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ToniDO
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test_latexformula

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\begin{align*}f(e^{\gamma},n)= & f(e^{\sum_{i=1}^{n}\gamma_{i}},n)=\prod_{i=1}^{n}f(e^{\gamma_{i}},1)=\prod_{i=1}^{n}f(e^{\delta},1)\phi(e^{\delta-\gamma_{i}})\\= & f(e^{\delta},1)^{n}\phi(e^{n\delta-\sum_{i=1}^{n}\gamma_{i}})=f(e^{\delta},1)^{n}\phi(e^{n\delta-\gamma})\end{align*}
\begin{align*}(\bar{i}_1^{p,q})^* \circ S_{E_2} \circ (\bar{r}^*)_0^{n-p,n-q}=(\bar{i}^{p,q})^* \circ S_{E_2} \circ (\bar{r}_1^*)_0^{n-p,n-q}=0.\end{align*}
\begin{align*}f(g) = \frac{g^2}{4 \pi^2} +...\end{align*}
\begin{align*}K_{2 i t}(x) = \frac{1}{2} \cosh(t \pi)^{-1} \int_{-\infty}^\infty \cos(x \sinh \zeta) e\Bigl( -\frac{t \zeta}{\pi} \Bigr) \, d\zeta\end{align*}
\begin{align*} M := \sup_{t \in (-T_-,T_+)} \|(u(t),\partial_t u(t))\|_{H^{\sigma-\frac{1}{2}, \frac{1}{2}}\times H^{\sigma-\frac{1}{2},-\frac{1}{2}}} < \infty.\end{align*}
\begin{align*}v \in H^{h}_0(\Omega) \sum_{0\leq|\alpha|,|\beta|\leq h} \int a_{\alpha, \beta}D^\beta v D^\alpha w = f(w) w \in H^h_0(\Omega).\end{align*}
\begin{align*}\| \hat K - 1 \|_{L^p(\hat \mu)} = o (1).\end{align*}
\begin{align*}\tilde{p}_j(x)=\theta(p_j(x)-\bar{u}_j)+\bar{u}_j, \theta=\min\Big\{\Big|\frac{M-\bar{u}_j^n}{M_j-\bar{u}_j^n}\Big|,\Big|\frac{m-\bar{u}_j^n}{m_j-\bar{u}_j^n}\Big|,1\Big\},\end{align*}
\begin{align*}{\cal{L}}_{\theta} = -{1 \over 2}e^{-2(\tilde{U}+\tilde{\Phi})} (\partial_0 \Psi)^2 \ .\end{align*}
\begin{align*}\sum_{m_1}\lambda_{1}(m_1)e\left(\frac{dm_1}{c}\right)W_{\eta H}\left(\frac{m_1-n}{H}\right)=\frac{H}{c}\sum_{m_1}\lambda_{1}(m_1)e\left(-\frac{\bar{d}m_1}{c}\right)W^{\star}_{\eta H}\left(\frac{m_1n}{c^2},\frac{m_1H}{c^2}\right),\end{align*}
\begin{align*}\delta_c (m) (a) :=~& x \cdot_1 m - m \cdot_1 x = x \cdot_2 m - m \cdot_2 x, m \in C^0_{com} (\mathfrak{g}, M) x \in \mathfrak{g}, \\\delta_c (h_1, \ldots, h_n ) :=~& ( \delta_1 h_1, \ldots, \underbrace{\delta_1 h_i + \delta_2 h_{i-1}}_{i-}, \ldots, \delta_2 h_n),\end{align*}
\begin{align*}\begin{array}{rcl}d\left(\hat{G}_{(5)} +\hat{H}\hat{C}_{(2)}\right) & = & 0\, ,\\& & \\d\left(\hat{\overline{G}}_{(5)} +\hat{H}\hat{\overline{C}}_{(2)}\right) & = & 0\, ,\\\end{array}\end{align*}
\begin{align*}A^{++}e^{nr}=e^{n+1,r}\quad A^{+-}e^{nr}=\frac{s-r+1}{n+s-r+1}e^{n,r+1}\end{align*}
