harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{rll}{P_{j}\,}&{=\,\frac1{\phi_{j}(b(j))}\phi_{j}(X)\,-\,\frac{\phi_{j}^{\prime}(b(j))}{\phi_{j}(b(j))^{2}}\bigl(X-b(j)\bigr)\phi_{j}(X)\qquad}&{j=0,\ldots,m}\\{}&{}&{}\\{N_{j}\,}&{=\,\frac1{\phi_{j}(b(j))}\,\bigl(X-b(j)\bigr)\phi_{j}(X)\qquad}&{j=0,\ldots\,(m-1)}\\{}&{}&{}\\{N_{j}^{+}\,}&{=\,T_{j}\,N_{j}\qquadN_{j}^{-}\,=(1-T_{j})\,N_{j}}\\\end{array}
\int_{{\calM}_{\beta}}R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}=\alpha\int_{{\calM}_{\beta_{H}}}R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}+8\pi(1-\alpha)\int_{\Sigma}R_{\mu\nu\lambda\rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}+O((1-\alpha)^{2})~~~,
ADE:\quad\sum_{y}\mu_{ij}\left(2x-\frac{c_{i}+c_{j}}{2},y\right)q^{y}=-\frac{1}{2}q^{2x-\frac{c_{i}+c_{j}}{2}}\left\langle\lambda_{j},\sigma^{x}\gamma_{i}\right\rangle\;.
d\tilde{g}.{\tilde{g}}^{-1}=d\tilde{\chi}_{1}\tilde{X}^{1}+e^{-\tilde{\chi}_{1}}d\tilde{\chi}_{2}\tilde{X}^{2}+e^{-\tilde{\chi}_{1}}d\tilde{\chi}_{3}\tilde{X}^{3}.
{\frac{\delta(\stackrel{(0)}{S_{0}}+g\stackrel{(1)}{S_{0}}+g^{2}\stackrel{(2)}{S_{0}}+...)}{\delta\varphi}}(\stackrel{\smash{(0)}}{R}_{\alpha}^{i}+g\stackrel{\smash{(1)}}{R}_{\alpha}^{i}+g^{2}\stackrel{\smash{(2)}}{R}_{\alpha}^{i}+...)=0.
b_{\kappa}^{(L)}=\frac{ie^{-\pi\kappa/2}}{\pi\sqrt{2}}\,\left\{\int_{z_{0}}^{\infty}F_{L}(-z,\kappa)\,dz+\frac12\left(\Gamma(i\kappa)\vphantom{\frac12}\left(\frac{mz_{0}}{2}\right)^{-i\kappa}-\Gamma(-i\kappa)\left(\frac{mz_{0}}{2}\right)^{i\kappa}\right)\phi_{M}(0,-z_{0})\right\}.
\mathrm{for~\sqrt{\frac{\phi^2-1}{2}}<\psi~<~\sqrt{\phi^2-1}~}\quad:\qquad\Sigma_{+}=\stackrel{4}{\mathrm{\Large~\#~}}_{_{_{_{\mathrm{\scriptsize~\!\!\!\!\!\!\!\!\!~i{=}1~}}}}}S^{3}\congS^{2}\timesS^{1}\;.
{\calH}_{O(N)}=\intd^{d}x\Bigl[{\frac{1}{2}}\sum_{i}(\partial_{\mu}\Phi_{i})^{2}+{\frac{1}{2}}r\sum_{i}\Phi_{i}^{2}+{\frac{1}{4!}}u\sum_{ij}\Phi_{i}^{2}\Phi_{j}^{2}\Bigr],
{\calL}(x)=\overline{{\Psi}}(x)\left[i\gamma^{\mu}\partial_{\mu}-eA^{\mu}(x)\gamma_{\mu}\right]\Psi(x)-\frac{1}{4}F^{\mu\nu}(x)F_{\mu\nu}(x)-\frac{\lambda}{2}\left[\partial_{\mu}A^{\mu}(x)\right]^{2}\;.
\deltaL_{_{(2)}}=\delta\alpha_{i}\frac{\partial\Omega}{\partial\alpha_{i}}\dot{\lambda}+\delta\alpha_{i}\frac{\partialH}{\partial\alpha_{i}}=\epsilon_{s}v^{s}(\Omega_{_{(1)}}+R_{_{(2)}}\dot{\lambda})=\epsilon_{s}v^{s}\Omega_{_{(1)}}\;.
{\frac{X_{1}^{2}}{X_{3}^{2}}}={\frac{1}{s}}\;,\quad{\frac{X_{2}^{2}}{X_{3}^{2}}}={\frac{t}{s}}\;,\quad{\frac{2X_{1}\cdotX_{3}}{X_{3}^{2}}}={\frac{1+s-t}{s}}\;,\quad{\frac{2X_{2}\cdotX_{3}}{X_{3}^{2}}}={\frac{1-s-t}{s}}\;.
\Gamma[A]=-iN\,\mathrm{Tr\,Ln}\,S^{-1}-\Omega\,\frac{B^{2}}{2}-\,\frac{N}{2}\int\!d^{3}x\,A^{\mu}(x)D_{\mu\nu}^{-1}(-i\partial_{x})A^{\nu}(x)+\widetilde{\Gamma}[A],
\Omega_{0}(z)=\Omega_{0}^{A}(z)\,\Theta_{A}=I^{AB}X_{B}(z)\,\Theta_{A},\quad\mathrm{where~}X_{A}=\langle\Theta_{A},\Omega_{0}\rangle=\int_{\widetilde{\Theta}_{A}}\Omega_{0},
\|(t_{0},t_{1},t_{2},t_{3})\|^{2}=c\sum_{0}^{3}\|t_{i}(+\infty)\|^{2}+\int_{0}^{+\infty}\sum_{0}^{3}\left(\|t_{i}(s)\|^{2}-\|t_{i}(+\infty)\|^{2}\right)ds.
