Datasets:
prithivMLmods
/
Latex-KIE

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\begin{align*}\frac{2}{W}({\cal P}{\overline{\cal P}}+\frac{U\overline{U}}{4})\chi^a(z, \overline{z})=E\chi^a(z, \overline{z}).\end{align*}
\begin{align*}\{{ A^{a}_{T,1}}(x) ,{ A^{b}_{T,2}}(y)\}^*= \epsilon_{ab} \frac{\partial_3}{\nabla^2} \delta(x-y)\end{align*}
\begin{align*}x^{*}\simeq-{3G\over8\ln({3G\over8})},\end{align*}
\begin{align*}E=\sum_{\rm all~faces}\epsilon(a_i,a_j,a_k,a_l)~.\end{align*}
\begin{align*}A_{a_{2}...a_{d+1}} \rightarrow \lambda ^{\frac {d-1}{2}}A_{a_{2}...a_{d+1}}\end{align*}
\begin{align*}x^{\mu} (\zeta^{A}) = (\tau + f(\sigma), \rho(\sigma), z(\sigma),\varphi(\sigma)) \, ,\end{align*}
\begin{align*}\omega_{\pm}^{2} = q^{2} + (A\pm B) m^{2}\ .\end{align*}
\begin{align*}A_{00}(x)=2-2\sin^2(\pi\chi)-f(x^0)\ ,\qquad A_{ij}(x)=-f(x^0)\delta_{ij}\ ,\qquad B(x)=-f(x^0)\ ,\end{align*}
\begin{align*}R=-\frac{\kappa^3}4h_{\mu\nu}h_{\nu\rho}\partial^2h_{\rho\mu}+\ldots,\end{align*}
\begin{align*}M^2=8e^{K} |C^3|^2=8e^{K}|T_{-}|^{2}.\end{align*}
\begin{align*}\{ Q, P \}_{qp} = 1 \qquad\qquad \hbox{hence} \qquad\qquad\{ q, p \}_{QP} = 1,\end{align*}
\begin{align*}\alpha \beta=\frac{2qs}{q+r+s} \end{align*}
\begin{align*}\Psi = \left(\begin{array}{c} \phi \\ \phi^{\ast} \\ A_{k} \end{array} \right).\end{align*}
\begin{align*}{\tilde {(\Gamma_a C^{-1})}} = (-1)^{T(T-1)/2}\Gamma_a C^{-1}\end{align*}
\begin{align*}d\Omega_{3,k}^2 = d\chi ^2 +\frac{\sin^2\! \left( \sqrt{k}\chi\right)}{k} \left( d\theta ^2 + \sin^2 \!\theta d\omega^2\right).\end{align*}
\begin{align*}\int \frac{dk_{0} }{2\pi i}\ \frac{1}{k_{0} +\xi + ik_{0} \delta} =\frac{1}{2} {\rm sgn} (\xi ), \end{align*}
\begin{align*}D^I_{mn} (U^{\dagger}) = D^{I *}_{nm} (U) = (-)^{n-m} D^I_{-n-m} (U) .\end{align*}
\begin{align*}\psi'' + [3 {\cal H} - 2 \frac{\varphi''}{\varphi'}] \psi' + [ 4 {\cal H}' - 4 {\cal H} \frac{\varphi''}{\varphi'}] \psi - \partial_{\alpha}\partial^{\alpha} \psi=0.\end{align*}
\begin{align*}\frac{\dot{a}}{a} = c_{1}\tanh{\mu t},~{\rm and}~~~\frac{\dot{c}}{c}= c_{2}\tanh{\mu t},\end{align*}
\begin{align*}{\rho}_{\parallel}(t) \equiv P \rho (t) P\equiv P U(t) P {\rho}_{0} P U^{+}(t) P .\end{align*}
\begin{align*}\langle\psi(y)\bar{\psi}(x)\rangle=(-)\frac{a}{H+am\Gamma_{5}}\gamma_{5}=(-)\frac{a(H+am\Gamma_{5})}{H^{2}+(am\Gamma_{5})^{2}}\gamma_{5}.\end{align*}
