harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
m_{\lambda, \rho}( w_1,w_2) = \exp \left( \frac{1- \rho}{2}\log(1+ \lambda w_2 P') - \right. \left. \frac{1}{2}\log \left( 1+ \lambda (1- \lambda \rho) w_2 P' + \lambda\rho (1-\lambda \rho) w_1 P' \right) \right),
(dS_{l})_{h}(H_{n}(w)) =\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}S_{l}(H_{n}(y)+\varepsilon\cdot H_{n}(w)) =\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}S_{l}(H_{n}(y+\varepsilon\cdot w)) =\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}f_{n}(y+\varepsilon\cdot w) =(df_{n})_{y}(w) =\xi.
W={W}(x_0,{t}_0,u_0)+ \epsilon^{\alpha+\gamma}{{y}}{\upsilon}+\frac{1}{n!}\epsilon^{n\alpha+\beta}{\tau}_{n-1}{\upsilon}^n + \epsilon^{\alpha(k+2)}A_{k+2}{\upsilon}^{\alpha(k+2)}+\dots \, .
(4d+2)E_3|_{E_3}= -D_{aff}|_{E_3}- 2dE_1|_{E_3}- (2d+2)\sum_{i=1}^k E_i^p|_{E_3}- (2d+1)E_2|_{E_3}&= -C_0 - 2dC_1 - (2d+2)\sum_{i=1}^k f_i - (2d+1)C_2.
W_2^2(\mu,\bar{\mu}) =\big(m(t)-\bar{m}(t)\big)^2+\Big(\sqrt{\sigma(t)}-\sqrt{\bar{\sigma}(t)}\Big)^2 &\leq \big(m_x(t)-\bar{m}(t)\big)^2+|\sigma_{xx}(t)-\bar{\sigma}(t)| & \leq \frac{4}{\gamma^2-4\omega^2}(\omega |x_0|+|v_0|)^2+\frac{16}{\beta(\gamma^2-4\omega^2)}=\frac{4}{\gamma^2-4\omega^2}\Big[(\omega |x_0|+|v_0|)^2+\frac{4}{\beta}\Big],
k(\omega)=k|_{\omega_{0}}+\frac{\partial k}{\partial\omega}|_{\omega_{0}}(\omega-\omega_{0})+\frac{1}{2}\frac{\partial^{2}k}{\partial\omega^{2}}|_{\omega_{0}}(\omega-\omega_{0})^{2}+...+\frac{1}{p!}\frac{\partial^{p}k}{\partial\omega^{p}}|_{\omega_{0}}(\omega-\omega_{0})^{p}+...
\kappa_{\omega}= \frac{2 \Gamma ( \Delta_{\omega})}{\pi \Gamma ( 1 - \Delta_{\omega})}\left( \frac{\sqrt{\pi}\Gamma \left( \frac{1}{2 - 2 \Delta_{\omega}}\right)}{2 \Gamma \left( \frac{\Delta_{\omega}}{2 - 2 \Delta_{\omega}}\right)}\right)^{2 - 2 \Delta_{\omega}}\, .
f_s(0) = 1,f^{(1)}_s(0) = 0,f^{(2)}_s(0) = s,f^{(3)}_s(0)= -2s, &f^{(4)}_s(0) = 3s(2+s),f^{(5)}_s(0) = -4s(6+5s), & f^{(6)}_s(0) = 5s(24+26s+3s^2),f^{(7)}_s(0) = -6s(120+154s+35s^2), &f^{(8)}_s(0) = 7s(720+1044s+340s^2+15s^3),\ldots.
\delta S_{0}\left[ \bar{\Psi}_a , \Psi_a, A^\mu_a \right] = \int dx \left[ - \xi \epsilon_a \left(\partial \cdot A_a \right){\,\lower 1.0pt\vbox{\hrule \hbox{\vrule height 0.3 cm \hskip 0.28 cm \vrule height 0.3 cm}\hrule}\,}+ J_{a}^{\mu}\partial_{\mu}- ie \left(\bar{\eta}_a \psi_a -\bar{\psi}_a \eta_a \right) \right] \Lambda_a \, ,
\begin{cases}\|E_{\eta,\ell}^{ex}\|_{L^{p}(\Omega\backslash\bar{D})}\leq C\{\|\nu\times H_{\eta,\ell}\|_{L^{p}(\partial D)}+\|\nabla\times H_{\eta,\ell}\|_{L^{p}(D)}\}, \|H_{\eta,\ell}^{ex}\|_{L^{2}(\Omega\backslash\bar{D})}\leq C\{\|\nu\times H_{\eta,\ell}\|_{L^{p}(\partial D)}+\|\nabla\times H_{\eta,\ell}\|_{L^{p}(D)}\},\end{cases}
\frac{1}{\sqrt{- g^{\prime}}}\partial_{\mu}( \sqrt{- g^{\prime}}g^{\prime \mu \nu}\partial_{\nu}\phi ) + \frac{1}{\chi}\left[ \frac{d V_{1}}{d \phi}- \frac{1}{\chi}\frac{d V_{2}}{d \phi}+ \frac{\alpha}{M_{p}}P ( \varphi ) e^{\alpha \phi / M_{p}}\right] = - \frac{h \gamma}{\sqrt{2}M_{p}}\overline{{{\Psi}}}^{\prime}\Psi^{\prime}\varphi \frac{e^{\gamma \phi / M_{p}}}{\chi^{3 / 2}};
\begin{array}{l}{g_{\tau \tau}= \partial_{\tau}X^{M}\partial_{\tau}X^{N}G_{M N}= G_{0 0}= - e^{2 V}}\\{g_{\sigma \sigma}= \partial_{\sigma}\rho \partial_{\sigma}\rho G_{\rho \rho}+ G_{\theta \theta}= ( \rho^{\prime})^{2}e^{- 2 V}\Delta^{- 1}+ \rho^{2}\Delta}\\ \end{array}
