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ToniDO
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test_latexformula

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\begin{align*} &\mathbf a_2 = [0,\; \ldots,\; 0,\; p(S_{m-1})C''_1,\; p(S_{m})C''_2,\; p(S_{m+1})C''_3,\; p(S_{m+2})C''_4,\; 0,\; \ldots,\; 0]^T,\\ &\mathbf a_1 = [0,\; \ldots,\; 0,\; w(S_{m-1})C'_1,\; w(S_{m})C'_2,\; w(S_{m+1})C'_3,\; w(S_{m+2})C'_4,\; 0,\; \ldots,\; 0]^T.\end{align*}
\begin{align*}\delta\iota_t(x)=X_{\delta g_t}(\iota_t(x))\\\dot{\iota}_t(x)=X_{h_t}(\iota_t(x)).\end{align*}
\begin{align*}\mu_{\beta\alpha}=\frac{e^{\sigma_\alpha}}{e^{\sigma_\beta}}\end{align*}
\begin{align*}\mathcal{L}_{X_{het}}L(x,y,z)= -2\varepsilon x^2 y^{2(2\lambda -1)}\left[(y^2-\lambda x^2)^2 +y^2 z^2 + \lambda ^2 x^2 z^2 \right] \leq 0, ~\forall (x,y,z)\in\mathbb{R}^3.\end{align*}
\begin{align*}P^{}\left( x_{A},x_{B}|\mu_{A},\mu_{B}\right) =\frac{1}{\mu_{A}\mu_{B}}\exp\left[ -\left( \frac{x_{A}}{\mu_{A}}+\frac{x_{B}}{\mu_{B}}\right) \right] ,\end{align*}
\begin{align*}\P(S_t \geq a) \ = \ \P(|B_t| \geq a).\end{align*}
\begin{align*}-\frac1{2q} + \left(1 - \frac1s\right)\frac{p}{p-2} =-\frac12 +\frac1{2p}+\left(1 - \frac1s\right)\frac{p}{p-2} \le -\frac14 + \frac1{2p} < 0\end{align*}
\begin{align*}\sum_{x\in \eta_{A_J^c}} g_J(x)=\sum_{x\in \eta_{A_J^c}}\sum_{j\in J} \tilde{l}_\alpha(|y_{c}(B_j)-x|)=\sum_{j\in J}\sum_{x\in \eta_{A_J^c}}\tilde{l}_\alpha(|y_{c}(B_j)-x|)=\sum_{j\in J}\tilde{\Psi}_{\eta_{A_J^c}}(y_{c}(B_j)),\end{align*}
\begin{align*} \frac{\partial}{\partial \upsilon_j} f(\upsilon) = -2\pi i \cdot \left\langle A^{-T}(\tau) \langle x\rangle , \frac{\partial}{\partial \upsilon_j}\Phi^{-1}(\upsilon+\tau) \right\rangle \cdot e_{\tau}(x,\upsilon) = -2\pi i \cdot (\phi_{\tau}(\upsilon)\cdot x)_j \cdot e_{\tau}(x,\upsilon) .\end{align*}
\begin{align*}\begin{array}{rcl}& & T(z) = -\frac{1}{4} \partial \phi(z) \partial \phi(z) + i \alpha_0 \partial^2 \phi(z)~,\\& &\langle\phi(z) \phi(w)\rangle= -2\ln (z - w)~,\\& &c = 1 - 24\alpha^2_0~.\end{array}\end{align*}
\begin{align*}\rho=n\left( 1+e/c^{2}\right) \end{align*}
\begin{align*} {\bf e_a}= (1/2){\bf w_a}\, ,\end{align*}
\begin{align*} (T_{\alpha}S)(x) = \alpha_i S(L_i^{-1}(x)) + R_i (\alpha_i, \phi), \end{align*}
\begin{align*}\lim_{\epsilon\rightarrow 0}\int_{\widetilde\Omega\times G_\Lambda^T}|\nabla(d_\epsilon-d)|^2=0.\end{align*}
