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ToniDO
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test_latexformula

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\begin{align*} g(h^T(X,Y),Z)=-(Z\mu)g(X,Y). \end{align*}
\begin{align*}\theta(n_s^{-1}(\lambda))T_{n_s^{-1}} - T_{n_s^{-1}}\theta(\lambda)& = \sum_{k = 0}^{\langle \nu(\lambda),\alpha\rangle - 1}\theta(n_s^{-1}(\lambda)\mu_{n_s}(k)n_s^{-2})c_{n_s,k}\\& = \sum_{k = 1}^{\langle \nu(\lambda),\alpha\rangle}c_{n_s,k}\theta(\mu_{n_s}(-k)n_s^{-2}\lambda).\end{align*}
\begin{align*}xx'+x'y+yy'=2n\,.\end{align*}
\begin{align*} f = \sum_{j=1}^\infty \theta_j\varphi_j = \sum_{j=1}^\infty \left\{\int_{[0, 1]} f(x) \varphi_j(x)\, \mathrm{d}x \right\} \varphi_j.\end{align*}
\begin{align*}\begin{aligned} & (i)~ \mbox{supp}~g(\omega) \subset [-M, M], K>\frac{M}{\varepsilon_0}\bigg(1+\frac{1}{\varepsilon_0}\bigg), \\& (ii)~ ||f_0||_{L^{\infty}} < \infty, {\mathcal M}(L^+_{\gamma_0}(0)) \geq {\mathcal M}_*(\varepsilon_0, \gamma_0).\end{aligned}\end{align*}
\begin{align*}\sum_{k=1}^{p-1}\frac{1}{k2^k}\sum_{j=1}^{k-1} \frac{2^j}{j}\equiv_{p} 0,\end{align*}
\begin{align*}\varpi=2r\omega-iF_{A_K}^+,\end{align*}
\begin{align*}f = -(X+A+\Phi) u\end{align*}
\begin{align*}H_2(n,{\textstyle \frac{1}{2}}) \ge \frac{\sqrt{2\pi}}{e^2}n^{1/2} \frac{(n-1)!}{2^{n+1}}(1-o(1)) =\frac{\sqrt{2\pi}}{e^2}n^{1/2} E(n,{\textstyle \frac{1}{2}})(1-o(1))\;.\end{align*}
\begin{align*}III_1&\leq \int_{\big( B(g, 4 C_2 r) \big)^c} \int_{B(g,\eta r)}\Big|K_j(\tilde g,\tilde{\tilde g})-K_j(\tilde g,g)\Big| \, d\tilde{\tilde g} \,d\tilde g\\&\quad+ \int_{B(g, 4 C_2 r) \backslash B(g, c_* \eta r)} \int_{B(g,\eta r)}\Big|K_j(\tilde g,\tilde{\tilde g})-K_j(\tilde g,g)\Big| \, d\tilde{\tilde g...
\begin{align*} \sum_{y\in [x, \infty)} (-1)^{\rho(y)} = 0;\end{align*}
\begin{align*} (1+(-\Delta)^{1/2})u=F(u), \end{align*}
\begin{align*} F_*^e( ju^{q-1}v^{q-1}(f+uv))= \bigoplus_{i\in\Delta_e}f_{(i,j)}F_*^e(iu^{q-1}v^{q-1}) \oplus uv F_*^e(j) ,\end{align*}
\begin{align*} &(-1)^{k+1}G^{k+1}_m(x) \\ =& \int_{0}^{x}(-1)^{k}zG^{k}_m(z) dm(z) + x\int_{x}^{y}(-1)^{k}G^{k}_m(z) dm(z) \\ &+ x\int_{y}^{1}(-1)^{k}G^{k}_m(z) dm(z). \end{align*}
\begin{align*}(\Delta +\mu -q)u=f\; \mathrm{in}\; M,(\partial_\nu u\mp ia)u=\varphi\; \mathrm{on}\; \partial M,\end{align*}