\begin{align*}\Gamma^{(2)}(p) = p^2 + M^2 + {c g_3^2 M^2 \over (p\circ p + 1/\Lambda^2)^{d-4}} + \ldots\end{align*}
\begin{align*}W = \sqrt{\frac{M+\mbox{\boldmath $\sigma$} \cdot \mbox{\boldmath $\nabla$}}{(M^{2} - \nabla^{2})^{\frac{1}{2}}}}\end{align*}
\begin{align*}\overline{l} = l - {\lambda \over 4 \pi } H\end{align*}
\begin{align*}\{\phi_a(x),\phi_b(y)\}=M_{ab}(-iD(x-y)),\quad a,b=1,\ldots,4\end{align*}
\begin{align*}\psi (y) = C_1 F({p+1 \over 4}, {p+1 \over 4}; 1 + {p \over 2}; y) + C_2 F({p+1 \over 4}, {p+1 \over 4}; {1 \over 2}; 1-y),\end{align*}
\begin{align*}{\bf \omega}(\alpha) = \cos(\alpha) {\bf 1} + \sin(\alpha){\bf \gamma}_0,\qquad{\bf \gamma}(\alpha) = \sin(\alpha) {\bf \gamma}_1 +\cos(\alpha){\bf \gamma}_2,\end{align*}
\begin{align*}\psi = \left( \begin{array}{cc}I_n & \chi \\begin{align*}.7em]0 & I_n\end{array} \right).\end{align*}
\begin{align*}S\,(g)={\frac{1}{2}}\int d^2 x \sqrt{-\eta_+}\;\eta_+^{\mu\nu}\:tr\left(\partial_\mu g\:\partial_\nu \tilde g\right)+\Gamma_{WZ}(g)\end{align*}
\begin{align*}(\alpha^{2}_{opt})^{ev}=\sqrt{\frac{2\mid\beta\mid^{2}tanh\mid\beta\mid^{2}+2\mid\beta\mid^{2}cos2\theta_{1}+1}{2\mid\beta\mid^{2}tanh\mid\beta\mid^{2}-2\mid\beta\mid^{2}cos2\theta_{1}+1}}(\alpha^{2}_{opt})^{o},\end{align*}
\begin{align*}\left( -\frac{d^{2}}{dx^{2}}+e^{2x}\right) \psi _{E}(x)=E\,\psi_{E}(x)\;.\end{align*}
\begin{align*}\psi^{-}_{++\dot q}={1\over 4i\rho^{++}}W^{\underline{\mu}}v^{-}_{\underline{\mu}\dot q}\end{align*}
\begin{align*}P_{1} = {1\over R} B^{-1} L_{2} B ,\qquad P_{2} = -{1\over R} B^{-1} L_{1} B ,\end{align*}
\begin{align*}r\rightarrow\rho:\quad \rho^2=r^2\left[1-\frac{1}{d-2}\epsilon(H_0(r)+H_1(r))\right].\end{align*}
\begin{align*}G_\Psi^a = \sum_{k=1}^n (\Psi_k )_i^\dagger T_{ij}^a (\Psi_k )_j - \sum_{k=n+1}^N (\Phi_k )_i^\dagger T_{ji}^a (\Phi_k )_j~.\end{align*}
\begin{align*}c_0 = {1\over \sqrt{2}}, ~~~~~~~~ c_n = {m \over \sqrt{m^2 + (\omega_n- n)^2}}.\end{align*}
\begin{align*}\psi \to \psi ' ~=~ e^{i\alpha} ~ \psi \quad . \end{align*}
\begin{align*}\begin{array}{l}AB = B A^{\star} \\ A^{2} + B B^{\star} = I.\end{array}\end{align*}
\begin{align*}{\cal L} = \int d^4 x {\sqrt -g} \left[ -R + 2 \partial_{\mu}\phi \partial ^{\mu}\phi + 2 \partial_{\mu}\varphi \partial^{\mu}\varphi + f( \phi , \varphi ) F^2 - V( \phi , \varphi ) \right] ,\end{align*}
\begin{align*}\sinh (ma^{\pm })=\frac{A\pm \cos (\alpha )}{\sin (\alpha )}\quad ;\quad A=\frac{4m}{M_{0}\beta ^{2}}\quad .\end{align*}
