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OleehyO
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latex-formulas

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\begin{align*}G^R (k) = { N^2 \epsilon^{2(\Delta - 4 )}\over 8\pi^2(\Delta - 3)!^2 2^{2\Delta - 5} } (q^2)^{\Delta - 2}\left[ \ln{q^2} - i \pi\, \mbox{sgn}\, \omega\right]\,, \;\;\;\; \Delta = 2, 3, 4, ...\end{align*}
\begin{align*}{\ddot \phi } + F(t,x)\phi \equiv H\phi=0 \, .\end{align*}
\begin{align*}\zeta(-1)=\frac{1}{2\pi i}\int\limits_{C}^{}\frac{(-z)dz}{z^3\left(1-e^{-z}\right)}.\end{align*}
\begin{align*}V_N(T)=\frac{1}{N-1}\sum_{i=1}^{N-1}\bigl|h_2(z_0^{(i+1)},T)-h_2(z_0^{(i)},T)\bigr|\end{align*}
\begin{align*}\delta g_{rr} =0\ ,\quad \delta g_{\mu r}=0\ ,\quad \delta g_{\mu\nu} = -2A'\tilde r e^{2A}\eta_{\mu\nu}\ ,\end{align*}
\begin{align*}P^{\rm b.t}_{{\bf k}_{\perp}} = \sum_{n = 1}^{\infty} (-1)^{n+1}e^{- 2n S_{{\bf k}_{\perp}}} = \frac{1}{ e^{2S_{{\bf k}_{\perp}}}+ 1},\end{align*}
\begin{align*}\overline T{^{\mu\nu}}K_{\mu\nu}^{\ \ \,\rho}=\perp^{\!\rho}_{\,\mu}\overline f{^\mu}\, ,\end{align*}
\begin{align*}\left. +\, {\rm Tr} [\Gamma_\nu D_\nu (\Gamma_\mu D_\mu - \Gamma_D \partial_D)^{-1} \, \Gamma_D ]^{(-)} \;-\; (+) \to (-) \right\}\;,\end{align*}
\begin{align*}[x^\mu ,x^\nu ]_\star \equiv x^\mu \star x^\nu -x^\nu \star x^\mu = \mathrm{i} \theta^{\mu\nu} .\end{align*}
\begin{align*}K = J - J^{N} = - \frac{k}{2}\int d^{2}\vec{x} \partial^{i}[x_{i}A_{j}A^{j}- A_{i}x_{j}A^{j}]\end{align*}
\begin{align*}[R^{(k-1)}_0,t_+]=[y^{(0)},R^{(k)}]+ \ldots + [y^{(k-p)},R^{(p)}]\end{align*}
\begin{align*}Z_{\nu} = \det (I_{\nu +i -j}(V \Sigma m)) ,\end{align*}
\begin{align*}DX_{a}(t,\tau) = - \theta_{a}(t) + \tau {\dot{x}}_{a}(t) \ ,\end{align*}
\begin{align*}\vec{p}_a + \vec{p}_b - {\tilde L}_p (E_a \vec{p}_b + E_b \vec{p}_a)/c- \vec{p}_c -\vec{p}_d + {\tilde L}_p (E_c \vec{p}_d + E_d \vec{p}_c)/c\simeq 0~.\end{align*}
\begin{align*}f(x)=\begin{cases} a_1x & x < 0\\ a_2 x & x \geqslant 0.\end{cases}\end{align*}
\begin{align*}F(k,\ell) = {\rm tr} \left[\gamma_\mu S(k) \hat{\Gamma}_\mu^{\rm T}(k,\ell) S(\ell)\right]. \end{align*}
\begin{align*}S= S_0+\int\,dt\,[y^*_\mu\{y^\mu,\tilde\chi_\alpha\}c^\alpha+\phi^*_\beta\{\phi^\beta,\tilde\chi_\alpha\}c^\alpha+\lambda^*_\alpha\dot c^\alpha]\end{align*}
\begin{align*}\int_{{\bf R}^2} d^2 x {\cal L}_{Hopf} = \frac{\imath |\beta|^2}{4}\int^{2\pi}_{0} d\theta \; c(\lambda, \beta,\gamma,\dot\gamma,\theta)\; ,\end{align*}
\begin{align*}k_i \to -k_i, U^+ \to - U^+, U^- \to U^-\end{align*}
