Datasets:
prithivMLmods
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Latex-KIE

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\begin{align*}{d \phi \over d \tau} = a \dot\phi = - {m_{\rm Pl}^2 \over 4 \pi} a H'\left(\phi\right),\end{align*}
\begin{align*}\Phi' \equiv \Phi|_{\theta^+ = -\eta \bar{\theta}^-}= \phi + \theta \psi_+ - \eta \bar{\theta} \psi_- -\theta \bar{\theta} (\eta F +i D_1\phi)\end{align*}
\begin{align*}y^2=(x-a_1)\cdots(x-a_{2g+2}).\end{align*}
\begin{align*}I_{pk}\left( M_{p}\right) =\int _{M_{p}}^{\infty }\, d\omega \, \omega \left( \omega +M_{p}\right) ^{-1/2}\left( \omega -M_{p}\right) ^{-1/2}\left( \omega +\mu \right) ^{-k}\left( \omega -\mu \right) ^{k}e^{-2k\omega }.\end{align*}
\begin{align*}\xi=\xi_1+\xi_2,\qquad\psi=2(\mu_1\xi_2-\mu_2\xi_1)/(\mu_1+\mu_2),\end{align*}
\begin{align*}f^a_\mu \equiv \frac{\partial x^a}{\partial q^\mu}\mid_{q^\perp=0},~~ f^\mu_a \equiv g^{\mu\nu} \eta_{ab} f^b_\nu,\end{align*}
\begin{align*}\dot{\phi}^{2}e^{2\alpha\phi/M_{p}}=const\end{align*}
\begin{align*}a^{B,F}_k |0_-\rangle~=~0~=~b^{B,F}_k |0_- \rangle~~.\end{align*}
\begin{align*}\hat\beta_g^{(l)} = 0 \quad \mbox{for } \quad l \geq 2 \ .\end{align*}
\begin{align*}{d g \over d \ln \mu} = g^2 -\nu^2~.\end{align*}
\begin{align*}[\partial_\mu,x^\nu]=\delta_\mu^\nu. \end{align*}
\begin{align*}G(p,p_\perp)=\frac{1}{\mathcal{M}(p^2+p_\perp^2+M^2)},\end{align*}
\begin{align*}\sum_{\mu}{}'A_{\mu} \equiv A_0-\sum_{j=1}^n A_j.\end{align*}
\begin{align*}Z(\alpha,\lambda)=\int[dx\,dp\,d\psi\,d\bar{\psi}]e^{S(\alpha,\lambda)}\end{align*}
\begin{align*}\lambda_2(a)=2 p a~ \frac {I_1(p z_{IR})K_1(p a)-I_1(p a)K_1(pz_{IR})} {I_2(p a) K_1(p z_{IR})+I_1(p z_{IR})K_2(p a)}\end{align*}
\begin{align*}Z\left[ J\right] \equiv \langle \Phi \mid T\left(\exp \left\{i\int d^4xJ\left(x\right) A^0\left(x\right) \right\} \right) \mid\Phi \rangle, \end{align*}
\begin{align*}\Omega\equiv \sum_A r_A\,(d\psi_A+\cos\theta_A\,d\psi_A)=\frac{i}{4}\sum_A \left(\xi_A^*d\xi_A-\xi_Ad\xi_A^*-\zeta_A^*d\zeta_A+\zeta_Ad\zeta_A^*\right), \end{align*}
\begin{align*}B_{\mu\nu}\rightarrow -B_{\mu\nu}\, ,\hspace{1.2cm}B_{\mu}\rightarrow -B_{\mu}\, .\end{align*}
\begin{align*}\eta_i=\left\{\begin{array}{cc}0 \, , &\hspace{1cm} 0\leq \hat{b}_i \leq 1/2, \\1 \, , & \hspace{1cm} 1/2\leq \hat{b}_i < 1 .\end{array} \right.\end{align*}
\begin{align*}\Phi _{2}=x^{0}-\zeta \tau =0~. \end{align*}