\begin{align*}\Lambda^{+}_{0}(x) N_{1,\varphi}^{+}(x) + N_{1,\varphi}^{-}(x) = N_{\varphi}(x), \; x\in {\mathbb R}.\end{align*}
\begin{align*} \alpha_z &= g\circ(f\circ\alpha_x\circ (\overline{f}\ast\det))\circ (\overline{g}\ast\det)\\ &= (g\circ f)\circ\alpha_x\circ ((\overline{f}\circ\overline{g})\ast\det). \end{align*}
\begin{align*}\begin{array}{l} \beta(1_b)(1_a) = r\cdot 1_a,\\ \beta(1_b)(x) = r_x\cdot 1_a,\\ \beta(1_b)(y) = r_y\cdot 1_a,\end{array}\end{align*}
\begin{align*}dF_{\gamma} (x) := e^{-\gamma x} dF(x) / f^*(\gamma), x \geq 0,\end{align*}
\begin{align*} g_{1}^*\cdot g_{2}^*=X(g_{1}, g_{2})\, (g_{1}g_{2})^*,\end{align*}
\begin{align*}\log m = -a\left(\frac{1}{2}\right)^{n_w} + \log m_0\end{align*}
\begin{align*}E^m(g, \phi) = L^m(\phi), \phi \in \Phi\,, \end{align*}
\begin{align*}H^\dagger=H\quad{\rm for}\quad q=e^{i\pi/r},\ r=2,3,\dots\end{align*}
\begin{align*}X_-(t)Y_-(t)= \sum_{k+\ell\leq s}\sum_{i:\lambda_i<0 } \xi^{(k,\ell)}_i(t)\mathcal{E}^{(k,\ell)}_i(t).\end{align*}
\begin{align*} \partial(\phi) = [\partial,\phi] = \partial\circ\phi - (-1)^{|\phi|}\phi\circ\partial=\partial\circ\phi - (-1)^{|\phi|\cdot|\partial|}\phi\circ\partial.\end{align*}
\begin{align*}-{^\ast}A_\mu^2J^{\mu 1}+{^\ast}A_\mu^1J^{\mu 2}.\end{align*}
\begin{align*}(g_{ij}) = \left(\begin{array}{ccc} -1 & - \cosh \vartheta & 0 \\ - \cosh \vartheta & -1 & 0 \\ 0 & 0 & 1 \end{array} \right)\end{align*}
\begin{align*}K(\gamma^1_p,\gamma^2_p):= \operatorname{Tr}_g\left(\gamma\left(\mathcal{A}\!\cdot\!\gamma^1_p,\mathcal{A}\!\cdot\!\gamma^2_p\right)\right),\end{align*}
\begin{align*} \nabla_{z \partial z} S^\textnormal{coh}(\tau,z)z^{-\mu}z^\rho = 0 \end{align*}
\begin{align*}&w:=\widehat{\textsf{c}}(\theta),&&\Theta(w):= \widehat{\textsf{c}}^{-1}(w),&&K(w):=\frac{\mathsf{K}(\Theta(w))}{\textsf{c}(\Theta(w))}.\end{align*}
\begin{align*}[f^{0}(\vec{x}),\phi(\vec{y})]=-\frac{g}{m}\phi(\vec{x})\delta(\vec{x}-\vec{y}),\end{align*}
\begin{align*}T_4&{}:=X_4, &T_3&{}:=X_3, & T_2&{}:=X_2-\frac{q^4}{(q^2+1)(q+q^{-1})} X_3^2 X_4^{-1}, & T_1&{}:= X_1-\frac{q^2(q+q^{-1})}{q^2-1}X_2 X_3^{-1}.\end{align*}
\begin{align*}F_2(z)\frac{dF_1(z)}{dz}- F_1(z)\frac{dF_2(z)}{dz}=F_0(z),z\in\mathbb R,\end{align*}
\begin{align*}\sum_i N_i \int_{B_i} \omega_j =\int_{B_j} \sum_i N_i \,\omega_i=\tau \ ,\end{align*}
