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OleehyO
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latex-formulas

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\begin{align*} F^{2}_{\nu \theta} \propto \sin \theta \sinh^2 \phi\; \rightarrow \; \frac{ac}{b}\,=\,-\frac{1}{\bar{g}}\end{align*}
\begin{align*}\hat{A}_\alpha (t) \left|\alpha, 0, t \right> = 0,\end{align*}
\begin{align*} - \epsilon^{q}_{bk} \hat{R}_{ib,am} \epsilon^{q}_{cm} K_{ac} =(\epsilon^{q})_{kb}^{t} \hat{R}_{ib,am} (\epsilon^{q})_{mc}^{t-1} K_{ac}= \hat{R}^{\epsilon} \, _{ik,ac} K_{ac} = K_{ik}^{\epsilon}\end{align*}
\begin{align*}h_p(z)=\lim_{t\to\infty}\, -\frac{1}{t}\,\nu_p(F^t(z)).\end{align*}
\begin{align*}\Bigl[\;{\cal L}(u)\; ,\;\Phi(v)\;\Bigr]=0. \end{align*}
\begin{align*}\left(\begin{array}{cc}\hat h+\Sigma & \beta \Delta ^{\dagger }\beta C \\-C\,\Delta & \hat h+\Sigma\end{array}\right) \left(\begin{array}{c}u \\v\end{array}\right) =E\left(\begin{array}{c}u \\v\end{array}\right) \end{align*}
\begin{align*}{\delta^{(0)} V_-^\alpha = i V_+^\alpha \bar \epsilon \lambda^* - i\Sigma V_-^\alpha.}\end{align*}
\begin{align*}(z_{1},z_{2}) ~\rightarrow ~(\bar z_{2}, -\bar z_{1})\end{align*}
\begin{align*}\Phi^{a}~=~\dot{N}^{a}~+~\chi^{a}(q^{i},p_{i},N^{a})~,\end{align*}
\begin{align*}\int_Cd^nk{P(k,p)\over{(k^2+i\epsilon)[(k+p^2+i\epsilon]\zeta^\alpha}},\ 0\leq\alpha\leq2,\end{align*}
\begin{align*}S_{BH} = {\frac{A}{4\hbar G_d}} \quad ,\end{align*}
\begin{align*} 2 \Delta \tilde \sigma = -e^{-2\tilde\sigma}- 4\pi (a_n -1) \delta^2(z)\end{align*}
\begin{align*}g_{str}=g_{YM}^2,\;\;\;\left({R\over l_{str}}\right)^4=g_{YM}^2N\end{align*}
\begin{align*}f_{12}^2 = -f_{21}^2 = f_{13}^3 = -f_{31}^3 = 1 \,\,.\end{align*}
\begin{align*}\mu(A)=\lim_{N\to\infty}\frac{\#(A\cap\mathcal{B}_N)}{\#(X\cap\mathcal{B}_N)}\end{align*}
\begin{align*}y_2 ( x ) = \log ( x ) F ( \alpha, \beta; 1; x ) +\sum_{n = 1}^{\infty} a_n x^n \ .\end{align*}
\begin{align*}\epsilon \equiv {E \over M + m} = {E \over 2m_0}.\end{align*}
\begin{align*}\frac{m}{2}\,\dot{K} \sim -\,\frac{\lambda^{2}}{12\pi}\,\frac{Ke^{2\lambda \tau}\,(Ke^{2\lambda \tau}+2)}{(Ke^{2\lambda \tau}+1)^{2}} \,,\end{align*}
\begin{align*}i\frac{\partial\psi}{\partial t} = -i\mbox{\boldmath$\alpha$}\cdot\nabla\psi + \beta m\psi -g\mbox{\boldmath$\alpha$}\cdot{\bf A}_i\lambda_i\psi + g A^0_i\lambda_i\psi\end{align*}
\begin{align*}\Delta\sigma ^2=l^2 \left( \Delta \psi^2+4\rho^2\sin^2{\Delta \theta \over 2}\right)\end{align*}