\begin{align*}\{ D_\alpha, D_\beta \} = 2i \delta_{\alpha\beta}D_-\;, \;\;\;[D_\alpha, D_-] = 0\;.\end{align*}
\begin{align*}k = m\omega_+ \omega_-, \hskip 1.5cm B = m(\omega_+ - \omega_-)\end{align*}
\begin{align*}\alpha^\pm_i=(i-1)t \pm \theta\,,\qquad \beta^\pm_i=r - (s-i)t \mp \theta\,.\end{align*}
\begin{align*}\delta f = \eta C, ~~~~~\delta C = i \eta^{\dag} f,\end{align*}
\begin{align*}\eta=\frac{\bar{g}^2}{6\pi^2}\, \frac{\omega_{{\rm max}}}{M} \; .\end{align*}
\begin{align*}\bar{\psi}^{c\alpha}_{\phantom{c}a}\ =\ -\ \frac{\delta W}{\delta\zeta^{\phantom{\rho}}_{\alpha a}}\end{align*}
\begin{align*}F(ze^{\pi i})=-e^{\nu \pi i}F(z)+o\left( z^{|{\mathrm{Re}}\nu |-1}\right) .\end{align*}
\begin{align*}\lambda_{\rm bare} =\lambda_{\rm brane} + V_7 \lambda_{\rm bulk}\ .\end{align*}
\begin{align*}T_1^{(2)}=-\,\frac{1}{2}\,\bigl(\eta^1\bigr)^2\end{align*}
\begin{align*}a^{N-6}+\frac{1}{\omega} (A_{N-7}\alpha^3 \beta^3 b)^{5/8}/\alpha^2 \beta^2 b^2=0, \end{align*}
\begin{align*}<H> = \frac{1}{Z}\int DA \Psi^* H \Psi\end{align*}
\begin{align*}\dot{r}^2+V(r)=0;\;\;\;\;\;\;\;\;V(r)=g^{rr}(E^2g^{tt}+g_{\phi\phi}),\end{align*}
\begin{align*}\Phi^a K_i K_a{}^{b\,i} =\Phi_c K^{ac\,i} K_a{}^b{}_i - \nabla_a \left( \nabla_c \Phi^c\gamma^{ab} - \nabla^b \Phi^a \right)\,.\end{align*}
\begin{align*}A_{1}\sim p\sim 1/\tau \, ,\quad A_{2}\sim p^{2}\sim 1/\tau ^{2}\, ,\quad A_{3}\sim 1/\tau ^{3}\: .\end{align*}
\begin{align*}\frac{d^2\tilde F_0}{d\tilde z^2}+\tilde z\frac{d\tilde F_0}{d\tilde z}+(1-\frac{1}{4\tilde z^2}+\frac{\mu^{2}}{\tilde z^{4}})\tilde F_0=0\end{align*}
\begin{align*}(\langle B| + \langle C|) \times (|B\rangle + |C\rangle).\end{align*}
\begin{align*}Gr^{(k)} (H) = \{ W \in Gr (H)|\ \dim(\mbox{Coker}(p_+ |_W)) -\dim (\mbox{Ker} (p_+ |_W)) =k\}.\end{align*}
\begin{align*}\Omega_\xi^{-1}(s,0){\delta\over{\delta\xi^\mu(s)}}\Omega_\xi(s,0)=-ig\,F_\mu[\xi|s].\end{align*}
\begin{align*}V^{(2)}(\xi_1,\xi_2)=\frac{ V(\xi_1) V(\xi_2) -\hat{g} (V_2(\xi_1)+V_2(\xi_2))}{1+c-\hat{g}(\xi_1+\xi_2)},\end{align*}
\begin{align*}(vw)^M=\Lambda_L^{2M}\ , \ v^M = t \ .\end{align*}
\begin{align*}\partial_{X}T^{X}_{X}+\Gamma^{t}_{tX}T^{X}_{X}-\Gamma ^{t}_{tX}T^{t}_{t}=0\end{align*}
\begin{align*}\{\Psi_{K,n}(z,\bar z, Z,\bar Z, t) =e^{-i\left(E_n+K^2/2\mu\right)t} e^{i{\bf K}\cdot{\bf R}}\psi_n(z,\bar z)\}.\end{align*}
\begin{align*}\Pi^{\mu\nu\alpha\beta} {\Pi_{\alpha\beta}}^{\sigma\lambda} = \Pi^{\mu\nu\sigma\lambda}.\end{align*}