\begin{align*}\widehat{\tilde K_\delta}(\xi,\tau) = \phi(|\xi|)\tilde\psi\left(\frac{|\xi|-\tau}{\delta}\right) .\end{align*}
\begin{align*}E_{ql}=2e^{-2\Phi }\partial _{n}\Phi =-\partial _{n}X\,\end{align*}
\begin{align*}\left\{H_{n},H_{m}\right\}_{1} = \int dx\;\frac{dX}{dx} = 0\;,\end{align*}
\begin{align*}\underline{\mathfrak{h}}_{x}={\rm Span\,}\left\{ \xi_{M',H}(x);\, \xi\in\mathfrak{h}\, \right\}.\end{align*}
\begin{align*}\prod_{i=1}^m \left(p^{km}-p^{k(i-1)}\right),\end{align*}
\begin{align*}&M''(t) M(t) - \frac{\omega+3}{4} (M'(t))^2 \geq M(t)\left( M''(t) - (\omega +3)\left( ||u_t||^2 + b\int_{0}^t ||u_s(s)||^2_{L^2(\mathbb{H}^n)} ds \right) \right)\\&=M(t) \left( -(\omega+1) ||u_t||^2_{L^2(\mathbb{H}^n)} - (\omega +3)b\int_{0}^t ||u_s(s)||^2_{L^2(\mathbb{H}^n)} ds -2I(u)\right),\end{align*}
\begin{align*} f^-_M(z_{M+1},\dots,z_n) = f(u_2, \dots, q_i^{-M+1} u_2, z_{M+1},\dots, z_n) \end{align*}
\begin{align*}g_1&=3e_0-e_1-2e_2-e_3-e_4-e_5,&g_2&=3e_0-e_1-e_2-2e_3-e_4-e_5,\\g_3&=3e_0-e_1-2e_2-e_4-e_5,& g_4&=3e_0-e_1-2e_3-e_4-e_5.\end{align*}
\begin{align*}\tau^a=\tau^b=0.\end{align*}
\begin{align*} \langle \mu, y(T, \cdot) \rangle = \langle \mu, y^0 \rangle + \mathcal{O}\left(T^{1/2} |\mu|_2 |y^0|_{H^2} \right). \end{align*}
\begin{align*}i_{\Gamma}^* A := (I, \Gamma)^T A (I,\Gamma) = B + C\Gamma + \Gamma^T C^T + \Gamma D \Gamma^T.\end{align*}
\begin{gather*} \mathfrak{R}_{w}(1,\beta)=\mathfrak{R}_{1 \times w}(0,\beta). \end{gather*}
\begin{align*}H_{Cal}^{(1)} ={1 \over 2} \sum_{i=1}^N \left[ -\partial_i^2 + x_i^2 \right] + \sum_{j < i}^N \frac g {(x_i-x_j)^2} + \frac{\gamma}{\tau_2}\ ,\end{align*}
\begin{align*}\mu^\alpha{}_i dz^i = 0, \qquad \alpha = 1, \ldots, m-n.\end{align*}
\begin{align*}\int^{+\infty}_{A_{3,r}}\frac{t^5}{(27+t^{12})^{2/3}}dt=\frac{\pi}{6}\int^{+\infty}_{\sqrt{r}}\eta_1(it)^4dt\end{align*}
\begin{align*} i(\overline{\beta}t-1)= \sqrt{|\beta|^2-1}\,(\sin\theta- \cos\theta) + i(\cos \theta+\sin \theta - 1). \end{align*}
\begin{align*}\int_r^{\mu-\mu^{1/3}} |H^{(1)}_\mu(r')|^q (r')^\rho \,dr' \leq r^{q/2} |H^{(1)}_\mu(r)|^q \int_r^\infty (r')^{\rho-q/2} \,dr' = \frac{r^{\rho+1}}{q/2-\rho-1} |H^{(1)}_\mu(r)|^q \,.\end{align*}
\begin{align*}H=\frac A{abc}e^1\wedge e^2\wedge e^3=A\sigma ^1\wedge \sigma ^2\wedge\sigma ^3,\end{align*}
\begin{align*}O_i(Q):=\left\langle\prod_{\ell\in I_i}\tau^{(a_\ell,b_\ell)}_{d_\ell}\sigma_1^{k_1(i)}\sigma_2^{k_2(i)}\sigma_{12}\right\rangle^{\mathbf{s}^{\Gamma_{0,k_1(i),k_2(i),1,I_i}},o}\end{align*}
