harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac 1{q'}\left(\sum_{i'=1}^{i-1}k_{i'}+ j\right)\leq\sum_{i'=1}^{i-1}\sum_{j'=1}^{k_{i'}}\alpha_{V_{i',j'}(G)}+\sum_{j'=1}^j\alpha_{V_{i,j'}(G)}\leq \frac 1{q'}\left(\sum_{i'=1}^{i-1}k_{i'}+ j\right) +\alpha_{\max}(G).
H_{h_\alpha}T_gf = \frac{1}{h_\alpha}H(h_\alpha T_gf) &= \frac{1}{h_\alpha}H(\exp(-\alpha(z)\rangle) T_g(h_\alpha f)) &= \frac{1}{\exp(\alpha(z)) h_\alpha}T_gH(h_\alpha f) &= T_g H_{h_\alpha},
d s_{1 1}^{2}= h_{0}^{- 2 / 3}( - d t^{2}+ d x_{1}^{2}+ d x_{2}^{2}) + h_{0}^{1 / 3}( d x_{3}^{2}+ \cdots + d x_{9}^{2}+ d x_{1 1}^{2}) ,
\int_0^{\Delta}\left(1-(1-s)^p\right)^{\frac{n-1}{p}}\mathrm{d}s=\int_0^{\Delta}\left(p(1-\sigma(s))^{p-1}s\right)^{\frac{n-1}{p}}\mathrm{d}s\leq \int_0^{\Delta}(ps)^{\frac{n-1}{p}}\mathrm{d}s= \frac{1}{n-1+p}(p\Delta)^{\frac{n-1+p}{p}}
\Gamma \lbrack A \rbrack = \int_{0}^{\infty}{\frac{d T}{T}}\mathrm{e}^{- m^{2}T}\int{\cal D}x \mathrm{e x p}\left[ - \int_{0}^{T}\! \! \! d \tau \left({\frac{1}{4}}{\dot{x}}^{2}+ i e A_{\mu}\dot{x}^{\mu}\right) \right]
\varphi'=\left(e_{n_{j-1}+1}^{\varphi_{n_{j-1}+1}}\cdot \ldots \cdot e_{n_j}^{\varphi_{n_j}}\right)^{-1}\varphi =e_{n_j+1}^{\varphi_{n_j+1}}\cdots e_{n_\ell}^{\varphi_{n_\ell}}
e^1 = \begin{pmatrix}2 0 0 2 \end{pmatrix}, \,e^2 = \begin{pmatrix}0 1 1 -1 \end{pmatrix}, \,e^3 = \begin{pmatrix}2 2 2 0 \end{pmatrix}, \,e^4 = \begin{pmatrix}0 0 0 0 \end{pmatrix}.
U = \left( \begin{array}{c c c c}{{- \sqrt{\frac{r_{0}}{2 M}}}}&{{\sqrt{\frac{2 M}{r_{0}}}}}&{{0}}&{{0}}\\{{\sqrt{\frac{r_{0}}{2 M}}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{\sqrt{\frac{r_{0}}{2 M}}}}\\{{0}}&{{0}}&{{\sqrt{\frac{2 M}{r_{0}}}}}&{{\sqrt{\frac{r_{0}}{2 M}}}}\\ \end{array}\right) \, .
< \varphi ( y_{1}, t_{1}) \cdots \varphi ( y_{n}, t_{n}) > = \frac{< 0 |{\cal{T}}[ \varphi ( y_{1}, t_{1}) \cdots{\cal{D}}\cdots \varphi ( y_{n}, t_{n}) ] | 0 >}{< 0 |{\cal{D}}| 0 >}
{f^{[2]}\left(x_0,x_{1},x_2 \right)}:=\begin{cases}\frac{f^{[1]}(x_0,x_1) - f^{[1]}(x_1,x_2)}{x_0 - x_2}, \text{if~$x_0\neq x_2$}, \frac{d}{dx_0}f^{[1]}(x_0,x_1), \end{cases}, \ \x_0, x_1,x_2\in\mathbb{R}.
(\frac{\partial}{\partial t}+\widetilde{\Box}^{\pm}_{b,m})(e^{-t\widetilde{\Box}^{\pm}_{b,m}}u)=0,\ \ \forall u\in\Omega^{0,\pm}(X,E),\ \ \forall t>0, &\lim_{t\rightarrow0^+}(e^{-t\widetilde{\Box}^{\pm}_{b,m}}u)=Q^{\pm}_mu,\ forall u\in\Omega^{0,\pm}(X,E).
\partial_x \left(A_t^{-1}(x)\right) = \frac{1}{1-\gamma_x\left(A_t^{-1}(x),t\right)}, \partial_x^2 \left(A_t^{-1}(x)\right) = \frac{\gamma_{xx}\left(A_t^{-1}(x),t\right)}{\left(1-\gamma_x\left(A_t^{-1}(x),t\right)\right)^3},
\left( \begin{array}{cccc}(E-m) 0 -P_{z}-\pi_{-}0 (E-m) -\pi_{+}P_{z}-P_{z}-\pi_{-}(E+m) 0 -\pi_{+}P_{z}0 (E+m)\end{array}\right) \left( \begin{array}{c}\varphi_{+}\varphi_{-}\chi_{+}\chi_{-}\end{array}\right) =0
L=\frac{1}{2}L_{1}\frac{dQ_{1}}{dt}^{2}+\frac{1}{2}L_{2}\frac{dQ_{2}}{dt}^{2}+m\frac{dQ_{1}}{dt}\frac{dQ_{2}}{dt}-\frac{Q_{1}^{2}}{2C_{1}}-\frac{Q_{2}^{2}}{2C_{2}}
r_i(x) = \Im\left[a_{t_ii}(x)^{\left(\frac{2\ell_in_i}{m_i}-n_i + \frac{2n_i(m_i+t_i)-2t_i}{m_i}\right)}h_{k_ii}(x)^{-2}\exp\int \frac{2n_i}{m_i}\frac{a_{(t_i-1)i}(x)}{a_{t_ii}(x)}dx\right]
\sum_{z\in\mathbb{Z}^+}2\lambda_z^p \sin^2\theta_{f,z}=\sum_{z\in\mathbb{Z}^+}\left( 1-\frac{1}{\sqrt{\lvert\hat{g}\left(2\lambda_z\right)\rvert^2\left(\lambda_z^2-m^2\right)+1}}\right)\lambda_z^p\displaybreak[0] &\leq\sum_{z\in\mathbb{Z}^+}\lvert\hat{g}\left(2\lambda_z\right)\rvert^2 \lambda_z^{p+2}
B \hat{A}^{-1}\hat{A}_s \hat{A}^{-\top}B^\top = B (\hat{A}+ \hat{A}^\top - A)^{-1}B^\top \le c \, B \hat{A}_s^{-1}B^\top \le c \, B A^{-1}B^\top \le c \, S \le c \, \hat{S}.
