x), (x+1) \hat Q_n(x)-(\alpha +x+2) \hat Q_{n}(x+1) =-(\alpha+1) Q_n(x+1).
Z_D(t)=\frac{1}{(4\pi t)^{d/2}}\left\{|D| -\frac{\sqrt{\pi}}{2}|\partial D|\,t^{1/2}+\sum_{j=3}^N c{_j}(D) t^{j/2}+ \mathcal{O}(t^{{(N+1)}/2})\right\},
\sigma_{; \alpha}^{; \alpha}= 2 - \frac{1}{6}g_{\gamma \delta}R \sigma^{; \gamma}\sigma^{; \delta}+ \frac{1}{2 4}g_{\gamma \delta}R_{; \rho}\sigma^{; \gamma}\sigma^{; \delta}\sigma^{; \rho}+ O ( \sigma^{2}) .
0=\frac{{\partial^2_{y}{\partial^n_x u}}}{u^s_y +{u}_y}\bigg|_{y=0}= \partial_y g_n|_{y=0}+ 2(g_n \eta_2 - 2 (\partial^{-1}_y g_n )\eta_2^2)|_{y=0}\qquad+ \partial^{-1}_y g_n \left(\frac{u_{yyy}^s +{u}_{yyy}}{(u^s_y + \tilde{u}_y)}\right)\bigg|_{y=0},
\mathcal{T}:=\Big\{T\in [0,\delta \epsilon^{-2}]:\|\zeta_{\alpha}-1\|_{H^s}\leq 5\epsilon,~~\|\mathfrak{F}\|_{H^{s+1/2}}\leq 5\epsilon, ~~\|D_t\mathfrak{F}\|{H^s}\leq 5\epsilon,\forall ~t\in [0,T] \Big\}
\mathcal{L}_{\gamma_{coop}}(s) =\frac{1}{(1+\frac{s}{\lambda_{eq}})^{N_c-1}\prod_{n=N_c}^N(1+\frac{s N_c}{\lambda_{eq}n})}&=\frac{\frac{N!}{N_c!N_c^{N-N_c}}\lambda_{eq}^N}{(s+\lambda_{eq})^{N_c}\prod_{n=1}^{N-N_c}[s+(1+\frac{n}{N_c})\lambda_{eq}]},
[ N_{i},{\cal P}_{j}] = i \, \delta_{i j}\left( \kappa \sinh \frac{{\cal P}_{0}}\kappa - \frac{\vec{{\cal P}}\,{}^{2}}{2 \kappa}\, e^{{\cal P}_{0}/ \kappa}\right) , \quad [ N_{i},{\cal P}_{0}] = i{\cal P}_{i}\, e^{{\cal P}_{0}/ \kappa}.
\phi_{\Phi_k, \beta}(x, y)= \begin{cases}|\{n \leq x; \ \ \frac{\phi(n)}{n^{\delta}}\leq y\}| k , |\{n \leq x; \ \ \frac{\phi(n) \alpha_{k}(n)}{n^{\delta}}\leq y\}| k . \end{cases}
\mathcal{E}(\tilde{i},\tilde{j},\tilde{m},\mathcal{C}_n)=\Big\{\big(\mathbf{u}(\tilde{i}),\mathbf{v}(\tilde{i},\tilde{j},\tilde{m}),\mathbf{Y}\big)\in\mathcal{T}_\epsilon^{n}(p_{U,V,Y})\Big\},
\triangle \left( \hat{Q}_{\pm}\right) =\hat{Q}_{\pm}\otimes I + q^{\pm T}\otimes \left( \hat{Q}_{\pm}-\hat{\epsilon}_{\pm}I \right) \in U_{q}(\widehat{sl_2}) \otimes\mathcal{B}^{\epsilon}_q (\widehat{sl_2}) \ .
\chi_{p_2,q_2,q_1,q_0}=\Big[\prod_{j\in p_2}\chi_{1,j}^{2}\Big] \Big[\prod_{j\in q_2}\chi_{2,j}^{2}\chi_{1,j}^{1}\Big] \Big[\prod_{j\in q_1}\chi_{2,j}^{1}\chi_{1,j}^{0}\Big] \Big[\prod_{j\in q_0}\chi_{2,j}^{0}\Big]\,.
\Delta E_{00}^{*}=\sqrt{(\frac{\Delta L^{\prime}}{k_{L}S_{L}})^{2}+(\frac{\Delta C^{\prime}}{k_{C}S_{C}})^{2}+(\frac{\Delta H^{\prime}}{k_{H}S_{H}})^{2}+R_{T}\frac{\Delta C^{\prime}}{k_{C}S_{C}}\frac{\Delta H^{\prime}}{k_{H}S_{H}}}
F^{\textup{GW}}(y,q) = F(z,\tau) G^{\textup{GW}}(y,q) = G(z,\tau) \widetilde{G}^{\textup{GW}}(y,q) = - \frac{1}{2}\left( y \frac{d}{dq}\right) G(z,\tau) H^{\textup{GW}}(y,q) = \left( q \frac{d}{dq}\right) F(y,q)
\int_{0}^{\infty}e^{-\beta t}p_D(t,x_1,y_1)p_D(t,x_2,y_2) \d t \geq \int_{t_0}^{\infty}e^{-\beta t}p_D(t,x_1,y_1)p_D(t,x_2,y_2) \d t \geq c_1 \int_{t_0}^{\infty}e^{-\beta t}e^{-2\mu_1 t}\d t = \frac{c_2 e^{-(\beta +2\mu_1)t_0}}{\beta+2\mu_1}.
