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ToniDO
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test_latexformula

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\begin{align*}\langle \mu^{\prime}(w_{(3)}^{\prime}),u\rangle&=\langle w_{(3)}^{\prime},\Phi(u)\rangle\end{align*}
\begin{align*} &U(y)=o\left(|y|^{\frac{n-2}{n}\left(\frac{2(n-2)}{n}\lambda+2\right)}\right)\\ &V(y)=o\left(|y|^{\frac{2(n-2)}{n}}\right). \end{align*}
\begin{align*}\tilde u & =\beta u-w \cr\tilde v & =\alpha u -v \cr\tilde c & = \Big(e+{cw+dv\over u}\Big){ e^{u\over2}s(v)s(w)\over s(v-\alpha u)s(w-\beta u)}- {cw+dv\over u} +\beta c+\alpha d\end{align*}
\begin{align*} \left\{\begin{array}{cl} (-\Delta)^{\alpha/2} u(x) = v(x)^q & \mathbb{R}^n, \\ (-\Delta)^{\alpha/2} v(x) = u(x)^p & \mathbb{R}^n. \end{array}\right.\end{align*}
\begin{align*}p_{B^{\prime}}^{}\left( x_{B}^{\prime}|\mu_{B}^{\prime},\sigma_{B}^{\prime}\right) =\frac{1}{\sqrt{2\pi\sigma_{B}^{\prime2}}}\exp\left( -\frac{\left( x_{B}^{\prime}-\mu_{B}^{\prime}\right) ^{2}}{2\sigma_{B}^{\prime2}}\right) ,\end{align*}
\begin{align*}\begin{aligned}C(x,\rho)=&\cos \rho (x-a)+\frac{1}{\rho}\left(q_{1}(x)-q_{1}(a)\cos \rho(x-a)\right)\\&-\frac{1}{\rho}\int\limits_{a}^{x}q_{1}^{\prime }(t)\cos \rho (x-t)dt+\frac{1}{\rho}\int\limits_{a}^{x}q_{0}(t)\sin \rho (x-t)dt.\end{aligned}\end{align*}
\begin{align*}\vec A_\mu(x)\!=\!\partial_\mu\vec n(x)\wedge\vec n(x)+C_\mu(x)\vec n(x)+\vec W_\mu(x),\end{align*}
\begin{align*}\overline a-\underline a=u(\bar x,-1) -u(\underline x,-1)+2\Lambda A^{1+\gamma}-2A|\bar x|^2-2A|\underline x|^2\le A+2\Lambda A^{1+\gamma}.\end{align*}
\begin{align*}\int_{{\cal M}^{1+1}} j_1 \wedge{\hat {\cal A}}_1 =\int_{\partial {\cal M}^{1+1}} {\tilde {\cal A}}_1.\end{align*}
\begin{align*}\check{H}(\Phi )=\mathrm{tr}\,g(\boldsymbol{\mu +}\left( \mathbf{K}^{\ast }\mathbf{K}-\mathbf{I}_{B}\right) /2), \end{align*}
\begin{align*}\lim_{n \rightarrow \infty} \frac 1 n {\mathbb E}\left ( \left ( f(S_n) \right)^2 \right) = \bar \Lambda(f,f) \, .\end{align*}
\begin{align*}X_2 = \{ (3, 4), (2, 3), (1, 2, 3), (1, 2, 4, 3), (1, 3, 2), (1, 3, 4), (1, 3)(2, 4), (1, 4, 3), (1, 4, 2, 3)\},\end{align*}