\begin{align*}\widetilde H_{T} = H_T + H_{T}^{(1)} + H_{T}^{(2)},\end{align*}
\begin{align*}\left\{Q(r,t),P(s,t)\right\} = \delta(r-s) \; . \end{align*}
\begin{align*}H = \frac{1}{2M} \sum_{m=1}^{D-1} \sum_{i,j = 1}^{N} (p_{ij}^m)^2+ M'^5 \sum_{m,n=1}^{D-1} \sum_{i,j = 1}^{N} |[X^m,X^n]_{ij}|^2\ .\end{align*}
\begin{align*}X_L = x_L - \frac{1}{2} p_L(\sigma + \tau) +\frac{1}{2}\sum_{k \neq 0}, {\tilde a}_k e^{-ik(\sigma + \tau)}, \end{align*}
\begin{align*}S_1= \phi^*_i\;R^i_{\alpha}\;C^{\alpha} +\sum_{A:k_A=1}S_A^0\xi^A\end{align*}
\begin{align*}S = {\textstyle{g^2\over2}} \int dt \int d^3{x} \int d^3{y}\left\{ \, -\dot{X}^{ax} (\gamma) \frac{1}{4\pi \mid x-y \mid} \dot{X}^{ay} (\gamma) - X^{ax} (\gamma) \, \delta(x-y) \, X^{ay} (\gamma) \, \right\} \end{align*}
\begin{align*}Z(g) = Z_0 + Z_1 \delta + O(\delta^2),\end{align*}
\begin{align*}\psi(x,E,\ell_1,\ell_2,\ell_3)=x^{\ell_1}+O(x^{\ell_1+3}),\end{align*}
\begin{align*}{\cal V}(\rho)={1\over4}\rho^4+{A\over2}\rho^2+B\rho\ ,\end{align*}
\begin{align*}\{A,B\} = \sum_i (\partial_{r_i} A \cdot \partial_{p_i}B -\partial_{p_i} A \cdot \partial_{r_i}B)\end{align*}
\begin{align*}f^2\left(\sum_bn_bX_b\right)f_0(W_{1L},\ldots,W_{nL},g^2)\end{align*}
\begin{align*}0 < \lambda = {e^2 \over 2 \pi c \kappa} < 1\end{align*}
\begin{align*}\vert {\vec K}_i \rangle \,\,\leftrightarrow \,\,\vert \vec k , u_i \rangle\,,\end{align*}
\begin{align*}- ( \omega' - \Omega_0 m)^2 + ( \frac{1}{r^2} m^2 + k^2 + p^2 ) = 0,\end{align*}
\begin{align*}\langle \xi e^{-2\phi}c\partial c\partial^2 c \rangle =1.\end{align*}
\begin{align*} \left(1 - (-1)^I\gamma^{0\parallel 2}\right)\theta^I = 0 ,\end{align*}
\begin{align*}R_l(m^2,r) = R_l(r) (1 + (ma)^2 \chi_l(r) + O(ma)^4).\end{align*}
\begin{align*}p\equiv |\vec{p}|=\sqrt{a^2+b^2}\end{align*}
\begin{align*}{\cal K}_I(A,\bar A)= {1\over 32\pi^2}\left({A\over \Lambda}\right)^{-4}\ln {A\over \Lambda} {\bar A\over \Lambda}+c.c.,\end{align*}
\begin{align*}<G(\tau,x,y)> = \int DA_\mu \; G(\tau,x,y) \; \exp\{-i\int dx \frac{1}{4} F^2 +\frac{\lambda}{2} (\partial A)^2\},\end{align*}
\begin{align*}\{\partial_\mu , \theta^\nu \}= \delta_\mu^\nu\end{align*}
\begin{align*}\varepsilon^{(0)}(R)={\cal A}(1/2)=-\frac{2}{(2\pi)^{D/2}}\left(\frac{m}{R}\right)^{D/2}F\left(\frac{D}{2};mR\right),\end{align*}
\begin{align*}\partial_0 C= [A_0,C]=0 \,.\end{align*}