\sum\limits_{y\neq 0,x}[(2+h\chi_4(y)+\overline{h}\overline{\chi_4}(y))(2+h\chi_4(x-y)+\overline{h}\overline{\chi_4}(x-y))] &=4(q-3)-4h\chi_4(x)-4\overline{h}\overline{\chi_4}(x)-2\chi_4(g)\varphi(x)J(\chi_4,\chi_4)+2\chi_4(g)\varphi(x)\overline{J(\chi_4,\chi_4)}.
p_0^{(-1)}(x)= \frac{|\Omega|^{-1/2}}r \, , u_0^{(-1)}(x,s) = |\Omega|^{-1/2}\Big(\frac{x}{sr^3}- \chi\Big( \frac{\nu}r \Big)\, e^{-\nu\sqrt{s}}\, \Big( \frac{x}{sr^3}- \frac{x\cdot\nabla\nu}{sr^3}\, \nabla\nu\Big)\Big).
D^{(a)}(\underline{z}) = (1-|a\alpha|^2) |u^{(a)}_{\underline{z}}|^2 - \Big( 1-a\alpha-a\frac{a-\overline{\alpha}}{1-a\overline{\alpha}}\Big) |u^{(a)}_{\underline{z}}| - \Big( 1-a\overline{\alpha}-a\frac{a-\alpha}{1-a\alpha}\Big) |u^{(a)}_{\underline{z}}| + |u^{(a)}_{\underline{z}}|(1+a^2-a\overline{\alpha}-a\alpha).
\mathop{\lim}\limits_{\varepsilon_2 \to \varepsilon_1}\mathop{\sup}\limits_{(\xi,j) \in K\times S}P\left ({\|{u^{\varepsilon_2}\left({t,s,\xi ,j}\right)-u^{\varepsilon_1}\left({t,s,\xi,j}\right)}\|\ge \eta}\right) = 0.
\sum_{\substack{a\leq\sqrt{\rho N}(a,n)=1}}\mu^2(a)\!\!\!\!\!\sum_{\substack{b\leq N/a (a,b)=1 b\equiv\psi(a)\mod{n}}}\!\!\!\!\!\mu^2(b) =\frac{CN}{n}\prod_{p\mid n}\left(\frac{p^2}{(p-1)(p+2)}\right) &\cdot\left(\frac{\log N}{2}+\frac{\log \rho}{2}+\gamma+3\kappa+\sum_{p\mid n}\frac{\log p}{p+2}\right)+O(N^{3/4+\epsilon})
\left<{\cal O}\left[ \vec{\xi}{}\right] \right>_{\vec{\xi}}= \frac{\int_{\vec{\xi}( 0 ) = \vec{\xi}( 1 )}^{}{\cal D}\vec{\xi}e^{- S}{\cal O}\left[ \vec{\xi}{}\right]}{\int_{\vec{\xi}( 0 ) = \vec{\xi}( 1 )}^{}{\cal D}\vec{\xi}e^{- S}}.
s' = f^*_{\hat{\kappa}(f^{\mathrm{an}}(\zeta))}(s) \otimes_{\hat{\kappa}(f^{\mathrm{an}}(\zeta))}1_{\hat{\kappa}(\zeta)}\in H^0(Y, f^*(L))_{\hat{\kappa}(\zeta)}\setminus \{0 \},
d(f_2)= \sum\limits_{\substack{d|N 1\leq d\leq \sqrt{N}}}\frac{\varphi(\gcd(d,N/d))}{\gcd(d,N/d)}\max{\left(\frac{N}{d}(N-2),\left(\frac{N}{d}-d\right)(N-1)\right)}& + \sum\limits_{\substack{d|N d\geq \sqrt{N}}}\frac{\varphi(\gcd(d,N/d))}{\gcd(d,N/d)}\frac{N}{d}(N-2).
\{I_1\}\cup \emptyset \cup (B|_{I_1 \cap I_2})_{\rm max}\cup \{\{2\}\}\cup(B|_{(I_1 \triangle I_2) \setminus \{i_1, i_2\}})_{\rm max}&=\{\{1, 2, 3, 4\}, \{3\}, \{4\}, \{2\}, \{2, 5\}\}
F_1(a,b) =(1-a)^2\left(h_1^2+\frac{\partial_x\xi (a,b)}{\gamma}\right), F_2(a,b) =(1-b)^2\left(h_2^2+\frac{\partial_y\xi(a,b)}{1-\gamma}\right), F(a,b) =\left(F_1(a,b),F_2(a,b)\right)
\tilde{E}_{n\delta}=\sum_{\begin{array}{c}k_1p_1+...+k_mp_m=n 0<k_1<...<k_m\end{array}}\frac{\left ( q^{(\alpha,\alpha)}-q^{-(\alpha,\alpha)}\right )^{\sum_ip_i-1}}{p_1!\;...\;p_m!}(E_{k_1\delta})^{p_1}...(E_{k_m\delta})^{p_m}
(\rho_1^{r1},\rho_1^{r2};\sigma_1^{r1},\sigma_1^{r2};\psi_1)(\rho_2^{r1},\rho_2^{r2};\sigma_2^{r1},\sigma_2^{r2};\psi_2)(\rho_3^{r1},\rho_3^{r2};\sigma_3^{r1},\sigma_3^{r2};\psi_3)
\Gamma_{g . f .}+ \Gamma_{\phi \pi}\longrightarrow \Gamma_{g . f .}+ \Gamma_{\phi \pi}+ \chi \int \left\{\frac 1 2 \bar{c}B + ( \partial_{\xi}\xi_{A}) m \bar{c}\varphi_{2}- ( \partial_{\xi}\hat{\xi}_{A}) m \bar{c}\hat{\varphi}_{2}\right\}\; .