\begin{align*}C_n=\frac{(-\sqrt{2})^{n-1}}{2^n}\sum_{k=1}^{n-1}(-1)^{k}\binom{n-1}{k}(\sqrt{2}-1)^{2(n-1-k)}S_{k+1}.\end{align*}
\begin{align*}\Pi_{T}L_{\Omega}=L_{\Gamma}\Pi_{T}\quad\Pi_{T}H_{\Omega}=\Pi_{T}-L_{\Gamma}\Pi_{T}=\left( I-L_{\Gamma}\right) \Pi_{T}=H_{\Gamma}\Pi_{T}. \end{align*}
\begin{align*}\begin{pmatrix}n_1-2&1& 0 & 0 & \ldots & 0 & 0 & 0\\-1&-n_2/2& 1 & 0 & \ldots & 0 & 0 & 0\\0&-1& n_3 & 1 & \ldots & 0 & 0 & 0\\0&0& -1 & -n_4/2 & \ldots & 0 & 0 & 0\\\vdots & \vdots & \vdots & \vdots & \ldots & \vdots & \vdots & \vdots\\0&0&0&0& \ldots & -n_{2k-2}/2 & 1 & 0\\0&0&0&0& \ldots & -1 & n_{2k-1} & 1\\0&0&0&0& \ldots & 0 & -1 & (2-n_{2k})/2\\\end{pmatrix},\end{align*}
\begin{align*}\varphi((u^{\beta}_{s,t})^*u^{\alpha}_{i,j})=\frac{\delta_{\alpha,\beta}\delta_{i,s}\delta_{j,t}(Q_{\alpha})_{i,i}^{-1}}{\mathrm{tr}(Q_{\alpha})}\mathrm{~and~}\varphi(u^{\beta}_{s,t}(u^{\alpha}_{i,j})^*)=\frac{\delta_{\alpha,\beta}\delta_{i,s}\delta_{j,t}(Q_{\alpha})_{j,j}}{\mathrm{tr}(Q_{\alpha})}\end{align*}
\begin{align*}L_{DR}(A,B)\approx -{1\over 4}Q^\mu (P^{-1})_{\mu\nu}Q^\nu.\end{align*}
\begin{align*}w^\mu = \frac{1}{\zeta}v^\mu \end{align*}
\begin{align*}dc_1 &= b_{1} + b_{3} - b_{3}b_{2}+b_{3}b_{2}b_{1} \\dc_2 &=-y - b_{1} - b_{3} +b_{2}b_{3} - b_{1}b_{2}b_{3}. \end{align*}
\begin{align*}\langle \widehat{u}g, \widehat{u}f_0 \rangle_{L^2 (\mathbb{R}^2)} + \langle \widehat{v}g, \widehat{v}f_0 \rangle_{L^2 (\mathbb{R}^2)} = \gamma \langle g,f_0 \rangle_{L^2 (\mathbb{R}^2)},\end{align*}
\begin{align*}W_{2}\left( {u,\xi }\right) =\exp \left\{ {S_{n}\left( -{u,\xi }\right)-S_{n}\left( -{u,}\alpha _{2}\right) }\right\} \left\{ {e^{-u\xi}+\varepsilon _{n,2}\left( {u,\xi }\right) }\right\} , \end{align*}
\begin{align*}\left( \gamma ^1\gamma ^2\gamma ^3\partial _z\ +\gamma^1\gamma^2\gamma ^0\partial _0\right) \Phi =\lambda \Phi ,\quad \Psi =\gamma^1\gamma ^2\Phi\end{align*}
\begin{align*} u(0,t) = u(1,t), \mbox{two other boundary conditions on }u, \end{align*}
\begin{align*} \frac{V[U_1]}{U_1} - \frac{V[U_2]}{U_2} = Q^*(\Phi) \Phi' , U_2 V[U_1] - U_1 V[U_2] = Q^*(\Phi) ( U_2U_1' - U_1 U_2'). \end{align*}
