harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\epsilon^2_{v}\geq \|V_l - \hat{V}\|^2_F \geq \|\mathbf{V}_{11}- \hat{\mathbf{V}}_{11}\|^2_F + \|\mathbf{v}_{21}- \hat{\mathbf{v}}_{21}\|^2_{2}= \|\check{E}\|^2_F + \|e\|^2_{2}.
\frac{\partial(M_{h(s)},\Lambda_{h(s)})}{\partial{s}}=(\sum_{\lambda=1}^{m}\frac{\partial t_{\lambda}}{\partial s}\frac{\partial M_t}{\partial t_{\lambda}},\sum_{\lambda=1}^{m}\frac{\partial t_{\lambda}}{\partial s}\frac{\partial{\Lambda_t}}{\partial t_{\lambda}})
\frac{d}{dy}\Re f_M(\Re q_M(\pm \theta) + iy; \theta) \begin{cases}< 0, y > \Im q_M(\theta), > 0, y \in (0, \Im q_M(\theta)), < 0, y \in (\Im q_M(-\theta), 0), > 0, y < \Im q_M(-\theta). \end{cases}
\Psi_{n}(1,t) = \int_{\mathbb{T}}\rho_{n}^{0}(y) - \langle \rho_{n}\rangle~dy - \int_{0}^{t}\rho_{n}u_{n}(1,s)~ds begin{align*}1ex] = - \int_{0}^{t}\rho_{n}u_{n}(1,s)~ds = - \int_{0}^{t}\rho_{n}u_{n}(0,s)~ds = \Psi_{n}(0,t).
\begin{array}{l}{{g_{\tau \tau}= \partial_{\tau}X^{M}\partial_{\tau}X^{N}G_{M N}= G_{0 0}= - e^{2 V}}}\\{{g_{\sigma \sigma}= \partial_{\sigma}\rho \partial_{\sigma}\rho \, G_{\rho \rho}+ G_{\theta \theta}= ( \rho^{\prime})^{2}e^{- 2 V}\Delta^{- 1}+ \rho^{2}\Delta}}\\ \end{array}
\sum_{t=-\infty}^{\infty}q^{(t+1)(\frac{1}{2}-s)}c_{t,l}(1) = -q^{-l(\frac{3}{2}-s)}\zeta_F(1) \frac{1-q^{\frac{1}{2}-s}}{1-q^{-\frac{1}{2}-s}}.
\Delta \Pi = \Phi_0 \textnormal{in}\mathfrak{N}\ \Leftrightarrow \ \Delta q \Pi = q\Phi_0 \textnormal{in}\mathfrak{N}_* \ \Leftrightarrow \k^{-1}((\Delta^{\#}q \Pi)^{-1}t (\Delta^{\#}q \Pi))k = (\Delta^{\#}q \Pi)^{-1}t (\Delta^{\#}q \Pi) \textnormal{in}G
{{f}_{{{I}_{\max}}}}\left({{I}_{\max}}\right) =\frac{d{{F}_{{{I}_{\max}}}}\left({{I}_{\max}}\right)}{d{{I}_{\max}}}&=\sum\limits_{n=1}^{N}{\prod\limits_{k=1,k\ne n}^{N}{{{f}_{{{I}_{n}}}}\left({{I}_{\max}}\right)}}{{F}_{{{I}_{k}}}}\left({{I}_{\max}}\right)
n=\int_{\{|v|>G\}}\frac{G^{\kappa_0}}{G^{\kappa_0}}f\,dv+\int_{\{|v|\leq G\}}f\,dv\leq \frac{1}{G^{\kappa_0}}\int_{\mathbb{R}^3}|v|^{\kappa_0}f\,dv+\|f\|_{L^\infty(\mathbb{R}^3)}(2G)^3.
D^+_n = \{X_n > 0 \},\,\,\, D^-_n = \{X_n < 0 \},\,\,\, D^0_n = \{X_n = 0 \}, \,\, n \geq 0, D_n = D^+_n (D^+_{n-1})^c \cup D^-_n (D^-_{n-1})^c \cup D^0_n(D^0_{n-1})^c, \,\, n \geq 1,
\hat{H}= \frac{\hat{p}^{2}}{2 M}- \frac{\Delta}{2}( | 1 > < 1 | - | 2 > < 2 | ) + \omega a^{\dagger}a + g ( a^{\dagger}e^{- i k x}+ a e^{i k x}) ( | 1 > < 2 | + | 2 > < 1 | )
\textnormal{(II)}= p\int (u_k^p+ku_k^{p-1}) (\nabla a,\nabla (\eta^2))\;dx-\int ((p-1)u_k^p+pku_k^{p-1}) (\nabla a,\nabla (\eta^2))\;dx \;\;\;\;+\int u \eta^2 ((p-1)u_k^p+pku_k^{p-1}) \;dx = \int u_k^p (\nabla a,\nabla (\eta^2))\;dx+\int u\eta^2 \left ( (p-1)u_k^p+pku_k^{p-1}\right )\;dx.