\sum_{j=1}^m\sigma_{jj}\frac{\partial^2g}{\partial s_j^2}(\mathbf{s}) = \frac{1}{2}f'(w)\sum_{j=1}^m(1-p_j)+f''(w)\sum_{j=1}^m(1-p_j)s_j^2 &=\frac{m-1}{2}f'(w)+wf''(w)-f''(w)\sum_{j=1}^mp_js_j^2.
{\rm char}_{A^\ell C^k|_{V^p}}(x)=\begin{cases}{(x^{p^{n-\nu_p(k)-1}}-1)}^{p^{\nu_p(k)}}, \nu_p(k)<n-1;{(x-1)}^{p^{n-1}-pj}{(x^p-1)}^{j}, \nu_p(k)\ge n-1.\end{cases}
c_1(\mathcal E_m') =3mL+D_1+D_2, c_2(\mathcal E_m') =3m^2L^2+2mL\cdot (D_1+D_2)+c_2(X) + D_1^2 + D_2^2 + D_1\cdot D_2, c_3(\mathcal E_m') =m^2L^2\cdot (D_1+D_2)+m L \cdot (D_1^2+D_2^2+D_1\cdot D_2) &+c_2(X)\cdot (D_1+D_2)-c_3(X) + D_1^3 + D_2^3 + D_1^2\cdot D_2 + D_1\cdot D_2^2.
(f(x_{j-1})-f(x_j))-((f(x_i)-l_{i-1}+1)-(f(x_i)+l_i-1)) &=(f(x_{j-1}-f(x_{i-1})+(f(x_i)-f(x_j))-l_i-l_{i-1}+2 &>(l_{i-1}-1)+(l_i-1)-l_i-l_{i-1}+2=0.
\|\mathcal{T}_t(f)\|_{A^p}\leq \frac{2^{\frac4p-1}}{t^{\frac2p}(1-t)^{\frac2p}}\left(\left(\int_{\mathbb{D}}\left|f(y)\right|^{p(m-1)}d\mu(y)\right)^\frac1{p(m-1)}\right)^{m-1}= 2^{\frac4p-1}t^{-\frac2p}(1-t)^{-\frac2p}\|f\|_{A^{p(m-1)}}^{m-1}.
\sigma_x=L_{\lambda^i\rho^j(e)}\rho= L_{(L_{\rho(e)}\rho^{-r})^i\rho^j(e)}\rho=L_{\rho^{j-ir}(e)}\rho= L_{\rho(e)}^{j-ir}\rho =(\lambda\rho^{r})^{j-ir}\rho =\lambda^{j-ir}\rho^{1+r(j-ir)}=\lambda^{j-ir}\rho^{1+jr-ir^2}
\lim_{t\to\infty}\frac{t^2-n(t)}{t\cdot\log^bt^2}= \lim_{n\to\infty}\frac{(\lambda_n+o(1))^2-n}{(\sqrt n+o(\sqrt n))\cdot\log^b(\sqrt n+o(\sqrt n))^2}= \lim_{n\to\infty}\frac{a\sqrt n\cdot\log^bn+o(\sqrt n)}{(\sqrt n+o(\sqrt n))\cdot\log^b(\sqrt n+o(\sqrt n))^2}=a.
\Psi_{k,t}^{\ast}(x)=e^{itx}(u_{k,k}^{\ast}(t)e^{i2\pi kx}+u_{k,-k}^{\ast}(t)e^{-i2\pi kx}+u_{k,-(k+1)}^{\ast}(t)e^{-i2\pi(k+1)x}+\varphi_{k,t}^{\ast}(x)),
2 \pi \int_{\Sigma}\langle \hat{\phi}( z )^{2}\rangle d^{2}z = 2 \pi \int_{\Sigma}{\cal G}( z , z ) d^{2}z = - \frac 1 2 \ln \det ( ( - \nabla_{\Sigma}^{2}+ m^{2}) \epsilon^{2}) .
\frac{N}{\beta}\log m(N^{-1/4}\beta) = \frac{1}{2!}\beta N^{1/2}\mu_2 + \frac{1}{3!}\beta^2 N^{1/4}\mu_3 + \frac{1}{4!}\beta^3 (\mu_4-3\mu_2^2) + O(N^{-1/4}).
\int_{\mathcal B_1^-(m)}(c,d)_p\chi_{\psi}(c) \chi_{\delta}(c)|c|^{-3/2}_p\mathbf h_p(h) dh = \sum_{j>-m}\mathrm{vol}(\mathcal R_{2j}) = (1-p^{-1})^2\sum_{j>-m}p^{-2j}= p^{2m-2}\frac{1-p^{-1}}{1+p^{-1}}.