\begin{align*}\tilde{s}^{2}(x,y)=-ie^2\int\limits_0^1\frac{dx^\nu}{ds}\,(d_\nu(y-x(s)) +d_\nu(x(s) - x))\end{align*}
\begin{align*}x(\eta )\equiv z(\eta )+\frac{\pi \nu }{2}-\frac{\pi }{4}, \quad y(\eta )\equiv z(\eta )-\frac{\pi \nu }{2}-\frac{\pi }{4}.\end{align*}
\begin{align*}\tau=\frac {\theta}{2 \pi} + i \frac { 4 \pi}{g^2} ,\end{align*}
\begin{align*}Q^2_{\rm grav}= R^2 + {1 \over 12} \left( F_{\mu \nu} F^{\mu \nu} \right)_{T^2}+{5 \over 36 N_V} \left( F_{\mu \nu} F^{\mu \nu} \right)_{G_{(N_V)}}~.\end{align*}
\begin{align*}\sqrt{2} M_{j} = \pm \sum_{i} \frac{\nu_{i}\phi(a_{i})}{a_{i}^{2}-m_{j}^{2}}\mbox{det}(a_{i}^{2} - m^{2})^{1/r_{i}} \prod_{j\ne i} (a_{j}^2-a_{i}^2)^{1-2r_{j}/r_{i}} \Lambda_{N=2}^{\frac{4(N_{c}-1)-2N_{f}}{r_{i}}}.\end{align*}
\begin{align*}\widehat L^{\mu\nu}(q) = \left (\eta^{\mu\nu}q^2- q^{\mu}q^{\nu}\right)\hat h(q^2)\end{align*}
\begin{align*}d_i'=d_i\prod_{p\in P}p^{-\nu_p(d_i)}i\in I.\end{align*}
\begin{align*}(Q(\lambda )\psi)({\vec x}) =\int d{\vec t}\int d{\vec y} \prod _{i=1}^N R_{\lambda +\beta_i-1}(t_i,x_i|t_{i-1}y_i)\psi({\vec y}) ~.\end{align*}
\begin{align*}U(N)^{2}\times U(2N)^{n-3}\times U(N)^{2}\end{align*}
\begin{align*} v_{\mu}^{(1)}=-\frac{1}{2\pi\kappa}H_{\mu}^{(2)}+G_{\mu}^{(1)} \;\;\;, \;\;\; v_{\mu}^{(2)}=-\frac{1}{2\pi\kappa}H_{\mu}^{(1)}+G_{\mu}^{(2)} \;\;,\end{align*}
\begin{align*}T^{\alpha \nu}= \frac{\partial u^{\alpha}}{\partial x^{\mu}} T^{\mu\nu}.\end{align*}
\begin{align*}S={A\over 4}=\pi|Z_{fix}|^{\alpha}, \end{align*}
\begin{align*}i \frac{\partial}{\partial t} \Psi(\phi; t ) =H \left [ \frac{1}{i} \frac{\delta}{\delta \phi}, \phi \right ] \Psi(\phi; t) \; .\end{align*}
\begin{align*}N_R = \sum_{n=1}^{\infty} n(a_n^i{}^\dagger a_n^i +s_n^a{}^\dagger s_n^a), ~~~~~ N_L = \sum_{n=1}^{\infty} n(\tilde{a}_n^i{}^\dagger \tilde{a}_n^i + \tilde{s}_n^a{}^\dagger \tilde{s}_n^a),\end{align*}
\begin{align*}{\cal F}(Y)~\approx~ \frac{i}{2\pi} \, Y^2 \log {Y^2} \, +\ldots \qquad\mbox{for }Y\to\infty\end{align*}
\begin{align*} \Omega =DA + A \wedge A \doteq DX^M \wedge DX^N \Omega _{MN} .\end{align*}
\begin{align*}{\cal U}'' + \frac{2}{r} {\cal U}' - ({\cal U})^2 ={1\over 4} (\Phi ')^2\end{align*}
\begin{align*} \widetilde H(K)=\widetilde H_0(K)-\widetilde V, \end{align*}
\begin{align*}\phi=-\ln\left(\Lambda r\right) .\end{align*}
\begin{align*}4\frac{\xi_a\bar\xi_b}{\zeta^2}\equiv i\epsilon_{abc}n^c+n_an_b+\eta_{ab}\qquad n_a\equiv\frac{\zeta_a}{\zeta} \end{align*}
\begin{align*}L_0 = \frac{1}{k} \sum_{n=-\infty}^{\infty} :T^a_{-n} T^b_{n} :\delta_{ab},\end{align*}
\begin{align*}H\equiv v=\frac{m_{2}}{\tau}\sqrt{(p_{1})^{2}+(p_{2})^{2}}.\end{align*}