\begin{align*}\sum_{q>0} \{ \hat{\rm G}_{+}(q) [\hat{v}_{\rm H,+}(q) ,\hat{\tilde{\rm G}}_{\pm}(p)]_{-} + \hat{\rm G}_{-}(q)[\hat{v}_{\rm H,-}(q),\hat{\tilde{\rm G}}_{\pm}(p)]_{-} \} \mp {\hbar}^2\frac{e^2{\rm L}^2}{8{\pi}^2} p \hat{v}_{\rm H,\mp}(p)\end{align*}
\begin{align*}\sum_{\nu=1}^\infty\frac\mu{\nu(\mu+\nu)}=H_\mu\,,\mu\in\mathbb{C}\backslash\mathbb{Z^-}\,,\end{align*}
\begin{gather*}{\rm Hilb}\big(6T_n^{!},t\big)=\sum_{k=0}^{n-1} {n \brace n-k } t^{k}.\end{gather*}
\begin{align*}\lambda_j(0)=\frac{1}{n+1}, 1\leq j\leq n+1. \end{align*}
\begin{align*}\frac{ W_{ j - 1 } W_j }{ \sqrt{n} } = \frac{ W_{ j -1 } \eta_j }{ \sqrt{n} }+ \frac{ g^{\prime \prime } (0) }{ 2 g^\prime ( 0 ) } \frac{ W_{ j -1 } \eta_j^2 }{ n } + O ( n^{ - 3/2 } ) .\end{align*}
\begin{align*}d_{2(j+1)} \ =\ ad_{2(j+1)-1} + bd_{2(j+1)-2}&\ =\ ad_{2j + 1} + bd_{2j}\\&\ =\ (a^2+b)d_{2j} + abd_{2j-1}.\end{align*}
\begin{align*}X_{L}=\left( c_{1}+2\psi _{I}\int T^{I}\left( t\right) dt\right) +\left(T^{I}\left( t\right) Y_{I}^{\alpha }\left( x^{\beta }\right) \right)\partial _{\alpha }+\left( a\left( x^{\beta },t\right) u+b\left( x^{\beta},t\right) +c_{2}u\right) \partial _{u}, \end{align*}
\begin{align*}\Delta E_{\rm F}^{(2)}=\frac{1}{(4\pi)^2}\int_0^\infty \frac{dq\, q^2}{(2\pi)^2}\tilde{V}(q)\tilde{V}(-q) \left[-2+\sqrt{1+\frac{4m^2}{q^2}}\, \ln \frac{\sqrt{1+\frac{4m^2}{q^2}}+1} {\sqrt{1+\frac{4m^2}{q^2}}-1}\right] \end{align*}
\begin{align*} \frac{x_i}{g} (x_1-x_k)=\begin{cases}x_i & (i = k)\\0 & (i \neq k).\end{cases}\end{align*}
\begin{align*} \boldsymbol{y}_{\mathrm{t}} = \left(\boldsymbol{G}_{\mathrm{t}}\boldsymbol{\Theta}_{\mathrm{t}}\boldsymbol{H}_{\mathrm{s,t}}+\boldsymbol{H}_{\mathrm{d,t}}\right)\boldsymbol{s}_{\mathrm{t}}+\boldsymbol{z},\end{align*}
\begin{align*}H(p)\;=\; \sup_{ s \in C}\, \langle s, p \rangle.\end{align*}
\begin{align*}c([\hat{W}],F)& \leq c([\hat{W}],L^1)+||\bar{H}||_{C^0}\\& = c([\hat{W}],H)+||H||_{C^0}\\ & <2(\sum_{i=1}^n(R_i-\epsilon)\cdot |e_i|+||G||_{C^0}).\end{align*}
\begin{align*}\frac{\hat g}{\sqrt{1 + (a u)^4}} \ll 1,\qquad g^2_{\rm eff} \sim \hat g \gg 1.\end{align*}
\begin{align*}\mathcal{M}^{r}(t)\geq \mathcal{M}^{r}(\delta)+(r-1)\theta_{1}\varrho_{0}^{1-\sigma}\int_{\delta}^{t}(\mathcal{M}^{r}(s))^{\sigma+1}ds,0<\delta<t<t_{0}<T_{\mathrm{gel}}.\end{align*}