\begin{align*}Z_{N}={\rm const.}{\rm det}_{jk}e^{\epsilon(-j+k)}I_{-j+k}(N/\lambda)={\rm const.}{\rm det}_{jk}I_{-j+k}(N/\lambda).\end{align*}
\begin{align*}P=a,\qquad\kappa =b\,R -1.\end{align*}
\begin{align*}\Pi\,\Psi\,=\,\psi\,u\,,\qquad\psi\,=\,{1\over3}\,(\,\Psi^1\,+\,\Psi^2\,+\,\Psi^3\,).\end{align*}
\begin{align*}{\partial W_{N+M+1}^{(N)} \over \partial X_i}=(-1)^{i+1}Y^{(N)}_{N+M+1-i}, \hskip10mm i=1,2,\cdots,N.\end{align*}
\begin{align*}W\supset\frac{1}{\tilde{N}_c}\textup{Tr}(q\tilde{q})(\sum_{k=1}^{N_c-\tilde{N}_c}\psi_k)-\sum_{k=1}^{N_c-\tilde{N}_c}\psi_ke_k\tilde{e}_k+\mu\Lambda\sum_{k=1}^{N_c-\tilde{N}_c}x_k\psi_k.\end{align*}
\begin{align*}V(r)=\mu_2/\mu_6=(2\pi\sqrt{\alpha^\prime})^4{\equiv}V_*\ .\end{align*}
\begin{align*}[\rho_n(s_i^2)\psi_n] = tyx [\rho_n(s_i)\psi_n] + (ty)^2[\psi_n] .\end{align*}
\begin{align*}g_6 \rightarrow 1/g_6\ ,\quad\alpha' \rightarrow \alpha' g_6^2\ .\end{align*}
\begin{align*}\Phi=\,{-1\over2i\pi}\,G^0_1G^0_2G^0_3\,\beta_1\beta_2\beta_3\,\left[\,V^0_{12}\,\psi_{12}\,+\,V^0_{23}\,\psi_{23}\,+\,V^0_{31}\,\psi_{31}\,\right]\end{align*}
\begin{align*}-2\Phi \rightarrow \Phi,~~~ T \rightarrow \sqrt 2 T, ~~~-R \rightarrow R.\end{align*}
\begin{align*}{\cal L}=-{1\over 12} [2g^{\mu\nu}h_{ij}{\cal D}_\mu q^i{\cal D}_\nu q^j+F^{ab}_\Lambda F_\Lambda^{ab}+2g^2 {\cal P}^u_\Lambda{\cal P}^u_\Lambda]\, \varepsilon_{cdef}e^c e^d e^e e^f\end{align*}
\begin{align*}\left( g^{*};k_{1}/n_{1},\ldots ,k_{r}/n_{r}\right)\end{align*}
\begin{align*}D_{++}^{(CPV)ab}(x)=D_{++}^{(CPV)}(x)\delta^{ab} = - {i\delta^{ab}\over (2\pi)^2} \int d^2k\,e^{ikx} {\partial\over \partial k_-} CPV\left({1\over k_-}\right)=-{i\delta^{ab}\over 2} |x^-|\delta(x^+)\ , \end{align*}
\begin{align*}S_{\rm Landau}(x) = \frac{\gamma\cdot x}{4\pi\left|x\right|^3} G(\left|x\right|e^2),\end{align*}
\begin{align*}\left( 1+ 2 \frac{\alpha}{M^4} X \right) \sqrt{2 X} =\frac{m}{\sqrt{12 \pi G}}~.\end{align*}
\begin{align*}\beta_R \equiv \left. m_R \frac{\partial e_R^2}{\partial m_R}\right|_{e^2} =\beta_1 e_R^4 + \beta_2 e_R^6 + \cdots\end{align*}
\begin{align*}\langle C^a(x) \bar C^b(y) \rangle = i \int {d^4k \over i(2\pi)^4} {-k^2 \delta^{ab}-v \epsilon^{ab} \over (-k^2)^2+v^2} e^{i k(x-y)} .\end{align*}
\begin{align*}S_{\rm CFT}\to S_{\rm CFT}+\int d^{d}x\; \phi \Phi(\phi) .\end{align*}
\begin{align*}ds^2=\left(1-{2m \over r}\right)d\tau ^2+\left(1-{2m \over r}\right)^{-1}dr^2+r^2d\Omega^2\end{align*}