F_{i}\mapsto x_{i i}\left( \sum_{j = 1}^{i}x_{j i}\partial_{x_{j i}}- \sum_{j = 1}^{i - 1}x_{j i - 1}\partial_{x_{j i - 1}}\right) - \sum_{j = i + 1}^{n}x_{i j}\partial_{x_{i + 1 , j}}+ \sum_{j = 1}^{i - 1}x_{j i}\partial_{x_{j i - 1}}+ \mu_{i}x_{i i}.
\max_{P(t), \t \in [0,D]}\int_0^D \frac{1}{2}\log\left(1 + P(\tau)\right)d\tau \mbox{s.t.}\int_0^{t}a e^{b \tau}P(\tau) d \tau \leq T_{\delta}e^{bt}, \forall t &\int_{0}^{t}P(\tau) d\tau \leq \sum_{i=0}^{h(t)}E_i, \forall t
\int_0^t |E(X(s),s)| \, ds\leq \sum_{i=0}^{n-1}\int_{t_i}^{t_{i+1}}\ |E(X(t),t)|\dt\leq & C T\left(P^{\frac43\gamma+\frac{\beta}{3}}+ P^{\frac53+\frac{\beta}{3}-\eta}\log P+P^{\beta-\frac{1}{3}-\gamma+\eta}\right).
h_{T,S}R_{T,S}= h_TR_T{\prod_{v\in S-S_{\infty}}R_v(\hat{T}) (\log N(v))^{r_v(\hat{T})}\frac{[H^0(\hat{\mathbb{Z}},H^1(I_v,\hat{T}))][Ext^{1}_{\hat{\mathbb{Z}}}(\hat{T}^{I_v},\mathbb{Z})]}{[H^1(K_v,T)]}}.
1-\xi_i^w(0) =\min_{-1<z_1<\ldots<z_{i-1}<1}\max_{\substack{p\in S p(z_1)=\ldots=p(z_{i-1})=0}}\frac{\int_{-1}^1(1-t)p(t)w(t)dt}{\int_{-1}^1p(t)w(t)dt}\geq \ge\frac{m}{M}\min_{-1<z_1<\ldots<z_{i-1}<1}\max_{\substack{p\in S p(z_1)=\ldots=p(z_{i-1})=0}}\frac{\int_{-1}^1(1-t)p(t)dt}{\int_{-1}^1p(t)dt}=\frac{m}{M}(1-\xi_i^u(0)).
\frac{\partial F}{\partial t}(x,t)=\alpha(\frac{\partial^{2}F}{\partial x_{1}^{2}}(x,t)+\frac{\partial^{2}F}{\partial x_{2}^{2}}(x,t)+\frac{\partial^{2}F}{\partial x_{3}^{2}}(x,t))+\beta Q(x,t)
I=-\frac{1}{16\pi G}\int d^3x\sqrt{g}(R-4(\partial\phi)^2 +2\Lambda e^{b\phi})+\frac{1}{8\pi G}\int d^2x \sqrt{h}K +\frac{1}{8\pi G}\int d^2x\sqrt{h}\frac{c_0}{l_{\rm eff}},
\partial_t\dot{\Delta}_j B-\mathrm{div}\big((\widetilde{\sigma}(\dot{S}_m a))\dot{\Delta}_j\nabla B\big) &\quad=\dot{\Delta}_jg+\dot{\Delta}_j( v\cdot\nabla B)+\dot{\Delta}_j\mathrm{div}((\widetilde{\sigma}(a)-\widetilde{\sigma}(\dot{S}_m a))\nabla B)+\mathrm{div}([\dot{\Delta}_j,\widetilde{\sigma}(\dot{S}_m a)]\nabla B).
\begin{aligned}\tau^{(2)}(G) :=\lbrack\tau_0(G);(\lbrack\tau_0(H);\tau_1(H)\rbrack)_{H\in\mathrm{Lyr}_1(G)}\rbrack, \varkappa^{(2)}(G) :=\lbrack\varkappa_0(G);(\lbrack\varkappa_0(H);\varkappa_1(H)\rbrack)_{H\in\mathrm{Lyr}_1(G)}\rbrack,\end{aligned}
y^{(m)}_{r_1,\ldots,r_k}=(\log R)\frac{\varphi(VP_f)}{VP_f}\Bigl(\prod_{i=1}^k\frac{\varphi(r_i)}{r_i}\Bigr)F_{r_1,\ldots,r_k}^{(m)}+O\left(\frac{F_{max}\varphi(VP_f)\log X}{VP_fD_0}\right),
k \left( 1 6 z^{3}h_{0}^{\prime}( z ) + 8 z^{2}h_{0}( z ) - 4 z^{2}q^{\prime}( z ) + 2 z q ( z ) \right) = k \left( \frac{1}{6}z^{2}A^{\prime}( z ) - \frac{1}{1 2}z A ( z ) \right) .