ds^{2}~=~\frac{1}{\left(f_{1}f_{2}f_{3}\right)^{2/3}}\left[-dt^{2}~\left(1-\frac{r_{0}^{2}}{r^{2}}\right)\right]~+~\left(f_{1}f_{2}f_{3}\right)^{1/3}~\left[\left(1-\frac{r_{0}^{2}}{r^{2}}\right)^{-1}~dr^{2}+r^{2}d\Omega_{3}^{2}\right]~,
dg^{\mu\nu}=i\hbar(det(-g))^{\frac{1}{2}}(R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R)d\tau-i\hbar\kappa(det(-g))^{\frac{1}{2}}T^{\mu\nu}d\tau+\hbar{\calE}^{\mu\nu;ac}\circdB_{ac}
\frac{1}{A^{a}B^{b}}=\sum_{s>0}\frac{1}{(A+B)^{a+b-s}}\left\{{\binom{a+b-s-1}{a-s}}\frac{1}{A^{s}}+{\binom{a+b-s-1}{b-s}}\frac{1}{B^{s}}\right\}\,
M_{12}=\left(\frac{i\hbar}{2}\right)\sum_{i=1}^{m}\left(\frac{\partial}{\partialq_{1}^{i}}\frac{\partial}{\partialp_{2}^{i}}-\frac{\partial}{\partialp_{1}^{i}}\frac{\partial}{\partialq_{2}^{i}}\right).
\begin{array}{l}{R(\theta-\theta^{\prime})[1\otimesK^{+}(\theta)]R(\theta+\theta^{\prime})[1\otimesK^{+}(\theta^{\prime})]=}\\{{[}1\otimesK^{+}(\theta^{\prime})]R(\theta+\theta^{\prime})[1\otimesK^{+}(\theta)]R(\theta-\theta^{\prime})}\\\end{array}
R=R_{\Sigma}+2R_{\mu\nu}n_{i}^{\mu}n_{i}^{\nu}-R_{\mu\nu\lambda\rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}-(\chi_{i\mu}^{\mu})^{2}+\chi_{i}^{\mu\nu}\chi_{i\mu\nu}
S_{KR}=\int\Bigl(\frac{Q^{2}}{6}e^{\bf\Phi}\epsilon_{abc}{\bfe}^{a}{\bfe}^{b}{\bfe}^{c}-\frac{\alpha^{\prime}}{8}Q\bigl({\bf\Omega}_{L}-{\bf\Omega}_{YM}\bigr)\Bigr)
S_{\bar{\psi}_{r}^{-}\bar{\psi}_{r}^{+}}=S_{\psi_{r}^{+}\psi_{r}^{-}}=S_{\bar{\Phi}_{r}^{-}\bar{\Phi}_{r}^{+}}=S_{\Phi_{r}^{+}\Phi_{r}^{-}}=\frac{M_{r}C_{D}C\Gamma_{11}}{P^{2}+M_{r}^{2}},
h_{nMN}=C_{\xi}^{*(-n)}\frac{g_{0MN}}{||g_{0MN}||}=\frac{1}{([-n]_{q^{2}}!(1-q^{2})^{-n})^{\frac{1}{2}}}C_{\xi}^{*(-n)}\frac{g_{nMN}}{||g_{0MN}||}
\frac{d\rho_{0}}{d\Lambda}=\frac{\Lambda\,T^{2}}{2\pi}\,g\left(\frac{\Lambda}{T}\right)\left[{\frac{6\,\lambda_{{4}}+12\,\rho_{{0}}\lambda_{{6}}}{{\Lambda}^{2}+4\,\rho_{{0}}\lambda_{{4}}}}\right]\frac{1}{2\lambda_{4}}\;,
\begin{array}{rcl}{\partial_{n}}&{=}&{S^{-1}(f_{(1)}^{l})<e_{n},f_{(2)}^{l}>e_{l}}\\{}&{=}&{S^{-1}(f_{(1)}^{\alpha})<e_{n},f_{(2)}^{\alpha}>e_{\alpha}}\\{}&{=}&{S^{-1}(f^{\alpha})<e_{n},f^{\beta}>e_{\alpha}e_{\beta}}\\{}&{=}&{S^{-1}(f^{\alpha})e_{\alpha}e_{n}}\\{}&{=}&{Ee_{n},}\\\end{array}
{\bf{j}}_{\bf{c}}(t,{\bf{r}})=\pm{\sqrt{2\lambda}}\frac{2{\bf{\hat{c}}}({\bf{\hat{c}\cdotr}})-{\bf{r}}}{\lbrackr^{2}\mp2t\sqrt{2({\bf{c\cdotr}})^{2}-c^{2}r^{2}-c^{4}t^{2}}\,\rbrack^{1/2}}\.
\tilde{\alpha}_{-m}^{(A)}\rightarrow\frac{1}{\sqrt{m}}\tilde{\alpha}_{-m}^{(A)}\,\,,\,\,\tilde{A}_{-r}\rightarrow\frac{1}{\sqrt{r}\gamma}\tilde{A}_{-r}\,\,,\,\,\tilde{B}_{-s}\rightarrow\frac{1}{\sqrt{s}\gamma}\tilde{B}_{-s}\,.
e^{iF(q,Q;t)}=\frac1{2\pi}\intdP\;e^{-iQP}\;e^{iF_{2}(q,P;t)}=e^{i\left(q-q_{c}(t)\right)p_{c}(t)+iS_{c}(t)}\;\delta\left(Q-q+q_{c}(t)\right).