T^{4-brane,l}_{MN}=\Lambda_l A(z_1,z_2)\delta(\vec{n}\cdot\vec{z})\left(\begin{array}{rrrrrr}1 -1 -1 -1 -{\rm cos}^2\varphi_l -{\rm sin}\varphi_l{\rm cos}\varphi_l -{\rm sin}\varphi_l{\rm cos}\varphi_l -{\rm sin}^2\varphi_l\end{array}\right),
\theta_{3}[exp(-\sqrt{3}\pi)]=\theta_{3}\{q[\frac{1}{4}(\sqrt{6}-\sqrt{2})]\}=\sqrt{2\pi^{-1}K[\frac{1}{4}(\sqrt{6}-\sqrt{2})]}=2^{-1/6}3^{-1/8}\pi^{-1/2}\beta(\frac{1}{3})^{1/2}
h_{\alpha}(x,t)=\frac{1}{2^{1-\alpha/2}t^{\frac{1-a_1-a_2}{2}\alpha+a_2}}W_{-\alpha/2, 1-\alpha/2}\left(-\frac{2^{\alpha/2}|x|}{t^{\frac{1-a_1-a_2}{2}\alpha}}\right)
\begin{array}{l}{{( H^{n})_{W}= H^{n}+ \frac{1}{2 4}( 2 n - 1 ) n ( n - 1 ) H^{n - 2}}}\\{{+ \frac{1}{5 7 6 0}( 2 0 n^{2}- 4 8 n + 7 ) \prod_{k = 0}^{3}( n - k ) H^{n - 4}}}\\{{+ \frac{1}{2 9 0 3 0 4 0}( 2 8 0 n^{3}- 1 5 9 6 n^{2}+ 1 8 7 4 n - 9 3 ) \prod_{k = 0}^{5}( n - k ) H^{n - 6}+ . . .}}\\ \end{array}
x=[\begin{matrix}(A^{T}P_{B}^{\perp})^{+}&(B^{T}P_{A}^{\perp})^{+}\end{matrix}][\begin{matrix}e\\f\end{matrix}]=(A^{T}P_{B}^{\perp})^{+}e+(B^{T}P_{A}^{\perp})^{+}f.
e^{{\cal{G}}_{\mathrm{a c}}\{{\phi}\}}= \frac{\int{\cal{D}}\{\phi^{\prime}\}e^{- S_{0}\{\phi^{\prime}\}- S_{\mathrm{i n t}}\{\phi^{\prime}+{\phi}\}}}{\int{\cal{D}}\{\phi^{\prime}\}e^{- S_{0}\{\phi^{\prime}\}}}.
\begin{aligned}\mathbf{A}\mathbf{d}_{hx}=0, &{{\mathbf{A}}^{T}}\mathbf{d}_{hy}+\mathbf{C}d_{hx}+\mathbf{d}_{hs}=0, &\bar{\mathbf{X}}\mathbf{D}\mathbf{d}_{hs}+\bar{\mathbf{S}}\mathbf{D}^{-T}\mathbf{d}_{hx}=0,\end{aligned}
E_{q - \mathrm{v i b - r o t}}( v , J ) ={\frac{1}{2}}\left( [ v + 1 + b \gamma ( J ) ]_{q ( J )}+ [ v + b \gamma ( J ) ]_{q ( J )}\right) h c \nu_{\mathrm{v i b}}+ \frac{h^{2}}{8 \pi^{2}I}J ( J + 1 ) ~ .
n\epsilon_n^2 = n\cdot n^{-\frac{2(\alpha \wedge \beta)+2p}{1+2\alpha+2p}}= n^{\frac{1+2\alpha - 2(\alpha \wedge \beta)}{1+2\alpha+2p}}= \epsilon_n^{-\frac{1+2\alpha-2(\alpha\wedge\beta)}{(\alpha\wedge\beta)+p}},
\underline{t}(w)=t_0+\frac{\lambda}{\kappa(c_{+0}-c_{-0})}\frac{w^3}{6}-\left(\frac{\lambda(4\kappa-3l)}{\kappa(c_{+0}-c_{-0})^2}+\frac{\xi}{\kappa(c_{+0}-c_{-0})}\right)\frac{w^4}{24}+\mathcal{O}\left(w^5\right).
S_{II}\sim \frac{\sigma_0}{\pi}\left( \frac{\sigma_0}{2\pi}\right)^{2\nu -1}\sum_{n \neq 0}^{}\left( \frac{r}{|n|}\right)^{2\nu -1}\frac{{\cos}^2 \left( - \frac{2\pi}{\sigma_0}|n| \tau_0 - \frac{\pi}{2}\nu - \frac{\pi}{4}\right)}{{\cos}^2 \left( - r\tau_0- \frac{\pi}{2}\nu - \frac{\pi}{4}\right)}~ |\phi_n(\sigma_0)|^2 ~S_I
\prod_{i=1}^n\mu_i^{-w_i}\prod_{j=0}^{w_i-1}(-\frac{w_i}{d}D_\lambda-\frac{1+\alpha_i}{d}-j)\omega_\mu=\prod_{i=1}^n\left(-\frac{d}{w_i}\mu_i\right)^{-w_i}\prod_{j=0}^{w_i-1}(D_\lambda+\frac{1+\alpha_i+dj}{w_i})\omega_\mu,
X^{(i)}=\left( \left( \frac{2}{3}\right)^7, \left(- \frac{1}{3}\right)^2 \right)^P, X^{(ii)}=\left( \left( \frac{2}{3}\right)^8, - \frac{4}{3}\right)^P, &X^{(iii)}=\left( \frac{4}{3}, \left( \frac{1}{3}\right)^8 \right)^P, X^{(iv)}=\left( \left( \frac{4}{3}\right)^2, \left( \frac{1}{3}\right)^6, - \frac{2}{3}\right)^P,
e(U,U') = e(V,N(u_0)) + e(V,U' - N(u_0)) + e(U - V, U') &\ge (s-1)|V| + (t-1)(n-|N(u_0)|) + \delta(n-|V|) &\ge (s-1)|V| + (t-1)(n-t+2) + (s-1)(n-|V|) &\ge n(t+s-2) - (t-1)(t-2) &\ge n(t+s-2) - (s+t-2)^2.
