harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Gamma(\pi) \ge\frac{1}{N}\sum_{\ell=1}^{N}\Bigg(\lim_{t\to\infty}\frac{\sum_{i=1}^{R_\ell^\pi(t)}\rho_\ell({T_{\ell i}^\pi})^2}{2t}+\lim_{t\to\infty}\frac{c_\ell R_\ell^\pi(t)}{t}\Bigg).
f\circ(g\circ x)=\sum_{i<\alpha}r_{i}e^{\gamma_{i}\circ(g\circ x)}=\sum_{i<\alpha}r_{i}e^{(\gamma_{i}\circ g)\circ x}=\left(\sum_{i<\alpha}r_{i}e^{\gamma_{i}\circ g}\right)\circ x=(f\circ g)\circ x.
\mathcal{V}\left(\overline{u},\overline{v}\right) =q(1)d\lambda_1^2e^{\lambda_1z}-q(1)c\lambda_1e^{\lambda_1z}+s\lambda_1e^{\lambda_1z}-sq(1)e^{2\lambda_1z}begin{align*}0.2cm] =P(c,\lambda_1)q(1)e^{\lambda_1z}-sq(1)e^{2\lambda_1z}begin{align*}0.2cm] =-sq(1)e^{2\lambda_1z}\leq0.
(B_1)_{32}=q_1\left( -p_1+\frac{q_2}{t_1}+1 \right)+\left( p_2q_2-\theta^\infty_1-\frac{t_2}{t_1}\right), (B_1)_{33}=-q_2\left( p_2+\frac{q_1}{t_1}\right)-q_1+\theta^\infty_1-\theta^\infty_2+\frac{t_2}{t_1}.
I_1 = \min\{I(X_1;Y_1),\, I(X_1;Y_2)\}+ \min\{I(X;Y_1|X_1,W_1),\, I(X;Y_2|X_1)\}, &\le I(X_1;Y_1) + I(X;Y_2|X_1) &= h(Y_1) - h(Y_1|X_1) + h(Y_2'|X_1) - h(Y_2'|X),
\left [ \delta^{a b}+C.\frac{\delta^2 \Gamma[\phi]}{\delta \phi^a\delta \phi^b}\right ]^{-1}(\mathbf{q},\mathbf{-q}) = \left [ \delta^{a b}+ 2 q^2 \left \{\delta^{ab}V' + 2 \phi^a\phi^b V'' + q^2 \delta^{ab}\right \}\right ]^{-1}
|\mu(T^{k(h_n+c_n)}(I_{n,a}) \cap B) - \mu(I_{n,a})\mu(B)| =|\sum \limits_{i=0}^{r_{n}}\mu(T^{k(h_n+c_n)}(I^{[i]}_{n,a}) \cap B) - \mu(I^{[i]}_{n,a})\mu(B)|.
\frac{\Delta V}{V_{0}}=\frac{(1+\epsilon_{11}+\epsilon_{22}+\epsilon_{33}+\epsilon_{11}\cdot\epsilon_{22}+\epsilon_{11}\cdot\epsilon_{33}+\epsilon_{22}\cdot\epsilon_{33}+\epsilon_{11}\cdot\epsilon_{22}\cdot\epsilon_{33})\cdot a^{3}-a^{3}}{a^{3}}
\left. \delta m \right\vert_{P V}= \left. \delta m \right\vert_{d i m}- \left. \delta m \right\vert_{d i m , \mid \mu_{r e g}\mid \to \infty}= \left. \delta m \right\vert_{d i m}+ \frac{g^{2}}{2 \pi}s i g n ( \mu_{r e g}) \ .
s(P:_{M}\mathfrak{a}) =s(\mathfrak{p}M:_{M}\mathfrak{a})=s(\mathfrak{p}:_{R}\mathfrak{a})M\leq P+(0:_{M}\mathfrak{a}) &=\mathfrak{p}M+((0:_{M}\mathfrak{a}):_{R}M)M &=(\mathfrak{p}+(0:_{R}\mathfrak{a}))M.
\lambda_1 = \inf_{0\neq u \in W_0^{1,2}(M)}\Bigg\{\frac{\int_M | \nabla u |_g^2 \d \mu_g}{\int_M | u|^2_g \d\mu_g}\ \ \Big| \ \u \neq 0, \ \u \in W_0^{1, 2}(M,g) \Bigg\},
\begin{array}{rcl}G(x_1,x_2) = \displaystyle -\frac{g^2}{2!}\int d^4x_3\int d^4x_4~A, A = \displaystyle <0|T\left(\varphi_1\varphi_2(\chi\star\varphi\star\pi+\pi\star\varphi\star\chi)_3 \right. & \displaystyle\times\left.(\chi\star\varphi\star\pi+\pi\star\varphi\star\chi)_4 \right)|0>, \end{array}
t_{12}t_{23}+c_{12}t_{23}c^{-1}_{12}t_{23}=t_{23}t_{12}+t_{23}c_{12}t_{23}c^{-1}_{12},\quad[t_{12}+c_{12}t_{23}c^{-1}_{12},t_{23}]=0, t_{23}t_{12}+c_{23}t_{12}c^{-1}_{23}t_{12}=t_{12}t_{23}+t_{12}c_{23}t_{12}c^{-1}_{23},\quad[t_{23}+c_{23}t_{12}c^{-1}_{23},t_{12}]=0,
L ={\frac{1}{2}}( \partial_{\mu}\Phi ) ( \partial^{\mu}\Phi ) +{\frac{1}{2}}( \partial_{\mu}\Psi ) ( \partial^{\mu}\Psi ) +{\frac{1}{2}}m^{2}\Phi^{2}+{\frac{1}{2}}\omega^{2}\Psi^{2}+{\frac{\lambda_{\Phi}}{2 4}}\Phi^{4}+{\frac{\lambda_{\Psi}}{2 4}}\Psi^{4}+{\frac{\lambda_{\Phi \Psi}}{4}}\Phi^{2}\Psi^{2}~ ,
\partial_{a_{1}}\ldots \partial_{a_{n}}\delta_{n}\phi = \epsilon_{a_{1}}^{\bar{a}_{1}}\ldots \epsilon_{a_{n}}^{\bar{a}_{n}}\bar{\partial}_{\bar{a}_{1}}\ldots \bar{\partial}_{\bar{a}_{n}}( \rho_{0}^{\bar{c}}\bar{\partial}_{\bar{c}}\phi ) +{\cal O}( \phi^{2}) .