\Psi_{\mathrm{Pf}}(z_{1},\cdots,z_{N})=\mathrm{Pfaff}\left({\frac{1}{z_{i}-z_{j}}}\right)\prod_{i<j}(z_{i}-z_{j})^{q}\,\mathrm{e}^{-{\frac{1}{4}}\sum|z_{i}|^{2}}
\begin{array}{rl}{\bar{\cal{H}}\,m_{(2,1;4,2)}=}&{\varepsilon_{(2,1;4,2)}\,m_{(2,1;4,2)}+2\beta\,m_{(3,1;3,2)}-4\beta\,m_{(3,2;2,2)}+4\beta\,m_{(2,1;3,3)}}\\{}&{+8\beta\,m_{(2,1;3,2,1)}+24\beta\,m_{(2,1;2,2,2)}+4\beta\,m_{(2,1;4,1,1)}\,.}\\\end{array}
\mathbf{B}(\mathbf{x})=\frac{1}{4\pi}\int\rho_{m}(\mathbf{x}^{\prime})\frac{\mathbf{r}}{r^{3}}\,d^{3}x^{\prime}\,-\frac{1}{4\pi}\int\mathbf{J}_{e}(\mathbf{x}^{\prime})\,{\bf\times}\,\frac{\mathbf{r}}{r^{3}}\,d^{3}x.
\begin{array}{rl}{}&{<0|\cdots\pi_{1}^{+}\pi_{2}^{+}\pi_{5}^{-}\stare^{\pmip_{+}\cdotx_{5}}\star\pi_{5}^{-}\cdots|0>}\\{=}&{<0|\cdots(D_{15}\pi_{2}^{+}+D_{25}\pi_{1}^{+})\stare^{\pmip_{+}\cdotx_{5}}\star\pi_{5}^{-}\cdots|0>}\\{=}&{(D_{15}\stare^{\pmip_{+}\cdotx_{5}}\starD_{25}+(1\leftrightarrow2))<0|\cdots|0>,}\\\end{array}
\begin{array}{rcl}{}&{}&{Z_{m}^{n}Z_{m}^{n}\proptoZ_{m}^{n}~,}\\{}&{}&{S_{m}wS_{m}\proptoS_{m}~,~~~\forallw\inA_{2m}~,}\\{}&{}&{Y_{m}wY_{m}\proptoY_{m}~,~~~\forallw\inA_{n}~,}\\{}&{}&{Y_{m}Y_{m}\not=0~.}\\\end{array}
\theta(g)=\left[\delta_{j}^{i}+\frac{1}{2!}c_{jk}^{i}g^{k}+\frac{1}{3!}c_{jk_{1}}^{h_{1}}c_{h_{1}k_{2}}^{i}g^{k_{1}}g^{k_{2}}+\frac{1}{4!}c_{jk_{1}}^{h_{1}}c_{h_{1}k_{2}}^{h_{2}}c_{h_{2}k_{3}}^{i}g^{k_{1}}g^{k_{2}}g^{k_{3}}+\ldots\right]dg^{j}X_{i}\quad,
\frac{1}{(q)_{\infty}}\prod_{n=1\atopn\equiv0,\pmi(\mathrm{mod}2k)}(1-q^{n})=\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k-1}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k-1}^{2}+n_{i}+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-2}-n_{k-1}}(q^{2};q^{2})_{n_{k-1}}},
\langle\Omega_{i_{1}i_{2}...i_{2r}}(x)\Omega_{j_{1}j_{2}...j_{2r}}(y)\rangle=P_{Y_{(2r)}}^{i_{1}i_{2}...i_{2r},j_{1}j_{2}...j_{2r}}r!(1+\frac{r(r-1)}{N^{2}}+\mathcal{O}(\frac{1}{N^{4}}))((x-y)^{2})^{-2r}
[\tilde{S}_{+},\tilde{S}_{-}]_{P.B.}=-2iH(S_{3})\tilde{S}_{3}+iS_{+}\displaystyle\frac{\partialA_{4}}{\partialS_{3}}\bar{C}_{3}C^{+}-iS_{-}\displaystyle\frac{\partialA_{4}}{\partialS_{3}}\bar{C}_{3}C^{-}~,~~~[\tilde{S}_{3},\tilde{S}_{\pm}]_{P.B.}=\mpi\tilde{S}_{\pm}~,
h^{-1}(y)=1+g\chi_{1}(y)+g^{2}(\frac12{\chi_{1}^{2}+\chi_{2}})+g^{3}({\frac16\chi_{1}^{3}+\frac12\chi_{2}\chi_{1}+\frac12\chi_{1}\chi_{2}+\chi_{3}})\,.
q_{ch}(x;\xi)=\frac12\left(\begin{array}{cc}{1+\frac{\xi_{3}}{\vert\xi\vert}}&{\frac{\xi_{1}-i\xi_{2}}{\vert\xi\vert}}\\{}&{}\\{\frac{\xi_{1}+i\xi_{2}}{\vert\xi\vert}}&{1-\frac{\xi_{3}}{\vert\xi\vert}}\\\end{array}\right).
10\tilde{\Gamma}\left(\{\rho_{i}\}\right)=\intd^{4}x\left[-\tilde{V}\left(\{\rho_{i}\}\right)+\frac{1}{2}\tilde{Z}_{ij}\left(\{\rho_{i}\}\right)\partial_{\mu}\rho_{i}\partial^{\mu}\rho_{j}+...\right]
{\mathrmRe}\partial_{p^{2}}\Gamma_{\bar{c}_{+}c_{-}}(p^{2})\big|_{p^{2}=\kappa_{W}^{2}}=1\quad{\mathrmRe}\partial_{p^{2}}\Gamma_{\bar{c}_{Z}c_{Z}}(p^{2})\big|_{p^{2}=\kappa_{Z}^{2}}=1\quad{\mathrmRe}\partial_{p^{2}}\Gamma_{\bar{c}_{A}c_{A}}(p^{2})\big|_{p^{2}=\kappa_{A}^{2}}=1
exp\bigg(ip\cdotX\bigg)(\tau)exp\bigg(iq\cdotX\bigg)(\tau^{\prime})\sim(\tau-{{\tau}^{\prime}})^{2\alpha^{\prime}G^{ij}p_{i}q_{j}}\cdotexp\bigg(-{\frac{1}{2}}\Theta^{ij}p_{i}q_{j}\bigg)exp\bigg(i(p+q)\cdotX\bigg)(\tau^{\prime})+\dots
L_{1}=\dot{r}p_{r}+\dot{\theta}p_{\theta}+\dot{z}p_{z}+\dot{\xi}p_{\xi}-\frac{1}{2}p_{r}^{2}-\frac{1}{2}\frac{1}{r^{2}}p_{\theta}^{2}-\frac{1}{2}p_{z}^{2}-\frac{1}{2}\omega^{2}r^{2}-\xi\left[p_{z}+gp_{\theta}\right]-\lambda^{1}p_{\xi}-\lambda^{2}\left[p_{z}+gp_{\theta}\right]\\\,
I_{E}[g,\Phi]=-\frac{1}{16\pi}[\int_{M}d^{4}x\sqrt{g}\{\Phi(R-2\Lambda)-\omegag^{\mu\nu}\frac{(\partial_{\mu}\Phi)(\partial_{\nu}\Phi)}{\Phi}\}+2\int_{\partialM}d^{3}x\sqrt{h}\Phi(K-K_{0})]
A^{0}(x)=\int_{-\infty}^{\infty}d\tau\a^{0}(x,\tau)=\-\frac{e}{4\piR}\\frac12\int_{-\infty}^{\infty}d\tau\\Bigl(\varphi(t-\tau-R/c)+\varphi(t-\tau+R/c)\Bigr)=-\frac{e}{4\piR}
[\psi({\bfr}),\psi^{\dagger}({\bf{r^{\prime}}})]=\delta({\bfr-r^{\prime}}),~[\psi({\bfr}),\psi({\bf{r^{\prime}}})]=[\psi^{\dagger}({\bfr}),\psi^{\dagger}({\bf{r^{\prime}}})]=0.