\begin{align*}V[z(r)] = {B_0\over A_0}\left[ {1\over 2r}\left({B_0'\over B_0}-{A'_0\over A_0}\right) +b(r)\right]\,.\end{align*}
\begin{align*}\left( \begin{array}{c} m \\ r \end{array} \right) = \left( \begin{array}{c} m \\ m-r \end{array} \right),\end{align*}
\begin{align*}T(s):=\max \{0;\min\{s;1\}\},\, s\in \mathbb{R}g(s):=\max\{0;-2s+1\},\,s\in\mathbb{R}^+_0.\end{align*}
\begin{align*} & (k+3)^r-2(k+2)^r+(k+1)^r+T((k+2)^r-(k+1)^r) \\ \geq & \sum^{r-1}_{t=0} \frac {r!T!}{2\cdot t! (r+T-2-t)!}((k+1)^t+(k+2)^t).\end{align*}
\begin{align*} A+B+C=(q-\ell+1)\prod_{i=2}^{s-1}\left(q- p_{i-1}+1\right), \end{align*}
\begin{align*}\varphi^{[B;m]}_{add}(\tau,u,v,t):=\frac{1}{2}e^{2\pi i(m+\frac{1}{2})t}\sum\limits^{2m}_{j=0} R^{[B]}_{j+\frac{1}{2},m+\frac{1}{2}}(\tau,v)(\Theta^{-}_{j+\frac{1}{2},m+\frac{1}{2}}+\Theta^{-}_{-(j+\frac{1}{2}),m+\frac{1}{2}})(\tau,2u)\end{align*}
\begin{align*}^F V = \bigoplus_{p \in \frac{1}{2} \mathbb Z} F_pV/F_{p-\frac{1}{2}}V\end{align*}
\begin{align*}gu'' = c_2 = \begin{pmatrix}a\\b\\c\\d\end{pmatrix}. \end{align*}
\begin{align*}{\cal X}^+\cdot\Gamma_0\cdot{\cal X}=\gamma_0,\end{align*}
\begin{align*}\delta P^\alpha = -{i\over 2}\;(D_\beta D_\alpha \Lambda^-) P^\beta+ {1\over 2} (N-2) (\partial_- \Lambda^-) P^\alpha \;.\end{align*}
\begin{align*}v_{b^*}'(b^*) &= - \theta C^1_{b^{*}} W^{(\theta)}(b^*) + \beta Z^{(\theta)}(b^*) \\&\quad+ \lambda \left[\left( \theta \dfrac{{C^{(\theta,r)}(b^*;w)} + \rho_{b^{*}}^{(\theta)}(b^*;w)}{Z^{(\theta)}(b^*)} -w(0)\right)W^{(\theta)}(b^*) - \rho^{(\theta)}_{b^*}(b^*;w'_{+}) \right]=1 .\end{align*}
\begin{align*} \lim_{k \rightarrow +\infty} \;\mathcal{Q}_m( x_{[m]}^{k} ) - \mathcal{Q}_m^{k}(x_{[m]}^{k})=0.\end{align*}
\begin{align*}-(p_i-2)F^{p_i-4}(\nabla \phi(x_i)) \mathcal{Q}_\infty \phi (x_i) - F^{p_i-2}(\nabla \phi(x_i)) \Delta_F ( \phi(x_i)) \ge\qquad\\ \Lambda_{p_i}^{p_i}(\Omega) |u_{p_i}(x_i)|^{p_i-2} u_{p_i}(x_i). \end{align*}
\begin{align*} \Vert u_{i+1}-u_i\Vert&<\tau\Vert u_i-u_{i-1}\Vert+\tilde\varepsilon\\&<\tau^2\Vert u_{i-1}-u_{i-2}\Vert+\tau\tilde\varepsilon+\tilde\varepsilon\\&<\ldots\\&<\tau^{i+1}\cdot\Vert u_0-p\Vert+\tilde\varepsilon\cdot\sum_{k=0}^{i-1}\tau^k\\&<\tau^{i+1}d+\frac{\tilde\varepsilon}{1-\tau}\end{align*}