\begin{align*}J^{0ij} (p) = -i \pi~ {\rm Tr}\, [X^i, X^j] e^{ipX}= \frac{1}{2} (2 \pi)^2 \epsilon^{ij} \delta (p) \;.\end{align*}
\begin{align*}v^4_4=1;~~v^a_\mu=e^{-\sigma}\delta^a_\mu;~~{\it det}~V=e^{-4\sigma}\end{align*}
\begin{align*}S_0^L\left[ \Phi ^i\right] =\int d^DxL\left( \Phi ^i(x),\partial_{\mu _1}\Phi ^i(x),\cdots ,\partial _{\mu _1}\cdots \partial _{\mu _s}\Phi^i(x)\right) ,\end{align*}
\begin{align*}P_\alpha = \left( \begin{array}{c} \alpha_1 \\\alpha_2 \\\alpha_3 \end{array}\right)\begin{array}{ccc} (\bar\alpha_1 & \bar\alpha_2 & \bar\alpha_3) \\ & & \\ & & \end{array} \end{align*}
\begin{align*}h^{1}({\cal S}, {\cal O}_{\cal S}(-i)) = i-1.\end{align*}
\begin{align*} ds_{11}^2=e^{2\alpha\varphi}ds_{10}^2+e^{2\beta\varphi}(dz+ \mathcal{A}_M dx^M)^2,\end{align*}
\begin{align*}N_{12}^{b}(x,a,b,d)|_{m=0}=-\frac{1}{a^{d-3}b}\frac{h(d)}{2^{d}}\Gamma(d-3)\left(\zeta(d-3,\frac{x}{a}+1)+\zeta(d-3,-\frac{x}{a}+1)\right).\end{align*}
\begin{align*}M^2_{L,R}=-\frac{1}{2}P^\mu P_\mu =\frac{1}{2}(P^{25}_{L,R})^2+\frac{2}{\alpha '}(N_{L,R}-1).\end{align*}
\begin{align*}{\cal B_C}(W) = \omega_{\cal C}(L) b^\frac{\nu(L)-|L|}{2} N^{-\nu(L)}\end{align*}
\begin{align*}\xi^{\mu\nu}{\cal G}_{\mu\nu}= n\;\; \mbox{and}\;\; \xi^{\mu\nu}\eta^{\alpha}_{A}{\cal G}_{\mu\alpha} =0\end{align*}
\begin{align*}\langle \xi_N,\eta_N\rangle_N=\sum_{j=1}^{2g}\, (\xi_N,\eta_N\,u_j^{\dagger})\,u_j, \qquad \xi_N,\eta_N\in {\bf C}^N \end{align*}
\begin{align*}n_{\mu} \, X^{\mu} (\tau , \sigma) = 2 \alpha' (n_{\mu} \, p^{\mu})\, \tau \, , \end{align*}
\begin{align*}f(x)\sim\sum _{n=o}^\infty a_n x^n \hspace{1cm} (x\rightarrow 0_+).\end{align*}
\begin{align*}H= - \frac{1}{2} \sum_{j=1}^{L} \{ ( \sigma_{x}^{j}\sigma_{x}^{j+1} + \sigma_{y}^{j} \sigma_{y}^{j+1} )(1-U \tau_{z}^{j+1}) + ( \tau_{x}^{j} \tau_{x}^{j+1} +\tau_{y}^{j} \tau_{y}^{j+1} ) (1- U \sigma_{z}^{j}) \}\end{align*}
\begin{align*}l ~=~ \int_{\cal C} | {\rm Tr}_{U(N)} ~ {{ \Delta e^{\mu}_{ a}}\over{d \lambda}}{{ \Delta e_{\mu b} }\over{d \lambda}} d\xi^{a} d \xi^{b} |^{1/2} d \lambda \quad .\end{align*}
\begin{align*}{\cal A}_{\mu} = {\cal A}_{\mu}^i \lambda^i,\end{align*}
\begin{align*}\Lambda_{ {\bf{k}} }({\bf{q}}) = \sqrt{ {\bar{n}}_{ {\bf{k}} + {\bf{q}}/2 }(1 - {\bar{n}}_{ {\bf{k}} - {\bf{q}}/2 }) }\end{align*}
\begin{align*}(u-u_0)^2 z^{\prime \prime}(u) + \left(2 (u-u_0)^2 \frac{y_1^\prime(u)}{y_1(u)} + (u-u_0) p(u) \right) z^\prime(u) = 0.\end{align*}