R e ( i \Delta_{F}( x ) ) ~ = ~{\frac{1}{8}}[ \epsilon ( x^{+}) + \epsilon ( x^{-}) ] N_{0}( m \sqrt{x^{2}}) ~ - ~{\frac{1}{4 \pi}}[ \epsilon ( x^{+}) - \epsilon ( x^{-}) ] K_{0}( m \sqrt{x^{2}})
c_{i,j}=\dfrac{4}{\pi^2}\int_Df(x,y)T_i(x)T_j(y)\omega(x,y)dxdy= \dfrac{4}{\pi^2}\int_{0}^{\pi}\int_{0}^{\pi}f(\cos\theta_x,\cos\theta_y)\cos i\theta_x \cos j\theta_y d\theta_x d\theta_y,
(V_{i}\otimes V_{j})\otimes V_{k}\cong\bigoplus_{l,m}H_{i,j}^{l}\otimes H_{l,k}^{m}\otimes V_{m}whileV_{i}\otimes(V_{j}\otimes V_{k})\cong\bigoplus_{m,n}H_{i,n}^{m}\otimes H_{j,k}^{n}\otimes V_{m}
F'_{n-1}\sum_{\substack{y_{i,i}1\leq i\leq n-2}}\chi_1(y_{1,1}\cdots y_{n-2,n-2})\chi_{n-1}\left(-\frac{y_{n-1,n-1}}{y_{n-2,n-2}}\right)\prod_{i=2}^{n-2}\chi_i\left(-\frac{1}{y_{i-1,i-1}}\right) \times \psi\left(\frac{1}{y_{1,1}\cdots y_{n-2,n-2}}-\sum_{i=2}^{n-1}y_{i-1,i-1}-\frac{y_{n-2,n-2}}{y_{n-1,n-1}}\right).
S_3(t_0) =\frac{1}{2}(X_4-X_3)\cdot (X_3-X_2)^\perp(t_0)=\frac{1}{2}(X_4-X_2)\cdot (X_3-X_2)^\perp(t_0) &=\frac{d_2}{2}(X_4-X_2)\cdot (X_2-X_1)^\perp(t_0)=d_2 S_2^4(t_0)\le 0.
(\Phi \circ \beta) (T_i)(y x_j) = x_j (\Phi \circ \beta) (T_i) (y) \j \ne i,i+1, (\Phi \circ \beta) (T_i)((x_i+x_{i+1})y) = (x_i+x_{i+1}) (\Phi \circ \beta) (T_i)(y) (\Phi \circ \beta) (T_i)(x_ix_{i+1}y) = x_i x_{i+1}(\Phi \circ \beta)(T_i)(y).
W_{d_{1},g_{1}}W_{d_{2},g_{2}}(X,g) =\displaystyle \sum_{(X,s) \in \mathcal{G}}W_{d_1,g_1}(X,s)\beta_{(X,s)}(W_{d_2,g_2}(Xs,s^{-1}g)) = \sum_{s \in G}1_{\Omega s^{-1}}(X)W_{d_1,g_1}(X,s)\beta_{(X,s)}(W_{d_2,g_2}(Xs,s^{-1}g)).
j^{1}( x , t ) \Bigm |_{x = 0}= \frac{1}{2}( \psi_{1}^{\dagger}( 0 ) \psi_{1}( 0^{-}) - \psi_{2}^{\dagger}( 0 ) \psi_{2}( 0^{-}) + \psi_{1}^{\dagger}( 0^{-}) \psi_{1}( 0 ) - \psi_{2}^{\dagger}( 0^{-}) \psi_{2}( 0 ) ) = 0 ,
c_{\phi}={\frac{i g_{Y M}^{2}C_{2}( R )}{3 2 \pi^{2}}}\sum_{j = 1}^{n}{\frac{1}{j}}\sim{\frac{i g_{Y M}^{2}C_{2}( R )}{3 2 \pi^{2}}}\operatorname{l n}n
AG \geq \frac{2}{n}\phi G^{2}-\bigg(2K+\frac{12\left\vert b\right\vert}{n}\bigg)\phi G-\frac{R\left\vert \nabla\phi \right\vert^{2}}{C_{1}\phi^{\frac{1}{2}}}G^{\frac{3}{2}}-\frac{C_{1}\phi^{\frac{1}{2}}G^{\frac{3}{2}}}{R}.
J^{(N)}\left(N; 1, \ldots , 1 \right)= 2\; \mbox{i}^{1-2N}\pi^{N/2}\frac{\Gamma\left({\textstyle{N\over2}}\right)}{\prod m_i}\int\limits_0^{\infty}\ldots \int\limits_0^{\infty}\prod\mbox{d}\alpha_i\; \delta\left( \alpha^T \|c\| \alpha-1 \right) .