\gamma_3 = 2 \alpha_1 \alpha_2 - 2\beta_3 = -2q^4s+12q^5s-4q^5s^2-4q^6-16q^6s+10q^6s^2+8q^7-4q^7s^2-2q^8+ 2q^8s \overset{(a)}{\le}-2q^4s+12q^5s \overset{(b)}{\le}4 q^4 \, ,
w ( l , \mu ) \: = \int_{- \infty}^{+ \infty}\! \! d p \int_{\mu}^{\infty}\! \! d x \mathrm{e}^{\textstyle - l p^{2}- \sqrt{x}l}= \frac{2 \sqrt{\pi}}{l^{5 / 2}}\mathrm{e}^{\textstyle - \sqrt{\mu}l}\left[ 1 + \sqrt{\mu}l \right] ,
\psi ( \rho , w ) ={\frac{1 - \rho}{\sqrt \rho}}\exp \left({\frac{\ln^{2}\rho}{2 \ln w}}\right) \prod_{n = 1}^{\infty}{\frac{( 1 - w^{n}\rho ) ( 1 - w^{n}/ \rho )}{( 1 - w^{n})^{2}}}\, .
\left(\frac{c_1+\lambda_1}{w_1}+\cdots+\frac{c_{|\mathcal{S}|}+\lambda_{|\mathcal{S}|}}{w_{|\mathcal{S}|}}\right)\frac{\left(\sum_{j \in \mathcal{S}}w_jd_j^{ne}\right)^2}{B}=\left(|\mathcal{S}|-1\right)\sum_{j \in \mathcal{S}}w_jd_j^{ne}.
D_{\mu}^{AC}J^{\mu\,C}= \partial_{\mu}J^{\mu\, A}-{ig\over 2}d^{ABC}[A_{\mu}^{B}, J^{\mu\, C}]_{\rm M}+{g\over 2}\,f^{ABC}\{A_{\mu}^{B}, J^{\mu\, C}\}_{\rm M}= 0
(b^2+c_1^2-d^2)\cos^2\theta+(c_2^2-d^2)\sin^2\theta+2c_1c_2\cos\theta\sin\theta+d^2 = \left(\frac{b^2+c_1^2-c_2^2}{2}\right)\cos 2\theta+c_1c_2\sin 2\theta+\frac{b^2+c_1^2+c_2^2}{2}\leq \sqrt{\left(\frac{b^2+c_1^2-c_2^2}{2}\right)^2+c_1^2c_2^2}+\frac{b^2+c_1^2+c_2^2}{2}\leq 1.
\forall i=1,2\,,\,n \in [N_{k-1},N_k] :\:\max \left\{\left\| X^i_n - Y_{k-1}\right\| , \left\| X^i_n - Y_k \right\|\right\}\, \leq \,R_{k}\, .\footnote{Unless stated otherwise, the notation $\| \cdot \|$ is used to denote the${\ell}_{\infty}$norm.}
m_{A}^{\phantom{A}B}= \frac{1}{2 !}({\cal P}_{+}\Gamma^{M N})_{A}^{\phantom{A}B}C_{M N}+ \frac{1}{6 !}({\cal P}_{+}\Gamma^{M_{1}\ldots M_{6}})_{A}^{\phantom{A}B}Z_{M_{1}\ldots M_{6}}^{+}.
S_{int}=\frac{1}{g^2}\frac{p^2}{p-1}\left[\frac{2\pi p\ln p}{p^2-1}\right]^{\frac{1}{2}}\int d^{D-1}y\sum_{\ell=3}^{p+1}\frac{p\ !}{\ell\ !(p+1-\ell)!}\sum_{n_1}\cdots\sum_{n_\ell}A_{n_1n_2\dots n_\ell}\prod_{i=1}^\ell\psi_{n_i}(y),
H_{0}= \int d^{2}x_{\perp}d x^{-}\{{\frac{\imath}{2}}\bar{\xi}\gamma^{-}\stackrel{\leftrightarrow}{\partial}_{-}\xi +{\frac{1}{2}}( F_{1 2})^{2}-{\frac{1}{2}}a_{+}\partial_{-}\partial_{k}a_{k}\}
{}^{4}\omega_{\check{A}}( \tau , \vec{\sigma}){\frac{\buildrel o}{\Lambda ( \tau , \vec{\sigma})}}\longrightarrow S ( \Lambda ( \tau , \vec{\sigma}) ) \,{}^{4}\omega_{\check{A}}( \tau , \vec{\sigma}) S^{- 1}( \Lambda ( \tau , \vec{\sigma}) ) - \partial_{\check{A}}S ( \Lambda ( \tau , \vec{\sigma}) ) S^{- 1}( \Lambda ( \tau , \vec{\sigma}) ) .
\int(f_ng_n-fg) = \int(f_n-f)g+\int f(g_n-g) \leqslant \left\|f_n-f\right\|_{L^q}\left\|q_n\right\|_{L^p}+\left\|f\right\|_{L^q}\left\|g_n-g\right\|_{L^p}\rightarrow 0.