\begin{align*} C = C_0 \cup D_1 \cup \dots \cup D_4, \end{align*}
\begin{align*}\phi_1-\frac{\phi_1^7}{7}+3 \phi_1^5 \phi_2^2- 5 \phi_1^3 \phi_2^4+ \phi_1 \phi_2^6=\gamma;\hspace{0.4cm} \phi_2 - \phi_1^6 \phi_2 +5 \phi_1^4 \phi_2^3 -3\phi_1^2 \phi_2^5+\frac{\phi_2^7}{7} =\gamma_\perp\end{align*}
\begin{align*}H'_{q^d,P^\alpha}:=\mathrm{Gal}\left(K_{q^d,P^\alpha}^{G_{q^d,P}}/\mathbb{F}_{q^d}K_{q,P^\alpha}^{G_{q,P}}\right).\end{align*}
\begin{align*}H= \frac{1}{P_+}\left[ (P_i)^2+ \left\{X^i,X^j\right\}^2 \right],\end{align*}
\begin{align*}\log(\kappa(\tfrac{\eta}{n}, \tfrac{1}{\eta})) = -\log(A(\tfrac{\eta}{n})) + \log(B(\tfrac{\eta}{n})) = - \log( A(\tfrac{\eta}{n}, \tfrac{1}{\eta})) - 2\log(\tanh(\tfrac{\eta}{2})) + \log(h(\eta)),\end{align*}
\begin{align*}&R(z)^*=\i\int _0^\infty e^{- \i\overline{z}t}T(t)^*\d t\\ &R(\overline{z})= \i\int_{-\infty}^0 e^{\i\overline{z}t}T(t)\d t= \i\int_0^\infty e^{-\i\overline{z}t}T(-t)\d t.\end{align*}
\begin{align*}\tau=\tau(t),~~\xi=\xi_1(x,t), \\\end{align*}
\begin{align*}\Vert (I-R_{A-q,B-q,\beta})(x_{n})\Vert&=2\beta\Vert (P_{A-q}-P_{B-q}(2\beta P_{A-q}-I))(x_{n})\Vert \\&=2\beta\Vert P_{A-q}(x_{n})-P_{B-q}(2\beta P_{A-q}(x_{n})-x_{n})\Vert \\&=2\beta\Vert P_{A}(x_{n}+q)-q-P_{B}(2\beta P_{A}(x_{n}+q)-q-x_{n}+q)+q\Vert \\&=2\beta\Vert y_{n}-P_{B}(2\beta y_{n}-x_{n})\Vert.\end{align*}
\begin{align*}\phi_f(G) = \inf_d \frac{\phi\bigl( \overline{\overline{G} \boxtimes K_d} \bigr)}{d}.\end{align*}
\begin{align*} y^l\frac{\partial N^k_{\,\,j}}{\partial y^l}=N^k_{\,\,j}.\end{align*}
\begin{align*}E\big(\sum_{i\in U_x} \frac{\delta_i y_i}{p(x)} \big) - \sum_{i\in U_x} y_i = p(x)^{-1} \sum_{i\in U_x} \big(p_i - p(x)\big) y_i \neq 0\end{align*}
\begin{align*}\sigma DL_g(h)_{1j}=h_{1j} + \frac{\alpha}{2}R_{jk1u}h^{ku},\end{align*}
\begin{align*}0\le\frac{d}{dx}H(x+if(x))=H'(x+if(x))(1+if'(x)).\end{align*}
\begin{align*}\partial_tu^{N,\alpha}=\tfrac{1}{2}\partial_x^2u^{N,\alpha}+F_N^\alpha(u^N)+\sigma_\beta^\alpha\partial_x\xi^\beta,\end{align*}
\begin{align*} \lfloor x \rfloor + \Big\lfloor x + \frac1n \Big\rfloor + \ldots + \Big\lfloor x + \frac{n-1}n \Big\rfloor = \lfloor nx \rfloor. \end{align*}
\begin{align*}ag = \underbrace{ac}hc = c\underbrace{bh}c = ch\underbrace{ac} = chcb = gb\,.\end{align*}