{\cal{A}}_{n o n r e l}^{c l a s s}\, = \, \cdots \int^{\infty}d^{3}k \left( \, \cdots \, \frac{i}{( k^{0}+ p^{0}) - ( \vec{k}+ \vec{p})^{2}/ 2 m}\, \right) \, \cdots
u_x=\frac{1}{\kappa}[s_1X_x\tilde{u}_{\tilde{x}}-r_2X_x\tilde{v}_{\tilde{x}}+r_2(s_{1x}v+s_{2x}u+s_{3x})-s_1(r_{1x}u+r_{2x}v+r_{3x})],
d_p(\nu,\nu')\leq\sqrt[p]{\sum_{1\leq i,j\leq L}\varrho^p(u_i,u_j)\cdot\pi(u_i,u_j)}<\sqrt[p]{L^2K^p \frac{\widetilde\varepsilon^p}{K^pL}}=\widetilde\varepsilon\leq\varepsilon.
E_D(\mu_1(T_1),\ldots,\mu_n(T_n))=[(1-p_{\epsilon(k_{1,1})})T_{k_{1,1}}\cdots T_{k_{1,q_1}}1_B] \otimes \cdots \otimes [(1-p_{\epsilon(k_{m,1})})T_{k_{m,1}}\cdots T_{k_{m,q_m}}1_B].
J^{\mu}= \left[ \left( 1 - \frac{\partial_{z}^{2}}{M_{r}^{2}}\right) \left( \not \! \partial \phi + i F \right) \right] \gamma^{\mu}\psi + \frac{1}{M_{r}^{2}}\, \delta_{1}^{\mu}\left\{- \partial_{z}( \! \not \! \partial \phi \not \! \partial \psi ) + i F \, \partial_{z}\! \not \! \partial \psi - i \, ( \partial_{z}F ) \! \not \! \partial \psi \right\}\; .
F_{(2)}= (q-1)(q+1) F_{(2^2)}= (q-1)^2 q \left(q+ (q+1) \right) F_{(2^3)}= (q-1)^3 q^3 \left(q^3+ 2q^2(q+1)+ q(q+1)^2+ (q^2+q+1)(q+1)\right) F_{(2^4)}= (q-1)^4 q^6 \left(q^6+ 3q^5[2] + 3q^4[2]^2 + q^3[2]^3 + 2q^3[3]! + 2q^2[2][3]! + q[3][3]!+ [4]!\right).
\left(1-q^{nN}\right) A_n = \sum_{j=1}^{r}\left( d^{j-1}\sum_{\substack{\alpha < a(r) w(\alpha)=j-1}}q^{\alpha}+ d^{j}\sum_{\substack{\alpha < a(r) w(\alpha)=j}}q^{\alpha}\right) (-1)^{j+1}A_{n-j}+ \sum_{j=1}^r \sum_{l=1}^r \sum_{k=0}^{\min(j-1,l-1)}c_{k,j}b_{l-k,j}q^{jN(n-l)}(-1)^{l+1}A_{n-l}.
\delta_{\epsilon}A_{\;\;\;(\sigma )}^{\alpha \beta}=\varepsilon^{\alpha\beta \gamma \delta}\partial_{\gamma}\epsilon_{\delta (\sigma )}\equiv \stackrel{(0)}{R}_{\;\;\;(\sigma )}^{\prime \alpha \beta \;\;\;\delta (\rho)}\epsilon_{\delta (\rho )},\;\delta_{\epsilon}B_{\alpha}^{\;(\sigma)}=0,
\|C(C\star C^2)\|_{L^1}\leq \|C^2\|^2_{L^1}\leq \frac{c}{m_t^4}{\bf 1}_{d=2}+ \frac{c}{m^2_t}{\bf 1}_{d=3}, &\|C(C\star C)\|_{L^1}\leq \|C^2\|_{L^1}\|C\|_{L^1}\leq \frac{c}{m_t^4}{\bf 1}_{d=2}+ \frac{c}{m_t^{3}}{\bf 1}_{d=3}.
\langle \chi_{T}( k ) | \chi_{T}( k^{\prime}) \rangle = \lim_{n \to \infty , n^{\prime}\to \infty}n^{2 k^{2}}{n^{\prime}}^{2{k^{\prime}}^{2}}\langle e^{i k^{\prime}. X \left( ( n + n^{\prime}- 2 ) \frac{\pi}{4}\right)}e^{i k . X ( 0 )}\rangle_{C_{n + n^{\prime}- 2}}.
\|(fg)_\Phi\|_{\dot{B}^{\sigma_1+\sigma_2-\frac{n-1}{p},\frac{1}{2}}_{p,1}}\lesssim \|f_\Phi\|_{\dot{B}^{\sigma_1,\frac{1}{2}}_{p,1}}\|g_\Phi\|_{\dot{B}^{\sigma_2,\frac{1}{2}}_{p,1}}.