W_{p}[\Sigma]={\frac{1}{2^{2p}p!}}\int_{\Sigma}\delta_{[j_{1}...j_{2p}]}^{[i_{1}...i_{2p}]}R_{\\\j_{1}j_{2}}^{i_{1}i_{2}}...R_{\\\j_{2p-1}j_{2p}}^{i_{2p-1}i_{2p}}~~~
L^{(2)}=-(\delta_{abc}^{\mu\nu\gamma}\,\partial_{\mu}\,\Gamma_{\nu}^{ab}\,\tau_{\gamma}^{c}+\delta_{ab}^{\mu\nu}\,\Gamma_{c\mu}^{a}\,\Gamma_{\nu}^{cb}+T_{a}^{\mu}\,\tau_{\mu}^{a}+\Sigma_{ab}^{\mu}\,\Gamma_{\mu}^{ab}),
\Phi(\omega;\lambda)\simeq{\frac{2\omega^{D-2}}{(4\pi)^{(D-1)/2}}}\sum_{n=0}^{\infty}\left[{\frac{a_{n}(\lambda)}{\Gamma\left({\frac{D-1}{2}}-n\right)}}\omega^{-2n}+{\frac{b_{n}(\lambda)}{\Gamma\left({\frac{D-2}{2}}-n\right)}}\omega^{-(2n+1)}\right]~~~.
V^{(0)}(\Phi)=\sum_{\alpha}(G_{\alpha}^{-1}+\deltam^{2})\Phi_{\alpha}^{2}+\frac{g+\deltag}{4!}(\sum_{\alpha}\Phi_{\alpha}^{4}+3\sum_{\alpha,\beta}\Phi_{\alpha}^{2}\Phi_{\beta}^{2})
<q|\hat{q}^{\alpha}|\psi>=q^{\alpha}\psi(q)\\,\\<q|\hat{p}_{\alpha}|\psi>=\frac{-i\hbar}{g^{1/4}(q)}\left[\frac{\partial}{\partialq^{\alpha}}+\frac{i}{\hbar}A_{\alpha}(q)\right]g^{1/4}(q)\psi(q)\\.
{\calR}_{a0c}^{b}=\partial_{0}\Omega_{ac}^{b}-\partial_{c}\Omega_{a0}^{b}+\Omega_{00}^{b}\Omega_{ac}^{0}+\Omega_{d0}^{b}\Omega_{ac}^{d}-\Omega_{0c}^{b}\Omega_{a0}^{0}-\Omega_{dc}^{b}\Omega_{a0}^{d}
\begin{array}{c}{n=2j=j_{1}+j_{2}-J}\\{n_{1}(b)=n_{2}(c)=j+m}\\{n_{2}(b)=n_{1}(c)=j-m}\\{n_{1}(a)=j_{1}-j+m_{1}-m}\\{n_{2}(a)=j_{1}-j-m_{1}+m}\\{n_{1}(d)=j_{2}-j+m_{2}+m}\\{n_{2}(d)=j_{2}-j-m_{2}-m}\\\end{array}
K_{\nu}(z)=\frac{1}{2}\biggl(\Gamma(\nu)\Bigl(\frac{z}{2}\Bigr)^{-\nu}\bigl(1+{\calO}(z^{2})\bigr)+\Gamma(-\nu)\Bigl(\frac{z}{2}\Bigr)^{\nu}\bigl(1+{\calO}(z^{2})\bigr)\biggr)
{\left(\Gamma^{0}\right)}^{-1}=\Gamma^{0},\quad\left(\Gamma^{i}\right)^{-1}=-\Gamma^{i},\quad\left(\Gamma_{11}\right)^{-1}=\Gamma_{11},\quad{\left(\Gamma^{i}\right)}^{T}=(-1)^{i}\Gamma^{i}
b_{n}=-\frac{\left(2E-2g+1+2\left(n+1\right)\right)\left(E^{2}-E+2g-g^{2}+\frac{1}{4}-\left(n+1\right)^{2}\right)}{\left(2E-2g+1+2n\right)\left(2n+3\right)\left(n+1\right)}.
\delta_{\left[abd\right]}^{\left[ijk\right]}\equiv\delta_{a}^{i}\delta_{b}^{j}\delta_{d}^{k}-\delta_{a}^{i}\delta_{d}^{j}\delta_{b}^{k}+\delta_{b}^{i}\delta_{d}^{j}\delta_{a}^{k}-\delta_{b}^{i}\delta_{a}^{j}\delta_{d}^{k}+\delta_{d}^{i}\delta_{a}^{j}\delta_{b}^{k}-\delta_{d}^{i}\delta_{b}^{j}\delta_{a}^{k}\,.