\sum_{\substack{1\le d < \infty (d,mnhk)=1}}W\left( \frac{cd}{Q}\right) = W\left(\frac{c}{Q}\right) +\frac{\phi(mnhk)}{mnhk}\int_{1}^{\infty}W\left( \frac{cx}{Q}\right)\,dx + \int_{1}^{\infty}W\left(\frac{cx}{Q}\right)\,dE.
\frac1{|Q|}\int_Q b^{-}(x)dx \le \frac1{|Q|}\int_Q |M_Q(b)(x)- b(x)| &\le \bigg(\frac1{|Q|}\int_Q |b(x)-M_Q(b)(x)|^q dx\bigg)^{1/q}&= |Q|^{\beta/n}\bigg\{\frac1{|Q|^{\beta/n}}\bigg(\frac1{|Q|}\int_Q |b(x)-M_Q(b)(x)|^q dx\bigg)^{1/q}\bigg\}&\le{C}|Q|^{\beta/n}.
\langle T_{d-i}^*VQ^\frac{1}{2}h,Q^\frac{1}{2}h' \rangle=\langle Q^\frac{1}{2}Ph,Q^\frac{1}{2}S_{d-i}h' \rangle= \langle QS_ih,h'\rangle=\langle T_iQ^\frac{1}{2}h,Q^\frac{1}{2}h'\rangle,
(\boldsymbol{\mathcal{A}}*\boldsymbol{\mathcal{V}}_n)_{i_1,i_2,:,:}=\sum_{k=1}^{N}\boldsymbol{\mathcal{A}}_{i_1,k,:,:}(\boldsymbol{\mathcal{V}}_n)_{k,i_2,:,:}=\sum_{k=1}^{N}\boldsymbol{\mathcal{A}}_{i_1,k,:,:}({V}_{i_2})_{k,:,:}=(\boldsymbol{\mathcal{A}}*{V}_{i_2})_{i_1,:,:}
e^{\mu\nu\sigma\tau}e_{\mu\kappa\lambda\rho}=-\left\vert\begin{array}[c]{ccc}\delta_{\kappa}^{\nu}\delta_{\lambda}^{\nu}\delta_{\rho}^{\nu}\delta_{\kappa}^{\sigma}\delta_{\lambda}^{\sigma}\delta_{\rho}^{\sigma}\delta_{\kappa}^{\tau}\delta_{\lambda}^{\tau}\delta_{\rho}^{\tau}\end{array}\right\vert ,
\sum_{\substack{m_1^2 +m_2^2 +p_3^2 +p_4^2 = n m_1,m_2,p_3,p_4 \in \mathcal{I}}}\rho_j(m_1)\rho_k(m_2) =\int_{0}^1 f_j(\alpha)f_k(\alpha)f_0(\alpha)^2e(-n\alpha) \, d\alpha.
H^{\prime}(I_{1},...I_{N},\phi_{1}...,\phi_{N})=\sum_{i=1}^{N}\omega_{i}I_{i}-\frac{K}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}\sqrt{I_{j}I_{i}}(I_{j}-I_{i})sin(\phi_{j}-\phi_{i}),
\mathcal{L}_{\mathcal I_{i,j}}(s) \exp\big(\int^{\infty}_{0}\mathbb{E}([1\exp({{s\upsilon_{i,j}P_ihr^{-\alpha}}})])\xi_{i,j}\lambda_{i,b}2\pi r dr \big), \exp(-\xi_{i,j}\lambda_{i,b}\pi \mathbb{E}(h^\sigma) \Gamma(1-\sigma)[s\upsilon_{i,j}P_i]^{\sigma}).
A_{777,\ell}= \int_{C/\ell}^{\pi -C/\ell}\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}}\sin \phi d \phi
I = \int_{\mathbb R_+^4}\left(\int \varphi(x)K(x,x_1,x_2)\,dx+\int \psi(X)K(X,x_1',x_2')\,dX\right) \,d\pi_t(x_1,x_1')\,d\pi_t(x_2,x_2') &\quad\leq \int_{\mathbb R_+^4}W_2^2\left(K(\cdot,x_1,x_2),K(\cdot,x_1',x_2')\right) \,d\pi_t(x_1,x_1')\,d\pi_t(x_2,x_2'),
\big(f_{0,d},f_{1,d},f_{2,d}\big) = \alpha_{d-\mu_1}\big(A_{0,\mu_1},A_{1,\mu_1},A_{2,\mu_1}\big)+\beta_{d,-\mu_2}\big(B_{0,\mu_2},B_{1,\mu_2},B_{2,\mu_2}\big),
\left| \frac{\gamma_2((j+1)/n;X) -2\gamma_2(j/n;X)+ \gamma_2((j-1)/n;X)}{2\gamma_2(1/n;X)}\right| \leq r(j), \frac{1}{n}\sum_{j=1}^n r(j)^2 \rightarrow 0, n \rightarrow \infty,
\int_{|z|\leq R}|z|^{-\beta q}dz \leq C \int_{\{H\in \overline{\mathbb{a}^{+}}:|H|\leq R\}}|H|^{-\beta q}|H|^{n-l}dH & \leq C\int_{0}^{R}r^{-\beta q}r^{n-1}dr & \stackrel{\beta<\frac{n}{q}}=CR^{-\beta q+n}\leq C.
