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OleehyO
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latex-formulas

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\begin{align*}\overline{\varphi (\theta +i\pi )}=-\varphi (\theta )\end{align*}
\begin{align*}{\cal H}_\Lambda =\Lambda _{\overline{0}}\widehat{\otimes }{\cal H}_\Lambda^{res}. \end{align*}
\begin{align*} S = \frac{1}{2\kappa^2} \int{ d^4x \sqrt{-g} \left( R + \frac{1}{6m^2} R^2 \right) }.\end{align*}
\begin{align*}z^{(k)}=z+r_k(a,b)k=1,2,\ldots\end{align*}
\begin{align*}\frac12 \sum_{k} g_k \left({2k \atop k}\right) r(t)^{k} =t.\end{align*}
\begin{align*}[{\cal D}_{--},{\cal D}_{-A}]=0,\end{align*}
\begin{align*}{\cal O}=\frac{T_R^6}{P_R} \otimes \frac{T_L^6}{P_L} \otimes \frac{T_L^{16}}{G}.\end{align*}
\begin{align*}A_{r}(t,r,\theta,\varphi)=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}b_{l}(r)Y_{lm}(\theta,\varphi)e^{i\omega t},\end{align*}
\begin{align*}m_C^2 = \alpha g^2 \langle i \bar C^a(x) C^a(x) \rangle ,\end{align*}
\begin{align*}H = {1 \over 2} [ {p}^2 + {w^2 \over v^2} + B v] (2L) \;.\end{align*}
\begin{align*}\delta _Kj_T^{0M_1\cdots M_p}=\sum_{k=1}^p\partial _{\mu_k}\left[ \frac 1{p!}\epsilon ^{0\mu _1\cdots \mu _k\cdots \mu _p}\cdot \phi_{,\mu _1}^{M_1}\cdots \delta _K\phi ^{M_k}\cdots \phi _{,\mu_p}^{M_p}\right] \quad .\end{align*}
\begin{align*}[h,\,e_{\pm}^{}]=\pm e_{\pm}^{}\,,\qquad [e_{+}^{},\,e_{-}^{}]=2h \,\end{align*}
\begin{align*}\Phi(n,p) = {4 |C_n| \over\hbar}~{\rm sin}~[({2\pi \over L})^2~{\hbar p n t \over 2}]\end{align*}
\begin{align*}\int \; d\mu[\varphi] \; \exp \Big( F (\varphi) \Big) \; \geq \;\exp \left( \; \int \; d\mu[\varphi] \; F (\varphi) \; \right) \; \; ,\end{align*}
\begin{align*}U_t({T},{D}) = \sum_{k=0}^{\lfloor (t-1)/2 \rfloor}\,c_k^{(t)}\;T^{t-2k-1}\,(-D)^k\end{align*}
\begin{align*}S_{L}= \int d^{2}\sigma \sqrt{\hat{g}}(\frac{1}{2} {\hat{g}}^{ab}\partial_{a}\eta \partial_{b}\eta +\hat{R}\eta +\mu e^{\eta}),\end{align*}
\begin{align*}B_{\mu\nu}(t)=\sum_{\rho=0}^N\frac{\eta_{\mu\rho}\eta_{\nu\rho}}{\sqrt{8\omega_\mu}}\left[\left(1+\frac{\omega_\mu}{\Omega_\rho}\right)\rm{e}^{i\Omega_\rho t}+\left(1-\frac{\omega_\mu}{\Omega_\rho}\right)\rm{e}^{-i\Omega_\rho t}\right]\;.\end{align*}
\begin{align*}\begin{array}{lcl}{\cal L}_A^{2n+1} a &=& e^{i\phi} \rho^{2n+1}a^\dagger \\{\cal L}_A^{2n} a &=& \rho^{2n}a \\ \end{array}\end{align*}
\begin{align*}\widetilde{A} = \left(\begin{array}{cc} A & 0 \\ 0 & 0\end{array}\right).\end{align*}