I = \int dx [ \Omega^{\prime}\rho^{\prime}- \frac{\mu}{8}e^{2 \rho}\Omega^{1-\lambda -\gamma/4}-\frac{1}{4}\Omega f^{\prime 2}+\frac{1}{4}e^{\chi f-2 \rho}\Omega^{1+\gamma/4 -\epsilon/2}A^{\prime 2}] .
\begin{aligned}\Delta K_{m,\epsilon}= d_0 \left(\frac{\kappa_m d_1|q|^2}{(|q|^2+\epsilon)^{\kappa_m+1}}\right)= -\frac{\kappa_m(\kappa_m+1)}{(|q|^2+\epsilon)^{\kappa_m+2}}d_0|q|^2\wedge d_1|q|^2+\frac{8\kappa_m\beta_n}{(|q|^2+\epsilon)^{\kappa_m+1}}=:A+B.\end{aligned}
\begin{aligned}W_Y(z) =2\int_{w_-}^{w^+}\zeta'\mathrm{d}w'\frac{\rho_Y(\delta_U+w^{\prime 2})}{z-\delta_U-w^{\prime 2}}&=\frac{1}{4-q}\left(\frac{qt_2}{t_3}+2z\right)-f_{\mathrm{sing.}}\left(\sqrt{z-\delta_U}\right)-f_{\mathrm{sing.}}\left(-\sqrt{z-\delta_U}\right)\ ;\end{aligned}
m_{S}^{2}= \frac{V^{'^{\prime}}}{G^{'^{\prime}}}|_{x = 0}= \frac{\mathrm{e}^{G_{0}}}{G^{'^{\prime}}}\frac{( f^{'})^{2}}{G^{'^{\prime}}}|_{x = 0}= \alpha m_{3 / 2}^{2}
2 \sqrt{\pi}\sum_{J=2}^N U_{J b}\Phi^{(J)}_0 = n_b 2 \pi \; ,\; \; \; \; n_b \; \in \; \mbox{Z\hspace{-1.35mm}Z}\; \; ,\; \; \; \; \forall \; \; b \; = \; 1,2, \; ... \; N \; .
\{\hat{S}^{\mu \nu}, \hat{S}^{\lambda \rho}\}'=\Delta^{\mu \lambda}\hat{S}^{\nu \rho}-\Delta^{\nu \lambda}\hat{S}^{\mu \rho}+\Delta^{\nu \rho}\hat{S}^{\mu \lambda}-\Delta^{\mu \rho}\hat{S}^{\nu \lambda}
\begin{array}{ccc}\Delta (E_i)=E_i\otimes 1+K_i\otimes E_i, \varepsilon (E_i)=0, S(E_i)=-K_{-i}E_i, \Delta (F_i)=F_i\otimes K_{-i}+1\otimes F_i, \varepsilon (F_i)=0, S(F_i)=-F_iK_i, \Delta (K_{\alpha_i^{\vee}})=K_{\alpha_i^{\vee}}\otimes K_{\alpha_i^{\vee}}, \varepsilon (K_{\alpha_i^{\vee}})=1, S(K_{\alpha_i^{\vee}})=K_{-\alpha_i^{\vee}},\end{array}
\frac{d}{dt}\Big|_{t = 0}d_g(\exp_{x^{0}}(t z), \exp_{x^{0}_*}(t z_*)) = 0, \frac{d^2}{dt^2}d_g(\exp_{x^{0}}(t z), \exp_{x^{0}_*}(t z_*)) = O((d_g(x^{0}, x^{0}_*) + |t||z_*|_g) |z_*|_g^2) .
\int \|A\|^s d\mu_d(A) = \int \left\|A_1\ldots A_d\right\|^s d\mu(A_1)d\mu(A_2)\cdots d\mu(A_d) &\leq d^{1+sd}\left(\int\left\|BAB^{-1}\right\|^s d\mu(A)\right)\left(\int \|A\|^sd\mu(A)\right)^{d-1}&\leq d^{1+sd}C_{s,d}\left(e^{\mathbf{M}(\mu,s)}+\varepsilon\right)\left(\int \|A\|^sd\mu(A)\right)^{d-1}
\begin{aligned}w_d =\sum_{i_1=0}^{p-1}\cdots\sum_{i_8=0}^{p-1}(y_1^{p^s})^{i_1}(y_2^{p^s})^{i_2}\cdots (y_8^{p^s})^{i_8}h_{i_1i_2 \cdots i_8}(y_1^{p^{s+1}},y_2^{p^{s+1}},\cdots, y_8^{p^{s+1}}) &\in \mathbb{F}_p[y_1^{p^s},y_2^{p^s},\cdots, y_8^{p^s}]\backslash \mathbb{F}_p[y_1^{p^{s+1}},y_2^{p^{s+1}},\cdots, y_8^{p^{s+1}}] ,\end{aligned}
Y^2=\left(\begin{array}{cc}Y_1^2+Y_2 Y_3 Y_1 Y_2 + Y_2 Y_4 Y_3 Y_1 + Y_4 Y_3 Y_3 Y_2 +Y_4^2 \end{array}\right)=\left(\begin{array}{cc}I_p 0 0 I_{n-p}\end{array}\right).
\int_{\frac{1}{2}}^{1+\varepsilon_T}\big(T^{\max(1-\sigma,\varepsilon)/6}(T^{\max(1-\sigma,\varepsilon)/6}+|a|^{\max(1-\sigma,\varepsilon)/3})\big)\frac{T^{\sigma}}{T^{1/2}}d\sigma=\mathcal{O}\left(( |a|^{1/6}+1)T^{1/2+\varepsilon}\right).