\begin{aligned}U=A_{1}^{\dagger}C_{1}+L_{A_1}S_1,\V=D_{1}B_{1}^{\dagger}+S_2R_{A_1},\X=A_{2}^{\dagger}C_{2}+L_{A_{2}}D_{2}B_{2}^{\dagger}+L_{A_{2}}U_{1}R_{B_{2}}, &Y=A_{3}^{\dagger}C_{3}+L_{A_{3}}D_{3}B_{3}^{\dagger}+L_{A_{3}}U_{2}R_{B_{3}},\Z=A_{4}^{\dagger}C_{4}+L_{A_{4}}D_{4}B_{4}^{\dagger}+L_{A_{4}}U_{3}R_{B_{4}},\end{aligned}
\begin{cases}\psi_t-\Delta \psi=\delta^* \sigma e^{\delta^* t},\ \psi\geq 0, |x|\in [c_k^- t, c_k^+t], \; t>T_1, \psi=0, |x|\in \{c_k^-t, c_k^+t\}, \; t>T_1, \psi=0, |x|\in [c_k^-t, c_k^+t], \; t=T_1.\end{cases}
\sum_{r = 0}^{\lfloor 10 \log \log \log T \rfloor}( 1 + \log \log T)^{r}e^{-(1-3 \varepsilon) V^2/ \log \log T}\ll (1 + 10 \log \log \log T) e^{10 \log \log \log T \log ( 1+ \log \log T)}e^{-(1-3 \varepsilon) V^2/ \log \log T}\ll e^{(\log \log \log T)^3}e^{- (1- 3 \varepsilon) V^2/ \log \log T}
\begin{aligned}\|\phi\|^2_{\mathcal H}= \int_{\mathbb{R}^2}|\nabla\phi|^2\,+\,\int_{\mathbb{R}^2}\Big[\,A_+\big({t^+}^2-{U^+}^2\big)-B\big({t^-}^2-{U^-}^2\big)\,\Big]|\phi^+|^2 &\,+\,\int_{\mathbb{R}^2}\Big[\,A_-\big({t^-}^2-{U^-}^2\big)-B\big({t^+}^2-{U^+}^2\big)\,\Big]|\phi^-|^2,\end{aligned}
x_3^2 d_u(C\langle 1, y_r, v_s y_s \rangle) = d_u(x^2 C\langle 1, y_r, v_s y_s \rangle) &\subset d_u(D_1/(v_t e_t) \{x_3^2 \}\oplus D_r \langle x_3^2 y_r \rangle )=\{0\},
\eta \geq \frac{d-1}{d^2}(1-\cos(\theta_0)) \geq \frac{1}{2d}\frac{\theta_0^2}{5}= \frac{(1-\lambda)^4\sigma^4}{10\cdot 2708^2d}\geq \frac{(1-\lambda)^8}{8\cdot 10^{11}d^7}.
\frac{1}{T_{n}-T_{n-k}}\ = \ \sum^{\infty}_{p=0}\frac{1}{T_{n+kp}}-\sum^{\infty}_{p=0}\frac{T_{n+kp}^{2}-T_{n+kp+k}T_{n+kp-k}}{T_{n+kp}\left(T_{n+kp}-T_{n+kp-k}\right)\left(T_{n+kp+k}-T_{n+kp}\right)}.
- z_i G_{h/2}= h \partial_i G_{h/2}, z_i z_j G_{h/2}= h^2 \partial_i\partial_j G_{h/2}- h \delta_{ij}G_{h/2}, - z_iz_jz_k G_{h/2}= h^3 \partial_i\partial_j\partial_k G_{h/2}- h\big(\delta_{ij}z_k + \delta_{ik}z_j + \delta_{jk}z_i\big)G_{h/2},
\mathcal{B}_2=-\frac{1}{8}\left(\frac{1}{8}-\frac{1+v^2}{15}+\frac{v^2(1+v^2)}{15(1-v^2)^2}\right),\mathcal{B}_4=-\frac{v^6(v^6-4v^4+5v^2-10)}{504(1-v^2)^4}+\frac{31}{16128}.
\mathrm{Tr}(\Omega(c)(1+D^2)^{1-\frac{z+p}{2}}) = \frac{2}{\Gamma\left(\frac{z}{2}\right)}\int_0^\infty s^{z-1}\mathrm{Tr}(\Omega(c)(1+D^2)^{1-\frac{p}{2}}e^{-s^2(1+D^2)})\,ds.
K_{p,q}:= \Big\{(z,w)\in\mathbb C^n\times\mathbb C^m:0<\sum_{j=1}^n|z_j|^{2p_j}=\sum_{j=1}^m|w_j|^{2q_j}<1\Big\}, L_{p,q}:= \Big\{(z,w)\in\mathbb C^n\times\mathbb C^m:\sum_{j=1}^n|z_j|^{2p_j}<\sum_{j=1}^m|w_j|^{2q_j}=1\Big\}.
-\beta\,E^\dagger(x_n)+\Omega(x_n) =\frac{\beta}{\tilde{\beta}}\,\bigl(-\tilde{\beta}\,E^\dagger(x_n)+\Omega(x_n)\bigr)+\left(1-\frac{\beta}{\tilde{\beta}}\right)\,\Omega(x_n) &\geq\frac{\beta\,\tilde{c}}{\tilde{\beta}}+\left(1-\frac{\beta}{\tilde{\beta}}\right)\,\Omega(x_n),
\sum_{m=1}^\infty \frac{\sin^2(\pi m y)}{m^2 - n^2/y^2}\frac{\sqrt{x + \sqrt{x^2 + m^2}}}{\sqrt{x^2 + m^2}}= - \frac{1}{2 x^{3/2}}{\rm Re}\int_1^\infty \frac{dz}{\sqrt{z^2 - 1}}\frac{\sqrt{1 + i \sqrt{z^2 - 1}}}{z^2 + n^2/(x^2 y^2)}\frac{1}{F(x,y,z)}
Q_{n+1}(x) = \frac{\sqrt{x}}{2}\sum_{j=0}^n{n\choose j}x^j (1- (-1)^{n-j}) x^{\frac12(n-j)}= \sqrt{x}\sum_{j=0}^n{n\choose j}x^j \delta_2^{n-j-1}x^{\frac12(n-j)}
\begin{gathered}e^{S}_{\pm}= \pm \frac{1}{4\sqrt{\beta}}[S] + \frac{1}{8\beta}[S]^2, e^{S_j}_{\pm}= \pm \frac{1}{4\sqrt{\beta}}[S_j] + \frac{1}{8\beta}[S_j]^2 . \end{gathered}
J_2^2 \le \iint_{{\mathcal{R}}_R}\big(v(x)-v(y)\big)^2\,\big(\tau(x)+\tau (y)\big)^2\,\,u_2(x)\,u_2(y)\,k(x-y)\,dx\,dy &\qquad\qquad\cdot\iint_{{\mathcal{R}}_R}\big(\tau(x)-\tau (y)\big)^2\,v^2(y)\,u_2(x)\,u_2(y)\,k(x-y)\,dx\,dy.