\begin{align*}{\textstyle\bigwedge^i} (W\oplus V)=\sum_{n=0}^i {\textstyle\bigwedge^n(W)\otimes \bigwedge^{i-n}}(V) .\end{align*}
\begin{align*}\gamma_{g,9}:|D_{\rm inv}\rangle \rightarrow {1\over\sqrt{D}} \sum_\alpha \sum_{\beta,\gamma} c_{\alpha\beta} {\overline c}_{\gamma\alpha} |d_\beta\otimes{\overline d}_\gamma \rangle={1\over \sqrt{D}} \sum_\alpha |d_\alpha\otimes{\overline d}_\alpha\rangle=|D_{\rm inv}\rangle~,\end{align*}
\begin{align*}\phi(r)=\phi_{\infty}+{D\over r_s}\ln\left( 1-{r_s\over r}\right).\end{align*}
\begin{align*}\begin{array}{lcr}D^{++}=u^{+ \alpha}{\partial\over{\partial u^{-\alpha}}};\qquad D^{--}=u^{- \alpha}{\partial\over{\partial u^{+\alpha}}}\\2D^{++}=[D^0,D^{++}];\qquad -2D^{--}=[D^0,D^{--}]\\D^0=[D^{++},D^{--}]=u^{+ \alpha}{\partial\over{\partial u^{+\alpha}}}-u^{- \alpha}{\partial\over{\partial u^{-\alpha}}}\end{array}\end{align*}
\begin{align*}a_0 \leq 1\;\;\;,\;\;\; 0< \beta \leq {24-d\over 48}\;\;\;;\end{align*}
\begin{align*}\vec \phi = \left(\begin{array}{c}\sin \frac{\theta}{2} \\ 0 \\ \cos \frac{\theta}{2}\end{array} \right),\end{align*}
\begin{align*}c_k^{(t)}=\binom{t-k-1}{k}.\end{align*}
\begin{align*}{\partial V_{\rm eff} \over \partial \sigma} \mid_{\sigma=\sigma_0}=0. \end{align*}
\begin{align*}H_{\hat{\chi}}=0, \quad \chi=\pi/2.\end{align*}
\begin{align*}e^\mu _a \,e_{b \mu }\,=\,\eta_{a b}, \quad 0\,\leq a, \,b \,\leq D-1, \quad \mu \,=\,0,1, \ldots , D-1\,{.}\end{align*}
\begin{align*}\Biggl( \phi_1(x), {\phi_2(x)\choose\phi_3(x)},{\phi_4(x)\choose\phi_5(x)} \Biggr) \rightarrow \Biggl( c\, e\sp{\lambda x} \phi_1(x), U {e\sp{-\lambda\sp\prime x}\phi_2(x)\choose e\sp{-\lambda\sp\prime x}\phi_3(x)},V {e\sp{\lambda\sp{\prime\prime}x} \phi_4(x)\choose e\sp{\lambda\sp{\prime\prime}x} \phi_5(x)} \Biggr)\end{align*}
\begin{align*}Y^-_{j,j+1}(q^{-2r})|_{q = 0} = -\Pi_{j,j+1}^-(0) = -(|+-><+-|)_{j,j+1} \quad(j=1,\dots,N-1).\end{align*}
\begin{align*}F_{MN}=\partial _{M}A_{N}-\partial _{N}A_{M}+\left[ A_{M},A_{N}\right],\quad R_{MN}^{ab}=\partial _{M}\omega _{N}^{ab}-\partial _{N}\omega_{M}^{ab}+\left[ \omega _{M},\omega _{N}\right] ^{ab}.\end{align*}
\begin{align*}S_{\rm L}(\tau) = Q \Psi_{\rm L} + 2 \pi i \tau k =\bar{Q} \bar{\Psi}_{\rm L} + 2 \pi i \bar{\tau} k,\qquad\tau = \frac{\theta}{2 \pi} + \frac{4 \pi i}{e^2},\end{align*}
\begin{align*}i \, \partial_{t} \, \psi = \frac{1}{2 m}\left[-\nabla^{2} +V\right] \psi \end{align*}