H_{m n}\equiv H_{m n p}v^{p}={\frac{2 \delta L_{D B I}}{\delta H^{* m n}}}={\frac{H_{m n}^{*}-{\frac{1}{2}}t r ( H^{*})^{2}H_{m n}^{*}+ ( H^{*})_{m n}^{3}}{L_{D B I}}}.
\widehat{P}_{n}^{(\alpha , \beta )}(x)=-\frac{1}{2}(x-b)P_{n-1}^{(\alpha , \beta )}(x)+\frac{b P_{n-1}^{(\alpha , \beta )}(x)-P_{n-2}^{(\alpha , \beta )}(x)}{2n-2+\alpha+\beta}
(g(\xi_0,\xi),(\xi_0,\xi))=\Big(g\,g^{-1}\Big(\frac{dx_0}{dt},\frac{dx}{dt}\Big),g^{-1}\Big(\frac{dx_0}{dt},\frac{dx}{dt}\Big)\Big) = \Big(\Big(\frac{dx_0}{dt},\frac{dx}{dt}\Big),g^{-1}\Big(\frac{dx_0}{dt},\frac{dx}{dt}\Big)\Big)
F(X_u)X_{u,v}+ DF(X_u)\mathbb{X}_{u,v}& - F(X_u)f(X_u)(v-u) - F(X_u)\sigma(X_u)B_{u,v}^H - \{F(X_u)\sigma(X_u)\}'\mathbb{B}^H_{u,v}= o(|v-u|).
S_{\mathrm{Q t}}= \frac{\kappa}{2 \pi}\int d^{2}x \/ \sqrt{- g}\left[ \frac 1 4 R \psi - \frac{1}{1 6}( \nabla \psi )^{2}- \omega \left( \nabla \phi \right)^{2}- \frac{1 - \omega}{2}\phi R \right] .
\mathbf{X}=\left( \begin{array}{cccc}1 -1 i i -i -i 1 -1 1 1 -i i -i i -1 -1\end{array}\right), \quad\mathbf{Y}=\left( \begin{array}{cccc}-1 1 i i -i -i -1 1 -1 -1 -i i -i i 1 1\end{array}\right).
\varphi_r^*g = \frac{1}{(1-2r^2)^2}\begin{pmatrix}dx dy \end{pmatrix}B^t\begin{pmatrix}2-4w_1^2 4w_1w_2 4w_1w_2 2-4w_2^2\end{pmatrix}B\begin{pmatrix}dx dy\end{pmatrix},
({\mathfrak{S}}^{r,t}_{g,d})^{n-1}:= \left\{[C,x,y] : \exists \,\ell \in G^r_{d}(C) \,\,\mbox{with}\,\, h^0\left( C,\ell\otimes\mathcal{O}_C\left({}-x-(n-1)y \right) \right) \geq r+1-t \right\}.
x^{\prime}=\frac{k^{2}x}{x^{2}+y^{2}+z^{2}},y^{\prime}=\frac{k^{2}y}{x^{2}+y^{2}+z^{2}},z^{\prime}=\frac{k^{2}z}{x^{2}+y^{2}+z^{2}}
\Upsilon (x^\ast,\omega^\ast,y,\eta) & := I-\overline{\Gamma(x^\ast,\omega^\ast,y,\eta)}e^{2\pi i\omega^\ast(x^\ast-y)}M_{-(y-x^\ast)/\beta(\omega^\ast)}T_{(\eta-\omega^\ast)\beta(\omega^\ast)}D_{\beta(\omega^\ast)/\beta(\eta)}.
f^*_1 \le f^*_{2}\le \dots \le f^*_{\lfloor \frac{\xi}{2}\rfloor + 1}f^*_{j}\le f^*_{\xi+1-j}\ \ 1 \le j \le \tfrac \xi 2 t^*_1 \le t^*_{2}\le \dots \le t^*_{\lfloor \frac{d-c}{2}\rfloor + 1}t^*_{j}\le t^*_{d-c+1-j}\ \ 1 \le j \le \tfrac{d-c}2 \, .
U^m := \left(U_{1/n}, U_{2/n}, \ldots, U_{m/n}\right)^T &= \left(\log \left( \frac{\hat{\gamma}_p(1/n;X)}{\gamma_p(1/n;X)}\right), \log \left( \frac{\hat{\gamma}_p(2/n;X)}{\gamma_p(2/n;X)}\right), \ldots, \log \left( \frac{\hat{\gamma}_p(m/n;X)}{\gamma_p(m/n;X)}\right)\right)^T,
\mathbb{TS}(\mathbb C^2,m+1)= \operatorname{S}^{m+1}\mathbb{C}^2\oplus \operatorname{S}_{m,1}\mathbb{C}^2=\operatorname{S}^{m+1}\mathbb{C}^2 \oplus (\wedge^2\mathbb{C}^2\otimes \operatorname{S}^{m-1}\mathbb{C}^2).