\begin{align*}K_q:=\{v\in H_q\,|\, \mathcal{L}(v,w)=0\quad\forall\, w\in H_q\}.\end{align*}
\begin{align*}t^{T} \frac{1}{\sqrt{\mathbb{V}(D_n)}} \begin{pmatrix}D_n-\mathbb{E}(D_n)\\D'_n-\mathbb{E}(D'_n)\\\end{pmatrix}\stackrel{D}{\rightarrow} N(0,1) \end{align*}
\begin{align*}\sigma(L)=\bigcup_{k \in [-\pi, \pi]^d}\sigma(L(k)).\end{align*}
\begin{align*}\langle M,\alpha\rangle=\sum_{\gamma\in\Pi_{\delta}}K_{\delta,\gamma}f(e^{\delta},1)\phi(e^{\delta-\gamma})\langle\gamma,\alpha\rangle.\end{align*}
\begin{align*}{\cal R}^{({\rm W})\alpha} _{XY}=0\,,\qquad \mbox{if }\quad r>1\,.\end{align*}
\begin{align*} u_{1r}=u_r-u, \ \ u_{2r}=g(\cdot,u_r)-w.\end{align*}
\begin{align*}\sin f=\frac{2K^{1/2}r^n}{Kr^{2n}+1},~~~~~~\cos f=\frac{Kr^{2n}-1}{Kr^{2n}+1}\end{align*}
\begin{align*} \Vert u_i-Tu_i\Vert&<(\lambda v\lambda\psi.\varphi_{i-1}(v))(u_i,\varphi_i)\\ &=\varphi_{i-1}(u_i).\end{align*}
\begin{align*}g = \exp(a_V(g)V) \exp(a_X(g)X)\exp\left( a_U(g)U + \sum a_{ij}(g) X_i^j\right)\end{align*}
\begin{align*}u^2-Dv^2=A^\prime m, |m|\leqslant 2\varepsilon\sqrt{ab}+1.\end{align*}
\begin{align*}d^{k-1}(x)\xi = (i)^{k-1} \sum_{j=0}^{k-1} \binom{k-1}{j}(-1)^jD^{k-1-j}xD^j\xi \end{align*}
\begin{align*}f\in L^{\infty}( (0,T) ),\,\, g(0)=0,\,\, g'\in L^{2}( (0,T) ),\,\,\alpha\in L^{1}( (0,T) ),\,\,\sqrt{t}\beta(t)\in L^{2}( (0,T) ).\end{align*}
\begin{align*}\dim V_1 \cap V_2= k \vert S_1 \cap S_2 \vert+1-g =k(s-1)+1-g\ .\end{align*}
\begin{align*}j !! \doteq \begin{cases} j(j-2)\cdots 1 & \mbox{if} \ \ j \ \ \mbox{is odd}, \\ j(j-2)\cdots 2 & \mbox{if} \ \ j \ \ \mbox{is even}. \end{cases}\end{align*}
\begin{align*}L(\phi^{\vee},z_1,\dots,z_n;x,y)=\sum_{d\geq 0}x^{-d}\sum\limits_{a\in A_{+,d}} \mu(a)a(z_1)\dots a(z_n)\langle a \rangle^y\end{align*}
\begin{align*} \frac{1}{r+1} \leq \frac{1}{r+1}\frac{2(r-3)}{r-2} = \frac{r-3}{\tau}.\end{align*}
\begin{align*}\bar {a}_1 \bar {a}_2 = \tau(a_1, a_2)\overline{a_1a_2} \ \ \ \ \ \forall a_1, a_2 \in A,\end{align*}
\begin{align*}\det\left(\begin{array}{ccc}\scriptstyle{d_{i,j}} & \scriptstyle{N-1} & \scriptstyle{2N-d_{N,i}} \\& & \\\scriptstyle{d_{N,j} d_{i,j}} & \scriptstyle{2N}& \sum_{l \not= i,N} \scriptstyle{d_{N,l} d_{i,l}} \\& & \\\scriptstyle{d_{N,j}} & \scriptstyle{N-1} & \scriptstyle{2N-d_{N,i}} \\\end{array}\right)= 0,\end{align*}