\begin{align*}u \leq_{j_{1}} u_{1} \leq_{j_{2}} \dotsb \leq_{j_{r}} u_{r} = v.\end{align*}
\begin{align*}\sum_{i=1}^{N}\|v_i(t_0)\|_2 \leq 2\sum_{i=1}^{N}\|v_{i,0}\|_2.\end{align*}
\begin{align*}|vacuum\rangle \cong |0\rangle_f \otimes \Phi_0(q).\end{align*}
\begin{align*}\hat{g}_m(x) := \frac{1}{m^2-1}\sum_{i=1}^m w_{\psi_m}(\zeta_i) \cdot (-\log \delta(x,x)_{\zeta_i})\end{align*}
\begin{align*} \omega_{k,m}(\pi) :=(\lambda_{M}+1-k)\cdot\prod_{i=1}^{M-1} (\lambda_{i}-\lambda_{i+1}+1-m).\end{align*}
\begin{align*}D \equiv \left \{ \sigma_{+}, \sigma_{-} \vert \Sigma_{-}\le \sigma_{-} < \infty \quad ; \quad\Sigma_{+} \le \sigma_{+} < \infty \quad ; \quad \Sigma_{-} - \Sigma_{+}\ge 0 \right \}\end{align*}
\begin{align*} \partial_Q \mathcal{X} (Q) = A(Q) \mathcal{X} (Q) \end{align*}
\begin{align*}\ddot Q_0(t) = (1-t)^{n-\frac{3}{2}} - \frac{t^2(1-t)^{n-\frac{7}{2}}}{2}\end{align*}
\begin{align*}\sum_{\substack{m,v \in \mathcal{M} \\ mp=v}} f(m)f(v) \le \sum_{\substack{m', v' \in \mathcal{M}' \\ |pm'/v'-1| \le3/T }} r(m')r(v').\end{align*}
\begin{align*}C_\mu (t,f) =0 \end{align*}
\begin{align*} d(u,x)+d(v,y)&\ge (1-\varepsilon)\big(d(u,v)-d(x,y)\big)+\varepsilon \big(d(u,x)+d(v,y)\big)\\ &\ge(1-\varepsilon)\big(d(u,v)-d(x,y)\big)+2(1-\varepsilon)d(x,y)\\ &= (1-\varepsilon)\big(d(u,v)+d(x,y)\big).\end{align*}
\begin{align*}\mathcal{J}_3 \! = \! \bigl\{&\emptyset, \{1\}, \{2\}, \{1,1\}, \{1,2\}, \{2,1\}, \{2,2\}, \\&\{1,1,1\},\! \{1,1,2\},\!\{1,2,1\},\!\{1,2,2\},\!\{2,1,1\},\!\{2,1,2\},\!\{2,2,1\},\!\{2,2,2\}\bigr\}.\end{align*}
\begin{align*}G'_{mn} = G_{mn} - \frac{ G_{m1} G_{1n} + B_{m1} B_{1n} }{G_{11}} \;,\;\;\;B'_{mn} = B_{mn} - \frac{G_{m1} B_{1n} + B_{m1} G_{1n}}{G_{11}} \;,\end{align*}
\begin{align*}\Pi(p^2,\mu) \simeq \frac{b_0}{8\pi^2} \left[\log\frac{z_1}{z_0}+ \log m z_0 - \frac{1}{4} \log\mu z_0 - \frac{1}{4} \log\mu z_1 + \frac{m^2 z_0^2}{8} \left( \log\frac{z_1}{z_0}-\frac{1}{2} \right) \right] \;.\end{align*}
\begin{align*}&\frac{\partial}{\partial t} f_\sigma - \Delta f_\sigma - 2 \, (1-\sigma) \, \Big \langle \frac{\nabla H}{H},\nabla f_\sigma \Big \rangle \\&+ 2 \, \sum_{i=1}^n \frac{H^{\sigma-1}}{\mu-\lambda_i} \, (D_i \mu)^2 - \sigma \, |A|^2 \, f_\sigma \leq 0\end{align*}