\quad\,\|((Q_\pi^{-\frac{1}{2}})_{jj}u_{ij}^{(\pi)}D^{\frac{1}{p}})\|_{CR_p[L^p(\mathbb G)]}&= \max \{\|(D^{\frac{1}{p}}\sum_{j=1}^{n_\pi}(Q_\pi^{-1})_{jj}(u_{ij}^{(\pi)})^*u_{ij}^{(\pi)}D^{\frac{1}{p}})^{\frac{1}{2}}\|_{p,\mathsf H},\|(\sum_{j=1}^{n_\pi}(Q_\pi^{-1})_{jj}u_{ij}^{(\pi)}D^{\frac{2}{p}}(u_{ij}^{(\pi)})^*)^{\frac{1}{2}}\|_{p,\mathsf H}\}.
f(\lambda )=\prod_{b=1}^{\mathsf{N}}\frac{\lambda^{2}-{\zeta_{b}^{(0)}}^{2}}{{\zeta_{a}^{(0)}}^{2}-{\zeta_{b}^{(0)}}^{2}}\mathbf{A}(\eta /2)+2\left( 4\lambda^{2}-\eta^{2}\right) \,\prod_{b=1}^{\mathsf{N}}\lambda^{2}-{\zeta_{b}^{(0)}}^{2},
0 ~\leq~ y\sqrt{ab}- x^2 begin{align*}.5em] = ~ x\sqrt{ab}- x^2 + y \sqrt{ab}- x\sqrt{ab}begin{align*}.5em] = ~ x(\sqrt{ab}- x) + \sqrt{ab}(y - x) begin{align*}.5em] \leq ~ (x+\sqrt{ab}) (y-x),
U_\sigma f''(x) \ge \Big| D_v f'' (x) - \frac{f''(x+hv)-f''(x)}{h}\Big| & = \Big| D_v \bar g(x) + D_vf'(x) - \frac{f'(x+hv)-f'(x)}{h}+ \frac{\bar g(x)}{h}\Big| & \ge \frac{1}{2}\big| D_v g(x) \big| - \Big| D_vf'(x) - \frac{f'(x+hv)-f'(x)}{h}\Big| - \frac{|g(x)|}{2h}\, .
Z_{v}^{X}\left( - \frac{1}{\tau}\right) = 2^{- \frac{b_{2}( X )}{2}}\left( \frac{\tau}{i}\right)^{- \frac{\chi ( X )}{2}}\cdot \sum_{u \in H^{2}( X , Z_{2})}( - 1 )^{u \cdot v}Z_{u}^{X}( \tau ) .
{j_{d}}_{1}={j_{d}}_{2}^{\prime \prime \prime}, \quad{j_{d}}_{2}={j_{d}}_{2}^{\prime}, \quad{j_{d}}_{3}={j_{d}}_{2}^{\prime \prime}, \quad{j_{d}}_{1}^{\prime}={j_{d}}_{3}^{\prime \prime \prime}, \quad{j_{d}}_{3}^{\prime}={j_{d}}_{3}^{\prime \prime}, \quad{j_{d}}_{1}^{\prime \prime}={j_{d}}_{1}^{\prime \prime \prime},
\gamma =\psi_{0}\vee \phi_{0}\vee \hat{\psi}_{1}\vee \psi_{1}\vee \phi_{1}\vee \hat{\psi}_{2}\vee \psi_{2}\vee \phi_{2}\vee \dots \vee \hat{\psi}_{n-1}\vee \psi_{n-1}\vee \phi_{n-1}\vee \psi_{n}
\left\langle P_h(\cdot,s),E(\cdot,w)\right\rangle =\frac{2\pi^{\overline{w}+\frac{1}{2}}h^{\overline{w}-\frac{1}{2}}\sigma_{1-2\overline{w}}(h)}{\zeta(2\overline{w})(4\pi h)^{s-\frac{1}{2}}}\frac{\Gamma(s+\overline{w}-1)\Gamma(s-\overline{w})}{\Gamma(\overline{w})\Gamma(s)},
\frac{1}{(\lambda(\tau))^{1/4}}-(\lambda(\tau))^{1/4}=\frac{1}{2}(\frac{\eta(\frac{\tau}{4})}{\eta(\tau)})^{4}=2\frac{\theta_{4}^{2}(\frac{\tau}{2})}{\theta_{2}^{2}(\frac{\tau}{2})}
\begin{aligned}0 =d_A^*B+\tau^1(B\centerdot\theta_1) &=d_A^*(\xi\otimes\omega)+\tau^1(\xi\otimes\omega\centerdot\theta_1) &=-d_A\xi\centerdot\omega+\xi\otimes d^*\omega+\xi\otimes\tau^1(\omega\centerdot\theta_1). \end{aligned}
{\cal O}_{\pi^{*}{\cal S}}( A \sigma |_{\pi^{*}{\cal S}}+ f F ) ={\cal O}_{\pi^{*}{\cal S}}( A \sigma |_{\pi^{*}{\cal S}}) \otimes{\cal O}_{\pi^{*}{\cal S}}( f F ) ={\cal O}_{\pi^{*}{\cal S}}( A \sigma |_{\pi^{*}{\cal S}}) \otimes \pi^{*}{\cal O}_{{\cal S}}( f ) .
{{f}_{{{I}_{y}}}}\left({{I}_{y}}\right)=\frac{{{\gamma}_{2}}I_{y}^{{{m}_{2}}{{\gamma}_{2}}-1}}{{{\left({{{\Omega}_{2}}}/{{{m}_{2}}}\; \right)}^{{{m}_{2}}}}\Gamma \left({{m}_{2}}\right)}\exp \left( -\frac{{{m}_{2}}}{{{\Omega}_{2}}}I_{y}^{{{\gamma}_{2}}}\right)
\frac{\Lambda_{{\rm b}\perp}\left(a=-a_l\right)}{{\cal E}_l}={\rm B}_l=\frac{{\cal C}_{11}{\cal C}_{44}a_l^2+\left({\cal C}_{13}^2+2{\cal C}_{13}{\cal C}_{44}-{\cal C}_{11}{\cal C}_{33}\right)a_l+{\cal C}_{33}{\cal C}_{44}}{{\rm E}_l}
\sum\limits_{i=1}^{n-1}\sum\limits_{k\neq i}^n 2^{|k-i|}\frac{x_i-x_n}{x_i-x_k}= \sum\limits_{i=1}^{n-1}2^{n-i}+ \sum\limits_{i<k}^{n-1}\Bigl( 2^{k-i}\frac{x_i-x_n}{x_i-x_k}+ 2^{k-i}\frac{x_k-x_n}{x_k-x_i}\Bigr), = \sum\limits_{j=1}^{n-1}(n-j)2^j = 2^{n+1}-2n-2.