\begin{aligned}\frac{d}{dt}\int \nabla^2 u \cdot \nabla^3 \varrho dx+\int |\nabla^3 \varrho|^2 dx &=\int |\nabla^2{\rm div}u|^2 dx+\int \nabla^2{\rm div}u \cdot \nabla^2(\varrho{\rm div}u+u\nabla \varrho)dx+\int \nabla^2 S_2 \cdot \nabla^3 \varrho dx &+\int [\mu \Delta \nabla^2 u+(\mu+\nu)\nabla^3{\rm div}u]\nabla^3 \varrho dx.\end{aligned}
\sum_{j = s+k_0}^{\infty}\sum_{I\in \mathcal N (j,s,t) \,:\, I\subset J}\lvert I\rvert \lesssim 2^{s}\sum_{j = s+k_0}^{\infty}\sum_{I\in \mathcal N (j,s,t) \,:\, I\subset J}\lVert B_{j-s}\mathbf 1_{I}\rVert_1 \lesssim{2^{s}}\lVert b \mathbf 1_{J}\rVert_1 \lesssim 2^{s}\lvert J\rvert.
0= \max_{\xi\in\partial\tilde{g}(z)}\xi^T f(z, \bar{u}(\bar{t})) = \min_{v\in U(z)}\max_{\xi\in\partial\tilde{g}(z)}\xi^T f(z, v) = \max_{\xi\in\partial\tilde{g}(z)}\min_{v\in U(z)}\xi^T f(z, v).
1!C(r,1,q) = -H(r,q) 2!C(r,2,q) = H(r,q)^2 - H(r,q^2) 3!C(r,3,q) = H(r,q)^3 - 3H(r,q)H(r,q^2) + 2H(r,q^3) 4!C(r,4,q) = H(r,q)^4 + 8H(r,q)H(r,q^3) - 6H(r,q)^2H(r,q^2) + 3H(r,q^2)^2 - 6H(r,q^4)
f \star g (\gamma) = \int_{\gamma_{2}\in G_{s(\gamma)}}f(\gamma\gamma_{2}^{-1})g(\gamma_{2}) = \int_{\gamma_{1}\in G^{r(\gamma)}}f(\gamma_{1})g(\gamma_{1}^{-1}\gamma) = \int_{(\gamma_{1},\gamma_{2})\in m^{-1}(\gamma)}f(\gamma_{1})g(\gamma_{2})
P_{z_0}( \prod_{j=0}^{n}\{Z_j = z_j\}, \tau> n) \ = \ \prod_{j=0}^{n-1}P(z_j,z_{j+1}) &\ = \ \lambda_c^n \prod_{j=0}^{n-1}\varphi_c(z_j)Q(z_j,z_{j+1}) \frac{1}{\varphi_c(z_{j+1})}& \ = \ \lambda_c^n \varphi_c (z_0) Q_{z_0}( \prod_{j=0}^{n}\{Z_j =z_j\}) \frac{1}{\varphi_c (z_{n})},
C\sum_{k=1}^\ell |(\tilde{q}-q,\phi_k)_{L^2(M )}^2\le \frac{\ell\lambda_\ell}{\epsilon}+ \ell e^{c\lambda_\ell^2}e^{c\epsilon}\|\mathcal{N}(\widetilde{q})-\mathcal{N}(q)\|^2
\tau_{r_{q_y}+1}^{v_{q_{y-1}}+1}\cdot \tau_{r_{q_y+1}}^{-(v_{q_y}+1)}=\tau_{r_{q_1}+1}^{v_{q_{y-1}}-v_{q_y}}i_{k_{p_0+1}}=v_{q_{y-1}}-v_{q_y}
E_{0}= \frac{\left| \mu_{e f f}^{2}\right|^{1 / 2}}{1 6 \pi^{2}r}\sum_{n = 1}^{\infty}\frac{1}{n^{3}}\left\{\eta^{3}\left[ \frac{H_{1}\left( \eta \right)}{2 \eta}- \frac{H_{2}\left( \eta \right)}{\eta^{2}}- \frac{H_{3}\left( \eta \right)}{2 \eta}+ \frac{\eta}{6 0 \pi}\right] - \eta N_{2}\left( \eta \right) + \eta^{2}N_{3}\left( \eta \right) \right\}_{\lambda = 1}.
\Gamma_{0}( b_{i}\to b_{f}+ \gamma ) ={\frac{2}{3 \pi}}\left({\frac{M ( N_{i})^{2}- M ( N_{f})^{2}}{2 \hbar M ( N_{i})}}\right)^{3}A_{H}( N_{i}) \;{\frac{2 J_{f}+ 1}{2 J_{i}+ 1}}\;{\frac{N_{f}}{N_{i}}}.
H =H_{x}+H_{y}&=-\frac{1}{2}D_{x}^{\mu_x}-\frac{1}{2}D_{y}^{\mu_y}+ \frac{1}{2}m^{2}x^{2}+\frac{\alpha_{x}+\beta_{x}R_{x}}{2 x^{2}}+ \frac{1}{2}n^{2}y^{2}+\frac{\alpha_{y}+\beta_{y}R_{y}}{2 y^{2}}.
