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OleehyO
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latex-formulas

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\begin{align*}\begin{array}{l}\lbrack q^{-},\alpha _0^{+}]=i\,, \\\left[ \alpha _n^{-}\,,\alpha _m^{+}\right] =n\,\delta _{n+m,0}\,\,\,, \\\left[ s_n\,,s_m\right] =\left( \frac k2-1\right) \,n\,\delta _{n+m,0}\,\,\,,\\\left[ L_n^{\prime }\,,L_m^{\prime }\right] =(n-m)L_{n+m}^{\prime }\end{array}\end{align*}
\begin{align*}\nabla _{\alpha}V_{\beta}-\nabla _{\beta}V_{\alpha}=0.\end{align*}
\begin{align*}u(x,t)=-2\frac{\partial ^2}{\partial x^2} \ln W =2 \frac{({W'}^2-WW'')}{W^2}~~,\end{align*}
\begin{align*}x=\frac{\overline{x}}{\underline{x}}\hskip 40pt \overline{x}, \underline{x}\in\mathbb{Z},\mathrm{gcd}(\overline{x},\underline{x})=1.\end{align*}
\begin{align*} {}^{(i)}\langle V| = {}^{(i)}\langle W^*|(-q)^{-D}, \qquad |V^*\rangle^{(i)} = (-q)^{-D} |W\rangle^{(i)}.\end{align*}
\begin{align*}\gamma _{m_{\mu }}\equiv \mu \frac{\partial m_{\mu }}{\partial \mu }\end{align*}
\begin{align*}\Delta _{H(K)}f(x)=\sum_{i=1}^{n}(\tau _{i}-\tau _{i-1})\left( \Delta_{K}^{(i)}\psi \right) (x_{\mathcal{P}}^{\prime }\mathbf{).} \end{align*}
\begin{align*}\hat{k}^{\hat{\mu}}\partial_{\hat{\mu}}=\partial_{\underline{x}}\, .\end{align*}
\begin{align*}S(p) = {1\over p\!\!\!\slash} + 2i\pi p\!\!\!\slash n(|p^{0}|) \delta(p^{2})\end{align*}
\begin{align*}\mu _{c}={\frac{(-1)^{k}}{2G_{k}}}.\end{align*}
\begin{align*}{\hat u}(\eta,{ \bf x}) = \int \frac{d^3k}{(2 \pi)^{3/2} } (u_k^+ a_ke^{ikx} + u_k^{* +} a_k^{\dagger} e^{-ikx}),\end{align*}
\begin{align*}\pi^{\mu \nu} = \frac{\partial{\cal L}}{\partial {\dot{B}}_{\mu \nu}} = \frac{1}{2}(H^{0 \mu \nu} - m \epsilon^{0 \rho \mu \nu}A_{\rho})\end{align*}
\begin{align*}\varphi (k,\lambda) = \lim_{m\rightarrow 0}[{1\over m} \overline\psi ((\lambda -1) k) \psi (\lambda k) ]\,\,\end{align*}
\begin{align*}\frac{ \partial \epsilon }{ \partial \tau } + \frac{3}{\tau}(\epsilon + p) = 0 \>.\end{align*}
\begin{align*} h^*= \sum_{p\in P_1}\nu_p(\underline{T})\log(p) + \frac{1}{2}\sum_{p\in P_2}\nu_p(\underline{D})\log(p)\end{align*}
\begin{align*}[{\hat\rho}^\mu,{\hat\rho}^\nu]_\circ\rightarrow[{\hat\beta}^\mu,{\hat\beta}^\nu]=2\beta^\mu\beta^\nu,\end{align*}