A = q_\nu(\lambda), \Delta_i A = q_\nu(\lambda^{i+}) -q_\nu(\lambda), B = \prod_{t=1}^s\sum_{\square\in \lambda}{\binom{c_{\square}}{2}}^{r_t}, \Delta_i B = \prod_{t=1}^s\sum_{\square\in \lambda^{i+}}{\binom{c_{\square}}{2}}^{r_t}-\prod_{t=1}^s\sum_{\square\in \lambda}{\binom{c_{\square}}{2}}^{r_t}.
\frac{2\sqrt{\pi}\,c^{\frac{1}{2}-s}}{\Gamma(s)}\cdot\frac{\Gamma\left(s-\frac{1}{2}\right)}{1+\frac{1}{\pi p}}\,\zeta_{p^{\prime}}(2s-1)=\frac{\pi}{\sqrt{c}\left(1+\frac{1}{\pi p}\right)}\,\frac{1}{s-1}+\frac{\pi}{\sqrt{c}\left(1+\frac{1}{\pi p}\right)}\left(2C_{p^{\prime}}^{(1)}-\log\left(4c\right)\right)+O\left(s-1\right).
P_i=\frac{\left(\alpha_{b_i}c_i+d_{b_i}\right)\tau}{N_s}=\frac{c_i\tau/\ln(2)}{-\tilde{c}_2'(\alpha_{b_i})\tau NN_s\sum_{j=i}^{N}\left(\lambda_j\right)-\mu_i+\nu_i}-\frac{1}{h_i}
-C_{0}\sum^{N}_{i=1}\frac{a^{2N}_{i}}{2 \Delta^{2}_{i}}\left(1 - \sum_{i \neq j}\frac{a_{j}}{a_{i}- a_{j}}\right) = \frac{C^{\prime}_{1}\pi^{2N}}{2^{2N+1}}\sum^{N}_{i=1}\sum_{i \neq j}\frac{(a_{i}+ a_{j})^{2N}}{(a_{i}- a_{j})^{2}\gamma_{i}\gamma_{j}}+ \frac{1}{2}A^{(1;1)}_{1}\sum^{N}_{i=1}a^{2}_{i}.
\lim_{j\to \infty}{\min \big\{\overline{F}_\epsilon^j(u):u\in H^s(\Omega),\int_{\Omega}{u\,dx}=0\big\}}&=\min \big\{F_\epsilon(u):u\in H^s(\Omega),|u(x)|\le 1\; \forall x\in \Omega, \int_{\Omega}{u\,dx}=0\big\}
F =\sum_{j=1}^{+\infty}\frac{1}{j!}\int_{\mathbb X^j}g_j(x_1,\ldots,x_j) N^{T,M}(dx_1) \cdots N^{T,M}(dx_j) &=\sum_{j=1}^{+\infty}\frac{1}{j!}\int_{([0,T]\times[0,M])^j}g_j(x_1,\ldots,x_j) N(dx_1) \cdots N(dx_j)
\sigma^{+}= \ \left( \begin{array}{c c}{{0}}&{{1}}\\{{0}}&{{0}}\\ \end{array}\right) \ , \ \sigma^{-}= \ \left( \begin{array}{c c}{{0}}&{{0}}\\{{1}}&{{0}}\\ \end{array}\right) \ , \ \sigma^{0}= \ \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) \ .
\Psi^{-1}(\mathbf x) = \left( 0, \frac 12, \frac 12 \right) + \frac{\frac{1}{\sqrt 2}\left(x_1, \frac{x_2-x_3}{2}, \frac{x_3-x_2}{2}\right)}{\frac{1}{\sqrt 2}- \frac{x_2+x_3-1}{\sqrt 2}}&= \left( \frac{x_1}{2-x_2-x_3}, \frac{1-x_3}{2-x_2-x_3}, \frac{1-x_2}{2-x_2-x_3}\right).
\bigg|\sum_{\substack{x<n\le y \gcd(n,M)=1}}\chi(n)\bigg|\hskip 0.1cm\le \hskip 0.1cm 2^{\omega(M)-\omega(g)}\left(\dfrac{1}{3\log 3}\sqrt{k}\log k+\dfrac{13}{2}\sqrt{k}\right).
\P\bigl(\tau(\beta)\leq t+h\bigr)\leq C_{p}\bigl(\beta^{p/2}+h^{p}\beta^{-p/2}+h^{p/2}\beta^{-p/2}\bigr)= C_{p}\bigl( h^{p/8}+ h^{7p/8}+ h^{3p/8}\bigr).