\begin{align*}h_t(\alpha_t)=\Phi_t(\alpha_t)=\alpha_t\exp[(t-1)u(\alpha_t) ].\end{align*}
\begin{align*} K := \{ x \in \mathbb{R}^{m} \;|\; Bx=b, ~x \in \{0,1\}^{m} \},\end{align*}
\begin{align*}Q_0 (\sigma) = 0,Q_2 (\sigma) = 0, \\ &Q_0(\sigma) = -u \sigma_u + 4g_2 \sigma_{g_2} + 6g_3 \sigma_{g_3} + \sigma , \\ &Q_2(\sigma) = -\tfrac{1}{2}\sigma_{u,u} - \tfrac{1}{24} g_2 u^2 \sigma + 6 g_3 \sigma_{g_2} + \tfrac{1}{3} g_2^2 \sigma_{g_3} \end{align*}
\begin{align*}\alpha_{2n}= 2^{2n} [2n+1]_q \beta_{2n}(q),n\in \mathbb{N}_0,\end{align*}
\begin{align*}\psi_1(u,v,w)=\alpha(v/u)+\beta(w/u) \psi_2(u,v,w)=\alpha(v/u)+\beta(w/v).\end{align*}
\begin{align*} B(t)-A(t) &= \varphi(-t)^2(G(t)H(-t)-G(-t)H(t))+2\varphi(t)^2G(t)H(t)\\ &= \varphi(-t^2)\dfrac{2t\psi(t^{10})}{\varphi(t^2)}+2\varphi(t)\varphi(t^5)\\ &= 2t\phi(t)\psi(t^{10})+2\varphi(t)\varphi(t^5)\\ &= 2\psi(t^2)\phi(t^5)\\ &= 2\varphi(t^4)^2\frac{\phi(t^5)}{\varphi(t^2)}\\ &= 2\varphi(t^4)^2(G(t)G(t^4)-tH(t)H(t^4))\end{align*}
\begin{align*}C_\nu(\omega):=\int_{-\infty}^0\exp\Bigg(\int_u^0\widetilde{A}_\nu(\tau)d\tau\Bigg)\widetilde{\mathbf{f}}(t)du,\end{align*}
\begin{align*}\varphi=-\nabla \cdot (\frac{w}{|w|}|w|^{q-1})\end{align*}
\begin{align*} u_t=u^{-r}(\triangle_g u+u^q). \end{align*}
\begin{align*} \mathcal{H}^s_\delta(A)=c_s \inf\left\{\sum_{k=1}^\infty diam(V_k)^s \,|\,diam(V_k)\le\delta, A\subset \bigcup_{k=1}^\infty V_k\right\} \end{align*}
\begin{align*}M_{\pm}(x;z)=I+\mathcal{P}^{\pm}\left(M_{-}(x; \cdot) R(x ;\cdot)\right) , z \in \mathbb{R},\end{align*}
\begin{align*} t(x',\xi',D_n)f:=\int_0^\infty{t}(x',\xi',y_n)f(y_n)\,dy_nf\in\mathcal{S}_+,\end{align*}
\begin{align*},\,\,a\in \mathcal{A}\vee \mathcal{B}^{\prime }\end{align*}
\begin{align*}\|u_{\epsilon}\|_{4^*}^{4^*}&=R^{4^*}\int_{\mathbb{R}^{N}}\varphi^{4^*}(x)(\frac{\epsilon}{\epsilon^{2}+|x|^{2}})^{N}dx\\&=R^{4^*}\int_{\mathbb{R}^{N}} \varphi^{4^*}(x)\frac{\epsilon^{N}}{(\epsilon^{2}+|x|^{2})^{N}}dx.\end{align*}
\begin{align*}|U_3(x)| &\leq C\int_{|y|>2|x|}\left(\frac{|x|}{|y|}+\frac{1}{|a||y|^2}\right)|f(y)|\,dy \\&\leq C(|x|^{-1}+|a|^{-1}|x|^{-3})\int_{|y|>2|x|}|y||f(y)|\,dy=o(|x|^{-1})\end{align*}