\begin{align*}B=\left|\begin{array}{cc}\alpha & \alpha +s \\-\alpha +s & -\alpha\end{array}\right| ,\end{align*}
\begin{align*}- 2\Bigl(e^{\psi} \partial_1\partial_{\bar{1}}+e^{\psi} \partial_2 \partial_{\bar{2}}+\partial_3 \partial_{\bar{3}} \Bigr) Z =(4\pi)^{1/2} \kappa e^{\psi} \rho_3\ ,\quad e^{\psi} \rho_3 =\sum_{j=1}^N \delta^6(x^m - x^m_j) \ .\end{align*}
\begin{align*}\Delta E(V+B)^{\rm ren} =0 \,.\end{align*}
\begin{align*}\begin{array}{c}a^{A*}=a^{A}~~~~b^{A*}=b^{A}~~~~\omega_{AB}{}^{*}=\omega_{AB}~~~~\lambda^{*}=\lambda~~~~T^{\dagger}=-T\\{}\\\bar{\rho}_{i}=\rho^{i\dagger}\gamma^{0}~~~~~~~\bar{\varepsilon}_{i}=\varepsilon^{i\dagger}\tilde{\gamma}^{0}\end{array}\end{align*}
\begin{align*}{i_c}_2={i_c}_2',\quad {i_c}_3={i_c}_2'',\quad {i_c}_3'={i_c}_3'',\quad{i_b}_3'={i_b}_3'',\quad{i_b}_1'={i_b}_3''',\quad {i_b}_1''={i_b}_1''', \end{align*}
\begin{align*}(f \star g) (p^{(3)}) =\frac{1}{ (2 \pi)^{13}} \int dp^{(1)}\, dp^{(2)}\,\delta^{26} (p^{(1)} + p^{(2)} - p^{(3)}) f (p^{(1)}) g (p^{(2)})\end{align*}
\begin{align*}O(d,d\,|{\bf Z}) \setminus O(d,d)\, / \,O(d)\times O(d).\end{align*}
\begin{align*}{\eta}_{1,n} = \frac{1}{2{\rm L}} \exp\{ - \frac{i}{2} e_1e_2I_1(x_{-},x_{+}) - \frac{i}{2} (E_{1,n} + \frac{1}{2}(e_1 + e_2)b) x_{+}\end{align*}
\begin{align*}K=e^{\frac{1}{2\pi i}\oint dz \xi(z) G^-(z)}\end{align*}
\begin{align*} \lim_{t\to\infty}\frac{1}{t} \log |\underline{x}_t| = \lim_{t\to\infty}\frac{1}{t} \log |\underline{y}_t| = h^*.\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*}
\begin{align*}{\rm Tr} g^{\rm adj}(x)~=~ \left| {\rm Tr}~g(x)\right|^2-1\end{align*}
\begin{align*}Y_3^{\rm triangle}(R)=-\frac{8}{\sqrt3}\Lambda^2L_3\left[R^2\Lambda^2-R^4 \Lambda^4-\frac{4\sqrt3}{45}R^5\Lambda^5+{\cal O} \left(R^6\Lambda^6\right)\right].\end{align*}
\begin{align*}\left( \hat \Gamma_{\hat \mu\hat \nu}\right) ^{\hat \alpha\hat \beta}\left( \hat \Gamma^{\hat \rho\hat \sigma}\right) _{\hat \alpha\hat\beta} =16\delta_{[\hat \mu}^{[\hat \rho}\delta_{\hat \nu]}^{\hat \sigma]}\ .\end{align*}
\begin{align*}\gamma^2 = - u v w +[(\zeta X+\zeta Z - \zeta Y),XYZ]+ \zeta^2(X^2+Y^2+Z^2) -2 \zeta^3\end{align*}
\begin{align*}[h_{j,j+1}, S^{\pm}] = [h_{j,j+1}, S^{z}] = 0\ \ \\ \forall j = 1,\ldots ,N-1 \end{align*}