\alpha_i^+(\omega)\ne 0, \alpha_j^-(\omega) \ne 0, \gamma_2(\omega)=0 \textnormal{and}\gamma_1(\omega)=\lim_{n\to \infty}\left(\sum_{l \in \mathbb{Z}^*}^{}x_{l}(\omega) \textbf{1}_{|x_l(\omega)|>\frac{1}{n^2}}\right)
\frac{\partial\Gamma_{11}^{2}}{\partial v}-\frac{\partial\Gamma_{21}^{2}}{\partial u}+\Gamma_{21}^{2}\Gamma_{11}^{1}+\Gamma_{22}^{2}\Gamma_{11}^{2}-\Gamma_{11}^{2}\Gamma_{21}^{1}-\Gamma_{12}^{2}\Gamma_{21}^{2}
M_G^{p,0}(0,T):= \{\eta=\sum_{j=0}^{N-1}\xi_j(\omega)I_{[t_j,t_{j+1})}(t): N\in\mathbb{N},\ 0\leq t_0\leq t_1\leq \cdots\leq t_N\leq T, & \ \xi_j\in L_{G}^p(\Omega_{t_j}),\j=0,1\cdots,N\}.
c_4 = -31 \left(s^4 + 42 s^2 - (32200/93) s - 147\right) c_2 = 2^8 \big(s^8 + 84 s^6 - (3472/3) s^5 + 1470 s^4 - 48608 s^3 + 53508 s^2 + 170128 s + 21609\big) c_0 = (2^{20}\cdot7/3) s (s^2 + 7)^3 (s^2 + 63) d = s^4 + 42 s^2 + (1736/3) s - 147.
\mathsf{P}_{\mathrm{SIR},g_{2}}\left(\lambda_{\mathrm{BS}}\right)= \mathbb{P}\left(\frac{P_{\mathrm{BS}}H_{\mathrm{U}_{0},\mathrm{BS}_{0}}g_{2}\left(d_{0}\right)}{\underset{\mathrm{BS}_{i}\in\Pi_{\mathrm{BS}}^{\dagger}}{\sum}P_{\mathrm{BS}}H_{\mathrm{U}_{0},\mathrm{BS}_{i}}g_{2}\left(d_{i}\right)}>\tau\right).
\frac{1}{n}\sum_{j=1}^m \sum_{k,l=1}^n \|x_{j,k,l}\|\cdot \|y_{j,l,k}\|\leq \frac{1}{n}\sum_{j=1}^m \sum_{k,l=1}^n \|X_j\|\cdot \|Y_j\|=n\sum_{j=1}^m\|X_j\|\cdot \|Y_j\|\leq nC\|h\|,
a\ \ast_{\sigma\circ\tau}\a' = \sigma\circ\tau((\sigma\circ\tau)^{-1}(a)\ast(\sigma\circ\tau)^{-1}(a')) = \sigma(\tau(\tau^{-1}(\sigma^{-1}(a))\ast\tau^{-1}(\sigma^{-1}(a')))) = \sigma(\sigma^{-1}(a) \ast_\tau \sigma^{-1}(a')) = a \{(\ast_\tau)}_\sigma \a' a,a' \in [n] \sigma, \tau \in S_n,
c_{x\sigma}=\vartheta a_{r(x)\sigma}\vartheta^{-1},\ \ \ \phi_x=\vartheta \phi_{r(x)}\vartheta^{-1},\ \ pi_x=-\vartheta \pi_{r(x)}\vartheta^{-1}, \ \x\in \Lambda_R, &\vartheta \Omega_L=\Omega_R,
N_{22}^{+}(x,y,a,b)=-\frac{1}{8ab}\sum_{n=1}^{\infty}\cos (\frac{2n\pi x}{a})\sum_{k=0}^{\infty}\frac{e^{-2\left( 1-\frac{y}{b}+k\right) \frac{n\pi}{a}b}}{\left( 1-\frac{y}{b}+k\right)}.
\begin{cases}p_0+m_0q_0 \leq (n-3-\beta) +m_0 \beta, s_0\geq s_{j-1}\,\,\,\mbox{for}\,\, 1\leq j \leq k, \Delta x \sim \rho \sigma^{m_0+\dots +m_{k}}\cdot \sigma^{1-m_0}\sim \Delta y \cdot \sigma^{1-m_0}\geq \Delta y .\end{cases}
\Vert\Psi(u)\Vert_{g}^{2}=\int_{M}b(x)qu^{2}\Psi(u)d\mu_{g}-\int_{M}b(x)q^{2}u^{2}\left( \Psi(u)\right)^{2}d\mu_{g}& \leq C\int_{M}u^{2}\Psi(u)d\mu_{g},
\pi_{r_0,x_0}:= \pi(x,t) - c_{r_0,x_0}(t) = \frac{1}{3}|v(x,t)|^2 + p_1(x,t) + p_2(x,t), p_1(x,t) := \textnormal{p.v.}\int_{B_{3r_0}(x_0)}K(x - y) : (v \otimes v)(y, t) \,dy, p_2(x,t) := \int_{\mathbb{R}^3 \setminus B_{3r_0}(x_0)}(K(x - y) - K(x_0-y) : (v \otimes v)(y, t) \,dy, \mathcal{E}_\gamma(s) := \sup_{r \in [\gamma,r_0]}E_r(s).
f = \left( 1 - \frac{\rho_{0}}{\rho}\right)^{\frac{1 - k^{2}}{1 + k^{2}}}, r^{2}= \rho^{2}\left( 1 - \frac{\rho_{0}}{\rho}\right)^{\frac{2 k^{2}}{1 + k^{2}}}, e^{\phi - \phi_{0}}= \left( 1 - \frac{\rho_{0}}{\rho}\right)^{\frac{2 l}{1 + k^{2}}}
\sum_{\substack{n\in\mathcal{A}(x) n\equiv v_0\pmod{W}}}\mathbf{1}_\mathcal{P}(L(n))w_n=\sum_{\mathbf{d},\mathbf{e}\in\mathcal{D}_k}\lambda_{\mathbf{d}}\lambda_{\mathbf{e}}\sum_{\substack{n\in\mathcal{A}(x) n\equiv v_0\pmod{W}[d_i,e_i]|L_i(n)\forall i}}\mathbf{1}_{\mathcal{P}}(L(n)).
S_1 = 0 \longrightarrow D \longrightarrow E^i_1 \longrightarrow \cdots \longrightarrow E^1_1 \longrightarrow C \longrightarrow 0 \mbox{and}S_2 = 0 \longrightarrow D \longrightarrow E^i_2 \longrightarrow \cdots \longrightarrow E^1_2 \longrightarrow C \longrightarrow 0.