\begin{align*}t_f-t_i=\int_{r_i}^{r_f}\,du\,\frac{\pm 1}{\sqrt{2\left[E_r-V(u)-\frac{L^2}{2u^2}\right]}}\ \ ,\ \ L=\frac{\varphi_f-\varphi_i}{g^2(t_f-t_i)+\int_{t_i}^{t_f}\,\frac{dt}{r^2(t)}}\ \ \ ,\end{align*}
\begin{align*}E_n \rightarrow {2\pi nT_0\over mN}\; .\end{align*}
\begin{align*}0\leqslant \nu_p\left( \binom{n}{m}\right) \leqslant \left\lfloor\frac{\log n}{\log p}\right\rfloor-\nu_p(m).\end{align*}
\begin{align*}N^{r,s}_0=\Delta^{(\ell)}_{r,-s}-\Delta^{(j)}_{r,s}={rs\over K}+\Delta^\ell_\ell-\Delta^j_j{}~.\end{align*}
\begin{align*}Q=nc-\frac{\theta}{2\pi} c^{2}P=e^{+\phi_{0}}(n-a_{0}e^{+\phi_{0}}P)\; ,\end{align*}
\begin{align*}{\rm Li}_3(i/\sqrt3)=\frac{1}{8}\sum_{n>0}\frac{(-1/3)^n}{n^3}+\frac{i}{\sqrt3}\sum_{n\ge0}\frac{(-1/3)^n}{(2n+1)^3}\end{align*}
\begin{align*}S\left( p\right) =\frac{1}{p\hspace{-0.2cm}/ A\left(p^{2}\right) -B\left( p^{2}\right) }\end{align*}
\begin{align*}{\gamma\over{2-\gamma}}{G'}^2+{{GG'}\over\tau}+GG''=k{{\gamma-1}\over{2-\gamma}}\end{align*}
\begin{align*}g_{ij}=g_1(\delta_i^1\delta_j^1-\delta_i^0\delta_j^0) +g_2(\delta_i^2\delta_j^2 +\delta_i^3\delta_j^3),\end{align*}
\begin{align*}\left[ L , \left(L^{2}\right)_{\geq 1} \right] = - \left(2J_{1} +J_{0}^{2} - J_{0,x}\right)_{x} + \partial^{-1} \left(2J_{0}J_{1} +J_{1,x}\right)\end{align*}
\begin{align*}\Delta^{\{a}\Delta^{b\}}=0, \quad V^{\{a}V^{b\}}=0,\quad\Delta^{\{a}V^{b\}} +V^{\{a}\Delta^{b\}}=0.\end{align*}
\begin{align*}\left\{\begin{array}{lcl}\Sigma &=& r^2 + a^2\cos^2\theta\\\Delta &=& r^2 + a^2 + Q^2 - 2Mr\;. \end{array}\right.\end{align*}
\begin{align*} T_7:=\{G_r\sp{(f)},(\sigma_7)\sb{-r}\}.\end{align*}
\begin{align*} z_t = \begin{pmatrix} {x}_t \\ {y}_t \end{pmatrix} =U_t({T},{D})\, \begin{pmatrix} {x}_1^\prime \\ {y}_1^\prime \end{pmatrix}- {D}\, U_{t-1}({T},{D})\, \begin{pmatrix} {x}_0^\prime \\ {y}_0^\prime \end{pmatrix}+ \begin{pmatrix} {x}^* \\ {y}^* \end{pmatrix}\end{align*}
\begin{align*}\phi_{an}T_0\left(M_{1,\nu_2},Z_1\right)=T_0\left(M_{1,\nu_2},Z_1\cup N_1\right)T_0(N_1).\end{align*}
\begin{align*}\kappa=V_{0\mu}\dot V_{1}^{\mu}=-\dot V_{0\mu}V_{1}^{\mu}~,\end{align*}
\begin{align*}N_1 = \sum\limits_{i=1}^{N_p} n_i \quad .\end{align*}
\begin{align*}|m|\simeq 1.7 \cdot 10^4 \; Gev,\end{align*}
\begin{align*}H_{pv}=-R/2\,(\pi/2) \,(a_1^2 +16/5\,a_3^2+4096/729\,a_5^2),\end{align*}
\begin{align*} C_{IJK}\tilde h^J \tilde h^K = H_I(y) \,,\end{align*}
\begin{align*}H = -\beta^i\frac{\partial\Gamma}{\partial\overline q^i} = \beta^i J_i = \sum_iH_i.\end{align*}