\begin{align*}= \{\chi^{\mu\nu}\} +\left(\begin{array}{cccc}((p^2 - q^2)a + 2pqb) & (pa + qb) & (pb -qa) & ((p^2 - q^2)a + 2pqb) \\(pa + qb) &0 & 0 & (pa + qb) \\(pb -qa) & 0 & 0 & (pb -qa) \\((p^2 - q^2)a + 2pqb) &( pa + qb) &( pb -qa) & ((p^2 - q^2)a + 2pqb)\end{array}\right).\end{align*}
\begin{align*}b_{\underline{n}\sigma}^\dagger= \frac{\pi/L_\perp}{\sqrt{8\pi^3 n}}b_{\underline{p}\sigma}^\dagger\,,\;\;a_{\underline{m}}^\dagger= \frac{\pi/L_\perp}{\sqrt{8\pi^3 m}}a_{\underline{q}}^\dagger\,,\;\;\end{align*}
\begin{align*}\frac{1}{AB}\frac{d}{dr}(A\sqrt{1+B^2\dot r^2})=-\frac{\kappa_5^2}{6}(2\rho+3p),\end{align*}
\begin{align*}\partial _{y}\phi =m\sum_{i=0}^{r}\sqrt{2n_{i}/e_{i}^{2}}e_{i}\exp(e_{i}\cdot \phi ). \end{align*}
\begin{align*} 0<|\underline{x}_t| \leqslant H(x_t)=\max(|\underline{x}_t|,|\overline{x}_t|) \leqslant C\, |\underline{x}_t| \end{align*}
\begin{align*}a={2\over\sqrt{D-1}}{{D-3}\over{D-2}}.\end{align*}
\begin{align*}\left[ A_{i},A_{j}\right] =i\varepsilon _{ijk}A_{k}, ~~\left[ B_{i},B_{j}\right] =i\varepsilon _{ijk}B_{k}, ~~ \left[ A_{i},B_{j}\right] =0.\end{align*}
\begin{align*}{j^{(1)}}^\mu_5 \; := \; \overline{q} \; \gamma^\mu \gamma_5 \; q \; ,\end{align*}
\begin{align*}\tau_n (x) = \sigma_n(y(x)) \ , \ n=2,3,\ldots , N \ ,\end{align*}
\begin{align*}\hat{G}(x)=\frac{1}{2\Gamma(\alpha)}\left[ e^{i\frac{\pi}{2}\alpha}x_+^{\alpha-1}+e^{-i\frac{\pi}{2}\alpha}x_-^{\alpha-1}\right]\end{align*}
\begin{align*}g = \displaystyle \prod_{j=1}^{n} g_j^{(1)} = \displaystyle \prod_{j=1}^{n}\Biggl({{a_0^{(j)} +e^{-{k_j}(r+it)}}\over{\bar a_0^{(j)}+ e^{{k_j}(r+it)}}}\Biggr)\end{align*}
\begin{align*}L(z^iz^j-Q_{ij}z^jz^i)=(z^iz^j-Q_{ij}z^jz^i)L\ . \end{align*}
\begin{align*}\frac{dT}{dx} = - 2\pi n\int\limits_0^\infty {bdb} \Pi \left( b \right)\frac{\int\limits_0^\infty {P\left( {b,\nu } \right)h\nu d\nu } }{\int\limits_0^\infty {P\left( {b,\nu } \right)d\nu } }\end{align*}
\begin{align*}\int^{2\pi}_{0} d\sigma X^{i}(\sigma)\hat{P}_{i}(\sigma) =x^{i}_{0}\hat{p}^{i}_{0} +\frac{g_{ij}}{2\sqrt{\alpha'}} \sum_{n=1}^{\infty}\left\{ \left( \chi_{n}^{i}\alpha^{j}_{-n} +\bar{\chi}^{i}_{n}\tilde{\alpha}^{j}_{-n} \right) +\left(\bar{\chi}_{n}^{i}\alpha^{j}_{n} +\chi_{n}^{i} \tilde{\alpha}^{j}_{n}\right) \right\}~.\end{align*}
\begin{align*}\omega_{2n-1}(\tilde{A})=\sum_{k=0}^{2n-1} \omega_{2n-1-k}^{k}(A,c),\end{align*}
\begin{align*} \lim_{t\to\infty} \frac{1}{t}\,\log \max(|x_t|,|y_t|) = \log |\alpha|.\end{align*}