\hat{b}(s) := \min_{\omega \in \Omega}\biggl\{a_{10}a_{20}+ \sum_{t:\omega_t=1}a_{2p^t_2}\bigl(s_{1p^t_1}- s_{1p^{t-1}_1}\bigr) + \sum_{t:\omega_t=2}a_{1p^t_1}\bigl(s_{2p^t_2}- s_{2p^{t-1}_2}\bigr)\biggr\},
\begin{cases}\frac{-\frac{d}{dt}w_{t,1}(u)}{w_{t,1}(u)\int_{\mathbb{T}}\lambda(u,v)w_{t,3}(v)dv}=\exp\left(-F_W(t,u)+G_W(t,u)\right), &\frac{-\frac{d}{dt}\sum_{k=1}^2w_{t,k}(u)}{\psi(u)w_{t,2}(u)}=\exp\left(-G_W(t,u)+H_W(t,u)\right), &\frac{-\frac{d}{dt}\sum_{k=1}^3w_{t,k}(u)}{\phi(u)w_{t,3}(u)}=\exp\left(-H_W(t,u)\right)\end{cases}
[Conf_3(B)]=[B^3]-[x=y]-[x=z]-[z=x]+3[x=y=z]-[x=y=z]= &=[B_{\{\{1\},\{2\},\{3\}\}}]-[B_{\{\{1,2\},\{3\}\}}]-[B_{\{\{1,3\},\{2\}\}}]-[B_{\{\{1\},\{2,3\}\}}]+2[B_{\{\{1,2,3\}\}}] \,.
\int_\Omega \Big|\frac{q-\mu_h}{C\,h_K}\Big|^{p_{\mathcal T}'(\cdot)}\,dx =\sum_K \int_K \Big|\frac{q-\mu_h}{C\,h_K}\Big|^{p_\mathcal T'(\cdot)}\,dx\leq \frac{1}{C'}\sum_K\bigg( \Big(\int_K |\nabla q|^{p'(\cdot)}+1\Big)\,dx+h_K^{n+2}\bigg) \leq \frac{1}{C'}\bigg(\int_\Omega \Big(|\nabla q|^{p'(\cdot)}+1\Big)\,dx+1\bigg)=1,
\int_{V^\gamma_\eta}D_{V^\alpha}K_{V^{\gamma}_{\eta}}\big(l(z,w,\eta),z^{\prime};\bar{\zeta},\bar{\zeta^{\prime}}\big)\zeta^\mathbf a\phi_{\mathbf a,\mathbf b}(\zeta^\prime)dV(\zeta,\zeta^{\prime}) = \pi^{-1}\big(\gamma\cdot(\mathbf a+\mathbf 1)\big)\phi_{\mathbf a,\mathbf b}(z^{\prime})e^{(\gamma\cdot(\mathbf a+\mathbf 1))(w\bar\eta-|\eta|^2)}z^{\mathbf a}.
\langle b^1, \eta^1_k\rangle - \frac{1}{2}\langle \div_{x^1}(a^1), \eta^1_k\rangle - \langle \langle a^1, \eta^1_k\rangle, \nabla_{x^1}\log \Lambda_1\rangle = \sum_{j = 1}^n \gamma_j \langle \Psi^{j, 1}_k, \eta_k^1\rangle + \sum_{i, j = 1}^n \rho_{ij}\partial_{y_j}\log \Lambda_1 \langle \Psi^{i, 1}_k, \eta_k^1\rangle;
\partial_s a_s^t = a_1^t - a_0^t, \partial_t a_s^t = s(a_1 - a_{1}^{\prime}) +(1-s) (a_0 - a_{0}^{\prime}), \textrm{and}\partial_t \partial_s a_s^t = a_1 - a_0 - (a_{1}^{\prime}- a_{0}^{\prime}).
\square_j^{(2^{l}+p 2^{l-\Delta})}(\theta) = 1- \sqrt{\frac{2^l+p2^{l-\Delta}}{2^l+p2^{l-\Delta}+j+1}}e^{i\psi_{2^{l}+p 2^{l-\Delta}+ j}(\theta ) -i\psi_{2^{l}+p 2^{l-\Delta}}(\theta )-ij\theta}= 1- \left( 1 + \Theta \right) e^{i A_{j+ 2^{l}+p 2^{l-\Delta}}^{(2^{l}+p 2^{l-\Delta})}(\theta)},
\P (G_0v_{||E||^{1/3}}(d, 0, E + 2 |E|)) \leq \sum_{k=1}^{\infty}\P (G_0v_{||E||_1^{1/3}}(d, k ||E||_1^{1/3}, E)) \, \P \left( \sup_{j \geq 0}d |\widetilde{W}_{2 |E|_j}| \geq ||E||_1^{1/3}(k-1) \right).
\|\theta-\frac{1}{2}+(m_n^{(1)}+\ell q_n)\alpha\|\leq \left(\left[\frac{a_{n+1}+3}{2}\right]+\left[\frac{a_{n+1}}{6}\right]+\frac{1}{2}+\frac{1}{2q_n}\right)\|q_n\alpha\| \leq \begin{cases}(\frac{2}{3}a_{n+1}+2+\frac{1}{2q_n})\|q_n\alpha\|, a_{n+1}\geq 7 (a_{n+1}-\frac{1}{2}+\frac{1}{2q_n})\|q_n\alpha\|, 4\leq a_{n+1}\leq 6.\end{cases}
\frac{(-q;q^2)_\infty}{(q^2;q^2)_\infty}\sum_{j\geq 0}q^{3j^2+2j}(1-q^{2j+1}) = \sum_{j\geq 0}q^{3j^2+2j}\frac{(-q;q^2)_{j}}{(q^2;q^2)_{2j}}\frac{(-q^{2j+3};q^2)_\infty}{(q^{4j+4};q^2)_\infty}.