\ \dot{s}\epsilon(z)\dot{s}(e-\varepsilon(1))\lambda_i = \varepsilon(-z^{-1}) \dot{s}(\varepsilon(-z)- \varepsilon(z^2-z)) \lambda_i \ = \sum_{k,\nu}g^i_{k,\nu}(\varepsilon(-z^{-1}(1+x^i_{k,\nu}))-\varepsilon(z^{-1}(z-1)^{-1}(x^i_{k,\nu}-z+1)) )\lambda_k.
U(T) \simeq \frac{\Delta n^2}{8}\, T{,}\;\; F(T) \simeq -\frac{\Delta n^2}{8}\, T\left [\ln (aT)-c\right ]{,}\;\; S(T)\simeq \frac{\Delta n^2}{8}\left [\ln (aT)+c+1\right ]{,}
\begin{aligned}B(n,q;\chi_{1},\ldots,\chi_{5})=\sum\limits_{a=1 \atop (a,q)=1}^{q}C_{1}(\chi_{1},a)C_{3}(\chi_{2},a)\ldots C_{3}(\chi_{5},a)e\left(-\frac{an}{q}\right)\end{aligned}
\begin{aligned}|h(x+ \delta u(\tau)) - h(x_a+\delta u(\tau))| \le L_r |x-x_a| &\le L_r(|x-z_1|+\gamma |\epsilon_1(\eta_a,\tau)|) &\le L_r(|x-z_1|+\gamma \delta L_r ( 3 + L_r)).\end{aligned}
\frac{1}{V}W = \frac{1}{8 \pi}\left( M^{2}- \mu^{2}\ln ( M^{2}/ a^{2}) - 8 \pi \frac{\alpha_{R}}{\beta^{2}}\left( \frac{M^{2}}{a^{2}}\right)^{\beta^{2}/ 8 \pi}\right)
\left[ a^{2}\dot{\phi}_{0}^{2}\partial_{y}\frac{1}{a^{2}\dot{\phi}_{0}^{2}}\partial_{y}a^{2}- \frac{2 \kappa}{3}a^{2}\dot{\phi}_{0}^{2}+ m^{2}\left( 1 + 2 \sum_{\sigma}\lambda_{\sigma}\delta ( y - y_{\sigma}) \right) \right] Y^{( 1 )}= 0 ,
\begin{array}{c c c c}{X_{0}^{0}= \tau ,}&{X_{0}^{1}= \sigma ,}&{\vec{X}_{0}= \vec{X}_{0}\left( \sigma , \tau \right) ,}&{\sqrt{- \det \eta_{\mathrm{m n}}\left( X_{0}\right)}= 1 .}\\ \end{array}
\Psi_{w_r}(1)-\Phi(r) \geq \frac{1}{1-k}\left(\Phi(r)-\theta-C\left(r^{1-\frac{n}{p^*}}+\omega(r)\right)\right) + \theta - \Phi(r) = \frac{k}{1-k}(\Phi(r)-\theta)- C\left(r^{1-\frac{n}{p^*}}+\omega(r)\right).
\begin{array}{l c}{{R ( \alpha_{i}) = \alpha_{\sigma ( i )},}}&{{R ( \alpha_{i}\alpha_{j}) = \alpha_{\sigma ( i )}\alpha_{\sigma ( j )},}}\\{{R ( \alpha_{i}\alpha_{j}\alpha_{k}) = \alpha_{\sigma ( i )}\alpha_{\sigma ( j )}\alpha_{\sigma ( k )},}}&{{\cdots ;}}\\ \end{array}
\begin{array}{ll}4p^{3m}= 4p^{2m}+4p^m(n_{1,1,0}+n_{-1,-1,0})+2(p^{\frac{d}{2}}+1)^2p^{m}(n_{1,1,1}+n_{-1,-1,1}) +2(p^{\frac{d}{2}}-1)^2p^{m}(n_{1,-1,1}+n_{-1,1,1})+2(p^d+1)^2p^m(n_{1,1,2}+n_{-1,-1,2}) &+2(p^d-1)^2p^m(n_{1,-1,2}+n_{-1,1,2}). \end{array}
\dot{N}(t) \geq -\|\lambda(N(t),F(t),S(t),\cdot)\|_{\mathrm{L}^\infty(\Omega)}\int_{\Omega}u(x,t)\, dx=-\|\lambda(N(t),F(t),S(t),\cdot)\|_{\mathrm{L}^\infty(\Omega)}N(t),
\varepsilon_{0}=\frac{m^{3}}{3\pi}-\frac{m^{3}}{2\pi}\frac{1}{\Gamma(-1/2)}\sum_{n=1}^{m}\frac{B_{2n}}{(2n)!}\left(\frac{2eB}{m^{2}}\right)^{2n}\Gamma(2n-3/2)+O\left[\left(\frac{2eB}{m^{2}}\right)^{2(m+1)}\right] .
F_{14}(a_1,a_1,a_1,b_1,b_2,b_1;c_1,c_2,c_2;x_1,x_2,x_3) = \sum_{m,n,p=0}^\infty\frac{(a_1)_{m+n+p}(b_1)_{m+p}(b_2)_n}{(c_1)_m(c_2)_{n+p}}\frac{x_1^mx_2^nx_3^p}{m!n!p!},
\Gamma_{a a}^{( 2 )}( k ) = \coth{\frac{\beta k^{0}}{2}}\Bigl [ \Gamma_{a r}^{( 2 )}( k ) - \Gamma_{r a}^{( 2 )}( k ) \Bigr ] , \qquad \Gamma_{a r}^{( 2 )}( k ) = \Gamma_{r a}^{( 2 ) *}( k ) .
w' = (s_k \cdots s_n \cdots s_k u_1)\cdot(u_2 s_{k-1}\cdots s_1 u_{n,k}^{-1}) = (u_2 s_k \cdots s_n \cdots s_k u_1) \cdot (s_{k-1}\cdots s_1 u_{n,k}^{-1}).