\begin{align*}S=-\frac{1}{16\pi G}\int_{\cal M}d^{n+2}x\sqrt{-g} \left (R-\frac{n(n+1)}{l^2}\right) +\frac{1}{8\pi G}\int^{\partial \cal M^+} _{\partial \cal M^-}d^{n+1}x\sqrt{h}K.\end{align*}
\begin{align*}ds^2 = \sum^{k} (d\rho_{i}^2 + \rho_{i}^{2} d\psi_{i}^{2}) +d\bar{s}_{d-2k}^2,\end{align*}
\begin{align*}n_I{\bf k}_I\cdot{\bf x}\,=\,n_R{\bf k}_R\cdot{\bf x}= {\bf k}_T\cdot{\bf x} \;\;\mbox{at}\; x_1=0.\end{align*}
\begin{align*}\mid \Psi \rangle = (L_{-2} + \frac{3}{2} L_{-1}^{2}) \mid\tilde{\chi} \rangle\end{align*}
\begin{align*}\partial_\mu\partial^\mu\chi=0,\end{align*}
\begin{align*}f(x)=\begin{cases} \frac{3}{2} x+\frac{3}{2} & x < -1\\ 0 & -1 \leqslant x \leqslant 1\\ \frac{3}{2} x-\frac{3}{2} & x > 1\end{cases}\end{align*}
\begin{align*}\int dx \phi^n_\star=\int dp_1 dp_2 \ldots dp_n \exp(-\frac{i}{2}\sum_{i<j}(p_i)_\mu\theta^{\mu \nu}(p_j)_\nu) \phi(p_1)\phi(p_2)\ldots\phi(p_n) \delta(p_1+p_2+\ldots+p_n)\end{align*}
\begin{align*}\Phi\,\rightarrow\,\Phi+\frac{2\pi}{\beta}\quad ,\quad \Phi\,\rightarrow\,-\Phi\,,\end{align*}
\begin{align*}u = \Phi(\chi_x,\chi_{xx},\chi_{xxx}) \equiv\frac{\chi_{xxx}}{\chi_x} - \frac{\chi_{xx}^{\;2}}{\chi_x^2} + 2\alpha \chi_{xx} + \alpha^2 \chi_{x}^2 + 2 \alpha \varepsilon +\frac{\varepsilon^{2}}{\chi_x^2} \end{align*}
\begin{align*}g_{+}(s',t)=\frac{f_{-}(s',t)}{s'^{2n-1}}\end{align*}
\begin{align*}\left\{\phi\left(x^0,{\bf x}\right),\pi\left(x^0,{\bf x'}\right)\right\}_{\rm P.B.}=\delta_{{\bf x},{\bf x'}}.\end{align*}
\begin{align*}\{\psi^n, \psi^m\}_D=-i\eta^{nm},\end{align*}
\begin{align*}{\sigma}_2 (|z| \to \infty) \approx \frac{ 2{\hbar}ce^{-\frac{\pi}{2}{\beta}} } {| {\Gamma}(1+i{\beta}) |}z^{-1/4} \cos\left(\frac{z}{2{\hbar}c} + \beta {\rm ln}\frac{z}{{\hbar}c}+ \delta \right).\end{align*}
\begin{align*}[a^{m},a^{\dagger n}]=\sum _{g=0}^{\infty} \frac{1}{2^{2g}(2g+1)!} \prod_{k=0}^{2g} (m-k)(n-k) (a^{m-2g-1} a^{\dagger n-2g-1} )_{W} . \end{align*}
\begin{align*}u_{0,l}(r)=\left( \begin{array}{l}\phi _{0,l,1}(r) \\ 0\end{array}\right) ,{\rm \;}{}l\geq 1;\;\;\;u_{0}^{II}(r)=\left( \begin{array}{l}\phi _{0,1}^{ir}(r) \\ 0\end{array}\right) ,\;{}l=0. \end{align*}
\begin{align*}\hat f_\sharp\,C_\lambda(X)= C_{\lambda}(X).\end{align*}
\begin{align*} z_t=F^t(z_0)=\mathrm{M}^t\,z_0^\prime+z^*.\end{align*}
\begin{align*}\gamma^0 S^+(p,p_\perp)\gamma^0=S(p,p_\perp)\end{align*}
\begin{align*}G^{+}(x,x')=G^{+}_{R}(x,x')+\langle \varphi (x)\varphi (x')\rangle ^{(b)}, \end{align*}
\begin{align*}(\gamma_t\gamma_\alpha \, p^\alpha - m \, \gamma_t) \, \Psi = 0\end{align*}
\begin{align*}X_1 = \sqrt{\alpha} \sinh( r_+ \bar \phi - r_- t),\end{align*}