\Lambda(\mathsf{A},\mu) :=\lim_{n\to\infty}\frac{1}{n}\int \log\|A_{x_n}\cdots A_{x_1}\|d\mu\left[(x_i)_{i=1}^\infty\right] &=\inf_{n \geq 1}\frac{1}{n}\int \log\|A_{x_n}\cdots A_{x_1}\|d\mu\left[(x_i)_{i=1}^\infty\right]
\frac{1+\sum_{i=0}^hk_1(i)}{k_1(0)+r}a^{\widehat{Q}^{+r}}+\frac{1+\sum_{i=0}^hk_2(i)}{k_2(0)+s}b^{\widehat{Q}^{+s}}= 1 + \sum_{j=2}^h\left(\frac{k_1(j)}{k_1(0)+k_1(j)}a^{\widehat{Q}_j}+\frac{k_2(j)}{k_2(0)+k_2(j)}b^{\widehat{Q}_j}\right).
\sum_{\substack{0\leq e_1\leq\ldots\leq e_n}}{\rm vol}(K_{e_1,\cdots, e_n})Y^{e_1+\ldots+e_n}= \frac{1-Y}{1-q^nY}\prod_{i=1}^n\frac{1+q^iY}{1-q^{n+i}Y},
F_n= -\nabla a^n\cdot\nabla \bar{u}^n-u_F^n\cdot \nabla \bar{u}^n-u_F^n\cdot \nabla u_F^n-\bar{u}^n\cdot \nabla u_F^n-\bar{u}^n\cdot \nabla \bar{u}^n &+a^n\Delta u_F^n+(1+a^n)(B_F^n\cdot \nabla B_F^n+B_F^n\cdot \nabla \bar{B}^n+\bar{B}^n\cdot \nabla B_F^n+\bar{B}^n\cdot \nabla \bar{B}^n),
z^{(j)}_i=\left(\frac{(\theta-1)z^{(j+1)}_i+\sum_{l=1}^{q-1}z^{(j+1)}_l+1}{\theta+ \sum_{l=1}^{q-1}z^{(j+1)}_l}\right)^k, \ \ 1\leq i \leq q-1, \ \ 0\leq j \leq m-1
\begin{cases}a_{k\,1}z_1 + a_{k\,2}z_3+ \cdots +a_{k\,n}z_{2n-1}=0 a_{k\,1}z_2+ a_{k\,2}z_4 +\cdots +a_{k\,n}z_{2n}= b_kz_{2n+1}\end{cases}\mbox{for}k=1, \ldots, n-r,
\phi(h_1x_1h_2x_2) = \phi(h_1h(x_1{\theta}h_2\circ x_2))~ = T\circ (x_1{\theta}h_2\circ x_2) = (T\circ x_1)~\circ~(T\circ x_2)~~~~~~~(~(x_1, x_1{\theta}h_2) \in U) = \phi (h_1x_1)\phi (h_2x_2).
\sum_{i=1}^d \frac{q^{-(d-i)}}{\omega(d-i,q)}= \sum_{i=0}^d \frac{q^{-(d-i)}}{\omega(d-i,q)}- \frac{q^{-d}}{\omega(d,q)}= \frac{1-q^{-d}}{\omega(d,q)}.
(K_{H}^{\ast}\varphi )(s)= \,c_{H}\Gamma \left( H+\frac{1}{2}\right)\left( D_{T^{-}}^{\frac{1}{2}-H}\varphi (s)\right) (s) & +c_{H}\left( \frac{1}{2}-H\right) \int_{s}^{T}\varphi (t)(t-s)^{H-\frac{3}{2}}\left( 1-\left( \frac{t}{s}\right)^{H-\frac{1}{2}}\right) dt.
|| \sqrt{A_x}\, \partial_x w ||^2_{L^2_1}+ || \sqrt{A_z \over \varepsilon\left(z\right)}\, \partial_z w ||^2_{L^2_1}- \left( \partial_x\left(A_x \partial_x w\left(\cdot ,z_\iota\right)\right) - \partial_x \left(A_x \partial_x \xi_2^\prime\right) , w\left(\cdot ,z_\iota\right)\right)_{L^2_2}=0.
\begin{array}{l}\lim\limits_{t\rightarrow\infty}\mathbb{E}^{\hat{\psi}_0}[\psi_\tau^{t,e}]= \mathbb{E}^{\hat{\psi}_0}[F(e^*_0, e^*_{\varsigma_k- \varsigma_{k-1}}...,e^*_{\varsigma_k- \varsigma_1},\tau,J_\tau,\kappa,T_\tau^{sel})],\end{array}
II_{\varepsilon}=r^{-\alpha/2}\int_{|v-x^r|>\varepsilon r^{-1}}\left(\delta_{D^{r}}(v)^{\alpha/2}{\bf 1}_{D^r\cap B(Q^r, 1/2)}(v)-\delta_{D^{r}}(x^{r})^{\alpha/2}\right)\frac{dv}{|x^r-v|^{d+\alpha}}.
D_{x}^{2}u\left( \bar{x}\right) =\left( \sin(\sigma)I+\cos(\sigma)D_{\bar{x}}^{2}\bar{u}(\bar{x})\right) \cdot\left( \cos\left(\sigma\right) I_{n}-\sin\left( \sigma\right) D^{2}_{\bar{x}}\bar{u}(\bar{x})\right)^{-1}& :=F_{\sigma}(D_{\bar{x}}^{2}\bar{u}(\bar{x}(x))).