\widetilde{F}^{(1)}(x)=-\int~du\,dv\,dz~\Phi^{j}(u){{\omega}^{-1}}_{jk}(u,v){X^{-1}}^{kl}(v,z)~\{\top_{l}(z),F(x)\}_{(A,\pi,\varphi,\pi_{\varphi})}.
\int_{0}^{\infty}dkk^{{\mu}-1}\mathrm{sin}(ak+\frac{b}{k})={\pi}\left(\frac{b}{a}\right)^{\frac{\mu}{2}}\{\mathrm{cos}\frac{{\mu}{\pi}}{2}J_{\mu}(2\sqrt{ab})-\mathrm{sin}\frac{{\mu}{\pi}}{2}N_{\mu}(2\sqrt{ab})\}
{\calL}_{\mathrm{kin}}^{(0)}=e^{-\phi}k^{-{\frac{1}{2}}}\sqrt{\dot{X}^{\mu}\dot{X}^{\nu}\bigl(g_{\mu\nu}-k^{2}A_{\mu}^{(2)}A_{\nu}^{(2)}\bigr)-k^{2}{\dot{S}}^{2}-2k^{2}\dot{S}{\dot{X}}^{\mu}A_{\mu}^{(2)}}\,.
H=\sum_{i=1}^{N-1}\left(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}+\frac{q+q^{-1}}{2}\left(\sigma_{i}^{z}\sigma_{i+1}^{z}-1\right)\right)+\frac{q-q^{-1}}{2}\left(\sigma_{1}^{z}-\sigma_{N}^{z}\right).
\begin{array}{ccl}{det_{q}(TKT^{\dagger})}&{=}&{P_{-}(T_{1}K_{1}T_{1}^{\dagger})\hat{R}^{(3)}(T_{1}K_{1}T_{1}^{\dagger})P_{-}}\\{\,}&{=}&{P_{-}T_{1}T_{2}K_{1}\hat{R}^{(3)}K_{1}T_{2}^{\dagger}T_{1}^{\dagger}P_{-}}\\{\,}&{=}&{(det_{q}T)\,(det_{q}K)\,(det_{q}T^{\dagger})\;.}\\\end{array}
Z_{k}(\lambda_{1}^{2},\lambda_{2}^{2},q^{2};\lambda^{2})=\int_{0}^{1}dzz^{k}ln\left(\frac{q^{2}z(1-z)+\left(\lambda_{1}^{2}-\lambda_{2}^{2}\right)z-\lambda_{1}^{2}}{-\lambda^{2}}\right),
{\calR}_{4}={\calA}^{2}~=~\left(\begin{matrix}{-3}&{1}&{-2}&{-2}&{0}&{4}\\{-2}&{1}&{-2}&{0}&{4}&{4}\\{0}&{0}&{1}&{0}&{4}&{0}\\{0}&{0}&{1}&{-1}&{2}&{0}\\{0}&{0}&{-1}&{1}&{-1}&{0}\\{-1}&{0}&{1}&{-2}&{2}&{1}\\\end{matrix}\right)
{\calE}\;=\;-{\frac{1}{8\pid}}\;\left[Li_{2}({\calR}^{2})-Li_{2}(-{\calR}^{2})\right]\\;=\;-{\frac{1}{8\pid}}\;[2\;Li_{2}({\calR}^{2})-{\frac{1}{2}}Li_{2}({\calR}^{4})]\.
{\calE}=\frac1{g_{YM}^{2}}\intd^{2}x\\left[\e^{-a|T|^{2}}|D_{\alpha}T|^{2}+\frac14e^{-b|T|^{2}}(1+c|T|^{2})F_{\alpha\beta}F_{\alpha\beta}+V(T)\\right],
v_{m_{1}m_{2}\ldotsm_{n}}^{(n)}=v_{m_{2}\ldotsm_{n}m_{1}}^{(n)},\;\;\;\;\;\;(Tr\;V)^{\dagger}=Tr\;V,\;\;\;\;\;\;v_{m_{1}\ldotsm_{n}}^{(n)\;*}=v_{m_{n}\ldotsm_{1}}^{(n)}
\partial_{\hat{y}}[(\hat{y}^{2}-1)(\hat{y}^{2}-\alpha^{2})\partial_{\hat{y}}\hat{f}(\hat{y})]+2i\omega\zeta\partial_{\hat{y}}\hat{f}(\hat{y})+2(\hat{y}^{2}-\beta_{D}^{2})\hat{f}(\hat{y})=0.
V=-{\frac{1}{16}}|Z_{1}^{0}-Z_{2}^{0}|^{4}(Z_{1}^{0}-Z_{3}^{0})^{2}(\bar{Z}_{1}^{0}-\bar{Z}_{3}^{0})(Z_{2}^{0}-Z_{3}^{0})(\bar{Z}_{2}^{0}-\bar{Z}_{3}^{0})^{2}.
b^{(n)}(x^{2})=\int_{-\infty}^{x^{2}}d\lambda\,\int_{-\infty}^{\lambda}d\lambda^{\prime}\,H^{\prime\prime}(\lambda^{\prime})\,b^{(n-1)}(\lambda^{\prime})\quad\mathrm{for~n~=~1,~2,~3,~\dots~.}
\begin{array}{l}{\qquadx^{\mu}(\tau,\sigma)=X^{\mu}-\frac1{\pi}(P^{\mu}+(P_{y}P)P_{y}^{\mu})\tau+}\\{\qquad\qquad\qquad\fraci{2\sqrt\pi}\sum\frac1n[\bar{\alpha}_{n}^{\mu}e^{i2n(\tau+\sigma)}+\alpha_{-n}^{\mu}e^{-i2n(\tau-\sigma)}],}\\{\qquadp^{\mu}(\tau,\sigma)=\frac1{\pi}P^{\mu}+\frac1{\sqrt\pi}\sum[\bar{\alpha}_{n}^{\mu}e^{i2n(\tau+\sigma)}-\alpha_{-n}^{\mu}e^{-i2n(\tau-\sigma)}],}\\\end{array}
\begin{array}{ll}{\mathrm{Tr}\gamma_{k,6}=\mathrm{Tr}\gamma_{k,6^{\prime}}=0,}&{k\neq4,8}\\{\mathrm{Tr}\gamma_{4,p}=\mathrm{Tr}\gamma_{8,p}=32,}&{p=6,6^{\prime}}\\\end{array}
{\calG}^{AB}\=\\mathrm{diag}\big(\{\calG}^{00}\,,\,{\calG}^{ab}\\big)\=\\mathrm{diag}\big(\\frac{1}{p^{2}}\,,\,\frac{1}{p^{2}-M_{W}^{2}}\,,\,\frac{1}{p^{2}-M_{W}^{2}}\,,\,\frac{1}{p^{2}}\\big)\\.