\begin{align*}\sum_{k=0}^{M}[4k-1]\frac{(aq^{-1},q^{-1}/a,q^{-1}/b,cq^{-1},dq^{-1},q^{-1};q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\bigg(\frac{bq^7}{cd}\bigg)^k\equiv0\pmod{[n]}.\end{align*}
\begin{align*}{\cal H}_1=a^3+i(1+b^3),\quad {\cal H}_2=(1+a^4)+ib^4.\end{align*}
\begin{align*} \int_\Sigma e^{- p \, r} = O (1).\end{align*}
\begin{align*}\frac{ d^3 x / d \mu^3 }{d x / d \mu} = \frac{3}{2} \frac{ d^2 x / d \mu^2 }{(d x / d \mu)^2} - \frac{1}{2} \frac{dx}{d \mu} \left ( \frac{1}{x^2} + \frac{1}{x(1-x)} + \frac{1}{(1-x)^2} \right ). \end{align*}
\begin{align*}a_k = \rho + \tau, a_{\ell} = \rho - \tau,\end{align*}
\begin{align*}I_{1,j}=\sum_{k=1}^d(\nabla \varphi_{j}^{(k)}\cdot \nabla \theta_{\epsilon},\varphi_{j}^{(k)}),\ \ \ I_{2,j}=(i(\mathbf{c}\cdot \nabla)\varphi_j,\theta_{\epsilon}\varphi_j),\ \ (\varphi_j=(\varphi_j^{(1)},\cdots,\varphi_j^{(d)}))\end{align*}
\begin{align*}J^1_pP=\{\gamma_p\in L(T_xX,T_pP)\mid T_p\pi\!\cdot\!\gamma_p=id_{T_xX}\},\end{align*}
\begin{align*}H\ \propto \; \frac{\partial}{\partial \lambda} \ell g F(\lambda,t)|_{\lambda = \frac{i}{2}}\end{align*}
\begin{align*}\left\{\begin{array}{rl}u_t-Lu=0 & \hbox{ on } (0,\infty)\times P,\\ u(0) = \Phi_l& \hbox{ on } P. \end{array} \right.\end{align*}
\begin{align*}\tilde{\lambda}_1&=\sum_{\substack{r_0<x^\xi\\ (r_0,W_0)=1}}\frac{\tilde{y}_{r_0}\mu^2(r_0)}{\phi(r_0)}=\xi\log{x}+O(\log\log{x})\gg \xi\log{x}.\end{align*}
\begin{align*}a_i\Lambda F_i +\mu_i (\phi)=c_i\end{align*}
\begin{align*}_{2}F_{1}\binom{a,b}{c}(z)=(1-z)^{-a}{_{2}F_{1}}\binom{a,c-b}{c}\left(\frac{z}{z-1}\right).\end{align*}
\begin{align*}S(z) = |\mathbb{S}^1| \int_{0}^{\frac \pi 2} \sin \theta \left[ \frac 1 {\cos^3(\theta/2)} B\left(\frac {|z|} {\cos (\theta/2)} , \cos \theta \right) - B(|z|, cos \theta) \right].\end{align*}
\begin{align*}\int_a^{r_2} (r_2-r_1)^\zeta dr_1= \frac{(r_2-a)^{1+\zeta}}{1+\zeta}.\end{align*}
\begin{align*} \lim_{t \to \infty} t^{- \kappa_1} \P_x \left( \tau_C > t \right) = \frac{\Gamma \left( \kappa_1 + \frac{d}{2} \right)}{\Gamma \left( 2 \kappa_1 + \frac{d}{2} \right)} \left( \frac{|x|^2}{2} \right)^{\kappa_1} M(x), \end{align*}
\begin{align*}F(x)=\sum_{i} h_{i}e_{i}+h_{\Omega}(x)x=\nabla h_{\Omega}(x)+h_{\Omega}(x)x.\end{align*}