\begin{align*}\phi(q)=\sum^{\infty}_{n,m=-\infty}q^{n^2+2nm+3m^2}\end{align*}
\begin{align*}s_{23}r_{13} = s_{13}r_{12} = 0,\end{align*}
\begin{align*} 0&\leq h(x)-h(y)-\langle\nabla h(y),x-y\rangle+\tfrac{1}{2}(\|x\|^2-\|y\|^2-2\langle y,x-y\rangle)\\&=h(x)-h(y)-\langle\nabla h(y),x-y\rangle+\tfrac{1}{2}\|x-y\|^2\leq \|x-y\|^2.\end{align*}
\begin{align*} \begin{array}{lll}Z=a_1X_1+a_2X_2+a_3X_3\ \\\\Y=b_1X_1+b_2X_2+b_3X_3\ \\\\X=c_1X_1+c_2X_2+c_3X_3,\end{array}\end{align*}
\begin{align*} e^{-Cr^4}\leq \mathbb P[ Z_{f_1}\cap D(0,r)=\emptyset]\leq e^{-cr^4}.\end{align*}
\begin{align*}0<|\alpha_j qu-p_j|<\frac{1}{H^\varepsilon(u)|q|^{\frac{d}{r}+\varepsilon}}\quad\mbox{for all integers~~} j=1,2,\ldots, r\end{align*}
\begin{align*}(a,b) = g(t_1, t_2).\end{align*}
\begin{align*}\gamma = \max\left\{ 1- \alpha + l, \, \alpha+l \right\}\end{align*}
\begin{align*}F(T, t) = \mathbb{E}\sum_{\substack{x \in V_{\sigma} (\mathbf{X}_0(T))\\x \geq \{\sigma_1 t\}}} f(\mathbf{X}_0(T), x).\end{align*}
\begin{align*}z\mapsto \frac{(e^{i\alpha}+|a|^{2})z+a(1-e^{i\alpha})}{\bar{a}(1-e^{i\alpha})z+(1+|a|^{2}e^{i\alpha})},\end{align*}
\begin{align*}\epsilon\,=\,\frac{\alpha}{2}\,\pi^2\,\left(\frac{r_s}{R}\right)^2\,{,}\end{align*}
\begin{align*}\zeta(z) =-\int\wp(z)dz = \frac{1}{z} +\sum_{\omega\neq 0} \{ \frac{1}{(z-\omega)}+\frac{1}{\omega}+\frac{z}{\omega^2} \},\end{align*}
\begin{align*} \theta_{0,\epsilon}(z,t,x)=\psi_{\mu,\lambda^{\epsilon}_{c}}(t,x)e^{\lambda^{\epsilon}_{c}z}-A\psi_{\mu,\lambda^{\epsilon}_{c}+\gamma}(t,x) e^{(\lambda^{\epsilon}_{c}+\gamma)z}\end{align*}
\begin{align*} \begin{cases} v\left(t,0-\right)=v\left(t,0+\right) \\ p\left(t,0-\right)=p\left(t,0+\right) \end{cases} \begin{cases} v\left(t,m-\right)=v\left(t,m+\right) \\ p\left(t,m-\right)=p\left(t,m+\right) \end{cases}\mbox{for a.e. }t\geq 0 \,.\end{align*}
\begin{align*}\mathbb{E}\sum_{k=1}^n(\tau_k-x)_+=\mathbb{E}\sum_{k=1}^n(m_k-F_n(x))_+(n/4)(2\ln n)^{\frac{1}{2}}\\=2\pi(n/4)(2\ln n)^{\frac{1}{2}} D_n(F_n(x)/2)\to(\pi/2)e^{c_0-x}.\end{align*}
\begin{align*} T(y, z)=\frac{y}{1-\frac{z}{1-z}T(y, z)}=\frac{y(1-z)}{1-z-zT(y, z)}. \end{align*}