\begin{align*}\int_{Jac \Sigma}\iota_{\xi}\omega\wedge\overline{\iota_{\xi}\omega}\wedge t^{r-1}\end{align*}
\begin{align*}U(\Lambda) |\Psi(p)\rangle = |\Psi(p_\Lambda) \rangle \; . \end{align*}
\begin{align*}G^{T}_{\mu \nu }=-\frac{ig_{\mu 0}g_{\nu 0}}{(\theta ^{2}k^{2}_{0}-|\mathbf{k}|^{2}+i\varepsilon )}\end{align*}
\begin{align*}Y(R,R) = 6\pi R^2 +\frac{90 \kappa}{\pi R^2} +{\cal O} \left( e^{-\pi R^2} \right)\end{align*}
\begin{align*}B_{\mu\nu} = -\frac i2 \eta_{ab} [e^a_{\mu}, e^b_{\nu}]\end{align*}
\begin{align*}\phi(x,r)={6\over \sqrt{k^2+6}}\,Q(x)+k\log{r}.\end{align*}
\begin{align*}\overline{G}_{\sigma}^{(n)} (\zeta', x^{2}\zeta_{2n})^{\varepsilon'\,\varepsilon_{2n}}=\sigma\overline{G}_{\sigma}^{(n)} (\zeta_{2n},\zeta')^{\varepsilon_{2n}\,\varepsilon'}\prod_{j=1}^{2n-1} \frac{\zeta_{2n}}{\zeta_j}.\end{align*}
\begin{align*}\Big\langle F_1(Q(x_1))F_2(Q(x_2))\dots F_p(Q(x_p))\Big\rangle \;, \end{align*}
\begin{align*}\tau = \frac{\theta}{2\pi} + i \frac{2\pi}{e^2}\ .\end{align*}
\begin{align*}\begin{array}{lll}d_v & = & 4\sum_{n=1}^{k-1}n + (2N-4k+1) k=2Nk-k(2k+1) , \\ d_H &= & [\frac{1}{2}\times 2 \sum_{n=1}^{k-1} ( (2n) ^2+2n(2n+2) )+\frac{1}{2}( 2(2k) ^2+ (2N-4k-2) 2k(2k+1) ) + 2k] \\ & - & [2\sum_{n=1}^{k-1}(n(2n-1) +n(2n+1) ) + (2N-4k-1) k(2k+1)+2k(2k-1) ]\\ & = & k.\end{array}\end{align*}
\begin{align*}I_1\equiv t={y\over x},\;\;\;I_2\equiv\sigma=2\ln x+W(x,\,y).\end{align*}
\begin{align*}f_{12}^{\left(1\right)} =-\frac{2\pi}{\kappa}\left( J_{0} - \rho \frac{Q}{A}\right),\end{align*}
\begin{align*} \xi(J_3) = g(J_3)-g(J_3 - 1) ~.\end{align*}
\begin{align*}(T_a - T^{\ast}_a)_{bc} = - i F_{abc}\end{align*}
\begin{align*}H(x) =\frac{(2\alpha-\gamma)\alpha^{2}k^{2}((kx+1)^{2\alpha+3\gamma+1}-1)}{12\pi(2\alpha+3\gamma+1)}\end{align*}
\begin{align*}\left\{ \begin{array}{c}(m-1)\partial _A\partial _Bx^j+L^i{}_{jk}(x,y)\partial _Ax^i\partial _Bx^j=0\\ (n-1)\partial _A\partial _By^b+C^a{}_{bc}(x,y)\partial _Ay^a\partial _By^b=0 \end{array}\right\}\end{align*}
\begin{align*}\{ b_k, b_{k'}^{\dagger} \} = \{ d_k, d_{k'}^{\dagger} \} =\delta_{kk'},\end{align*}
\begin{align*}\int d^4 x \;\sqrt{-g}\; h_{uv} \, D_{\mu} q^u D^\mu q^v = \int \left( u \star \bar u + v \star \bar v + e \star \bar e + E \star \bar E \right), \end{align*}
\begin{align*}\Phi^\alpha,~~\epsilon_\alpha = 0,~~|\alpha| = m\end{align*}