\frac{1}{2}\epsilon^{i j}( \partial_{i}{\cal A}_{j}^{a}- \partial_{j}{\cal A}_{i}^{a}+ f^{a b c}{\cal A}_{i}^{b}{\cal A}_{j}^{c}) = - \frac{i}{\kappa}\sum_{p = 1}^{N - 1}T_{( p )}^{a}\delta^{2}({\bf r}-{\bf r}_{p}) .
\underline{f_1}+\underline{f_2}:=\underline{f_1\big|_{U_1\cap U_2}+f_2\big|_{U_1\cap U_2}},\underline{f_1}*\underline{f_2}:=\underline{f_1\big|_{U_1\cap U_2}f_2\big|_{U_1\cap U_2}}.
q(x)= -\Big(2a_1a_2+4a_3+a_4\Big)+\Big(12a_2+3a_5-a_1(6a_3+2a_4)\Big)x &+\Big(8a_3+3a_4+4a_6+a_1(6a_2+2a_5)\Big)x^2-\Big(4a_2+a_5-2a_1a_6\Big)x^3.
\sum\limits_{n=1}^{\infty}\frac{G\left( n,\chi\right) G\left(in,\chi\right)}{1-e^{-2\pi n}}n^{-2N-1}=-\frac{k\left( 2\pi/k\right)^{2N+1}}{4}\sum\limits_{m=0}^{2N+2}\left( -i\right)^{m}\frac{B_{m}\left(\chi\right)}{m!}\frac{B_{2N+2-m}\left( \chi\right)}{\left(2N+2-m\right) !}.
{S_1}[\chi,\phi,{g_c}]={1\over{2\gamma}}\int{d^2}\xi\sqrt{g_c}{{g_c}^{ab}}{\partial_a}\chi{\partial_b}\chi-{1\over{2\gamma}}{\eta_c}\int{d^2}\xi\sqrt{g_c}{\chi^2}-{1\over{4\gamma}}{k_c}\oint{d}{s_c}{\phi^2}.
(\Omega_{\tau})_{0,k}(x_0(0,\tau),\ldots,x_{n-1}(0,\tau) ) =(\Omega_{\tau})_{0,k}(0,\alpha_1(\tau)\ldots,\alpha_{n-1}(\tau) ) &=\lambda \cdot \sum_{1\leq i \leq n-1, i \neq n-k}\frac{\alpha_{k}(\tau)}{\alpha_i(\tau) \alpha_{k-i}(\tau)}\alpha_{i}(\tau)\alpha_{k-i}(\tau) &=\lambda(n-2)\alpha_k(\tau).
[\begin{matrix}1&3&2\\ 2&3&1\end{matrix}]\oplus[\begin{matrix}1&6\\ 0&1\end{matrix}]=[\begin{matrix}1&3&2&0&0\\ 2&3&1&0&0\\ 0&0&0&1&6\\ 0&0&0&0&1\end{matrix}].
_\pm \langle \{\lambda \}, \pm q|\prod_{i=1}^{N}\Psi^{\pm}(z_i) |\pm(q- N \beta) \rangle =\prod_{i<j}(z_i -z_j)^{\beta}\prod_i^N (z_i)^{q- N \beta}J_{\{\lambda \}}^{(\beta)}(z_1,...z_N) \ ( j_{\lambda}^{\beta})^{-1/2}
U_kp=\left(\begin{array}{ll}\frac{1}{p^T Hp}Hp \frac{1}{p^T Bp}Bp\end{array}\right) \left(\begin{array}{cc}\hat{m}_{1,1}\hat{m}_{1,2}\hat{m}_{1,2}\hat{m}_{2,2}\end{array}\right)\left(\begin{array}{c}1 1 \end{array}\right)=Hp-Bp.
\varphi(d_{1}(^{\ast}b)d_{2}n) =\psi(d_{1}(^{\ast}b)d_{2}n) &=\psi(d_{1}\overline{N}(^{\ast}b)(d_{2}n)) &=d_{1}\psi(\overline{N}(^{\ast}b)(d_{2}n)) &=d_{1}\overline{N'}(^{\ast}b)(d_{2}n) &=d_{1}(^{\ast}b)d_{2}\varphi(n)
U_1(e_1)=\begin{pmatrix}1 e_1 0 0 0 1 0 0 0 0 1 -e_1 0 0 0 1\end{pmatrix}, U_2(e_2)=\begin{pmatrix}1 0 e_2 0 0 1 0 -e_2 0 0 1 0 0 0 0 1\end{pmatrix}, U(e_1,e_2) = U_1(e_1) U_2(e_2),
\langle |F_k|^2 , \chi_{\mathcal{R}}\rangle = \frac{3}{\pi}\cdot \langle 1, \chi_{\mathcal{R}}\rangle + O(\delta) + O \Big ( (C/\delta)^2 (\log k)^{\varepsilon}\prod_{p \leq k}\Big ( 1 - \frac{\tfrac12 (|\lambda_f(p)| - 1)^2}{p}\Big ) \Big ).
AS' =(A^k{T'}^{k+1}A) A^k{T'}^{k+1}=(A^k{T'}^k)A^k{T'}^{k+1}=A^k v^{k+1}B^k A^{2 k}{T'}^{k+1}=A^k{T'}^{k+1}(B^k A^k)^{k+1}&= S^{k+1}=A^k{T'}^{k+1}=S'.