\begin{align*}g_L=\sqrt2m_L\,e^{\omega_3\delta},\quad g_M= {m_M}\,e^{2\omega_3\delta}/2,\quad g_S= {m_S}\,e^{4\omega_3\delta}/4\sqrt2, \end{align*}
\begin{align*}C_{\ell}\simeq\frac{2}{\pi}\int\frac{dk}{k}k^3\left[\left(\Theta_{\omega=0}+\Psi_{\omega=0}\right)\jmath_{\ell}(kd)+v_{\gamma\omega=0}\jmath '_{\ell}(kd)\right]^2 .\end{align*}
\begin{align*} I + II = III\end{align*}
\begin{align*}\beta_{n,q}(l,i,p):=\sum_{k=q}^{2n-1}c_{n,q,k}(q-n-1-k)^{i-p}B_{l+p+1}(k+n+1-q),\end{align*}
\begin{align*}M_d^2 R_d' = \frac{d^2 R_d}{\gcd(R_d, r_d^2)} = [q, d^2].\end{align*}
\begin{align*}L:=\widehat{K_v^{\mathrm{ur}}}\end{align*}
\begin{align*}Q_{A}(u,\bar{u})=\langle P,u|\hat{A}|P,u\rangle\;.\end{align*}
\begin{align*}h_{(j,N)}^{-} & =1_{A_{j}^{(N)}}g_{-}-1_{A_{j}^{(N)}}\mathbb{E}\left(g_{+}\;|\;A_{j}^{(N)}\right) \\& \leqslant\phi_{(j,N)}f\leqslant1_{A_{j}^{(N)}}g_{+}-1_{A_{j}^{(N)}}\mathbb{E}\left( g_{-}\;|\;A_{j}^{(N)}\right) =h_{(j,N)}^{+},\end{align*}
\begin{align*}u^{(a,\delta,\lambda)}(t,x):= \lambda^{-\frac{1}{\nu-1}} \phi^{(a,\delta)}(\lambda^{-1}t, \lambda^{-1}\delta x). \end{align*}
\begin{align*}A= \left( u+2\,M \right) ^{2}{u}^{2},\end{align*}
\begin{align*}\nu_0=1-\frac{q_0-5}{5q_0}, \end{align*}
\begin{align*}\left({-\hbar^2\over 2}\triangle + {\cal U}\right)\Psi = E \Psi\end{align*}
\begin{align*}f_\beta \left( T_a(z)\right) =T'_a(z)f_\beta (z)-\frac{\partial T_a(z)}{\partial \beta _a}\frac{\delta \beta _a}{\epsilon }\ ,\end{align*}
\begin{align*}T_{\;\;a}^{i}=T_{\;\;ak}^{i}y^{k}, \end{align*}
\begin{align*} R_9 & = 27 \, \lambda_1 \lambda_2 (\lambda_1-1) (\lambda_2-1), \\ S_9 & = (\lambda_1+1)(\lambda_1-2)(2\lambda_1-1) (\lambda_2+1)(\lambda_2-2)(2\lambda_2-1) \;.\end{align*}
\begin{align*}\nu_{d,p}(\mathbb G_2^c) \quad\leq\frac{1}{\delta_d} e^{ \gamma_2 M_d^2 + \gamma_2 p} d^{-5 \xi_2} \kappa_2= \kappa_2 d^{-5 \xi_2 + \tau_1 + \tau_2 + \gamma_2 p / \log d}\end{align*}
\begin{align*}\tilde H = \int H + {1\over2}\int[ dxdy\theta_1(x)M_{11}(x,y)\theta_1(y)+ \theta_2(x)M_{22}(x,y)\theta_2(y)]\end{align*}
\begin{align*}||\widetilde Q_T(f,g)||^2_{L^2(dP)}=||\widetilde Q_T(g,f)||^2_{L^2(dP)}.\end{align*}
\begin{align*}\sum\nolimits y^dx^{des(\sigma)} &= 1 + \sum\limits_{d=1}^\infty y^d(1+(d-1)x)\\&= \frac{1}{1-y} + xy^2\sum\limits_{d=1}^\infty (d-1)y^{d-2}\\&= \frac{1-y+xy^2}{(1-y)^2.}\end{align*}