\begin{align*}A_q = \int^{\tau_f}_{\tau_i} L_q d\tau + \Gamma_\xi, \end{align*}
\begin{align*}\varepsilon(p,q)=\varepsilon(q,-p-q).\end{align*}
\begin{align*}A_\mu=\frac{1}{m\cosh\mu\sinh\mu}\partial_{[+}A_{-]}\ .\end{align*}
\begin{align*} x_t = \mathcal{T}_t^{(1)}+\mathcal{T}_t^{(0)}+{x}^* \end{align*}
\begin{align*}\alpha= { L \over 2},~~~~~~ -{ L+2 \over 2}.\end{align*}
\begin{align*}\langle {\cal O}^A(x) {\cal O}^B(y) \rangle = \frac{ \delta^{AB} }{\left| x-y \right|^{\Delta_A + \Delta_B} }\end{align*}
\begin{align*}g_S=1+N^2(h-1)+\frac{\sum_{i=1}^v o_i}{2}.\end{align*}
\begin{align*}\frac{p^{2}-m^{2}}{\sqrt{2}p^{+}} =\frac{2p^{+}p^{-}-\omega_{p}^{2}}{\sqrt{2}p^{+}}\end{align*}
\begin{align*}<a_1> \; \propto \; \frac{i x^4}{4} F([1/2, 2/3],[5/3],x^6) |^{x=1/2+i \sqrt{3}/2}_{x=1} \neq 0,\end{align*}
\begin{align*}\langle x^2\rangle_{GH} = \left\{ \begin{array}{l}\zeta + \mbox{ lower orderterms for }\zeta \to \infty,\\O(|\zeta|^{\frac12}) \mbox{ for } \zeta \to -\infty \end{array} \right.\end{align*}
\begin{align*}\partial_{\mu} J_{\mu 5} (x) = 2\bar{q}(x) M \gamma_{5} q(x) + A^{II}(x)\end{align*}
\begin{align*}H_F = \frac{m}{2}\sum_j(\psi_j^\dagger \psi_j - \psi_j\psi^\dagger_j) = mQ_F\end{align*}
\begin{align*}\sigma (\tau) = \sum_{i=-l}^m k_i |\tau-\tau_i| + k_c \tau + c ~,~\,\end{align*}
\begin{align*}\varsigma^{*} W_C= W_1(\rho)\pm W_2(\chi)\end{align*}
\begin{align*}\mu_k^{N,M}= \sum_{j=1}^{J^{N,M,k}} q_j^{N,M,k}\delta_{(u_j^{N,M,k},y_j^{N,M,k})}, \ \ \ \ k= 1,...,K^{N,M},\end{align*}
\begin{align*} \mathcal{T}_t^{(1)}={x}_1^\prime \, \sum_{i_1=0}^{\lfloor (t-1)/2 \rfloor}\,c_{i_1}^{(t)}\,T^{t-2i_1-1}\,(-D)^{i_1},\mathcal{T}_t^{(0)}={x}_0^\prime \sum_{i_0=1}^{\lfloor t/2 \rfloor}\,c_{i_{0}-1}^{(t-1)}\,T^{t-2i_0}\,(-D)^{i_0-1}.\end{align*}
\begin{align*}\Phi = Vol_B + \sum_{i}e_i \wedge \theta_i,\end{align*}
\begin{align*}\omega(G)\le 4-n_{h}-n_{u}-n_{\tilde{u}}-d+n\,.\end{align*}
\begin{align*}\omega (t)= {2\pi \alpha \over {t_2 - t_1}} {(t - t_1)}.\end{align*}
\begin{align*}\sigma \xi^2 = \frac{\sigma}{m^2} \approx 2 \frac{m}{g}\,.\end{align*}
\begin{align*}2 \varepsilon \, \epsilon_{abcde} \left[\delta R^{ab} +{2(1-\varepsilon)\over l^2} e^a \delta e^b\right] e^c e^d =0 .\end{align*}
\begin{align*}V_{\parallel,3}(t)= -i\Xi(\lambda,t)(P_{1}+P_{2})+\Xi(\lambda_{3},t)P_{3}\end{align*}
\begin{align*}V_{(--)}^i= {1 \over 2} e^{2W}r^{(++)\underline{m}}l^i_{\underline{m}}\end{align*}