\begin{align*}{\cal C}'=U^T {\cal C}U\,.\end{align*}
\begin{align*}V_{\rm eff}(\sigma)=-N\sigma[{1 \over g_0^2}+ {1 \over 8\pi}{\rm e}^{\gamma\rho}(\ln{\sigma \over \Lambda^2}-1)], \end{align*}
\begin{align*}f_{\bf k}({\bf x},t) = n^{(m-2)/2}\,\left[ 2\omega (2\pi)^m \right]^{-1/2}e^{i(n{\bf k}\cdot{\bf x}- \omega t)}\end{align*}
\begin{align*}q = n_1 \, (1,0,0,0) + n_2 \, (1,0,0,0)^1 \, ,\end{align*}
\begin{align*}\frac{i}{\pi} ( \beta_1 - \beta_2) g_{RL} ( \beta_1) g_{RL} (\beta_2) =f_{RL} ( \beta_1) g_{RL} ( \beta_2) - g_{RL} ( \beta_1) f_{RL} (\beta_2),\end{align*}
\begin{align*}D_{z^a}\phi = \partial_{z^a}\phi+(A_{+z^a}-A_{-z^a})\phi= \partial_{z^a}\phi+\mathcal{A}_{z^a}\phi,\end{align*}
\begin{align*}\widetilde h^{C_\nu}(k) = \widetilde h_0^{C_\nu}(k)-\widetilde v^{C_\nu},\end{align*}
\begin{align*}lA_1+nB_1= l U^n_l Str\left(\ldots \phi^{i_{2k-1}'}\ldots \underbrace{\partial_{a_l}\phi^{j_l}\phi^{i_{2k}'}}\ldots \right)+n U^n_l Str\left(\ldots \phi^{i_{2k-1}'}\ldots \underbrace{\phi^{i_n}\phi^{i_{2k}'}}\ldots \partial_{a_l}\phi^{j_l}\ldots \right)\end{align*}
\begin{align*} \delta\bar\theta^A(\sigma) = -\bar\epsilon^A + \bar\kappa^B(\sigma)({\bf1}\,\delta^{BA} + \Gamma^{BA}(\sigma)) + f^A{}_{BC}\Lambda^B(\sigma)\bar\theta^C(\sigma) \,.\end{align*}
\begin{align*}w_{\{\alpha\}}^{(n)}(t_1) ~=~ \sum_\beta ~ U_{\alpha\beta}(t_1,t_0)w_{\{\beta\}}^{(n)}(t_0) \quad . \end{align*}
\begin{align*}{\mathcal H} \;\equiv\; -\partial_t^2\,+\,2H,\end{align*}
\begin{align*}ds_{(4)}^{2}=-B(r)dt^{2}+A(r)dr^{2}+r^{2}(d\theta^{2}+\sin ^{2} \theta d \phi^{2}). \end{align*}
\begin{align*}\delta\psi_{\mu } = \hat\nabla_\mu\epsilon= \nabla (\omega) \epsilon +{1 \over4\sqrt 3 }\Bigl(\Gamma^{\rho \sigma} \Gamma_\mu +2 \Gamma^\rho \delta_\mu{}^ \sigma\Bigr)\tilde F_{\rho \sigma }\epsilon.\end{align*}
\begin{align*}\delta L=\left( 1-n\right) \,\,\partial _\tau \left( \bar{\varepsilon}_A\Gamma \left( p\right) \theta _A\right) .\end{align*}
\begin{align*} \ddot x=-2xy^2+{(p_\alpha^2+p_\beta^2)x(x^4+10x^2y^2+5y^4)-2p_\alpha p_\beta y(y^4+10x^2 y^2+5x^4)\over\left(x^2-y^2\right)^4}\end{align*}
\begin{align*}\int \sqrt{g}\ \widehat{R}=\int \sqrt{g}\ \left( R-\frac{d}{d-2}\Lambda\right) , \end{align*}
\begin{align*}J_7\ =\ t^2 \partial_t \ + \ t u \partial_u\ - \ n t \ ,\ J_8\ =\ t u\partial_t \ + \ u^2 \partial_u\ - \ n u\ ,\end{align*}
\begin{align*} G(z)=\sum\limits_{n=0}^{\infty} \sum\limits_{(i_1j_1),\ldots,(i_nj_n)} G_0(z)V_{i_1j_1}G_0(z)V_{i_2j_2}\ldots G_0(z)V_{i_nj_n}G_0(z),\end{align*}
\begin{align*}L = \sqrt{-g} [k {\bf R}^2 + (1/G) R + \Lambda],\end{align*}