\begin{align*}L^{ca}{}_i K^{b}{}_{c}{}^i + L^{cb}{}_i K^{a}{}_{c}{}^i = 2 L^{ab}\,.\end{align*}
\begin{align*}\max_{|z_j| = 3^j/\log T} |G(z_1,z_2,z_3,z_4)| \ll T (\log T)^{k^2 - 4},\end{align*}
\begin{align*}a_{\bar s}^n(\omega) =F_{s _{n-1}} (\omega)\circ F_{s _{n-2}}(\theta \omega) \circ \cdots \circ F_{s_0}(\theta ^{n-1}\omega)(p(\theta ^n\omega)).\end{align*}
\begin{align*}\int f\frac{d}{ds}\mu_{st}&=\mathbb{E}[f'(Z_{st})U]\\&=\mathbb{E}[f'(Z_{st})\mathbb{E}[U|Z_{st}]]\\&=-\int f(Z_{st})\nabla(\mathbb{E}[U|Z_{st}]\mu_{st})\end{align*}
\begin{align*}\lim_{\delta \rightarrow 0}\delta \sum_{i=\ell }^{n}\xi _{i}\approx\int\limits_{\tau -\left( n+1\right) \delta }^{\tau -\left( \ell +1\right)\delta }\left( \alpha +\beta p\left( t\right) \right) dt \end{align*}
\begin{align*} \phi[1] = Y_{1} X_{1}^{-1} \phi Y_{1} X_{1}^{-1} \end{align*}
\begin{align*}M:=\begin{pmatrix} ( g_1, h_{1}^{(1)} h_{1}^{(2)} )_{\Omega}& \dots & ( g_m, h_{1}^{(1)} h_{1}^{(2)} )_{\Omega}\\\vdots & \vdots & \vdots\\ ( g_1, h_{m}^{(1)}h_{m}^{(2)} )_{\Omega} & \dots & ( g_m, h_{m}^{(1)} h_{m}^{(2)} )_{\Omega}\end{pmatrix}\end{align*}
\begin{align*}h_{\alpha,me}= \{\lambda \in E_r\mid \langle \lambda+\rho , \alpha^\vee\rangle =me\}.\end{align*}
\begin{align*}c_n=\Bigg\{\begin{array}{ll}\frac{1}{4\omega_1}=\frac{1}{8\pi} & n=2, \\\frac{-1}{4\omega_3} & n=4,\\\frac{1}{2(4-n)(2-n)\omega_{n-1}} & n=3,$ $n\geq 5,\end{array}\end{align*}
\begin{align*}u_{-1}^*=\|D u_{-1} \|_{\infty}.\end{align*}
\begin{align*}\nabla_U V = T_U V + \nu \nabla_U V,\end{align*}
\begin{align*}\left( x,v\right) \in\Omega_{-}=\left\{ e_{\pm}>-\min\beta,v<0\right\},\end{align*}
\begin{align*}\lim_{n\to\infty} (h_n^j h_n^k)^{-\frac{1}{2}} \Big|B_n^j(R) \cap B_n^k(R)\Big|=0,\end{align*}
\begin{align*}P_{|\Omega}^s(x,y)\ :=\ -\left.\frac{\partial}{\partial\nu_y}\right. G^s_{|\Omega}(x,y), x\in\Omega, y\in\partial\Omega.\end{align*}
\begin{align*}& \Big|\int_{[- \pi, \pi] \times \mathbb{T}^4} e^{ it\Delta } \phi_1 \overline{e^{it\Delta}\phi_2} e^{\pm i t |\nabla|} \phi_3\, dtdx\Big|\\& \leq \|e^{ it\Delta } \phi_1 \|_{L_t^2 L_x^4} \|e^{ it\Delta } \phi_2 \|_{L_t^2 L_x^4} \|e^{\pm i t |\nabla|} \phi_3 \|_{L_t^{\infty} L_x^2}\\& \lesssim N^{\varepsi...