\begin{align*}P_{1}{d\Phi^1\over dr}= -2 \ell \left(r\right)^{\Lambda\Sigma}{\rm Im}{\cal N }_{ \Lambda \Sigma,\Gamma\Delta}f^{\Gamma\Delta}_{~~XY}\, {e^U\over r^2}\end{align*}
\begin{align*}\frac{B_{36}}{T}=\tan\theta_1,\quad\frac{B_{47}}{T}=\tan\theta_2,\quad\frac{B_{58}}{T}=\tan\theta_3,\quad(0\leq\theta_i\leq\pi),\end{align*}
\begin{align*}b_{00}=2\Bigl(\psi(1)+\psi(\mu)-\psi({\textstyle{\frac{1}{2}}}\tilde{\eta}_{o})-\psi(\mu-{\textstyle{\frac{1}{2}}}\tilde{\eta}_{o})\Bigl).\end{align*}
\begin{align*}\nu_p(x_1')+\nu_p(T)\not=\nu_p(x_0')+\nu_p(D)\end{align*}
\begin{align*}{\hat P}^{\mu}=P^{\mu}+\lambda^{\mu}\int\nolimits V(x)\delta(\lambda x-\tau)d^4x\end{align*}
\begin{align*}\mathcal{A}_N:=\mathrm{End}(\mathcal{H}^N)\end{align*}
\begin{align*}Q_i = p_y \, p_{\overline{\eta}} + \eta \, \sigma_i\ ,\ \ \ \ Q_i^2 = 0\ ,\ \ \ \ \epsilon_Q = 1 \; . \end{align*}
\begin{align*}[{\cal L}_\alpha,{\cal J}_\beta]=-\frac{i}{2}(1+\nu R)(P\gamma)_{\alpha\beta}+(1+\nu R)\Gamma_{\alpha\beta}+2i(PJ)\epsilon_{\alpha\beta},\end{align*}
\begin{align*}\left.\frac{dV}{d\sigma}\right|_{\sigma\rightarrow +0} \rightarrow -\frac{1}{L}\, .\end{align*}
\begin{align*}\phi_\epsilon({\bf x}) = \phi(\epsilon,{\bf x}).\end{align*}
\begin{align*} \Gamma^{\sf t}_{ij} = a\left(\frac{da}{d{\sf t}}\right)\gamma_{ij}\;, ~~~\Gamma^{i}_{{\sf t}j} = \frac{1}{a}\left(\frac{da}{d{\sf t}}\right)\delta^i_j \;,~~~\Gamma^{i}_{jk}=\frac{\gamma^{il}}{2}(\partial_j\gamma_{lk}+\partial_k \gamma_{lj}-\partial_l\gamma_{jk}) \;.\end{align*}
\begin{align*}C_{r}= {AdS}_{d+1} \cap \{{X^0}^2+{X^{d+1}}^2 = r^2 + 1\}\end{align*}
\begin{align*}\Delta X \sim |\Delta r| = \alpha' |\Delta U| .\end{align*}
\begin{align*} a_j=0, \quad j=3,\, 7/2,\, 4,\, 5/2,\, 5,\, \ldots\end{align*}
\begin{align*}\nu_p(x_1')-\nu_p(x_0')-\nu_p(T)\not=\nu_p\bigl(\frac{t-1}{2}\bigr)\end{align*}
\begin{align*}P_i \to P_i - e\hat{A}_i\ \ {\bf{\longrightarrow}}\ \ P_i \to P_i - e\hat{A}^{tot}_i\ ,\end{align*}
\begin{align*}{\cal B} (r) = \ell (-\ell + 2ikr_0)\end{align*}
\begin{align*}\Phi =\frac 12\ln (\Delta _1^{s_1}\Delta _2^{s_2}\cdots \Delta_r^{s_r})\end{align*}
\begin{align*}\Delta_0(0) - \Delta_0(x^2) = -\frac{\xi_0}{8\pi}\left|x\right|,\end{align*}
\begin{align*}E_{pm}=-\frac{\pi}{8Ar{\gamma}}({\cal P}^2 + {\cal M}^2).\end{align*}
\begin{align*} j_{++}^{nm}=a_n^\dagger a_m, \hspace{1cm} j_{--}^{nm}=b_n^\dagger b_m, \hspace{1cm} j_{+-}^{nm}=a_n^\dagger b_m, \hspace{1cm} j_{-+}^{nm}=b_n^\dagger a_m,\end{align*}
\begin{align*}H = \partial_u \partial_v \left( k + \tilde{k} \right) .\end{align*}