C_{i+1}\frac{L_{i+2}+1}{[i+2]_{q}}C_{i+1}=\frac{[i+1]_{q}^{2}}{[i]_{q}[i+2]_{q}}C_{i+1}\frac{L_{i+1}^{(2)}+1}{[i+1]_{q}}C_{i+1}-\frac{q^{i}}{[i]_{q}[i+2]_{q}}C_{i+1}.
\left(-\Box+m^{2}\right)\psi(r)=e^{2}\nu^{2\epsilon}(3-2\epsilon)\int\frac{d^{D}p}{(2\pi)^{D}}e^{ipr}\int\frac{d^{D}k}{(2\pi)^{D}}\frac{M(k^{2})}{k^{2}+m^{2}}\frac{1}{(p-k)^{2}}\;\;\;.
{\calL}=\frac12\left({\bfE}^{2}-{\bfB}^{2}\right)\left[1+({\bf\theta}\cdot{\bfB})\right]-\left({\bf\theta}\cdot{\bfE}\right)\left({\bfE}\cdot{\bfB}\right)+{\calO}(\theta^{2})+A_{\mu}J_{\mu},
S_{\mathrm{bdry}}^{2}=\intdx^{0}\left({\frac{i}{2}}(\varsigma\widetilde{D}_{0}\overline{{\varsigma}}-\overline{{\varsigma}}\widetilde{D}_{0}\varsigma)+\beta\overline{{\beta}}+{\frac{i}{2}}(p^{\prime}\widetilde{D}_{0}\overline{{p}}^{\prime}-\overline{{p}}^{\prime}\widetilde{D}_{0}p^{\prime})+\gamma\overline{{\gamma}}\right)
G^{+}(z_{1})\gamma(z_{2})=-\frac{(\gamma\gamma)}{z_{12}^{2}}+\left[\frac{2}{3(1+c_{1})}(G^{-}\gamma\gamma\gamma)-\frac{2}{1+c_{1}}(J_{w}\gamma\gamma)+(\gamma^{\prime}\gamma)\right]\frac{1}{z_{12}}\;.
\Big\langle\Big\langle\left(\intd^{4}xh_{\mu\nu}f_{\mu\nu}^{V}\right)^{2}\Big\rangle_{V}\Big\rangle_{YM}=\intd^{4}x\intd^{4}yb_{\mu}(x)b_{\nu}(y)\langle\langlek_{\mu}(x)k_{\nu}(y)\rangle_{V}\rangle_{YM}\rightarrow\intd^{4}x{\frac{1}{2}}m_{b}^{2}U_{\mu}^{2}(x),
{S_{2}=-T_{2}\intd^{2}\xi({\frac{1}{2}}\sqrt{-\gamma}\gamma^{ij}\partial_{i}X^{M}\partial_{j}X^{N}\Omega_{s}(\phi)g_{MN}+{\frac{1}{2}}\varepsilon^{ij}\partial_{i}X^{M}\partial_{j}X^{N}B_{MN}),}
\begin{array}{l}{\mathrm{H}_{2j}^{\left(k\right)}=\varepsilon^{j}\left(1-\varepsilon\right)^{j}}\\{\mathrm{H}_{2j+1}^{\left(k\right)}=\varepsilon^{j}\left(1-\varepsilon\right)^{j+1}}\\\end{array}\;\forall\;0\leqj\leq\frac{k-1}2
\zeta_{0}(\nu)=-{\frac{\nu\varrho^{-2\nu}}{\pi}}\int_{\mu}^{\infty}d\omega\int_{C_{+}}dz{\frac{2z^{2}}{(z^{2}+\omega^{2})^{\nu+1}}}\breve{\Psi}(\omega;z)e^{i\epsilonz}~~~,
H_{A\dot{A}}^{\\\B\dot{B}}=2\left(H_{A}^{\\B}\varepsilon_{\dot{A}}^{\,\dot{B}}+H_{\dot{A}}^{\\\dot{B}}\varepsilon_{A}^{\\B}\right)+\frac{i}{2}\left(\nabla_{A}H_{\dot{A}}^{\\\B\dot{B}}+\nabla_{\dot{A}}H_{A}^{\\\B\dot{B}}\right)
\nonumber\sigma_{1}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right)\quad\sigma_{2}=\left(\begin{matrix}{0}&{-i}\\{i}&{0}\\\end{matrix}\right)\quad\sigma_{3}=\left(\begin{matrix}{1}&{0}\\{0}&{{-1}}\\\end{matrix}\right)\quad\sigma_{4}=\left(\begin{matrix}{1}&{0}\\{0}&{1}\\\end{matrix}\right)
[L_{\alpha},L_{\beta}]_{{}_{+}}=4i(J\gamma)_{\alpha\beta},\quad[J_{\mu},J_{\nu}]=-i\epsilon_{\mu\nu\lambda}J^{\lambda}\quad[J_{\mu},L_{\alpha}]=\frac{1}{2}(\gamma_{\mu})_{\alpha}{}^{\beta}L_{\beta},
\Lambda\frac{\partial}{\partial\Lambda}V(\phi,\Lambda)=\frac{1}{2}\int\!d^{d}xd^{d}y\int\!\frac{d^{d}k}{(2\pi)^{d}}e^{ik(x-y)}\frac{2}{\Lambda^{2}}K^{\prime}(\frac{k^{2}}{\Lambda^{2}})(\frac{\delta^{2}V}{\delta\phi(x)\delta\phi(y)}-\frac{\deltaV}{\delta\phi(x)}\frac{\deltaV}{\delta\phi(y)}),
\partial_{s_{0}}^{3}{\calF}_{mirror}\sim-\frac{c_{11}}{2a\epsilon}\Big[-\frac{8}{a}+\frac{1}{a+m}+\frac{1}{a-m}-\sum_{n_{1}=1}^{\infty}\gamma_{n_{1}}\frac{\Gamma(4n_{1}+1)}{4^{n_{1}}c_{1}}\Big(\frac{m}{a}\Big)^{n_{1}}\Big(\frac{\Lambda}{a}\Big)^{3n_{1}}\Big].