\sum_{r=0}^{n+k}\sum_{s=0}^{r}(-1)^{s}\tilde{E}_{s,n}\tilde{H}_{r-s,n}(u|t)^{n-s}(u|\tau^{n-s+r}t)^{k-r+s}=(u|t)^{N}+(-1)^{n}q\;.
\widehat{P}(t, \omega) =A(\omega)e^{-\frac{1}{2}\sigma^{2}q(\omega)t}+\frac{rK}{\omega}\int_{t}^{T}(\mathcal{B}(s))^{\omega}e^{\frac{1}{2}\sigma^{2}q(\omega)(s-t)}\mathrm{d}s,
J(x, y) := \kappa(x,y) j(|x-y|) = \begin{cases}\kappa(x,y) |x-y|^{-d-\alpha}\psi (|x-y|)^{-1}\beta \in [0, \infty), \kappa(x,y) |x-y|^{-d-\alpha}{\bf 1}_{\{|x-y| \le1\}}\beta = \infty. \end{cases}
\begin{aligned}\langle d\lambda,\mu\rangle_{L^{2}\Lambda^{k}(\Omega)}=\langle\lambda,\delta\mu\rangle_{L^{2}\Lambda^{k-1}(\Omega)}+\int_{\partial\Omega}(\lambda)\wedge(\ast\mu) =\langle\lambda,\delta\mu\rangle_{L^{2}\Lambda^{k-1}(\Omega)}+\int_{\partial\Omega}\mathrm{tr}(\lambda)\wedge\ast\boldsymbol{n}(\mu),\end{aligned}
\partial \Phi^n_{U}(f) [\phi] = \int_U f(u, v) (\partial_u \phi)^n(u) \, \mathrm{d}u \mathrm{d}v = \int_{\mathcal{I}}\left( \int_{\mathbb{R}}f(u, v) \, \mathrm{d}v \right) (\partial_u \phi)^n(u) \, \mathrm{d}u = \Psi^n (\eta f) [\partial_{\Sigma}\phi]
\psi^C_{i,j}=\begin{cases}\sum_{k=1}^{i}\left( \prod_{\ell=i+1}^j \alpha_{k,\ell}\right) \partial_{{k}}\mbox{if}j \ne 2n+1-i, \sum_{k=1}^{i}\left ( \prod_{\ell=i+1}^n (x_{k}-x_\ell)(x_{k}+x_\ell)\right) x_{k}\partial_{{k}}\mbox{if}j=2n+1-i.\end{cases}
\langle F\rangle_{{\rm ML}, \hat g}= \langle F(\cdot-\tfrac{Q}{2}\omega)\rangle_{{\rm ML},g}\times \exp\big( \tfrac{1+6Q^2}{96\pi}S^{{\rm cl},0}_{{\rm L}}(\hat g,g)+\beta S^{{\rm cl}}_{\rm M}(\hat g, g)\big)
\omega^{(1)}= \left(\omega^{(1)}- \sum_{1 \le i_1 < \alpha^\prime+N-m-1}\omega^{(1)}_{i_1}x^{i_1}\right) + \sum_{1 \le i_1 < \alpha^\prime+N-m-1}\omega^{(1)}_{i_1}x^{i_1}
= \; ( - 1 )^{N}\frac{1}{( 2 \pi )^{N + l}}\left( \frac{1}{2 L}\right)^{N + l}\exp \left( \frac{1}{2}[ N - l ]^{2}I_{2}\; + \; \frac{1}{2}[ N + l ]^{2}I_{3}\right) \; .
H^{2}= - \frac{k}{a^{2}}+ \frac{\omega_{4}M}{a^{4}}- \frac{3 \omega_{4}^{2}Q^{2}}{1 6 a^{6}}- \frac{1}{L^{2}}+ \frac{4 \pi}{3 M_{p}^{2}\rho_{0}}\left( \rho_{0}+ \rho_{\mathrm{b r}}\right)^{2},
L:=\exp\left\{\sum_{i=1}^{\infty}\frac{2\mu_{i}Z_{i}-\mu_{i}^{2}}{2\sigma_{i}^{2}}\right\},L^{(m)}:= \exp\left\{\sum_{i=1}^{m}\frac{2\mu_{i}Z_{i}-\mu_{i}^{2}}{2\sigma_{i}^{2}}\right\}, m\in \mathbb{N}.
F_{N}(a,b;t) =1-\frac{b(1-aq)(1-q^{N+1})(1-tq^{N+1})}{(1-tq^N)(1-bq^{N+2})(b-aq)}+ \frac{(1-aq)(1-q^{N+1})(b-aqt)}{(1-tq^N)(b-aq)(1-bq^{N+2})}F_{N}(aq,b;t).
\begin{array}{c}eC \supset g_{\alpha}d_{\alpha}g_{\alpha}^{-1}C = (Ug_{\alpha}) d_{\alpha}(Ug_{\alpha})^{-1}C = (h \psi_{\alpha}) d_{\alpha}(h \psi_{\alpha})^{-1}C =h (\psi_{\alpha}d_{\alpha}\psi_{\alpha}^{-1}) h^{-1}C\,.\end{array}
e_1{\phi}_{(1)}^{\rm self}+ e_2{\phi}_{(2)}^{\rm self}=\frac{1}{8{\rm L}}(e_1^2 + e_2^2) (x_{+}^2 + x_{-}^2)+ \frac{1}{4{\rm L}}(e_1^2 - e_2^2) x_{+}x_{-}+ \frac{{\rm L}}{4}(e_1^2 + e_2^2),
\frac{\partial}{\partial t}\nabla R =\nabla\big((1+\frac{\alpha'}{2}R)\Delta R+\frac{\alpha'}{2}|\nabla R|^2+R(R+\frac{\alpha'}{4}R^2-r)\big) &=\big(1+\frac{\alpha'}{2}R\big)\nabla\Delta R+\frac{\alpha'}{2}\nabla R\Delta R+\frac{\alpha'}{2}\nabla|\nabla R|^2+2R\nabla R+\frac{3\alpha'}{4}R^2\nabla R-r\nabla R.