A-B =\sum_{j=2}^{i}g_{i,j}\left(\int_{x_{i+1/2}-x_j}^{x_{\alpha_{1}+1/2}}(v-x_{\alpha_{1}})dv-\int_{x_{i-1/2}-x_{j-1}}^{x_{\alpha_{2}+1/2}}(v-x_{\alpha_{2}})dv\right) +{\cal O}(\Delta x^3).
\Gamma ( \phi_{0}) = \int d^{3}{\bf x}\left\{\frac{1}{2}( \partial_{i}\phi_{0})^{2}+ \frac{1}{2}\mu_{R}^{2}\phi_{0}^{2}+ \frac{\lambda_{R}T}{4}\phi_{0}^{4}+ \frac{( N + 2 6 )}{9}\frac{\lambda_{R}^{3}\zeta ( 3 )}{2^{9}\pi^{4}}\phi_{0}^{6}, \right\}.
\mathbf{P}_f=\! \! \begin{bmatrix}f(-N_{\rm cp}) f_1(-N_{\rm cp}) \!\! \cdots \!\! f_V(-N_{\rm cp}) f_1(-N_{\rm cp}) f_2(-N_{\rm cp}) \!\! \cdots \!\! f_{V+1}(-N_{\rm cp}) \vdots \vdots \!\! \!\! \vdots f_V(-N_{\rm cp}) f_{V+1}(-N_{\rm cp}) \!\! \cdots \!\! f_{2V}(-N_{\rm cp})\end{bmatrix},
X(k_{1},k_{2},k_{3})=\frac{1}{2}[\hat{X}(k_{1},k_{2},-k_{3})+\hat{X}(k_{1},-k_{2},k_{3})+\hat{X}(-k_{1},k_{2},k_{3})-\hat{X}(-k_{1},-k_{2},-k_{3})].
\langle F_{k,m}, F_{k,m}\rangle = \frac{1}{\pi}\int_{\mathbb{H}/\Gamma_0(2)}|F_{k,m}(x+iy)|^2 y^k \frac{dx dy}{y^2}&= \frac{1}{\pi}\sum_{i=1}^3 \int_{\mathcal{F}}\left|F_{k,m}|_{\alpha_i^{-1}}(x+iy) \right|^2 y^{k-2}dx dy.
(y_1, \dots, y_{g-1}) \longmapsto \left( \,\prod_{i=1}^{g-1}\frac{1-\kappa_1y_i}{1-\kappa_2y_i}, \,\,\prod_{i=1}^{g-1}\frac{1-\kappa_3y_i}{1-\kappa_4y_i},\dots, \prod_{i=1}^{g-1}\frac{1-\kappa_{2g-1}y_i}{1-\kappa_{2g}y_i}\,\right).
\widetilde{\mathcal{A}}_1\;:=\;\left\{A=A(t,x)\left|\!\begin{array}{c}\mathrm{div}_x A=0\;\textrm{for a.e.}\;t\in\mathbb{R}, A=A_1+A_2\textrm{such that, for $j\in\{1,2\}$,}A_j\in L^{a_j}_\mathrm{loc}(\mathbb{R},L^{b_j}(\mathbb{R}^3,\mathbb{R}^3)) a_j\in(4, +\infty],b_j\in(3, 6),\frac{2}{\,a_j}+\frac{3}{\,b_j}<1\end{array}\!\!\right.\right\}
{\frac{\beta^{2}}{1 6 \pi}}( \tilde{\mu}^{2}( \varphi_{0}) - \tilde{m}_{0}^{2}) \int{\frac{d \tilde{p}}{f^{2}( \tilde{p}) ( 2 f ( \tilde{p}) - \tilde{m}_{2})}}= 1 \; .
\int_{-1}^{1}\frac{(4x^{2}-4xy-1+\rho^{2})dy}{\pi\sqrt{1-y^{2}}((1-\rho^{2})^{2}-4xy\rho(1+\rho^{2})+4\rho^{2}(x^{2}+y^{2}))}=\frac{4x^{2}-1-\rho^{2}}{(1+\rho^{2})^{2}-4x^{2}\rho^{2}}.
A ( k , \Delta E ) = \frac{e^{i ( \Delta E + \omega ) \tau_{0}}}{i}\, \frac{\bar{g}}{\sqrt{( 2 \pi )^{3}2 \omega}}\, \left( \frac{1}{\Delta E + \omega}- \frac{1}{\Delta E + \gamma ( \omega - k_{z}v )}\right) \; .
\mbox{Res}\left( J_{R}^{(1)}(s),s=0 \right) = \frac{1}{4}\begin{pmatrix}-d_{1}d_{0}^{2}-d_{0}i\big(d_{0}^{2}d_{1}^{2}+2d_{0}d_{1}+ \frac{7}{12}\big) id_{0}^{2}d_{1}d_{0}^{2}+d_{0}\end{pmatrix}.
v_1(y) = 3 x^{\frac{1}{3}}y^{\frac{1}{3}}- \frac{T^2 c m p}{4 \pi^2 y n_1^2}v_2(y) = -3 x^{\frac{1}{3}}y^{\frac{1}{3}}- \frac{T^2 c m p}{4 \pi^2 y n_1^2},
(-\Delta)^sw(x)+c(x)w(x) = (-\Delta )^s v (x)+c(x) v(x) + (-\Delta)^s (w-v)(x) \geqslant -C \big (1 + u(a) +(\theta d)^{-n-2s-2}\big ) &\geqslant -C \big ( (\theta d)^{-n-2s-2}+u(a) \big )
\frac{\Im \eta(\xi)}{c_{N,s}}= -\int_{\S^{N-1}}a(-\theta) \int_0^\infty [\sin (r \xi \theta) - r \xi \theta 1_{[0,1]}(t)]r^{-1-2s}\,dr\,d\theta &= \int_{\S^{N-1}}a(\theta) h_s(\xi \cdot \theta) \,d\theta = \int_{\S^{N-1}}a_{odd}(\theta) h_s(\xi \cdot \theta) \,d\theta
\mu_H\circ ((a_H \otimes b_H)\otimes c_H) \circ (\rho_{p,q}(G/H)\otimes Id_r)\circ \rho_{p+q, r}(G/H) & = \mu_H\circ ((a_H \otimes b_H)\otimes c_H) \circ (Id_p\otimes \rho_{q,r}(G/H))\circ \rho_{p,q+r}(G/H) + (-1)^{rq}\mu_H\circ ((a_H \otimes b_H)\otimes c_H)\circ (Id_p \otimes \tau_{r,q}(G/H) \circ \rho_{r,q}(G/H))\circ \rho_{p,q+r}(G/H).