\begin{align*}A^a(z)\Phi_i(0)={2\over k+2c_V(H)}{\bar t^a_i\over z}\Phi_i(0) .\end{align*}
\begin{align*}(ab)_{ZZ'} = \sum_{{Z''} \atop {Z \sim Z' \sim Z''}} (a)_{ZZ''}(b)_{Z''Z'}\end{align*}
\begin{align*}( \Gamma_1, \Gamma_2 ) \mapsto( \Gamma_1, \Gamma_2 ) \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right) \ .\end{align*}
\begin{align*}L: (x,y,z)\in { M} \mapsto L(x,y,z)\in { M}\end{align*}
\begin{align*}[I_+,I_-]=2I_3(-1)^{I_3+p},\quad[I_3,I_\pm]=\pm I_\pm\end{align*}
\begin{align*}{\cal L}=\bar{\psi}(i\!\not{\!\partial}+J)\psi +\frac{1}{\sqrt{N}}\bar{\psi}\not{\!\!A}\psi -\frac{1}{\sqrt{N}}\sigma\bar{\psi}\psi -\frac{1}{2G_S}\sigma^2 +\frac{1}{2G_V}A_\mu^2 -\frac{1}{2\xi}(\partial^\mu A_\mu)^2\ ,\end{align*}
\begin{align*}L_{Yuk}(\overline{\Psi^{\prime}}\Psi^{\prime}\varphi \,;\, \phi)=-\frac{h}{\sqrt{2}}\overline{\Psi}^{\prime}\Psi^{\prime} \varphi \frac{e^{\gamma\phi/M_{p}}}{\chi^{3/2}}= -\frac{h}{4}\overline{\Psi^{\prime}}\Psi^{\prime}\varphi\left[\frac{M^{4}+V_{1}(\phi)}{V_{2}(\phi)}\right]^{3/2}e^{\gamma\phi/M_{p}}. \end{align*}
\begin{align*}\partial_+ \tilde {J} {^{R} _3} ={g \over 2 \pi} \partial _- A _3\; .\end{align*}
\begin{align*}G_c = G_R \cdot ( 1 + 2 f') - ( 1 + 2 f') \cdot G_A - 2 G_R \cdot \Sigma_{off} \cdot G_A , \end{align*}
\begin{align*}\bar{x} = \int_{-\infty}^\infty {dt \over \epsilon \sqrt{\pi}} \, e^{-t^2 / \epsilon^2} x(t)\end{align*}
\begin{align*}S\leq \kappa LS_B=\sqrt{1+{{2\Lambda L^2}\over{n(n-1)}}}S_B.\end{align*}
\begin{align*}\lim_{\phi_0\to0} X_3[\phi_0,m(\mu),\lambda(\mu)] = 0.\end{align*}
\begin{align*}\oint_{\alpha_{i}}d\Omega_{0}= \oint_{\alpha_{i}}d\Omega_{k}=\oint_{\alpha_{i}^{*}}d\Omega^{*}_{0}=\oint_{\alpha_{i}^{*}}d\Omega_{k}^{*}=0,\end{align*}
\begin{align*}{C(1,1)\over2^{5/2}}={\rm Cl}_2(4\alpha)-{\rm Cl}_2(2\alpha)\end{align*}
\begin{align*}LG/H_{M}=G\times \Omega G/H_{M}=(G/H_{M})\times \Omega G\, \, ,\end{align*}
\begin{align*}({\bar 8},1;{\bar 8},1),~~(1,8;1,8)\end{align*}
\begin{align*}{\cal D}\!R^{\otimes}+2\epsilon_{AB}\bar\psi_A\wedge\rho_B=0,\end{align*}
\begin{align*}\begin{array}{c}PR'\left(\sqrt{z\over w}\right)\left(q^{S_3}\otimes {E\over z}+{E\over w}\otimes q^{-S_3}\right)=\\=\left(q^{S_3}\otimes {E\over w}+{E\over z}\otimes q^{-S_3}\right)PR'\left(\sqrt{z\over w}\right)\end{array}\end{align*}
\begin{align*}V\sim{{D-2}\over{D-3}}{\widetilde{\Lambda}} \exp(-2A)~.\end{align*}