\begin{align*} I_3&\leq \frac\lambda4\iint_{Q_t} \widetilde{u}_x ^2 dxds+C\iint_{Q_t} (\widetilde{\theta}^2+\widetilde{\rho}^2) dxds.\end{align*}
\begin{align*}f''(x) = a+b'x+c'x^q,\end{align*}
\begin{align*}e^f_\mu = \left( {1 \over 2} \left(e^{[+2]}_\mu + e^{[-2]}_\mu\right) , {1 \over 2} \left(e^{[+2]}_\mu - e^{[-2]}_\mu\right)\right)\end{align*}
\begin{align*} Ric(g_{0})=\Lambda g_{0},\end{align*}
\begin{align*}g_{\sf t t}=-1 ~~\mbox{and}~~ g_{ij}=a(\sf t)^2\gamma_{ij} \;,\end{align*}
\begin{align*}h^k\sum_{|\alpha |=k}\|D ^\alpha u\|\ge \epsilon_0\varepsilon^{-\delta }\|r^{\mu k}u\|\forall k=1,\dots ,m\end{align*}
\begin{align*}\psi_x = &\left(\begin{matrix} -iz & u \\ \bar u &iz \end{matrix} \right) \psi \\\psi_t = & i\left(\begin{matrix}-[2z^2 +|u|^2] & -2izu+u_x \\-2 i z\bar u - \bar u_x & 2z^2 +|u|^2\end{matrix}\right)\psi \end{align*}
\begin{align*}M_{b_1} M_{b_2} \cdots M_{b_n} = \begin{pmatrix} A_{n-1} & A_n \\ B_{n-1} & B_n\end{pmatrix},\end{align*}
\begin{align*}\Xi(\beta(q))-\Xi(q)=r(q)\end{align*}
\begin{align*}f(\xi)-\psi_{\Gamma(x)}(\xi)=\frac{1}{2}\phi(2x)+\frac{1}{2}\phi(2\xi)-\phi(x+\xi)\geq0,\end{align*}
\begin{align*}\sin ^{2}\theta g_{1}(\nabla^{^{M_1}}_{U}V,Z)=-g_{1}(\mathcal{T}_{U}w\phi Z,V)+g_{2}((\nabla\pi _{\ast })(U,\varphi V),\pi _{\ast }(wZ)),\end{align*}
\begin{align*} \left\{ \begin{array}{lll}y^2 & = & \alpha xy + \beta yx \\ x^2 & = & 0\end{array}\right.\ \mbox{is \ Koszul}\ \Leftrightarrow \ \alpha = \beta.\end{align*}
\begin{align*}N_{2j}+L_{2j}\cdot \tfrac{B(j,1)}{B(j)} + \tfrac{B(j,3)^p}{B(j)^p}-\tfrac{B(j-1,3)}{B(j-1)}= 0.\end{align*}
\begin{align*} (q_1-b_1)q_{-1}+q_{-1}q_1=E_0, \end{align*}
\begin{align*}\Theta_{,j}^{(\alpha)}(t)=\left[l*\theta_{,j}^{(\alpha)}\right](t)=\int_{0}^{t}\theta_{,j}^{(\alpha)}(\tau)d\tau.\end{align*}
\begin{align*}\langle \alpha_X(\xi), \eta\rangle_A = \langle \alpha_X(\xi), \alpha^2_X(\eta)\rangle_A = \alpha_A(\langle \xi, \alpha_X(\eta)\rangle_A) = \langle \xi, \alpha_X(\eta)\rangle_A.\end{align*}
\begin{align*}\big\langle x\big\vert\, \exp -i\left[\left(-d_{x}^2-4q p_u x\right)s\right]\, \big \vert x\big\rangle= \left({1 \over {(4\pi is)^{1/2}}}\right)\;\exp -4i\left(q p_u xs + {1 \over 3} q^2 p_u^2 s^3\right).\end{align*}