\begin{align*}\overline{\psi}^{(1)}(x) P_+ \psi^{(1)}(x) \; \longleftrightarrow \;\frac{1}{2\pi} c^{(1)}:e^{-ia \varphi^{(1)}(x)}:_{M^{(1)}}:e^{-ib \varphi^{(2)}(x)}:_{M^{(2)}} \; e^{i\frac{\theta}{2}} \; ,\end{align*}
\begin{align*}S_{--}(q_0t_0,p_1t_1) = -p_{1i} \tilde {q}_1^{i} +\int_{\gamma_1}(\tilde {p}_i d\tilde {q}^i- H(\tilde {q}(\tau),\tilde{p}(\tau)))\ d\tau \end{align*}
\begin{align*}\langle T_{i}^{k}\rangle _{{\mathrm{(sub)}}}^{(R)}= \frac{1}{2\pi \xi ^2}\left( \zeta -\frac{1}{12}\right) {\mathrm{diag}}(1,-1), \end{align*}
\begin{align*}J^a = l^a + S^a = l^a + {1 \over 2} \sigma ^a\end{align*}
\begin{align*}Z_d\;=\; \int \prod_{i=1}^d d U_i\exp \left\{ N\beta \sum_{i>j=1}^d{\rm Tr} \, \left[ U_i U^\dagger_j + U_j U_i^\dagger\right]\right\}\;.\end{align*}
\begin{align*}\zeta^{LLT+TR}_{2}(0)=\left(\frac{\alpha}{A}\right)(Pr^2)-\left(\frac{a_{1}-2}{A}\right)(Qr^2)+\left[\frac{2}{3A}(3\alpha-2A)+\frac{a_{1}-2}{A}\right]\ .\end{align*}
\begin{align*}dp\wedge dq,\quad\epsilon_{{\underline a}}s\left(\frac{p^2}{2} +\frac{\epsilon_{{\underline a}}\alpha^2}{2q^2}-{\cal E}_{\underline a}\right)\approx 0,\end{align*}
\begin{align*}c_\alpha^A \ \Psi\;=\;0, \quad {\rm for \ all} \quad \Psi \in {\mathcal H}_0\ .\end{align*}
\begin{align*}Vol(X)\;\geq\;c_{d}\cdot p\end{align*}
\begin{align*}\int d\sigma \partial_\sigma X^-(\sigma) = 2\pi R \nu,\end{align*}
\begin{align*} C \, = \, \begin{pmatrix} T & -D \\ 1 & 0 \end{pmatrix}.\end{align*}
\begin{align*}p(r)=\frac{1}{r^3}\int_{\infty}^{r}{\varepsilon (t)t^2dt}=\frac{2}{r^2}\int_{\infty}^{r}{p_\perp (t)tdt},\end{align*}
\begin{align*}\lim_{m\rightarrow \infty} I_{rel} = -i \frac{g}{2} \delta^2(\tilde{p})\,.\end{align*}
\begin{align*}\Sigma^{(2)}_{n,k_\alpha}:\quad F_{n,k_\alpha}(s,v)=f_{n,k_\alpha}+ig_{n,k_\alpha}=0.\end{align*}
\begin{align*} A_k = a_k + 2n_k - n_{k-1} - n_{k+1} \geq 0\;, \quad k=1,\ldots, N-1\;.\end{align*}
\begin{align*}W(R,T) \simeq \exp\left[- \sigma T R\right] \end{align*}
\begin{align*}\frac{ -i\langle T a^{\dagger t}_{ {\bf{k}} }(-{\bf{q}})a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle} = \sum_{n}\mbox{ }exp(\omega_{n}t)\mbox{ }\frac{ -i\langle T a^{\dagger n}_{ {\bf{k}} }(-{\bf{q}})a^{\dagger}_{ {\bf{k}}^{'} }({\bf{q}}^{'}) \rangle }{\langle T 1 \rangle}\end{align*}