\int d t G ( x_{1}, t_{1}; 0 , t ) G ( 0 , t ; x_{2}, t_{2}) = - \int \frac{d \omega}{2 \pi}e^{- i \omega ( t_{1}- t_{2})}e^{- i \hat{k}( x_{1}+ x_{2})}\frac{1}{( i \hat{k}- 2 \sigma_{1})^{2}}.
\int_0^T f^2(x,t)\,dt = \int_0^T |\mathcal{W}[y+d] - \mathcal{W}[y] - \omega|^2(x,t)\,dt &\le\rho^2_{y(x,\cdot)}(\|d(x,\cdot)\|_{\infty,T}) \cdot\|d_t(x,\cdot)\|^2_{L^p(0,T)},
\Gamma = \pi A_{h}\frac{(\omega^2 - m^2)^{3/2}}{m (\omega - m\phi_{4})}\sqrt{\frac{r_{1}r_{2}r_{3}}{r_{4}}}\frac{1}{(e^{2\pi m \sqrt{\frac{r_{1}r_{2}r_{3}}{r_{4}}}}- 1)}\frac{1}{(e^{\frac{\omega - m}{T_{H}}}- 1)}\frac{d^3k}{(2\pi)^3},
F^{a_1,a_2,\ldots, a_{2s-1},a_{2s}}:=f^{a_1a_2}(f^{a_3a_2})^*f^{a_3a_4}(f^{a_5a_4})^*\cdots f^{a_{2s-1}a_{2s}}(f^{a_1a_{2s}})^*.
\mathrm{vol}(\vec{v}) =\int_{K}\sqrt{1+\sum h_{ij}^{2}+(\mathrm{det}(h_{ij}))^{2}+\ldots}&\geq \int_{K}\sqrt{1+2\sigma_{2}(\vec{v})+\sigma_{2}(\vec{v})^{2}}&\geq \int_{K}(1+\sigma_{2}(\vec{v}))\geq 2\mathrm{vol}(K)=\mathrm{vol}(H)
h ( \underline{{\theta}},{\underline{{z}}}) = \prod_{1 \le i < j \le n}F ( \theta_{i j}) \prod_{i = 1}^{n}\prod_{j = 1}^{m}\tilde{\phi}( \theta_{i}- z_{j}) \prod_{1 \le i < j \le m}\tau ( z_{i}- z_{j})
(h\cdot\rho'-\pi^* \mathrm{Av}(h\cdot\rho')\rho)(x)=h(x)\rho'(x) - \left(\int_{s^{-1}(x)}h(t(g))\overrightarrow{\rho'}\right)c(x)\rho'(x) &= \left(\int_{s^{-1}(x)}h(x)c(t(g))\overrightarrow{\rho'}- \int_{s^{-1}(x)}c(x)h(t(g))\overrightarrow{\rho'}\right)\rho'(x),
\frac{ds^2}{l_s^2}= \frac{u^{7-p}}{R^{7-p}}\Big[ - f d\tilde{t}^2 + (d\tilde{x}^1)^2 \Big] + (d\tilde{x}^2)^2 + \cdots + (d\tilde{x}^p)^2+ f^{-1}du^2 + u^2 d\Omega_{8-p}^2
\lambda_x^\perp(\widehat{\rho}_x(X))(u) = [\rho_x(X), Y_1]_x = [\rho_x(X), \widehat{Y}_1]_x, \lambda_x^\perp(\widehat{\rho}_x(X))(v) = [\rho_x(X), Y_2]_x = [\rho_x(X), \widehat{Y}_2]_x.
I_2=\frac{R}{4\pi i}\int\limits_{\Lambda}^{-\Lambda}\frac{(-y^2)\,i\, dy}{i\sqrt{y^2+\omega_0^2}}\frac{\cosh(R\sqrt{y^2+\omega^2_0})}{\sinh(R\sqrt{y^2+\omega^2_0})}dy=-\frac{1}{2\pi}\int\limits_{0}^{\Lambda}y\, d\left[\ln\sinh\left(R\sqrt{y^2+\omega_0^2}\right)\right].
v(\rho_2)-v(\rho_1) = \int b(\rho_2) c(\rho_2) - b(\rho_1) c(\rho_1) \mu_W(dw) & = \int (b(\rho_2)- b(\rho_1)) c(\rho_2) \mu_W(dw) - \int b(\rho_1) (c(\rho_2)-c(\rho_1)) \mu_W(dw) & = \int \int_{\rho_1}^{\rho_2}b(\gamma) c(\gamma) d\gamma c(\rho_2) \mu_W(dw)- \int b(\rho_1) (c(\rho_2)-c(\rho_1)) \mu_W(dw),
H^{\varepsilon}w^{\varepsilon}(x,t) \ := \frac{\partial w^\varepsilon}{\partial t}- L^{\varepsilon}w^\varepsilon \ = \ \Big(\frac{\partial u}{\partial t}(x^\varepsilon,t) - \Theta \cdot \nabla \nabla u(x^\varepsilon,t) \ + \ \phi^\varepsilon(x^\varepsilon,t)\Big)|_{_{\, x^\varepsilon = x - \frac{b}{\varepsilon}\, t}},
c_{q_iq_j}(tq,tp) = c_{q_iq_j}(tq,0)+ c_{q_iq_j,p_k}(tq,0)p_k+a^{(1)}_{ij,kl}(q,p)p_kp_l, c_{p_ip_j}(tq,tp) = c_{p_ip_j}(0,tp)+ c_{q_k,p_ip_j}(0,tp)q_k+a^{(2)}_{ij,kl}(q,p)q_kq_l
q_{\infty}(\mathcal F_1^{c'}, \mathcal F_2^{c'}, \mathbf T^{'})-q_{\infty}^{*}=f_{1,K_1^c+K_1^b,\infty}\left(\frac{1}{K_1^b}\sum_{n\in \mathcal F_1^{b'}}a_n-\frac{1}{F_1^b}\sum_{n\in \mathcal F_1^{b*}}a_n\right)K_1^b>0.