\begin{align*} \sqrt{e^{-2\pi t} \sin^2{\pi\sigma} + (1 \pm e^{-\pi t}\cos{\pi\sigma})^2} = \sqrt{1 + e^{-2\pi t} \pm 2 e^{-\pi t}\cos{\pi\sigma}} &\geq \sqrt{1 + e^{-2\pi t} - 2 e^{-\pi t}}\\ &= \sqrt{(1 - e^{-\pi t})^2} \\ &= 1 - e^{-\pi t}.\end{align*}
\begin{align*}\frac{1}{|{\mathcal K}|}\sum_{k\in {\mathcal K}}S_{hk}(x) =BC(h)x\{1+O(1/\log x)\}\ . \end{align*}
\begin{align*} \frac{\partial \textbf{\textit{W}}}{\partial t} -(B-Q)\textbf{\textit{W}}=-\textbf{\textit{Y}}. \end{align*}
\begin{align*}z_a=-z_a^{tr},z_az_b+z_bz_a=-2\delta_{ab}I,2\leq a,b\leq 7-r.\end{align*}
\begin{align*}a_i^{({\rm eff})}(x)= {\theta_i \over L_i}- (Q +1) \tilde x_i+\epsilon_{ij} \partial_{x^j}\int \! d \vec y G(\vec x, \vec y)b^{(1)}(y) -{e \over \mu c}\epsilon_{ij} \partial_{x^j}\int \! d \vec y G_p(\vec x, \vec y)J_0(y)\,. \end{align*}
\begin{align*} G_2= \left( \begin{array}{cccccc} 1 & 0 & 1 & 0 & 1 & 1\\ 0 & 1 & 0 & 1 & 1 & 1\\ \end{array} \right). \end{align*}
\begin{align*} \Delta_n \leqslant\frac{A}{n \bar{B}_n^{3/2}} \sum\limits_{i,j=1}^n\left(|\mu_{ij}+\bar{a}_{ij}^\ast|^{3} + E|\bar{X}_{ij}-\bar{a}_{ij}|^3 \right)+\frac{1}{n} \sum\limits_{i,j=1}^n P(|X_{ij}-\mu_{ij}| \geqslant b_n)+ \Theta_n + \Upsilon_n\end{align*}
\begin{align*}\{a_n\}=\{0,\frac 13,1,\frac 43,\frac 43,2,2,\frac 73,\frac 73,\frac73,3\ldots\}\end{align*}
\begin{align*}\left\{\begin{aligned}&A=-\frac{p'(x^{*})}{k_1(x^{*})^{k_1-1}},\\&\bar{\theta}_2=-\frac{K}{A(x^{*})^{k_1}+p(x^{*})}=\frac{Kk_1}{k_1p(x^{*})-p'(x^{*})x^{*}}.\\\end{aligned}\right.\end{align*}
\begin{align*}(\alpha,\beta)=\{t\alpha+(1-t)\beta:t\in(0,1)\}.\end{align*}
\begin{align*}\frac{\partial X^\mu}{\partial \sigma}\left(\frac{\partial}{\partial \sigma}\frac{\partial L}{\partial \frac{\partial X^\mu}{\partial \sigma}}+\frac{\partial}{\partial \tau}\frac{\partial L}{\partial \frac{\partial X^\mu}{\partial \tau}}\right)=0\end{align*}
\begin{align*}(2a)^{\pi(a)} + c_{\pi(a) - 1}(2a)^{\pi(a) - 1} + c_{\pi(a) - 2}(2a)^{\pi(a) - 2} + \cdots c_1(2a) = -c_0.\end{align*}
\begin{align*}\left. \frac{\partial^{2}}{\partial x^{2}}(W_{s}(s,x)+G(s,x,W(s,x),W_{x}(s,x),W_{xx}(s,x),\bar{u}(s)))\right\vert _{x=\bar{X}^{t,x;\bar{u}}(s)}\geq0.\end{align*}
\begin{align*} h(a) \circeq -\log g(a) = -\log \Gamma(aM + 1) + \sum_{i=1}^{d+1} \log \Gamma(a \gamma_i + 1) - a \sum_{i=1}^{d+1} \gamma_i \log x_i. \end{align*}