<0|N|0>_{\rm loc}=\int{d^3\vec{q}\over (2\pi)^3}{1\over 2\omega(\vec{q})}\left[|J_{\mu}^{\rm cl}(\vec{q})|^2-{1\over M^2}\left|J_{\mu}^{\rm cl}(\vec{q})q^\mu\right|^2\right]
\int_{\mathcal{Y}^d}\widetilde{m}^{q-1}|\nabla_y\widetilde{m}|^2 dy \leq \int_{\mathcal{Y}^d}\widetilde{m}^{q+1}|\nabla_yV|^2dy\leq \sup_{x,y}|\nabla_yV |^2\int_{\mathcal{Y}^d}\widetilde{m}^{q+1}dy.
a_2' :=28\log((1.264)^3\beta_3)>60\log\left(1.264\cdot \frac{1}{\sqrt[3]{2}}\sqrt{c}\right)+8\log ((1.264)^3 \beta_3)\geq &\geq \varrho |\log \gamma_2|-\log |\gamma_2|+8h(\gamma_2).
\frac{\partial}{\partial t}(\begin{matrix}\rho\\j\\E^{t}\end{matrix})+\nabla\cdot(\begin{matrix}j\\ \frac{1}{\rho}j\otimes j+pI\\ (E^{t}+p)\frac{1}{\rho}j\end{matrix})=(\begin{matrix}0\\f\\ \frac{1}{\rho}j\cdot f\end{matrix})
\omega \|\psi_{>}\|_{L^2}^2 =\mathcal E_{\mu, \epsilon}(\psi, \psi, \psi, \psi_>) &=\mathcal E_{\mu, \epsilon}(\psi_<+\psi_\sim+\psi_>,\psi_<+\psi_\sim+\psi_>, \psi_<+\psi_\sim+\psi_>, \psi_>).
\frac{\partial}{\partial t_{\vec s,k}}(L(t,-z)\alpha, L(t,z)\beta) = \frac{1}{z}(\tilde{T}_{\vec s,k}\star_t L(t,-z)\alpha, L(t,z)\beta)-\frac{1}{z}(L(t,-z)\alpha, \tilde{T}_{\vec s,k}\star_t L(t,z)\beta) = 0
\delta^2 \geq \int_0^1 (e^{\phi(x)/2}- C_\alpha^{1/2}e^{-\alpha x/2})^2 \, dx = C_\alpha \int_0^1 e^{-\alpha x}\biggl(\frac{1}{C_\alpha^{1/2}}e^{\{\phi(x) + \alpha x\}/2}- 1 \biggr)^2 \, dx &\geq C_\alpha e^{-\alpha}\int_0^1 \biggl(\frac{1}{C_\alpha^{1/2}}e^{\{\phi(x) + \alpha x\}/2}- 1 \biggr)^2 \, dx.
\int_{C_i(A)}f(x) \,\mu(dx)= \displaystyle \displaystyle \int_{C_i(A)}f(x)h(x)\, dx= \displaystyle \displaystyle \int_{A}\displaystyle f\left(\frac{N}{y+i}\right)\,h\left(\frac{N}{y+i}\right)\frac{N}{(y+i)^2}\,dy.
f(y_{1},...,y_{n})=\frac{1}{(2\pi)^{n/2}\sqrt{det(\Sigma)}}exp(-\frac{1}{2}[y_{1}-\mu_{1},...,y_{n}-\mu_{n}]\Sigma^{-1}[y_{1}-\mu_{1},...,y_{n}-\mu_{n}]^{T})
A\left([0.1,0.2),\left\lfloor \frac{2 \times 10^J}{k}\right\rfloor; (x_n)_n \right) = \sum_{j=0}^{J}{\#\{n: 10^j \leq a_n < 2 \times 10^j \}}= \sum_{j=0}^{J}{\left(\left\lfloor \frac{10^j}{k}\right\rfloor + O(1) \right)},
\|\tfrac{d}{d\tau}|_{\tau=0}\Pi_+^{\psi_+ + \phi_+ + \tau (\zeta_++{\mathfrak a}_+),T}(v)\|_\delta^T \leq c \|v\|_\delta^T \cdot \|\zeta_++{\mathfrak a}_+\|_{*,\delta}\leq c(\|\psi_+\|_{*,\delta}+\|\psi_-\|_{*,\delta}+ \|\phi_+\|_{*,\delta}+ \|\phi_-\|_{*,\delta})\|\zeta_++{\mathfrak a}_+\|_{*,\delta}.
A\left( x\right) =\left( \sinh x-x\cosh x\right)^{2}\cosh x>0, B\left( x\right) =x\left( x\cosh x-\sinh x\right) \sinh^{2}x>0 C\left( x\right) =-2x^{2}\cosh x+x\sinh x+\cosh x\sinh^{2}x>0,
\langle L(\phi), \eta_{R_3,\xi_j}\varphi_{0,j}\rangle =\langle h, \eta_{R_3,\xi_j}\varphi_{0,j}\rangle+c_j \int_{\mathbb{S}^2_{\lambda}}\chi_{R_1,j}|\varphi_{0,j}|^2+c_0\int_{\mathbb{S}^2_{\lambda}}\chi_{R_1,j}\varphi_{0,j}.
\frac{n-2}{4n}\int_0^\infty\frac{(n+4t)^2}{(1+t)^{n-2}}t^{\tfrac{n}{2}}dt = \frac{n-2}{4n}\int_0^\infty\frac{(n-4)^2+8(n-4)(1+t)+16(1+t)^2}{(1+t)^{n-2}}t^{\tfrac{n}{2}}dt = \frac{n-2}{4n}\Big[(n-4)^2B(\tfrac{n}{2}+1,\tfrac{n}{2}-3)+8(n-4)B(\tfrac{n}{2}+1,\tfrac{n}{2}-4)+16B(\tfrac{n}{2}+1,\tfrac{n}{2}-5)\Big],
S_{E}^{v o l}= \sum_{a}\frac{V_{a}}{4 G_{d}}+ \frac{\beta}{1 6 \pi G_{d}}\sum_{a}\int_{M_{a}^{d - 2}}F \wedge \Psi - \frac{\beta}{8 \pi G_{d}}\int_{\partial \Sigma_{\infty}}J_{D}^{i}d \sigma_{i}.