\sum_{l\geq0}\sum_{\midm\mid\leql}\int_{1}^{\infty}d\lambdau_{\lambda,l,m}(r,\theta,\phi)\overline{{u}}_{\lambda,l,m}(r^{\prime},\theta^{\prime},\phi^{\prime})=\Omega(k)\delta^{3}(k-k^{\prime}).
ds^{2}=\left({\frac{\mu}{D-3}}\right)^{2}{\frac{1}{\rho^{2}}}\left[-dt^{2}+d\rho^{2}\right]+\mu^{2}d\Omega_{D-2}^{2},\\\\\A_{t}=-{\frac{\mu}{D-3}}{\frac{1}{\rho}},
\sigma^{\pm}=b_{0}|y|\left[{\frac{M^{2}}{4\alpha}}\left(1\pm\left(1+{\frac{4\alpha\Lambda_{b}}{3M^{5}}}\right)^{\frac{1}{2}}\right)\right]^{\frac{1}{2}}\equivk_{\pm}b_{0}|y|.
E_{0}=-\frac{1}{6\pi}\frac{1}{L^{3}}\left\{(mL)^{3}+3\pi^{-1/2}(mL)^{3/2}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{3/2}}\,K_{-3/2}(2nmL)\right\}\,.
S=\intd^{2}x\left\{\frac{1}{g^{2}(1+\varphi\bar{\varphi})^{2}}\partial_{a}\bar{\varphi}\partial^{a}\varphi-\frac{4i\theta}{(1+\varphi\bar{\varphi})^{2}}\epsilon^{ab}\partial_{a}\bar{\varphi}\partial_{b}\varphi\right\}
q^{-{\frac{1}{2}}}\chi_{\frac{1}{2}}(q)=1+q+q^{2}+q^{3}+2\,q^{4}+2\,q^{5}+3\,q^{6}+4\,q^{7}+5\,q^{8}+6\,q^{9}+8\,q^{10}+9\,q^{11}+12\,q^{12}+O(q^{13})
S_{jk}=\frac{g^{2}(\mu_{j},\lambda_{k})}{f(\mu_{j},\lambda_{k})}e^{\phi^{\kappa}(\mu_{j})+\phi^{\nu}(\lambda_{k})}+\frac{g^{2}(\lambda_{k},\mu_{j})}{f(\lambda_{k},\mu_{j})}e^{\phi^{\kappa}(\lambda_{k})+\phi^{\nu}(\mu_{j})}
\tilde{A}=\sum_{k}\tilde{V}_{i,i,k,k}\int_{\mu^{-2}}^{\infty}\frac{ds}{s}e^{-m_{k}^{2}s}+\sum_{k}\tilde{V}_{i,k,l}\tilde{V}_{k,l,i}\int_{\mu^{-2}}^{\infty}ds\,e^{-m_{k}^{2}s}.
\Phi^{A}(\theta)=\phi^{A}+\pi^{Aa}\theta_{a}-\lambda^{A}\theta^{2},\qquad\frac{\delta}{\delta\Phi^{A}(\theta)}=\frac{\delta}{\delta\phi^{A}}\theta^{2}-\theta^{a}\frac{\delta}{\delta\pi^{Aa}}-\frac{\delta}{\delta\lambda^{A}}
g(AdS;soliton)={r^{2}}dt\otimesdt+{\frac{L^{2}}{r^{2}}}{(1-{\frac{{r_{0}}^{n}}{r^{n}}})^{-1}}dr\otimesdr+{\frac{r^{2}}{L^{2}}}{(1-{\frac{{r_{0}}^{n}}{r^{n}}})}d\phi\otimesd\phi+{r^{2}}\sum_{i=1}^{n-2}\;d\theta_{i}\otimesd\theta_{i}.
\overbrace{\underbrace{1,\dots,1}_{r-1},0,\underbrace{1,\dots,1}_{p-r-1},0}^{p},\dots,\overbrace{\underbrace{1,\dots,1}_{r-1},0,\underbrace{1,\dots,1}_{p-r-1},0}^{p},\dots\,.
A(1,2,3,4)={\frac{1}{256}}e^{E_{3}^{(4)}-p_{2,\mu}(\theta_{23}\sigma^{\mu}\bar{\theta}_{23})}(\theta_{43}\theta_{23})\bar{\theta}_{43}^{2}(\bar{\theta}_{23}\bar{\theta}_{13})\theta_{13}^{2}.\qquad
\psi_{n_{1}>1}=\frac1{\sqrt{\pik_{s}}}\frac{2^{n_{s}-2}\Gamma\left(n_{s}\right)}{n_{1}(1-n_{1})}a_{s}^{-m}z_{S}^{2(n_{1}-1)}g_{S}^{-l}\left(\frac{k}{k_{s}}\right)^{2-n_{s}}(k_{s}\eta)^{2n_{1}}
\intd^{D}x\sqrt{-G}\,i\bar{\Psi}_{M}\Gamma^{MNP}D_{N}\Psi_{p}\sim\int_{-1/K_{p}}^{1/K_{p}}dy{\calW}^{-{\frac{1}{2}}}\intd^{p+1}x\,i\bar{\psi}_{\alpha}\gamma^{\alpha\beta\delta}\partial_{\beta}\psi_{\delta}.