\begin{align*}\hat{l} = 0; \hat{B}_{i\bar{j}} = 0;D_{\hat{\mu}\hat{\nu}\hat{\sigma} \hat{\tau}}=0.\end{align*}
\begin{align*}\not \! d \,=\, \sigma_\mu D_\mu\end{align*}
\begin{align*}g_{\mu}=\langle phys |A_{\mu} M^{\dagger}(k)|\psi \rangle\end{align*}
\begin{align*}\psi_{\pi\varphi\ldots}(\stackrel{\rightharpoonup}{x_1},\ldots, \stackrel{\rightharpoonup}{x_N})\rightarrow(-1)^{\#\pi-{\rm indices}}\psi_{\pi\varphi\ldots}(-\stackrel{\rightharpoonup}{x_1},\ldots, -\stackrel{\rightharpoonup}{x_N})\end{align*}
\begin{align*}z^{\mu}_{\tau}(\tau ,\vec \sigma )=(\sqrt{ {g\over {\gamma}} }l^{\mu}+g_{\tau {\check r}}\gamma^{{\check r}{\check s}}z^{\mu}_{\check s})(\tau ,\vec \sigma ),\end{align*}
\begin{align*}\bar{a}_i^be_b^\mu D_{\mu L}|a\rangle=0\,,\quad(a_i^b\bar{a}_i^b-h_i)|a\rangle=0\,,\quad\bar{a}_i^b\bar{a}_i^b|a\rangle=0\,,\quad\varepsilon^{ij}a_i^b\bar{a}_j^b|a\rangle=0\,,\end{align*}
\begin{align*}u=\nu_p(D),v=\nu_p(T)\end{align*}
\begin{align*}<\bar{0}|\eta_{\hat{\alpha}\hat{\beta}} :\bar{T}^{\hat{\alpha}\hat{\beta}}:|_{P_0}|\bar{0}> = -<\bar{0}|:\bar{T}^{\hat{1}\hat{1}}:|_{P_0}|\bar{0}> = -\frac{1}{\sinh^2(\chi)} <\bar{0}|:T^{\hat{0}\hat{0}}:|_{P_0}|\bar{0}>\end{align*}
\begin{align*}\left( D_{i}\right) _{\;\;b}^{a}=\delta _{\;\;b}^{a}\partial_{i}+gf_{\;\;bc}^{a}A_{i}^{c},\;\left( D_{i}\right) _{b}^{\;\;a}=\delta_{b}^{\;\;a}\partial _{i}-gf_{\;\;bc}^{a}A_{i}^{c}. \end{align*}
\begin{align*}\frac{p_r^2}{g_{rr}} - \frac{ {g'}_{tt}}{- \cal D} \left( p_\phi + \frac{g_{t \phi } + \Omega_0 g_{\phi \phi}}{{g'}_{tt}} E \right)^2 = - \left( \frac{ E^2 }{ {g'}_{tt}} + V \right),\end{align*}
\begin{align*}\left(\ref{kotencpnkosoku}\right)=\int^\infty_{-\infty}d\lambda e^{-i\lambda}\prod^{N+1}_{\alpha=1}{1\over\rho\theta_\alpha-i\lambda}\ .\end{align*}
\begin{align*}\left\{\bar D_{a}\, , \bar D_{b} \right\}= 2im^2\bar C_{ab} \, .\end{align*}
\begin{align*}\frac{P^2}{2}+\omega^2 X^2=iD+\tau D^2. \end{align*}
\begin{align*}\left( K_{\alpha} +\frac{1}{4} K_ {\gamma} \right)^{\cdot} +4 \dot{\hat{\alpha}} \left( K_{\alpha} + \frac{1}{4} K_{\gamma} \right) =0.\end{align*}
\begin{align*}A_{{\alpha}[p]} \equiv A_{{\alpha}_1....{\alpha}_p} \ .\end{align*}
\begin{align*}x_0^2=x_1^2+x_2^2 +x_3^2 +l^2.\end{align*}
\begin{align*}S^{SR}_{mm',nn'}=\frac{1}{u} (-1)^{G+G'+(u-1)(m+n)}e^{-i\pi(u+1)mn/u} e^{-i\pi(u-1)(m-2m'+u)(n-2n'+u)/u},\end{align*}
\begin{align*}\mu_k^{N,M} = {\rm argmin}_{\mu\in W_M(z_k^{N,M})}\{\tilde G(\mu, z_k^{N,M}) + \nabla \zeta^{N,M} (z_k^{N,M})^T \tilde g (\mu , z_k^{N,M}) \}.\end{align*}