M(t, D)= A_1 A_2 A_3\begin{pmatrix}1 1 1\end{pmatrix}+ \dfrac{1}{A_1 A_2 A_3}\begin{pmatrix}-A_1^2 A_2^2 A_3^2 A_1^2 -A_2^2 A_3^2 A_1^2 A_2^2 -A_3^2\end{pmatrix}D-\dfrac{1}{t}M_{-1}(D)
\tilde{\lambda}_\ell^{1,\pm}= \inf\{\int_{\Omega_\ell^{\pm}}A\nabla u.\nabla u \,: \, u\in H^1(\Omega_\ell^{\pm}),\, \int_{\Omega_\ell^{\pm}}u^2 = 1, u=0 \text{on}\gamma_\ell^{\pm}\cup\Gamma_\ell^\pm\}\,.
\Delta_{MAC}= \frac{1}{2}\log\left(\frac{1 + (1+\sqrt{2}+10)^2}{1 + 1+2+100}\right) \approx 0.29, \Delta_{BC}= \frac{1}{2}\log\left(\frac{1+1+2+100}{1+100}\right) \approx 0.02,
P_{\tilde{g}}\Phi=\Delta^2_{\tilde{g}}\Phi + \mathrm{div}_{\tilde{g}}\left(4S_{\tilde{g}}(\nabla\Phi,\cdot) \right) -\frac{(n-4)}{(n-2)^2}|\mathrm{Ric}_{\tilde{g}}|^2_{\tilde{g}}\Phi,
\left\langle (x-\alpha-j)_{j+1}p(x),q(x)\right\rangle_{\lambda,j,\ell}= \left\langle (x-\alpha-j)_{j+1}{\bf u}^{M}, p(x)q(x)\right\rangle begin{align*}2mm]= \left\langle p(x), (x-\alpha-j)_{j+1}q(x)\right\rangle_{\lambda,j,\ell}.
\underset{j}{\lim}\, \underset{i}{\lim}\ (\Gamma^{A,B,C}(u_iv_j)(X,Y)Z) = \underset{j}{\lim}\ (\Gamma^{A,B,C}(uv_j)(X,Y)Z) & = (\Gamma^{A,B,C}(uv)(X,Y)Z).
\P^{0,0}_{2T,{\rm cl}}(X(t)=x)= \begin{cases}\frac{\displaystyle{\binom{t}{(t+x)/2}\binom{2T-t}{\{(2T-t)+x\}/2}}}{\displaystyle{\binom{2T}{T}}}, \mbox{if $(t, x) \in \Lambda_{2T}$}, 0, \mbox{otherwise}.\end{cases}
\mathcal{I}_{1}\leqslant\frac{\sum_{\substack{n=1}}^{K}\mathbb{P}(S_{(n)}>\mathit{xy})\mathbb{P}(\eta=n)}{\sum_{\substack{n=1}}^{K}\mathbb{P}(S_{(n)}>x)\mathbb{P}(\eta=n)}\leqslant\max\limits_{\substack{1\leqslant n\leqslant K n\in{\rm supp}(\eta)}}\frac{\mathbb{P}(S_{(n)}>\mathit{xy})}{\mathbb{P}(S_{(n)}>x)},
\begin{aligned}\phi_x^+ =-\frac{E_A-E_{C}}{3\delta}\bar{\xi}^{A}_x-\frac{c_-}{3\delta}\bar{\xi}^{B}_x, \phi_x^- =\frac{E_A-E_{C}}{3\delta}\bar{\xi}^{A}_x+\frac{c_+}{3\delta}\bar{\xi}^{B}_x,\end{aligned}
A_{n,\nu}\left( t\right) :=\sup_{\widetilde{U}_{1:n}\leq s<\widetilde{U}_{t_{n}:n}}\frac{n^{\nu}\left\vert \widetilde{\alpha}_{n}\left(s;t\right) -\widetilde{B}_{n}\left( s\right) \right\vert}{s^{1/2-\nu}}=O_{\mathbb{P}}\left( 1\right) ,
\begin{cases}\big\langle \nabla_{e_1}A(e_2) -\nabla_{e_2}A(e_1)-A[e_1,e_2],e_1\big \rangle= \cos \phi \, J(e_2^h)(r), \big\langle\nabla_{e_1}A(e_2) \nabla_{e_2}A(e_1)-A[e_1,e_2],e_2\big \rangle= -(G-4r^2)\cos\phi \sin\phi &-J(\eta^h)(r)\sin\phi-J(e_1^h)(r)\cos\phi.\end{cases}
\prod_{p}^{}\left(1+\frac{b(p)}{p^{s}}+\cdots+\frac{b(p^{m})}{p^{ms}}+\cdots\right) &=\zeta(s-2)L(s,\chi)\prod\limits_{n=1}^{j}L(s-2,sym^{2n}f)L(s,sym^{2n}f\otimes\chi) &=G_{j}(s), (\mbox{say})
g_{1}(\nabla^{^{M_1}}_{Z}W,X) =\cos^{2}\theta g_{1}(\nabla^{^{M_1}}_{Z}W,X)-g_{1}(\mathcal{H}\nabla^{^{M_1}}_{Z}\omega\phi W,X) &+g_{1}(\nabla^{^{M_1}}_{Z}\omega W,\mathcal{B}X)+g_{1}(\nabla^{^{M_1}}_{Z}\omega W,\mathcal{C}X)+g_{1}(W,\varphi Z)\eta(X).
\begin{array}{l}{\widehat \phi}^{(n,m)}(\omega)={\widehat B}^{(n-2,m)}(\omega)\,F^{(m)}(\omega)\,,{\widehat \phi}^{(n,0)}(\omega) ={\widehat B}^{(n-2,1)}(\omega)\,F^{(0)}(\omega)\,,\end{array}
\overline{r}(\overrightarrow{\pi})=\sum_{k=1}^{t/2}k(\ell(\pi_{k})-\ell(\pi_{t+1-k})),\text{for}\overrightarrow{\pi}=(\pi_{1},\pi_{2},\cdots,\pi_{t})\in\mathcal{P}^t, \mathcal{\overline{P}}^t \mathcal{\hat{P}}^t.