\begin{align*}\psi_{g_0}(x,t)=\exp(-i\frac{m}{2}g_0^2t)\exp(-img_0x)\psi^\sigma_{\frac{(2mV_0)^{1/2}}{\omega}}(\omega (x-g_0t))\end{align*}
\begin{align*}E^2(R<<k)=\frac{2}{k}[j(j+1)-m^2+(m+\frac{kN}{2})^2 R^2]\end{align*}
\begin{align*}\{R(z,\omega) - 1\}[a^{\dag}(m),a^{\dag}(n)]|k\> = \frac {2}{\sqrt{mn}}(z^{-m}\omega^{-n}-\omega^{-m}z^{-n})\end{align*}
\begin{align*}m^{3}-\frac{1}{2}\alpha\, m^{2}+\frac{1}{32}\alpha \beta^{2}=0. \end{align*}
\begin{align*}\sigma_t=i \Leftrightarrow z_t\in\Omega_i.\end{align*}
\begin{align*}e^i_\alpha(\xi_1,\xi_2)\Psi^{\alpha j} +(-1)^{\epsilon(i)\epsilon(j)} e^j_\alpha(\xi_1,\xi_2)\Psi^{\alpha i}=0\,. \end{align*}
\begin{align*}\tilde{\rho} = \alpha^*_a \rho^a_{\;\;b} \alpha^a\end{align*}
\begin{align*}R(u_1,u_2,u_3)^{j_1j_2j_3}_{i_1i_2i_3}\stackrel{\tau}{=}R(u_3,u_2,u_1)^{j_3j_2j_1}_{i_3i_2i_1},\end{align*}
\begin{align*}ln\biggl({\zeta+\epsilon\over{\sqrt{\zeta^2+i\epsilon\eta^2}}}\biggr).\end{align*}
\begin{align*}\Delta _{L}\left( \theta ^{a}\right) =M_{\,\,\,b}^{a}\otimes \theta ^{b} .\end{align*}
\begin{align*}\int_{C}d\theta (\Pi S(\theta _{i}-\theta )-1)a_{n-1}(\theta _{1},..\theta_{n-1})e^{imu\exp \theta }\end{align*}
\begin{align*}Z'_a(\beta) |B \rangle = {R'}^{b}_{a}(\beta)Z'_b(-\beta) |B \rangle\end{align*}
\begin{align*}S=\int d^4x \, \sqrt{g(x) } \left\{ \left[ \frac{\Lambda}{8 \pi G} + \frac{1}{2} \mu^2(x) \right] - \frac{1}{8 \pi G} R(x) \right\} + S_1 + S_2 ,\end{align*}
\begin{align*}\big[\big(C_1^{[r]}\big)_{2V,2V}\big]_{\Lambda=0}=\frac{1}{P^2}\Big\{\sum_{n=1}^rB_n^{[r]}(\Lambda=0)\frac{t(P)}{u_{r,2n+1}}B_n^{[r]}(\Lambda=0)\Big\}.\end{align*}
\begin{align*}\begin{array}{ccccccc}{\cal L}_\mathbf{I} K^x & = &\epsilon^{xyz}K^yW^z_\mathbf{I} & ; &{\cal L}_\mathbf{I}\omega^x&=& \nabla W^x_\mathbf{I}\end{array}\end{align*}
\begin{align*} \ q_j^{N,M,k}> 0, \ \ j=1,..., J^{N,M,k}, \ \ \ \ \sum_{j=1}^{J^{N,M,k}} q_j^{N,M,k}=1,\end{align*}
\begin{align*}X=\bigcup_{t\geqslant 0} F^{-t}(\partial\Omega)\hskip 40pt\partial\Omega=\bigcup_{i\in I}\partial\Omega_i.\end{align*}
\begin{align*}\hat{B}_L (f_B^{(\tau)}) \cdot 1 \hat{\tau}_3 \cdot \hat{B}_R (f_B^{(\tau)}) = 1 \, \hat{\tau}_3 ,\end{align*}
\begin{align*}[a^{(p)}_{i_p},(a^{(p^\prime)}_{i_{p^\prime}})^\dagger]=\delta_{i_{p^\prime}i_p}\delta ^{p^\prime p},\end{align*}
\begin{align*}\delta\lambda=- \frac{e^{\phi}}{2}\Gamma^{M}\partial_{M}\tau \, \varepsilon^* +\frac{e^{\frac{1}{2}\phi}}{24}\Gamma^{MNP}G_{MNP}\, \varepsilon\end{align*}