\begin{align*}L_\delta \omega_\delta=l_{1,\delta}+l_{2,\delta}+R_\delta(\omega_\delta),\end{align*}
\begin{align*} \nu_B = \nu_{t-1}, \ \ \nu_C = \nu_{r-1}.\end{align*}
\begin{align*}P_{0,p}(x)=P_p(x)+O(p^{-\infty})\,,\:\:p\to\infty,\end{align*}
\begin{align*}d\zeta_h:=&\left\|\frac{2}{1-r^2}\,(dr,\,h\,d\psi)\right\|\end{align*}
\begin{align*}\int_{0}^{L} dz\, u_{n}(z)u_{n'}^{*}(z)= \delta_{n,n'},\end{align*}
\begin{align*}\frac{B^i \tilde{B}_{{j}} - (\det M) (M^{-1})^{i}_{{j}}}{\Lambda^{2N_c-1}} + m^i_{j} = 0.\end{align*}
\begin{align*} C_{\ell,p} \; : \; y^2 = x^p - \ell C'_{\ell,p} \; : \; y^2 = x^p - 2\ell\end{align*}
\begin{align*}\bold{D}_Q^{\ast}(F)=\sum_{\beta,J_1,J_2}(-D)_{J_1}S_{-J_2}\left(\frac{\partial Q^{\beta}}{\partial u_{J_1;J_2}}F_{\beta}\right).\end{align*}
\begin{align*}|\sigma(J)| = |j(\tau_\sigma)|\ge c_7 |\tau_\sigma-\zeta|^3.\end{align*}
\begin{align*}R_{\mu \nu} -2\nabla_{\mu}\nabla_{\nu}\Phi=0\end{align*}
\begin{align*} Y [1]=\begin{vmatrix} Y_{1} & Y_{0}\\\lambda_{1} X_{1} & {\boxed{\lambda_{0} X_{0}}}\end{vmatrix}=\delta_{Y}^{e}[1]\end{align*}
\begin{align*}e^{ih} &=(u,v)e^{i[x,x^*]}e^{ia}e^{-ib}\\&=(u,v)e^{ic}e^{i[x,x^*]+ia}e^{-ib}\\&=(u,v)e^{ic}e^{i([x,x^*]+a-b)}.\end{align*}
\begin{align*} \frac{d}{dt}\Biggl[ {\cal H}(q(t),u(t),\mu(t),p(t),p_{\alpha}(t))-(1-\alpha)p_{\alpha}(t)\cdot{_a^CD_t^{\alpha}}q(t) \Biggr]= 0\end{align*}
\begin{align*} 1-\exp\left(-\frac{\delta_{\rm I}^2}9\binom{\rho n}{2}p_{\rm I}\right)\end{align*}
\begin{align*} \pi(x, z) = \#\left\{ n \leq x:p \not | \; n \; \forall p \leq z \right\} . \end{align*}
\begin{align*} C^{(\omega)}=\omega \log \left(1+\frac{P}{2 \omega}\right).\end{align*}
\begin{align*}\xi_v = -e^{-1-\frac{v}{2}}.\end{align*}
\begin{align*}{\cal G}_{z 4}\left(\rho,z\right)={\displaystyle\frac{1}{4\pi A{\cal C}_{66}}\sum_{l=1}^{4}{\frac{\Lambda_{\rm{c4}}\left(a=-A_l\right)}{\sqrt{A_l\rho^2+z^2}\prod_{j=1,(j\neq l)}^{4}{\left(A_j-A_l\right)}}}}\end{align*}
\begin{align*}- (6k+2) + 2 h = - 6 l + 2 r\end{align*}
\begin{align*} \int_{a_i}^{a_{i+1}}|V(x,t)|^2=\left|\begin{pmatrix} S_{d,1}(t,t_{1,ex}(a_i))S_c(t_{1,ex}(a_i),t_{1,en}(a_{i+1})))S_d(t_{1,en}(a_{i+1}),0)v_{1,0}(x)+R_{1,i}\\S_{d,2}(t,t_{2,ex}(a_i))S_c(t_{2,ex}(a_i),t_{2,en}(a_{i+1})))S_d(t_{2,en}(a_{i+1}),0)v_{1,0}(x)+R_{2,i}\end{pmatrix}\right|^2\end{align*}