r^{2}=\frac{(H^{2}k_{2}^{2}-k_{1}^{2})(c_{1}^{2}+2Hc_{1}k_{2})}{r^{2}k_{2}^{2}}\left(r^{2}-\frac{k_{2}^{2}}{c_{1}^{2}+2Hc_{1}k_{2}}\right)\left(r^{2}+\frac{k_{2}^{2}}{H^{2}k_{2}^{2}-k_{1}^{2}}\right),
\begin{array}{l}{W_{3}=\frac{1}{2}\int(\partial_{j}\delta_{ca}+\hat{\Gamma}_{jc}^{a})\hat{A}_{a}(\partial_{j}\delta_{cb}+\hat{\Gamma}_{jc}^{b})\hat{A}_{b}}\\{\equiv\frac{1}{2}\int\hat{A}_{a}[(-\partial_{j}+\hat{\Gamma}_{j})(\partial_{j}+\hat{\Gamma}_{j})]_{ab}\hat{A}_{b}}\\\end{array}
T_{2}^{\mu\nu}=\frac{1}{Eq^{2}}\left(p^{\mu}p^{\nu}-(p^{\mu}q^{\nu}+p^{\nu}q^{\mu})\frac{q\cdotp}{q^{2}}+g^{\mu\nu}\frac{(p\cdotq)^{2}}{q^{2}}\right)
L_{2}=e^{\alpha\phi/M_{p}}\left[-\frac{b_{g}}{\kappa}R(\omega,e)+\frac{b_{k}}{2}g^{\mu\nu}\phi,_{\mu}\phi,_{\nu}-hm\overline{{\Psi}}\Psie^{\frac{1}{2}\alpha\phi/M_{p}}\right]
F_{ij}dx^{i}\mathrm{\tiny~\wedge\,~}dx^{j}=F_{\alpha\beta}dx^{\alpha}\mathrm{\tiny~\wedge\,~}dx^{\beta}+2F_{\alphap}dx^{\alpha}\mathrm{\tiny~\wedge\,~}dx^{p}+F_{pq}dx^{p}\mathrm{\tiny~\wedge\,~}dx^{q}.
W_{\gamma}=W-W_{F}=2\pi\alpha\biggl(\int_{C_{1}}d{x_{\mu}}\int_{C_{2}}d{{x^{\prime}}_{\nu}}+\int_{C_{2}}d{x_{\mu}}\int_{C_{1}}d{{x^{\prime}}_{\nu}}\biggl)D^{\mu\nu}(x,x^{\prime})\,.
\frac{1}{N}V[\sigma]=\frac{\sigma^{2}}{2\lambda_{B}}+\frac{|\sigma|}{(4\pi)^{3/2}a^{2}}\left[\Gamma\left(-\frac{1}{2},\frac{\sigma^{2}}{\Lambda^{2}}\right)+2a^{2}\sigma^{2}\Gamma\left(-\frac{3}{2},\frac{\sigma^{2}}{\Lambda^{2}}\right)\right].
\begin{aligned}{{}\big[M^{(8)},W^{(8)}\big]}&{{}=\alphaM^{(8)}+\tilde{\alpha}W^{(8)}\,,}&{\big[W^{(8)},W^{(8)}\big]}&{{}=\betaM^{(8)}+\tilde{\beta}W^{(8)}}\\{{}\big[M^{(5)},W^{(5)}\big]}&{{}=\gammaM^{(5)}+\tilde{\gamma}W^{(5)}\,,}&{\big[W^{(5)},W^{(5)}\big]}&{{}=\deltaM^{(5)}+\tilde{\delta}W^{(5)}\,.}\\\end{aligned}
\Psi_{k}(p)=\frac{\sqrt3}{\sqrt{2M(E_{k}+jM/k)}}\left(\begin{array}{c}{(E_{k}-jM/k)/\sqrt{3}}\\{p^{y}+ip^{x}}\\{p^{y}-ip^{x}}\\{(E_{k}+jM/k)/\sqrt{3}}\\\end{array}\right)\phi(p)\;,
L=\left[-\frac{\Lambda_{B}}{8\piG_{B}}-\frac{R}{16\piG_{B}}+c_{B}^{1}R^{2}+c_{B}^{2}R_{\mu\nu}R^{\mu\nu}+c_{B}^{3}R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}\right]~~~.
h=\frac{8\pi}{15}l_{p}^{\prime6}\sum_{n=1}^{N}\sum_{n_{0}=-\infty}^{\infty}\frac{Q_{n}}{\left(|\vec{r}-\vec{r}_{n}|^{2}+(x_{11}-x_{n11}+n_{0}R_{11}^{\prime})^{2}\right)^{3}},
\begin{array}{c}{P=Q\,(\sigma,\sigma^{\prime},\tau),}\\{\sigma=\Box\,q_{\bot}^{2}=\Box\,q^{2}-(qd)^{2}\;;\;\sigma^{\prime}=\Box\,q_{\bot}^{2}=\Box\,q^{2}-(q^{\prime}d)^{2}}\\{\tau=\Box\,(q_{\bot}q_{\bot}^{\prime})=\Box\,(qq^{\prime})-(qd)(q^{\prime}d).}\\\end{array}
\int_{C}{d^{n}}{\eta}D_{c}^{(n)}(\eta,{\eta})=-\sum_{N=0}^{N_{0}}\sum_{\alpha=0}^{N}\frac{1}{\lambda_{N}}Y_{\alpha}^{N}({\eta})Y_{\alpha}^{N}({\eta})\int_{C}{d^{n}}{\eta}
S_{\mathrm{part}}^{(\mathrm{N})}=S_{\mathrm{part},0}^{(\mathrm{N})}+\intdt\,\sum_{\alpha=1}^{N}\,Q_{\alpha}^{a}\left(A_{j,\alpha}^{a}\dot{x}_{\alpha}^{j}+A_{0,\alpha}^{a}\right)+S_{SU(2)}^{(\mathrm{N})}\,,
(\alpha\,f_{1}+\beta\,f_{2})^{*}={\bar{\alpha}}\,f_{1}^{*}+{\bar{\beta}}\,f_{2}^{*},\quad(f_{1}\,f_{2})^{*}=f_{2}^{*}\,f_{1}^{*},\quad(f^{*})^{*}=f