\begin{align*}\begin{array}{l}v_{k+1}=W\left[ \left( 2F(u_k,v_k)-F(u_k,v_{k-1})\right) \,,u_k\right] , \\u_{k+1}=\bar W\left[ \left( 2\bar F(u_k,v_k)-\bar F(u_{k-1},v_k)\right)\,,v_k\right] .\end{array}\end{align*}
\begin{align*}{\rm Tr}\,\left( \partial _{\alpha }M^{-1}\partial_{\beta}M\right) = - 4{\rm Tr}\,\left( P_{\alpha }P_{\beta }\right) \end{align*}
\begin{align*}F\ddot{F}-F^2-2\dot{F}^2=d{\cal G} ~~\mbox{and}~~F^4-\dot{F}^2-F^2= 2{\cal G},\end{align*}
\begin{align*}v_{x^3}=\left| \frac{k^0}{k^3}\right| \approx 1+ \theta B,\end{align*}
\begin{align*}\frac{dS}{dt} = \frac{8 \pi^{2}}{\tau g^{2}}~~,\end{align*}
\begin{align*}\varphi_{a} \rightarrow \varphi^{\prime}_{a} (\varphi_{b})\end{align*}
\begin{align*}\begin{array}{l}a_{ij}^{{\bf 4}}= \frac{1}{g}\sum\limits_{\gamma=1}^{r}r_{\gamma }\chi_{\gamma }^{R_M \oplus {\bf 1}^2 }\chi_{\gamma }^{(i)}\chi_{\gamma }^{(j)*} \\a_{ij}^{{\bf 6}}=\frac{1}{g}\sum\limits_{\gamma=1}^{r}r_{\gamma }\chi_{\gamma }^{R_M \oplus R_M \oplus {\bf 1}^2 }\chi_{\gamma }^{(i)}\chi_{\gamma }^{(j)*} \\\end{array}\end{align*}
\begin{align*}h_p(z_0,T)= \frac{\nu_p(z_0)-\nu_p(z_T)}{T} \approx h_p(z_0)\end{align*}
\begin{align*}F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+[A_{\mu},A_{\nu}]\end{align*}
\begin{align*}{\cal G}\frac{}{}\mid_{p_2=-p_1} =\int {\cal D} k \frac{i}{[-(p+k)^2 -i\eta]^2}\frac{1}{(-k^2-i\eta)^{-\frac{1}{2} +\frac{\epsilon}{2}}}\;,\end{align*}
\begin{align*}f_3(l,l,b) = -\frac{1}{(4\pi)^h\Gamma(h)}\frac{1}{\gamma}[1 - \frac{\gamma}{h-1} + ...]\;.\end{align*}
\begin{align*}{\cal F}_{{\rm F}=0} = (1-{\rm N}^2) \frac{e_{+}^2}{2{\pi}^2{\hbar}^2} \sum_{n>0} \frac{1}{n}\sin(\frac{2\pi}{\rm L} n(x-y)) = (1-{\rm N}^2) \frac{e_{+}^2}{2\pi{\hbar}^2} ( \frac{1}{2} \epsilon(x-y)- \frac{1}{\rm L} (x-y) ).\end{align*}
\begin{align*}\Lambda^q\mbox{Tor}_1^X({\cal O}_S, {\cal O}_T)\to\mbox{Tor}_q^X({\cal O}_S, {\cal O}_T).\end{align*}
\begin{align*}\lambda^{\mu} Y_{,\mu} - a \, X - \dot{z}^{\mu} X_{,\mu}+ \epsilon \, \lambda^{\mu} X_{,\mu} = 0 \end{align*}
\begin{align*}-\frac{i}{ \pi} \int_0^\infty dt \int_0^{\pi/2} d \theta \frac{ t^2 \left[ g(-i t) - g(i t) \right] \cos \theta}{(t^2 + z^2 \cos^2 \theta)^{3/2}} =-\frac{i}{\pi} \int_0^\infty dt \frac{t [ g(-it) - g(it)] }{(t^2 + z^2)},\end{align*}