\begin{align*}\tau^i=-f^2\frac{\epsilon^{ijk}}{\sqrt{h}}\partial_j\omega_k,\end{align*}
\begin{align*}\bar{\delta_f} A_\mu = f^{\alpha} F_{\alpha\mu} \;.\end{align*}
\begin{align*}V(\bar{\phi})=- \left[{\cal L}_{1}(\bar{\phi})+{\cal L}_{ct}(\bar{\phi}) \right]-\frac{T}{\Omega}\log Z'(K).\end{align*}
\begin{align*}g_{\mu\nu} - \eta_{\mu\nu} =O( r^{-b}) \; \; \; \; \; g_{\mu\nu,\lambda} =O( r^{-b -1})\end{align*}
\begin{align*} Z = \sum_{A} \Omega(A) e^{-\beta A} = \sum_{A,N} \Omega(A,N) e^{-\beta A}.\end{align*}
\begin{align*}\sum_i n_i=\, \int_{K3}trR^2\, =\, 24\end{align*}
\begin{align*}\tilde{Q}^{2}_{+}\left| \left. a,1/2\right| n\right\rangle =2\left( \frac{\gamma ^{2}}{2}+M\cos \nu _{n}+M\right) \left| \left. a,1/2\right| n\right\rangle\end{align*}
\begin{align*}H_{k,i}(L):= \int_{S^1}{\rm res\,} \left(L_d(L)\right)_{ii}^{k/{p}},\qquad\forall\,i=1,\ldots, 2s,\quad k=1,2,\dots\,.\end{align*}
\begin{align*}\lbrack{d^2\over dr^2} -{\lambda\over r^2}\rho_3-{\lambda^2\over r^2}+{2E\gamma\over r}+ k^2\rbrack \hat\Phi=0,\end{align*}
\begin{align*}Tr\{\Gamma_{5}f(\frac{(\gamma_{5}D)^{2}}{M^{2}})\}=Tr\{\Gamma_{5}f(\frac{(H/a)^{2}}{M^{2}})\}=n_{+} - n_{-}\end{align*}
\begin{align*}\zeta_j = \prod_{i=1}^N x_i^{\alpha_{ij}},\end{align*}
\begin{align*}[J^a(y),O(z,\bar z)]=[\bar J^{\bar a}(\bar y),O(z,\bar z)]=0.\end{align*}
\begin{align*}\delta \alpha _2=2f_{abc}\left( \partial _\nu B^{*\nu \mu }\right)\eta ^a\eta ^bA_\mu ^c,\end{align*}
\begin{align*}{\cal L}=\frac{1}{2}\left\{ \partial _\mu\varphi_i\,\partial^\mu\varphi_i+i \bar \psi_i \gamma^\mu \partial_\mu \psi_i + f_i f_i +2 f_i\,\frac{\partial{ W}}{\partial \varphi_i} - \frac{\partial^2 {W}}{\partial\varphi_i \partial\varphi_j}\, \bar\psi_i\psi_j \right\}\;.\end{align*}
\begin{align*}I = \int ds\, v^\dagger({\bf r},s)v({\bf r},s) + \sum_P S^\dagger_P({\bf r}) S_P({\bf r}) \,\, .\end{align*}
\begin{align*}E_2 \equiv <\!2|H|2\!> = \int d^3x <\!2|T_{00}|2\!> \end{align*}
\begin{align*}(C^{-1} T^{-1})_{\beta \alpha },\qquad(C^{-1} T^{-1}i \gamma_5 )_{\beta \alpha } , \qquad(C^{-1} T^{-1} \gamma_\lambda i \gamma_5 )_{\beta \alpha },\end{align*} \begin{align*}(C^{-1} T^{-1} \gamma_{\lambda}\tau_n)_{\beta \alpha }, \qquad(C^{-1} T^{-1} {\mbox{\small$\frac12$}} \Sigma_{\lambda \rho}\tau_n )_{\beta \alpha},\end{align*}
\begin{align*}\bar{a}_1 \equiv e^\varphi\tilde{a} \, .\end{align*}
\begin{align*}|\psi\rangle \equiv |\Psi_{e\bar e}\rangle\end{align*}