\begin{align*} |G_0(x)| \leq \kappa \begin{cases} |x|^{-2},&~|x | < R, \\ e^{-m c |x|},&~ |x| \geq R. \end{cases}\end{align*}
\begin{align*} \operatorname{Cap}_{\mathcal{G}_1,q'}(E)=\inf\left\{\int_{\mathbb{R}^{N+1}}|f|^{q'}dxdt: f\in L^{q'}_+(\mathbb{R}^{N+1}), \mathcal{G}_1*f\geq \chi_E\right\}, \end{align*}
\begin{align*}w=\beta _1\beta _2\dots\beta _{p_w}\alpha _ws_w,\end{align*}
\begin{align*}\begin{aligned}dY^2(t)=&\eta^2(t)dt+dW^2(t),\end{aligned}\end{align*}
\begin{align*}\varphi_a(z)=\dfrac{a-P_a(z)-s_aQ_a(z)}{1-\langle z,a\rangle}, z\in \mathbb{B}_n,\end{align*}
\begin{align*}\delta \phi = \phi \epsilon + \epsilon \zeta\end{align*}
\begin{align*} [\mathbf{h}'_{{3i+2}},\mathbf{h}_{{3i+2}}] =1. \end{align*}
\begin{align*}\langle T^{*}(a^{*}_{2}),a_1\rangle = \langle a^{*}_{2},T (a_1)\rangle.\end{align*}
\begin{align*}(X^k, Y^k) &\sim \prod_{i=1}^k p(x_{i}) p(y_{i})\\&= \prod_{i=1}^k\left(\prod_{n=1}^N p(x^{(n)}_{i})\prod_{\ell=1}^L p(y_{\ell i})\right). \end{align*}
\begin{align*}P(x,y) = P^i{}_j(x,y)\, \partial_{x^i} \otimes dy^j,\end{align*}
\begin{align*}\varkappa_1=\mp {\frac{1}{4}}(2\pi )^{1-d}\omega_{d-1}\int_{\partial X}(1-V)_+^{\frac{d-1}{2}}\,dS'\sqrt{g''}\,dx''\end{align*}
\begin{align*}\left \{ \begin{array}{l} iw_t +w_{xx} =-\lambda |v|^{p-2} v, x\in \mathbb{R} , \ t\in \mathbb{R},\\ \\ w(x,0) =\psi (x).\end{array}\right.\end{align*}
\begin{align*}f= \frac{h}{\prod_{j=1}^N \vert s_j\vert^2 (-\log \vert s_j\vert)^{1+\alpha}}.\end{align*}
\begin{align*}x_i=H(x_{i-1},e_i).\end{align*}
\begin{align*}Z\left({\cal A},G,p,t^{(A)}_{n,l}\right) = \sum_{\{r^{(A)}_{n,l}\}}Z\left({\cal A},G,p,r^{(A)}_{n,l}\right) \prod_{A=1}^p \prod_{l=1}^{\infty}\prod_{n=-\infty}^{+\infty} \left[t_{n,l}^{(A)}\right]^{r^{(A)}_{n,l}}~.\end{align*}
\begin{align*} \delta_X^{} \phi(m)~=~X(\phi(m)) \, - \, T_m^{} \phi \, (X_M^{}(m)) \mbox{for $\, m \in M$}~.\end{align*}
\begin{align*}\frac{1}{t^s}\rho(t,t^s \kappa(s)) = A t^{1-s} + {\rm Im}(e^{i\pi(1-\beta_1)} \kappa(t)) + B t^{2-s} + C t^s|\kappa(t)|^2+\end{align*}
\begin{align*}E(\varepsilon_{0} \psi;k,R) + \varepsilon_{0}^{4} \leq - \varepsilon_{0}^{2} + C_{3} \varepsilon_{0}^{4} \leq - \frac{ \varepsilon_{0}^{2} }{2} =: -C_{1} < 0. \end{align*}
\begin{align*}\sum_{\tau \in D(n-1)} \prod_{j=1}^{n-1} \dfrac{1}{1-\zeta^{j-\tau(j)}}=\dfrac{1}{n}\left(\dfrac{n-1}{2}! \right)^{2}, \end{align*}