\begin{align*}\frac{d}{dt} \delta \Delta = - \delta \Delta ^\lambda \Gamma ^\mu _{\lambda \nu } \dot{q}^\nu + 2 \dot{q}^\nu S _{\nu \lambda }{}^\mu \delta _h q^\lambda .\end{align*}
\begin{align*}X^{\mu}(\tau, \sigma) = \sum_{n=-\infty}^{\infty} X_{n}(\tau) e^{in\sigma}\end{align*}
\begin{align*}t^2 \parallel \!\! H \!\! \parallel_{\infty} \; < \; 1 \; .\end{align*}
\begin{align*}({n\omega \over 2}\Bigr ) \sinh \Bigl ({m\omega \over 2}\Bigr)\end{align*}
\begin{align*}\int_{M^{d-2}} F \wedge \Psi = \Psi_{f} V_{d-4} \int_{M^2} F,\end{align*}
\begin{align*}\partial_\mu A^\mu = 0\quad {\rm or} \quad \partial_\mu \tilde{A}^\mu = 0.\end{align*}
\begin{align*}E^a_b \left(\partial \bar A^{b} - \bar \partial A^{b}+f_{cd}^{b}A^{c}\bar A^{d}\right)=0\,\,\,.\end{align*}
\begin{align*}V'(0) = 0, \; \; \; V''(0) = \rho, \; \; \; U(0) = 0, \; \; \; U''(0) = 0.\end{align*}
\begin{align*}ig \sin(\xi p \tilde{q}) \theta_{\mu \nu} \end{align*}
\begin{align*}\Sigma\le & \inf\limits_{\Psi}({\bf h}^{D}(k_0)\Psi,\Psi) \le \inf\limits_{\phi} (h^{D_1}(k_0)\phi,\phi) +\inf\limits_{\psi} (h^{D_2}(K-k_0)\psi,\psi)\\ = & z_1(k_0) + z_2(K-k_0). \end{align*}
\begin{align*}z_1 = y_4z_3-y_1z_4,\qquad z_2=-y_2z_3-y_3z_4\end{align*}
\begin{align*}S_{matter}=16\pi\int d^4x\sqrt{-g}\;e^{a\psi}L_{matter},\end{align*}
\begin{align*}\Gamma_{hol} \; \to \; \Gamma_{hol} \,+\, i \,( \pi \tau n_1^2 \, + \, 2 \pi n_1 \alpha ) \;.\end{align*}
\begin{align*}a_n=\frac{\alpha_n}{\sqrt{n}},\qquad a^{\dag}_{n}=\frac{\alpha_{-n}}{\sqrt{n}}.\end{align*}
\begin{align*}g(\varphi,t)=e^{\lambda t}\alpha h(\beta \varphi)\end{align*}
\begin{align*}\psi = e^{ie\int_C dx^j A_j} \varphi, \end{align*}
\begin{align*}\check{t_{21}}s_{n}^{\psi}(\rho,\varphi,\phi)=e^{i\psi+\frac{nh}{2\pi}}s_{n}^{\psi}(\rho,\varphi,\phi). \end{align*}
\begin{align*}J^\mu=\frac{1}{g}\epsilon^{\mu\nu\rho}\partial_\nu b_\rho\end{align*}
\begin{align*}\psi^{(0)}(\vec{a},\vec{x}) \equiv\left\{ a\psi^{(0)}(x_1),a\psi^{(0)}(x_2), a\psi^{(0)}(x_3) \right\},\end{align*}
\begin{align*}|1\rangle^H_w \equiv {|1\rangle_a|1\rangle_b + |2\rangle_a|2\rangle_b\over \sqrt{2} }.\end{align*}
\begin{align*}\Sigma+\eta > &\frac{1}{M}\sum\limits_{l=1}^M\Big((h^{D_1}(k_0)\phi_l,\phi_l) \|\psi_l\|_{L^2}^2+(h^{D_2}(K-k_0)\psi_l,\psi_l) \|\phi_l\|_{L^2}^2\Big)\\\ge & (z_1(k_0) +z_2(K-k_0)) \,\, \frac{\sum\limits_{l=1}^M \|\phi_l\|_{L^2}^2 \|\psi_l\|_{L^2}^2 }{M}=z_1(k_0) +z_2(K-k_0).\end{align*}
\begin{align*}\hat{c}^\#_{g_\lambda/h_\eta/h^\prime_{\eta^\prime}} = \hat{c}_{g_\lambda} - (\hat{c}_{h_\eta}-\hat{c}^\#_{h^\prime_{\eta^\prime}})\end{align*}