\begin{align*}z_t=\alpha^t c_1\mathbf{w}_1+\beta^t c_2\mathbf{w}_2\end{align*}
\begin{align*}{\bar G}^{(\zeta )}(\beta )={(\beta +i\pi )\sinh\left( {\displaystyle \pi\beta\over \displaystyle 2\pi +\zeta}\right) \over\Gamma \left( 1- {\displaystyle\pi-i\beta \over \displaystyle 2\pi+\zeta}\right)\Gamma \left( 1- {\displaystyle\pi+i\beta \over \displaystyle 2\pi+\zeta}\right)}\ ,\end{align*}
\begin{align*}\sigma^{\beta}=\sigma-\frac{\lambda}{24\beta^2}\end{align*}
\begin{align*}\begin{array}{rl}{\rm gravity}: & (3,3) + 2(2,3) + (1,3) \\{\rm tensor}: & (3,1) + 2(2,1) + (1,1) \\{\rm vector}: & (2,2) + 2(1,2) \\{\rm hyper}: & 2(2,1) + 4(1,1).\end{array}\end{align*}
\begin{align*} {\rm Ad} U_{\pi}(g)\circ\pi=\pi\circ\alpha_{L(g)}\end{align*}
\begin{align*} E_2(R,\theta) = 2E_1 + k R^{-3} \cos(\theta_+ - \theta_-),\end{align*}
\begin{align*}\bar{\eta}_\gamma \gamma^* {\bf\bar{\alpha}} \, \bar{\eta}_\gamma^{-1}+ \bar{\eta}_\gamma d\bar{\eta}_\gamma^{-1} = {\bf\bar{\alpha}}.\end{align*}
\begin{align*}-\nabla^2=-\nabla_c^2-\epsilon\nabla_1-\epsilon^2\nabla_2-\cdots\;,\end{align*}
\begin{align*}(\Gamma^{(s)}_{\sigma})_{j\bar{\jmath}} \sim \delta_{j\bar{\jmath}}(j(j+1)-6)^{-1},\end{align*}
\begin{align*}\frac{1}{\beta} =\frac{N}{\beta N} = \int \frac{dp\,d\lambda}{2\pi}\,\theta\biggl( \varepsilon_F -\frac{p^2 }{2} - V(\lambda) \biggr)\ . \end{align*}
\begin{align*}\mu'' - \partial_{\alpha}\partial^{\alpha} \mu - \frac{s''}{s}\mu=0.\end{align*}
\begin{align*}\Vert z_t\Vert_p=|\alpha|_p^t\Vert c_1\mathbf{w}_1\Vert_p,\end{align*}
\begin{align*}\Gamma^x = \mathbf{E}(\theta^x) - \mathbf{I}^\dagger(D_{\theta^x}) \# .\end{align*}
\begin{align*}a_{i,0} (x,\theta) \ = \ \sum_{p=0}^{k+i} a_{i,0,p}\ x^p\ +\ \theta\sum_{p=0}^{k+i-1}\overline{a}_{i,0,p}\ x^p\end{align*}
\begin{align*}\frac{\delta S_{f}}{\delta f^{a}(x)}\vert_{\tilde{f}}=0,\end{align*}
\begin{align*}S=S^0+\phi ^*_i Z^i{}_{a}(\phi ) c^a+\sum_{k=0}c^*_{a_k}Z^{a_k}{}_{a_{k+1}}c^{a_{k+1}}+\ldots \ ,\end{align*}
\begin{align*}0=F_u^{\prime\prime}+\frac{2}{r}F_u^\prime \left[1+rL_-^\prime(kr)\right]-\frac{s^\prime}{s+\frac{\omega}{m}} \left[F_u^\prime+F_u L_-^\prime(kr)\right] +W(r,-m) F_u -\left[K,F_u\right] \end{align*}
\begin{align*}{{2\epsilon^3}\over{35}} - {{57}\over{14}} - {1\over{20}}\,{{{\rm Bi}'(-\epsilon)}\over{{\rm Bi}(-\epsilon)}}\end{align*}
\begin{align*}\zeta_V(s)=\sum_{(n)}(\lambda _{(n)}^2 + m^2)^{-s},\end{align*}
\begin{align*}\mu' = \alpha^2 x^2 (\bar K + (\bar U - \frac{q^2}{x^2})) \ .\end{align*}
\begin{align*}L^{(1)}=\left(a_i-\eta^{\mu}A_{\mu i} \right) {\dot{y}}^i-H(y), \end{align*}