CB.({h}^{l}B) \equiv (a_{l+1}h^{l+1}+a_{l-1}h^{l-1}+\cdots+a_{1}h)B a_{l+1}=2(l+2) k =1,3,\ldots,k-2\ (k-1) a_{l-k}=\frac{1}{2}\bigg((8-\mu){l\choose k}4^{k}+6{l \choose k+1}4^{k+1}+{l\choose k+2}4^{k+2}\bigg).
S = \bigl\{[y_1, y_2, \dots , y_n] \mid y_k \in X^2 \mbox{for some}k, 2 \le k \le n-1; \ \y_i \in X \mbox{for all}i \ne k \bigr\}, S' = \bigl\{[y_1, y_2, \dots , y_n] \mid y_2 \in X^2, \y_i \in X \mbox{for all}i \ne 2 \bigr\}.
-\mathcal{R}_{i,j}\triangle_i=|\nabla_{d_{i,j}}+\nabla_{D_{i,j}}|^2\mathcal{R}_{i,j},\,\,\,\,-\mathcal{R}_{i,j}\triangle_j=|\nabla_{d_{i,j}}-\nabla_{D_{i,j}}|\mathcal{R}_{i,j}
K_0 = e^{\textup{i}\pi b^2}A\, D - \sin^2 \pi b^2 \, C^2 , &K_1 = \frac{e^{-\textup{i}\pi b^2}A^2 + e^{\textup{i}\pi b^2}D^2}{4\sin^2 \pi b^2}-B\, C -\frac{1}{2}\cos \pi b^2 \, C^2
\left[\frac{1}{2}\mathfrak{a}^{T}K\mathfrak{a},\prod\limits_{l = 1}^mf_{l}^{T}\mathfrak{a}\right] = \sum\limits_{i = 1}^m\prod\limits_{l = 1}^mf_{l}^{T}(JK)^{\delta_{il}}\mathfrak{a}.
f_{\vec{k}}( \Omega ) ~ = ~ - \sum_{l = 0}^{\infty}~ ( 2 l + 1 ) ~ \int d r ~ r^{2}~ U ( r ) ~ \frac{f ( r )}{r}~ j_{l}( k r ) ~ i^{l}\int d \theta \sin \theta P_{l}( \cos \theta ) \int d \phi ~ Y_{l m}( \theta , \phi ) ~ ,
\g_0:=-\,\mathcal{G}_{a_0}(t_0,x_0)>0\,,\ \ \ \mathrm{where}\ \ \ \mathcal{G}_{a}(t,x):=( \varphi_t-{\mathcal{L}}_{a}\varphi) (t,x)\,, \ (a,t,x)\in \mathcal{A}(x) \times (0,\infty)\times \mathbb{R}_+^n\,.
\begin{aligned}J_1(v_1(\cdot),u_2(\cdot))-J_1(u_1(\cdot),u_2(\cdot))\geq \mathbb{E}\int_0^TH_{1v_1}(t)(v_1(t)-u_1(t))dt = \mathbb{E}\int_0^T\mathbb{E}\Big[H_{1v_1}(t)\big(v_1(t)-u_1(t)\big)|\mathcal{F}_t^1\Big]dt.\end{aligned}
X = \frac{L_{-}+ L_{+}}{\nu}= z^{2}- \left( 1 + \frac{H}{2 \nu^{2}}\right) , P = \frac{2 ( L_{-}- L_{+})}{i}= \frac{1}{i}\left[ 2 z \frac{d}{d z}+ 1 \right] .
\sum_{i=2n-1}^{2n}\sum_{j=1}^{2n-2}\left| D_iD_j u(p) - D_iD_j u(q) \right| \leq C \left( d^{\frac{1}{\beta}-1}\sup_{B_\beta(0,1)}|u| + \int_0^d \frac{\omega(r)}{r}dr + d^{\frac{1}{\beta}-1}\int_d^1 \frac{\omega( r)}{r^{1/\beta}}dr \right),
\int_{-\infty}^{\infty}\widehat{R}_\varphi f(\theta,p)p^kdp = \sum_{|\alpha|=k}\frac{k!}{\alpha!}c_{\alpha_3}\gamma_{\alpha_1,\alpha_2}\cos^{\alpha_1}{\theta}\sin^{\alpha_2}{\theta}&= \sum_{\alpha_3=0}^k C(k,\alpha_3) c_{\alpha_3}\sum_{j=0}^{k-\alpha_3}C(k-\alpha_3,j)\gamma_{j,k-\alpha_3-j}\cos^{j}{\theta}\sin^{k-\alpha_3-j}{\theta}.