H_{X}= \lim_{x \uparrow 1}H_{X}(R_{X}x) = \lim_{x \uparrow 1}\left( \, \log Z_{X}(R_{X}x)- P_{X}(R_{X}x) \, \right) = \lim_{x \uparrow 1}\left( \, \left(\, \log Z_{X}(R_{X}x)+ \log(-\log x) \, \right)- \left( \, P_{X}(R_{X}x)+ \log(-\log x) \, \right) \, \right) = \log C_{X}+ \gamma- B_{X}.
R_{l}^{\mathrm{W K B}}( r ) \propto \frac{1}{\sqrt{\kappa ( r )}}\left\{A e^{- \int_{r_{2}}^{r}d r^{\prime}\kappa ( r^{\prime})}+ B e^{\int_{r_{2}}^{r}d r^{\prime}\kappa ( r^{\prime})}\right\}, \kappa ( r ) \equiv \sqrt{2 \mu ( V_{\it e f f}( r ) - E )}
\omega^{(2,1)}(\xi)= \sum_{n=0}^{\infty}\left(\nabla I_{n}+\nabla^{2}I_{n,1}^{2}\right)\xi^{n}= (1-\xi)I(\xi)+(1-\xi)^{2}I_{1}^{2}(\xi) = (1-\xi)\left(I(\xi)-2I_{1}^{2}(\xi)+(3-\xi)I_{1}^{2}(\xi)\right),
K_n(q) = \frac{(q + 1) \prod_{i=0}^{n-1}(q^{2n+1-i}+ 1) \prod_{i=1}^{\lfloor n/2 \rfloor - 1}(q^{2n+1-2i}+ 1)}{\prod_{i=1}^{\lfloor n/2 \rfloor}(q^{2i+2}+ 1)}= (q^{2n+1}+ 1) (q^{n+2}+ 1) (q+1) \prod_{i=1}^{\lfloor n/2 \rfloor}\frac{(q^{2n+1-i}+ 1)(q^{2n-i}+ 1)^2}{(q^{2i+2}+ 1)}.
\partial_t r(t,u) = \frac{1}{2\gamma}\partial^2_{uu}\big[\tau(r, e)\big], \partial_t e (t,u) = \partial_u \big[\mathcal D(r,e) \partial_u \beta^{-1}(r,e) \big] + \frac 1{4\gamma}\partial^2_{uu}\left(\tau(r,e)^2\right).
I_{cc}=2 \int_{0}^{\infty}\frac{z^c_1(\beta)(z^c_1)'(\beta)z^c_2(\beta)}{(z^c_1(\beta)^2+z^c_2(\beta)^2)^2}d\beta = I_{cc}^{1}+ I_{cc}^{2}+ I_{cc}^{3}+ I_{cc}^{4},
x^2\frac{d^2u(x)}{dx^2}+x\frac{du(x)}{dx}-\left[\lambda^2-\frac{k^2\alpha x}{1-x}+\frac{k^2\beta x}{(1-x)^2}+\frac{\ell(\ell+1)x}{(1-x)^2}\right]u(x)=0,
T_1 = - C_*(p) \sum_{j> j_0}\sum_{m\ge j}p^{j(j-1)/2}q^{j-1}\xi_{m-j+1}p^m n^m\sum_{r=0}^{k-j}{k-j \choose r}p^{m(k-j-r)}q^{mr}e^{-n p^{k-j-r}q^r}.
(n+1)\sum_{k=0}^{N}\tbinom{N}{k}(-1)^{k}\mathfrak{d}_1\dot{\phi}(n+k) =\sum_{k=0}^N\tbinom{N}{k}(-1)^k\dot{\psi}(n+k) &+ N\sum_{k=0}^{N-1}\tbinom{N-1}{k}(-1)^k\mathfrak{d}_1\dot{\phi}(n+k+1),
\begin{array}{rl}\hskip 0.3cm\nabla_{X_1^h}Y_2^{v}= \nabla_{Y_2^v}X_1^{h}=\frac{c^2f_2^vY_2(f_2)^vX_1(f_1)^h}{(1+c^2(f_2^v)^2b_1^h)}(gradf_1)^h +\big(X_1(\ln f_{_{1}})\big)^hY_2^v, \end{array}
\mathfrak{F}_{\lambda}(e_i) = \frac{\dim(V_i)}{\dim(\mathcal{C})}\, \chi_{i^*}\mathfrak{F}_{\lambda}^{-1}(\chi_i) = \frac{\dim(\mathcal{C})}{\dim(V_i)}\, e_{i^*}(i = 0, \dotsc, m).
\mathcal{F}_n(u;-l) =\lim_{s\to -l}\frac{1}{\Gamma(s)(e^{2\pi i s}-1)}\int_{C_\varepsilon}t^{-l-1}\mathcal{G}_n(u,t)dt &=\sum_{m=0}^\infty\sum_{j=0}^{n}{n \brack j}B_l^{(-m-j)}(n)\,\frac{u^m}{m!}.