R ={1 - \tau\over 1 + \tau}=\left({\rho_0 (1 + t^2) + 2 \tilde{\rho}_0 t - 4 t\rho^+\over \rho_0 (1 + t^2) + 2 \tilde{\rho}_0 t + 4 t \rho^-}\right)^{1/2}.
p^{19}=\left[\begin{array}{ccccc}1 1 3 3 2 2 2 1 1 3 3 3 2 2 1 \end{array}\right],p^{20}=\left[\begin{array}{ccccc}1 1 2 3 1 2 2 1 2 3 3 3 3 1 2 \end{array}\right],p^{21}=\left[\begin{array}{ccccc}1 1 2 3 2 2 2 1 2 3 3 3 3 1 1 \end{array}\right],
u_\infty(x)=\frac{1}{\ell}\int_0^\ell u_0(y)\,dy\; \frac{2\delta\,e^{-2\delta x/\ell}}{1-e^{-2\delta}}+\frac{\rho N}{\delta}\Big(\frac{2\delta\,e^{-2\delta x/\ell}}{1-e^{-2\delta}}-1\Big),0\le x\le\ell.
\left[\begin{array}{ll}p_{n-1}p_{n}q_{n-1}q_{n}\end{array}\right]\cdot\left[\begin{array}{ll}1 0 0 a_{n+1}\end{array}\right]\cdot\left[\begin{array}{l}\lambda_{n+1}-\lambda_n \lambda_{n+1}-\lambda_{n-1}\end{array}\right] = \left[\begin{array}{l}0 0\end{array}\right]
M_{i}(q_{1},\ldots ,q_{N}):=L_{iN}\left( \begin{smallmatrix}1 0 0 q_{N}\end{smallmatrix}\right)_{i}\cdots L_{i2}\left( \begin{smallmatrix}1 0 0 q_{2}\end{smallmatrix}\right)_{i}L_{i1}\left( \begin{smallmatrix}1 0 0 q_{1}\end{smallmatrix}\right)_{i}
\alpha_{+}(s):=\angle \gamma_{-t_*}(s) \gamma_{t_*}(s) p, \,\,\,\,\, \alpha_{-}(s):=\angle \gamma_{t_*}(s) \gamma_{-t_*}(s) p, \tilde\alpha_{+}(s):=\tilde\angle \gamma_{-t_*}(s) \gamma_{t_*}(s) p, \,\, \,\,\,\tilde\alpha_{-}(s):=\tilde\angle \gamma_{t_*}(s) \gamma_{-t_*}(s) p.
\|\Box_k^{\alpha_1}f\|_{M_2}= \left(\sum_{l\in \Gamma_k^{\alpha_2,\alpha_1}}\|\Box_l^{\alpha_2}\Box_k^{\alpha_1}f\|^q_{L^{\infty}}\right)^{1/q}\lesssim 2^{jn\alpha_2/2}\left(\sum_{l\in \Gamma_k^{\alpha_2,\alpha_1}}\|\Box_l^{\alpha_2}\Box_k^{\alpha_1}f\|^q_{L^{2}}\right)^{1/q}.
P_1 =R_0\cup S_0\cup T_0 \cup U_0\cdots R_{\frac{d}{2}-2}\cup S_{\frac{d}{2}-2}\cup T_{\frac{d}{2}-2}\cup U_{\frac{d}{2}-2}\cup R_{\frac{d}{2}-1}\cup S_{\frac{d}{2}-1}\cup T_{\frac{d}{2}-1}& = \cup_{i=0}^{\frac{d}{2}-1}R_i\bigcup \cup_{i=0}^{\frac{d}{2}-1}S_i\bigcup \cup_{i=0}^{\frac{d}{2}-1}T_i\bigcup \cup_{i=0}^{\frac{d}{2}-2}U_i,
\frac{\partial\pi_i^A(\boldsymbol{d}_i;\!\boldsymbol{d}_{-i})}{\partial d_i}\!\! = \!\!1\! - \! \frac{\gamma\!\left( \!1\!-\! \epsilon_i\! \sum_{j \in \mathcal{I}}{\frac{1}{\epsilon_j}}\right) \!-\! \gamma \epsilon_i \!\sum_{j \in \mathcal{I}}{\frac{1}{\epsilon_j}}\!-\! z_i}{2\epsilon_i\left( 1+ \gamma \sum_{j \in \mathcal{I}}{\frac{1}{\epsilon_j}}\right)}=0
\psi_1 = \frac{3}{2 \sqrt{2}}( f_1 + f_2 ) - \frac{1}{\sqrt{2}}f \ , psi_2 = \frac{2}{3}\bar{\phi}+ \frac{1}{4 \sqrt{2}}( f_1 + f_2 ) + \frac{1}{2 \sqrt{2}}f .
\phi ( x ) \sim{\frac{\alpha_{0}}{B}}\int_{\partial M}d^{d}x^{\prime}\left( \frac{z}{z^{2}+ | \vec{x}- \vec{x}^{\prime}|^{2}}\right)^{\Delta}\int_{0}^{\epsilon}d x_{0}^{\prime}{z}^{\Delta - d - 1}J ( x^{\prime}) .
U_{\infty'}^{\tau'}= U_{0'}= \varphi^{-1}U_0 \varphi = \varphi^{-1}U_\infty^\tau \varphi ={\varphi^{-1}\tau^{-1}\varphi}U_{\infty'}{\varphi^{-1}\tau \varphi}= U_{\infty'}^{\varphi^{-1}\tau \varphi},
\operatorname*{l i m}_{{\mathrm{\boldmath ~ \varepsilon ~}}\to 0}\left( \frac{\mathrm{{\boldmath ~ \varepsilon^i ~}}\mathrm{{\boldmath ~ \varepsilon^j ~}}}{{\mathrm{\boldmath ~ \varepsilon ~}}^{2}}\right) = \frac{1}{3}\delta^{i j}.