\times \exp \left( 2 \sum_{b , b^{\prime}= 1}^{N}\sum_{j , j^{\prime}= 1}^{n_{b}, n_{b^{\prime}}}\Big ( \delta_{n}( x_{j}^{( b )}) \! - \! \delta_{n}( y_{j}^{( b )}) , \Big [ ( \frac{e \pi}{\pi \! + \! g N})^{2}\tilde{Q}+ g \tilde{C}\Big ] \Big [ \delta_{n}( x_{j^{\prime}}^{( b^{\prime})}) \! - \! \delta_{n}( y_{j^{\prime}}^{( b^{\prime})}) \Big ] \Big ) \right)
r_1 = (b-c) \{a^2+(b+c)^2 \}(R_{1212}R_{1313}-R_{1213}{}^2), & r_2 = 2(a+d)(b^2-c^2) (R_{1212}R_{1313}-R_{1213}{}^2), & r_4 = (b-c) \{(b+c)^2+d^2 \}(R_{1212}R_{1313}-R_{1213}{}^2).
{h_{\varepsilon,\infty}}(x) |\overline{\nabla}U\big(r_{\varepsilon,\infty}(\xi)\cdot\xi\big)|^2 \det\!\big(\nabla^2h_{\varepsilon,\infty}(x)\!+\! h_{\varepsilon,\infty}(x)I\big)\!=\!\gamma_{\varepsilon}f(x)\widehat{\psi}_\varepsilon(h_{\varepsilon,\infty}),
\rho(R,z)=\frac{\partial_{z}\partial_{z}\Phi}{4\pi G}+\frac{\partial_{R}(R\partial_{R}\Phi)}{4\pi GR}=\frac{M_{0}R_{0}/z_{0}}{4\pi(R^{2}+R_{0}^{2})^{3/2}}H(z_{0}-|z|),
S = \int \, d^D \, x \, \sqrt{g}\left[ R(g) - \partial_{\mu}\, \varphi \, \partial^{\mu}\, \varphi - \sum_p \ \frac{1}{2}\, \frac{1}{(p+1)!}\e^{\lambda_p \, \varphi}\, (d \, A_p)^2 \right] \, .
h(\sqrt{b}X_2^n+Z_2^n) =h(\sqrt{b}(X_2^n+Z_3^n)+\sqrt{1-b}\tilde{Z}^n) \geq\frac{n}{2}\log\left[ 2^{\frac{2}{n}h(\sqrt{b}(X_2^n+Z_3^n))}+ 2\pi e(1-b) \right]
P(C,L/F):=\min\{\mathrm{N}_{F/\mathbb{Q}}~\mathfrak{p}\colon \textup{$\mathfrak{p}$ unramified in $L$, $[\tfrac{L/F}{\mathfrak{p}}]=C$, $\mathrm{N}_{F/\mathbb{Q}}~\mathfrak{p}$ a rational prime}\}.
\mathrm{\cfrac{d}{dt}}\, F(t, x) \Big|_{t = \tau_{\pm}^r (x)}= D\varphi \big( X(\tau_{\pm}^r (x), x) \big) \cdot \dot X (\tau_{\pm}^r (x), x) = 8 X_1 (\tau_{\pm}^r (x), x) X_2 (\tau_{\pm}^r (x), x) \not= 0,
u = \sum_{j=1}^{r+2}(-1)^{n-r-3+j}C(2n-4-r,n-r-3+j)e_{j,n-r-2+j}v = \sum_{j=1}^{r+2}(-1)^{j-1}C(r,j-1) e_{n-r+j-2,j}.
\lim_{\lambda\rightarrow\infty}\sup_{i\in I}\max_{z\in V(G_i)}\mathbf{P}^{G_i}_z\left(\max_{x,y\in V(G_i)}\max_{0\leq t\leq Td(G_i)^{\beta}}\frac{d(G_i)^{-(\beta-\alpha)}\left|L^{G_i}_{t}(x)-L_t^{G_i}(y)\right|}{\sqrt{\tilde{d}_{G_i}(x,y)^{\beta-\alpha}\left(1+\ln \tilde{d}_{G_i}(x,y)^{-1}\right)}}\geq \lambda\right)=0
\langle \mathcal{V}_{f,\overline{g}},\mu_j \rangle = \langle T_{-1}\mathcal{V}_{f,\overline{g}},\mu_j \rangle = \langle \mathcal{V}_{f,\overline{g}},T_{-1}\mu_j \rangle = - \langle \mathcal{V}_{f,\overline{g}},\mu_j \rangle.
|\Gamma_{\ell,0,0}|=(q-1)^3(q^2+q+1)(q+1)q^{3\ell+6}&|\Gamma_{\ell,\ell,0}|=(q-1)^3(q+1)^2q^{4\ell+6}|\Gamma_{\ell,\ell,\ell}|=(q-1)^3(q^2+q+1)(q+1)q^{3\ell+6}.