\begin{align*}\;\;\;\;\;(((2)(11')(32'))\pm((2)(12')(31'))), \;\; \;\;(((3)(11')(22'))\pm((3)(12')(21'))) \; \}\end{align*}
\begin{align*}X_{-1}(A(t))\cong X_{-1}(A(0))=:X_{-1},\end{align*}
\begin{align*} &2 \leq q, r < \infty;& &\frac{1}{q} + \frac{d}{r} = \frac{d}{2} - 1.&\end{align*}
\begin{align*} \hat\lambda q- \breve G(\hat\lambda,\nabla\hat f,\hat m)=\hat\lambda q-\breve F( \hat\lambda,\hat m)\ge\lambda q-\hat F( \lambda,\hat m) \ge \lambda q- \breve G(\lambda,\nabla f,\hat m)\,.\end{align*}
\begin{align*}\omega_{Z,Z'}=R_Z\otimes\left[R_{\binom{5,3,1}{4,2,0}}+R_{\binom{5,3,0}{4,2,1}}+R_{\binom{4,3,1}{5,2,0}}+R_{\binom{4,3,0}{5,2,1}}+R_{\binom{5,2,1}{4,3,0}}+R_{\binom{5,2,0}{4,3,1}}+R_{\binom{4,2,1}{5,3,0}}+R_{\binom{4,2,0}{5,3,1}}\right].\end{align*}
\begin{align*}T=\begin{pmatrix} 1 & 0& 0& 0 & .... \ 0 & 0\\ 1 & 1 & 0& 0 & .... \ 0& 0 \\ 0 & 1 & 1 & 0 & ....\ 0& 0\\ &\ddots & \ddots & \ddots &\ddots& \\ 0 &0 & 0 & \ldots & 1&0\\ 0 & 0 & 0 & \ldots & 1& 1 \end{pmatrix}\end{align*}
\begin{align*}H''= \bigg\{ x \notin H' :\, \frac{x\cdot z^+}{|z^+|} \leq \frac 12 \, R^+\bigg\}\,, && H'''= \bigg\{ x \notin H' :\, \frac{x\cdot z^+}{|z^+|} > \frac 12 \, R^+\bigg\}\,,\end{align*}
\begin{align*}A^-(x)&=\frac{-x}{1-x}+\frac{x}{(1-x)^2}A(x)-\frac{2x^2}{1-x}A(x)+x\big(C(x)-1\big)-\frac{x^3(1-3x)}{(1-x)^3(1-2x)}.\end{align*}
\begin{align*}L(p) = b\frac{\left(16/3\right)^{p/2}}{4-p} + \frac{1}{36}(p/2+2)(p/2+3)\end{align*}
\begin{align*}&\delta f(x_1,\cdots,x_{n+1})\\&:= \sum_{i=1}^{n+1}(-1)^{i+1}\{\alpha^{n-1}(x_i),f(x_1,\cdots,\hat{x_i},\cdots,x_{n+1})\}\\&+\sum_{1\leq i<j\leq n+1}f([x_i,x_j],\alpha(x_1),\cdots,\hat{\alpha(x_i)},\cdots,\hat{\alpha(x_j)},\cdots,\alpha(x_{n+1}))\end{align*}
\begin{align*}\phi(x)=F\left(\frac{c}{F^{-1}(x)}\right)\end{align*}
\begin{align*}\sum_{j=1}^m\left|{\textbf E}(f_j\cdot \phi_l)\right|^2\le \frac{ \sum_{j=1}^m\|f_j\|_2^2}{ n}\end{align*}
\begin{align*}N \subseteq Q \sqsubseteq R Q,R \prec (H_\theta,\in) \ \implies \ \ R \cap (Q \cap \omega_2) = Q \cap \omega_2.\end{align*}
\begin{align*}\pi_{f, n} = \zeta_{(f)}^{\varphi^{-(n+1)}} (\zeta_{p^{n+1}}-1).\end{align*}
\begin{align*}\Gamma_2^{-1}(z\,|\,a_1,a_2) = (2\pi)^{-z/2a_1}a_2^{1+{}_2 S_0 (z\,|\,a_1, a_2)}G\Bigl(\frac{z}{a_1}\,\Big|\,\frac{a_2}{a_1}\Bigr).\end{align*}
\begin{align*}(\P^\ast)\supseteq \{\pm e_i: i=1, \dots, d\},\end{align*}
\begin{align*}\breve{J}W=\breve{J}TW+\breve{J}QW. \end{align*}
\begin{align*}\lim_{t\to+\infty}\left|w(t)-e^{-tA}x_{0}\right|_{D(A^{1/2})}e^{\gamma t}=0\quad\forall \gamma<\delta.\end{align*}
\begin{align*}\Big\langle Q[\bar{\psi},\psi,A,h] \Big\rangle_0 \; := \;\int d\mu_Q[A] d\mu_C[h^\prime] \; \;Q^\prime [G,\frac{\delta}{\delta a},A,h^\prime ]\end{align*}
\begin{align*}\begin {array}{lcr} P(W_2)=(y^2+x^3+z^6+ \mu xyz)+\\w(a_0 z^{n+3}+a_1 yz^n+a_{n-1} z ^{{7 \over2}}x^{{(n-6)\over 2 }} y^{{1\over 2 }} ) + w ^2\sum\limits _{i=0}^{n-4} a_{i+1}z^{2(n-i)} x^i. \end{array}\end{align*}