[\begin{matrix}I&C\\ -C^{t}&I\end{matrix}]^{-1}=[\begin{matrix}(I+CC^{t})^{-1}&-(I+CC^{t})^{-1}C\\C^{t}(I+CC^{t})^{-1}&-C^{t}(I+CC^{t})^{-1}C+I\end{matrix}]
\Pi^*(s)=\frac{b_0(s,Y)}{\sigma^2_0(s,Y)}+\frac{\mathbb{E}[D_s\delta_Y(y)|\mathcal{F}_s]_{y=Y}}{\sigma_0(s,Y)\mathbb{E}[\delta_Y(y)|\mathcal{F}_s]}_{y=Y}, 0 \leq s \leq T < T_0.
n_{i}^{\prime\prime}=\frac{J_{i}}{J_{i}+J_{j}}=\frac{P_{i}(p_{i}^{\prime}-p_{i}^{\prime\prime})}{P_{i}(p_{i}^{\prime}-p_{i}^{\prime\prime})+P_{j}(n_{j}^{\prime}-\frac{1}{\phi}n_{j}^{\prime\prime})}
C_{k}\left( \mathbf{q,l,j}\right) :=\left( -1\right)^{\left\vert\mathbf{q}\right\vert}\binom{\mathbf{q}}{\mathbf{l}}\binom{\mathbf{q}}{\mathbf{j}}1_{j_{1}=0}1_{2\left\vert \mathbf{q}\right\vert -k=\left\vert\mathbf{l}\right\vert +\left\vert \mathbf{j}\right\vert >0},
\begin{array}{lcl}dd^cU_m^j = p_{2\star}[p_1^\star(\eta T) \wedge \tau^\star(dd^cK_m^j)]+p_{2\star}[p_1^\star(dd^c(\eta T))\wedge \tau^\star(K_m^j)] &\ +p_{2\star}[p_1^\star(d(\eta T)) \wedge\tau^\star(d^cK_m^j)]-p_{2\star}[p_1^\star(d^c(\eta T))\wedge\tau^\star(dK_m^j)] &= p_{2\star}[p_1^\star(\eta T) \wedge \tau^\star(dd^cK_m^j)]+R_m^j.\end{array}
\begin{cases}\Big(\tilde\tau+sign(U_s''(a))z^2\Big)^2\frac{d}{dz}W+i\frac{d^3}{dz^3}\Big(\big(\tilde\tau+sign(U_s''(a))z^2\big)W\Big)=0, \lim\limits_{z\rightarrow-\infty}W~=~0,\quad\lim\limits_{z\rightarrow+\infty}W~=~1.\end{cases}
\left<x,x\right> \otimes \left<y,y\right> \ = \ \bigl( \sum_i x_i^*x_i\bigr) \otimes\bigl(\sum_jy_j^*y_j\bigr) \quad \ = \ \sum_{i,j}x_i^*x_i \otimes y_j^*y_j \ = \ \sum_{i,j}(x_i \otimes y_j)^* (x_i \otimes y_j).
{\cal J}= \frac{1}{4 \pi}\int \left( \frac{( | R_{z}|^{2}+ | R_{{\bar{z}}}|^{2}) ( 1 + | z |^{2})^{2}}{( 1 + | R |^{2})^{2}}\right) \frac{2 i \mathrm{d}z \mathrm{d}{\bar{z}}}{( 1 + | z |^{2})^{2}}
L^X(z) := \begin{pmatrix}L_{++}^X(z) L_{+-}^X(z) L_{-+}^X(z) L_{--}^X(z)\end{pmatrix}= \begin{pmatrix}1 F^X(z) 0 1\end{pmatrix}\begin{pmatrix}K_+^X(z) 0 0 K_-^X(z) \end{pmatrix}\begin{pmatrix}1 0 E^X(z) 1\end{pmatrix}.
\int \sqrt{g}~ \widehat{R}^{\mu \nu \lambda \rho}\widehat{R}_{\mu \nu \lambda \rho}, \qquad \int \sqrt{g}~ \widehat{R}^{\mu \nu}\widehat{R}_{\mu \nu}, \qquad \int \sqrt{g}~ \widehat{R}^{2}.
W ( t ) = \exp i{\frac{e}{\hbar c}}\int \! \int d \vec{x}d \vec{y}J_{0}^{( 1 )}( \vec{x}, t ) G_{\mathrm{b o x}}( \vec{x}, \vec{y}) \partial_{k}a_{k}^{( 1 )}( \vec{y}, t )
\zeta (x) = \left[ \begin{array}{c}\left.1 \middle/\sigma (x)\right. b (x) \end{array}\right] \mbox{and}G \left(z,(a,q)\right) = \left[ \begin{array}{c}0 -\frac12 \, g \, a^2 \, z_1 - g \, a \, z_2 \end{array}\right]
\phi^2+\phi*\phi = (s^2(\ell-1)+t^2(m-1)+(\ell-1)(m-1))\delta_{(0,0)}+(s^2(\ell-1)+(\ell-2)(m-1)+2t(m-1))\chi_A +(t^2(m-1)+(\ell-1)(m-2)+2s(\ell-1))\chi_B +(1+(\ell-2)(m-2)+2st+2s(\ell-2)+2t(m-2))\chi_C.
\left[\pi(z_{n}), \pi(z_{n+1})\right]^{r}\pi(z_{n+1})\left[\pi(z_{n}), \pi(z_{n+1})\right]^{s}u =C\pi(z_{n+1})\left[\pi(z_{n}), \pi(z_{n+1})\right]^{r+s}u,