T^{\,n}\Bigl(\sum_{s=1}^{j_{m}+j}z_{s}\Bigr)-x_{l}=T^{\,kd_{j_{m}+j}+i}\, \Bigl(\sum_{s=1}^{j_{m}+j}z_{s}\Bigr)-x_{l}=T^{\,i}\,\Bigl(\sum_{s=1}^{j_{m}+j}z_{s}\Bigr)-x_{l}\qquad\hbox{by (i)}\smash[b]{=T^{\,i}\,\Bigl(\sum_{s=1}^{j_{m}+j-1}z_{s}\Bigr)-x_{l}+T^{\,i}z_{j_{m}+j}.}
\partial^{\alpha}u_j(x)=\left\{\sum_{k = 1}^3 \beta_k \, \partial^{\alpha}\mbox{$\mathfrak Z$}_{jk}(x,0)+ \left(\int_{\partial \mbox{$\mathfrak D$}}u \cdot n^{( \mbox{$\mathfrak D$})}\, do_z\right) \, \partial^{\alpha}E_{4j}(x)\right\}+\partial^{\alpha}\mbox{$\mathfrak F$}_j(x),
P_{{\cal S}}( A ) = \sum_{j = 1 , 2}P_{{\cal S}_{j}/{\cal S}}P_{{\cal S}_{j}}( A ) \pm 2 \sqrt{P_{{\cal S}_{1}/{\cal S}}P_{{\cal S}_{1}}( A ) P_{{\cal S}_{2}/{\cal S}}P_{{\cal S}_{2}}( A )}\cosh \theta ( A ;{\cal S},{\cal S}_{j}) \; .
\partial_y(Q^{(\kappa+1)}_nW^{(\kappa+1)})=G^{(\kappa)}_n(\partial_y^{n+1}\,W^{(\kappa+n+1)}(y))=G^{(\kappa)}_n(G^{(\kappa)}_{n+1})^{-1}Q^{(\kappa)}_{n+1}W^{(\kappa)}.
\begin{array}{lll}\mu\circ ((f\otimes\eta)\otimes\eta) = \mu\circ (id_M\otimes \eta)\circ ((f\circ\eta)\otimes id_I) &= \rho_M\circ ((f\circ\eta)\otimes id_I) &= (f\circ \eta)\circ \rho_I & \mbox{(by naturality of $\rho$)}\end{array}
\mathbb E\left[\langle \tilde Z_n^N(i), \tilde Z_n^N(j)\rangle_{\mathcal H}\right]= \mathbb E\left[\mathbb E\left[\langle \tilde Z_n^N(i), \tilde Z_n^N(j)\rangle_{\mathcal H}|\mathcal F_{(i-1)\Delta_n}\right]\right] = \mathbb E\left[\langle \mathbb E\left[\tilde Z_n^N(i)|\mathcal F_{(i-1)\Delta_n}\right], \tilde Z_n^N(j)\rangle_{\mathcal H}\right]=0.
V_1\colon y^2=((3a-b-3)x-a+3)(1+(a-3)x(1-x)), V_2\colon x^3+y^3+1+a(x^2y+xy^2+x^2+x+y^2+y)+bxy=0, V_3\colon y^2=-((a+1)x^3+(2a+b)x^2+4ax+4)(x^3+ax^2+ax+1).
F_{n}(a,q;x)=(-1)^{n}q^{-n(n-1)/2}\sum_{j=0}^{n}\frac{q^{jn}(q^{-n};q)_{j}}{(q;q)_{j}}\,\,_{2}\phi_{1}(q^{j-n},q^{j+1};q^{-n};q,q^{-j}a)\, x^{j}.
\alpha_j = - \sum_{i=0}^J{\sf A}_{ij}^{-1}T^{\frac{\gamma}{2}+j}\int_{-\log T}^\infty e^{(\frac{\gamma}{2}+j)\tau_1}(T-t) (G,e_i)_\rho d\tau_1 \lesssim T^{\frac{\gamma}{2}+j}\int_0^T (T-t)^{-1+J-j+{\sf c}_1}dt \lesssim T^{\frac{\gamma}{2}+J+{\sf c}_1}.
R_0\leq\min\{I(U;S_1),I(U;S_2)\}, &R_0+R_i\leq\min\{I(U;S_1),I(U;S_2)\}+H(S_i|U),i=1,2, &R_0+R_1+R_2\leq\min\{I(U;S_1),I(U;S_2)\}+H(S_1,S_2|U)
\begin{aligned}B^m= \left.(-1)^m\rho^{m+2}{n-m \choose k-m}\dfrac{k+n-2m}{n-m}\left[\frac{\phi'^{k-m}}{D^{k+1}}\phi^{n-m-1}\right]\right|_{u=|\nabla u|^2=0}= (-1)^m\rho^{m+2}{n-m \choose k-m}\dfrac{k+n-2m}{n-m}\phi'^{k-m}(\rho)\phi^{n-m-1}(\rho).\end{aligned}
H_{j,k}(x_{j,0}^{-})^{m_j}v^{+}= (x_{j,0}^{-})^{m_j}H_{j,k}v^{+}+[H_{j,k},(x_{j,0}^{-})^{m_j}]v^{+}&=(x_{j,0}^{-})^{m_j}H_{j,k}v^{+}+\sum_{r=0}^{m_j-1}(x_{j,0}^{-})^{r}[H_{j,k},x_{j,0}^{-}](x_{j,0}^{-})^{m_j-r-1}v^{+}.
S_{6}:=[\begin{matrix}1&1&1&1&1&1\\ 1&1&z&z&z^{2}&z^{2}\\ 1&z&1&z^{2}&z^{2}&z\\ 1&z&z^{2}&1&z&z^{2}\\ 1&z^{2}&z^{2}&z&1&z\\ 1&z^{2}&z&z^{2}&z&1\\ \end{matrix}]\in H(3,6)
\Delta = \sum_{j=1}^2 \sum_{p,q}\left(\frac{\partial^2}{\partial ({\rm Re}\,\sigma_{pq}^{jj})^2}+ \frac{\partial^2}{\partial ({\rm Im}\,\sigma_{pq}^{jj})^2}\right) + 4 \sum_{j\neq l}\sum_{p,q}\frac{\partial^2}{\partial\sigma_{pq}^{jl}\sigma_{pq}^{jl*}}\ .