\bigl| \partial^\alpha_x\, \partial^\beta_y\, K_\phi (x, y) \bigr|^2\le \bigl( \partial^\alpha_x\, \partial^\alpha_y\, K_\phi (x, y)\big|_{x=y}\bigr)\, \bigl( \partial^\beta_x\, \partial^\beta_y\, K_\phi (x, y)\big|_{y=x}\bigr)
i^G_{M_{S_1}}([M_{S_2}, \mu_{S_2}])(\pi)= (\mathrm{Ind}^G_{P_{S_1}}\circ [M_{S_2}, \mu_{S_2}] \circ ( \delta_{P_{S_1}}\otimes \mathrm{Jac}^G_{P^{op}_{S_1}}))(\pi) \otimes [1][| \cdot |^{\langle \rho_G, \mu_{S_1}\rangle - \langle \rho_G, \mu \rangle}],
(p-2) \Delta^N_{\infty,X}\zeta^k_j(X^k_j,Y^k_j,t^k_j) + \Delta_X \zeta^k_j(X^k_j,Y^k_j,t^k_j) \geq (p-2)\Lambda(\nabla_X^2 \zeta^k_j(X^k_j,Y^k_j,t^k_j)) + \Delta_X \zeta^k_j(X^k_j,Y^k_j,t^k_j) = (p-2)\Lambda(-H^X_k) + \textrm{tr}(-H^X_k) = (p-1)\Lambda(-H^X_k) + \sum^{m-1}_{i=1}\lambda_i(-H^X_k).
\bar u_p(x) = (d_g(x,p)^{-\mu + 1}+ a - d_g(x,p)^{-\mu + \delta})^{\frac{n-2}{\mu - 1}}0 < d_g(x,p) < r_1/2, \bar u_p(x) = 1 d_g(x,p) > r_1.
[ \pi_{\mu}, \pi_{\nu}] = 0 , \quad [ \pi_{\mu}^{\dag}, \pi_{\nu}^{\dag}] = 0 , \quad [ \pi_{\mu}, \pi_{\nu}^{\dag}] = i \hbar \partial_{\mu}\Gamma_{\nu}= i \hbar \partial_{\mu}\Gamma_{\nu}
P_t(A)v = P_t(\lambda)v = \sum_{i=0}^{\lfloor \frac{t}{2}\rfloor}\Theta (d^i) \cdot \lambda^{t-2i}v = \sum_{i=0}^{\lfloor \frac{t}{2}\rfloor}\Theta (d^i) \cdot \Theta(d^{\alpha(t-2i)})v = \sum_{i=0}^{\lfloor \frac{t}{2}\rfloor}\Theta(d^{t\alpha + i(1-2\alpha)})v.
\min_{\nu_1,\nu_2,\cdots,\nu_M}\Bigg\{\Bigg|\Bigg|\sum_{m=1}^{M}\nu_m \nabla_{\overline{\pmb{\theta}}}\hat{\mathcal{L}}^m(\overline{\pmb{\theta}})\Bigg|\Bigg|^2_2 \Bigg| \sum_{m=1}^{M}\nu_m = 1, \nu_m \geq 0 \Bigg\},
\sum_{\ell=\ell^+_*}^{L^+_j+1}\varepsilon^{-2}L^{-2d}_K 2^{-js(1+\alpha)}\ell^{-2\beta-1}\lesssim \varepsilon^{-2}L^{-2d}_K 2^{-js(1+\alpha)}\ell_*^{-2\beta}={\rm rhs}(\ref{intermediate}).
{{M}^{\kappa \mu s}_{\gamma \left| \theta \right.}}(s)=\frac{{{m}^{m}}{{\mu}^{\mu}}{{\left( 1+K \right)}^{\mu}}{{\left( \mu \left( 1+K \right)-\overline{\gamma}s \right)}^{m-\mu}}}{\left ( a(s)+b(s) \cos{\theta}\right )^{m}}
\|F_{il}(x_{il},u_l) - F_{il}(\tilde x_{il}, u_l)\| = \sqrt{| c_{il}'(g_{il}) - c_{il}'(\tilde g_{il})|^2+|p'_l(u_l)(s_{il}-\tilde s_{il})|^2}\cr \le \sqrt{L_{il}^2 |g_{il}- \tilde g_{il}|^2+\bar p_l^2 |s_{il}-\tilde s_{il}|^2}.
g_i^kg_j^l\,=\, g_{\sigma_i(j)}^{\alpha_k(l)}\,g_{\gamma_j(i)}^{\beta_l(k)}f_j^lf_i^k\,=\, f_{\gamma_j(i)}^{\beta_l(k)}\,f_{\sigma_i(j)}^{\alpha_k(l)}f_{\sigma_{\gamma_j(i)}(s)}^{\alpha_{\beta_l(k)}(m)}\,g_i^k(T_j^l)\;=\; g_{\gamma_{\sigma_j(s)}(i)}^{\beta_{\alpha_l(m)}(k)}\,\,f_s^m(T_j^l)
{\partial_{x}}\sigma (x,t)={\partial_{t}^{2}}u(x,t),\;\;x\in \mathbb{R},\;t>0, \sigma (x,t)+\tau \,{}_{0}^{C}\mathrm{D}_{t}^{\alpha}\sigma(x,t)=\varepsilon (x,t)+{}_{0}^{C}\mathrm{D}_{t}^{\alpha}\varepsilon(x,t),\;\;x\in \mathbb{R},\;t>0, \varepsilon (x,t)=\mathcal{E}_{x}^{\beta}u(x,t),\;\;x\in \mathbb{R},\;t>0.