\mathcal{L}= \frac{4}{9}F_{( \alpha \beta \gamma ) \nu}F^{( \alpha \beta \gamma ) \nu}+ \frac{2}{3}F_{( \alpha \beta \gamma ) \nu}F^{( \alpha \beta \nu ) \gamma}- F_{( \alpha \beta \mu )}{}^{\mu}F^{( \alpha \beta \nu )}{}_{\nu}\ ,
\mathcal{L}(\Phi) = \Phi_t - \displaystyle\sum\limits_{i,j=1}^{n-1}a_{ij}\Phi_{{x_{i}}{x_{j}}}+ \displaystyle\sum\limits_{i=1}^{n-1}a_{i}\Phi_{x_{i}}=\tilde f{\Phi\big|}_{t=0}={\Phi_0}
\lambda_n (\sum\limits_{j=1}^{p}\vert \beta_{0,j}+ z_{j}/ \sqrt{n}\vert^q - \vert \beta_{0,j}\vert^q) \rightarrow\lambda_{0}q \sum\limits_{i=1}^{p}z_{j}sgn(\beta_{0,j}) \vert \beta_{0,j}\vert^{q-1}
P(M_k \mid{\bf y}_n)<\frac{1}{1+O_p(1) n^{\frac{3}{2}(|k|-|t|)}\frac{P(M_t)}{P(M_k)}}= \frac{n^{-\frac{3}{2}(|k|-|t|)}\frac{P(M_t)}{P(M_k)}}{n^{-\frac{3}{2}(|k|-|t|)}\frac{P(M_t)}{P(M_k)}+O_p(1)}=n^{-\frac{3}{2}(|k|-|t|)}\frac{P(M_k)}{P(M_t)}O_p(1),
( \rho_{1}^{r 1}, \rho_{1}^{r 2}; \sigma_{1}^{r 1}, \sigma_{1}^{r 2}; \psi_{1}) ( \rho_{2}^{r 1}, \rho_{2}^{r 2}; \sigma_{2}^{r 1}, \sigma_{2}^{r 2}; \psi_{2}) ( \rho_{3}^{r 1}, \rho_{3}^{r 2}; \sigma_{3}^{r 1}, \sigma_{3}^{r 2}; \psi_{3})
\langle\phi_{1}(x_{1})\phi_{2}(x_{2})\phi_{3}(x_{3})\rangle=\frac{C_{123}}{x_{12}^{\Delta_{1}+\Delta_{2}-\Delta_{3}}x_{23}^{\Delta_{2}+\Delta_{3}-\Delta_{1}}x_{13}^{\Delta_{3}+\Delta_{1}-\Delta_{2}}},
\begin{aligned}3^{l_1+1}x^{r + 6}+3a(r) x^{21}+ 3b(r) x^{39}+ 3 a(r)^2 x^{33}+ 2 \cdot 3a(r)b(r) x^{51}+ 3 b(r)^2 x^{69}+ 3 \cdot 3^{2l_1}x^{2 r+3}+ 2 \cdot 3^{l_1+1}a(r) x^{18 + r}+ 2 \cdot 3^{l_1+1}b(r) x^{36 + r}\in \langle F \rangle. \end{aligned}
V^2\circ\gamma(s) = |\nabla V|^2(p) (s^2 + \frac{1}{3}(Ric(\gamma'(0),\gamma'(0)) - \epsilon n)s^4 + O(s^{6})), |\nabla V|^2\circ\gamma(s) = |\nabla V|^{2}(p)( 1 + (Ric(\gamma'(0),\gamma'(0)) - \epsilon n)s^2 + O(s^4)).
\langle \;{\cal B}({\{x\}}) \; \rangle^\theta_0 \; \; := \; \;e^{i \theta \frac{Q_5({\scriptstyle{\cal B}})}{2N}}\lim_{\tau \rightarrow \infty}\langle{\cal U}_\tau ({\scriptstyle{\cal B}}) \;{\cal B}({\scriptstyle\{x\}}) \rangle_0 \; .
J_{\mu\nu}= 2i\big\{e_{[\mu}{}^{\alpha\dot\alpha}e_{\nu]}{}^\beta{}_{\dot\alpha}\widehat A^{(1)}_\alpha\star \widehat A^{(1)}_\beta + \mbox{h.c.}\big\}_{z=0}+ L_{\mu\nu}+\mbox{$W$-terms},
\int_0^\infty \mu(d\xi) = \frac{1}{2\sqrt{\pi}}\int_0^\infty e^{-\xi}\xi^{-3/2}d\xi = \infty, \int_0^\infty \xi \mu(d\xi) = \frac{1}{2\sqrt{\pi}}\int_0^\infty e^{-\xi}\xi^{-1/2}d\xi = \frac{1}{2}.
\underline U_t(t,x)= K_2\Theta\frac{|x|\underline h'(t)}{\underline h^2(t)}\preceq K_2\Theta \frac{\underline h'(t)}{\underline h(t)}= \frac{K_1K_2\Theta}{{\gamma}-1}\underline h(t)^{1-\gamma}.
\begin{array}{lll}I = \left[ \frac{1}{t}\dot{h}(t)\right]_{t_{0}}^{\theta}+(\alpha+1)\int_{t_{0}}^{\theta}\frac{1}{t^{2}}\dot{h}(t)dt = C +\frac{1}{\theta}\dot{h}(\theta) +\frac{(1+\alpha )}{\theta^{2}}h(\theta )+2(1+\alpha)\int_{t_{0}}^{\theta}\frac{1}{t^{3}}h(t)dt.\end{array}
\bigcup_{(\ell, j, j') \neq (0,j,j)}R_{\ell j j'}^{(II)}= \bigcup_{\begin{subarray}{c}\ell\neq 0 |\sqrt{j}- \sqrt{j'}| \leq C \langle \ell \rangle\end{subarray}}R_{\ell j j'}^{(II)}.
\pi_{\alpha}^{( 0 )}* H^{( 1 )}* \pi_{\alpha}^{( 0 )}= \frac{i \hbar}{2}\lambda_{\alpha}< \Pi , \pi_{\alpha}^{0}( F^{B}- F^{0}) \pi_{\alpha}^{0}> + \frac{i \hbar}{2}\sum_{\gamma \neq \alpha}\lambda_{\gamma}< \Pi , S^{\gamma \alpha}S^{\alpha \gamma}> .