\begin{align*}\sum_{k=1}^\infty (1+|z_k|)^{-mq}\int_{D(z_k,2r)} |f(z)(1+|z|)^m|^qe^{-\frac{\alpha q}{2}|z|^2}d\mu(z)\end{align*}
\begin{align*} {W(\Delta_1,z) W(\Delta_2,z) = \sum_{\Delta_3}z^{d-l(\Delta_1)-l(\Delta_2)+l(\Delta_3)}C_{\Delta_1\Delta_2}^{\Delta_3} W(\Delta_3,z) },\end{align*}
\begin{align*}\overline{P}_\alpha,\overline{Q}_\alpha \;\; (\alpha = 1,\dots, m)\end{align*}
\begin{align*}f\, \omega = \frac{dh}{h} \wedge \xi + \eta,\end{align*}
\begin{align*}T_xPV = PT_xV + A_xV\end{align*}
\begin{align*} \begin{cases} (\dfrac{\partial}{\partial t} + iP)u= f \\ u(0)= g \end{cases} \end{align*}
\begin{align*}J_{\nu}(x)=\frac{2^{\nu+1}x^{-\nu}}{\sqrt{\pi}\Gamma\left(\frac{1}{2}-\nu\right)}\,\intop_{1}^{\infty}\frac{\sin(xt)}{(t^{2}-1)^{\nu+\frac{1}{2}}}\,dt,\,\,\,\,\,|(\nu)|<\frac{1}{2},\,x>0,\end{align*}
\begin{align*} \{f_1, \ldots, f_n\}_{(AG, A^*G)} =& \Pi^{\sharp}\big|_M (d_A f_1 \wedge \cdots \wedge d_A f_{n-1}) f_n\\=& \Pi^{\sharp} ( d (\beta^* f_1) \wedge \cdots \wedge d (\beta^* f_{n-1}))\big|_M f_n\\=& \Pi (\beta^* f_1 , \ldots, \beta^* f_{n-1}, \beta^* f_n)\big|_M\\=& (-1)^{n-1} \big( \beta^*\{f_1, \ldots, f_n\} \big) \big|_M\\=& (-1)^{n-1} \{f_1, \ldots, f_n\}_M.\end{align*}
\begin{align*}\textrm{span}\{\xi_y-\xi_z : y, z \in E^{*}\}=X;\end{align*}
\begin{align*} \phi ( x_1 , \dots , x_k ) = \sum_{ i =1 }^k {\rm Tr}_i ( x_ i ). \end{align*}
\begin{align*}A(y_n...y_1x) - A(y_n...y_1x') = A(1^{n+k}0...) - A(1^n0^\infty) = c_{n+k} - c_n \ ,\end{align*}
\begin{align*}\begin{aligned}&\frac{1}{\tau^k}[f(x)-f(\breve{x}^{k})]-\frac{1}{\tau^{k-1}}[f(x)-f(\breve{x}^{k-1})]+(u-\widetilde{u}^k)^TF(\widetilde{u}^k)\\&+c_1^k(A(x-\widetilde{x}^k))^T(A\breve{x}^{k-1}-b)\geq(v-\widetilde{v}^k)^TQ^k(v^k-\widetilde{v}^k), ~\forall u,\end{aligned}\end{align*}
\begin{align*}\psi^{\prime}(z^{\prime},\bar{z}^{\prime},|{\bf p}^{\prime}|)=(\bar{c}z+\bar{d})\psi(z,\bar{z},|{\bf p}|)\end{align*}
\begin{align*}M_1 = B/(x), M_2 = B/(y), M_3 = B/(z),\end{align*}
\begin{align*}\begin{cases}\varphi_x(L,t)=\psi_x(L,t)=0, & \,\, (0,T),\\ a\varphi_{xx}(0,\cdot)+\frac{1}{c}\psi_{xx}(0,\cdot)=0, & \,\, (0,T), \\ a\varphi_{xx}(L,\cdot)+\frac{1}{c}\psi_{xx}(L,\cdot)=0, & \,\, (0,T),\end{cases}\end{align*}
\begin{align*}m\left(S\otimes \right)\Delta = m\left(I\otimes \right)\Delta=\eta\cdot\epsilon\end{align*}
\begin{align*}\tilde{g}^{\mu\nu}:=\sqrt{-g}\: g^{\mu\nu}\, ,\end{align*}