\begin{align*}ds_{brane}=-(1+2 \Phi_0) dt^2+e^{2\alpha_0(t)}(1-2 \Psi_0) \delta_{ij} dx^i dx^j \ ,\end{align*}
\begin{align*}\Bigl(X_{AB}^{ab}(x,y)\Bigr)=\left(\begin{array}{cc}0&-\,m^2\,\delta^{ab}\\\delta^{ab}&\frac{1}{2}\,g\,f^{abc}\,(D_i\pi^i)^c\end{array}\right)\,\delta(x-y)\,.\end{align*}
\begin{align*}P(x) = <0|J^p(x)(J^p(0))^+|0> \; ; \; P'(x) = <0|J^{p'}(x)(J^{p'}(0))^+|0>\end{align*}
\begin{align*}\delta {\hat \xi}^{{\hat \imath}}=\delta^{{\hat \imath}6}[-\Lambda^{(0)}+\frac{m}{2}(2\pi\alpha^\prime){\rho}^{(0)}-\frac{m}{2}(2\pi\alpha^\prime)\sigma^{(0)}\omega^{(0)}]\, ,\end{align*}
\begin{align*}R=\left(\prod_{n\geq 0}^{\rightarrow} \exp_{q_{\alpha}}\left( (q-q^{-1})e_{\alpha+n\delta} \otimes e_{-\alpha-n\delta}\right)\right)\cdot\end{align*}
\begin{align*}\sinh^2{\Lambda\Delta x\over 2} < a_2 = {\sqrt{6}\over 4} \ ,\end{align*}
\begin{align*}{\beta_h}(g) = - { g^3 \over 16\pi^2} \left( 3 N_c - N_f (1- \gamma) \right) \, ,\qquad \gamma(g) = - {g^2 \over 8 \pi^2} {N_c^2 -1 \over N_c} + O(g^4), \end{align*}
\begin{align*}u\partial_u\left(f^2uj\right)+\partial_\phi\left(f^2g\right)=0.\end{align*}
\begin{align*}E_{i}=\epsilon_{ijk}[D_{j}[A],C_{k}] + [D_{i}[A],\phi].\end{align*}
\begin{align*}(\mu_{k^{N,M}}^{N,M}, z_{k^{N,M}}^{N,M})\in \{(\mu_k^{N,M}, z_k^{N,M}),\ k=1,...,K^{N,M} \}, \ \ N=1,2,..., \ \ M=1,2,...,\end{align*}
\begin{align*}\Psi_l(p_1,\ldots,p_N):=|U_{\rho_l}(k_0)|^{-1/2}\,\chi_{U_{\rho_l}(k_0)}^{}\Big(\sum\limits_{l=1}^n p_l\Big)\,\phi\Big(\sum\limits_{i=1}^n p_i;p_1,\ldots,p_n \Big)\,\psi\Big(\sum\limits_{i=n+1}^N p_i;p_{n+1},\ldots,p_N\Big), \end{align*}
\begin{align*}A_{2m+1,0}^T{1\over\sqrt2}+\sum_{n=1}^\infty A_{2m+1,2n}^T (-1)^n=0.\end{align*}
\begin{align*}F_{k}(x,1-x)=\left\{\begin{array}{l}\left[x(1-x)\right]^{\beta+k}\\\left[x(1-x)\right]^{\beta+k}(2x-1)\ ,\end{array}\right.\end{align*}
\begin{align*}F_r={F\over r}\left({-{2\over 3}\atop r-{1\over 3}}\right)(-1)^{r-{1\over3}}\,,\end{align*}
\begin{align*}j_{top}^\alpha = \frac{R^2}{k \sqrt{\alpha^\prime}} \epsilon^{\alpha \beta} \partial_\beta y .\end{align*}
\begin{align*}\rho_{\mu}^{a}(t)=\int_{0}^{1}sds\epsilon_{\mu\nu}\gamma^{\mu}(t)\sqrt{g(s\gamma(t))}J^{a}(s\gamma(t))\end{align*}
\begin{align*}\epsilon_{i} = \left(e^{\frac{i}{2} a r M \gamma_{r}}\right)_{ij} \left(\delta_{jk} + \frac{i}{2} a x^{\alpha} \gamma_{\alpha} (M_{jk} - i\delta_{jk} \gamma_{r}) \right) \xi_{k}\end{align*}
\begin{align*}W_{ \nu}(\zeta) \sim\frac{2^{i\nu }\Gamma\left(\frac{d-1}{2}-i\nu\right)\Gamma(i\nu)} { 2(2\pi)^{\frac{d+1}{2}}}\ \ \zeta^{-\frac{d-1}{2}+i\nu} \ \ \ \ \ \ \ \ \makebox{forIm} \ \nu < 0\end{align*}