\partial_{k}S_{k}^{I}[ \phi ] ={\frac{1}{2}}{\frac{\delta S_{k}^{I}[ \phi ]}{\delta \phi}}\partial_{k}G_{k}{\frac{\delta S_{k}^{I}[ \phi ]}{\delta \phi}}-{\frac{1}{2}}{\mathrm{T r}}\left[ \partial_{k}G_{k}{\frac{\delta^{2}S_{k}^{I}[ \phi ]}{\delta \phi \delta \phi}}\right] .
\propto\pi(\{y_{ij}\}_{i=1,j=1}^{N,M_{i}},\{\theta_{li}\}_{i=1,l=1}^{N,K},\sigma^{2},\{\alpha_{l}\}_{l=1}^{K},\{\beta_{lb}\}_{l=1,b=1}^{K,P},\{\omega_{l}\}_{l=1}^{K})
\mathfrak{U}=\left\{\left[\begin{array}{cc}M \chi_{\underline{\beta}}\underline{\beta}^{t}N\end{array}\right]\mid M\in M_{n}(\mathfrak{a}),\ (\eta M)^{t}+M=0,\ \underline{\beta}\in\mathcal{B}^{n},\N\in\mathfrak{U}(\chi)\right\},
C(\mathbb{HP}^n_q):=C(S^{4n+3}_q/SU_q(2)):= C(S^{4n+3}_q)^{SU_q(2)}:= \big\{b\in C(S^{4n+3}_q)\;|\; \delta_{C(S^{4n+3}_q)}(b)= b\otimes 1\big\}.
\lim_{k\rightarrow \infty}\int_{\mathbb{S}^n}\psi_kdg_0=\int_{\mathbb{S}^n}\psi_0 dg_0,\|\psi_0\|_{H^{\frac{n}{2}}(\mathbb{S}^n)}\leq \liminf_{k\rightarrow \infty}\|\psi_k\|_{H^{\frac{n}{2}}(\mathbb{S}^n)}.
=\Re\left\langle \Pi^{l}\overline{\Pi^{k}}\varphi,\Pi^{l}\Pi^{k}\varphi\right\rangle \geq-\frac{1}{2}(||\Pi^{l}\overline{\Pi^{k}}\varphi||_{2}^{2}+||\Pi^{l}\Pi^{k}\varphi||_{2}^{2}).
f_{\gamma_{i}}(\gamma_{i}) = \frac{1+K_{i}}{\overline{\gamma}_{i}}e^{-K_{i}-\frac{(1+K_{i})}{\overline{\gamma}_{i}}\gamma_{i}}I_{o}\left(2 \sqrt{\frac{K_{i}(1+K_{i})}{\overline{\gamma}_{i}}\gamma_{i}}\right)
\frac{1-2^{n-1}+3^{n-1}-4^{n-1}+5^{n-1}-6^{n-1}+\dots}{1-2^{-n}+3^{-n}-4^{-n}+5^{-n}-6^{-n}+\dots}= -\frac{1\cdot 2\cdot 3 \cdot \dots(n-1)(2^n-1)}{(2^{n-1}-1)\pi^n}\cos\left(\frac{n\pi}{2}\right) \frac{1-3^{n-1}+5^{n-1}-7^{n-1}+\dots}{1-3^{-n}+5^{-n}-7^{-n}+\dots}= \frac{1\cdot 2\cdot 3 \cdot \dots (n-1)(2^n)}{\pi^n}\sin\left(\frac{n\pi}{2}\right).
0 = 4 a \left( \frac{d \bar{\phi}}{d A}\right)^{2}- \frac{2}{P ( A )}\frac{d P ( A )}{d A}\frac{d \bar{\phi}}{d A}- \frac{\bar{f}_{0}^{2}}{2 P^{2}( A )}+ \frac{2 \bar{Q}+ \mu e^{- 4 a \bar{\phi}}/ \bar{Q}}{4 P ( A )},
f_{\min,\Delta(n,r)}- f_{\min,\Delta_n}= g_{\min,\Delta(n,r)}- g_{\min,\Delta_n}\le \frac{(1+\epsilon){d \choose 2}}{r^2(x^*_{\min})^2}g_{\min,\Delta_n}= \frac{(1+\epsilon){d \choose 2}}{r^2(x^*_{\min})^2}(f_{\min,\Delta_n}- \hat f_{\min}).
\frac{(-q;q^3)_\infty(-q^2;q^3)_\infty}{(q;q^3)_\infty(q^2;q^3)_\infty}=\sum_{n\geq 0}\frac{q^{n^2}(-q;q)_{n}}{(q;q)_{n}(q;q^2)_{n+1}}
A(p) - 1 = \frac{-e^2\xi}{4\pi^2p^2}\int_0^\infty dq\, \frac{1}{A(q)}\left(\frac{p^2A(p) - q^2A(q)}{p^2-q^2}- \frac{p^2A(p) + q^2A(q)}{2pq}\ln\left|\frac{p+q}{p-q}\right|\right),
s = \sum_{Q}m_{Q}^{2}S_{Q}( p ) , \; r = \sum_{Q}m_{Q}^{2}R_{Q}( p ) , \; \tilde{s}= \sum_{Q}m_{Q}^{4}S_{Q}( p ) , \; \tilde{r}= \sum_{Q}m_{Q}^{4}R_{Q}( p )
1=B_{2k-1}^g=b_k \cdot b_{k+1}b_{k+2}\cdots b_{g-k}\cdot b_{g+1-k}c_{g+1-k}, \ 1\leq k\leq r; &1=B_{2k}^g=b_{k+1}b_{k+2}\cdots b_{g-k}\cdot c_{g-k}, \ 1\leq k\leq r.