\eta (x,t) = \mbox{Re}\left ( \frac{- 3 A_0^2}{2 i q (2 - q^2 - 3 i q)}\exp i (\omega t - q x)\int_{-\infty}^{\infty}\phi_K (\xi) \eta_D^2 (\xi) \eta_{q}(\xi)\mbox{d}\xi\right )
\Pi ( q^{2}) = -{\frac{1}{4 8 \pi^{2}}}\left[{\frac{1}{\bar{\epsilon}}}+{\frac{1}{2}}\ln \left({\frac{\mu^{2}}{4 k T}}\right) \right] +{\frac{1}{1 6 \pi^{2}}}\int_{0}^{1}d x x \sqrt{1 - x^{2}}\ln N_{+ +}\left({\frac{x \sqrt{q^{2}}}{2}}\right) ,
\| A - A_m\| = \Big \| \sum_{j=1}^\infty (M_{\varphi_{j,\frac{1}{m}}}A - M_{\varphi_{j,\frac{1}{m}}}A M_{\psi_{j,\frac{1}{m}}}) \Big \| &= \Big \| \sum_{j=1}^\infty M_{\varphi_{j,\frac{1}{m}}}AM_{1-\psi_{j,\frac{1}{m}}}\Big \| &\to 0
I_j =(eRe)\ast_f f_j \ast_f (eRe)=eRf_jRe= &= f_jRf_jRf_j+f_jRf_jRf_0+f_0Rf_jRf_j+f_0Rf_jRf_0= &= f_jRf_j+f_jRf_0+f_0Rf_j+f_0Rf_jRf_0 .
\Omega_{\pi^{+}, \pi^{-}, m , n}^{( 2 )}={\cal P}_{\pi^{+}, \pi^{-}, m}^{( 1 )}( z ) \tilde{\cal O}_{\pi^{+}, \pi^{-}, n}^{( 1 )}( \bar{z}) d z \wedge d \bar{z}
\alpha_j^{(t)}=\frac{\sum_{m=1}^{K}{f(y_{t-1}|Z_{t-1}=m,\mathcal{I}_{t-2})}\alpha_m^{(t-1)}p_{m,j}}{\sum_{m=1}^{K}{f(y_{t-1}|Z_{t-1}=m,\mathcal{I}_{t-2})}\alpha_m^{(t-1)}},
{\tilde{\gamma}_{5}}={\frac{\epsilon^{c d}}{4 \sqrt{g}}}{\hat{\epsilon}^{E F}}\partial_{c}X^{\mu}\partial_{d}X^{\nu}N_{E}^{\lambda}N_{F}^{\rho}\frac 1 4 [ \gamma_{\mu}, \gamma_{\nu}] [ \gamma_{\lambda}, \gamma_{\rho}] .
\rho_{n}\left( k\right) \leq\rho_{n}\left( n/\lambda_{0}\right) =\frac{{\displaystyle\sum\limits_{m=0}^{n}}a_{m,n}\left( \frac{\lambda_{0}}{n}\right)^{2m}}{{\displaystyle\sum\limits_{m=0}^{n}}\left( m+1\right) a_{m,n}\left( \frac{\lambda_{0}}{n}\right)^{2m}}=:\rho_{n}^{\operatorname{I}}.
(\mathcal{A}(G^{k,s})y)_{w}=[(x^{e})^{\frac{t}{k}}]^{s}[(\mu^{-1}x^{e})^{\frac{1}{k}}]^{k-ts-1}& =\mu^{\frac{ts}{k}}(\mu^{-1}x^{e})^{\frac{k-1}{k}}& =\mu^{\frac{ts}{k}}y_{w}^{k-1}.
p(t,x):=\pi^{-\frac{d}{2}}|x|^{-d}\textnormal{H}_{12}^{20}\left[ \frac{1}{4}t^{-\alpha}|x|^{2}\ \Big|\begin{array}{cc}(1,\alpha) (\frac{d}{2},1) (1,1)\end{array}\right].
Q_{\alpha A}=T_A^{\,\,\,\,a}Q_{\alpha a},\quad Q_{\dot{\alpha}\bar{A}}=T_{\bar{A}}^{\,\,\,\,\dot{a}}Q_{\dot{\alpha}\dot{a}},\quad T_{\bar{A}}^{\,\,\,\,\dot{a}}=\left( T_A^{\,\,\,\,a}\right)^{*}
P_n(t,\beta)= P_n(t,\alpha=0,\beta)= t^n+ \sum_{r=1}^{n-1}t^{r}\left( \sum_{k=0}^{n- 1-r}{r \over n}{n \choose k+r}{n-r-1 \choose k}(1+\beta)^{n-r-k}\right),
Q^{\{3\}}_{\{a,t\}}(m(q))=q^{-a/12+t^2/(4a)}\exp\left(-\sum^{\infty}_{n=1}q^n\sum_{\scriptsize\begin{array}{cc}AB=n B\equiv \pm (a-t)(mod 2a)\end{array}\normalsize}\frac{(-1)^A}{A}\right).
\begin{cases}({J}_\lambda \Phi)_0 =0, ({J}_\lambda \Phi)_1(\theta) =\displaystyle\int^0_\theta e^{\lambda(\theta-\tau)}\phi_1(\tau)d\tau,\hskip 10pt \theta\in [-r, 0],\hskip 10pt \hbox{for}\hskip 10pt \Phi\in{\cal X},\end{cases}
\widetilde{f}= \sum_{I<J, |I|<d}V_I\overline{Y}^I + \overline{Y}^J + \sum_{J<I, |I|<d}pV_I\overline{Y}^I & \sum_{|I|\geq d}\left\{\sum_{K<J, |I|<d}b_{IK}V_K + b_{IJ}+ \sum_{J<K, |K|<d}b_{IK}pV_K\right\}\overline{Y}^I.
\frac{\partial f}{\partial x_1}(x) = \sum_{|I| = d}b_I I_1 x_1^{I_1 - 1}x_2^{I_2}\cdots x_n^{I_n}\ge b_{(1, d - 1, 0, \ldots, 0)}\cdot 1 \cdot x_2^{d - 1}> 0.