\mathfrak{S}_{w^{({\boldsymbol{n}})}}\big(1,q,q^2,\ldots\big)=q^{\sum{n_i \choose 3}}C_{n_{1}+n_{2}}^{(n_1)}(q) \times C_{n_{1}+n_{2}+n_{3}}^{(n_1+n_2)}(q) \times C_{n}^{(n-n_{p})}(q).
\frac{S O ( 1 , 5 )}{S O ( 5 )}\subset \frac{S O ( 5 , 5 )}{S O ( 5 ) \times S O ( 5 )}, \quad \frac{S O ( 4 , 5 )}{S O ( 4 ) \times S O ( 5 )}\subset \frac{S O ( 5 , 5 )}{S O ( 5 ) \times S O ( 5 )}.
\widetilde{vol}=\frac{(s+1)}{2^{2s+s^{2}-1}}\frac{1}{y_{1}\cdots y_{s}}\mathrm{d}x_{1}\wedge\mathrm{d}y_{1}\wedge\cdots\wedge\mathrm{d}x_{s+1}\wedge\mathrm{d}y_{s+1}
T^{\mu}_{\ \mu\rho\sigma}=-p^{2}T_{1}\Biggl[\left(p_{\rho}p_{\sigma}-p^{2}g_{\rho\sigma}\right) \mp\frac{1}{4}\left( \varepsilon_{\rho\lambda}p^{\lambda}p_{\sigma}+\varepsilon_{\sigma\lambda}p^{\lambda}p_{\rho}\right)\Biggl] \, .
\widehat{R}_{k}( \mathrm{\boldmath \lambda \unboldmath}) = \left. \frac{1}{( 2 \pi )^{k}}\frac{\partial^{k}}{\prod_{p = 1}^{k}\partial J_{p}}Z_{k}( \mathrm{\boldmath \lambda \unboldmath}+{\bf J}) \right|_{{\bf J}= 0}\ ,
\sum_{n\geq1}nq^{\frac{n^2+n}{2}}=\frac{\left(q^2;q^2\right)_{\infty}}{\left(q;q^2\right)_{\infty}}\sum_{n\geq1}\frac{(-1)^nq^n}{1-q^n}+\sum_{n\geq0}\left(\frac{\left(q^2;q^2\right)_{\infty}}{\left(q;q^2\right)_{\infty}}-\frac{\left(q^2;q^2\right)_{n}}{\left(q;q^2\right)_{n+1}}\right).
(\partial_t\Phi_{t,\rho(t)}|_{t=t^*}\circ\Phi_{t^*,\rho(t^*)}^{-1})'\left(\frac{\varphi}{\Phi_{t^*,\rho(t^*)}'}\circ \Phi_{t^*,\rho(t^*)}^{-1}\right) &=\left(\frac{\varphi}{(\Phi_{t^*,\rho(t^*)}')^2}\circ \Phi_{t^*,\rho(t^*)}^{-1}\right)\partial_t\Phi_{t,\rho(t)}'|_{t=t^*}\circ\Phi_{t^*,\rho(t^*)}^{-1}
\varphi \big( g^{j}_1 g_3 \big) \varphi \big( g^{i}_1 g_2 \big) \overset{(\ref{eq:8})}{=}\varphi (g_3) \varphi \big( g^{j}_1 \big) \varphi \big( g^{i}_1 g_2 \big) \overset{(\ref{eq:5})}{=}\varphi (g_3) \varphi \big( g^{j}_1 \big) \varphi \big( g^{i}_1 \big) \varphi (g_2) \overset{{\bf A3}}{=}\varphi (g_3) \varphi (g^{n}_1) \varphi (g_2) \,.
{1 \over \sum\limits_{n',k'}\exp\!\left[{-{(E_{n'}-E)^2 - (E_n -E)^2 \over 2 \varepsilon^2}}\right]}< \exp\!\left[{-{(E_{n}-E)^2 - (E_{n_0}-E)^2 \over 2 \varepsilon_0^2}}\right]
\omega(x,y,z)+\omega(y,z,x)+\omega(z,x,y) = \varphi([x,y],z)-\rho(z)\varphi(x,y)+\varphi([y,z],x)-\rho(x)\varphi(y,z)+\varphi([z,x],y)-\rho(y)\varphi(z,x) = \partial_{\rho}(\varphi)(x,y,z)=0.
U_{i}= \left( \begin{array}{c c}{{A_{i}}}&{{E_{i}}}\\{{C_{i}}}&{{D_{i}}}\\ \end{array}\right) , \quad V_{i}= \left( \begin{array}{c c}{{B_{i}}}&{{F_{i}}}\\{{G_{i}}}&{{H_{i}}}\\ \end{array}\right) ,
X+Y+\Psi(X,Y) = \sum_{m=1}^\infty{(-1)^{(m-1)}\over m}\sum_{\vbox{\hbox{$k_i+l_i\geq 1$}\hbox{$k_i\geq 0, l_i\geq 0$}}}{[X^{k_1}Y^{l_1}\cdots X^{k_m}Y^{l_m}]\over k_1! l_1! \cdots k_m! l_m!},
I=\langle\langle E'(\lambda- \mu)(f^n(t,x,\lambda)-f^n(t,x,\mu)), \nu^{\delta,\varepsilon}_{(t,x)}(\lambda)\rangle, \sigma^{\delta,\varepsilon}_{(t,x)}(\mu)\rangle +{\cal R}_{\varepsilon,\delta,n}^\lambda+{\cal R}_{\varepsilon,\delta,n}^\mu.
\underline{W}_j:= \overline{g}_{jm,\alpha}\overline{g}_{m,\alpha}\sum_{i=0}^{m-1}\sum_{\ell=1}^{m}G(S_{jm-i},S_{jm+\ell}), & \underline{\hat{W}}_j:= \overline{g}_{jm,\alpha}\overline{g}_{m,\alpha}\sum_{i=m}^{jm-1}\sum_{\ell=1}^{m}G(S_{j m-i},S_{jm+\ell}),