\begin{align*}\mathfrak G^b_{n,j}(\theta)&\equiv \frac{1}{\sqrt n}\sum_{i=1}^n \left(m_j(X_i^b,\theta)-\bar m_n(\theta)\right)/\sigma_{P,j}(\theta)\\&=\frac{1}{\sqrt n}\sum_{i=1}^n(M_{n,i}-1)m_j(X_i,\theta)/\sigma_{P,j}(\theta).\end{align*}
\begin{align*}Ric(\omega_t)=t \omega_t+(1-t)\hat{\omega}_0,\end{align*}
\begin{align*}\mathrm{Tr} \tilde{\mathcal{B}}=(n-1)q^{-\frac{1}{2}}-nq^{\frac{1}{2}}+q^{\frac{2n-1}{2}}\end{align*}
\begin{align*}c_{\mathbb{H}}(\psi(C)) = \inf_{\xi\in \mathbb{M}} \Phi_{\psi(C)} (\xi) \leq c_{\mathbb{H}}(C).\end{align*}
\begin{align*}\lambda_k=k(k+N-2),\ \ k=0,1, \ldots\end{align*}
\begin{align*}\frac{1}{s}+m\left\{\frac{r}{s} \right\}+(x-1)\left\{\frac{s}{s}\right\} = \frac{1+mr}{s}=1 \, ,\end{align*}
\begin{align*}\left((q^{p-1}+\delta[p-1]\widehat{V}_{i})(\widehat{T}_{i}+\beta(\gamma+\delta \widehat{X}_{i+1}))(g)\right)(x)=\left((q^{p}+\delta[p]\widehat{V}_{i+1})(g)\right)(x). \end{align*}
\begin{align*}-\left(\int_{\Sigma}\overline{q}^2d\sigma_\Sigma\right)L+\left(-\int_{\Sigma}\overline{q}^2d\overline{\sigma_\Sigma}+\int_{\Sigma}Ad\sigma_\Sigma+\sum_{i=1}^n\int_{\gamma_i}k^{\infty,s}_{\gamma_i,\Sigma}d\overline{s}\right)+O(L^{-\frac{1}{2}})=2\pi\frac{\chi(\Sigma)}{\sqrt{L}}.\end{align*}
\begin{align*}F_{k,m} := \big\{ \max_{j \le 2km} |S_j| \le \sqrt{m} \big\} \Longrightarrow \inf_{j,\ell \in [1,2km]} \{ G(S_j,S_\ell) \} \ge (4 C m)^{-1} \,.\end{align*}
\begin{align*}S_{j}(x,t)=\sum\nolimits_{k\in\mathbb{S}_{j}}a_{k}(t)\Psi_{k,t}(x),\text{}S(x,t)=\sum\nolimits_{k\in\mathbb{S}}a_{k}(t)\Psi_{k,t}(x).\end{align*}
\begin{align*}\Phi'''(z) = \exp\left(-\frac{z^2}{2} \right) \bigl(H'''(z) - 3 z H''(z) + 3 (z^2-1) H'(z) + z(3-z^2) H(z) \bigr)\end{align*}
\begin{align*} x_i^2 = \epsilon^2 = 1, 1 \leq i \leq n,\end{align*}
\begin{align*}\widehat{G}(0)\frac{N}{w}\sum_{d \mid v} \frac{\mu(d)}{d}\sum_{(a,w)=1} e \left( \frac{ad \overline{m\ell_1\ell_3 v}}{w} \right) = \widehat{G}(0)\mu(w)\frac{N}{w} \frac{\varphi(v)}{v},\end{align*}
\begin{align*}\frac{\partial \tilde{x}_j}{\partial \zeta_k}=\delta_{j,k}-(t-s)^2a_{jk}(t,s,y,\zeta),\end{align*}
\begin{align*}\langle v, \Phi^t w\rangle =\sum_{xy} (\Phi^t)_{xy}v(x)w(y)\leq \frac{K^2\rho^{2t}}{n} \|v\|_1 \|w\|_1.\end{align*}
\begin{align*}-\left( 2\alpha+1\right) \delta \,\acute{}\,^{2}+\delta \delta \,\acute{}\,\acute{}-2\delta^{2}\left( 1+\kappa \lambda \right) =0. \end{align*}