\begin{align*}0.2/T^ \leq \|u(t)\|_1 \leq 1/T^, T^=7.5.\end{align*}
\begin{align*}t_h\circ (m_h\circ a_1\circ m_h^{-1})\circ (g_d\circ m_h)= a_2\circ t_h^d\circ (g_d\circ m_h).\end{align*}
\begin{align*}\alpha_{1,2}=-\arctan (k/S_{1,2}) \, \ \ .\end{align*}
\begin{align*}{}S \left[ {\bf g} \right] = S^G \left[ {\bf g} \right] +S^R \left[ {\bf g} \right] ,\ S^R \left[ {\bf g} \right] = \int_{\cal T} dt L^R \left[ {\bf g}, t \right).\end{align*}
\begin{align*}\qquad L^4 = {N\kappa\over 2 \pi^{5/2}} = 2 g_{{\rm YM}}^2 N \alpha'^2\ ,\end{align*}
\begin{align*}A'=\bigcup_{i=1}^{h^{N-1}}(a_i+\eta{\overline Q'}).\end{align*}
\begin{align*}&\int_{ \{-t'_3\le\Psi<-t'_4\}}|F_{t_0,\tilde{c}}|^2_h\\= & \int_{\{z\in M:-\Psi(z)\in (t'_4,t'_3]\}}|\tilde F|^2_h+\int_{\{z\in M:-\Psi(z)\in (t'_4,t'_3]\backslash N\}}|F_{t_0,\tilde{c}}-\tilde F|^2_h\\&+\int_{ \{z\in M: -\Psi(z)\in(t'_4,t'_3]\cap N\}}|F_{t_0,\tilde{c}}|^2_h.\end{align*}
\begin{align*}cl(n,1) \,=\, 2^{2^{n-1}-n}.\end{align*}
\begin{align*}S_{Q}:=|\wp|^{\frac{2}{Q}}Q^{\frac{Q-2}{Q}}(Q-2)\left(\frac{\Gamma(Q/2)\Gamma(1+Q/2)}{\Gamma(Q)}\right)^{\frac{2}{Q}}.\end{align*}
\begin{align*}F^{(m)}_r(s)=\sum_{j=0}^r(-1)^{r-j}q^{(r-j)(r-j-1)}{n-j\brack n-r}{n-s\brack j}\, c^j,\end{align*}
\begin{align*}\frac{X(s)^{\perp}}{2}&=\frac{1}{2}(X(s)-g^{ij}\langle X(s),X_i(s)\rangle X_j(s))\\&=\sqrt{2}\left(\frac{1}{2}+\frac{s}{2}f-s^2\left(\left(\frac{\partial f}{\partial\theta_1}\right)^2+\left(\frac{\partial f}{\partial\theta_2}\right)^2\right)\right)(e^{i\theta_1},e^{i\theta_2})+T_2(s),\end{align*}
\begin{align*}&\lim_{n}\frac{1}{\phi(n)}[S_{a_n,\phi(n)}(i,j;\omega)-S_{a_n,\phi(n)}(i;\omega)p(i,j)]\\=&\lim_{n}\frac{1}{\phi(n)}\sum_{k=a_n+1}^{a_n+\phi(n)}\mathbf{1}_{\{i\}}(\xi_{k-1})[p_{k}(i,j)-p(i,j)]=0\ \ a.e.\end{align*}
\begin{align*}v_R=E_N [\psi_{\beta_{\delta}^+}\circ f_N] e^{N\varphi \sigma_3}M^{-1}\quad\Sigma_3^+.\end{align*}
\begin{align*} X(t)&= x+\int_0^t A(s,\overline{X}(s))ds+\int_0^t B(s,\overline{X}(s))dW(s) \\ &\quad\ +\int_0^t \int_{\{|z|<1\}} H(s,\overline{X}(s),z)\widetilde{N}(ds,dz)\\&\quad\ +\int_0^t\int_{\{|z|\ge1\}} J(s,\overline{X}(s),z)N(ds,dz).\end{align*}
\begin{align*}\frac{1}{\tilde{\kappa}^2}\frac{1}{CN^4[\tilde{t}\mid\frac{1}{2}]}\frac{d^2A}{d\tilde{t}^2}=\frac{-3}{\tilde{\kappa}^2}\frac{A}{CN^6[\tilde{t}\mid\frac{1}{2}]}+A-A^3,\end{align*}