\begin{align*}\frac{4q^2+12q+15+ 8p(q+2)+4p^2}{4\{2\}}b^2\ + \frac{(3+2p)[q^2+3q+3+2p(q+1)]}{2}b\bigg]\ ,\end{align*}
\begin{align*}s(x)=x^2-T'x+D'p^{u-2v}\mathrm{with}\frac{ds(x)}{dx}=2x-T'.\end{align*}
\begin{align*} (t^{k+1}_k)^{a_k+1}|a_1,\ldots,a_{N-1}\rangle =0\;.\end{align*}
\begin{align*}\tilde{\cal H}_1({\bf r}_1, {\bf r}_2) = -\frac{2}{m}\frac{f'(r_{12})}{f(r_{12})}\frac{{\bf r}_{12}}{r_{12}}\frac{\partial}{\partial{\bf r}_1}\end{align*}
\begin{align*}R(C_{L},D_{L})\;Q(K_{\star})= Q(K_{\star})\;R(c_{L},d_{L}).\end{align*}
\begin{align*}\quad \lim_{\sigma \rightarrow \infty }Q_{2s+2t+1}^{\Sigma|2s,2t+1} \sim e^{-s\sigma }\end{align*}
\begin{align*}\langle {\bf p}\, \vert {\bf x} ,t \rangle =\langle {\bf p}\, \vert U(I, ({\bf x},t) \vert {\bf 0} ,0 \rangle =e^{i {\bf p} \cdot {\bf x} - i \omega_m ({\bf p}\, ) t} \langle {\bf p}\, \vert {\bf 0} ,0 \rangle \end{align*}
\begin{align*}D_{++}^{(ML)cd}(x) =D_{++}^{ML}(x) \delta^{cd} = {i \delta^{cd}\over \pi^2}\int d^2k\, e^{ikx} {k_+^2\over (k^2 + i \epsilon)^2}= {\delta^{cd}\over \pi}{(x^-)^2\over (-x^2 + i\epsilon)}\ \end{align*}
\begin{align*}[J^{x\bar{x}},Q^{\pm i}]=\pm\frac{1}{2}Q^{\pm i}\,,\quad[J^{x\bar{x}},Q^\pm_i]=\mp\frac{1}{2}Q^\pm_i\,,\end{align*}
\begin{align*}{\bar{D}}_{\dot{\alpha}}=\left( -\partial _{\dot{\alpha}}-\theta ^{\alpha}\Gamma _{\alpha \dot{\alpha}}P_{\mu }\right)\end{align*}
\begin{align*}\int d^2\theta\left(W_\alpha^2+ Y\phi \bar Y\right)+\int d^4\theta\left(\bar\phi \phi +\bar Y Y\right)\, .\end{align*}
\begin{align*}\delta I_{D}=\int [\delta e^{a}{\cal E}_{a}+\delta \omega^{ab}{\cal E}_{ab}]=0,\end{align*}
\begin{align*}r_k=\frac{1}{1+p^k}k=1,2,\ldots\end{align*}
\begin{align*}V_3 \equiv V(\eta_3) = \Delta_3 f(\eta_3) F(\eta_3)\end{align*}
\begin{align*}\delta _{Ra}\delta _{Rb}+\delta _{Rb}\delta _{Ra}=0,\;a,b=1,2, \end{align*}
\begin{align*}\check{G}_{\mu\nu}= -8\pi G_{5}\check{T}_{\mu\nu}.\end{align*}
\begin{align*}Z_2 = -2\tau T R\Omega (\Omega-1) (1+x^{2T})^{\Omega-2} {x^{2(2T-1)}}\end{align*}
\begin{align*}\begin{array}{rlrl}1^* :& a\mapsto \epsilon(a)&\mu^*(f_1,f_2):& a \mapsto f_1\otimes f_2\circ\Delta(a) \\\epsilon^*:& f \mapsto f(1)& \Delta^*(f):& (a_1,a_2) \mapsto f(\mu(a_1,a_2))\\S^*(f):& a \mapsto f(S(a))&~&~\end{array}\end{align*}
\begin{align*}\det(M_{tN}) = \det(\tilde{M}_{tN}) = \prod_{r=1}^{t} \det \left( A + \omega^r D + \omega^{-r} E \right)\end{align*}
\begin{align*}S = \int d^dx(\frac{1}{2} (\partial_\mu \vec\Phi(x))^2 -\frac{U}{4} (\vec\Phi^2(x)-C^2)^2)\end{align*}